Atomic Molecular Optical Physics by Hertel C Schulz, Volume 2

Graduate Texts in Physics

Ingolf V. Hertel Claus-Peter Schulz

Atoms, Molecules and Optical Physics 2 Molecules and Photons - Spectroscopy and Collisions

Graduate Texts in Physics

For further volumes: www.springer.com/series/8431

Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advancedlevel undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field.

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Professor Susan Scott Department of Quantum Science Australian National University Science Road Acton 0200, Australia [email protected]

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Professor Martin Stutzmann Walter Schottky Institut TU München 85748 Garching, Germany [email protected]

Ingolf V. Hertel r Claus-Peter Schulz

Atoms, Molecules and Optical Physics 2 Molecules and Photons – Spectroscopy and Collisions

Ingolf V. Hertel Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie Berlin, Germany

Claus-Peter Schulz Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie Berlin, Germany

ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Texts in Physics ISBN 978-3-642-54312-8 ISBN 978-3-642-54313-5 (eBook) DOI 10.1007/978-3-642-54313-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2014953011 © Springer-Verlag Berlin Heidelberg 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my Wife Erika IVH To my Wife Gudrun CPS

Preface

The two textbooks on Atomic, Molecular and Optical (AMO) physics presented here aim at providing something like the canonical knowledge of modern atomic and molecular physics together with a first entry into optical physics and quantum optics. All of these topics constitute a vital area of active and highly productive research in physics. And in spite of, or perhaps even because of its remarkable history the field continues to constitute an indispensable basis for any more profound understanding of nearly all branches of modern physics, physical chemistry and partially even biological and material sciences. Specifically the latter appear to become more and more based on genuine molecular concepts. We want to address on the one hand advanced students of physics and physical chemistry, who have to study these topics within their respective curricula. At the same time, however, these textbooks should be useful to all those who discover in different contexts that they miss some essential basics from this field and who seek for suitable means to acquire that knowledge. Of course, we also address quite specifically Ph.D.-students or young researchers who start for the first time their own activities in the field – or just want to know more about it. They will find here reliable knowledge and stimulating challenges for their own work. We thus have tried not only to provide the essential basics for working with these topics, but whenever possible also to inform the interested reader about the present state-of-the-art and to allow her or him a glimpse on today’s cutting edge research. The general remarks as well as details about formats, notation, units, and typography outlined in the preface to Vol. 1 (H ERTEL and S CHULZ 2014) are equally valid for the present Vol. 2. In the following we just give a guide through the contents, as the readers will find many topics and details far beyond the standard textbooks and routine teachings on AMO science. Chapter 1 resumes the discussion of light (comprising in the broadest sense the whole electromagnetic spectrum) and photons, which are key themes of these two volumes. The focus is here on lasers (one of the most important tools of modern AMO physics), Gaussian beams, polarization and nonlinear processes. In the following Chap. 2 the emphasis is on the properties of photons and coherence. In addition to discussing some basics of quantum optics and its applications, we also vii

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Preface

complete now the theory of photon induced transitions (Chap. 4, Vol. 1) and introduce field quantization which allows us to treat spontaneous emission. After these preparations we are ready to enter into molecular physics and modern molecular spectroscopy. We start in Chap. 3 with diatomic molecules, the most simple prototypes, and enhance our view with ‘real’ polyatomic molecules in Chap. 4, including a brief excursion into the subject of symmetries. While along this way we have already encountered various comparatively simple examples of molecular spectroscopy, Chap. 5 leads us deep into a variety of sophisticated modern methods of spectroscopy. The field is dominated today by laser based methods, but we also gain some insights e.g. into the possibilities of photoelectron spectroscopy. Three quite detailed Chaps. 6–8 are devoted to the present status in the physics of electronic, atomic, molecular and ionic collisions (including collisional ionization) – a topic of great practical importance and with demanding intellectual challenges, both from an experimental and theoretical view point. Chapter 9 gives a down to earth manual for using the density matrix. It also gives a brief look on the theory of measurement, including a powerful method to analyze radiation patterns from anisotropically populated mixtures of excited states. Finally, making use of these concepts, an introduction of the optical B LOCH equations is given in Chap. 10, which again addresses many exciting facets of quantum optics with interesting examples. A possible extension of the two volumes published now is under consideration. Such a Vol. 3 would approach modern research even more closely and illuminate some particularly hot and rapidly developing areas such as ultra-cold matter and quantum gases, ultrafast dynamics, attosecond physics, cluster spectroscopy and similar themes. We wish all our readers an exciting and stimulating reading as well as efficient understanding and successful learning. In several readings, we have tried to produce text, mathematical formulas and figures as free from errors as possible. Clearly, this can only be an approximation process. Thus, we encourage our readers to kindly communicate any critical comments, errors or simply even typos which they may discover – and to make suggestions for improvements wherever it appears advisable. We shall correct such errors at the web-site http://staff.mbi-berlin.de/AMO/bookhomepage/ if and as soon as they become known to us. As further reading we recommend for comparison or for a deeper look into some specialties the textbooks listed below. Berlin Adlershof January 2014

Ingolf V. Hertel Claus-Peter Schulz

Preface

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Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics.

References ATKINS, P. W. and R. S. F RIEDMAN: 2010. Molecular Quantum Mechanics. Oxford: Oxford University Press, 2nd edn. B ERGMANN, L. and C. S CHAEFER: 1997. Constituents of Matter – Atoms, Molecules, Nuclei and Particles. Berlin, New York: de Gruyter, 902 pages. B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics 64. Berlin, Heidelberg: Springer, 3rd edn., 343 pages. B ORN, M. and E. W OLF: 2006. Principles of Optics. Cambridge University Press, 7th (expanded) edn. B RANSDEN, B. H. and C. J. J OACHAIN: 2003. The Physics of Atoms and Molecules. Prentice Hall Professional. B RINK, D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3rd edn., 182 pages. D EMTRÖDER, W.: 2010. Atoms, Molecules and Photons. Berlin, Heidelberg, New York: Springer, 2nd edn. D EMTRÖDER, W.: 2008a. Laser Spectroscopy, vol. 1: Basic Principles. Berlin, New York: Springer, 4th edn., 457 pages. D EMTRÖDER, W.: 2008b. Laser Spectroscopy, vol. 2: Experimental Techniques. Berlin, New York: Springer, 4th edn., 697 pages. D RAKE, G. W. F., ed.: 2006. Handbook of Atomic, Molecular and Optical Physics. Heidelberg, New York: Springer. E DMONDS, A. R.: 1996. Angular Momentum in Quantum Mechanics. Princeton: Princeton University Press, 154 pages. H ERTEL, I. V. and C. P. S CHULZ: 2014. Atoms, Molecules and Optical Physics 1; Atoms and Spectroscopy, vol. 1 of Springer-Textbook. Berlin, Heidelberg: Springer, 1st edn. L OUDON, R.: 2000. Quantum Theory of Light. Oxford, New York: Oxford University Press, 3rd edn. M UKAMEL, S.: 1999. Principles of Nonlinear Optical Spectroscopy. Oxford: Oxford University Press, 576 pages. S TEINFELD, J. I.: 2005. Molecules and Radiation – 2nd Edition, An Introduction to Modern Molecular Spectroscopy. Mineola: Dover Edition. W EISSBLUTH, M.: 1978. Atoms and Molecules. Student Edition. New York, London, Toronto, Sydney, San Francisco: Academic Press, 713 pages.

Acknowledgements

Over the past years, many colleagues have encouraged and stimulated us to move forward with this work, and helped with many critical hints and suggestions. Most importantly, we have received a lot of helpful material and state of the art data for inclusion in these textbooks. We would like to thank all those who have in one or the other way contributed to close a certain gap in the standard textbook literature in this area – that is at least what we hope to have achieved. Specifically we mention Robert Bittl, Wolfgang Demtröder, Melanie Dornhaus, Kai Godehusen, Uwe Griebner, Hartmut Hotop, Marsha Lester, John P. Maier, Reinhardt Morgenstern, Hans-Hermann Ritze, Horst Schmidt-Böcking, Ernst J. Schumacher, Günter Steinmeyer, Joachim Ullrich, Marc Vrakking und Roland Wester; their contributions are specifically noted in the respective lists of references. Of course, there all other sources are documented which we have used for informations and which have provided data which we have used to generate the figures in these books. One of us (IVH) is particularly grateful to the Max Born Institute for providing the necessary resources (including computer facilities, library access, and office space etc.) for continuing the work on this book after official retirement. Special thanks are expressed to the Wilhelm und Else H ERAEUS foundation for sponsoring a Senior Professorship at Humboldt Universität zu Berlin.

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Contents of Volume 2

1

Lasers, Light Beams and Light Pulses . . . . . . . . . . . 1.1 Lasers – A Brief Introduction . . . . . . . . . . . . . . 1.1.1 Basic Principle . . . . . . . . . . . . . . . . . . 1.1.2 FABRY-P ÉROT Resonator . . . . . . . . . . . . 1.1.3 Stable, Transverse Modes and Diffraction Losses 1.1.4 The Amplifying Medium . . . . . . . . . . . . 1.1.5 Threshold Condition and Stationary State . . . . 1.1.6 Laser Rate Equations . . . . . . . . . . . . . . 1.1.7 Line Profiles and Hole Burning . . . . . . . . . 1.2 Gaussian Beams . . . . . . . . . . . . . . . . . . . . . 1.2.1 Diffraction Limited Profile of a Laser Beam . . 1.2.2 FAUNHOFER Diffraction . . . . . . . . . . . . . 1.2.3 Ray Transfer Matrices . . . . . . . . . . . . . . 1.2.4 Focussing a Gaussian Beam . . . . . . . . . . . 1.2.5 Measuring Beam Profiles with a Razor Blade . . 1.2.6 The M 2 Factor . . . . . . . . . . . . . . . . . . 1.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Polarization and Time Dependent Intensity . . . 1.3.2 Lambda-Quarter and Half-Wave Plates . . . . . 1.3.3 S TOKES Parameters, Partially Polarized Light . 1.4 Wave-Packets . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Description of Laser Pulses . . . . . . . . . . . 1.4.2 Spatial and Temporal Intensity Distribution . . . 1.4.3 Frequency Combs . . . . . . . . . . . . . . . . 1.5 Measuring Durations of Short Laser Pulses . . . . . . . 1.5.1 Principle . . . . . . . . . . . . . . . . . . . . . 1.5.2 Correlation Functions . . . . . . . . . . . . . . 1.5.3 Interferometric Measurement . . . . . . . . . . 1.5.4 Experimental Examples . . . . . . . . . . . . . 1.6 Nonlinear Processes in Gaussian Laser Beams . . . . . 1.6.1 General Considerations . . . . . . . . . . . . .

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1.6.2 Cylindrical Geometry (2D Geometry) 1.6.3 Conical Geometry (3D Geometry) . 1.6.4 Spatially Resolved Measurements . . Acronyms and Terminology . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Coherence and Photons . . . . . . . . . . . . . . . . . . . . 2.1 Some Basics for Quantum Optics . . . . . . . . . . . . . 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.1.2 First-Order Degree of Coherence . . . . . . . . . 2.1.3 Quasi-Monochromatic Light . . . . . . . . . . . 2.1.4 Temporal or Longitudinal Coherence . . . . . . . 2.1.5 Higher-Order Degree of Coherence . . . . . . . . 2.1.6 Photon “Bunching” Experiments . . . . . . . . . 2.1.7 Spatial or Lateral Coherence . . . . . . . . . . . 2.1.8 Astronomical Interferometry . . . . . . . . . . . 2.1.9 H ANBURY B ROWN -T WISS Stellar Interferometer 2.1.10 Bunching and Anti-Bunching . . . . . . . . . . . 2.2 Photons, Photon States, and Radiation Modes . . . . . . 2.2.1 Towards Quantization of the Radiation Field . . . 2.2.2 Modes of the Radiation Field . . . . . . . . . . . 2.2.3 Density of States and Black Body Radiation . . . 2.2.4 Number of Photons per Mode . . . . . . . . . . . 2.2.5 The Multi-Mode Field and Energy . . . . . . . . 2.3 Field Quantization and Optical Transitions . . . . . . . . 2.3.1 Second Quantization and Photon Number States . 2.3.2 The Electric Field Operator . . . . . . . . . . . . 2.3.3 G LAUBER States . . . . . . . . . . . . . . . . . . 2.3.4 Addendum for Multi-Mode States . . . . . . . . . 2.3.5 Interaction Hamiltonian for Dipole Transitions . . 2.3.6 Perturbation Theory and Spontaneous Emission . 2.3.7 Spontaneous Emission in a Cavity . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Diatomic Molecules . . . . . . . . . . . . . . . . . . . . 3.1 Characteristic Energies . . . . . . . . . . . . . . . 3.1.1 Hamiltonian . . . . . . . . . . . . . . . . . 3.1.2 Electronic Energy . . . . . . . . . . . . . . 3.1.3 Vibrational Energy . . . . . . . . . . . . . . 3.1.4 Rotational Energy . . . . . . . . . . . . . . 3.2 B ORN O PPENHEIMER Approximation . . . . . . . 3.2.1 Molecular Potentials . . . . . . . . . . . . . 3.2.2 General Form of Molecular Potentials . . . 3.2.3 Nuclear Wave Functions . . . . . . . . . . . 3.2.4 Harmonic Potential and Harmonic Oscillator

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3.2.5 M ORSE Potential . . . . . . . . . . . . . . . . . . 3.2.6 VAN DER WAALS Molecules . . . . . . . . . . . . 3.3 Nuclear Motion: Rotation and Vibration . . . . . . . . . . 3.3.1 S CHRÖDINGER Equation . . . . . . . . . . . . . . 3.3.2 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . 3.3.3 Population of Rotational Levels and Nuclear Spin . 3.3.4 Specific Heat Capacity . . . . . . . . . . . . . . . . 3.3.5 Vibration . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Non-Rigid Rotor . . . . . . . . . . . . . . . . . . . 3.3.7 D UNHAM Coefficients . . . . . . . . . . . . . . . . 3.4 Dipole Transitions . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Rotational Transitions . . . . . . . . . . . . . . . . 3.4.2 Centrifugal Distortion . . . . . . . . . . . . . . . . 3.4.3 S TARK Effect: Polar Molecules in an Electric Field 3.4.4 Vibrational Transitions . . . . . . . . . . . . . . . 3.4.5 Vibration-Rotation Spectra . . . . . . . . . . . . . 3.4.6 RYDBERG -K LEIN -R EES Method . . . . . . . . . . 3.5 Molecular Orbitals . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Variational Method . . . . . . . . . . . . . . . . . 3.5.2 Specialization for H+ 2 . . . . . . . . . . . . . . . . 3.5.3 Charge Exchange in the H+ 2 System . . . . . . . . . 3.5.4 MOs for Homonuclear Molecules . . . . . . . . . . 3.6 Construction of Total Angular Momentum States . . . . . . 3.6.1 Total Orbital Angular Momentum . . . . . . . . . . 3.6.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Total Angular Momentum . . . . . . . . . . . . . . 3.6.4 H UND’s Coupling Cases . . . . . . . . . . . . . . . 3.6.5 Reflection Symmetry . . . . . . . . . . . . . . . . 3.6.6 Lambda-Type Doubling . . . . . . . . . . . . . . . 3.6.7 Example H2 – MO Ansatz . . . . . . . . . . . . . . 3.6.8 Valence Bond Theory . . . . . . . . . . . . . . . . 3.6.9 Nitrogen and Oxygen Molecule . . . . . . . . . . . 3.7 Heteronuclear Molecules . . . . . . . . . . . . . . . . . . 3.7.1 Energy Terms . . . . . . . . . . . . . . . . . . . . 3.7.2 Filling the Orbitals with Electrons . . . . . . . . . . 3.7.3 Lithiumhydrid . . . . . . . . . . . . . . . . . . . . 3.7.4 Alkali Halides: Ionic Bonding . . . . . . . . . . . . 3.7.5 Nitrogen Monoxide, NO . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Polyatomic Molecules . . . . . . . . . 4.1 Rotation of Polyatomic Molecules 4.1.1 General . . . . . . . . . . . 4.1.2 Spherical Rotor . . . . . .

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4.1.3 Symmetric Rigid Rotor . . . . . . . . . . . 4.1.4 Asymmetric Rigid Rotor . . . . . . . . . . . 4.2 Vibrational Modes of Polyatomic Molecules . . . . 4.2.1 Normal Modes of Vibration . . . . . . . . . 4.2.2 Energies and Transitions of Normal Modes . 4.2.3 Linear, Triatomic Molecules AB2 . . . . . . 4.2.4 Nonlinear Triatomic Molecules AB2 . . . . 4.2.5 Inversion Vibration in Ammonia . . . . . . 4.3 Symmetries . . . . . . . . . . . . . . . . . . . . . 4.3.1 Symmetry Operations and Elements . . . . 4.3.2 Point Groups . . . . . . . . . . . . . . . . . 4.3.3 Eigenstates of Polyatomic Molecules . . . . 4.3.4 JAHN -T ELLER Effect . . . . . . . . . . . . 4.4 Electronic States of Some Polyatomic Molecules . . 4.4.1 A First Example: H2 O . . . . . . . . . . . . 4.4.2 Hybridization – sp 3 Orbitals . . . . . . . . 4.4.3 Electronic States of NH3 . . . . . . . . . . 4.4.4 sp 2 Hybrid Orbitals Forming Double Bonds 4.4.5 Triple Bonds . . . . . . . . . . . . . . . . . 4.5 Conjugated Molecules and the H ÜCKEL Method . . Acronyms and Terminology . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Microwave Spectroscopy . . . . . . . . . . . . . . . . . 5.3 Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . 5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 F OURIER Transform Infrared Spectroscopy . . . . 5.3.3 Infrared Action Spectroscopy . . . . . . . . . . . 5.4 Electronic Spectra . . . . . . . . . . . . . . . . . . . . . 5.4.1 F RANCK -C ONDON Factors . . . . . . . . . . . . 5.4.2 Selection Rules for Electronic Transitions . . . . 5.4.3 Radiationless Transitions . . . . . . . . . . . . . 5.4.4 Rotational Excitation in Electronic Transitions . . 5.4.5 Classical Emission and Absorption Spectroscopy . 5.5 Laser Spectroscopy . . . . . . . . . . . . . . . . . . . . 5.5.1 Laser Induced Fluorescence . . . . . . . . . . . . 5.5.2 REMPI for a ‘Simple’ Triatomic Molecule . . . . 5.5.3 Cavity Ring Down Spectroscopy . . . . . . . . . 5.5.4 Spectroscopy of Small Free Biomolecules . . . . 5.5.5 Other Important Methods . . . . . . . . . . . . . 5.6 R AMAN Spectroscopy . . . . . . . . . . . . . . . . . . . 5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.6.2 Classical Interpretation . . . . . . . . . . . . . .

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5.6.3 Quantum Mechanical Theory . . . . . . . . 5.6.4 Experimental Aspects . . . . . . . . . . . . 5.6.5 Examples of R AMAN Spectra . . . . . . . . 5.6.6 Nuclear Spin Statistics . . . . . . . . . . . . 5.7 Nonlinear Spectroscopy . . . . . . . . . . . . . . . 5.7.1 Some Basics . . . . . . . . . . . . . . . . . 5.7.2 An Example . . . . . . . . . . . . . . . . . 5.8 Photoelectron Spectroscopy . . . . . . . . . . . . . 5.8.1 Experimental Basis and the Principle of PES 5.8.2 Examples . . . . . . . . . . . . . . . . . . . 5.8.3 TPES, PFI, ZEKE, KETOF, MATI . . . . . 5.8.4 PES for Negative Ions . . . . . . . . . . . . 5.8.5 PEPICO, TPEPICO and Variations . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 6

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338 342 343 345 348 349 353 355 355 358 362 364 366 372 376

Basics of Atomic Collision Physics: Elastic Processes . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Integral and Total Cross Sections . . . . . . . . 6.1.2 Principle of Detailed Balance . . . . . . . . . . 6.1.3 Integral Elastic Cross Sections . . . . . . . . . 6.2 Differential Cross Sections and Kinematics . . . . . . . 6.2.1 Experimental Considerations . . . . . . . . . . 6.2.2 Collision Kinematics . . . . . . . . . . . . . . 6.2.3 Mass Selection of Atomic Clusters . . . . . . . 6.3 Elastic Scattering and Classical Theory . . . . . . . . . 6.3.1 The Differential Cross Section . . . . . . . . . 6.3.2 The Optical Rainbow . . . . . . . . . . . . . . 6.3.3 The Classical Deflection Function . . . . . . . . 6.3.4 Rainbows and Other Remarkable Oscillations . 6.4 Quantum Theory of Elastic Scattering . . . . . . . . . . 6.4.1 General Formalism . . . . . . . . . . . . . . . 6.4.2 Angular Momentum and Impact Parameter . . . 6.4.3 Partial Wave Expansion . . . . . . . . . . . . . 6.4.4 Semiclassical Approximation . . . . . . . . . . 6.4.5 Scattering Phase Shifts at Low Kinetic Energies 6.4.6 Scattering Matrices for Pedestrians . . . . . . . 6.5 Resonances . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Types and Phenomena . . . . . . . . . . . . . . 6.5.2 Formalism . . . . . . . . . . . . . . . . . . . . 6.5.3 An Example: Electron Helium Scattering . . . . 6.6 B ORN Approximation . . . . . . . . . . . . . . . . . . 6.6.1 Scattering Amplitude and Cross Section in FBA 6.6.2 RUTHERFORD Scattering . . . . . . . . . . . . 6.6.3 B ORN Approximation for Phase Shifts . . . . .

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383 383 385 387 389 393 393 396 400 402 402 403 405 409 418 418 421 422 425 428 432 436 436 438 441 444 445 446 447

xviii

7

8


Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

448 449

Inelastic Collisions – A First Overview . . . . . . . . . . 7.1 Simple Models . . . . . . . . . . . . . . . . . . . . . 7.1.1 Reactions Without Threshold Energy . . . . . 7.1.2 The Absorbing Sphere Model . . . . . . . . . 7.1.3 An Example: Charge Exchange . . . . . . . . 7.1.4 M ASSEY Criterium for Inelastic Collisions . . 7.2 Excitation Functions . . . . . . . . . . . . . . . . . . 7.2.1 Impact Excitation by Electrons and Protons . . 7.2.2 Electron Impact Excitation of He . . . . . . . 7.2.3 Finer Details in e− + He Impact Excitation . . 7.2.4 Electron Collisions with Rare Gases . . . . . 7.2.5 Electron Impact at Atomic Mercury – The F RANCK -H ERTZ Experiment . . . . . . . . . 7.2.6 Molecular Excitation by Electron Impact . . . 7.2.7 Threshold Laws for Excitation and Ionization 7.3 Scattering Theory for the Multichannel Problem . . . 7.3.1 General Formulation of the Problem . . . . . 7.3.2 Potential Matrix and Coupling Elements . . . 7.3.3 The Adiabatic Representation . . . . . . . . . 7.3.4 The Diabatic Representation . . . . . . . . . . 7.4 Semiclassical Approximation . . . . . . . . . . . . . 7.4.1 Time Dependent S CHRÖDINGER Equation . . 7.4.2 Coupling Elements . . . . . . . . . . . . . . . 7.4.3 Solution of the Coupled Differential Equations 7.4.4 L ANDAU -Z ENER Formula . . . . . . . . . . . 7.4.5 A Simple Example: Na+ + Na(3p) . . . . . . 7.4.6 S TÜCKELBERG Oscillations . . . . . . . . . . 7.5 Collision Processes with Highly Charged Ions (HCI) . 7.5.1 Above-Barrier Model . . . . . . . . . . . . . 7.5.2 An Experiment on Electron Exchange . . . . 7.5.3 HCI Collisions and Ultrafast Dynamics . . . . 7.6 Surface Hopping, Conical Intersections and Reactions Acronyms and Terminology . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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453 453 453 455 456 457 460 460 461 464 465

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466 468 470 472 472 477 478 480 484 484 486 487 489 492 496 499 501 504 506 506 510 511

Electron Impact Excitation and Ionization . . . . . . . . . . . 8.1 Formal Scattering Theory and Applications . . . . . . . . . 8.1.1 Close-Coupling Equations . . . . . . . . . . . . . . 8.1.2 Theoretical Methods and Experimental Examples . 8.2 B ORN Approximation for Inelastic Collisions . . . . . . . 8.2.1 FBA Scattering Amplitude . . . . . . . . . . . . . 8.2.2 Cross Sections . . . . . . . . . . . . . . . . . . . . 8.2.3 B ORN Approximation and RUTHERFORD Scattering

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515 515 516 520 525 525 527 528


xix

8.2.4 An Example . . . . . . . . . . . . . . . . . . . . Generalized Oscillator Strength . . . . . . . . . . . . . . 8.3.1 Definition . . . . . . . . . . . . . . . . . . . . . 8.3.2 Expansion for Small Momentum Transfer . . . . 8.3.3 Explicit Evaluation of GOS for an Example . . . 8.3.4 Integral Inelastic Cross Sections . . . . . . . . . . 8.4 Electron Impact Ionization . . . . . . . . . . . . . . . . . 8.4.1 Integral Cross Sections and the L OTZ Formula . . 8.4.2 SDCS: Energy Partitioning Between the Electrons 8.4.3 Behaviour at the Ionization Threshold . . . . . . 8.4.4 DDCS: Double-Differential Cross Section and the B ORN -B ETHE Approximation . . . . . . 8.4.5 TDCS: Triple-Differential Cross Sections . . . . . 8.4.6 Electron Momentum Spectroscopy (EMS) . . . . 8.5 Recombination . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Direct and Dielectronic Recombination . . . . . . 8.5.2 The Merged-Beams Method . . . . . . . . . . . . 8.5.3 Some Results . . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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529 530 530 531 533 534 534 537 539 540

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544 549 558 563 563 564 565 566 568

The Density Matrix – A First Approach . . . . . . . . . 9.1 Some Terminology . . . . . . . . . . . . . . . . . . . 9.1.1 Pure and Mixed States . . . . . . . . . . . . . 9.1.2 Density Operator and Density Matrix . . . . . 9.1.3 Matrix Representation for Selected Examples 9.1.4 Coherence and Degree of Polarization . . . . 9.2 Theory of Measurement . . . . . . . . . . . . . . . . 9.2.1 State Selector and Analyzer . . . . . . . . . . 9.2.2 Interaction Experiment with State Selection . 9.3 Selected Examples of the Density Matrix . . . . . . . 9.3.1 Polarization Matrix and S TOKES Parameters . 9.3.2 Atom in an Isolated 1 P State . . . . . . . . . . 9.4 Angular Distribution and Polarization of Radiation . . 9.4.1 Formulation of the Problem . . . . . . . . . . 9.4.2 General Discussion . . . . . . . . . . . . . . 9.4.3 Details of the Evaluation . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3

9

10 Optical B LOCH Equations . . . . . . . . . . . . . . . 10.1 Open Questions . . . . . . . . . . . . . . . . . . 10.2 Two Level System in Quasi-Monochromatic Light 10.2.1 Dressed States . . . . . . . . . . . . . . . 10.2.2 R ABI Frequency . . . . . . . . . . . . . . 10.2.3 Rotating Wave Approximation . . . . . .

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573 575 575 581 582 585 588 588 590 596 596 602 611 611 616 619 623 623

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625 625 629 629 630 631

xx


10.2.4 The Coupled System . . . . . . . . . . . . . . . . 10.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 M OLLOW Triplet . . . . . . . . . . . . . . . . . 10.3.2 AUTLER -T OWNES Effect . . . . . . . . . . . . . 10.4 Quantum Systems in Strong Electromagnetic Fields . . . 10.4.1 Temporal Evolution of the Density Matrix . . . . 10.4.2 Optical B LOCH Equations for a Two State System 10.5 Excitation with Continuous Wave (cw) Light . . . . . . . 10.5.1 Relaxed Steady State . . . . . . . . . . . . . . . 10.5.2 Saturation Broadening . . . . . . . . . . . . . . . 10.5.3 Broad Band and Narrow Band Excitation . . . . . 10.5.4 Rate Equations . . . . . . . . . . . . . . . . . . . 10.5.5 Continuous Excitation Without Relaxation . . . . 10.5.6 Continuous Excitation with Relaxation . . . . . . 10.6 B LOCH Equations and Short Pulse Spectroscopy . . . . . 10.6.1 Excitation with Ultrafast Laser Pulses . . . . . . . 10.6.2 Ultrafast Spectroscopy . . . . . . . . . . . . . . . 10.6.3 Rate Equations and Optical B LOCH Equations . . 10.7 STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Three Level System in Two Laser Fields . . . . . 10.7.2 Energy Splitting and State Evolution . . . . . . . 10.7.3 Experimental Realization . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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633 635 635 637 639 639 640 642 643 643 645 646 647 648 649 649 652 653 657 657 659 661 665 666

Appendix A First B ORN Approximation for e + Na(3s) → e + Na(3p) A.1 Evaluation of the Generalized Oscillator Strength . . . . . . . . A.2 Integration of the Differential Cross Section . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

669 669 672 672 672

Appendix B Guiding, Detecting and Energy Analysis of Electrons and Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 SEM, Channeltron, Microchannel Plate . . . . . . . . . . . . . . B.2 Index of Refraction, Lenses and Directional Intensity . . . . . . B.3 Hemispherical Energy Selector . . . . . . . . . . . . . . . . . . B.4 Magnetic Bottle and Other Time of Flight Methods . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

673 673 678 680 683 685 686

Appendices

Appendix C Statistical Tensor and State Multipoles . . . . . . . . C.1 Multipole Expansion of the Density Matrix . . . . . . . . . . C.2 State Multipoles and Expectation Values of Multipole Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Recoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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687 687

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690 692 694


Appendix D Optical Pumping . . . . . . . . . . . . . . . . . D.1 A Standard Case: Na(3 2 S1/2 ↔ 3 2 P3/2 ) . . . . . . . D.2 Multipole Moments and Their Experimental Detection D.3 Optical Pumping with Two Frequencies . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

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695 695 698 700 702 702

Index of Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

703

Index of Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715


1

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview, History and Magnitudes . . . . . . . . . . . . . . . . 1.1.1 Quantum Nature of Matter . . . . . . . . . . . . . . . . . 1.1.2 Orders of Magnitude . . . . . . . . . . . . . . . . . . . . 1.2 Special Theory of Relativity in a Nutshell . . . . . . . . . . . . . 1.2.1 Kinematics and Dynamics . . . . . . . . . . . . . . . . . 1.2.2 Time Dilation and LORENTZ Contraction . . . . . . . . . 1.3 Some Elementary Statistics and Applications . . . . . . . . . . . 1.3.1 Spontaneous Decay and Mean Lifetime . . . . . . . . . . 1.3.2 Absorption, LAMBERT-BEER Law . . . . . . . . . . . . 1.3.3 Kinetic Gas Theory . . . . . . . . . . . . . . . . . . . . 1.3.4 Classical and Quantum Statistics – Fermions and Bosons 1.4 The Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Photoelectric Effect and Quantization of Energy . . . . . 1.4.2 COMPTON Effect and Momentum of the Photon . . . . . 1.4.3 Pair Production . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Angular Momentum and Mass of the Photon . . . . . . . 1.4.5 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . 1.4.6 PLANCK’s Radiation Law . . . . . . . . . . . . . . . . . 1.4.7 Solar Radiation on the Earth . . . . . . . . . . . . . . . . 1.4.8 Photometry – Luminous Efficiency and Efficacy . . . . . 1.4.9 X-Ray Diffraction and Structural Analysis . . . . . . . . 1.5 The Four Fundamental Interactions . . . . . . . . . . . . . . . . 1.5.1 COULOMB and Gravitational Interaction . . . . . . . . . 1.5.2 The Standard Model of Fundamental Interaction . . . . . 1.5.3 Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 The Electron . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Particles in Electric and Magnetic Fields . . . . . . . . . . . . . 1.6.1 Charge in an Electric Field . . . . . . . . . . . . . . . . 1.6.2 Charge in a Magnetic Field . . . . . . . . . . . . . . . .

1 1 2 5 10 10 13 14 15 17 18 20 26 26 28 30 30 31 31 34 37 40 43 44 46 48 49 51 52 53 xxiii

xxiv

2


1.6.3 Cyclotron Frequency and ICR Spectrometers . . . . 1.6.4 Other Mass Spectrometers . . . . . . . . . . . . . . 1.6.5 Plasma Frequency . . . . . . . . . . . . . . . . . . 1.7 Particles and Waves . . . . . . . . . . . . . . . . . . . . . 1.7.1 DE BROGLIE Wavelength . . . . . . . . . . . . . . 1.7.2 Experimental Evidence . . . . . . . . . . . . . . . 1.7.3 Uncertainty Relation and Measurement . . . . . . . 1.7.4 Stability of the Atomic Ground State . . . . . . . . 1.8 BOHR Model of the Atom . . . . . . . . . . . . . . . . . . 1.8.1 Basic Assumptions . . . . . . . . . . . . . . . . . . 1.8.2 Radii and Energies . . . . . . . . . . . . . . . . . . 1.8.3 Atomic Units (a.u.) . . . . . . . . . . . . . . . . . 1.8.4 Energies of Hydrogen Like Ions . . . . . . . . . . . 1.8.5 Correction for Finite Nuclear Mass . . . . . . . . . 1.8.6 Spectra of Hydrogen and Hydrogen Like Ions . . . 1.8.7 Limits of the BOHR Model . . . . . . . . . . . . . 1.9 STERN-GERLACH Experiment and Space Quantization . . 1.9.1 Magnetic Moment and Angular Momentum . . . . 1.9.2 Magnetic Moment in a Magnetic Field . . . . . . . 1.9.3 The Experiment . . . . . . . . . . . . . . . . . . . 1.9.4 Interpretation of the STERN-GERLACH Experiment 1.9.5 Consequences of the STERN-GERLACH Experiment 1.10 Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Magnetic Moment of the Electron . . . . . . . . . . 1.10.2 EINSTEIN-DE-HAAS Effect . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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54 54 56 57 57 58 61 63 64 65 67 67 68 68 69 69 70 70 71 72 75 77 78 79 79 81 84

Elements of Quantum Mechanics and the H Atom . . . . . . 2.1 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Limits of Classical Theory . . . . . . . . . . . . . . 2.1.2 Probability Amplitudes in Optics . . . . . . . . . . 2.1.3 Probability Amplitudes and Matter Waves . . . . . 2.2 SCHRÖDINGER Equation . . . . . . . . . . . . . . . . . . 2.2.1 Stationary SCHRÖDINGER Equation . . . . . . . . 2.2.2 HAMILTON and Momentum Operators . . . . . . . 2.2.3 Time Dependent SCHRÖDINGER Equation . . . . . 2.2.4 Freely Moving Particle – The Most Simple Example 2.3 Basics and Definitions of Quantum Mechanics . . . . . . . 2.3.1 Axioms, Terminology and Rules . . . . . . . . . . 2.3.2 Representations . . . . . . . . . . . . . . . . . . . 2.3.3 Simultaneous Measurement of Two Observables . . 2.3.4 Operators for Space, Momentum and Energy . . . . 2.3.5 Eigenfunctions of the Momentum Operator p . . . 2.4 Particles in a Box – And the Free Electron Gas . . . . . . . 2.4.1 One Dimensional Potential Box . . . . . . . . . . .

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87 87 87 88 89 90 91 91 92 94 95 95 99 100 101 102 103 103


2.4.2 Three Dimensional Potential Box . . . . . . . . 2.4.3 The Free Electron Gas . . . . . . . . . . . . . . 2.5 Angular Momentum . . . . . . . . . . . . . . . . . . . 2.5.1 Polar Coordinates . . . . . . . . . . . . . . . . 2.5.2 Definition of Orbital Angular Momentum . . . . 2.5.3 Eigenvalues and Eigenfunctions . . . . . . . . . 2.5.4 Electron Spin . . . . . . . . . . . . . . . . . . 2.6 One Electron Systems and the Hydrogen Atom . . . . . 2.6.1 Quantum Mechanics of the One Particle System 2.6.2 Atomic Units . . . . . . . . . . . . . . . . . . . 2.6.3 Centre of Mass Motion and Reduced Mass . . . 2.6.4 Qualitative Considerations . . . . . . . . . . . . 2.6.5 Exact Solution for the H Atom . . . . . . . . . 2.6.6 Energy Levels . . . . . . . . . . . . . . . . . . 2.6.7 Radial Functions . . . . . . . . . . . . . . . . . 2.6.8 Density Plots . . . . . . . . . . . . . . . . . . . 2.6.9 Spectra of the H Atom . . . . . . . . . . . . . . 2.6.10 Expectation Values of r k . . . . . . . . . . . . . 2.6.11 Comparison with the BOHR Model . . . . . . . 2.7 Normal ZEEMAN Effect . . . . . . . . . . . . . . . . . 2.7.1 Angular Momentum in an External B-Field . . . 2.7.2 Removal of m Degeneracy . . . . . . . . . . . 2.8 Dispersion Relations . . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

xxv

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104 105 107 107 109 109 114 117 117 118 119 119 121 122 123 124 126 126 127 128 129 130 131 134 135

Periodic System and Removal of Degeneracy . . . . . . . 3.1 Shell Structure of Atoms and the Periodic System . . . . 3.1.1 Electron Configuration . . . . . . . . . . . . . . . 3.1.2 PAULI Principle . . . . . . . . . . . . . . . . . . 3.1.3 How the Shells are Filled . . . . . . . . . . . . . 3.1.4 The Periodic System of Elements . . . . . . . . . 3.1.5 Some Experimental Facts . . . . . . . . . . . . . 3.2 Quasi-One-Electron System . . . . . . . . . . . . . . . . 3.2.1 Spectroscopic Findings for the Alkali Atoms . . . 3.2.2 Quantum Defect . . . . . . . . . . . . . . . . . . 3.2.3 Screened COULOMB Potential . . . . . . . . . . . 3.2.4 Radial Wave Functions . . . . . . . . . . . . . . 3.2.5 Precise Calculations for Na as an Example . . . . 3.2.6 Quantum Defect Theory . . . . . . . . . . . . . . 3.2.7 MOSLEY Diagrams . . . . . . . . . . . . . . . . 3.3 Perturbation Theory for Stationary Problems . . . . . . . 3.3.1 Perturbation Ansatz for the Non-degenerate Case . 3.3.2 Perturbation Theory in 1st Order . . . . . . . . . 3.3.3 Perturbation Theory in 2nd Order . . . . . . . . .

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137 137 138 138 139 140 142 144 145 146 148 149 150 152 159 161 161 162 163

xxvi


3.3.4 Perturbation Theory for Degenerate States . . . . . 3.3.5 Application of Perturbation Theory to Alkali Atoms Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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164 165 167 168

Non-stationary Problems: Dipole Excitation with One Photon . . . 4.1 Electromagnetic Waves: Electric Field, Intensity, Polarization and Photon Spin . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Electric Field and Intensity . . . . . . . . . . . . . . . . 4.1.2 Basis Vectors of Polarization . . . . . . . . . . . . . . . 4.1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . 4.1.4 Angular Momentum of the Photon . . . . . . . . . . . . 4.2 Introduction to Absorption and Emission . . . . . . . . . . . . . 4.2.1 Stationary States . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Optical Spectroscopy – General Concepts . . . . . . . . 4.2.3 Induced Processes . . . . . . . . . . . . . . . . . . . . . 4.2.4 Spontaneous Emission – Classical Interpretation . . . . . 4.2.5 The EINSTEIN A and B Coefficients . . . . . . . . . . . 4.3 Time Dependent Perturbation Theory . . . . . . . . . . . . . . . 4.3.1 General Approach . . . . . . . . . . . . . . . . . . . . . 4.3.2 Perturbation Ansatz for Transition Amplitudes . . . . . . 4.3.3 Transitions in a Monochromatic Plane Wave . . . . . . . 4.3.4 Dipole Approximation . . . . . . . . . . . . . . . . . . . 4.3.5 Absorption Probabilities . . . . . . . . . . . . . . . . . . 4.3.6 Absorption and Emission: A First Summary . . . . . . . 4.4 Selection Rules for Dipole Transitions . . . . . . . . . . . . . . 4.4.1 Angular Momentum and Selection Rules . . . . . . . . . 4.4.2 Transition Amplitudes in the Helicity Basis . . . . . . . . 4.4.3 Transition Matrix Elements and Selection Rules . . . . . 4.4.4 An Example for E1 Transitions: The H Atom . . . . . . . 4.5 Angular Dependence of Dipole Radiation . . . . . . . . . . . . . 4.5.1 Semiclassical Picture . . . . . . . . . . . . . . . . . . . 4.5.2 Angular Distributions from Quantum Mechanics . . . . . 4.6 Strength of Dipole Transitions . . . . . . . . . . . . . . . . . . . 4.6.1 Line Strength . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Spontaneous Transition Probabilities . . . . . . . . . . . 4.6.3 Induced Transitions . . . . . . . . . . . . . . . . . . . . 4.7 Superposition of States, Quantum Beats and Jumps . . . . . . . . 4.7.1 Coherent Population by Optical Transitions . . . . . . . . 4.7.2 Time Dependence of Optically Excited States – Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Quantum Jumps . . . . . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 170 170 171 174 175 176 176 177 178 181 184 186 186 187 188 189 190 193 196 196 198 200 201 203 204 206 212 212 213 215 217 217 220 224 225 226


xxvii

5

Linewidths, Photoionization, and More . . . . . . . . . . . . 5.1 Line Broadening . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Natural Linewidth . . . . . . . . . . . . . . . . . 5.1.2 Dispersion . . . . . . . . . . . . . . . . . . . . . 5.1.3 Collisional Line Broadening . . . . . . . . . . . . 5.1.4 DOPPLER Broadening . . . . . . . . . . . . . . . 5.1.5 VOIGT Profile . . . . . . . . . . . . . . . . . . . 5.2 Oscillator Strength and Cross Section . . . . . . . . . . . 5.2.1 Transition Rates Generalized . . . . . . . . . . . 5.2.2 Oscillator Strength . . . . . . . . . . . . . . . . . 5.2.3 Absorption Cross Section . . . . . . . . . . . . . 5.2.4 Different Notations – Radiative-Transfer in Gases 5.3 Multi-photon Processes . . . . . . . . . . . . . . . . . . 5.3.1 Two-Photon Excitation . . . . . . . . . . . . . . 5.3.2 Two-Photon Emission . . . . . . . . . . . . . . . 5.4 Magnetic Dipole and Electric Quadrupole Transitions . . 5.5 Photoionization . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Process and Cross Section . . . . . . . . . . . . . 5.5.2 BORN Approximation for Photoionization . . . . 5.5.3 Angular Distribution of Photoelectrons . . . . . . 5.5.4 Cross Sections in Theory and Experiment . . . . . 5.5.5 Multi-photon Ionization (MPI) . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 227 227 232 233 234 236 238 238 238 240 242 244 245 248 250 254 255 256 260 261 265 270 271

6

Fine Structure and L AMB Shift . . . . . . . . . . . . . . . 6.1 Methods of High Resolution Spectroscopy . . . . . . . 6.1.1 Grating Spectrometers . . . . . . . . . . . . . . 6.1.2 Interferometers . . . . . . . . . . . . . . . . . . 6.1.3 D OPPLER Free Spectroscopy in Atomic Beams 6.1.4 Collinear Laser Spectroscopy in Ion Beams . . . 6.1.5 Hole Burning . . . . . . . . . . . . . . . . . . 6.1.6 D OPPLER Free Saturation Spectroscopy . . . . 6.1.7 R AMSEY Fringes . . . . . . . . . . . . . . . . 6.1.8 D OPPLER Free Two-Photon Spectroscopy . . . 6.2 Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . 6.2.1 Experimental Findings . . . . . . . . . . . . . . 6.2.2 Magnetic Moments in a Magnetic Field . . . . . 6.2.3 General Considerations About LS Interaction . 6.2.4 Magnitude of Spin-Orbit Interaction . . . . . . 6.2.5 Angular Momentum Coupling . . . . . . . . . . 6.2.6 Terminology for Atomic Structure . . . . . . . 6.3 Quantitative Determination of Fine Structure . . . . . . 6.3.1 FS Terms from D IRAC Theory . . . . . . . . . 6.3.2 Fine Structure of the H Atom . . . . . . . . . . 6.3.3 Fine Structure of Alkali and Other Atoms . . . .

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273 274 274 278 282 283 284 285 288 289 293 293 294 295 296 297 301 303 303 306 307

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xxviii


6.4

7

Selection Rules and Intensities of Transitions . . . . . . . . . 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Transitions Between Sublevels vs. Overall Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Some Useful Relations for Spectroscopic Practice . . 6.5 L AMB Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Fine Structure and L AMB Shift for the Hα Line . . . . 6.5.2 Microwave and RF Transitions – D OPPLER Free . . . 6.5.3 Experiment of L AMB and R ETHERFORD . . . . . . . 6.5.4 Precision Spectroscopy of the H Atom . . . . . . . . 6.5.5 LAMB Shift in Highly Charged Ions . . . . . . . . . 6.5.6 QED and F EYNMAN Diagrams . . . . . . . . . . . . 6.5.7 On the Theory of the L AMB Shift . . . . . . . . . . . 6.6 Electron Magnetic Moment Anomaly . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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310 310

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310 313 316 316 317 317 319 322 324 326 331 336 337

Helium and Other Two Electron Systems . . . . . . . . . . . 7.1 Introduction and Empirical Findings . . . . . . . . . . . 7.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 He I Term Scheme . . . . . . . . . . . . . . . . . 7.2 Some Quantum Mechanics of Two Electrons . . . . . . . 7.2.1 HAMILTON Operator for the Two-Electron System 7.2.2 Two Particle Wave Functions . . . . . . . . . . . 7.2.3 Zero Order Approximation: No e− e− Interaction . 7.2.4 The He Ground State – Perturbation Theory . . . 7.2.5 Variational Theory and Present State-of-the-Art . 7.3 PAULI Principle and Excited States in He . . . . . . . . . 7.3.1 Exchange of Two Identical Particles . . . . . . . . 7.3.2 Symmetries of Spatial and Spin Wave Functions . 7.3.3 Perturbation Theory for (Singly) Excited States . 7.3.4 An Afterthought . . . . . . . . . . . . . . . . . . 7.4 Fine Structure . . . . . . . . . . . . . . . . . . . . . . . 7.5 Electric Dipole Transitions . . . . . . . . . . . . . . . . 7.6 Double Excitation and Autoionization . . . . . . . . . . . 7.6.1 Doubly Excited States . . . . . . . . . . . . . . . 7.6.2 Autoionization, FANO Profile . . . . . . . . . . . 7.6.3 Resonance Line Profiles . . . . . . . . . . . . . . 7.7 Quasi-two-Electron Systems . . . . . . . . . . . . . . . . 7.7.1 Alkaline Earth Elements . . . . . . . . . . . . . . 7.7.2 Mercury . . . . . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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341 342 342 343 344 344 345 346 348 350 351 351 352 355 358 360 362 365 365 366 369 371 371 372 374 374

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xxix

8

Atoms in External Fields . . . . . . . . . . . . . . . . . . . . 8.1 Atoms in a Static Magnetic Field . . . . . . . . . . . . . 8.1.1 The General Case . . . . . . . . . . . . . . . . . 8.1.2 ZEEMAN Effect in Low Fields . . . . . . . . . . . 8.1.3 PASCHEN-BACK Effect . . . . . . . . . . . . . . 8.1.4 Do Angular Momenta Actually Precess? . . . . . 8.1.5 In Between Low and High Magnetic Field . . . . 8.1.6 Avoided Crossings . . . . . . . . . . . . . . . . . 8.1.7 Paramagnetism . . . . . . . . . . . . . . . . . . . 8.1.8 Diamagnetism . . . . . . . . . . . . . . . . . . . 8.2 Atoms in an Electric Field . . . . . . . . . . . . . . . . . 8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . 8.2.2 Significance . . . . . . . . . . . . . . . . . . . . 8.2.3 Atoms in a Static, Electric Field . . . . . . . . . . 8.2.4 Basic Considerations about Perturbation Theory . 8.2.5 Matrix Elements . . . . . . . . . . . . . . . . . . 8.2.6 Perturbation Series . . . . . . . . . . . . . . . . . 8.2.7 Quadratic STARK Effect . . . . . . . . . . . . . . 8.2.8 Linear STARK Effect . . . . . . . . . . . . . . . . 8.2.9 An example: RYDBERG States of Li . . . . . . . 8.2.10 Polarizability . . . . . . . . . . . . . . . . . . . . 8.2.11 Susceptibility . . . . . . . . . . . . . . . . . . . 8.3 Long Range Interaction Potentials . . . . . . . . . . . . . 8.4 Atoms in an Oscillating Electromagnetic Field . . . . . . 8.4.1 Dynamic STARK Effect . . . . . . . . . . . . . . 8.4.2 Index of Refraction . . . . . . . . . . . . . . . . 8.4.3 Resonances – Dispersion and Absorption . . . . . 8.4.4 Fast and Slow Light . . . . . . . . . . . . . . . . 8.4.5 Elastic Scattering of Light . . . . . . . . . . . . . 8.5 Atoms in a High Laser Field . . . . . . . . . . . . . . . . 8.5.1 Ponderomotive Potential . . . . . . . . . . . . . . 8.5.2 KELDISH Parameter . . . . . . . . . . . . . . . . 8.5.3 From Multi-photon Ionization to Saturation . . . . 8.5.4 Tunnelling Ionization . . . . . . . . . . . . . . . 8.5.5 Recollision . . . . . . . . . . . . . . . . . . . . . 8.5.6 High Harmonic Generation (HHG) . . . . . . . . 8.5.7 Above-Threshold Ionization in High Laser Fields Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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377 377 377 380 384 386 388 392 394 396 399 399 399 400 401 402 405 405 407 409 411 413 414 418 418 420 421 422 427 432 432 434 434 436 438 439 441 442 444

9

Hyperfine Structure . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . 9.2 Magnetic Dipole Interaction . . . . . . . . . . . . 9.2.1 General Considerations and Examples . . 9.2.2 The Magnetic Field of the Electron Cloud

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447 447 452 452 453

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xxx


9.2.3 Nonvanishing Orbital Angular Momenta . . 9.2.4 The FERMI Contact Term . . . . . . . . . . 9.2.5 Some Numbers . . . . . . . . . . . . . . . . 9.2.6 Optical Transitions Between HFS Multiplets 9.3 ZEEMAN Effect of Hyperfine Structure . . . . . . . 9.3.1 Hyperfine Hamiltonian with Magnetic Field 9.3.2 Low Magnetic Fields . . . . . . . . . . . . 9.3.3 High and Very High Magnetic Fields . . . . 9.3.4 Arbitrary Fields, BREIT-RABI Formula . . . 9.4 Isotope Shift and Electrostatic Nuclear Interactions . 9.4.1 Potential Expansion . . . . . . . . . . . . . 9.4.2 Isotope Shift . . . . . . . . . . . . . . . . . 9.4.3 Quadrupole Interaction Energy . . . . . . . 9.4.4 HFS Level Splitting . . . . . . . . . . . . . 9.5 Magnetic Resonance Spectroscopy . . . . . . . . . 9.5.1 Molecular Beam Resonance Spectroscopy . 9.5.2 EPR Spectroscopy . . . . . . . . . . . . . . 9.5.3 NMR Spectroscopy . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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457 458 459 460 461 462 462 464 467 471 471 473 477 480 482 482 484 487 491 492

10 Multi-electron Atoms . . . . . . . . . . . . . . . . . . . . . 10.1 Central Field Approximation . . . . . . . . . . . . . . 10.1.1 Hamiltonian for a Multi-electron System . . . . 10.1.2 Centrally Symmetric Potential . . . . . . . . . . 10.1.3 HARTREE Equations and SCF Method . . . . . 10.1.4 HARTREE Method . . . . . . . . . . . . . . . . 10.1.5 THOMAS-FERMI Potential . . . . . . . . . . . . 10.2 HARTREE-FOCK Method . . . . . . . . . . . . . . . . 10.2.1 PAULI Principle and SLATER Determinant . . . 10.2.2 HARTREE-FOCK Equations . . . . . . . . . . . 10.2.3 Configuration Interaction (CI) . . . . . . . . . . 10.2.4 KOOPMAN’s Theorem . . . . . . . . . . . . . . 10.3 Density Functional Theory . . . . . . . . . . . . . . . 10.4 Complex Spectra . . . . . . . . . . . . . . . . . . . . . 10.4.1 Spin-Orbit Interaction and Coupling Schemes . 10.4.2 Examples of Complex Spectra . . . . . . . . . . 10.5 X-Ray Spectroscopy and Photoionization . . . . . . . . 10.5.1 Absorption and Emission from Inner Shells . . . 10.5.2 Characteristic X-Ray Spectra – MOSLEY’s Law 10.5.3 Cross Sections for X-Ray Ionization . . . . . . 10.5.4 Photoionization at Intermediate Energies . . . . 10.6 Sources for X-Rays . . . . . . . . . . . . . . . . . . . 10.6.1 X-Ray Tubes . . . . . . . . . . . . . . . . . . . 10.6.2 Synchrotron Radiation, Introduction . . . . . .

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495 496 496 497 498 500 501 503 503 506 508 509 510 512 512 514 519 520 522 524 527 530 530 531


xxxi

10.6.3 Synchrotron Radiation, Quantitative Relations 10.6.4 Undulators and Wigglers . . . . . . . . . . . 10.6.5 Free Electron Laser (FEL) . . . . . . . . . . . 10.6.6 Relativistic THOMSON Scattering . . . . . . . 10.6.7 Laser Based X-Ray Sources . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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536 540 542 543 543 544 546

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551 551 553 554 554 556 557 557

Appendix B Angular Momenta, 3j and 6j Symbols . . . . . . B.1 Angular Momenta . . . . . . . . . . . . . . . . . . . . . B.1.1 General Definitions . . . . . . . . . . . . . . . . B.1.2 Orbital Angular Momenta – Spherical Harmonics B.2 Coupling of Two Angular Momenta: CLEBSCH-GORDAN Coefficients and 3j Symbols . . . . . . . . . . . . . . . . B.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . B.2.2 Orthogonality and Symmetries . . . . . . . . . . B.2.3 General Formulae . . . . . . . . . . . . . . . . . B.2.4 Special Cases . . . . . . . . . . . . . . . . . . . B.3 RACAH Function and 6j Symbols . . . . . . . . . . . . . B.3.1 Definition . . . . . . . . . . . . . . . . . . . . . B.3.2 Orthogonality and Symmetries . . . . . . . . . . B.3.3 General Formulae . . . . . . . . . . . . . . . . . B.3.4 Special Cases . . . . . . . . . . . . . . . . . . . B.4 Four Angular Momenta and 9j Symbols . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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559 559 559 562

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564 564 565 566 567 568 568 569 570 571 571 572 572

Appendix C Matrix Elements . . . . . . . . . . . . . . . . C.1 Tensor Operators . . . . . . . . . . . . . . . . . . . C.1.1 Definition . . . . . . . . . . . . . . . . . . C.1.2 WIGNER-ECKART Theorem . . . . . . . . . C.2 Products of Tensor Operators . . . . . . . . . . . . C.2.1 Products of Spherical Harmonics . . . . . . C.2.2 Matrix Elements of the Spherical Harmonics C.3 Reduction of Matrix Elements . . . . . . . . . . . .

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575 575 575 576 578 579 580 582

Appendices Appendix A Constants, Units and Conversions . A.1 Fundamental Physical Constants and Units A.2 SI and Atomic Units . . . . . . . . . . . . A.3 SI and GAUSS Units . . . . . . . . . . . . A.4 Radian and Steradian . . . . . . . . . . . A.5 Dimensional Analysis . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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xxxii


C.3.1 Matrix Elements of the Spherical Harmonics in LS Coupling . . . . . . . . . . . . . . . . . . . . . . . C.3.2 Scalar Products of Angular Momentum Operators . C.3.3 Components of Angular Momenta . . . . . . . . . C.4 Electromagnetically Induced Transitions . . . . . . . . . . C.4.1 Electric Dipole Transitions . . . . . . . . . . . . . C.4.2 Electric Quadrupole Transitions . . . . . . . . . . . C.4.3 Magnetic Dipole Transitions . . . . . . . . . . . . C.5 Radial Matrix Elements . . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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583 585 586 587 588 588 589 590 592 592

Appendix D Parity and Reflection Symmetry . . . . . . . . . . . . . . D.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Multi-electron Systems . . . . . . . . . . . . . . . . . . . . . . D.3 Reflection Symmetry of Orbitals – Real and Complex Basis States D.4 Reflection Symmetry in the General Case . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

593 593 594 595 599 603 603

Appendix E Coordinate Rotation E.1 EULER Angles . . . . . . E.2 Rotation Matrices . . . . E.3 Entangled States . . . . . E.4 Real Rotation Matrices . References . . . . . . . . . . .

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605 605 606 609 610 611

Appendix F Multipole Expansions and Multipole Moments F.1 Laplace Expansion . . . . . . . . . . . . . . . . . . . F.2 Electrostatic Potential . . . . . . . . . . . . . . . . . F.3 Multipole Tensor Operators . . . . . . . . . . . . . . F.3.1 The Quadrupole Tensor . . . . . . . . . . . . F.3.2 General Multipole Tensor Operators . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

613 613 614 616 617 619 621 622

Appendix G Convolutions and Correlation Functions G.1 Definition and Motivation . . . . . . . . . . . . G.2 Correlation Functions and Degree of Coherence G.3 Gaussian Profile . . . . . . . . . . . . . . . . . G.4 Hyperbolic Secant . . . . . . . . . . . . . . . . G.5 LORENTZ Profile . . . . . . . . . . . . . . . . . G.6 VOIGT Profile . . . . . . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

623 623 625 626 627 628 628 629 629

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . . . . .


xxxiii

Appendix H Vector Potential, Dipole Approximation, Oscillator Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.1 Interaction of the Field of an Electromagnetic Wave with an Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . H.1.1 Vector Potential . . . . . . . . . . . . . . . . . . . H.1.2 Intensity . . . . . . . . . . . . . . . . . . . . . . . H.1.3 Static Magnetic Field . . . . . . . . . . . . . . . . H.1.4 Relation Between Matrix Elements of p and r . . . H.1.5 Ponderomotive Potential . . . . . . . . . . . . . . . H.1.6 Series Expansion of the Perturbation and the Dipole Approximation . . . . . . . . . . . . . . . . . . . . H.2 Line Strength and Oscillator Strength . . . . . . . . . . . . H.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . H.2.2 THOMAS-REICHE-KUHN Sum Rule . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

631

. . . . . .

. . . . . .

. . . . . .

631 631 632 633 634 634

. . . . . .

. . . . . .

. . . . . .

635 636 636 639 641 641

Appendix I FOURIER Transforms and Spectral Distributions of Light I.1 Short Summary on FOURIER Transforms . . . . . . . . . . . . . I.2 How Electromagnetic Fields are Written . . . . . . . . . . . . . I.3 The Intensity Spectrum . . . . . . . . . . . . . . . . . . . . . . I.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.4.1 Gaussian Distribution . . . . . . . . . . . . . . . . . . . I.4.2 Hyperbolic Secant . . . . . . . . . . . . . . . . . . . . . I.4.3 Rectangular Wave-Train . . . . . . . . . . . . . . . . . . I.4.4 Rectangular Spectrum . . . . . . . . . . . . . . . . . . . I.4.5 Exponential and LORENTZ Distributions . . . . . . . . . I.5 Fourier Transform in Three Dimensions . . . . . . . . . . . . . Acronyms and Terminology . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

643 643 646 648 649 650 651 652 652 653 655 657 658

Appendix J Continuum . . . . . . . . . . . . . . . . . . . . . J.1 Normalization of Continuum Wave Functions . . . . . J.2 Plane Waves in 3D . . . . . . . . . . . . . . . . . . . . J.2.1 Expansion in Spherical Harmonics . . . . . . . J.2.2 Normalization in Momentum and Energy Scale Acronyms and Terminology . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

659 659 661 661 662 663 663

Index of Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

665

Index of Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

679

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

About the Authors

Ingolf V. Hertel was born in 1941 in Dresden, 1967 Diplom in Physics, Universität Freiburg, Ph.D. thesis at the University of Southampton UK, 1969 Dr. rer. nat. Universität Freiburg, Assistent University Mainz, 1970 Associate Professor University Kaiserslautern, 1978 Full Professor for Experimental Physics Freie Universität Berlin, 1986 Full Professor Universität Freiburg, Extended Research Periods at JILA University of Colorado Boulder USA and Orsay France, 1992 to 2009 Director at Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy in Berlin-Adlershof, 1993 to 2009 also Full Professor FU Berlin, since 2010 Wilhelm und Else Heraeus Senior Professor for the Enhancement of Teachers Education at Humboldt Universität zu Berlin.

Claus-Peter Schulz was born in 1953 in Berlin, 1981 Diplom in Physics TU Berlin, 1987 Dr. rer. nat. Freie Universität Berlin, Postdoc at JILA University of Colorado Boulder USA, 1988 Assistent Universität Freiburg, since 1993 Staff Scientist at Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy in Berlin-Adlershof, Extended Research Periods at Université Paris-Nord and Orsay France as well as at JILA Boulder USA.

xxxv

1

Lasers, Light Beams and Light Pulses

Light plays a key role in optics and spectroscopy – using the term “light” in a general sense for all electromagnetic radiation from the terahertz to the hard X- and γ -ray spectral range. So far, we have tacitly assumed that the wave nature of light is well described by plane, monochromatic waves (Sect. 4.2, Vol. 1). The spatial and temporal profile has not yet played a role. We shall abandon these restrictions now.

Overview

We still use a classical wave description of light, but start in Sect. 1.1 with a brief introduction into the physics of lasers – certainly the most important tools of modern optics and spectroscopy. In Sect. 1.2 Gaussian light beams are explored, and their manipulation and measurement is illustrated. Section 1.3 gives a precise definition of polarization and describes some experimental tools for its characterization. Wave-packets are discussed in Sect. 1.4, with focus on short pulses as interesting examples from current research. Section 1.5 introduces “correlation functions” and describes methods for determining short pulse durations. Finally, in Sect. 1.6 we explore some characteristics of intense laser fields – a topic of great importance in present research – and thus transcend classical, linear spectroscopy.

1.1

Lasers – A Brief Introduction

It is difficult to imagine modern science, technology or even every day life without lasers. For modern atomic and molecular physics they are essential as they are of course for the optical sciences. A wealth of literature on lasers exists, about which we cannot provide a comprehensive overview. A small selection is found in S IEG MAN (1986), H ODGSON and W EBER (2005), M ILLONI and E BERLY (2010). The history of lasers is a show case for how innovation is based on serendipity in science. The theoretical foundations have been laid by E INSTEIN in 1916 with his ground breaking discussion of induced emission for an alternative derivation of P LANCK’s radiation law. More than 40 years passed before T OWNES and his collaborators (G ORDON et al. 1955) built the first ammonia maser (microwave © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_1

1

2

1


amplification by stimulated emission of radiation), a molecular amplifier for microwave radiation. Without dwelling in the historical details, we mention that the first published proposal (based on quantitative estimates) to use this scheme also for the infrared and visible spectral range was made by S CHAWLOW and T OWNES (1958). The first solid state laser, the ruby laser, was demonstrated by M AIMAN (1960). And the first gas laser was the famous helium-neon laser, which was brought to oscillation by JAVAN et al. (1961) at 1.1 µm. It is unclear who first introduced the term laser (light amplification by stimulated emission of radiation). One finds, however, a number of papers already in 1961 which use this term, while S CHAWLOW – also in 1961 – titled a Scientific American article: “Optical masers – These devices generate light in such a manner as to open a whole realm of applications for electromagnetic radiation – Salient feature of light they produce is that its waves are all in step” (B LOEMBERGEN and S HAWLOW 1981). A felicitous characterization and prognosis! Soon also the first liquid lasers were explored and tuneable dye lasers have been made available more or less simultaneously by S CHÄFER et al. (1966) and S OROKIN and L ANKARD (1966). The progress in laser physics and technology during the past five decades and the spectroscopic and technological applications of lasers are spectacular and continuing. We mention a few key aspects: 1. Today a broad spectral range is covered by laser systems, from the microwave region into the X-ray region, in many instances combined with excellent tunability of wavelengths. 2. The stability and monochromaticity achieved today allows one to maintain some standard light frequencies to a precision of a few Hz (i.e. with an accuracy on the order of 1015 ) for very long times. 3. Conversely, phase controlled light pulses in the visible spectral range of a few fs are provided today on a stable and reliable basis for experiments. The shortest laser pulses available today (typically in the soft X-ray region) have a durations of several attoseconds (1 as = 10−18 s). 4. A historic overview on peak intensities accessible in the focus of a pulsed laser beam is given in Fig. 1.1. To put this into perspective, we recall that a 100 W incandescent light bulb emits about 120 cd, i.e. 0.176 W sr−1 or ca. 2.1 mW cm−2 in a distance of 1 m.1 Note (right hand scale in Fig. 1.1) the extremely high ponderomotive potentials of free electrons associated with such laser pulses (see Sect. 8.5 in Vol. 1). With the new large scale facilities (e.g. the European Laser Institute, ELI) one hopes to reach 1024 W cm−2 in the coming years, and thus to enter into ultra relativistic optics.

1 As discussed in Sect. 1.4.8, Vol. 1, 1 cd at 555 nm corresponds a “radiant intensity” of (1/683) W sr−1 . In contrast, a relatively weak tunable dye laser generates 100 mW CW output power without problems. Even without focussing, this implies much more than the 0.63 W cm−2 needed to saturate the Na D transition – as we shall show in Sect. 10.5.2. The same laser might, well focussed, provide already 1.3 × 107 W cm−2 . Presently, large scale high intensity short pulsed lasers facilities achieve about 1020 to 1021 W cm−2 .


3

focussed intensiy / Wcm-2

non linear QED 1025

2.35 × 1028: λc e0 E0 = 2me c2

ultra relativistic optics

1 TeV ELI

1020

relativistic optics 8.5 × 1018: Up = mc 2 @ λ = 800nm 3.51×1016:

1015

EH (in H atom at a0)

1 MeV

strong laser fields

chirped pulse amplification (CPA) 1 eV mode synchronization (mode locking) 3.69 × 1011: x 0 ~ a0 @ λ = 800nm

1010

Q switching

1960

1970

1980

1990

2000

2010

ponderomotive potential Up @ λ = 800nm

1.1

Fig. 1.1 Progress in generating highest intensities in laser pulses since the realization of the first laser systems. Marked in red are physical phenomena connected to the respective intensities. Black labels indicate methodological advances. The trend curve (full, black), adapted from the inventor of “Chirped pulse amplification, CPA” Mourou (S TRICKLAND and M OUROU 1985) had to be modified somewhat downward to match reality

1.1.1

Basic Principle

The fundamental concept for a laser, schematically illustrated in Fig. 1.2, is essentially the same as that for any generator of high frequency electric or magnetic fields. The three key elements are: 1. the amplifier which generates the oscillation (out of noise initially) and compensates for losses, 2. the resonator which suppresses all unwanted frequencies, and

RF generator (a)

feedback resonator

feedback

power source

L

C

resonator

L R1= 100%

RF output

amplifier

La amplifier medium

laser (b)

laser beam output R2 < 100%

energy

(pump)

Fig. 1.2 Scheme of (a) an RF generator and (b) a laser. Key elements are in both cases amplifier, resonator and feedback mechanism – as well as of course a power source

4

1


3. the feedback which returns most of the amplified signal to the entrance of the amplifier. Thus, a signal once generated is amplified repeatedly during multiple turnarounds through the system – if, and only if, it has the desired frequency. Eventually a balance between the gain in the amplifier and the unavoidable losses in the resonator (as well as due to the desired output) is achieved. The losses of the resonator are characterized by the quality factor (Q factor). It is defined by the damping of a once excited electric field amplitude ωt i + c.c., (1.1) E(t) = E0 exp iωt − 2 2Q with ω = 2πν being the (angular) resonance frequency of the resonator. Correspondingly, the radiation energy W stored in the (passive) resonator and the intensity I ∝ W which propagates within the resonator decays as I (t) = I0 exp(−ωt/Q) = I0 (t) exp(−t/τr ),

(1.2)

if it was I0 at time t = 0. Thus, one may say that the average lifetime of a photon in the resonator is τr = Q/ω.

(1.3)

The spectrum of this damped intensity is described by a L ORENTZ distribution (see Appendix I.4.5 in Vol. 1) with the resonator bandwidth ωr (FWHM). It is related to the Q factor by Q = ω/ωr = ν/νr ,

(1.4)

and ωr = 2πνr = 1/τr . Since usually the laser is designed to provide some output, typically these ‘useful’ losses determine the quality of the resonator and its bandwidth. Correspondingly, an amplifier is characterized by its amplification α. Passing through the amplifier medium, a signal increases with time t or distance z = ct as I (t) = I0 (t) exp(αt) = I0 (z) exp(αz/c), α=

1 dI , I dt

and

with α(ω) = 1/s.

(1.5) (1.6)

Laser activity is expected if the amplification is larger than the losses: α > ω/Q = 1/τr .

1.1.2

(1.7)

FABRY-P ÉROT Resonator

Generally speaking, laser resonators are specially designed FABRY-P ÉROT interferometers (FPI). We have already introduced this spectroscopic tool in Sect. 6.1.2, Vol. 1. Here we briefly summarize its characteristics, as far as relevant for using

1.1


5

it as a laser resonator. As indicated Fig. 1.2, in the simplest case a FABRY-P ÉROT resonator consists of two mirror plates positioned parallel at a distance L. These mirrors are polished with highest precision. One of them has a reflectivity of R1 = 100 % (as good as possible), while the other has a somewhat smaller R2 < 100 % in order to transmit the fraction (1 − R2 ) of the intensity stored inside the resonator for external use. Depending on the specific application, the mirrors may be planar or curved (for details see Sect. 1.1.3). The two key parameters of the resonator are: 1. The free spectral range (FSR) according to (6.9) in Vol. 1 determines the distance (in frequency) of two maxima of transmission. It is identical to the inverse turnaround time Tr :2 νFSR = c/2L = 1/Tr .

(1.8)

In wavenumbers this corresponds to ¯νFSR = 1/2L. For whole numbers z, called order of the interference or index of the mode, light of the frequencies ν(z) = zνFSR

(1.9)

is transmitted with highest probability. At these – and only at these – frequencies may electromagnetic energy be stored inside the resonator: as standing waves. They are called the longitudinal modes of the resonator. 2. The finesse F of a FABRY-P ÉROT interferometer is the ratio of the free spectral range to the √ transmitted FWHM νr of the intensity in the passive resonator. With R = R1 R2 the finesse is found to be √ π R νFSR . (1.10) = F= νr 1−R The spectral transmission profile of the resonator is described by an A IRY function (6.14), Vol. 1. For a finesse F 5 the latter may well be approximated by a series of L ORENTZ profiles of FWHM νr . With (1.9) and (1.10) the Q factor (1.4) of the resonator becomes Q=

ν ν =F = F × z, νr νFSR

(1.11)

with the order of interference z according to (1.9). By the way, the Q factor is identical to the resolution (6.17), Vol. 1 of the FPI when it is used as a spectrometer, with F corresponding to an effective number of interfering beams. Finally, the effective lifetime (1.3) of a photon in the resonator is given by τr =

F 1 zF zF FL = = = = , ωr ω0 2πν0 πc 2πνFSR

2 More precisely, we would have to use here the respective group velocity v

(1.12)

g (see Sect. 8.4.4, Vol. 1) in the different media instead of the speed of light c in vacuum. To keep things simple we assume, however, n ≡ 1 for the index of refraction inside the whole resonator.

6

1


with ωr = 2πνr . To give two typical numerical examples: due to its low amplification, a He-Ne laser allows only little output coupling (typically R 99 %, F 300), while for a pulsed excimer laser the contrary is the case (R = 30 %, F = 2.5). At a resonator length of 1 m this leads to an effective photon lifetime of τr = 330 ns and 2.6 ns, respectively. Of course, several additional types of losses have to be taken into account beyond output coupling. Absorption in the amplifier medium or in optical elements is one of them. An absorbing medium of length L1 (absorption coefficient μ, see Eq. (4.21) in Vol. 1) causes losses per resonator turnaround time Tr which are ∝ exp(−2μL1 ) or, recalculated per time t I /I0 = exp −2μL1 (t/Tr ) = exp −μ(L1 /L)ct = exp[−t/τa ], (1.13) with τa being the effective photon life time due to absorption.

1.1.3

Stable, Transverse Modes and Diffraction Losses

Of particular importance are losses due to diffraction. We shall see that finally they are responsible for the spatial profile of the electromagnetic field in the resonator. They define the radial mode structure of the light. Up to now we have treated the FABRY-P ÉROT resonator simply as if there were no lateral boundaries: the field was described by infinitely extended plane waves. In reality, the mirrors and the amplifier medium have finite diameters: we speak about an “open resonator”. Even by geometrical optics one may easily visualize, how light rays move out of the resonator after a few turnaround passages when they are not normally incident onto the mirrors. In an early, pioneering publication KOGELNIK and L I (1966) were able to show that the FABRY-P ÉROT resonator with plane parallel mirrors does not represent an ideal resonator configuration in this respect. However, a range of “stable” mirror configurations exists for which even non-paraxial rays remain inside the resonator after multiple reflections from both end-mirrors. For these 0 < (1 − L/r1 )(1 − L/r2 ) < 1,

(1.14)

must hold, with the resonator length L, and the radii r1 and r2 giving the curvature of the end mirrors. This criterion is illustrated schematically in the stability diagram Fig. 1.3. From a wave-optical point of view diffraction occurs at all beam limiting apertures. During each round trip the light field is thus confined twice, at the end mirrors. Its energy content is reduced accordingly. One may visualize this process rolled out along the optical z-axis as shown in Fig. 1.4. Mathematically, the steady state in the resonator is obtained by requiring that the profile of the field in the ρϕ plane (perpendicular to the optical axis), Ej +1 (ρ, ϕ) = e−κ+iδ Ej (ρ, ϕ),

1.1


7

Fig. 1.3 Stability diagram for laser resonators according to KOGELNIK and L I (1966). Instable regions are hatched. One sees that a setup with plane parallel mirrors corresponds to the boundary between stable and unstable regions, and the same holds for the opposite extreme of two concentric mirrors. The confocal resonator plays a special role in the centre of stable and unstable regions

is reproduced from reflection to reflection – apart from the unavoidable overall losses (described by the damping factor κ) and an overall phase shift δ. Quantitatively, one may understand Ej +1 (ρ, ϕ) as the diffraction pattern of Ej (ρ, ϕ) arising from the limiting aperture Sj . It may be calculated by F RAUNHOFER diffraction theory. We shall discuss the latter in Sect. 1.2.2. Here we just communicate a few important results without going into the details of the calculation: In addition to the longitudinal mode structure discussed in Sect. 1.1.2 (standing waves in z-direction) radial intensity profiles evolve in the ρϕ plane of an open resonator. Just as freely propagating waves they are of transverse nature (E ⊥ B ⊥ k), so called transverse modes which are characterized as TEMij , i and j indicating the number of nodes in ρ and ϕ direction, respectively. Some examples are sketched schematically in Fig. 1.5(a). The fundamental mode TEM00 has a Gaussian intensity distribution as a function of the radial distance ρ. We shall discuss Gaussian beams in the next section in detail. The ‘doughnut’ profile, characterized as TEM01∗ , is a end mirror left

S0

ρ

right

S1

ρ

left

S2

ρ

right

S3 z

j=0

1 L

2 L

3 L

intensity after j half-round-trips through the resonator

Fig. 1.4 Schematic illustration of the formation of a radial beam profile by iterative diffraction at the beam limiting apertures (e.g. at the end mirrors of the resonator). Sj indicates the beam limiting areas

1

(a)


(b)

TEM00

TEM01*

TEM10

TEM01

TEM20

TEM02

losse per round trip / %

8

TEM 10 pp 00

100 10 10

1 00

20 cf

0.1 0.01 0.4

0.8 1.2 F = w2 / λ L

Fig. 1.5 (a) Mode structure in a resonator with cylindrical symmetry for the lowest TEM resonator modes; shown is the intensity profile (very schematic) in a plane perpendicular to the optical z-axis; (b) diffraction losses for some of these modes as a function of the F RESNEL number F , in a FAB RY-P ÉROT resonator with plane parallel mirrors (pp), dashed lines, and for a confocal resonator (cf), full lines

linear combination of the TEM01 mode shown in the figure and a degenerate one turned by 90◦ . Of key interest are the diffraction losses for the different modes. They are determined by the F RESNEL number w2 (1.15) λL of the resonator, with w being the radius of the beam limiting aperture and L again the length of the resonator. Some examples are shown in Fig. 1.5(b). As expected, the diffraction losses decrease rapidly with increasing F RESNEL number. As a typical order of magnitude we estimate for a He-Ne laser F 0.8–3.2 (resonator lengths L = 50 cm, beam limiting gas discharge tube w = 0.5–1 mm, λ = 632.8 nm). Figure 1.5(b) shows that at these conditions diffraction losses for a TEM00 mode are negligible for a confocal resonator. However, as already mentioned, a plane parallel mirror setup is very unfavourable: it is unstable and subject to high diffraction losses. We note in particular, that in any case higher modes are subject to significantly higher losses as compared to the fundamental mode. It is due to this fact that in almost all laser systems only the fundamental mode TEM00 is active: for all other modes amplification usually is not sufficiently high to compensate for the losses. When modelling a specific laser system quantitatively, all losses have to be accounted for – including diffraction losses (effective photon lifetime due to diffraction τd ) and absorption (effective photon lifetime due to absorption τa ). In total, all losses due to the different mechanisms multiply according to the scheme I = I0 exp(−t/τr ) × exp(−t/τa ) × exp(−t/τd ) . . . . The inverse, effective photon lifetimes thus add and the resulting overall intensity becomes F=

I = I0 exp(−t/τe ) with 1/τe = 1/τr + 1/τa + 1/τd + · · · .

(1.16)

1.1


1.1.4

9

The Amplifying Medium

Spontaneous as well as radiation induced optical transitions have been discussed in detail in Chap. 4, Vol. 1. To describe these processes quantitatively in a laser medium, we shall continue to use rate equations as introduced heuristically in Sect. 4.2, Vol. 1.3 We recall that the transition probabilities are proportional to the spectral intensity distribution I˜(ω) = u(ω)/c ˜ (with u(ω) ˜ being the energy density per unit angular frequency ω = 2πν), while the overall strength of absorption and induced emission is characterized by the E INSTEIN B coefficients. Note, that in Chap. 4, Vol. 1 the bandwidth of the light was generally assumed to be much larger than the bandwidth ωb (FWHM) of the absorption line. However, in an active laser system we typically have to consider radiation of well defined direction and polarization, whose angular frequency ω is nearly in resonance with a transition frequency ωba of the medium, and whose intrinsic bandwidth ωlight is usually small or at least comparable to ωb . Thus we have to use now a frequency ˜ dependent absorption coefficient B(ω) as introduced in (5.12), Vol. 1 for an excited state which just decays by spontaneous emission, i.e. is represented by a L ORENTZ profile: ωb2 /4 gb λ3 ˜ B(ω) = ga 2πh ωb2 /4 + ω2

with ω = ωba − ω.

(1.17)

Here ωb = A is the FWHM of the absorption or emission line, A the E INSTEIN coefficient for spontaneous emission, and λ = 2πc/ω the wavelength. Let us, for simplicity and only for the moment, assume that indeed ωlight

ωb , i.e. the linewidth of the laser line (at angular frequency ω) is negligible compared to the width of the absorption profile. Then the total intensity I (t) of the laser radiation is relevant for the induced transition probability. According to (5.10), Vol. 1 the transition rate per atom is Rba = Bba gL (ω)

I (t) I (t) ˜ = B(ω) . c c

Let the population density of the upper laser level be Nb , that of the lower laser level Na . One defines population inversion as N = Nb − Na .

(1.18)

As we know, radiation emitted due to induced transition corresponds in frequency, direction and polarization exactly to that of the inducing radiation. Hence, starting with a finite inversion N > 0, amplification is expected and the photon density4 3A

more profound rationalization of these rate equations will be presented in Sect. 10.5.

recall: The dimension of I is Enrg T−1 L−2 , that of ω is Enrg, for c it is L T−1 so that Nω has the dimension (number of photons) L−3 . 4 We

10

1


Nω = I /cω in the resonating mode changes with time: dNω I (t) ˜ = B(ω)N . dt c

(1.19)

Correspondingly Nb decreases and Na increases. This may also be written as dI ˜ = ωB(ω)NI = α(ω)I dt

(1.20)

with the amplification factor defined in (1.6). Inserting (1.17) it becomes α(ω) =

N gb λ 2 c . ga 2π 1 + (2ω/A)2

(1.21)

Up to now we have tacitly assumed that the linewidth in the upper laser level is ωb = A, i.e. is determined exclusively by radiative decay. If this is not the case, the line profile would have to be replaced correspondingly, e.g in the case of collision broadening by α(ω) =

N gb λ2 c A = cσba (ω)N, ga 2π ωb 1 + (2ω/ωb )2

(1.22)

with ωb given by (5.18), Vol. 1. We have introduced here a cross section for absorption σba (ω) which has a maximum value σba (ωba ) =

gb λ 2 A . ga 2π ωb

(1.23)

For a purely radiative linewidth ωb = A and with gb /ga = 3 (p ← s transition) this maximum is 0.477λ2 . In the case of inhomogeneous line broadening (i.e. if one has to average over atoms with different absorption frequencies) the L ORENTZ profile in (1.22) has to be replaced correspondingly, e.g. by a G AUSS distribution (5.21), Vol. 1 in the case of D OPPLER broadening. We now also account for the fact that the light intensity will in general not be purely monochromatic. Clearly, (1.20) also holds for the intensity distribution I˜(ω) in (angular) frequency space. Thus, if we just consider the amplifier medium without accounting for losses, I˜(ω) will change when passing through the amplifier according to (1.24) I˜(ω) = I˜0 (ω) exp α(ω)z/c = I˜0 (ω) exp σba (ω)Nz , where I˜0 (ω) is the distribution at the entrance of the amplifier. As gain profile G(ω) of an amplifier medium in a resonator one defines the relative change after one full round trip: G(ω) =

I˜(ω) = exp α(ω)2L/c = exp σba (ω)N 2L . I˜0 (ω)

(1.25)

1.1


11

Fig. 1.6 Gain narrowing by a L ORENTZ type amplification profile (black) α(ω). The amplified (red) intensity profile I˜(ω) as a function of frequency is much narrower

α(ω) I(ω) -3 -2

-1

0

ω / Δωu

1

2

3

Remarkable about this formula is the fact that a highly amplified intensity profile may look rather different from the line profile (1.22): in the centre of the line (where amplification is high) the exponential factor leads to particularly high intensity, while in the line wings amplification is very small. This leads to a significant narrowing of the line, so called gain-narrowing as indicated in Fig. 1.6. Considering that in an active laser system the gain profile is in addition convoluted with the resonator profile (which deliberately induces losses for unwanted frequencies) one understands why the spectral distribution of the radiation generated in a laser is much narrower than both, the profiles of the amplifying medium or that of the resonator. So far we have not mentioned the role of spontaneous emission for the laser process. It always occurs as a byproduct. In an amplifier medium it will also be amplified. This may – especially in long stretched geometries with high amplification – lead already in one passage (i.e. even without mirrors) to substantial light intensities. This amplified spontaneous emission (ASE) is often used to provide an intensive light source with properties similar to that of laser radiation. Prominent examples are the well known nitrogen-laser, as well as many of the so called X-ray and free electron lasers (see Sect. 10.6.5 in Vol. 1). ASE may, however, be very troublesome, e.g. when intense, short laser pulses are to be generated. Typically, one first tries to build up sufficient population inversion for amplification, before in a second step the laser action is triggered. During this building-up process ASE may already lead to premature destruction of population inversion and formation of an unwanted background signal.

1.1.5

Threshold Condition and Stationary State

It is now straight forward to derive criteria for the possibility of laser action. To this end one has to account for amplification, according to (1.20), as well as for the losses, according to (1.16), and obtains for the intensity in the laser resonator: I I dI = α(ω)I − = cσba (ω)NI − . dt τe τe

(1.26)

For simplicity we assume that the amplifier medium fills the whole resonator.

12

1

Fig. 1.7 Illustrating the derivation of laser rate equations

Lasers, Light Beams and Light Pulses Pb

Nb σba I ħω

Pa

Na

A

γb γa

Amplification after one full turnaround is obtained only if dI /dt ≥ 0. From this follows the threshold condition for laser action: N =

1 . σba (ω)cτe

(1.27)

We remember that the effective photon lifetime τe may contain several contributions according to (1.16), and cτe is something like an ‘effective path length’ for a photon. In the volume σba cτe at least one atom has to be in the upper state for laser action to occur. Note that the actual intensity I of the laser field in the resonator has dropped out of (1.27). Clearly, under active laser operation I will be finite and depend on the details of the operating conditions as we shall discuss in a moment. However, for stationary laser operation we must always demand dI /dt = 0. Hence, when the laser is active, (1.27) will also be satisfied. The important message is thus: Population inversion under stationary state lasing conditions is identical to threshold inversion. Strictly speaking, one also has to consider the increase of the radiation field due to spontaneous emission. Since that is emitted, however, into the full solid angle and with the full bandwidth ωb , only a negligible fraction falls into the active laser mode. Spontaneous emission thus plays a role only in the starting phase of laser action, and possibly as noise (see, however, the above discussion on ASE).

1.1.6

Laser Rate Equations

In a two level system population inversion may only be reached for short times, and stationary laser operation is not possible with such a scheme. Hence, in efficient laser media typically three or four levels are involved in the overall process. Rate equations for a laser process have to account for all losses and gains. They must describe the temporal evolution of the population densities for all levels involved as well as for the radiation intensity I (t) in the resonator according to (1.26). A schematic is shown in Fig. 1.7, and the overall balance for upper (b) and lower (a) laser level reads: σba (ω)I dNb = Pb − N − ANb − γb Nb dt ω dNa σba (ω)I = Pa + N + ANb − γa Na . dt ω

(1.28) (1.29)

1.1


13

Fig. 1.8 Four level laser scheme

P Nb pump Na

γa >>

σba I ħω

These rate equations take into account that upper and lower laser levels may in principle be populated with rates Pb and Pa , respectively, from other levels, and decay in turn with rates γb and γa into other levels. Note the dimensions are L−3 for Na,b , T−1 L−3 for Pa,b , Enrg L−2 T−1 , L2 for σba , Enrg for ω, and T−1 for γa,b . Ideally, in the stationary state one would like to have Na = 0 and N = Nb or at least Na Nb . For this to happen, one tries to achieve Pa = 0 and γa σba I /ω, so that the lower laser level is rapidly depleted during laser action. The ruby laser, the first laser actually realized by M AIMAN (1960), was a three level laser. More convenient, and realized in most of today’s laser systems, are four (or more) level laser schemes which allow to establish nearly ideal conditions. Such a scheme is sketched in Fig. 1.8. Highly simplified one may assume that γb = 0 and γa is indeed very large,5 so that Na 0 and N Nb . With A = 1/Tb and Pb = P one obtains from (1.28) and (1.29): dN σba (ω)I =P − N − N/Tb . dt ω

(1.30)

For stationary conditions dN/dt = 0 must hold, and N is given by the threshold inversion (1.27). Inserting this into (1.30) one obtains for the laser intensity I = P ωcτe −

ω . σba Tb

(1.31)

Hence, with rising pump rate the intensity in the laser resonator rises – as one would expect – even though the population inversion remains constant (the excess energy is essentially converted into useful output). If the resonator is tuned to the maximum of the line profile, with (1.23) the intensity becomes I = P ωcτe −

4π 2 cωb , 3λ3

(1.32)

where we have assumed that the relaxation time Tb of the upper state is determined exclusively by spontaneous radiative decay (ATb = 1). Thus, the laser intensity I – as far as externally controllable – depends on the pump rate P and on the resonator 5 Interestingly, in a He-Ne laser the lower laser level is depopulated by collisions with the walls surrounding the gas discharge. Hence, the diameter of the plasma tube must not become too large.

14

1


losses (via τe ), while the negative term with the effective linewidth ωb of the excited level is system specific. We note that the loss term is ∝ λ−3 ∝ ν 3 , and point out again that this explains why building a laser becomes increasingly difficult as the wavelength decreases. For laser operation high above threshold one may neglect the loss term in (1.32), and the intensity becomes I P ωcτe . It now looks as if one just has to increase τe to obtain more intensity – i.e. one may think about increasing the reflectivity R of the mirrors in the resonators. Albeit this holds within the resonator, and is even used in various spectroscopic applications (see e.g. Sect. 5.5.3), for the laser output intensity Iout we may have to reconsider such strategy: only a fraction (1 − R) is coupled out; and it is the very fact that power is coupled out which leads to a reduction of photon lifetime. With (1.16), (1.12) and (1.10) one estimates: √ Iout P ωcτe (1 − R) < P ωcτr (1 − R) = P ω RL. (1.33) The output intensity depends of course on the population rate P for the upper laser level and on the length L of the resonator (here assumed equal to the length of the amplifying medium). And interestingly enough, by increasing the reflectivity of the output mirror one indeed gains some output power – however, this only holds as long as other losses (such as diffraction or absorption) can be neglected, i.e. if 1/τe 1/τr (see Eq. (1.16)).

1.1.7

Line Profiles and Hole Burning

Figure 1.9 summarizes what we have learned so far about amplification, line profiles and mode structure in active and passive laser resonators, and combines it with the concept of homogeneous and inhomogeneous line broadening introduced in Sect. 5.1, Vol. 1. We emphasize the specificities of homogeneous vs. inhomogeneous gain profiles in respect of laser operation. We also recall hole burning in a D OPPLER profile which was introduced in Sect. 6.1.5, Vol. 1. Figure 1.9(a) shows a typical line profile as a function of angular frequency ω of an amplifying medium (FWHM = ωb ) – still without laser activity. The dash dotted red line marks the threshold amplification necessary for laser action. The positions of the resonance frequencies (vertical dashed grey lines) and the free spectral range ωFSR of the laser resonator are indicated. Two resonance frequencies are emphasized in red: at these positions amplification is above threshold and lasing is possible. Complementary to this, Fig. 1.9(b) shows the transmission of the passive resonator. The bandwidth ωr of these transmission lines is a function of the finesse F of the resonator according to (1.12). Figure 1.9(c) illustrates the situation for the active laser in the case of homogeneous line broadening (e.g. natural linewidth, pressure broadening etc.). Since all atoms or molecules contribute to the amplification with the same spectral profile, and since the population inversion under operating conditions is equal to the threshold inversion, the whole amplification profile is attenuated such that the operating


σba (ω)ΔN

(a)

15

resonator frequencies

threshold

amplification profil passive Δω b

(b) transmission

1.1

resonator modes passive Δ ωr Δω FSR

ω

Δω FSR σba (ω)ΔN

(c)

homogeneous

σba (ω)ΔN

(e)

inhomogeneous

threshold

operating point

threshold

ω

A Δω b

Δω FSR

I (ω)

(d)

amplification profile active ω

resonator modes active

ω

Δω FSR

ω

I (ω) (f ) resonator modes active ω

Fig. 1.9 Amplification profiles ∝ σba (ω)N and longitudinal modes in passive and active resonators for homogeneous and inhomogeneous line profiles; for details see text

point corresponds to threshold amplification. As illustrated in Fig. 1.9(d) this is possible for only one longitudinal laser mode, since for all other modes the amplification is below threshold as shown in Fig. 1.9(c). The thus generated, monochromatic intensity distribution is sketched in Fig. 1.9(d). As illustrated in Fig. 1.9(e), the situation is completely different for inhomogeneous line broadening (e.g. D OPPLER broadening in a gas discharge, broadening due to statistically distributed environments in solid state media). Since now the amplification is different for different groups of atoms or molecules, each having their own, individual gain profile (e.g. corresponding to their individual velocity), they do not influence each other. The bandwidths of these individual group profiles typically correspond to the natural (or collision broadened) linewidth – in Fig. 1.9(e) marked as A. Now the laser may start to oscillate at all resonator mode frequencies for which the passive gain is higher than the laser threshold. At these and only at these frequencies population inversion is depleted down to threshold inversion. One finds the typical “hole burning”, which we have met already in Sect. 6.1.5, Vol. 1. In this case, the output power of the laser is distributed over several longitudinal modes, as depicted in Fig. 1.9(f). As already mentioned above, each mode has a linewidth which is much narrower than the width of the passive resonator modes, amplification profile or even the hole-widths burned into the latter. Note also, that for reasons of clarity linewidths and frequency distances shown in Fig. 1.9 are not presented to scale. In particular,

16

1


in the case of inhomogeneous line broadening, typically many resonator modes may be amplified. But also for homogeneous line profiles, in general several resonator modes are in principle ready for laser activity and will compete with each other – a situation which may lead to instable laser operation. It is now obvious that a simple laser setup as sketched in Fig. 1.2(b) on p. 3 will only in exceptional cases lead to strictly monochromatic radiation. As a rule, a rather broad mixture of lines from many longitudinal modes will emerge. They compete for population inversion and may strongly fluctuate. Thus, one needs additional means to achieve stability and to limit the bandwidth, in order to finally obtain a monochromatic, coherent and parallel output corresponding to the ideal of laser radiation. For further details the reader is referred to the specialized literature. Section summary

• The key elements of a laser are (i) an optical amplifier – based on population inversion, (ii) a resonator to suppress all unwanted frequencies – typically a FABRY-P ÉROT, and (iii) feedback which returns a major fraction of the amplified signal back to the entrance of the amplifier – which is achieved by the resonator mirrors. • Q factor (a large number), angular frequency ω, and average photon lifetime τr in the resonator are related by τr = Q/ω. The bandwidth of the passive resonator is ωr = 1/τr = ω/Q. The linewidth of an active laser mode is typically much narrower due to gain narrowing. • A FABRY-P ÉROT is characterized (i) by its free spectral range (or the turnaround time for light Tr ) νFSR = c/2L = 1/Tr , which is identical to the distance of longitudinal modes in the resonator, and (ii) by its finesse F which determines the resonator bandwidth: ωr = ωFSR /F. • Resonator stability depends on the curvature and distance of the mirrors. Substantial losses for different (transverse) TEMn modes may arise from diffraction at the mirrors which act as finite apertures of radius w in the resonator. These losses are smallest for the TEM00 mode and increase with the F RESNEL number F = w 2 /λL. • Laser threshold is reached when the gain compensates the losses −1 N = σba (ω)cτe , with the inversion density N and the absorption cross section σba (ω). This threshold population inversion is also maintained during active laser action: additional pump energy is converted into laser output. • Homogeneous and inhomogeneous line broadening leads to very different longitudinal mode patterns. Of particular interest is the hole burning which occurs in inhomogeneous amplifier media.

1.2

Gaussian Beams

1.2

17

Gaussian Beams

We now have to focus in some more detail onto the spatial properties of the light which is generated by a laser. Typically, one obtains light which inside the resonator corresponds more or less to the (desired) TEM00 fundamental mode. When such light propagates freely in space, it becomes what is known as a Gaussian beam. The reader may find some of the following text and formulas known from Vol. 1 or elsewhere, or even trivial. We need, however, a solid basis to avoid misunderstandings later on. We start with the spatial profile of a Gaussian beam, and specify some terminology such as beam radius, beam divergence and complex beam parameter. We then introduce the so called ABCD matrices, important, simple tools for the whole of laser physics. We discuss focussing and widening of laser beams – and how to measure the width. Finally, we shall introduce the so called M 2 factor as a practical measure of beam quality.

1.2.1

Diffraction Limited Profile of a Laser Beam

Naively, one may imagine a light beam as a (narrow) cone of small but finite divergence angle θe , which is filled with a quasi-monochromatic plane wave of electromagnetic radiation at an angular frequency ω and wavelength λ (wave vector |k| = ω/c = 2π/λ). This would imply a constant amplitude over the whole cross section, dropping to zero outside the cone. But even if the eye may perceive a laser beam as something like that, such a sharply limited energy distribution is not compatible with the general wave equation:6

1 ∂2 − 2 2 E(x, y, z, t) = 0. c ∂t

(1.34)

A trivial solution is of course the well known plane wave (4.1), Vol. 1. We have used it throughout Vol. 1 when describing the interaction of electromagnetic radiation with atoms. However, such a wave field extends laterally to infinity, and if one tries to cut out of it a light beam – e.g. by a circular aperture of radius w0 , the cut out ‘beam’ forms immediately a characteristic diffraction pattern as we have seen it in the laser resonator (Fig. 1.4) and a divergent bundle of light with smooth boundaries emerges: Diffraction simply washes out all sharp contrast.

Field Distribution and Beam Parameters For a more general description of the electromagnetic wave we introduce an envelope function E0 (r, t) for the electric field amplitude, which allows it to depend 6 For

simplicity we only consider wave propagation at the speed of light c, i.e. in vacuum.

18

1


(slowly) on position in space and time, and an arbitrary phase φ0 relative to the envelope. We thus generalize (4.1), Vol. 1:7 E(r, t) =

i ∗ E0 (r, t)ei(kr−ωt+φ0 ) e − E0 (r, t)e−i(kr−ωt+φ0 ) e∗ . 2

(1.35)

Somewhat more compact we write E(r, t) =

i − E (r, t)e − E + (r, t)e∗ 2

∗ with E + (r, t) = E0 (r, t)e−i(kr−ωt+φ0 ) = E − (r, t) .

(1.36)

For mathematical convenience, we shall focus on the E + (r, t) component (positive sign for iωt) and on E0 (r, t). But we emphasize again, that E(r, t) is a vectorial observable, and as such a real, measurable quantity – with important consequences as discussed in Chap. 4, Vol. 1. In Sect. 1.3.1 we shall explore additional, surprising aspects of this fact. In contrast E ± (r, t) are complex functions, and the envelope function E0 (r, t) may also be complex and vary (slowly) with position r and time t. The most ‘beam like’ solution of the wave equation (1.34) starts from a plane wave, say propagating in z-direction (kr = kz), but allows for a spatial profile. For the moment we ignore the explicit dependence of E0 (r, t) on time and defer this aspect to Sect. 1.4. Thus, we set t = 0, insert the ansatz (1.35) into (1.34), and obtain for the spatial envelope E0 (x, y, z): ∂E0 ∂ 2 E0 ∂ 2 E0 ∂ 2 E0 = 0. −2ik + + 2 ∂z ∂x 2 ∂y 2 ∂z

(1.37)

:= 0 in SVE approximation The so called slowly varying envelope (SVE) approximation demands δE0 δE0 δE0 E0 , ,

, δx δy δz λ

(1.38)

so that the overall character of the plane wave does not get lost. Specifically, for paraxial propagation the change of the amplitude in z direction is assumed to be particularly small, and one neglects the second derivative in respect of z completely, as indicated in (1.37). This differential equation can then be solved by the ansatz

x2 + y2 E0 (x, y, z) = E0 exp −iP (z) exp −ik . 2q(z)

(1.39)

that this is identical to (4.1), Vol. 1 for φ0 = 0 and E0 (r, t) = E0 = const. For linearly polarized light, e = e∗ , we have E(r, t) = eE0 (r, t) sin(kr − ωt).

7 Note

1.2

Gaussian Beams

19

This is already written in cylinder symmetrical form, i.e. laterally the field depends only on ρ = x 2 + y 2 , the distance from the optical axis. After some mathematical manipulations this leads to rather simple ordinary differential equations for P (z) and q(z). Without entering into details we communicate the key results: With suitable boundary conditions one obtains P = i ln(1+z/izR ) and the so called complex beam parameter q(z) = z + izR .

(1.40)

It characterizes the Gaussian beam by just one real parameter, the R AYLEIGH length zR , and may be rewritten as 1 λ 1 i 1 = −i − = . q z(1 + (zR /z)2 ) zR (1 + (z/zR )2 ) R(z) πw 2 (z)

(1.41)

This suggests to describe the beam profile locally by two variables, the radius of curvature R(z) = z 1 + (zR /z)2 ,

(1.42)

(which gives the surfaces of constant phase in the far field z zR ) and the beam radius w(z) = w0 1 + (z/zR )2 .

(1.43)

According to (1.41), its smallest value at z = 0 is the so called zR λ zR = , beam waist w0 = w(0) = π 2k

(1.44)

and the R AYLEIGH length may be written as zR =

πw02 kw02 = . λ 2

(1.45)

Thus, a Gaussian beam may also be characterized by w0 rather than by zR . With these definitions we can now rewrite (1.39) and define the field amplitude for a Gaussian beam: ikρ 2 E0 ρ2 × exp − × exp iφG (z) , × exp − E0 (ρ, z) = 2 2R(z) w(z) 1 + (z/zR )2 (1.46) and specifically for z = 0: E0 (ρ) = E0 exp −(ρ/w0 )2 . (1.47) The maximum field amplitude E0 / 1 + (z/zR )2 on the beam axis obviously depends on z. Radially the field drops to 1/e (36.8 %) of its maximum value at a distance ρ = w from the axis (beam radius). At the beam waist (z = 0, w(0) = w0 ) the radius of curvature is R(0) = ∞ (plane wave). The maximum curvature (minimum |R|) is found at z = ±zR with 1/|R| = 1/2zR ; the beam radius there is

20

1


√ w(zR ) = 2w0 . For larger |z| the radius of curvature increases again, approaching R(z) → z. The first factor in (1.46) ensures conservation of total energy in the beam, the second describes the actual radial G AUSS profile. The third factor exp[−ikρ 2 /2R(z)] contains a phase in the exponent and essentially quantifies the deviation of a Gaussian beam from a plane wave. It is also called F RESNEL factor.8 In the far field it approaches unity and the Gaussian beam practically becomes a plane wave. The last factor in (1.46) is also interesting; it contains the so called G OUY phase: φG (z) = − arctan(z/zR ).

(1.48)

This is a phase which the wave ‘collects’ in comparison to a plane wave due to the fact that the curvature of the wavefront changes. The G OUY phase changes from π/2 to −π/2 when z runs from −∞ to ∞ and is defined so that the overall phase vanishes for z = 0. Exactly 50 % of the overall phase change are collected between z = ±zR . Recently, the G OUY phase gains increasing importance in the context of ultrafast laser pulses and has been determined experimentally for the first time by L INDNER et al. (2004).

Intensity and Power With (1.46) the cycle averaged intensity (I.19), Vol. 1 of the Gaussian beam is:

2 2 1 ρ . (1.49) I (ρ, z) = ε0 c E0 (ρ, z) = I0 (z) exp −2 2 w Thus, the beam radius w gives the distance ρ = w from the z axis at which the intensity is 1/e2 (13.5 %) of its maximum value. This definition corresponds to the international norm ISO 11146. The total power Ptot in the beam is obtained by integration over the whole radial profile: Ptot = I0 (z)2π 0

∞

2 I0 (z)πw(z)2 I0 πw02 ρ ρdρ = = . exp −2 w 2 2

(1.50)

I0 is the overall cycle averaged maximum of the intensity – on the beam axis at the waist. With this and w(z) as defined in (1.43) we may rewrite (1.49) to show the intensity dependence on z explicitly:

−2ρ 2 I0 2Ptot with I0 = I (ρ, z) = exp 2 . 2 1 + (z/zR )2 w0 (1 + (z/zR ) ) πw02

(1.51)

Figure 1.10 illustrates this profile graphically. The red line in Fig. 1.10(a) indicates where the intensity I (ρ, z) has dropped to 1/e2 and defines asymptotically the beam divergence θe . The 3D plot in Fig. 1.10(b), a blow up of the centre part in (a), may write the exponent also ikρ 2 /2R = iπ × ρ 2 /λR and recognize the F RESNEL number according to (1.15).

8 We

1.2

Gaussian Beams

(a) k

21

4 ρ/w0 2

ρ/w0 R 10

θe w (z) 20

0.8 0.6 0.0

0.4

R /w0 20 10

-10

1.0

2.0 1.0

1/e2 of the maximum intensity; waist at z= 0

- 20

I / I0

z/w0

2w0

(c)

(b)

0.2

- 1.0 - 10 10 - 20

20 z/w 0

- 2.0 -10

w0 -5

0

0 5

10

z/w 0

b = 2z R

Fig. 1.10 Profile and parameters in a strongly focussed Gaussian beam w0 = 2λ, zR = 2πw0 . (a) Geometry in the vicinity of the beam waist w0 , (b) 3D plot of the intensity profile around the focus, (c) radius of the wavefront as a function of z/w0 . Note the different scales for the ρ and z-axes

shows some kind of a ‘dog bone’ structure. Indicated is also the confocal√parameter b = 2zR . Note that with (1.43) at z = ±zR the beam radius is w = 2w0 so that the beam area πw 2 = 2πw02 has doubled. Thus, the intensity decreases within the confocal parameter in both directions to 50 % of its maximum. Figure 1.10(c) shows R, the radius of curvature of the wave front, as a function of z/w0 . Notice the pronounced minima of |R| = 2zR at z = ±zR .

Far Field and Beam Divergence In the far field z zR the beam radius (1.43) grows linearly with z (just as in geometrical optics) lim w(z) =

|z|→∞

w0 2|z| λ|z| |z| = = . zR kw0 πw0

(1.52)

The beam intensity profile (1.51) then approaches (kw0 )2 (ρkw0 )2 . lim I (ρ, z) = Ptot exp − z→∞ 2πz2 2z2

(1.53)

For small angles θ ρ/z 1 this may also be expressed as the radiation power emitted into the solid angle element 2πθ dθ : P (θ ) = Ptot

(θ kw0 ) 2 (kw0 )2 exp − 2π 2

(1.54)

22

1


with [P (θ )] = W/sr. Thus, the divergence angle of the Gaussian beam is given by θe =

2 w0 λ = = . kw0 zR πw0

(1.55)

There, the intensity has dropped to 1/e2 of its maximum, the field strength to 1/e. The corresponding solid angle divergence is δΩe = πθe2 =

λ2 . πw02

Useful Formulas and Experimental Verification For some purposes it is convenient to introduce an additional parameter √ a(z) = w(z)/ 2

(1.56)

(1.57)

at which the intensity has dropped to 1/e, which we shall call G AUSS radius. Its minimal value at the waist is related to the R AYLEIGH width by a(0) = λzR /2π = zR /k, cf. (1.44), and the beam intensity is written as I (ρ, z) = I0 (z) exp −ρ 2 /a 2 . The beam diameter at half maximum intensity (FWHM) is given by √ √ d1/2 = 2 ln 2a = 1.665a = 2 ln 2w = 1.177w.

(1.58)

(1.59)

The maximum intensity on the beam axis in terms of the total power Ptot in the beam is obtained from (1.50): I0 (z) =

Ptot 2Ptot Ptot = = 0.693 . πa 2 πw 2 π(d1/2 /2)2

(1.60)

This is easy to memorize for the G AUSS radius: the maximum intensity of a Gaussian beam is equal to the intensity in a hypothetical, cylindrical rod with radius a (assumed as constant). Often one is interested in the fraction P (ρ) of the total power Ptot which is contained in the centre of the beam between ρ = 0 and ρ. By integrating (1.58) one obtains the useful formula I (ρ) P (ρ) = Ptot 1 − (1.61) I0 and the average intensity Iav = I0

2 ρ w2 1 − exp −2 . w 2ρ 2

(1.62)

1.2

Gaussian Beams

23 ln( I / I0) 0

I / I0

(a)

-1

(b )

d1/2 a

-2

d1/2

w

-3

a

-4 w -2

0

ρ /mm

-5 2

-2

0

ρ / mm

2

Fig. 1.11 Radial intensity profile of a continuous dye ring laser. Experimentally determined points (•) and G AUSS fit ( ) plotted (a) on a linear and (b) on a logarithmic intensity scale. The quantities FWHM = d1/2 , G AUSS radius a, and beam radius w are marked to scale Table 1.1 Intensity and power in a Gaussian beam

ρ

I (ρ)/I0

P (ρ)/Ptot

Iav /I0

0

1

0

100 %

0.83a = 0.588w √ a = w/ 2 √ w = 2a √ 2a = 2w

1/2 = 50 %

50 %

72.1 %

1/e = 36.8 %

63.2 %

63.2 %

= 13.5 %

86.5 %

43.2 %

1/e4 = 18.3 %

98.7 %

24.5 %

1/e2

A typical, experimentally determined lateral profile of a laser beam is shown in Fig. 1.11. The comparison of the measured data points with the fit according to (1.58) shows that the G AUSS distribution is quite realistic, albeit by no means perfect. Finally, for practical purposes some numbers are collected in Table 1.1. We see that only 63.2 % of the total power pass through a circular aperture of radius a around the axis, while ≈99 % pass through one with radius 2a. Also given in the table is the intensity Iav averaged over these apertures. All these quantities are, however, only relevant if one studies processes which depend linearly on the light intensity. We shall come back to this in Sect. 1.6.

1.2.2

FAUNHOFER Diffraction

It is instructive to compare the angular divergence of a Gaussian beam with that of a plane wave diffracted at a circular aperture of radius w0 . We have already touched this theme in the context of laser modes and diffraction losses in Sect. 1.1.3. To obtain a quantitative result, we now use the opportunity for a short detour into classical wave optics (see e.g. B ORN and W OLF 2006). Let us consider the diffraction of a plane wave at a circular aperture. We treat again just the E + component of the field (1.36). With the H UYGEN -F RESNEL principle, one may describe the diffraction of a plane wave with amplitude ES+ at a pla-

24

1

Fig. 1.12 Geometry for F RAUNHOFER diffraction


η

diffraction pattern at the detector ψ x y

ξ w0

φ

S

r0 r

ρ

D s

θ

dρ

z

dφ diffracting surface S

nar object with a surface S as superposition of spherical waves.9 The field detected at a point r D at the time t = 0 is E + (r D , t = 0) =

iES+ 2λ

T (ρ, ϕ) S

exp(ikr) ρdρdϕ. r

(1.63)

The geometry is sketched in Fig. 1.12. We choose cylinder coordinates z, ρ and ϕ for the computation. The object is described by a transmission function T (ρ, ϕ) – in the most general case complex – which may modify phase and amplitude ES+ of the incoming wave. One reads the distance r between a point on S – characterized by ρ and ϕ – to the detection point D – characterized by s and ψ – from Fig. 1.12: r 2 = (x − ξ )2 + (y − η)2 + z2 , while r02 = x 2 + y 2 + z2 xξ + yη ξ 2 + η2 . so that r 2 = r02 1 − 2 + r02 r02 With the scalar product of the two radial vectors ρ on the diffracting surface, and s on the detector plane one writes xξ + yη = ρ · s = ρs cos(ψ − ϕ). We replace ξ 2 + η2 = ρ 2 and expand r for w0 r0 xξ + yη ρ 2 ρs cos(ψ − ϕ) ρ2 r = r0 1 − 2 + r − + . (1.64) 0 r0 2r0 r02 r02 For small diffraction angles θ = s/r0 holds, so that finally the spherical wave in (1.63) may be written: exp(ikr0 ) ρ2 exp(ikr) → . exp −ikρθ cos(ψ − ϕ) exp −ik r r0 2r0 9 We

refrain here from presenting the full derivation according to K IRCHHOFF, and assume for simplicity the wave to encounter the object perpendicularly.

1.2

Gaussian Beams

25

The last factor is the F RESNEL factor already known from the field amplitude (1.46) of the Gaussian beam. Again the F RESNEL number F , (1.15), comes into play; depending on the characteristic dimensions ( w0 ) of the diffracting surface and the distance (r0 z) to the detector one distinguishes F RAUNHOFER diffraction F RESNEL diffraction with r0 z w0

for F = w02 /r0 λ 1/π for F

and

= w02 /r0 λ 1/π,

in both cases.

Relatively easy to evaluate is F RAUNHOFER diffraction since the F RESNEL factor becomes exp(−ikρ 2 /2r0 ) 1. It yields the well known A IRY diffraction pattern. The electric field (1.63) at the observation point D in the detector plane at distance z is then given by 2π w0 iE + E + (r D ) = S eikz ρdρ T (ρ, ϕ)e−ikρθ cos(ψ−ϕ) dϕ. (1.65) 2λz 0 0 One has to evaluate this integral for the given geometry and transmission function T , and obtains the intensity of the diffraction pattern ∝ |E + (r D )|2 . The simplest case is a circular aperture of radius w0 with T (ρ, ϕ) ≡ 1. One makes use of the properties of the B ESSEL functions of 1st order: x 2π 1 iu cos ϕ inϕ e e dϕ and xJ (x) = uJ0 (u)du. (1.66) Jn (u) = 1 2πin 0 0 Comparison with (1.65) for n = 0 with u = kθρ leads to i + ikz 2π 1 2 kθw0 + E (r D ) = ES e uJ0 (u)du 2 λz kθ 0

(1.67)

kw 2 J1 (kw0 θ ) i . = ES+ exp(ikz) 0 2 z kw0 θ The power diffracted per solid angle finally becomes

(kw0 )2 2J1 (kw0 θ ) 2 P (θ ) = Ptot . 4π kw0 θ

(1.68)

In Fig. 1.13 this is compared with a Gaussian profile (1.54). Even though the divergence is of similar magnitude in both cases, the Gaussian beam is clearly better collimated. A Gaussian beam is the best possible approximation to the hypothetical light ray in geometrical optics. Its divergence angle θe is related to its minimal radius w0 (at intensity I = I0 /e2 ) and its wave vector k or wavelength λ by (1.55). This may, somewhat sloppily, be interpreted as a kind of uncertainty relation: for the transverse photon momentum

26

1

Fig. 1.13 Comparison of the angular dependence of a Gaussian beam (red), with a beam waist radius w0 , and the far field of a plane wave (grey), diffraction limited by an aperture of radius w0 . The two profiles are normalized to each other at θ = 0

Lasers, Light Beams and Light Pulses 1.0

P(θ )

0.5

θe=2/kw0

1/e2

- 3.83 -6

-4

-2

0

3.83 2

4

6 kw0 θ

k⊥ = kθe one finds k⊥ w0 ≥ 2. Generally, one defines a so called beam parameter product BPP ≡ w0 θe as a measure for the quality of a light beam (specifically for lasers). For a Gaussian beam we have BPP = w0 θe = 2/k = λ/π,

(1.69)

while for the aperture limited plane wave according to (1.68) the divergence angle w0 θe ≥ 3.83/k = 1.22

λ 2

and sAiry = θe z = 1.22

zλ 2w0

(1.70)

holds. The characteristic quantity here is the radius sAiry of the central diffraction pattern (the so called A IRY disc) on the detector screen.10 We shall come back to these expressions again in Sect. 2.1.7 in the context of lateral coherence of natural light and laser radiation.

1.2.3

Ray Transfer Matrices

Before we can efficiently treat the manipulation of laser beams (focussing, defocussing, deflecting etc.) we have to introduce an important tool of geometrical optics, applicable also to Gaussian beams: the so called ray transfer matrices or ABCD matrices. They are used for ray tracing of paraxial light rays (with sin θ ∼ = θ ), i.e. they allow to describe in a simple manner the propagation of light rays through an optical arrangement. These ABCD matrices are efficient tools e.g. for designing laser systems and for computing the properties of laser resonators – typically describing the propagation of well collimated Gaussian beams through a number of optical elements such as mirrors, lenses, and prisms. 10 Note

that this formula is the basis of the well known R AYLEIGH criterium for the diffraction limited resolving power of optical instruments. It states that sAiry is the smallest distance between two objects that can be resolved by an instrument with an effective limiting aperture of diameter D = 2w0 and focal length f = z.

1.2

Gaussian Beams

27

Fig. 1.14 ABCD matrices operating on a light ray. (a) Ray translation through a distance d. (b) Refraction at an interface between two materials

(a)

(b)

d ray'

ray

ρ' ray' ρ ρ' ray θ

θ

ρ

θ'

A paraxial light ray that propagates at a (small) angle θ in a distance ρ from the optical axis, is described by a ray vector: ρ θ

ray

corresponding to the scheme optical axis

z

In Fig. 1.14 the change of a ray into ray is illustrated for two important cases: translation and refraction. Such a change upon transition of the ray through the optical setup corresponds to a linear transformation of the ray vector,11 described by an ABCD matrix: ρ A B ρ = . (1.71) θ C D θ Specifically, for a Gaussian beam the complex beam parameter (1.41) changes under such transformation from q to q (here without proof; see e.g. KOGELNIK and L I 1966): q =

Aq + B Cq + D

or

1 C + D/q . = q A + B/q

(1.72)

The most important simple cases are collected in Table 1.2. One verifies ρ 1 d ρ ρ + θd that case 1 = = θ 0 1 θ θ describes a translation, while case 2

ρ θ

=

1 0

ρ ρ = nθ n θ n n 0

represents the S NELLIUS diffraction law for small angles, n θ = nθ (paraxial rays). Note that in the latter case ρ = ρ does not change. The propagation of a ray through several optical devices is described by the product of the respective ABCD matrices. For example, the ray vector after translation and refraction by a thin lens is derived by multiplying matrix 1 with matrix 3 from 11 In

a mathematical sense the ray vector is simply the position vector expressed by ρ, the distance from the optical axis (z-axis), and θ = ρ/z, the polar angle. The pictographs used in the following are not always correct in that sense, but quite instructive.

28

1


Table 1.2 The most important, simple ABCD matrices Case

Description

1

translation through distance d in medium with index of refraction n

ABCD 1 dn

2

refraction by a plane surface with index of refraction left n, right n

3

thin lens of focal length f

4

reflection from a concave mirror

5

refraction at a spherical surface

0

1

1

0

0

n n

Scheme n d

n n′

1

0

− f1

1

1

0

− R2

1

f

R

1

0

n n

− nn−n R

f

n n′

R

Table 1.2:

1 − f1

0 1

1 0

d 1

1 ρ = 1 − θ f

d 1−

d f

ρ + θd ρ = . θ − ρ+θd θ f

(1.73)

One verifies e.g. for any ray coming out of the focal point (d = f and ρ = 0), that after passing the lens the exit angle is always θ = 0, i.e. all such rays leave the lens parallel – as well known from geometrical optics. Conversely, a ray entering the lens parallel to the optical axis (θ = 0) always leaves the lens at ρ = ρ with θ = −ρ/f , independently of d, i.e. the ray is refracted into the focal point. We may also describe ray propagation behind the lens to a position d if we know the ray vector in front of the lens:

1 0

d 1

1 − f1

0 1

1 − df ρ = θ −1

f

ρ − ρd ρ f +d θ = . (1.74) θ −ρ +θ 1

d

f

One verifies e.g. that rays which are parallel to the axis before entering the lens (θ = 0) cross the axis behind the lens at d = f . Finally, the product of matrix 1 with matrix 3 and again matrix 1 allows one to describe imaging by a lens. Further multiplication with the matrices for additional optical pathways and other elements allows one to easily describe even complex optical setups very efficiently. We shall now apply this technique to the manipulation of a Gaussian beam with lenses.

1.2

Gaussian Beams

1.2.4

29

Focussing a Gaussian Beam

Focussing and de-focussing, expanding and concentrating Gaussian beams is very important in experimental practice. In principle, very wide beams with large radii may be generated as well as tightly focused ones with extremely high intensity at the focal point. In any case, however, the product of beam radius w and divergence angle θe remains constant, and is according to (1.69) for a Gaussian beam θe w = λ/π . To familiarize ourselves with the tools we let a Gaussian beam propagate freely, say, emerging from a laser system. Let us assume that we start with an ideally parallel beam (R = ∞) of radius wL . With the R AYLEIGH length zR = πwL2 /λ the complex beam parameter according to (1.41) is λ i 1 = −i =− . 2 q z πwL R

(1.75)

It changes with distance travelled according to (1.72). With the propagation matrix 1 in Table 1.2 we find at a distance z = d 1 C + D/q 1/q = = , q A + B/q 1 + z /q which can be rewritten in the form (1.41) as i 1 1 λ 1 − = −i = . 2 2 2 2 q πw 2 z (1 + zR /z ) zR (1 + z /zR ) R Thus we find R = z (1 + zR2 /z2 ) and w = wL 1 + z2 /zR2 , expressions which are completely equivalent to (1.42) and (1.43). We see that the matrix method reproduces indeed the behaviour of a Gaussian beam as it propagates along the optical axis – including the correct divergence angle θe = λ/πwL corresponding to (1.52). We now want to focus this laser beam by a collecting lens of focal length f . Let the lens be positioned at z = 0, and we state now more precisely that the assumed initial parallelism implies RL f . In geometrical optics such a “plane wave” is assumed to be focused in the focal point of the lens, whereby the initial plane wave front (RL = ∞) is converted into a spherical wave with a radius of curvature R = (z − f ). In reality, however, due to diffraction the focal point becomes a circular disc of radius w0 , as sketched in Fig. 1.15. The complex beam parameter in front of the lens is given by (1.75) as in the previous example. According to (1.72) we use this time the product matrix (1.74) to obtain the new complex beam parameter at a distance z behind the lens: −1/f − iλ/(πwL2 ) 1 C + D/q . = = q A + B/q (1 − zf ) − z iλ/(πwL2 )

(1.76)

30

1

Fig. 1.15 Focussing a parallel Gaussian beam


2z 'R

wL √2 w'0

w'(z' )

z'

w'0 2θ'e

Laser beam far from its waist R' = f R L= ∞

f

Without going into further details we concentrate onto the focal point of the lens, z = f , in order to determine the waist w0 = w (f ).12 The result is simply πw 2 1 1 1 λ = − i 2L = − i q f R λf πw02 with the latter equality again according to (1.41). Comparing real and imaginary parts we obtain the waist radius w0 of the focused beam w0 =

fλ , πwL

(1.77)

while the radius of curvature of the wave front becomes R (f ) = f . For practical √ purposes we also note the relation with the 1/e intensity G AUSS radius a = w/ 2 a0 =

fλ , 2πaL

(1.78)

and recall that the maximum intensity (1.60) in the beam centre which may be obtained by this kind of focussing is I0 = Ptot /πa02 . Inserting (1.77) into (1.45) gives for the focused beam the R AYLEIGH length zR and the confocal parameter (the distance around the focus for which the intensity on axis is above 50 % of the maximum): b = 2zR =

2πw02 2λf 2 = . λ πwL2

Table 1.3 presents numerical examples for three different focal lengths f and two different initial beam diameters wL , all at the 800 nm wavelength of the Titaniumsapphire laser – the workhorse of ultrafast spectroscopy. closer look into the algebra shows, however, that the smallest cross section is found at z = f z02 /(zR2 + f 2 ), i.e. slightly in front of the focal point of the lens. There the radius of curvature even approaches R → ∞. Since typically we start with a relatively large beam radius wL the R AYLEIGH length is very large, zR = πwL2 /λ f , so that the small difference does not play a practical role. 12 A

1.2

Gaussian Beams

31

Table 1.3 Beam radius w0 (at 1/e2 intensity) and confocal parameter b = zR in the focus of a lens with focal length f . An initially parallel Gaussian beam is assumed, at λ = 800 nm with radius wL

f/mm

wL /mm

wL /f

w0 /µm

b

100

0.4

0.004

64

31 mm

100

2

0.02

13

1.3 mm

50

0.4

0.008

31

8 mm

50

2

0.04

6.4

318 µm

10

0.4

0.04

6.4

318 µm

10

2

0.2

1.2

12 µm

Conversely, one may parallelize a diverging laser beam by putting a lens of focal lengths f at a distance f behind its waist. As illustrated in Fig. 1.16 the lens converts the curved wave front into an essentially plane wave. Starting with a beam waist w0 at z = 0 and a radius of curvature RL = −f the complex beam parameter at the waist is 1 λ 1 =− −i . q f πw02 We use again (1.72), this time with the product matrix (1.73). After a brief calculation we obtain the new complex beam parameter directly behind the lens: π 1 C + D/q = −i 2 w02 . = q A + B/q f λ Beam radius and radius of curvature immediately behind the lens are thus w0 =

fλ πw0

and R = ∞,

respectively.

As expected, this is just the inverse of the focussing process according to (1.77). We have to be aware that the ‘new’ laser beam thus generated has now its waist just behind the lens. From there it will propagate in a slightly divergent manner. According to (1.55) the new divergence angle of the ‘parallelized’ Gaussian beam

w'0

z=0 Laser beam close to its waist

w'(z' )

θe z'

w0 R= - f f

Fig. 1.16 De-focussing of a Gaussian beam

f

θ'e θ'e R' = ∞

32

1


Table 1.4 Beam radii of Gaussian beams as a function from the beam waist for different initial waist radii w0 according to (1.43). The R AYLEIGH length zR and the far field divergence angle θe are related to w0 by (1.45) and (1.55), respectively. The numbers are given for the Titaniumsapphire laser wavelength λ = 800 nm Nr.

w/mm Beam radius at 1/e2 Intensity at z (distance to waist)

Beam parameter w0 /mm

zR /mm

θe /mrad

1m

2m

20 m

100 m

1000 m

1

10

400 × 103

2.5 × 10−2

10

10

10

10.3

27

2

3

35 × 103

3 × 10−2

3

3

3.4

9

85

3

1

3.9 × 103

0.25

1

1.1

5.2

25

255

4

0.3

353

0.84

0.9

1.7

16

85

850

5

0.1

39

2.5

2.5

5

51

254

2546

6

0.03

3.5

8.5

8.5

16

170

850

8500

7

0.01

0.4

25

25

50

250

2550

25 465

is θe =

λ λπw0 w0 = . = πw0 πf λ f

(1.79)

This divergence angle may even be interpreted geometrically as indicated in Fig. 1.16: it corresponds to the divergence one would expect from classical ray optics imaging an extended light source with a radius w0 to infinity. To give a feeling for the divergence of Gaussian beams, in Table 1.4 numerical examples are collected for different initial beam radii w0 (equivalently R AYLEIGH lengths zR or divergence angles θe ). The beam radii w at different distances z from the waist illustrate clearly how the beams diverge – one may often observe this by laser illuminations in the nightly skies of big cities (thanks to R AYLEIGH scattering in dry weather, or even better in humid air due to M IE scattering as discussed in Sect. 8.4.5, Vol. 1). These numbers are astonishing indeed, at least at first glance. Compare e.g. beam 1, 3, and 5 in Table 1.4. Beam 1 may initially appear as a rather broad ‘brush’ of light. Note, however, that it widens only by a factor of 3 over the distance of 1km, and it would thus be well suited e.g. in telecommunication or metrology. In contrast, beams 3 or 5 look on the laser table nicely thin and ‘laser-like’. They expand, however, very rapidly and get a rather large radius in some distance. Beam 5 will hardly be useable in a neighbouring lab at 20 m distance, while beam 6, finally, is a highly focussed beam generating very high intensity in its focal point, but with a R AYLEIGH length of 0.4 mm, its on axis intensity decays rapidly to less than 50 % outside the confocal length of 0.8 mm around the focus. For transporting beams over wide distances one first has to expand them. To this end one uses telescope systems as sketched in Fig. 1.17. In the destination where the beam is to be used, one may place a second telescope system – more or less identical to the first one – in order to re-compress the beam. Even if one wants to

1.2

Gaussian Beams

33

Fig. 1.17 Telescope systems for expanding a Gaussian beam. The top setup according to K EPLER (two collecting lenses) contains an aperture, positioned in the focus, for beam improvement (spatial filter). The lower setup according to G ALILEI (dispersing lens and collecting lens) can be used even for very high intensities

f1

f2

2w1

2w2

2w0

aperture

lens 1

lens 2 f2 2w2

- f1 2w1

lens 1 lens 2

generate a particularly tight focus, it is recommendable to first expand the beam: according to (1.77) the minimal radius w0 achievable in the focus of a lens is inversely proportional to the radius wL of the laser beam in front of the focusing lens. As illustrated in Fig. 1.17 one may construct telescope systems essentially in two ways: either according to K EPLER with two collecting lenses, arranged con-focally or according to G ALILEI with one dispersing and one collecting lens, the focus of the latter matching the backward focus of the dispersing lens. From the geometry shown in Fig. 1.17 one reads the beam expansion from radius w1 to w2 in both cases to be w2 = w1 |f2 /f1 |.

(1.80)

The divergence angle of the expanded beam is again estimated from (1.55): θ2 =

λ . πw2

(1.81)

Strictly, this only holds if the radius of the phase surfaces at lens 2 exactly matches f2 – which may always be achieved by fine adjustment of the lens positions. Both types of telescope systems have their specific merits. Between two collecting lenses a real focal-point exists, into which one may e.g. place a small, circular aperture (diameter typically 3 to 5 × w0 ) as indicated in Fig. 1.17. With such a spatial filter one often may improve the lateral profile of not completely perfect Gaussian beams considerably. The (lateral and longitudinal) adjustment of such an aperture requires some experimental skills. On the other hand, for very high laser intensities – as often encountered when working with femtosecond laser pulses – one must avoid a real focus where ionization and electric breakdown in air may occur, and destroy the laser pulse. In such cases only the G ALILEI type of telescopes is applicable.

34

1

Fig. 1.18 Signal from a Gaussian beam measured with the knife edge method. One determines a from the y0 position for a signal ratio of 0.24 and 0.76 – as indicated


0.5

-2

1.2.5

1.0

P( y0) ____ P tot

-1

0 y0 /a = 1

1

2 y0 / a

Measuring Beam Profiles with a Razor Blade

If one already knows that the radial beam profile is Gaussian, or is least described well enough by (1.58), one may determine the radii a or w according to (1.59) by the rather simple, practical “knife-edge” method. One mounts a razor blade onto a precision optical table, which can be moved perpendicularly into the beam, say in ydirection. One registers the power P (y0 ) which reaches a detector positioned behind it as a function of y. If the beam is covered from y = y0 to y = ∞ this amounts to (with ρ = x 2 + y 2 ) P (y0 ) =

Ptot πa 2

y0

−∞

dy

+∞ −∞

2 x + y2 . dx exp − a2

(1.82)

This expression may be integrated in closed form and gives P (y0 ) 1 = √ Ptot a π

2 1 y0 y erf +1 exp − 2 dy = 2 a a −∞

y0

(1.83)

with the error function erf(y0 /a). P (y0 /a)/Ptot is shown in Fig. 1.18, being 0.2398 and 0.7602 for y0 /a = √ ∓0.5, respectively. At these two positions one reads the G AUSS radius a = w/ 2 as marked in Fig. 1.18.

1.2.6

The M 2 Factor

In practice, of course, laser beams are never completely perfect. More or less pronounced deviations from the ideal Gaussian beam profile are the rule. This is very important when describing the quality of a laser system. Most concisely this is expressed by the beam parameter product (BPP) according to (1.69): the larger the BPP, the poorer the beam quality. The international norm ISO 11146 defines as a quantitative characteristic of a laser beam with a divergence angle θe and a beam radius w0 the so called M 2 factor by θe = M 2

λ . πw0

(1.84)

1.3

Polarization

35

One may thus write M 2 as the ratio of ideal to real BPP, or real (θe ) to ideal divergence angle (λ/πw0 ): M2 =

θe BPP BPP θ e w0 = = = . λ/π λ/(πw0 ) λ/π BPPideal

For a Gaussian beam M 2 = 1, while M 2 1.2 is a typical value for a very good, real laser system. Section summary

• Gaussian beams are as close as one can get to mimic a classical “light ray” – they are, so to say, the diffraction limited version of a plane wave. Their spatial profile is characterized by a single,√ real parameter, the R AYLEIGH length zR . It is related to the waist radius w0 = zR λ/π of the beam at which the intensity has dropped to 1/e2 of its maximum I0 . • Along the propagation direction z, the intensity decreases to I0 /2 at z = ±zR . The beam divergence angle in the far field is θe = w0 /z0 = λ/πw0 . This leads to the so called beam parameter product BPP = θe w0 (= λ/π for a Gaussian). The divergence of an arbitrary light beam is characterized by the factor M 2 = BPP/BPPideal ≥ 1. • A brief excursion into F RAUNHOFER diffraction theory shows that even the well known A IRY pattern of a plane wave, diffracted by a circular aperture, is less well focused than the Gaussian beam. • Beam propagation in an optical setup is readily described with the help of the complex beam parameter q(z) = z + izR and so called ray transfer matrices (also ABCD matrices). • Focussing a (nearly) parallel Gaussian beam with a lens of focal length f from an initial beam radius w leads to a new waist radius w0 = f λ/πw. Hence, the wider the beam originally is, the tighter it can be focussed. Telescope systems are important tools for laser beam manipulation. • The razor blade method allows an easy determination of beam radii.

1.3

Polarization

The electric field vector (1.35) of a light beam is expressed in terms of amplitude E0 , polarization vector e and the exponential propagation terms exp[∓i(kz − ωt)]. We emphasize again, that E(r, t) is a real observable, and both exponential terms are necessary for its description. The field amplitude E0 = E0 (ρ, z, t) may depend (slowly) on ρ, and on z as we have just seen, as well as on t as we shall discuss in Sect. 1.4. The vector character of light is fully described by its polarization – an important property of light. It determines e.g. the selection rules for optical transitions as discussed in Chap. 4, Vol. 1. In the present section we resume and extend the dis-

36

1


cussion on polarization from Sect. 4.1, Vol. 1. We begin with fully polarized light and discuss some specific temporal aspects of interest for nonlinear processes. We then present several useful tools and recipes for the preparation and analysis of optical polarization. Finally, we introduce the S TOKES parameters and the degree of polarization for a realistic description of incompletely polarized light. Depending on the case, one may use a Carthesian basis (ex ey ez ), or alternatively the helicity basis (e+1 e−1 ez ) to describe unit polarization vectors (see Sect. 4.1.2 in Vol. 1). We recall that the general unit vector for elliptically polarized light, propagating parallel to the +z axis, may be written as eel = e−iδ cos βe+1 − eiδ sin βe−1 .

(1.85)

The ellipticity angle β describes the degree of ellipticity and the alignment angle δ gives the direction of the ellipse in respect of ex .

1.3.1

Polarization and Time Dependent Intensity

It is interesting to note that the ellipticity of polarized light introduces a temporal dependence into the light intensity – on a time scale of its oscillation period. With (1.36), the electric field is a real quantity, and the intensity becomes 2 ε0 c + 2 E (r, t)e + E − (r, t)e∗ I (t, β) = ε0 cE(r, t) = 4 ε0 c − + E (r, t)E (r, t) + sin(2β) Re E + (r, t)2 , = 2

(1.86) (1.87)

with (ε0 c)−1 = 376.7 . In the following we consider CW light, so that ∗ E + (r, t) = E0 (r) exp i(ωt − kr − φ0 ) = E − (r, t) , and for a plane wave, the most simple case, at a given position in space (say kr = 0) we obtain from (1.87) I (t, β) =

ε0 c 2 E0 1 + sin(2β) cos(2ωt − 2φ0 ) . 2

(1.88)

Obviously, the intensity oscillates rapidly with twice the light frequency ω – except for LHC or RHC polarized light, where β = 0 or = π/2, respectively, and sin 2β = 0. The oscillations are most pronounced for linearly polarized light, where β = ±π/4 and sin(2β) = 1.

1.3

Polarization

37

In linear spectroscopy these rapid oscillations are fortunately without significance: they cancel out with time, and the cycle averaged intensity becomes

1 I (t, β) = Tc

Tc /2

−Tc /2

I (t, β)dt =

ε0 c − E (r, t)E + (r, t) 2

(1.89)

E2 = ε0 c 0 = I0 , 2 independent of β. Above considerations are not restricted to plane waves: for a Gaussian beam one simply has to replace I0 by the profile given in (1.51). When studying nonlinear (multi-photon) processes the situation is different: higher order averages determine the experimentally observable signal. As an example, for not too high intensities the multi-photon excitation and ionization rates are proportional to I N , according to the general formula (5.43) introduced in Vol. 1. This suggests that in such a case, where N photons are simultaneously absorbed, the cycle averaged N th power of the intensity (1.87) might be relevant. One finds (S HCHATSININ et al. 2009) N ω I (t, β) = I0N 2π

2π/ω

N 1 + sin(2β) cos(2ωt) dt

0

2 N N sin2K β cos2N −2K β = I0 K K=0 1 , = I0N cosN (2β)PN cos(2β) N

(1.90)

with PN (x) being the L EGENDRE polynomial of N th order. One easily verifies that I N (t, β) decreases with increasing ellipticity – the higher N the more pronounced. Indeed, recent experiments have shown that in the strong field of intense femtosecond laser pulses the efficiency of multi-photon ionization decreases rapidly and correlates in some cases surprisingly well with I N (t, β) (H ERTEL et al. 2009; S HCHATSININ et al. 2009). By the way: the relative simplicity of these expressions is essentially due to using the helicity basis. It allows one to describe linearly, elliptically and circularly polarized light in the same coordinate system. In contrast, in the literature one usually changes the coordinate system and chooses the z-axis to be parallel to the polarization vector for linear polarization, while for circular polarization z is assumed parallel to the propagation direction. Not only does this choice make the results easily confusing, it also does not allow to change the polarization continuously between the two extremes. Finally, we mention that the more general case of short pulsed laser radiation may be treated in much the same manner as CW light discussed so far. The first term in (1.87) will also vary with time (albeit slowly in the framework of the SVE approximation), while the second term oscillates rapidly – subject to the magnitude

38

1


of the ellipticity angle β. Thus, the relations (1.90) for the cycle averaged N th power of the intensity will also depend on time.

1.3.2

Lambda-Quarter and Half-Wave Plates

We shall now familiarize ourselves with some experimental tools for manipulation of polarized light. We start with the transformation of linearly into circularly polarized light. As discussed in Sect. 4.1.2, Vol. 1, circularly polarized light may be written as superposition of two linear components, oscillating perpendicularly to each other with a phase difference of ±π/2, manifested in (4.12) and (4.13), Vol. 1. The standard tool for generating such conditions is a so called λ/4 plate. That is a thin, very plane ground, birefringent plate (typically made of quartz, magnesium fluoride, or calcite). It has different indices of refraction nf and ns > nf for two crystal axes, a so called fast (f) and a slow (s) axis. The phase velocity for light with field vector parallel to these axes (i.e. light propagating perpendicularly to them) is vf = c/nf and vs = c/ns , with vf > vs , respectively. The wavelengths for slow and fast axes are correspondingly λf > λs , and the two orthogonal field components develop a phase difference while passing through the plate. Specifically, for a λ/4 plate this will be 90◦ , i.e. the thickness of the plate d is given by λ d × (ns − nf ) = . 4

(1.91)

The transformation of linearly polarized light into circularly polarized light is sketched schematically in Fig. 1.19: linearly polarized light enters perpendicularly onto the λ/4 plate, its E vector being aligned at 45◦ between fast and slow axis. Its components, Ex and Ey , are thus parallel to one of the two crystal axes. The figure illustrates how at the exit (out) of the λ/4 plate a shift of λ/4 between the components arises. Together, these two components represent an electric field E vector which rotates around the propagation axis z, as indicated in the figure. The illustration shows σ + (LHC) light. Right hand circularly polarized σ − light (RHC) would be generated if initially the linearly polarized E vector was aligned at −45◦ . Fig. 1.19 Schematic illustration of light passing through a λ/4 plate. It generates σ + light from linearly polarized light. To generate σ − light the initial E(z = 0) vector has to be aligned at −45◦ rather than at +45◦ as shown here

fast axis

z || k

x

x

λ /4

E (z = ℓ ) y

out

d

Ex Ey

y slow axis

x

45°

E (z = 0)

Ex

k in

Ey

y

1.3

Polarization

Fig. 1.20 Short laser pulses (about 7.5 fs FWHM at 800 nm) after passing a λ/4 plate of (a) first-order and (b) second-order

39

(a)

E(t) / E0

1

(b)

E(t ) / E0

1

t / fs 0

-10

-1

10 fast axis slow axis

t / fs -10

0

10

-1

A practical warning appears in order: λ/4 plates must be adjusted very carefully, in respect of perpendicular incidence as well as for proper alignment at 45◦ . A simple check is done by rotating a linear polarizer very fast around the axis of the circularly polarized beam (at best with a motor). The transmitted light – detected by a synchronized oscilloscope – must not show any variation of intensity. A useful rule about the orientation of the light applies: The sense of rotation of the circularly polarized light behind a λ/4 plate is obtained by imagining to turn the incident, linearly polarized E vector on the fastest possible way into the slow axis. This rule is independent of the direction from which the beam is viewed. For the materials used as λ/4 plates typically ns − nf 1. Hence, a λ/4 is much thicker than λ/4 – otherwise it would be a very fragile object. We also mention that the plates are usually not cut exactly parallel, but rather with a very small wedge so that interferences from reflections are avoided. Equivalent phase shifts may also be generated by plates with a thickness 5d, 9d, etc. One speaks of λ/4 plates of higher order, which are much more stable. When working with ultrafast light pulses one must, however, be very careful with such plates. On the one hand thicker plates may lead to unwanted nonlinear effects. On the other hand one has to realize that e.g. a pulse at 800 nm of 7.5 fs duration (today not unusually short) consists of only a few cycles! A comparison of the temporal profiles behind a λ/4 plate of first and higher order shown in Fig. 1.20 illustrates the consequences for ultrashort laser pulses very clearly. The absolute phases are shifted in respect of the envelope of the two field components. Hence, one expects from a first-order λ/4 plate a circularly polarized pulse of essentially the same temporal shape as its linearly polarized parent. In contrast, second (and higher) order plates will lead to serious distortions of the pulse shape. Note, that for continuous light beams such considerations do not play a role. Quarter wave plates have the disadvantage that they fulfill the condition (1.91) only for exactly one wavelength. Alternatively, one may exploit the phase shift due to total reflection inside a prism. The difference of the phase shift for light polarized perpendicular and parallel to the plane of incidence is used.13 Specifically, the so called F RESNEL rhomb sketched in Fig. 1.21 is often used. For an index of reflection of n = 1.51 (glass) a phase difference of 45◦ is achieved with a rhomb angle of to B ORN and W OLF (2006) (Eq. (1.61)) the phase difference δ = δ⊥ − δ is given by tan(δ/2) = cos θi sin2 θi − 1/n2 / sin2 θi , where n is the index of refraction of the medium, and θi is the angle of incidence. 13 According

40

1

Fig. 1.21 Fresnel rhomb and sense of rotation of the circular polarization generated from linearly polarized light after passing the rhomb

elliptic

Lasers, Light Beams and Light Pulses circular

slow fast axes

linear 54º37'

in out top view

side view

54◦ 37 (which is equal to the angle of incidence). The two reflections thus lead to an overall phase difference of π/2. The figure illustrates also the sense of rotation of the circularly polarized light obtained for two possible alignments of the incident E vector. According to the above mentioned rule, the vertical axis of the rhomb is the “slow axis”. The F RESNEL rhomb has the great advantage that it can be used over a wide range of wavelengths. A disadvantage of F RESNEL rhombi is the displacement of the beam. F RESNEL rhombi are often used as pairs, turned by 180◦ , thus forming a λ/2 plate in which the beam displacement is compensated. In any case, half-wave plates (F RESNEL rhombi or birefringent crystal plates) are also very useful optical devices. Instead of (1.91) they satisfy λ d(ns − nf ) = , 2

(1.92)

and are used, e.g. for rotating the plane of linearly polarized light as illustrated in Fig. 1.22. We assume the fast axis of the λ/2 plate to be aligned at an angle α in respect of the E vector of the incoming light beam. As shown in Fig. 1.22(a), one may again consider the two components of E in respect of the slow and fast axis. After passing half the plate (i.e. λ/4 path difference) the beam is elliptically polarized as shown in Fig. 1.22(b). After a full passage through the λ/2 plate a phase difference of π has been accumulated, i.e. the electric vector along the slow axis has now the opposite sign as initially. Hence, the full E vector has been rotated by an angle δ = 2α as shown in Fig. 1.22(c). A full rotation of the λ/2 plate from α = 0 to 2π thus rotates the polarization vector through 4π ; in between the E vector is four times parallel to its original direction. An alternative application of a λ/2 plate is to change LHC light into RHC and vice versa.

(a)

E(0) E(0)

α

fast

(b)

fast

fast

δ = 2α

E(λ /4)

slow

(c)

E(λ /2)

slow

slow

Fig. 1.22 Rotation of the polarization plane by means of a λ/2 plate. (a) E vector at the entrance into the plate, (b) at half distance, (c) after full passage

Polarization

41

Fig. 1.23 S OLEIL -BABINET compensator

t

slow

slo

w

fas

1.3

hI h II fast

Finally we mention as a particularly flexible device the S OLEIL -BABINET compensator, consisting of two birefringent crystal wedges with their optical axes aligned at 90◦ to each other, i.e. their slow and fast optical axes are interchanged. One of the wedges may be moved such that the optical path length through it changes as indicated in Fig. 1.23. In summary the optical path difference is s = (ns − nf )(hI − hII )

(1.93)

with the path lengths hI and hII though the two plates. One may thus change s from negative to positive values continuously, typically from −λ/4 up to 2λ. The devices are built so that this optical path length is constant over a sufficiently large area of the plate as needed for beams with finite radii. With even more comfort, electro-optical devices are used (e.g. ADP) with a birefringence that may be varied continuously by applying a high electric field (linear electro-optical effect). These so called P OCKELS cells are of particular importance in ultrafast laser technology.

1.3.3

S TOKES Parameters, Partially Polarized Light

Definitions of the S TOKES Parameters Alternatively to describing the polarization of light by polarization vectors, traditionally one uses the three S TOKES parameters which are directly accessible to the experiment (introduced 1852 by George Gabriel S TOKES): P1 =

I (0◦ ) − I (90◦ ) I (0◦ ) + I (90◦ )

(1.94)

P2 =

I (45◦ ) − I (135◦ ) I (45◦ ) + I (135◦ )

(1.95)

P3 =

I (RHC) − I (LHC) . I (RHC) + I (LHC)

(1.96)

42

1


They characterize the relative intensity differences I (ep ) of a light beam in respect of the three pairwise orthogonal polarization vectors defined in Sect. 4.1.2, Vol. 1. For their measurement one needs in principle six different filters, each of which transmits exactly one of these different polarizations ep . For 0◦ , 45◦ , 90◦ and 135◦ this is simply a linear polarization filter,14 for the analysis of circular polarization a combination of a λ/4 plate and a linear polarizer. The S TOKES parameters may easily be rewritten in terms of the parameters β and δ, defining the general unit vector eel of polarization for elliptically polarized light according to (1.85): one projects eel onto the 4 linear polarization vectors ep according to (4.7) and (4.9), Vol. 1, or onto (e− , e+ ) for linear and circular polarization, respectively. The S TOKES parameters are then given by the difference of the absolute squares of these amplitudes: P1 = |eel · ex |2 − |eel · ey |2 = cos 2δ sin 2β 2 2 P2 = eel · e 45◦ − eel · e 135◦ = sin 2δ sin 2β

(1.97)

P3 = |eel · e− | − |eel · e+ | = − cos 2β.

(1.99)

2

2

(1.98)

Degree of Polarization In all previous considerations we have assumed that the light beams or wave fields were essentially monochromatic, plane waves – modified if necessary for a spatial variation of the amplitudes within the framework of the SVE approximation, leading e.g. to Gaussian beams. In physical reality, however, we often deal with (i) only quasi-monochromatic and (ii) only partially polarized light beams or wave fields. We shall approach a full description of these facts in several steps. Presently, we simply introduce as an easy to measure quantity the degree of polarization of light: (1.100) P = + P12 + P22 + P32 with 0 ≤ |P| ≤ 1. For fully polarized light – which may be described by eel according to (1.85) – the degree of polarization is P = 1, as one verifies easily by inserting (1.97)–(1.99) into (1.100). For many laser sources P ∼ = 1 is indeed a good description, while natural light is most often unpolarized, i.e. one finds P = 0. This holds e.g. for diffuse illumination by daylight, for incandescent light bulbs, or generally – expressed in more scientific terms – for black body radiators (or cavity radiators). The three S TOKES parameters completely characterize the polarization state of light. One also speaks of a S TOKES vector P = (P1 , P2 , P3 ) of the light, its magnitude P being given by (1.100). 14 As

linear polarizers one uses specially cut arrangements of prisms, exploiting birefringence and total reflection, such as the N ICOL or G LAN -T HOMPSON prism. Alternatively, thin film polarizers are used, exploiting interference effects and special material properties.

1.3

Polarization

43

One may also define a linear degree of polarization P12 : 0 ≤ P12 = + P12 + P22 ≤ 1.

(1.101)

The limits follow from 0 ≤ P3 ≤ 1 and (1.100). If one finds in a measurement of linear polarization P12 < 1, this may have two reasons: either the light as a whole is not fully polarized (P < 1) and/or it contains a fraction of circular polarization (P3 = 0). The physical origin of incomplete polarization will be addressed in several steps. Briefly, the nice and clean description of light as an electromagnetic wave expressed by (1.35) – a single function in space and time – is an idealization. In reality, the overall phase φ0 of the wave (1.35) is not stable over longer periods. Rather, it may be considered constant only over a finite, so called coherence time, to be discussed in Chap. 2. According to the H EISENBERG uncertainty relation this leads to a finite bandwidth, and thus to quasi-monochromatic light. The phase relation between wave-packets with orthogonal pairs of polarization vectors is also correlated only over times on the same order of magnitude. Hence, polarization is no longer complete. As a first step towards a fully realistic description we shall introduce in Sect. 1.4 the superposition of waves with different frequencies, leading to wave-packets. In Chap. 2 we shall then attempt a more precise definition of quasi-monochromaticity and approach a quantum mechanical description of the states of light. In order to quantitatively describe also the polarization states one has to introduce a statistical description of light. The necessary tools are provided by the density matrix which will be treated in Chap. 9. There we shall finally come back again to polarization.

Measuring the Degree of Polarization In real experiments one has to account for the fact that also the analyzer, by which polarization is determined, is not always perfect. Quite generally one may describe (anl) (anl) (anl) an analyzer also by a S TOKES vector P (anl) = (P1 , P2 , P3 ). An ideal analyzer would thus be characterized by a degree of polarization P (anl) = 1, for a realistic analyzer one expects 0 < P (anl) < 1. In Sect. 9.3.1 we shall formally derive this type of description. We note already here a plausible, very useful relation: I (pol) = (I0 /2) 1 + P · P (anl) (anl) (anl) (anl) . = (I0 /2) 1 + P1 P1 + P2 P2 + P3 P3

(1.102)

It describes the intensity transmitted when a light beam of intensity I0 with a S TOKES vector P = (P1 , P2 , P3 ) passes through such a polarizer (ignoring possible unspecific absorption processes). Obviously, in the case of completely unpolarized light with P ≡ 0 one half of the intensity I0 is transmitted through the analyzer, independent of its alignment: an ideal analyzer always suppresses just one of the two polarization components.

44

1


As an example we consider a polarization measurement with an ideal analyzer for linearly polarized light. It does not distinguish between RHC and LHC light, (anl) so that with (1.99) P3 = 0 and cos 2β (anl) = 0 while sin 2β (anl) = 1; it transmits, however, linearly polarized light along its axis of polarization to 100 %, so that (anl)

(anl)

P (anl) = (P1 )2 + (P2 )2 = 1, according to (1.100). If the analyzer axis of polarization is rotated through an angle δ (anl) in respect of the x-axis one obtains (anl) (anl) from (1.97) and (1.98) for P1 = cos 2δ (anl) and P2 = sin 2δ (anl) , respectively. Let the linear polarization of the light studied be described by P1 = P12 cos 2δ

and P2 = P12 sin 2δ,

(1.103)

with the alignment angle δ and the linear degree of polarization P12 according to (1.101). We insert all this into (1.102) to obtain the signal intensity: 1 I (pol) = I0 1 + P12 cos 2 δ (anl) − δ . 2

(1.104)

By varying the alignment angle δ (anl) of the analyzer one may thus determine P12 as well as the alignment angle δ, for which the signal becomes largest. P12 < 1 implies poor polarization – or a circularly polarized background. In both cases the ratio of minimum (Imin ) to maximum signal (Imax ) is Imin I (π/2 + δ) 1 − P12 = . = Imax I (δ) 1 + P12

(1.105)

For P12 = 1 and polarization along the analyzer axis, (1.104) is equivalent to the well known M ALUS’s law I (δ (anl) ) = I0 cos2 δ (anl) . Section summary

• Polarization is described most flexibly in the helicity basis, with the general unit vector for elliptic polarization eel = e−iδ cos βe+1 − eiδ sin βe−1 , with the ellipticity angle β and the alignment angle δ. • On a sub-cycle time scale the light intensity depends crucially on the ellipticity angle β. While the (1st order) cycle averaged intensity is independent of β, higher order averages depend strongly on β. This is highly relevant for processes such as MPI. • λ/4 and λ/2 plates are useful devices for the generation and manipulation polarized light. • The S TOKES parameters, compact as S TOKES vector P = (P1 , P2 , P3 ), give an even more general description of polarization, including partially polarized light. The signal from a light beam with polarization P passing through an analyzer described by P (anl) is given by I (pol)=(I0 /2) 1 + P · P (anl) .

1.4

Wave-Packets

45

1.4

Wave-Packets

1.4.1

Description of Laser Pulses

In reality, a monochromatic electromagnetic wave – with only one frequency ω (or ν), and only one wave vector k – is in many respects a crude simplification, be it as a plane wave or a single mode Gaussian beam. Our first step towards a more realistic model is a so called wave-packet. As already mentioned in Sect. 1.2.1, the field envelope E0 (r, t) in (1.35) may vary as a function of space and time. Thus, we now express it as a linear superposition of plane waves, which in the most general case involves a 3D F OURIER transform (see Appendix I.5 in Vol. 1): 1 E + (r, t) = E0 (r, t)e−i(kr−ωt+φ0 ) 2 1 + (k)e−i(kr−ωt) d3 k. E ⇒ (2π)3

(1.106)

+ (k)/(2π)3 is the (generally complex) amplitude for the different wave vectors k E (and correspondingly different angular frequencies ω = kc). In Sect. 1.2 we have considered only the dependence of the field on position r. Now, we take the opposite approach and focus on the variation with time t. To make things not too complicated, we consider quasi-monochromatic light in vacuum, composed of plane waves, propagating in only one direction, with different angular frequencies ω = kc from a narrow bandwidth δω |ω − ωc | around a carrier frequency ωc δω. The latter condition indicates that we stay well within the limits of the SVE approximation. We shall return to the dependence on r in the next subsection. We assume now that these waves propagate along the z-axis. We substitute kz = ωz/c for kr and express the amplitudes as a function of ω only, replacing 3 k → E(ω)dω. We further simplify (1.106) by considering the field at one E(k)d point in space, say kr = 0. We write the time dependence of the field envelope as an (inverse) F OURIER transform,15 i.e. we replace i E + (r, t) = − E0 (r, t)e−i(kr−ωt+φ0 ) ⇒ 2 ∞ 1 + (ω)eiωt dω E + (t) = E0 (t)ei(ωc t−φc ) = E 2π −∞ ∞ −iφc 1 − ωc )eiωt dω. =e E(ω 2π −∞ 15 For

(1.107) (1.108)

more details about F OURIER transforms and spectral distributions see Appendix I in Vol. 1.

46

1


Fig. 1.24 Evolution of the electric field (grey) and its envelope (red) for a short, Gaussian pulse

1.0 E(t ) ____ E0 -3 -2

ϕc

h(t )

-1 -0.5

1 2 3 t/τ

The last identity is the frequency shift relation for F OURIER transforms (I.22), Vol. 1. The phase shift φc refers here to the maximum of the carrier envelope 1 E0 (t) = E0 h(t) = 2π

∞

−∞

iωt dω. E(ω)e

(1.109)

The maximum field amplitude is E0 , and h(t) is a real envelope function, here nor malized such that h(0) = 1. If it is symmetric in respect of t = 0, E(ω) is also real and symmetric around ω = 0. The carrier frequency ωc enters via (1.108), which is to be inserted into (1.35) to describe the full time dependence of the electric field vector E(t). It is important to note that this construction does not describe a continuous light beam but rather a wave-packet, i.e. a light pulse of finite extension in space and time. With today’s ultrafast laser systems one may generate such F OURIER transform limited light pulses without problems, typically with pulse durations between picoseconds (ps) and some femtoseconds (fs) – with a trend towards even shorter, attosecond (as) pulses. Experience shows that the temporal profile of ultrafast laser pulses can often be described very well by a Gaussian, in analogy to their spatial profile (1.47): h(t) = exp −(t/τG )2 .

(1.110)

As obvious from Fig. 1.24, the relative phase φc of the carrier oscillation becomes an increasingly significant parameter as the pulse duration gets shorter. We still remain, however, within the limits of SVE. The temporal profile of the cycle averaged intensity (I.19), Vol. 1 is16 2 ε0 c E0 h(t) = I0 exp −2(t/τG )2 , 2 √ with a FWHM t1/2 = 2 ln 2τG = 1.177τG .

I (t) =

16 We

(1.111) (1.112)

follow here the convention of the laser community with τG denoting the time at which the intensity has decreased to 1/e2 .

1.4

Wave-Packets

I(t )

(a)

47

(b)

1.0 exp[-(1.665t)2 ]

sech 2 (1.763t) 1.0

0.5

1.0

1.0

10-1 10-2

0 -2

-1

0

1

2

10-3 -2

t / Δ t 1/2

-1

0

1

2

Fig. 1.25 Comparison of a Gaussian (red) and a sech2 (grey) intensity distribution on (a) linear and (b) logarithmic scale. The FWHM t1/2 are in both cases identical, the time scale is measured in units of this FWHM

Alternatively one often uses the squared hyperbolic secant for the temporal intensity distribution, albeit mathematically less convenient (see Appendix I.4.2, Vol. 1): 2 ε0 cE02 2 I (t) = I0 sech (t/τs ) = 2 et/τs + e−t/τs √ with a FWHM a t1/2 = τs 2 ln( 2 + 1) = 1.763τs . 2

(1.113) (1.114)

To obtain the same FWHM as in the Gaussian profile (1.111), one thus has to set τs = 0.668τG . In Fig. 1.25 both time profiles are compared on a linear as well as on a logarithmic scale. The main difference is in the wings: for large |t| the Gaussian distribution decays significantly faster. The spectral properties of these pulse shapes are discussed extensively in Appendix I.3, Vol. 1. One key result is the relation between the F OURIER transform E(ω) of the field envelope and the spectral intensity profile: 2 ε0 c + 2 ε0 c E (ω) = E(ω − ωc ) . I˜(ω) = 4π 4π

(1.115)

We emphasize that I˜(ω) is not the F OURIER transform of I (t). For a Gaussian pulse the spectrum is

I0 ω − ωc 2 ω − ωc 2 ε0 c 2 ˜ = 2 exp − , (1.116) E exp − I (ω) = 2 0 ωG ωG 2ωG ωG √ √ with ωG = 2/τG and a FWHM of ω1/2 = 2 2 ln 2/τG = 2.3548/τG , (1.117) which is centred at the carrier frequency ωc . For F OURIER transform limited pulses we note the general rule, the shorter the pulse, the broader the spectrum,

48

1

Fig. 1.26 Comparison of the squared F OURIER transform (spectrum) for F OURIER limited Gaussian and sech2 pulses. The frequency difference ν = ν − νc of these profiles is measured here in units of 1/t1/2

Lasers, Light Beams and Light Pulses 1.0

( F [ exp(-(t/τG)2)] ) 2

( F [ sech(t/τS)]) 2

0.5 0.315 0.441 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

Δν Δ t1/2

which is ultimately a consequence of the uncertainty relation. More compact this is expressed by the so called time-bandwidth product t1/2 ν1/2 : With ν1/2 = 2πω1/2 one finds for a Gaussian pulse t1/2 ν1/2 =

2 ln 2 = 0.441. π

(1.118)

For practical purposes we communicate this relation in units of wavenumbers and wavelengths: ¯ν1/2 t1/2 = 14710, cm−1 fs

and

λ1/2 /nm = 1.471 × 10−3

(λ/nm)2 . t1/2 /fs

(1.119) (1.120)

A typical short, but not too short laser pulse may have t1/2 = 100 fs, a bandwidth of ¯ν1/2 150 cm−1 (or λ1/2 10 nm at 800 nm). For the hyperbolic secant (1.113) the spectral intensity distribution (I.47), Vol. 1 is

ε0 c E02 2 ω − ωc ˜ I (ω) = , sech π ωs2 ωs where ωs =

2 1.1224 and the FWHM ω1/2 = 1.763ωs = . πτs τs

(1.121) (1.122)

With (1.114) the time-bandwidth product becomes ν1/2 t1/2 = 0.315,

(1.123)

to be compared with (1.118) for the Gaussian. In Fig. 1.26 both spectral distributions are compared, illustrating the narrower time-bandwidth product of sech.

1.4

Wave-Packets

1.4.2

49

Spatial and Temporal Intensity Distribution

The full position and temporal intensity distribution of a Gaussian light pulse is obtained from the stationary expression (1.51) by replacing the maximum intensity I0 there with I (t) according to (1.111):

2

2 t I0 ρ exp −2 (1.124) I (ρ, z, t) = exp −2 w τG 1 + ζ2 with w 2 = w02 1 + ζ 2 , w02 = zR λ/π and ζ = z/zR . The overall maximum intensity I0 can be related to the total energy Wtot in the pulse by integration over time and space. First, integration over the time gives the so called fluence, in units [F ] = J/cm2 : ∞ π I0 (1.125) I (ρ, t)dt = τ F (ρ, z) = exp −2(ρ/w)2 . 2 G 2 1 + ζ −∞ Integration over the whole cross section of the beam as in (1.50) gives the pulse energy Wtot , and the maximum intensity becomes 3/2 Wtot 2 Wtot I0 = = 0.83 . (1.126) 2 π τG w02 t1/2 d1/2 Wtot can readily be measured, as well as the beam width w0 (see e.g. Sect. 1.2.5). With Z0 = (ε0 c)−1 = 376.7 one may then also derive the maximum field amplitude (4.2), Vol. 1 E0 (0, 0) = 2I0 Z0 . For a temporal dependence according to sech2 instead of (1.126) the relation between pulse energy Wtot and intensity becomes with (I.44), Vol. 1 I0 =

1.4.3

Wtot Wtot = 0.78 . 2 2 πτs w0 t1/2 d1/2

(1.127)

Frequency Combs

With the N OBEL prize for H ALL and H ÄNSCH (2005) a broad scientific community has become aware of frequency combs as tools for high precision calibration of optical frequencies. In the present context they are an interesting, special kind of wave-packets and just one more example. In our treatment of Gaussian light pulses, we have so far described exactly one isolated pulse with a duration t of typically a few fs. This situation may indeed be realized in the laboratory without problems, typically with a repetition rate of some Hz to several kHz as needed for any specific experiment. However, the preparation

50

1 ~ E(ω)


ωc= mcωr +ω0

ωr ω0

Δωb

ω Fig. 1.27 Spectrum of a frequency comb with a carrier frequency ωc , a free spectral range (turnaround time) ωr and an ‘offset’ ω0 (after U DEM et al. 2002)

of such pulses usually involves initially a continuous sequence of mode-coupled laser pulses. This is easily understood by remembering the basic laser setup in an active resonator as introduced in Fig. 1.2(b). As described in Sect. 1.1.2, characteristic for a FABRY-P ÉROT resonator is the longitudinal mode structure. The mode distance in the frequency domain is νFSR according to (1.8) – typically 10 MHz to 100 MHz, and the frequency of a T I :S APPH laser is ν 380 THz (at 800 nm). The longitudinal mode index (1.9) is thus a very large integer (on the order of z = 107 ± 105 ). In contrast to the situation in a narrow band CW laser as illustrated in Fig. 1.9, for short pulse lasers one uses an amplifier medium with a rather broad bandwidth ν. This allows, in principle, the generation of short, FT limited pulses, for which the time-bandwidth product is tν 0.3, according to (1.118) and (1.123). Hence, many longitudinal modes will be generated which have to be mode synchronized. This implies constructive interference in the active medium, which fills only a small region in the resonator. The synchronization is achieved by so called active or passive mode locking. The key to mode locking is that the amplifying process favours the highest intensities. Thus, whenever the pulse passes through the amplifier, its maximum is amplified most strongly: once a pulse like structure is formed, it gets shorter and shorter with each turnaround through the resonator. The turnaround time in a resonator is Tr = 1/νFSR with the free spectral range defining the pulse repetition frequency νr = νFSR . The corresponding angular frequency is 2πvg , 2L where we now use correctly the group velocity vg (instead of c, the vacuum speed of light). We shall see in a moment why this is important. The central angular carrier frequency may be written as ωc = zc ωr + ω0 and the angular frequencies of any laser mode is ωr =

ωz = (z + zc )ωr + ω0 .

(1.128)

This is graphically illustrated in Fig. 1.27. The ‘offset’ ω0 with 0 ≤ ω0 < ωr accounts for the fact that the mode frequencies are not necessarily equal to an integer multiple of the resonator frequency. More about this in a moment.

1.4

Wave-Packets

51 2ϕ

ϕ Δt1/2 ~ Tr / 5

field envelope

field amplitude ϕ Δt1/2 ~ Tr / 10

2ϕ

field envelope

field amplitude

0

0.5

1

1.5

2

t /Tr

Fig. 1.28 Example of frequency combs with two different bandwidths ωb of the laser amplifier (ωb for the lower trace is twice that of the upper one). For practical reasons we have only summed over a couple of modes in this model calculation, hence the remaining wiggles around zero signal which disappear in reality

The electric field of such a mode structure is given by the superposition of all modes. Depending on the amplification profile of the laser, the modes will have different intensities (for simplicity we assume a Gaussian intensity profile ωb as indicted in Fig. 1.27). Thus, the field amplitude is ∞ exp i (z + zc )ωr + ω0 t) E(t) ∝ Re

(1.129)

z=−∞

× exp −4 ln 2[zωr /ωb ]2 . Contributions to this sum come only from those modes which are amplified – still a large number, typically on the order of 105 . This F OURIER series with many discrete modes obviously replaces in the present case the F OURIER integral (1.107). Figure 1.28 shows two model frequency combs. As already discussed in the context of Fig. 1.24, for very short pulses the relative phase of the carrier wave in respect of the envelope may play an important role. Here we have now such a case. Since the carrier wave propagates with the phase velocity, the envelope however with the group velocity, a little extra phase shift φ = ω0 /Tr is accumulated for each full turnaround in the resonator, as illustrated in Fig. 1.28. When working with frequency combs and “passion for precision”, as cultivated by N OBEL laureate Ted H ÄNSCH (2005), one important issue is to measure this offset or to make the frequency comb stable enough so that the offset can be compensated completely. It was one of the crucial observation to find out that the short laser pulses generated from these frequency combs may contain several octaves in their frequency spectrum – and nevertheless remain coherent in phase. This opens

52

1


unprecedented perspectives for the precision measurement of light frequencies: the frequencies may in this manner be determined quasi by counting their oscillations in a given time interval. For a detailed discussion we refer to an instructive Nature article by U DEM et al. (2002) and further references given there. Section summary

• We construct wave-packets for short pulses as F OURIER transforms of plane, monochromatic waves. Contributions come from a relatively narrow bandwidth ωb around a carrier frequency ωc ωb . The most used temporal intensity profiles are Gaussian and sech2 . • The spectrum of the pulse is proportional to the squared F OURIER transform of the field amplitude ε0 c 2 I˜(ω) = E(ω) . 4π • Temporal width and the width of the spectrum of short pulses are inversely proportional. The time-bandwidth products of F OURIER limited pulses are: t1/2 ν1/2 = 0.441 (Gaussian) and ν1/2 t1/2 = 0.315 sech2 . • The maximum intensity I0 in a pulse with Gaussian spatial and temporal profile is related to the total pulse energy Wtot by 2 I0 = 0.83Wtot / t1/2 d1/2 . • Frequency combs are a fascinating and important tool for modern high precision spectroscopy. We have given here only a brief introduction.

1.5

Measuring Durations of Short Laser Pulses

1.5.1

Principle

We now give a brief introduction into the measurement of ultrashort laser pulses. The pico-, femto-, or even attosecond time scales, studied in research today, are much too short for direct time resolved recording with electronic techniques. One has to resort to optical methods, essentially comparing optical pathways of the light. In Fig. 1.29 the principle of such a measurement is sketched, exemplified by the scheme for recording an autocorrelation function (the pulse is compared with a copy of itself). First, the pulse is split into two equal parts, e.g. by a semi-transparent mirror. Both parts then propagate along different pathways, with a variable time delay δ built into one of them. In practice this is achieved by an optical delay line which simply involves a longer optical path, e.g. with the help of a M ICHELSON or M ACH -Z EHNDER interferometer. Both beams are then superposed again and finally

1.5


Fig. 1.29 Schematic principle for determining an autocorrelation function; f (t) stands for the intensity or the field strength of the pulse, depending on the detection scheme

53

f(t)

detector

×

2f(t)

∫

f(t + δ) delay

detected, exploiting some suitable linear or, more often, nonlinear optical effect. As we shall see, one typically determines correlation (or autocorrelation) functions. The signal may e.g. depend on the square of the total field or of the total intensity. As we shall see in a moment, this effectively leads to a multiplication of the fields or intensities, as indicated by [×] in Fig. 1.29, and finally one integrates over several cycles, as symbolized by the boxed integral [ ]. Practical examples will be described at the end of this section.

1.5.2

Correlation Functions

We briefly summarize here what has been communicated about correlation functions in Appendix G, Vol. 1. The determination of pulse profiles (in the most simple case just of the FWHM) is an important application. Ideally, one correlates the pulse shape f1 (t) to be measured, with a second pulse shape f2 (t) which is well known. The functions f1 and f2 may be the field amplitudes or the intensity or other characteristic observables of the pulse. In an actual measurement one delays the pulse to be measured in respect of the reference pulse by a well defined, variable time δ, multiplies both and then integrates over a sufficiently long time t. The signal detected as a function of the delay time δ is thus given by17 ∞ G(δ) = (f1 f2 )(δ) = f1∗ (t)f2 (t + δ)dt. (1.130) −∞

G(δ) is called cross-correlation function or first-order correlation function. For example, for two Gaussian pulses with 1/e2 decay time τ1 and τ2 , respectively, one finds (properly normalized): G(δ) = (f1 f2 )(δ) =

2δ 2 2 . exp − 2 π(τ12 + τ22 ) τ2 + τ12

(1.131)

2 The delay time at which the cross-correlation function reaches 1/e of its maximum

value is

17 For

τ22 + τ12 .

later use, we generalize this expression to include also complex functions.

54

1


Often the pulse to be determined is also used as reference pulse (as indicated in the schematic Fig. 1.29) – which of course requires a detailed knowledge of the pulse shape if one wants to extract quantitative information. In that case one speaks of an autocorrelation function. The pulse ‘inquires’ about itself, so to say, at a later time how far it still remembers its own history. Specifically for a Gaussian intensity pulse, with a FWHM t1/2 , the autocorrelation function is again a Gaussian with a FWHM of

auto = t1/2

√ 2t1/2 .

(1.132)

For a pulse whose intensity is described by sech2 (t/τs ) the situation is somewhat more complicated, as outlined in Appendix G.4, Vol. 1. The squared hyperbolic secant does not have the nice property of reproducing itself as autocorrelation function. One finds for the autocorrelation function a FWHM of

auto t1/2 = 1.542t1/2 .

(1.133)

Thus, it is slightly broader than a Gaussian autocorrelation function generated by a pulse with the same FWHM.18 Albeit mathematically not exact, one may, in practice, use G(t) ∝ sech2 [t/(1.542τs )] for the autocorrelation function as a reasonable approximation (see Appendix G.4 in Vol. 1).

1.5.3

Interferometric Measurement

In the schematic setup, introduced in Fig. 1.29, the multiplier is a key element for measuring pulse durations. It indicates superposition of the pulse to be measured with the reference pulse. Naturally, interference effects play an important role at this point, though in practice one often tries to avoid them. Reliable detection of interferences requires high spatial and temporal stability of the beam guiding optics. In standard measurements one thus tries to average over an extended area so that interferences cancel. If one actually is interested in detecting such interferences, special care must be taken for the stability of the source and the setup. Also, one has to superpose both beams as parallel as possible in order to be able to localize the inference patterns well enough. Such an experiment is called an “interferometric setup”. In the following we shall start with the discussion of such a setup and only with hindsight consider averaging processes in time and space. Typically one uses multiphoton processes for detection which occur efficiently in the field of intense, bandwidth limited laser pulses. Harmonic generation from the fundamental oscillation frequency of the wave is most commonly exploited, especially second harmonic when back transforming a sech2 (fitted to an experimentally determined autocorreauto /1.542, i.e. seemingly shorter lation function) the width obtained for the generating pulse is t1/2 auto than that derived from a Gaussian fit, t1/2 /1.414. This is probably the reason why the sech2 pulse shape is very popular among experimentalists in spite of its less friendly mathematics. 18 Conversely,

1.5


55

generation (SHG). During the passage of intensive laser pulses (angular frequency ωc ) through certain (nonlinear) optical crystals a fraction of the light is converted into light with angular frequency 2ωc . This is just a consequence of the nonlinear response of the crystal to electromagnetic radiation: the polarization and the index of refraction depend on intensity, so that the pulse shape is distorted and contains higher harmonics. Alternatively, one may resort to multi-photon excitation or ionization of atoms and molecules. Quite generally, N -photon processes occur to a good approximation with a probability proportional to the N th power of the light intensity (see Sect. 5.3 in Vol. 1). As indicated in the schematic Fig. 1.29, the electric field at the detector originates from superposition (interference) of the fields in the two beam parts – displaced in time by δ. The signal S(δ) at the detector is then the cycle averaged N th power of the intensity (1.90). For simplicity we assume circularly polarized light (β = 0 or π/2) so that N S(δ) = I (δ) N ε0 c N − = E (t) + E − (t + δ) E + (t) + E + (t + δ) (1.134) 2 2 N ε0 c N + 2 E (t) + 2 Re E − (t)E + (t + δ) + E + (t + δ) = 2 N I0N Tav /2 2 = h (t) + 2h(t)h(t + δ) cos(ωc δ) + h2 (t + δ) dt, (1.135) Tav −Tav /2 where h(t) is the envelope function of the field amplitude (1.109). For short pulses of duration τ the angle brackets . . . refer to averaging over a sufficiently long time ±Tav /2 with Tav (τ + δ). In addition, the experiment averages over variations of the phase differences ωc δ, which may occur statistically from pulse to pulse due to experimental instabilities, or due to spatial averaging over an extended detector area etc.19 We may account for this averaging by one further integration over delay times equivalent to one period of the carrier frequency Tc = 2π/ωc : 1 S(δ) = Tc

Tc /2 −Tc /2

S(δ)dδ.

This will now be specialized for some examples as illustrated in Fig. 1.30.

19 Note,

here we do not refer to the absolute stability of the “carrier envelope phase” φc according to Fig. 1.24, which does not enter into the cycle averaged expression (1.135). The signal is only influenced by fluctuations of the relative phase φ = ωc δ between the two time delayed pulses.

56 2

1

N=1

8

S(δ)

N=2


S(δ)

120

N=4

S(δ)

100 80

5 1

60

S(δ) S(δ)

1 0 -2

0

2

0

40 20 0

-2

0

2

S(δ) -2

0

2

δ / τG Fig. 1.30 Autocorrelation functions S(δ) of order N as function of the delay time δ. Calculated from (1.135) for a Gaussian pulse; red S(δ): interferometric stability, black S(δ): averaged over phase fluctuations. The signal is normalized at large delay times to S(δ → ∞) = 1

N =1 Let us start by assuming that we have a detector (photodiode, electron multiplier, thermopile) which simply registers the total intensity linearly, i.e. only one photon is involved in the detection process. Then (1.135) essentially describes YOUNG’s classical double slit interference experiment. The crucial interference term in (1.135) is determined by the autocorrelation function of h(δ) according to (1.130). Normalized to the signal at δ/τG 1 one finds ∞ h(t)h(t + δ)dt. (1.136) S(δ)/S(∞) = 1 + cos(ωc δ) −∞

For a Gaussian envelope (1.110) this gives according to (1.130) and (1.131)

1 δ 2 (cos ωc δ). S(δ)/S(∞) = 1 + exp − 2 τG

(1.137)

This expression is illustrated in Fig. 1.30, panel N = 1: YOUNG’s double slit experiment thus measures the autocorrelation function of the electric field. In principle, one could use such an experiment to determine the pulse duration – if the phase φ = ωc δ was sufficiently stable (any fluctuations would have to be small ∂(φ) π ). This is, however, in the usual simple setup not the case, and the experiments averages over δ for a number of periods. Consequently, cos(ωc δ) → 0 and S(δ)/S(∞) → 1, i.e. the measured signal becomes completely structureless (black line in Fig. 1.30, panel N = 1).

N =2 One avoids this destruction of the autocorrelation function in the phase averaged signal by a nonlinear detection scheme. A popular method is to exploit SHG which originates from two photons of frequency ωc . The signal depends quadratically (N = 2) on the laser intensity. In the case of a Gaussian envelope one may again

1.5


57

integrate (1.135) in closed form, and obtains as autocorrelation function (2nd order in the field amplitude) 2 S(δ) 3δ 2 δ = 1 + 4 exp − 2 cos ωc δ + exp − 2 1 + 2 cos2 ωc δ . (1.138) S(∞) 4τG τG This is sketched in Fig. 1.30, panel N = 2. The maxima are now massively enhanced due to the squared dependence of the signal on intensity, equivalent to |E(t)|4 (red trace). Without specific provision for an interferometric measurement, the phase fluctuations will, here too, wash out the interference structures. Averaging over at least one period makes the first cos(ωc δ) term disappear, while the second, cos2 ωc δ averages to 1/2. Hence S(δ)/S(∞) = 1 + 2e−(δ/τG ) . 2

(1.139)

Clearly, in this case the averaged signal remains a function of the time delay δ (black line in Fig. 1.30, panel N = 2). With such a measurement the autocorrelation function (1.131) of the laser intensity may be determined even if the phase ωc δ fluctuates slightly from pulse to pulse. To be specific: the averaged signal (1.139) corresponds to the autocorrelation function 2nd order in the field amplitude, and the exponential term is just the autocorrelation function 1st order of the intensity. Independent of the line profile (1.110), the phase averaged signal for the case N = 2 is given by: ∞ 2 h (t)h2 (t + δ)dt S(δ) = 1 + 2 −∞ ∞ 4 . (1.140) S(∞) −∞ h (t)dt Equation (1.140) is the basis for standard analysis20 of short pulses by SHG detection.

N =4 The interference signal expected in a four photon process as a function of the delay time δ is shown in Fig. 1.30, panel N = 4, again for two coherent Gaussian laser beams. It represents essentially the 4th order autocorrelation function of the laser field. Again, the full red line predicts the interference signal if the detector effectively averages only over phase fluctuations δ τ . The black bell shaped curve represents the average in case of strong phase fluctuations. Note that the maximum of the signal (also called the four photon coherence signal) is 50 times higher than the background at δ τ . In Table 1.5 the results discussed above and some more are summarized. Note the strong enhancement of the maximum as the order N increases, while at the same time the overall width t1/2 of the phase averaged signal decreases. 20 However, in a real experiment the two time delayed beam parts typically intersect at a very small angle, and the SHG signal is detected at the angle bisector. For this geometry we have to add the specific phase matching conditions to (1.134). Due to this arrangement the measured signal is then free of background (S(δ → ∞) = 0).

58

1


Table 1.5 Determination of correlation functions of different order by multi-photon processes with a different number N of photons involved. Reported are theoretical predictions for the phase averaged signal S(x)/S(∞) as function of delay time x = δ/τG , for the FWHM t1/2 , for the signal maxima S(0)/S(∞), – and for comparison also the maxima S(0)/S(∞) predicted for an interferometric measurement N

S(x) for Gaussian intensity profile 2 e−2x

(with

x2

S(0)

= δ 2 /τG2 )

1

1

2

2 1 + 2e−x

3

1 + 9e−4x

4

1 + 18e−2x + 16e−3x

5

2 1 + 100e−12x /5

6

1 + 36e−5x

1

1.665

8

1.442

32

35

1.257

128

131

1.132

512

462

1.009

2048

10

2 /3

2 /2

2 + 25e−8x /5

2 + 5e−9x /10

+ 200e−3x + 225e−8x 2

2 /3

1 2

3 2

S(0)

∞

1 2 /3

t1/2 /τG (FWHM) √ 2 ln 2 = 1.177 √ 2 ln 2 = √ √ 3 ln 2

Generalization In our examples we have assumed so far Gaussian envelopes of the field amplitude. Important limits of (1.135) may, however, be formulated quite generally. For example, at large delay times one may neglect all terms containing products of functions taken at different times t and t + δ, respectively. Thus, for large δ only the N th power of the first and the last term contribute and we obtain for δ τ

simply S(δ → ∞) = 2 × (I0 )N .

The other limiting case is δ = 0, which leads according to (1.135) to the maximum signal S(δ = 0) = (4 × I0 )N , so that 4N S(δ = 0) = S(δ → ∞) 2

(1.141)

independent of the pulse shape. This relation describes the maximum signal (at δ = 0) in the case of interferometric stability, i.e. one requests fluctuations ∂(ωc δ)

π/2 – in the observation volume and over the total observation time. Conversely, if the phase fluctuation is large (i.e. ∂(ωc δ) π ) – and that is indeed the case for most experimental setups which do not take special care for stabilization – the averaging may be modelled independently from the line profile. We still assume that the pulse duration is long compared to the period of oscillation, i.e. that ωc 1/τ . We may then assume h(t) h(t + δ) over the phase averaging time and do the averaging prior to the integration over time t. Independent of the pulse shape, one obtains then for N and hence (1.142) δ τ always S(0) = 2N I0N 1 + cos(φ) π (2N )! φ S(0) 1 = 22N −1 dφ = cos2N . (1.143) S(∞) 2π −π 2 2(N !)2 Note that this expression reproduces the values explicitly derived for Gaussians as shown in Table 1.5.

1.5

~ I(λ)

1


(a))

1

59

(b)) (b

1.0

( ) (c)

spectrum c

Δλ ≈

22 4 22.4nm

0.5

0 0.5

sech2

FWHM 118 fs

0.1

Gaussian 0 1000 1050 wavelength λ / nm

0 1100 - 200 Δν1/2 Δt1/2 = 0.45

200

- 100

0

100

delay time δ / fs

Fig. 1.31 Experimentally determined (a) spectral and (b, c) temporal intensity distribution of a nearly F OURIER limited laser pulse, with ν1/2 = 6.25 THz and t1/2 = 74 fs according to S CHMIDT et al. (2010)

1.5.4

Experimental Examples

Figure 1.31 shows a nice experimental example for the determination of a short pulse duration. A mode locked laser pulse generated in a diode pumped crystal made out of a new material, Yb:LuScO3 , was investigated by S CHMIDT et al. (2010).21 The experimentally determined amplification profile I˜(λ) in Fig. 1.31(a) may be fitted surprisingly well by a Gaussian or a sech2 profile and has a bandwidth (FWHM) of ca. 22.4 nm (corresponding to ν1/2 = 6.25 THz). The autocorrelation function Fig. 1.31(b) has been measured in a pump-probe scheme as just discussed (N = 2). From the fit of the experimental data with a sech2 (t/τs ) distribution one finds for the autocorrelation function a FWHM ≈ 118 fs, which according to (1.133) corresponds to a pulse width of t1/2 = 74 fs. The logarithmic display Fig. 1.31(c) allows an instructive comparison of the two pulse shapes described in Sect. 1.4.1. In the present case excellent agreement is found with a suitably fitted G AUSS profile. The time-bandwidth product measured corresponds to ca. 0.45, to be compared with the theoretical value 0.315 according to (1.123). One may consider this pulse as nearly F OURIER limited, in particularly so as the choice of the profile for fitting the experimental data is not actually compelling (see also footnote 18). Figure 1.32 shows the realization of a typical interferometric measurement.22 The experimental setup used here consists of a very small, stable M ICHELSON interferometer and detection by SHG (N = 2) in a thin BBO crystal. To compensate dispersion effects two beam splitters are used in this special setup. This makes the setup applicable also for very short laser pulses (<10 fs). The experimental measurement shown here has been obtained with a pulse from a T I :S APPH laser (800 nm) having a pulse duration of ca. 19.5 fs. One sees a nice interference pattern, symmetric in respect of δ = 0, and rather close to the expected ratio S(0):S(∞) = 1:8 21 We 22 We

thank Uwe Griebner for providing us with the original, measured data.

thank Günter S TEINMEYER (2010) for kindly letting us have his experimental data and sketch of the setup as well as for helpful discussions.

60

1

filter

am

input

S( δ ) S(∞)

detector

SHG

be

Fig. 1.32 Interferometric determination of the autocorrelation function of an ultrashort laser pulse (FWHM t1/2 = 19.5 fs), kindly provided by S TEINMEYER (2010)


8

sp lit te rs

delay 1 -150

-100

-50

0

50

100

150

0

delay time δ / fs

(cf. Fig. 1.30, panel N = 2). The side maxima originate from satellite pulses due to incomplete compensation of mode turnaround times in the laser resonator. Section summary

• The duration (FWHM) t1/2 of ultrashort laser pulses can be measured by optical time delay δ. This is achieved by different optical path lengths s1 − s2 = cδ for two equal fractions of the laser beam. Effectively one multiplies their electric fields (or intensities) f1 (t) and f2 (t + δ), and integrates over a time t1/2 . • Mathematically such a measurement is expressed as first-order autocorrelation function: ∞ f1∗ (t)f2 (t + δ)dt. G(δ) = f1 (δ) f2 (δ) = −∞

• The autocorrelation function of a Gaussian time profile has a FWHM √ auto = 2t1/2 . t1/2 • If an N -photon process is used for detection, the measurable signal (1.135) is a more complicated function of field (or intensity) and δ (summarized in Table 1.5 for N ≤ 6). Depending on the experimental setup, one may record an averaged signal which allows a robust determination of the pulse width, or an interferometric signal with fast oscillations corresponding to the cycle time of the light. • The higher the order N of the processes used for detection, the narrower the temporal distribution of the measured profile.

1.6

Nonlinear Processes in Gaussian Laser Beams

61

1.6


1.6.1

General Considerations

As long as processes are studied which depend linearly on intensity, averaging over the temporal and spatial profile of a laser beam leads to signals which are simply proportional to the averaged intensity: the cross section or absorption coefficient does not depend on intensity. However, in the case of nonlinear processes the situation is more complicated as just illustrated, and some basics about multi-photon processes have been discussed in Sects. 5.3 and 8.5, Vol. 1. Since the interaction of free atoms, molecules and cluster with intense laser fields is an important theme in modern laser based science, we now take a closer look at multi-photon ionization to illustrate the consequences of nonlinearity. (N ) Let N be the particle density of the target, σba the relevant cross section for N photon ionization, and Φ = I /ω the photon flux (dimension L−2 T−1 ). According to (5.43), Vol. 1 the rate (dimension T−1 ) for the MPI process is: (N )

(N )

Rba = σba Φ N = sN I N (ρ, z, t) with sN = σ (N ) /(ω)N

(1.144)

(N )

dimension of σba : L2N TN −1 , dimension of sN :

L2N TN −1 Enrg−N .

Such processes may be investigated efficiently with short laser pulses. Let us consider a Gaussian beam whose intensity depends on time and position according to (1.124). To compute the expected experimental signal one has to integrate over time and detection volume. Also, one must account for the fact that the initial target density N0 may change significantly during the pulse, since ionization depletes the initial state. Hence, dN(ρ, z, t) = −N(ρ, z, t)sN I N (ρ, z, t)dt, and integration over the whole laser pulse leads to N(ρ, z) = N0 (ρ, z) exp −sN τN I N (ρ, z) .

(1.145)

For abbreviation we write I (ρ, z, 0) = I (ρ, z) and introduce an effective N photon pulse duration τN =

∞

−∞

exp −2N (t/τG )2 dt =

π τG . 2N

(1.146)

We define a saturation intensity Is = (τN sN )−1/N

(1.147)

62

1


at which the initial number of target species in the ground state has decreased to 1/e. We can now rewrite (1.145): N . N(ρ, z) = N0 exp − I (ρ, z)/Is

(1.148)

In the focus of a Gaussian beam (1.124) the intensity is I (0, 0, 0) = I0 . Thus, if this maximum intensity I0 becomes equal to the saturation intensity Is , the target density N0 in the centre of the laser focus has decreased to 37 % of its initial value, while 63 % of the atoms (or molecules) are ionized. The total measurable signal is obtained by integration over the whole target volume V detected by the experiment (ignoring possible reductions due to detection efficiencies): S

I0 Is

= N0 V

I (ρ, z) N dV 1 − exp − . Is

(1.149)

We insert the dependence of the intensity on position according to (1.124), write u = (I0 /Is )/(1 + ζ 2 ), choose cylinder coordinates for the integration, with dV = 2πρdρdz and integrate in a first step in radial direction:

∞ ρ2 (1.150) dS(u) = 2πN0 dz ρdρ 1 − exp −uN exp −2N 2 w 0 ∞ πN0 w02 zR 1 + ζ 2 dζ = ρdρ 1 − exp −uN exp −ρ 2 . N 0 The radial integral may be written in closed form as πN0 w02 zR 1 + ζ 2 γ + ln uN + E1 uN dζ, (1.151) 2N ∞ with E1 (x) = − Ei(−x) = 1 (e−xt /t)dt (see e.g. W EISSTEIN 2004, Ei(x) being the so called exponential integral) and γ = 0.5772157 E ULER’s constant. Alternatively one may expand the integrand in (1.150) into powers of ρ and then integrate: dS(u) =

∞

dS(u) = −

πN0 w02 zR uj N dζ 1 + ζ2 (−1)j 2N j j!

(1.152)

j =1

=

∞ πN0 w02 zR

2N

(−1)j −1

j =1

1−j N (I0 /Is )j N 1 + ζ2 dζ. j j!

For sufficiently low intensities, typically I0 /Is < 0.5, the first term of the series dominates, and independent of the geometry the signal becomes ∝ (I0 /Is )N as expected. The situation is more complicated if the intensity becomes comparable to saturation intensity. The series expansion converges then only very slowly and further considerations are required which will be explicated in the following. Two experimental geometries are distinguished.

1.6


63

S /S (1) 100

I0 / Is

1 0.1

10

( I0 / Is ) 5

100 d z

10-4

ρ

10-6

tar

(I0 / Is ) 8

0.01

ge t

10-8

Fig. 1.33 Multi-photon ionization signal S as a function of intensity I , measured in units of the saturation intensity Is in a log − log plot. The strictly cylindrical geometry is sketched in the inset. The full lines are computed according to (1.153) with S ∝ I 5 and ∝ I 8 . The experimental points ++ are data for the ionization of C60 → C+ 60 (red) and C60 (grey) obtained with laser pulses at 800 nm and a pulse duration of 27 fs (S HCHATSININ et al. 2006)

1.6.2

Cylindrical Geometry (2D Geometry)

Most simple is the strictly cylindrical 2D geometry where the target is a thin sheet of thickness d zR which is traversed perpendicularly by the focussed laser beam (R AYLEIGH length zR ). This may be realized by a molecular beam whose angular divergence is limited by a slit. The geometry is illustrated in the inset of Fig. 1.33. In this case the ζ dependence in (1.151) can be ignored and the signal is

N πN0 w02 d I0 I0 + E1 . γ + N ln S(u) = 2N Is Is

(1.153)

We recall that for low intensities this expression is ∝ (I0 /Is )N . Conversely, for sufficiently high intensity, say I0 /Is > 3, the E1 term may be neglected and the ln term dominates. Figure 1.33 illustrates this signal for a 5 and an 8 photon ionization process. The experimental example chosen is MPI of C60 by focussed 800 nm (1.55 eV) laser pulses of 27 fs duration (FWHM) according to S HCHATSININ et al. (2006). The double logarithmic display documents the power law S ∝ I N and allows one to directly extract the exponent N as slope for low intensities. The generation of ++ 5 C+ 60 (red) obviously requires 5 photons (S ∝ I ), the double ionization (C60 grey) 23 8 shows approximately a S ∝ I behaviour. This agrees well with the ionization potentials of C60 (7.56 eV 5ω) and of C+ 60 (11.8 eV 8ω), respectively. For intensities I0 > Is one clearly recognizes the saturation like behaviour. It sets in when in the centre of the laser beam most of the neutral target molecules are ionized. The 23 Figure 1.33 is scaled dimensionless, i.e. the intensities have been normalized to the saturation intensity Is , while the ion signal S is normalized to the signal S(I0 /Is = 1).

64

1


S / arb. un.

4 C602+

Is

2

C60+ 0 0.2

1

2

10

I 0 / 1014 Wcm-2 Fig. 1.34 Multi-photon ionization signal S as a function of the intensity I as in Fig. 1.33 – now, however in a lin − log plot. The experimental data (signal S in arb. un. vs. maximum I0 in the focus in W cm−2 ) are not rescaled here. Slightly different saturation intensities are found for C+ 60 and C++ 60 (red and grey arrow, respectively)

volume (here a circular disk) within which I (ρ, z) ≥ Is grows with increasing maximum intensity I0 . Thus, the increase in signal above saturation in the centre results from an increase in effective volume. This simple geometry can be evaluated easily. Alternatively to the log − log display, one may plot the signal S linearly versus log(I0 ) and extrapolate the data at high intensity to zero signal, as suggested by H ANKIN et al. (2001). With (1.153) one reads the saturation intensity Is as intersection of this line with the intensity axis (apart from a small shift due to γ and the influence of the E1 term, negligible at high intensities). This is illustrated in Fig. 1.34 for the example just discussed. The experimental data are not scaled. Clearly, in this special example the satura++ tion intensities for C+ 60 and C60 differ only little from each other. This is a somewhat puzzling result if one assumes sequential ionization according to the scheme + − C60 + N1 ω → C+ 60 + e as a first, and C60 + N2 ω as a second step (with N1 = 5 and N2 = 8): obviously the physics is somewhat more complex! From the saturation intensities thus determined one may in principle determine the MPI cross section by simply inverting (1.147). With (1.146) and (1.144) one obtains −1 −N sN = τN Is

−1 −N and σ (N ) = (ω)N τN Is .

(1.154)

This is, of course, only valid if the power law holds up to saturation. Even if this is not strictly correct, the relation may serve as a first-order approximation for obtaining an estimate for an ‘equivalent’ MPI cross section. For the record we note another important fact: the measured saturation intensities depend on the pulse width. This is of course not surprising since saturation implies depletion of the initial target state which clearly is a function of time. Providing (1.144) is strictly valid, we obtain from (1.147) for a given cross section at two different pulse widths τ1 and τ2 the ratio of the respective saturation intensities: Is1 /Is2 = (τ2 /τ1 )1/N .

(1.155)

1.6


1.6.3

65

Conical Geometry (3D Geometry)

To achieve the highest possible intensities one has to focus the laser beam tightly. Typically in such experiments, the extension d of the target volume in z-direction ‘seen’ by the detector gets larger than the R AYLEIGH length, d zR . The simple cylindrical geometry sketched in Fig. 1.33 is thus no longer applicable. Instead, one has to integrate (S PEISER and J ORTNER 1976) over the full ‘dog bone’ geometry of the focus illustrated in Fig. 1.10(c). Let us first get a rough estimate of the volume Vs within which the intensity is larger than the saturation intensity Is : At sufficiently high I0 the boundary of Vs is located in the far field region so that the intensity (1.124) on the beam axis is, reasonably accurate, I √ I0 /ζ 2 . The extension of the ‘saturated dog bone’ in z-direction is thus zs = zR I0 /Is , from which we obtain Vs =

dV

zs

−zs

w2 πw 2 dz = π 20 2 zR

zs

z2 dz =

0

πw02 zR I0 3/2 . 3 Is

The thus derived dependence (of the observed signal) on I 3/2 is characteristic for saturated processes in 3D geometry. A clean evaluation has to account also properly for the radial expansion – as it turns out, this does not change the overall result. Strictly speaking one also has to account for the regions outside the saturation volume Vs . This implies integration of (1.151) or (1.152) over all z from −∞ to +∞. One finds that the integral exists if N > 3/2. But it is not completely trivial to evaluate it, since the series (1.152) converges rather slowly. It has been evaluated for the first time by C ERVENAN and I SENOR (1975). One chooses a desired precision by the parameter L (typically L ≡ 3) and integrates for small (I0 /Is )N ≤ L the series, for large (I0 /Is )N > L the expression (1.151) were then the exponential integral may be neglected. With u ≡ I0 /Is the signal for uN > L is ∞ u3/2 S(u) 1/N −j 1/N 1/N = + H u/L . a(j ) u/L + G u/L S(1) V (N ) j =0

For lower intensities I0 /IsN < L one finds ∞

S(u) 1 (−1)j −1 Nj = u V (N j ) S(1) V (N ) j j!

with

j =1

V (m) =

(−1)m Γ (1/2)π Γ (m − 1)Γ (5/2 − m) ∞

a(j ) = (−1)j

Γ (1/2) (−L)n Γ (j + 1)Γ (1/2 − j ) (2(N n + j ) − 3)nn! n=1

66

1 S

1000

3D geometry ( I / I s) 3/2 ln ( I / I s)

100

2D geometry

1 0.1

10 0.01

( I / I s) 5


100

I / Is

10-4 ( I / I s) 8

10-6

Fig. 1.35 Multi-photon ionization signal S as a function of the intensity I measured in units of the saturation intensity Is in a log − log plot – as in Fig. 1.33. In contrast, however, here this simulation of the signal represents an integration over the full volume of the laser beam (3D geometry). For comparison, also shown are the corresponding (I0 /Is )3/2 traces (dashed lines), and the ln(I0 /Is ) fits (thin lines) reproduced from Fig. 1.33

1 G(u) = (γ + ln L)(1 + 2/u)(1 − 1/u)1/2 3 2 H (u) = N (1 − 1/u)3/2 + 6 1/u − 1 − arcsin(1 − 1/u)1/2 . 9 In Fig. 1.35 the behaviour of this function for the full 3D geometry is sketched. One recognizes that this leads to a faster rise of the signal in the saturation region in comparison to the 2D geometry. In practice the observed signals are even more complicated. As already mentioned, sequential multi-photon ionization is observed in many cases, but may be complicated by more delicate processes, involving e.g. non-sequential processes where several electrons are involved. In addition, in molecules fragmentation processes may occur, further complicating the observed signals. In clusters interaction of the field with a micro-plasma induced within the cluster can lead to a wealth of processes which evolve independently of each other. Due to their specific intensity dependence these processes may in turn impress characteristic structures onto the temporal and spatial distribution of the ionization yield. In any case, the interaction of atoms, molecules and nano-particles with intense laser pulses is an exciting field of modern research which develops at very fast pace.

1.6.4

Spatially Resolved Measurements

Time and position dependent structures of these processes have been studied in a nice, still relatively simple experiment by S TROHABER and U ITERWAAL (2008). It is summarized in Fig. 1.36. MPI of Xe atoms is investigated with a short laser pulse (800 nm, 50 fs). (a) A combination of a narrow slit (y-coordinate) with energy analysis of the ions by time of flight (TOF, x-coordinate) leads to a 3D image of the


(a)

V3

lens

laser beam d repeller

MCP ion trajectory V2

V1 = 0

67

slit

1.6

V= 0

(d)

VR

V0 = (x0 /d )VR x0 optical axis

x z

5.1

S(TOF)

TOF(x)

5.0 N(x) 4.9 1.5

Xe+

(c)

TOF/μs

2.0

ion signal / arb. un.

(b)

y

Xe2+ Xe3+ Xe4+

x / mm - 30

0

30 μm

Fig. 1.36 Experiment with spatially resolved detection of multi-photon ionization for Xe atoms in the focus of a Gaussian beam according to S TROHABER and U ITERWAAL (2008). (a) Experimental setup, (b) dependence of the TOF on charge and origin x of ions, (c) Xeq+ signal for different charges q as a function of x (signal traces) and (d) as a 3D plot of the signal strength in respect of the xy plane

different charge states observed. Figure 1.36(b) shows the TOF as a function of the positions x where the ions have been created, for Xeq+ ions of charge state q = 1–4. The saturation intensity for these MPI processes is highest for q = 4 and obviously chosen in the present experiment such that the all other charge states are already bleached out in the centre of the laser focus. The smaller q, the lower the saturation intensity – which leads to detection of these ions from the outer zones of the laser beam: the smaller q the farther away from the focus the respective maximum signal is seen. Figures 1.36(c) and (d) give a 2D and 3D overview, respectively, of the observed ion signals. Ultimately (d) is something like a direct, nonlinear image of the laser intensity. One may directly compare it to the ‘dog bone’ intensity distribution in Fig. 1.10. Section summary

• While in linear spectroscopy the spatial and temporal profile of the laser beam is usually irrelevant, it plays a crucial role when nonlinear processes, such as N photon ionization, are studied. • At low intensities, one expects a signal ∝ I N (except for more complex processes, such as non-sequential MPI). Characteristic in high intensity Gaussian beams is a depletion of the initial target state and hence saturation of the ob-

68

1


served signal, if the maximum intensity I0 in the beam is higher than a process dependent saturation value Is . For I0 > Is it is geometry rather than physics that determines the signal. • Assuming N th power dependence of the rate, the saturation intensity −1/N I s ∝ τ N σ (N ) depends on the N photon ionization cross section σ (N ) , and on the effective N photon pulse duration τN = π/(2N )τG , where τG characterizes the 1/e2 duration of the Gaussian temporal profile of the laser pulse. • In the saturation region MPI the dependence of the signal on intensity is different for cylindrical geometry (2D) – where it is essentially proportional to N ln(I0 /Is ) – and in conical geometry (3D) where it a rises proportional to (I0 /Is )3/2 .

Acronyms and Terminology ADP: ‘Ammonium dihydorgen phosphate’, crystal, birefringent, piezoelectric, used also in nonlinear optics. ASE: ‘Amplified spontaneous emission’, may occur in (long) optical amplifier media with high gain. BBO: ‘Beta barium borate’, crystal, birefringent, excellent nonlinear optical properties, piezoelectric. BPP: ‘Beam parameter product’, characterizing the quality of a laser beam (see Chap. 1, Eq. (1.69)). c.c.: ‘complex conjugate’. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. FPI: ‘FABRY-P ÉROT interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2 in Vol. 1). FSR: ‘Free spectral range’, of an optical interferometer (see Sect. 6.1.2 in Vol. 1). FT: ‘F OURIER transform’, see Appendix I in Vol. 1. FWHM: ‘Full width at half maximum’. LHC: ‘Left hand cicularly’, polarized light, also σ + light. MPI: ‘Multi-photon ionization’, ionization of atoms or molecules by simultaneous absorption of several photons. RF: ‘Radio frequency’, range of the electromagnetic spectrum. Technically, one includes frequencies from 3 kHz up to 300 GHz or wavelengths from 100 km to 1 mm; ISO 21348 (2007) defines the RF wavelengths from 100 m to 0.1 mm; in spectroscopy RF usually refers to 100 kHz up to some GHz. RHC: ‘Right hand cicularly’, polarized light, also σ − light.

References

69

SHG: ‘Second harmonic generation’, doubling of a fundamental frequency, for infrared or visible light typically by methods of nonlinear optics. SVE: ‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38)). TEM: ‘Transversally electric and magnetic’, modes of an electromagnetic wave. Ti:Sapph: ‘Titanium-sapphire laser’, the ‘workhorse’ of ultra fast laser science. TOF: ‘Time of flight’, measurement to determine velocities of charged particles, and consequently their energies (if the mass to charge ratio is known) or their mass to charge ratio (if their energy is known).

References B LOEMBERGEN , N. and A. L. S HAWLOW: 1981. ‘The N OBEL prize in physics “for their contribution to the development of laser spectroscopy” ’, Stockholm. http://nobelprize.org/nobel_prizes/ physics/laureates/1981/. B ORN , M. and E. W OLF: 2006. Principles of Optics. Cambridge: Cambridge University Press, 7th (expanded) edn. C ERVENAN , M. R. and N. R. I SENOR: 1975. ‘Multi-photon ionization yield curves for Gaussian laser-beams’. Opt. Commun., 13, 175–178. E INSTEIN , A.: 1916. ‘Strahlungs-Emission und -Absorption nach der Quantentheorie’. Verh. Dtsch. Phys. Ges., 18, 318–323. G ORDON , J. P., H. J. Z EIGER and C. H. T OWNES: 1955. ‘Maser – new type of microwave amplifier, frequency standard, and spectrometer’. Phys. Rev., 99, 1264–1274. H ALL , J. L. and T. W. H ÄNSCH: 2005. ‘The N OBEL prize in physics: for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/2005/. H ANKIN , S. M., D. M. V ILLENEUVE, P. B. C ORKUM and D. M. R AYNER: 2001. ‘Intense-field laser ionization rates in atoms and molecules’. Phys. Rev. A, 6401, 013405. H ÄNSCH , T. W.: 2005. ‘N OBEL lecture: Passion for precision’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/2005/hansch-lecture.html. H ERTEL , I. V. I. S HCHATSININ, T. L AARMANN, N. Z HAVORONKOV, H.-H. R ITZE and C. P. S CHULZ: 2009. ‘Fragmentation and ionization dynamics of C60 in elliptically polarized femtosecond laser fields’. Phys. Rev. Lett., 102, 023003. H ODGSON , N. and H. W EBER: 2005. Laser Resonators and Beam Propagation, vol. 108 of Springer Series in Optical Sciences. Berlin: Springer, 2nd edn., 824 pages. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. JAVAN , A., W. R. B ENNETT and D. R. H ERRIOTT: 1961. ‘Population inversion and continuous optical maser oscillation in a gas discharge containing a He-Ne mixture’. Phys. Rev. Lett., 6, 106–110. KOGELNIK , H. and T. L I: 1966. ‘Laser beams and resonators’. Appl. Opt., 5, 1550–1567. L INDNER , F., G. G. PAULUS, H. WALTHER, A. BALTUSKA, E. G OULIELMAKIS, M. L EZIUS and F. K RAUSZ: 2004. ‘Gouy phase shift for few-cycle laser pulses’. Phys. Rev. Lett., 92, 113001. M AIMAN , T. H.: 1960. ‘Optical and microwave-optical experiments in ruby’. Phys. Rev. Lett., 4, 564–566. M ILLONI , P. W. and J. H. E BERLY: 2010. Laser Physics. Hoboken: Wiley, 832 pages. S CHÄFER , F. P., W. S CHMIDT and J. VOLZE: 1966. ‘Organic dye solution laser’. Appl. Phys. Lett., 9, 306–309. S CHAWLOW , A. L. and C. H. T OWNES: 1958. ‘Infrared and optical masers’. Phys. Rev., 112, 1940–1949.

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S CHMIDT , A. et al.: 2010. ‘Diode-pumped mode-locked Yb:LuScO3 single crystal laser with 74 fs pulse duration’. Opt. Lett., 35, 511–513. S HCHATSININ , I., H.-H. R ITZE, C. P. S CHULZ and I. V. H ERTEL: 2009. ‘Multi-photon excitation and ionization by elliptically polarized, intense short laser pulses: Recognizing multi-electron dynamics and doorway states in C60 vs. Xe’. Phys. Rev. A, 79, 053414. S HCHATSININ , I., T. L AARMANN, G. S TIBENZ, G. S TEINMEYER, A. S TALMASHONAK, N. Z HAVORONKOV, C. P. S CHULZ and I. V. H ERTEL: 2006. ‘C60 in intense short pulse laser fields down to 9 fs: excitation on time scales below e-e and e-phonon coupling’. J. Chem. Phys., 125, 194320. S IEGMAN , A. E.: 1986. Lasers. Sausalito: University Science Books, 1283 pages. S OROKIN , P. P. and J. R. L ANKARD: 1966. ‘Stimulated emission observed from an organic dye, chloro-aluminum phthalocyanine’. IBM J. Res. Dev., 10, 162–163. S PEISER , S. and J. J ORTNER: 1976. ‘3/2 power law for high-order multi-photon processes’. Chem. Phys. Lett., 44, 399–403. S TEINMEYER , G.: 2010. ‘Interferometric determination of the autocorrelation function of a sub 20 fs laser pulse’. Private communication. S TRICKLAND , D. and G. M OUROU: 1985. ‘Compression of amplified chirped optical pulses’. Opt. Commun., 56, 219–221. S TROHABER , J. and C. J. G. J. U ITERWAAL: 2008. ‘In situ measurement of three-dimensional ion densities in focused femtosecond pulses’. Phys. Rev. Lett., 100, 023002. U DEM , T., R. H OLZWARTH and T. W. H ÄNSCH: 2002. ‘Optical frequency metrology’. Nature, 416, 233–237. W EISSTEIN , E. W.: 2004. ‘En-function’, Wolfram Research, Inc., Champaign, IL, USA. http:// mathworld.wolfram.com/En-Function.html, accessed: 9 Jan 2014.

2

Coherence and Photons

In the year 1900 Max P LANCK postulated – at the beginning very reluctantly – an energy packet W = hν, today known as the “photon”. In 1905, the famous “annus mirabilis” of E INSTEIN, classical physics finally broke down: E INSTEIN explained the photoelectric effect based on P LANCK’s quantum of action h, he also formulated the theory of special relativity, declared the equivalence of mass and energy, and presented an atomistic explanation of B ROWN’s motion. However, only in the middle the 1950ies – nearly 50 years later – quantum optics came to life and remains a very active field of modern research until now. The present chapter gives a first introduction into some of its basics.

Overview

After the previous extensive exploration into the wave character of light, the present chapter focuses on its particle properties and on the statistical properties of photons. In Sect. 2.1 concepts such as “quasi-monochromatic” and “partially coherent” light will be defined and exemplified by simple models for a laser and a classical light source. We shall familiarize ourselves with the fundamental experiments, beginning with the famous “Hanbury B ROWN T WISS experiment”. In Sect. 2.2 we shall try to find a pragmatic approach to the quantum mechanical description of photon states – giving an introduction for “pedestrians” so to say. Finally, we shall in Sect. 2.3 apply the new tools to the theory of absorption and emission of light – this time with explicit consideration of the quantum nature of photons. This will allow us for the first time to derive the basic formulas for spontaneous emission – as opposed to the previous, hand waving introduction of this inherently quantum mechanical phenomenon.

© Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_2

71

72

2

2.1

Some Basics for Quantum Optics

2.1.1

Introduction


Ground breaking work on the quantum statistics of light has been carried out in 1954 and the following years. Of fundamental importance are the experiments of R. Hanbury B ROWN1 and R.Q. T WISS (1954, 1956a, 1956b, 1958). Roy J. G LAUBER was one of the pioneers of theoretical quantum statistics (see e.g. G LAUBER 1963) and received the N OBEL prize for his work 2005 – together with John H ALL and Ted H ÄNSCH as already noted in the context of precision spectroscopy and frequency combs. The work of G LAUBER provides much of the essential theoretical background for the present chapter. We start by describing a continuous light source, be it a laser beam whose light is not strictly monochromatic, be it a completely chaotic light source such as an incandescent bulb, our sun, or a fluorescent lamp. It has a finite bandwidth δωc around a central frequency ωc , and is called quasi-monochromatic, if δωc ωc .

(2.1)

We shall see that the concept of quasi-monochromaticity is closely related to coherence or partial coherence onto which this section will focus, and which we shall meet time and again later on. For further details we refer the interested reader to the standard work of L OUDON (2000), by which much of this chapter has been inspired, as well as to W EISSBLUTH (1989) and the more recent monograph by L AMBROPOULOS and P ETROSYAN (2007) who also give many further references. We now deal with continuous light beams which can no longer be described in a neat analytic form as wave-packets. Nevertheless, these light beams are still capable to generate typical interference structures, similar to those reported for light pulses in Sect. 1.5. The property that both have in common is quantified by the degree of coherence.

2.1.2

First-Order Degree of Coherence

Correlation functions have already been introduced in Sect. 1.5.2, and more details are found in Appendix G.2, Vol. 1. Now we shall use these correlation functions to characterize the coherence properties of electromagnetic radiation. For the field amplitude E + (t) = (E − (t))∗ as defined by (1.36) we write2

1 The

experiment is usually referred to as “Hanbury B ROWN -T WISS Experiment”, but one should know that “Hanbury” is a first name, and the second author’s name is “Twiss”. that this definition differs slightly from L OUDON (2000) (δ → −δ) who uses a somewhat unconventional definition of the F OURIER transform.

2 Note

2.1


G(1) (δ) = E − (t)E + (t + δ) ∞ E − (t)E + (t + δ)dt = −∞

=

1 Tav

Tav /2

−Tav /2

73

(2.2) for a pulse, and

E − (t)E + (t + δ)dt

for a CW source.

(2.3) (2.4)

The mode of averaging . . . depends on the specific case. Note that the averaging time Tav for the CW case has to be sufficiently long, so that G(1) (δ) does no longer change when Tav is extended. As just defined, the dimension of G(1) (δ) depends on the case (pulse or CW). Thus it is advantageous to introduce the dimensionless first-order degree of temporal coherence: E − (t)E + (t + δ) E − (t − δ)E + (t) = = g (1) (−δ)∗ E − (t)E + (t) E − (t)E + (t) with 0 ≤ g (1) (δ) ≤ 1.

g (1) (δ) =

(2.5)

In general g (1) (δ) is complex and |g (1) (δ)| gives a quantitative measure of coherence. It determines how far the field E + (t) and its displaced image E + (t + δ) may be separated in time and still have a memory of each other. If they fully overlap g (1) (0) = 1 and the light is said to be fully coherent, if they are far apart g (1) (∞) = 0 the light is incoherent. In the case of a wave-packet (1.107), with an envelope E0 h(t) according to (1.109), the degree of coherence with (2.2) becomes simply ∞ ! ∞ (1) iωc δ g (δ) = e h(t)h(t + δ)dt h2 (t)dt. (2.6) −∞

−∞

Note that h(t) is an analytic, square integrable function, representing a pulse or a finite sequence of pulses. Using the W IENER -K HINCHIN theorem (I.17), Vol. 1 for the F OURIER transform of auto-correlation function of the field, we may write the intensity spectrum: ∞ ε0 c + 2 ε0 c ∞ −iωδ I˜(ω) = e dδ E − (t)E + (t + δ)dt. (2.7) E (ω) = 4π 4π −∞ −∞ For normalization we can use the fluence of the light source ∞ ∞ ε0 c ∞ − F= I (t)dt = E (t)E + (t)dt = I0 h2 (t)dt, 2 −∞ −∞ −∞

(2.8)

with I0 = ε0 cE02 /2. Inserting this into (2.7) and applying the definition (2.5) of g (1) (δ) we obtain ∞ F g (1) (δ)e−iωδ dδ. (2.9) I˜(ω) = 2π −∞

74

2


Thus, the intensity spectrum is given by the inverse F OURIER transform of the firstorder degree of temporal coherence of a light source. ∞ It is also useful to recall the units [I˜(ω)] = J s m−2 , while −∞ I˜(ω)dω = F (which is easily verified from Eq. (2.9)) has indeed the unit [F ] = J m−2 . Specifically for a Gaussian pulse with the field amplitude (1.110), one finds from the convolution (2.6) g (1) (δ) = eiωc δ e

− 12 ( τδ )2 G

.

(2.10)

With this and (2.9) the intensity spectrum of a Gaussian pulse follows (see also Appendix I.4.1 in Vol. 1):

I0 F ω − ωc 2 ω − ωc 2 ˜ = 2 exp − (2.11) I (ω) = √ exp − ωG ωG ωG π ωG √ with F = π/2I0 τG , and ωG = 2/τG . In contrast, for CW light the evaluation may not be that trivial, since any realistic model will have to describe a stationary light source as a random ensemble of wavepackets. It turns out to be more convenient to average in frequency space. To this end, we adapt the spectral intensity distribution (2.7) appropriately, and replace the integrals (2.3) by averages (2.4): −Tav /2 ε0 c ∞ −iωδ 1 ˘ I (ω) = e dδ E − (t)E + (t + δ)dt 4π −∞ Tav −Tav /2 ∞ I g (1) (δ)e−iωδ dδ (2.12) = 2π −∞ Tav /2 ε0 c − 1 with I = I (t)dt. (2.13) E (t)E + (t) = 2 Tav −Tav /2 We have used here the symbol I˘(ω) (unit [I˘(ω)] = W s m−2 ) for the intensity spectrum ∞ of the CW light, in order to indicate its different definition. From this follows ˘ −∞ I (ω)dω = I , which is now the average intensity of the stationary source, measured in units [I ] = W m−2 (rather than the fluence as in the case of a pulse). We finally invert (2.9) and (2.12) and find the useful relations by which thefirstorder degree of coherence can be derived in both cases (properly normalized) as inverse F OURIER transform of the spectrum. For pulsed and CW sources we obtain 1 ∞ ˜ (2.14) I (ω)eiωδ dω and g (1) (δ) = F −∞ 1 ∞ ˘ = (2.15) I (ω)eiωδ dω, respectively. I −∞ The next two subsections are devoted to quasi-monochromatic light beams, with their degree of coherence g (1) (δ) and their interference properties.

2.1


Fig. 2.1 (a) Illustration of the wave-packet described by (2.16), (b) schematic representation of the model for a stationary, quasi-monochromatic laser composed of such wave-packets (for visual clarity we have drawn Tc much too large; in reality, of course, we have Tc τi )

2.1.3

75 E(t)

τi

(a) t

ti

E(t)

τ1

Tc

τ2

(b) τ3

τ4 t

t1

t2

t3

t4

Quasi-Monochromatic Light

We recall that the laser pulses which we have discussed in the previous chapter have been introduced in Sect. 1.4.1 as coherent superposition of plane waves from a limited frequency range of a FWHM ω1/2 . Such a light pulse has a finite duration τ ∝ 1/ω1/2 . Alternatively we have described in Sect. 1.4.3 periodic pulse trains as a F OURIER series. Obviously, neither of these two descriptions can lead to a realistic model of a quasi-monochromatic and continuous laser beam, since such a CW laser radiates effectively from t = −∞ to t = +∞ without obvious intermission (at least for a couple of hours or days). With some effort and good electronics the frequency may be stabilized for a long time to a few Hz. Still it cannot be modelled by a plane (or Gaussian) continuous wave – nor by any kind of a wave-packet. Such a CW light beam has to be modelled with “stationary and ergodic statistical properties, so that ensemble averages over the probability distribution are equivalent to long-time averages over the beams in a single experiment” (L OUDON 2000). Let us imagine – as a simple model3 – a laser beam to be composed of a large number of rectangular wave trains (see Appendix I.4.3 in Vol. 1) of constant amplitude but different, finite durations. One such wave-packet is illustrated in Fig. 2.1(a) as a function of time at a fixed position in space (without loss of generality we choose again r = 0 ⇒ kz = 0). Thus, in our standard notation (1.36) we have E0 ei(ωc t−ωc ti −φi ) for ti < t < ti + τi , + Ei (t) = (2.16) 0 else, and the intensity is as usual I0 = ε0 cE02 /2 in the wave-packet and zero outside. The pulse begins at t = ti , it has a duration τi and its relative phase φi is statistically distributed. To make things not too complicated we assume, however, that the period Tc = 2π/ωc (or its wavelength λc ) is constant. Such a wave-packet may typically contain 108 to 1011 periods. The spectral intensity distribution of this pulse is given 3 Similarly one has to treat any chaotic light, with large phase and intensity fluctuations, as e.g. emitted from a collision or D OPPLER broadened gas discharge, an incandescent bulb or an ensemble of excited atoms – even if the spectrum may be different, the bandwidth larger and the coherence time to be introduced here correspondingly shorter.

76

2


by (I.53), Vol. 1: I˜i (ω) =

I0 τi2 τi (ω − ωc ) sin x with sinc x = sinc2 . 2π 2 x

(2.17)

A real quasi-monochromatic light beam, which extends over large times and distances, can now be modelled by many such pulse trains as indicated in Fig. 2.1(b). They may, of course, also overlap each other. The frequency bandwidth in a CW laser is usually determined by mechanical and thermal instabilities of the experimental setup, such as vibrations of the mirrors, collision processes in the amplifier medium, dust particles accidentally passing the laser beam etc. These processes occur completely statistically, let us assume with a constant average rate 1/τc . We further assume that such events after the times τ1 , τ2 , . . . , τi just change the phase φ1 , φ2 , . . . , φi . The amplitude is kept constant. The probability that such a wavepacket has a duration between τi and τi + dτi , is described by an exponential distribution as outlined in our elementary introduction to statistics, Sect. 1.3.1 in Vol. 1: w(τi )dτi =

1 −τi /τc e dτi . τc

(2.18)

The average time between the phase changes is τc . We call it coherence time. The corresponding length of the wave-packet sketched in Fig. 2.1(b) is the so called coherence length c = τc c.

(2.19)

The whole light beam is described by this statistical distribution of individual wave-packets. Each of them is characterized by a spectral distribution according to (2.17) and an arbitrary statistical phase φi . We emphasize again, that this continuous light beam cannot be described by any kind of coherent, linear superposition of waves. Its overall spectral distribution is found as the statistical average of the individual spectral distributions for all possible durations τi of the wave-packets and re-normalization according to (I.32), Vol. 1. With (2.17) and (2.18) the integration can be carried out in closed form:

∞ τi (ω − ωc ) I0 dτi w(τi )τi2 sinc2 I˘(ω) = I˜i (ω) = 2πτc 0 2 =

ω1/2 I τc I = , 2 2 2 π 1 + (ω − ωc ) (τc ) 2π (ω − ωc ) + (ω1/2 /2)2

(2.20)

with a FWHM ω1/2 = 2/τc = 2c/c . Thus, one finds a L ORENTZ profile. It is normalized here so that the integration over all frequencies gives the local average intensity I = I0 = ε0 cE02 /2 of the laser beam (assumed to be independent of time). The profile is characterized by the coherence time τc . The maximum of the spectral intensity distribution (intensity per angular

2.1


77

Table 2.1 Coherence time τc and first-order degree of coherence g (1) (δ) for different spectral distributions with FWHM ω1/2 Lorentziana Gaussianb Rectangle

Spectrum

ω1/2

g (1) (δ) × e−iωc δ

g (1) (τc )

[τc2 (ω − ωc )2 + 1]−1 exp[−τc2 (ω − ωc )2 ] 1 for − τπc ≤ ω − ωc ≤ τπc

2/τc √ 2 ln 2/τc

exp[−|δ|/τc ]

1/e

exp[−(δ/τc )2 ]

1/e

2π/τc

sinc(πδ/τc )

0

a Note

that this definition for the L ORENTZ profile differs slightly from (5.8), Vol. 1, used there for spontaneous emission with a FWHM of ω1/2 = 1/τnat √ b Coherence time and the usual Gaussian time are related by τ = 2τ c G

frequency) at ω = ωc is I˘(ωc ) =

2I . πω1/2

(2.21)

For the first-order degree of temporal coherence (2.5) one obtains g (1) (δ) = eiωc δ e−|δ|/τc

(2.22)

for the L ORENTZ profile (2.20), as can easily be verified with (2.12). By way of example, a continuous dye laser (often used in spectroscopy) may provide an intensity I 1 W cm−2 with a typical bandwidth of ν1/2 1 MHz. Coherence time and coherence length are then τc 320 ns and c 100 m, respectively. The peak spectral intensity is I˘(ωc ) 5 × 10−8 W cm−2 s. We may compare this to the spectral intensity of the sun at 555 nm which according to (1.85), Vol. 1 is I˘(ωc ) 3.5 × 10−12 W s cm−2 (at the surface of the sun!). The above description of a quasi-monochromatic light beam is just one possible model. In principle, one has to start from a detailed analysis of a given experimental situation. A variety of wave-packets differing from those shown in Fig. 2.1 are conceivable. In any case, g (1) (δ) and the spectral distribution I˘(ω) are related by (2.12)–(2.15). If e.g., the radiation source is mainly D OPPLER broadened, it will be characterized by a distribution of frequencies corresponding to a Gaussian with statistically distributed phases. The corresponding degree of coherence will be the same√as that derived for the Gaussian pulse (2.10) and the coherence time is then τc = 2τG . The definition of a coherence time τc (or the coherence length c = cτc ) must, inevitably, be somewhat arbitrary. We shall use the time for which g (1) (τc ) = 1/e, unless it passes through g (1) (τc ) = 0 at a finite delay time, in which case that time is taken. In Table 2.1 we summarize the spectra and first-order coherence properties for three important cases of quasi-monochromatic light. Their first-order degree of coherence is plotted in Fig. 2.2 as a function of delay time δ. They are compared with strictly monochromatic light (I˜(ω) ∝ δ(ω − ωc )) which – in contrast to the three statistical light sources – shows no fluctuations at all, i.e. |g (1) (δ)| ≡ 1 holds independent of δ.

78

2

|g (1)(δ )| 1.0

Fig. 2.2 Magnitude of the first-order degree of coherence, |g (1) (δ)|, for chaotic light with a coherence time τc . Compared are light sources with Gaussian, Lorentzian and rectangular spectral profiles; they are confronted with a fully coherent wave (infinite coherence time) |g (1) (δ)| = 1

2.1.4

Coherence and Photons fully coherent GAUSS

LORENTZ 0.5

rectangular

1/e

-3 -2

-1

0

1

2

δ/τc

Temporal or Longitudinal Coherence

To develop the concept of coherence further, we return to interference experiments as discussed in Sect. 1.5.3 and apply the just defined first-order degree of coherence. This will also be a useful preparation for later discussions of polarization and state distribution in atoms (Chap. 9). Let us take a closer look on first-order coherence observed e.g. in YOUNG’s double slit experiment, or in a M ICHELSON interferometer. Here, as a first step, we idealize the light beam and assume it to be parallel (e.g. originating form a point like source). In Fig. 2.3 the key elements of such an experiment are illustrated very schematically. The electric field E(r, t) is split into two parts, E(r 1 , t) and E(r 2 , t), i.e. by a double slit in the diffraction experiment, or with the help of a beam splitter in the interferometer experiment. Both rays A and B propagate along different optical pathways s1 and s2 , respectively – be it due to diffraction, changes of the index of refraction or just due different distances. This leads to a time delay δ between the two partial beams. Finally, both parts are superposed and interfere – effectively at different individual times, t1 and t2 . Using the terminology (1.36), we write E + (r, t) = a1 E + (r 1 , t1 ) + a2 E + (r 2 , t2 ) . collimator lense

double slit or beam splitter r

point source

interference ray 1

s1

r1 , k

r2 , k f

parallel light rays

s2

ray 2

light detector

optical delay

Fig. 2.3 Very schematic layout of an interference experiment with two parallel light rays originating from a point like light source; the brace on the right just indicates that the two rays are made to interfere – it does not sketch a light path

2.1


79

The prefactors a1 and a2 account for the fact the ray A and B are only a fraction of the beam and may even be further reduced before reaching the detector. If only ray A or only ray B were present, the signals would be 2 ε0 c 2 + a1 E (r 1 , t1 ) = a12 I or 2 2 ε0 c 2 + a2 E (r 2 , t2 ) = a22 I, respectively, I2 = 2 I1 =

(2.23)

with I being the averaged total intensity of the original beam. For a first-order process (N = 1) we write the averaged time dependent intensity (1.134) at the detector as ε0 c − I (r, t) = E (r, t)E + (r, t) = I1 + I2 + I12 . (2.24) 2 The interference term in which we are mostly interested, is given by I12 = C 2 E − (r 1 , t1 )E + (r 2 , t2 ) + E + (r 1 , t1 )E − (r 2 , t2 ) ε0 c a1 a2 , (2.25) = 2C 2 Re E − (r 1 , t1 )E + (r 2 , t2 ) with C 2 = 2 s1 s2 s1 − s2 . t1 = t − , t2 = t − = t1 + δ, and δ = c c c The expected pattern is a function of the relative phase ωδ between rays A and B. Since partially coherent light with a coherence time τc has a bandwidth ω 1/τc of different frequencies, one expects the interference structure to smear out when ωδ ≥ π , i.e. if δ ≥ πτc . The detector usually integrates over times τc . For a quantitative evaluation we have to keep in mind, that in any model of quasimonochromatic light the electric field will be described as a statistical ensemble of many individual wave-packets Ei+ (r, t), e.g. as described by (2.16). Thus, we have to average the interference term in (2.24) temporally – or to find the ensemble average. It turns out that this is done most conveniently in frequency space: we rewrite the interference term (2.25) by using the (inverse) F OURIER transform (1.106), with + (k) being independent of the direction of k (parallel light with k = ω/c): E I12 = 2

C 2π

2 "

dω

# + ∗ i(k·r −ωt ) + −i(k ·r −ω t ) 1 1 E 2 2 . (2.26) (ω) e ω e dω E i j

− (ω)E + (ω ) is affected by the statistical averaging over waveObviously, only E i j packets. We also observe that each of the wave-packets i and j carries its own statistical phase φi or φj , respectively – as exemplified by (I.51), Vol. 1. Thus, these complex quantities are distributed at random on a circle in the complex plane – and hence they average out over the whole ensemble. Only those terms which are caused by the same wave-packet i = j contribute to (2.26). Somewhat laxly one says: Each photon interferes only with itself.

80

2


− (ω)E + (ω ) according to (I.51), Vol. 1 contains the function We also note that E i i exp[i(ω − ω )ti ]. Averaging over the statistically distributed starting times ti (i.e. integrating over all times ti ) lets all terms with ω = ω disappear. Thus + ∗ + + 2 (ω) E (ω) . ω = 2πδij δ ω − ω E E (2.27) i j i In summary, the ensemble average of the interference term (2.26) is simply + 2 C2 (ω) Re dωeik·(r 1 −r 2 ) eiω(t2 −t1 ) E (2.28) I12 = 2 i 2π = 2a1 a2 Re dωeik·(r 1 −r 2 ) eiω(t2 −t1 ) I˜i (ω) , where we have used (I.32), Vol. 1. With k ⊥ (r 1 − r 2 ) in our model geometry Fig. 2.3, and with the time delay t2 − t1 = δ we obtain the sought-after interference term as: I12 = 2C 2 Re E − (r 1 , t1 )E + (r 2 , t2 ) = 2a1 a2 Re eiωδ I˘(ω)dω = 2 I1 I2 Re g (1) (δ) . (2.29) I˘(ω) = I˜i (ω) is the ensemble averaged intensity spectrum. In the last step, using (2.15), we have identified the resulting integral as the first-order degree of (longitudinal) coherence and use the abbreviations (2.23). For quasi-monochromatic light with a carrier frequency ωc the first-order degree of coherence always assumes the form ±|g (1) (δ)| exp(iωc δ) (see the examples given in Table 2.1). Thus, inserting (2.29) into (2.24) we obtain the interference signal (first-order) as a function of the delay time δ: I (δ) = I1 + I2 ± 2 I1 I2 g (1) (δ) cos ωc δ. (2.30) We emphasize that the above derivation is characteristic for any kind of quasimonochromatic light composed of an ensemble of wave-packets with statistically distributed phases. The |g (1) (δ)| is the quantitative measure for temporal coherence we have been looking for. One calls this property temporal coherence or longitudinal coherence, since coherence time and coherence length are directly related by c = cτc : c gives the distance by which a wave-packet may be displaced from its image so that the degree of coherence decreases to 1/e. Some Examples In practice, interference fringes often show less contrast than expected from (2.30) where Re[g (1) (0)] = 1. Instrumental imperfections or spatial incoherences can be responsible, as we shall discuss Sect. 2.1.7. To quantify this reduction of contrast one introduces a parameter visibility,

V=

Imax − Imin , Imax + Imin

(2.31)

2.1


S(δ ) S(∞) 1.2

81 ^

=2π /ω c 1+2V √I1 I2 / (I1 + I2 )

1.0 0.8 1- 2V √I1I2 / (I1+ I2 ) 0.6 -4

-3

-2

-1

0

1

2

3

π∆ ν δ

Fig. 2.4 Interference pattern for two rays with a rectangular spectrum according to (2.32). The dotted line represents suitably processed experimental data, measured at the CHARA high resolution stellar interferometer array, extracted from TEN B RUMMELAAR et al. (2005). The full black line is proportional to sinc(πνδ)

by which the interference term in (2.30) has to be multiplied. Visibility can be measured by registering I (δ) in a delay scan. It may contain valuable information about the light source as we shall discuss in Sects. 2.1.7–2.1.8. In the following we assume for an ideal interferometric measurement I1 = I2 and normalize the signal to the uncorrelated limit I (∞). Finally, the expressions given in Table 2.1 have to be inserted. First we consider a rather broad band CW light source which is passed through a narrow-band spectral filter, as done e.g. in stellar interferometry. If the filter has a rectangular profile with a bandwidth ω as described in Appendix I.4.4, Vol. 1, we obtain from (2.14) and (I.56), Vol. 1 Re[g (1) (δ)] = sinc(ωδ/2) cos ωc δ. The normalized interference signal is thus I (δ) = 1 + 2V sinc(ωδ/2) cos ωc δ. I (∞)

(2.32)

Figure 2.4 shows such an interference pattern. For reference we note that the fringes vanishes for the first time at ωδ/2 = πνδ, with ν given in frequency units. The fringes are caused by the cos ωc δ term in (2.32) and depend on the phase difference ωc δ = k(s1 − s2 ) between rays 1 and 2. The contrast clearly changes with delay time and is given by V × |g (1) (δ)|. It has its maximum for δ = 0 where |g (1) (δ)| = 1, while it disappears for long delay times. To compare the above theoretical derivation with some real experiment, we show in Fig. 2.4 a “fringe scan” extracted from one of the first publications of the CHARA optical/infrared interferometric array located on Mount Wilson, CA (TEN B RUMMELAAR et al. 2005). The spectra were taken in the K band at 2.133 µm with one of their 15 very long baseline interferometers. The agreement with (2.32) is impressive, albeit – as expected with such simple modelling – not perfect. We shall come back to these experiments in Sect. 2.1.8. For the quasi-monochromatic light model introduced in the previous subsection, the spectrum I˘(ω) is a L ORENTZ distribution (2.20). Its first-order degree of coher-

82

2


ence g (1) (δ) is given by (2.22). Thus, the interference pattern (2.30) becomes I (δ) = 1 + 2V e−|δ|/τc cos ωc δ. I (∞)

(2.33)

We recall: the coherence time τc corresponds here to the average duration of the wave-packets which define the temporal properties of the quasi-monochromatic light. If the light originates from a D OPPLER broadened (Gaussian) source, with (2.10) the interference pattern (2.30) becomes I (δ) − 1 ( δ )2 = 1 + 2V e 2 τG cos ωc δ. I (∞)

2.1.5

(2.34)

Higher-Order Degree of Coherence

To extend the concept “degree of coherence” introduced in Sect. 2.1.2 one defines a general degree of coherence of N th order as g (N ) (r 1 , t1 , . . . r N , tN , . . . r 2N , t2N ) =

E − (r

) . . . E − (r

(2.35)

)E + (r

) . . . E + (r

1 , t1 N , tN N +1 , tN +1 2N , t2N ) , + 2 + 2 + [|E (r 1 , t1 )| . . . |E (r N , tN )| . . . |E (r 2N , t2N )|2 ]−1/2

with |E + (r N , tN )|2 ≡ E − (r N , tN )E + (r N , tN ) ∝ I (r N , tN ). For details the interested reader is referred to the specialized literature (see e.g. L OUDON 2000; G LAUBER 2006). In the following we refer again to the dependence on time t only – which may be replaced by t − rk/ω if the r is explicitly needed – and discuss some basic aspects of the particularly important second-order degree of temporal coherence g (2) (δ) =

I (t)I (t + δ) E − (t)E − (t + δ)E + (t)E + (t + δ) = , I (t)2 E − (t)E + (t)2

(2.36)

where the brackets . . . again imply the same kind of averaging as in (2.2)–(2.4). The symmetry relation is now somewhat simpler than (2.5): g (2) (δ) = g (2) (−δ).

(2.37)

However, while for the first-order degree of coherence the limits 0 ≤ g (1) (δ) ≤ 1 hold, no general upper limit exists for g (2) (δ). Still, one may show that 0 ≤ g (2) (δ)

and for δ = 0 :

1 ≤ g (2) (0) ≤ ∞.

2.1


83

The latter follows from the C AUCHY-S CHWARZ inequality,4 which leads to I 2 = I (t)2 ≤ I (t)2 . Physically g (2) (δ) represents the correlation function of the light intensity, i.e. it answers the question whether fluctuation in the light intensity is completely random or whether there is some kind of memory effect. In a quantum picture the photon flux, I (t)/ω ∝ w(t) is proportional to probability for a photon to arrive (at time t) per unit of time, and g (2) (δ) gives the probability w(t)w(t + δ) for two photons to arrive with a specific time delay: do the photons arrive completely at random or perhaps with an enhanced probability to come in pairs? At first thought this appears a strange question. Why should it be more probable to find two photons at once than at random – if the light is otherwise completely chaotic? We cannot go into details of the statistics of chaotic light sources, but let us glance over the key arguments. We recall the model of a chaotic light source presented in Sect. 2.1.3 and assume the light to originate from many atoms. They all contribute with their individual electric field Ej+ (t), each characterized by its own statistical phase. The overall field is thus given by E + (t) = Ei+ (t). i

Using this expression one derives the second-order correlation function (i.e. the nominator in Eq. (2.36)) G(2) (δ) = I (t)I (t + δ) = E − (t)E − (t + δ)E + (t)E + (t + δ) . (2.38) Since uncompensated phases cancel out statistically in the averaging process, again only those terms are kept, where the field from each atom is multiplied by its own conjugate complex (at time t or t + δ). However, since now the products of four amplitudes are involved, and a large number of atoms participates, the remaining, dominant terms are those which arise from two pairs of atoms i = j : − Ei (t)Ei+ (t)Ej− (t + δ)Ej+ (t + δ) = Ii (t) Ij (t + δ) and (2.39) − Ei (t)Ei+ (t + δ)Ej− (t + δ)Ej+ (t) = Ei− (t)Ei+ (t + δ) Ej− (t + δ)Ej+ (t) . (2.40) As the average intensity I¯ in a stationary source is independent of time, the first line is simply = I¯2 . In the second line we recognize the first-order correlation function (2.5) and its complex conjugate. Thus, in its normalized form (2.36) becomes 2 g (2) (δ) = 1 + g (1) (δ) .

4 The

(2.41)

C AUCHY-S CHWARZ inequality may be written in an easy to comprehend relation between

N dimensional vectors: |a · b|2 ≤ |a|2 · |b|2 . If one chooses the intensities I (tj ) as components of

a vector a and 1 as components of b, the latter relation follows immediately.

84

2

|g (2) (δ) |

2


GAUSS rectangular

LORENTZ 1 classical wave (fully coherent)

-2

-1

0

δ/τc

1

2

Fig. 2.5 Second order degree of temporal for chaotic light. Compared are light sources with a spectral distribution of L ORENTZ, G AUSS and rectangular type. All are assumed to have equal coherence time τc , i.e. the delay time δ is given in units of the coherence time. They are contrasted with a classical source of radiation such as a CW laser of very large coherence length or an RF generator

Specifically, for light with a Lorentzian or Gaussian type of spectrum the secondorder degree of temporal coherence (2.36) becomes (2.42) g (2) (δ) = 1 + exp −2|δ|/τc , and (2.43) g (2) (δ) = 1 + exp −2δ 2 /τc2 = 1 + exp −δ 2 /τG2 , respectively. These functions are illustrated schematically in Fig. 2.5. We recall that the correlation times τc are related to the respective spectral distributions by Table 2.1. An important limiting case is the classical continuous, constant and coherent wave, e.g. a highly stabilized RF generator or an ideal CW laser. In that case I (t)I (t + δ) ≡ I¯2 , there are no intensity fluctuations and g (2) (δ) = 1. It may sound somewhat surprising at first sight, but in a fully coherent radiation source, such as an ideal laser, the photons are distributed as randomly as possible! Quite generally, for long delay times δ there are no correlations in the statistical intensity fluctuations and thus g (2) (δ) → 1

2.1.6

always holds for δ τc .

Photon “Bunching” Experiments

The proposal of Hanbury B ROWN and T WISS (1954) for “A new type of interferometer for use in radio astronomy” marks the beginning of quantum optics (G LAUBER 2006). In their pioneering investigations, correlations in the intensity of an extended light source were measured for the first time – both in a table top laboratory exper-

2.1

(a)


85

point source

ρ ( δ )∝

(b)

g (2) ( δ )- 1

wavelength filter, FPI collimator beam splitter

P2

P1 pulse amplifiers THC PHA

0

paper-tape computer

-6

-4

-2

0

2

4

6

time delay δ / ns

Fig. 2.6 Photon bunching experiment according to P HILLIPS et al. (1967). (a) Schematic of the experimental setup with two photo-multipliers P1 and P2, time to height converter (THC) and pulse height analyzer (PHA). (b) Observed true two photon coincidence rate (normalized) as a function of time delay between the two photons; light filtered with a 3 cm FPI; the maximum is ρ(0) ∼ 17.3 %

iment (1956a, 1958) and for light from a star (1956b). In such an HBT experiment the intensity of chaotic light is registered by two spatially separated detectors whose signal is then correlated – in contrast to YOUNG’s double slit experiment where the electric field amplitudes of the light are superposed. However, before we can discuss HBT type experiments, we shall have to introduce spatial or lateral coherence in Sect. 2.1.7. Conceptually somewhat more straight forward are so called photon “bunching” experiments – a number of which were performed in the years following the original HBT experiment. One with particular nice data by P HILLIPS et al. (1967) is sketched schematically in Fig. 2.6(a). Quasi-monochromatic light from a mercury spectral lamp passes a narrow band filter and then an FPI to select the 435.8 nm line and reduce the bandwidth, i.e. to increase the coherence time. The light beam is then strongly collimated by pin holes with diameters of 0.3 mm and 2 mm, separated by 1.5 m before it reaches a beam splitter which provides two branches of equal light intensity. Two separate photo-multipliers P1 and P2 are setup to detect individual, single photons, which are recorded after amplification and clipping as pulses with a rise time of less than 2 ns. The time delay between these pulses is registered by the combination of a time to pulse height analyzer (THC) and a multichannel pulse height analyzer (PHA). Data storage and communication with a computer was at that time still done by punched paper-tape. The experiment thus determines coincidence rates for counting a photon in branch 2 after a time δ when a photon has been registered in branch 1 (or vice versa). If the individual count rate at P1 is R1 and at P2 it is R2 (in this experiment some 104 counts/ s). The coincidence rate is R1 (t) × R2 (t + δ) × δ, in this exper-

86

2


Fig. 2.7 Interference from two sources

iment <10 s−1 with δ (here some ns) being the time resolution of the electronics. One subtracts the statistical coincidence rate R1 × R2 × δ (corresponding to the coincidence rate for δ → ∞). In summary, one records the true coincidence rate, which properly normalized is ρ(δ) =

R1 (t) × R2 (t + δ) R1 (t) × R2 (t)

2 − 1 ∝ g (2) (δ) − 1 = g (1) (δ) .

A typical result is shown in Fig. 2.6(b). The frequency bandwidth of the FABRYP ÉROT filter was in this case ca. ν1/2 160 MHz, in fair agreement with 208 MHz gleaned from the correlation measurement. The maximum of the normalized true coincidence was found to be ρ(0) ∼ 17.3 % – the authors attribute the fact that it is not = 1 to the finite temporal resolution of the electronics, but also to finite lateral coherence (a compromise had to be found between a reasonable count rate and low angular divergence of the beam). But clearly, the experiment shows beyond any doubt, that the probability to register two photons at the same time is significantly higher than expected by purely random coincidences (observed at long delay times) – in full agreement with the considerations outlined in Sect. 2.1.5.

2.1.7

Spatial or Lateral Coherence

So far, in our discussion of coherence experiments we have assumed strictly parallel, quasi-monochromatic light rays. (In the experiment just explained, this was approximated well enough by the high efforts to collimate the light.) Now we also give up this usually somewhat unrealistic assumption. Even laser beams have a finite angular divergence, as we have discussed in Sect. 1.2. The problem with this fact and the measurement of interference patterns is, that different incident angles lead also to a phase difference, and hence, to shifted interference fringes. Let us start with a rough estimate of this effect, before we enter into a more rigorous treatment. We consider two point like sources at very large distance. Their (quasi-monochromatic) light is assumed to be parallel and to be diffracted by a YOUNG’s double slit arrangement, the slit distance being B. As sketched in Fig. 2.7, the first interference minimum from source S1 (red fringe pattern) is found at an angle ϑmin for which λ/2 Bϑmin . Source S2 , which is seen under an angle ϑc , generates its own interference pattern (grey) with its main maximum at an angle ϑc . Sketched is a partial overlap for both fringe patterns. If the two sources were still further apart, so that

2.1

Some Basics for Quantum Optics source with collimator

extended source

lens

ϑ0 / 2

2w

2w 0

double aperture, beam splitter etc. r1

ϑd /2

87

k'

ray 1

∆r B = | ∆r |

ϑd /2 ϑd /2

f

beam combining unit, interference

r2 ray 2

k

k' k

Δk

s1

s2

Δk detector optical delay

Fig. 2.8 On spatial coherence: very schematic diagram of an interference experiment with slightly diverging light rays 1 and 2 from an extended source (uniform disc angular diameter ϑd )

ϑc = ϑmin , the maximum from S2 would fully coincide with the first minimum from S1 : hence, the fringe patterns would disappear: interference structures can only be discerned if ϑc ≤

λ . 2B

(2.44)

The light is said to be spatially or laterally coherent if ϑc is smaller than this limit. Even though the assumed limit is somewhat arbitrary, clearly this spatial coherence or incoherence will influences the fringe visibility discussed in Sect. 2.1.4 for temporal (or longitudinal) coherence. To obtain a quantitative understanding we now consider an extended, stationary light source of diameter D0 (= 2w0 ) which is collimated by a lens with a focal length f and a (useable) diameter D(= 2w), as sketched in Fig. 2.8. The initial divergence of this “beam” is given by ϑ0 ≈ D0 /f (angular diameter), quite analogous to the situation for a Gaussian beam according to Fig. 1.16, if we identify the disc radius w0 with the beam waist and the Gaussian divergence angle θe with ϑ0 /2. For not too large aperture angles ϑ0 , the (full) angular divergence ϑd after collimation is wϑd w0 ϑ0 ,

(2.45)

if the lens is used up to a diameter 2w. With this more realistic description of a light beam we have to modify our treatment of the interference experiment presented in Sect. 2.1.4. The following derivation is completely independent of the origin of the two slightly divergent light rays. The source-collimator arrangement (grey shaded area in Fig. 2.8) may e.g. be replaced by a distant star that emits light with a small divergence angle ϑd (“uniform disc angular diameter” equivalent to its diameter divided by its distance). The light may be collected by two different mirrors or telescopes placed at a distance B(= 2w). In the context of astronomical interferometry this distance is called baseline. We shall come back to this context in Sect. 2.1.8.

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2


Comparison of Figs. 2.3 and 2.8 shows that we now have to treat interferences of plane waves with wave vectors k i around the mean wave vector k. In analogy to the averaging over frequencies, we now have to sum in addition over the contributions from all k i . As before the contributions from superpositions belonging to different k i and k j statistically average out: as before “each photon interferes only with itself”. The key question is now whether, and to what extent, the interference patterns from different k i disturb each other. We start again from (1.106) and write the electric fields propagating from r 1 and being detected at time t1 = t − s1 /c as (2.46) Ei+ (r 1 , t1 ) = Ei+ (ω)ei(kr 1 −ωt1 +k i ·r 1 ) dω. The propagation vectors k i of the individual wave-packets is now written with reference to the central wave vector k k i = k + k i .

(2.47)

The interference term I12 ∝ Ei+ (r 1 , t1 )Ej− (r 2 , t2 ) – after summation over different wave-packets and exploiting the “one photon interferes only with itself” rule – becomes in analogy to (2.28) I12 = 2

C2 Re 2π

2 dω Ei (ω) ei[k·(r 1 −r 2 )−ω(t1 −t2 )+k i ·(r 1 −r 2 )] .

(2.48)

The averaging . . . must include the angular divergence reflected in k i · (r 1 − r 2 ). We write the distance vector r 1 − r 2 = r, with |r| = B, and account for the fact that r is per definition perpendicular to k, hence k · r = 0 holds. The delay time is again given by δ = t1 − t2 = (s1 − s2 )/c. Thus, (2.48) becomes (2.49) I12 = 2a1 a2 Re dωeiωδ I˜i (ω)eik i ·r . The averaging . . . is greatly simplified by assuming ki ωc /c = k, i.e. keeping it constant at its average value. This is a reasonable approximation for narrow band radiation ω1/2 ωc , so that k does not change significantly over the spectral distribution I (ω). Then the averaging in (2.49) can be factorized. To evaluate the angular part determined by exp(k · r), we read from Fig. 2.8 for small angular divergence ϑd , that the projection of k onto the drawing plane is essentially parallel to r so that k · r = |k|B cos ϕ = u cos ϕ,

with u = |k|B = kBθ

(2.50)

representing the polar angle θ at which the light from the source enters, while ϕ is the azimuthal angle of k in respect of r = B in a plane perpendicular to k. We recall that we consider a light source with a small angular diameter ϑd , e.g. a collimated disc or a distant star, with an intensity distribution I (θ, ϕ) = I (u/kB, ϕ).

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89

The averaging over the angular part in (2.49) (essentially over a cone with 0 ≤ θ ϑd /2 or 0 ≤ u kBϑd /2), properly normalized, may be written g

(1s)

(x) = Re e

ik i ·r

= Re

= Re

2π udu 0 I (u/kB, ϕ)eiu cos ϕ dϕ 2π 0 I (u/kB, ϕ)ududϕ

I (ξ, η)eik(pξ +qη) dξ dη

dξ dηI (ξ, η)

(2.51)

.

In the second line, the integrals are just rewritten from cylindrical coordinates u, ϕ into a Cartesian ξ η plane perpendicular to the average wave vector k (for details see B ORN and W OLF 2006, Chap. 10.4). In analogy to (2.29), g (1s) (x) is called degree of spatial coherence (or spatial correlation function), and (2.51) represents the VAN C ITTERT-Z ERNICKE theorem according to which the degree of spatial coherence is equal to the normalized F OURIER transform of the intensity distribution of the source. We specialize now to an “uniform disc” model for the light source, with constant emission I (u/kB, ϕ) for 0 ≤ θ ≤ ϑd /2, independent of ϕ and obtain 1 g (1s) (x) = eik i ·r = πu2d

x

udu 0

2π

eiu cos ϕ dϕ.

(2.52)

0

The prefactor 1/πu2d ensures proper normalization. The double integral here is the same as that encountered in Sect. 1.2.2 where we have derived the diffraction pattern from a uniform circular aperture.5 With (1.66) for n = 0 it can be expressed by the first-order B ESSEL function J1 (x), g (1s) (x) =

2J1 (x) x

with x = kBϑd /2 = πϑd B/λ

(2.53)

as illustrated in Fig. 2.9. For x = 3.83 (2.53) reaches zero and interference structures disappear: one says that the light is laterally coherent if πϑd B < 3.83 or λ

ϑd <

1.22λ . B

(2.54)

This may be compared to our initial, crude estimate (2.44) – giving the right order of magnitude. We recognize the second inequality as the famous R AYLEIGH criterium for the angular resolution of optical instruments, if we interpret B as diameter of the objective lens. One may also convert this into lateral resolution by setting ϑd = w0 /f , where w0 is the smallest object that can be resolved and f the focal length 5 Note,

however, that here the diameter B of the entrance pupil replaces the radius of the aperture w0 there: essentially, (2.52) describes how the diffraction pattern from the source affects the interference patterns in the experimental scheme Fig. 2.8.

90

2

Fig. 2.9 Absolute value of the degree of spatial coherence (2.53) as a function of x = πBϑd /λ, with the baseline B, the angular diameter of the light source ϑd and wavelength λ; note that 3.83/π = 1.22 so that the first diffraction minimum is found at ϑd = 1.22λ/B


|g (1s) (x)|

1.0

0.5 3.83

0

2

4

6 8 x= π B ϑd / λ

10

of the objective. Note that for larger opening angles ϑd , as often encountered in optical instruments, the lateral coherence condition is

sin ϑd < 1.22λ/B.

The final evaluation proceeds as in Sect. 2.1.4. Thus, the overall interference patterns given by I (δ) = I1 + I2 + 2 I1 I2 Re g (1) (δ) g (1s) (πϑd B/λc ) cos ωc δ,

(2.55)

replacing (2.30). The visibility of the interference fringes (2.31) is thus determined by |g (1s) |. For I1 = I2 we obtain at δ = 0: Imax − Imin (1s) 2J1 (x) with x = πϑd B/λc . V= = g (x) = (2.56) Imax + Imin x We recall that our derivation is for a circular disc light source with a uniform angular diameter ϑd , such as a distant star. A systematic measurement of the visibility V as a function of baseline B at well defined wavelengths λc thus allows one to extract ϑd as will be discussed in the next subsection. To summarize: the interference structure is lost not only for long delay times |δ| τc – as a consequence of an optical path difference larger than the coherence length, |s1 − s2 | c . It also disappears for a light beam with too large lateral extension or to large angular divergence (2.54). This can be rewritten for a light beam of a half divergence angle θe = ϑd /2, a radius w = B/2 (see Fig. 2.8) and with 3.83 4: w<

2 λ := wcoh . πθe θe k

(2.57)

Light is considered coherent if the left inequality holds. We have defined here (somewhat arbitrary) wcoh , a spatial (lateral) coherence radius of a light source. Correspondingly, for a source of radius w we call θe = λ/πw the coherence angle. This description implies that all wave-packets originating from a cross section 2 are considered coherent: their respective interference patterns do not disπwcoh

2.1


91

turb each other significantly. Hence, 2 Acoh = πwcoh =

λ2 λ2 = πθe2 δΩe

(2.58)

is the coherence area of a light source, where δΩe = πθe2 is the solid divergence angle of the beam. Correspondingly, for a given width w of a source, we call δΩe = λ2 /πw 2 the coherence (solid) angle. We finally combine the lateral coherence area (2.58) with the longitudinal coherence length according to (2.20) – slightly arbitrarily and for a Lorentzian frequency distribution – and define a coherence volume Vcoh = Acoh 2c =

4cλ2 . ω1/2 δΩe

(2.59)

Photons are considered as coherent if they originate from a cylindrical volume extending from +c to −c in k direction around the center of the beam with of radius wcoh (beam waist) with a solid divergence angle δΩe . For a “beam” of light derived from a chaotic (or natural) source, these considerations just imply that phase fluctuations within the so defined coherence volume are small enough so that interference structures are not disturbed significantly. For a freely propagating, stationary laser beam the definition of a coherence volume comes even more naturally: Let the radial profile of the beam be Gaussian, and the frequency profile Lorentzian. The lateral coherence radius is identified as the beam waist w0 (we recall that according to Table 1.1 86 % of the total power flows through the corresponding cross section). On the other hand, the relation of w0 to the divergence angle (1.55) is identical to that of wcoh according to (2.57). And the (longitudinal) coherence lengths Table 2.1 is the same as just assumed. In summary, expressions (2.57)–(2.59) also hold for a Gaussian laser beam with a Lorentzian spectral profile.

2.1.8

Astronomical Interferometry

A direct application of the concept of spatial coherence just developed, is the lateral or angular characterization of extended light sources emitting at far distances: this is exploited by astronomical interferometry which dates back to M ICHELSON and P EASE (1921) who mounted a steel beam of initially 6 m length with four mirrors on top of a 2.5 m diameter telescope on mount Wilson, California, in order to determine the lateral degree of coherence of stellar light. The scheme is sketched in Fig. 2.10. If one changes the distance B (the so called baseline) between the two light receiving mirrors M1 and M2 , according to (2.54) interference is only observable for ϑd ≤ 1.22λ/B. This allows one to determine the angular diameter ϑd at which the object studied is seen (e.g. a disc like star, double stars). If the distance of the star is known, one may thus determine its diameter. The resolving power of such an astronomical interferometer depends on the fringe spatial frequency B/λ: the larger it is, the smaller divergence angles can

92

2

Fig. 2.10 Scheme of the original M ICHELSON stellar interferometer


ϑd

li astr ght from ono a (nea mical s n o rly p aral urce lel) bas eline B M 3

M

M

1

ϑd

4

dete

ctor M

2

telescope mirror

be determined. For example, for a baseline of B = 20 m and observation of visible light an angular divergence of about θ = 0.007 = 7 mas can still be resolved.6 M ICHELSON , his coworkers and his successors determined quite a number of angular diameters in that way. Immense technical and methodological progress has been made since M ICHELSON’s ground breaking work, now nearly a century ago. The interested reader is referred to the excellent review by M ONNIER (2003) as a starting point. The most dramatic advances seem to have been made during the past decade – at least thats how it looks from the outside of this specialized field of research (i.e. to the authors of this textbook). A hole flock (at least a dozen with more to come) of very powerful optical/infrared interferometric arrays (as opposed to single baseline interferometers) has started operation during the past years, exploiting all advanced techniques one might dream of in this context (including adaptive optics, fast high precision optical delay lines, low noise high speed VIS and near IR detectors, highly sensitive digital imaging, advanced control and evaluation algorithms, fast computers). Figure 2.11(a) schematically illustrates the design of modern stellar interferometers, which may be compared to the original M ICHELSON setup Fig. 2.10. Key elements are the two light receiving telescopes, the beam guiding (“relay”) optics, the delay lines and the beam combining unit. Note that the effective baseline B used for interferometry is the “projected baseline” (perpendicular to the direction of the incident radiation) – as opposed to the distance between the two telescopes b. Today, the world’s largest telescopes, the two 10 m diameter K ECK telescopes in Hawaii as well as the four 8.5 m telescopes at the European southern observatory in Chile (ESO) can ‘of course’ be combined to interferometric setups (the latter up to a baseline of 100 m) – even if only for rather limited observation times. Specialized sites such as CHARA on Mount Wilson provide a facility for astronomical 61

milli-arcsecond = 1 mas = 2π × 10−3 /(60 × 60 × 360) = 4.848 × 10−9 rad.


telescope 1

93

source

(c) from telescope 2

(b) from telescope 1

om d e et l a rica y

ge

d te ec n e oj l i pr a s e B b

l

f so rom ur ce

(a)

f so rom ur ce

2.1

telescope 2

single beam pixel splitter detectors

+-

baseline b

beam combination unit

from telescope 1

from telescope 2 with delay

variable delay line

detector array

amplifier

δ delay scan

Fig. 2.11 Schematic of modern astronomical interferometers adapted from M ONNIER (2003). (a) Overall layout with the telescopes, the beam guiding optics, delay line and beam combination. Two types of beam combination schemes are shown: (b) image plane interference (similar to YOUNG’s double slit setup), and (c) pupil plane where the collimated beams are brought to interference by a beam splitter

interferometry in the optical/infrared spectral range. Its 6 collecting telescopes with diameters of “only” 1 m each, are arranged in a “Y” configuration and can be combined to a total of 15 baselines, ranging from b = 31 to 331 m. The limiting angular resolution is specified with 0.65 mas in the NIR and 0.15 mas in the VIS – that corresponds to about the diameter of Nils Armstrong’s helmet on the moon, if directly viewed from the earth. Anyone who ever adjusted a laser system on a laboratory table may vaguely imagine the technological challenges to stabilize and manipulate an interferometer mirror setup over distances of more than 300 m with the necessary sub-wavelengths distance control and angular alignment precision! Laser metrology makes it possible. These facilities are by now extremely productive, with measuring angular diameters of astronomical objects as well as in interferometric image reconstruction. B OYAJIAN et al. (2012) point out that 8231 stellar objects with known angular diameters were listed as of July 2004. However, of these the angular diameters for only 24 main sequence stars had been determined with an accuracy of better than 5 %, thus giving hope for quantitative modelling. Their 2012 paper alone reports angular diameters for 44 main sequence stars with a precision of better than 4 %! Figure 2.12 illustrates typical data obtained in this work for two arbitrary examples. Plotted are the visibilities at λ = 2.14 µm, derived from temporal interference patterns of the type shown in Fig. 2.4 (after suitable calibration). The individual data points are measured at the 15 baselines of CHARA. Since the projected baseline B depends on the inclination of the observed star, which changes with time due to earth rotation, the number of data points is much greater than 15 and allows for sufficient precision. The data sets for each star are then fitted by functions similar to (2.56) (see Fig. 2.9). As illustrated in Fig. 2.4, quite different parts of the spatial correlation function (2.56) are exploited in these measurements, depending on the respective

94

2


1.0

visibility

0.8 0.6

HD146233

0.4 0.2

HD4614

0.0 80 100 120 B λ-1 / 10 6 rad -1

140

160

Fig. 2.12 Examples of visibilities for two stellar objects as determined by B OYAJIAN et al. (2012) at the CHARA interferometric array. The (red) data points were taken at a series of values (baseline/wavelength) = Bλ−1 . Note that the full black curves are not straight lines; rather they are fits by functions essentially of the type (2.56), from which the disc diameter angle is derived: ϑLD = 1.623 ± 0.004 mas for HD4614 and ϑLD = 0.780 ± 0.017 mas for HD146233

angular diameters of the stars, here exemplified for ϑLD = 1.623 ± 0.004 mas and ϑLD = 0.780 ± 0.017 mas.7 We cannot close this topic without at least mentioning radio-frequency interferometry. Radio astronomy is a very powerful and highly developed area of modern astronomy, with hundreds of modern facilities worldwide, operating at wavelengths between 1.3 mm and several metres. Baselines of radio-frequency interferometers must be much larger than optical or infrared interferometers to allow detection of the same angular diameters ϑd λ/B. However, radio-frequencies have the great advantage that amplitudes and phases can be recorded directly, while at optical wavelengths typically only cycle averaged intensities can be detected. Hence, amplitudes have to be superposed locally in a beam combining unit to record interference patterns. In contrast, radio frequency interferometry correlates the amplitudes electronically, and no local superposition on the detector is needed. The signals (amplitudes and phases) may be collected anywhere in the world and be brought to “interfere” later on by a mathematical algorithm in a powerful computer. This concept is realized in very long baseline (radio) interferometry networks, e.g. in the global mm-VLBI array in which several dozens of the most powerful radio telescopes of the world co-operate – including very large single dishes such as the 100 m diameter telescope at Effelsberg (Germany) and the 305 m telescope at Arecibo (Puerto Rico), as well as a number of large radio telescope arrays. All what needs to be done is to record simultaneously the electric field of a particular frequency and direction received from space, store it on a tape, and provide that with an accurate time marker – based essentially on synchronized atomic clocks (or masers). With baselines of more than 10000 km an angular resolutions of about 7 While (2.56) is exact for a uniform disc (UD), astronomical models also account for limbdarkening (LD), which in the present case leads to a correction of about 2 %.

2.1


95

50 micro-arcsec can be obtained at 3 mm wavelength – which in spite of the much longer wavelengths is a factor of three better than today’s best optical resolution. At present, the transport of data by magnetic tape is still a bottle neck. But research is underway to use fast optical networks (the next generation internet) for rapid data transfer into the central processing computer. Located anywhere in the world – it constitutes, so to say, a very flexible “beam combining unit”. Of course, different information from space is carried by optical/infrared vs. radio-frequency emission. Thus, both types of interferometry are complementary, and progress will continue. Even space based interferometry is discussed, both for the optical and for the radio-frequency range.

2.1.9

H ANBURY B ROWN -T WISS Stellar Interferometer

One may determine the degree of lateral coherence also by measuring the secondorder correlation function, i.e. by recording the intensity correlations and exploiting (2.41). Hanbury B ROWN and T WISS have suggested for the first time such an experiment in 1954. They tested it in a laboratory setup with a spectral lamp (1956a, 1958) and performed the first successful astronomical measurement determining the angular diameter of Sirius (1956b) based on a measurement of intensity correlation. In principle, such kind of measurement is much more flexible than the interferometry just described – one simply has to record intensities at two detectors, separated by a baseline B, and to determine the correlation g (2) (B) ∝ I (r 1 )I (r 2 ) = I (r 1 )I (r 1 + B) between these signals according to (2.36). With (2.41) one derives the first-order degree of coherence g (1) (πϑd B/λ), the same quantity as measured by an interferometer. But with such technique there is no need for a highly stable setup, precisely adjusted to a fraction of a wavelength over long distances, and even the telescopes do not require high quality as long as one can resolve the object studied. The baseline B can easily be varied and may, in principle, be chosen very long as the signal can be registered at widely separated locations. The setup originally used by B ROWN and T WISS (1956b) is shown in Fig. 2.13(a). They actually used two standard search light mirrors of 1.56 m diameter as telescopes. The normalized second-order correlation function is recorded as a function of the projected baseline distance B. One expects a signal corresponding to Fig. 2.5, convoluted with the experimental resolution. As an example, in Fig. 2.13(b) the normalized signal g (2) (B) − 1 is plotted for the star Sirius, which was the test object of B ROWN and T WISS (1956b). From the fit shown in the graph an angular diameter of 63 mas was determined. Hanbury B ROWN continued a successful carrier as a radio astronomer, but still made several contributions to measuring stellar diameters based on his method in the optical spectral region. However, according to DAVIS and L OVELL (2003), “with rapid improvements in the technology of the phase-correlation interferometer, Hanbury’s intensity interferometer did not survive as a technique for the measurement of the angular sizes of radio sources. As Hanbury later remarked, he had spent two years ‘building a steamroller to crack a nut’.”

96

2

(a)

light from star (nearly parallel)

(b)

g (2) - 1 1.0

B

P1

P2

0.5

b telescope (1)

telescope (2)

δ

× amplifier (1)


0

amplifier (2) integrator & detector

0

5

10

projected baseline B / m

Fig. 2.13 Hanbury B ROWN -T WISS stellar interferometer. (a) Experimental setup according to B ROWN and T WISS (1956b). The two “telescopes” where standard searchlight mirrors of 1.56 m diameter and the baseline was varied up to B 10 m. The detector also acts as integrator. (b) Comparison of experiment and theory for a measurement of the angular diameter of Sirius determining the angular diameter to be 63 mas; adapted from B ROWN and T WISS (1956b)

(a)

point source

(b)

extended source

P2

ϑd /2 delay δ

variable

beam spliter

P1

B P2

P1

Fig. 2.14 Two varieties of the basic concept for Hanbury B ROWN -T WISS type experiments. In each case the signals from the two detectors P1 and P2 are correlated (see text). (a) Radiation (photons or other particles) originate from a point like source with limited temporal coherence; the total flux is split into two equal branches, one of which can be delayed by a variable time δ. (b) The two branches originate from an extended source with limited lateral coherence, corresponding to a phase difference πϑd B/λ

To fully appreciate the impact that the HBT concept and its realization had, as a starting point of quantum optics, let us look again at its essential ingredients. Figure 2.14(a) gives a highly simplified schematic of the photon bunching experiment (temporal/longitudinal coherence) introduced in Sect. 2.1.6. The incoming

2.1


97

light, highly collimated from a point like source, is split into two equal branches and is detected by two photo-multipliers. One records the probability for detecting one photon at detector (P1) and another photon at the other detector (P2) – with some time-delay δ, corresponding to a phase difference ωδ. This time delay between pairs of photons is measured electronically (see e.g. Fig. 2.6). Figure 2.14(b) shows the scheme of the original HBT experiment with an extended light source. It differs from (a) by the fact that now the lateral extension of the source (angular diameter ϑd ) creates a phase difference πϑd B/λ between the two detectors as explicated in Sect. 2.1.7. In this case, the baseline B is varied. In both cases one measures the second-order correlation function (2.38). And in either case one finds (in an ideal experiment) for statistical light sources that the correlated signal at δ = 0 (or at b = 0) is twice that for δ → ∞ (or for b → ∞, respectively) – provided the detectors are sufficiently fast, i.e. their response time is much shorter that the temporal coherence time of the source (for that purpose the light is passed through a narrow band pass filter prior to detection). In contrast, a fully classical source, such as an ideal, intense CW laser beam, shows no enhanced second-order correlation at any time – the photons are distributed completely at random as we shall discuss in Sect. 2.2. As recently pointed out by K LEPPNER (2008), the Hanbury B ROWN -T WISS effect is one of those rare occasions where a classical explanation is quite straight forward, while at first sight it appears to contradict intuition from a quantum point of view: As we have seen in Sect. 2.1.7 the HBT effect arises essentially from the statistical fluctuations of the amplitudes of the radiation.8 Hanbury B ROWN actually started as a radio engineer and was quite familiar with noisy signals. The mathematician T WISS helped him to work out the theory for his experiment on a fully classical basis. However, from a quantum point of view the experiment was completely puzzling and started a vivid and controversial discussion: a photon is either here or there. Why should the probability of finding one simultaneously at each of the detectors (g (2) (δ) = 2) be higher than the statistical probability for random coincidence? But the experiment shows, even at low count rates, that if a photon is registered at (P1) the probability to register at the same time a photon at (P2) is twice that (g (2) (δ) = 2) for purely random arrival of completely uncorrelated beams. The answer to this puzzle is quite simple: photons are bosons, so they may occupy the same phase space – and in a chaotic sources they have indeed a clear tendency to bunch, rather then to occupy all modes equally. Consequently, for electrons and other fermions one may expect the opposite: anti-bunching as we shall see in the next subsection. 8 We also recall that the second-order degree of coherence (autocorrelation function of the intensity) is efficiently used for measuring the duration of femtosecond pulses (Sect. 1.5). There, nonlinear processes such as SHG are applied to detect a signal which is proportional to the square of the intensity (compare Table 1.5, for N = 2 with (2.43)). Note, however, that the results differ in the prefactor of the exponential (2 vs. 1), owing to the fully coherent nature of the laser pulse vs. the chaotic light source assumed here.

98

2


2.1.10 Bunching and Anti-Bunching G LAUBER developed the quantum theory of light which also explains the HBT effect, consistent with the particle nature of photons (a summary of his work is available as G LAUBER 2007). But intensity interferometry has in the mean time successfully been adapted for other particles, exploiting the advances with fast imaging detector arrays. In high energy heavy ion and particle collisions, two particle correlations between protons, pions, or even photons again, are studied to obtain information on the “space-time geometry” of such collisions (BAYM 1998). For fermions one expects and observes anti-bunching (H ENNY et al. 1999; H ASSELBACH 2010). Fermions cannot occupy the same phase space, they avoid each other and this can indeed be observed experimentally. In this context it is appropriate to mention that even in photon correlations one may encounter situations where anti-bunching is observed (K IMBLE et al. 1977): if a single atom fluoresces while being excited by a not too intense radiation field, this atom will have zero probability for emitting a second photon immediately after it has just decayed from its excited state into its ground state. Even more decisive is the experiment of G RANGIER et al. (1986). They prepared a genuine single photon source: photons emerging from an atomic cascade are detected only when triggered by the first photon in the cascade. As expected, they observe strong anti-correlation between the triggered detection on both sides of a beam splitter. We shall come back to further experiments of this type in Chap. 10. A relatively new field for fascinating applications of the HBT effect appears to be – quite unexpectedly – the physics of ultracold quantum gases, where complex phases and structures are revealed by such experiments. We cannot go into details here. We show, however, one particularly neat experiment on ultracold helium, which bears out the difference between fermions and bosons. J ELTES et al. (2007) prepared in a magnetic trap ultracold, metastable 3 He∗ or alternatively 4 He∗ at 0.5 µK. The trapped samples were approximately Gaussian ellipsoids of 110 × 12 × 12 µm3 size. The atoms are released from the trap by turning off the magnetic field – the atoms fall under the influence of gravity and the cloud expands. They are detected by a position-sensitive detector (micro-channel plate and delay-line anode) that detects single atoms. The single atom signal simply reflects the overall shape of the (expanded) cloud. However, the two particle coincidences allow in principle to determine a full 3D second-order correlation function g (2) (x, y, z) of the particle positions – analogous to the (one dimensional) schematic shown in Fig. 2.14(b) for photons. The best resolution is obtained in z-direction (determined by the arrival time of the atoms at the detector). The results for g (2) (0, 0, z) shown in Fig. 2.15 give a very clear and impressive picture of anti-bunching in 3 He∗ (fermions) and of bunching for 4 He∗ (bosons). Section summary

• The intensity spectrum of a light pulse is given by the (inverse) F OURIER transform of the first-order degree of temporal coherence of the field amplitude.

Some Basics for Quantum Optics 1.05 4He*

g (2)(∆z)

Fig. 2.15 Boson and fermion two particle correlation from an ultracold gas of 3 He∗ (fermions, grey symbols) or 4 He* (red symbols). If two atoms originate from the same position in space (z = 0), very clear anti-bunching or bunching is observed for fermions and bosons, respectively. Adapted from J ELTES et al. (2007)

99

1.00

1.00 correlation

2.1

3He*

0.95

0

1 2 atom separation

3 ∆z / mm

• With (2.16)–(2.20) we have modelled a “quasi-monochromatic”, stationary light beam. Its coherence properties are described by the first-order degree of coherence. This is summarized in Table 2.1. • Using these classical concepts we have quantified the conditions for coherent interference of electromagnetic as observed e.g. in YOUNG’s double slit experiment. As a general rule, in quasi-monochromatic (chaotic) light each photon interferes only with itself. • The interference fringes can be expressed by the first-order degree of coherence. The characteristic patterns observed as a function of time delay between the interfering beams depend on the spectral characteristic of light source. The overall visibility V contains valuable information about the lateral coherence of such a light source. • Higher order correlation functions are defined. The (normalized) second-order degree of coherence g (2) [δ] describes the correlation of field intensities at different positions in time and space. It also gives the probability to detect two photons in (delayed) coincidence. For chaotic light g (2) can be expressed in terms of g (1) . • Thus, the first-order degree of coherence can be derived from intensity correlations. This photon bunching was first observed by Hanbury B ROWN and T WISS. Although classically well understood, the HBT effect is conceptually more difficult to reconcile with the particle nature of photons, and has started quantum optics in the mid 1950ies. • Spatial (lateral) coherence complements the concept of temporal (longitudinal) coherence. Lateral coherence is lost for extended light sources at too large divergence angles since interference patterns from different parts of the source cancel each other. • Lateral coherence is used in astronomical interferometry to determine the angular diameters of stars. Historically, the HBT effect was also exploited for this purpose. Today powerful facilities for (amplitude) interferometry are

100

2


used almost exclusively: arrays of interferometers in the visible and infrared spectral range, worldwide antennae networks in the radio-frequency region. • The HBT effect can only be observed since photons are bosons. Today it is applied successfully also to other particles, including fermions for which antibunching is observed.

2.2

Photons, Photon States, and Radiation Modes

In this section we prepare the quantization of the electromagnetic radiation field, in the following section we shall actually present the essential steps. As throughout this book, we shall do this in a heuristic manner with focus on understanding the physics – for which we may sacrifice some mathematical strictness. By no means do we intend to give a stringent introduction into quantum electrodynamics and quantum optics – for which a rich literature exists. Among the references to this chapter the ambitious reader finds several fine textbooks for further reading (L OUDON 2000; G LAUBER 2007; G RYNBERG et al. 2010; M ANDEL and W OLF 1995; W EISSBLUTH 1978; M ILLONI and E BERLY 2010). Up to now, we have treated light as a completely classical radiation field. For the interaction of matter with light we have used the semiclassical approach presented in Chap. 4, Vol. 1: atoms are treated quantum mechanically, the electromagnetic field classically. For a laser beam this turns out to be a rather correct description, even though we know that light has also particle properties manifested by photons. In fact, a laser beam contains a very large number of photons. We shall clarify in Sect. 2.2.4 what precisely that means. And we shall see, that it is this very fact which makes the semiclassical description a very good approximation. On the other hand, photon counting experiments as discussed in the previous section call for a quantum mechanical interpretation – even though a classical explanation was possible in the cases discussed so far. For at least two reasons the introduction of a fully quantized description of the field appears to be compelling: one is spontaneous emission which in the semiclassical approach occurs only as a kind of afterthought and cannot really be understood. However, spontaneous emission is a key phenomenon in many areas of physics. The second reason is of a more fundamental – one might say aesthetical – nature: to document energy conservation for radiation induced processes. Clearly, energy is needed to excite an atom, and conversely, it cannot be lost when the atom is deexcited. The semiclassical picture does not account for this explicitly; energy comes from somewhere and gets lost to somewhere. In contrast, the fully quantized description will connect absorption and emission with the annihilation and creation of a photon, respectively, and thus expresses energy conservation explicitly.

2.2.1

Towards Quantization of the Radiation Field

Before going into details, let us get our bearings with the overall picture. Quantum mechanics, as we have used it so far, is essentially particle wave mechanics in

2.2


101

the S CHRÖDINGER picture. Historically, particles (electrons, atoms, nuclei) existed a long time before the invention of S CHRÖDINGER, D IRAC or K LEIN -G ORDON equations. With photons,the situation is exactly opposite: the wave equation for photons, i.e. for electromagnetic radiation (based on M AXWELL’s equations) existed a long time before the photon was discovered (or should we say, was “invented” as a concept?). Thus, we know the wave equation for photons already. What is required at this point is a genuine quantization of the field. We have to find a common framework for describing the states of electrons and those of photons – and their interaction. There is, however, one major difference between electrons and photons: while the former are fermions the latter are bosons. For electrons the PAULI principle holds and any state can only be occupied by one electron. In contrast, many photons can, in principle, occupy any given photon state. Thus, the programme is as follows: We first recall the basic properties of photons and introduce the concept of photon states. Secondly, we take a more detailed look at the photon wave functions, called modes of the electromagnetic field. Thirdly, we introduce a convenient scheme of book-keeping for photons, called second quantization, which characterizes photon states by their occupation numbers.9 In the usual S CHRÖDINGER picture the electromagnetic field itself (as an observable) is then represented by a time independent operator, all time dependence will be cast into the evolution of the photon states. We begin by recalling the well known experimental facts about photons. From the photoelectric effect we know that the energy in the electromagnetic fields is quantized in well defined packets of Wph = ω,

(2.60)

associated with the particle photon. The photon travels (in vacuum) with the speed of light c and has no rest mass. We may attribute to it a relativistic mass mph = ω/c2 . The momentum of the photon, also known from experiment (C OMPTON effect) is p ph = k

with pph =

h = h¯ν . λ

(2.61)

Finally, photons have an intrinsic angular momentum, the photon spin S, with a spin quantum number S = 1. This too is based on experimental evidence (B ETH 1936), as reported in Sect. 4.1.4, Vol. 1. Photon states |k, e may be characterized by the photon’s propagation vector k and its polarization e according to Sect. 1.3. One may introduce a photon spin operator S and its components, in particularly Sz and express the photon states in the helicity basis |eq , where the z-axis is chosen parallel to k. The states |e± refer to circularly polarized light, and the usual angular momentum algebra applies: 2 S |eq = 2 S(S + 1)|eq with S = 1,

(2.62)

9 Actually, a similar scheme can be applied to the electronic states of atoms. But that scheme is much simpler since these states can only be occupied or not be occupied.

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2


Sz |eq = q|eq with q = ±1 and

(2.63)

eq |eq = δqq .

(2.64)

Alternatively, we may use basis states for linearly polarized photons, 1 |ex = − √ |e+1 − |e−1 2

i and |ey = √ |e+1 + |e−1 , 2

(2.65)

following (4.7) in Vol. 1. With (2.63) one verifies that these linearly polarized states are eigenstates of Sz2 : Sz2 |ex = 2 |ex and Sz2 |ey = 2 |ey

(2.66)

but not of Sz . Rather, the expectation value of Sz becomes zero for the |ex as well as for the |ey state. This too is confirmed by experiment. And in complete analogy to (1.85), the most general, elliptically polarized photon may be represented by |e = e−iδ cos β|e+1 − eiδ sin β|e−1 .

(2.67)

We emphasize that (2.63) includes only two states, with q = ±1, i.e. angular momentum components ± in z-direction. Even though in conventional angular momentum algebra (2.62) and (2.63) would formally define three substates (with q = 0, ±1), the particle “photon” exists only with the two angular projections, q = ±1. This somewhat unusual behaviour reflects the transverse nature of the polarization of light and the fact that photons do not have a rest mass, i.e. always propagate with the speed of light. Classically we had associated the spherical basis vectors with three oscillators: two of them oscillating in the xy plane (q = ±1) while the third oscillates along the z-axis. The corresponding radiation characteristics are described in Chap. 4, Vol. 1. Somewhat loosely one might say that the photon state with q = 0 does not propagate along the z-axis.

2.2.2

Modes of the Radiation Field

The photon states |k, eq discussed above correspond to a single photon with polarization eq , wave vector k, and frequency ω = k/c. A realistic light beam consists of many photons and a range of wave vectors. As a complete basis set for constructing any “photon wave function” one could e.g. take all plane waves with all possible values of k – as we have already shown in the previous Sect. 2.1. A quasimonochromatic, stationary light beam with a mean wave vector k c would contain a narrow range of angular frequencies δω around ωc (or δk = δω/c around the magnitude of the wave vector kc ) and have an angular distribution of wave vectors in an angular range δθ (or solid angle δΩ = πδθ 2 , respectively). Of course, photons are in principle unbound particles, so that we would have to deal with an infinite number of basis states and an infinite number of energies. To avoid these complications, one usually switches to a very large but finite normalization volume (in real position space), say a cubic cavity of edge length L, and

2.2


103

kz /(2 π /L)

Fig. 2.16 Two dimensional cut through k space, divided into a grid of unit length 2π/L. A light beam is characterized by the probability to find a certain wave vector k i around the mean wave vector k, and by the number of photons populating this cell

10

k kc

δθ

5

δk

kx / (2 π /L) -5

0

5

10

perfectly conducting walls where the electric field must vanish. Mathematically this is equivalent to introducing periodic boundary conditions10 kx = mx

2π , L

ky = my

2π , L

kz = mz

2π L

(2.68)

with mx , my , mz = 0, 1, 2, 3, . . . . Thus, a countable number of bound states emerges, called modes of the radiation field, to each of which we attribute a photon state |k, e. The whole k space is thus divided into very small but finite cells as sketched in Fig. 2.16. The size of the cells is determined by kx = ky = kz = 2π/L, so that k = kx ky kz = 3

2π L

3 .

(2.69)

Only a finite number of these modes is needed to describe a quasi-monochromatic light beam with an average wave vector k c – as indicated by the red dashed area in Fig. 2.16. Each mode is characterized by its wave vector k (and polarization) and may be occupied by any number of photons. We note an important consequence of this structure of k space. With p = k we may write 3 3 3 2π pL 3 p L 3 3 . (2.70) k= 3 3 = L L h3 Obviously (2.69) and (2.70) can only hold simultaneously if the size of a unit cell in phase space is 3 pL3 = h3 .

10 The

(2.71)

treatment given here is quite analogue to that for electrons in a 3D box, presented in Sect. 2.4.2, Vol. 1 – except that there fermions (spin 1/2) were described and each cell in k space was only filled by at most two electrons (of opposite spin).

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2


It is important to point out here that in the preceding paragraph one key assumption has been made which is crucial (albeit plausible) for the following considerations: periodic boundary conditions in k space (2.68) for the electromagnetic field. – One may, of course, also turn the arguments around and define (2.71) as the fundamental theorem: the minimal cell size in phase space is h3 . This, together with the well defined energy ω of a photon, may be seen as the key paradigm beyond the quantization of the electromagnetic wave field. It will turn out to have decisive consequences, e.g. in the context of spontaneous emission. We note that the size of the box, L3 , which is our reference volume, does not necessarily refer to a real physical situation. Usually it is just a mathematical construct introduced to avoid an infinite number of photon states with which one would otherwise have to deal, and one simply has to choose L just large enough so that the grid is sufficiently fine for describing the properties of the radiation field applied. On the other hand, there are situations where the normalization volume really refers to a genuine physical geometry, e.g. to a laser resonator or any type of optical cavity in which light may be confined. The genuine modes of this cavity will have to be used if one wants to describe an experimental situation quantitatively. Laser theory is one such application. Another field is the so called “cavity QED” which we shall touch briefly in Sect. 2.3.7. If the size of the cavity becomes comparable to the wavelength of the radiation studied one finds that even spontaneous emission is substantially modified. Later on we shall need an expression for the number of modes in a specified range of k vectors with a given polarization. This can now easily be derived. The number of modes dmke between k = (kx , ky , kz ) and k + dk = (kx + dkx , ky + dky , kz + dkz ) is obtained by dividing the volume element in k space dkx dky dkz by the size of the unit cell (2.71): dmke =

dkx dky dkz L3 2 = k dkdΩ = ρ(k, e)dkdΩ. (2π/L)3 (2π)3

(2.72)

With ω = kc we may also refer this to the angular frequency interval dω: dmωe =

L3 ω2 dωdΩ = ρ(ω, e)dωdΩ. (2πc)3

(2.73)

The values dmke and dmωe give the number of modes with polarization e propagating into a solid angle dΩk and with wave vectors between k and k + dk, or angular frequencies between ω and ω + dω, respectively. The expressions ρ(k, e) =

k2 dmke = L3 , dkdΩ (2π)3

ρ(ω, e) =

dmωe ω2 = L3 , dωdΩ (2πc)3

ρ(ν, e) =

dmνe ν2 = L3 3 dνdΩ c

(2.74) and

(2.75) (2.76)

2.2


105

are called mode density. The mode density obviously depends on the square of the wavenumber or frequency. After having identified the radiation field as a discrete and countable set of modes, as a final step one extends the normalization volume L3 to values so large (essentially to infinite) that the usual continuous spectrum is effectively recovered, i.e. L is chosen large enough to obtain a sufficiently fine mesh kx,y,z = 2π/L in k space to describe the problem at hand to any degree of accuracy needed. This allows one to finally replace all necessary summations over spectral modes by an integration over the solid angle Ω and k (or ω). With the mode densities (2.74) and (2.75) just derived we may thus write symbolically k

... →

L3 (2π)3

. . . k 2 dkdΩ = k,Ω

L3 (2πc)3

. . . ω2 dωdΩ.

(2.77)

k,Ω

As far as the radiation field is spatially isotropic one may carry out the angular integration and obtains k

... →

L3 2π 2

. . . k 2 dk = k

L3 2π 2 c3

. . . ω2 dω

(2.78)

ω

for each specified polarization e. If one investigates optical transitions induced by a well collimated radiation source, such as a laser beam, typically (2.78) cannot be used and (2.77) must be applied. We have already mentioned this aspect in our semiclassical treatment of light induced transitions in Chap. 4, Vol. 1. Note that the mode density derived here is proportional to the normalization volume L3 . Fortunately, as we shall see below, all measurable properties which we shall compute are densities of some kind, i.e. have to be evaluated per volume. Thus, L3 will drop out of the final results.

2.2.3

Density of States and Black Body Radiation

We take here a little detour back to black body radiation. Dividing ρ(ν, e) given in (2.74) by L3 , and multiplying it by 8π (integration over the full solid angle and summation over the two polarization directions) leads to the density of states (per volume) as introduced in Sect. 1.3.4, Vol. 1. For photons one usually refers to frequency space: 8π 2 ν dν. (2.79) c3 Inserting this into the B OSE -E INSTEIN distribution (1.63), Vol. 1 for a black body radiator in thermal equilibrium, we obtain the spectral photon density: g(ν)dν =

8πν 2 dν N(ν)dν = 3 . c exp(hν/kB T ) − 1

(2.80)

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2


The chemical-potential of the massless particle photon has been set here m ¯e =0 – it takes no energy to split or unite photons in statistical interactions (e.g. with the surrounding walls) as long as the total energy remains constant. Note that the ν 2 factor in the nominator prevents divergence for hν → 0 (i.e. avoids the so called infrared catastrophe). Integration of (2.80) over all (positive) frequencies gives a finite value for the photon density in the black body, N = 16(kB T )3 π ζ (3)/(hc)3 , with the R IEMANN function ζ (x). N amounts to about 20 photons/cm3 at 1 K. We recall now that P LANCK’s law describes the spectral energy density of the photons, i.e. it is obtained from (2.80) by multiplication with the photon energy hν. Comparison with (1.81), Vol. 1 shows that we have indeed derived P LANCK’s law.

2.2.4

Number of Photons per Mode

We still have to establish a quantitative relation between the number of photons in a specific mode and the intensity I of the electromagnetic field – or its electric field strengths E. The photon states |e discussed above refer to a single photon in a specific mode k, e. In reality, however, a light source such as a laser beam, is characterized by many photons per mode. How is that number of “photons per mode” determined? Let us start with the total number of all photons Ne with polarization e in the normalization volume L3 – assuming it is completely filled with radiation of intensity I = cu at a photon energy ω: Ne = u

I L3 L3 = . ω c ω

(2.81)

More specific, in an interval ω to ω + dω of angular frequencies we find Ne (ω)dω = u(ω) ˜

L3 I˜(ω) L3 dω = dω ω c ω

(2.82)

photons, with u(ω) ˜ = I˜(ω)/c being the spectral radiation density and I˜(ω) intensity spectrum (per unit angular frequency). Considering the finite divergence angle δΩ of a light beam, the number of photons with polarization e in a frequency interval dω per solid angle dΩ is N (ω, Ω; e)dωdΩ =

I˜(ω) L3 dωdΩ. δΩ cω

(2.83)

Finally, we recall dmωe , the number of modes (2.73) in a range dωdΩ of frequencies and solid angles. With this we obtain the number of photons per mode: Nke =

I˜(ω) (2πc)3 I˜(ω) λ3 N (ω, Ω; e)dωdΩ . = = dmω,e ωcδΩ ω2 δΩ c

(2.84)

2.2


107

We thus have worked out a relation between the quantum mechanically relevant number of photons per mode and the measurable intensity per solid angle and angular frequency. To be even more specific: with (2.21) for the maximum I˜(ωc ) of a Lorentzian spectral distribution (FWHM = ω1/2 = 2/τc ) we obtain Nke =

λ3 I = 2τc λ2 . ω1/2 δΩ πc ωδΩ 2I

(2.85)

As expected, Nke is independent of the normalization volume (both the number of photons and the mode density grow linearly with L3 ). But it is proportional to the coherence time and inversely proportional to the divergence angle δΩ of the light source. It is instructive to look at some numbers Nke for some typical light sources. Let us, e.g. take an ideal laser beam with a Gaussian radial profile and a Lorentzian spectrum. In this case, the divergence angle of the source is diffraction limited, i.e. δΩ = δΩe and with (2.59) we identify the coherence volume Vcoh . Thus, for a diffraction limited beam we can write Nke =

I Vcoh . cω

(2.86)

As I /(cω) is the photon number density, this relation can be read as: the number Nke of photons per mode is equivalent the number of photons in the coherence volume of the beam. We recall: for a laser beam Vcoh is simply its geometrical waist cross section πw 2 (at 1/e2 width) multiplied by 2c = 2cτc . A slightly different situation is encountered for a chaotic radiation source. Let it have a small but finite diameter d = 2w and radiate with a total power P isotropically into the full solid angle δΩ = 4π . Coherent emission from its effective area πw 2 occurs into a solid angle δΩe = πθe2 = λ2 /πw 2 , with an intensity I = P /πw 2 . Thus, (2.85) may be written Nke =

P δΩe 4λ2 P = 2τc . ω 4π πw 2 ω ω1/2 4π

The second equality states that the number of photons per mode is equivalent to the number of photons emitted coherently (i.e. into a solid angle δΩe ) during twice the coherence time. Table 2.2 summarizes characteristic parameters for some typical radiation sources: total power, lateral extension, wavelength, bandwidth and relative bandwidth. From these one calculates coherence (half) angle, coherence time and coherence lengths according to Table 2.1 as well as the rate of coherent photon emission Pcoh /ω and the number of photons per mode Nke according to (2.85). Among the sources compared are two essentially chaotic ones (spectral lamp and atoms at rest) and three quasi-monochromatic sources with rather long coherence times. The “spectral lamp” could be a typical, commercially available device, here

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2


Table 2.2 Characteristic parameters of five typical light sources: Total power of emitted light P , beam waist or source radius w, wavelength λ, FWHM of the spectral distribution δλ1/2 and δν1/2 , coherence (half) angle δθe , coherence time τc , coherence length c , coherently emitted photon rate Pcoh /ω, number of photons per mode Nke Source

P/W w/ mm λ light (total)

spectral lamp

0.5

atoms at rest

10−6

0.05

780 nm

CW dye laser

1

0.5

590 nm 1.2 × 10−6 1 MHz

TiSa laser pulse

2.0 × 1010

0.1

microwave oscillator

103

100

derived from the above parameters: Source δθe / rad = λ/πw

λ/ nm

590 nm 10−3

5

1.2 × 10−5

ν1/2

ν1/2 /ν = λ/λ

860 MHz

1.7 × 10−6

5.9 MHz

1.5 × 10−8 2 × 10−9

800 nm 19

8.8 × 103 GHz

2.3 × 10−2

3 cm

100 Hz

1 × 10−8

0.3

τc / s−1 = 1/πν1/2

c / m = c × τc

Pcoh /ω s−1 Photons/ s

Nke

0.11

5.3 × 108

0.4 2.5

spectral lamp

3.8 × 10−5

3.7 × 10−10

atoms at rest

5 × 10−3

5.4 × 10−8

16

2.3 × 107

CW dye laser

3.8 × 10−4

3.8 × 10−7

95

3 × 1018

1.9 × 1012

TiSa laser pulse

2.6 × 10−3

3.6 × 10−14

1 × 10−5

8 × 1028

5.8 × 1015

0.1

3.2 × 10−3

106

1.5 × 1026

1024

microwave oscillator

emitting at the Na wavelength. The “atoms at rest” might e.g. be a B OSE -E INSTEIN condensate, assuming 105 excited 87 Rb atoms to emit at the 780 nm, with the natural width of this resonance line. These two sources are assumed to emit isotropically into the full solid angle 4π . The (half) angle θe indicates the maximum angle within which the light can be considered as coherent. The other sources are highly directional and are assumed spatially coherent over their full cross section. The dye laser is operating CW in the yellow spectral range, with reasonable stabilization and a bandwidth as often used in spectroscopy. The pulsed source (ca. 1 mJ with a temporal FWHM of 50 fs at 800 nm) represents a standard femtosecond Titanium-sapphire laser setup, with a beam focused moderately to w = 100 µm. We assume the whole pulse to represent one mode of radiation – due to the short pulse duration with a rather broad bandwidth. Finally, we also compare with a classical radiation source, a microwave oscillator. The characteristic quantity Nke derived here, the number of photons per mode, gives of course an average value if many modes are needed to describe the spectrum and the angular profile of a source. Note that these sources represent rather different types of radiation: For the spectral lamp Nke is very small so that most modes do not contain any photon at all; the atomic source shows already a significant probability to find one or even more photons per mode; the highly coherent laser sources as well as the microwave source contain a very large number of photons per mode, and the field can be considered as essentially classic.

2.2


2.2.5

109

The Multi-Mode Field and Energy

We may now explicitly write down the field variables of a multi-mode electromagnetic radiation field, i.e. its vector potential A(r, t), and its electric E(r, t) and magnetic field B(r, t). They are related to each other as described in Appendix H.1.1, Vol. 1. We focus again on the electric field vector. Following (1.35) we write now E(r, t) =

i − + eq Ekq (t)eikr − e∗q Ekq (t)e−ikr 2

(2.87)

kq

with

− − −iωt (t) = Ekq e Ekq

+ + iωk t and Ekq (t) = Ekq e .

(2.88)

The summation has to be carried out over all occupied field modes. In a classical description Ekq is the field amplitude in each mode, with wave vector k and polarization q. As discussed previously, random phase fluctuations will have to be included if one wants to describe a stationary light beam. We now have to make the translation to quantum mechanics, i.e. we are looking for the field operator. A good starting point is the total energy W stored in the electromagnetic field. By inserting (2.87) into (1.86) we obtain the intensity I and the energy density u = I /c. Integration over the whole normalization volume L3 eventually leads to 2 W = L3 ε0 cE(r, t) ε0 − ε0 − + + = L3 Ekq (t)Ekq (t) = L3 Ekq Ekq . 2 2 kq

(2.89) (2.90)

kq

This convincingly clear result is essentially a consequence of the confinement to a large normalization volume with periodic boundary conditions: squaring (2.87) leads to a double sum over kq and k q . However, with integration over L3 one finds that exponential terms of the type exp[i(k − k )r] lead to delta functions, so that only contributions from terms diagonal in k remain. Finally, with (2.88) the time dependence also drops out. Section summary

• In this section several conceptual steps were taken to familiarize ourselves with the notion of photon states, and to prepare the quantization of the electromagnetic field. • After recalling the quantum properties of photons, we introduced modes of an electromagnetic field by assuming a large but finite normalization volume L3 and demanding periodic boundary conditions. As a consequence we find that 3 pL3 = h3 is the smallest size of a phase space cell.

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2


• This concept allowed us to specify the number of modes, respectively to determine the density of modes in k space. • An important quantity which connects the classical view of a continuous electromagnetic field and the quantum description of photons is the (average) number of photons Nke per mode with a specified wave vector k and polarization e. Quantitatively, Nke is related by (2.85) with intensity of the radiation, its frequency spectrum, and its angular divergence. • For several characteristic radiation sources Table 2.2 summarizes the relevant parameters, coherence properties and numbers of photons per mode. • Finally, we have – for the general case of a quasi-monochromatic light source with finite divergence angle – rewritten the electric field vector and the energy contained in the radiation field, using the language of field modes introduced here.

2.3

Field Quantization and Optical Transitions

2.3.1

Second Quantization and Photon Number States

So far we have not really quantized the field yet. In order to do so, one needs some quantum mechanical tools: the matrix formulation of the harmonic oscillator and second quantization. The latter is a clever method of book keeping for the population of states with particles – here of photon states with photons. We start with the total energy (2.89) of the electromagnetic field, and consider one single mode k, q populated. The field energy in this mode is Wkq = L3

ε0 − ε0 − + + E (t)Ekq (t) = L3 Ekq Ekq 2 kq 2

+ + iωk t with Ekq (t) = Ekq e .

(2.91)

In the following we drop the indices kq for simplicity of writing, and introduce new variables: L3 ε0 − L3 ε0 − + E (t) + E (t) and P = −i E (t) − E + (t) . (2.92) Q= 2ω 2 The inverse relations are 1 − (ωQ + iP ) E (t) = 3 L ε0

+

and E (t) =

1 L3 ε0

(ωQ − iP ).

(2.93)

With this the total energy (2.91) is written as W=

1 2 P + ω2 Q2 . 2

(2.94)

2.3


111

This expression looks very familiar: it is mathematically identical to the energy of the harmonic oscillator in classical mechanics: W=

px2 mω2 2 + x . 2m 2

The key idea is now, to identify the oscillations of the electromagnetic radiation field with the harmonic oscillator, and use the rules sketched in Chap. 2, Vol. 1 to translate the field modes into quantum mechanics. For this, Q and P must be canonical conjugate coordinates. With (2.92) we find P˙ = −ω2 Q

˙ = P, and Q

and the partial derivatives of the energy (2.94) are ∂W = ω2 Q = −P˙ ∂Q

and

∂W ˙ = P = Q. ∂P

This set of equations are the classical H AMILTON equations in one dimension with Q and P being indeed canonical conjugates. Thus, W := H represents the Hamiltonian of the electromagnetic field! What follows is the decisive step in the quantization process: canonical conjugates are replaced by operators which obey the commutation rule P ] = i. [Q,

(2.95)

We now rewrite the relations (2.93) in dimensionless form by multiplying them with L3 ε0 /(2ω): 1 + i P) (ωQ aˆ = √ 2ω

1 − iP ). and aˆ + = √ (ωQ 2ω

(2.96)

With (2.95) one verifies that these operators obey the simple commutation rule

a, ˆ aˆ + = aˆ aˆ + − aˆ + aˆ = 1,

(2.97)

+ 1. aˆ aˆ + = aˆ + aˆ + 1 = N

(2.98)

which may be recast into

Here we have introduced the so called number operator = aˆ + a. N ˆ The Hamiltonian of the electromagnetic field (2.94) takes now the form

(2.99)

112

2

Fig. 2.17 Energy level diagram for photons in a mode of the electromagnetic radiation field. Indicated it the effect of photon creation and annihilation operators, aˆ + and a, ˆ respectively, onto the number states |N


energy W

number of photons +1

^+

a ^

a

-1

≈ 5ħ

/2

2

3ħ

/2

1

ħ

/2 0

0

F = ω aˆ + aˆ + aˆ aˆ + = ω aˆ + aˆ + 1 H 2 2 + 1 . = ω N 2

(2.100) (2.101)

, aˆ and aˆ + is straight forward, The derivation of the algebra for the operators N based on the commutation rule (2.97). It can be found in all quantum mechanics is text books. We thus only summarize here the results. The number operator N + Hermitian (while aˆ and aˆ are not). It has eigenstates |N with integer numbers N as eigenvalues: |N = N |N N with N = 0, 1, 2, . . . and N | N = δN N .

(2.102) (2.103)

With this the eigenvalues of the Hamiltonian (2.101) follow immediately: F |N = WN |N H where WN = (N + 1/2)ω for N = 0, 1, 2, . . . .

(2.104) (2.105)

We recognize the well known eigenenergies of the harmonic oscillator. One interprets N as the number of photons present in the particular resonator mode under consideration. Since photons are bosons, the mode may be populated with any number N photons. This is illustrated in the energy diagram Fig. 2.17. The levels of the harmonic oscillator are equally spaced, and the lowest energy is given by ω/2, the “zero point energy”. Excitation of the N th harmonic of the classical oscillator corresponds to a state occupied by N photons. The eigenstates |N of the number operator (and the harmonic oscillator) may be generated from the vacuum state |0 by repetitive application of the operator aˆ + for which √ aˆ + |N = N + 1|N + 1. (2.106)

2.3


113

Hence, aˆ + is called creation operator. In contrast, the operator aˆ reduces the photon number by one, when applied to a number state a|N ˆ =

√ N |N −1,

(2.107)

and hence aˆ is called annihilation operator. N such operations lead to the vacuum state |0 for which the relation a|0 ˆ ≡0

(2.108)

must hold, since a nonexisting photon cannot be destructed any further. The inverse scheme starts with √ generate any number state √ the vacuum state, from √ which one can + N by (aˆ ) |0) = N !|N . The factors N + 1 and N in (2.106) and (2.107), respectively, make sure that the number states are correctly normalized as stated by (2.103). Obviously, the number operator (2.99) counts the number of photons N in the mode under consideration. This number is increased or decreased by one when the ˆ respectively, acts on the photon states. Writing the Hamiltonian operator aˆ + and a, in the form (2.101) implies simply counting the occupation number. This procedure is called second quantization and may be applied to other quantum objects as well. The evolution of the photon states with time |ψN (t) is obtained from the trivial time dependent S CHRÖDINGER equation F ψN (t) = i δ ψN (t) , H δt

(2.109)

ψN (t) = e−iWN t/ |N = e−i(N + 12 )ωk t |N .

(2.110)

which is solved as usual by

Note that in all this discussion the S CHRÖDINGER picture is used. All time dependence of the radiation field is now cast into the time dependence of the photon states. The operators aˆ and aˆ + are not time dependent.

2.3.2

The Electric Field Operator

Finally, we come back to the key question: how to quantize the electromagnetic field? Comparing the definition (2.96) for annihilation and creation operators with the classical field quantities (2.93) leads us immediately to operators for the electric

114

2


field (we resume now showing the indices kq):11 2ω k − + = 2ωk aˆ + . = aˆ kq and E E kq kq L3 ε0 L3 ε0 kq

(2.111)

Again, in the S CHRÖDINGER picture these operators are independent of time. They have to be inserted into (2.87) in place of their classical counterparts. Thus, the electric field operator may be written as 2ωk i t) = E(r, (2.112) aˆ k ukq (r) − aˆ k+ u∗kq (r) 2 ε0 kq

with ukq (r) = L−3/2 eq exp(ikr).

(2.113)

Since creation and annihilation operators are defined dimensionless, one easily ver = V m−1 . We point ifies that the unit of the electric field operator is indeed [E] t) is a Hermitian operator (while its constituents aˆ k and aˆ + are not). out that E(r, k We also mention here, that (2.112) is sufficiently flexible to adapt the quantization formalism for any specific experimental situation by an appropriate change of the modes (2.113) – e.g. for application to quantum optics in a cavity. To obtain the field energy one has to insert (2.112) into (2.89) and evaluate it in full analogy to the classical considerations. However, now the commutation rule (2.97) must be observed. This finally leads to a sum of Hamiltonians (2.101) for all modes:

1 + + + F = 1 H ωk aˆ kq aˆ kq + aˆ kq aˆ kq = ωk aˆ kq aˆ kq + . (2.114) 2 2 kq

2.3.3

kq

G LAUBER States

It is important to realize that the photon number states introduced above do not represent coherent light. Rather, coherent light must be described by a linear superposition of many number states as shown for the first time by G LAUBER (1963). For a single mode, these so called G LAUBER states (also coherent photon states) are given by 1 αN |α = exp − |α|2 |N (2.115) 2 (N !)1/2 N

11 We

mention that G LAUBER (1963) uses time dependent field operators (H EISENBERG picture) and a slightly different notation. He writes (in esu) “the positive frequency part of the electric field operator” E (+) (r, t) = i ω/2ak uk (r)e−iωk t . k

2.3


115

α ∗N 1 and α| = exp − |α|2 N |. 2 (N !)1/2 N

Let us have a brief look at the properties of G LAUBER states. We first note that they are normalized, α ∗N α N α|α = exp −|α|2 = 1, N!

(2.116)

N

as the sum corresponds to the exponential function exp(|α|2 ). They are, however, not orthonormal, rather we have 1 2 1 2 1 2 1 2 α ∗N β N ∗ exp − |α| − |β| − α β , α|β = exp − |α| − |β| 2 2 N! 2 2 N

and for the absolute squared of this scalar product one obtains α|β2 = exp −|α − β|2 .

(2.117)

This set of coherent states is thus overcomplete, i.e. there are more coherent states than number states |N . From (2.117) we see, however, that two G LAUBER states get nearly orthogonal, if |α − β| 1. Applying the photon annihilation operator onto (2.115), we obtain with (2.107) 1 αN a|α ˆ = exp − |α|2 N 1/2 |N − 1 2 (N !)1/2

(2.118)

N

α N −1 1 = α exp − |α|2 |N − 1 = α|α. 2 ((N − 1)!)1/2 N

ˆ One G LAUBER states are thus eigenstates of the photon annihilation operator a. may extract photons of a G LAUBER state without changing that state. Conversely, |α is not an eigenstate of the photon creation operator. It is important to note, that with (2.111) single mode G LAUBER states are also eigenstates of the field operator − (but not of its conjugate counter part). Measuring electromagnetic fields usually E implies that photons are registered, i.e. a photon is extracted from the radiation field. If the field can be described by a G LAUBER state |α, the detectable probability − |α ∝ α. This is in essence what makes amplitude will thus be proportional to α E G LAUBER states coherent. Since the characteristic parameter α can also be complex, α may be seen to represent phase and amplitude of the electromagnetic field. The expectation values of the annihilation and creation operators in a G LAUBER state follow from (2.118) and by applying (2.106) onto (2.115), respectively: α|a|α ˆ =α

and α|aˆ + |α = α ∗ .

(2.119)

116

2


Finally we have to obtain a relation between a G LAUBER state and the intensity of the radiation. Let us first note that the population of photon number states |N in a G LAUBER state is given by a P OISSON distribution 2 |α|2N . pN = N |α = exp −|α|2 N!

(2.120)

The expectation value of the photon number operator (2.99) in a state |α, i.e. the mean photon number N in a G LAUBER state, is α ∗N α N |α = exp −|α|2 N = |α|2 . N = α|N N!

(2.121)

N

In this context we recall a well known property of the P OISSON distribution: its standard deviation is given by 2 |α − α|N |α2 α|N = |α| = N . (2.122) N = |α α|N This is actually very good news for all of our following discussion. A look at Table 2.2 shows that for lasers – the light source typically used todayin spectroscopy – the number of photons per mode is extremely large. And since N /N = 1/ N , the relative width of the distribution of photon numbers is very small (e.g. for the dye laser mentioned in Table 2.2 on the order of 10−6 ). Thus, for all intents and purposes in spectroscopy, we may represent the ideal, coherent G LAUBER state by a pure number state |N , where N represents the average number of photons N per mode according to (2.84). With this – extremely good – approximation the derivation and application of optical transition probabilities given below can be accomplished without any mathematical difficulties. Although the G LAUBER states considered here refer to a single occupied mode only, they provide a good description for a sufficiently intense, quasimonochromatic and well collimated radiation field, such as a laser beam (one may even adapt the modes (2.113) suitably). Quantitatively, we derive the expectation value of the electric field operator (2.112) with the help of (2.119) ωk ikr ∗ ∗ −ikr . (2.123) with Ck = α|E|α = iCk αe · e − α e · e 2L3 ε0 At very high (classical) intensities when representing a G LAUBER states by a single photon number state with N = N , with (2.121) we can set with sufficient accuracy √ √ |α| = N N N + 1. (2.124) Comparing this to the spatial part of the classical field according to (1.35) i E(r) = E0 eeikr − e∗ e−ikr 2

(2.125)

2.3


117

we can relate the amplitude E0 = 2Ck α of the classical, single mode field to the number of photons in the mode: √ √ I0 3 |E0 | N N + 1 |α| = = (2.126) L . 2Ck cω In the last step we have used the standard relation I0 = ε0 c|E0 |2 /2 between intensity and field amplitude. We mention that this relation is equivalent to (2.86), with N being the number of photons in the (presently thus defined) coherence volume L3 . In the general case of radiation with a finite bandwidth and divergence angle, N has to be identified with (2.84).

2.3.4

Addendum for Multi-Mode States

As discussed above the average number of photon per mode in a typical laser beam may be very high and a classical, coherent radiation field will be described by G LAUBER states. However, in principle the photon number states |0, |1, . . . , |N . . . can be found also with quite different populations – and thus represent different coherence properties of the radiation field. This is a key theme of modern quantum optics. Here we just add a few remarks relevant to multi-mode states as needed to describe any realistic radiation field in some detail. These states are typically written as products of single particle states.12 As the most simple case we discuss here only products of pure number states which by the arguments given in the preceding subsection can be a valid description of a quasi-monochromatic laser beam: {Nkq } = |N1 |N2 . . . |Ni | . . . = |N1 N2 . . . Ni . . .. (2.127) Such a state describes an electromagnetic field with N1 , N2 , . . . , Ni , . . . photons in modes characterized by k 1 , k 2 , . . . , k i , . . . and polarization vectors e1 , e2 , . . . , ei , . . . The respective creation and annihilation operators generate or annihilate one photon in a specific mode according to the scheme aˆ k+i qi {Nkq } = aˆ k+i qi |N1 . . . Ni . . . = Ni + 1|N1 . . . Ni + 1 . . . aˆ k i qi {Nkq } = aˆ ki qi |N1 . . . Ni . . . = Ni |N1 . . . Ni − 1 . . . aˆ ki qi |N1 . . . 0 . . . ≡ 0.

(2.128)

Since each of these operators acts only onto one of the modes, the commutation rule (2.97) is now generalized by + + aˆ kq , aˆ k+ q = δkk δqq while [aˆ kq , aˆ k q ] = aˆ kq (2.129) , aˆ k q ≡ 0. 12 A different situation is encountered with so called entangled states – an interesting subject but beyond our present scope. See also Appendix E.3 in Vol. 1.

118

2


The multi-mode number states are orthonormalized: N1 N2 . . . Ni . . . |N1 N2 . . . Ni . . . = δN1 N1 δN2 N2 . . . δNi Ni . . . .

(2.130)

Their total Hamiltonian is given by (2.114). We now have all necessary tools to describe a more or less quasi-monochromatic light beam with small or even larger divergence in quantum mechanical terms. We must, however, keep in mind – as described in detail in Sects. 2.1.1–2.1.7 – that an arbitrary classical radiation field is not a simple linear superposition of plane waves. Neither can we describe the quantized radiation field by a linear superposition of |{Nkq } states – except in the special case of a fully coherent state. As in the classical case (Sect. 2.1.4), the field is defined by a distribution E − (k)E + (k ) of amplitudes and frequencies (or wave vectors). In the classical case this distribution was found to be diagonal in k. This also holds for the quantum description. It will be sufficient to specify the probability amplitudes for the states |N1 00 . . . 0, |0N2 0 . . . 0, |00N3 . . . 0, . . . , |00 . . . Ni . . . 0, . . .

(2.131)

in a range of relevant modes k i qi where Ni refers to the average number of photons in that particular mode. The proper quantum mechanical tool for the necessary book keeping is the density matrix. Chapter 9 will give an introduction into the density matrix formalism. Quantitative treatments can become rather involved (see e.g. M UKAMEL 1999).

2.3.5

Interaction Hamiltonian for Dipole Transitions

As in the semiclassical treatment the interaction energy is dominated by the dipole energy. As an excellent approximation one neglects again the wavelength dependence of the electric field on the position r within the atom, since at least for the IR, VIS and UV spectral range the wavelength is large compared to atomic dimensions, k · r 1. Thus, exp(ik · r) 1 and we shall limit the discussion here exclusively to electric dipole (E1) transitions (a generalization, if needed, can be obtained following the corresponding considerations in Sect. 5.4, Vol. 1). As already emphasized, we use the S CHRÖDINGER picture with a time independent perturbation,13 and translate the semiclassical treatment of radiation induced transitions (Chap. 4 in Vol. 1) into the fully quantized description. With the field operator (2.112) the interaction Hamiltonian between atom and field is given in

13 One

could also use the H EISENBERG picture with a time dependent field operator and time independent states. The final result would be the same.

2.3


119

Fig. 2.18 Schematic of a level system

ħΔω ħωba

|b

>

ħω

|a

>

analogy to (4.55), Vol. 1 by14 (r) = er · E = −D · E =i U

k

=i

eCk Daˆ k − D† aˆ k+ ,

k

ωk er · eaˆ k − e∗ aˆ k+ 3 2L ε0 ωk with Ck = 2L3 ε0

and the dipole transition operators D=r ·e

(2.132)

and D† = r · e ∗ ,

for absorption and emission of a photon with polarization e, respectively. For convenience of writing we have here again pulled the elementary charge e out from the electron dipole moment D = −er.15 As expected, in contrast to (4.55), Vol. 1 the interaction Hamiltonian (2.132) is now time independent and documents energy conservation: in this fully quantized picture energy is simply exchanged between atomic and photonic states. , we point Before we derive the matrix elements of the interaction Hamiltonian U out that each relevant mode k, e in the sum (2.132) contains two parts: the first part (with aˆ k ) destroys a photon (in a mode with the wave vector k and the polarization vector e) and corresponds to absorption, while the second part (with aˆ k+ ) generates a photon (in a corresponding mode) and describes emission. We recall that in the semiclassical description of the electromagnetic field (2.87) these two terms correspond to the positive and negative frequency part, respectively. Without interaction between field and atomic system the eigenstates of the total system may be written as product states |ψ; {Ni } of atomic states |ψ and photon states |N according to (2.127). If the spectral intensity distribution of the radiation is close to a resonance of the atomic system – as sketched in Fig. 2.18 – a two level system is a usually a good approximation. The eigenfunction of |ψ then corresponds to either upper or 14 As in the semiclassical description we apply the dipole length approximation. In dipole velocity approximation the quantized perturbation reads e v (r) = i U p · eaˆ k + e∗ aˆ k+ . 2L3 ε0 ωk me ke

With exact eigenfunctions both approximations lead to the same transition probabilities. 15 If

more than one active electrons are involved, one has to replace the position vector r by a sum over all r i for the active electrons.

120

2


lower state, |b and |a, respectively, while N1 , N2 . . . Ni . . . defines the number of photons in the modes k 1 , k 2 . . . k i . . . The matrix elements of the interaction Hamiltonian (2.132) are given by: N1 N2 . . . Ni . . . U a; {Ni } = b; N1 N2 . . . Ni . . . |er · E|a; b; Ni eCk Dba N1 . . . Ni . . . |aˆ k |N1 . . . Ni . . . =i (2.133) k

† Dba N1 . . . Ni . . . |aˆ k+ |N1 . . . Ni . . . . − The latter rearrangement is possible since r acts only onto the atomic part, while act only onto the photon part of the system. The matrix aˆ k and aˆ k+ (and thus E) elements of the dipole transition operators for absorption and emission are the same as (4.57), Vol. 1, elaborated for the semiclassical treatment in Sect. 4.3.4, Vol. 1: † and Dab = r ab · e∗ = r ∗ba · e∗ = D∗ba with r ba = b|r|a = ψb∗ (r)rψa (r)d3 r = r ∗ab .

Dba = r ba · e

(2.134)

Since r has odd parity, |b and |a must have different parity. According to (2.128), the operators aˆ k+ and aˆ k create or annihilate a photon of one specific mode and polarization. And the photon states are orthogonal according to (2.130). The matrix is thus zero unless N = Ni ± 1 holds for one of the photon states, element of the U i while all others are the same before and after the transition. For one single occupied mode, i.e. for Nke photons with momentum k and polarization e, the nonvanishing matrix elements (2.133) may be written in compact form: |aNke = b| D|aNke − 1|iaˆ k |Nke bNke − 1|U Dba eCk Nke = i

(2.135a)

|bNke = a| aNke − 1|U D|bNke − 1|iaˆ k |Nke Dab eCk Nke = i

(2.135b)

|bNke = a| aNke + 1|U D† |bNke + 1| − iaˆ k+ |Nke † Dab eCk Nke + 1 = −i

(2.135c)

|aNke = b| bNke + 1|U D† |aNke + 1| − iaˆ k+ |Nke † Dba eCk Nke + 1. = −i

(2.135d)

Here Ck is the field normalization constant used in (2.132), originating from proper calibration of the total field energy (2.114). When deriving (2.135a)–(2.135d) from (2.133) we have used (2.106) and (2.107). The somewhat abstract number Nke of photons per mode can be related by (2.84) to the (measurable) spectral intensity distribution I˜(ωk ).

2.3


absorption ^

〈b –1|U |a

|b 〉

â

〉

ħω

(a)

^

|a 〉

(c)

|b 〉

â

(b)

â+

emission

〈a +1|U |b 〉

^

〈a –1|U |b 〉

121

ħω

^

〈b +1|U |a |a 〉

ħω

â+

〉

(d)

ħω

|b 〉 |a 〉 |b 〉 |a 〉

Fig. 2.19 Interaction matrix elements between atom and field, schematically; (a)–(d) refer to equations (2.135a)–(2.135d), respectively

The physical interpretation of the matrix elements is schematically explained in Fig. 2.19. As summarized for the semiclassical treatment in (4.53), Vol. 1, only two of these matrix elements are relevant within the framework of 1st order perturbation theory: • Figure 2.19(a) symbolizes absorption (annihilation) of a photon, accompanied by excitation of the system from a lower state |a into an upper state |b according to (2.135a), • Figure 2.19(c) symbolizes emission (creation) of a photon, accompanied by de-excitation of the system from an upper state |b into a lower state |a according to (2.135c). The other two nonvanishing matrix elements correspond to so called “virtual deexcitation” (Fig. 2.19(b)) and “virtual excitation” (Fig. 2.19(d)) processes by absorption and emission of a photon, respectively. These processes are not energy conserving and do not play a role in 1st order perturbation theory, as we shall see in a moment. However, they are of crucial importance in the description of higher order processes, such as multi-photon excitation or ionization in strong fields, as well as for R AMAN scattering and other nonlinear processes. We finally note that – if necessary – Dba may be modified appropriately as in the semiclassical description to describe other types of transitions, such as E2 and M1.

2.3.6

Perturbation Theory and Spontaneous Emission

We shall use again 1st order perturbation theory to describe E1 transitions. Even though this approach has its limitations, to be discussed at the end of this section, we shall be able now to derive a rate for spontaneous emission, which was not possible with the semiclassical approach. In any case, the following, fully quantized treatment of radiation induced transitions will form the basis for later, more rigorous treatments, e.g. in Chap. 10. We consider an effective two level system with the atomic states |a and |b A be being nearly in resonance with the radiation as indicated in Fig. 2.18. Let H

122

2


F that of the free field (2.114). The stationary the Hamiltonian for the free atom, H S CHRÖDINGER equation for the unperturbed atom is A |b = ωb |b H

A |a = ωa |a, and H

(2.136)

with a transition frequency ωba = ωb − ωa > 0, while F |Nke = Nke ωk |Nke H

(2.137)

describes a state of Nke photons in a mode k with polarization e. We start our derivation again with a single occupied field mode and sum later on over all field modes. That is possible without problems due to the orthogonality relation (2.130). The corresponding time dependent S CHRÖDINGER equation i

∂|ψ(t) A + H 0 + U )ψ(t) = (H F + U )ψ(t) = (H ∂t

(2.138)

according to (2.132). has to be solved with the interaction U A + H F + U for atom, field and interaction Note that the full Hamiltonian H is still time independent. Hence, energy conservation holds in this fully quantized S CHRÖDINGER picture – in contrast to the semiclassical radiation theory (4.40), Vol. 1, where the interaction was time dependent. Thus, we have to find stationary solutions of (2.138). We shall do this indeed in Chap. 10. For the moment we are simply interested in all possible transitions which are induced by switching the interaction on. Quite generally, one may expand |ψ(t) into a series of unperturbed eigenfunctions of the system: ψ(t) = cj N (t)|j N e−i(ωj +N ω)t . (2.139) Nj

Here cj N is the probability amplitude for finding N photons in the field while the atom is found in state |j . For simplicity of writing we have dropped again the indices k and e for the photon states N and for the angular frequency ω of the field. We insert (2.139) into (2.138), multiply from the left with bN | or aN |, and obtain two sets of differential equations for cbN and ca N , respectively: dcbN (t) i |aN ei[(N −N )ω+ωba ]t =− ca N bN |U dt N

i dca N (t) |bN ei[(N −N )ω−ωba ]t . =− cbN aN |U dt

(2.140)

N

We have exploited the fact that only matrix elements between different atomic states are non-zero. We insert now the matrix elements (2.135a)–(2.135d). Since only the terms with N = N ± 1 are non-zero, two types of exponential factors appear in (2.140):

2.3


123

• energy conserving terms exp[±i(ω − ωba )t] and • non-energy conserving terms with exp[±i(ω + ωba )t]. As we have seen already in Sect. 4.3.2, Vol. 1, in a perturbation treatment these terms are weighted with resonance denominators of the type 1/(ω − ωba ) and 1/(ω + ωba ). We now focus on the nearly resonant situation |ω| = |ω − ωba | ωba ,

(2.141)

but allow nevertheless for small detuning ω, as indicated in Fig. 2.18. In typical spectroscopic applications we shall have to account for detuning on the order of 108 s−1 , which are to be compared with transition frequencies on the order of 1015 s−1 . Thus, to a very good approximation the non-resonant terms 1/(ω + ωba ) can be neglected. This approximation is called rotating wave approximation (RWA), since the terms exp[±i(ω − ωba )t] imply, so to say, that the system follows the field in phase, while the others rotate in the opposite sense and thus average out with time.16 Consequently, this simplifies (2.140) substantially. Inserting (2.135a)–(2.135d) for the matrix elements this leads for the two level system to a simple set of two coupled equations eCk √ N + 1 Dba ca N +1 ei(ωba −ω)t eCk √ N + 1 D∗ba cbN e−i(ωba −ω)t , c˙a N +1 = − c˙bN =

(2.142) (2.143)

√ with Ck ∝ ω being the field normalizing constant in (2.132). To derive the absorption probability we now assume, as in the semiclassical case, that at time t = 0 all atoms are in the lower state |a. We also assume that the photons in the mode k with polarization e are represented sufficiently well by a photon number state |N . Thus, our initial conditions are ca N (0) = 1

and cj N (0) ≡ 0 for all j, N = a, N .

(2.144)

In 1st order perturbation theory one assumes in addition that ca N 1 remains constant. Thus, (2.142) may be integrated directly to obtain the probability amplitude cbN −1 (t) for finding |b N − 1, i.e. for a transition of the system into the excited state |b by annihilation of a photon. During this process one of the originally N photons in mode k is absorbed, so that in complete analogy to the classical case (4.58), Vol. 1 we have √ ei(ωba −ω)t − 1 eCk N cbN −1 (t) = . (2.145) Dba i(ωba − ω) 16 Originally

this terminology was coined by microwave and radio frequency spectroscopy (EPR and NMR), where this phase reflects indeed a real physical rotation of the spin, induced by the exciting field.

124

2


The transition probability per unit of time is again |cbN −1 (t)|2 /t. This leads to a transition rate (N )

dRba k =

2π 2 2 πωk e2 | e Ck | Dba |2 Nke g(ωk ) = 3 Dba |2 Nke g(ωk ) 2 L ε0

(2.146)

induced by the Nke photons in the mode. We have now reintroduced the indices for polarization e and wave vector k of the radiation. With g(ωk ) we identify again the line profile as introduced in Sect. 4.3.5, Vol. 1. Integrated over all frequencies it is normalized to unity. Completely equivalent one assumes for the de-excitation process |b → |a initial conditions cbN (0) = 1 and cj N (0) ≡ 0 for all j, N = a, N .

(2.147)

By integration of (2.143) one derives the probability amplitude for finding the system in a state |aN + 1. From this the transition rate for a ← b by emission of a photon into the mode k, e is obtained: (N )

dRab k =

πωk e2 | Dab |2 (Nke + 1)g(ωk ). L3 ε0

(2.148)

So far the derivation was completely analogous to the semiclassical approximation. Now we have to recall, however, that there are dmωe modes per frequency interval dωk and solid angle element dΩ, with dmωe given by (2.73). We thus find the absorption or emission probability into given solid angle dΩ element by integration over all available angular frequencies of the electromagnetic radiation inducing the transition. This leads to a rate +∞ ω2 πe2 L3 (N k ) dωk | D |2 Nke ωk g(ωk ) dRba = dRba dmωk e = 3 2 dΩ 3 ba L ε0 (2πc) −∞ = dΩ

2 πe2 ωba | Dba |2 Nke ωba . ε0 2 (2πc)3

(2.149)

In the last step we have assumed that the line profile of the transitions is very narrow, g(ωk ) = δ(ωk − ωba ), compared to the spectral bandwidth of the radiation which induces the transition. We shall present in Chap. 10 a simple recipe to modify the result for narrow band radiation. We see now, that in the final step the normalization volume L3 has happily dropped out, since normalization of the field operator cancels versus mode density. We have written (2.149) in a manner to show the essential ingredients: apart from the numerical prefactor we recognize the mode density per angle and volume Dba , and (2.75), the dipole transition moment projected on the polarization (2.134), the total photon energy Nke ωba , with Nke being the number of photons in the mode k, e prior to absorption or emission with an angular frequency corresponding to the transition frequency ωba .

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For emission we obtain correspondingly dRab = dΩ

2 πe2 ωba | Dab |2 (Nke + 1)ωba . ε0 2 (2πc)3

(2.150)

We point out that this differs from (2.149), valid for absorption, by the replacement Nke → (Nke + 1). As we shall see in a moment, this is crucial for spontaneous emission. According to (2.149) it is evident that the atom can only be excited from the lower state |a into the upper state |b if Nke > 0 – that is, if at least one photon of the frequency ωba is present in the field mode k, e. The absorption process reduces this number of photons by exactly one. Let us assume now that the number of photons √ Nke in the mode k, e is very high – so high that the relative uncertainty 1/ Nke about that number is negligible. Then Nke may be set equal to its average value according to (2.84) for all relevant modes. We insert this value – as indicated by [ ] – into (2.149):

˜ 2 I (ωba ) (2πc)3 πe2 ωba 2 Rba = ωba dΩ | D | ba 2 ωba cδΩ ωba ε0 2 (2πc)3 beam ˜ πe2 2 I (ωba ) = dΩ| D | . ba δΩ ε0 c2 beam We point out that the mode density and the single photon energy cancel out. The spectral intensity I˜(ωba ) is a measurable source parameter. E.g., its value is given by (2.21) for a laser tuned into resonance ωba – if its overall bandwidth is much larger than the linewidth of the transition. We now recall that | Dba |2 = |r ba · e|2 depends on the propagation direction of the light. For any reasonable laser beam, well collimated to δΩ 1, we may consider |r ba · e|2 to be constant for all populated wave vectors k in the beam. Under such conditions the angular dependence of I˜(ωba )/δΩ may be considered a delta function (beam) and the integration of (2.149) over all solid angles yields the total absorption probability Rba =

˜ 4π 2 α I˜(ωba ) πe2 2 I (ωba ) = | | D | Dba |2 I˜(ωba ) = Bba ba 2 c c ε0

(2.151)

with the fine structure constant α = e2 /4πε0 c and Bba = 4π 2 αc| Dba |2 /, the E INSTEIN coefficient for the specific sub-transition b ← a induced with polarization e. We note that this expression is completely identical to (4.63), Vol. 1, derived in our previous semiclassical treatment. The rate Rba (dimension T−1 ) is – as already mentioned earlier – by a factor of 3 larger than usually given in textbooks, since we have derived the expression for a laser beam, rather than for isotropic radiation. Of particular interest is now the emission of a photon in the transition |b → |a. The factor (Nke + 1) in (2.150) suggests to distinguish between induced and spontaneous emission: the induced emission probability is taken proportional to Nke ,

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that is to the number of photons in the relevant modes prior to the emission process. A comparison of (2.149) and (2.150) shows, that this probability is identical to the absorption probability. Thus, for one well defined upper and one well defined lower state |b and |a Rab = Rba .

(2.152)

We note in passing, that the photons generated by induced emission appear by definition exactly in the mode by which they are created – as we have partitioned the whole radiation field (in k space) into well defined, discrete modes and treated them independently prior to integration. This confirms what in the semiclassical treatment has simply be assumed: radiation due to induced emission agrees in frequency and direction exactly with the inducing field. The factor (Nke + 1) in (2.150) implies that emission may occur even if there is initially no field present, or more precisely, if initially the field is describe by the vacuum state with Nke = 0 (the initial state being |b 0). There is an additional finite, albeit small probability for the de-excitation process |b → |a, involving the emission of a photon. This transition may be seen as induced by the vacuum field. This notion may appear somewhat difficult to accept and we shall come back to it in the next subsection. In any case, according to (2.150) the resulting spontaneous transition probability for emission of a photon into a mode k with a frequency ωba and polarization e into a solid angle dΩ is: (spont)

dRab

=

3 2 αωba πe2 ωba 2 | D | ω dΩ = | Dab |2 dΩ. ab ba ε0 2 (2πc)3 2πc2

(2.153)

We recall that we had “gleaned” this expression as (4.67), Vol. 1 for spontaneous emission earlier on. Above derivation supplements the proof. However, we cannot confine the derivation to just one mode: all empty modes of frequency ωba do indeed contribute to the process. The angular distribution of this radiation and its polarization is described by | Dba |2 = |r ba · e|2 (for details we refer to Sect. 4.5 in Vol. 1). While our earlier treatment of spontaneous emission was essentially guesswork, it is now firmly based on the quantized radiation field. The integration over all solid angles gives again the characteristic factor 8π/3|r ba |2 , so that the total spontaneous emission rate becomes 4α 1 (spont) (spont) 3 Rab = dRab = 2 |r ba |2 ωba = Aab = . (2.154) τ 3c ab 4π e Aab refers here to spontaneous decay of a specific excited sub-state |b into the specific lower sub-state |a and fully confirms (4.109), Vol. 1, onto which we have based up to now all discussions and applications of the A coefficients. To obtain the overall natural lifetime τnat of a level, one has to sum this expression also over all final states. Detailed evaluation of the A and B coefficients for specific transitions

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127

and the generalization to degenerate levels has already been presented in Sect. 4.4, Vol. 1. We thus have achieved not less than an ab initio derivation of spontaneous emission and of the relation between the E INSTEIN coefficients. We can now trace the origin of the well known ω3 dependence of spontaneous emission: It arises from the mode density (2.75), which is ω2 /(2πc3 ) per unit volume and per angular frequency – and from the photon energy ω. In the treatment of induced probabilities both terms cancel against the number of photons per mode and the spectral intensity per frequency interval. Thus, the induced rate (2.151) does not depend directly on ω – except through the resonance condition g(ωk ) = δ(ωk − ωba ) in (2.149). Finally, we also have to realize the limitations of the present treatment for emission and absorption of electromagnetic radiation: we only have used 1st order perturbation theory. For induced processes we shall correct this to some extend in Chap. 10. We shall show there that the set of coupled equations (2.142) and (2.143) may be solved exactly, as long as the RWA holds, and spontaneous emission can be neglected. For intense (but not too intense) laser fields and times short compared to the natural lifetime this is indeed an excellent approximation. For very high intensities – as available today with state-of-the-art ultra fast, high power laser systems, the rotating wave approximation breaks down, and similarly perturbative approaches are of limited value only. Hence, special strong field approximations or brute force numerical methods must be applied. As for spontaneous emission, it is of fundamental importance for any more rigorous treatment of radiative problems, and warrants further efforts beyond 1st order perturbation theory. The problem is, that the initial conditions (2.147) are, at a closer look, not strictly valid if spontaneous emission is to be included: All empty modes are present at time zero. It is important to realize that these empty modes are not nothing, but represent the vacuum field which is (almost) always present. Indeed, the vacuum field associated with these many unoccupied modes close to resonance leads to a broadening of the excited states which we know as natural linewidth. A quantitative treatment can be achieved in 2nd order perturbation theory – but is somewhat involved, and we refrain from presenting it here. The result is, as already assumed in Chap. 5, Vol. 1, that g(ω) in (2.146) and (2.148) can no longer be treated as a δ-function. Rather, it has to be described by a L ORENTZ profile with a linewidth (FWHM) ωnat = Aab = 1/τnat . With today’s techniques the bandwidth of lasers used in spectroscopy may easily be kept much below that value. This implies that also our assumptions for deriving the relevant expression (2.151) for induced processes do no longer hold. However, as we shall show in Chap. 10 this particular problem may be cured by a small modification.

2.3.7

Spontaneous Emission in a Cavity

In order to obtain a quantized form of the radiation field we have introduced a very large but finite normalization volume. This has led us to discrete radiation modes – still infinitely many, but countable. And all these modes ‘own’ a characteristic vac-

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uum field. It is this very vacuum field that we hold responsible for spontaneous emission. Many other physical phenomena are also caused or influenced by the vacuum field (one speaks of radiative corrections). We recall the L AMB shift or the g − 2 anomaly of the electron magnetic moment treated in Chap. 6, Vol. 1, for which radiative corrections are held responsible. Nevertheless, the vacuum state is not a really trivial concept. It is by no means empty space, its eigenenergy being hωk /2 according to (2.114) – for each mode k. Quantum electrodynamics deals with the problem of this infinite energy by its specific recipe for “re-normalization”. But of course, one might pose the question: How real is this vacuum field? Is it perhaps just a mathematical construct made to give the right answers for spontaneous emission? What may happen, if one forcefully chooses experimental conditions so that the normalization volume cannot be made infinitively large? What if we confine any potential radiation to a small volume and let it interact there with an excited atom? What does “vacuum state” mean in such a case? The idea of such an experiment has been around for some time (see e.g. P URCELL 1946; K LEPPNER 1981). However, it became feasible only by modern lasers and sophisticated experimental techniques. The first experiment of this type was performed by G OY et al. (1983) – and finally led, loosely speaking, to the N OBEL prize for H AROCHE and W INELAND (2012). They studied Na RYDBERG atoms, prepared by two photon resonant excitation in the 23s state, and investigated its spontaneous decay to the 22p state in a microwave cavity. The dipole transition moment between such high n levels is very high, while at the same time the spontaneous transition probability in free space is very small, being proportional to ∝ ν 3 . In the case discussed here the transition frequency is ν = 341 GHz (as one easily verifies with the quantum defects of Na given in Table 3.4, Vol. 1). The spontaneous decay rate between the 23s and 22p levels is only τ22p23s = 150 s−1 . At this transition wavelengths (λ = 0.88 mm) super-conducting microwave resonators with extremely high finesse can be built, through which an atomic beam can pass without problems. Figure 2.20(a) shows a very schematic summary of the experimental setup. A low density sodium beam passes through the microwave cavity where the atoms are excited in a two photon process by 2 collinear, pulsed (5 ns) dye laser beams, entering the resonator perpendicular to the Na-beam and to the resonator mode. The RYD BERG atoms are detected by field ionization in a parallel plate capacitor to which (after the laser pulse) a ramp voltage is applied as indicated in Fig. 2.20(b). The ionization process is monitored very efficiently by detecting the ejected electrons with an electron multiplier. Atoms with the lower ionization potential WI (23s) are ionized at a lower field strength (i.e. earlier) than those with higher ionization potential WI (22p). In the detected ionization signal, Fig. 2.20(c), atoms in the 23s state are recorded first, 22p atoms appear later. The black signal trace is taken with the cavity out of resonance for the 22p ← 23s transition. The cavity may be tuned into resonance mechanically, but fine adjustment is done by a small electric field in the cavity which can S TARK shift the 22p levels slightly. The actual experiment is carried out with only a few (1–3) excited atoms in the cavity, so that they do not influence each other. The red line shows the signal with the cavity tuned into reso-


(a) tuning LHe-cooled (5.7K) Nb microwave cavity

ramp voltage

WI

e–

electron multiplier

22p

245cm-1

1

(b)

23s

234cm-1

electric field

Na beam

pulsed laser beams, perpendicular

129 ramp voltage

2.3

2

3

t ramp / μs cavity in resonance

(c)

cavity off resonance 23 s

22p

Fig. 2.20 Experiment of G OY et al. (1983) documenting spontaneous transitions between Na 23s → 22p levels in a microwave cavity. (a) Setup very schematically, (b) ramp voltage tuning to detect 23s or 22p levels, (c) experimental signal taken from Fig. 3a in G OY et al. (1983), showing the signal measured when the cavity is off resonance (black line) and on resonance (red line)

nance: a surprisingly intense signal originates from atoms in the 22p state – which is attributed to spontaneous emission in the cavity. This is fascinating result! To fully understand it we first have a closer look at the cavity. It is made of very precisely machined and highly polished, niobium spheres with 20 mm diameter and 26 mm radius of curvature, arranged in nearly confocal configuration at L = 25 mm distance. The Gaussian mode sustained by the cavity at λ = 0.88 mm has a waist w = 1.9 mm and a total volume of Vcav = Lπw 2 /4 = 70 mm3 . Nb becomes super-conducting at 9.2 K and the cavity is liquid He cooled to ca. 5.7 K. This ensures that the surface of the cavity is highly conducting and together with very good polishing this leads to a very high quality factor Q of the cavity (see Eq. (1.11)) on the order of 106 . In addition, the cooling ensures that black body background radiation cannot lead to induced transitions: according to (1.63), Vol. 1, B OSE -E INSTEIN statistics gives a population of the N = 1 mode of [exp(ω/kB T ) − 1]−1 0.06 relative to the vacuum state N = 0 so that radiation induced processes can be neglected compared to spontaneous transitions which are caused by the vacuum field. We emphasize the fact that no external or background microwave field is involved in this experiment. The atoms spent only about 2 µs in the resonant cavity mode. In the free field case this would lead to a maximum of 150 s−1 ×2 µs 3 × 10−4 transitions. How then can we understand the observed enhancement of spontaneous emission by about at least 5 orders of magnitude? From our derivation of spontaneous emission in the free field case we recall now, that one crucial parameter was the mode density per unit volume, according to (2.75) 2 /(2πc)3 . This has entered directly into the spontaneous emission rate ρfree (ω) = ωba (2.153). In the final step, integration over all angles leads to multiplication by a

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factor of 8π/3, and another factor of 2 for the two possible polarization modes. We now compare this to the situation in the cavity. The mode density per angular frequency, polarization and volume is ρcav (ω) = 1 mode/(Vcav ωr ), where Vcav is the cavity volume and ωr = ωr /Q with the Q factor of he cavity according to (1.11). Integration over solid angles is obsolete in this case, since the cavity sustains only two modes (of different polarization) which are resonant with the transition. In summary, one has to replace 2 × (8π/3)ρfree (ω) in free space by 2 × ρcav (ω) in the resonant cavity case. Thus, an enhancement of the spontaneous radiation probability ρcav (ω) Q (2πc)3 1 3Q 3 = λ = 2 (8π/3)ρfree (ω) Vcav ωba (8π/3)ωba Vcav 4π is expected. In the present case this enhancement factor is on the order of 103 , so that the transition probability is changed from 150 s−1 to 3 × 105 s−1 so that during the passage time of 2 µs a substantial fraction of the Na atoms in the initial 23s state decays by spontaneous transitions into any of the 22p substates – as shown by the experimental result Fig. 2.20(c). In conclusion, this experiment documents that the vacuum field is not just a theoretical construct, it is real and can be manipulated in a finite cavity – leading to an observable modification of the spontaneous transition probability – also reflected in the respective “natural” lifetime. A number of additional question arise from these findings: e.g. what happens to the emitted photon? Can spontaneous emission also be suppressed? What about other effects caused by the vacuum field in a cavity? As it turns out, in the experiment described here the Q factor of the cavity is not high enough to store the emitted photon long enough for subsequent reabsorption. In the mean time, experiments have been reported in which oscillatory energy between the cavity and a single atom in it has been observed. And, yes, spontaneous emission can also be quenched in a cavity of the right dimensions if the vacuum field is not in resonance. Also changes of atomic energy levels and modifications of the L AMB shift have been observed. Cavity quantum electrodynamics has become a very active and productive topic of modern research as e.g. summarized in a nice review by WALTHER et al. (2006). We also mention that such effects play an important role in nano-optics, another area of cutting edge research. Section summary

• We have quantized the electromagnetic field, based on the preceding introduction of discrete, countable modes of the electromagnetic radiation field in a large, but finite normalization volume L3 . • To this end we have introduced in (2.92) new variables P and Q as linear combinations of the components E − and E + of the electric field. The mode energy was then recognized as formally equivalent to the harmonic oscillator, with P and Q being canonic conjugate coordinates.

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131

• We have applied the standard quantization scheme, replacing these variables P ] = i. Back transformation led by operators with the commutation rule [Q, + us to operators aˆ and a, ˆ the so called creation and annihilation operators, for which the commutation rule [a, ˆ aˆ + ] = 1 holds. • The Hamiltonian for a single mode of the free electromagnetic field is F = ωk aˆ + aˆ + 1/2 . H

• • •

•

•

•

•

• •

It has eigenstates |N , with N representing the number of photons in that particular mode of energy ωk . The mode energies correspond to those of the harmonic oscillator WN = (N + 12 )ωk . =aˆ + aˆ counts the number N of photons in a mode and the The operator N creation and annihilation operators aˆ + and a, ˆ respectively, were found to raise or decrease N by one. The total Hamiltonian of the free field is the sum of the Hamiltonians for all modes. This scheme of writing the Hamiltonian is called second quantization. + and E − of the electric field operator (2.112) are proThe components E + portional to aˆ and a, ˆ respectively. In the S CHRÖDINGER picture all operators are independent of time. All time dependence of the problem, being ∝ exp[−iWN t/], has been cast onto the states. We have briefly introduced G LAUBER states (2.115), representing a coherent electromagnetic field. They are eigenstates of the negative component of the − . Some of their properties have been described in electric field operator E Sect. 2.3.3. The interaction between the atom and field has been written in full analogy (r) = er · E, independent of time. Thus, to the semiclassical treatment as U the problem is now formulated energy conserving: energy is just exchanged between the atom and the electromagnetic field. Transition probabilities were treated again in 1st order perturbation theory. The number of photons per mode, Nke , appears in the transition rates. We have identified it with the mean number of photons per mode as derived in the previous section. Specifically, for de-excitation processes the rate was found to be proportional to Nke + 1. This implies that de-excitation is possible even if there is no external field. This has allowed us to derive a quantitative expression (2.154) for spontaneous emission. We thus have concluded that spontaneous transitions are induced by the vacuum field. The vacuum state is not simply nothing! Its energy is ωk /2 in each mode, and the vacuum field is a physically present field. Experimentally this may be verified in a high Q cavity, where the vacuum field can be manipulated. Spontaneous emission is found to be enhanced or suppressed in such a cavity, depending on whether the mode is on or off resonance with the transition.

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Acronyms and Terminology chemical-potential: ‘In statistical thermodynamics defined as the amount of energy or work that is necessary to change the number of particles of a system (by 1) without disturbing the equilibrium of the system’, see μ in Sect. 1.3.4, Vol. 1. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. E2: ‘Electric quadrupole’, transitions induced by the interaction of a quadrupolar charge distribution with the electromagnetic radiation field. EPR: ‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2 in Vol. 1). ESO: ‘European southern observatory’, in Chile, hosting four of today’s largest telescopes of the world, with 8.5 m diameter each. esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3 in Vol. 1). FPI: ‘FABRY-P ÉROT interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2 in Vol. 1). FWHM: ‘Full width at half maximum’. HBT: ‘Hanbury B ROWN and T WISS’, experiment, to determine the lateral correlation of light by a second-order interferometric measurement (see Sect. 2.1.6). IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. NIR: ‘Near infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1.4 µm according to ISO 21348 (2007). NMR: ‘Nuclear magnetic resonance’, spectroscopy, a rather universal spectroscopic method for identifying molecules (see Sect. 9.5.3 in Vol. 1). QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. RF: ‘Radio frequency’, range of the electromagnetic spectrum. Technically, one includes frequencies from 3 kHz up to 300 GHz or wavelengths from 100 km to 1 mm; ISO 21348 (2007) defines the RF wavelengths from 100 m to 0.1 mm; in spectroscopy RF usually refers to 100 kHz up to some GHz. RWA: ‘Rotating wave approximation’, allows to solve the coupled equations for a two level system in a strong electromagnetic field in closed analytical form (see Sect. 10.2.3). SHG: ‘Second harmonic generation’, doubling of a fundamental frequency, for infrared or visible light typically by methods of nonlinear optics. UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). VIS: ‘Visible’, spectral range of electromagnetic radiation. Wavelengths between 380 nm and 760 nm according to ISO 21348 (2007).

References

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VLBI: ‘Very long baseline interferometry’, worldwide network of radio telescopes for interferometry.

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3

Diatomic Molecules

The step from atom to molecule takes us onto a higher, significantly more complex level of understanding the structure of matter. Although the properties of atoms play an important role when describing molecules, we are faced with a significantly more intricate task than simply adding atomic properties. In this chapter we identify the most important molecular phenomena and introduce suitable methods for understanding them.

Overview

This chapter outlines the basic concepts of molecular physics as exemplified for diatomic molecules. We begin with some energetic considerations in Sect. 3.1 and introduce in Sect. 3.2 the B ORN -O PPENHEIMER approximation – the basis of all molecular physics. Molecular rotation and vibration are treated in Sect. 3.3 followed by an elaboration on dipole transitions in Sect. 3.4. Elements of the molecular orbital concept are presented in Sect. 3.5 while Sect. 3.6 focusses on angular momentum coupling and the famous H UND’s cases. While all the previous discussion was focussed on homonuclear molecules, the chapter ends by introducing the specificities of heteronuclear diatomic molecules in Sect. 3.7. The content of this chapter is essential for understanding most of the following ones – it is one of the keystones within these textbooks. The reader should thus familiarize him- or herself very thoroughly with all topics discussed here. A broad range of methods is available today for obtaining detailed information about the structure and dynamics of molecules. A key role plays spectroscopy in all spectral ranges: from radio frequency (NMR) via the microwave range (EPR and rotational spectroscopy), the FIR and NIR (vibrations), the visible and ultraviolet (electronic transitions) to finally X-ray spectroscopy (chemical shifts of inner shell transitions). Absorption and emission of electromagnetic radiation is used in a wide variety of methods and has led to a wealth of information without which our present understanding of molecules would not be conceivable. Additional information is obtained e.g. from X-ray and neutron diffraction, which give very direct insight into the spatial structure of molecules. Scattering © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_3

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Diatomic Molecules

experiments and more recently also ultrafast methods have revealed important structural information, but they also allow in addition to study the dynamics of molecular systems, i.e. the evolution of processes within or between molecules when interacting with each other or with photons. We shall treat these themes in some detail in Chaps. 6–8, and some special aspects will be mentioned in Chap. 10. In the present chapter we shall develop the key concepts for understanding molecules as exemplified for diatomic molecules, the simplest molecular systems. Based on these concepts we shall explain the most important experimental findings for homonuclear (A2 ) and heteronuclear (AB) diatomic molecules.

3.1

Characteristic Energies

The large difference in mass between electrons (me ) and atomic nuclei (M) me 10−3 . . . 10−5 M is the basis for the most important approximations in molecular physics. The relevant forces which keep molecules together and are responsible for their interactions and spectroscopic properties, are again – as for atoms – of purely electromagnetic nature.1 Since the C OULOMB force acting on electrons is identical to that acting on nuclei, the velocity of the atomic nuclei is typically much smaller than that of their electrons. Thus, nuclei stay essentially fixed in space while electrons move around them very fast. Equilibrium distances RAB are usually found in a rather narrow range from 0.075 nm to 0.18 nm. For example for the hydrogen molecule (H2 ) RHH = 0.07417 nm, for oxygen (O2 ) ROO = 0.12074 nm, for nitrogen (N2 ) RNN = 0.10976 nm and for carbon monoxide (CO) RCO = 0.11282 nm. In some few cases also larger distances are observed, as e.g. in K2 with RKK = 0.3923 nm or in I2 with RII = 0.2668 nm. Polyatomic molecules may have a broad variety of geometries. One example of quite specific symmetry is methane (CH4 ), its atoms being positioned in tetrahedral form as sketched in Fig. 3.1 with an equilibrium distance2 of RCH = (0.108595 ± 0.00003) nm. Fig. 3.1 Tetrahedral structure of methane

H H H

C H

1 Notwithstanding this fundamental fact, there is a notable, brave quest to observe influences of weak interaction in special instances by state-of-the-art high precision spectroscopy. 2 In chemical terminology the equilibrium distance is usually called “bond distance” or “bond length”. The value given here for CH4 is based on state-of-the-art quantum chemical calculations and comparison with precision infrared and R AMAN spectra of different isotopologues of CH4−x Dx according to S TANTON (1999).

3.1

Characteristic Energies

137

Fig. 3.2 Molecular coordinates for a diatomic molecule: The nuclear coordinates for atom A and B are indicated by capital letters, the coordinates (here r 1 , r 2 and r 3 ) of the individual electrons by lower case letters

z MA r3 O r2

RB MB

3.1.1

RA

r1 y

x

Hamiltonian

In this chapter we concentrate on diatomic molecules and start with deriving the Hamiltonian. We use atomic units (a.u.), and relative coordinates in respect of the centre of mass (O) of the system as illustrated in Fig. 3.2. Translational motion of the molecule as a whole does not play a role in the following discussion. It simply reflects the thermal motion in an ensemble. The kinetic energy of the two nuclei and the N electrons is3 me 2 Tn = − ∇R 2M¯

and Tr =

N 1 2 − ∇ ri , 2

(3.1)

i=1

respectively, with the electron coordinates r i , the internuclear distance R = R B − ¯ here of the two nuclei R A , and the reduced mass M, M A MB M¯ = . M A + MB

(3.2)

The C OULOMB potential for all particles is V (r, R) = −

N i=1

N N ZA ZB ZA ZB e2 − + + . (3.3) |r i − R A | |r i − R B | |r i − r k | R i=1

i,k=1 i
The sum of these energies gives the total Hamiltonian = Tn (R) + Tr (r) + V (r, R), H

(3.4)

where now r is considered to represent the entirety of the coordinates of all electrons. With this follows the S CHRÖDINGER equation (3.5) Tn (R) + Tr (r) + V (r, R) Ψ (r, R) = W Ψ (r, R). 3 We

ignore here the kinematic shift (which is anyhow problematic for multi-electron systems) and ¯ ¯ me . assume (with reasonable accuracy) m ¯ e = me M/(m e + M)

138

3

Diatomic Molecules

In this form it is also valid for polyatomic molecules, if R is taken to represent the entirety of all nuclear coordinates.

3.1.2

Electronic Energy

Let us try to obtain an estimate for the electronic energy of such a system. For clarity we switch back to SI units. We identify the extension of the electron orbitals r with the bond length R0 of the molecule. For molecular hydrogen (H2 ) as an example one finds r R0 0.074 nm. The uncertainty relation gives us an estimate for the momentum of the electrons p = /r, and an average kinetic energy Tr = p 2 /(2me ), which we may insert into the well known Virial theorem 2Tr = −Ve for −1/r potentials. Thus, the binding energy of the electrons can be estimated roughly as We = Tr + Ve = −Tr −p 2 /(2me ) = −2 / 2me r2 7 eV. Typically, We is found in an energy range of several eV, and the electronic spectra are expected in the ultraviolet and visible range of the electromagnetic spectrum – quite comparable to electronic transitions within atoms. More interesting are the nuclear degrees of freedom. We distinguish vibrations of the atomic nuclei within a molecule, relative to each other, and rotation of the whole nuclear structure – considered as fixed in a 0th order approximation.

3.1.3

Vibrational Energy

Since atoms and electrons are bound to each other, the force on the nuclei must be of the same order of magnitude as that on the electrons. Let us assume this force to be harmonic, i.e. F = −kR. The (angular) vibrational frequency of the nuclei is ¯ with the reduced mass M¯ on the order of the nuclear masses. then ωv = k/M, The frequency of the electron is given by a force constant of similar magnitude so √ energies of vibrational motion and electronic that ωe = k/me . The corresponding √ energy are Wv = ωv = k/M¯ and We = ωe = k/me , respectively. Thus, for nuclear motion and electronic energy we find Wv /We the ratio of vibrational ¯ and since me /M¯ 10−2 vibrational energies will be on the order of me /M, magnitude ¯ e 0.1 eV. Wv me /MW (3.6) Transitions are found in the infrared spectral range, e.g. for HCl at ν¯ = 1/λ = ν/c 3000 cm−1 = 0.37 eV or λ 3.33 µm.

3.2

B ORN O PPENHEIMER Approximation

139

Fig. 3.3 Rotation of a diatomic molecule

z

ω

O R0 ~

3.1.4

1Å

Rotational Energy

Let us consider a diatomic molecule, rotating around an axis perpendicular to the molecular axis as indicated in Fig. 3.3. We denote the angular momentum operator , the corresponding rotational quantum number is N . The of the molecule by N 2 = 22 is standard rules for angular momenta apply (Appendix B in Vol. 1). N the squared angular momentum in the first rotationally excited state (N = 1), the ¯ 2 Mr ¯ 2 . Thus, we estimate the moment of inertia in the ground state is I0 = MR 0 rotational energy: N 2 me = We 2 2 ¯ ¯ 2MR Mr M¯ 2

WN =

10−4 –10−3 We 1 meV to 10 meV.

(3.7)

The corresponding transitions are in the far infrared and microwave spectral range (¯ν = 1 cm−1 to 10 cm−1 ). Section summary

• We have introduced the molecular Hamiltonian (3.4) and derived some rough estimates for typical molecular energies, being several eV for the electronic part, 0.1 eV for the molecular vibration and in the 1 meV –10 meV region for rotational motion.

3.2


3.2.1

Molecular Potentials

The big difference between electronic and nuclear energies suggests to separate electronic and nuclear motion. This was first proposed by B ORN and O PPEN HEIMER (1927)who expanded the contributions of the nuclei to the Hamiltonian 4 ¯ They found that nuclear vibrations correspond to 2nd, rointo a series of me /M. tation to 4th order, while 1st and 3rd order disappear. The B ORN -O PPENHEIMER (BO) approximation turns out to be an excellent approximation and forms the basis of all molecular structure theory.

140

3

Diatomic Molecules

The first step is a product ansatz composed by the wave functions of electrons φ(r 1 , r 2 , . . . , r N ) and those of atomic nuclei ψ(R), Ψ (r 1 , r 2 , . . . , r N , R) = φ(r 1 , r 2 , r 3 , . . . , r N )ψ(R) ≡ φ(r)ψ(R),

(3.8)

where r refers to the entirety of electronic and R to all nuclear coordinates. The key idea is to consider – on a timescale relevant for the rapid motion of the electrons – the nuclei as fixed in space. The electronic part of the Hamiltonian (3.4) at a fixed value of R is − Tn (R) = Tr + V (r; R) , (el) = H (3.9) H and the corresponding S CHRÖDINGER equation is now written as (el) φγ (r; R) = Tr + V (r; R) φγ (r; R) = Vγ (R)φγ (r; R). H

(3.10)

For a start, BO approximation considers R simply as a ‘parameter’. The semicolon in the electronic wave function φγ (r; R) and in the potential V (r; R) is meant to emphasize this assumption. But in the back of our minds we have to remember that electronic and nuclear coordinates are, strictly speaking, nonseparable: in the potential V (r, R) given by (3.3), the r and R coordinates are nicely intertwined – except for the C OULOMB repulsion terms of which ZA ZB /R can be treated as a simple additive constant to the electronic energy. In the framework of the BO approximation, (3.10) is solved for each, fixed value of R independently. This leads to a set of electronic quantum numbers γ . The electronic energy Vγ (R) for each set of γ is a continuous function of R, called molecular potential (in the diatomic case), and potential hypersurface (in the general, multidimensional case). As the velocities of the electrons are orders of magnitude higher than those of the nuclei, the electrons orbit many times around the nuclei before these have moved significantly. Considering R as fixed is thus a very reasonable approximation for solving (3.10). All preceding discussion holds for any number of atoms. In the following we shall, however, concentrate for simplicity on the diatomic case, the generalization being straight forward. Then R stands for the relative coordinate R = R A − R B of the two atomic nuclei, and the electronic energy will depend exclusively on the internuclear distance R = |R|. By solving (3.10) for a range of R, one thus obtains for each γ a molecular potential Vγ (R), along with the corresponding electronic wave function φγ (r; R).4 In analogy to atoms, the indices α, β, γ define a set of quantum numbers which characterize the electronic charge cloud. The electronic wave functions φγ form again a complete, orthonormal set (3.11) φγ |φγ = φγ∗ (r; R)φγ (r; R)d3 r = δγ γ . 4 Depending

on the required accuracy, r refers usually only to the coordinates r i of the most important electrons, e.g. to the valence electrons, and one ignores the inner shells.

3.2


Fig. 3.4 Schematic overview of a molecular potential the bonding (full red line) and antibonding case (dashed line). At large internuclear distances R an attractive, albeit very weak polarization interaction is always dominant

141

COULOMB repulsion of nuclei ∝1/R Vγ (R) not overlapping electronic clouds => antibonding VAN DER WAALS attraction ∝- 1/ R 6

Vγ (∞)

overlapping electron clouds => molecular bonding

De united atoms

3.2.2

R0 equilibrium distance

R

separated atoms

General Form of Molecular Potentials

Electronic energies, for diatomic molecules called potentials Vγ (R), may be obtained from (3.10) for any set of electronic quantum numbers as a continuous function of R. We postpone to Sect. 3.5 a more detailed discussion about how exactly to compute them. For the moment we just give a qualitative overview. With (3.3) we may write the potentials derived from the electronic S CHRÖDINGER equation (3.10) Vγ (R) = Uγ (R) +

ZA ZB , R

(3.12)

now again in a.u. The C OULOMB repulsion of the nuclei dominates at short internuclear distances R while the term Uγ (R) is due to the electrons and is dominantly attractive if the molecular electron configuration is bonding. A typical molecular potential has several characteristic regions as illustrated in Fig. 3.4, for a bonding and for an antibonding diatomic potential. At large distances the polarizability of the atomic electron clouds leads to an attractive (albeit usually very weak) VAN DER WAALS potential ∝ −R −6 , as discussed in Sect. 8.3, Vol. 1. The repulsive C OULOMB interaction plays an important role in a systematic treatment of molecular orbitals – even though the limit of united atoms (more precisely atomic nuclei) is never reached. We shall come back to this in Sect. 3.5.4. In between these limiting cases the interaction of the electron shells is particularly strong. Its nature determines whether two atoms can form a molecule or not: as we shall proof in Sect. 3.5 a strong overlap of the electronic charge clouds leads to bonding, little or no overlap to antibonding potentials. As sketched in Fig. 3.4 the bond energy (or dissociation energy) De and the equilibrium distance R0 correspond to the potential minimum, zero energy referring here to completely separated atoms, Vγ (∞) = 0.

142

3

Diatomic Molecules

The behaviour of the interaction potentials discussed here is of general nature, and pertains also to large molecules and their coordinates.

3.2.3

Nuclear Wave Functions

In BO approximation, the nuclear wave functions can be evaluated without problems if the potentials are known. We rewrite the Hamiltonian (3.4): (el) . = − 1 ∇ 2R + H H 2M¯

(3.13)

The S CHRÖDINGER equation (3.5) for the molecule as a whole is then φγ (r i ; R)ψvN (R) = − me ∇ 2R φγ (r i ; R)ψvN (R) + H e φγ (r i ; R)ψvN (R) H 2M¯ = Wγ vN φγ (r i ; R)ψvN (R),

(3.14)

with Wγ vN = W being the total energy of the molecule. The indices vN refer to the nuclear motion as we shall see in a moment. Now let us have a detailed look at the components of this S CHRÖDINGER equation. The electronic term is indeed (el) acts (el) φγ ψvN = Vγ (R)φγ ψvN , since H given by (3.10), here explicitly as H as differential operator only on the electronic coordinates. However, the kinetic ¯ acts on both factors of the product energy operator for the nuclei, −∇R2 /(2M), φγ (r i ; R)ψvN (R)φγ ψvN . The S CHRÖDINGER equation (3.14) for the whole system thus becomes: me 2 (3.15) ∇ R ψvN + Vγ (R)φγ ψvN H φγ (r i ; R)ψvN (R) = φγ − 2M¯ me 2 me − ψvN ∇ R φγ + 2(∇ R φγ )(∇R ψvN ) (3.16) 2M¯ 2M¯ = Wγ vN φγ (r i ; R)ψvN (R). So far, the derivation is still completely correct. Now, in B ORN -O PPENHEIMER approximation one simply neglects ∇R φγ completely, i.e. the terms (3.16) are dropped: the electronic wave function φγ (r i ; R) changes only very little with the nuclear distance R (in particular so near equilibrium distance R0 ). Hence, ∇R φγ (r i ; R) is neglected, and even more so the second derivative. To obtain a more quantitative feeling we consider the following: The magnitude of ∇R φγ is of the same order of magnitude as ∇r i φγ , since the same regions of the molecule are involved in the evaluation of the gradients. For clarity we use now SI units, write the electron momentum pe = ∇r i φγ , and compare 2 2 2 2 p2 me me pe2 ∇ R φγ ∇ r i φγ e = Vγ , ¯ ¯ ¯ ¯ 2m 2M 2M 2M M M¯ e

3.2


143

i.e. we neglect an energy on the order of 10−3 . . . 10−5 Vγ . As a matter of fact, one finds that the B ORN -O PPENHEIMER approximation is a surprisingly excellent approximation, far beyond what one might expect from this estimate! Thus, we have to solve for the nuclear motion (now back to a.u.) 1 2 φγ − ∇ R ψvN + Vγ (R)φγ ψvN = Wγ vN φγ ψvN . 2M¯ By multiplying this from the left with φγ∗ , integrating over r, and using the orthogonality of the electronic wave functions (3.11), we obtain the S CHRÖDINGER equation for the nuclear wave function ψvN (R) and its eigenvalues Wγ vN : −

me 2 ∇ R ψvN (R) + Vγ (R)ψvN (R) = Wγ vN ψvN (R). 2M¯

The Hamiltonian for the nuclear motion is thus n = − me ∇ 2R + Vγ (R). H 2M¯

(3.17)

(3.18)

The R dependent eigenvalues Vγ (R) of the electronic S CHRÖDINGER equation (3.10) thus constitute the potential of the S CHRÖDINGER equation for the motion of the nuclei (3.17).

3.2.4

Harmonic Potential and Harmonic Oscillator

Our next goal is to understand the nuclear motion and describe it in quantum mechanical terms. Traditionally, several steps of approximation are taken: again one tries to separate different types of motion. A first step is to expand the potential into a TAILOR series and see how far the lowest order approximation leads. To study the oscillations of the molecule, the potential is expanded around its equilibrium distance R0 : 2 dVγ 1 2 d Vγ Vγ (R) = Vγ (R0 ) + (R − R0 ) + (R − R0 ) + ··· . 2 dR R=R0 2 dR R=R0 At the potential minimum dVγ /dR|R=R0 = 0, and, neglecting higher terms, we obtain a harmonic potential 1 Vγ (R) = Vγ (R0 ) + k(R − R0 )2 2

(3.19)

with the force constant k = d2 Vγ /dR 2 R=R . 0

According to classical mechanics such a potential leads to harmonic oscillations with an angular frequency ¯ ω0 = k/M. (3.20)

144

3

Fig. 3.5 Harmonic oscillator: Potential energy (full black line), total energy (dashed, black lines) and eigenfunctions Rv (R) for the vibrational sates v = 0 . . . 4 (full, red lines, shifted in height for clarity)

Diatomic Molecules

Wv / ħω0

v=4

4.5 v=3 3.5

v=2

2.5 v=1 1.5

v=0

0.5 -4

-2

0

2

(R - R0 ) / l

The harmonic oscillator is probably one of the best treated objects on all levels of physics education. We just summarize the essential results. In quantum mechanics one obtains the wave functions Rv (R) and the energy eigenvalues Wv from the one-dimensional S CHRÖDINGER equation:

¯ 2 Mω 2 d 2 0 2 − Rv (R) = Wv Rv (R). (R − R + ) (3.21) 0 2 2M¯ dR 2 Introducing a characteristic length l (typically between 0.1 and 0.25a0 ) ¯ 0 ) and setting x = (R − R0 )/ l, l = /(Mω one may write (3.21) in dimensionless form:

2 d 2Wv 2 + − x Rv (x) = 0. ω0 dx 2

(3.22)

(3.23)

Solutions are the H ERMITE functions hv (x) which are related to the H ERMITE polynomials Hv (x) of degree v by Rv (x) ≡ hv (x) = exp −x 2 /2 Hv (x). Table 3.1 H ERMITE functions for vibrational states v = 1 . . . 4

v 0 1

hv (x) 1 −x 2 /2 √ 4πe √ 2 −x 2 /2 √ 4 π xe

2

√ 1√ e−x

3

√ 1√ xe−x

4

2 /2

2 π

(2x 2 − 1)

2 /2

3 π

2

√1√ e−x 6 π

2 /2

(2x 2 − 3)

(4x 4 − 12x 2 + 3)

3.2


145

In normalized form they may be derived from dv (−1)v hv (x) = exp −x 2 . exp x 2 /2 √ v v dx 2 v! π

(3.24)

The corresponding energy eigenvalues of the harmonic oscillator are Wv = ω0 (v + 1/2)

for v = 0, 1, . . . .

(3.25)

The wave functions for the four lowest levels are sketched in Fig. 3.5 and summarized in Table 3.1. Note that they alternate between being symmetric and antisymmetric in respect of the equilibrium distance R0 . We also emphasize the finite extension of the ground state wave functions (v = 0, zero point oscillation), which is a pure G AUSS function. The corresponding zero point energy of the ground state is Wmin = ω0 /2. Figure 3.6 shows an example for a rather high vibrational quantum number v = 20. One sees that in this case the probability density at the boundaries increases substantially – corresponding fully to the longer time a classical harmonic oscillator spends at the classical turning point. Fig. 3.6 Harmonic oscillator in the v = 20 state: potential energy (full black lines on the left and right), total energy (dashed black line) and eigenfunction (red dashed). The full red line gives the square of the wave functions and hence the density probability as a function of internuclear distance R

3.2.5

Wv / ħω0

21.0

v = 20

20.5 -6 -4

-2

0

2

4

(R - R0 ) / l

M ORSE Potential

It is instructive to compare the ideal harmonic potential with a realistic molecule. We choose CO for whose ground state an experimentally very well determined, so called RYDBERG -K LEIN -R EES (RKR) potential is available (we shall come back to the RKR method in Sect. 3.4.6). In Fig. 3.7 the experimental data points are indicated by crosses. For each measured vibrational state one pair of such data point exists, one for the inner and one for the outer turning point. Some vibrational levels are indicated. The harmonic approximation of this potential, the dashed red line in Fig. 3.7(a), is obviously of very limited value, and only allows a description of the lowest vibrational levels (v = 0, 1). There are a number of approaches for an analytic approxi-

146

3

Diatomic Molecules

mation to reality. Often the potential introduced by P.M. M ORSE (1929) is used: (3.26) VM (R) = De e−2a(R−R0 ) − 2e−a(R−R0 ) . This M ORSE potential offers already some flexibility with three parameters De (bond energy), R0 (equilibrium distance) and a (some kind of stiffness of the potential). It reproduces the limits correctly and is rather simple to handle in computations. For CO the fit to the experimental data by a M ORSE potentials (full red line in Fig. 3.7) looks surprisingly good.5 As indicated in Fig. 3.7 one usually defines zero energy for the separated atoms in the electronic ground state, i.e. Vγ (∞) = 0, and obtains for the potential minimum Vγ (R0 ) = −De . Alternatively one also uses the form 2 VM (R) = De 1 − e−a(R−R0 ) , which differs from (3.26) just by a shift of zero energy to −De . The true ground state bond energy required to dissociate the molecule (from its vibrational ground state v = 0 as indicated in Fig. 3.7) is in any case D00 = De − ω0 /2. V(R) / eV

R / nm

0.10

0.14

0.18

0.22

0.26

0.3

0.10

0.14

0.18

0 harmonic

(a)

0

( ) -2

-2 Lennar -Jones

v = 28 -4

-4

v = 28 v = 27

-6

-6 De = 11.11eV

-8

-8 v =1

- 10

0 D0 =

10.91eV

-10

v=0

Fig. 3.7 Potential of the CO ground state. Crosses give the experimentally determined points of the RKR potential according to F LEMING and R AO (1972), M ANTZ et al. (1971). A M ORSE potential has been fitted to it (full, red line) with a bond energy De = 11.108 eV and an equilibrium distance R0 = 0.11282 nm. This potential may be compared to (dashed red lines) (a) a harmonic potential of equal vibrational frequency in the ground state or (b) a 12, 6 L ENNARD -J ONES potential 5 However,

experimental precision is so far advanced that the M ORSE potential does not suffice state-of-the-art requirements. The accuracy of the data points given in the literature is in some instances as good as 1 ppm.

3.2


147

¯ Table 3.2 Potential parameters for some characteristic diatomic molecules: Reduced mass M, equilibrium distance R0 , dissociation energy D00 = De − ω0 /2 in respect of the ground vibrational state, its vibrational energy ω0 , and its dipole momenta Dγ v Molecule

¯ u M/

R0 / nm

D00 / eV

ω0 / eV

H+ 2

0.504

0.1052

2.651

0.2714

H2

0.504

0.07414

4.478

0.5156

D2

1.0071

0.07415

4.556

0.37095

7 Li1 H

0.8812

0.15957

2.4287

0.16853

19.6256

1 H35 Cl

0.9796

0.12746

4.433

0.3577

3.6979

N2

7.0015

0.109768

9.759

0.28888

14 N16 O

7.466

0.115077

6.497

0.23260

O2

7.997

0.120752

5.115

0.19295

12 C16 O

6.8562

0.112832

11.09

0.26573

Na2

11.4949

0.30788

0.720

0.01955

23 Na35 Cl

13.870

0.23608

4.23

0.0448b

Cl2

17.4844

0.1987

2.479

0.0687

Dγ v /10−30 C m

0.52943 0.3662 30.025

As far as not otherwise mentioned, according to H UBER and H ERZBERG (1979) a L OVAS b R AM

et al. (2005)

et al. (1997)

For general orientation and further reference, Table 3.2 summarizes the relevant potential parameters for some characteristic diatomic molecules. The harmonic approximation to the M ORSE potential is VM (R) = −De 1 − a 2 (R − R0 )2 + · · · and with (3.19) and (3.20) the stiffness a is related to the spring constant k close to the potential minimum and to the fundamental frequency ω0 by k = 2De a 2

and ω0 =

¯ k/M¯ = a 2De /M.

(3.27)

The anharmonicity of the potential can be reflected as a decrease of the spring constant k with increasing v. Consequently angular frequency ωv and vibrational energy ωv decrease with v: the difference between neighbouring vibrational levels is smaller for higher energies. For the same reason the average nuclear distance increases with vibrational excitation:

R(v + 1) > R(v) > · · · > R0 .

This is also the physical cause for thermal expansion of solids: with increasing temperature the average vibrational energy increases, hence the average value of v increases as well, and so does R.

148

3

Diatomic Molecules

Note, however, that the number of bound states in such a “potential well” is always finite – even though the energy spacing between neighbouring levels gets very small close to the dissociation limit. We recall that, in contrast, the number of bound electronic states in an atom is infinite. This is entirely due to the particular nature of the 1/r C OULOMB potential. All other (bonding) potentials support at most only a finite number of bound states.

3.2.6

VAN DER WAALS Molecules

True chemical bonding due to formation of molecular orbitals will be discussed in Sect. 3.5. There are, however, also other, weaker forces by which atoms can be attracted to each other. C OULOMB attraction between atoms and polar molecules is one of them. Another important, attractive interaction is the so called VAN DER WAALS (vdW) interaction, or dispersion interaction, which has already been introduced in Sect. 8.3, Vol. 1 and was briefly mentioned in Sect. 3.2.2. It acts even between neutral atoms and molecules without a dipole, and is caused by mutual polarization of the electron charge clouds. The vdW interaction can be estimated according to (8.93), Vol. 1 as V (R) ∝ −αA αB R −6 with αA and αB being the polarizabilities of the two interacting atoms A and B. It is approximately additive and acts pairwise. The vdW interaction leads to (among other things) deviations from the ideas gas law. As well known, the state equation of real gases is given by the VAN DER WAALS equation a (3.28) p + 2 (V − b) = RT . V While the parameter b (the so called co-volume) reflects the finite extension of atoms and the repulsive part of the intermolecular potentials and decreases the available volume, the term a/V 2 (the so called cohesive pressure) is attributed to VAN DER WAALS forces at large distances which effectively increase the external pressure p. The vdW interaction is about a factor 102 to 103 smaller than typical bond energies. For example, for the non-bonding rare gas system Ar· · · Ar it amounts to about 0.08 eV in the minimum, while Cl2 (in the periodic table directly next to Ar) is chemically bound, albeit very weakly, with De 2.48 eV. VAN DER WAALS interaction is typically of the same magnitude as thermal energies at room temperature 203.6 cm−1 . kB × 293 K 0.025 eV =

(3.29)

Hence, to study it in detail one has to work at very low temperatures. Helium, for instance, may be liquified at low temperatures (4.3 K) in spite of its closed 1s 2 shell. At sufficiently low temperature He gas forms clusters or little droplets. At any rate, rare gases have a strong tendency to form atomic clusters at low temperatures. A popular method to generate such objects is adiabatic expansion of the rare gases in dense atomic beams.

3.2


149

V(R) / 10-4 Eh

He2

1.5 Tang-Toennies Lennard-Jones (12,6)

1.0 0.5 0.0 - 0.5

De = 0.348 4

8

10

12 R / a0

R 0 = 5.62

Fig. 3.8 Interaction potential between two He atoms. We compare the nearly exact TANG -T OEN NIES potential (TANG et al. 1995, red line) with a L ENNARD -J ONES 12, 6 potential (3.30) fitted to = 0.9465 meV) and it (black line). Note that only two free parameters De = 0.34784 × 10−4 Eh ( R0 = 5.62a0 determine the L ENNARD -J ONES potential. Both potentials are – on the scale used here – nearly indistinguishable

In this manner diatomic, very weakly bound combinations of practically all atoms may be generated, even if they do not form chemically bound molecules. For these so called VAN DER WAALS molecules, in particular for rare gas pairs, the interaction is usually approximated very well by a so called L ENNARD -J ONES potential, also called 12, 6 potential: V (R) =

De 2De − . 12 (R/R0 ) (R/R0 )6

(3.30)

It contains only two free parameters, R0 and De , and is thus not suitable for modelling potentials of chemically bound molecules – as documented by the dashed red line in Fig. 3.7(b). However, 12, 6 potentials are quite appropriate to describe VAN DER WAALS molecules. In the normalization used here, R0 is the equilibrium distance (also called VAN DER WAALS contact distance) and De is the depth of the potential well. For the example of the He-He system, Fig. 3.8 documents excellent agreement between the TANG -T OENNIES potential, which has been determined experimentally with extremely high precision (TANG et al. 1995), and the 12, 6 potential fitted to it.. The graphical accuracy of Fig. 3.8 hardly allows to see any difference. The experimental data are, however, much more precise and allow decisive conclusions about the He2 ‘molecule’. For many years it was not clear whether the simplest rare gas dimer forms a bound state at all. To obtain an estimate we recall the one dimensional potential box. According to (2.52) in Vol. 1 the lowest state of a particle with mass ¯ 2 ). For such a state to exist the potential depth M¯ has the energy W1 = h2 /(8ML ¯ h2 > 1. With De = 0.348 × 10−4 (all in a.u.) must at least be W1 , or De L2 × 8M/ the effective extension of the He-He potential would have to be L > 6.24a0 , which appears not unrealistic in view of Fig. 3.8. This is of course only a very rough estimate which cannot replace an exact computation. In diffraction experiments with

150 Table 3.3 VAN DER WAALS radii of some important atoms

3

Diatomic Molecules

Element

vdW radius/ nm

H

0.120

He

0.140

C

0.170

N

0.155

O

0.152

P

0.180

S

0.180

ultracold He atomic beams the existence of the He2 molecule has been established beyond any doubt. According to G RISENTI et al. (2000) the presently most accu+0.3 rate value for the bond energy of its one existing state is D00 = (1.1−0.2 ) mK, i.e. −7 10 eV(!), its average size is R = (5.2 ± 0.4) nm(!). Clearly, He2 is a quite pathological case of a molecule! By a combination of VAN DER WAALS contact distances between different partner atoms one estimates the so called VAN DER WAALS radii of individual atoms. Table 3.3 presents examples for some important elements. A graphical overview has already been communicated in Fig. 3.3, Vol. 1. Section summary

• We have introduced the B ORN -O PPENHEIMER approximation as fundamental for our quantitative understanding of molecules. It is based on the fact that electrons move much faster than nuclei. In a classical picture, electrons circle on their orbits many times before the nuclei have significantly changed their position R. • Hence, the electronic S CHRÖDINGER equation (3.10) can, to a very good approximation, be solved for the electronic coordinates only, with R being treated as a freely variable parameter. The electronic energy Vγ (R) for a set of electronic quantum numbers γ is a function R, called molecular potential (hypersurface for polyatomic molecules). • Characteristic shapes of Vγ (R) are shown for the diatomic case in Fig. 3.4. Nuclear motions occurs on these potentials. • The harmonic potential and the harmonic oscillator were found to be only a very crude approximation, describing the near equilibrium region and the lowest vibrational levels of a diatomic molecule reasonably well. • The M ORSE potential (3.26) was shown to be a useful analytic form for a chemically bound diatomic molecule, characterized by dissociation energy De , equilibrium distance R0 and molecular stiffness a related to the fundamental frequency ω0 by (3.27). • Even if there is no chemical bond, neutral atoms experience a long range attractive force caused by mutual polarization, the so called VAN DER WAALS interaction ∝ −1/R −6 . Based on this and a short range repulsion term ∝ 1/R 12 the L ENNARD -J ONES (12, 6) potential is used to describe non-bonding

3.3

Nuclear Motion: Rotation and Vibration

151 Z

Z

M Θ

Θ

R = RA – R B

ζ

MA R A = (M B / M ) R O

O

Y

Y

Φ

X MB

R B = (MA / M ) R

X

Φ

Fig. 3.9 The ‘reduced nuclear particle’: one transforms the two body problem by introduction of a reduced molecular mass M¯ = MA MB /M (with M = MA + MB ) into an effective one particle problem whose motion is described by the relative nuclear position coordinate R

interactions, e.g. between rare gas atoms. He2 was seen to be a very special case of a weakly bound molecule with De 10−7 eV.

3.3


3.3.1

S CHRÖDINGER Equation

With our present understanding of molecular potentials it is possible to study the nuclear motion in some more detail. To avoid confusions, we switch back to SI units in the following discussion. The reduced mass M¯ = MA MB /M moves on a spherical symmetric electronic potential Vγ (R). By introducing the relative coordinate R = R A − R B instead of the individual nuclear coordinates R A = R MB /M and R B = R MA /M (with M = MA + MB ) one replaces the vibrating nuclear mass, so to say by one ‘reduced nuclear particle’ as illustrated in Fig. 3.9.6 This leads again to a one particle S CHRÖDINGER equation very similar to that for the hydrogen atom. Merely the potential 1/r has to be replaced by Vγ (R), the electronic eigenenergy which now depends on the distance R between the nuclei. In complete analogy to (2.108) in Vol. 1 we write (3.17) now: R + H rot + Vγ (R) ψγ vN (R) = Wγ vN ψγ vN (R) n ψγ vN (R) = H (3.31) H 2 2 R = − 1 ∂ R 2 ∂ and H rot = N . with H ¯ 2 ∂R 2M¯ R 2 ∂R 2MR The quantum number v characterizes the radial motion (vibration). The angular of the reduced nuclear particle is a conserved quantity – momentum operator N commutes according to the usual angular just as L for the electron of the H atom. N 6 The electron coordinates too are transformed and referred to the molecular axis (in Fig. 3.9 the ζ axis).

152

3

Diatomic Molecules

n , N z ] = 0 and [H n , N 2 ] = 0 holds. momentum rules with the Hamiltonian. Thus, [H 2 is described in analogy to the H atom by In the position representation N

1 ∂ ∂ 1 ∂2 2 2 sin Θ + 2 N = − sin Θ ∂Θ ∂Θ sin Θ ∂Φ 2 and its eigenfunctions are the spherical harmonics YN MN (Θ, Φ) 2 YN MN (Θ, Φ) = 2 N(N + 1)YN MN (Θ, Φ), N

(3.32)

with eigenvalues 2 N(N + 1), where N is an integer. In analogy to the electron orbital angular momentum in atomic hydrogen, N and MN are now the quantum numbers of the molecular orbital angular momentum, i.e. of the molecular rotation. We characterize the angular coordinates of the molecular axis with capital Greek letters Θ and Φ (to be distinguished from lower case θ and ϕ, which we shall continue to use for electronic coordinates). The wave function of the nuclear motion ψγ vN (R) may thus be written as a product of spherical harmonics YN MN (Θ, Φ) and a radial part: ψγ vN (R) = R −1 Rγ vN (R)YN MN (Θ, Φ).

(3.33)

Insertion into (3.31) leads to the one dimensional S CHRÖDINGER equation

2 d 2 − + Veff (R) Rγ vN (R) = Wγ vN Rγ vN (R) (3.34) 2M¯ dR 2 for the radial motion (vibration), characterized by the vibrational quantum number v. The effective potential is here Veff (R) =

2 N(N + 1) + Vγ (R). ¯ 2 2MR

(3.35)

The reduced mass M¯ moves in this effective potential. In principle, one may solve (3.34) for each N numerically – as usual for reasonable physical boundary conditions. One can thus obtain rotational-vibrational energies Wγ vN and the radial eigenfunctions Rγ vN (R). It is, however, useful to simplify the problem even further.

3.3.2

Rigid Rotor

One exploits again the different time and energy scales: The molecule oscillates rapidly in comparison to the rotation, as we have seen. Thus, as a good first approximation, one may assume the nuclear distance R to be fixed. As sketched in Fig. 3.10 the molecule may assume any arbitrary polar Θ and azimuthal angle Φ in space. One may identify this fixed value of R with the equilibrium distance in

3.3


153

Fig. 3.10 Rigid rotor in space

z R

Θ MA

ω y

Φ

MB x

the lowest vibrational state. The S CHRÖDINGER equation (3.31) may then indeed be fully separated, and we obtain the eigenvalue equation for the rigid rotor: rot YN MN (Θ, Φ) = H

2 N Y (Θ, Φ) ¯ 2 N MN 2MR

= WN YN MN (Θ, Φ).

(3.36)

Using (3.32), the rotational energy thus becomes WN =

2 N(N + 1) = hcBN(N + 1) ¯ 2 2MR

(3.37)

with the rotational constant (unit cm−1 ) B=

2 1 2 1 = , ¯ 2 hc 2I hc 2MR

(3.38)

¯ 2 being the molecular moment of inertia. The resulting energy scheme is I = MR sketched in Fig. 3.11. In wavenumbers the rotational energies are simply F (N) = WN / hc = BN (N + 1).

(3.39)

The rotational constant B is (with the reduced mass known) a very sensitive measure for the bond length R0 of the molecule. Table 3.4 summarizes typical values of important molecules. The subscript in I0 , B0 and ω0 indicates that the constants given here refer to the lowest vibrational state. We see that spectra from the infrared to the microwave spectral region are expected. We must emphasize at this point that the image of a diatomic molecule as a rotating dumbbell has now to be replaced by the quantum mechanical picture: The spherFig. 3.11 Term scheme of the rigid rotor

N 4

WN / Bhc +

20 8

3

–

12 6

2 1 0

+ – +

4

6 2 0

154

3

Diatomic Molecules

Table 3.4 Rotational and vibrational constants and corresponding absorption wavelengths for several characteristic diatomic molecules. Note that only molecules with a finite dipole moment are infrared active Molecule

2 (2I0 )−1 / eV

B0 / cm−1

λrot

ω0 / eV

λvib / µm

IR active

H+ 2

3.641 × 10−3

29.4

170 µm

0.2714

4.568

no

H2

7.356 × 10−3

59.32

84 µm

0.5156

2.405

no

D2

3.708 × 10−3

29.90

167 µm

0.37095

3.342

no

LiH

9.184 × 10−4

7.4065

675 µm

0.16853

7.357

yes

HCl

1.2945 × 10−3

10.440

479 µm

0.3577

3.466

yes

N2

2.467 × 10−4

1.9896

2.51 mm

0.2888

4.292

no

O2

1.7827 × 10−4

1.4377

3.48 mm

0.19295

6.426

no

CO

2.384 × 10−4

1.9225

2.60 mm

0.26573

4.666

yes

NO

2.103 × 10−4

1.696 1

2.95 mm

0.23260

5.330

yes

Na2

1.913 × 10−5

0.15427

3.24 cm

0.01955

63.4

no

NaCla

2.6938 × 10−5

0.21725

2.30 cm

0.0448

27.68

yes

Cl2

3.02 × 10−5

0.243

2.05 cm

0.0687

18.0

no

As far as not otherwise mentioned according to H UBER and H ERZBERG (1979) a For

the isotopologue 23 Na 35 Cl according to R AM et al. (1997)

N= 0

N = 1 py

N = 1 pz

N = 1 px

N = 3 MN = 3

real states

Fig. 3.12 The rigid rotor in space: the probability for finding the molecular axis aligned at Θ and Φ, is given by the absolute square of the spherical harmonics (or linear combinations of these); here shown for N = 0 and N = 1, the latter having been excited by linearly polarized light; also shown is the distribution for N = 3, MN = 3 as an example where the molecule has been excited by circularly polarized light

ical harmonics YN MN (Θ, Φ) describe the probability amplitudes for the angles Θ and Φ, and |YN MN (Θ, Φ)|2 sin ΘdΘdΦ is the probability to find the molecular axis aligned between (Θ, Φ) and (Θ + dΘ, Φ + dΦ). We have discussed this already in detail in Sect. 2.5.3, Vol. 1. As |YN MN |2 does not depend on Φ this probability is rotational symmetric around the Z-axis. Alternatively the real basis YN |MN | (Θ) may represent the excitation probabilities more conveniently – if linearly polarized light has been used for excitation into the NMN states. In Fig. 3.12 the corresponding probabilities are shown for N = 0 and N = 1. The image of a rotating molecule

3.3


155

Table 3.5 Rotational temperatures for some diatomic molecules Molecule

H2

D2

HCl

N2

O2

CO

Cl2

Trot / K

93

47

15

2.89

2.08

2.77

0.4

cannot be recognized in these probability distributions. In particular, the rotational ground state with N = 0 has an isotropic angular distribution. This is also true for higher N if all 2N + 1 substates are equally populated, e.g. in the thermal equilibrium. Only for large values of N and MN the wave function may reflect the classical image of a molecule rotating around the z-axis, depicted in Fig. 3.3. Such a wave function (e.g. N = 3, MN = 3) can be prepared by absorption of several circularly polarized photons. If the molecule is excited by linearly polarized light, states with positive and negative MN are populated with equal probability – independent of which coordinate system is used to describe this situation – and the expectation z ≡ 0. In summary, a truly rotating molecule requires z disappears, N value of N rather special excitation conditions.

3.3.3

Population of Rotational Levels and Nuclear Spin

We want to address now the population probabilities of rotational states in diatomic molecules. Apart from their significance for molecular spectroscopy they play an important role in statistical mechanics and thermodynamics. Of course they depend on the rotational energies and on the degeneracies of the levels. They are a function of temperature and provide the basis for the derivation of specific heat capacities. If we compare the rotational constants in Table 3.4 with the thermal energy, say at room temperature, kB T 0.025 eV, we see that rotational energies (3.37) are usually small in comparison to thermal energies. Thus, at room temperature, (3.29), many rotational levels are populated. We define a characteristic rotational temperature Trot for a molecule by kB Trot =

2 = Bhc. 2I

(3.40)

This rotational temperature plays an important role when determining the specific heat capacity as we shall see in a moment. Typical values are given in Table 3.5. In analogy to n levels in the atomic case ( being the orbital angular momentum), the degeneracy of rotational levels is gN = 2N + 1. The population probability of a rotational level N is given by a B OLTZMANN distribution w(N, T ) =

(2N + 1) gN WN N (N + 1)Trot = , exp − exp − ZN kB T ZN T

(3.41)

156

3

w(N)

w(N) CO

ortho H2

0.6

0.06

(a)

0.04

(b)

0.4 para H2

0.2

0.02 0.00 0

20

10

Diatomic Molecules

N 30

0.0 0

1

2

3

4

5

N

Fig. 3.13 Relative population of rotational levels N in diatomic molecules at room temperature (293 K) in thermodynamic equilibrium: (a) CO – nuclear spin statistics is irrelevant in this case since the two nuclei are distinguishable particles; (b) for H2 (nuclear spin I (1 H) = 1/2) – here states with even and odd N are populated differently (red: para H2 ↑↓, grey: ortho H2 ↑↑), see text

with the partition function ZN (T ) =

∞

(2N + 1)e−WN /kB T

(3.42)

N =0

→ 0

∞

(2N + 1)e−N (N+1)Trot /T dN =

T Trot

for T Trot .

As examples we consider carbon monoxide and molecular hydrogen. For the heteronuclear molecule CO the atoms are distinguishable, and nuclear spin statistics does not play a role. Inserting the (very low) rotational temperature Trot = 2.77 K into (3.41) we obtain the population probabilities depicted in Fig. 3.13(a). We notice that due to the degeneracy (2N + 1) of rotational levels the energetically lowest level N = 0 is by no means the most populated one. In contrast to heteronuclear molecules, for homonuclear molecules one has to account for the indistinguishability of atomic nuclei. If the constituent nuclei are fermions, the PAULI principle demands the total wave function (nuclear spin × nuclear rotation × electronic state) to be antisymmetric in respect of nuclear exchange. For bosons the wave function has to be symmetric. In respect of the rotational wave function YN MN (Θ, Φ), exchange of two nuclei is equivalent to inversion at the centre of mass. Exchange symmetry is thus identical with parity and is given by (−1)N , as indicated by + and − in Fig. 3.11. Correspondingly, in the case of two fermions the nuclear spin function of the molecule must be odd for even N , while it must be even for odd N . For bosons the opposite holds. For bosons with nuclear spin I = 0, as e.g. in the case of He or O, the nuclear spin function is always symmetric, thus, the remainder of the wave function has to be symmetric too. Particularly clear is the case of H2 , being a fermion pair with the nuclear spins I = 1/2. The electronic ground state is characterized by 1 Σg+ (see Sect. 3.6) which


Fig. 3.14 Relative fraction of ortho and para H2 in a gas of hydrogen molecules as a function of absolute temperature; red: para – grey: ortho

157

1.0 ortho H2 (↑↑)

0.75 w(T )

3.3

0.5 para H2 (↑↓)

0.25 0.0 0

100

200

T /K

is symmetric in respect of reflection at a plane through the nuclear axis but also in respect of exchange of the nuclear coordinates. The nuclear spin function may be a singlet or a triplet (analogous to the electron states in excited atomic He, see Chap. 7, Vol. 1). To obtain an overall antisymmetric wave function for even rotational states (N = 0, 2, 4 . . . ) the nuclear spin function must be antisymmetric, i.e. be a singlet. One speaks of para hydrogen (p-H2 ). In contrast, ortho hydrogen (o-H2 ) is characterized by rotational states with N = 1, 3, 5, . . . belong to nuclear spin triplets. The population density (3.41) has to be multiplied then by the spin state degeneracy gS = (2S + 1), i.e. with gS = 3 and = 1 for triplet and singlet states, respectively. The resulting populations for p-H2 and o-H2 at room temperature are depicted in Fig. 3.13(b). Transitions between p-H2 and o-H2 are very improbable under normal conditions in a gas. One may almost speak of two different species which are stable over days, once generated. In thermodynamic equilibrium, the respective probabilities w(T ) to find either species is shown in Fig. 3.14 as a function of temperature. For very low temperatures only para H2 is found (rotational ground state) while at higher temperatures the ratio between p-H2 and o-H2 approaches the value 1:3 reflecting the ratio of the spin state degeneracy gS . One can generate para H2 from normal H2 gas by cooling it below 20 K and bringing it into contact with a catalyzer, e.g. Fe III containing substances, which support collision induced singlet-triplet transitions (magnetic interaction), and thus achieving thermodynamical equilibrium. As shown in Fig. 3.14 for low temperatures this equilibrium corresponds to pure para-H2 . After this process one heats the gas carefully up again without catalyzer(!). Only states of the para system will then be populated thermally, e.g. states with N = 0, 2, 4 . . . . It takes many hours, even days before the thus prepared p-H2 (without catalyzer) returns into thermodynamic equilibrium. As already mentioned, nuclear spin statistic plays an important role in the interpretation of rotationally resolved electronic or R AMAN spectra. We shall come back to this in Sect. 5.6.6.

158

3.3.4

3

Diatomic Molecules

Specific Heat Capacity

At this point, a brief discussion of CV (T ) for diatomic molecules is thus in order. The relevance of the above discussion for the specific heat capacity CV of molecules is evident: For very low temperatures T Trot rotation cannot be excited and thus, does not contribute to the specific heat capacity. Hence the temperature dependence of this macroscopically relevant quantity is a direct consequence of the rotational structure, while at higher temperatures molecular vibration enters. In the following discussion we assume to be well above the critical temperature (1.71), Vol. 1 for B OSE -E INSTEIN condensation, so that B OLTZMANN statistics describes the molecular gas well. In statistical thermodynamics one derives the characteristic observables from the partition function Z(T ) =

∞ i=1

Wi gi exp − kB T

(3.43)

where gi is the degeneracy of level i with the energy Wi . The summation is over all energy levels, if necessary, it has to be replaced by an integration and gi becomes the density of states. The average internal energy of a system with N molecules may be expressed in terms of the partition function as ∞

U =

W ∂ ln Z N kB T 2 ∂Z N − i N = kB T 2 . Wi gi e kB T = Z Z ∂T ∂T

i=1

From this one derives the molar heat capacity (specific heat capacity per mol at constant volume). Setting N kB = R (general gas constant) it is given by

∂ RT 2 ∂Z ∂U ∂ 2 ∂ ln Z RT = . = CV = ∂T V =const ∂T ∂T ∂T Z ∂T

(3.44)

From this expression we first derive the specific heat of an ideal gas due to translational motion. The partition function of a freely moving atomic particle without internal energy is given by the M AXWELL -B OLTZMANN distribution: Z(T ) ∝ 0

∞

π kB T 3/2 mv 2 dv = v 2 exp − . 2kB T 2 m

When inserting this into (3.44) we obtain the well known contribution from translational motion,

CV 3 = , R 2

(3.45)

reflecting the equipartition law with an average energy kB T /2 per molecule for each of the three translational degrees of freedom.

3.3

Nuclear Motion: Rotation and Vibration Cv / R

3.5

3.5

H2 thermal eq ili ri m

3.0

CO vi ration 3.0

T/K

2.5

2.5

2.5 CO rotation

(a) 2.0 1.5

159

2.0

H2 3:1 ortho-para mixt re 0

200

400

1.5

(c) 0

2000 4000 6000

( ) 0

10

20

Fig. 3.15 Molar heat capacity CV for diatomic molecules (in units of the general gas constant R). At very low temperature only the translational degrees of freedom are active (CV = 1.5R). The rotational contribution rises up to R; (a) for H2 thermal equilibrium conditions are compared with a mixture of ortho to para H2 of 1:3 as found at higher temperatures; (b) shows the rotational contribution for CO and (c) illustrates how CV rises again at higher temperatures due to vibrational excitation, in the limit again by R

Next we evaluate the rotational contribution to CV for a diatomic molecule. If spin statistics does not play a role (heteronuclear molecules) the partition function is given by (3.42). If it does, one has to include the nuclear spin degeneracy factor gS : ZN (T ) =

∞ N =0

N (N + 1)Trot . gS (N )(2N + 1) exp − T

For example, for the hydrogen molecule in thermal equilibrium we set gS (N ) = 2 − (−1)N , while for para H2 we have to set gS (N ) = (1 + (−1)N )/2 and for ortho H2 gS (N ) = 3(1 − (−1)N )/2. To evaluate (3.44) we have to resort to numerical evaluation. Fortunately, since the rotational energies increase ∝ N 2 , higher rotational levels are progressively less populated and one obtains reasonable accuracy with a moderate number of N in the sum. In the limit of high temperatures, T Trot , rotational excitation of a diatomic molecule always contributes R. The results are depicted for H2 and CO in Fig. 3.15(a, b). H2 with Trot = 93 K features a rotational contribution which becomes noticeable at rather high temperatures. CV is shown for H2 in thermal equilibrium (i.e. in the presence of a catalyzer) and for a mixture 1:3 of ortho to para H2 – corresponding to normal H2 at room temperature quickly cooled without a catalyzer. Both functions of course converge for higher temperatures and have a limiting value of CV = R, corresponding to an internal energy for the two degrees of rotational freedom of 2 × kB T /2 per molecule. In contrast, CO with Trot = 2.77 K reaches this limit already at rather low temperatures.

160

3

Diatomic Molecules

In Fig. 3.15(c) the contribution of vibrational excitation to CV is shown for the example of CO. Using the harmonic oscillator model the partition function is Zv (T ) =

0 exp(− kω ) ω0 1 BT +v = exp − , ω 2 kB T 1 − exp(− kB T0 ) v=0

∞

(3.46)

and with (3.44) the contribution of vibrational excitation to the molar heat capacity is exp(−Tvib /T ) ω0 CV Tvib 2 , with Tvib = . (3.47) = R T kB [1 − exp(−Tvib /T )]2 In the limit of high temperatures T Tvib the denominator becomes (Tvib /T )2 and again CV → R (we remember the two degrees of freedom for vibration mentioned in Sect. 1.3.3, Vol. 1, i.e. per molecule 2 × kB T /2 energy). For multi-atomic molecules with several vibrational degrees of freedom, (3.47) is the starting point for evaluating the specific heat capacity. One has to sum (3.47) over all vibrational frequencies. If the molecule possesses a quasi-continuum of vibrational levels (e.g. a large organic molecule) one has to multiply (3.47) with the density of states and to integrate over all frequencies. For solid state materials the lattice contribution to the specific heat is derived in just the same manner (theories of E INSTEIN and D EBYE).

3.3.5

Vibration

Vibrational energies are typically two to three orders of magnitude larger than rotational energies (see Table 3.4), so that one may separate the two forms of motion in a 1st order approximation as indicated above. Typically, the anharmonicity of the potentials is not too dramatic, so that for low vibrational excitation one may indeed assume that the average bonding lengths is constant, R = R0 , and the rotational energy derived for the rigid rotor can just be inserted into the radial S CHRÖDINGER equation (3.34) for the nuclear motion. One replaces 2 N (N + 1) ¯ 2 2MR

in (3.35) by WN =

2 N (N + 1) = B0 hcN(N + 1). ¯ 2 2MR 0

The total energy Wγ vN of the rotating oscillator may thus be understood as the sum of this vibrational energy Wγ v in the potential Vγ (R) and the rotational energy WN . One has to solve the radial wave equation without centrifugal potential:

2 d 2 − + Vγ (R) Rv (R) = Wγ v Rv (R). (3.48) 2M¯ dR 2 More explicitly, the total energy of a ro-vibrational state with vibrational and rotational quantum numbers v and N , respectively, is given by Wγ vN = Vγ (R0 ) + WvN .

(3.49)

3.3


Fig. 3.16 Qualitative, schematic picture of the electronic energies (potentials) Vγ (R) and the total energy Wγ vN = Wγ + WvN for some bound molecular states

161 W V2(R) W2vN R0(2)

V1(R)

W1υN V0(R)

R0(1) W0υN

R

R0(0)

Fig. 3.17 Blow up from one electronic state: vibrational and rotational energies of a harmonic oscillator with rigid rotor (for clear visibility the distances between rotational levels are massively enlarged)

vibration Wv = (v+1/2)ħω0 v =3

rotation

WN = B h c N ( N+1) v =2

v =1

v =0

ħω0 /2

N= 4 3 2 1 0

potential minimum

Vγ (R0 ) refers here to the minimum of the potential in the electronic state γ , while in the most simple approximation (harmonic oscillator + rigid rotor) the energy of the nuclear motion is given by 1 WvN = Wv + WN = ω0 v + (3.50) + B0 hcN(N + 1). 2 Qualitatively we obtain the picture shown in Fig. 3.16: for bound electronic states (ground state and several excited states), γ = 0, 1, 2 . . . , the molecular potentials (γ ) Vγ (R) are sketched with their equilibrium distances R0 . The nuclear motion occurs in these potentials. The energies of vibration and rotation WvN = Wv + WN are just added to the respective electronic energies at equilibrium distance. On an enlarged scale Fig. 3.17 illustrates for one electronic state the vibrational and rotational energies WvN = Wv + WN according to (3.50). Real values for vibrational ω0 and rotational B0 hc = 2 /(2I0 ) energies have been reported in Table 3.4 for a selection of molecules.

162

3

Diatomic Molecules

Table 3.6 Nonlinearity parameters for our set of diatomic reference molecules: ωe xe (Anharmonicity), αe (change of B with vibrational state) and centrifugal term De in comparison to the harmonic vibrational frequency ωe Molecule

ωe / cm−1

ωe xe / cm−1

H+ 2

2321.7

66.2

905400

50370

H2

4401.21

121.33

1824330

91800

D2

3115.50

61.82

912660

32336

LiH

1405.65

23.20

225258

6491

HCl

2990.946

52.8186

317582

9209

N2

2358.57

14.324

59906

519

O2

1580.19

11.98

43100

477

CO

2169.756

13.288

57908

524.8

184

NO

1904.2

14.075

50121

534

34

Na2

159.124

0.7254

4638.0

26.19

NaCla

364.684

1.776

6537.37

48.709

Cl2

559.7

2.67

7319.5

45.5

Be / MHz

αe / MHz

De / kHzb

9.3506

If not mentioned otherwise according to H UBER and H ERZBERG (1979), L OVAS et al. (2005) a For

the isotopologue 23 Na35 Cl according to R AM et al. (1997)

b Not

to be confused with the minimum potential energy De

Of course, the simple model of a harmonic oscillator with a rigid rotor is only a beginning of a realistic description. For a given potential Vγ (R) one may (still assuming a rigid rotor) solve the radial S CHRÖDINGER equation (3.34) without problems numerically – using the standard recipes for stable solutions as for the bound electron in atoms. Solutions Wv found in that manner may be expanded for N = 0 and any anharmonic potential into a series of the type 1 1 2 − ω e xe v + + ··· . G(v) = Wv / hc = ωe v + 2 2

(3.51)

For spectroscopic reasons one typically gives the term positions in wavenumbers, with the harmonic oscillation frequency (in wavenumbers) ωe = ω0 / hc = ω0 /2πc = ν0 /c,

(3.52)

and the so called anharmonicity constant ωe xe ωe . Table 3.6 gives anharmonicity constants ωe xe for the characteristic molecules known already from Table 3.4. The manner in which these parameters are written may appear somewhat arbitrarily. The terminology is historic and in agreement with the standard compendium on molecular physics (H ERZBERG 1989): ωe is not an angular frequency, ωe xe is one parameter (it is rather small indeed, justifying the series expansion (3.51)), and the parameter αe as well as De (written in calligraphic fonts for distinction) will be explained in the next subsection. (The latter has nothing to do with the bond energy De introduced in Sect. 3.2.5.)

3.3


163

Fig. 3.18 Schemati illustration of centrifugal distortion

Fc=ħ 2 N(N+1) /μRc3

Rc

Fr =−k(Rc−R0)

ω

For a M ORSE potential according to Sect. 3.2.5 one finds (here without proof) hcωe xe =

2 ω02 2 = a2 4De 2M¯

or ωe xe = ωe2

hc . 4De

(3.53)

We recall again, that in any potential well the number nmax of bound states is finite. Specifically for a M ORSE potential one finds that nmax ≤ 2De /ω0 − 1. For instance, in the case of CO the number of bound vibrational states is 41.

3.3.6

Non-Rigid Rotor

So far we have used the rigid rotor model. The radial S CHRÖDINGER equation (3.34) was solved for fixed nuclear distance R ≡ R0 , with the centrifugal term 2 N (N + ¯ 2 ) considered to be constant. This assumption allows one to treat rotation 1)/(2MR and vibration as separate, uncoupled forms of motion. Rotation is just an additive constituent of the total energy Wγ νN . For a real molecule one has to account for the effective potential (3.35) Veff (R) =

2 N(N + 1) + Vγ (R) ¯ 2 2MR

as a function of R and to solve the full radial S CHRÖDINGER equation (3.34). This leads to further modifications of the energy. One characteristic effect is centrifugal distortion by rotation, which leads to a coupling of rotation and vibration. Without explicitly solving the radial equation we try to estimate this effect. Rotational energy (3.37) leads to a centrifugal force Fc = −

2 N(N + 1) dWN = . ¯ 3 dR MR

As sketched in Fig. 3.18 it is balanced by the restoring force of the harmonic potential, ¯ 02 (Rc − R0 ), Fr = −k(Rc − R0 ) = −Mω

164

3

Diatomic Molecules

¯ 2 according to (3.20). At the new equilibrium posiwith the force constant k = Mω 0 tion Rc we must have Fc (Rc ) + Fr (Rc ) = 0, hence 2 ¯ 02 (Rc − R0 ) = N(N + 1) and Mω ¯ c3 MR 2 N(N + 1) 2 N (N + 1) > R Rc ≈ R0 1 + with

1. 0 M¯ 2 ω02 R04 M¯ 2 ω02 R04

The molecule stretches with higher rotational energy and the rotational constant becomes smaller. The overall rotational energy of the stretched molecule is the sum of rotational energy and potential (stretch) energy: WN =

¯ 2 2 N(N + 1) Mω 0 (Rc − R0 )2 . + ¯ c2 2 2MR

Inserting Rc , expanding around R0 and neglecting terms of higher order leads to WN =

2 4 N(N + 1) − N 2 (N + 1)2 + · · · ¯ 2 ¯ 3 ω2 R 6 2MR 2 M 0 0 0

Bhc N (N + 1) − De hc N 2 (N + 1)2 .

(3.54)

To obtain an estimate we insert B from (3.38): De =

4 1 1 4(Bhc)3 4B 3 4B 3 = = = . ¯ 2 )2 hc 2 ω2 hc 2ω02 (MR (Wv / hc)2 ωe2 0 0

Comparison for CO, NO and NaCl on the basis Table 3.6 shows rather good agreement with the spectroscopically determined values of De . A precise treatment of the S CHRÖDINGER equation (3.34) requires at least one more term. Typically one writes: Wγ vN / hc = Te

electronic term 1 1 2 − ω e xe v + + ωe v + 2 2

(3.55) vibration with anharmonicity

+ Be N(N + 1) − De N 2 (N + 1)2 rotation with stretch correction 1 N(N + 1) vibrational-rotational coupling. − αe v + 2 (γ )

(0)

The electronic term corresponds to Te = (Vγ [R0 ] − V0 [R0 ])/ hc. The last term, vibrational-rotational coupling, may be understood as a change of the rotational constant B due to the anharmonicity of the vibration and the resulting increase of R.

3.3


165

Often one extends the expansion even further: F (N) = Bv N(N + 1) − De N 2 (N + 1)2 1 1 2 + γe v + with Bv = Be − αe v + + ··· 2 2

(3.56) (3.57)

describes rotation, while the vibrational terms are written as 1 1 2 1 3 − ω e xe v + + ω e ye v + + ··· . G(v) = ωe v + 2 2 2

(3.58)

The spectroscopic accuracy today is often so high that expansions up to the 6th or even 10th power of (v + 12 ) become meaningful (see e.g. M ANTZ et al. 1971; L E ROY 1970).

3.3.7

D UNHAM Coefficients

Somewhat more general, today the eigenvalues of the S CHRÖDINGER equation (3.34) are often written as a series expansion in both quantum numbers v and N :

k 1 i WvN = Yik v + N(N + 1) . hc 2

(3.59)

This formulation has first been used by D UNHAM (1932) already in the early years of quantum mechanics. D UNHAM coefficients Yik – not to be confused with the spherical harmonics – have become more and more accepted during the past decades. For comparing different spectroscopic literature it is useful to note the following equivalences and relations: Y10 ωe ∝ 1/M¯ 1/2 Y20 −ωe xe ∝ 1/M¯

Y11 −αe ∝ 1/M¯ 3/2 Y21 γe ∝ 1/M¯ 2

Y30 ωe ye ∝ 1/M¯ 3/2 Y40 ωe ze ∝ 1/M¯ 2

Y02 −De ∝ 1/M¯ 2 Y12 −βe ∝ 1/M¯ 5/2

Y01 Be ∝ 1/M¯

Y03 −He ∝ 1/M¯ 3 .

(3.60)

Although these are approximations, they are sufficient for most comparisons. The exact expressions are given in the original work of D UNHAM (1932). We only note here that due to anharmonicity, the vibrational terms are slightly displaced in respect of the potential minimum (M ANTZ et al. 1971) by G(v) + Y00 with Y00 = Be /4 + αe ωe /12Be + (αe ωe )2 /144Be3 − ωe xe /4.

166

3

Diatomic Molecules

¯ which allows one to comImportant is also the dependence on the reduced mass M, pare spectra for different isotopologues. Section summary

• We have developed an understanding of nuclear motion. In summary, energy levels of a diatomic molecule may be expressed as a sum of electronic, vi(γ ) brational and rotational energy, Wγ vN / hc = Vγ [R0 ]/ hc + ωe (v + 12 ) + B0 N (N + 1) to 1st order approximation (rigid rotor with rotational constant B0 and harmonic oscillator with eigenfrequency ωe , both given in wavenumbers). • We have discussed higher order approximations, obtaining reasonable estimates for the changes due to centrifugal stretch and anharmonicity in a M ORSE potential. Even more general one expands the total energy (3.59) as a series in powers of [v + 1/2]i [N(N + 1)]k with the so called D UNHAM coefficients. • We also have made a brief excursion into statistical thermodynamics in Sect. 3.3.3, acquainting ourselves with the population of rotational energy levels at low temperatures. Based on this information we have derived in Sect. 3.3.4 specific heat capacities CV (T ) for diatomic molecules as a function of temperature T . At very low temperature only the kinetic degrees of freedom can be excited and CV = 3/2R, while at temperatures T Trot = hcB/kB the specific heat capacity rises to 5/2R. At still higher temperatures, around T Tvib = ω0 /kB vibration can be excited and the specific heat approaches its limiting value CV = 7/2R.

3.4

Dipole Transitions

Which transitions can be induced by electromagnetic radiation in molecules? To answer this question we essentially follow the treatment of atomic transitions in Chap. 4, Vol. 1, specifically as outlined in Sect. 4.3.4, Vol. 1 for E1 transitions. We have, however, to account now explicitly for the interaction of N electrons (i) of charge −e and the charges +eZk of several nuclei (k) with the electromagnetic field. Hence, the relevant electric dipole operator is now given by D(R, r) =

N nu k=1

Zk R k −

N

ri

N r i · e, · e = ZR −

i=1

(3.61)

i=1

where again r represents all electronic, R all Nnu nuclear coordinates, and e the unit polarization vector of the radiation. In principle this expression holds for any number of nuclei. Specifically for a diatomic molecule with Nnu = 2, the electric field acts on a distance weighted, average charge (see Fig. 3.9) = (ZA RA + ZB RB )/(RA + RB ). Z

(3.62)

3.4

Dipole Transitions

167

The transition probability into a state |a = |γ vN M from a state |b = |γ v N MN is proportional to the squared dipole transition matrix element Dba = γ v N MN |D|γ vN M = Ψb∗ (r, R) D(R, r)Ψa (r, R)d3 Rd3 r. By inserting (3.33) we obtain ∗ Dba = D(R, r) YN M (Θ, Φ)R −1 R∗γ v N (R)φγ∗ (r; R) N

× φγ (r; R)R −1 Rγ vN (R)YN M (Θ, Φ) d3 Rd3 r.

(3.63)

The evaluation of this matrix element is not a completely trivial task. We thus postpone electronic transitions to Chap. 5, and begin with rotational and then vibrational transitions within one electronic state (γ = γ ). These spectra in the infrared and microwave region may be observed by absorption (possibly also by induced emission). Spontaneous emission spectra are not observable due to the ν 3 factor in the E INSTEIN coefficients Aab ∝ Bab ν 3 .

3.4.1

Rotational Transitions

In pure rotational transitions the vibrational state v and the electronic state γ remain unchanged. With d3 R = R 2 dR sin ΘdΘdΦ and eR = R/R being the unit vector of the relative nuclear coordinate, one may separate the dipole transition matrix element (3.63) for a transition N ← N into a radial and an angular part. We make use of the molecular symmetry around the internuclear axis – i.e. we let the electron coordinates r i = (ξi , ηi , ζi ) refer to the molecular axis (ζi eR ). With (3.61)–(3.63), the radial part of the integration can be cast into N 2 3 2 Dγ v = ζi φγ (r; R) d r i Rγ v (R) dR. (3.64) ZR − i=1

The contributions for the other two components ξi and ηi disappear in this symmetry when averaging over all r i . Obviously Dγ v = −eDγ v

is the permanent dipole moment

of the molecule in the state |γ v (see e.g. Appendix F.2 in Vol. 1, specifically Eq. (F.10)). With this abbreviation, the angular part of the integration in the dipole transition matrix element is given by DN N = Dγ v YN∗ M (Θ, Φ)eR · eYN MN (Θ, Φ) sin ΘdΘdΦ. (3.65) N

The important message from this expression is: pure rotational spectra can only be observed for molecules with a permanent dipole moment, but not for homonuclear molecules such as H2 or N2 . Table 3.4 explicitly emphasizes the infrared active

168

3

Diatomic Molecules

heteronuclear molecules. The selection rules are derived in analogy to those for and m in the case of atoms, described in detail in Sect. 4.4, Vol. 1. We just have D = e · r there, by D = Dγ v e R · e to replace the electronic dipole transition operator here. The permanent dipole moment of the molecule D = −eDγ v eR is parallel to the molecular axis eR . As in the atomic case, the three components of the molecular unit operator eR are now expressed in terms of the renormalized spherical harmonics eR = {C1−1 (Θ, Φ), C10 (Θ, Φ), C11 (Θ, Φ)}. We can now evaluate (3.65), which leads to selection rules in full analogy those for atomic transitions between different m states: N = ±1 and MN = 0, ±1.

(3.66)

For absorption, N = N + 1 ← N , the transition energy in wavenumbers is ν¯ rot =

WN +1 − WN = B (N + 1)(N + 2) − N(N + 1) = 2B(N + 1). hc

(3.67)

For induced emission, N − 1 ← N , one finds correspondingly = (WN − WN −1 )/ hc = 2BN . ν¯ rot

(3.68)

The spectral lines for a rigid rotor all have the same distance 2B in wavenumbers. The measurement of such a spectrum is, however, not trivial since it involves a broad spectral range from microwave to the FIR spectral range. Tabulated values usually refer to a number of different measurements. In Fig. 3.19 we show an artificially synthesized spectrum of rotational lines for CO in the vibrational ground state v = 0, as derived from transition frequencies published by L OVAS et al. (2005). It is quite instructive to consider the intensities in these line spectra. The absorption probability RN MN N MN for specific transition |N MN ← |N MN between the orientation states is proportional to the square of the rotational dipole matrix element (3.65), which according to (4.79) and (C.28) in Vol. 1 is (for a polarization vector e = eq ) 2 (3.69) RN MN N M ∝ |Dγ v |2 N MN |C1q |NMN 2 2 N 1 N N 1 N 2 = |Dγ v | 2N + 1 (2N + 1) × . MN q MN 0 0 0 The C1q (Θ, Φ) are the renormalized spherical harmonics characterizing the polarization (q = 0, ±1) of the incident light, and N = N ± 1 is the rotational quantum number of the upper state. The 3j symbols (: : :) have been introduced in Appendix B, Vol. 1. In view of the fact that many rotational states are populated, as discussed in Sect. 3.3.3, we also have to include induced emission in our evaluation of the line spectrum. The population probabilities for the specific rotational substates |N MN and |N MN are given by the B OLTZMANN factor7 exp(−N (N + 1)Trot /T ), where 7 Note that for each individual state |NM

N the statistical weight is gN = 1, while the factor 2N + 1 used in (3.41) refers to all MN states of a rotational level N .

Dipole Transitions

Fig. 3.19 The rotational absorption spectrum of CO in the FIR and mm spectral range, synthesized from the data given by L OVAS et al. (2005). The lines have a distance of 2B = 3.84 to 3.76 cm−1 corresponding to the rotational constant B0 = 1.9225 cm−1 according to Table 3.4. The absorption has been derived from (3.71)

169 100%

relative transmission

3.4

0

20

40

60

80

100

ν rot / cm-1

T is the temperature of the sample and Trot the rotational temperature of the molecule studied (see Table 3.5). Now, the probability RN MN N MN for the absorption process |N MN ← |N MN is exactly equal to the induced transition probability RN MN N MN for |N MN → |N MN . On the other hand, the B OLTZMANN factor for the upper rotational level is somewhat smaller than for the lower states, so that for each individual rotational state the relative net absorption for the incident radiation I is 2 I (N MN N MN ) ∝ |Dγ v |2 N MN |C1q |NMN (3.70) I × exp −N(N + 1)Trot /T − exp −N N + 1 Trot /T . For the overall absorption we have to sum over all upper and lower orientation states. With (3.69) we obtain the relative absorption signal for a single rotational line N ← N : I (N M NMN ) I (N N ) N = I I MN MN

∝ |Dγ v |2 exp −N(N + 1)βr − exp −N N + 1 βr 2 N N 1 N × 2N + 1 (2N + 1) × M 0 0 0 N

1 q

MN MN

N MN

2 .

Exploiting the orthogonality relation (B.42), Vol. 1, and with (B.53), Vol. 1 for evaluation of the remaining 3j symbol we finally obtain a simple expression for the overall absorption probability: N + 1 −N (N+1)Trot /T I (N N ) ∝ |Dγ v |2 e − e−(N +2)(N +1)Trot /T . I 3 This is presented in Fig. 3.19 for the vibrational ground state of CO.

(3.71)

3

3.4.2

/ (N +1) (WN+1 - WN )

Fig. 3.20 Centrifugal distortion in CO. Shown is a quadratic fit to the spectroscopic data from L OVAS et al. (2005). The ordinate intercept gives 2B, while the parameters De and H in (3.72) are derived from the slope and the slight curvature of the line connecting the data points

/ cm-1

170

3.84

Diatomic Molecules

CO v=0

3.83 3.82 3.81 0

400

1200

800

(N+1)2

Centrifugal Distortion

In a more precise analysis of the data one notices, however, that with increasing N the distance between neighbouring lines decreases slightly. This documents the centrifugal distortion discussed in Sect. 3.3.6. For a quantitative evaluation of the experimental material we write the rotational energy WN / hc = BN (N + 1) − De N 2 (N + 1)2 + HN 3 (N + 1)3 . With this, one easily verifies WN +1 − WN = 2B − (4De − 2H)(N + 1)2 + 6H(N + 1)4 . hc(N + 1)

(3.72)

Figure 3.20 shows this expression, plotted as a function of (N + 1)2 , for CO in the v = 0 state. To a first approximation this gives a straight line with the ordinate intercept 2B while the slope is −4De . The parameters derived from this fit correspond to the values given in Table 3.6. The curvature of the fit even allows one to derive the parameter H = 0.1715(4) Hz. Today rotational lines are determined with highest precision by F OURIER transformation IR spectroscopy (FTIR). An example is shown in Fig. 3.21. We shall describe FTIR spectroscopy in Sect. 5.3.2 in detail.

Fig. 3.21 Single rotational transition line, N = 1 ← 0, for H35 Cl in the v = 0 state, recorded by F OURIER transform spectroscopy in the FIR adopted from K LAUS et al. (1998). The extreme precision allows to resolve even the hyperfine structure (35 Cl has a nuclear spin 3/2)

H 35Cl v = 0 N=1 ← 0 F=

3/2 ← 3/2 625.90

5/2 ← 3/2

1/2 ← 3/2 625.92 625.94 frequency / GHz

3.4

Dipole Transitions

3.4.3

171

S TARK Effect: Polar Molecules in an Electric Field

We are now prepared to treat the S TARK effect in polar molecules, i.e. to evaluate the energies of rotational levels in an external static electric field. In Sect. 8.2, Vol. 1 we have discussed the S TARK effect in atoms at some lengths. We have found there that it is a weak effect for low lying electronic terms, while for highly excited RYDBERG states it may become substantial due to interaction with other, near neighbouring RYDBERG states. For polar molecules we expect the largest S TARK effect for the lowest rotational levels, since the energies (3.37) rise quadratically with the rotational quantum number, and hence neighbouring levels interact less and less as N increases. Let the external electric field E be parallel to the molecular z-axis (E eR ). With Vel (R) = eE · eR Dγ v and (3.69) the interaction matrix elements (8.53), Vol. 1 are γ vN MN |Vel |γ vN MN = eE Dγ v N MN |C10 (Θ)|N MN .

(3.73)

Using the same formulas (8.58) and (8.59), Vol. 1 as in the atomic case we have for each rotational N > 0 two nonvanishing matrix elements γ vN MN |Vel |γ vN MN

⎧ ⎪ ⎪ (N +1)2 −MN2 ⎨ (2N +1)(2N +3) = eE Dγ v δMN MN δN N ±1 × 2 ⎪ N 2 −MN ⎪ ⎩ (2N −1)(2N +1)

(3.74) for N = N + 1 for N = N − 1

while for N = 0 only N = 1 is meaningful. The diagonal matrix elements disappear and the S TARK effect becomes quadratic – just as in the atomic case with already removed degeneracy. In analogy to (8.63), Vol. 1 we obtain (0)

WN MN = WN MN − WN MN = |eE Dγ v |2

N MN |C10 |N MN 2 N

WN − WN

.

(3.75)

Inserting (3.74), the two terms in the sum give WN MN =

|eE Dγ v |2 f (N, MN ), Bhc

with f (N, MN ) = and

N(N + 1) − 3MN2 1 2 N(2N + 3)(2N − 1)(N + 1)

= −1/6 for N = 0.

Figure 3.22 illustrates this expression for N = 4.

(3.76)

172

3

103 f(N,M )

Fig. 3.22 S TARK splitting in a polar molecule

Diatomic Molecules

5 0

|M | = 0 1 N=4 2 3

-5 -10

4

To obtain a feeling for the order of magnitude of the S TARK effect, we consider again CO as an example. In its ground state it has a permanent dipole moment |D| = eDγ v = 0.3662 × 10−30 C m and its rotational constant is B0 = 1.9225 cm−1 . For an electric field strength of 1 kV / cm, which is still conveniently achievable in the laboratory, one obtains a relative S TARK shift |eE Dγ v |2 WN MN = 2 f (N, MN ) 10−6 f (N, MN ). WN MN B0 h2 c2

(3.77)

Considering the very high precision of microwave spectroscopy this is experimentally quite detectable. In electromagnetic radiation fields the dynamic S TARK effect leads to significant splittings already at rather moderate intensities I . In the spirit of Sect. 8.4.1, Vol. 1 we identify E 2 in (3.77) with the average field E 2 = I /ε0 c and obtain WN MN /WN MN 3.5 × 10−10 f (N, MN ) × I / W cm−2 for CO. In the field of a short pulse lasers the S TARK splitting of such a molecule becomes substantial. For the N = 4, MN = 4 state in CO, we find e.g. a lowering of the energy by about 3 % at a very moderate intensity of I = 1010 W cm−2 . We finally mention that the S TARK effect is also the basis for aligning molecules in high electric fields or by laser pulses.

3.4.4

Vibrational Transitions

To evaluate the dipole transition matrix elements (3.63) for vibrational excitation (or de-excitation) we follow Sect. 3.4.1 – still within one electronic state. We have to replace the permanent dipole moment Dγ v = −eDγ v by the transition dipole moment and (3.65) now becomes Dγ v ←v = R∗γ v (R)Dγ (R)Rγ v (R)dR (3.78)

N 2 3 − with Dγ (R) = ZR ζi φγ (r; R) d r i .

(3.79)

i=1

Once more, we have exploited the symmetry of the diatomic molecule and chosen electronic coordinates with ζi eR : the dipole moment, −eD, and its derivatives are parallel to the molecular axis. We may thus expand Dγ (R) around the equilibrium

3.4

Dipole Transitions

distance R0 :

173

∂ Dγ Dγ (R) = Dγ (R0 ) + (R − R0 ) + · · · . ∂R R0

(3.80)

This has to be inserted into (3.78) for a transition v ← v. The first (constant) term disappears due to the orthogonality of the radial wave functions Rv (R). Thus, the linear term dominates for dipole induced (E1) vibrational transitions: ∂ Dγ Dv ←v = (3.81) R∗v (R)(R − R0 )Rv (R)dR + · · · . ∂R R0 The evaluation leads to the following vibrational selection rules: • Vibrational transitions are only possible if ∂ Dγ /∂R|R0 = 0. Since in homonuclear molecules the dipole moment is zero, and also its derivative disappears, vibrational transitions within one electronic state are not possible. Diatomic gases (H2 , N2 , O2 , etc.) are transparent in the infrared spectral region. In contrast, CO, HCl etc. are strong IR absorbers since ∂ Dγ /∂R|R0 is large in these cases. • For a pure harmonic oscillator v = ±1 holds.8 The spectroscopy in this case is relatively simple, since the energy difference between two vibrational states is always independent of v: G = ωe (v + 1 + 1/2) − (v + 1/2) = ωe . When the demand for precision is higher, one has to account for the anharmonicity of the potential and possibly also has to include higher terms in the series expansion (3.80). In consequence the selection rule v = ±1 does no longer strictly apply, transitions with v = ±2, ±3, . . . become weakly allowed, and the separation of the energy levels changes with v. The population of vibrational states in thermodynamic equilibrium is determined by the B OLTZMANN distribution: Tvib g(v) Wv ω0 exp − with Tvib = Nv = exp − . (3.82) Zv kB T T kB Here Zv is the partition function, for the harmonic oscillator given by (3.46). Nv decreases monotonically with v, since in contrast to rotation the vibrational states of a diatomic molecule are non-degenerate, i.e. g(v) = 1 holds. We note that at room temperature for small diatomic molecules ω0 kB T 0.025 eV holds (cf. Table 3.4 for some examples). Hence, at normal conditions and thermodynamic equilibrium essentially only the ground vibrational state (v = 0) is populated. 8 One

readily verifies this for the lowest levels by inserting the H ERMITE functions from Table 3.1 into (3.81).

174

3

N''

20

N'

19

R branch

band origin

transmission

P branch

Diatomic Molecules

19 20

1 0 1← 0

2← 0 0 1

CO

×100 2000

v' = 1 ← v'' = 0

4000 2050

2100

_

2150

2200

2250

ν / cm-1 Fig. 3.23 Vibration-rotation band for CO, in the electronic ground state for the v = 1 ← v = 0 transition, at a temperature T = 293 K, simulated with HITRAN (ROTHMAN et al. 2009) data for the R and P branch; the inset shows also (weak) second harmonics lines, v = 2 ← v = 0

3.4.5

Vibration-Rotation Spectra

However, pure vibrational transition are not allowed! Simultaneously (3.65) has to be applied (with Dγ v replaced now by Dγ v ←v ). And as no E1 transitions occur without change of the rotational quantum number N (parity conservation). In summary, we have now three selection rules: v = ±1 (as well as ± 2 in the anharmonic case – very weak), N = ±1,

and MN = 0, ±1.

(3.83)

According to H ERZBERG (1989), the upper levels are designated with v N , the lower ones with v N . As an introductory example, Fig. 3.23 shows the infrared absorption spectrum of CO in the ground state (v = 1 ← v = 0). One recognizes a typical band structure with many lines. The inset (not rotationally resolved) gives a feeling for the importance of higher harmonics, showing the v = 2 ← v = 0 band which amounts to less than 1 % of the total absorption. The spectrum (so called sticks spectrum) shown has been generated synthetically from data of the HITRAN data bank. In this impressive collection of molecular spectra more one finds nearly three million spectral lines of presently 47 molecules with 120 isotopologues – a real treasure for analysts and spectroscopists, who want to search for traces of molecules, e.g. in the earth atmosphere.

3.4

Dipole Transitions

175

The band structure shown in Fig. 3.23 is caused by a combination of vibrational and rotational transitions. With (3.56)–(3.58), the difference energies ν¯ (in wavenumbers) for N v ← N v transitions are ν¯ N v ← N v = F N + G v − F N − G v . (3.84) The most important characteristics of the vibration-rotation bands may already be recognized without any higher order terms in (v + 1/2) and N . In the following we just account for the anharmonicity. In principle, the vibration-rotation spectra comprise three ‘branches’ of transitions – we write them ν¯ = P (N), ν¯ = Q(N) and ν¯ = R(N ). Of these, however, the Q branch is forbidden for pure vibration-rotation spectra in diatomic molecules: 1. P branch with N = −1 (i.e. N = N − 1): P N = ωe − 2ωe xe v + 1 − 2BN for N = 1, 2, 3, . . . .

(3.85)

2. Q branch with N = 0 forbidden for diatomic molecules): Q N = ωe − 2ωe xe v + 1 for N = 0, 1, 2, 3, . . . .

(3.86)

3. R branch with N = +1 (i.e. N = N + 1): R N = ωe − 2ωe xe v + 1 + 2B N + 1 for N = 0, 1, 2, . . . . (3.87) Note that due to anharmonicity the band origin is not exactly at ωe , as one recognizes in Fig. 3.23 for the CO spectrum: ωe = 2169.756 cm−1 according to Table 3.6, however, the origin (between P and R branch) is located at 2143.24 cm−1 , corresponding to (3.86). Figure 3.24 illustrates schematically how the band structure arises due to rotation. In respect of a (hypothetical) pure vibration spectrum (Q branch) with Q(N) = ωe the lines in the R branch have higher (R(N ) > ωe ), those in the P branch lower energies (P (N ) < ωe ). The Q branch with N = 0 is missing due to the overall parity conservation. Somewhat more precisely: this is due to the dipole moment Dγ = −eD(R)eR being parallel to the molecular axis. Its changes are, according to (3.78)–(3.81), responsible for vibrational excitation – one speaks about “parallel” transitions. They are the only ones possible in a diatomic molecule. However, already for a triatomic linear molecule this may be different as we shall see in Sect. 4.2.3.9 The intensity distribution in the vibration-rotation absorption bands may be derived in complete analogy to our considerations for pure rotational transitions in Sect. 3.4.1. It arises again essentially from the thermal population. However, induced emission usually does not play a role since initially the final vibration state is practically unpopulated. 9 Also, in transitions between different electronic states a

Q branch may be possible – if the photon angular momentum is transferred to the electronic charge cloud. We come back to this aspect in Sect. 5.4.4.

176

3

Diatomic Molecules

Fig. 3.24 How the bands arise: P , (Q) and R branches in a vibration-rotation spectrum, the Q band is forbidden for a diatomic molecule within one electronic state

N' = 5 4 3 1 2 0

v' = 1

ΔN = 1

Δ N = −1 ΔN = 0

N'' = 5

P branch

Q branch

4 3 1 2 0

R branch

v'' = 0

_ ν

As an example we show in Fig. 3.25 the infrared absorption spectrum of HCl, which has a large dipole moment is thus infrared active. The spectrum is again based on a simulation with the help of the HITRAN data bank. It also illustrates that isotopologues have to be considered when evaluating such spectra. If higher accuracy is required, one has to account for contributions from higher order terms in (v +1/2) and N to the energies F (N ) as well as to G(v). Also, the dependence of Bv on the vibrational state according to (3.57) has to be included. Thus,

2600

2700

3015

3030 3045 3059 3073

2900 _ ν / cm-1

2998 2981 2945 2963

2844

2906

2866 band origin

2800

2926

the deeper minima originate from H35Cl, the lesser minima from H37Cl

2799 2822

2776

transmission

2752

2703

R branch

2728

2678

2626

2652

P branch

3000

HCl 3100

Fig. 3.25 Vibration-rotation spectrum of HCl, simulated according to HITRAN (ROTHMAN et al. 2009) for absorption at room temperature (sticks spectrum)

3.4

Dipole Transitions

177

the spectra become quite complicated, since the term distances are no longer equal. When evaluating experimental data one uses a nice trick to separate the rotational constants of the upper and lower vibrational states: in suitably chosen differences of spectral lines, one or the other of these constant drops out. One finds from (3.56) and (3.58): R(N − 1) − P (N + 1) = 4B − 6De (N + 1/2) − 8De (N + 1/2)3 (3.88) R(N ) − P (N) = 4B − 6De (N + 1/2) − 8De (N + 1/2)3 . (3.89) By plotting the data correspondingly, one obtains by suitable fits the four parameters B and De for the upper, and B and De for the lower state – in the same manner as discussed in Sect. 3.3.6 for pure rotational spectra. By measuring several vibrational states one can obtain in a similar manner the anharmonicity. In a so called B IRGE -S PONER plot the differences between experimentally determined energies of adjacent vibrational lines (band origin) are plotted as a function of (v + 1). With (3.58) up to second order G v + 1 − G v = ωe − 2ωe xe v + 1 + · · · , (3.90) holds – which is of course identical to (3.86). The slope of this curve gives the anharmonicity ωe xe , the intercept with the ordinate yields the ground state frequency ωe . Specifically for a M ORSE potential one may estimate the bond energy with the thus determined parameters from (3.53) De = hc

ωe2 . 4ωe xe

As for the pure rotational lines, different isotopologues have different vibrationrotation frequencies. In addition to the change of the moment of inertia I0 , now the shift of the vibrational frequency ω0 = k/M¯ is significant. As a rule, the (absolute) isotope shift of the vibrational transitions is larger than that for the rotational absorption line.

3.4.6

R YDBERG -K LEIN -R EES Method

Before turning to the ab initio (i.e. the quantum mechanical) computation of molecular potentials we briefly want to introduce a standard method to determine potentials directly from the experimentally measured spectra. The method developed by RYDBERG, K LEIN and R EES already in the early years of quantum mechanics, the so called RKR method, uses measured vibration-rotation spectra in a semiclassical ansatz. It tries, so to say, to invert the solutions of the S CHRÖDINGER equation for energies and nuclear wave functions. One has to know the vibrational term energies G(v) and the rotational constants B(v) as good as possible, for as many v as possible. They are then considered as continuous functions of v and one determines

178

3

Diatomic Molecules

from these the classical turning points Rmax (v, N = 0) and Rmin (v, N = 0). From these one may then construct the potential as illustrated in Fig. 3.7. One may show rigorously that Rmax =

f 2 − f/g + f

and Rmin =

f 2 − f/g − f

(3.91)

holds. The functions f and g are derived from v dv 1 f = (Rmax − Rmin ) = √ √ 2 2hcM¯ vmin G(v) − G(v ) √ 1 1 2hcM¯ v B(v )dv 1 = g= . − √ 2 Rmax Rmin G(v) − G(v ) vmin

(3.92)

(3.93)

Evaluation of these integrals is not completely trivial. The interested reader is referred to the literature (e.g. M ANTZ et al. 1971; F LEMING and R AO 1972, where one also finds references to the original work). Section summary

• E1 transitions between vibrational and rotational states within one electronic state are studied by absorption in the infrared and microwave region, respectively. Spontaneous emission is not observed at these frequencies as a consequence of the ν 3 factor for spontaneous emission. • They require a permanent molecular dipole moment (i.e. do not occur in homonuclear molecules). Vibrational transitions depend on the change of the dipole moment with distance. • Pure rotation spectra in polar molecules obey selection rules for the angular momentum, N = ±1 and its projection MN = 0, ±1. To 1st order the lines are equally spaced by 2B, with B being the rotational constant. Centrifugal distortion adds small terms ∝ (N + 1)2 and more. • A quadratic S TARK effect in polar molecules is weak but observable due to the high accuracy of microwave spectra. In intense laser field it becomes important. • In harmonic approximation, only v = ±1 transitions are allowed. Some weak higher harmonic lines may be observed due to anharmonicity. • Vibration-rotation bands have, according to (3.85)–(3.87) in principle three branches, P , Q, and R for N = −1, 0, and 1, respectively (in absorption). The Q branch is not allowed in diatomic molecules within one electronic state. • Relatively simple formulas (3.88)–(3.90) can be used for extracting anharmonicities from observed spectra. The RYDBERG -K LEIN -R EES method inverts in principle the S CHRÖDINGER equation and derives the molecular potential from measured vibration-rotation spectra.

3.5

Molecular Orbitals

179

3.5

Molecular Orbitals

3.5.1

Variational Method

Let us now discuss the electronic part of the S CHRÖDINGER equation (3.10). It is most instructive to explain the basic concepts used in typical quantum chemical methods for the example of H+ 2 , the simplest of all molecules. The extension to multi-electron systems is relatively straight forward, even though it may become rather elaborate. With the coordinates defined in Fig. 3.26 the Hamiltonian (again in a.u.) becomes: el = − 1 ∇ 2r − 1 − 1 + 1 . H 2 rA rB R

(3.94)

The variational principle has been used already in Sect. 7.2.5, Vol. 1. Even in its simplest form, it provides a good first approximation for the molecular potentials, i.e. for the electronic energies as a function of R, and for the wave functions. One chooses a suitably parameterized “trial function” φ(r i ; R) and rewrites the S CHRÖDINGER φ = W φ as equation H ∗ φd3 r φ H W = min ∗ 3 . φ φd r

(3.95)

In this formulation φ(r; R) doesn’t even need to be normalized. The best eigenvalue W is obtained by variation of φ (i.e. by changing the parameters which define φ) such, that W becomes a minimum. Formally, the “functional” W (φ) is minimized !

by searching for ∂W (φ)/∂φ = 0.

MOs from LCAO As a trial wave function for the molecular orbitals (MO) one often uses a linear combination of atomic orbitals (AOs), called MO from LCAO: φ(r; R) =

2 i

(3.96)

ci Φi (r j ).

j =1

We have to sum over a suitable number i of atomic orbitals Φi (r j ) as well as over both atomic nuclei j = A, B. In the most simple case the Φi are eigenfunctions of the H atom, however, localized now at the two different atoms and rewritten Fig. 3.26 Coordinates for the H+ 2 molecule

rA

e-

rB

r B

A O

R

180

3

Diatomic Molecules

such that the coordinates refer to the same origin O as indicated in Fig. 3.26. For different masses MA and MB , the origin O is shifted towards the heavier mass. With M = MA + MB the electronic coordinates are r B = r − (MA /M)R

and r A = r + (MA /M)R.

(3.97)

(In the case of several electrons r, r A and r B represent all electron coordinates.) With the LCAO “trial” wave function (3.96) the trial energy becomes ( ∗ ∗ ( c Φ H c k Φk d 3 r > Wtrue , " = (i i ∗ i ∗ ( k 3 i c i Φi k c k Φk d r where Wtrue is the true eigenvalue. Since the coefficients ci are just (complex) numbers, we may interchange summation and integration and obtain ( ( ∗ c ck Hik with the (3.98) " = (i (k i∗ i k ci ck Sik Φk d3 r, molecular Hamiltonian in the atomic basis Hik = Φi∗ H (3.99) and the so called overlap integral Sik =

Φi∗ Φk d3 r.

(3.100)

Note, that for normalized AOs, Φi , we have Sii = 1, but Sik vanishes only for orbitals i = k which are localized on identical atoms. We have to find now the minimum of ", which is as close as possible to the true energy eigenvalue W within the given class of functions. At this minimum, ∂" ∂" = ∗ =0 ∂ci ∂ci must hold for all i. We rewrite (3.98) as ci∗ ck Hik = " ci∗ ck Sik , i

k

i

k

and differentiate in respect of ci∗ . With ∂"/∂ci∗ = 0 we obtain (Hik − "Sik )ck = 0

(3.101)

k

after some reordering for the optimal set {ck }. This homogeneous system of linear equations has nontrivial solutions only if det(Hik − "Sik ) = 0.

(3.102)

This is called the characteristic equation for ". The solutions "γ (i.e. the roots of a polynomial) are the sought-after energies of the electronic system. To obtain the MOs one inserts a specific solution "γ (for a given state γ ) into the system of linear γ equations (3.101). The solutions are the coefficients {ck } to be inserted into (3.96).

3.5

Molecular Orbitals

181

Fig. 3.27 Overlap (hashed) of 1s atomic H orbitals localized at the two nuclei A and B of the H+ 2 molecule, respectively

3.5.2

ΦB

ΦA

A

R

B

Specialization for H+ 2

The most simple approach to the lowest MOs is to superpose just two 1s atomic hydrogen orbitals, each centred at one of the two protons: on proton A: ΦA = Φ1s (rA ) = on proton B:

ΦB = Φ1s (rB ) =

1 a0 π

1 a0 π

1/2 1/2

e−rA /a0

A

B

e−rB /a0

A

B

Since ΦA and ΦB are normalized, SAA = SBB = 1 holds and we just have to compute the overlap integral ∗ S(R) = SAB (R) = Φ1s (rA )Φ1s (rB )d3 r (3.103) indicated in Fig. 3.27. This two centre integral has limiting values S = 0 and R→∞ S = 1. R→0

To compute Hik we exploit the fact that ΦA and ΦB are eigenfunctions of the H atoms. We indicate this in the Hamiltonian (3.94):

A =H

1 1 1 1 2 el = − e − H − + . 2 r rB R

A B =H

We define matrix elements Hik between two orbitals (where i, k = A or B): ∗ el Φ1s (rk )d3 r. Hik = Φ1s (ri )H For symmetry reasons HAA = HBB and HAB = HBA . Explicitly 1 1 ∗ ∗ A Φ1s (rA )d3 r A − Φ1s (rA )H (rA ) Φ1s (rA )d3 r A + HAA = Φ1s rB R W1s Φ1s (rA )

1 = W1s + − R

∗ Φ1s (rA )

1 1 Φ1s (rA )d3 r A = W1s + − C(R). rB R

(3.104)

182

3

Fig. 3.28 Energy scheme for H+ 2 : increase and decrease of the 1s energies for the 1σu∗ and 1σg MO, respectively

Diatomic Molecules

H AA – H AB 1– S W1s A

ɕu =1σ*u ɕg = 1σg

W1s B

H AA + H AB 1+ S

The so called C OULOMB integral C(R) is positive and −C(R) represents simply the interaction of the electron charge distributed around nucleus A with the positive charge of nucleus B – it disappears for large R and compensates for small R the C OULOMB repulsion of the nuclei. The matrix element HAB = HBA is called the resonance integral: HAB =

∗ el Φ1s (rB )d3 r Φ1s (rA )H

=

∗ Φ1s (rA )

1 1 − + HB Φ1s (rB )d3 r R rA

W1s Φ1s (rB )

1 1 ∗ S(R) − Φ1s (rA ) Φ1s (rB )d3 r R rA

1 S(R) − K(R) = HBA . HAB = W1s + R = W1s +

(3.105)

K(R) is a kind of exchange integral. For not too small R one finds HAB < 0, since S(R) decreases with increasing R and the term −K(R) dominates. In summary, with Sii = 1 we obtain for the characteristic equation (3.102) the determinant HAA − " HAB − "S (3.106) HAB − "S HAA − " = 0. It has two solutions for ": HAA − HAB and 1−S HAA + HAB "g = . 1+S

"u =

(3.107) (3.108)

With HAB < 0 this leads to the energy scheme sketched in Fig. 3.28. Inserting these values for " into the system of Eqs. (3.101) allows one to compute the coefficients cA and cB : 1 (g) (g) cA = cB = √ 2(1 + S)

1 (u) (u) and cA = −cB = √ . 2(1 − S)

(3.109)

3.5

Molecular Orbitals

Fig. 3.29 The two lowest LCAO-MOs for H+ 2 constructed from Φ1s (rA ) and Φ1s (rB ). The finite electron density between the atomic nuclei A and B for the φg -MO leads to molecular bonding

183 | g |2

| u |2

g

u

A

B

A

B

H2+

We thus find two different (lowest) LCAO-MOs for H+ 2 : one which is symmetric in respect of inversion r → −r, the so called even or “gerade” state (g) and one which changes its sign upon inversion, the odd or “ungerade” state (u): 1 Φ1s (rA ) + Φ1s (rB ) φg = √ 2(1 + S(R)) 1 φu = √ Φ1s (rA ) − Φ1s (rB ) . 2(1 − S(R))

(3.110) (3.111)

Figure 3.29 illustrates these two MOs (more about the g − u symmetry in Sect. 3.5.4). The electron charge density becomes −e ∗ ∗ Φ1s (rA ) ± Φ1s (rB ) Φ1s (rA ) ± Φ1s (rB ) 2(1 ± S) 2 2 −e Φ1s (rA ) + Φ1s (rB ) = 2(1 ± S) ∗ e ∗ ∓ Φ1s (rA )Φ1s (rB ) + Φ1s (rB )Φ1s (rA ) , 2(1 ± S)

−e|φg,u |2 =

(3.112)

overlap of the atomic orbitals

where the sign ± refers to the (g) and (u) state, respectively. These two wave functions (MOs) have cylinder symmetry around the molecular axis. We summarize their key properties: • The φg = 1σg orbital with the energy "g (R) has even inversion symmetry, i.e. the sign of the wave function does not change upon inversion r → −r. It describes the electronic ground state of H+ 2 and is a bonding orbital. • The φu = 1σu∗ orbital with the energy "u (R) has odd inversion symmetry, i.e. the sign of the wave function changes upon inversion r → −r. It is an antibonding orbital, indicated by labelling it *.

184

3

Diatomic Molecules

The negative sign of HAB is responsible for the bonding of the symmetric orbital, hence with (3.105) eventually the exchange integral is 1 ∗ (rA ) Φ1s (rB )d3 r. (3.113) K(R) = Φ1s rA Crucial for the bonding of an orbital is the overlap of Φ1s (rA ) and Φ1s (rB ), i.e. the second term in the charge density (3.112). This is illustrated in Fig. 3.29. For H+ 2 the energies "g and "u according to (3.107) and (3.108) may explicitly be computed as a function of R. This leads to a first approximation for the potentials, the equilibrium distance R0 , and the bond energy De . If we insert the integrals (3.104) and (3.105) we find: "g,u (R) =

HAA ± HAB W1s + = 1±S

1 R

− C(R) ± [(W1s + R1 )S(R) − K(R)] 1 ± S(R)

"u (R) = W1s +

C(R) K(R) 1 − + R 1 − S(R) 1 − S(R)

(3.114)

"g (R) = W1s +

C(R) K(R) 1 − − . R 1 + S(R) 1 + S(R)

(3.115)

For the φg orbital, the finite charge density between the atoms is clearly visible in Fig. 3.29. This leads to bonding of the H+ 2 molecule. The two centre integrals S(R), C(R) and K(R) can be evaluated analytically in elliptic coordinates. We just communicate the results (in a.u.):

1 2 −R K(R) = [1 + R]e−R S(R) = 1 + R + R e , 3 C(R) =

1 1 − (1 + R)e−2R . R

As documented in Fig. 3.30 one obtains with this most simple LCAO ansatz (dashed red line) already a reasonable guess for the potential in the bonding ground state 1σg 2 Σg+ state, with R0 0.132 nm and De 1, 77 eV. The potential for the repulsive, antibonding state is also plotted in Fig. 3.30. Interestingly, for the H+ 2 molecule a complete analytical solution is possible (the only molecule at all for which this can be done). The Hamiltonian Hel can be separated in confocal elliptic coordinates. One obtains for the ground state R0 = 0.106 nm and De = 2.79 eV, the experimental values R0 0.106 nm and De 2.65 eV are very close (full red line in Fig. 3.30).

3.5.3

Charge Exchange in the H+ 2 System

The symmetry properties of the H+ 2 molecule give rise to a number of remarkable interference phenomena which may directly be observed experimentally. They relate

3.5

Molecular Orbitals

185

W (R)/Eh

2Σ +

u

3

0.10

1σu* LCAO 1σg LCAO

0.05 0.00

1

2

De

R0

2 1

3

4

- 0.05 - 0.10

W (R )/eV

exact

5

R / a0 -1 -2

2Σ +

g

exact

-3

Fig. 3.30 Potentials for the H+ 2 molecule: dashed red lines give LCAO orbital energies in the simplest form. The full red lines show for comparison the exact potentials according to S HARP (1971). For the bonding ground state (1σu and 2 Σg+ ) LCAO gives only a rough first order approximation. For the repulsive state, exact (2 Σu+ ) potential and approximation (1σu∗ ) are practically identic. The energy zero has been fixed for the dissociated atoms in their ground state

to the fact that the molecular states may have even or odd symmetry. In both cases the probability to find the electron at one or the other proton is equal. Quantum mechanically the two positions can, strictly speaking, not be distinguished. However, if the nuclei are separated by a collision or a photoinduced dissociation process, at some point during the separation process the electron has to ‘decide’ at which of the nuclei it wants to remain for good.

Charge Exchange in the Collision Process H+ + H The classical example are collisions between a proton and a hydrogen atom, first investigated by L OCKWOOD and E VERHART (1962). A fast proton beam passes through a target gas of H atoms. If the two atomic nuclei come close enough (“close encounter”), for a short time an H+ 2 molecule is formed. At this point one can no longer distinguish which nucleus carries the electron, the charge cloud may thus be exchanged between the two protons. Schematically one may distinguish two processes: " H + H+ elastic collision (a) (3.116) H + H+ → H+ 2 # H+ + H charge exchange (b) This is illustrated in Fig. 3.31 by ‘snapshots’ as seen in the centre of mass system. After the collision one detects the charge exchange by detecting the newly formed, fast H atoms, which have been scattered into a specific (small) angle θ . In this particular experiment θ = 3◦ . In the experiment one first selects all particles scattered at the angle θ by an aperture, thus discriminating scattered from unscattered particles. The respective signal, S(a) + S(b) , is detected by a particle multiplier (see Appendix B.1). Then one deflects from the scattered particle beam all protons with the help of an electric field, and detects only the fast H atoms (signal S(b) ). In

186

3

Fig. 3.31 Charge exchange in H+ + H collisions schematic, with the momenta of the particles p A,B and p A,B before and after the process; prior to collision (a), the electron charge is attached to one of the protons; during the collision (b) temporarily an H+ 2 molecule is formed; after the collision the system is found either in configuration (c) or (d)

H e-

(a)

Diatomic Molecules H+ pB

pA H2+ p' A

t p'B H

(c) p'B

e- p'A H+

( )

eH+

( ) p'B

p'A

eH

this manner one determines that a close encounter has happened. The probability for charge exchange during such an interactions is thus we = S(b) /(S(a) + S(b) ). At the time of the close encounter one can, in principle, not distinguish whether the system H+ +H moves on the potential curve belonging to φg (1σg ) or to φu (1σu ). We have a situation very similar to a “double slit interference experiment”. The probability amplitudes for both possibilities have to be added and this leads to typical interference structures. Let us have a closer look at this process. Prior to the collision (t → −∞, or R → ∞) the electron is localized at proton A, as sketched in Fig. 3.31(a). With (3.110) and (3.111) we may express this in the molecular H+ 2 picture: 1 Φ1s (rA ) ≡ √ (φg + φu ). 2

(3.117)

If we want to describe the temporal evolution of the system, we have to account for the different time dependence of the two states involved: "g (R) "u (R) t and φu (R, t) = φu exp −i t . φg (R, t) = φg exp −i The energies "g,u (R) for g and u states (i.e. the potentials shown in Fig. 3.30) split as the two protons approach each other. Correspondingly, the initial wave function (3.117) evolves with time: 1 φ(t) √ φg exp(−i"g t/) + φu exp(−i"u t/) . 2 With Wgu = "g (R) − "u (R) and W = ("g (R) + "u (R))/2 we rewrite this:

Wgu t Wgu t iW t 1 + i(φg − φu ) sin exp − . (3.118) φ(t) √ (φg + φu ) cos 2 2 2

3.5

Molecular Orbitals

187

Fig. 3.32 Electron exchange probability we in H + H+ collisions experimentally observed by L OCKWOOD and E VERHART (1962); plotted is the probability of charge exchange as a function of the inverse relative velocity (kinetic energy of the proton 0.5 to 50 keV)

we

20.1

3.92 1.57 0.78 keV h h h ⎯⎯ ⎯⎯ ⎯⎯ 〈aWgu〉 〈aWgu〉 〈aWgu〉

1.0 0.8 0.6 0.4

h

⎯⎯ 〈aWgu〉

0.2

h

⎯⎯ 〈aWgu〉

0 0

1

H+ + H → H + H+ 3º exchange scattering

2 1/v / 10-6 m-1 s

3

The phase factors sin(Wgu t/2) and cos(Wgu t/2) appear and disappear in opposition and φ(t) oscillates between 1 √ (φg + φu ) Φ1s (rA ) 2

and

1 √ (φg − φu ) Φ1s (rB ). 2

Correspondingly, during the close encounter, the electron ‘oscillates’ between proton A and proton B back and forth. Strictly speaking, the ansatz (3.118) is valid only if the energy splitting Wgu between the g and u state is constant. But for a rough guess we may replace Wgu t by an average value Wgu t → Wgu t Wgu tcol = Wgu × a/v

(3.119)

where tcol = a/v is an effective interaction time, a an effective interaction length, and v the relative velocity of the interacting particles. In more detail, we expect to obtain a reasonable approximation with Wgu t =

∞

−∞

Wgu (R)dt =

1 v

∞

−∞

Wgu (R)dR = aWgu /v.

(3.120)

Somewhat oversimplified we have identified the relative velocity of the particles with v = dR/dt, i.e. we assume a straight line trajectory along the internuclear axis. After the collision the wave function (3.118) of the system may again be recast into a superposition of the atomic orbitals Φ1s (rA ) and Φ1s (rB ): aWgu 1 aWgu 1 + iΦ1s (rB ) sin π . lim φ(t) = Φ1s (rA ) cos π t→∞ h v h v

188

3

Diatomic Molecules

The probability to find the electron after the collision at A or B is given by the square of the respective amplitudes. The probability for electron exchange is thus aWgu 1 . we = sin2 π h v

(3.121)

Correspondingly, we expect an oscillatory behaviour of the exchange probability as a function of 1/v. This is exactly what one observes in the experiment, as documented in Fig. 3.32 by the original data. We see very pronounced maxima and minima for the exchange probability, even though, due to finite angular resolution, the minima and maxima do not reach 0 and 1, respectively, as predicted by (3.121). The model predicts maxima of charge exchange for πaWgu 1 1 = n+ π h v 2

i.e. for

h 1 1 = n+ . v aWgu 2

(3.122)

In the experiment one reads between the maxima or between the minima a difference of (1/v) 6.6 × 10−7 m−1 s on the inverse velocity scale (with only a slight variation). This corresponds to aWgu = h/(1/v) 6.27 eV nm. From the potential energy diagram Fig. 3.30 one estimates an average distance between 1σg and 1σu∗ potential of about Wgu 10 eV and an effective interaction length of about 12a0 = 6.3 nm – in plausible agreement with the experiment. We note, however, that the first maximum is observed at a phase 2.4 and not at π/2 1.57 as predicted by (3.122) – which shows the limitations of such a simple model.

Photo-Dissociation of H+ 2 Induced by Ultrashort Laser Pulses Another very nice example for such charge oscillations is a more recent experiment by K LING et al. (2006), studying the laser induced dissociation of D+ 2 (see ). State-of-the-art ultrafast (FWHM 5–7 fs), inalso K REMER et al. 2009, for H+ 2 tense (1 × 1014 W cm−2 ) laser pulses (at λ 800 nm) are used with only a few oscillation cycles to first ionize D2 molecules, and then to dissociate the D+ 2 system. We cannot enter into the details of this sophisticated experiment. It combines a velocity map imaging (VMI) detection system for measuring the kinetic energy of the D+ fragment ion with phase stabilizing technique for ultra short pulses. Figure 3.33(a) and (b) show some experimental results together with a model calculation in Fig. 3.33(d) and (e). The measured D+ ion signal is plotted in Fig. 3.33(a) as a function of the kinetic energy. While for this overview spectrum the carrier envelope phase φc is not stabilized, the asymmetry determination shown in Fig. 3.33(b) is only possible with phase stabilized pulses. We have already mentioned in Sect. 1.4.1 the importance of this carrier envelope phase φc for very short pulses. It is again illustrated in the electric field profiles sketched in Fig. 3.33(c), where the electric field ∝ exp −(t/τG )2 cos(ωc t − φc )

3.5

Molecular Orbitals

189

kinetic energy of D+ / eV

measured asymmetry 12 10

02 0.2

(a)

(b)

calculated relative electron delocalization 0.5

8

-5 0 laser pulse amplitude

0

6

5

10

(d)

0

4

15

t / fs

2 0

signal

(c) laser pulse shape

0

1

2

3

4 ϕc / π

- 0.2 - 0.5

A A B

(e)

B charge asymmetry (schematic)

Fig. 3.33 Experiment and model calculation for the dissociation of D+ 2 by phase stabilized, ultrashort laser pulses (5 fs) according to K LING et al. (2006) (for details see text)

is shown. The light is linearly polarized. In the experiment one selects preferentially such dissociation processes for which the molecular axis is aligned parallel to the laser field. Let us think these molecules to be align parallel to the vertical axis in the paper plane. All three pulses have the same carrier frequency ωc and the same pulse duration characterized by a Gaussian as discussed in Sect. 1.4.1. They differ only in respect of the carrier envelope phase. While for φc = 0, 2π etc. the field points at the maximum of the carrier ‘up’ in respect of the molecule, its direction points ‘down’ for φc = π, 3π etc. For φc = π/2, 3π/2 etc. the highest field strengths are of equal magnitudes in up and down direction. The key issue is now, to observe whether the phase has an influence on the dissociation process. The results shown in Fig. 3.33(b) give clear evidence that it does! Plotted is the asymmetry of the measured ion signal (SA − SB )/(SA + SB ) for ions which leave the D+ 2 in the direction ‘up’ (SA ) or ‘down’ (SB ), respectively. This asymmetry (colour code) is plotted as a function of both, the kinetic energy (vertical axis) and the phase φc (horizontal axis). The influence of the phase is amazingly clear for kinetic energies between ca. 2 eV and 8 eV, while for lower energies no asymmetry is observed. These processes too may be understood on the basis of the potential energy diagram Fig. 3.30 for the H+ 2 molecule. We cannot analyze here the quite complex reaction processes which lead to dissociation and generate the rather substantial relative kinetic energy of the two nuclei. But obviously, the laser pulse produces a range of wave-packets which describe the two dissociating atomic nuclei with different initial kinetic energies. Low kinetic energies may be attributed to processes where only the repulsive 1σu potential of the D+ 2 is populated. One may describe this fully within the B ORN -O PPENHEIMER approximation and speaks about an “adiabatic” process: the two atomic components of the molecule just separate on the 1σu potential. Since in this orbital (3.111) the electron has equal probability to be localized at atom A or B, no asymmetry is observed. This is different for higher kinetic

190

3

Diatomic Molecules

energies. Under the influence of the laser field the 1σu and 1σg are strongly coupled and the negative charge oscillates between the two states. Very similar to the charge exchange collisions discussed above, this implies oscillation of the electron density between the two atoms A and B, indicated in the (very schematic) sketch Fig. 3.33(e) as a function of time for the separating protons. The result of corresponding model calculations is shown in Fig. 3.33(d) and verifies these considerations quantitatively. In the example shown (φc = 0), for large times the electron charge is found preferentially at atom B (down). The detector then detects preferentially D+ ions ejected up (ion A). This depends strongly on the carrier envelope phase φc as shown in the 2D plot Fig. 3.33(b). In the model calculation too, for φc = π the directions A (up) and B (down) a just reversed (not shown here) and one expects preferentially to observe the D+ ion in direction B. This is verified in the experiment.

3.5.4

MOs for Homonuclear Molecules

In the following we treat the building-up principles for molecular orbitals for diatomic molecules. This concept corresponds to the filling of the n shells in the atomic case (Sect. 3.1 in Vol. 1), and leads to rules which may be considered a periodic system for molecules.

Symmetry and Angular Momentum Atomic electrons move in a spherically symmetric potential. Thus the electronic wave functions (orbitals) may be separated into a radial and angular part – as we have done it so far very successfully: φnm (r) = Rn (r)Ym (θ, ϕ). In contrast, linear molecules have to be described in cylindrical symmetry. The potential in the electron Hamiltonian (3.94) does explicitly depend on the polar angle θ (via rA and rB ) and the atomic ansatz does no longer work. 2 2 el , L ] = 0, and the Hence, L no longer commutes with the Hamiltonian, [H orbital angular momentum quantum number is no longer a good quantum number. This does not really surprise us, as we know this situation already from the S TARK effect (see Sects. 8.2.3 and 8.2.8 in Vol. 1). We may actually view the H+ 2 molecule as a particular, limiting case of the S TARK effect: an H atom in the (very strong) electric field of a proton! Here, as in Sect. 8.2.8, Vol. 1, the projection of L onto the molecular axis (taken el , L z ] = 0. Thus, the eigenvalues of as z-axis) is still a conserved quantity, i.e. [H z are still m as in the atomic case, and m remains a good quantum number. L Adapted to the natural symmetry one writes the electronic wave functions for diatomic molecules in cylindrical coordinates (ρ, z, ϕ), with ρ = r sin θ and

3.5

Molecular Orbitals

191

z = r cos θ (again in a.u.): (±λ)

φel

√ (ρ, z, ϕ) = ∓φγ λ (ρ, z) exp(±iλϕ)/ 2π

z φ with L el

(±λ)

(±λ)

(ρ, z, ϕ) = ±λφel

(3.123)

(ρ, z, ϕ).

We use the standard phase convention and assume that φγ λ (ρ, z) is normalized to unity. The new quantum number introduced here is λ = |m |, since the sign of m has no influence on the energy (just as in the case of the S TARK effect). The angular part of the electronic wave functions may thus be constructed as introduced by (D.6), Vol. 1 for the real representations of the spherical harmonics. The quantum number λ is used to characterize one electron wave functions (MOs) for diatomic molecules. It thus replaces, so to say, the quantum number for atoms. Following the atomic designation s, p, d etc. for orbitals the following notation is used: λ

0 σ

1 π

2 δ

3 φ

...

(3.124)

Correspondingly, to characterize the total molecular state with several electrons one uses capital Greek letters. The quantum number for the projection of the total angular momentum on the molecular axis is called Λ (Lambda), its values are designated as follows: Λ

0 Σ

1 Π

2

3 Φ (

...

(3.125)

Since angular momenta( are added vectorially, Λ ≤ λi holds, just corresponding to the atomic case L ≤ i . Another allowed symmetry operation with homonuclear molecules is (as we have already discussed above) inversion at the centre of mass, r → −r. In cylinder coordinates this corresponds to z → −z and ϕ → ϕ + π . The states are thus distinguished according to their parity “g” (even) and “u” (odd): even: φg (r) = φg (−r) odd: φu (r) = −φu (−r). Even though LCAO-MOs are not the very best approximation for estimating R0 and De , they provide a good 1st order guide for the construction of molecular orbitals. Important is in this context that the atomic orbitals involved in constructing a particular MO must all have the same symmetry in respect of the molecular axis. Only in this case the overlap integral does not disappear Sik = 0. For the construction of MOs from atomic s and p electrons the following possibilities exist (for compact writing we use somewhat loosely |s, |px , |py , |pz for AOs and |σg,u and |πg,u etc. for MOs; antibonding orbitals are designated with an

192

3

(a)

σg (ns) y +

+ + s±s +

z

y

–

–

+ –

– z

–

y –

x

+

–

–

y –

π*g (np) z

σg (np)

–

+

z x

+

(c)

pz ± pz

–

+

–

σ*u (ns)

πu (np)

+

–

+

z

x

+

+

py ± p y

+

–

y

(b)

x

z

Diatomic Molecules

–

z

+

+

+

x σ*u (np) –

+ z

Fig. 3.34 Construction of molecular orbitals from |s and |p atomic orbitals: (a) |s ± |s, (b) |py ± |py (y ⊥ plane), (c) |pz ± |pz . Red shaded are the positive, dark-grey the negative areas of the wave function. In each case the upper MOs are bonding, the lower ones antibonding (denoted by *)

asterisk *): |s + |s → |σg and |s − |s → σu∗ |pz + |pz → σu∗ and |pz − |pz → |σg |py + |py → |πu and |py − |py → πg∗ . The πu and πg∗ orbitals are each twofold degenerate, as they may also be obtained from |px + |px and |px − |px , respectively. Figure 3.34 illustrates this orbital construction scheme. Antibonding orbitals always have a nodal plane perpendicular to the molecular axis (indicated in Fig. 3.34 by dashed lines). It is also important to note, that the bonding σ orbitals have even symmetry, while the bonding π orbitals have odd symmetry. It should be noted at this point that such schemes of molecular or atomic orbitals as sketched in Fig. 3.34 may be somewhat misleading at times. They typically represent the magnitude of the wave function (or its absolute square) at given angles θ and ϕ in space seen from the origin, for a fixed value of the radial distance. Therefore, they do not necessarily reflect the spatial extension of the orbitals which would require at least a contour plot such as shown in Fig. 3.29.

Correlation Diagrams To obtain an overview of the relative position of the energy levels one may consider the two limits: “united atom” on the one hand, where the two nuclei are arbitrarily close to each other with a united charge and “separated atoms” on the other hand, with infinite distance of the two constituents and no interaction. In between, the

3.5

Molecular Orbitals

193

Fig. 3.35 Correlation of atomic orbitals and molecular orbitals between united atom and separated atoms. Positive regions of the wave functions are pink shaded, negative ones grey. Antibonding MOs are designated by an asterisk *. Depending on whether the molecular nature of the MOs or the LCAO view is more relevant one writes them in one or the other notation

united atom

molecular orbital

separated atoms

p

3p σ*↔ σu* 2p

p+p

d

3d π* ↔ π*g 2p

p-p

p

2p π ↔ π u 2p

p+p

s

2s σ ↔ σg 2p

p-p

p

2p σ*↔ σu* 1s

s-s

1s σ ↔ σg 1s

s+s

s

Fig. 3.36 Correlation diagram for the orbital energies of diatomic molecules. The abscissa corresponds to the nuclear distance, the ordinate reflects (very schematically) the potential energies. The dashed lines indicate the true MO energies at equilibrium distance for a number of specific molecules. The notation of the MOs corresponds on the left with the united atom, on the right with the separated atoms and in the middle one just counts the different symmetries and λ from bottom to top

united atom (R = 0) 3p 3s

2p 2s

1s

molecule 3 σu* 1π*g

3p π 3p σ* 3s σ 2p π 2p σ* 2s σ

separated atoms (R → ∞)

π*g 2p

3 σg 1 πu

2σu*

1σu*

σ*u 1s

1σg H2 He2 R

πu σg 2p σ*u 2s

2p

2s

σg 2s

2 σg

1s σ

σ*u 2p

1s

σg 1s Li2

F2 B2 N2 O2 C2 + + N2 O2

molecular orbitals are formed as just discussed. We summarize the two perspectives in Fig. 3.35. The energetic association of the MOs to united and separated atoms leads to so called correlation diagrams, which allow a semi-quantitative discussion of the energetics for the different MOs. Figure 3.36 shows schematically the potential energy

194 Fig. 3.37 Ordering of energies for molecular orbitals. For the lighter molecules the scheme (a) holds, for O2 and heavier molecules the ordering within the 2p levels changes and (b) is valid. Dashed levels indicate the degeneracy of a state. The total number of states in the isolated atoms A and B is identical to the total number of MOs

3 3 * 2p

* g

1 3 g

3 * 2p

2p

2s

2s

1s

1s

B

A

1

2s

1s A

2 * 2 g 1 * 1 g

Diatomic Molecules

(a)

1 *g 1 3 g 2 * 2 g 1 * 1 g

2p

2s

1s B

( )

as a function of the nuclear distance for the general homonuclear, diatomic case. On the left and on the right one marks the energies of the corresponding atomic states (ignoring the C OULOMB repulsion of the two nuclei). The connections between the respective energies for united atom and separated atoms leads to a prediction about the trend for potential energies of the molecular orbitals. A few rules have to be observed: 1. Only orbitals with the same quantum number λ are connected. 2. Parity must also be conserved (g ↔ g and u ↔ u). 3. Potential curves with equal symmetry don’t cross each other. The non-crossing rule 3 has already been derived in Sect. 8.1.6, Vol. 1. From the correlation diagram Fig. 3.36 one may read the energies for different MOs as a function of internuclear distance. All MOs correlated to the 1s, 2s and 2p AOs in the limit of separated atoms are shown for the general case. For infinite distance they correspond to the three respective atomic energies, in the case of the united atom to eight. Also indicated in Fig. 3.36, by vertical dashed lines, are the orbital energies for some important diatomic molecules from the first row of the periodic system of elements. If one is interested in the details for a specific molecule one has, of course, to draw such a correlation diagram starting with the correct numerical data for this particular system. One finds e.g., that for light atoms up to N2 the 1πu orbital is energetically lower than the 3σg orbital, while for O2 and larger molecules this reverses as summarized in Fig. 3.37(a) and (b), respectively.

Filling the Orbitals Just as in atoms, the orbitals for different electrons must differ by at least one quantum number as a consequence of the PAULI principle. When building up molecules, each of the singly degenerate orbitals σg,u may be filled with 2, each of the two fold degenerate πg,u orbitals with 4 electrons. When filling the orbitals correspondingly, one obtains the H+ 2 molecule, the H2 and He2 etc.; a kind of periodic system of

3.5

Molecular Orbitals

Fig. 3.38 Filling the molecular orbitals with electrons for the smallest, homonuclear diatomic molecules; shown is the occupation of the MOs in the electronic ground state; the arrows indicate the spin orientation of the electrons

195 3 σu* 1π*g 3 σg 1πu 2σu* 2 σg 1σu* 1σg

+

H2

H2

+

He2

He2

Li2

Be2

B2

C2

molecules emerges. Slightly different notations are used in the literature: e.g. the σu∗ 2px MO is also referred to as 2px σu∗ or simply just as 3σu∗ . The highest occupied molecular orbital in any given molecule is called HOMO, the lowest unoccupied molecular orbital is called LUMO. Figure 3.38 illustrates schematically how the lowest orbitals are filled for the molecules H+ 2 up to C2 . As in the periodic system of elements, there are some irregularities, here e.g. for B2 . The electron configuration of some further molecules is given in Table 3.7 along with a summary of bonding properties. The 1σg orbital is bonding and the 1σu∗ + orbital is antibonding. Thus, H+ 2 , H2 and He2 are stable molecules, since they own more bonding than antibonding electrons in their ground state. Among these, H2 has most bonding electrons, and consequently the smallest equilibrium distance R0 and the highest bond energy De . More general, the strength of a bond may be estimated from the so called bond order, which is defined as the difference between the number of bonding and anti+ bonding electrons divided by 2, i.e. (n − n∗ )/2. H+ 2 and He2 have bond order 1/2, for H2 it is 1, and He2 with the bond order 0 should not form a stable molecule at all – according to this simple rule. As we have already discussed in Sect. 3.2.6 one observes indeed only a very weakly bonded VAN DER WAALS molecule. The same scheme can be used to derive the periodic system of molecules for other homonuclear diatomic molecules. As mentioned above, the energetic ordering of the MOs changes between N2 and O2 (see correlation diagram Fig. 3.36 and term scheme Fig. 3.37). As documented in Table 3.7 the bond lengths beyond Li2 are all + larger than for H+ 2 , H2 and He2 due to the more extended valence orbitals (n = 2). One particularly interesting point is the fact that O2 has two unpaired electrons, a consequence of H UND’s rules which we have introduced already for atoms (see in particular Chap. 7 in Vol. 1). It says, that among otherwise identical electron configurations, states with the highest total spin have the lowest energies: electrons are first filled into all energetically degenerate orbitals before one of the orbitals is doubly occupied. The electrons of the singly occupied orbitals have the same orientation. In the case of oxygen they form a triplet (S = 1). Thus, O2 is paramagnetic, in contrast e.g. to N2 . For an approximative treatment of the potentials for multi-electron molecules one usually may confine the efforts to a few outer electrons, the valence electrons, i.e. to the highest occupied orbitals. The other electrons are typically strongly localized

196

3

Diatomic Molecules

Table 3.7 Filling the MO shells in the most simple homonuclear, diatomic molecules (electron configuration in the ground state). The bond order is (n − n∗ )/2, R0 is the bond distance. Dissociation energies De and term energies Te refer to the potential minimum; usually Te = 0 for the neutral ground state, see (3.55) Molecule Electron config.

n−n∗ 2

H+ 2

(σg

H2

(σg 1s)2

1

He+ 2

(σg 1s)2 (σu∗ 1s)1 (σg 1s)2 (σu∗ 1s)2 He2 (σg 2s)2 He2 (σg 2s)2 (σu∗ 2s)2 He2 (σg 2s)2 (σu∗ 2s) . . . (πu 2px )2 (σg 2pz ) Be2 (πu 2px )2 (πu 2py )2 Be2 (πu 2px )2 (πu 2py )2 . . . (σg 2pz ) Be2 (πu 2px )2 (πu 2py )2 . . . (σg 2pz )2 N2 (πg∗ 2px )(πg∗ 2py ) N2 (πg∗ 2px )2 (πg∗ 2py )2 N2 (πg∗ 2px )2 (πg∗ 2py )2 . . . (σu∗ 2pz )2

1/2

He2 Li2 Be2 B2 C2 N+ 2 N2 O2 F2 Ne2 a See

1s)1

1/2

0 1 0 2 2 2 1/2 3 2 1 0

State De / eV 2Σ + g 1Σ + g 2Σ + u 1Σ + g 1Σ + g 1Σ + g 3Σ − g 1Σ + g 2Σ + g 1Σ + g 3Σ − g 1Σ + g

R0 / nm Te / cm−1

2.65

0.1052 125 443

4.48

0.074

0

2.47

1.08

178 400

0.00095a 0.297a

0

1.07

0

0.267

not observed 3.0

0.159

0

6.32

0.1243 0

8.85

0.112

9.90

0.1098 0

5.21

0.121

0

1.66

0.141

0

125 744

not observed

Sect. 3.2.6

to the atomic nuclei and contribute only little to the molecular bonding. Of course, here as with atomic structure calculations the general rule holds, that in a rigorous computation the results become the better the more MOs (which are or possibly might be occupied) are accounted for. Section summary

• We have learned how molecular orbitals can be constructed as linear superpositions of atomic orbitals (MO from LCAO). Using the variational method to minimize the energy, one has to determine the Hamiltonian matrix (3.99) and the overlap integrals (3.100) in the atomic basis. This leads to the characteristic equation (3.102) from which the eigenenergies of the orbitals are derived. • We have detailed this for the simplest molecule, H+ 2 . By including only the two 1s AOs localized on either of the protons, the two energetically lowest MOs are found; they have even (1sσg ) and odd (1sσu∗ ) inversion symmetry (r → −r). The 1sσg is bonding, the 1sσu∗ antibonding (repulsive potential). A surprisingly good first guess for the molecular potential is obtained as shown in Fig. 3.30. • These two molecular potentials give rise to interesting interference phenomena in charge exchange collisions and in ultrafast photo-dissociation. • The systematic construction of MOs for a range of homonuclear, diatomic molecules starts by writing the wave function (3.123) in cylinder coordinates, the key quantum number λ being the projection of the angular momentum

3.6

Construction of Total Angular Momentum States

197

onto the internuclear axis. We have constructed a few characteristic MOs from s and p atomic orbitals (Fig. 3.34). • MOs must have a finite electron density in between the atoms to be bonding. • Correlation diagrams for MOs give an overview of the energetics; they connect the energies of the united atom with those for separated atoms (Fig. 3.36). Based on this one may derive some kind of periodic system for homonuclear, diatomic molecules (Table 3.7), which allows to make predictions about the strength, symmetries and spin properties of the molecular bonding. As a prominent example, the electronic ground state of molecular oxygen is found to be a triplet, hence O2 is paramagnetic.

3.6


3.6.1

Total Orbital Angular Momentum

z is a conserved For the total orbital angular momentum too, only the z-component L quantity. It is obtained as the sum of the z-components of the individual MOs: z = zi with eigenvalues ML = L mi . (3.126) L i

i

The designation Λ = |ML | with the term notation Σ, Π , etc. for Λ = 0, 1, 2 has already been introduced above. We note here in addition, that molecular orbitals have a well defined reflection symmetry in respect of planes through the molecular axis. This reflection symmetry must remain conserved when composing the total wave function. When combining several MOs to an overall molecular state positive and negative mi may contribute in different combinations, so that the states are somewhat more complex than suggested by (3.126).

3.6.2

Spin

As in the atomic case the total spin is the sum of the individual spins of the electrons S i with | S= S| = S(S + 1) i

and results in a multiplicity 2S + 1 of the overall electronic states. The projection of S onto the molecular axis is called Σ = Ms with positive and negative values in contrast to Λ. For each S there are 2S + 1 values Σ = S, S − 1, S − 2, . . . , −S. Caution: do not mistake this spin quantum number Σ for a Σ state (total angular moment projection Λ = 0)!

198

3

Diatomic Molecules

In analogy to the term designation for atoms, molecular states are denoted 2S+1

(3.127)

Λg,u .

We recall briefly and symbolically the construction of spin states which has been discussed extensively in Vol. 1 (here for one and two electron systems): Doublet

Singlet

One electron

Two electrons

S=

1 2,

MS = ± 12

Triplet

S = 0, MS = 0

1/2 |↑ = |χ1/2

S = 1, MS = 1, 0, −1 |↑↑ = |χ11

|↑↓−|↓↑ √ 2

= |χ00

−1/2

|↑↓+|↓↑ √ 2

(3.128)

= |χ10

|↓↓ = |χ1−1

|↓ = |χ1/2

Of course, as in the atomic case, the PAULI principle has to be observed, i.e. the total wave function must be antisymmetric. For a system with two active electrons – e.g. for the H2 molecule – this implies – as for the He atom: for singlet states (two antiparallel spins with one antisymmetric spin function, 2nd column in Eq. (3.128)) the spatial electron wave function must be symmetric; for triplet states (two parallel spins, three symmetric spin functions, 3rd column in Eq. (3.128)) the spatial wave function must be antisymmetric.

3.6.3

Total Angular Momentum

The projection of the total electronic angular momentum J e of a molecular state onto the molecular axis (z-axis) is usually designated as Ω. Figure 3.39 indicates schematically how Ω is constructed as sum of orbital angular momentum L (Λ when projected onto the z-axis) and total spin S (Σ when projected onto the z-axis): Ω = |Λ + Σ|.

(3.129)

Thus, instead of the indices g or u according to (3.127), for more complex states one often finds Ω as an index. An additional complication comes with the rotation of the molecule: all relevant angular momenta have to be combined to a total angular momentum J of the whole system, applying the general rules for angular momentum coupling (see Appendix B in Vol. 1). Depending on the coupling between orbital angular momentum, spin and nuclear rotation several different possibilities exist – quite similar to the atomic case where we had to couple the orbital angular momentum, electron spin and nuclear spin. For molecules all these complications exist too, and rotation makes them even a little bit more complex. In Table 3.8 we summarize the angular momenta and

3.6

Construction of Total Angular Momentum States 3∆

Ω Σ =1

Λ

3

3∆

Λ

2

Ω

3∆

Λ

1

199

Ω Σ=0 Σ = −1

Fig. 3.39 Electronic angular momentum coupling in diatomic molecules for the example of Λ = 2 ( state) with S = 1 (triplet): only the components of the angular momenta in the direction of the molecular axis are good quantum numbers Table 3.8 Angular momenta and quantum numbers of diatomic molecules Kind of angular momentum

Operator

Quantum number Total

projection onto z-axis

Orbital angular momentum

L S

L

Λ

S

Σ

Electron spin Electron total angular momentum Nuclear rotation Total angular momentum Total without spin

Je = L + S N J = L + S +N K =L+N

Je

Ω

N

0

J

Ω

K

Λ

quantum numbers used in the following, essentially according to the notation of H ERZBERG (1989).10

3.6.4

H UND’s Coupling Cases

According to H ERZBERG (1989) one distinguishes several, so called H UND’s coupling cases, which are of key importance for the interpretation of rotational bands in electronic spectra. Here we can give only a brief first introduction and refer the interested reader to specialized literature, e.g. to H OUGEN (2001). H UND’s Case (a) As illustrated in Fig. 3.40(a) one assumes the coupling between molecular rotation and electronic orbital angular momentum L or spin S to be very weak. In contrast, the coupling of the orbital angular momentum to the internuclear axis is strong and the spin couples to the thus generated internal magnetic field parallel to the molecular axis. The situation corresponds to the electric analogue to the PASCHEN -BACK effect. As for the not rotating molecule we have Ω = |Λ + Σ|, together with the nuclear rotation N forms the total angular momentum and Ω J. Values of J < Ω are not possible and we have: J = Ω, Ω + 1, Ω + 2, . . . . 10 The

(3.130)

finite number of letters in the alphabet limits the choices, unfortunately. H OUGEN (2001) e.g. use the letters R instead of N , and N instead of K.

200

3

(a)

J

N

(b)

K

Ω Σ

Λ L

S

N

J

(c)

J S

Diatomic Molecules

N

L

K

N

Ω Je

Λ L

(d)

L

S

Fig. 3.40 H UND’s cases (a), (b), (c) and (d) for coupling angular momenta in diatomic molecules

Because of (3.129) in principle J may also be half integer (if the spin is half integer). In this coupling case the total energy (3.49) for a 2S+1 ΛΩ state is found to be (here without proof): Wγ vN / hc = Te + G(v) + F (J ) with F (J ) = Bv J (J + 1) − Ω 2 . (3.131) The electronic term energy Te may in addition be split by fine structure (cf. Eq. (6.36), Vol. 1): Te = T0 + aΛΣ.

(3.132)

Apart from the lowest state and the missing values J < Ω there is no difference between (3.131) and the rigid rotor (3.37), if one replaces N by J . For each fine structure level of an electronic state a specific ladder of rotational states exists. H UND’s Case (b) If the orbital angular momentum is zero, or if the electron spin is coupled only very weakly to the orbit, there is no coupling of the spin to the nuclear axis. As illustrated in Fig. 3.40(b) we have in that case practically just a couples with the projection of the orbital rigid rotor, whose angular momentum N The quantum number K replaces now the rotational angular momentum Lz to K. quantum number. For Σ states it is even identical with N . Only at the very end one has to account for fine structure coupling. With the usual rules the total angular momentum becomes: J = K + S, K + S − 1, . . . , |K − S|.

(3.133)

The energy of the rotational states is determined essentially by the nuclear rotation, possibly showing a small (2S + 1) fold FS splitting. For the particularly simple case of a 2 Σ state the rotational energy becomes 1 γ K for J = K + 2 2 γ 1 = B(v)K(K + 1) − (K + 1) for J = K − , 2 2

F (K) = B(v)K(K + 1) +

where the small constant γ is due to FS interaction.

(3.134)

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201

H UND’s Case (c) If the coupling of the orbital angular momentum to the internuclear axis is weak, the spin-orbit coupling of the electron cloud is not broken. As shown in Fig. 3.40(c) an electronic total angular momentum J e is formed, which as a whole couples to the internuclear axis. The projection onto z is again called Ω. This component of the electronic angular momentum finally couples again with the to the overall total angular momentum rotation N J . In its final result this case is hardly different from H UND’s case (a) and the energy terms and possible angular momentum quantum numbers J are indeed described by (3.131) and (3.130), respectively. H UND’s Case (d) Figure 3.40(d) shows the rather simple vector diagram for the case that the coupling of the orbital angular momentum L with the internuclear axis is very weak (e.g. for highly excited electrons). On the other hand one assumes that to the coupling of the molecular rotation N L is strong. One then observes primarily the energy spectrum of the rigid rotor, F (N) = B(v)N (N + 1), with where, however, the coupling of N L leads to a (2L + 1) fold splitting of each rotational level. L and S couple strongly with each H UND’s Case (e) Finally, it is also possible that other and form an electronic total angular momentum J e . However, the interaction with the internuclear axis is weak. This leads to a situation very similar to case (d). Except that the spitting of the rotational levels is now (2Je + 1) fold. In H ERZBERG (1989) one still reads about this case that – albeit thinkable – it has never been observed. This has changed by now, e.g. it has clearly been detected in the case of moderately high lying RYDBERG states of diatomic molecules such as O2 . Also, when discussing low energy collision processes between atoms one has to account for such a possibility, e.g. if collision induced fine structure transitions are observed. In the spectroscopic practice there are, of course, all kinds of transitions between the five pure cases introduced here. Higher precision requires, finally, also the inclusion of hyperfine interaction – if the atomic constituents have a nuclear spin. In Fig. 3.21 we have already seen an example of that. The analysis of such interactions follows essentially that for atoms (see Chap. 9 in Vol. 1). H UND’s cases have to be extended correspondingly.

3.6.5

Reflection Symmetry

For a range of questions it is important to know the reflection symmetry of the electronic states with respect of a plane through the molecular axis. For single orbitals this is rather clear and one can extend the considerations presented in Appendix D, Vol. 1 – replacing the quantum numbers q used in Appendix D.4, Vol. 1, or M

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Diatomic Molecules

in (D.14), Vol. 1 by λ. For multi-electron systems, the construction of states with well defined symmetries becomes somewhat more involved and we restrict the discussion here to a few important examples. We denote the reflection symmetry operator in respect of an ab plane as σˆ v (ab). In cylindrical coordinates these reflections correspond to coordinate replacements: σˆ v (xz) :

ϕ → −ϕ

(3.135a)

σˆ v (yz) :

ϕ →π −ϕ

(3.135b)

σˆ v (xy) :

z → −z.

(3.135c)

Inversion (which determines g or u symmetry) is achieved by ıˆ :

z → −z

and ϕ → ϕ + π.

(3.136)

For the simplest MOs these symmetries can be gleaned from Fig. 3.34. It is instructive, however, to elucidate this on the basis of wave functions for the orbitals. Typically, one uses real combinations of (3.123) as introduced with (D.6) in Vol. 1. For σ orbitals (λ = 0) we write (somewhat laxly in kets) |σ = φel (ρ, z, ϕ) =

φσ (ρ, z) , √ 2π

(3.137)

while there are two types of π orbitals (λ = 1): |πx =

φπ (ρ, z) cos ϕ exp(+iϕ) + exp(−iϕ) = φπ (ρ, z) √ √ 2 π π

(3.138a)

|πy =

φπ (ρ, z) sin ϕ exp(+iϕ) − exp(−iϕ) = φπ (ρ, z) √ . √ 2 πi π

(3.138b)

The angular part is normalized to unity in all cases. By applying (3.135a) and (3.135b), we see that σ orbitals have always positive reflection symmetry (no dependence on ϕ): σˆ v (xz)|σ = +|σ and σˆ v (yz)|σ = +|σ . In contrast, the πx,y orbitals have opposite reflection symmetries: σˆ v (xz)|πx = +|πx and σˆ v (yz)|πx = −|πx σˆ v (xz)|πy = −|πy and σˆ v (yz)|πy = +|πy . For multi-electron systems, we have to account for the indistinguishability of the electrons and the PAULI principle. We illustrate this now for a few examples.

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203

Example H2 Molecule In the most simple case we only have to fill one orbital with two electrons, as e.g. the 1sσ orbital for the ground state of the H2 molecule. With the notations (3.127) and those given in Fig. 3.35 (close to the united atom), the total electronic wave function is written: 1 + 0 Σ = χ φ1sσ (ρ1 , z1 ) · φ1sσ (ρ2 , z2 ) . (3.139) g 0 Since the spatial part is obviously symmetric in respect of exchange of the electrons, the spin function |χSMS must be antisymmetric, i.e. be a singlet. As for both electrons λ = 0 holds, also the total orbital angular momentum remains Λ = 0, i.e. we describe a Σ state. Obviously this function is also symmetric in respect of inversion, and finally, positive reflection symmetry for both 1sσ orbitals leads to overall positive reflection symmetry. This is indicated by the superscript + sign. The bound ground state of H2 is a singlet X 1 Σg+ state (the letter X is just used for numbering the states, and one start with X for the ground state, the letters A, B, . . . follow for excited states). Next we keep one electron in the 1sσ orbital and bring the other one in the next higher 2pσ ∗ orbital (σu∗ 1s in MO notation). The configuration |1sσ 2pσ ∗ may be combined to two different spatial functions: √ φ1sσ (ρ1 , z1 )φ2pσ (ρ2 , z2 ) ∓ φ1sσ (ρ2 , z2 )φ2pσ (ρ1 , z1 ) / 2. (3.140) Both wave functions have again Σ + reflection symmetry (they inherit this property from the constituents), but both change their sign upon inversion (due to the involvement of a 2pσ ∗ orbital). We also see that one of the functions is antisymmetric, the other symmetric in respect of exchange of the two electrons. The spin function thus has to be triplet or singlet, respectively. Closer inspection shows that the resulting |3 Σu+ state is repulsive (one bonding and one antibonding orbital). The other state does not lead at all to a low lying molecular state. Next we have a look at the configuration |1sσ 2pπ (two bonding orbitals with λ1 = 0 and λ2 = 1). We may construct from this e.g. an odd (u), bonding state, symmetric in respect of electron exchange. With Λ = λ1 + λ2 = 1 we obtain from (3.138a): 1 0 √ Πu = χ |1sσ 2pπx + |2pπx 1sσ / 2 (3.141) 0 0 iϕ = χ0 φ1σg (z1 ρ1 )φ1πu (z2 ρ2 ) e 2 + e−iϕ2 √ + φ1σg (z2 ρ2 )φ1πu (z1 ρ1 ) eiϕ1 + e−iϕ1 /(2 π ). With (3.136) we verify that we have indeed constructed a (u) state: for both electrons φ(zρ) = φ(−zρ) and replacing both azimuthal angles ϕ → ϕ + π brings an overall minus sign. Reflection symmetry in this case is positive in respect of the xz plane (πx like – just replace both ϕ → −ϕ). In contrast, rotation of the molecule around the z-axis by π/2 generates a state whose reflection symmetry is πy like, i.e. negative. The electronic problem is, however, completely symmetric in respect of the z-axis: both

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Diatomic Molecules

states |1 Πu+ and |1 Πu− are degenerate. Thus, one usually omits the symmetry designation ± for molecules with Λ = 0. We shall, however, discuss the limitations of this view in Sect. 3.6.6.

Example O2 Molecule What is, however, the situation for Σ states with Λ = 0? They may e.g. be constructed from two π orbitals with opposite orientation. According to Table 3.7 the ground state of O2 is an example. The system is well bonded by the electrons in the completely filled N2 of the inner electrons. In addition, two πg∗ orbitals have to be filled (configuration (πg∗ 2p)2 with λ1 = λ2 = 1). We first note that the overall spatial wave function is in any case even (g) in respect of inversion. We start again with (±λ) the complex orbitals (3.123) and use now the MO notation |φel = |πg∗ 2p±1 . According to H UND’s rules we expect the lowers state to be a triplet. If we ignore M spin orbit interaction the three symmetric spin wave functions |χ1 S lead to three degenerate states. The spatial wave function must be antisymmetric in respect of electron exchange, hence: 3 − MS ∗ √ π 2p−1 π ∗ 2p1 − π ∗ 2p1 π ∗ 2p−1 / 2 Σ = χ g g g g g 1 # 1 1 i(ϕ2 −ϕ1 ) e − e−i(ϕ2 −ϕ1 ) = χ1MS φ1πg∗ (z1 ρ1 )φ1πg∗ (z2 ρ2 ) √ π 2 2i √ MS φ1π ∗ (z1 ρ1 )φ1π ∗ (z2 ρ2 ) sin(ϕ2 − ϕ1 ) /(π 2). (3.142) = χ 1

g

g

The terms e±i(ϕ2 −ϕ1 ) make sure that the angular momenta compensate each other, so that z1 + L z2 )3 Σg− ≡ 0 (L and Λ = λ1 − λ2 = 0. For inversion symmetry we note that again for both electrons φ(zρ) = φ(−zρ), but when replacing both azimuthal angles ϕ → ϕ + π the π is compensated and the sign does not change. We have indeed constructed a Σg state. Reflection in respect of the xz plane (i.e. ϕ1 → −ϕ1 and ϕ2 → −ϕ2 ) changes the sign, hence, the overall reflections symmetry of the state is negative. It is important to note that in this case rotation around the z-axis by an arbitrary angle δ (i.e. ϕ1 → ϕ1 − δ and ϕ2 → ϕ2 − δ) does not change anything about the eigenfunction! This is obviously due to the fact that the two orbital angular momenta just compensate each other. Thus, for Σ states reflection symmetry is a good quantum number which characterizes the state. This statement also holds for the singlet state complimentary to (3.142): 1 + 0 √ Σ = χ φ1π ∗ (z1 ρ1 )φ1π ∗ (z2 ρ2 ) cos(ϕ2 − ϕ1 ) /(2 π). (3.143) g 0 g g It has positive (g) inversion symmetry and positive reflection symmetry, again independent of the alignment of the reflection plane through the molecular axis. The exchange symmetry for the two position coordinates (1 and 2) is now positive (cos function). Hence, the spin function must be antisymmetric and we have a singlet

3.6


205

state – as it turns out its electronic energy is significantly higher than that of the |3 Σg− state. In contrast, the Π , . . . etc. states are degenerate in respect of the ± symmetry. Still with the same electron configuration (πg∗ 2p)2 we construct the remaining two singlet states, with spatial wave functions which are again symmetric in respect of electron exchange: 1 0 ∗ √ g = χ π 2p−1 π ∗ 2p1 + π ∗ 2p1 π ∗ 2p−1 / 2 g g g g 0 # 0 1 i(ϕ2 +ϕ1 ) −i(ϕ2 +ϕ1 ) ∗ ∗ +e e = χ0 φ1πg (z1 ρ1 )φ1πg (z2 ρ2 ) √ 2 πi √ (3.144) = χ00 φ1πg∗ (z1 ρ1 )φ1πg∗ (z2 ρ2 ) sin(ϕ2 + ϕ1 ) /(2 π). This is a 1 g state with Λ = λ1 + λ2 = 2 due to the identical signs of ϕ2 and ϕ1 . It has negative reflection symmetry in respect of the xz plane. However, in this case there exists another, orthogonal but energetically degenerate state 1 g when the previous one is rotated by π/4 around the z-axis (i.e. ϕ1 → ϕ1 +π/4 and ϕ2 → ϕ2 + π/4), for which |1 g ∝ cos(ϕ2 +ϕ1 ) holds. Its reflection symmetry is now positive, while the electron exchange symmetry as well as inversion symmetry remains also positive. In total, we have identified 6 states based on the configuration (πg∗ 2p)2 . They illustrate very clearly why only for the Σ states reflection in respect of a plane through the molecular axis is independent of the alignment of this plane. We shall come back to the electronic states of O2 in Sect. 3.6.9. We shall see there that we have found indeed the lower lying states X 3 Σg− , a 1 g and b 1 Σg+ – in this energetic ordering, and again in agreement with H UND’s rules (highest multiplicity has lowest energy, for equal multiplicity highest total angular momentum is lowest, see Sect. 7.3.3 in Vol. 1).

3.6.6

Lambda-Type Doubling

At the end of this discussion we allude to a spectroscopically important phenomenon, the so called lambda-type doubling. In principle, it is relevant for H UND’s cases (a) and (b) whenever Λ = 0 (i.e. for Π , etc. states) and the rotational quantum number is large. So far we have neglected the coupling between rotation and electronic orbital angular momentum. Such coupling arises within diatomic molecules for states with Λ = 0 which – without rotation – are doubly degenerate and may have positive or negative reflection symmetry as just discussed. Let us consider the most simple example of a 2 Π state which arises from a single, filled π orbital in the valence shell. The energy of 2 Π + and 2 Π − state is completely identical as long as the molecule does not rotate. The x- and y-axes may be defined completely arbitrary (perpendicularly to the molecular axis z) and the πpy and πpx orbitals are fully equivalent. However, if the molecule rotates, this symmetry is removed. Assume it rotates around its y-axis. We may then quite intuitively imagine

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3

Diatomic Molecules

that the py orbital (dumbbell, rotationally symmetric around the y-axis) is hardly influenced by this rotation. In contrast, the electron in a px orbital (dumbbell, perpendicular to the y- and z-axis) is exposed quite evidently to a centrifugal force. This leads to a splitting which is called Λ-type doubling. Even if this is an energetically very small contribution, it is well observable with modern spectroscopic precision. The effect may become quite significant in dissociation processes where e.g. diatomic molecules are ejected from a larger complex. The rotational distribution of these fragments may then have a pronounced asymmetry in respect of the two molecular rotation axes.

3.6.7

Example H2 – MO Ansatz

We come back once more to the computation of electronic wave functions of diatomic molecules and focus in some detail on the H2 molecule with its two equivalent electrons. We shall use what we have worked out in Sect. 3.5.2 for H+ 2 , and summarize in passing the findings of Sect. 3.6.5. The overall wave function must antisymmetric in respect to electron exchange (PAULI principle). For the two active S 2 (with Σ = 0 or 1 projected onto the molecuelectrons the total spin is S = S1 + L2 (with Λ = |λ1 ± λ2 |). lar axis), the total orbital angular momentum is L= L1 + The spatial wave functions may be constructed from even or odd orbitals according to (3.110). In analogy to the He atom, the total wave functions of H2 are singlets with MS = 0 :

φS (1, 2) = φ+ (1, 2)χ00 (1, 2),

triplets with MS = −1, 0, 1 :

and

φT (1, 2) = φ− (1, 2)χ1MS (1, 2)

(3.145) (3.146)

with symmetric φ+ (1, 2) and antisymmetric spatial function φ− (1, 2). For the lowest states we include in our evaluation only the 1σg and 1σu MOs. For brevity we write the respective spatial wave functions φg (i) and φu (i), with i = 1 or 2 depending on which electron is placed into this orbital. The ground state (3.139) with two electrons in the bonding 1σg orbital is written in short notion: X 1 Σg+ :

φX (1, 2) = φg (1)φg (2)χ00 .

(3.147)

If one of the electrons is in the antibonding 1σu MO one may obtains as indicated in (3.140) b3 Σu+ :

M 1 φb (1, 2) = √ φg (1)φu (2) − φg (2)φu (1) χ1 S , 2

(3.148)

for which a detailed calculation shows that it is indeed the lowest lying, repulsive triplet state and correlates in the separated atom limit with both atoms in the 1s ground state. In contrast, the second combination with one electron in a 1σu MO, 1

Σu+ :

1 φ3 (1, 2) = √ φg (1)φu (2) + φg (2)φu (1) χ00 2

will even be higher in energy according to H UND’s rules.

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207

This holds a fortiori for 1

Σg+ :

φ4 (1, 2) = φu (1)φu (2)χ00

where both electrons are in antibonding orbitals. In the separated atom limit, both combinations do not correlate with the atoms in their 1s ground state and only contribute to higher lying bound and unbound states. In a similar manner the higher lying molecular states are constructed from more complex MOs. The principles are those treated in Sect. 3.6.5. An overview of the H2 potentials computed with quantum chemical methods gives Fig. 3.41. It also in+ cludes its ions H− 2 and H2 . As already mentioned, the states are usually ‘labelled’ with capital letters, the ground state with X, the next higher with A, B, C etc. In a number of cases, due to the historical development, this is not followed consequently. Often one uses the lower case alphabet for states which have been found later. As H2 is of particular importance (and its potentials form a rather complex manifold) a section of the energy diagram is shown once again in Fig. 3.42 on an enlarged scale. − Note that no stable negative ion (anion) H− 2 exists. The lowest energy of H2 lies − above the bound, neutral ground state of H2 . Thus, H2 can decay spontaneously into H2 + e− – if it is at all ever formed in the first place. One observes such states as resonances in the scattering of low energetic electrons by H2 . The observed feature looks very similar to autoionization which we have treated in Chap. 7, Vol. 1. We shall come back to this in Chap. 8. One more, closely related, process should be mentioned at this point, the so called predissociation. We note the quite remarkable potential maxima for a number of H2 states, as seen in Fig. 3.42 e.g. for the states designated with I , i and h. They are a consequence of avoided crossings – which generate local maxima in the potentials. These maxima make it possible to form vibrational states which lie above the dissociation limit. Although these states are strictly speaking not stable states, they still live for some time since the potential barrier prevents immediate decay. The probability for dissociation of such molecular states by “tunnelling” depends on the height and width of the barrier. A few words on how to calculate such potentials are in order at this point. For the H2 molecule one may in principle follow the same procedure as in the H+ 2 case, and use suitably chosen trial functions as approximate solutions of the S CHRÖDINGER equation. The electronic Hamiltonian is now (in a.u.) 1 1 + H = H1 + H2 + (3.149) r12 R i = − 1 ∇ 2i − 1 − 1 . with H 2 rAi rBi ‘New’ as compared to H+ 2 is in particular the repulsive term with r12 , the distance of the two electrons, while rAi and rBi are the distances of electron i from atomic nucleus A and B, respectively, and R is again the distance between the nu-

208

3

Diatomic Molecules 24

24

+

22

V Π u 4f

2Σ+

B'' 1Σu 4pσ

22

1

+

u 2pσu

D'' 1 Π u 5p

20

D' 1 Π u 4p

H2

20

+

g 3Σg 3dσ

+

B' 1Σu3pσ

H++H(1s)

+

18

X 2Σg 1sσg

+

e 3Σu2pσ +

16 h

d 3Π u 3p

m 3Σu 4fσ 3 i Π g 3d 3sσ g

3Σ+

16 H(1s)+H(2ℓ) 14

14 potential energy V(R ) / eV

5ℓ 4ℓ H(1s)+H(3ℓ)

-

I Π g 3d

12 1

+

B Σu 2pσ

+

C +e- 2Σg

10

H(2s)+H (1s 2) D 1Π u 3p and J,j 1,3∆ g 3d δ 12

C 1 Π g 2p

m 3Σu 4fσ

+

a 3Σg 2sσ + H 1Σg 3sσ +

+ c +e- 2Σg

H2

8

10

c 3Π u 2p +

E,F 1Σg 2sσ + 2pσ2

8

+

6

6

b 3Σu 2pσ H(1s)+H(1s)

+

b +e- 2Σg

H2

4

4

10

-

2

5

0

X 1Σg 1sσ

+

0

H(1s)+H

+

X + e - 2Σ g

0.1

-

(1s2) 2

H2

0.2 0.3 nuclear distance R / nm

0 0.4

0.5

0.6

Fig. 3.41 Potentials for the most important states of the H2 molecule (for comparison also for the + 1 + anion H− 2 and the cation H2 ) after S HARP (1971). The equilibrium distance of H2 in its X Σg electronic ground state is R0 = 0.07416 nm, the dissociation energy D0 = 4.476 eV. The b 3 Σu+ state is repulsive. For brevity we omit the description 1sσ . Note that the ion pair p + H− (1s 2 ) is formed at 17.5 eV

clei. The molecular potential for a state φγ is found by minimizing (3.95). In the R ITZ variational method one may construct φγ as linear combination of MOs φg and φu . The MOs needed for the ansatz (3.147) may e.g. be found by first solving

3.6


209

17

17 V 1 Π u 4f

H2+

+

m 3Σu 4fσ +

g 3Σg 3dσ

16 D1 X

d 3Π u 3p

2Σ+

g

J,j D'' 1 Π u 5p

1,3∆

i 3Π g 3d

g 3d δ

I Π g 3d

D' 1 Π u 4p

+

h 3Σg 3sσ B''

1Σ + 4pσ u + B' 1Σu3pσ

14

16

Π u 3p

15 potential energy V(R ) / eV

H(1s) +H(3ℓ)

15 H(1s) +H(2ℓ)

14

+ +

H 1Σg 3sσ

C 1 Π g 2p 3

a Σg 2sσ 13

e

3Σ+ 2pσ

13

u

+

E,F 1Σg 2sσ + 2pσ2

12

H2

c 3Π u 2p

12

+

B 1Σu 2pσ +

b 3Σu 2pσ 11

11 0.05

0.1

0.15

0.2

0.25

0.3

0.35

nuclear distance R / nm

Fig. 3.42 Enlarged part of the potential energy diagram for H2 Fig. 3.41 after S HARP (1971). Note the interesting states designated E, F and H with double minima, and the “predissociating” states with double minimum potential marked I , i and h

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Diatomic Molecules

the S CHRÖDINGER equation for H+ 2 i φ(r i ) = W φ(r i ). H However, it is also possible and quite instructive to start with LCAO-MOs as a first approximation, in the most simple case with (3.110). Explicitly for the ground state (without normalization) this leads to: φX(S) (1, 2) ∝ Φ1s (rA1 )Φ1s (rB2 ) + Φ1s (rA2 )Φ1s (rB1 ) (3.150) + Φ1s (rA1 )Φ1s (rA2 ) + Φ1s (rB1 )Φ1s (rB2 ) . (3.151) The two parts (3.150) and (3.151) of this wave function may be identified with a covalent and an ionic contribution, respectively. The covalent part represents a relatively uniform distribution of the two electrons at both atoms. For large distances between the atoms, the covalent wave function eventually merges with the description of two separated atoms H + H. In contrast, the ionic part localizes both electron either at atom A, i.e. Φ1s (rA1 )Φ1s (rA2 ), or at atom B, i.e. Φ1s (rB1 )Φ1s (rB2 ). This corresponds thus to the situation H− + H+ and H+ + H− , respectively. Asymptotically the thus described state is found for H2 (in its vibrational ground state) at an energy WI + D00 − WEA 17.32 eV, where WI is the ionization potential of the H atom, with D00 being the dissociation energy of H2 in its vibrational ground state, and WEA stands for electron affinity of the H atom. Looking at Fig. 3.41 tells us that this is far above the H + H asymptote. Interestingly, the ionic state contributes to the excited B 1 Σg+ state which is much weaker bound than the X 1 Σg+ ground state of H2 . To finally evaluate the energy according to (3.95) with the simple trial function (3.150)–(3.151) one again introduces the coordinate transformation (3.97) in order to compute the integrals which evolve. We refrain from presenting here the details of the calculation – in particular because the results of this procedure is not really convincing: on obtains with this ansatz R0 = 0.08 nm and De = 2.68 eV – which has to be compared with the precisely calculated and experimentally determined values R0 = 0.07414 nm and D00 + ω0 /2 = 4.747 eV (see Table 3.2).

3.6.8

Valence Bond Theory

Instead of starting from MOs, one may simply use a covalent charge distribution as trial function. One chooses, e.g. for the H2 ground state (not normalized) φX(S) (1, 2) = Φ1s (rA1 )Φ1s (rB2 ) + Φ1s (rA2 )Φ1s (rB1 ) × χ00 (1, 2) (3.152) and computes the energy of the X 1 Σg+ state from WX(S) =

|φX(S) φX(S) |H . φX(S) |φX(S)

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211

Correspondingly the ansatz 1 φb(T ) (1, 2) = Φ1s (rA1 )Φ1s (rB2 ) − Φ1s (rA2 )Φ1s (rB1 ) × χM (1, 2) S leads to the repulsive potential of the non-bonding 3 Σu+ state. This so called valence bond theory, first introduced by H EITLER and L ONDON, is less complicated than the MO method and gives, in the case of H2 , even somewhat better results that the simple MO ansatz just discussed.11 With today’s state-of-the-art computers and quantum chemical methods one may obtain nearly ‘exact’ ab initio molecular potentials, wave functions and other properties – at least for small molecules, without recurring to the above discussed approximations. Today, such programmes use large, well tested MO or AO basis sets and have already been discussed in Vol. 1 for atomic multi-electron systems. Sophisticated H ARTREE -F OCK methods (HF-SCF) with configuration interaction (CI) and various corrections are applied. We refrain from listing the different acronyms for a large variety of elaborate approximations. Today, for larger molecules DFT is used more and more, which ultimately is based on variational principles for the electron density, similar to those sketched here for MOs. A range of very powerful computing programmes are available on a commercial basis for various application areas (e.g. G AUSSIAN 2013; M OLPRO 2012; T URBOMOLE 2010; GAMESS 2010).

3.6.9

Nitrogen and Oxygen Molecule

As examples for more complex, diatomic, homonuclear molecules – in comparison to H2 in a 1s 2 configuration – the potentials for N2 and O2 are shown in Figs. 3.43 and 3.44, respectively. According to Table 3.7, N2 is the most stable of the small molecules. K and L shells are completely filled, which include all bonding MOs based on 2p atomic orbitals. In the X 1 Σg+ ground state 6 bonding 2p electrons participate in the bonding and the spins are saturated (singlet state); N2 represents, so to say, a closed molecular shell. Correspondingly, N2 is also chemically very inert. The first excited state, A 3 Σu+ state lies with 6 eV rather high above the ground state, and the ionization potential of 15.6 eV belongs to the highest ones among all molecules. In contrast, the anion is not stable at all. Its ground state X 2 Πg is drawn in Fig. 3.43 as grey line and lies 1.6 eV above the neutral N2 ground state. As (short-lived) resonance it may, however, be observed very clearly in electron scattering. We shall come back to this in Sect. 8.1.2. Corresponding to the atomic configuration of the O atom in the ground state, 2s 2 2p 4 3 P, one expects for the O2 molecule a more complex electronic structure. According to Table 3.7, two antibonding πg∗ electrons have to be added to the very stable N2 core (1 Σg+ ) with its spatially symmetric wave functions. In Sect. 3.6.5 we 11 Which

isn’t really surprising here, as we have seen that the ionic component of the MOs leads to completely wrong asymptotic states.

212

3

Diatomic Molecules

28 N(2Do) + N+(3P) 2

u

26 C

2Σ +

u

N(4So) + N+(1D)

2Σ –

u

2∆

24

4

u

N(4So) + N+(3P)

u

4Σ –

u

22

B 2Σ u+

4∆

u

D2

4Σ +

u

20

18

g

+

N2

–

A 2 Πu

+

N

16

X 2 Σ g+ C' 3

potential energy V(R ) / eV

14

3

C

+ 3Σ g

N(4So) + N(2Do)

u

a1

10

a'

6

N(4So) + N (2P o)

b' 1Σu

5

8

N(2Do) + N(2Do)

u

+

E 12

+N

g

u 5Σ

w 1∆u

1Σ –

u

+ g

7Σ

–

+ u

N(4So) + N (3P) N(4So) + N(4So)

3∆

u

–

B' 3Σu B 3Π g

+

A 3Σu

–

N2

4

X2

N2

g

2 X 1Σg+ 0 0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36


Fig. 3.43 Potentials for the nitrogen molecule and its ions according to G ILMORE (1965). The potential of the unstable N− 2 anion has been adjusted to fit the experimental observations from electron scattering resonances (see Sect. 8.1.2)

3.6


213

24 O(3P) + O+(2Do) 22 2Σ

2Φ

O(1D) + O+(4So)

u

– g

2Δ

6Σ +, 6Σ + , 6

g

20

+ g

4Σ

–

2Σ

b 4Σ g

g

4

+

+ g

+

and 4Σu

X2 g

1Σ

12

+ u

O(1D) + O(1S)

10

1

O(3P) + O(1S)

1Δ

u

O2

6

O(1D) + O(1D)

B

g

3Σ

3

3

g

+

O(3P) + O(1D)

– u

5 1

u

u

5Σ +

u

g

5Δ

A'

4

5

+

g and Σg – + + 5Σ , 3Σ and 1Σ u u g

A 3Σ u

O(3P) + O(3P) 3Δ

u

b

c 1Σ u

2

–

o

O(3P) + O (2P )

u

+ 1Σ g

–

O2

2 a 1Δg 0

g

O2

a4 u

14

8

6

O(3P) + O+(4So)

g

2Σ

A2 u


u,

g

18

16

u

–

X 3Σ g

0.08

X2 0.12

g

0.16

0.20

0.24

0.28

0.32

0.36


Fig. 3.44 Potentials for the oxygen molecule and its ions according to G ILMORE (1965); minima of the A, A and c states have been modified according to J ENOUVRIER et al. (1999)

214

3

Diatomic Molecules

have already discussed that the ground state X 3 Σg has highest multiplicity according to H UND’s rules. The next higher states (built from the same atomic electron configurations) are the a 1 g and b 1 Σg+ states. As already mentioned, oxygen gas O2 in its ground state X 3 Σg is paramagnetic, due to its two electrons with parallel spin, S = 1. We also note that a stable anion − − exists, O− 2 in the ground state (in contrast to H2 and N2 ). Nevertheless, the anion − decays when vibrationally excited: O2 → O2 + e; conversely, it may also be generated in a vibrationally excited state, again as a resonance in electron scattering by O2 . The overviews on the potentials of N2 and O2 presented here (as well as that on NO in Fig. 3.53) are based on the very systematic work of G ILMORE (1965). Spectroscopic data have been evaluated there in the spirit of the RKR method (see Sect. 3.4.6), but also quantum chemical computations have been included. Even though since then many new measurements and calculations have been published, we have added only a few modifications to this data compilation, since a comparable critical overall evaluation of all available data has – up to now – not been reported. Section summary

• The electronic states of diatomic molecules are characterized by projections of total angular momenta (sums over all active electrons) onto the internuclear axis. • In the simplest cases (e.g. for H UND’s case a) the projection of the total orbital angular momentum, Λ, and total spin, S, are good quantum numbers. States with Λ = 0, 1, 2 etc. are designated by Σ, Π, etc., while the spin quantum numbers S = 0, 1/2, 1 etc. lead to singlets, doublets, triplets etc. In analogy to atoms, typically the notation is 2S+1 Λg,u is used. • The five H UND’s cases differ by the strength of the interaction between internuclear field, orbital, spin and rotational angular momentum. The ensuing different angular momentum coupling schemes lead to spectroscopically relevant differences in the energy spectra. • In addition to angular momenta, molecular states are characterized by their symmetries. For homonuclear, diatomic molecules, the relevant symmetry operations are inversion in respect of the origin, and reflection at planes through the molecular axis. • For multi-electron systems (exemplified by H2 and O2 ) positive or negative reflection symmetry lead to different energies for Σ states – and only for these. For other values of Λ, states with ± reflection symmetry are degenerate if the molecule does not rotate. However, high rotational levels show a small splitting, the so called Λ-type doubling. • Potential energy diagrams for H2 , O2 , and N2 show many interesting structures. In all three cases negative ions (anions) may be observed as short-lived resonances in electron scattering, while only the O− 2 anion has a stable vibrational ground state.

3.7

Heteronuclear Molecules

215

• N2 is the most strongly bound diatomic molecule (De 9.9 eV) and has also a particularly high ionization potential. H2 and N2 have rather isolated electronic ground states, while O2 has three lowest states separated by less than 1.7 eV and ordered according to H UND’s rules: the X 3 Σg− ground state (paramagnetic), the a 1 g and the b 1 Σg+ state.

3.7


3.7.1

Energy Terms

We generalize now the considerations on molecular orbitals outlined in Sect. 3.5. For diatomic molecules with two different atomic nuclei inversion symmetry is no longer relevant, and the molecules can no longer be characterized as even or odd. As a rule, the covalent bonding energy is weaker as for homonuclear molecules, and HAA = HBB . We recall the simplest approximation for orbital energies (3.107) and (3.108) in the case of homonuclear molecules "g,u = HAA ± HAB . For the sake of simplicity we have set the overlap integral S = 0 – which will change the overall trends only when both atoms come very close. For heteronuclear molecules the secular equation (3.106) thus is (HAA − ")(HBB − ") − HAB HBA = 0. With HAB HBA = |HAB |2 the energy eigenvalues become now (HAA − HBB )2 HAA + HBB "± = ± + |HAB |2 2 4 HAA + HBB HAA − HBB 4|HAB |2 ± = 1+ . 2 2 (HAA − HBB )2 For many heteronuclear molecules |HAA − HBB | |HAB | holds (in particular for those with very different nuclear charges). Then the two eigenvalues are approximately "+ HAA +

|HAB |2 HAA − HBB

and "− HBB −

|HAB |2 . HAA − HBB

(3.153)

Figure 3.45 illustrates the raising and lowering of the atomic orbital energies. We have assumed here that HAA > HBB . Instead of the even and odd functions (3.110) and (3.111), respectively, we obtain now for the orbitals (±)

(±)

φ± = cA ΦA + cB ΦB

216

3

Fig. 3.45 Schematic of raising and lowering of atomic orbital energies for heteronuclear, diatomic molecules due to the resonance integral HAB (3.105)

Diatomic Molecules |HAB |2 ___________

ε+ HAA |HAB |2 ___________

HBB

HAA − HBB

B

HAA − HBB

εA

Fig. 3.46 Orbital energies for heteronuclear diatomic molecules

6σ* 2 * 2p

2p

5σ 1 4σ *

2s

2σ * 1s A (+)

with

cA

(+) cB

cA

cB(−)

1s

1σ B

HAB HAA − HBB =

1 and "+ − HAA HAB

(−)

−

2s

3σ

−

HAB HAB =

1. "− − HAA HAA − HBB

The molecular orbitals are thus rather similar to the AOs, i.e. φ+ ΦA and φ− −ΦB . The lower level correlates as expected with ΦB .

3.7.2

Filling the Orbitals with Electrons

A direct consequence of the asymmetric molecular orbitals is an unequal distribution of charge in the molecule. Consequently, heteronuclear diatomic molecules have a permanent dipole moment. In comparison to the homonuclear molecules Fig. 3.37 the building-up of MOs is somewhat more specific. For molecules made of atoms with similar nuclear charges (e.g. CO) one obtains MOs from LCAO as schematically indicated in Fig. 3.46. Since there is no longer a difference between g and u, the designation is done simply by consecutively numbering the levels for each λ. The respective correlation diagrams are based on this scheme, and in the limit of separated atoms two different energies appear for the atomic orbitals 1s, 2s, 2p etc. Apart from this the same non-crossing rules are valid. A typical correlation diagram is shown in Fig. 3.47 – to be compared to Fig. 3.36. As for the homonuclear molecules the states with higher λ but equal lead to higher

3.7

Heteronuclear Molecules 3d δ

3d

3dσ* 3p π 3pσ 3sσ* 2p π

3p 3s 2p 2s

217 3π 7σ*

3d π

σ*3sA

6σ* 2π 5σ* 4σ* 1π 3σ

π 2pB

2σ*

σ*1sB

π 2pA σ*2sB

1sσ 1s united atom (R = 0)

2pB 2pA 2sB 2sA

σ2pB σ2pA

σ2sA

2pσ 2sσ*

3sA

1sB 1sA

σ1sA

1σ molecule R

separated atoms ZA > ZB (R → ∞)

Fig. 3.47 Correlation diagram for heteronuclear diatomic molecules Table 3.9 MOs for heteronuclear diatomic molecules Molecule Expected electronic configuration

Bond-order

2S+1 Λ

Observed R0 nm

De eV

0.13309

4.69

BeO

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4

2

1Σ +

BeF

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )1

2 12

2Σ +

0.13610

≈6.0

BNa

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)3 (5σ )1

2

3Π

0.13291

3.99

BO

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )1

2 12

2Σ +

0.12045

8.40

BF

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2

3

1Σ +

0.12625

7.89

CN+

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4

2

1Σ

0.11729

4.93

CN

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )1

2 12

2Σ +

0.11718

7.83

CN−

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2

3

0.114

≈10

CO+

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )1

2 12

0.11151

8.47

CO

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2

CF

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2 (2π)1

NO+

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2

NO

(1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2 (2π)1

2Σ +

3

1Σ +

0.11283

11.11

2 12

2Π

0.12718

5.75

3

1Σ +

0.10632

11.00

2 12

2Π

0.11508

6.61

R0 and De as far as not otherwise mentioned according to H UBER and H ERZBERG (1979) a LI

and PALDUS (2006)

energies. We have discussed this in the context of the S TARK effect in Sect. 8.2.7, Vol. 1. For large internuclear distances the energetic ordering is thus σ < π < δ. However, due to the necessary correlation with the united atom scheme this trend may be inverted in the MO region. In Fig. 3.47 this is particularly evident for the 1π , 5σ , 2π and 6σ states. In Table 3.9 the electron configurations, equilibrium distances R0 and the bond energies De of the ground state for a number of heteronuclear diatomic molecules are summarized.

218

3

Diatomic Molecules

It should be noted that the scheme communicated here gives only a first overview of potential MOs. In practice one has to investigate the energetics and each atomic configuration in detail to find a suitable MO basis set for describing a particular molecule. As already mentioned for homonuclear molecules, today a number of sophisticated methods and efficient computer programmes are available for this task. To get somewhat more specific, we shall in the following discuss a few interesting examples in more detail.

3.7.3

Lithiumhydrid

LiH is quite a remarkable molecule – albeit not available in a gas bottle. A number of specialities may be studied with it. In recent years it has even attracted interest in the context of B OSE -E INSTEIN condensation of ultracold atoms (see e.g. C ÔTE et al. 2000; J UARROS et al. 2006), not least because of its large dipole moment which makes one expect specific long range effects. One may e.g. apply two-photon processes to generate such molecules by photo-association from ultracold atoms. The constituents of the LiH molecule have the configuration 1s 2 2s (Li) and 1s (H), whereas the closed 1s 2 does not participate in the bonding. These electrons fill the 1σ MO localized at the Li nucleus. A comparison of the first excited atomic states 1s 2 2p (Li) and 2s, p (H) with 1.848 eV and 10.2 eV excitation energy, respectively, shows that a correlation diagram would not lead us very far in this case. According to the above considerations, the active MOs will be determined by the easy to excite AOs of Li. The ground state is essentially given by the 2s electron of Li which interacts with the 1s electron of the hydrogen atom: the atomic orbitals 2s (Li) and 1s (H) form a 2σ MO. In Fig. 3.48(a) the potentials for some states of LiH are collected from several sources. Most clear is the situation for the triplet states 3 Σ, where the PAULI principle provokes a spatial repulsion of the σ MOs from n − 1s. A closer inspection shows (not recognizable in Fig. 3.48) that here too long range dispersion interaction leads to a weak attraction. For the a 3 Σ + state this is of importance in a distance of 1 − 2 nm and is of significant relevance for the interaction among ultracold atoms. More complicated are the conditions for the singlet states for which in addition to the symmetric, neutral AO configuration Li0 H0 also the ion pair Li+ H− must be accounted for. The hypothetical energy of the ion pair is indicated in Fig. 3.48(a) by the dashed red line. As seen, the asymptotic energy of this ion pair lies below the ionization energy of Li (5.39 eV) by just the electron affinity of the H atoms, WEA = 0.756 eV. A suitable MO which accounts for this, will be characterized by a significantly asymmetric charge distribution. In the most simple approach one realizes this by a linear combination of a 2s and a 2pz AO of lithium (a |σ 2s and a |σ 2p AO, in respect of the internuclear axis z). This is sketched in Fig. 3.49.12 Such a combination of atomic orbitals with different 12 We

remind the reader that such schematic polar plots of orbitals just display the magnitude of the wave function under a given polar angle θ at fixed radial distance. They do not give a realistic image of the spatial extension of the orbital.

3.7


219

Li++H

V R) / eV V( 5

Li++ H

(a) well depth ca. 34.5 meV

4

–

dipole moment / ea0 WEA

(b)

R

4 X state

3 Σ+

3

3

B 1Π +

2 A

1

A state

2

Li(2p 2P) + H(1s 2S)

1Σ +

1 Li+H–

0

Li(2s 2S) + H(1s 2S)

a3Σ+

0 B state

−1

-1 X 1Σ +

−2

-2 0

0.2

0.4

0.6

0.8

0.2

1.0

0.4

0.6

0.8

1.0

R / nm

Fig. 3.48 Potentials (a) and dipole moments (b) of the LiH molecule. Ground state and excited state A 1 Σ + are strongly influenced by the ionic Li+ H− configuration. Triplet states and B state are essentially covalent (adapted from PARTRIDGE and L ANGHOFF 1981; Y IANNOPOULOU et al. 1999; C ÔTE et al. 2000) Fig. 3.49 Formation of an s-p hybrid orbital

z

+

-

+

z

+

√3 √ 1 _ |σ 2s〉 |σ 2p〉 2 2 hybrid: = |2spσ 〉 –

+

z

angular momenta is called a hybrid orbital. According to PAULING (1931), who first introduced this important concept of hybridization, the bond will be strongest for a wave function which has the largest value in the direction of the bond. In the √present case one easily verifies that this implies superposition coefficients 1/2 and 3/2 for the |σ 2s and |σ 2p orbital, respectively, obviously resulting in a quite asymmetric shape of the orbital. We recall that such type of orbital was already encountered in Sect. 8.2.8, Vol. 1 in the context of the linear S TARK effect for the 2p, 2s states of atomic hydrogen. When a molecule is formed, the very strong electric field of the two atomic nuclei acts correspondingly. The phenomenon always occurs if ns and np states have similar energies. A more detailed treatment of hybridization will be given in Sect. 4.4.2.

220

3

Diatomic Molecules

For LiH one consequence is a significant displacement of the charge distribution, away from the Li atom towards the H atom. This in turn leads to a large dipole moment for the X state as documented in Fig. 3.48(b). It is remarkable that it reaches its maximum at (0.2 to 0.3) nm, i.e. just there where the hypothetical potential curve for Li+ H− (red dashed) is particularly close to the true (computed) potential of the X 1 Σ + ground state. Obviously the dominantly ionic character of the orbitals involved leads to the bonding. Hence, the singlet state lies here (contrary to the usually valid H UND’s rules) lower than the triplet state which has no ionic character. To avoid misunderstandings: the dashed potential for the ionic Li+ H− bonding does not represent a real state at small distances: the corresponding MOs are, so to say, an integral part of the X 1 Σ + ground state, but to some extend also of the excited A 1 Σ + state.13 In the region of the “avoided crossing” at about (0.4 to 0.6) nm the A state has even a larger dipole moment as seen in Fig. 3.48(b): one reads a maximum of 4.2ea0 at 0.5 nm, corresponding to a charge delocalization of more than 40 %. Obviously the sign of this charge displacement reverses for the A state at small internuclear distances, i.e. when the H atom dives fully into the excited 2p orbital of the Li atom. The potential of the A state has another remarkable property: it has a negative value of ωe xe , i.e. in contrast to the usually found behaviour (as described e.g. by a M ORSE potential) it is flat at the bottom and becomes steeper at higher energies – just another consequence of the locally very ionic character of the molecular bond. Interesting is also the B 1 Π state, which correlates with the excited 2px or 2py AOs. The overlap with the 1s H orbital is minimal. Due to the π character of the orbitals involved there is, however, no avoided crossing with the hypothetical ionic potential curve (the ground state of Li− has the configuration 1s 2 2s 2 ). This gives rise to the very flat shape of the B state shown in Fig. 3.48(a). The well depth is, however, sufficient to sustain 3 stable vibrational states (PARTRIDGE and L ANGHOFF 1981), which could possibly be used as intermediate states for the proposed molecular B OSE -E INSTEIN in this molecule (J UARROS et al. 2006; D ULIEU and G AB BANINI 2009). Also a number of higher excited singlet and triplet states are known, some with interesting shapes of their potential minima. The computation requires extended basis sets up to f orbitals (Y IANNOPOULOU et al. 1999).

3.7.4

Alkali Halides: Ionic Bonding

The alkali halides (i.e. LiF, NaCl, NaI, KBr etc.) are prototypical examples for ionic molecular bonding. As crystal solids some of them are also of considerable practical importance. Their preparation as free molecules for spectroscopic purposes requires principle, one may let a Li+ cation collide with an H− anion in a scattering experiment. For large distances, the interaction potential of such a system is indeed given by the red dashed line (∝ −1/r) and remains – for sufficiently high kinetic energies – a good first approximation also at smaller distances. Transitions at the (avoided) crossings may indeed be induced in such collision processes.

13 In


Fig. 3.50 Several 1 Σ + potential curves for NaCl as a typical example for ion bonding: the red dashed curved shows the potential for a hypothetical, purely ionic system Na+ + Cl− which converges for large R towards 1.522 eV. The ground state is dominated by the ionic bonding. At about 1 nm one sees the classical avoided crossing with the excited A 1 Σ + state which becomes covalent for small internuclear distances

221 4 3

NaCl 1Σ +

Na(3p) + Cl(3p)5

2 potential energy / eV

3.7

1

Vγ (∞) =1.522 eV

Na+ + Cl

–

A1Σ+

0 -1

Na+Cl

Na(3s) + Cl(3p)5

–

-2 -3

X1 Σ +

D e= 4.23eV 0.2 0.4 0.6 0.8 R0 = 0.236079 nm

RH 1.0

1.2

1.4 R / nm

some efforts (e.g. evaporation at high temperatures). Nonetheless a wealth of publications exists about these interesting molecules. They have played a key role in understanding elementary chemical reactions in molecular beam studies, for which H ERSCHBACH, L EE and P OLANYI (1986) have been honoured with the N OBEL prize in chemistry. The key theme is reactive processes of the type A + BC → A+ + (BC)− → A+ B− + C,

(3.154)

where A is typically an alkali atom and BC a halogen containing molecule (e.g. Br2 , CCl4 etc.). The rather low ionization potential WI of the alkali atoms and the high electron affinity WEA of the halogens lead to a positive energy balance for the electron jump indicated as intermediate in the reaction (3.154). For this to happen efficiently, the reaction partners have to approach each other close enough, say to a distance RH . Once the ion pair is formed, the C OULOMB attraction of the ions leads to intense interaction. The whole process is called harpooning: the electron of atom A is, so to say, fired with a harpoon at the molecule BC to catch it. In real scattering experiments one measures the angular distribution of reactants and/or reaction products after the interaction process. From the harpooning radius RH one obtains an esti2 . One may illustrate the electron jump mate for the reaction cross section σr πRH in principle already by a diatomic molecule, as shown by Fig. 3.50 for the example NaCl. The configuration of AOs in this system is 3s (Na) and 3s 2 3p 5 (Cl). The 3s valence electron of the Na atom is only weakly bound and is given off easily to complete the 3p shell in Cl. The ionization potential of Na is WI = 5.1391 eV, the electron affinity of Cl is WEA = 3.617 eV. At infinite distance one thus needs an energy Vγ (∞) = WI − WEA = 1.522 eV

222

3

Diatomic Molecules

to form the ion pair Na+ and Cl− . However, at shorter distances energy is gained by the C OULOMB attraction and the ionic potential is expected to be Vγ (R) = Vγ (∞) −

e2 , 4πε0 R

which is plotted as red dashed line in Fig. 3.50. At distances R < RH 1 nm C OULOMB interaction and bond energy for Na+ + Cl− are more than compensated and the electron may ‘jump’ from Na to the Cl atom. At internuclear distances R < RH one thus expects spontaneous formation of the ion pair Na+ Cl− , and consequently of a bound molecule. At very small distances the electron densities of the closed shells of Na+ and Cl− begin to overlap – what finally leads to a strongly repulsive potential just as for rare gases. The overall result is the potential14 for a rather well bound molecule in the X 1 Σ + ground state (full black line) with De = 4.2303 eV at R0 = 0.236 079 nm. Comparison of the hypothetical ionic potential with the true potential (semi-empirically computed on the basis of spectroscopic data) shown in Fig. 3.50 illustrates impressively that the ionic bonding model reproduces the situation very well between R0 and RH . Consequently, the dipole moment of NaCl is large: DX = 30 × 10−30 C m (see Table 3.2). At equilibrium distance R0 this corresponds an effective charge displacement of about 0.8e! At RH we see the aforementioned avoided crossing with the excited A 1 Σ + state. Obviously, the latter is dominated by covalent orbitals up to RH – beyond which the ionic character takes over. The potential thus forms a small well in which several vibrational states can be bound. For energetic reasons, the next higher state, correlating with the excited 3p electron of the Na, can not cross the ionic state. The relations found in the case of NaCl are rather typical for all alkali halides – except for spin-orbit interaction which naturally increases with the atomic number and can no longer be neglected for large Z. For atomic iodine it amounts already to 0.9078 eV, much larger that the respective exchange interaction. We discuss now the consequences and some more details for NaI on the basis of relatively recent ab initio computations by A LEKSEYEV et al. (2000). The electronic configuration of valence electrons in this calculation included 9 + 7 electrons in the atomic orbitals: 2s 2 2p 6 3s (Na) and 5s 2 5p 5 (I). For the highest of these outer shell LCAO-MOs this is illustrated schematically in Fig. 3.51. The strongly bound X 1 Σ + ground state is dominated by the . . . σ 2 π 4 ionic configuration. The lowest excited states are 1 Π and 3 Π with the covalent MO configuration . . . σ 2 π 3 σ ∗ . They differ only little in energy, since exchange interaction between the σ ∗ and π orbitals is very weak. As indicated in Fig. 3.51, these 14 For small distances and around the potential minimum we have used the potential derived analytically from spectroscopic data reported by R AM et al. (1997). For larger distances the semiempirical valence bond calculations of C OOPER et al. (1987) appeared more plausible: experimental data were available up to v = 8 only (ca. 350 meV). It does not seem reasonable to assume that the extrapolation of such data to about 4 eV excitation energy gives reliable information in the vicinity of RH .

3.7


3s

223 σ*

σ* antibonding

π

Na

π 4 "lone pairs"

5p

σ bonding

σ NaI

I

Fig. 3.51 MOs from AOs for the alkali halogenide molecules. Due to their very different orbital energies the two “lone pairs” of π electrons are localized at the I atom and do not participate in bonding 3

NaI

0+

2 potential energy / eV

Fig. 3.52 The most important potential curves for NaI in Ω symmetry according to A LEKSEYEV et al. (2000). The red dashed curve gives again the potential for the hypothetical ionic system Na+ + I− , which for large R converges to 5.105 eV. In the present case this leads to two avoided crossings at RH1 = 7.2 and RH2 = 12.5a0 . These in turn create two quasi bound, excited states A0+ and B0+

B0+

Na(2P

1/2)

+

–

Na+ + I

I(2P

3/2)

1 Na(2S1/2) + I(2P1/2)

A0+ 0 -1

Na+I

Na(2S1/2) + I(2P3/2)

–

-2

X 0+ D e= 0.4 0.6 3.026eV R0 = 0.2836Å

RH1 0.8

RH2 1.0

1.2

1.4 R / nm

orbitals are localized in great distance from each other, at the Na and I atom, respectively. The next states, A1 Σ + and 3 Σ + states, which are formed with the . . . σ π 4 σ ∗ MOs, lie somewhat higher, as the excitation of the σ orbital requires more energy than exciting a “lone pair” of electrons out of the π orbitals. At this stage of the computation one obtains a picture very similar to that in the case of NaCl in Fig. 3.50: with pronounced curve crossings as a consequence of the ionic influence. However, one now has account for spin-orbit interaction and diagonalize the whole Hamiltonian again with it included. This leads now to a range of Ω ± states which arise from Λ + Σ as we have discussed formally already in Sect. 3.6.3. Figure 3.52 shows only three of these states, which can be accessed by optical excitation from the ground state. According to A LEKSEYEV et al. (2000), the above mentioned Λ states with S = 0 and 1 are responsible for the X0+ ground state and five more Ω states (2(I ), 1(I ), 1(II), 0− (I ) and A0+ ) which correlate asymptotically with Na(3 2 S1/2 ) + I (5 2 P3/2 ). For symmetry reasons, of those only the A0+ state interacts with the ionic configuration and leads to an avoided crossing. In addition, there are the states (1(III), 0− (II) and B0+ ) which correlate for R → ∞ with Na(3 2 S1/2 ) + I(5 2 P1/2 ). Of those, again only the B0+ state interacts with the ionic configuration and avoids the crossing. Those two avoided crossings give rise to two excited, quasi-bound states, A0+ and B0+ , as clearly seen in Fig. 3.52. For completeness we also have

224

3

Diatomic Molecules

drawn another state, 0+ (IV) according to C OOPER et al. (1987), which correlates with Na(3 2 P1/2 ) + I(5 2 P3/2 ) which is, however, not bonding since the ionic potential does not cross this state. The diagonalization of the spin-orbit interaction hardly influences the ground state. It is, however, responsible for a lowering or raising of the asymptotic potentials by −1/3 and +2/3 of the atomic spin-orbit splitting (0.9426 eV). Thus, the overall bonding energy of the X0+ ground state becomes De = 3.02631 eV. The bond-length is R0 = 0.2836 nm. NaI has played a key role in the development of femto-chemistry. With ultrashort laser pulses one may generate wave-packets on the repulsive part of the excited A0+ state potential (just above the minimum of the X0+ at R0 , see Fig. 3.52). These wave-packets may oscillate between the left and right border of the A0+ potential, just as a classical oscillator. However, these vibrational states are not completely stable. Speaking figuratively: at the avoided crossing the potential well has a leak, through which during each oscillations a certain fraction of the wave-packet ‘flows out’. With femtosecond pump-probe techniques one may follow this process. For his seminal work studying such transition states in chemical reactions with femtosecond spectroscopy Ahmed Z EWAIL (1999) received the N OBEL prize in chemistry.

3.7.5

Nitrogen Monoxide, NO

Before ending this discussion of diatomic molecules we briefly mention the NO molecule. It is of great importance for several disciplines of science and technology, e.g. in biology and physiology, in atmospheric chemistry, in technical combustion processes and many more. NO is a very reactive radical but may nevertheless be stored and transported in bottles. It has a dipole moment (i.e. it is infrared active) and has a well known electronic structure. In molecular physics it is thus a very popular reference object for which countless spectroscopic studies have been carried out. In the context of multiphoton ionization processes in molecules it is valued almost as a kind of ‘Drosophila’ of molecular physics (similar to Na in atomic physics). Figure 3.53 gives an overview of the most important potentials for the states of neutral NO and the lowest states of the NO+ cation. The data presented here are again adapted from G ILMORE (1965). Corresponding to the AO configuration . . . 2s 2 2p 3 (N) and . . . 2s 2 2p 4 (O) of its constituents, the ground state of NO is characterized by a (1σ )2 (2σ )2 (3σ )2 (4σ )2 (1π)4 (5σ )2 (2π)1 MO configuration. Obviously, NO has one unpaired 2π ∗ electron in its valence shell. This is the cause of its high reactivity as a radical. NO is paramagnetic and its ground state must be a X 2 Π state. What can be illustrated very clearly by this example is the difference between the so called molecular RYDBERG states and valence states. If only the outer, antibonding 2π ∗ valence orbital15 is excited, then the excited molecule will be stronger 15 See

Fig. 3.46 for the ordering of orbitals.

3.7


225

1Σ+ and 3Σ+

22

7Σ +

5 Σ+

1Δ

20

N+(3P)+O(3P)

N(4So)+O+(4So)

1Σ -

A1

3 Σ-

18 w 3Δ b3

16


NO+ 14

a 3Σ+

12 X 1Σ +

4 Σ-

o

N(4S )+O(1S) o

10

S

F 2Δ

(from top to bottom) and H 2Σ

H' 2 2 Σ+

E 2Σ + 8

N(2P )+O(3P)

2Σ+, M 2Σ+, K 2

2Φ

D 2Σ +

G 2Σ

C2

i

4

B' 2Δ i

o

N(4S )+O(1D)

4Δ

i

4Σ+ and 6Σ+

6

o

N(4S )+O(3P)

6 A 2Σ + b 4Σ 4

-

5

3

B2

r

a4

i

o

N(2D )+O(3P)

and 5 Σ

o

-

o

N(4S )+O (2P )

NO

NO-

2 X 3Σ

-

0 X2

r

(1 )2 (2 )2 (3 )2 (4 )2 (1 )4 (5 )2 (2 )1

-2 0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32


Fig. 3.53 Potentials for nitrogen monoxide and its ions according to G ILMORE (1965)

0.36

226

3

Diatomic Molecules

bonded than the ground state (smaller R0 , more rigid potential). In this manner a whole series of potentials is generated, A 2 Σ + , C 2 Π , D 2 Σ + , E 2 Σ + , with nearly identical shape of their potentials. As seen in Fig. 3.53, these states converge towards the ground state X 1 Π of the NO+ cations and reflect already its potential: the outer electron is far away from the ion core of the molecule, and does not participate in the bonding. It behaves like an atomic RYDBERG electron, hence one speaks about molecular RYDBERG states. In contrast, the states a 4 Π , B 2 Π , b 4 Σ − show neither a great similarity to the neutral nor to the ionic ground state. They are significantly weaker bound than either of those: they are generated by exciting electrons from lower lying, bonding MOs with subsequent rearrangement of the whole electron configuration and a softening of the bonding. One speaks of valence states. As states of equal symmetry must not cross, a whole series of avoided crossings arises, both for RYDBERG and valence states. These are clearly seen in Fig. 3.53 at energies between 7 eV and 9 eV. As also indicated in Fig. 3.53, here too an anion exists. NO− is, however, not very stable (electron affinity WEA = 24 meV). It may be observed readily as resonance in low energy electron scattering. We shall come back to this briefly in Sect. 7.2.6. Section summary

• MOs for heteronuclear molecules are constructed essentially by the same principles as those for homonuclear molecules. Of course, the even (g) vs. odd (u) symmetry is no longer relevant and correlation diagrams become somewhat more complicated. • The scheme for filling the orbitals, shown in Table 3.9, is consequently less transparent and less rigorous than for homonuclear molecules. • In principle, for all heteronuclear diatomic molecules ionic bonding is important – the strength of its influence depending on the specific system. Caused by this ionic character of the bonding, all have a permanent dipole moment, i.e. they are microwave and infrared active. • We have looked into the potential energy diagrams of several specific examples of heteronuclear, diatomic molecules, LiH, the alkali halides, NaCl and NaI and finally NO, the ‘Drosophila’ of molecular physics. A number of interesting properties have been discussed. We recall specifically the “harpooning” mechanism in molecular reactions involving an intermediate ionic complex.

Acronyms and Terminology AO: ‘Atomic orbital’, single electron wave function in an atom; typically the basis for a rigorous structure calculation. a.u.: ‘atomic units’, see Sect. 2.6.2 in Vol. 1. BO: ‘B ORN O PPENHEIMER’, approximation, the basis when solving the S CHRÖ DINGER equation for molecules (see Sect. 3.2). CI: ‘Configuration interaction’, mixing of states with different electronic configurations in atomic and molecular structure calculations, using linear superpositon of S LATER determinants (see Sect. 10.2.3 in Vol. 1).

Acronyms and Terminology

227

DFT: ‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3 in Vol. 1). E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. EPR: ‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2 in Vol. 1). FIR: ‘Far infrared’, spectral range of electromagnetic radiation. Wavelengths between 3 µm and 1 mm according to ISO 21348 (2007). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6 in Vol. 1). FTIR: ‘F OURIER transform infrared spectroscopy’, (see Chap. 5, p. 298ff.). FWHM: ‘Full width at half maximum’. good quantum number: ‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the H AMILTON operator (see Sect. 2.6.5 in Vol. 1)’. HF: ‘H ARTREE -F OCK’, method (approximation) for solving a multi-electron S CHRÖDINGER equation, including exchange interaction. HITRAN: ‘High-resolution transmission molecular absorption database’, http:// www.cfa.harvard.edu/hitran (ROTHMAN et al. 2009). HOMO: ‘Highest occupied molecular orbital’. IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). isotopologue: ‘Molecules that differ only in their isotopic composition’, http://en. wikipedia.org/wiki/Isotopologue. LCAO: ‘Linear combination of atomic orbitals’, linear superposition of atomic, single electron wave functions to form a molecular orbital (MO). LUMO: ‘Lowest unoccupied molecular orbital’. MO: ‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation. NIR: ‘Near infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1.4 µm according to ISO 21348 (2007). NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. NMR: ‘Nuclear magnetic resonance’, spectroscopy, a rather universal spectroscopic method for identifying molecules (see Sect. 9.5.3 in Vol. 1). RKR: ‘RYDBERG -K LEIN -R EES’, precise method for determining molecular potentials from spectroscopic data. SCF: ‘Self-consistent field’, method for solving coupled integro-differential equations iteratively. SI: ‘Système international d’unités’, international system of units (m, kg, s, A, K, mol, cd), for details see the website of the Bureau International des Poids et Mésure http://www.bipm.org/en/si/ or NIST http://physics.nist.gov/cuu/Units/index. html. vdW: ‘VAN DER WAALS’, interaction between atoms or molecules, ∝ R −6 .

228

3

Diatomic Molecules

VMI: ‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B).

References A LEKSEYEV , A. B., H. P. L IEBERMANN, R. J. B UENKER, N. BALAKRISHNAN, H. R. S ADEGH POUR , S. T. C ORNETT and M. J. C AVAGNERO : 2000. ‘Spin-orbit effects in photodissociation of sodium iodide’. J. Chem. Phys., 113, 1514–1523. B ORN , M. and R. O PPENHEIMER: 1927. ‘Zur Quantentheorie der Molekeln’. Ann. Phys. Berlin, 84, 0457–0484. C OOPER , D. L., S. B IENSTOCK and A. DALGARNO: 1987. ‘Mutual neutralization and chemiionization in collisions of alkali-metal and halogen atoms’. J. Chem. Phys., 86, 3845–3851. C ÔTE , R., M. J. JAMIESON, Z. C. YAN, N. G EUM, G. H. J EUNG and A. DALGARNO: 2000. ‘Enhanced cooling of hydrogen atoms by lithium atoms’. Phys. Rev. Lett., 84, 2806–2809. D ULIEU , O. and C. G ABBANINI: 2009. ‘The formation and interactions of cold and ultracold molecules: new challenges for interdisciplinary physics’. Rep. Prog. Phys., 72, 086401. D UNHAM , J. L.: 1932. ‘The energy levels of a rotating vibrator’. Phys. Rev., 41, 721–731. F LEMING , H. E. and K. N. R AO: 1972. ‘A simple numerical evaluation of Rydberg-Klein-Rees integrals – application to X1 Σ + state of 12 C16 O’. J. Mol. Spectrosc., 44, 189–193. GAMESS: 2010. ‘General atomic and molecular electronic structure system’, Gordon research group at Iowa State University, USA. http://www.msg.chem.iastate.edu/gamess/, accessed: 9 Jan 2014. G AUSSIAN: 2013. ‘Gaussian 09 rev. D’, Gaussian, Inc., Wallingford, CT, USA. http://www. gaussian.com/, accessed: 9 Jan 2014. G ILMORE , F. R.: 1965. ‘Potential energy curves for N2 , NO, O2 and corresponding ions’. J. Quant. Spectrosc. Radiat. Transf., 5, 369–389. G RISENTI , R. E., W. S CHÖLLKOPF, J. P. T OENNIES, G. C. H EGERFELDT, T. KÖHLER and M. S TOLL: 2000. ‘Determination of the bond length and binding energy of the helium dimer by diffraction from a transmission grating’. Phys. Rev. Lett., 85, 2284–2287. H ERSCHBACH , D. R., Y. T. L EE and J. C. P OLANYI: 1986. ‘The N OBEL prize in chemistry: for their contributions concerning the dynamics of chemical elementary processes’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/laureates/1986/. H ERZBERG , G.: 1989. Molecular Spectra and Molecular Structure, vol. I. Diatomic Molecules. Malabar, Florida: Krieger Publishing Company, 660 pages. H OUGEN , J. T.: 2001. ‘The calculation of rotational energy levels and rotational line intensities in diatomic molecules (version 1.1)’, NIST. http://physics.nist.gov/DiatomicCalculations, accessed: 9 Jan 2014. H UBER , K.-P. and G. H ERZBERG: 1979. Constants of Diatomic Molecules. New York: Van Nostrand Reinhold. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. J ENOUVRIER , A., M. F. M ERIENNE, B. C OQUART, M. C ARLEER, S. FALLY, A. C. VANDAELE, C. H ERMANS and R. C OLIN: 1999. ‘Fourier transform spectroscopy of the O2 Herzberg bands – I. Rotational analysis’. J. Mol. Spectrosc., 198, 136–162. J UARROS , E., K. K IRBY and R. C ÔTE: 2006. ‘Laser-assisted ultracold lithium-hydride molecule formation: stimulated versus spontaneous emission’. J. Phys. B, At. Mol. Phys., 39, S965–S979. K LAUS , T., S. P. B ELOV and G. W INNEWISSER: 1998. ‘Precise measurement of the pure rotational submillimeter-wave spectrum of HCl and DCl in their υ = 0, 1 states’. J. Mol. Spectrosc., 187, 109–117. K LING , M. F. et al.: 2006. ‘Sub-femtosecond control of electron localization in molecular dissociation’. Science, 312, 246–248.

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K REMER , M. et al.: 2009. ‘Electron localization in molecular fragmentation of H2 by carrierenvelope phase stabilized laser pulses’. Phys. Rev. Lett., 103, 213003. L E ROY , R. J.: 1970. ‘Molecular constants and internuclear potential of ground-state molecular iodine’. J. Chem. Phys., 52, 2683–2689. L I , X. Z. and J. PALDUS: 2006. ‘Singlet-triplet separation in BN and C2 : Simple yet exceptional systems for advanced correlated methods’. Chem. Phys. Lett., 431, 179–184. L OCKWOOD , G. J. and E. E VERHART: 1962. ‘Resonant electron capture in violent protonhydrogen atom collisions’. Phys. Rev., 125, 567–572. L OVAS , F. J., E. T IEMANN, J. S. C OURSEY, S. A. KOTOCHIGOVA, J. C HANG, K. O LSEN and R. A. D RAGOSET: 2005. ‘Diatomic spectral database (version 2.1)’, NIST. http://physics.nist.gov/ Diatomic, accessed: 9 Jan 2014. M ANTZ , A. W., J. K. G. WATSON, K. N. R AO, D. L. A LBRITTON, A. L. S CHMELTEKOPF and R. N. Z ARE: 1971. ‘Rydberg-Klein-Rees potential for the X 1 Σ + state of the CO molecule’. J. Mol. Spectrosc., 39, 180–184. M OLPRO: 2012. ‘Molpro quantum chemistry package’, H.-J. Werner, Universität Stuttgart, Germany, and P. J. Knowles, Cardiff University, UK. http://www.molpro.net/, accessed: 9 Jan 2014. PARTRIDGE , H. and S. R. L ANGHOFF: 1981. ‘Theoretical treatment of the X 1 Σ + , A 1 Σ + , and B 1 Π states of LiH’. J. Chem. Phys., 74, 2361–2371. PAULING , L.: 1931. ‘The nature of the chemical bond. . . ’ J. Am. Chem. Soc., 53, 1367–1400. R AM , R. S., M. D ULICK, B. G UO, K. Q. Z HANG and P. F. B ERNATH: 1997. ‘Fourier transform infrared emission spectroscopy of NaCl and KCl’. J. Mol. Spectrosc., 183, 360–373. ROTHMAN , L. S. et al.: 2009. ‘The HITRAN 2008 molecular spectroscopic database’. J. Quant. Spectrosc. Radiat. Transf., 110, 533–572. S HARP , T. E.: 1971. ‘Potential-energy curves for molecular hydrogen and its ions’. At. Data, 2, 119–169. S TANTON , J. F.: 1999. ‘A refined estimate of the bond length of methane’. Mol. Phys., 97, 841– 845. TANG , K. T., J. P. T OENNIES and C. L. Y IU: 1995. ‘Accurate analytical He-He van-der-Waals potential based on perturbation-theory’. Phys. Rev. Lett., 74, 1546–1549. T URBOMOLE: 2010. ‘Quantum chemistry (QC) program package’, COSMOlogic GmbH & Co. KG, Leverkusen, Germany. http://www.cosmologic.de/QuantumChemistry/main_qChemistry. html, accessed: 9 Jan 2014. Y IANNOPOULOU , A., G. H. J EUNG, S. J. PARK, H. S. L EE and Y. S. L EE: 1999. ‘Undulations of the potential-energy curves for highly excited electronic states in diatomic molecules related to the atomic orbital undulations’. Phys. Rev. A, 59, 1178–1186. Z EWAIL , A. H.: 1999. ‘The N OBEL prize in chemistry: for his studies of the transition states of chemical reactions using femtosecond spectroscopy’, Stockholm. http://nobelprize.org/nobel_ prizes/chemistry/laureates/1999/.

4

Polyatomic Molecules

Diatomic molecules, treated in the previous chapter, provide only a first step into the world of real molecules. Anyone who wishes to obtain an idea about that world should at least browse through the present chapter – even though it is, admittedly, somewhat demanding. The spectroscopy of triatomic and polyatomic molecules is largely determined by their symmetry, and all concepts introduced in Chap. 3 have to be generalized. The motion of, say, Nnu atomic nuclei is characterized by three rotational and 3Nnu − 6 vibrational degrees of freedom (aside from the trivial three translational motions). Correspondingly, the description of electronic states becomes significantly more complex.

Overview

Section 4.1 presents the arbitrarily shaped, rigid rotor, and Sect. 4.2. introduces normal coordinates used to treat vibrations. Section 4.3 addresses the symmetries of point groups as a key principle to classify polyatomic molecules. The specialties of the electronic structure of polyatomic molecules are introduced in Sect. 4.4 by way of example, starting with H2 O. We then introduce the important concept of orbital hybridization by discussing sp 3 MOs for CH4 and NH3 (Sect. 4.4.2). Finally, in Sect. 4.5 conjugated hydrocarbon systems are introduced. A simple, quantitative description is provided by the H ÜCKEL approximation (HMO).

4.1

Rotation of Polyatomic Molecules

4.1.1

General

We first recall some classical mechanics and focus on the case of an arbitrarily shaped, rigid rotor. We assume that the Nnu atomic nuclei of the molecule with masses m1 , m2 , . . . , mk , . . . , mN are positioned in 3D space at R 1 , R 2 , . . . , R k , . . . , R Nnu which are constant in respect of their distance from the centre of mass and their relative orientation within the molecule. The moments of inertia of such a © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_4

231

232

4


molecule are described by a second rank tensor, the inertia tensor I with the 3 × 3 components Iij =

Nnu

mk R 2k δij − Rk,i Rk,j ,

(4.1)

k=1

where Rk,1 = Xk , Rk,2 = Yk and Rk,3 = Zk . If the molecule rotates around an arbitrary axis with an angular velocity ω, its angular momentum (denoted by N as in Chap. 3) is N = Iω. In respect of an axis in arbitrary direction, characterized by a unit row or column $ vector, e$ R = R /R and eR = R/R, respectively, the moment of inertia is IR = e $ R IeR =

3 1 Ri Iij Rj . R2

(4.2)

ij =1

There is always one body fixed coordinate system, let us call its axes abc, in which the inertia tensor becomes diagonal. By convention, the thus defined principle moments of inertia are ordered: Ia ≤ I b ≤ I c .

(4.3)

In respect of an axis e$ R = (a1 b1 c1 ) in the body fixed coordinate system, with 2 2 2 a1 + b1 + c1 = 1, the moment of inertia (4.2) becomes IR = Ia a12 + Ib b12 + Ic c12 . √ We divide this by IR and write a1 / IR = a so that

(4.4)

1 = Ia a 2 + Ib b 2 + Ic c 2 .

(4.5)

In the body fixed coordinate system, this equation describes √ degree√sur√ a second face – more precisely: an ellipsoid with principal axes √ 1/ Ia , 1/ Ib and 1/ Ic , the so called ellipsoid of inertia. The distance |R| = a 2 + b2 + c2 of any point on √ that surface from the origin is just 1/ IR , if IR is the moment of inertia in respect of that axis in the direction R. This is sketched in Fig. 4.1. In respect of the space fixed coordinate system XY Z the body fixed system with axes abc is characterized by the E ULER angles αβγ which we have introduced in Appendix E, Vol. 1. The direction of the angular momentum N of this rigid rotor is now specified by two quantities (instead of one in the case of a diatomic molecule): the usual projection M on a space fixed Z-axis, and in addition by the projection K onto a body fixed axis, e.g. onto the c axis. The quantum mechanics of the rigid, extended rotor has already been developed within one year after the emergence of quantum mechanics (1927) by R ABI and others. We just summarize the most important results without entering into details.

4.1


233

Fig. 4.1 Ellipsoid of inertia and E ULER angles αβγ

Z c

β

M

N

K

b

γ

α Y',Y''

X

α X' β X''

Y

γ a

We have to expand the notation of Sect. 3.3.2 for the components of the angular moX , N Y and N Z , as well as in mentum operator in respect of the space fixed system, N respect of the three body fixed, orthogonal principle axes of the ellipsoid of inertial, a , N b and N c , which relate to the square of the angular momentum operator by N 2 = N X2 + N Y2 + N Z2 = N a2 + N b2 + N c2 . N

(4.6)

2 and its components obey the usual angular momentum algebra and commutation N rules in the space fixed as well as in the body fixed coordinate system. One may find Z and N c : 2, N eigenstates |N MK which are simultaneously eigenstates of N 2 |N MK = 2 N(N + 1)|NMK N Z |N MK = M|NMK, N

with N = 0, 1, 2, . . . ,

c |N MK = K|N MK and N

(4.7)

with M = 0, ±1, ±2, . . . , ±N and K = 0, ±1, ±2, . . . , ±N. In analogy to the spherical components of the angular momenta J± = ∓(Jx ± iJy )/ √ 2 used for atoms in Vol. 1, one may compose the corresponding components of X and N Y or of N a and N b :1 N √ a ± iN b )/ 2. ± = ∓(N (4.8) N A close analysis shows, however, that the matrix elements in the body fixed coordinate system correspond to the conjugate complex operators of those in the space fixed system (see e.g. VAN V LECK 1951). Thus, with (B.15) in Vol. 1 we have: ± |N MK = ∓ N(N + 1) − M(M ± 1) /2 N (M ± 1)K N (4.9) ± |N MK = ∓ N(N + 1) − K(K ∓ 1) /2. (4.10) N M(K ∓ 1)N 1 Note that we use, here too, the orthonormalized operators in contrast to the combinations N a ±iN b

often used in the literature.

234

4


The eigenstates |NMK correspond to eigenfunctions DN MK (αβγ ) in position space which we have met already in (E.1), Vol. 1. The Hamiltonian is advantageously written in respect of the body fixed axes abc – as the energy does not depend on the orientation M of the angular momentum in respect of the space fixed axes (unless there is an external field present), while in respect of the axes of the ellipsoid of inertia the energy is given by 2 2 2 N rot = 1 Na + b + Nc . H 2 Ia Ib Ic

(4.11)

In the most general case this leads to three rotational constants A=

2 , 2Ia hc

B=

2 2Ib hc

and C =

2 , 2Ic hc

(4.12)

sorted according to (4.3) as A ≥ B ≥ C.

4.1.2

Spherical Rotor

The simplest case is clearly the spherical rotor (Ia = Ib = Ic = I ) describing symmetric molecules such as CH4 , SF6 and the like. The S CHRÖDINGER equation becomes 2 rot |NMK = N(N + 1) |N MK H 2I with rotational energies

WN =

2 N(N + 1) = Bhc N (N + 1) 2I

(4.13)

(4.14)

quite analogous to (3.37). However, we have now a (2N + 1)2 fold degeneracy – in contrast to the linear molecule which the degeneracy is only (2N + 1).

4.1.3

Symmetric Rigid Rotor

Still relatively clear is the rigid symmetric rotor (short: symmetric top), characterized by a well defined, at least threefold symmetry axis, with two identical principle moments of inertia. The prolate top (cigar shaped) symmetric rotor Ia < Ib = Ic and thus A > B = C. Examples are methyl-chloride Cl−CH3 , chloroform CHCl3 or propyne CH3 C≡CH. Alternatively, the oblate top (pancake like) is characterized by Ia = Ib < Ic and thus A = B > C. Examples are ammonia NH3 as well as all planar molecules such as benzene C6 H6 . The respective ellipsoids of inertia are shown in Fig. 4.2.

4.1


235

c

(a)

(b)

K

N

N a

c

b K b

a

and projection K onto the Fig. 4.2 (a) Prolate rotational ellipsoid with angular momentum N symmetry axis, here a. (b) Oblate rotational ellipsoid with symmetry axis c

(a) WNK (hc) -1 / arb.un.

Fig. 4.3 Energy terms of the rigid symmetric rotor (symmetric top): (a) prolate top, for B = 1 arb.un., A = 2 arb.un., (b) oblate top, for B = 2 arb.un., A = 1 arb.un.

(b)

prolate

oblate

N= 3

20 N= 10

2

3 2

1 0 0 |K | = 0

1 0 1

2

3

0

1

2

3

To solve the S CHRÖDINGER equation one rewrites the Hamiltonian (4.11) suitably, e.g. for the prolate symmetric top as 2 2 2 c2 + Na = N + 1 − 1 N b + N a2 , rot = 1 N H 2Ib 2Ia 2Ib 2Ia 2Ib

(4.15)

so that the S CHRÖDINGER equation can be solved in closed form. Using (4.7) we obtain

2 2 2 1 1 rot |N MK = N(N + 1) + K |N MK. (4.16) − H 2Ib 2 Ia Ib The rotational energy of the prolate top is thus given by WN K = Bhc N (N + 1) + (A − B)hcK 2 ,

(4.17)

while for the oblate top one finds correspondingly WN K = Bhc N (N + 1) + (C − B)hcK 2 .

(4.18)

The rotational energy obviously depends now also on the projection |K| = 0, 1, onto the figure axis. The resulting energy terms . . . , N of the angular momentum N are illustrated in Fig. 4.3 as a function of N and K. According to (4.17) and (4.18), respectively, the prefactor of K 2 is positive for the prolate rotor (A > B = C), and

236

4

Fig. 4.4 Energy terms (schematic) for the asymmetric top as compared to the prolate top (left) and the oblate top (right)

prolate A = 2, B = C = 1


asymmetric

oblate A = B = 2, C = 1

A=2>B>C=1 N Ka

N Ka Kc 220

2 2

221

N Kc 2 0 2 1

211 212 2 1 2 0

202

1 1 1 0

110 111 101

0 0

000

2 2

1 0 1 1

0 0

negative for the oblate rotor (A = B > C). This may be visualized from Fig. 4.2 by recalling from (4.12) that for a given value of N the axis with highest rotational energy corresponds to the lowest moment of inertia. For the prolate top the rota is as parallel as possible to the tional energy is largest if the angular momentum N symmetry axis (here a), i.e. for K = N ; in contrast, for the oblate top the energy is is perpendicular to the symmetry axis (here c), i.e. for K = 0 when it highest if N N lies within the ab plane. In both limiting cases, (4.18) becomes identical to the corresponding expression (3.37) for the linear rigid rotor, with I = Ib . may assume different orientations in space, expressed in (4.16) by the Finally, N quantum number M. Terms with K = 0 are thus 2N + 1 fold degenerate, those with |K| > 0 however 2(2N + 1) fold since K may be positive or negative. We finally note (without proof) that a planar molecule with at least one threefold symmetry axis is an oblate top for which Ic = 2Ia = 2Ib , i.e. A = B = 2C holds. In contrast, nonlinear triatomic species as e.g. water, H2 O, are asymmetric top molecules which we shall discuss next.

4.1.4

Asymmetric Rigid Rotor

In the general case with Ia = Ib = Ic , the asymmetric rigid rotor (short: asymmetric top), simply rewriting the Hamiltonian as done in the last subsection does no longer help: The energies and eigenfunctions of the asymmetric top cannot be given in closed form – not even in the case of a rigid rotor. Nevertheless, many important molecules belong to this class, e.g. those with a twofold symmetry axis such as H2 O. One may, however, obtain a qualitative estimate for this situation by simply interpolating the energy terms between prolate and oblate top. This is illustrated in Fig. 4.4. The asymmetric top is characterized by three rotational constants A > B > C which are associated to the three principle axes abc of the ellipsoid of inertia

4.1


237

(Ia < Ib < Ic ). The prolate top (A > B = C) and the oblate top (A = B > C) are limiting cases in this notation. The twofold degeneracy for K > 0 in the case of a symmetric top is now removed. One speaks about Ktype doubling in analogy to Λ type doubling to which we have been introduced in the context of electronic states of diatomic molecules Sect. 3.6.6: it follows directly from breaking the symmetry and is characterized here by two quantum numbers Ka and Kb . In the limit of the prolate rotor Ka becomes what we have called K, the projection of the angular momentum onto the axis of the smallest moment of inertia; in the case of the oblate top Kc becomes the projection onto the axis with the largest moment of inertia. Clearly, each of the |N Ka Kc states is still 2N + 1 fold degenerate, since here too, the an may have 2N + 1 orientations in space (again described by the gular momentum N quantum number M). The exact computation of the eigenstates and eigenenergies is somewhat elaborate. In principle the Hamiltonian (4.11) is written as + 2 − 2 rot = α N 2 + βN c2 + γ N H , + N

(4.19)

+ and N − are the operators defined in (4.8). The constants α, β and γ are where N different linear combinations of A, B and C, chosen according to the ratios A : B : C in such a way that the constant γ describing the deviations from the symmetric top becomes as small as possible. The eigenstates |N MΓ of the Hamiltonian may then 2 and N c : be written as linear combination of those for the symmetric top, i.e. of N |NMΓ =

N

fN K |N MK.

K=−N

With this ansatz one may diagonalize the Hamiltonian (4.19) for each N , using the matrix elements (4.9) and (4.10). For larger values of N this leads to increasingly complicated expressions. Moreover, the rigid rotor is only a first approximation. In order to meet spectroscopic accuracy one has to include centrifugal distortion, vibration-rotation coupling and coupling with the electronic angular momentum in the spirit of H UND’s cases, possibly also hyperfine interaction. Also, vibronic couplings may play an important role, as we shall discuss in connection with the JAHN T ELLER effect in Sect. 4.3.4. Numerical methods and approximations have been developed for this purpose and extended reviews and monographs have been written on this subject. Today one interprets the experimental spectra by direct numerical comparison of a suitably parameterized ansatz and with the help of sophisticated simulation programmes. We cannot go into details here, but sketch as an example the situation for the H2 O molecule. Pure rotational spectra are found in the sub-millimetre range, but only few experimental data are available. An overview of the rotational level scheme for 0 ≤ N ≤ 3 is shown in Fig. 4.5. It has been obtained (within the highly simplified model of the rigid rotor) from the convenient computer programme PGOPHER (2013) in the online version.

238 300

WN K K (hc) -1 / cm -1 a c

Fig. 4.5 Rotational energy levels of the H2 O molecule for angular momenta N ≤ 3. The terms are arranged from left to right according to increasing Kc and decreasing Ka derived with the help of PGOPHER (2013)

4 330 331

NKa Kb

321 322 200

312 313 303

220 221 211

100 110 111 0

Fig. 4.6 Geometry of the H2 O molecule


212 202

101

000

O m n 8 c

0.00347nm 0. 09 a 1 57 7 5 18 9 1 0 n ° 0 4. 4 7 4 m 0. H

b

H

The parameters used in this computation are derived from the H2 O geometry, shown in Fig. 4.6 and the known atomic masses. The three moments of inertia are Ia = 0.00632 u nm2 , Ib = 0.01154 u nm2 , and Ic = 0.01786 u nm2 ; due to the light H atom, the corresponding rotational constants are rather large: A0 = 27.880591 cm−1 , B0 = 14.5216246 cm−1 , and C0 = 9.27774594 cm−1 (B ERNATH 2002b). Section summary

• Polyatomic molecules have in general 3 rotational degrees of freedom. From the inertia tensor (4.1) one obtains the moments of inertia for a given axis in the direction eR by IR = e$ R IeR . In diagonalized form I is expressed by the three principle moments of inertia, Ia ≤ Ib ≤ Ic . They may be visualized √ by Ia , the√ellipsoid of inertia (4.5) whose three principle axes are given by 1/ √ 1/ Ib , 1/ Ic , respectively. The rotational constants (4.12) are A ≥ B ≥ C, correspondingly. of an arbitrarily shaped rotor and its components • The angular momentum N b , N c in respect of the principle axes obey the usual commutation rules a , N N and eigenvalue equations for angular momenta. The total energy of the rotor (4.11) is given by the sum of the energies of the three components. • For the spherical (or symmetrical) rotor with three identical moments of inertia (molecules such as CH4 ) the energy is simply WN = Bhc N (N + 1), just as in the case of a diatomic molecule. The degeneracy is now, however, (2N + 1)2 fold.

4.2

Vibrational Modes of Polyatomic Molecules

239

• The symmetric top too can be solved in closed form, leading to energies (4.17)–(4.18) for the prolate (A > B = C) and oblate top (A = B > C), respectively. They depend on the total angular momentum N and on its projection onto the figure axis, |K| = 0, 1, . . . , N . Terms with K = 0 are 2N + 1 fold, with |K| > 0 they are 2(2N + 1) fold degenerate. • The asymmetric top is more complicated but may be understood as general case between prolate and oblate top.

4.2


We note at this point that with present state of computation and computerized visualization techniques a host of sources in the internet provide useful, illustrative information on molecular vibrations. Specifically, for small polyatomic molecules we mention I MMEL (2012), who relates the normal modes of many common small molecules to their infrared spectra. A host of useful links are collected at J MOL (2011) to web-pages using Jmol: an open-source Java viewer for chemical structures in 3D (which appears to become the standard for 3D molecular visualization).

4.2.1

Normal Modes of Vibration

A molecule consisting of Nnu atoms has 3Nnu degrees of freedom since each atom can move into three spatial directions. Of these 3Nnu degrees of freedom three describe the overall motion of the molecule (translation of the centre of mass). Rotation of the molecule (in the general case) is described by three more degrees of freedom (in the case of a linear molecule by two degrees of freedom). This leaves us with 3Nnu − 6 degrees of freedom (3Nnu − 5 for linear molecules) to describe the internal motion, i.e. the vibrations of the molecules. Each atomic nucleus may oscillate around its equilibrium distance. We designate the (3Nnu − 6) relative displacement coordinates – measured in a body fixed system – as ξi . They describe all internal motion of the Nnu atoms in the molecule. For small oscillations around the equilibrium we may, just as in the case of a diatomic molecule, expand the potential into a series: V (ξ1 . . . ξNnu ) = V0 +

3N nu −6 i

3Nnu −6 ∂V ∂ 2 V 1 ξ + ξi ξj . i ∂ξi ξi =0 2 ∂ξi ∂ξj 0

(4.20)

i,j

We set V0 = 0, i.e. we choose the absolute minimum of the potential as zero energy. Since we have expanded around the equilibrium the first partial derivatives are ∂V /∂ξi |ξi =0 = 0, and the total energy is written as the sum of kinetic (T ) and

240

4


potential energy (V ): 3Nnu −6 3Nnu −6 1 ∂ 2 V 1 2 ˙ W =T +V = mi ξi + ξi ξj . 2 2 ∂ξi ∂ξj 0 i

(4.21)

i,j

In the next step we introduce mass weighted coordinates qi =

√ mi ξi

(4.22)

with: and the so called Hessian matrix V Vij =

∂ 2 V . ∂qi ∂qj 0

(4.23)

Die Hessian matrix is real, symmetric (Vij = Vj i ) and positive definite, since the potential has a minimum for qi = 0. With this the energy (4.21) becomes W=

3Nnu −6 3Nnu −6 1 1 q˙i2 + Vij qi qj , 2 2 i

(4.24)

i,j

or more compact with (3Nnu − 6) dimensional row and column vectors, q $ and q, respectively: 1 1 W = q˙ $ · q˙ + q $ V q. (4.25) 2 2 In general Vij = 0, and the cross terms qi qj in the sum (4.24) do not vanish: in general, the vibrations are coupled. One thus searches for a new coordinate system Qi , in which this coupling is is symmetric and real, an orthogonal matrix2 A exists removed. Since the matrix V which diagonalizes it: A = Ω. −1 V A

(4.26)

− λ It is found as usual by solving the characteristic equation det(V 1) = 0. The . Since V is positive are the eigenvalues λi of V elements of the diagonal matrix Ω definite, all its eigenvalues are also positive, thus we write λi = ωi2 . The eigenvectors are then found by solving the system of linear equations of V − ωi2 V 1 ai = 0 for each eigenvalue ωi2 . The transformation matrix is the matrix of the eigenvectors: = a1 A 2 That

a2

...

ai

...

a Nnu .

1. is a matrix for which Aˆ $ = Aˆ −1 or equivalently Aˆ $ Aˆ =

4.2


241

Normal coordinatesQ are now defined by q = AQ

or

$ q. −1 q = A Q=A

(4.27)

Inserting this into (4.25), one obtains with a few lines of algebra 1 ˙ 2 1 1 ˙$˙ = Q + Q$ ΩQ W= Q Qi + ωi2 Q2i . 2 2 2

(4.28)

i

Hence, the vibrational energy can be written by a single sum over i. That is equivalent to saying that motions in these coordinates are decoupled. The Qi describe 3Nnu − 6 independent, harmonic oscillators. We identify (4.28) as the classical H AMILTON function, with the canonical pairs of position and momentum coordinates, Qi and Pi = ∂Ti /∂ Q˙ i = Q˙ i , respectively: H=

i

(Ti + Vi ) =

1 2 Pi + ωi2 Q2i . 2

(4.29)

i

In the classical H AMILTON equations of motion, ˙ i = ∂H Q ∂Pi =⇒

∂H and P˙i = − = −ωi2 Qi = Q¨ i ∂Qi ¨ i + ωi2 Qi = 0, Q

the different coordinates are completely decoupled, and the individual solutions are Qi (t) = Qi (0)e±iωi t . The motions in the coordinates Qi are thus simple, harmonic vibrations of frequency ωi . They are called normal modes of the molecule. Back transformation into the original (mass weighted) molecular coordinates is done by superposing all normal modes q = AQ or qj = Aj k Qk . (4.30) k

Such a normal mode describes in the general case complex (harmonic) motions of all atoms in a molecule. If only one normal mode Qi is excited, we find qj = Aj i Qi , i.e. all atoms j oscillate with equal frequency ωj and in phase (if Aj i = 0): a normal mode is delocalized over the whole molecule. Conversely, so called local modes for which dominantly one bond vibrates, may be constructed by judiciously superposing several normal modes.

242

4.2.2

4


Energies and Transitions of Normal Modes

Starting point of the quantum mechanical description is the H AMILTON function (4.29). The H AMILTON operator for the (decoupled) normal modes is = H

i H

i = − with H

i

2 d 2 1 + ωi2 Q2i . 2 2 dQi 2

(4.31)

The S CHRÖDINGER equation with this Hamiltonian is separable, the eigenfunctions may be written as product of eigenfunctions of harmonic oscillators ) Rv1 v2 ... (Q) = Rvi (Qi ) = Rv1 (Q1 ) · Rv2 (Q2 ) · . . . · RvNnu −6 (Q3 ) (4.32) i

with 3Nnu − 6 (or −5) factors for all Qi . For each coordinate one vibrational quantum number vi describes the energy and the total energy becomes W= (vi + 1/2)ωi . (4.33) i

For large molecules this leads to substantial internal energies already at thermal excitation. Even the zero point energy may be high: The famous “football” molecule, C60 , e.g. has 174(!) normal modes (the lowest mode energy corresponds to ν¯ 500 cm−1 ). The calculation leads to a zero point energy on the order of Wmin 5.4 eV (we mention the N OBEL prize 1996 honouring the discovery of C60 by C URL, K ROTO, and S MALLEY). The selection rules for dipole transitions are derived in analogy to the considerations in Sect. 3.4. In harmonic approximation vi = ±1 for all Qi , and the dipole transition matrix element (3.78) is now Dγ v ←v = Rv v ... (Q)Dγ (Q)Rv1 v2 ... (Q)dQ1 dQ2 . . . . 1 2 The permanent dipole moment of the molecule −eDγ (Q), in an electronic state γ (3.79), now depends on all normal coordinates Q$ = (Q1 Q2 . . . Qi . . . ). A series expansion around Qi = 0 gives in analogy to (3.80) ∂ Dγ Qi + · · · . Dγ (Q) = Dγ (Q0 ) + (4.34) ∂Qi 0 i

As discussed in Sect. 3.4.4, here too a normal mode Qi is infrared active (with vi = vi ± 1), if and only if ∂ Dγ /∂Qi |0 = 0. All other vibrational modes remain unchanged in such a one-photon transition.

4.2


243

Fig. 4.7 Normal modes of an AB2 molecule

mB

mA

mB

q1

q2

q3

q4

q5

q6

Efficient programmes are available today for computing normal modes of polyatomic molecules by diagonalization of the Hessian matrix. Important simplifications are achieved by considering the molecular symmetry. For the various point groups according to which molecules may be classified (see Sect. 4.3) group theory provides suitable tools when determining the normal modes. We refrain from treating these procedures in detail and just summarize in the following a few results for instructive examples. It must be pointed out, however, that the harmonic approximation just discussed, is again only a first order approximation. Today’s spectroscopic accuracy affords much higher quality of the theory – and correspondingly allows highly precise structural determinations for even rather complex molecules.

4.2.3

Linear, Triatomic Molecules AB2

The coordinates qi and masses mA,B of a linear AB2 molecule are specified in Fig. 4.7. It has 3Nnu − 5 = 4 internal degrees of freedom, and thus four normal modes: two vibrations along the molecular axis and two bending modes perpendicular to it (one in the paper plane and one perpendicular to it). The normal coordinates are plausible by realizing that the centre of mass of the molecule must remain at rest during normal motions: • The symmetric stretch mode, Q1 =

q1 − q3 √ , 2

√ has an eigenfrequency ω1 = k/mB . The atom A remains at rest. The vibrational frequency ω1 corresponds to an atom B mounted on a fixed wall through a spring constant k. • The asymmetric stretch mode, √ q1 − 2 mB /mA q2 + q3 Q3 = , √ 2 + 4mB /mA √ has the eigenfrequency ω3 = k(1/mB + 2/mA ). If the√central atom is very heavy (mA mB ) the eigenfrequency approaches ω3 ≈ k/mB with the coor√ dinate Q3 ≈ 12 (q1 + q3 )/ 2. This corresponds again to vibration of the lighter atom B in respect of a solid wall (infinitely heavy atom A).

244

4


• The two equivalent bending modes are represented by Q2 with Q2 =

mA q5 = −q6 = −q4 . 2mB

The orthogonal motion (perpendicular to the paper plane) is otherwise completely analogous. The bending frequency is independent of the other two normal modes. As a prominent example we take a closer look at carbon dioxide, CO2 , which is of continuing general interest, not least due to its eminent relevance for atmospheric physics and chemistry, for astrophysics and many other areas. The equilibrium bond length C=O amounts to R0 = 0.1166 nm, the experimentally determined eigenfrequencies of the normal modes are ν¯ 1 = 1 285.4 cm−1 (symmetric stretch), ν¯ 2 = 667.4 cm−1 (bending vibration) and ν¯ 3 = 2 349.2 cm−1 (asymmetric stretch), according to RODRIGUEZ -G ARCIA et al. (2007).3 The vibrational energy terms are characterized by five quantum numbers: v1 v2 v3 r (HITRAN terminology), but often only v1 v2 v3 are mentioned. The normal modes and their harmonics are denoted by v1 , v2 and v3 as described above. The angular momentum quantum number accounts for the fact that the two bending modes – albeit degenerate – may combine to effective rotations around the molecular axis (if added with suitable phases) with different energies. This angular momentum quantum number may assume values || = v2 , v2 − 2, v2 − 4, . . . 1 or 0. Finally, the index r characterizes the perturbation in case of F ERMI resonances (see below). The rotation constants for CO2 are rather small, e.g. B000 1 = 0.38714044 cm−1 , the rotational population at room temperature is thus substantial and leads to broadening of the vibrational bands. The two oxygen atoms in CO2 are electro-negative, i.e. they carry a small negative charge, while the carbon atom is slightly positive. However, for symmetry reasons in equilibrium position the dipole moment vanishes, Dγ (Q) = 0. The symmetric stretch vibration Q1 conserves the symmetry; it is thus not infrared active. The other modes, Q2 and Q3 , however, break the symmetry and lead to a change of the permanent electric dipole moment ∂ D/∂Qi |0 = 0. Hence, they are infrared active. A summary of the lowest, experimentally determined vibrational levels is shown in Fig. 4.8. When comparing the symmetric and asymmetric stretch frequencies with the √ = k/m and ω3 = normal mode analysis given above, one notices that with ω 1 B √ kM/mA mB the ratio should be ω3 /ω1 = 1.915, while the experimentally determined value is 1.828. The reason for this is the near degeneracy of the states (100 0) and (020 0) with otherwise equal symmetry. Such a so called F ERMI resonance causes an interaction of these two terms: the states are mixed and repel each other as we have seen it already in a number of cases in the vicinity of avoided crossings. the older literature (e.g. H ERZBERG 1991) one finds for ν¯ 1 = 1 388.3 cm−1 and for 2¯ν2 = 1 285.5 cm−1 . This leads to an interchange of the states (100 0) and (020 0).

3 In

4.2


Fig. 4.8 Vibrational levels of CO2 . The red, full and dashed lines indicate two Fermi pairs (see text). Grey arrows indicate IR active transitions

245

W (v1,v2,v3,ℓ ) (hc) -1 / cm -1

2000

1000

(1110)

(0330) (0310)

(1000)

(0001)

9.4

µm

10

.4

µm

(0200) (0220) (011 0) (0000)

0

Something similar happens between the states (111 0) and (033 0). The two F ERMI resonance pairs are marked red in Fig. 4.8. F ERMI resonances are a rather common phenomenon in atomic and molecular physics. Also indicated in Fig. 4.8 are some infrared transitions (grey arrows). Of specific importance are those at 9.4 µm and 10.4 µm wavelengths: these are the most important transitions in the CO2 laser which still is a work horse in material processing technologies (welding, cutting, drilling of massive material). Due to the closely spaced rotational lines the CO2 laser (exploiting several isotopes) is tuneable over a spectral range from 9.2 µm to above 11 µm – quasi continuously. The infrared spectroscopic data for CO2 are excellently documented (e.g. in the HITRAN data bank by ROTHMAN et al. 2009). As an example, we show in Fig. 4.9 a simulated absorption spectrum for the (weak) vibration-rotation transition (11102) ← (00001) in CO2 .4 The Q branch makes this spectrum particularly interesting. It is observed here in addition to the P and R branch which we know already from diatomic molecules. For the diatomic molecules (see Sect. 3.4.5) we had excluded N = 0 transitions for parity reasons. Albeit CO2 is also a linear molecule, for this bending vibration the change of the dipole moment is perpendicular to the molecular axis. And it is this change which according to (4.34) is responsible for transitions. Hence, its average over YN∗ M (Θ, Φ)YN M (Θ, Φ) does not disappear in this case, and the transition is allowed (so called “perpendicular” transition).

4.2.4

Nonlinear Triatomic Molecules AB2

In this case there are 9 − 6 = 3 vibrational modes Qi . We discuss as the most prominent example the water molecule, H2 O. Here too a small charge displacement is observed: oxygen is charged slightly negative, the two hydrogen atoms slightly 4 HITRAN notation. See Sect. 5.2.4, Vol. 1 how to convert spectroscopic line strengths into spectra.

246

4 P branch

Q branch

Polyatomic Molecules R branch

transmission

1.0

0.9

0.8

0.7 1880

1900

1920 _

ν / cm-1

1940

1960

1980

Fig. 4.9 Simulated transmission spectrum for the infrared bands of the (weak) vibrational transition (11101) ← (00001) in 12 C16 O2 , obtained from HITRAN-W EB (2012), using the HITRAN databank (296 K, 1 atm, opt. path 1 m, app. resolution 0.1 cm−1 ). Note that for this perpendicular transition one observes – in addition to the P and R branch – also a Q branch

ν1

Fig. 4.10 Normal modes of H2 O: symmetric stretch, Q1 , bending mode, Q2 , and asymmetric stretch, Q3

Table 4.1 Normal modes of the water molecule (ground state) for the three most important isotopologues

H+

H2

16 O

ν2

O -

H+

O -

H+

ν3 H+

O H+

ν¯ 1 / cm−1

ν¯ 2 / cm−1

ν¯ 3 / cm−1

3657.053

1594.746

3755.929

HD 16 O

2723.68

1403.48

3707.47

D2 16 O

2669.40

1178.38

2787.92

H+

positive. The three vibrational modes are sketched in Fig. 4.10. We refrain here from a mathematical description. For all three modes ∂ Dγ /∂Qi |0 = 0 holds, they are infrared active. However, as a detailed calculations show, their absorption cross sections are rather different, with σ (ν1 ):σ (¯ν2 ):σ (¯ν3 ) 0.07:1.47:1.00. A look at Fig. 4.10 makes it plausible that the change of dipole moment is smallest for excitation of the symmetric stretch mode, ν¯ 1 . The eigenfrequencies for the three most important isotopologues of H2 O are summarized in Table 4.1. The term energies of the lowest vibrational levels are shown in Fig. 4.11. It gives already a feeling of the high complexity in the overall vibrational spectrum of the water molecule: the validity of the harmonic approximation is rather limited and numerous higher harmonic bands are observed. As in the case of diatomic molecules, superposed onto the vibrational excitation is the rotational structure. It is evident from the discussion in Sect. 4.1.4 that these bands are much more complicated than for diatomic molecules. The coupling of vibration and rotation, centrifugal

4.2


Fig. 4.11 Normal mode energy levels of H2 O. Red lines denote the fundamental excitations, above them the corresponding harmonics are shown. In the three columns on the right the observed combination tones are displayed. Spectroscopic data from T ENNYSON et al. (2001) were used for this graph

WNK K (hc) -1 / cm -1 a c (210)

(111)

(031)

8000 (200) (050) (002) (130) (101) (120) (021) (040) 6000 (110) (011) (030) (001) 4000 (100) (020) (010)

2000

transmitted intensity / arb.un.

0

Fig. 4.12 Fourier transform absorption spectrum of vibration-rotation bands in the H2 16 O molecule in a small section of the visible spectrum according to C ARLEER et al. (1999). One sees overtones of the fundamental vibrational modes with the corresponding rotational structure

247

(000)

0.10

0.08

0.06

0.04 16870

16880 16890 wavenumber / cm-1

16900

distortion, anharmonicity etc. leads to a multitude of absorption (or emission) lines in spectral range from the near infrared (from 1 595 cm−1 or 6.3 µm with maxima around (3400 to 3900) cm−1 ) over the whole visible range (12 500 to 25 000 cm−1 ) and into the near UV range. Presently more than 20 000 vibration-rotation transitions have been measured with high precision and are well analyzed (T ENNYSON et al. 2001). To illustrate the complexity of such spectra, we show in Fig. 4.12 a small section of the visible spectral range (high harmonics), obtained from FT absorption spectroscopy. Evidently water plays an important role also as “greenhouse gas”, since it is responsible for a multitude of significant absorption bands over the whole spectrum of the sun (see e.g. B ERNATH 2002a).

4.2.5

Inversion Vibration in Ammonia

Level Splitting by Tunnelling The ammonia molecule has a pyramidal structure as sketched in Fig. 4.13(a). The bond distance N-H is R0 = 0.10124 nm, and the bond angle α = 106.67◦ differs

248

(a)

Q2 0 - Re

0.

1 10

24

(b) nm

N 106.67o H

H H

N

ν1

H H H N

ν3

H H H

4


N

ν2

H H H N

ν4

H H H

Fig. 4.13 NH3 : (a) Coordinate system: x is the distance of the N atom from the (H-H-H) plane, Re = 0.0383 nm is the equilibrium distance of the inversion mode; (b) shows the normal modes, of which ν3 and ν4 are each twofold degenerate

only slightly from the ideal tetrahedral angle 109.47◦ .5 The equilibrium distance of the inversion coordinate x is Re = R0

cos2

α 1 2 α − sin = 0.0381 nm. 2 3 2

NH3 has, as illustrated in Fig. 4.13(b), 3 × 4 − 6 = 6 normal modes: the symmetric stretch vibration with the eigenfrequency ν¯ 1 = 3 336.6 cm−1 , the inversion vibration (also called umbrella mode, in analogy to an umbrella turning itself inside out in strong wind) with ν¯ 2 = 950 cm−1 , the asymmetric stretch ν¯ 3 = 3 443.8 cm−1 , and the bending vibration ν¯ 4 = 1 626.8 cm−1 . The latter two are each twofold degenerate. We want to study the important umbrella mode in some detail. The slightly negative N atom at the top of the pyramid may oscillate in respect of the positively charged H3 plane – considering the mass ratios one should say more precisely: the H atoms oscillate in respect of the nearly space fixed N atom. As in the case of a real umbrella, the N atom in the ammonia molecule can flip its position, so that in Fig. 4.13(a) it could also lie under the H3 plane. The barrier for this inversion is ca. 0.3 eV (2 400 cm−1 ) which amounts to about three times the excitation energy of the inversion mode. Due to a tunnelling process inversion may occur already in the vibrational ground state. We investigate this process in a one dimensional model to see what influence the tunnelling process may have on wave functions and energy eigenvalues. We consider the motion of a representative particle with the reduced mass M¯ = 3mH mN /(3mH + mN )

5 One

also finds slightly different values for R0 and α in the literature. For a recent discussion see YACHMENEV et al. (2010).

4.2


(a)

V(Q 2) / cm-1 N N

H H

249 H H

(b)

H

H 597

4000

+

v2 = 3

511.4

-

3.5ν 2

3000

35.7 284.7

+

0.793

+

+ -

-

Vb

2000

1000 - Re

-0.4

v2 = 2

2.5ν 2

+ -

-0.8

v2 = 4

4.5ν 2

1.5ν 2

v2 = 1

ν 2/2

v 2 =0

Re

0.0

0.4

-0.8

Q2

Fig. 4.14 Cut through the NH3 potential hypersurface in the direction of the inversion vibration (red lines) and the corresponding harmonic approximation (grey). The term positions for the ν2 mode are indicated by horizontal lines – (a) exact values accounting for tunnelling splitting (in cm−1 ), (b) harmonic approximation

in a potential with a barrier. According to DAMBURG and P ROPIN (1972) one may use a model potential6 2 (4.35) V (x) = k Re2 − x 2 / 8Re2 ¯ 2 and the sketched in Fig. 4.14(a) as dashed, red line. The force constant k = Mω 2 two equilibrium distances x = ±Re are derived from the experimental data. V (x) is a good approximation in the region of the vibrational ground state of the ν2 mode. Close to the barrier height it is, however, no longer correct. In Fig. 4.14(a) we thus have drawn V (x) slightly corrected by eye (full red line). In the classical picture a particle of energy W < Vb cannot cross the barrier. The quantum mechanical treatment of the tunnelling process is done without difficulties by numerical integration of the one-dimensional S CHRÖDINGER equation for the mass M¯ moving along the inversion coordinate x on a potential hypersurface as good as possible. The term positions shown in Fig. 4.14(a) correspond the experimentally determined values which can be fully reproduced by theory. As a 1st order approximation V (x) according to (4.35) may be used. But a good physical understanding of the tunnelling process and the ensuing level splitting is obtained already from considering eigenfunctions of harmonic oscillators around 6 Note that the position x

of N in respect of the H3 plane is here not mass scaled. The corresponding √ ¯ Mx.

normal coordinate is given by Q2 =

250

4


the two equilibrium positions. In the vibrational ground state where the two minima are separated by a high barrier, we first assume that the oscillation is localized fully in the left or right potential minimum. Harmonic potentials according to Fig. 4.14(b) are fitted to each minimum as indicated by grey lines in Fig. 4.14(a). In this 0th order approximation eigenfunctions correspond to Table 3.1. The respective energies are plotted in Fig. 4.14(b) and characterized by vibrational quantum numbers v2 = 0, 1, . . . . Each term is twofold degenerate, corresponding to the two possible positions. Since the two positions are physically completely equivalent, we cannot distinguish them at all. Thus we have to combine them to symmetric and antisymmetric states – quite analogue to the electronic eigenfunctions of the H+ 2 molecule (see Sect. 3.5.2). In other words, the potential V (x) has even symmetry, i.e. the parity and the Hamiltonian H = 0 and we of the system commute: [H , P] operator P and H by symmetric and antisymmetric linear define common eigenstates of P combinations: + v = √1 v (r) + v (l) 2 2 2 2

1 (r) (l) and v2− = √ v2 − v2 . 2

(4.36)

Specifically, for the vibrational ground state the respective wave functions R+ (x) and R− (x) are constructed from ¯ 1 − Mω 2 2 R(l) (x) = √ e 2 (x+Re ) 4 π

¯ 1 − Mω 2 2 and R(r) (x) = √ e 2 (x−Re ) 4 π

1 so that R± (x) = √ R(l) (x) ± R(r) (x) 2 depending on whether the molecule is close to the left or the right minimum, as sketched in Fig. 4.15. In perturbation theory this approach leads to the removal of degeneracy. The perturbation potential is the difference between inversion potential V (x) and the two isolated, harmonic oscillator potentials (red and gray lines in Fig. 4.14(a), respectively). In the barrier region it is negative, and we note (insets in Fig. 4.15) that the antisymmetric wave function R− (x) has there a lower probability than the symmetric eigenfunction R+ (x). Thus, for the corresponding eigenenergies one finds − + > Wv2 , i.e. the energies of the antisymmetric states lie higher than those for the Wv2 symmetric states – as illustrated in Fig. 4.14(a). Since the probability in the barrier region is the higher the closer the eigenenergy of the states Wv2 comes to the barrier, higher lying states will split more. This trend obviously continues above the barrier. Experimentally best determined is the splitting in the ground state where the two modes are separated by the so called inversion frequency: ν0 =

ω0 W0− − W0+ = = 23.870 GHz. 2π h

(4.37)

4.2


Fig. 4.15 Realistic construction of the symmetry adapted ground state wave functions for the double minimum potential of NH3 . The two localized functions, R(l) (x) and R(r) (x) (black lines) are combined symmetric and antisymmetric, R+ (x) and R− (x) (red lines), respectively. On the left an enlarged section of the symmetric wave function in the classically forbidden region around x = 0 is shown

251

R(x)

H H H

N

N

R(l)(x)

H H H

R(r)(x) 0.1

R- (x)

- 0.2

0.1

0.1

R+ (x)

+

- 0.1

0.1

- 0.8

x - 0.4 - Re

0.4

0.8

Re

- 0.1

As the wave functions R+ and R− have different parity and the molecule has a permanent dipole moment along the x-coordinate, E1 transitions may be induced between these states. For the ground state the transition frequency in ammonia lies in the range of millimetre waves with λ0 = 12.56 mm or ν¯ 0 = 0.796 cm−1 .

Temporal Evolution of the Wave Function One may ask the question: how fast does the molecule tunnel? Let us assume at time t = 0 we have prepared the molecule in one of the two minima, say in the right one. With (4.36) this molecular state is then described by R(t = 0) ≡ v (r) = √1 v + + v − . 2 2 2 2 This is, however, not an eigenstate of the Hamiltonian but a wave packet. Its temporal evolution (specifically for v2 = 0) is given by R(t) = √1 e−iW0+ t/ v + + e−iW0− t/ v − . 2 2 2 With little algebra and (4.37) one finds for the probability density: R(x, t)2 = cos2 (ω0 t/2)R(r) (x)2 + sin2 (ω0 t/2)R(l) (x)2 . The following picture emerges: at time t = 0 the probability density is concentrated in the right potential well, at time t = π/(2ω0 ) we have equal distribution between left and right and at t = π/ω0 the density has “tunnelled” completely into the left potential well. Thereafter the process is inverted and after t = 2π/ω0 the initial state is recovered. The three H atoms of the ammonia oscillate with a frequency ν0 = 23.870 GHz (i.e. during 42 ps) from one side of the nitrogen atom to the

252 Fig. 4.16 Scheme of an NH3 maser setup. In a strong inhomogeneous electrostatic field both states, |v2+ and |v2− are deflected differently so that population inversion may be generated

4

Polyatomic Molecules microwave resonator

molecular beam inhomogeneous electric field

other. Hence ν0 is called inversion frequency and corresponds exactly to the energy difference between the two slightly split ground states |v2+ = 0 and |v2− = 0. T OWNES and co-worker (G ORDON et al. 1955) were first to show that for the example ammonia amplification is possible by stimulated emission. Their maser (Microwave Amplification by Stimulated Emission of Radiation) was the predecessor of the laser. As explained in Sect. 1.1 amplification by stimulated emission requires a population inversion among the levels of the transition of interest. This is not possible in thermodynamic equilibrium. In the present case we have ω0 kT at room temperature, i.e. nearly equal population of both states. However, in a molecular beam both states may be separated spatially very efficiently using an inhomogeneous electric fields (S TARK effect) as schematically indicated in Fig. 4.16. The process is based on the fact that both states have an opposite electric permanent dipole moment. They may thus be separated in an inhomogeneous electric field (analogous to the S TERN -G ERLACH experiment treated in Sect. 1.9, Vol. 1 where the two electron spin orientations with opposite magnetic dipole moment were separated in an inhomogeneous magnetic field). In this manner, one deflects the molecules in the energetically higher lying state |v2− from the beam into a microwave resonator. The latter is tuned onto the inversion frequency and allows one to detect the amplified microwaves. Section summary

• There are 3Nnu − 6 degrees of freedom for the vibrational motions in polyatomic molecules. Normal modes Qi are such linear combinations of the co√ ordinates qi (conveniently mass scaled by mi ) that diagonalize the Hamiltonian as shown by (4.31). • The total vibrational energy is the sum of the energy in all normal modes, the nuclear motion is described by products of the respective wave functions. • Optical transitions (typically in the infrared) can be induced in such modes that change the permanent dipole moment of the molecule. • As examples of triatomic molecules we have discussed the CO2 (linear) and H2 O. In both cases the normal modes are: the symmetric stretch, the bending mode (lowest frequency) and the asymmetric stretch (highest frequency). In the case of CO2 the bending mode is two fold degenerate and allows for a type of rotational motion, compensating for the fact that as a linear molecule CO2 has only two degrees of rotational motion. • A quite particular molecule is the ammonia, NH3 . The symmetry of its nearly ideal tetrahedral shape leads to two indistinguishable minimum positions,

4.3

Symmetries

253

with the N atom on top or below the H3 plane. Consequently, the six normal vibrational modes include the so called “umbrella” mode in which the N atom may tunnel through this plane. This mode is split into a symmetric and an antisymmetric state, of which the symmetric one has the lowest energy. The splitting in the ground state is 23.870 GHz and is identical to the tunnelling frequency. • The ammonia maser uses this inversion transition. It exploits the fact that symmetric and antisymmetric ground states have opposite permanent electric dipole moments. They may be separated in a molecular beam, passing through an inhomogeneous electric field so that population inversion can be achieved.

4.3

Symmetries

Understanding the structure of small molecules may be simplified substantially by considering their symmetry properties. With the help of (mathematical) group theory the number and properties of normal modes and molecular orbitals can be derived with relative ease even for rather complex molecules. Symmetry considerations are particularly useful for the spectroscopy of polyatomic molecules and the determination of allowed and forbidden transitions. Many textbooks are devoted to molecular symmetries and their applications (E NGELKE 1996; B UNKER and J ENSEN 2006, to mention just two), but also a few excellent web-pages (e.g. G OSS 2009; W IKIPEDIA CONTRIBUTORS 2013, 2014). Here we communicate only some basic notations which the reader will often encounter the literature on molecular spectroscopy and in the present book.

4.3.1

Symmetry Operations and Elements

Symmetry operations in molecular physics are linear transformations of a molecule in space. They transfer equivalent atoms into each other and the molecule as a whole into a geometry which cannot be distinguished from the initial one. One defines the following eight elements of symmetry groups:7 E Identity (no change) or rotation through 360◦ (E originating from the German word Einheit = unit or unity). Cn Rotation in respect of a symmetry axis through an angle 2π/n with n = 2, 3, . . . . Applying this operation k times is denoted by Cnk . With these definitions the identity becomes E = Cnn . σ Reflection at a plane – for which σ σ = σ 2 = E holds. 7 Note: Quantum mechanical operators associated with these symmetry elements will be designated

n for the operator rotating a wave function by a caret () on top of the element symbol, e.g. C through an angle 2π/n.

254

4


σh Special reflection at (horizontal) plane perpendicular to the principle axis of symmetry (axis with the highest n). σv Special reflection at a (vertical) plane containing the principle axis of symmetry. σd Diagonal or dihedral reflection, a special case of σv in which the vertical plane bisects two twofold symmetry axes perpendicular to the principle symmetry axis. ı Inversion or point reflection at the origin. Sn Rotation-reflection or improper rotation: a rotation through 2π/n with successive reflection in a plane perpendicular to that rotation axis. Thus Sn = σh Cn = Cn σh . Also S2 = ı as well as ıσh = C2 and ıC2 = σh holds.

4.3.2

Point Groups

The above symmetry operations may be combined to mathematical groups. They are called point groups since each symmetry operation leaves at least one point in space invariant. According to S CHÖNFLIES one distinguishes the following point groups, which may be identified in a unique manner from the symmetry properties of a molecule – as illustrated in Fig. 4.17.

Groups of Low Symmetry C1 This trivial group contains all molecules which have no symmetries at all; e.g. CHFClBr or 1,2-dibromo-1,1-dichloroethane (C2 H2 Br2 Cl2 ). Ci =S2 contains only ı as symmetry operation; e.g. 1,2-dibromo-1,2-dichloroethane (C2 H2 Br2 Cl2 ). Cs Only mirror symmetry in respect of a single plane; e.g. nitrosyl chloride (O=N−Cl). Rotation Groups Cn Only rotations though an angle 2π/n around an axis (Cn1 , Cn2 , . . . , Cnk , . . . , Cnn = E), i.e. n elements, including E; e.g. hydrogen peroxide (H2 O2 ) has C2 symmetry. Cnh Rotation though 2π/n and reflection, σh , in a plane perpendicular to the rotational axis and in addition inversion (for n = 2) or n − 1 improper rotations (for n > 2), i.e. 2n elements; e.g. boric acid (B(OH)3 ) belongs to the C3h group. Cnv Rotation though 2π/n as well as n reflections, σv , in planes parallel to the rotational axis, i.e. 2n elements; e.g. the water molecule (H2 O) has C2v , ammonia (NH3 ) has C3v symmetry. Dihedral Groups Dn One Cn axis (principle axis) and n C2 axes perpendicular to the principle axis. Dnh As Dn , with an additional reflection plane, σh , perpendicular to the principle axis; e.g ethene (C2 H4 ) belongs to the group D2h and benzene, the simplest aromatic ring (C6 H6 ) has D6h symmetry. Dnd As Dn , and in addition n reflection planes σd parallel to the principle axis; e.g. ethane (C2 H6 ) has D3d symmetry.

4.3

Symmetries

255 Molecule

y

n linear ?

D ∞h

y

i?

n

two or more Cn with n >2 ?

y C∞v

y

n

i?

n

y

Td

C n? n

C2 ┴ to Cn with largest n?

n

Cs

y

σ?

n

y Th

y

C 3?

Dnh

y

n Oh

y

3C4? n Ih

σh ?

Ci

n Dnd

y

σh?

nσd?

y

y

i?

n

C1

Cnh

n

n

nσυ ?

y

Cnv

Dn n Cn

n

S2n ?

y

S2n

Fig. 4.17 Decision tree for determining the molecular symmetry in the notation of S CHÖNFLIES

Special Point Groups C∞v Linear molecules without inversion centre; e.g. HCl, N2 O. D∞h Linear molecules with inversion centre; e.g. H2 , CO2 . Improper Rotation Groups Sn Contains only the symmetry operation Sn . Note: S2 corresponds to Ci , and if n is odd, Sn = Cnh . Only the groups S4 , S6 , S8 , . . . are genuine groups; e.g. the somewhat exotic molecule 1,1 ,1 ,1 -methanetetrayltetrabenzene (4 benzene rings attached to one carbon atom) belongs to S4 .

256

4


Tetrahedral Groups T The group of genuine tetrahedral rotations: E, 4C3 , 4C32 , (axes from corners to the middle of opposite faces), 3C2 (axes from the middle of the edges to the middle of the opposite edges), in total 12 symmetry elements. Td As T and in addition the 6σd reflections and the respective 6S4 operations, in total 24 symmetry elements; e.g. methane (CH4 ). Th As T and in addition all operations that arise from multiplication the former with inversion ı, in total again 24 symmetry elements; e.g. C60 Br24 . Octahedral Group O The group of true cube rotations: E, 8C3 (four 3-fold axes through the corners of the cube, with the three elements C31 , C32 and C33 = E, i.e. two new elements for each axis), 6C4 and 3C2 elements (3 coinciding C2 and C4 axes from the middle of the faces to the opposite faces, with the elements C21 and C41 , C42 = C21 , C42 , and C44 = E, i.e. one C2 element and two C4 elements for each axis), and finally 6C2 (axes from the middle of the edges to the middle of the opposite edges), in total 24 symmetry elements. Oh Complete octahedral group: O and in addition the operations due to multiplication with inversion ı (improper rotations) lead to 48 symmetry elements; e.g. SF6 . Icosahedral Group The icosahedral group contains • The true icosahedron consisting of 20 identical equilateral triangular faces, 30 edges and 12 vertices (corners). • The dodecahedron consisting of 12 identical, regular pentagonal faces, with three meeting at each of the 20 vertices, and with 30 edges. • The truncated icosahedron (soccer ball shape), which has 12 identical, regular pentagonal faces, 20 identical, regular hexagonal faces, 60 vertices and 90 edges. One may view it as constructed from the true icosahedron whose 12 vertices have been truncated so on third of each edge is cut, which created 12 identical, regular pentagons and leaves the original 20 triangles as 20 identical, regular hexagons. It has the full symmetry of the icosahedron. In respect of the symmetries one distinguishes: I Ih

Rotations and improper rotations with 60 symmetry elements. An example is the smallest fullerene C20 which consists of just 20 pentagons. As I and in addition the operations which arise from multiplication with ı, i.e. a total of 120 symmetry elements; the most prominent example is the Buckminster-Fullerene C60 .

4.3

Symmetries

257

Classes of Symmetry Operations in a Group The above point groups contain different, specific sets of symmetry operations consisting of elements given in Sect. 4.3.1, including powers of the rotational operations. These operations represent linear transformations in 3D space. They replace the rotational matrices in the O(3) group, and can be described by 3 × 3 matrices, in simpler cases by 2 × 2 matrices or by just one (complex) number, typically ±1. The trace of the matrix of a symmetry operation is called its character. If a symmetry operation X may be obtained from an operation X by a similarity transformation X = Y −1 XY , the two operations are called conjugated. The complete set of conjugated symmetry operations in a point group is said to belong to a class of symmetry operations. The number of classes in a point group is called its dimension.

4.3.3

Eigenstates of Polyatomic Molecules

From atomic physics we are familiar with the full three dimensional rotation group (called O(3) or SO(3), respectively, depending on whether inversion is included or the group is redistricted to pure rotation). This symmetry group determines the behaviour of angular momenta of atoms. Obviously, for polyatomic molecules this full freedom of rotation has to be replaced by the symmetry operations of the point groups. Each point group may be represented by a set of irreducible representations Γi . And the number of irreducible representations in point group is equal to its dimension (i.e. to the number of different classes of symmetry operations). The irreducible representations Γi replace, so to say, the angular momentum states with quantum numbers LM for states (or m for orbitals) in atomic physics – which are the irreducible representations of the O(3) group. All point groups described above are sub groups of O(3). We have already obtained a taste of the ensuing complications in the context of diatomic molecules. However, there it was possible to derive the electronic states in a rather straight forward manner by projecting the spherical harmonics onto the symmetry axis. In the case of nonlinear tri- and polyatomic molecules the situation is even more complex. The symmetry groups help with book keeping. As mentioned above, applying a certain class of symmetry operations onto an irreducible representation (a molecular state, an orbital, a normal mode) corresponds to a linear transformation in space. It turns out, that the entirety of characters for all symmetry operations in a group onto one irreducible representation Γi fully characterizes the symmetry properties of this Γi . Hence, the characters of all irreducible representations of a point group are summarized in a so called character table. For the simplest groups the symmetry operations have only the characters −1 and +1, indicating whether upon this particular operation, the irreducible representation Γi (i.e. the state, orbital, normal mode) does or does not change its sign, respectively. For degenerate states one does not recognize this result immediately, since in this case linear combinations of the different degenerate states are transformed (by a 2 × 2, 3 × 3 etc. matrix).

258

4


Table 4.2 Notation of irreducible representations of the 3D point groups according to M ULLIKEN (1955). Cn stands for rotation around the principle axis, C2 and C2 for rotation around one or two axes, respectively, perpendicular to it; note that the subscripts for B symbols depend on the special point group and include in some cases also 3 Symbol Degeneracy Symmetry

Indices

A

1

+ in respect of Cn

B

1

− in respect of Cn

E

2

sup ,

T

3

sub 1, 2

G, H

4, 5

Symmetry in respect of ı

σh

C2

C2 σv

sub g, u +, − +, − +, − or

+, −

in I and Ih

Since the thirties of the past century one characterizes the irreducible representations (i.e. the molecular states) by a notation in letters and indices according to M ULLIKEN (M ULLIKEN 1966), summarized in Table 4.2. As in atomic physics, electronic orbitals are written in lower case letters, total states in capital letters. Vibrations are usually (but not always) characterized by lower case letters, while the letter t is replaced by f . For the point groups C∞v and D∞h the notation Σ, Π, , . . . is maintained as in Chap. 3. The degeneracy of these molecular states and vibrations describes the number of possibilities to obtain the same symmetry properties for different arrangements in space. It corresponds to the 2L + 1 fold degeneracy of the |LM angular momentum states with different projections M in respect of the z-axis in the O(3) group. The letters 2S+1 A, 2S+1 B etc. with their superscripts (sup) and indices (sub) replace the denotation 2S+1 L of atomic physics (2 S, 1 P etc.), while the multiplicity 2S + 1 of states with well defined electron spin S is still included in the established form. In detail the rules given in Table 4.2 for denoting nonlinear molecules are not always unambiguous. Thus, already M ULLIKEN (1955) recommends “that in general every author, in referring to Bi or Bip species8 shall define these species clearly in each paper in terms of the specific geometry of the molecules he is discussing and in so doing, that he shall make an effort to follow previous usage if such exists and there is no strong reason to change”. This is all best explained with some examples at hand and we begin with the rather simple point group C2v , to which e.g. the water molecule H2 O belongs. Its electronic structure will be discussed in Sect. 4.4.1. Its symmetry operations are illustrated in Fig. 4.18: a rotation around the twofold symmetry axis z, characterized by C2 , and reflection in respect of the xz and the yz planes, characterized by σv (xz) and σv (yz), respectively. Table 4.3 represents the so called character tables, of the relevant point groups, here C2v for the full symmetry shown in Fig. 4.18, and Cs for the case that one of the bonds is stretched, and one is compressed as in the asymmetric stretch vibration. In the top left cell the point group is specified, here C2v and Cs , respectively. Below that, the character tables as such are in the present case 4 × 4 and 2 × 2 matrices. The first column 8 With

i = 1, 2, or 3 and p = u or g.

4.3

Symmetries

259 z y O

σ'v

H

RO

–

+ H

x

C2

O + H

θ

D

H

H σv

Fig. 4.18 Geometry and symmetry of the H2 O molecule. On the left the definition of the molecular parameters is shown, on the right the coordinates and symmetry operations of the point group C2v are indicated (following M ULLIKEN 1955; different authors occasionally use different labelling of the axes, e.g. C HAPLIN 2013). D indicates the permanent dipole moment of H2 O along the z-axis due to excess electron charge at the O atom (−0.8e, |D| = 2.35 D)

Table 4.3 Character tables of the point groups C2v and Cs with basis functions C2v E C2 σv (xz) σv (yz)

Cs E σh

A1

1 1

1

1

z

x 2 , y 2 , z2

A 1 1

A2

1 1

−1

−1

Rz

xy

A 1 −1 yz, xz

B1

1 −1 1

−1

x, Ry xz

B2

1 −1 −1

1

y, Rx yz

x 2 , y 2 , z2 , xy

gives the M ULLIKEN symbols for the irreducible representations Γi of the group (i.e. the possible total states, orbitals or normal modes), following the scheme given in Table 4.2. In the first row the possible symmetry operations, including the unit operation E, are listed. The main content of the table are the so called characters of the respective symmetry operations for irreducible representations given in column one. The C2v and Cs are among the simplest cases, where all representations are singly degenerate. In this terminology 3 A1 (second row in C2v Table 4.3) characterizes a fully symmetric triplet state. An electron configuration, e.g. {. . . (nb2 )2 . . . } (fifth row in C2v Table 4.3), characterizes a system which contains two electrons (spin up and spin down) in a (spatially non-degenerate) nb2 orbital which changes its sign upon 180◦ rotation around the z-axis as well as upon reflection on the xz plane. The character table also contains information about how the Carthesian basis vectors xyz, rotations around them Rx , Ry , Rz , and certain quadratic combinations – all shown in the two last columns of the C2v Table 4.3 – transform under these symmetry operations: e.g. polar as well as axial vectors do not change under the respective rotations, while x, y, z (representing polar vectors) change their sign under reflection through a plane perpendicular to them, and Rx , Ry , Rz change their sign for each reflection at a plane which contains them.

260

4 y

H

Table 4.4 Character table of the D2h point group, e.g. ethylene: aligned as suggested by M ULLIKEN (1955) D2h E C2 (z) C2 (y) C2 (x) ı Ag

1

1

B1g

1

B2g

1

B3g


C

x

H z

C

H

. Coordinates xyz

H

σv (xy) σv (xz) σv (yz) 1

x 2 , y 2 , z2 x 2 , y 2 , z2

1

1

1

1

1

1

−1

−1

1

1

−1

−1

Rz xy

−1

1

−1

1

−1

1

−1

Ry xz

1

−1

−1

1

1

−1

−1

1

Rx yz

Au

1

1

1

1

−1 −1

−1

−1

B1u

1

1

−1

−1

−1 −1

1

1

z

zx 2 , z3 , y 2 z

B2u

1

−1

1

−1

−1 1

−1

1

y

yz2 , y 3 , x 2 y

B3u

1

−1

−1

1

−1 1

1

−1

x

xz2 , x 3 , y 2 x

xyz

Fig. 4.19 Octahedron in a cubic environment

z

y

Oh x

An example with significantly more variety is the point group D2h whose character table is shown in Table 4.4. Here eight symmetry operations are relevant, in addition to rotation around the C2 (z) axis, σv (xz), and σv (yz) reflections, there are two more C2 axes (x and y), one more reflection σv (xy) as well as inversion ı (leading to additional subscripts g and u). The symmetry properties of the states (orbitals, normal modes) are now characterized as shown in the first column: A states are again always symmetric in respect of rotation (to all rotations as it turns out), and a 3 Ag state would stand for a totally symmetric triplet state. In contrast, the different symmetries of the B states in respect of rotation are now characterized by subscripts 1, 2, and 3. The symmetry properties of basis vectors, rotations, quadratic and cubic combinations is here indicated by writing the respective expressions into the last three columns. We see now e.g. that inversion changes the sign of the polar basis vectors (x, y, z) but not that of axial vectors, i.e. rotations (Rx , Ry , Rz ). As a last example of character tables we shall now discuss the full octahedral group Oh , a somewhat more complex and general case. It is relevant not only for molecular physics but also describes the structure of mixed crystals with cubic lattice. Its geometry is sketched in Fig. 4.19: consider a cation positioned in the centre of an octahedron, its ligands located at the 8 vertices, i.e. in the centres of the cube

4.3

Symmetries

261

Table 4.5 Character table of the Oh point group Oh

E 8C3 6C2 6C4 3C2 ı

6S4 8S6 3σh 3σd

A1g 1 1

1

1

1

1

1

1

1

1

A2g 1 1

1

−1

−1

−1

1

1

−1 1

Eg

2 −1

0

0

2

2

0

−1 2

T1g

3 0

−1

1

−1

3

1

0

T2g

3 0

1

−1

−1

3

−1 0

A1u 1 1

1

1

1

−1 −1 −1 −1 −1

A2u 1 1

−1

−1

1

−1 1

−1 −1 1

0

0

2

−2 0

1

T1u 3 0

−1

1

−1

−3 −1 0

1

1

T2u 3 0

1

−1

−1

−3 1

1

−1

Eu

2 −1

0

x 2 + y 2 + z2 (3z2 − r 2 , x 2 − y 2 )

0

−1 −1 (Rx , Ry , Rz ) −1 1

(xy, yz, zx)

−2 0 (x, y, z)

surfaces. The 10 different classes of symmetry operations (dimension 10) in the Oh group are listed in the first row of the character Table 4.5, a 10×10 matrix. The numbers in front of the symmetry operations give the numbers of elements in that class. The Oh group obviously contains singly (A1g , A2g , A1u , A2u ), doubly (Eg , Eu ) and triply (T1g , T2g , T1u , T2u ) degenerate irreducible representations. The degeneracy is directly recognizable from the first column for the character of E operating onto a given representation: since the unit operator is a matrix with diagonal elements = 1 only, for a 3 × 3 matrix (triple degenerate Γi ) the character is 3, for a 2 × 2 matrix (double degenerate Γi ) and so on. The irreducible representation A1g in the first row is again totally symmetric since all characters are +1, while A2g is antisymmetric in respect of two rotational axes. For electron configurations {. . . (na2g )4 . . . } the quantum number n labels different orbitals of equal character (in analogy to n in atomic physics), and specifically the e2g orbitals are antisymmetric in respect of two rotations, and symmetric in respect of inversion. The designation (na2g )4 implies that the orbitals are filled with 2 × 2 = 4 electrons – according to the PAULI principle this corresponds to the maximum of the doubly degenerate state. In the last two columns we find again information on how the Carthesian basis vectors (x, y, z), the respective rotations (Rx , Ry , Rz ) and the quadratic functions, bilinear in x, y, z behave under these transformations: they are associated with the irreducible transformations, which transform in the same manner. For the Oh group according to Table 4.5 the basis vectors x, y, z are obviously associated with T1u . This association will turn out (in Sect. 5.4.2) to be decisive for the allowing dipole induced transitions. The most prominent molecule in this group is sulfur hexafluoride, SF6 , in which the six valences of sulfur bond one fluor atom each. The sulfur atom in the centre has equal distance R0 (SF) = 0.156 nm of all 6 fluor atoms at the vertices of the octahedron. The totally symmetric stretch vibrations (equal elongation or contraction of all S-F distances) has an eigenfrequency ωe = 769 cm−1 (similar to Cl2 ). Such a geometry is not completely trivial as we shall see in the following section. Here it is based on completely filled orbitals. The atomic electron configuration for the

262

4


valence electrons of S is 3s 2 3p 4 , the electron configuration (see e.g. TACHIKAWA 2002) of the octahedral SF6 is (1t2u )6 (5t1u )6 (1t2g )6 leading to the electronic ground state 1 A1g . Obviously, in addition to the 6 valence electrons of the sulfur, two electrons of each fluor atom (electron configuration 2s 2 2p 5 ) contribute to the 16 bonding electrons. The molecule SF6 is often used in technical applications as electron ‘quenching’ gas, since it readily attaches an extra electron (electron affinity between 0.4 and 1.5 eV). This extra electron fills one 6a1g orbital, and the SF− 6 anion maintains the octahedral structure albeit somewhat expanded with R0 (SF) = 0.1732 nm. Its overall electronic structure is 2 A1g . We shall come across more character tables in later context: The D6h group will be discussed when describing the electronic orbitals of benzene in Sect. 4.5, and in Sect. 5.5.2 we shall get to know the D3h group in connection with the interesting spectroscopy of the Na3 molecule. The character tables of all symmetry groups relevant in molecular physics are well documented in the text books and internet pages mentioned in the introduction to the present section. One also finds there additional information: correlation tables allow one to switch from one symmetry group to another one (e.g. if the symmetry is reduced due to a distortion of the arrangement of the atomic nuclei). From product tables one may derive the combinations of different symmetry operations as well as which transitions are dipole allowed or forbidden.

4.3.4

J AHN -T ELLER Effect

In the context of these symmetry considerations we discuss now an important effect (JTE) which was first treated by JAHN and T ELLER (1937). They investigated “whether the energy of a degenerate electronic state should depend linearly or not upon nuclear displacements” and found what is now called the JAHN -T ELLER (JT) theorem: “All nonlinear nuclear configurations are therefore unstable for an orbitally degenerate electronic state.” And they continue: “Thus if we know of a polyatomic molecule that the nuclei in the equilibrium configuration do not all lie on a straight line, then we know at the same time that its ground electronic state does not possess orbital degeneracy.” Later reformulations of these original statements such as nonlinear molecules with degenerate electronic states will undergo distortion and assume a lower symmetry and energy, thus removing the degeneracy are somewhat misleading from present perspectives. The key point is, that an electronic degeneracy of potential surfaces determined in the framework of the B ORN -O PPENHEIMER approximation leads to a coupling between nuclear and electronic coordinates (so called vibronic coupling) which calls for a special treatment. The literature on this subject is quite extensive (see e.g. B ERSUKER 2001, and references given there), and we can give here only a brief introduction.

4.3

Symmetries

263

The JTE is of rather fundamental importance, especially as it bears out the limits of the BO approximation quite clearly. Already in 1934 had R. R ENNER investigated the situation for linear, triatomic molecules and found, that degenerate electronic states split when the bending vibration is excited: today this is called the R ENNER -T ELLER effect. We have already seen a related behaviour for diatomic molecules, called there avoided crossing of two potential curves with equal symmetry. However, if the molecule has more than one relevant nuclear degree of freedom such crossings of reduced dimensionality are possible. For example, in the case of two degrees of vibrational freedom the corresponding potential surfaces may cross in one point. This leads to a so called conical intersection, which we shall meet later again. Conical intersections represent, however, not potential minima of the adiabatic potential surfaces. Rather, they are instable in respect of the nuclear coordinates as stated by the JT theorem. Such vibronic couplings play a role also if the electronic states are not degenerate but lie very close together. This situation is called pseudo-JAHN -T ELLER effect (PJTE). All three effects are characterized by vibronic couplings which require an extension of the usual BO approximation. The JTE is best explained by way of example. Molecular complexes with equilateral sided octahedral geometry Oh (see Sect. 4.3.3), such as compounds of metal atoms with several d electrons, may be subject to stretching or compression. Let us discuss the doubly charged copper cation (Cu2+ , electron configuration 3d 9 ) which forms complexes of the Cu(H2 O)2+ 6 type. We first recall the angular dependence of the d orbitals (Sect. 2.5.3 in Vol. 1). In atomic physics one normally uses the complex representation of these orbitals (eigenfunctions of orbital angular momentum with quantum numbers L and M). More appropriate for the present situation are real orbitals, i.e. linear combinations (Table D.2, Vol. 1) designated by dz2 , dxz , dyz , dx 2 −y 2 , dxy which have good quantum numbers L = 2 and |M| = 0, 1, 1, 2, 2, respectively: t2g :

√ 3 dxz = 2 xz r 1 dz2 = 2 3z2 − r 2 2r

√ dxy = dx 2 −y 2

3

xy √ 3 = 2 x2 − y2 . 2r r2

√ 3 dyz = 2 yz r

(4.38)

Alternatively and symmetry adapted to the octahedral case, one may combine the latter two as: √ √ eg : (2dz2 + dx 2 −y 2 )/ 5 and (2dz2 − dx 2 −y 2 )/ 5. (4.39) As a result we have three t2g and two eg orbitals, as sketched in Fig. 4.20. Electrons in eg orbitals are thus clover shaped and aligned towards the 6 ligands at the vertices of the octahedron, while the t2g orbitals (also clover shaped) point towards the 8 edges, i.e. exactly in between the ligands. If one assumes that the ligands are anions, one expects a stronger repulsion for the eg electrons in comparison to the t2g electrons. Hence, the five d levels, which are degenerate in isotropic O(3)

264 Fig. 4.20 Angular dependence of the five d orbitals in real, Oh symmetry adapted representation; plotted are the moduli of the wave functions, coloured according to the sign of the wave functions

4

t2g :

Fig. 4.21 JAHN -T ELLER effect with compounds of Cu2+ (electron configuration 3d 9 ). If the orbitals eg and t2g of the Oh symmetry group (middle) would be filled with electrons a spin distribution as indicated by the little red arrows would result; three electrons in the eg orbitals may, however, be placed in two different, energetically degenerate ways (indicated by the dashed arc with an arrow); thus, JAHN -T ELLER splitting leads to contraction and stretching of the octahedron, respectively, and D4h symmetry

z

eg :

z

O(3)

Polyatomic Molecules z

2d d z 2 + dx 2- y 2

2d dz 2 – d x 2 - y 2

y

y

x

x z

dxy

z

dxz

dyz

y

y

y

x

x

x

Oh

D4h

z

z

y

y

x stretch

x

2 degnerate states b1g (dx 2 - y 2)

5 d orbitals (degenerate)

eg

2dz 2 d x 2- y 2 dxy dxz dyz

a1g (dz 2)

}δ

1

∆

t2g

b2g (dxy) eg (dxz ,dyz)

} δ2

space, are now replaced by two higher lying eg and tree lower lying t2g levels as illustrated in Fig. 4.21 on the left. One may easily imagine how these atomic orbitals form bonding MOs – e.g. with the p electrons of the O atoms from the six water ligands in the Cu(H2 O)2+ 6 complex. Figure 4.21 illustrates in addition how in the special case of a Cu2+ cation in the centre, the nine 3d electrons are to be distributed to the orbitals. For the higher, twofold degenerate eg orbital only three more electrons are available, leading to an electron hole. The three electrons may thus arrange their spin orientation in two energetically completely equivalent ways, ↑↓ + ↑ and ↑ + ↑↓, respectively. According to perturbation theory such states split in the presence of a suitable interaction potential (here provided by the molecular environment). In the present case the Oh

4.3

Symmetries

265

symmetry is contracted or stretched (depending on the chemical environment) leading to D4h symmetry as sketched in Fig. 4.21 on the right. The eg orbitals of the Oh group correlate in that situation with the a1g and b1g orbitals of the D4h group, the t2g orbitals with b2g and two degenerate eg orbitals (now also in respect of the D4h group). The stretch indicated in Fig. 4.21 leads to an energy for the a1g orbital which is lower by δ1 in respect of b1g , while eg is below b2g by an energy δ2 , where δ1 > δ2 . In case of a compression the orbital energy levels would just be inverted. We thus have here a special case of the JTE splitting and symmetry change of degenerate states. Interestingly, the corresponding nickel complex, Ni(H2 O)2+ 6 does not show such a JT displacement: the electron configuration of Ni is 3d 8 , so that the eg orbitals are filled with only one electron each. No alternative, energy equivalent distribution of the electrons is possible. Consequently, there are no degenerate states and hence no JTE occurs. Concluding this brief excursion into the interesting physics of the JAHN -T ELLER effect, we point out again that vibronic coupling between electronic and nuclear degrees of freedom is responsible for this effect. A consequent treatment is not possible within the B ORN -O PPENHEIMER approximation. As mentioned already in Chap. 3 and detailed in the following sections, the quantum mechanical standard methods based on the BO approximation compute the electronic structure of polyatomic molecules for fixed positions of the nuclei. Nuclear dynamics is considered to happen on the thus determined adiabatic potential energy surfaces (APES). In contrast, vibronic JT coupling cannot be treated in this manner as being just a small perturbation, even if the nuclear motion itself may be small. Rather, the APES which determine the vibrational frequencies and their anharmonicities are themselves controlled by vibronic couplings with other electronic states (B ERSUKER 2001). We shall come back to vibronic couplings (also called non-adiabatic couplings) in more detail in Sects. 5.5.2 and 7.3.3. Section summary

• Symmetries play an important role in classifying and characterizing the electronic and geometric structure of polyatomic molecules. The point groups introduced here are based on several elementary symmetry operations, i.e. inversion, rotations, reflections, and combinations thereof. Classes of symmetry operations include powers of the relevant rotations. • The number of these classes is identical to the number of different irreducible representations in each symmetry group. Essentially, they correspond to the angular momentum quantum numbers in atomic physics and characterize the symmetry of electronic orbitals, states, and vibrational modes. • The general rules for denomination of these irreducible representations is explained in Table 4.2, and their properties are summarized in the character table of each group, some of which have been communicated and discussed above. For non-degenerate states, the characters directly reflect their behaviour if the symmetry operations act on them.

266

4


• The JAHN -T ELLER (JT) theorem states that all nonlinear nuclear configurations are unstable for an orbitally degenerate electronic state. In such a situation, the molecule assumes a lower symmetry, thus avoiding the degeneracy. This is called JT effect. In modern language it refers to vibronic coupling which cannot be fully treated in BO approximation. As an example we have discussed compounds of Cu2+ which has an electron configuration 3d 9 .

4.4

Electronic States of Some Polyatomic Molecules

For diatomic molecules we have learned that molecular bonding occurs if the atomic orbitals of the atoms involved have sufficient overlap, i.e. if there is a significant amount of electron density in the region between the nuclei. This condition also holds for polyatomic molecules and determines their geometry. We may thus formulate as general bonding rule: Molecular bonding between atoms occurs in that direction in which the atomic orbitals forming the molecular orbitals overlap most. In the following we shall thus try to predict geometries of polyatomic molecules from known AOs or MOs. Conversely, we may glean information about the orbitals involved in the bonding from the observed molecular geometries.

4.4.1

A First Example: H2 O

We start with this extremely important molecule from which a lot of fundamental aspects about polyatomic molecules can be learned – even though H2 O may appear a rather simple example at first sight. Rotational and vibrational spectra of H2 O have already been introduced in Sects. 4.1.4 and 4.2.4, and were found to be quite complex. Consequently, in condensed phase, water is characterized by 68 anomalies of its physical properties – ultimately a consequence of its molecular structure and dynamics. For a comprehensive collection of facts on water in all its phases we refer to the well kept web-page of C HAPLIN (2013). A quote from it may stimulate further interest: “Water is the main absorber of the sunlight in the atmosphere. The 13 million tons of water in the atmosphere (∼0.33 % by weight) are responsible for about 70 % of all atmospheric absorption of radiation, mainly in the infrared region where water shows strong absorption. It contributes significantly to the greenhouse effect ensuring a warm habitable planet, but operates a negative feedback effect, due to cloud formation reflecting the sunlight away, to attenuate global warming.” The greenhouse effect, obviously, does not just have negative consequences as one might think when reading the daily newspapers. The issue is simply to maintain the optimal balance which has been reached by nature over many millions of years, and which should not be disturbed by mankind in a serious manner!

4.4


267

Let’s get back to the isolated H2 O molecule. We have already discussed in Sect. 4.3 that H2 O has C2v symmetry, with its character Table 4.3, and the definitions of coordinates and geometry shown in Fig. 4.18. The permanent dipole moment of the water molecule (|D| = 2.35 D = 7.84 × 10−30 C m = 0.0489 e nm), along the C2 symmetry axis z, is rather large (cf. Table 3.2) – with about −0.8e electron charge donated from the hydrogen atoms to the oxygen. To understand the molecular bonding in terms of MOs we recall that the oxygen atom has the electron configuration (1s)2 (2s)2 (2p)4 (see Sect. 10.4.2 in Vol. 1). In the energetically lowest ground state of O, two of the p electrons with opposite spin fill one of the three p orbitals, the other two p electrons are unpaired, i.e. they have equal spin and fill the remaining two p orbitals. This results in a total 3 P ground state in agreement with H UND ’s rules. At first glance one might suspect that the two unpaired electrons in orthogonal orbitals are the key to the bonding and structure of the water molecule, each forming an MO with a 1s electron of one of the H atom. A first guess of the bond angle would thus be 90◦ , reflecting the perpendicular nature of the real as px , py and pz AO. This has to be compared with an experimentally observed H−O−H angle of 104.474◦ . Indeed, even though these unpaired p orbitals are essential in forming the OH bonds of water, the reality is somewhat more complex. The correlation diagram between the MOs of the H2 O molecule (C2v symmetry) and the AOs of its constituents O and 2H is shown in Fig. 4.22. The H atom has an IP (= −electron binding energy) of WI = 13.606 eV, the first IP of O for the process O2s 2 2p 4 3 P → O2s 2 2p 3 4 S + e− is almost identical (see K RAMIDA et al. 2013), WI = 13.62 eV. Obviously it corresponds to the removal of one of the paired electrons in 2p AO. This “lone pair” of electrons (see also Sect. 3.7.4) does not contribute to the bonding. In C2v symmetry (Table 4.3) the respective MO it is denoted as 1b1 . As indicated in Fig. 4.22 on the right, it reflects the shape of a px AO (with reference to the coordinate system introduced in Fig. 4.18). It is filled with two electrons of opposite spin and is the HOMO of neutral H2 O in its ground state. The second IP of O allows an estimate of the 2py and 2pz energies and the excitation energy O+ 2s 2 2p 3 → 2s2p 4 positions the 2s orbital energy. The construction of the 3a1 , 1b2 and 2a1 MOs is evident from Fig. 4.22. They are all characterized by charge density along the OH bonds, and hence, all are bonding. In contrast, the closed 1s shell electrons (i.e. the 1a1 MO) do not participate notably to the chemical bonding.9 It is interesting to note that electron binding energies in liquid water are essentially the same (their absolute value just being 1.5 to 2 eV smaller due to the solvation shell formed around each molecule (see e.g. W INTER et al. 2004, 2007). 9 The electron binding energies of the water MOs given in Fig. 4.22 are derived from gas phase photoionization energies (see e.g. W INTER et al. 2004): WI = 539.9 eV (1a1 ), 32.6 eV (2a1 ), 18.8 eV (1b2 ), 14.8 eV (3a1 ), and 12.6 eV (1b1 ). According to KOOPMAN’s theorem (see Sect. 10.2.4 in Vol. 1) these values should correspond to the orbital energies W = −WI . The energies of the unoccupied MOs are taken from C HAPLIN (2013) 6 eV (4a1 ), 8 eV (2b2 ) and 28 eV (3b2 ) (RHF approximation).

268

4

W / eV

Polyatomic Molecules z

H 2O

y

3b2 y

20

unoccupied orbitals

O H

2b2

H

y LUMO

4a1 - WI (O 3P → O+ 4S)

0

- 20 O 2py

HOMO O 2px O 2pz

1b1 3a1

H 1s

x

O H

H 1s y

1b2 O 2s 2a1

y

≈ - 530

- 540

O 1s

occupied orbitals

y

1a1

y

O H

H

Fig. 4.22 Lowest MOs of the water molecule (C2v geometry); left: correlation with AOs for O and H and electron binding energies; right: schematic 3D contour plots of the these MOs derived from C HAPLIN (2013) and other web-sources. Note: for the 1b1 orbital the paper plane corresponds the xz molecular plane, for all others to the yz plane

In summary, the configuration of H2 O in its electronic ground state is (1a1 )2 (2a1 )2 (1b2 )2 (3a1 )2 (1b1 )2 in the terminology of the C2v group. All orbitals are populated with two electrons each, i.e. with paired electron spins. Interestingly, detailed quantum mechanical calculations for an isolated water molecule show a rather smooth density distribution of the electrons, not explicitly displaying the “lone pair” (1b1 )2 . For the properties of the water molecule, however, specifically for the structure of liquid water it plays an important role, enabling the tetrahedral coordination of liquid water and solvation shells. It should be emphasized that the structure of liquid water is presently still under strong discussion. For the spectroscopy of the free water molecule (in the gas phase) one needs to know the total energies of the states. As we know already from atoms, these depend only indirectly on orbital energies. The electronic total wave function is obtained here too as a linear combination of products of electron orbitals. They too have to obey the symmetry properties of the point group – C2v in the present case – and they are designated with capital letters. The total state of the H2 O molecule is a 1 A1 singlet state. Here too one writes the multiplicity (2S + 1) totally symmetric X is – as in as superscript before the state designation, with the total spin S, while X

4.4

Electronic States of Some Polyatomic Molecules ~ D'' 1A 2 ~ 1 D A1 ~ ~ F 1A1 C 1B1

W I = 12.6 eV 12 11

~ D

energiy / eV

10

~ B

9 8

~ F ~ D''

269

~ E' 1A' ~1 F A' ~ E' 1B2

~ E'

~ D'' 1A'' ~ C 1A'' ~ D 1A'

2 1A1

~ C

~ B 1A''

~1 B A2

~ A

2 1A'

~ A 1A''

7

~ A 1 B1

6

asymmetric (Cs)

symmetric (C2v)

5 0

1

2 3 σ / 10 -17cm-1

2

3 4 R(O-H1,2) / a 0

2

3 4 R(O-H1) / a0

5

Fig. 4.23 Left: photo-absorption spectrum of the H2 O molecule. Here the photon energy is given as ordinate, the absorption cross section σ as abscissa, in order to directly compare with potential energy diagrams. Middle: cut through the potential surface in C2v symmetry (both OH bonds are stretched). Right: Cs symmetry (only one OH bond is stretched). The potentials are sketched after the computation of VAN H ARREVELT and VAN H EMERT (2000). However, we have scaled the data published there such that the energetically observed spectra match the potentials

the case of diatomic molecules – just a spectroscopic abbreviation for the ground state (the tilde is set to distinguish the states from diatomic molecules). To understand the electronic structure of such polyatomic molecules, especially the excited states, one has to be aware that the electronic energies Wγ (R) now depend on more than one distance parameter R. Potential curves are replaced by Potential hypersurfaces on which the nuclear motion proceeds. Usually one illustrates these potential surfaces by cuts along the relevant coordinates or – alternatively – as contour plots in two dimensions. For the free water molecule we show in Fig. 4.23 and absorptions spectrum in the UV and VUV spectral range, from B etc. On the right of these spectra, 1 A1 ground state to the excited states A, the X and directly comparable, Fig. 4.23 shows two examples of cuts through the H2 O hyper-surface, demonstrating the complexity of the problem: in both cases, the potential is shown along the ROH coordinate. In the middle panel the C2v symmetry is conserved, i.e. both OH bonds are stretched symmetrically. In contrast, in the right panel only one bond is stretched and the symmetry is broken: what remains is only reflection symmetry σˆ v (zy) in respect of the zy plane. We learn again a little piece of group theory: the point group Cs has only two irreducible elements: A if the sign under reflection in respect of the molecular plane is conserved, and A if it changes. A comparison of the C2v and Cs character Tables 4.3 shows, that A1 and B2 become A due to the symmetry breaking, while A2 and B1 become A . We see the dramatic difference between symmetric and antisymmetric stretch: 1 B1 state appears to be bound, the second geometry while in the first case the A

270

4

Fig. 4.24 Energy scheme for sp 3 hybridization of the carbon 2s, 2p AOs as a basis for carbon chemistry, exemplified for CH4


4 H(1s) + C(sp3) 5S 8.56 eV - 25.31eV 4 H(1s) + C(2s) 2 (2px)(2py) 3P0

CH4

reveals that in actual fact it may dissociate very rapidly since the in Cs symmetry 1 A is strongly repulsive. This is reflected directly in the absorption spectrum A to A state (left panel in Fig. 4.23): the absorption bands for the transition from X state the is broad and unstructured. Because of the short lifetime of the excited A energy is not sharply defined and vibrations cannot be associated with the spectra. In contrast, of the higher excited states some are obviously stable when deformed (they have a well defined energy minimum). Consequently the absorption spectrum is increasingly more structured. Due to the large number of closely spaced rotational and vibrational spectra, however, as energy increases one observes again only more or less broad bands as energy increases.

4.4.2

Hybridization – sp3 Orbitals

The concept of hybrid atomic orbitals goes back to PAULING (1931), who received in 1954 the N OBEL prize for his work. It laid the foundation for a theoretical understanding of chemical bonding – and in particular so for organic chemistry. We have already introduced hybridization in the context of ionic bonding in LiH (Sect. 3.7.3) – a rather a simple case but already exploiting the basic idea: superposition of electronic wave functions (orbitals) with different orbital angular momenta leads to AOs with a pronounced directional quality. Bonding occurs for such superpositions and into that direction where the angular dependence of the electron density reaches its maximum. The concept works best if the energies of the participating orbital energies are not too different. But even that is not a stringent requirement if the overall energetic balance of the electronic rearrangement is favourable. This is readily explained for the most prominent example of the carbon atom, the basis of all organic chemistry. The C atom in its ground state has an electron configuration (1s)2 (2s)2 (2px )(2py ) 2 P0 . For the very important, so called sp 3 hybridization one of the 2s valence electrons has first to be moved into the empty 2pz orbital: C atom: (2s)2 (2px )(2py ) → (2s ↑)(2px ↑)(2py ↑)(2pz ↑). This reorganization within the atom requires energy – which is, however, more than compensated by the binding energy of organic compounds, as illustrated in Fig. 4.24 for CH4 as an example. As illustrated in Fig. 4.25, in principle four bonds may be realized with these orbitals: the 2px , 2py and 2pz orbitals point into the x-, y- and z-direction, while the 2s orbital has no directionality. Clearly, such four bonds are not equivalent. The

4.4


271

Fig. 4.25 n = 2 orbitals of the C atom, schematic

z 2pz 2s

y 2py

x

Fig. 4.26 The four hybrid sp 3 AOs according to (4.40)

2px

z

1

2

x

y

4 3

expected bonding angle between the p orbitals would be 90◦ – and not 109.47◦ as observed experimentally in the case of an ideal tetrahedron such as methane. Mathematically, one linearly superposes the familiar 2s and 2p AOs: 1 |1 = √ |2s − |2px + |2py + |2pz 4 1 |2 = √ |2s − |2px − |2py − |2pz 4 1 |3 = √ |2s + |2px − |2py + |2pz 4 1 |4 = √ |2s + |2px + |2py − |2pz . 4

(4.40)

These four new, hybrid sp 3 AOs are illustrated in Fig. 4.26. They are fully equivalent: in each case all four original atomic orbitals contribute with equal weight. One readily verifies that they are orthonormal if their constituents are orthonormal. √ The p components of the sp 3 hybrids may be written |j p = |j − |2s/ 4 with j = 1, 2, 3, 4; they determine the directions into which the orbitals are aligned. As the thus defined |j p are real, they may be viewed as vectors pointing into the direction of the bond. The angle θ between two of them, say between |1p and |2p , can thus be derived from the scalar product by cos θ =

1|2p 1 =− 3 1|1p 2|2p

=⇒ |θ | = 109.47◦ .

(4.41)

272 Fig. 4.27 The four hybrid sp 3 AOs in exploded view along the tetrahedral axes (diagonals of a cube)

4


1

2

4

3

z

x y

Fig. 4.28 Very schematic image of the four bonding σ MOs of methane, CH4 ; each orbital is occupied by two electrons; note that the vertical axis is a main symmetry axis of the Td group

109.5o 109.5o

H

C H

H

H

109.5o

Fig. 4.29 Contour plot of the total electron density, |ψ|2 (sum over all filled MOs) at the 99 % level; realistic shape of the electron density gleaned from I MMEL (2012)

109.5o

109.5o 109.5o

109.5o

109.5o

This is indeed the tetrahedral angle of an ideal tetrahedron. To describe a molecular bond, the four hybrid orbitals (4.40) must be superposed with suitable orbitals of neighbouring atoms, as introduced in Sect. 3.5.10 This leads to the expected tetrahedron as illustrated by Figs. 4.27, 4.28 and 4.29.

10 If

one would simply add the charge densities of the four hybrid orbitals ( would be a fully spherical charge distribution.

(

|ψj |2 ) the result

4.4


273

Table 4.6 Character table of the Td point group to which e.g. methane belongs Td

E 8C3 3C2 6σd

6S4

A1 1

1

1

1

1

A2 1

1

1

−1

−1

x 2 + y 2 + z2

E

2

−1

2

0

0

T1

3

0

−1

−1

1

(Rx , Ry , Rz )

T2

3

0

−1

1

−1

(x, y, z)

xyz

(x 2 − y 2 ), 3z2 − r 2 [x(z2 − y 2 ), y(z2 − x 2 ), z(x 2 − y 2 )] (xy, yz, zx)

(x 3 , y 3 , z3 ); [x(z2 + y 2 ), y(z2 + x 2 ), z(x 2 + y 2 )]

σ Bonding To describe the simplest hydrocarbon molecule methane, CH4 , one has to combine each of the four sp 3 hybrid AOs of the C atom with a 1s AO from one of the four H atoms. Thus, one constructs the following four MOs: |σ1 = C a|1s1 + b|1 |σ2 = C a|1s2 + b|2 (4.42) |σ3 = C a|1s3 + b|3 |σ4 = C a|1s4 + b|4 . √ Here |1sj is the 1s orbital of the j th H atom and C = 1/ a 2 + 2abS + b2 is a normalization constant with the overlap integral S (see Sect. 3.5). The shape of the resulting MOs (4.42) is quite similar to σ orbitals in diatomic molecules. Hence, this type of bonding is called σ bond. Again, these σ MOs may each be filled with two electrons. In the present case the four σ bonds are filled with four electrons from the C atom and with four electrons from the four H atoms. We obtain in this way the tetrahedrally shaped methane molecule as sketched schematically in Fig. 4.28. Note that the “sausage” type schemes of orbitals presented in Fig. 4.28 are typically used in chemistry. They are rather useful to emphasize the bond characters, but do not reflect the probability densities of electrons in a real molecule. As shown for methane in Fig. 4.29 the electron charge distribution is typically much smoother and indicates the directionality of the bonds only vaguely (if at all). For reference, in Table 4.6 we also communicate the character table of the Td point group to which methane belongs. As shown, it has five classes of symmetry operations and consequently five different irreducible representations, two of which (A1 and A2 ) are singly degenerate, one (E) is twofold and two (T2 and T3 ) are threefold degenerate. The ground state, illustrated in Figs. 4.28–4.29, is a totally symmetric A1 state. Sigma bonding also plays the key role in other organic molecules, wherever single bonds are involved – as illustrated in Fig. 4.30 for the example of C2 H6 (ethane). One may visualize the carbon atoms as being kept together by the overlap of two hybrid sp 3 wave functions (AOs). They jointly form σ MOs – again sketched in a very schematic, simplified picture as “sausage” type symbols.

274

4

Fig. 4.30 Top: σ bonding in ethane between two carbon atoms by sp 3 hybrid orbitals (black dots in the left panel indicate the unpaired electrons in sp 3 orbitals prior to bonding); bottom: structure of ethane (left: schematic, right: 3D view)

H C H

H H C H

H

C(sp3)

C(sp3)

H

Polyatomic Molecules H H C H

H C H H

σ bond

H 109.6o

H

C C 154 pm

H H

H

In a serious calculations the MOs have to be adapted to the molecular symmetry, in the case of C2 H6 this is D3d . In total, 9 orbitals have to be filled with two electrons each. Apart from the carbon core orbitals (which contribute very little to the bonding), 7 valence orbitals (bonding and antibonding) are constructed directly from the 2s, 2px , 2py , 2pz carbon AOs and the 1s hydrogen orbitals. They are filled with two electrons each and have a1g , a2u , eu , a1g and eg symmetry (the eu and eg orbitals are doubly degenerate, the latter is the HOMO).

4.4.3

Electronic States of NH3

The formation of molecular orbitals in the case of ammonia (NH3 ) occurs by a similar scheme. However, atomic nitrogen has the electron configuration (1s)2 (2s)2 (2p)3 and each of the three atomic orbitals 2px , 2py and 2pz is occupied by one unpaired electron. Thus, in a hybridized scheme one of the four sp 3 orbitals of N is already filled with two electrons which do not participate in the bonding (lone pair), the three other sp 3 orbitals contain one electron each – and readily accept one additional 1s electron from an H atom. Experimentally (and confirmed by theory) one observes a bond angle between the N−H bonds of ca. 106.7◦ , slightly

N H H H

106.7o

106.7o

Fig. 4.31 Geometry and schematic of molecular orbitals for the electronic ground state of NH3 . One recognizes the origin of the three bonding molecular orbitals from the hybridized 2spx,y,z orbitals of N and one 1s electron of atomic H. The orbital pointing upwards does not participate in the bonding and is occupied by a lone pair of electrons (indicated by two black dots)

4.4

Electronic States of Some Polyatomic Molecules z

( )

12 0°

2 ı3

sp2ı3

(a)

sp

90 ° y

275

x

x sp2ı1

pʌ

2

pʌ

sp2ı1 sp 2 ı

sp2ı2

y

Fig. 4.32 Schematic of the three carbon sp 2 hybrid σ AOs defined by (4.43) and the (hatched) pz (or pπ ) orbital, forming a double bond; (a) side view, (b) view from below

less than the ideal tetrahedral angle of ca. 109.5◦ , obviously as a consequence of electron repulsion by the lone pair. A schematic representation of the MOs involved in NH3 is shown Fig. 4.31. Ammonia belongs to the C3v symmetry group with the 1 elements A1 , A2 and E, the latter being two fold degenerate. The ground state, XA sketched in Fig. 4.31 has again total symmetry.

4.4.4

sp2 Hybrid Orbitals Forming Double Bonds

Planar molecules with a C=C double bond are explained by sp 2 hybrid orbitals which are formed by the 2s and two of the 2p AOs. Let the x-axis point into the direction of the double bond, the z-axis perpendicular to it. Then the three sp 2 AOs are: √ 2 sp σ1 = (1/ 3)|2s + 2/3|2px √ √ 2 sp σ2 = (1/ 3)|2s − 1/6|2px + (1/ 2)|2py (4.43) √ √ 2 sp σ3 = (1/ 3)|2s − 1/6|2px − (1/ 2)|2py . These sp 2 AOs too, have σ character. The bond angle between them is derived as in the case of sp 2 σ AOs. In analogy to (4.41) one obtains from the p components of the orbitals |sp 2 σ1 and |sp 2 σ2 for cos θ = −1/2, i.e. a bond angle |θ | = 120◦ . The three |sp 2 σj orbitals thus span an equilateral triangle in the xy plane. The fourth orbital is perpendicular to that plane and parallel to the z-axis as illustrated in Fig. 4.32. It is called π orbital and has negative reflection symmetry in respect of the xy plane. In C2 H4 (ethene or ethylene) the three sp 2 hybrid orbitals of each carbon atom lead to three σ bonds (one for each of the two H atoms and one for the other C atom) as illustrated in Fig. 4.33 – again very schematically. The remaining two pπ electrons (one from each C atom) combine to an additional bond with an electron charge distribution perpendicular to the σ bond and the xy plane. Considering the

276

4

Fig. 4.33 Double bond in ethylene consisting of one σ bond and one π bond (hatched)


z

π bond

C

C

x

y

H

H σ bonds

similarity in the corresponding diatomic case, see e.g. Fig. 3.34(a), one speaks of a π bond. The geometry of these orbitals sketched in Fig. 4.32 suggests that the double bond leads to D2h symmetry (Table 4.4) of the molecule. All four H atoms lie in the xy plane. In a serious MO calculation, five σ type valence orbitals (with ag , b3u , b2u , ag , b1g symmetry) are constructed from the 2s, 2px , 2py carbon AOs and the four 1s hydrogen orbitals, while the (bonding) π HOMO is derived from the remaining 2pz carbon AOs and has b1u symmetry.

4.4.5

Triple Bonds

Finally, a third type of hybridization is possible with carbon orbitals: the sp hybrid which is needed for all triple bonds. The most simple example is C2 H2 (ethyne or acetylene), a linear molecule with D∞h symmetry. The sp hybrid orbital is formed by from the 2s and the 2px AOs of carbon: √ |spσ1 = |2s + |2px / 2 √ |spσ2 = |2s − |2px / 2.

Fig. 4.34 Triple bond in acetylene. Top: separated atoms displaying C AOs: sp hybrid, px and py , respectively, together with the respective H(1s) AOs. Below: bound molecule with σ and π MOs. The area hashed red indicates the π MOs with different sign of the wave function (red and black)

z py C

s H

(4.44)

z

y

sp px

py sp

y

C

sp px

s sp

x

z x H

ı

ʌ C

ı

ʌ

H

C ıı ʌʌ

ʌ H

y

4.5

Conjugated Molecules and the H ÜCKEL Method

277

The two orbitals point into +z- or −z-direction, respectively. Together with corresponding AOs of neighbouring atoms one bonding σ orbital is available per neighbour (as well as an antibonding σ ∗ orbital contributing to higher excited states). The two remaining electrons in the 2px and 2py orbitals form – jointly with a neighbouring C atom – two additional π bonds. In summary, a triple bond is formed between two carbon atoms, as sketched in Fig. 4.34 for the example acetylene. The σ π double and the σ π 2 triple bonds are characterized by a certain rigidity which makes a rotation around the C=C or C≡C axis difficult. In contrast, the barrier for rotation around a σ single bond is low. This difference is important for understanding many properties of organic molecules, e.g. in the context of cis-trans isomerization. Section summary

• To describe the electronic structure of polyatomic molecules, symmetry adapted orbitals are required, from which the total state of the molecule is constructed. We have discussed some important examples. • Water belongs to C2v symmetry and has a bond angle of ca. 104.5◦ in its (overall totally symmetric) ground state 1 A1 . Its electron configuration is (1a1 )2 (2a1 )2 (1b2 )2 (3a1 )2 (1b1 )2 , of which the valence MOs (2a1 )2 (1b2 )2 (3a1 )2 are bonding while the (1b1 )2 is filled with a lone pair of electrons, which does not contribute to the bond. • sp 3 , sp 2 , and sp hybridization of AOs is an important concept. Specifically for C atomic orbitals it forms the basis for carbon chemistry. The sp 3 AOs (Sect. 4.4.2) lead to tetrahedrally bonded structures, as in the case of methane CH4 (with an ideal tetrahedral bond angle of 109.47◦ ), and essentially also in all linear hydrocarbon chains. • The structure of NH3 (Sect. 4.4.3) is also very close to tetrahedral, with a bond angle of 106.7◦ (slightly less than ideal due to one lone pair of electrons in the valence shell). • On the other hand, sp 2 hybrid AOs are the basis for all double bonds in organic compounds, leading to a bond structure with three symmetric σ bonds in a plane (bonding angle 120◦ , Sect. 4.4.4) and one perpendicular π bond. • Finally, sp hybrids (Sect. 4.4.5) provide the concept for triple bonds in organic chemistry: they consist, per carbon atom, of two σ bonds (in opposite direction) and two perpendicular π bonds.

4.5


The class of conjugated organic molecules consists of a chain of carbon atoms which are bound together by σ bonds (lying in one plane) based on sp 2 hybrid AOs and π bonds from the 2p orbital perpendicular to the former (in the following denoted by pz ). Examples are butadien, C4 H6 , or benzene, C6 H6 , a cyclic molecule. The basic structure of these molecules is determined by the properties of the sp 2 hybrid orbitals (such as the bonding angle of 120◦ , see Fig. 4.32). The additional π bonds are

278 Fig. 4.35 Benzene structure, very schematic: (a) the structure is defined by 18 σ sp 2 AOs localized at the C atoms, 6 of which form bonds with the 1s orbitals of the H atoms, while 12 combine to 6σ -σ bonds between the C atoms; (b) in addition there are 6 πpz orbitals, perpendicular to the plane of the ring, which in their energetically lowest configuration form a ring of non-localized π bonds

4

(a)


z C6

σ – bonding C

C

C

C

C

C

C'2

σv C2''

C2' z

(b) π – bonding

C C C

C C

C

delocalized in conjugated systems. The usual chemical notation −C=C−C=C−, which appears to suggest that the σ π double bonds are somehow clamped between certain pairs of C atoms, is certainly rather misleading. Specifically for benzene, the 6 C atoms are completely equivalent. They form a symmetric hexagonal ring (point group D6h , RC−C = 0.13902 nm, RC−H = 0.10862 nm, P LIVA et al. 1991), as indicated in Fig. 4.35. The π bonding electrons may move more or less freely between the C atoms and are not localized within the molecule – in contrast to electrons in σ bonds. Thus, in modern chemical literature benzene, and aromatic rings in general, are usually denoted by a hexagon with a ring inside to indicate the delocalized bonds, as shown here: . Figure 4.35 gives an image of a still rather naive LCAO approach: molecular orbitals are seen (a) as an addition of the carbon σ 2sp 2 hybrid orbitals with the hydrogen 1s orbitals, and (b) as addition of all π2pz orbitals. Strictly speaking, each of the thus constructed MOs can only be filled with two electrons (assuming all to be spin saturated). However, benzene has 4 × 6 + 6 = 30 valence orbitals (ignoring the 1s 2 carbon core electrons) which have to be filled into suitable MOs. To classify these, we recall that benzene belongs to the symmetry group D6h . The main C6 symmetry axis z is perpendicular to the plane of the ring. Some of the C2 and C2 axes are also indicated in Fig. 4.35. As documented in the character Table 4.7, the D6h group has 12 symmetry classes (including the unit operator) and hence 12 irreducible characters (types of states or orbitals). The σ orbitals shown in Fig. 4.35(a) together form a totally symmetric 2a1g wave function. And the π electron configuration in benzene shown in Fig. 4.35(b) has a2u character: no sign changes for rotation around the symmetry axes (C6 , C3 , C2 ), neither upon reflections through the planes containing the principle z-axis (σˆ d , σˆ v ); but all operations which somehow interchange top and bottom of the molecule change

4.5


279

Table 4.7 Character table for the point group D6h and basis functions D6h E 2C6 2C3 C2

3C2 3C2 ı

2S3 2S6 σˆ h

3σˆ d 3σˆ v

A1g 1 1

1

1

1

1

1

1

1

1

1

A2g 1 1

−1

1

1

−1 −1 Rz

x 2 + y 2 , z2

1

1

1

−1

1

1

B1g

1 −1

1

−1 1

−1

1

−1 1

−1 1

B2g

1 −1

−1 −1 1

−1

1

−1 −1

1

1

−1 1

E1g 2 1

−1

−2 0

0

2

1

E2g 2 −1

−1

2

0

0

2

−1 −1 2

A1u 1 1

1

1

1

1

−1 −1 −1 −1 −1 −1 z

A2u 1 1

1

1

−1

−1

−1 −1 −1 −1 1

B1u 1 −1

1

−1 1

−1

−1 1

−1 1

−1 1

B2u 1 −1

1

−1 −1

1

−1 1

−1 1

1

−1

E1u 2 1

−1

−2 0

0

−2 −1 1

2

0

0

E2u 2 −1

−1

2

0

−2 1

−2 0

0

0

−1 −2 0

1

0

0

(Rx , Ry ) (yz, zx) (x 2 − y 2 , xy)

0 1

(x, y)

the sign of the wave function. As we shall see in a moment, this orbital corresponds to the energetically lowest π level in benzene. To compute the binding energies and spatial distributions of the 2pz electrons in conjugated molecules one often uses as a first approximation a method developed already in 1931 by H ÜCKEL. In the end it is simply an application of the variational method to obtain optimal LCAO orbitals as already introduced in the context of H+ 2 (see Sect. 3.5.1). We illustrate the H ÜCKEL method by way of example for benzene and concentrate on the π orbitals of the six carbon atoms. Albeit very simple, this approximation leads to quite reasonable results for an estimate of the electronic structure and energies. As we have just discussed, the symmetry of the σ bonds in this planar molecule is different from that of the π bonds. One may thus separate their Hamiltonians (they have no joint non-diagonal elements) and derive the solutions for both types of orbitals independently. The σ bonds are substantially stronger than the π bonds. As it turns out, the energies of the π and π ∗ orbitals fall (essentially) into the band gap between σ and σ ∗ orbitals. Thus, it is mainly the π and π ∗ orbitals which determine the spectroscopic behaviour of conjugated molecules in respect of absorption and fluorescence in the VIS and UV spectral region. The approximations of the H ÜCKEL method may be summarized as follows: 1. pi |pj = δij , all overlap integrals are zero, |pi = α, the diagonal elements of the Hamiltonian correspond to the 2. pi |H atomic orbitals, |pj = βδij ±1 , only neighbouring orbitals interact with each other. 3. pi |H Here |pj stands for the 2pz orbital at C atom j . The values of α (C OULOMB integral) and β (resonance integral) are both negative and may be treated as parameters

280

4


Fig. 4.36 πpz orbitals placed into the middle of a benzene ring

z pz y Rj x

to be calibrated by the experiment. One starts with LCAO-MOs: |φ = cj |pj .

(4.45)

k

Since in benzene all C atoms are equivalent, |cj |2 must be identical for all j . The most simple procedure is now to write down the H AMILTON matrix and to diagonalize it according to the standard rules. Applying (3.102) to the 6 carbon atoms, the problem is described now by a 6 × 6 determinant for which we have to find the roots. With the rules 1.–3. just specified we obtain: α − ε β 0 0 0 β β α−ε β 0 0 0 0 β α − ε β 0 0 (4.46) = 0. 0 0 β α − ε β 0 0 0 0 β α−ε β β 0 0 0 β α −ε One readily verifies that the solutions are: ε0 = α + 2β,

ε±1 = α + β,

ε±2 = α − β,

ε3 = α − 2β.

(4.47)

Two of these energy levels (ε±1 and ε±2 ) are doubly degenerate. Alternatively one may derive a general solution for ring molecules with n carbon atoms and identical conjugated bonds from solving the system of linear equations (3.101) by exploiting symmetry properties of the system. As a starting point one defines a wave function Φ(r) for a carbon 2pz orbital, which is localized in the centre of the benzene ring as sketched in Fig. 4.36. The six atomic orbitals Φj are then given by Φj (r) = Φ(r − R j ) now be the effective Hamilwhere R j is the position vector of the C atom j . Let H tonian for the π electrons. For symmetry reasons it is invariant in respect of rotation and C n commute: of the ring through 360◦ /n. Mathematically this implies that H , C n ] = 0. [H (i.e. the energies ε) and C n have joint eigenfunctions. The eigenvalues of H Thus, H n . Without going into details of that may thus be derived from the eigenvalues of C consideration we obtain the solution (4.47) for the n = 6 eigenvalues, written in the

4.5


281

εk k =3 k =- 2 1e2u

α - 2β 1b2g

k =2 1e2u

LUMO

α + 2 β cos(4π / 6) = α - β α

k =- 1 1e1g

HOMO k=0

k =1 1e1g

α + 2 β cos(2π / 6) = α + β α + 2β

1a2u -3

-2

-1

0

1

2

3

k

Fig. 4.37 Orbital energies of the π electrons in benzene according to the H ÜCKEL model. The red dotted curve represents (4.48), the red dots give the orbital energies for k = −2 to +3, corresponding to the energy terms (full, horizontal red lines). The three lowest orbital are each occupied with two π electrons of different spin orientation (black arrows)

very plausible form 2π k with k = 0, ±1, ±2, 3. εk = α + 2β cos n

(4.48)

This is illustrated in Fig. 4.37. These molecular orbitals are filled with the six π electrons in the benzene ring – according to the PAULI principle – indicated in Fig. 4.37 by double arrows. The doubly degenerate 1e1g MOs (k = ±1) are obviously the HOMOs. If we fill instead higher lying levels, we obtain excited states of benzene. We note that the total contribution of the π orbitals to the ground state energy may be estimated from the occupied orbitals as επ = 2(α + 2β) + 4(α + β) = 6α + 8β. In contrast, the energy contribution of 6 localized π double bonds would be11 εloc = 6α + 6β. Thus, we find that delocalization leads to an additional bond energy 2β. The shape of the respective benzene MOs is obtained by explicitly evaluation of the coefficients for the AOs |pj in (4.45). They follow from the six solutions of the secular equation (4.46). They are cj +1 = e

2πi 6 k

cj ,

localized bonds we had (3.108) εg = (HAA + HAB )/(1 + S). Identifying HAA = α and HAB = β gives for one atom εg = α + β when the overlap integral S is ignored.

11 For

282

4 + -

-

-

+

+

e2u α-β

-

+

e1g α+β

-

+

b2g α-2β

+ +


-

+

+

a2u α+2β

Fig. 4.38 MO orbitals of the 6 benzene π valence orbitals; left: symmetry properties of the orbitals (D6h , Table 4.7); middle, black the designation of the orbitals follow the irreducible representations of D6h and middle, red their Hückel binding energies corresponding to (4.47); right the 3D contour plots of the MO orbitals have been gleaned schematically from I MMEL (2012) and represent the volume within which 90 % of the electron charge is confined

from which, trivially, cj +6 = cj emerges. The solutions of the coefficients {cj } are thus given by (k)

cj = e

2πi 6 k·j

(k)

c0 ,

(k) 2 (k) 2 for which cj = c0

holds, i.e. for all MOs |k the probability for finding an electron at a particular atom j is independent of j : the π electrons are symmetrically delocalized. The symmetry adapted wave functions of a single electron thus reads φk (r) =

cj(k) Φ(r − R j ) = c0(k)

6

e

2πi 6 k

j

Φ(r − R j ),

(4.49)

j =1

j

√ for k = 0, ±1, ±2, 3, with the normalizing constant c0(k) = 1/ 6. And explicitly, the eigenfunctions (4.49) are given by φ3 (1b2g ) ∝ −Φ1 + Φ2 − Φ3 + Φ4 − Φ5 + Φ6 φ±2 (1e2u ) ∝ e±

2π 3

i

Φ1 + e ±

φ±1 (1e1g ) ∝ e± 3 i Φ1 + e± π

4π 3

2π 3

i

i

Φ2 + e±2πi Φ3 + e±

Φ2 + e±πi Φ3 + e±

4π 3

8π 3

i

i

Φ4 + e ±

Φ4 + e ±

5π 3

10π 3

i

i

Φ5 + e±4πi Φ6

Φ5 + e±2πi Φ6

φ0 (1a2u ) ∝ Φ1 + Φ2 + Φ3 + Φ4 + Φ5 + Φ6 , ordered here according to their energy. In Fig. 4.38 on the left their symmetry properties are schematically sketched, corresponding to the irreducible representations of the point group D6h (cf. Table 4.7). A three dimensional impression of these six π orbitals is given in Fig. 4.38 on the right. Full 3D images of these orbitals may be viewed on several rather instructive web-sites, e.g. NASH (2004), or somewhat

4.5


Table 4.8 Benzene, occupied and first unoccupied MOs. If not otherwise indicated, all data are taken from D ELEUZE et al. (2001), Table II

MO

Type

283 # electr.

εHF/eV

Exp./eV

1b2g

π

0

8.79a /9.00b

2b1u

σ

0

7.82a

3e2g

σ

0

6.96a

3e1u

σ

0

6.14a

3a1g

σ

0

5.20a

1e2u

π

0

3.39a /3.48b

1e1g

π

4

−9.062

−9.24

3e2g

σ

4

−13.323

−11.4

1a2u

π

2

−13.538

−12.38

3e1u

σ

4

−15.843

−13.98

1b2u

σ

2

−16.704

−14.86

2b1u

σ

2

−17.376

−15.46

3a1g

σ

2

−19.100

−16.84

2e2g

σ

4

−22.294

−19.2

2e1u

σ

4

−27.506

−22.7c

2a1g

σ

2

−31.176

−25.9c

1s

σ

12

−290.3d

a VON N IESSEN et al. (1976) b D UFLOT et al. (2000) c G ELIUS et al. (1971) d M EDHURST et al. (1988) (the 1s AOs may in principle be combined

to σ MOs – which do however, not contribute significantly to the bonding)

more detailed and universal I MMEL (2012), where these images can be viewed and manipulated in 3D space with Chime and Jmol,12 respectively. The energies of the σ orbitals cannot be derived by a similarly simple treatment. We thus refer here only to the literature without going into much detail. Table 4.8 gives a summary of calculations and measurements on orbital symmetries and energies for all occupied MOs in benzene, together with a few of the excited ones. Most of this data has been taken from relatively recent work by D ELEUZE et al. (2001). They performed state-of-the-art ab initio calculations for ionization bands of several aromatic hydrocarbons and compared them with experimental data. The symmetries of the orbitals in the first column of Table 4.8 refer to the D6h point group (see Table 4.7). The next column indicates whether the orbital refers to a σ or π bond and the third column gives the occupation number of each orbital (all e orbitals are doubly degenerate). The valence orbitals are completely filled – up to the HOMO 1e1g orbital with spin saturated electrons. 12 Jmol is an open source Java viewer for molecular structures which appears now to be the generally accepted standard: It can be easily installed into most browsers.

284

4


MO energy εj / eV 10

HF Hückel 1b2g

1e2u 0

- 10

1e1g

energy of electronic states and transitions WI 1E 2g 3E 1E 2g 1u 3B 1B 2u 1u 3E 1B 1u 2u 3B 1u 1A

1g

1a2u

Fig. 4.39 Energy of molecular π type orbitals (left) and electronic states (right) in benzene. The energies of the HF molecular orbitals (heavy black lines) are those given in Table 4.8; they are compared with the (scaled) H ÜCKEL energies (thin red lines) according to (4.48). The state energies are the ab initio values of H ASHIMOTO et al. (1998) (experiment in brackets); singlets: 1 B2u 4.70 (4.90), 1 B1u 6.21 (6.20), 1 E1u 6.93 (6.94), 1 E2g 7.82 (7.80) eV, triplets: 3 B1u 3.89 (3.95), 3E 3 3 1u 4.53 (4.76), B2u 5.54 (5.60), E2g 7.02 (6.83) eV. The ionization energy of benzene in the 1 A ground state (gas phase) is W = 9.24378 ± 0.00007 eV (NIST 2011) 1g I

Since the π MOs are the most loosely bound orbitals they determine the ground state and the lowest excited states. Hence, the ground state electron configuration is described by (1a2u )2 (1e1g )4 1 A1g (as in the atomic case, in a closed shell system the ground state is a singlet state). To obtain the first excited states of benzene, one electron is raised from a 1e1g HOMO into an empty 1e2u LUMO, leading to a (1a2u )2 (1e1g )3 (1e2u ) electron configuration. Thus, we have to combine a 1e1g and a 1e2u orbital, each filled with an unpaired electron, in such a manner that the PAULI principle is conserved and at the same time a irreducible representation of the D6h point group emerges. Since both orbitals are two-fold degenerate, there are four different possibilities to comply with these conditions. In mathematical terms one writes these electronic states as direct product E1g ⊗ E2u = B1u + B2u + E1u . We thus have three states: B1u , B2u and the doubly degenerate E1u state. Singlet as well as triplet states can be formed. According to H UND’s rules, the latter is always lower in energy than the respective singlet state. Figure 4.39 summarizes the energetics of benzene. On the left, the orbital energies are sketched. The Hückel energies have been scaled such that the energy difference between 1b2g and 1a2u orbital is the same as that for the ab initio calculated HF values. Obviously, the Hückel method does rather well in predicting the relative distances between the four π levels. The energies of the overall states have been calculated by H ASHIMOTO et al. (1998) with multi-reference M ØLLER -P LESSET perturbation theory (MRMP, see Sect. 10.1.1, Vol. 1) with complete active space self-consistent field (CASSCF) reference – a rather rigorous ab initio approach. It


285

compares extremely well with experiment (values given in the figure caption in brackets). Note that direct optical transitions from the 1 A1g ground state are allowed to the 1 E1u state. They are observed experimentally as a strong absorption band while the other transitions are symmetry forbidden, but may be observed as weak transitions. As we have seen here, the H ÜCKEL method as presented above gives a valuable first order estimate for energies and structures. It is very simple and flexible, and gives reasonable estimates even for larger molecules and clusters, typically up to the HOMO. One may even generalize what has been exemplified for benzene to any kind of Nnu regularly ordered, identical atoms – and derive thus e.g. a one dimensional solid state model. The eigenfunctions then become B LOCH functions and instead of discrete eigenvalues one obtains energy bands. The method is used quite often in solid state physics, usually called extended H ÜCKEL tight-binding approach (the latter essentially being another name for LCAO). Section summary

• Delocalized double bonds in organic molecules can be treated to a good approximation by the H ÜCKEL method. This particularly simple approach for obtaining quantitative information on the structure and energy of bonding π MOs uses an LCAO approach which parameterizes the energies in terms of only two parameters. We have applied the method to benzene, C6 H6 , the simplest aromatic ring. The spectroscopy of benzene is essentially determined by the six valence π electrons and their orbitals. We have familiarized ourselves with the dihedral symmetry group D6h to which benzene belongs.

Acronyms and Terminology AO: ‘Atomic orbital’, single electron wave function in an atom; typically the basis for a rigorous structure calculation. APES: ‘Adiabatic potential energy surfaces’, potential energy hyper-surface determined in B ORN -O PPENHEIMER approximation. BO: ‘B ORN O PPENHEIMER’, approximation, the basis when solving the S CHRÖ DINGER equation for molecules (see Sect. 3.2). E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. FT: ‘F OURIER transform’, see Appendix I in Vol. 1. good quantum number: ‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the H AMILTON operator (see Sect. 2.6.5 in Vol. 1)’. HF: ‘H ARTREE -F OCK’, method (approximation) for solving a multi-electron S CHRÖDINGER equation, including exchange interaction. HITRAN: ‘High-resolution transmission molecular absorption database’, http:// www.cfa.harvard.edu/hitran (ROTHMAN et al. 2009).

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HMO: ‘H ÜCKEL molecular orbital method’, approximation for describing molecular orbitals in conjugated hydrocarbon molecules (Sect. 4.5). HOMO: ‘Highest occupied molecular orbital’. IP: ‘Ionization potential’, of free atoms or molecules (in solid state physics the equivalent is called “workfunction”). IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). isotopologue: ‘Molecules that differ only in their isotopic composition’, http://en. wikipedia.org/wiki/Isotopologue. JT: ‘JAHN and T ELLER’, have first treated in 1937 the symmetry breaking effect, now referred to by their names. JTE: ‘JAHN -T ELLER effect’, symmetry breaking effect first treated by JAHN and T ELLER in 1937. LCAO: ‘Linear combination of atomic orbitals’, linear superposition of atomic, single electron wave functions to form a molecular orbital (MO). LUMO: ‘Lowest unoccupied molecular orbital’. MO: ‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation. PJTE: ‘Pseudo-JAHN -T ELLER effect’, vibronic coupling for nearly degenerate molecular states, leading to symmetry breaking. RHF: ‘Restricted H ARTREE -F OCK’, assuming all spatial wave functions in a given closed shell to be equal when computing atomic wave functions. UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). VIS: ‘Visible’, spectral range of electromagnetic radiation. Wavelengths between 380 nm and 760 nm according to ISO 21348 (2007). VUV: ‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References B ERNATH , P. F.: 2002a. ‘Laser chemistry – water vapor gets excited’. Science, 297, 943–944. B ERNATH , P. F.: 2002b. ‘The spectroscopy of water vapour: Experiment, theory and applications’. Phys. Chem. Chem. Phys., 4, 1501–1509. B ERSUKER , I. B.: 2001. ‘Modern aspects of the Jahn-Teller effect theory and applications to molecular problems’. Chem. Rev., 101, 1067–1114. B UNKER , P. R. and P. J ENSEN: 2006. Molecular Symmetry and Spectroscopy. Ottawa: NRC Research Press, 2nd edn., 747 pages. C ARLEER , M. et al.: 1999. ‘The near infrared, visible, and near ultraviolet overtone spectrum of water’. J. Chem. Phys., 111, 2444–2450. C HAPLIN , M.: 2013. ‘Water structure and science’, London South Bank University. http://www. lsbu.ac.uk/water/, accessed: 9 Jan 2014. DAMBURG , R. J. and R. K. P ROPIN: 1972. ‘Rotational structure of the inversion spectrum of ammonia’. J. Phys. B, At. Mol. Phys., 5, 1861–1867.

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D ELEUZE , M. S., A. B. T ROFIMOV and L. S. C EDERBAUM: 2001. ‘Valence one-electron and shake-up ionization bands of polycyclic aromatic hydrocarbons. I. Benzene, naphthalene, anthracene, naphthacene, and pentacene’. J. Chem. Phys., 115, 5859–5882. D UFLOT , D., J. P. F LAMENT, J. H EINESCH and M. J. H UBIN -F RANSKIN: 2000. ‘Re-analysis of the K-shell spectrum of benzene’. J. Electron Spectrosc., 113, 79–90. E NGELKE , F.: 1996. Aufbau der Moleküle: Eine Einführung. Leipzig: Teubner, 339 pages. G ELIUS , U., C. J. A LLAN, G. J OHANSSON, H. S IEGBAHN, D. A. A LLISON and K. S IEGBAHN: 1971. ‘ESCA spectra of benzene and iso-electronic series, thiophene, pyrrole and furan’. Phys. Scr., 3, 237–242. G ORDON , J. P., H. J. Z EIGER and C. H. T OWNES: 1955. ‘Maser – new type of microwave amplifier, frequency standard, and spectrometer’. Phys. Rev., 99, 1264–1274. G OSS , J. P.: 2009. ‘Point group symmetry’, University of Newcastle upon Tyne, UK. http://www. staff.ncl.ac.uk/j.p.goss/symmetry/index.html, accessed: 9 Jan 2014. VAN H ARREVELT , R. and M. C. VAN H EMERT : 2000. ‘Photodissociation of water. I. Electronic structure calculations for the excited states’. J. Chem. Phys., 112, 5777–5786. H ASHIMOTO , T., H. NAKANO and K. H IRAO: 1998. ‘Theoretical study of valence and Rydberg excited states of benzene revisited’. J. Mol. Struct., Theochem, 451, 25–33. H ERZBERG , G.: 1991. Molecular Spectra and Molecular Structure, vol. II. Infrared and Raman Spectra of Polyatomic Molecules. Malabar: Krieger Publishing Company, 636 pages. HITRAN-W EB: 2012. ‘HITRAN on the Web’, Harvard-Smithsonian Center for Astrophysics (CFA), Cambridge, MA, USA, and V.E. Zuev Institute of Atmospheric Optics (IAO), Tomsk, Russia. http://hitran.iao.ru/molecule, accessed: 9 Jan 2014. I MMEL , S.: 2012. ‘Tutorials’, Darmstadt, Germany: Universität Darmstadt. http://csi.chemie.tudarmstadt.de/ak/immel/, accessed: 9 Jan 2014. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. JAHN , H. A. and E. T ELLER: 1937. ‘Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy’. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 161, 220–235. J MOL: 2011. ‘Websites using jmol’, Jmol Community. http://wiki.jmol.org/index.php/Websites_ Using_Jmol, accessed: 9 Jan 2014. J R ., R. F. C., H. W. K ROTO and R. E. S MALLEY: 1996. ‘The N OBEL prize in chemistry: for their discovery of fullerenes’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/laureates/ 1996/. K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’. http://physics.nist.gov/asd, accessed: 7 Jan 2014. M EDHURST , L. J., T. A. F ERRETT, P. A. H EIMANN, D. W. L INDLE, S. H. L IU and D. A. S HIRLEY: 1988. ‘Observation of correlation-effects in zero kinetic-energy electron-spectra near the N 1s-threshold and C 1s-threshold in N2 , CO, C6 H6 , and C2 H4 ’. J. Chem. Phys., 89, 6096– 6102. M ULLIKEN , R. S.: 1955. ‘Report on notation for the spectra of polyatomic molecules (the name of the writer was inadvertently omitted when this report was published)’. J. Chem. Phys., 23, 1997–2011. M ULLIKEN , R. S.: 1966. ‘N OBEL lecture: Spectroscopy, molecular orbitals, and chemical bonding’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-lecture. html. NASH , J. J.: 2004. ‘Visualization and problem solving for general chemistry’, Purdue University, Chemistry Department. http://www.chem.purdue.edu/gchelp/, accessed: 9 Jan 2014. NIST: 2011. ‘Chemistry webbook’. http://webbook.nist.gov/, accessed: 9 Jan 2014. PAULING , L.: 1931. ‘The nature of the chemical bond. . . ’ J. Am. Chem. Soc., 53, 1367–1400. PGOPHER: 2013. ‘A program for simulating rotational structure’, C. M. Western, University of Bristol, UK. http://pgopher.chm.bris.ac.uk, accessed: 9 Jan 2014. P LIVA , J., J. W. C. J OHNS and L. G OODMAN: 1991. ‘Infrared bands of isotopic benzenes – ν13 and ν13 of 13 C6 D6 ’. J. Mol. Spectrosc., 148, 427–435.

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5

Molecular Spectroscopy

In Chaps. 3 and 4 we have treated structure and properties of diatomic and polyatomic molecules, together with some basics on rotational and vibrational spectra. Now we shall deepen this first acquaintance and introduce also electronic transitions in molecules. Instrumental to modern spectroscopy are narrow band lasers and synchrotron radiation, covering together with microwave, sub-millimetre and radio frequency sources a spectral range of more than ten decades, ready for any conceivable applications in molecular spectroscopy – for which we shall present selected examples.

Overview

After a brief introduction in Sect. 5.1 we shall expand our knowledge about rotational (microwave) and vibrational (infrared) spectroscopy in Sects. 5.2 and 5.3, respectively, and supplement it with short excursions into infrared F OURIER transform spectroscopy (FTIR) and IR action spectroscopy. In Sect. 5.4 we turn to the spectroscopy of electronic transitions (VIS, UV and VUV) and present a few state-of-the-art methods of modern molecular spectroscopy. In Sect. 5.6 basics of R AMAN spectroscopy will be developed – a very important spectroscopic art, which may be said to reside in between electronic and vibrational spectroscopy. In Sect. 5.5.4 we illustrate the astonishing capabilities of today’s high resolution spectroscopy with sophisticated methods, as applied to larger, even biologically relevant molecules. Finally, in Sect. 5.8 we introduce the important field of photoelectron spectroscopy.

5.1

Overview

What we have learned about molecular structure in Chaps. 3 and 4, is based on more than a century of hard and delicate experimental work. As in atomic physics, spectroscopy was – and continues to be – the most important source of information onto which today’s understanding of molecules is built. © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_5

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Let us briefly recall some of the most important facts. In the framework of B ORN O PPENHEIMER approximation (i.e. almost always except for some notable exceptions), the total wave function Ψγ vN (r, R) of a molecular system may be written as product of electronic and nuclear wave functions, φγ (r; R) and ψγ vN (R), respectively: Ψγ vN (r, R) = φγ (r; R) × ψγ vN (R). Here r represents the entirety of all electronic, R of all nuclear coordinates. The nuclear wave function may usually again be factorized into vibration and rotation. In the case of a diatomic molecule one finds ψγ vN (R) =

Rγ vN (R) × YN MN (Θ, Φ), R

with R being the internuclear distance and Θ, Φ describing the polar and azimuthal angle of the internuclear axis. The indices characterize the electronic state (γ ), the nuclear vibration (v) and the angular momentum of nuclear rotation (N ). According to (3.49) and (3.50) the total energy of the system may be written as a sum of electronic, rotational and vibrational energy: γ (5.1) Wγ vN = Vγ R0 + Wv + WN . For abbreviation, the minima of electronic states are often denoted as Te = V (R0 )/(hc) and measured in wavenumbers, [Te ] = cm−1 . The potential energies (electronic) and relative term levels of vibrational and rotational energies are schematically summarized in Fig. 5.1. Of course, the three components of energy terms in (5.1) are, upon closer inspection, not completely independent of each other – as already discussed in Sects. 3.3.6 and 3.3.7 for diatomic molecules in some detail. Nevertheless, the energy scheme Fig. 5.1 provides a good illustration for the following discussion. For polyatomic molecules consisting of Nnu atoms this has to be generalized – in principle without problems, in practice often with substantial efforts: typically there will be 3Nnu − 6 vibrational degrees of freedom, i.e. v has to be replaced by the vibrational quantum numbers of these normal modes (Sect. 4.2), and the rotational energy will depend also on the alignment angle of the rotational axis in respect of the principle axes of the ellipsoid of inertia (Sect. 4.1). Finally, the R coordinate in Fig. 5.1 is to be read as one normal coordinate or a representative linear combination of spatial coordinates. As indicated in Fig. 5.1 by the black double arrows, spectroscopy distinguishes essentially three categories of transitions between bound molecular states: rotational spectra, vibration-rotation spectra and electronic band spectra, reflecting rotational, vibrational and electronic transitions in the molecule. Figure 5.2 shows again the electromagnetic spectrum – well known to us from Vol. 1 – this time from a molecular perspective, and indicates the spectral regions where the types of molecular spectra mentioned above should be expected. We see that the different molecular transitions correspond to rather different spectral re-

5.1

Overview Vγ(R )

WvN

291 higher electronic state

rotational levels WN'

V'(R ) v' = 3

lower electronic state

vibrational levels, Wv'

v' = 2 v' = 1 v' = 0

rot

R 0'

rot

vib-rot el-vib -

V'(R 0' )- V ''(R''0 )

V''(R )

rotational levels WN'' v'' v'' = 3 v'' = 2 v'' = 1 v'' = 0 R''0

vibrational levels, Wv''

R

Fig. 5.1 Composing the total energy of a diatomic molecule from rotation, vibration, and electronic energy. The heavy black double arrows indicate three different types of molecular spectra: electronic bands with rotational and vibrational transitions (el-vib-rot), vibration-rotation spectra (vib-rot) and pure rotation spectra (rot); traditionally (according to H ERZBERG 1989) upper levels are denoted by a single dash , lower levels by two dashes

rotationvibration bands

rotational bands

1mm MW

100μm FIR

10μm

electronic photoelectron bands spectroscopy UPS XPS X-ray VIS absorption NEXAFS X-ray EUV 1μm

MIR NIR

100nm UV

VUV

10nm

1nm

0.1nm

XUV

Fig. 5.2 The spectrum of electromagnetic radiation seen from the perspective of molecular spectroscopy. MW: microwave, FIR: far infrared, MIR: middle and NIR: near infrared (IR), VIS visible, UV: ultraviolet, VUV: vacuum ultraviolet, EUV: extreme ultraviolet, XUV: soft X-ray

gions, and hence, a broad variety of techniques and methods is applied for molecular spectroscopy, mostly exploiting E1 type transitions. Very important exceptions, not shown in Fig. 5.2, are M1 transitions as the basis of NMR and EPR spectroscopy (which we have already introduced in the context of hyperfine structure in Chap. 9, Vol. 1) and R AMAN spectroscopy (to be explained in Sect. 5.6). In the following sections we shall present a selection of powerful spectroscopic methods for the different spectral regions and categories of spectra. We shall illustrate these, as far as possible and instructive, with examples from recent literature.

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Section summary

Molecular energy levels are essentially given by the sum of electronic, vibrational and rotational energy. Correspondingly, spectroscopy in the different spectral regions of electromagnetic radiation probes different aspects of molecular structure: • Radio frequencies (kHz to some 100 MHz) induce nuclear spin resonance transitions. NMR spectroscopy exploits these with highly sophisticated methods for structural determination of large molecules. • Frequencies from about 1–100 GHz (λ 30 cm –3 mm) are called microwaves MW. Electron spin resonances EPR are induced by such frequencies. At least equally important are applications to rotational transitions in larger molecule as we have seen already in Chap. 3. • The infrared spectral region extends from microwaves to the visible spectral range at around 760 nm. In the far infrared (λ = 0.1 mm to 1 mm) rotational spectra of many molecules are found. Vibration-rotation spectra are typically associated with the centre of the IR range but extend often even down to the far red (λ = 700 nm to 1.4 µm). • The visible, ultraviolet and vacuum ultraviolet spectral regions are associated with electronic transitions and their band spectra. • Beyond the VUV region extend the XUV, X-ray and γ -ray regions. With high photon energies, inner shell electrons are studied. Photoelectron spectroscopy is probably the method most often applied (UPS: with VUV light; XPS: with X-ray radiation). But also X-ray absorptions spectroscopy (XAS, XANES, NEXAFS) provides useful information. As these methods address specific, localized atoms (e.g. in large organic molecules) such methods are very selective in respect of the geometry of the molecule studied.

5.2

Microwave Spectroscopy

We may keep this short as the key aspects have already been discussed in Sect. 3.4.1. Rotational transitions are studied almost exclusively in absorption, since the probability for spontaneous emission is extremely low, due to the low transition frequency and the ω3 factor for the E INSTEIN Aab coefficients (see Eq. (2.154)). A typical microwave spectrometer consists of a microwave source, a waveguide structure in which the gaseous molecular target to be studied is positioned, and a detector. As sources one uses magnetrons, reflection klystrons or travelling wave tubes. Long absorption paths (metres) are needed since the absorption cross sections are small and the target density has to be kept low (to avoid collisional line broadening). Alternatively one uses high quality (Q) resonators and determines the damping as a function of frequency – which is equivalent to a long absorption path. To improve the signal to noise ratio one applies modulation techniques combined with phase sensitive detection. When studying rotational lines, modulation may be achieved by

Microwave Spectroscopy 160MHz

ν+

gas jet nozzle FPI resonator

160MHz

ν

mol. beam

mixer

ν ν

mixer

Fig. 5.3 MB-MWFT spectrometer according to A NDERSEN et al. (1990). The supersonic molecular beam is introduced parallel to the resonator axis. The resonator (essentially an FPI) is tuned through the frequency range of interest by a motor and a microphone. The MB carrier frequency ν is modulated in various mixers and again demodulated in the detection system. The signal is finally recorded by an analogue to digital converter (A/D) and registered in a computer, which is also used for controlling the experiment

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157.5 MHz

MW source

PM

motor& microphone

mixer

5.2

A/D computer

an alternating electric field which varies the absorption frequency periodically by virtue of the S TARK effect. With such a setup one may determine rotational line positions with a resolution of 106 . As an alternative to gas cells, one uses pulsed supersonic molecular beams (MB), so that only a few vibrational levels are initially populated. In a F OURIER transform (FT) spectrometer the molecular beam enters a microwave resonator whose lengths is scanned to determine the FT of the absorption spectrum – which finally has to be inverted back into frequency space. FT spectrometers are also used in the infrared and visible spectral range and we shall come back to the principles in Sect. 5.3.2. A typical setup of such an MB-MWFT spectrometer is sketched schematically in Fig. 5.3. It has been described by A NDERSEN et al. (1990) and allows the determination of absorption lines with an accuracy of kHz, i.e. with a resolution of 107 . A supersonic molecular beam enters from a pulsed nozzle into the FPI microwave resonator, perpendicularly to the FPI axis (in more recent versions parallel). The resonator is scanned by a motor and a microphone. The microwave carrier frequency ω is ‘mixed’ with a sideband and then pulsed through a pin-diode. Several sophisticated microwave techniques are applied to feed the radiation into the FPI. The resonator has a very high Q value so that molecular absorption is detected very sensitively by back-reflection from the resonator. The back reflected signal is ‘down converted’ in two ‘mixer’ stages and finally detected at 2.5 MHz. The signal from an analogue to digital converter is registered in a computer which also controls the microwave frequencies and the resonator scanning. The power meter (PM) helps to optimize the resonator. With this type of setup numerous simple but also very complicated molecules and even clusters have been measured with extremely high precision. The rotation spectrum of a molecule is indeed determined mainly by its three principle axes of inertia and provides very accurate values for the overall nuclear distances. However, the extremely high precision of these measurements – in combination with state-

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1012 1011 1010

blow-up

(a)

(b)

0+404-303A 0+404-303E

-

H

H N

0 404-303A -

0 404-303E

10-8

signal (linear)

signal (log scale)

Molecular Spectroscopy 0+404-303A 6-4 8-6

H H C H

109 108 107 10240

10260

10280

10300

10286.0

10286.4

frequency / MHz

Fig. 5.4 Parts of the microwave absorption spectrum for p toluidine according to H ELLWEG (2008), obtained with an advanced version of the MW-MWFT shown in Fig. 5.3 (G RABOW et al. 1996, operating in a frequency range from 3 GHz to 25 GHz). (a) Overview (b) blow up of the rotational spectrum with the transitions 404 –303 of the methyl torsion mode A from the vinv = 0+ vibrational state. Shown are three hyperfine transitions 2F → 2F , 10–8, 6–4 and 8–6. The rectangular brackets mark the D OPPLER splitting of these lines

of-the-art of ab initio quantum chemical calculations – allows also for the determination of distortions and modifications of the equilibrium distances due to rotation, vibrational excitation, inner rotation of sub groups in the molecule, and hyperfine interaction. We illustrate this by a relatively recent measurement of H ELLWEG (2008) who studied p toluidine (C7 NH9 ). As shown in Fig. 5.4, it consists of a benzene ring with one amino and one methyl group. The rotational levels of this asymmetric rotor are characterized by NKa Kc (as introduced in Sect. 4.1.4 the rotational quantum number is N , its projection onto the principle axes of the ellipsoid of inertia are denoted by Ka and Kc ). This particular molecule has two extra specialties: the methyl group may rotate, leading to two series of A and E symmetry. In addition, the two H atoms of the amino group, which point out of the plane defined by the benzene ring, may undergo inversion vibrations. Figure 5.4(a) gives an overview of the MW spectrum for a frequency range of about 60 MHz. As in the case of NH3 (see Sect. 4.2.5) the structure leads to two split vibrational ground states vinv = 0+ and 0− . Figure 5.4(a) documents that the latter is populated only weakly, owing to the cooling in the supersonic jet beam. The impressive resolution is illustrated in the blow up Fig. 5.4(b) for part of this spectrum over less than 1 MHz with fully resolved hyperfine structure: nuclear spin I (I = 1 for 14 N) and rotational momentum N add to a total angular momentum F with quantum numbers F = N or N ± 1. One recognizes in Fig. 5.4(b) that all lines appear in pairs due to D OPPLER splitting. Albeit very small in the microwave region due to ν = νv/c, the very high resolution allows one to resolve and assign it: note that each line is observed as clearly split by ca. 100 kHz in contrast to the usual D OPPLER broadening. As the molecular beam enters the resonator coaxially (see Fig. 5.3), the microwave excites different velocity components of the beam depending on the MW travelling from left to right

Microwave Spectroscopy

295

201 50 101 (b)

0.9

111 (c)

0 NK K = 000 a c

18708.5 / MHz

Å

(d)

90.04o

1. 2

111.02o

25

Å

(c) F=1-1

(b)

72

1.6 88 Å

/ GHz

(a)

F=0-1

5.2

18709.5 80154 80158

Fig. 5.5 Microwave spectroscopy of the HO3 radical according to S UMA et al. (2005). (a) Term scheme for rotational levels, (b) microwave spectrum for one rotational transition (NKa Kc = 101 –000 , J = 1.5–0.5, F = 2–1), (c) double resonance (NKa Kc = 111 –000 , J = 0.5–0.5) with hyperfine splitting due to the proton (F = 1–1 and F = 0–1), observed by reduction of the maximum signal for transition (b) when tuning the mm wave through transition (c) as indicated in the term scheme (a); (d) geometry of HOOO determined by this experiment

or after reflection from right to left. With the internal translational temperature of the molecules in the supersonic beam being very low, both components are well resolved and can be assigned as indicated in Fig. 5.4(b). Supported by advanced quantum theory evaluation of these spectra allowed H ELLWEG (2008) to determine 44 well identified molecular parameters with high precision – thus demonstrating the high potential of microwave spectroscopy for structural assignments also in the case of larger molecular systems. Often microwave spectra are rather complex and difficult to analyze. Then it may be helpful to induce transitions by two (or even more) frequencies simultaneously (so called double resonance spectroscopy). In Fig. 5.5 we illustrate this important and highly selective method for the example of the HO3 radical. In Fig. 5.5(a) a part of the rotational level scheme of HOOO is sketched schematically (without hyperfine splitting). A ‘simple’ absorption line observed with cm waves is shown in Fig. 5.5(b). It arises from the rotational excitation NKa Kc = 101 ← 000 at 18709.1 MHz as indicated by the red arrow marked (b) in the term scheme Fig. 5.5(a). Keeping this frequency fixed, one records the double resonance signal shown in Fig. 5.5(c) by tuning a mm wave over a range from 80153 MHz to 80161 MHz as indicated by the grey arrow marked (c) in the term scheme. As shown in Fig. 5.5(c), two hyperfine components of the 111 ← 000 transition are detected by reduction of the single photon absorption line Fig. 5.5(b) at its maximum: the ground state is depleted if one hits the (second) resonance. (Note that the scanning range in Fig. 5.5(c) is much smaller than the levels linewidths indicated in Fig. 5.5(a).)

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Using such precision spectroscopy, S UMA et al. (2005) were able to determine the structure of this molecule with high accuracy. The key results are summarized in Fig. 5.5(d). The HO3 radical is very important for atmospheric photo-chemistry and we shall come back to it again later. Section summary

• Microwave radiation induces rotational transitions in small and large molecules (E1). Due to the ν 3 factor associated with spontaneous emission, only absorption spectra are observed. • Using various sophisticated methods such as up and down frequency conversion, frequency synthesizing, heterodyne detection and employing FT techniques or molecular beam targets, it has been developed today to an extreme precision, which allows one to determine molecular constants with highest accuracy. • In addition to the principle moments of inertia from which overall dimensions of a molecule may be extracted, also internal motions such as torsion or inversion vibrations may be gleaned from such spectra.

5.3

Infrared Spectroscopy

5.3.1

General

With infrared spectroscopy (partially also in the visible) one studies vibrationrotation bands, typically in absorption. To give an impression of the complexity that some of these spectra may involve, we show in Fig. 5.6 the line strengths of CO2 and H2 O over a wide spectral range from 400 nm to 17 µm. These sticks-spectra have been derived from GATS (2012), using the HITRAN spectral database.1 We cannot go into details, but we mention that parts of these spectra have already been discussed in Chap. 4. For CO2 this is indicated by the blow up in the inset, which corresponds to the (111 0) ← (000 0) transition for which a simulated absorption spectrum has been shown in Fig. 4.9 – there as a function of wavenumbers. For the water molecule, most of the bands seen in Fig. 5.6, in particular those at shorter wavelengths, are overtone and combination bands of the three normal vibrational modes of H2 O. We may compare the F OURIER transform spectrum Fig. 4.12, briefly discussed in Sect. 4.2.4, which corresponds to a tiny section of the visible region presented here (0.5913 µm to 0.5928 µm). This just emphasizes – specifically for the water molecule – the amazing richness of its visible and infrared spectrum. So much for the complexity of vibrational spectra. Nevertheless, for the vast majority of molecules IR spectra show very characteristic features and vibrationrotation spectroscopy provides an excellent tool for chemical analysis. Convention1 Note that these stick-spectra just communicates molecular line strengths. They do not yet represent absorption spectra (see Sect. 5.2.4 in Vol. 1).


Fig. 5.6 Spectroscopic line strengths for CO2 and H2 O (on a logarithmic scale), obtained from GATS (2012) as a function of wavelength over a wide spectral range; the unit on the vertical axis represents wavenumber per column density as explained in Sect. 5.2.4, Vol. 1. These spectra contain all known absorption lines in this wavelength region: 247 552 lines in the case of CO2 and 61 722 lines for H2 O. The little inset in the CO2 data, a blow up of the 5.05 to 5.3 µm region, may be compared to the absorption spectrum in Fig. 4.9

297 10- 18 line strength / cm -1 molecules-1 cm 2

5.3

CO2

10- 22 10- 26 10- 30 10- 18

H 2O

10- 22 10- 26 10- 30 0

5 10 wavelength / μm

15

ally one still uses so called glow bars as radiation source. These are graphite or SiC rods are heated by an electric current up to 1000 °C to 1500 °C and emit continuous IR radiation corresponding to P LANCK’s radiation law. The IR radiation is fed by mirrors through a typically rather long absorption cell. Usually, a reference beam passes through an identical, but empty absorption cell in order to register fluctuations in the IR source. Behind the cell the radiation is spectrally dispersed by a grating spectrometer. Detection is achieved by thermic devices (bolometers) or by IR sensitive photodiodes. Typically, the IR beam is modulated to distinguish the signal from the background of the apparatus (thermal radiation). But laser sources or laser diodes conquer more and more of the IR spectral regions and allow of course for better resolution and higher sensitivity. In particular for various applications in analytical sciences, absorption spectroscopy with tunable laser diodes appears to become a method of choice. Synchrotron radiation too is a frequently used source, in particularly so if broad tunability is needed. In addition, several free electron laser systems are available worldwide, designed specifically for the IR spectral region. They have proven their usefulness as an efficient tool in the spectroscopy of molecules and clusters. Generally speaking, infrared spectroscopy is a very important method in chemical and physical analytics. Numerous textbooks on the subject exist which treat the subject extensively (often together with R AMAN spectroscopy). Thus, we just mention here that IR spectroscopy is highly selective for specific chemical bonds – based on a wealth of data available in the literature. Various molecular groups in larger (specifically organic) molecules oscillate with very characteristic frequencies: one finds e.g. the stretch vibrations of the CH group around ν¯ 3000 cm−1 , NH at 3400 cm−1 and OH at 3600 cm−1 . In contrast, bending vibrations are usually found at wavenumbers below 1000 cm−1 . From characteristic vibrational spectra of a system the expert may already ‘see’ its essential structural features – even without

298

5

Fig. 5.7 M ICHELSON interferometer setup as FTIR spectrometer

mirror (fixed position)

probe in the focus


compensator beam splitter s/2 moving mirror IR source collimator

detector

detailed evaluation. And for quantitative analysis a number of powerful computer programmes are available, that allow to model such spectra and fit them to the experimental observations. Since we have presented in Chaps. 3 and 4 already a number of vibration-rotation spectra for diatomic and polyatomic molecules, we confine the present discussion to two particularly important and interesting special methods used in IR spectroscopy.

5.3.2

F OURIER Transform Infrared Spectroscopy

Standard spectrometers have the big disadvantage that the spectra are measured sequentially, i.e. the spectrometer is tuned through a range of wavelengths and the detector registers the absorption of the radiation – effectively as a function of time. Since absorption lines are typically narrow, most of the time no absorption signal is recorded during a measurement. Thus, the average signal is usually very low and requires long integration times (or many iterative measurements) to obtain a reasonable signal to noise ratio. F OURIER transformation IR spectrometer (FTIR) overcome this problem by analyzing and recording the whole spectrum at once with the help of a M ICHELSON interferometer. Thus, F OURIER transform spectroscopy belongs today to the most efficient methods of optical spectroscopy (in various frequency regions). We have already mentioned some applications previously, e.g. in Sects. 3.4.1 and 5.2. To understand FTIR spectrometry we may resort to what has already been discussed in Chaps. 1 and 2 (and Appendix I in Vol. 1) about F OURIER transform, interference and coherence.

Experimental Setup Sketched in Fig. 5.7 is the schematic of a typical setup with a M ICHELSON interferometer (see also Fig. 6.4 in Vol. 1). One uses a broad band light source (‘white light’) which after collimation to a parallel light beam hits a beam splitter (half transparent mirror plates) where it is divided into two essentially identical beams. Each of these beams travels on a different optical path. One of the mirrors is moveable over a distance s/2 and allows one to introduce an optical path difference s between

5.3


299

the two beams, corresponding to a temporal delay δ = s/c

(5.2)

(we assume here rays which are parallel to the optical axes). After the two beams have acquired this delay, their wave fronts are merged again through the beam splitter and are then focussed onto the probe of molecules to be studied. There, the spectrum of the probe is impressed onto the optical field. Finally, the thus modified white light is focussed onto the detector to interfere. By continuous variation of the spatial (respectively temporal) delay one obtains an interferogram which contains the whole spectral information about the probe.

Method: FT of a Spectrum To see this, we first note that the FTIR scheme corresponds essentially to the general setup of an interference experiment as discussed in Sect. 2.1.4. According to (2.24) the intensity registered at the detector consists of a constant background and an interference term (5.4) which depends on the optical delay time δ and the spectral intensity distribution I˘(ω) of the light source. The initial intensity I0 at the entrance slit of the interferometer may be written I0 = I˘(ω)dω. (5.3) √ It is split into two equal beams (a1 = a2 = 1/ 2) and with (2.29) the signal without probe, detected by the FTIR spectrometer, is I (δ) = Re 1 + eiωδ I˘(ω)dω. (5.4) When passing the probe the intensity distribution I˘(ω) in frequency space is mul˘ tiplied with the absorption profile of the probe. Thus, I˘(ω) becomes S(ω), which is nothing else but the absorption spectrum of the probe as it would be obtained by a dispersive monochromator when tuning its transmission frequency (for the same light source being absorbed by the same probe). Corresponding to (5.3) the detector ˘ now integrates over the spectrum S(ω): I (δ) = Re

˘ 1 + eiωδ S(ω)dω.

(5.5)

Thus, we obtain an experimental signal I (δ) which depends on the temporal delay δ. The first term of this integral gives a constant background that is practically identical to I0 (since from the whole spectrum of the source usually only a small fraction is absorbed). However, the second term depends explicitly on δ: iωδ ˘ Re S(ω)e dω = I (δ) − I0 . (5.6)

300

5 S(ω;t1)

1.0

0.5

w

wt1→ ∞ -6

-4


-2 0 2 (ω- ω0)/ w

wt1=2 4

6

Fig. 5.8 Comparison of a genuine L ORENTZ line profile (full, red line) with the truncated F OURIER back-transformed shape according to (5.8). The back-transformation has been constraint alternatively to two different time intervals: wt1 = 2πt1 ν1/2 = 1 (grey) and = 2 (red dashed)

The integral is nothing else but 2π times the real part of the inverse F OURIER trans˘ form (I.3), Vol. 1 of the spectrum S(ω). F OURIER back transformation gives2 1 ˘ (5.7) S(ω) = I (δ)e−iωδ dδ. π The big advantage of the FTIR method is that no separation of the frequency components by a dispersive monochromator is necessary. All spectral components are ‘multiplexed’, i.e. they are, so to say, registered simultaneously. Very important too is the fact that one works with an extended light source and a collimated beam. It turns out that in this case the lateral coherence (see Sect. 2.1.7) of the source is much less critical than in a conventional, dispersive spectrometer.

Resolution of the FTIR Spectrometer Of course, trees don’t grow to the sky. The resolution is limited by the optical path difference s which can be accessed by the spectrometer. Figure 5.8 illustrates this for ˘ = (w/2)2 /((w/2)2 + (ω − ω0 )2 ) which we consider a L ORENTZ line profile, S(ω) to be recorded by an FTIR spectrometer. Let its spectral FWHM in terms angular frequency be w (ν1/2 = w/2π on the frequency scale). Its F OURIER transform is I (δ) ∝ exp(−wδ/2). The back transformation t1 wδ −iωδ ˘ e exp − dδ (5.8) S(ω; t1 ) ∝ 2 0 can only be carried out up to a maximum delay time t1 = s/c as given by the maximum optical path difference s available by the spectrometer. Figure 5.8 shows the observed profiles for two different values of wt1 = 2πt1 ν1/2 in comparison with the original L ORENTZ profile – as one would see it by full back transformation with 2 Strictly mathematical the following equation is not completely correct: since we back-transform only the real part of the inverse FT, negative frequencies arise . . . which have to be ignored.

5.3


301

t1 → ∞. The line profiles shown in Fig. 5.8 are indeed very similar to those observed experimentally, as we have seen e.g. in Fig. 3.21. For wt1 ≥ π the signal ˘ S(ω; t1 ) is already very similar to the original. Thus, with ν1/2 1/(2t1 ) the resolving power of this spectrometer is ν s λ s 2t1 ν = 2 ν = 2s ν¯ = 2 = N × z λ ν1/2 c λ

(5.9)

– apart from a somewhat arbitrary prefactor which characterizes the separability of neighbouring spectral lines. This corresponds again to the key formula (6.4) already derived in Vol. 1 for the resolving power of all spectrometers based on interference. Here we have identified N = 2 as the number of interfering beams and z = s/λ as the order of interference.

Some Technicalities Today, the necessary F OURIER back transformation, i.e. the integration (5.7) over the measured signal I (δ), can be carried out more or less ‘on-line’ with a PC, using “fast F OURIER transform algorithms” and special processors built into the electronics of the experiment. The side maxima, illustrated in Fig. 5.8, caused by this evaluation step, may still pose a serious problem. To overcome this influence of finite optical delay, the measured interferograms are often apodized, i.e. they are convoluted (smoothed) with a function p(x) which has no or only weak side maxima, e.g. a triangular function. Albeit this may reduce the resolution, the overall contrast in the backconverted spectra can be improved in this manner. The step width when scanning s has also to be chosen with care. The so called N YQUIST sampling theorem demands that per period at least two points have to be known in order to reproduce a sine function. Finally, one has to account for the finite divergence angle of the beams, since (5.2) strictly holds only for coaxial rays. Modern, commercially available FTIR spectrometers have resolving powers from 1000 up to 1 000 000. To separate e.g. two lines at ν¯ 3300 cm−1 with a difference in wavenumbers of ¯ν = 0.1 cm−1 the mirror has to be translated by at least 5 cm, according to (5.9). Typical optical delay lines in FTIR spectrometers range from 10 cm up to 2 m. F OURIER transform spectrometers are particularly popular for investigations in the IR spectral range, and we have already mentioned applications in the microwave range. The mirrors in FTIR spectrometers have to be moved over rather long distances with a precision of distance and alignment to only fractions of a wavelength. Clearly, this is easier to realize for infrared and a fortiori for microwave radiation than in the visible. More and more the technique is, however, also used in the visible, ultraviolet and even VUV spectral region, employing highly sophisticated methods for the stabilization and beam control. As broad band light source one may use with advantage synchrotron radiation. Also for the spectroscopy of astronomical objects, for solar studies as well as atmospherical investigations, F OURIER spectrometry is a method of choice since these are all broad band radiations sources whose fine structure one wants to investigate.

302

5


An interesting variety which finds particularly broad interest in biology, is the combination of FTIR with microscopic positional resolution – a look at Fig. 5.7 confirms that already the standard setup is almost ideally suited for such applications.

5.3.3

Infrared Action Spectroscopy

If only few particles are available for a spectroscopic investigation, as e.g. in studies of dynamical processes in molecular beams, or IR spectroscopy of molecular clusters in a supersonic jet, the kind of absorption spectroscopy described above soon reaches its limits due to small absorption coefficients, combined with finite sensitivity and dynamical range of the detectors. Absorption spectroscopy, in general, has the disadvantage that signals are recorded as potentially small reduction of a high background (the incoming light). Fortunately, a number of methods have been devised to outwit this problem by direct detection of the products generated in the absorption process. Such methods may be very sensitive and even allow the detection of single particles. We shall further explore this in the following sections. Here we present the so called infrared action spectroscopy (IAS). We illustrate this method by experiments with HO3 – a radical already known to us from Sect. 5.2. It plays an important role in the photochemistry of the earths atmosphere as an intermediate species of critical reactions in ozone chemistry of the type H + O3 → HOOO → OH(v ≤ 9) + O2 OH(v) + O2 → HOOO → OH(v − 1) + O2

(5.10)

O + HO2 → HOOO → OH + O2 . According to M URRAY et al. (2007) in an altitude of 10 km to 15 km about 50 % of the atmospheric OH radicals may be stored as HOOO. It is thus important to be able to identify this radical by clear spectroscopic signatures. D ERRO et al. (2007) have created HO3 by photolysis in a supersonic molecular beam. In a gas mixture of O2 and Ar, ‘seeded’ with nitric acid (HONO2 ), OH is generated by photo-dissociation using an ArF excimer laser (192 nm) which intersects the beam closely behind the jet nozzle. OH reacts according to (5.10) with O2 to produce HOOO. In the supersonic expansion zone it is cooled and thus stabilized. An energy level diagrams is sketched in Fig. 5.9, also indicating the detection scheme. About 15 mm downstream an OH vibration in HOOO is excited in the IR. Since the excitation energy is higher than the HO−OO bond energy, the system becomes unstable. The vibrational energy may be redistributed from the OH vibration into other vibrational degrees of freedom. Such processes are called intra molecular vibrational relaxation (IVR). Finally, the system dissociates into OH and O2 . One detects the OH formed in this process by laser-induced fluorescence (LIF), as also indicated in Fig. 5.9 (full black arrows up = excitation laser, dashed grey arrows down = fluorescence).

5.3


303 34.0

energy / 103cm-1

OH A2Σ+ 32.0 8.0 6.0

UV probe, fixed vibrational _ pre2 νOH dissociation

5.0 _ 2.0

νOH

IR pump (tuned)

0.0

O2 a1Δg

N= 9 8 v= 1 0 LIF 9 8 v= 0 0

OH X 2ΠΩ +O2 X 3Σg

-2.0 HOOO X 2A''

Fig. 5.9 IR pump, UV probe scheme for the detection of OH vibrations in the HOOO radical according to D ERRO et al. (2007). After IR excitation of the OH stretch vibration νOH in the v = 1 (alternatively v = 2) state HOOO decays into the end products OH(v − 1)X 2 ΠΩ + O2 X 3 Σg− , thereby loosing one ωOH quantum. One detects the IR absorption by laser induced fluorescence (LIF) in the dissociated OH after UV excitation. The energy scale refers to the dissociation channel OH + O2 (note the scale change between 8000 and 32000 cm−1 )

This double resonance experiment illustrates quite impressively the high standard of the methods in modern molecular spectroscopy. As IR source (2.8 µm or 1.4 µm for νOH and 2νOH , respectively) optical parametric oscillators are used, pumped by an injection pumped Nd:YAG laser. For exciting the product OH radicals a frequency doubled dye laser is used, again pumped by a Nd:YAG laser. Thus, detection is done state selectively. Even the rotational distribution of the OH may be determined in this way. Scattered radiation has to be screened very well and the detection efficiency of the fluorescence detectors has to be very high. Figure 5.10 shows two typical IR vibration-rotation spectra, detected by action upon absorption as just explained. The rotational structure of this asymmetric top rotor is only partially resolved. Due to the large moment of inertia it is much narrower as we have seen in Sect. 3.4.5 for diatomic molecules. Note that P , R and Q branches are observed – we have already discussed in Sect. 4.2.3 the appearance of the Q branch in polyatomic molecules. The simulation of the spectra by D ERRO et al. (2007) shown in Fig. 5.10 is based on high quality quantum mechanical computations (MRCI), using the rotational constants determined by S UMA et al. (2005). It turns out that inclusion of additional parameters such as centrifugal distortions, spin-rotation and magnetic dipole coupling does not improve the overall excellent agreement any further. Quite remarkable is the unstructured, broad background in the spectra shown in Fig. 5.10, corresponding to the red dashed contribution of the simulations. It is attributed to the cis-configuration of HOOO (H atom and final O atom are on the same side). Obviously it is only very short-lived, so that the rotational structure is washed out.

5

intensity

304

(a)

_ 2 νOH

(b)

_

-10


νOH

-5 _ 0 IR Δν / cm-1

5

10

Fig. 5.10 IR action spectra for the HOOO radical in (a) the νOH and (b) 2νOH spectral region according to D ERRO et al. (2007). The experimental detection (black lines) was done at fixed probe laser wavelength on the P1 (4) transition of the OH A 2 Σ + ← X 2 Π (v = 1, 0) and (1, 1) bands, respectively. The simulation (grey, nearly indistinguishable from the experiment) accounts for trans-HOOO (full red) and cis-HOOO (dashed red). The relative wavenumbers refer to the band origins of the trans-HOOO species at ν¯ OH = 3569.30 ± 0.05 and 2¯νOH = 6974.18 ± 0.05 cm−1 , respectively

Fig. 5.11 Normal mode vibrations of trans-HOOO according to D ERRO et al. (2008)

ν1 = νOH OH stretch

ν2

ν3

OO end stretch

HOO bend

ν4

ν5

ν6

OOO bend

central OO stretch

torsion

In the mean time, the IR spectra of HO3 and DO3 have been determined in even more detail (see e.g. D ERRO et al. 2008; M C C ARTHY et al. 2012). The normal mode structure shown in Fig. 5.11 may illustrate what kind of details modern IR spectroscopy is able to reveal. Section summary

• By infrared (absorption) spectroscopy vibration-rotation transitions (E1) of molecules are studied. They are a rich source for obtaining detailed information on molecular structure with characteristic absorption bands that may be attributed to specific molecular bonds.

5.4

Electronic Spectra

305

• FT spectroscopy is a method of choice in the IR spectral region. It has two advantages compared to normal dispersion spectrometers: – Light is collected and analyzed simultaneously from all relevant frequency regions. One avoids zero signal while scanning over spectral regions without structure, the average light intensity registered is significantly higher, and the measuring time is reduced (so called F ELLGETT advantage). – A much larger opening angle of the beam collimator is acceptable in comparison to a grating spectrometer (JACQUINOT advantage). • Usually, absorption spectroscopy works as a so called ‘flop out’ technique, i.e. a small signal has to be measured on a large background. Sophisticated methods have been devised to overcome this problem, such as “action spectroscopy” were the result of the IR absorption is probed by a second photon in the UV on essentially zero background (effectively a ‘flop in’ technique).

5.4

Electronic Spectra

5.4.1

F RANCK-CONDON Factors

Electronic transitions do not only change the electronic quantum numbers3 (γ ← γ ), but as a rule also the vibrational (v ← v ) and rotational quantum numbers (N ← N ). The transition probability is then proportional to the squared dipole transition matrix element (3.63) with the dipole operator (3.61). For simplicity, in the following we suppress the change of rotational quantum numbers. We also focus on diatomic molecules, with an electronic wave functions depending only on the nuclear distance R. The extension to polyatomic molecules with R → R is in principle straight forward but in detail not trivial – with R representing the full nuclear coordinate space. In B ORN -O PPENHEIMER approximation the nuclear motion (vibration) in the initial (γ ) and final state (γ ) occurs now in different potentials (typically in absorption, the initial state will be the ground state γ = 0). Indexing the vibrational wave functions R∗γ v N (R)/R also with γ is thus essential. When evaluating the dipole transition matrix element Dγ v ←γ v according to (3.63) one may first integrate over electronic coordinates r i to obtain an electronic · e of the dipole Dγ ←γ . The nuclear component ZR E1 transition matrix element D(R, r) does not contribute in this procedure, since due to ortransition operator thogonality of the electronic wave functions φγ∗ (r i ; R)Rφγ (r i ; R)d3 r i = 0 vanishes. This holds independent of R as long as γ = γ . Thus, we just have to evaluate Dγ ←γ (R) = e · φγ∗ (r i ; R)r i φγ (r i ; R)d3 r i (5.11) i 3 We

continue to use the spectroscopic notation according to H ERZBERG as introduced in Chap. 3: lower state double primed , upper state single .

306

5


which depends on the nuclear coordinate R – quite similar as discussed in Sect. 3.4.4 for the case of pure vibrational transitions. The transition probability (and the absorption cross section) between two vibrational states v ← v thus become proportional to 2 2 ∗ |Dγ v ←γ v | = Rγ v (R)Dγ ←γ (R)Rγ v (R)dR . (5.12) Dγ ←γ (R) varies only slowly with R, and the vibrational wave functions are localized to the vicinity of R0 . Since, however, these radial wave functions between different electronic states are not orthogonal, expansion around R0 as in (3.80) does not lead us any further. Rather, we average Dγ ←γ (R) over R and use to 1st order its mean value Dγ ←γ (R): 2 2 | Dγ v ←γ v |2 = Dγ ←γ (R) × R∗γ v (R)Rγ v (R)dR .

(5.13)

The second factor is the famous F RANCK -C ONDON (FC) factor 2 ∗ FC γ v ← γ v = Rγ v (R)Rγ v (R)dR short

(5.14)

2 = γ v γ v .

It is the square of the overlap integral between the vibrational wave functions in the potentials Vγ (R) and Vγ (R) and determines the relative intensities of vibrational lines within one electronic transition γ ← γ . The FC factors may be computed explicitly in the harmonic approximation since the radial wave functions Rγ v (R) = hv (x) are then simply the Hermitian functions (3.24) with x given by (3.22). One may then show rather easily that for absorption the sum rule

FC γ v ← γ v = 1

(5.15)

v

holds, independent of the initial vibrational state. A corresponding relation holds also for emission which leads to the important rule: The lifetime of electronically excited molecular states depends only on the electronic state, not on the initial vibrational state. This so called C ONDON approximation has been introduced by Edward U. C ONDON (1928). He formulated the FC principle for the first time on a strict quantum mechanical basis. Originally it had been introduced by James F RANCK (1926) using semiclassical arguments. As all approximations, this one too has its limitations (so called non-C ONDON transitions), e.g. if several, overlapping molecular potentials interact (see also Sect. 5.4.3). At room temperature (where ω0 kT ) only the v = 0 level is populated in the electronic ground state. For the following discussion of absorption it may thus suffice to evaluate | Dγ v ←00 |2 according to (5.13). The electronic part | Dγ ←γ (R)|2

5.4

Electronic Spectra

Fig. 5.12 For illustration of the F RANCK -C ONDON principle: Transitions (here absorption) occur preferentially if the averaged overlap between vibrational wave functions before and after the transition is large. Specifically, this is the case for vertical transitions (bold arrow)

307 V(R) v' = 16

γ'

v' =8 v' =4 v' = 0

γ'' = 0

R'0

v'' =0 R''0

R

leads to more or less strict selection rules due to angular momentum conservation, just as in atomic physics. We shall come back to these in the next subsection. However, which vibrational levels are populated in absorption or emission depends very specifically on the FC factors (5.14) of each molecule. The rules emerging, the so called F RANCK -C ONDON principle, are no strict selection rules. Rather the FC factors lead to some kind of propensity rules which are essentially a consequence of the shape and relative position of the two potential wells involved in the electronic transition. We distinguish the two cases most often encountered:

Case (1): R0 < R0 Figure 5.12 illustrates the position of vibrational levels and the shape of the radial wave functions. In the vibrational ground state these are essentially (in the case of harmonic oscillators exact) G AUSS functions. Thus, the overlap between Rγ 0 (R) in the lower electronic state, with Rγ 0 (R) in the excited state, is very small in the case shown – hence the FC factor (5.14) is very small. As v increases, Rγ v (R) has its maximum at about the classical turning point of the vibration, and changes there only slowly, while for larger R it oscillates rapidly. Thus the overlap between Rγ 0 (R) and Rγ v (R) will be maximal when the classical turning point in the excited state is above the minimum of the ground state. This is called a vertical transition – thought to occur from Vγ (R0 ) in the ground state potential vertically up to the potential curve Vγ (R). For still higher v the FC factors decrease again since by integration over R they average out to zero owing to the rapid oscillations in Rγ v (R). This quantum mechanical picture is consistent with a classical consideration: the electronic excitation occurs very fast ( fs) so that the nuclei do not move during that time. Thus, the nuclear distance R as well as the nuclear momentum M¯ R˙ 0 remain unchanged during such vertical transitions. The overall dynamics of an optical excitation process for a larger molecule is schematically illustrated in Fig. 5.13. After the absorption process (black, wide arrow) one expects relaxation in the excited state (several arrows down, IVR). Finally, reemission of the radiation (pink, wide arrow down) occurs. The excited state lifetime of molecules is – compared to the transition frequency as well as to the vibrational frequency – rather long (typically 10−9 s). During this time excitation may be

308

5 W(R)

STOKES shift dissociation ivr (in polyat. molec.)

emission

D'e

16

v' = 0 fluorescence

absorption excite

D'0 absorption

excitation

A

X


18 20 / 1000 cm-1

T e' D''e D''0

v'' = 0

R

Fig. 5.13 F RANCK -C ONDON principle for an optical excitation of a molecule from its ground state X into an excited state A. In polyatomic molecules or in a relaxing medium (liquid) the initial vibrational excitation in the A state is rapidly converted (dissipated) and the molecule emits from the vibrational ground state of A. This leads to a characteristic red shift in respect of the absorption profile (S TOKES shift), as indicated in the inset for the example of rhodamine 6G in ethanol (excitation at 480 nm, equivalent to 20.8 × 1000 cm−1 )

redistributed, e.g. in a gas or liquid by collisions with other molecules or atoms, in polyatomic molecules also by IVR as we have already seen in Sect. 5.3.3. In thermal equilibrium the lowest vibrational state v = 0 of the electronically excited system γ will then be populated preferentially. Thus, reemission (fluorescence) usually occurs from there into the ground state (γ = 0). Intensities will thus be determined (again in C ONDON approximation) by 2 2 ∗ |Dγ 0←γ 0 | ∝ Rγ v (R)Rγ 0 (R)dR . The relative magnitude of these F RANCK -C ONDON factors for emission is estimated by the same arguments as for absorption: The most intensive lines are found close to a vertical transition (γ v ← γ 0) with high v , as indicated in Fig. 5.13 (wide, pink arrow). A direct transition to v = 0 is not very probable. The emission is thus shifted towards longer wavelengths, as indicated in the inset for the well known dye molecule rhodamine 6G in ethanol as solvent – as used in tuneable dye lasers. This red shift is called S TOKES shift. Absorption and emission spectra are approximately mirror images of each other.

Case (2): R0 R0 Here the strongest transitions are v = 0 ↔ v = 0, in absorption and emission. The spectrum is narrow and asymmetric, the red shift is small.

5.4

Electronic Spectra

309

(b)

energy

(a)

(c)

V'(R)

2

3

V'(R)

4 2

1 v' =0

3

4 2

V''(R) 3

R

3←0 2←0 1←0

4

V''(R)

4 2 1 v'' =0

2 1 v'' =0 0←0 1←0 2←0 3←0 _ ν / cm-1

3

1 v' = 0

1 v' =0 V''(R)

V'(R)

3

4

R

4←0 5←0 6←0 _ ν / cm-1

R

_

ν / cm-1

Fig. 5.14 Absorption spectra (schematic) as a function of equilibrium distances R0 : (a) R0 does not change during excitation (case 2); (b) R0 increases (case 1); (c) the equilibrium distance R0 in the excited state is shifted so much that excitation occurs dominantly into the dissociative continuum (also case 1)

Summary Figure 5.14 summarizes and specializes the FC principle of vertical transitions for different relative equilibrium positions of ground and excited states. The different forms of an electronic band spectrum allow already to glean important information about the change in equilibrium distance R0 during electronic excitation.

5.4.2

Selection Rules for Electronic Transitions

The F RANCK -C ONDON factors identify the most probable vibrational transitions between different electronic states. In addition, owing to conservation of angular momenta and parity, more or less rigid selections rules also hold for molecules – similar to those in atomic physics. We summarize some key rules here briefly. As for electric dipole transitions in atoms, quite generally the following selection rules for the total angular momentum J must hold: J = 0, ±1 but J = 0 J = 0

(5.16)

and MJ = 0, ±1. In addition, for small molecules (RUSSEL -S AUNDERS coupling) the total electron spin S is conserved in E1 transition, i.e. S = 0. As already discussed for atoms, E1 transitions between singlet and triplet system (intercombination lines) are forbidden.

310

5


Table 5.1 Allowed electronic E1 transitions for diatomic molecules (C∞v symmetry) Selection rule

For coupling case

total angular momentum

J = 0, ±1 J = 0 J = 0

all

electron spin

S = 0

all

Σ = 0

only (a)

orbital angular momentum

Λ = 0, ±1 Σ + ↔ Σ +, Σ − ↔ Σ −, Σ + Σ −

(a) and (b)

electron total angular momentum

Ω = 0, ±1 J = 0 forbidden if Ω = 0 → Ω = 0

only (a)

total angular momentum without spin

K = 0, ±1 K = 0 forbidden for Σ–Σ

only (b)

Generally speaking, the situation is more complex than for atoms. For each specific case the selection rules depend on the symmetry of the system, and on the coupling scheme between the angular momenta (orbital angular momenta of electrons, electron spins, nuclear rotation, nuclear spin). The simplest case is again C∞v symmetry, encountered in diatomic and other linear molecules. We summarize the most important rules for H UND’s coupling cases (a) and (b) in Table 5.1. These rules will be illustrated for some examples later on. Much more complex is the situation for polyatomic, nonlinear molecules. Transition probabilities are determined here by the molecular symmetry. Clearly, the dipole transition matrix element (5.11) will depend on the direction of the electric vector in respect of the molecular axes as well as on the symmetries of initial and final state. With the help of the group theoretical instruments addressed in Sect. 4.3 one may derive selection rules in a relatively simple manner, if the symmetry group of initial and final state are the same.4 An electric dipole transition is allowed if the direct product of the irreducible representation of initial electronic state (ΓGS ), final electronic state (ΓES ) and of the dipole operator (ΓD ) contains the totally symmetric irreducible representation (ΓSym ), i.e. if ΓGS ⊗ ΓD ⊗ ΓES ⊃ ΓSym .

(5.17)

This rule follows directly from (5.11) by application of group theory. It is intuitively evident since only if the product under the integral (5.11) contains at least some totally symmetric parts, the integral over all space will not vanish. Depending on the symmetry group the totally symmetric irreducible representation is designated by A, A1 , A1g , Ag or A, as indicated in the character tables of the respective symmetry groups. The irreducible group of the dipole operator – in essence the vector (x, y, z) – can also be found in the character table. For example, in the Oh octahedral group ΓD = T1u (according to Table 4.5), and for the C2v group 4 Obviously, that is not necessarily the case, since a change in electronic structure may also change the symmetry of the system.

5.4

Electronic Spectra

Table 5.2 E1 allowed transitions in the point group C2v

311 C2v

A1

A1

+

A2

A2

B1

B2

+

+

+

+

+

+

B1

+

+

B2

+

+

+

ΓD = A1 + B1 + B2 (Table 4.3) – with the + sign to be understood in terms of group theory: the whole dipole operator can only be expressed as a combination of the three irreducible representations. Different polarizations may thus induce quite different transitions – as in atomic physics but with a lot more variety. For all symmetry groups relevant to molecular physics one finds the triple products tabulated in the references quoted in Sect. 4.3. As a still rather simple example, Table 5.2 shows for the C2v symmetry group (H2 O being a member of it) which transitions are dipole allowed by a + sign.

5.4.3

Radiationless Transitions

In polyatomic molecules even intercombination lines (S = 0) may be weakly allowed, in particular when atoms with large Z are involved and the electronic Hamil(el) contains strong spin-orbit coupling terms. As in atomic physics these tonian H increase with Z 4 . However, in larger molecules, even small spin-orbit admixtures can facilitate radiationless transitions between singlet and triplet states, so called intersystem crossing (ISC), with relatively high transition rates. ISC plays an important role in many organic molecules, particularly so in dye molecules. This is illustrated schematically in Fig. 5.15. The singlet and triplet states involved in such processes are often simply denoted as Sn and Tn , respectively. By absorption of a photon with appropriate wavelength a transition is induced between the thermally occupied, low lying S1

IVR T1 ICS

v' = 0

fluorescence

V(R)

absorption

Fig. 5.15 Schematic of an intersystem crossing (ISC, horizontal, grey arrow) after optical excitation, competing with fluorescence (dashed arrow). In between excitation and ISC as well as thereafter, vibrational energy may further relax (IVR). At the end a very slow optical decay process leads back into the ground state (so called phosphorescence)

S0

v'' = 0 representative nuclear coordinate

312

5


vibrational states in the S0 ground state and the F RANCK -C ONDON region of the Sn states above it (as a rule into higher lying vibrational states). In the example shown in Fig. 5.15 only the ground state S0 , the first excited singlet state S1 and the lowest triplet state T1 are considered (as in atomic physics the T1 state minimum is somewhat lower than that of the S1 state and a T0 state does not exist). After the optical excitation, IVR processes (also radiationless) may redistribute energy among different vibrational degrees of freedom5 in the S1 state (we have already encountered IVR in Sect. 5.3.3). In addition, ISC transitions may occur from S1 → T1 – competing with spontaneous reemission from S1 → S0 (fluorescence) – typically on a time scale of several ns, depending on the strength of the spin-orbit interaction and the magnitude of the FC factors between the vibrational levels of S1 and T1 states. In the T1 state the initially high vibrational energy may again relax by further IVR processes, so that finally the system is found in the lowest vibrational states of T1 (in Fig. 5.15 indicated by v = 0). Back reactions into the S1 state are then no longer possible. In this manner excitation energy may be stored for a long time in the lowest triplet state T1 . Since E1 intercombination transitions are to 1st order forbidden, the transition probability into the S0 ground state is typically very small. If the excited state finally decays by radiation one speaks about phosphorescence. This is a very weak, long lasting radiation as mean lifetimes of T1 triplet states may be in the ms to min range. For completeness we mention other types of radiationless transitions which may occur, in addition to ISC, among different electronic states with the same multiplicity, e.g. Sn (v ) → Sn (v ). Such transition are called internal conversion (IC). They may occur by “surface hopping” close to avoided crossings (see Sect. 8.1.6 in Vol. 1) – as far as energetic position and FC factors permit. We speak about nonadiabatic transitions. We shall address further important processes in the context of inelastic collisions in Sect. 7.4. A specific kind of such transitions occur at conical intersections (see Sect. 7.6).

5.4.4

Rotational Excitation in Electronic Transitions

Up to now we have considered only changes in vibrational population during an electronic transition. Of course, the rotational state too may change as a look at Fig. 5.1 suggests. According to (3.55)–(3.58) for a diatomic molecule, the total transition energy is then composed of W/ hc = Te + G + F, 5 One

has to keep in mind that during the IVR process in an isolated molecule vibrational energy does of course not get lost – as one might infer from the display of the energy terms in Fig. 5.15. Energy (except in optical emission) is just redistributed among the many other vibrational degrees of freedom within the molecule. A flow of energy back into the ‘representative nuclear coordinate’ is – for statistical reasons – the less probable the larger the molecule.

5.4

Electronic Spectra

313

where Te , G and F represent the differences of the electronic, vibrational and rotational term energies, respectively (more precisely: the wavenumbers in cm−1 ). Neglecting centrifugal distortion one obtains for the latter according to (3.57):6 F = Bv N N + 1 − Bv N N + 1 . Note that the rotational constants Bv and Bv belong to different electronic states and may thus differ substantially. As selection rule N = 0, ±1 holds. In contrast to pure rotational spectra which we have treated in Sect. 3.4.5, now N = 0 is allowed in principle, since angular momentum may also be transferred to the electronic system. Thus, for electronic transitions three branches exist with different contributions from rotational transitions, F = P (N), Q(N) and R(N ), respectively: P branch N = N − 1: P N = −2Bv N − Bv − Bv N N − 1 Q branch N = N : Q N = − Bv − Bv N N + 1 (5.18) R branch N = N + 1: R N = 2Bv N + 1 − Bv − Bv N + 1 N + 2 . As in the case of pure vibration-rotation spectra (see Sect. 3.4.5) we expect the highest transition wavenumbers for the R branch, the lowest for the P branch (at least for small rotational quantum numbers N ). Since the difference (Bv − Bv ) may also assume significant positive as well as negative values, we expect large deviations from equidistance (2Bv ) between the lines – which we have seen as a first approximation in contrast to pure rotation and vibration-rotation spectra. The absorption spectrum for an electronic transition will thus look, as a rule, quite complicated. One observes vibrational bands with a characteristic rotational structure on top. If one plots the rotational quantum numbers N (for a specific vibrational transition) as a function of the transition wavenumber ν¯ one obtains parabolas, so called F ORTRAT diagrams. ν¯ = o + bN + cN 2 .

(5.19)

Their shape reflects the expressions R(N ), Q(N ) and P (N ). They are sketched in Fig. 5.16. From the form of these parabolas one may in principle glean the differences (Bv − Bv ) and thus the change of equilibrium distance from electronic ground to excited state. Today one typically simulates the measured spectra by multi-parametric fit-functions – ideally they are rotationally resolved by high resolution spectroscopy. One needs to know from theory the relative line strengths for rotational transitions (for linear molecules these are the so called H ÖNL -L ONDON factors) which have to be weighted with the thermal population of rotational states prior to the transition (see examples in Sects. 3.3.3 and 5.5.4). 6 In

the literature one often finds J for the rotational quantum number, instead of N .

314

5 B' < B''

N''

Molecular Spectroscopy B' > B''

B' = B'' N''

N''

R

P

Q

P

P

_ ν

Q

Q

R

R

_ ν

_ ν

Fig. 5.16 F ORTRAT diagrams for various relative magnitudes of the rotational constants. Dashed vertical lines indicate the positions of rotational band heads in the spectra from electronic transitions

5.4.5

Classical Emission and Absorption Spectroscopy

After these general considerations it is obvious that emission and absorption spectra from electronic transitions will usually be very complex – even in simple molecules. Absorption spectra are typically expected in the UV and VUV spectral range, emission spectra may also be found in the VIS region. Particularly in emission one expects superposition of many vibration-rotation bands with different v and v , each of them with typical structures from P , Q and R branches as just discussed. Additional complications arise from the superposition of many different electronic transitions, as we recognize already by looking at typical potential energy diagrams for diatomic molecules (see e.g. Chap. 3, p. 212 ff.). In hindsight, one can only be full of admiration for past generations of spectroscopist who identified and parameterized the majority of simple molecules by their spectra – long before the invention of lasers. Molecular spectroscopy has been and still is an active field of research for more than a century, and most of the ground breaking discoveries have been made and understood long before 1960. Instrumental for the accumulation and interpretation of a huge wealth of data has been the life’s work of the physicist Gerhard H ERZBERG – honoured in 1971 with the N O BEL prize in chemistry. The intricate detective work of whole generations of spectroscopists may be illustrated somewhat by Fig. 5.17. There, the emission spectrum of the diatomic PN molecule is shown. It has been generated in a gas discharge lamp and was analyzed in classical manner by a highly resolving grating spectrometer and recorded on a photo plate. The original paper published in the then leading journal was still written in German language – before H ERZBERG had to leave Germany and found a new home in Canada. The interpretation reproduced from his N OBEL prize lecture (H ERZBERG 1971) in the lower part of the figure is more or less self explaining. One obtains from it a glimpse of the experimental skill, spectroscopic intuition and combinatoric insight which were essential in those early days prior to the laser, to acquire the wealth of knowledge about molecules which today we just take for guaranteed.

5.4

Electronic Spectra

315

P

Q R rotation-resolved 0 - 0 band

267.71nm (P I)

0 -2 1 -3 2 -4 3 -5 4 -6 5 -7 ∆υ = -2

238.12nm (As I)

0 -1 1 -2 2 -3 3 -4 4 -4 5 -5 ∆v = -1

v' = 0

0 -0 1 -1

∆v = 0

1 -0 2 -1 2 -0 3 -2 3 -1 4 -3 4 -2 5 -4 5 -3 ∆v = 1

v' = 1 v' = 2 v' = 3 v' = 4 v' = 5

∆v = 2

Fig. 5.17 Emission spectrum from the A 1 Π → X 1 Σ + transition in the PN molecule according to C URRY et al. (1933), recorded in a high resolution 3m UV grating monochromator (for calibration atomic lines in the P I and As I spectra have been used). Middle: photographic record of the spectrum (in first order diffraction) displaying several vibrational bands. Top: (partially) rotation-resolved 0–0 band (second order diffraction, enlarged section) indicating the position of the P , Q and R band head. Bottom: scheme for interpretation of the bands as presented in the N OBEL prize lecture of H ERZBERG (1971)

Such high resolution emission and absorption spectra have remained to be very useful tools in molecular spectroscopy even after the invention of the laser. Of course the experimental techniques of registration have been improved substantially. Spectral photometers, CCD cameras and image amplifiers have replaced in the second half of the past century the direct recording by photographic plates. The case of PN (Fig. 5.17) may be considered a still surprisingly clear spectrum without overlapping bands, albeit of a somewhat exotic molecule. Figure 5.18 illustrates how complex it can get – in spite of micro densitometers – to evaluate the emission spectrum for one of the most important homonuclear diatomic molecules such as O2 , when several electronic transitions overlap each other. The bands shown in Fig. 5.18 are sections (VIS) of the so called H ERZBERG bands, forbidden for optical dipole transitions, which are important for atmospheric physics. We recall (Fig. 3.44) that O2 is built from two triplet 3 P atoms, from which a large variety of molecular states may be formed. The spectra shown here are observed in the flowing afterglow of a gas discharge in O2 -He. They are due to transitions from the closely spaced excited states c 1 Σu− , A 3 u and A 3 Σu+ (each in its vibrational ground state) into higher vibrational states of the X 3 Σu− ground state – or in one case into the first excited state

316

5 +

-

c 1Σu→X 3Σg b

1Σ + → g

∞ 100 50 30

X 0-0

3Σ g

0- 12

0- 11

0- 10 -

A' 3Δu( Ω =2) →X 3Σg 0 -11

0- 9 0 -10

+

OH MEINEL band heads

Molecular Spectroscopy 0- 8 0- 9

0- 7

0- 6

0- 8

-

A 3Σu → X 3Σg 0-10 0-9 0-8

intensity

20 15 10 A' 3Δu( Ω =1)→a 1Δg 5 0-9

0-8

0-7

0- 6

0- 5

0- 4

0- 3

0 750

700

650

600 550 wavelength / nm

500

450

400

Fig. 5.18 Section of the H ERZBERG bands of O2 in the visible spectral range. This emission spectrum from the afterglow of an O2 -He discharge was recorded by S LANGER (1978) behind a high resolution Echelle spectrometer, still using a photographic film and image amplifiers. It has been evaluated with a micro densitometer (note the essentially logarithmic sensitivity of the film, which complicates the quantitative evaluation)

o 1 g which lies only 1 eV above the X state. The readers may follow the interpretations given in Fig. 5.18 without difficulty by comparing them with Fig. 3.44. Here again, one has to admire the intellectual achievements of a generation of spectroscopists, which have solved a multitude of such delicate puzzles (using the relevant theory), and finally have transformed these data into flocks of molecular potentials of the type shown in Sects. 3.6 and 3.7. Such classical spectrometers are still used today and compete often remarkably well with much more expensive methods from laser spectroscopy – in particular so for analytical purposes. Detection and recording is done today usually with optical multi channel analyzers (OMA) that are read out and controlled by a computer. In addition FTIR spectroscopy is used more and more also in the VIS and UV spectral range. However, without further selection techniques the interpretation of unknown spectra of this type in absorption or emission, is even today still a challenging and complicated undertaking. Section summary

• Spectra of electronic transitions in molecules typically have a rather complex band structure which is determined by F RANCK -C ONDON factors |γ v |γ v |2 . These describe the overlap between the vibrational states |γ v and |γ v in the electronic states characterized by the quantum numbers γ and γ , respectively. In a classical picture they correspond to “vertical” transitions between the two potentials involved.

5.5

Laser Spectroscopy

317

• FC factors essentially give propensity rules, describing relative strengths of transitions. They are complemented by rather strict selection rules from angular momenta and symmetry. For E1 transitions in diatomic molecules these are summarized in Table 5.1, while for polyatomic molecules they general rule (5.17) holds. • Additional complexity in electronic E1 transitions arises from rotational transitions. P , Q and R branches (for N = −1, 0, and 1, respectively) may occur in electronic transitions. Since the rotational constants Bv and Bv may differ substantially between initial and final electronic states, a characteristic dependence of the transition wavenumber ν¯ on N (in absorption) is observed – to 1st order a parabola. Plotting N (¯ν ), so called F ORTRAT diagrams for the three branches, explains the typical rotational band heads observed in electronic spectra. • Classical absorption and emission spectroscopy with high resolution grating spectrographs has provided a wealth of information in the past – even thought the spectra are usually highly complex. It continues to be a very useful tool even today, in particular for analytical purposes.

5.5

Laser Spectroscopy

At this point, laser spectroscopy comes to help. The high monochromaticity of laser light, ideally, makes it possible to induce transitions between exactly one initial and one final vibration-rotation level in the initial and final electronic states, respectively. For any subsequent process this reduces the large number of intermediate states which otherwise might participate in a second step absorption or emission process. Thus, the complexity of the spectra is dramatically reduced and precision is substantially improved. As a rule, the price to pay for these advantages is considerably higher technical effort. Since the invention of the laser in 1960, and in particular since flexible, tuneable laser systems for a wide spectral range are available, numerous methods of laser spectroscopy have been devised – more or less sophisticated and efficient. They differ in the methods of preparing the species to be investigated and in the detection schemes for the photo-absorption processes exploited. The amount of data gained in this way is enormous and we cannot attempt here any kind of summary (we mention, however, the N OBEL prize to S IEGBAHN 1981). We shall simply give a brief survey on the most important methods and present a few, particularly interesting examples.

5.5.1

Laser Induced Fluorescence

Laser induced fluorescence (LIF) is probably the most often used method in laser spectroscopy. One excites with a narrow band laser and detects the emitted fluorescence. The most straight forward procedure is to tune the exciting laser and to detect

318

5


all emitted fluorescence. Whenever the exciting laser hits a resonance, fluorescence increases correspondingly. The result corresponds basically the classic absorption spectroscopy. The two obvious advantages of this procedure are (i) a substantially enhanced resolution due to the monochromatic laser radiation, (ii) the detection efficiency can be orders of magnitude higher than in classical absorption spectroscopy since the fluorescence is now detected on a negligible background. LIF is, roughly speaking, a flop in method. In contrast, in usual absorption spectroscopy the transmitted signal of the intense, incident radiation is just slightly reduced due to the resonance one is looking for (flop out experiment). Thus, LIF may also be combined very conveniently with D OPPLER free methods as we have discussed them in Vol. 1 on several occasions. The high sensitivity of the method allows one e.g. to perform such measurements on cold, supersonic molecular beams where the number of thermally populated vibration-rotation levels in the initial electronic state is kept low. This leads to very clear, easy to interpret spectra. Alternatively one may also set the excitation wavelength to a fixed value and analyze the spectral distributions of the emitted fluorescence. In this manner one obtains emission spectra from well defined vibration-rotation levels in an electronically excited state. The combination of both schemes, finally, gives very comprehensive and unambiguous sets of data (partially redundant if desired), which allow for a precise and unique determination of the electronic and nuclear structures of the molecules studied, even in the case of rather complex species. As a special, particularly simple and clear example we discuss the laser induced fluorescence of isolated iodine molecules. I2 is a molecule that has been studied with extremely high spectroscopic precision. Owing to its high molecular mass of 253.8 (very low D OPPLER broadening) and its negligible isotope abundance (spectroscopically unique), it is often used in spectroscopy as an excellent secondary wavelengths standard. Many thousands of absorption and emission lines of I2 have been tabulated with high precision. As shown in Fig. 5.19(a), spin-orbit splitting is very large.7 The X state dissociates (R → ∞) into two ground state atoms I(2 P3/2 ), the excited B state into one I atom in the 2 P3/2 and one in the excited 2 P1/2 state. The example shown here may even be presented as a demonstration experiment in the classroom for undergraduate students.8 As illustrated in Fig. 5.19, one ob3 serves the fluorescence after excitation of the B0+ u (or Πg ) state from the ground + 1 + state X0g (or Σg ). The experiment, sketched in Fig. 5.19(b), uses an excitation wavelength of 514.5 nm (green line from an argon-ion laser) to excite between well defined vibration-rotation levels. Due to an accidental coincidence both, the X(v = 0, N = 13) → B(v = 43, N = 12) as well as the X(v = 0, N = 15) → 7 Thus

the coupling of angular momenta in I2 is most appropriately described as H UND’s case (c), see Sect. 3.6.4. The often used classification by singlet and triplet looses its validity due to the strong spin-orbit splitting. Otherwise the transitions studied here would all be forbidden intercombination lines.

8 As

done by Hartmut H OTOP (2008) who made these data available for us.

5.5

Laser Spectroscopy

W / eV

laser excitation 514.5nm

1

_

I2

(a)

2

319

+

B 0u ( 3 Π g )

ν / 1000 cm-1 2P

3/2 +

2P

1/2

12

v' = 43 N' = 12;16 2P

3/2 +

+

40

laser

cell I2

514.5 nm

molecules

monochromator _

30 2P

14

ν

3/2

(b)

20

+

X 0 g (1Σ g)

16

v'' = 40 30

10

20 0 0.2

18

10 0 0.3

det.

(c)

v'' = 4 0.4

R 0 = 0.266 nm

0.5

v'' = 0

R / nm 20

fluorescence intensity

Fig. 5.19 Laser induced fluorescence from I2 after excitation of the B(v = 43, N = 12; 16) ← X(v = 0, N = 13; 15) transition; (a) potential energy diagram (essentially according to DE J ONG et al. 1997); the black arrow illustrates monochromatic, vertical absorption at λ = 514.5 nm (argon-ion laser), the two pink arrows indicate emission; (b) schematic of the experimental setup with an iodine cell from which the B(v = 43) → X(v ≥ 0) fluorescence spectrum (c) was obtained according to H OTOP (2008)

B(v = 43, N = 16) are excited. In the reemission process (two pink arrows in Fig. 5.19(a)), only transitions B(v = 43) → X(v ≥ 0) occur, with N = 0, ±1. The small rotational broadening is not resolved in the present experiment. Other electronic states (grey dashed in Fig. 5.19(a)) do not play any role in this experiment. The fluorescence spectrum Fig. 5.19(c) shows very clearly all vibrational levels of the electronic ground state with 40 v ≥ 0 (indicated in the figure up to v = 4 by black horizontal lines, spaced by 214 cm−1 ). The F RANCK -C ONDON factors |B(v = 43)|X(v )|2 decrease rapidly with increasing v , corresponding to the positions of the potentials. The alternating intensity of the lines too, may easily be understood, considering the alternating, approximately even or odd symmetry of the vibrational wave functions R0v (R) in respect of R0 (see Fig. 3.5): in one case the maxima in the B state (v = 43) are just above the vibrational maxima of the X(v ) states, in the other case maxima coincide with zero transits so that certain regions of the wave functions average out. We finally mention that LIF provides also a very efficient detection method for molecules or radicals with well known spectra. We have seen this already in the case of OH in Sect. 5.3.3: one tunes the exciting laser in resonance with a known transition of the species to be detected and registers their presence or formation via fluorescence. The method is not only very sensitive and allows the detection of low

320

(a)

5 W

(b) e-

e-

e-

AB++C

ABC+

hνpu

A+BC

hνpu ABC

ABC*

ABC Qi

AB++C hνpr2 AB*+C

hνpr1

AB*+C ABC*

WI

e-

W

ABC+ hνpr


AB+C A+BC Qi

Fig. 5.20 Scheme for (a) resonantly enhanced two-photon ionization spectroscopy (RTPI) and (b) depopulation spectroscopy by ionization or fragment detection

concentrations, it is also state selective. Hence, it is used with great success e.g. for the analysis of reactive collision processes or photoinduced reactions.

5.5.2

REMPI for a ‘Simple’ Triatomic Molecule

Next to LIF, the resonantly enhanced multi-photon ionization (REMPI) is one of the most often used, particularly sensitive methods of modern molecular spectroscopy with lasers. It comes in several versions, such as resonant two-photon ionization (RTPI, also R2PI). We have already encountered a non-resonant variety in connection with D OPPLER-free methods in Vol. 1, Sect. 6.1.8. The basic idea of REMPI is quite similar to LIF. Detection is now, however, achieved by ionization of the molecule with the help of one or more additional laser sources. This leads to an even higher detection probability since ions may be collected with nearly 100 % efficiency. By using time of flight or quadrupole mass spectrometers one may in addition exploit mass selectivity. This can be very useful e.g. when studying isotope mixtures or photoinduced fragmentation processes. Of course such methods are limited to investigation of molecules or clusters in the gas phase or at surfaces in the vacuum, since one wants to transport the ions or electrons generated with little losses to the detector. The scheme of RTPI is sketched in Fig. 5.20(a) for a triatomic molecule ABC. With the tunable “pump” photon hνpu different vibration-rotation levels may be populated within the FC region of the potential for electronically excited ABC∗ . Starting from this level, a “probe” photon hνpr (usually at a fixed frequency) ionizes the system. Mass selection may be necessary, if the ABC+ ion fragments or if the initial target is composed of several constituents. One finally detects the ions with high efficiency by a secondary electron multiplier (see Appendix B.1). In this manner one obtains an electronic molecular spectrums which is fully equivalent to the classical absorptions spectrum (including the typical rotational bands with P , Q and R branches) – except with much higher sensitivity. Owing to the latter, supersonic molecular beams may be used to cool the initial population of rotational and

5.5

Laser Spectroscopy

321

vibrational states dramatically, so that the spectra become much more transparent than those obtained from a gas cell or even a liquid. Alternatively one may also detect the photo-electrons – in more sophisticated experiments with analysis of their kinetic energy or even in coincidence with the ion (see also Sect. 5.8). The detection scheme has to be even more specific if the excited state decays during the laser pulse duration by unimolecular dissociation ABC∗ → AB + C, or internal conversion (IC) or other processes. As indicated in Fig. 5.20(b) there are several options to overcome this problem and gain additional information about the decay mechanism. One may, e.g. try to detect the reaction products directly by ionizing them with a suitable probe photon hνpr2 (we have met an example in Sect. 5.3.3). Alternatively one may study the depletion of the initial state of the ABC molecule by ionization with a probe photon hνpr1 of sufficiently high energy. This ion signal is then detected as a function of the pump photon energy hνpu . At each resonance the signal decreases, hence one speaks about depletion spectroscopy. One finds a wealth of beautiful examples in the literature for all these varieties of REMPI. They range from diatomic molecules, with NO being some kind of ‘Drosophila’ of molecular RTPI studies, up to small biomolecules with many atoms, for which we shall present some examples in Sect. 5.5.4. At this point we want to discuss in some detail the spectra of Na3 and Li3 which are already quite complex. They will allow us to familiarize ourselves with a number of interesting aspects in molecular spectroscopy. These triatomic molecules (owing to their relatively weak bonding they often are also considered to be metal clusters) are generated in supersonic molecular beams. A rare gas at high pressure (usually argon at 2–20 bar) flows through a temperature resistant cartridge (oven) with the hot metal atoms (vapour pressure 10 to 100 mbar) and expands together with this metal vapour through a narrow nozzle into vacuum (so called ‘seeded’ beam). This expansion leads to adiabatic cooling and partial condensation. The thus generated clusters have internal vibrational, rotational and translational temperatures of only a few degrees K – however a high overall velocity. By collimation through a set of “skimmers” (usually of conical shape) a well defined cluster beam is formed. In a second, differentially pumped chamber it is then crossed with the laser beams. From the interaction volume one extracts the ions and/or electrons which are then detected. Pioneering experiments with Na3 have already been performed in 1986 with pulsed, tunable dye lasers (D ELACRÉTAZ et al. 1986; B ROYER et al. 1986, 1987). Figure 5.20 shows a small selection from these data, illustrating the rich kind of spectroscopic information that demands adequate interpretation. D ELACRÉTAZ et al. (1986) have tried this, using half integer quantum numbers which they attributed to so called pseudorotation: The basic arrangement of the three Na atoms is an equilateral triangle, which is, however, JAHN -T ELLER distorted. Later, quantum mechanical calculations have shown that the Na3 system is even more complex as initially assumed; the interpretation given in Fig. 5.21(b) was not fully confirmed. It turns out that the JT features are superposed by a PJTE effect (see Sect. 4.3.4).

322

5


(a) (c) 500

(b)

550

600

9/ 7/ 5/ 3/ 1/ 2 2 2 2 2

650

Qs

nm

9/ 7/ 5/ 3/ 1/ 2 2 2 2 2

3/ 1/ 2 2

Qb 9/ 7/ 5/ 3 / 1/ 2 2 2 2 2

Qa 605 16500

610 16400

615 16300

620 16200

625 nm

16100

16000 cm-1

Fig. 5.21 RTPI spectra of Na3 , recorded with two pulsed dye lasers. (a) Survey spectrum in the visible spectral region. (b) Blow up for one electronic transition, with interpretation of the bands by half integer quantum numbers for the pseudorotation (according to D ELACRÉTAZ et al. 1986). (c) The three normal modes of trimer molecules such as Na3 and Li3 in D3h geometry Table 5.3 Character table of the point group D3h

1 σˆ h D3h

2Cˆ 3

2Sˆ3

3Cˆ 2

σˆ v

A1

1 1

1

1

1

1

A2 E

1 1

1

1

−1

−1 Rz

2 2

−1

−1

0

0

A1

1 −1 1

−1

1

−1

A2 E

1 −1 1

−1

−1

1

z

2 −2 −1

1

0

0

Rx , Ry

x 2 + y 2 , z2 x, y

x 2 − y 2 , xy

xz, yz

Since then quite a few theoretical and experimental investigations have been devoted to these problems. Highly resolved measurements and state-of-the-art quantum chemical calculations for the significantly simpler Li3 (K RÄMER et al. 1999; K EIL et al. 2000) have led to a complete understanding of the ro-vibronic structure of such systems and their interesting spectroscopy (see also B ERSUKER 2001). Without entering into full depth of the problems, we give a brief introduction to the not completely trivial phenomena connected with these symmetry issues. The starting point is an equilateral triangle in D3h symmetry. The character Table 5.3 lists the irreducible representations of this point group. The electronic ground state of Li3 belongs (as that of Na3 ) to a two fold degenerate E representation. As discussed in Sect. 4.3.4, this degeneracy is not allowed in such a molecule and is avoided by a JT distortion, lowering the energy minimum. The potential surface thus forms two sheets which conically intersect. In the vicinity of this intersection the state can no longer simply be written as a (B ORN O PPENHEIMER) product of electronic and nuclear wave function. An exact descrip-

5.5

Laser Spectroscopy

323

equilateral triangle ction) (at conical intersection)

W(Qb ,Qa)

D3h

obtuse triangle

Qb C2v

acute triangle

©=

© = 60o

© = 240

obtuse triangle

0o C2v

acute triangle C2v

WJT

obtuse triangle

© = 120o

© =180o

C2v

C2v Qa

Fig. 5.22 Adiabatic potential surfaces with conical intersection for a trimer (schematic). When linear and quadratic vibronic JAHN -T ELLER coupling terms are included one obtains for the lower potential the somewhat deformed ‘Mexican hat’ with three minima, separated by corresponding walls. The upper part of the potential just forms a cylinder symmetric cone. WJ T designates the JT reduction in respect of the position of the conical intersection. The minima correspond to an obtuse, the maxima to an acute triangle – the saddle points occur if the molecule forms an equal-sided triangle

tion requires the inclusion of ro-vibronic coupling and the joint treatment of all degrees of freedom. As evident from Fig. 5.21(c) the D3h symmetry remains conserved only by the symmetric stretch vibration Qs . However, bending vibration Qb and asymmetric stretch vibration Qa break this symmetry. The combination Qb sin φ ± Qa cos φ describes a so called pseudorotation in C2v geometry, as illustrated in Fig. 5.22. As a function φ the molecule changes its isosceles triangular shape between obtuse at the minima and acute at the saddle points of the potential. The whole motion constitutes a coordinated motion of all three atoms which feigns a kind of rotation of the whole molecule around its centre (some sort of ‘hula-hoop’ motion which is, however, not a true rotation).9 To obtain a feeling for the orders of magnitude for the case of Li3 : according to the potential calculation (very close to the experimental fits) in the potential minimum of D3h symmetry (equilateral triangle) the atomic distances are 5.428a0 2 E ground state, 5.551a0 in the excited A 2 E . The JT distortion leads in the X ◦ ◦ to acute isosceles triangles with 71.6 and 77.3 , respectively, for the potential and A state, respectively (the length of the minima in C2v symmetry for the X equal sides are 5.225a0 and 5.551a0 , respectively). The potential minima there are WJT / hc = −501.8 cm−1 and −787.2 cm−1 , respectively, measured in respect of the conical intersection. Finally, the barrier height (saddle point) between the three min and A states, respectively. ima is 72 cm−1 and 156 cm−1 in the X 9 We note that the pseudorotation does not occur on a perfect circle. This would only be the case if the potential minima and the saddlepoints would both lie on the same circle.

324 160 W / cm-1

Fig. 5.23 Cut through the potential surface of Li3 along the pseudorotation angle φ. The vibrational states 0, 0) in the electronic X(0, 0, 0) in ground state and A(0, the excited state show a tunnelling splitting (representation in D3h symmetry). Shown as dashed lines are the squared wave functions of the pseudorotation states (after K EIL et al. 2000)

5


~ A state

80

A 1'' E''

0 80

~ X state

A '2 E'

0

0 60 120 180 pseudorotational angle

240 ©/ o

300

360

The barriers between the minima shown in Fig. 5.22 hinder of course the pseudorotation in the vibronic ground state. The low barrier height leads to tunnelling splitting quite similar as we have seen it in Sect. 4.2.5 for the inversion vibration in NH3 . For the specific example of Li3 Fig. 5.23 shows a cut along the pseudorotation coordinate φ through the potential hypersurface Fig. 5.22. It also shows the energetic positions of the thus resulting pseudorotation states with E and A2 vi 0, 0) as well as in the electronically exbronic symmetry in the ground state X(0, 0, 0) state. The quantum numbers in brackets represent (here and in the cited A(0, following) the vibrational quantum numbers (vs , vb , va ) – see Fig. 5.21(c). Let us now have a look at some of the results from the impressive studies by K RÄMER et al. (1999) and K EIL et al. (2000) for Li3 . Figure 5.24 shows the spec2 E ← X 2 E transition, recorded according to the scheme RTPI (a). tra for the A Here the mass selected isotopologue (7 Li)3 , briefly 21 Li3 , has been detected. The s , vb , va ) ← overview spectrum Fig. 5.24(a) showing several vibrational bands A(v 0, 0) has been recorded with a pulsed dye laser. The heavily magnified, rotaX(0, 0, 0) ← X(0, 0, 0) tionally resolved section in Fig. 5.24(b) (upper part) for the A(0, band was measured using a tunable, CW single mode dye laser for the excitation step, together with an Ar-ion laser (also CW) for the ionization step. By using a quadrupole mass filter, different isotopologues were separated. The modelling of the RTPI spectra has been achieved with the help of reliable quantum chemical computations (lower part of Fig. 5.24(b)). The excellent agreement with the experimental data documents convincingly, that today – by a combination of modern laser spectroscopic methods, advanced laser systems, and state-of-the-art quantum chemistry – such a system is fully under control. In such a demanding evaluation one has, of course, to include that the system may also rotate as a whole, and that normal rotation and pseudorotation have to be combined to ro-vibronic states in D3h symmetry. Helpful in this context is the fact that the system is to 1st order a symmetric top which can be described by the quantum numbers N and Kc . Nevertheless, a unique identification of the lines is anything else but trivial. K EIL et al. (2000) thus have used an additional trick, a further step of spectroscopic sophistication: the optical-optical double resonance

5.5

Laser Spectroscopy

325

(7Li)3

(0,0,0)

(a)

(0,1,0) (1,0,0) (0,2,0) (0,0,1) 14600

14800

experimental spectrum

(1,1,0) (2,0,0) (0,3,0) (0,2,0) (1,2,0) (0,0,2) (0,4,0) (0,0,3) 15000

wavenumber / cm-1

Q branch

P branch

R branch

(b)

simulated spectrum 14568

14572

14576

14580

14584 cm-1

2 E ← X 2 E transition in Li3 . (a) Overview spectrum acFig. 5.24 RTPI spectrum of the A s , vb , va ) ← X(0, 0, 0). cording to K RÄMER et al. (1999) showing several vibrational bands A(v 0, 0) ← X(0, 0, 0) transition with fully resolved rotational struc(b) Magnified section for the A(0, ture according to K EIL et al. (2000). One may imagine the P , Q and R branches from the experiment in the upper half of (b). However, comparison with ab initio theory in the lower half of (b) at a “rotational temperature” of T = (8 + N/2) K documents a fully quantitative understanding of the system

(OODR). The principle of OODR is sketched in Fig. 5.25(a). One uses here two tunable CW dye lasers with different photon energies. With the pump photon hνpu a specific initial rotational state is marked by exciting from this level a well defined but otherwise uninvolved ro-vibronic level of the upper electronic state (here in the 0, 0) vibrational state). When scanning the pump photon one obtains the RTPI A(1, spectrum shown in Fig. 5.25(b). The line selected which will be kept fixed during the OODR experiment is marked by a black arrow. The pump photon beam (having the selected photon energy hνpu ) is now modulated mechanically at a rate f1 , so that the population of the thus marked lower rotational level is also modulated, accordingly. The probe photon, hνpr , too is modulated, albeit with a different frequency f2 . Again one detects the absorption processes by ionization of the excited state levels, using a third photon at a fixed energy from an Ar-ion laser. When tuning the probe photon one may now distinguish the ‘simple’ RTPI spectrum from the OODR spectrum: the former is modulated with the frequency f2 , the latter with f1 + f2 . It is easy to separate the two signals by a special electronic device, a so called lock-in-amplifier. 0, 0) ← X(0, 0, 0) band of 21 Li3 – A section of the rotationally resolved A(0, recorded as a function of hνpr – is shown for both methods of detection in

hνpr with f 2 ~ Li3(X )

RTPI signal

R (2,0,E' )

R (3,2,E' )

(c) R (3,3,E'' )

14837.0

Q (6,6,E' )

14835.5

(b)


R (3,2,A1' )

hνpu with f 1 marked initial state

detecting laser (Ar-ion)

RTPI

P (3,2,E' )

~ Li3( A )

hνpu

Q (8,8,A'2 )

OODR signal detected at f 1+ f 2

P (3,1, A''1 )

(a)

5 Q (1,1,E'' ) Q(3,2,E' ) Q (1,1, A''2)

326

OODR hνpr / cm-1

14572.25 14572.30 14575.25 14575.30 14579.00 14579.05

Fig. 5.25 Optical-optical double resonance at 21 Li3 according to according to K EIL et al. (2000). (a) The experimental scheme. Pump- and probe photon, hνpu and hνpr are modulated with the 0, 0) ← X(0, 0, 0), frequency f1 and f2 respectively. (b) Low resolution RTPI spectrum for A(1, recorded by tuning hνpu ; the frequency marked hνpu is kept fixed for the OODR spectrum, shown 0, 0) ← X(0, 0, 0) transitions. in (c) (lower trace) for which hνpr has been tuned through the A(0, 0, 0) ← X(0, 0, 0) The RTPI also shown (c) (upper trace) is also recorded by tuning through A(0, but with hνpu off (see text). The change of the rotational quantum numbers is marked as usual by P , Q and R, the quantum numbers in round brackets refer to (N , Kc , Γ ) refer to the initial rotational quantum numbers and to the initial ro-vibronic symmetry Γ . All lines in the RTPI spectrum are split by HFS. Note the dramatic simplification of the spectrum by OODR

Fig. 5.25(c): as RTPI and as OODR spectrum. The RTPI spectrum still shows the full complexity of ro-vibronic transitions – even though the Li3 beam is well cooled: it still contains numerous excited rotational levels. In contrast the OODR spectrum is dramatically more simple. One only sees now the isolated P , Q and R lines (corresponding N = −1, 0 and 1, respectively). They all start from a single rotational level, here the N = 3, Kc = 2, Γ = E state. The representation Γ refers here to the ro-vibronic symmetry of the initial state (to be distinguished from the electronic as well as from the vibrational symmetry for which the same letters are used). Instead of using the somewhat problematic half integer quantum numbers for pseudorotation, K EIL et al. (2000) use solutions of the full Hamiltonian for the JT problem. We cannot go into further details of this rather complex problem (and refer to the detailed presentation in the papers of K RÄMER et al. 1999; K EIL et al. 2000). We mention, however, that the potentials have been determined in the demanding MRCI approximation, and the full JT Hamiltonian has been solved in hyperspherical coordinates, including the interactions from pseudorotation and rotation (C ORIOLIS coupling). Finally we mention hyperfine interaction which at that lever of precision becomes relevant. The splitting of the rotational lines in the RTPI probe spectrum Fig. 5.25(c) will not have escaped the attentive reader. 7 Li has a nuclear spin of 3/2,

5.5

Laser Spectroscopy

327 gas inlet

(a)

piezo

high voltage

Jet

CW ring dye laser autoscan

on AOM

plasma

IV

pulse generator pump laser

(b)

L SM

data acquisition III resonator resonances

SM PD

II piezo ramp

off

threshold detector

oscillograph

I 0

computer

piezo generator

66 t /ms

132

Fig. 5.26 Scheme of a cavity ring down (CRD) experiment after B IRZA et al. (2002). (a) Core of the experiment is the FABRY-P ÉROT resonator between two high reflecting, piezo controlled spherical mirrors (SM). It encloses the plasma (jet) ejected from a pulsed slit nozzle which provides the molecular ions or radicals to be investigated. (b) Pulse sequences to control the experiment (see text)

which leads – beyond F ERMI contact interaction (see Sect. 9.2.4, Vol. 1) – with one unpaired electron spin and three valence electrons in 21 Li3 to a clearly measurable HFS splitting. In the OODR spectrum it is of course not observed since the pump photon marks just one specific initial hyperfine level.

5.5.3

Cavity Ring Down Spectroscopy

Another very efficient method to detect photo-absorption by molecules is based on the so called cavity ring down (CRD). It is used – alternatively to the REMPI methods – for species which are only available in low concentration (radicals, molecular ions) or to detect particularly weak absorption (E1 forbidden transitions). The idea is quite simple: light passes many times through the absorbing medium. Most efficiently this is achieved in a FABRY-P ÉROT resonator of as high as possible quality Q, i.e. with high finesse F (see Sect. 1.1.2). According to (1.16) in an empty resonator (length L) the average inverse lifetime of a photon is 1/τe = 1/τr + 1/τd – it is finite due to reflection (τr ) and diffraction losses (τd ). An absorbing medium reduces this lifetime to τe , with 1/τe = 1/τr + 1/τd + 1/τa . Thus, one fills a resonator with a short light pulse and measures the exponential decay of this filling as a function of the incident photon energy ω – with (τe ) and without the medium inside the resonator (τe ). The effective absorption life time τa is then obtained from 1/τa = 1/τe − 1/τe . One finally obtains the absorption coefficient μ from (1.13). The larger τe , the smaller absorption in the medium may be detected. Figure 5.26 illustrates this by way of example with the setup of John M AIER and collaborators (B IRZA et al. 2002). Clearly, the realization of such an experiment is by no means trivial. The molecular ions or radicals to be studied are generated

328

5


by a pulsed supersonic molecular beam with a slit nozzle (3 cm × 200 µm) with a built in plasma discharge. In the supersonic expansion they cool down to a rotational temperature of 20 K to 40 K. The centrepiece of the setup is the FPI-resonator, consisting of highly reflecting spherical mirrors (SS) at a distance of 32 cm, with a reflectivity R = 99.995 %. This resonator encloses the plasmajet in the vacuum. The photon lifetime in the empty resonator is τe = 27 µs, implying that the light pulse interact 25 000 times with the target. Together with the slit nozzle one obtains a remarkable effective total absorption length of ca. 760 m. Absorptions spectra are recorded with a continuous, tunable dye laser. As photons may be filled into the resonator only if the photon frequency is in resonance with the resonator, one has to couple the latter with the resonator of the dye laser (so called passive mode locking). At a fixed wavelength, the laser beam is then fed into the FPI by switching the AOM into the ‘on’ status, as indicated in Fig. 5.26(a). Then the length of the measuring resonator is tuned with a piezo crystal, driven by a triangular ramp voltage from a piezo generator, marked as curve I in Fig. 5.26(b). The scanning range of the piezo crystal corresponds to two free spectral ranges of the FPI, so that for each back and forth movement of the piezo, the resonator becomes four times resonant (curve II). Only one of these resonances is used for a real measurement (marked red in curve II). The resonances are detected in the transmission signal at the photodiode (PD). At a certain signal height (threshold detector) the laser beam is coupled out by the AOM (status ‘off’), and one now follows the decay of the transmitted signal. This procedure is repeated several times. Only during each second decay cycle the plasma discharge is ignited to measure τe with the absorbing molecule. It is then compared with the resonator lifetime τe (without plasma), see Fig. 5.26(b) curve III and IV). After 15 measurements each, with and without plasma, the laser wavelength is changed (autoscan). M AIER and his group have studied a large number of molecular ions with this method. Such measurements are of special astrophysical relevance, in particular those of unsaturated hydrocarbons. On the basis of these laboratory experiments, a number of spectra observed from interstellar matter have been identified as being due such molecules. The analysis is not trivial and requires good spectroscopic knowledge of this type of molecules. Often isotope substitution turns out to be a helpful tool in such studies. As a typical example the C6 H+ 4 cation may serve. Its structure is shown in Fig. 5.27(a) while (b) and (c) show parts of absorption spectra obtained by CRD in the vicinity of 604 nm (with different resolution). The simulations (dark grey) show convincing agreement between experiment and theory. C6 H+ 4 is an asymmetric rotor as indicated in Fig. 5.27(b) by displaying Ka and N . It turns out to be a nearly stretched, symmetric rotor. Only for Ka = 1 one recognizes a slight asymmetry. For further details we refer to the original papers.

5.5.4

Spectroscopy of Small Free Biomolecules

Among the most impressive achievements of modern laser spectroscopy is the structural elucidation of isolated amino acids, peptides and similar molecules of biolog-

5.5

Laser Spectroscopy

329 H +C H

(a) C 6H 4

+

H C C C C C H R branch Q branch

(b)

Ka = 4 Ka = 3 Ka = 2 246 Ka = 1 1 357 Ka = 0 0 2 46

(c) C6H4+

P branch

×

×

×

16540

16545

×

16550

16545 _

ν/

16546

16547

16548

cm-1

Fig. 5.27 (a) Structure of the C6 H+ 4 ion which is studied with the cavity ring down (CRD) method 2 A . Experiment (red) and at 604 nm. (b, c) Absorption spectra from the origin band 2 A ← X simulation (dark grey). (b) According to B IRZA et al. (2002) with an experimental resolution of 0.15 cm−1 ; peaks marked with × do not originate from C6 H+ 4 ; the simulation agrees best for a temperature of 40 K. (c) Section of the spectrum (R branch) with improved resolution (0.01 cm−1 ) according to K HOROSHEV et al. (2004); the rotational temperature is now 20 K

ical relevance and a multitude of their compounds and complexes. The interest in such spectroscopy arises from the possibility to study the intrinsic properties of these ‘building blocks of life’ free of the complex biological environment. This allows – at least in principle – a reductionistic approach to the construction principles of biological entities. In such an approach it is e.g. possible to add one by one additional components of a true biological environment such as single water molecules, and study the spectroscopic consequences. The big challenges in this field of research are, for one, to bring these big molecules without destruction into the gas phase. The second difficulty is to identify the true structures out of complex variety of possibilities by a rigorous, theory based evaluation of equally complex spectra. Astonishing progress has been made in the past two decades, uncovering a wealth of information (summarized e.g. in R IZZO et al. 2009; DE V RIES and H OBZA 2007). Recent advances may illustrate the possibilities and problems ahead (see e.g. B ISWAL et al. 2012; WASSERMANN et al. 2012). Typically, we are talking about molecules with more than a dozen atoms, which in addition may form a rather flexible structure: they do not only occur as different isomers, but also as different conformers. Thus, the observed spectra are usually very complicated and consist of superpositions of several components which mass

330 Fig. 5.28 Scheme for double resonance experiments at biologically relevant molecules according to Z WIER (2001): (a) UV-UV double resonance hole burning spectroscopy, (b) resonant ion dip infrared spectroscopy (RIDIRS), (c) S1 state fluorescence-dip infrared spectroscopy (S1 FDIRS) – details see text

5

(a)

(b)

UV-UV hole burning

RIDIRS


(c) S1 FDIRS

Mol+ +ehνUV2 (fixed)

hνUV2 (fixed)

dark hνIR states (probe) IC

S1 hνUV1 (fixed)

hνUV1 (fixed)

hνUV1 (fixed)

hνLIF

S0 hνUV (probe)

hνIR (probe)

spectroscopy cannot attribute to specific molecular structures. Smart and often time consuming detective work is required to obtain a clear, unambiguous picture of the structure and dynamics of such systems. In addition to elaborate, high resolution studies with as much selectivity as possible, one also studies different isotopomers or molecules with different substitutes. In addition, one always needs strong support by advanced quantum chemical calculations. For preparation of such species one typically uses ‘seeded’ supersonic molecular beams in which the species to be studied are added to a carrier gas (neon or argon) and thus cooled down to a few K. More and more MALDI (Matrix assisted laser desorption ionization TANAKA 2002) and ESI (Electro spray ionization F ENN 2002) based methods are applied, allowing to study also such biomolecules which cannot easily be vaporized. Detection is done again by LIF and REMPI – usually supplemented by different double resonance techniques. Three varieties are schematically illustrated in Fig. 5.28. The basic idea of optical-optical double resonance spectroscopy has already been introduced in Sect. 5.5.2 (see Fig. 5.25): there, certain rotational levels in the electronic ground state were marked. Here one marks specific molecules by fixing one laser (grey arrow) to a specific excitation energy hνUV1 and thus detects only one particular geometry and/or isomer either by RTPI with a second photon hνUV2 or by fluorescence hνLIF (black arrows). One then tunes a third laser, the probe photon (red arrow). Resonance occurs only for transitions within the selected molecular species, and the ion signal (or the fluorescence) is reduced whenever resonance conditions are met by the tunable laser (“dip” or “hole burning” spectroscopy). The transition from the electronic S0 ground state to the first excited state S1 is typically found in the UV range. The scheme according to Fig. 5.28(a) employs UV-UV double resonance spectroscopy for selection and detection, with photons hνUV1 and hνUV2 (in suitable cases one may even use the same frequency). The key of the method is to fix the pump photon hνUV1 (grey arrow) onto a well defined absorption resonance of the S1 ← S0 transition for one specific isomer or conformer.

5.5

Laser Spectroscopy

331

GG1 UV- UV GG2 UV- UV GG UV- RTPI

33000

33200

33400 33600 h νUV / cm-1

Fig. 5.29 UV-UV RTPI hole burning spectra for the base pair guanine-guanine according to A BO -R IZIQ et al. (2005). The base pair is generated in a supersonic molecular beam and detected by TOF mass spectroscopy. The simple RTPI spectrum obtained by just two UV photons is marked as GG. The two other spectra, GG1 and GG2, show hole burning spectra for the two dominant geometrical arrangement of the guanine molecules

The (usually) pulsed probe laser is timed to interact prior to the detection process with the molecule. Pioneering work on the spectroscopy of base pairs, the ‘letters’ of DNA, has been performed by DE V RIES and collaborators. As an example in Fig. 5.29 UVUV RTPI spectra are shown for a base pairs consisting of two guanine molecules. These may assume (among others) two different, nearly planar structures – as one concludes from the spectra and accompanying quantum chemical calculations. The comparison of the standard RTPI spectrum in Fig. 5.29 (GG) with the UV-UV hole burning spectra (GG1 and GG2), determined at different photon energies hνUV1 of the marking laser, show that both species have different, characteristic UV absorption spectra. Even if the two double resonance spectra are somewhat more noisy, it is rather plausible that both added together essentially reproduce the RTPI spectrum (red, GG). Complementary to UV-UV double resonance spectroscopy one may also apply RIDIRS as sketched in Fig. 5.28(b) to this system. Detection is identical to UVUV hole burning. However, tuning now an IR probe photon hνIR records the IR absorption spectrum in the S0 ground state – instead of the electronic absorption bands for the S1 ← S0 transition in the previous case. Results are summarized in Fig. 5.30. The recorded spectra of the two species GG1 and GG2 (the same species as those in Fig. 5.29) show very characteristic differences. As indicated by arrows in the sketches of the two structures, theory is able to attribute the bands arising to the different vibrational modes of the two base pairs (arrows, red, black, dark and light grey), and thus to identify the structure of these highly complicated, anharmonic and flexible entities. We cannot go any deeper into the details.

332

5


GG2

GG1

3400

3600 3800 h ν IR / cm-1

Fig. 5.30 Resonant ion dip infrared spectroscopy for two modifications of the guanine-guanine base pair according to A BO -R IZIQ et al. (2005). With the help of quantum chemical computations characteristic vibrations have been identified (sticks spectrum), to which the two IR absorption spectra can be assigned. The insets show the respective geometry of the two guanine-guanine base pairs. Full red circles correspond to O atoms, open, black circles indicate N atoms

Another interesting modification of double resonance spectroscopy, sketched in Fig. 5.28(c), deserves to be mentioned: S1 state fluorescence dip infrared spectroscopy (S1 FDIRS). It has been devised to study the vibrational modes of the excited S1 state. One detects in this case the excited species by LIF. The fluorescence signal changes when the IR probe laser (hνIR ) is tuned and hits a vibrational transition. Typically, the corresponding LIF signal is lost by IC to an unobserved “dark state”. Such spectroscopic determination of isolated biomolecules, their clusters and compounds as well as investigations of the photo induced dynamics can only be successful by close collaboration between sophisticated spectroscopic methods and quantum chemical calculations. A number of specialized research teams are active in this attractive field of modern research in molecular physics and chemistry. Often they combine their respective strengths. A particularly remarkable example is a study of tryptamine by B ÖHM et al. (2009). Tryptamine [2-(1H-Indol-3-yl)-ethylamine] with the sum formula C10 H12 N2 is an important by-product of metabolism and closely related to tryptophane, one of the three aromatic amino acids which are responsible for the fluorescence properties of proteins. Tryptamine exists in 9 low lying conformers in the S0 ground state. The spectroscopy of the excited S1 state is complicated in addition by conical intersections. B ÖHM et al. (2009) attack the problem with a broad range of experimental and theoretical methods – ranging from application of double resonance spectroscopy as just explained, through laser induced dispersed fluorescence (DF), to rotationally resolved absorption spectroscopy and their interpretation with the help of genetic algorithms – a very interesting method to handle the large number of data obtained from such spectra (see e.g. the review of M EERTS and

5.5

Laser Spectroscopy

(a)

333

0,0+332cm-1

(b) (c)

0,0+403cm-1

(d) - 20000 - 10000

0

experiment

(e) 0,0+412cm-1

simulation

(f)

experiment

(g)

simulation

(h)

experiment

simulation

- 40000 10000 relative frequency / MHz

0

40000

Fig. 5.31 Rotationally resolved absorption spectra of the S1 ← S0 bands in tryptamine above the origin (0, 0) according to B ÖHM et al. (2009). The experimental spectra (a), (c), and (e) are compared with simulations, (b), (d), and (f), respectively. The latter is combined of two species, (g) and (h), and illustrates the complexity of the problem

S CHMITT 2006). Figure 5.31 gives an impressive example for the potential of such modern methods. It accounts convincingly for consistency between experiment and theoretical model for such a large and complex molecule (23 atoms!). Here too, the interested reader has to be referred to the original sources for more detail.

5.5.5

Other Important Methods

A variety of other techniques are also employed in laser spectroscopy. Different degrees of sophistication are available for the study of electronic, photoinduced transitions in smaller or larger molecules. We just mention photo-fragment spectroscopy with cations and anions in electromagnetic traps (P ENNING trap, PAUL trap) as well as matrix isolation spectroscopy. Both methods are particularly well suited for rare or difficult to generate species. In matrix isolation spectroscopy one prepares the molecule, radical or cluster in a suitable fashion and deposits it into a rare gas matrix at low temperatures (typically a Ne matrix at 6 K). One may e.g. select a specific molecular or cluster size by a mass spectrometer and enrich the species in the matrix by deposition over a sufficiently long time (the thus deposited ions are usually neutralized in the matrix). One may use such a matrix to record more or less standard absorption spectra. The method thus features a kind of solid state spectroscopy with a variety of potential complications. Since, however, the interaction of the species studied with the rare gas matrix is usually very weak, one may obtain in this manner a good first order impression of unknown absorption spectra for interesting, but rare classes of molecules. Usually these spectra are only slightly shifted as compared to the free molecule. For a first ‘screening’ of new types of molecules this method has proven to be very valuable (s.z.B J OCHNOWITZ and M AIER 2008). Based on such knowledge one may obtain more detail by LIF or RTPI.

334

5


Section summary

• Laser spectroscopy is a key tool for unravelling molecular structure. A broad variety of methods is used today. They all exploit the selectivity, high resolution and sensitivity of modern, (mostly tunable) laser sources. • The most simple scheme is laser induced fluorescence as described in Sect. 5.5.1 – where the whole (integrated) fluorescence spectrum is observed to monitor absorptions spectra as a function of the exciting laser wavelength. In a more sophisticated version one also analyzes the wavelengths emitted (DF). • Multi-photon processes, in particular REMPI or more specific RTPI, are among the most frequently used and highly selective methods to analyze the electronic bands of small and larger molecules, as illustrated in Sect. 5.5.2 by high resolution spectroscopy for the interesting case of triatomic metal clusters. They are subject to JTE, the detailed understanding of which requires very sophisticated double resonance methods and a high level of quantum chemical theory. • CRD, discussed in Sect. 5.5.3, exploits the damping of laser radiation in a high Q resonator due to absorption within the molecule studied. The method is highly sensitive and suitable for very low concentrations of target molecules. • Many variations and sophistications of these and related methods have been devised for various spectroscopic applications. Perhaps most remarkable is today’s capacity in resolving and understanding the structure of band spectra for rather complex molecules, such as amino acids and even small peptides, the building blocks of life (Sect. 5.5.4).

5.6

R AMAN Spectroscopy

5.6.1

Introduction

In 1928 R AMAN (Nobel prize 1930) and simultaneously L ANDSBERG and M AN DELSTAM found when scattering non-resonant light by molecules several additional lines. Soon this finding developed into a very powerful type of molecular spectroscopy, which – complementary to IR spectroscopy treated in Sect. 5.3 – offers a direct access to determine vibrational frequencies of molecules, and also for solid state materials. Today, R AMAN spectroscopy belongs to the most important and frequently used spectroscopic tools, specifically for analytic purposes. Already in 1930 R AMAN obtained the N OBEL prize for his discovery. One observes three types of lines, the origin of which is explained by a so called JABLONSKY diagram in Fig. 5.32. One line with ω = ω, unshifted in respect of the incident radiation ω, is called R AYLEIGH line (elastic scattering), one set of lines is observed at lower energies ω = (ω − ωba ) < ω, the so called S TOKES lines, and finally one more set of (weaker) lines is observed at higher energies ω = (ω + ωba ) > ω, the so called anti-S TOKES lines. The shift of the lines

5.6

R AMAN Spectroscopy

335

STOKES k RAMAN k k

RAYLEIGH scattering

ν'= ν- νba ν ν ν

ν

fluorescence

Anti STOKES

ν' = ν + νba ν0 ν' ≤ ν = ν- νab ν0

IR absorption

νba = νb – νa = – νab

k k

}

b a

Fig. 5.32 Level scheme explaining the origin of a R AMAN spectrum (JABLONSKY diagram). Grey arrows correspond to the incident radiation frequency ν0 , red arrows illustrate the frequencies of the scattered light: ν0 − νj i S TOKES, ν0 R AYLEIGH and ν0 + νj i anti-S TOKES line. The horizontal red, dashed lines are occasionally considered to represent so called virtual intermediate states. However, real are only the black energy levels, denoted by i, j and k. For comparison, infrared absorption (black arrow up) and standard fluorescence (red arrow down) are also indicated

is independent of the wavelength of the incident light and is exclusively determined by suitable transition energies ωba = ωb − ωa = −ωab between so called R A MAN active (see below) vibrational and/or rotational levels of the molecule studied. R AMAN processes are, as evident from Fig. 5.32, two-photon processes which may start from any populated initial level of the molecule studied. Clearly, the scattering rate depends on the population density of the respective initial state |i. Thus, for S TOKES lines we expect significantly higher scattering rates than for anti-S TOKES lines, since in the former case the initial state |i = |a has lower energy than in the latter case where |i = |b, with |a being more densely populated than |b according to the thermal B OLTZMANN distribution. We have again to consider the rotational population and the change of the rotational quantum number N – just as in the case of IR absorption spectroscopy. The situation is, however, somewhat more complex. We focus on the most transparent case of a rigid, linear rotor. Since the R AMAN process is a two-photon transition during which an angular momentum 2 may be transferred, we have now the selection rule N = N − N = 0, ±2,

(5.20)

again with indicating the lower, and the upper levels. In analogy to P , Q and R branches in vibration-rotation spectra discussed in Sect. 3.4.5 (as well as in elec-

336

5

Fig. 5.33 Right: how the S, Q and O branches of rotational bands arise in a R AMAN spectrum (S TOKES region). Left: for comparison the origin of P and R branches in IR absorption spectroscopy of polar molecules


RAMAN scattering (STOKES)

IR absorption N' = 2 1 0

v' = 1

ΔN = 2 ΔN = 0 ΔN = - 2 N'' = 2 1 0 Q branch R branch O branch S branch 4B

ΔN = - 1 ΔN = +1 v'' = 0 P branch 2B

νba

ν- νba

tronic transition bands, Sect. 5.4.4) there are three branches: O N

− 2,

N

(5.21)

and S branch for

Q = N

= N

N

and

= N

+ 2, respectively.

The origin of these three branches is illustrated in Fig. 5.33 for the S TOKES lines. The differences and analogies to infrared absorption spectroscopy and electronic absorption and emission bands are evident: while these imply absorption or emission of a single photon, R AMAN spectroscopy is based on the absorption and emission of one photon in each step. We can now discuss a typical R AMAN spectrum for a diatomic molecule, very schematically sketched in Fig. 5.34. The characteristic vibration-rotation bands with their branches can be observed red shifted (S TOKES) or blue shifted (anti-S TOKES) in respect of the incident frequency. They may be compared to the pure IR absorption spectra discussed in Sect. 3.4.5. For the O branch the rotational quantum number in the final state is reduced by 2 with regard to the initial state. By considering the scheme shown in Fig. 5.32, one easily realizes that the S TOKES shift (in RAYLEIGH STOKES STOKES 2. harm. Q S

anti STOKES 4B

~ ~

O S

S O

ν - 2 νba

ν - νba

ν

Q

S

ν + νba νR

Fig. 5.34 R AMAN spectra of a diatomic molecule (schematic): S TOKES and anti-S TOKES vibrational R AMAN, each with S, Q and O branch; the pure rotational R AMAN spectrum left and right of the (elastic) R AYLEIGH line has only an S branch; on the very left (weaker) a S TOKES spectrum of the second harmonics (2νba ) is indicated

5.6

R AMAN Spectroscopy

337

respect of the incident line) is smaller for the O branch than for the Q and S branch. This also holds for the anti-S TOKES spectrum (N in both cases refers to the lower level); however, the ordering of the branches is just opposite. Note that (for diatomic molecules) pure rotational R AMAN spectra (Fig. 5.34 middle) have only S branches: for the S TOKES lines (ω < ω) one starts with a lower lying rotational level N and ends up in N = N + 2, corresponding to an S transition. The anti-S TOKES lines (ω > ω) arise when the process starts with a higher lying rotational level N and ends on lower level N = N − 2. According to (5.21) this transition corresponds again to the S branch.

5.6.2

Classical Interpretation

The popular classical interpretation considers the molecule time dependent in the oscillatory electric field (ω = 2πν) of the incident light E(t) = E 0 cos(ωt). The field polarizes the molecule and hence an oscillating electric dipole is induced that oscillates with the frequency of the light ν = ω/2π : D el = αE E = αE E 0 cos(ωt).

(5.22)

In the general case the polarizability αE is a tensor of rank 2. It is treated here, however, as a scalar (isotropic polarizability) and one expands αE around the equilibrium distance of the molecule R0 : dαE αE (R) = αE (R0 ) + (R − R0 ) + · · · . (5.23) dR R0 Without the field the molecule oscillates harmonic with angular eigenfrequency ωba = 2πνba around R0 : R(t) − R0 = R1 cos(ωba t).

(5.24)

Inserting this and (5.23) into (5.22) leads to:

dαE R cos(ω t) E0 cos(ωt) (5.25) Del (t) = αE E(t) = αE (R0 ) + 1 ba dR R0 dαE = αE (R0 )E0 cos(ωt) + R1 E0 cos (ω + ωba )t + cos (ω − ωba )t . dR R0 Thus, the field induced dipole oscillates at three frequencies, i.e. the wave with incident frequency ω is modulated by two side bands. This obviously corresponds just to the three types of scattered lines which we have discussed above: • an unshifted (elastic) R AYLEIGH line: Del (t) ∝ cos(ωt) • the S TOKES line: Del (t) ∝ cos[(ω − ωba )t] and • the anti-S TOKES line: Del (t) ∝ cos[(ω + ωba )t]

338

5


According to (5.25) a molecule is R AMAN active if and only if its polarizability changes with nuclear distance, i.e. if dαE /dR|R0 = 0 holds. This may well happen also in the case of homonuclear molecules. A permanent dipole moment is not required. R AMAN spectra may thus be recorded also for H2 , N2 and O2 – in contrast to pure IR absorption spectroscopy.

5.6.3

Quantum Mechanical Theory

The first approaches towards a quantum mechanical treatment of the R AMAN effect go back to Maria G ÖPPERT-M AYER (1931), then a student of Max B ORN. One of the most commonly quoted references is A LBRECHT (1961), who refers to B EHRINGER and B RANDMÜLLER (1956), who start with a more general formula without reference. Fortunately, the differential R AMAN cross section may readily be derived in 2nd order perturbation theory: two photons are involved, the incident and the scattered photon. We briefly sketch the key steps for such a calculation. It essentially follows that for multi-photon processes as outlined in Chap. 5, Vol. 1; explicitly we have expounded there two-photon excitation processes, and we expect an expression similar to (5.46), Vol. 1. However, in the present case of the (spontaneous) R AMAN effect, only the absorption of the incident photon ω is an induced process while the scattered photon ω is emitted spontaneously. The prefactors will thus be different. Since we want to understand spontaneous R AMAN scattering, we have to use now the quantized form of the interaction as presented in Sect. 2.3.6. Basically, we start with equations (2.140). But we can no longer restrict the treatment to quasi resonant excitation, so that the full set of coupled ODEs dck Nk (t) i |iNi ei[(Nk −Ni )ω+ωki ]t =− ci Ni kNk |U dt

(5.26)

i Ni

has to be solved, with |iNi being a state where the molecule under investigation is described by quantum numbers i, and Ni gives the number of photons in a mode of interest. The transition matrix elements are given by (2.135a)–(2.135d). They D = r · e and contain, quite familiar to us by now, the dipole transition operators D = r · e for the exciting and the emitted photon of frequency ω and ω , respectively. In principle we also have to account for different modes. But this ansatz is sufficient for the first step (1st order interaction with the incident quasi-monochromatic, well collimated photon beam of frequency ω). To obtain the 1st order solution, we assume that on the right hand side of (5.26) only the amplitudes ci Ni for one molecular state |i are finite, all others vanish. Both processes, absorption and induced emission from this state into any other atomic state |k have to be considered. Conveniently the matrix elements vanish unless Nk − Ni = ∓1, respectively. Thus, making explicit use of (2.135a)–(2.135d) we obtain eC † c˙k Nk = Nk + 1 Dki ci Nk +1 ei(ωki −ω)t − Nk Dki ci Nk −1 ei(ωki +ω)t ,

5.6

R AMAN Spectroscopy

339

with C = ω/2L3 ε0 being the field normalizing constant in (2.132). Here Nk 1 characterizes the intense laser field, so that Nk Nk ± 1 = N . We thus may drop the index Nk completely and assume on the right hand side ci = ci Nk ±1 = 1 in 0th order. After integration the excitation amplitudes in 1st order thus become ck (t) =

i(ωki +ω)t − 1 ei(ωki −ω)t − 1 eC √ † e . N Dki Dki − i(ωki − ω) i(ωki + ω)

(5.27)

In the next step we address spontaneous emission from the thus prepared intermediate state. We have to solve again the ODE (5.26), now for the final state amplitudes cf Nf (t) in 2nd order (replacing |kNk → |f Nf and |iNi → |kNk ) by inserting the 1st order amplitude (5.27) for all intermediate molecular states |k and all radiation modes. We may treat each radiation mode separately since each emission process changes the photon number in one mode only. Almost all of the many intermediate radiation modes are empty (except for the laser mode which we ignore in this step) so that for all initial modes Nk = 0 and for all final modes Nf = 1 (marked by dashed quantities if necessary). We then have to solve dcf (t) i |k0ei[ω +ωf k ]t =− f 1|U dt k

×

i(ωki +ω)t − 1 ei(ωki −ω)t − 1 eC √ † e N Dki Dka − i(ωki − ω) i(ωki + ω)

(5.28)

† |k0 = −ieC with f 1|U Df k according to (2.135a)–(2.135d). To integrate this we apply the usual procedure as detailed in Sect. 4.3.5, Vol. 1. For long times, only stationary exponential terms contribute to the integral. In the present case such situation arises only for the first exponential term in the square bracket of (5.28), which in combined form is

∝

† D Dki fk

ωki − ω

ei(ωki −ω+ω +ωf k )t .

(5.29)

When integrated it contributes for t → ∞ if and only if ω + ωf k + ωki − ω = !

ω − ω + ωf i = 0, i.e. for ω = (ω − ωf i ).

(5.30)

This corresponds indeed to the R AMAN condition, with the energy difference Wf − Wi = ωf i between the final (f ) and initial (i) state of the molecule.10 All other terms in (5.28) do not contribute to the integral at reasonable experimental conditions. However, at this point we have to extend our considerations slightly. emphasize that (5.30) holds for S TOKES and anti-S TOKES lines, since i and f in ωf i refer here to the initial and final molecular states, respectively. In contrast, in the JABLONSKY diagram Fig. 5.32 the indices b and a refer to upper and lower R AMAN levels, respectively.

10 We

340

5


ћω'

Fig. 5.35 Graphs for R AMAN scattering

ћω'

ћω

(a)

ћω

(b) f

k

i

f

k

i

Characteristic for perturbation theory in 2nd order is the summation over all possible intermediate states |k, in principle including the continuum. One may visualize this in terms of F EYNMAN type graphs as sketched in Fig. 5.35. Graph (a) corresponds precisely to (5.29) as just discussed: an incoming photon ω is absorbed so that the state |i changes into |k. Somewhat later the photon ω is emitted and |k becomes |f . Graph (b) corresponds to an additional term which still has to be added to (5.29): first the photon ω is emitted spontaneously and |i changes into |k while only in the second step ω is absorbed by |k which is transferred into |f . Further evaluation in the usual manner leads to the spontaneous R AMAN probability per mode,

dRRAMAN =

2

2 1 πωe2 δ ωf i − ω + ω πω e , N . . . L3 ε0 L3 ε0 2π

(5.31)

k

where we have abbreviated the action of the two terms just mentioned by k

=

† D f k Dki k

ωki − ω

+

† Df k D ki ωki + ω

.

We may compare (5.31) with (2.146) and (2.148) for absorption and emission, respectively. The terms in square brackets are identical, with N ∝ I (the incident laser intensity) and considering that for spontaneous emission Nk ≡ 0. The additional factor 1/2π arises, since only one line shape function is involved here, in the combined form δ(ωf i − ω + ω ). In the final evaluation we have to account for the mode densities, normalization etc. as detailed with Eqs. (2.149)–(2.153). We then have to integrate over the laser line (which removes the delta function) to obtain the probability dR¯ RAMAN for scattering into a given solid angle dΩ. This probability is linear in respect of the intensity of the incident photon source. Finally, the scattered R AMAN intensity dI = ω dR¯ RAMAN is obtained, from which the differential cross section for spontaneous R AMAN scattering follows: † † 2 dσf i Df k D dI e4 ω4 D f k Dki ki . = = + dΩ I ωki − ω ωki + ω (4πε0 )2 c4

(5.32)

k

It has the standard dimension L2 . If one wishes, one may replace the prefactor by re2 m2e ω4 , where re = α 2 a0 is the classical electron radius and me the electron mass. We emphasize the proportionality ∝ ω3 characteristic for spontaneous processes

5.6

R AMAN Spectroscopy

341

while the additional factor ω arises from the fact that the cross section is proportional to the scattered intensity dI . Note that this expression for dσf i /dΩ refers to well defined polarizations e and e for the exciting laser as well as for the spontaneously emitted photons, respectively. All quantities are written in SI units. In the theoretical literature (especially in early work such as A LBRECHT 1961) one often finds the total scattering intensity averaged over all polarizations and integrated over all angles which implies multiplication of our expression by 8π/9; also esu are used which implies multiplication by (4πε0 /e2 )2 . For quantitative evaluation of the dipole matrix elements in (5.32) one follows essentially the procedure described in Sect. 5.4.1. In addition to strict selection rules of the type given in (5.20), F RANCK -C ONDON factors determine again the scattering intensities – here between the initial state |i and the intermediate states |k as well as those between the latter and the final states |f . At this point, we note another important aspect about Chap. 8 in Vol. 1: the sum is nearly identical to the polarizability αE (for linear polarization e = e parallel to the z-axis) according to (8.94) as derived in Chap. 8, Vol. 1. This establishes the connection to the classical interpretation. Actually, one often starts the treatment of R AMAN scattering with the polarization tensor and may expand it around the equilibrium position R0 just as discussed in the semiclassical case, Sect. 3.4.4. This leads to a quantum mechanical justification of the rule presented there, which reads in a more general form: A molecule is R AMAN active with regard to a particular normal coordinate qi if and only if the electronic polarizability changes in this very coordinate. For more details we refer to the literature (e.g. A LBRECHT 1961; C HAN and S ILVI 1981, and references there). In practice, explicit evaluation of (5.32) or total cross sections can be very elaborate. One usually tries to find suitable approximations in order to reduce the number of intermediate states necessary for reasonable estimates of the cross sections. However, as outlined above, the R AMAN shift ωf i according to (5.30) as such depends only on the term energies of the initial and final vibration-rotation levels of the molecule studied. Explicit expressions are particularly transparent for diatomic molecules. The S TOKES shifts in the three branches are simply the differences of rotational and vibrational terms F (N) and G(v), according to (3.57) and (3.58), respectively, (in wavenumbers): DRASEKHARAN

O(N ) = ν¯ 10 + (2B1 − 4D1 ) − (3B1 + B0 − 12D1 )N

(5.33)

+ (B1 − B0 − 13D1 + D0 )N 2 + (6D1 + 2D0 )N 3 − (D1 − D0 )N 4 Q(N) = ν¯ 10 + (B1 − B0 )N (N + 1) − (D1 − D0 )N 2 (N + 1)2

(5.34)

S(N) = ν¯ 10 + (6B1 − 36D1 ) + (5B1 − B0 − 60D1 )N

(5.35)

+ (B1 − B0 − 37D1 + D0 )N 2 − (10D1 − 2D0 )N 3 − (D1 − D0 )N 4 .

342

5


Here B1 and B0 are the rotational constants in the excited and in the initial vibrational level, D1 and D0 are the corresponding rotational stretch corrections and ν¯ 10 = G(1 ← 0) is the vibrational frequency of the transition according to (3.86). With these expressions one obtains a good estimate of the structure of the spectra. Specifically, it becomes clear that the Q branch leads only to very small shifts of in the rotational R AMAN spectra, since in this case only the (small) differences of the rotational constants B and of the correction terms D in the two vibrational states involved enter in the spectral position. For larger N , however, one expects a quadratic growth of the line distances, while in the O and S branches the distances between neighbouring rotational lines are expected to be constant to 1st order. Cross sections for R AMAN scattering are notoriously very small. Since for all intermediate states |k as a rule ωka = ω holds – for the so called normal R AMAN spectroscopy one even has ωka ω – there are no typical resonance denominators in (5.32). In contrast, these resonance denominators make one-photon spectroscopy in absorption as well as in emission so efficient, as described in Chaps. 4 and 5, Vol. 1. In principle, in special cases it may happen that ωka ω (one then has to introduce a damping term iΓe into the respective denominator in (5.32)). One occasionally calls such processes “resonance R AMAN scattering”. However, such a situation may actually be better described as a special variety of optically induced fluorescence, specifically as laser induced dispersed fluorescence (DF).

5.6.4

Experimental Aspects

While the classical absorption and emission spectroscopy of molecules dates back to the 19th century and celebrated its big achievements already in the first half of the 20th century, the extremely successful history of R AMAN spectroscopy – exploring molecular structure and becoming one of the most important tools in analytical science – is closely related to the discovery of the laser. A number of important publications soon after the discovery of the effect in 1928 and the N OBEL prize for R AMAN in 1930 document that the extraordinary potential of R AMAN scattering has been realized already very early. However, the cross sections are extremely small, so that efficient highly intensive, quasi-monochromatic and well collimated laser beams were necessary before efficient use could be made of it. Fortunately, several fixed frequencies of the lasers in the visible and near UV spectral range were quite sufficient – already available in the 60th of the past century, e.g. from argon ion lasers. A typical setup for a Raman spectrometer shows Fig. 5.36 – essentially self explaining. While the principle of the measurement is quite straight forward, the big experimental challenge of R AMAN spectroscopy is, (i) to collect the scattered radiation efficiently, (ii) to separate it carefully from the (elastic) R AYLEIGH scattering which is orders of magnitude more intense, as well as from stray light of the incident beam which may be reflected at various parts of the spectrometer and (iii) finally in providing highest spectral resolution. A variety of R AMAN spectrometers are commercially available today, both for gas phase analysis as well as for solid state or surface targets – in the latter case even spatially resolved on a µm and sub

5.6

R AMAN Spectroscopy entrance slit

343

of the spectrometer L2 F2 L1

target cell

exit window CM

FPI

L = lenses F = filter LP = laser prism M = mirrors CM = collecting mirrors FPE = FABRY-PEROT etalon FPI = FABRY-PEROT interferometer

F1 FPE λ /2

Ar ion laser

LP + M

CM

Fig. 5.36 Schematic of a gas phase R AMAN spectrometer. The laser beam from the Ar ion laser is reflected several times within the target cell (containing the gas to be studied), thus allowing a sufficiently long interaction region. Very important is the design of the collecting mirrors (CM) for the R AMAN scattered light and direct it onto the spectrometer entrance slit

µm scale – which is of great practical relevance it the time of nano-, bio-, and similar technologies. In special cases, R AMAN scattering from solid surfaces may even be enhanced substantially. This is exploited in the so called “surface enhanced R AMAN spectroscopy” (SERS). Modern R AMAN spectrometers with highest resolution use interferometric analyzing methods (similar to IR spectrometers), in particular with F OURIER transform spectroscopy (see e.g. C HASE and R ABOLT 1994) and with highly stabilized single mode argon ion lasers, typically at 488 nm. A resolution on the order of 0.01 cm−1 can be reached – which requires, however, that the collection angle is reduced so that D OPPLER broadening can be avoided. For details of the various spectrometer types we refer the readers to specialized original literature and a number of relevant monographes.

5.6.5

Examples of R AMAN Spectra

In the following we want to discuss a few characteristic examples of spectra and start with the two most prominent molecules in the air surrounding us, N2 and O2 . Figure 5.37 shows the vibration-rotation R AMAN spectrum of N2 . The fundamental frequency is excited, for which according to Table 3.6 one computes with (3.86) that ν¯ 10 = G(1 ← 0) = 2329.92 cm−1 (within the linewidth this corresponds exactly to the observed position of the Q(0) line in Fig. 5.37; these values are continuously improved with new data from Raman spectroscopy). The spectrum shows nicely the constant line distances in the O and S branch, as expected according to (5.33), as well as the narrow Q branch. The blow up with high resolution, shown on the right of Fig. 5.37, documents very nicely the quadratic dependence of the R AMAN shift in the Q(N) branch, as predicted by (5.34). Interesting is the alternating line intensity due to nuclear spin statistics which we shall discuss in a moment.

344

5

Molecular Spectroscopy Q branch

N2

6

8

4

10 12

2

5 7

14 9

3

11 13

1 0 S(20) S(15)

2500

S(10)

S(5)

S(0) Q(0) O(5)

2400

O(10)

2300

O(15)

2330 2329 2328 2327

2200

STOKES shift νba / cm-1

νba / cm-1

Fig. 5.37 Vibration-rotation R AMAN bands for N2 for excitation of the fundamental vibration. The left spectrum, originally recorded by BARRETT and A DAMS (1968) with the 488 nm line of an Ar+ ion laser, has been shifted and stretched slightly in order to match recent high resolution data of B ENDTSEN and R ASMUSSEN (2000). The fully resolved Q branch (shown on the right) is taken from the latter work. The lower scales give the S TOKES shift of the lines with respect to the incident wavenumber. Also noted are the initial rotational quantum numbers N of the spectra for the O, Q and S branches. Due to nuclear spin statistics the intensity for odd N is only half that of even numbered lines (see Sect. 5.6.6)

Fig. 5.38 Vibration-rotation R AMAN bands for excitation of the fundamental vibration of O2 according to BARRETT and A DAMS (1968), recorded with the 488 nm line of an Ar+ ion laser. Scales as in Fig. 5.37. Due to nuclear spin statistics in O2 only odd N levels are observed (see Sect. 5.6.6)

O2

S(23) S(17) S(11)

1700

S(5)

Q(0) O(5) O(11) O(17)

1600 1500 STOKES shift νba / cm-1

O(23)

1400

We briefly discuss also the vibration-rotation Raman spectrum for the O2 molecule shown in Fig. 5.38. Here the values from Table 3.6 give ν¯ 10 = 1556.23 cm−1 corresponding exactly to the limit of the Q branch marked as Q(0) in Fig. 5.38.

5.6

R AMAN Spectroscopy

S(N) 70

60

50

40

345 30

R(N) 75

N

C

HC

55

35

10

10

15

15

20 35

30 55

40

50 60

70 S(N)

75 R(N)

H N

N

70

20

C

H

60 50 40 30 anti-STOKES shift / cm-1

20

10

0

10

20 30 40 50 STOKES shift / cm-1

60

70

Fig. 5.39 Highly resolved pure rotational R AMAN spectrum of s-triazine, recorded with the 488 nm Ar+ ion laser line according to W EBER (1979)

More recent measurements are available with even higher precision, however, only in tabulated form. Thus we communicate here only the first R AMAN spectrum for O2 reported in the literature (BARRETT and A DAMS 1968). As a new specialty, we note that in the electronic ground state of O2 obviously only rotational states with odd quantum numbers N are populated – again a consequence of nuclear spin statistics, to be discussed in the following subsection. Finally, Fig. 5.39 shows a highly resolved, pure rotation R AMAN spectrum (v = 0) for the polyatomic molecule s-triazine. As mentioned at the end of Sect. 5.6.1, pure rotational R AMAN spectra of diatomic molecules show only an S branch, both on the (N = 2) S TOKES as well as on the anti-S TOKES side. This is similar for any symmetric and asymmetric rotor. However, due to the possible change of the rotational projection quantum number K, now in addition on both sides of the R AYLEIGH line appears also an R branch (N = 1). However, we cannot indulge her into details.

5.6.6

Nuclear Spin Statistics

We have already discussed in Sect. 3.3.3 the influence of nuclear spin on the population of rotational levels in diatomic molecules. There, we have become acquainted with ortho and para hydrogen. The R AMAN spectra of N2 and O2 just discussed require further attention. The intensity change of 2:1 between even and odd rotational quantum numbers in the N2 spectra for even and odd rotational quantum numbers N , as well as the missing lines for even N in the case of O2 have the same origin. Such intensity changes may occur in molecules with two identical nuclei and are a consequence of the PAULI exclusion principle according to which the total wave function of fermions must be anti-symmetric with respect to exchange of these particles. For bosons, in contrast, it has to be symmetric.

346 Fig. 5.40 Illustration of the symmetry operations discussed in the text: the complete exchange of the nuclei A and B (red, horizontal arrow, top) is equivalent to the sketched sequence of symmetry operations

5 A↑ B↓


exchange of nuclei ^

PAΨ = ±Ψ

B↑

A↓

180o rotation around the z axis ^ C2 ψrot = (-1)N ψrot B↓ A↑ inversion of the electron system (g,u) B↓

B↓

A↑ reflection of the electron system (±) nuclear spin exchange A↑ B↑ ^ PS χsp = ± χsp

A↓

One may write the total wave function of a molecule as product of spatial and spin function with respect to electronic and nuclear coordinates r and R. For the symmetries discussed here, only the components relating to angular momenta (including the spin) need to be considered. We write: Ψ (r, R) = φel (r)ψrot (R)χsp (I ).

(5.36)

Here φel (r) describes the electronic wave function, (see Sect. 3.6) and ψrot (R) the nuclear wave function. We now focus on the angular components of these wave functions. Since we consider only exchange of the nuclei, the electron spin is only relevant as it defines the electronic wave function and thus the symmetry of the local environment of the nuclei: χsp (I ) thus refers exclusively to the nuclear spin. In the following we consider a sequence of symmetry operations which in total are equivalent to the exchange of the two atomic nuclei. In Fig. 5.40 we indicate the two nuclei by A and B (in reality they are of course indistinguishable). The electronic state φel (ˆr ) is characterized by the ellipse, its symmetry behaviour is indicated A ) by the red marks. For the complete exchange of the nuclei (exchange operator P A Ψ = ±Ψ P must hold, depending on whether we describe bosons or fermions. To realize such an exchange in detail for the wave function (5.36) we let the following symmetry operations occur (their effect onto a given molecular state is known): 1. Cˆ 2 : rotation of the whole molecule with regard to the z-axis (perpendicular to the molecular axis) through 180◦ . We obtain: Cˆ 2 ψrot = (−1)N ψrot .

(5.37)

Electronic and spin function have no phase factor. 2. ıˆ: Inversion of the electronic wave function: ıˆφel = ±φel .

(5.38)

5.6

R AMAN Spectroscopy

347

The + or − sign refers to g and u states, respectively (homonuclear molecule). 3. σˆ h : Reflection of the electronic wave function with regard to a plane through the nuclear axis: σˆ h φel = ±φel .

(5.39)

Specifically for Σ ± states the + or − sign has to be applied. S : Exchange of the two nuclear spins. 4. P S χsp = ±χsp . P

(5.40)

The symmetry of the nuclear spin function χsp (I ) = |IIIM follows the usual rules of angular momentum coupling. We refer to the individual nuclear spins by I , to the total nuclear spin by I and its projection is M. Since 0 ≤ I ≤ 2I with (2I + 1) substates in total (2I + 1)(2I + 1) different nuclear spin states exist, which are either symmetric or anti-symmetric – as one easily derives for each case. As illustrated in Fig. 5.40, the total sequence of symmetry operations 1–4 is fully equivalent to exchange of the two nuclei. Thus, in total: S χsp = ±Ψ. A Ψ = σˆ h ıˆφel (−1)N ψrot P P We now discuss the examples which we have encountered so far. 1H

2

Molecule

The electronic ground state is a 1 Σg+ state for which ıˆ as well as σˆ h has the eigenvalue +1. Thus, φel has no influence on the total symmetry. The nuclear spin of 1 H is I = 1/2, i.e. the molecule is built of two fermions and the total wave function has to be anti-symmetric. As for a two electron system we have an anti-symmetric singlet state with I = 0 and a symmetric triplet with I = 1. Thus, with (5.37) for even N the nuclear spin function has to be anti-symmetric (one state), while for odd N the nuclear spin function must be symmetric (three states). In a R AMAN spectrum (not shown here) the intensity ration of lines arising from even N to those from odd N must be 1:3. 14 N

2

Molecule

Here too the electronic ground state is a 1 Σg+ state and φel is without influence on the statistics. The nuclear spin of 14 N is I = 1, we are discussing bosons for which the total wave function must be symmetric. We now have11 one symmetric nuclear spin singlet with I = 0, one anti-symmetric triplet with I = 1 and one symmetric quintuple with I = 2 – a total of 9 nuclear spin states of which 6 are symmetric and 3 anti-symmetric. To construct an even total nuclear wave function we combine the symmetric nuclear spin states with even N , the anti-symmetric spin states belong to odd N . This explains the ratio of 2 : 1 between intensities for even and odd rotational quantum numbers N observed in the R AMAN spectrum Fig. 5.37. 11 One

easily verifies this by looking up the respective C LEBSCH -G ORDAN coefficients.

348 16 O

5 2


Molecule

In this case the electronic wave function in the ground state is 3 Σg− for which σˆ h ıˆφel = −φel . Again we have bosons as the nuclear spin of 16 O2 is I = 0. The total wave function has to be symmetric. Now, from two nuclear spins I = 0 we can construct only one symmetric singlet total nuclear spin state with I = 0 can be constructed. The positive total symmetry can thus only be realized with odd rotational quantum numbers N which compensate the negative symmetry of φel . Hence, there are only R AMAN lines in the spectrum Fig. 5.38 arising from odd N : in the electronic ground state of 16 O2 there are simply no other nuclear states providing the correct symmetry. This nuclear spin statistics has, of course, also consequences on other properties of O2 , e.g. on the temperature dependence of its heat capacity. Quite generally one may show (easy to verify for the three cases discussed here), that the degeneracies gs of symmetric states to ga for anti-symmetric states are related by I +1 gs . = ga I

(5.41)

Section summary

• R AMAN spectroscopy of molecular vibration and rotation belongs today to the most important analytical methods. • R AMAN scattering is a two-photon process: one photon (energy ω) is absorbed (induced process), one photon (ω ) is spontaneously emitted. Their energy difference is the difference of energies between initial and final vibrational-rotational level (typically within one electronic state). • Characteristic of a R AMAN spectrum are the S TOKES lines (ν < ν), the antiS TOKES lines (ν > ν) and the elastic R AYLEIGH line. • Strict selection rules hold for the rotational quantum number. In diatomic molecules it can change by N = 0, ±2, corresponding to Q, S and O branches, in polyatomic molecules also R and P branches occur. For vibration propensity rules hold, determined again by F RANCK -C ONDON type considerations. • A normal mode of a molecule is R AMAN active, if and only if that motion changes the electronic polarizability. • In homonuclear, diatomic molecules, nuclear spin statistics has a significant influence on R AMAN spectra – as discussed for H2 , N2 and O2 (see R AMAN spectra of the latter two, Figs. 5.37 and 5.38).

5.7

Nonlinear Spectroscopy

In all of the spectroscopic methods discussed so far,12 the observed signal depends linearly on the intensity of the incident electromagnetic radiation. This also holds for 12 With

the exception of pure emission spectroscopy where no incident light is involved.

5.7


349

LIF as well as for the just discussed, spontaneous (also called incoherent) R AMAN scattering. However, at higher laser intensities, easily reached with today’s laser sources (see Sect. 8.5 in Vol. 1), one may use nonlinear processes with advantage for certain applications in molecular spectroscopy. Generally speaking, nonlinear optics NLO and nonlinear spectroscopy are important areas of modern research. We cannot even give a rudimental introduction to it and refer to various reviews (e.g. W RIGHT et al. 1991; K NIGHT et al. 1990; D RUET and TARAN 1981) and monographs (e.g. B OYD 2008; S HEN 2003). Here we only present a few basics and indicate the potential of nonlinear methods in molecular spectroscopy by way of example.

5.7.1

Some Basics

Nonlinear processes arise from changes of the material under investigation due to its interaction with the electromagnetic radiation. In turn, the latter is influenced by these changes. In a perturbative approach, i.e. for not too high intensities, one describes this by a field induced polarization P: P = P(1) + PN L = ε0 χ (1) E + χ (2) EE + χ (3) EEE + · · · .

(5.42)

This expression thus replaces (8.79) in Vol. 1, where we have introduced the linear term P = P(1) . In analogy to χ (1) , the (linear) susceptibility, χ (2) and χ (3) are denoted as nonlinear susceptibilities of 2nd and 3rd order, respectively. The corresponding terms PN L are called nonlinear polarization. Note, however, that in the general case (i.e. for a non-isotropic medium) the susceptibilities χ (N ) of N th order are tensors of rank N + 1.13 Just as in the linear case described by (8.81), Vol. 1, the absolute magnitude of the polarization is proportional to the particle density N (i) in the initial state of the molecule studied (note that N (i) must not be confused with the rotational quantum number Ni of that state). According to (8.99), Vol. 1, the index of refraction n and the linear susceptibility are related by χ (1) = n2 − 1 (in isotropic media), with χ (1) being typically on the order of 1 in solid state materials. At low light intensities, the terms of higher order in (5.42) are comparatively very small. Unfortunately, there is no general rule for light intensities at which nonlinear processes might be expected. Each case needs to be considered individually (see also Chap. 10). are measured in units [χ (k) ] = mk−1 V−k+1 , i.e. only χ (1) is dimensionless. Note that (5.42) is an abbreviation. Explicitly, the components of the polarization vector are (2) (3) P i = ε0 χij(1) Ej + χij k Ej Ek + χij k Ej Ek E + · · ·

13 They

j

jk

j k

for i = x, y, z. Each index in the sums runs over x, y, and z.

350

5


However, to obtain a feeling for the order of magnitude of these quantities B OYD (1999) used an old argument which we follow.14 One assumes that linear and nonlinear terms in (5.42) should become of equal order of magnitude for electric field strengths on the order of the atomic field strength (8.141), Vol. 1, EH = e/(4πε0 o02 ), i.e. at intensities 3 × 1016 W cm−2 . Thus we obtain χ (2) χ (1) /EH

2 and χ (3) χ (1) /EH .

(5.43)

Using the polarizability αE for atomic hydrogen according to (8.77), Vol. 1 the linear susceptibility per atom according to (8.81), Vol. 1 is χa(1) 4π4a03 7.4 × 10−30 m3 . To determine the particle density N (i) we use the hydrogen VAN DER WAALS radius (see Fig. 3.3 in Vol. 1) with rH 0.120 nm and assume close-packed (1) spheres (74 % filling). Then the linear susceptibility becomes χ (1) = N (i) χa 0.76. Inserting this and EH into (5.43) we obtain χ (2) = 1.5 × 10−12 m V−1 and χ (3) = 2.9 × 10−24 m2 V−2 . When using the corresponding data for sodium with (8.82), Vol. 1, all estimates increase by a factor of about six. Including both, linear and nonlinear polarization, the general wave equation (1.34) has to be replaced by: 1 ∂PN L n2 ∂ 2 . (5.44) − 2 2 E(x, y, z, t) = 2 c ∂t c ε0 ∂t 2 This wave equation describes the propagation of electromagnetic radiation in matter, and thus must also be applied to devise and interpret spectroscopic experiments. Due to the terms of higher order in (5.42) one may in principle expect the formation of sums and differences of all frequency components contained in the incident electromagnetic wave(s). Hence, (5.44) forms together with (5.42) the basis of all nonlinear spectroscopy. The evaluation of these equations for the general case is not trivial. Strictly speaking, the susceptibilities χ (1) , χ (2) and χ (3) are tensors of rank two, three and four (see footnote 13). In practice, symmetry considerations usually help to simplify the problem dramatically. For example, χ (2) is different from zero only if the system has no centre symmetry – optically active crystals show indeed very typical and strong anisotropies which are used e.g. for generating the second and higher harmonics of laser light. However, if the target has centrosymmetry (as e.g. a molecular beam or an isotropic liquid), the χ (2) term drops out completely, and the χ (3) terms usually become quite transparent. The nonlinear polarization is in this case PN L = ε0 χ (3) · EEE and describes so called coherent four wave mixing (CFWM, short FWM) processes: three electromagnetic waves of different or equal frequencies (ω1 , ω2 , ω3 ) and polarization (e1 , e2 , e3 ) are incident and generate a time dependent polarization PN L , which in turn produces an electromagnetic field of frequency ω4 and polarization 14 Unfortunately he still used the old Gaussian esu, strangely in combination with the unit V. We use here of course SI units.

5.7


351

e4 – the desired signal which is emitted coherently with the incident radiation. In the most simple case all incident light waves have the same polarization. Then, χ (3) (−ω4 , ω1 , ω3 , −ω2 ) is simply a scalar function of the incident frequencies (positive sign indicates excitation, negative sign denotes stimulated emission), and the nonlinear polarization becomes PN L = ε0 χ (3) [E(t, z)]3 , with E(t, z) being the sum of the electric field vectors of the incident radiation. An ab initio computation of χ (3) contains expressions in analogy to (5.32), however, in this case in perturbation theory of 3rd order: one has to deal with triple sums over products of four dipole matrix elements, each between initial and final state and the various intermediate states. Each of them is multiplied by three G REEN’s propagators (resonance denominators) of the type (ω − ωkj + iΓkj )−1 , with ω = ω3 ∓ ω2 ± ω1 or ω3 ± ω2 ± ω1 where ωkj is the energy difference of the intermediate states and Γkj the average value of the natural lifetime of the levels j and k involved. The resulting sums may contain up to 48 terms for each level involved. Their evaluation is a rather demanding task, for which one uses F EYNMAN type diagrams according to Y EE et al. (1977), also called B ORDÉ (1983) diagrams (we have already used them to illustrate spontaneous R AMAN scattering, Fig. 5.35). For details in the context of the processes of interest here we refer e.g. to W ILLIAMS et al. (1994, 1995, 1997) and D I T EODORO and M C C ORMACK (1999). A variety of such FWM processes exist and may in principle be used for spectroscopy. In Fig. 5.41 a small selection is sketched, along with their respective energy level schemes. We follow the usual convention with ω1 and ω3 describing absorption, ω2 and ω4 emission processes. The incident frequencies are ω1 , ω2 and ω3 , while ω4 refers to the coherent signal generated in the process. Coherent anti-S TOKES R AMAN scattering (CARS) and coherent S TOKES R AMAN scattering (CSRS) are the coherent version of R AMAN scattering. They allow for a wealth of interesting, vibrationally selective applications in spectroscopy, particularly so in surface analytics. Degenerate four wave mixing (DFWM), i.e. the resonant variety of FWM where three equal frequencies are mixed to a signal of again the same frequency may – as absorption or LIF – be used for the study of electronic transitions. Two colour resonant four wave mixing (TC-RFWM) is in the ‘UP’ version a special kind of double resonance spectroscopy which we have met already several times in its linear version. The other diagrams in Fig. 5.41 are more or less self explaining, so that we refrain here from further discussion. Generally speaking, nonlinear methods are often useful in situations where linear methods fail. A typical application is suppression of a diffuse background, e.g. when studying molecular ions or radicals generated in a plasma, which in itself generates a lot of stray light. It is important to note that in all FWM processes sketched in Fig. 5.41, both energy and momentum conservation for the photons must hold: ω1 + ω3 = ω2 + ω4

(5.45)

k1 + k2 = k3 + k4.

(5.46)

352

5


electronically excited states

ω2

ωSTOKES

pump

ω3

ω2 ω4 ω1

e'

e

ω4 =

CSRS e' dump

e

ω3

i'

f probe

pump

pump

ω 1 = ω2

ω 1 = ω2

ω4

f

DFWM

ω4 ωvib

CARS

ω3

i

ω2 ω3

ωvib

ωvib COORS or LIF e

ω1

probe

ω1

ω0

ω4

i TC-RFWM, SEP

= ω3 i'

i TC-RFWM, UP

Fig. 5.41 Comparison of COORS (common ordinary old R AMAN scattering) or – in the resonant case – LIF, with different CFWM processes. Full red arrows correspond to the incident radiation, dashed red arrows mark the signal. CARS: ω1 = ω3 (pump and probe, respectively), ω2 = ω1 − ωvib = S TOKES and ω4 = ω1 + ωvib = anti-S TOKES signal; CSRS: ω1 = ω3 (pump and probe), ω2 = ω1 + ωvib = anti-S TOKES and ω4 = ω1 − ωvib = S TOKES signal; DFWM: four equal frequencies, starting from two degenerate states, i and i , ending in two degenerate excited states e and e ; TC-RFWM with stimulated emission pumping (SEP): ω1 = ω2 (pump), ω3 = ω4 (dump); TC-RFWM with double resonance from the ground state (UP): ω1 = ω2 (pump), ω3 = ω4 (probe)

The last relation with the wave vectors k j (kj = ωj /c = 2π/λj ) ensures the so called phase matching of the four waves involved. Phase matching is essential for coherent interaction of the waves to be mixed. It is the key to any nonlinear process and requires very careful alignment of the four light beams, i.e. a characteristic geometry as we shall illustrate in the next subsection. A detailed analysis shows (W ILLIAMS et al. 1997) that the signal intensities observed are given by expressions of the type 2 I4 ∝ N (i) × L2 × I1 I2 I3 (5.47) 2 2 × [Sie ]2 [Sef ]2 × L(ω1 , ω3 ) × G(e4 , e1 , e2 , e3 ; Ni , Ne , Nf ) . (This specific expression holds for TC-RFWM.) Here N (i) is the target density in the initial state, L the length of the interaction region, I1 , I2 and I3 are the three incident intensities and Sj k the line strengths of the transitions between the states j and k. As detailed in Appendix H.2, Vol. 1 these line strengths are proportional to the square of the reduced matrix elements of the dipole operator (H.32), Vol. 1 for the respective transition, Sj k ∝ |j Dk|2 . In the present case they depend on the vibrational and rotational quantum numbers, Ni , Ne , Nf , involved. The line

5.7


I1

(a)

353

I2

j jet

(b)

M

R pump

k3

I3 k4

k2

M

k1

n slit

oz

M

L P BS

Nd:YAG

jet

R

zle

R R valve HV

PM

oscillograph

dye laser

SF

probe

R

RF

pulse generator Nd:YAG

computer dye laser

PD R

Fig. 5.42 Experimental scheme for four wave mixing spectroscopy according to M AZZOTTI et al. (2008). (a) BOXCARS setup for achieving phase matching at the plasma slit source emerging from a supersonic molecular beam, with masks (M) for defining the beam positions. (b) Overall setup with two pulsed, tunable dye laser systems (in DFWM only one is used), a 1000 mm lens, beam splitters (BS), high reflectivity mirrors (R), a prism (P) and spatial filters (SF), jet setup (with high voltage, HV) for plasma generation), photo-detector for triggering (PD) of the experiment, and signal detecting photo-multiplier (PM), as well as the typical electronics for controlling and detection

profile L(ω1 , ω3 ) accounts for the velocity distribution of the target molecules (i.e. for D OPPLER broadening) as well as for the natural and collision induced linewidths of the transitions. Finally, G(e4 , e1 , e2 , e3 ; Ni , Ne , Nf ) describes the influence of the polarization of the incident and the detected radiation and depends of course on the angular momenta of the states involved, first and foremost on the rotational quantum numbers. Note that the signal – the result of a coherently amplified process – depends on the square of the target density N (i) and the interaction length L. Observation of FWM (or higher order) processes always requires the coherent interaction of many particles. In a single atom or molecule such processes are not relevant (even though the cross section may depend on a higher power of the radiation intensity). As a consequence, the signal is emitted in a highly collimated beam of radiation, which may readily be distinguished form a fluorescent background emitted into a large solid angle.

5.7.2

An Example

Figure 5.42 shows a typical experimental setup as used by M AZZOTTI et al. (2008), with a so called BOXCARS arrangement (a) by which the phase matching can be realized. The slit nozzle plasma discharge is already known to us from Sect. 5.5.3, where we have discussed its use together with the CRD method. Here one uses DFWM and TC-RFWM for electronic spectroscopy of radicals. Characteristic is the beam positioning with the help of a matrix M, which defines the entrance and

354 Fig. 5.43 DFWM spectra for C2 according to M AZZOTTI et al. (2008). (a) 0–0 d 3 Πg − a 3 Πu transition at Trot 140 K, (b) ditto but at Trot 40 K, (c) 1–0 d 3 Πg − a 3 Πu transition at Trot 100 K. The rotational temperatures Trot have been estimated by simulations (not shown here)

5


R

P

Trot = 140 K

(a)

Q

0-0

(b)

Trot = 40 K 514

0-0 515

516

R Trot = 100 K

517

P

(c)

Q

1-0 472

473 wavelength / nm

474

exit angles, as well as the divergence, by apertures – also for the signal to be detected (red arrow in Fig. 5.42(a)). In this particular setup the crossing angle of the beams is 1.7◦ , the beam diameter prior to the collection lens (L) ca. 2 mm, which allows for an overlap length of L 30 mm in the target (L enters the signal quadratically as mentioned above). The photomultiplier for detecting the signal is positioned at 4 m distance from the jet. Several spatial filters on the way (not shown in Fig. 5.42) allow a very good separation of signal and stray light from the plasma source. Figure 5.43 shows for the example of the d 3 Πg − o3 Πu transition in C2 , some typical spectra thus obtained with DFWM (all four waves have the same frequency and are jointly tuned). Without entering into details of the spectroscopy of the C2 molecule, we note the excellent signal to noise ratio. The rotational progressions in the R, Q, and P branches are characterized by a large variety of lines. For high rotational quantum numbers N one recognizes in the R branch a clear triplet structure due to the spin of the electronic states. Section summary

• Nonlinear processes in the interaction of matter with radiation fields arise from changes of the properties of the material studied, due to the radiation field. The resulting nonlinear polarization depends on powers E 2 , E 3 etc. of the incident electric field strength, and the re-emitted radiation thus mixes all incident frequencies, so that one may (at sufficiently high laser intensities) e.g. generate higher harmonics or various difference frequencies from the incident radiation. • For gas phase spectroscopy this may be exploited in sophisticated pumpprobe schemes, e.g. to suppress undirected stray background light. In such centrosymmetric systems (gas phase) the second order nonlinearity χ (2) vanishes and the third order χ (3) leads to a polarization which depends on three

5.8

Photoelectron Spectroscopy

355

coherent incident field amplitudes. If energy and momentum is properly conserved (the latter is called phase matching) these three field amplitudes may be mixed and create a fourth amplitude, the desired signal. The signal observed in this coherent four wave mixing (FWM) process depends on the squares of target density and interaction length, as well as on the product of the three incident intensities.

5.8


Photoelectron spectroscopy (PES) is another very important method in molecular spectroscopy. In recent years it has become a particularly versatile tool for characterizing solid state surfaces and molecules deposited onto surfaces. The origin of PES goes finally back to E INSTEINs interpretation of the photoelectric effect from 1915. Pioneering work has been carried out between 1957 and 1970, honoured by the N OBEL prize to Kai B LOEMBERGEN and S HAWLOW (1981). Electron spectroscopy experienced its heyday in the past three decades of the 20th century when it developed into a mature analytic method, based on a number of cutting edge technical achievements and sophisticated procedures. In a number of monographs (e.g. B ERKOWITZ 1979; P OWIS et al. 1995) a comprehensive picture of the field is presented. Some basics for understanding photoionization and examples of PES on atoms have already been treated in Sect. 5.5, Vol. 1. Here we want to introduce briefly into PES with molecules and sketch a few recent developments.

5.8.1

Experimental Basis and the Principle of PES

Figure 5.44(a) illustrates, very schematically, the experimental setup of an electron spectrometer. The photon energies used range from the ultraviolet spectral range (often only a few eV above the adiabatic ionization limit WI ) up to hard X-ray radiation. In the former case one speaks about UPS (ultraviolet photoelectron spectroscopy) by which only valence electrons can be studied, in the latter case of XPS (X-ray photoelectron spectroscopy) which allows to remove electrons also from the inner shell. Suitable radiations sources for UPS are gas discharge lamps as well as harmonics of laser sources. For XPS X-ray tubes can be used and in both cases, of course, synchrotron radiation is the ideal, flexible light source. For completeness we mention here that it is also possible to use electron impact ionization as primary process as shall be discussed in some detail in Sect. 8.4.6. Recently, very interesting perspectives are arising from HHG sources, where tightly focussed femtosecond laser pulses generate rather high harmonics of the laser radiation (see Sect. 8.5.6, Vol. 1). These sources promise also excellent temporal resolution for the study of dynamical processes.

356

5

(a) electron detector

eWkin

(b)

ħω

A+B+

X+

Wkin(max) AB+

X

D'0 WX + v' N' (min) ionization

electrons

W(R)

FC region

synchroton

probe gas(jet) liquid jet surface

electon lens

UV (UPS)

monochromator: ħω

ionization/excitation X-ray (XPS)


v' = 0

AB

A+B

WI

WV D''0

WX v'' N'' electron spectrometer

v'' = 0 R

Fig. 5.44 Photoelectron spectroscopy of molecules. (a) Experimental realization, schematic, with different sources for excitation, various target preparations, electron beam guidance, energy analysis (spherical capacitor) and electron detection. (b) Energy scheme with adiabatic ionization potential WI , and vertical ionization potential WV . From the measured kinetic energies of the emitted electrons Wkin and the photon energy ω one derives the energy levels of the ionic states according to (5.49)

As targets for PES one may use molecules in the gas phase (today preferentially prepared in a cold supersonic beam), but liquids beams may also be used (see Sect. 5.8.2). The broadest application of UPS and XPS today is, however, in surface physics. Photoelectrons are emitted – as discussed in the context of atomic ionization in Sect. 5.5, Vol. 1 – with a characteristic angular distribution. In the experiment one typically selects a small fraction of this distribution and may thus determine (if required) the anisotropy parameter β. We recall the discussion in Vol. 1 on the angular distribution (5.80) of photoelectrons.15 With γ being the angle between 15 Equation

(5.80), Vol. 1 holds for pure, linearly polarized light. If the light is not fully polarized one has to correct for the finite degree of linear polarization P12 of the source according to (1.101). |P12 | ≤ 1 is usually calibrated by the well known angular distributions from rare gases. The observed electron angular distribution is then (5.48) I (γ ) ∝ 1 + β P12 3 cos2 γ − 1 /2 , as one may derive using the theory of measurement sketched in Chap. 9.

5.8


357

(linear) polarization vector and ejected electron, one thus derives the anisotropy parameter β, usually as a function of photon energy. Note: If one is only interested in average cross sections, it is advisable to detect the electron yield at the so called magic angle γ = 54.76◦ – where the β dependent term in (5.48) vanishes. In Appendix B some details on the experimental methods are summarized. Today electrostatic setups are mostly used. The hemispherical analyzer sketched in Fig. 5.44(a) and explained in more detail in Appendix B.3 is particularly popular. As outlined in Appendix B.4, alternatively, time of flight electron spectrometers are also used. They are of particular advantage when working with pulsed laser sources. Finally, the electrons have to be detected, which in general is achieved by secondary electron multipliers (SEM) described in Appendix B.1. Today, often so called ‘imaging’ methods are used by which behind the electrostatic monochromators a broad section of the electron energy spectrum may be registered simultaneously (VMI). In more sophisticated schemes, angular and energy distributions are registered together in such imaging device. We have mentioned an example already in Sect. 5.5.5, Vol. 1. Figure 5.44(b) illustrates for the most simple case of a diatomic molecule the characteristic energetics of a UPS process. Energy conservation requires ω + AB γ v N → AB+ γ v N + e− (Wkin ), (5.49) where the quantum numbers γ , v and N refer as usual to the electronic state, to vibration and rotation, respectively. We assume here that γ and γ correspond to the neutral electronic ground state X and to its cation ground state X + , respectively. The kinetic energy of the electrons Wkin (to be determined experimentally) results from the ionizing photon energy ω, the adiabatic ionization potential WI of the molecule with respect to the lowest vibration-rotation level of γ , the total energy Wγ v N of the initial state16 and the excitation energy Wγ v N , in which the ion remains after the process: Wkin = ω − WI + Wγ v N − Wγ v N .

(5.50)

The schematic energy diagram Fig. 5.44(b) assumes v = 0 in the electronic ground state γ = X (broad arrow upward), typically with several occupied rotational levels N . The ion in this scheme is also generated in its electronic ground state γ = X + . For the relative position of the potentials as indicated, ionization leads to a distribution of several vibration-rotation states. They are populated in a similar fashion as in electronic excitation processes by dipole transitions (see Sect. 5.4.1). Again we expect that F RANCK -C ONDON (FC) factors (between initial ground state and final ionic state) play a key role for the respective ionization cross sections. Often the the binding energy of the emitted electron WB (γ v N ) = −(WI − Wγ v N ) is communicated. The literature is somewhat ambiguous about the sign. If one refers to the free electron after emission, the electron binding energy is of course negative.

16 Often

358

5


probability to reach the energetically lowest vibration-rotation state may be low – even vanishing. This is indicated in Fig. 5.44(b) by the heavy black upward arrow and the red bracket around the “FC region”. From the highest electron kinetic energy observed, Wkin (max), one determines the minimal excitation energy, Wγ v N (min), in the ion. One thus determines the so called “vertical” ionization potential WV = ω − Wkin (max) = WI − Wγ v N + Wγ v N (min) = −WBV ,

(5.51)

which obviously differs from the true ionization potential WI . Its negative value, WBV = −WV , is often called vertical binding energy. These quantities are, however, defined only somewhat loosely as one recognizes from Fig. 5.44(b), simply because the FC region has no sharp limit. Over all, photoelectron spectra for ionization, correspond essentially to absorption spectra for excitation – except that now the photon energy ω does not need to be tuned: the energy balance is automatically taken care of by the kinetic energy Wkin of the electron emitted according to (5.50) which also contains the spectroscopic information. Of course it is possible at high photon energies, that ionization occurs into different final electronic states of the ion. Just as in absorption spectroscopy, the spectra observed may thus become quite complicated. In addition we have to remember that in PES different electrons of the target may be ionized. In UPS studies different valence electrons with different binding energy may be ejected – each of them may lead to its own vibration-rotation bands. In XPS one usually focusses on ionization of electrons from inner shells: these may show characteristic energy shifts according to their different chemical environment. This has evolved to be a very valuable analytical tool. As exemplified below, this so called chemical shift can be exploited for chemical analysis of this very environment. According to S IEGBAHN this method is called electron spectroscopy for chemical analysis (ESCA).

5.8.2

Examples

We want to discuss here only a few, characteristic examples which illustrate the potential but also the limitations of PES. We start with the H2 O molecule with which we are already well familiar. The photoelectron spectrum of H2 O in the gas phase has been studied with synchrotron radiation for the first time by T RUESDALE et al. (1982), and somewhat later by BANNA et al. (1986) over a larger energy interval, 30 eV ≤ ω ≤ 100 eV. As discussed in Sect. 4.4.1 the electron configuration of the electronic ground state of H2 O is (1a1 )2 (2a1 )2 (1b2 )2 (3a1 )2 (1b1 )2 . With photon energies up to 100 eV only electrons from the four valence orbitals may be ionized, as one reads from Fig. 4.22. One thus detects according to the scheme in Fig. 5.44(b) ionic states (doublets) – each with one hole in one of the valence shells. The respective ionization processes may be described as: H2 O + ω → H2 O+ 1b1−1 2 B1 , v + e− (Wkin ) (5.52)

5.8


(b)

359 1b1

×5

electron signal / arb. un.

3a1

1b1

(a)

gas phase

gas phase

1b2 -20

-15

-10 1b2

2a1

1b1

(c) liquid phase -40

3a1

1b2

3a1

2a1 -30

-20 electron binding energy / eV

-10

Fig. 5.45 Photoelectron spectrum of the valence electrons of water. (a) H2 O in the gas phase at ω = 100 eV (b) with improved resolution according to G ODEHUSEN (2004); (c) measured with a liquid jet, i.e. at the surface of the liquid, at 60 eV according to W INTER et al. (2004). Clearly visible are the shifts (indicated by dashed lines) of the absorption maxima due to solvation energy in the liquid

H2 O + ω → H2 O+ 3a1−1 2 A1 , v + e− (Wkin ) H2 O + ω → H2 O+ 1b2−1 2 B1 , v + e− (Wkin ) H2 O + ω → H2 O+ 2a1−1 2 B2 , v + e− (Wkin ).

(5.53) (5.54) (5.55)

Figure 5.45(a) (and Fig. 5.45(b) with high resolution) shows characteristic photoelectron spectra from a recent measurement of G ODEHUSEN (2004) at ω = 100 eV with high resolution. The four main structures correspond without doubt to the processes (5.52)–(5.55) (for simplicity we simply use the initial orbitals for characterization). One usually plots the signal as a function of “binding energy” WB = Wkin −ω. One has to be aware, however, that in the ion too vibrational states v may be excited, the real binding energy should thus be determined according to (5.50) from −Wγ v N . In the present case such vibrational excitation obviously occurs: one clearly recognizes, specifically in the high resolution spectrum Fig. 5.45(b), clear vibrational structures within the main peaks 1b1 , 1a2 and 1b2 (we assume here that H2 O initially was prepared mainly in the vibrational ground state). Figure 5.45(a) and (b) illustrate also the limits of photoelectron spectroscopy. The typical bandwidth of the electron monochromators is a few meV at best, which allows usually for an analysis of vibrational structures, but excludes in most cases to resolve rotational structures. For comparison, in addition to the isolated H2 O molecule, Fig. 5.45(c) also shows photoelectron spectra for water at the liquid surface according to W INTER et al. (2004). One may record such spectra today with a very thin water beam (some µm diameter) with well focussed synchrotron radiation. In the liquid each water molecule may be thought to have its own solvation shell. The high relative permit-

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tivity of water,17 εopt 1.8, screens the C OULOMB potentials in H2 O by a factor (1 − 1/εopt ) and thus reduces the electron binding energy. This reduction by about 2 eV is well recognized in Fig. 5.45(c) in comparison to Fig. 5.45(a). The spectral structures are, however, also broadened in the liquid environment due to fluctuations, as compared to the gas phase. Vibrational structures can thus no longer be discerned. For H2 O in the gas phase, T RUESDALE et al. (1982) and BANNA et al. (1986) have also measured the trends of the ionization cross section and of β, the anisotropy parameter (see Sect. 5.8.1), as a function of photon energy. The latter contains valuable information on the properties of the orbitals ionized. As discussed in Sect. 5.5.3, Vol. 1, one finds β = 2 if the electron was originally in a pure atomic s orbital. For electrons in atomic p orbitals, β strongly depends on energy; for not too high energies one typically finds β < 0. Similar rules hold for σ and π electrons in molecules. However, computations of β for molecules are in general much more complicated than for atoms where (5.90), Vol. 1 gives a clear prescription for computations. On the one hand, electronic states are now defined with respect to the molecular structure – they are not simply spherical harmonics as for atoms. On the other hand one has to account for the nuclear motion in a suitable way, and one usually has to average over all orientations of the molecule. Anyway, for H2 O one finds (not shown here) that the structure associated with the 2a1 orbital has the largest value of β over a broad range of photon energies, i.e. it corresponds indeed to a kind of s type orbital. As a further example, illustrating the kind of complexity which may today be attacked by PES, we show in Fig. 5.46 the photoelectron spectrum from the valence electrons of the nucleobase cytosine, according to T ROFIMOV et al. (2006). Cytosine is one of the four ‘letters’ of the DNA alphabet. As illustrated in Fig. 5.46, in total 16 valence electrons have been identified for one particular tautomer by comparing the experiment with so called ‘sticks spectra’ modelled by theory. The thus determined valence orbitals are listed in the caption. We cannot enter here into the details of the rather sophisticated quantum chemical computation (ADC(3) stands for “third order algebraic diagrammatic construction” and OVGF for “outer valence G REENS functions” – additional labels denote the respective MO basis sets). Finally, Fig. 5.47 shows the photoelectron spectrum for ethyl trifluoroacetate in the gas phase. It is probably the most impressive molecule for studying the chemical shift in organic molecules by ESCA – originally investigated by S IEGBAHN and his collaborators (see e.g. G ELIUS et al. 1974), then still with rather poor energy resolution. Today ethyl trifluoroacetate has acquired the name “ESCA molecule” (T RAVNIKOVA et al. 2012) because it shows the distinctively different chemical shifts of four different carbon groups (CH3 , CH2 , C=O and CF3 ) in such unrivalled clarity, and with the shifts following so nicely the respective electron affinities of these groups. 17 We have to use the dynamic relative permittivity (dielectric constant) here, which is much smaller than the static one (εstat 80).


photo electron signal

5.8

361

experiment at ħω = 80 eV

(a)

sticks spectrum: OVGF/6-311++G**

NH2 N O H

12 11

N

7 10 8 9

5

photo electron simulation

6

4

1 = 21a(π5 ) 2 = 20a(π4 ) 3 = 19a(σN ) 4 = 18a(σN ) 5 = 17a(π3 ) 6 = 16a(π2 ) 7 = 15a(σO) 8 = 14a(σ) 9 = 13a(π1 ) 10 = 12a(σ) 11 = 11a(σ) 12 = 10a(σ) 13 = 9a(σ) 14 = 8a(σ) 15 = 7a(σ) 16 = 6a(σ)

2 31

theory: ADC(3)/6-31G

(b)

15 + 16

14 13

8 11 12 10

7

9

5

-22

2 3 1

4

6

-20 -18 -16 -14 -12 -10 electron binding energy / eV

-8

Fig. 5.46 Photoelectron spectrum of cytosine (2b) in the gas phase according to T ROFIMOV et al. (2006). (a) Experiment and OVGS sticks spectrum, (b) ADC(3) sticks spectrum and convolution with experimental line profile

O

H

H H

F

C

C

C

C O

H

H

F

F photoelectron yield

Fig. 5.47 C 1s photoelectron spectrum of the “ESCA” molecule ethyl trifluoroacetate according to T RAVNIKOVA et al. (2012), recorded with ω = 340 eV. The binding energy of the C 1s electron at the CH3 has been determined to be WB = −291.47 eV (for the pure carbon atom, C HANTLER et al. (2005) gives the K edge at 283.8 eV). The chemical shift Wchem (i.e. the binding energy with respect to the main peak from the CH3 group) is characteristic for the chemical environment as indicated by red arrows

-10

-8

-6

-4

-2

2 0 ΔWchem / eV

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T RAVNIKOVA et al. (2012) have recently had a fresh look at the C 1s electron spectrum of the ESCA molecule in the gas phase, using a photon energy of 340 eV at the new French third generation synchrotron SOLEIL, with high resolution (ca. 50 meV FWHM: photon source 36 meV, electron monochromator ca. 20 meV). The photoelectron yield is plotted in Fig. 5.47 as a function the chemical shift Wchem = WB − WB (CH3 ) with respect to the binding energy WB (CH3 ) of the 1s electron at the CH3 group. Surprisingly, the four clearly resolved peaks are still significantly broader than the 50 meV FWHM of the apparatus. They also show a pronounced asymmetry – but no structure is resolved. T RAVNIKOVA et al. (2012) have carried out state-of-the-art structure calculations (using G AUSSIAN) for the four peaks exploring two conformers of the ESCA molecule: the anti-anti structure shown in Fig. 5.47 (which has Cs symmetry) and the anti-gauche structure (C1 , i.e. no symmetry) which is obtained from the latter by changing the dihedral angle of the COCC planes from 180◦ to 80◦ (not shown here). The ratio of the two conformers was assumed to be 44:56, based on prior ab initio calculations. Including initial and final state effects, a proper F RANCK -C ONDON analysis of the ionization process, and accounting for the finite lifetime of the core holes, they obtain a nearly perfect fit to the data in Fig. 5.47. The asymmetry of the peaks is attributed to post-collision effects (i.e. to electronic and nuclear rearrangement in the ion, including dissociation). It should also be noted that the ratio of the (integrated) four peaks is about 1:0.91:0.90:0.78 (and not 1:1:1:1 as one would expect from their stoichiometric relation). This effect is attributed to multiple scattering processes of the ejected photoelectron. We mention at this point (without going into details) that such effects are exploited with advantage for structural analysis of solid state targets in X-ray absorption spectroscopy by NEXAFS and EXAFS. XPS (alias ESCA) in a variety of forms with different acronyms has been developed over the past decades to a very efficient and robust method for the chemical analysis of surfaces, coatings, and thin films (a nice survey has been given by R EIN ERT and H ÜFNER 2005). It may also be used in position sensitive versions, e.g. in connection with X-ray microscopy, based on synchrotron radiation. In addition to C 1s edge spectroscopy (first and foremost relevant for organic materials) K edges of O, N, S and other characteristic atom may be used to analyze their chemical environment. Even K and occasionally L edges of metals are often used for a quantitative analysis of surfaces and depth profiles of thin layers (in the latter case in connection with sputtering methods for surface etching). State-of-the-art equipment for such analysis are today available on a commercial basis in various designs (see e.g. K ELLY 2004).

5.8.3

TPES, PFI, ZEKE, KETOF, MATI

If one compares the photoelectron spectra shown above with optical spectra which we have seen in previous sections of this chapter, one notices the significantly poorer energy resolution of PES – essentially a consequence of the fundamentally different properties of electrons and photons: no electron monochromator can ever reach the

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363

resolution which is available today with optical spectroscopy. Nevertheless, energy selective electron detection is essential for the spectroscopy of ionization processes. Hence, numerous efforts have been made since the early days of PES to improve electron energy resolution or to combine PES somehow with the advantages of optical spectroscopy. One basic concept has finally turned out to be successful: to record only electrons that are ejected at the energetic threshold of each ionization process, starting from a well defined initial state with quantum numbers (γ v N ) and leading to the final quantum numbers (γ v N ) according to (5.49). Thus, if one detects only electrons of practically negligible kinetic energies, these transitions can be determined with optical precision by just tuning the ionizing wavelength. One speaks of threshold photoelectron spectroscopy (TPES). To achieve zero kinetic energy detection, one exploits the fact that photoelectrons with finite kinetic energy and momentum are emitted essentially into the whole solid angle of 4π . If one extracts them from the ionization volume with only a small electric field, and limits the detection angle, the collection efficiency will increase with decreasing initial kinetic energy (steradiancy discrimination). This technique can be applied with continuous (discharge lamps, X-ray tubes) or quasi-continuous light sources (synchrotron radiation in multi bunch mode). Early experiments of this kind, have already been performed by BAER et al. (1969), and a particularly efficient TPES detector has been reported by C VEJANOV and R EAD (1974). With later improvements by K ING et al. (1987) the scheme is still used today very successfully (S ZTARAY and BAER 2003; C OUTO et al. 2006; E LAND 2009); in these spectrometers sophisticated electron optics extract threshold electrons (from a nearly field free ionization volume) with high efficiency and image them according to their energy onto a position sensitive detector (with such devices one obtains today electron energy resolutions in the sub-meV range; see e.g. BAER et al. 2012). We shall come back to this in Sect. 5.8.5. Alternatively one works with pulsed laser sources (laser pulses from ns to fs, or synchrotron radiation in the single bunch mode). In this case the time of ionization is well defined and electrons are extracted with some delay from the ionization volume (pulsed field ionization, PFI). Ideally, the timing is set such that all electrons with finite kinetic energy have left the ionization volume and only zero kinetic energy electrons are collected. This method has been used for the first time by M ÜLLER D ETHLEFS et al. (1984) as zero kinetic energy (ZEKE) photoelectron spectroscopy. A schematic is shown and explained in Fig. 5.48. We note here in passing that one may also separate ions of different kinetic energy in a similar manner, e.g. after ionization and fragmentation of a molecule. Such dissociative ionization processes are – at high enough photon energies – important phenomena (see also Sect. 5.8.5). Here too, ions are allowed to first drift field free for some time, after which they are accelerated and extracted by a delayed voltage pulse. In this case one has, however, to compensate the ion drift along the propagation direction of the molecular beam – which is here typically of similar order of magnitude as the fragment energy (in contrast to electrons). Such kinetic energy analysis by time of flight (KETOF), with very high energy resolution, has been used for the first time by H AUGSTÄTTER et al. (1988, 1989, 1990), detecting slow Na+

5 'steradiancy' analyzer

field free cted) ot dete fast e (n

ħω

detection cone 123 μ metal shielding V<0

pulsed extraction field

electron detectuib (channel plates)

ionization


time of flight (TOF) spectrum ZEKE photoelectrons transmission = 100 % signal

364

1

2

3 TOF

slow (near ZEKE) photoelectrons (low transmission)

Fig. 5.48 Principle of detecting ZEKE photoelectrons and discriminating against nearly ZEKE electrons according to M ÜLLER -D ETHLEFS and S CHLAG (1991). In the example shown an extraction field is applied to the originally field free interaction region after 1 µs. Electrons with an energy of only 0.1 meV (v = 6 mm/µs) have then travelled already on a sphere of 6 mm diameter (the centre of this sphere being the ionization volume); a few of them fly paraxially and are detected in the TOF spectrum on positions 1 and 3 and may thus be discriminated easily against true ZEKE electrons detected at 2. Non-paraxial nearly ZEKE electrons cannot reach the detector at all, since their perpendicular velocity component is too large (steradiancy discrimination)

fragment ions after dissociative ionization of the Na2 as a function of photon energy ω. As mass analyzed threshold ionization (MATI) spectroscopy this has also been proposed as an alternative to ZEKE photoelectron spectroscopy (see Z HU and J OHNSON 1991; L EMBACH and B RUTSCHY 1996). Figure 5.49 compares for the example of a two-photon ionization process of pyrazine (a) the total ion signal from direct photoionization with (b) the corresponding MATI and (c) ZEKE spectra. Detailed experimental studies and theoretical considerations have shown that the astonishingly high detection sensitivity of ZEKE and MATI methods are based on the fact that photo-absorption does not directly lead into the continuum, but rather to long-lived RYDBERG states very close to the ionization threshold. These are then ionized by the pulsed electric field. ZEKE spectroscopy has been developed into an extraordinary powerful method (M ÜLLER -D ETHLEFS and S CHLAG 1998), which is used in a variety of version for many applications, and which today is even available on a commercial basis.

5.8.4

PES for Negative Ions

So far we have only treated ionization of neutral, isolated molecules, radicals and clusters for which PES provides direct access to the electron binding energies, and allows at the same time to derive spectroscopic information on electronic and nuclear structure of the ions created. Of course, this method may also be used for negative ions. Typically, anions can only be prepared with very low concentration and their spectroscopy often relies exclusively on PES. Usually the weakly bound outer electron may readily be detached with typical laser sources. The reaction (5.49) has

Photoelectron Spectroscopy 100 50 ion signal / arb. units

Fig. 5.49 Two colour pump probe ionization spectra at the first ionization threshold of pyrazine according to Z HU and J OHNSON (1991). The pump laser excites the origin of the S1 state, the probe laser ionizes directly (or excites high lying RYDBERG states: (a) total, direct ion signal as a function of the probe laser wavelength; (b) MATI ion signal at the first ionization threshold 000 and for three low lying vibrational states of the ion; (c) corresponding ZEKE photoelectron signal; for (b) as well as for (c) ions (or electrons) are generated by field ionization in a weak, pulsed electric field

365

(a)

direct photoionization

(b)

field ionization (MATI)

0

100

000

6a10

50

16b10

16b20

0

100 e- signal

5.8

000

(b)

ZEKE PES 6a10

50 16b10

16b20

0 227

226

225

224

223

222

probe laser wavelength / nm

now to be written as ω + AB− γ v N → AB γ v N + e− (Wkin ),

(5.56)

and the ionization potential WI in (5.51) has to be replaced by the electron affinity WEA of the neutral molecule or cluster studied. PES with anions (electron photo-detachment spectroscopy) is an active field of research for now at least 40 years and literature about it is extensive. The field has essentially been shaped by W.C. L INEBERGER and his many students and collaborators (see e.g. L INEBERGER and W OODWARD 1970; H OTOP and L INEBERGER 1985; E RVIN and L INEBERGER 1992; N EUMARK 2001, 2002; R IENSTRA -K IRACOFE et al. 2002; E LLIOTT et al. 2008; S HEPS et al. 2009) and a few other groups worldwide (e.g. C HA et al. 1992; YANG et al. 1987; L EE et al. 1991; TAYLOR et al. 1992; M ARKOVICH et al. 1994; C ASTLEMAN and B OWEN 1996; W RIGGE et al. 2003, and further references there). The spectroscopic methods differ only little from those for neutral atoms, molecules and clusters.18 Imaging detection methods (VMI) too are very popular these days in 18 There is, however, one key difference: anions do not come out of the bottle and have to be specifically prepared, and – as they carry a charge – may also be mass selected prior to the interaction with photons. Specifically for the study of clusters this is an essential advantage compared to neutral cluster beams. They usually have a broad distribution of cluster sizes, and mass selective detection after the interaction process does not really help, since usually very difficult to discriminate against fragments from larger clusters.

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anion spectroscopy (e.g. E LLIOTT et al. 2008) and a number of acronyms have been coined to characterize various techniques.

5.8.5

PEPICO, TPEPICO and Variations

For all the above discussed methods of PES one tacitly makes two essential assumptions: 1. The molecule (or cluster) studied remains intact – apart from the electron loss due to the ionization process (5.49) – or potential dissociation processes can be identified without doubt as being correlated with the observed electron. 2. One deals with one well known species of target molecules or may distinguish different species by different positions of the absorption bands (the kind of complications that arise when this is not the case have been indicated for the “ESCA” molecule on p. 361 f.). If one of these assumptions is not satisfied the electrons detected cannot be correlated to a specific ion, and consequently the photoelectron spectrum observed can provide only limited insight. At sufficiently high photon energy ω, even for the most simple case – a diatomic molecule – either a stable (vibrationally) excited parent-ion is generated as indicated in Fig. 5.44(b). Alternatively, the molecule might fragment during ionization if one assumes the potential minimum position for the AB+ ion to be shifted to a slightly larger R so that the FC region would extend above the dissociation limit (WI + D0 ) of the AB+ parent-ion, as sketched in Fig. 5.44. This so called dissociative ionization is a rather common process for larger molecules. Already for a triatomic molecule one expects at higher ω several open channels for such processes: (5.57) ω + ABC → ABC+ γ v N + e− (Welkin ) → AB + C+ (Wionkin ) + e− (Welkin ) → A + BC+ (Wionkin ) + e− (Welkin ) → AB+ + C(Wionkin ) + e− (Welkin ), . . . etc. Figure 5.50 illustrates the first and the second channel, here as a cut along one nuclear coordinate through the potential hypersurface of the triatomic ABC+ ion. We have to distinguish between the kinetic energy of the electron Welkin and the kinetic energy Wionkin of the fragments. The latter refers of course to the total relative energy in the centre of mass system AB + C+ of the dissociating molecular ion. The balance of energy (5.50) has to be extended slightly for such a process: Wγ v N . (5.58) Welkin + Wionkin = ω − (WI − Wγ v N ) − ( In the case of polyatomic molecules or clusters, Wγ v N is the sum of all internal energies of all molecular fragments (neutral and/or ionic) – in the example (5.57) these would be the vibration-rotation energies in ABC+ , or in AB or BC+ etc., respectively, depending on the channel studied. An unambiguous spectroscopy of

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367

W R) W( e-

ħω

FC region

Welkin(AB+C+)

Welkin

(ABC+)

v'N' ~ X+

AB+C+ +Wionkin

ABC+

D0'

RAB-C

Fig. 5.50 Dissociative ionization of a triatomic molecule. The F RANCK -C ONDON (FC) region (heavy, black arrow) is defined by the potential minimum of the ground state (not shown here) and extends in the present case over bound states of the ABC+ ions up to the dissociative limit D0 (electron energy Welkin (ABC+ )). But it is also sufficiently high to reach into the dissociation continuum of the open channel AB + C+ ; in that case, the C+ ion and the neutral fragment AB separate with a relative kinetic energy Wionkin , while the electron carries a kinetic energy Welkin (AB + C+ )

such a process is possible only if the ejected electron e− can be associated uniquely with a molecular ion ABC+ or fragment C+ or BC+ etc., respectively. If several channels are open, this can be achieved by coincident detection of ions and electrons – if possible with a full analysis of all energies and momenta (directions) of fragments and electrons. This is, of course, a big challenge which only in a few special cases has been achieved completely. Over the years, however, great progress has been made towards this goal. Since more than 40 years, energy selected ions are prepared and analyzed by photoelectron-photoion coincidence spectroscopy, PEPICO (see e.g. B REHM and VON P UTTKAMER 1967; E LAND 1972; DANBY and E LAND 1972; W ERNER and BAER 1975; JARVIS et al. 1999; S ZTARAY and BAER 2003; BAER et al. 2005; E LAND 2009). The basic concept, sketched in Fig. 5.51(a), has remained essentially the same as first used in the 1970s. However, modern time and position sensitive detection methods with fast electronics, sophisticated electron and ion optics and imaging methods as indicated in Fig. 5.51(b) have brought us rather close to the ideal of a state selective analysis of photoionization and fragmentation dynamics for some selected model systems. To understand the principle, one has to realize that these methods have been devised originally for rather weak light sources (gas discharge lamps with monochromators), and individual dissociative processes were detected with a very low count rate. Thus, each photoelectron detected can in principle be correlated to one specific ion. Due to its much lower velocity the latter is observed with a time delay in respect of the electron. The electron triggers, as in-

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5

ħω

(a) e- MCP

acceleration ion energy Ionen Masse and/or mass bzw. Energie analyser Analsator

e- energy analyser detecting electronics

extraction field

ion pulser

start

variable delay line

(b) Roentdek DLD40

ħω 40 V/cm hot eZEKE


MCP for ions

detecting electronics TAC or TDC

stop

PC (MCA, dig. osci.) optional deceleration ions

MCP for ions

esplit plate

Fig. 5.51 (a) Principle of PEPICO or TPEPICO experiments – in the spirit of E LAND (1972) and BAER (1979). Today, electrons and ions are usually detected by multi channel plates (MCP). The electron signal triggers the ion extraction and the timing (start). In earlier experiments, the difference between the flight times of the electron and the ion (stop) have measured by time– to-pulse-height conversion (TAC) with subsequent multichannel analyzers (MCA); today one digitalizes this time delay directly (TDC) and records it in a PC. (b) Modern version of electron (left) and ion optics (right) according to B ODI et al. (2009). The trajectories for ‘hot’ (pink) and threshold electrons (ZEKE, red) are separated by spatially resolved detection (RoentDek DLD40)

dicated in Fig. 5.51(a), a coincidence electronics which measures the time until the detection of an ion. In the most simple case one lets the electrons drift out of the interaction region with the photon ω in a very low electric field and thus limits the acceptance angle as already discussed in Sect. 5.8.3. Such a “steradiancy” analyzer selects preferentially very slow electrons (TPES) – emitted when the photon energy is just sufficient to excite a specific state of the ion. The electron signal then triggers a pulsed electric extraction field for the ions which were essentially at rest until this moment. The delay until detection identifies the mass of the ion which has been generated in this particular ionization event – with suitable ion optics one may even record the kinetic energy of the ion. Each event is registered in the memory of the PC according to its delay time. Subsequent events are added in the storage corresponding to their time delay until sufficiently good statistics has been reached. The whole procedure is repeated for each photon energy of interest, preferentially in rapid iteration to compensate for experimental fluctuations. In this manner a so called threshold photoelectron-photoion coincidence (TPEPICO) spectrum is built up. A wealth of


Fig. 5.52 TPES of Ar2 according to B ODI et al. (2009). The dark line shows the electron signal at nearly zero kinetic energy (ZEKE), recorded in coincidence with Ar+ 2 ions. The weak grey line represents the total ZEKE signal (without coincidence). The resolution of the photon energy is about 2 meV. The bottom curve represents the total ion signal (without energy discrimination). The peaks correspond to autoionization processes

369

ZEKE an ion signal / arb.un.

5.8

Ar2 TPES

Ar2+ ion yield 14.5

15.0

15.5 ħω / eV

results documents the efficiency of the TPEPICO method (see e.g. the review of BAER et al. 2005, and further references there). We mention also the fact that TPEPICO has also been used successfully in cases where – albeit no fragmentation was expected – the target contained several species which all could be ionized at a given photon energy. Specifically this is the case for neutral cluster beams where typically a broad mass distribution is created. In that manner during the 1980s and 1990s numerous ionization potentials of neutral atomic (Arn , Krn , K AMKE et al. 1989) and molecular clusters ((NH3 )n , (N2 O)n , K AMKE et al. 1988; G REER et al. 1990) have been determined using TPEPICO, to mention just a few examples. Over the years the methods have been improved systematically. While initially the time of flight was measured by time to amplitude converters (TAC) with subsequent recording in multi channel analyzers,19 in the mean time one uses direct, digital recording of the time difference between electron and ion signal in TDCs (time to digital converters) or digital oscillographs. Modern devices also register several subsequent hits (multi hit) or may start again during one measuring process (multi start) – an overall significant improvement in flexibility and reduction of dead-time losses (B ODI et al. 2007). This is particularly important for investigations with intense light sources, such a synchrotron radiation and laser systems, which allow high count rates. As indicated in Fig. 5.51(b), recent developments combine PEPICO methods with state-of-the-art imaging techniques (the acronym iPEPICO has been coined). In this setup the electron extraction optics is designed such that the slow threshold electrons (ZEKE) hit the position sensitive detector in the middle, while faster electrons are detected exclusively on rings around the axis and are thus energy analyzed. The high resolution and the ability to discriminate different masses is documented in Fig. 5.52 for the example of the Ar2 dimer molecule (TPES). The resolution is 19 In these storage devices, originally designed for nuclear physics experiments, events are added up and stored according to their pulse height.

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on the order of a few meV. The grey line in the upper half of the graph (signal without coincidence, i.e. from all Ar+ n cluster ions in the molecular beam) differs in this case only very little from the black line, the pure electron signal correlated with Ar+ 2 . Obviously the ionization cross section for higher clusters e.g. Ar3 , is very small in this energy range. Generally speaking, the possibility to correlate electron spectra to different fragment masses is a key feature of this kind of setup – but obviously very demanding and time consuming, considering the overall output from these sophisticated sources making full use of their potential. In contrast to the determination of ionization thresholds with TPEPICO, the more general PEPICO method requires a true electron energy analyzer for arbitrary energies. In earlier years, electrostatic fields were used for this purpose. However, the imaging techniques just described, with sophisticated electron optics, provide an interesting alternative (see also the summary on the reaction microscope in Appendix B.4). On the other hand, today one often uses pulsed light sources which are a priori ideally suited for time of flight methods to determine the electron energy. One must, however, keep the number of random coincidences sufficiently low. Hence, for a long time it appeared nearly impossible to perform PEPICO experiments with pulsed laser sources: as a rule, each laser pulse generates many ions and the assignment of the electrons detected to specific ions appeared impossible. For the first time S TERT et al. (1997, 1999) have overcome this problem by working at extremely low ion yields ( 1 per laser pulse). They used a two colour pump-probe scheme with strongly attenuated, high repetition femtosecond laser pulses to ionize molecules and clusters via a resonant intermediate state. This femtosecond time resolved electron ion coincidence (FEICO) method exploits a magnetic bottle (see Sect. B.4) by which electrons may be collected from a nearly 4π emission angle, while their energy is measured by electron TOF. Here too the detected electrons trigger a fast high voltage pulse to extract the ions. The method has opened for the first time the possibility to study dynamical process on a femtosecond time scale in photo-excited, neutral molecules and clusters by TOF in a mass selective manner. As an example we mention ammonia clusters (FARMANARA et al. 1999), which show an interesting fragmentation scheme. In a mass spectrum they are predominantly observed as protonated species: ˜ ωpr ωpu (NH3 )n (A) ˜ ˜ (NH3 )n (X) −→ (NH3 )n (A) → (5.59) ˜ + NH2 −→ · · · (NH3 )n−1 H(A) or

ωpr − (NH3 )+ n ˜ −→ (NH3 )n (A) e (Welkin ) + (NH3 )+ → n (NH3 )n−1 H+ + NH2 .

(5.60)

We cannot enter here into the details of the observed dynamics. However, Fig. 5.53 gives a brief survey for one fixed delay time (t = 0 fs) between pump (ωpu ) and probe pulse (ωpu ). Shown here are the raw data and some key results. They document that one can indeed observe mass selective photoelectron spectra from neutral molecular clusters using femtosecond pulsed lasers.

9.02

20

30 ion time of flight / μs

40

normalized e- signal

(c)

(b) n=2 n= 1 20

3

mass spectrum 5 6

4 3

2 40

electron spectra

4 60

Welkin / eV

10

9.54

1 0 Welkin(max)

AP/eV 9.2

2

NH4+

9.14

2

0

ion signal

(NH3)+

(a) raw data

(NH3)2NH4+

371

NH3NH4+

(NH3)3NH4+

4

(NH3)2NH4+

NH3NH4+


NH4+

electron time of flight / μs

5.8

7

5 80

100

M/u

Fig. 5.53 Electron ion coincidence (PEPICO) for the resonant two-photon, two colour ionization of (NH3 )n clusters in a FEICO experiment. According to (5.59) or (5.60) protonated (red) and unprotonated (pink) cluster ions may be generated. (a) Raw data: each registered delayed coincidence is represented by one point as a function of electron and ion time of flight. In vertical direction the electron energy varies, in horizontal direction the observed mass M. (b) Mass spectra are obtained by projection of all these data onto the TOF axis of the ions. The values of n refer to the size of the respective neutral parent cluster. (c) Electron spectra associated with a given ion are obtained by projection of the raw data for a particular ion onto the electron TOF axis. Of course, finally the TOFs and signal heights have to be converted, in the former case onto a mass scale M/u, in the latter case onto a the electron energy scale Welkin . The grey, horizontal arrows indicate in each case Welkin (max), the maximum electron kinetic energy observed. It corresponds the grey numbers in called appearance potential, AP = ω − Welkin (max)

In principle, this type of experiments may also be performed at synchrotron storage rings of the third generation. Highly repetitive single or even double pulses can be provided there, typically with a pulse duration of some ps. Pulse distances of some 100 ns allow a comfortable TOF analysis of electron energies. With these, similar methods as those just described can be used in the VUV spectral range (JARVIS et al. 1999) – even though the obvious difficulties of these techniques have prevented a big flow of data so far. Section summary

• The kinetic energy of photoelectrons Wkin depends on the energy of the ionizing photon and on the internal energies of the initial neutral and final ionic state according to (5.50). This relation forms the basis of photoelectron

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spectroscopy (PES). The probability of photoelectron emission is determined again by the respective FC factors. • We have seen that PES of molecules can provide very valuable information about the photoionization process as such, about electronic and vibrational structure of neutrals and ions, and give detailed information on MOs (both in inner and outer shells). • The energy levels of inner shell PESs, studied with XPS, are significantly influenced by their chemical environment. Hence, the so called ESCA, originally introduced by S IEGBAHN (1981) and his collaborators has developed into a very efficient and much used analytical method – not only for isolated molecules but also for molecules at solid state surfaces and thin films. • Combined with state-of-the-art quantum chemistry, PES is today a very powerful tool in molecular spectroscopy. A variety of specialized methods and techniques have been devised and were refined over the years, involving the selection of threshold electrons (TPES, ZEKE, MATI), using sophisticated electron optics and imaging techniques, as well as coincidence techniques for studying dissociative ionization, which allow one to correlate electrons to one specific fragmentation process (PEPICO, TPEPICO, FEICO).

Acronyms and Terminology AOM: ‘Acousto-optic modulator’, device to modulate and shift the frequency of light by diffraction in a B RAGG grating generated by sound waves (usually RF). BOXCARS: ‘Schematic geometry of a setup for nonlinear spectroscopy’, (see Fig. 5.42). CARS: ‘Coherent anti-S TOKES R AMAN scattering’, coherent version of R AMAN scattering. CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). CFWM: ‘Coherent four wave mixing’, nonlinear optical processes (see Sect. 5.7.1). conformer: ‘Special kind of isomers (same atomic composition but different molecular structure) having the same sequence of atoms but different geometrical arrangement, such as cis-trans isomers or different alignment with respect to rotation around an axis’, http://en.wikipedia.org/wiki/Conformational_isomerism. COORS: ‘Common ordinary old R AMAN scattering’. CRD: ‘Cavity ring down’, spectrometer (see Sect. 5.5.3). CSRS: ‘Coherent S TOKES R AMAN scattering’, coherent version of R AMAN scattering. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. DF: ‘(laser induced), dispersed fluorescence’. DFWM: ‘Degenerate four wave mixing’, nonlinear optical process (see Sect. 5.7.1). DNA: ‘Deoxyribonucleic acid’, large nucleic acid which contains the genetic code according to which living organisms are build.


373

E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. EPR: ‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2 in Vol. 1). ESCA: ‘Electron spectroscopy for chemical analysis’, see Sect. 5.51. ESI: ‘electro spray ionization’, method for bringing very large molecular ions into the gas phase (see Sect. 5.28). esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3 in Vol. 1). EUV: ‘Extreme ultraviolet’, part of the UV spectral range. Wavelengths between 10 nm and 121 nm according to ISO 21348 (2007). EXAFS: ‘Extended X-ray absorption fine structure’, X-ray absorption by inner shell electrons in a broad energy range above the respective X-ray absorption edge (as opposed to NEXAFS). FC: ‘F RANCK -C ONDON’, introduced an important approximation for optical transition between electronic states (see Sect. 5.4.1). FDIRS: ‘fluorescent-dip infrared spectroscopy’, (see Z WIER 2001). FEICO: ‘Femtosecond time resolved electron ion coincidence’, see Sect. 5.8.5. FIR: ‘Far infrared’, spectral range of electromagnetic radiation. Wavelengths between 3 µm and 1 mm according to ISO 21348 (2007). FPI: ‘FABRY-P ÉROT interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2 in Vol. 1). FT: ‘F OURIER transform’, see Appendix I in Vol. 1. FTIR: ‘F OURIER transform infrared spectroscopy’, see Sect. 5.3.2. FWHM: ‘Full width at half maximum’. FWM: ‘Four wave mixing’, nonlinear optical processes (see Sect. 5.7.1). HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9 in Vol. 1). HHG: ‘High harmonic generation’, in intense laser fields. HITRAN: ‘High-resolution transmission molecular absorption database’, http:// www.cfa.harvard.edu/hitran (ROTHMAN et al. 2009). IAS: ‘Infrared action spectroscopy’, special method to detect infrared absorption by particle detection (see Sect. 5.3.3). IC: ‘Internal conversion’, radiationless transition between different electronic states (see Sect. 5.4.3). iPEPICO: ‘Imaging photoelectron-photoion coincidence spectroscopy’, see also PEPICO, Sect. 5.8.5. IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). ISC: ‘Intersystem crossing’, radiationless transition between states with different total spin, typically between singlet and triplet states (see Chap. 5, Fig. 5.15). isomer: ‘Molecules with the same atomic composition but different molecular structure’, http://en.wikipedia.org/wiki/Isomer. isosceles triangle: ‘Triangle with two equal sides’, has two varieties: acute (all angles are <90◦ ) and obtuse (one angles is >90◦ ).

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isotopologue: ‘Molecules that differ only in their isotopic composition’, http://en. wikipedia.org/wiki/Isotopologue. isotopomer: ‘Molecules with the same number of isotopes of each element but differ in their position within the molecule’, http://en.wikipedia.org/wiki/ Isotopomers. IVR: ‘Intra molecular vibrational energy redistribution’, excess vibrational energy in one mode of a polyatomic molecule is redistributed among other vibrational modes. JT: ‘JAHN and T ELLER’, have first treated in 1937 the symmetry breaking effect, now referred to by their names. JTE: ‘JAHN -T ELLER effect’, symmetry breaking effect first treated by JAHN and T ELLER in 1937. KETOF: ‘Kinetic energy analysis by time of flight’, method for determining fragmentation energies after dissociative ionization. LIF: ‘Laser induced fluorescence’, radiation emitted from a quantum system after excitation by laser radiation (see Sect. 5.5.1). M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. MALDI: ‘Matrix assisted laser desorption ionization’, method for bringing very large molecular ions into the gas phase (see Sect. 5.28). MATI: ‘Mass analyzed threshold ionization’, see Sect. 5.8.3. MB: ‘Molecular beam’. MCA: ‘Multi channel analyzer’, electronic device, storing pulses according to their pulse height (originally used in nuclear physics). MCP: ‘Multi channel plate’, electron multiplier with many amplifying elements. MIR: ‘Middle infrared’, spectral range of electromagnetic radiation. Wavelengths between 1.4 µm and 3 µm according to ISO 21348 (2007). MO: ‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation. MRCI: ‘Multi reference configuration interaction’, high quality quantum chemical method for computing molecular potentials. MW: ‘Microwave’, range of the electromagnetic spectrum. In spectroscopy MW usually refers to wavelengths from 1 mm to 1 m corresponding to frequencies between 0.3 GHz to 300 GHz; ISO 21348 (2007) defines it as the wavelength range between 1 mm to 15 mm. MWFT: ‘Microwave F OURIER transform’, spectrometer (see Sect. 5.2). NEXAFS: ‘Near edge X-ray fine structure absorption, also XANES’, X-ray absorption by inner shell electrons close to the respective X-ray absorption edge. NIR: ‘Near infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1.4 µm according to ISO 21348 (2007). NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. NMR: ‘Nuclear magnetic resonance’, spectroscopy, a rather universal spectroscopic method for identifying molecules (see Sect. 9.5.3 in Vol. 1). ODE: ‘Ordinary differential equation’.


375

OMA: ‘Optical multichannel analyzer’, spectrometer which allows simultaneous registration of a whole spectrum. OODR: ‘Optical-optical double resonance’, spectroscopy with two photons, one kept fixed on a resonance transition, one tuning another part of the spectrum. PEPICO: ‘Photoelectron-photoion coincidence spectroscopy’, method to correlate a photoelectron with one specific fragment ion (see Sect. 5.8.5). PES: ‘Photoelectron spectroscopy’, see Sect. 5.8. PFI: ‘Pulsed field ionization’, electrons are extracted from the ionization volume with some time delay. PJTE: ‘Pseudo-JAHN -T ELLER effect’, vibronic coupling for nearly degenerate molecular states, leading to symmetry breaking. R2PI: ‘also RTPI, resonantly enhanced two-photon ionization spectroscopy’, special version of REMPI. REMPI: ‘Resonantly enhanced multi-photon ionization’, ionization of atoms or molecules by several photons with one resonant intermediate state. RF: ‘Radio frequency’, range of the electromagnetic spectrum. Technically, one includes frequencies from 3 kHz up to 300 GHz or wavelengths from 100 km to 1 mm; ISO 21348 (2007) defines the RF wavelengths from 100 m to 0.1 mm; in spectroscopy RF usually refers to 100 kHz up to some GHz. RIDIRS: ‘Resonant ion dip infrared spectroscopy’, (see Z WIER 2001). RTPI: ‘also R2PI, resonantly enhanced two-photon ionization spectroscopy’, special version of REMPI. SEM: ‘Secondary electron multiplier’, see Appendix B.1. SEP: ‘Stimulated emission pumping’, special kind of two colour resonant four wave mixing (see TC-RFWM). SERS: ‘Surface enhanced R AMAN spectroscopy’. SI: ‘Système international d’unités’, international system of units (m, kg, s, A, K, mol, cd), for details see the website of the Bureau International des Poids et Mésure http://www.bipm.org/en/si/ or NIST http://physics.nist.gov/cuu/Units/index. html. TAC: ‘Time to amplitude converter’, electronic device, same as time to height converter. tautomer: ‘Special isomers which readily interconvert by moving single atoms (e.g. H) or atomic groups’, http://en.wikipedia.org/wiki/Tautomer. TC-RFWM: ‘Two colour resonant four wave mixing’, nonlinear optical process (see Sect. 5.7.1). TDC: ‘Time to digital converter’, electronic device. TOF: ‘Time of flight’, measurement to determine velocities of charged particles, and consequently their energies (if the mass to charge ratio is known) or their mass to charge ratio (if their energy is known). TPEPICO: ‘Threshold photoelectron-photoion coincidence spectroscopy’, method to correlate photoelectrons of nearly zero kinetic energy with one specific fragment ion (see Sect. 5.8.5). TPES: ‘Threshold photoelectron spectroscopy’, PES of only those electrons which are emitted with nearly vanishing kinetic energy, i.e. at threshold of the process studied.

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UPS: ‘Ultraviolet photoelectron spectroscopy’. UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). VIS: ‘Visible’, spectral range of electromagnetic radiation. Wavelengths between 380 nm and 760 nm according to ISO 21348 (2007). VMI: ‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B). VUV: ‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007). XANES: ‘X-ray absorption near edge spectroscopy, also NEXAFS’, X-ray absorption by inner shell electrons close to the respective X-ray absorption edge. XAS: ‘X-ray absorption spectroscopy’, Used for to study the electronic states of inner shell electrons. XPS: ‘X-ray photoelectron spectroscopy’, see Sect. 5.8.1. XUV: ‘Soft x-ray (sometimes also extreme UV)’, spectral wavelength range between 0.1 nm and 10 nm according to ISO 21348 (2007), sometimes up to 40 nm. ZEKE: ‘Zero kinetic energy’, photoelectron spectroscopy (see Sect. 5.8.3).

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6

Basics of Atomic Collision Physics: Elastic Processes

In this and the two following chapters we present a perspective onto the world of electronic, atomic and molecular collision processes. It constitutes and active and productive field of modern physics, which is often neglected in academic education – in spite of a wealth of exciting questions and enormous practical relevance. In essence, atomic collision physics is concerned with the continuum of (free) atomic and molecular states, in contrast to standard spectroscopy which focusses on bound states.

Overview

The subject of this and the following two chapters is collisions between electrons, atoms, ions and molecules. We mostly refer here to examples from the particularly productive pioneering period between 1965 and 1990. However, when appropriate, we mention already state-of-the-art research. In Sects. 6.1 and 6.2 we familiarize ourselves with cross sections, and how they are measured, with collision kinematics and its applications. As far as scattering theory is concerned we shall refrain from rigid derivations and prefer easy to understand models. In Sect. 6.3 we introduce elastic scattering and its classical theory while Sect. 6.4 outlines the quantum theory of elastic scattering. A first glimpse on resonances is given in Sects. 6.5 and 6.6 introduces B ORN approximation for elastic scattering. The two following chapters will go into more depth and treat in particular also inelastic processes. We thus recommend to the reader to study the present chapter with particular care.

6.1

Introduction

Collision physics1 is concerned with the dynamics of interacting particles – in contrast to spectroscopy which focusses onto the structure of matter. We shall address 1 We shall use the terms “collision” and “scattering” physics essentially as synonyms – perhaps with some emphasis on the entrance and exit channels, respectively.


383

384

6


Fig. 6.1 Typical potential between two interacting particles with characteristic energy domains of collision physics (W > 0) and spectroscopy (W < 0)

V(R ), W scattering physics: free particles energy continuum dynamics cross sections

W> 0

0 W< 0

R

spectroscopy: bound particles discrete energies structure transition frequencies

here electron scattering (e− + atom, ion, molecule) as well as the so called “heavy particle” scattering, i.e. processes where atoms, ions and molecules interact with each other (an excellent, profound introduction into the latter area gives L EVINE 2005, in the following we shall refer again to the original literature and to reviews for more details). Figure 6.1 illustrates – somewhat schematically – the relation of energy domains to scattering physics (W > 0) and spectroscopy (W < 0) for the interaction of two particles. Clearly, the limits are not sharp and the most interesting phenomena and insights of modern atomic and molecular physics are often found where the fields overlap, as e.g. for autoionization and resonance scattering, with ultra cold atoms and molecules or in ultrafast physics and femtochemistry – all cutting edge research themes which to some extend bridge the two fields as far as atomic, molecular and cluster physics is concerned. Specifically in the development of new methods, today spectroscopy and scattering physics profit of each other in a very direct manner. It may suffice at this point to mention the application of state-of-the-art laser techniques and molecular beams, or multidimensional measuring and detection methods with sophisticated imaging devices – methods which both fields rely on since several years. Nevertheless, one should bear in mind the different methods, concepts and goals if one wants to assess the status of modern research in both fields. The most obvious distinction are the objects of the respective investigations: in spectroscopy one is usually interested in determining the positions of energy levels. This is possible today with a precision of which collision physics does not even dare to dream. The latter usually takes the energetic position of atomic or molecular levels as given and measures cross sections for specific dynamical processes. As a consequence of the different nature of such measurements for collision processes, measurements with an accuracy of a few percent have to be rated as great achievements. Nevertheless, during the history of physics, this kind of experiments has led time and again to key insights and breakthroughs. We recall the historic experiments of RUTHER FORD (atomic structure) and H OFSTÄDTER (structure of atomic nuclei, Nobel prize 1961), as well as the whole high energy physics which finally lives of collision pro-

6.1

Introduction

385

cesses. With Ugo FANO one may say that the “final goal of all low energy physics is elucidation of physico-chemical elementary processes in terms of wave mechanical concepts”. Beyond this somewhat puristic aspect, we emphasize the enormous practical significance which the precise knowledge of scattering and reaction cross sections has for practically all areas of physics and physical chemistry, in particular for plasma physics (and thus in particular for fusion research), for astrophysics, for atmospheric research or for radiation physics and chemistry.

6.1.1

Integral and Total Cross Sections

We distinguish several types of processes in a collision of two particles A and B (possibly in different initial and final states, |a and |b, respectively): A+B→A+B

elastic scattering

inelastic scattering A + B(a) → A + B(b)

(6.1) (6.2)

ionization

A + B → A + B+ + e−

(6.3)

reactive scattering

A + B → C + D + ··· .

(6.4)

To specify a few examples for these types of collisions for illustration, we mention: ionization by electron impact, e.g. e− + H → 2e− + H+ , a so called (e, 2e) process, dissociative attachment e− + AB → A− + B, charge transfer A + B+ → A+ + B or chemical reactions A + BC → AB + C. The variety of processes is gigantic and many of them are of considerable practical importance. Today, a multitude of methodical developments open unforeseen possibilities for studying such processes, many of them based on the continuous progress of laser physics and refined detection techniques for atomic and molecular particles. The key quantity to be determined is the so called cross section, which we specify as σel , σinel , σion and σreact for elastic, inelastic, ionizing, and reactive cross sections. At this point we do not yet distinguish according to the angle under which the colliding particles are scattered. This aspect will be introduced in Sect. 6.2. The so called integral cross section has been integrated over all scattering angles. The sum of all the different integral cross sections for a given pair of interacting particles is usually called the total cross section σtot = σel +

σinel +

σion +

σreact ,

(6.5)

even though there is occasionally some confusion in the literature between integral and total cross section. A total cross section is determined by an absorption experiment as sketched in Fig. 6.2. A beam of particles A (think e.g. of atoms, ions or electrons) with a flux Φ0 ([Φ0 ] = particles s−1 m−2 ) passes through a static target in a collision chamber. We recall from Sect. 1.3.2, Vol. 1 that the loss of flux for

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6


Fig. 6.2 Absorption experiment, schematic, for determining the total cross section of A + B collisions

collimating aperture source

target B scattering chamber detector for A flux Ф(d )

A beam flux Ф0

state and/or energy selector

x

0 Δx d

particles A over a distance x is given by ΦA = −ΦA (x)σtot NB x,

(6.6)

where NB is the density of the target gas ([NB ] = particles m−3 ). The thus introduced total cross section σtot for all the reactions A + B according to (6.1)–(6.4) has 2 if A and the dimension L2 . It may be thought to represent an interaction area πRint B interact over a distance Rint . In this simple form (6.6) is of course only valid if σtot NB x 1. For an extended or more dense interaction volume of length d the integration leads to the well know L AMBERT-B EER law ΦA (d) = ΦA0 e−dσtot NB = ΦA0 e−d/ l

with l = 1/(σtot NB )

(6.7)

being the mean free path length of particle A in the target gas B. To compare the terminology of collision physics with that of chemical reaction kinetics one writes (6.6) in symmetric form: −

ΦA = ΦA σ NB = σ vNA NB . x

(6.8)

We suppress here the index for σ , since the relation holds as well for all the partial cross sections, e.g. for elastic or reactive processes. The particle flux ΦA = vNA is derived from velocity v and density NA of the particle beam A. Its decrease per length, i.e. (minus) its derivative −dΦA /dx corresponds to the number of scattering events N˙ AB per time and volume. Thus, (6.8) is also valid for a gas mixture in a cell – providing the velocity v is replaced by the relative velocity vrel of the two particles involved. Hence, we may rewrite (6.8) as a rate equation N˙ AB = vrel σ NA NB = kAB NA NB ,

(6.9)

with the rate constant2 kAB = vrel σ which is measured in units [kAB ] = m3 s−1 . Chemical reactions are usually studied in gas cells or in the condensed phase, in each case in an environment of many particles with a distribution of velocities. The latter can often be described by a thermal (M AXWELL -B OLTZMANN) distribution fA (vA , MA ) and fB (vB , MB ), respectively, with velocities vA and vB and the respective masses MA and MB . A similar situation is encountered e.g. for collision has to distinguish the rate constant kAB defined here (unit m3 s−1 ) of the rate Rba (unit s−1 ) which we have introduced in Chap. 4, Vol. 1, in the context of light induced processes, which is defined per target atom.

2 One

6.1

Introduction

387

processes in plasmas or in atmospheric physics and chemistry. The relevant cross sections depend on the relative velocities vrel = |v A − v B | of the collision partners – to be described also by a M AXWELL -B OLTZMANN distribution. To obtain an effective rate constant one finally has to average (6.9) over all relative velocities: (6.10) kAB = vrel σ = frel (vrel )vrel σ (vrel )d3 v rel . If the cross section depends only weakly on the relative velocity on may pull σ (vrel ) = σ in front of the integral, so that: kAB vrel σ with vrel = vA 2 + vB 2 . (6.11) As mentioned in Sect. 1.3.4, Vol. 1, the average velocities of particles A and B in aM AXWELL -B OLTZMANN distribution at a temperature Θ are given by vA,B = 8kB Θ/πMA,B (with the B OLTZMANN constant kB ). If target and projectile are √ identical, according to (6.11) the average collision velocity is vrel = 2v – with v being the average velocity of these particles. However, as a rule the cross section σ depends on the relative velocity and from (6.11) one can only derive an average. Structures, such as resonances, which strongly depend on the energy, can only be detected with beam experiments in which the velocity of the projectile is preselected and the target is appropriately cooled. We shall illustrate this with an interesting example in Sect. 6.3.4.

6.1.2

Principle of Detailed Balance

At this point we derive a useful relation between cross sections for collision processes which may be considered the inverse of each other. Let us consider as an example the inelastic process (6.2) and its inverse and let us assume these processes to occur in a gas in thermodynamic equilibrium. For simplicity we take particle B to be very heavy3 as compared to A, the latter being e.g. an electron which carries essentially all available kinetic energy T prior to the (inelastic) excitation process σba (T )

A(T ) + B(a) −→ A(T − Wba ) + B(b),

(6.12)

while T − Wba is its kinetic energy after the process, and Wba is the excitation energy for the transition |b ← |a. The de-excitation process (also called “superelastic” process) is exactly the inverse: one starts with a kinetic energy T − Wba and the state |b as initial state: A(T − Wba ) + B(b)

σab (T −Wba )

−→

A(T ) + B(a).

(6.13)

3 One may drop this restriction without changing the result – the derivation becomes, however, much less transparent.

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6


Fig. 6.3 Schematic illustration of energy loss and energy gain, respectively, and energy redistribution by collisions in the energy distribution of a gas with M AXWELL -B OLTZMANN energy distribution according to (1.58), Vol. 1

f (T )

a→b

energy gain

b←a

energy loss

dT

dT T

T–Wba

T

The fact, that these two, complementary processes always exist is called micro reversibility. Let us now consider a gas mixture consisting of particles A and B with densities NA and NB , respectively. Let the distribution of kinetic energies of particles A be given by f (T ), hence dNA = NA f (T )dT particles A per unit volume have an energy between T and T + dT . The ensemble of all particles A looses energy by inelastic processes (6.12), it gains energy by de-excitation processes (6.13), as schematically illustrated in Fig. 6.3. In an energy range between T and T + dT according to (6.9) dN˙ ba = vkin σba (T )NA f (T )NB(a) dT

(6.14)

excitation processes occur per unit time and volume. They fill the energy distribution between (T − Wba ) and (T − Wba ) + dT and depopulate the energy range from T to T + dT . The corresponding number of de-excitation processes is dN˙ ab = vkin σab (T − Wba )NA f (T − Wba )NB(b) dT .

(6.15)

They fill the energy distribution between T and T + dT and depopulate it between (T − Wba ) and (T − Wba ) + dT . The velocity of the particles A before and after , respectively. The particle densities N the excitation process is vkin and vkin B(a) and NB(b) refer to particles B in state |a and |b, respectively. Now, in thermodynamic equilibrium the so called principle of detailed balance must hold: each individual process is kept in equilibrium by its inverse process. The energy loss by reaction (6.14) is kept in balance (for the sum of all processes) by the energy gain according to (6.15). Hence vkin σba (T )f (T )NB(a) = vkin σab (T − Wba )f (T − Wba )NB(b) .

(6.16)

In thermodynamic equilibrium (temperature Θ) the energy distribution is described by the M AXWELL -B OLTZMANN distribution according to (1.58), Vol. 1: 2 f (T )dT = √ π

1 kB Θ

3/2

√

T dT . T exp − kB Θ

(6.17)

6.1

Introduction

389

At the same time for the densities of particle B the B OLTZMANN statistics must hold NB(b) gb − Wba = e kB Θ NB(a) ga

(6.18)

levels, respectively. Inserting with the degeneracies gb and ga of the upper and lower√ (6.17) and (6.18) into (6.16) and considering that v ∝ T , we obtain the important relation (T − Wba )gb σa←b (T − Wba ) = T ga σb←a (T ),

(6.19)

which connects the cross sections for excitation and de-excitation of an atomic or molecular collision induced transition process. We may regard this relation as the ‘collisional analogue’ to the equivalence of induced emission and absorption, i.e. as corresponding to the E INSTEIN relation for the B coefficients according to (4.38) in Vol. 1. In contrast to optical processes, no energy resonance is needed for collisional excitation or de-excitation: the projectile looses or gains as much energy as needed for the excitation or de-excitation process. Of course, the respective cross sections are a function of energy, and excitation cross sections clearly disappear for kinetic energies below the threshold T < Wba . Another important difference between (6.19) and the optical equivalent (4.38), Vol. 1 is expressed by the energetic prefactors in the case of collision processes. They are a consequence of the changing particle velocity due to inelastic processes and ensure that the flux of particles A remains conserved during a collision. One word about the orders of magnitude of atomic and molecular cross sections. They may vary over many decades. Typical gas kinetic (elastic) cross sections are on the order of 10−19 m2 (see next section). But values from some 10−17 m2 or even larger, down to 10−25 m2 and less are observed – depending on the type of process and the kinetic energy during the interaction. In Chap. 7 we shall get to know some general trends for inelastic processes.

6.1.3

Integral Elastic Cross Sections

At sufficiently low energies and for non-reactive collision partners only elastic scattering occurs during which the kinetic energy T in the centre of mass system of the collision partners is conserved. In this case the order of magnitude of the cross section observed may be visualized intuitively: if one thinks of two billiard balls of radius R1 and R2 , the classical prediction for the elastic cross section is σel = π(R1 + R2 )2 . This is the gas kinetic cross section. If we consider one of the most simple scattering systems H + He, one estimates from the covalent radii for H and He according to Fig. 3.4 in Vol. 1 for the elastic cross section σel 0.4 nm2 = 4 × 10−19 m2 . A comparison with real experimental data, as presented in Fig. 6.4, shows that the order of magnitude is right – the details suggest, however, that further consideration is indicated!

390 Fig. 6.4 Elastic cross section for the scattering of H and D atoms by He atoms at thermal and supra-thermal energies as a function of the relative velocity u. Shown are results from a transmission experiment (see Fig. 6.2) with H and D atomic beams of very well selected velocity and a cooled He gas target. Open points are taken from G ENGENBACH et al. (1973), full points from T OENNIES et al. (1976)

6

Basics of Atomic Collision Physics: Elastic Processes σel / 10-16 cm 2

H + He D + He

50 45 40 35 30 1000

10000 u / ms-1

Fig. 6.5 H + He interaction potential as a function of the internuclear distance R according to G ENGENBACH et al. (1973) (the potential is essentially repulsive, note the somewhat unusual logarithmic scale with an additive constant). The classical turning point Rc depends on the total energy T (here given for u = 104 m /s). For comparison a hard sphere potential is indicated (red dashed line)

V(R ) + 2meV / meV

Atoms simply are not hard spheres; they have, so to say, a soft cover. Thus it becomes understandable that the cross section decreases with increasing velocity (or energy) in the supra-thermal energy range:4 A collision at higher energies probes the repulsive part of the interaction potential (see e.g. Fig. 6.1) at lower distances R. For the H + He system the interaction potential V (R) is shown in Fig. 6.5, and is compared with an idealized hard sphere. The classical turning point Rc depends strongly on the total energy T (i.e. on the initial kinetic energy in the centre of mass system): Rc is obtained from T = V (Rc ). At higher energies the elastic cross section decreases explicitly with the relative velocity u – and not with energy – as documented by the comparison of H + He and D + He in Fig. 6.4. This will be explained in Sect. 6.3.3. In contrast, for hard spheres (dashed red line in Fig. 6.5) one would expect a constant Rc (and thus a cross section independent of u and T ).

103

hard sphere

Rc

414 meV (10000 m/s)

102 H + He

T

10 V(R ) = 0 1 1.0

2.0 3.0 R / 0.1nm

4.0

note, that the root mean square relative velocity in a hypothetical gas of H + He (or D + He) atoms at room temperature corresponds to u 1750 m /s (or 1350 m /s).

4 We

6.1

Introduction

Fig. 6.6 Elastic scattering cross section for Na + Hg as a function of the relative velocity u according to B UCK et al. (1971). (a) log-log overview; (b) strongly enlarged relative change to show the so called glory oscillations

391 σel / 10-16 cm 2

Na+Hg

(a)

4

3 2.5

1000

u / ms-1

2000

(b)

∆σ ___el σel 2% 0% - 2% 1000

u / ms-1

2000

The fact that for still lower velocities the cross section decreases again is of quantum mechanical origin. We shall discuss this below in some detail. Here we just point out that for the system H + He (reduced mass M¯ = 4/5 u) the DE ¯ 0.38 nm at u 1300 m /s (where B ROGLIE wavelength is about λdB = h/Mu the maximum cross section is reached). Thus, we find the interesting ‘coincidence’ πλ2dB = 0.46 nm2 σel (max). For the system D-He at the same velocity λdB is smaller by a factor 3/5, and the cross section still increases with decreasing relative velocity, down to u (700 to 800) m s−1 . We show a few more examples of elastic cross sections, which illustrate that such scattering processes may lead to a variety of interesting structures. First the system Na + Hg in Fig. 6.6, where we recognize a series of oscillations as a function of relative velocity, the so called glory oscillations, which we shall explain later. The general trend – the cross section decreases with velocity – is explained in the same manner as in the previous example by a softness of the interaction potential. A different behaviour may be observed for the scattering of slow electrons e.g. by rare gases. In Fig. 6.7 the situation is shown for the system e− + He. This still looks very similar to in the case of H + He scattering which we have just discussed. One may glean from the data that the cross section first goes through a maximum before joining the general trend to decrease with increasing energy. However, as shown in Fig. 6.8, for e− + Ar, e− + Kr and e− + Xe one observes distinctively different structures as a function of energy. The cross section starts at finite values and passes through a minimum to climb a rather pronounced maximum before finally decaying again with energy. The pronounced minimum which is seen here, the so called R AMSAUER minimum is today well understood and often encountered in low energy electron scattering; in some rarer cases it may also be observed in heavy particle scattering. These observations can, however, not be understood in terms of classical models – we shall come back to this in Sect. 6.4.5. Also, when

392 Fig. 6.7 Elastic scattering of slow electrons by He atoms: older measurements and red points with error bars (partial wave analysis) according to A NDRICK and B ITSCH (1975); open, grey circles are absorption experiments from different authors according to BAEK and G ROSSWENDT (2003)

6

Basics of Atomic Collision Physics: Elastic Processes σel / 10-16 cm 2 e- + He 6 4 2 0 0

5

10

15

20

electron kinetic energy T / eV

Fig. 6.8 Scattering of low energy electrons by several rare gases according to S ZMYTKOWSKI et al. (1996). Clearly visible for Ar, Kr and Xe are the R AMSAUER minima at low energies

σel / 10-16 cm 2 Xe

40

30

Kr

20

10

e- + rare gas

Ar

He

Ne

0 1

10

100

electron kinetic energy T / eV

using the concept of a potential one should be very careful in the context of interactions between electrons and atoms: atoms themselves consist of electrons and may become transparent – as the initial rapid decrease of the cross section shows. Hence, the theoretical treatment of electron scattering has to be fully quantum mechanically, in contrast to the theory of heavy particle collisions which usually are treated in a semiclassical manner. We shall develop these theories in some detail later in this and the two following chapters. Section summary

• When studying elastic or inelastic cross section one observes a variety of interesting structures as a function of velocity or energy. We have shown some examples in Figs. 6.4–6.8. One may extract from such observations specific, microscopic information on the dynamics of a collision process. • Obviously, experiments in a gas cell where only rate constants are measured, such fine details are averaged out according to (6.10). The price for additional

6.2

Differential Cross Sections and Kinematics

393

information is a significantly higher experimental effort. For example, the absorption experiment sketched in Fig. 6.2 requires good selection of the initial velocity as well as cold target gas which may be considered at rest. For neutral particles one still uses mechanical selectors of the F IZEAU type. Supersonic molecular beams simplify the selection. For charged particles one usually uses electrostatic selectors or time of flight methods (see Appendix B), occasionally also magnetic selectors. • Finally, one has to bear in mind that elastic cross sections, as we have discussed them up to now, can only be determined with such ‘simple’ absorption experiments, if the kinetic energy is clearly below the thresholds for potential processes. At higher energies inelastic processes may e.g. discerned by energy measurements of the scattered particles corresponding to the process equation (6.12). Another possibility for identification of such processes is the detection of optical fluorescence from excited states.

6.2


6.2.1

Experimental Considerations

Atomic and molecular collision processes have been studied over the past decades in ever finer details, aiming at a precise microscopic understanding of the interaction processes. On this basis a quantitative comparison with sophisticated quantum mechanical models can be achieved. Beyond the measurement of total cross sections, as discussed in the previous section, the next logical step is the measurement of the angular distribution of scattering intensities for the different processes. In Fig. 6.9 a scheme for such a differential scattering experiment is presented. The number of particles N˙ which are scattered per unit time from a scattering volume VS = AP d into the solid angle Ω is given by N˙ = (ΦA NB AP d)I (θ )Ω.

(6.20)

As in Sect. 6.1.1 the flux of particles A in the projectile beam is ΦA = vNA , and NB is the particle density in the target gas B. The area of the projectile beam is AP , and d is the effective length from which particles are registered at the detector. The angular distribution I (θ ) characterizes the physics of the collision process, and is called differential cross section (often just DCS): I (θ ) =

dσ dσ = . dΩ dϕ sin θ dθ

(6.21)

The DCS has the dimension L2 per solid angle, [I (θ )] = m2 sr−1 . In the most general case it depends on the (polar) scattering angle θ as well as on the azimuthal angle ϕ. For statistically aligned projectiles and targets, however, I (θ ) is only a

Fig. 6.9 Beam gas experiment to determine differential cross sections. (a) Overall scheme with the scattering volume as seen by the detector. The dashed red lines indicate the finite angular resolution. (b) Simplified scheme to illustrate the necessary sin θ correction of the scattering volume

6


(a)

scattering chamber, target B

A beam

scattering volume

θ

394

Δθ

ape b

rtur

e

detector for scattered particles A

d = b / sinθ

(b)

b

θ

A beam

function of θ . Integration over all scattering angles leads again to the integral cross section (ICS) introduced in Sect. 6.1: π dσ dΩ = 2π σ= I (θ ) sin θ dθ. (6.22) 0 4π dΩ The differential cross section may refer to elastic, inelastic, ionizing or reactive processes, just as the ICS. We emphasize again that integral cross sections σ should not be called total cross section – as sometimes done. The total cross section is defined according to (6.5) as the sum of all possible integral cross sections. Experimentally one may keep the solid angle seen by the detector constant without problems by using angle limiting apertures. If the detector area is AD and its distance from the scattering volume is R, then Ω = AD /R 2 is the solid angle ‘seen’ by the detector.5 In contrast to Ω, the effective length d of the scattering volume in the beam-gas experiment Fig. 6.9(a) also depends on the scattering angle, as sketched schematically in Fig. 6.9(b). Hence, in such an experiment the observed scattering intensity has to be divided by sin θ to obtain the true angular distribution I (θ ). One immediately sees that this may lead to substantial uncertainties for small scattering angles. Experiments with crossed beams may overcome this problem – if they are setup properly. Figure 6.10 illustrates schematically such an experiment. Ideally, two well collimated, energy selected and (as far as possible state selected) particle beams cross – usually at right angles. The detector has to be constructed such that one of the collision partners A or B is detected, if necessary after suitable energy and state analysis. Important for such an experiment is, that the whole scattering volume is seen by the detector completely for each arbitrary scattering angle. For this, the 5 Note: this solid angle of detection must not be confused with the finite angular resolution of the setup which may be determined by a finite target – indicated in Fig. 6.9 by θ .

Differential Cross Sections and Kinematics energy and/ or state selector A

source A

395 optional: spectrograph, polarization analysis photon detector

de tec

tio nc on

e

beam dump

A beam B beam

6.2

energy and/or state analyzer

ert ure

scattering volume

ap

energy and/ or state selector B

detector for scattered particles

source B

Fig. 6.10 Scheme of a crossed beam scattering experiment. Note that the detector always sees the whole scattering volume, independent of the scattering angle, so that no sin θ correction is necessary

detection cone must be large enough, at the same time, however, provide the necessary angular resolution. In practice it is difficult to meet all these demands and often special corrections still have to be applied. For completeness we mention here that for electron scattering often the so called momentum transfer cross section is given for practical applications: dσ σemom = dΩ (1 − cos θ ) dΩ 4π π = 2π (1 − cos θ )I (θ ) sin θ dθ. (6.23) 0

From Fig. 6.11 one reads the magnitude of the momentum transfer (p e → p e ) of an electron e− during an elastic collision: pe = 2pe sin(θ/2). This momentum difference is transferred to the target B (atom, ion, molecule), which then has the momentum p B = −pe . Strictly speaking, one has to compute the energy balance in the centre of mass system (see next subsection). One finds that the target receives a ¯ kinetic energy (pe )2 /2MB = 4p 2 sin2 (θ/2)/2MB = 2T (M/M B )(1 − cos θ ). Here p is the relative momentum and T the initial kinetic energy of the electron in the centre of mass system, M¯ its reduced mass, and MB the target mass. The momentum Fig. 6.11 Momentum transfer during elastic scattering

pB

Δpe pe θ

pe'

396

6


transfer cross section, defined in (6.23), thus gives a measure for the average energy which is transferred per elastic collision onto the target. This energy may e.g. lead to the heating of a plasma.

6.2.2

Collision Kinematics

The assumption, particle B had infinite mass and could be assumed to be at rest before and after collision is of course not generally valid. We drop it now and present the correct treatment which considers the motion of both particles. According to classical mechanics the momentum of the centre of mass of two particles is independent of their interaction. All definitions remain valid if we refer to the relative motion of the collision partners. Only the relative coordinates of A and B are used to describe the dynamics. The necessary kinematic transformations are, however, of key importance for the correct interpretation of all heavy particle collisions. Only for collisions of electrons with atoms and molecules they may be disregarded. Figure 6.12 shows trajectories of the particles A and B. In laboratory coordinates (origin O), briefly lab-frame, they are described by the coordinates R A (t) and R B (t). The centre of mass (CM) moves on a straight line (dot-dashed). We recall the transformation from lab-frame to CM-frame as used when treating the H2 molecule. The Hamiltonian for the system of particles A and B with momenta p A and pB (masses MA and MB ), respectively, and the interaction potential V (R A − R B ) is given by HAB =

p2A p2 + B + V (R A − R B ). 2MA 2MB

(6.24)

It is transformed into the CM-frame by introducing the relative coordinate R and the relative velocity u R = RA − RB

˙A−R ˙ B = vA − vB and u = R

(6.25)

according to Fig. 6.12. With this, after a brief calculation (6.24) becomes HAB = Fig. 6.12 Schematic course of trajectories of two colliding particles A and B with coordinates R A and R B , respectively. Indicated is the motion of the centre of mass system on straight lines (black dash-dotted) and the motion of A and B prior to the collision (red and grey dashes, respectively)

p2 P2 + + V (R) 2M 2M¯

(6.26)

A R RA

O

CM RB

B

6.2


397

with the momentum of the CM P = MV = MA v A + MB v B = const,

(6.27)

the relative momentum ¯ p = Mu,

(6.28)

the total mass M and the reduced mass M¯ M = M A + MB

MA MB and M¯ = . M A + MB

(6.29)

The constant kinetic CM energy P 2 /2M may be subtracted from the total energy HAB (separation of the overall translation and relative motion) and we obtain (classically as well as quantum mechanically) for the energy of the relative motion in the CM system the H AMILTON operator: HCM =

p2 + V (R). 2M¯

(6.30)

The whole relevant dynamics is thus described by the relative canonical coordinates R and p. So far all this is in principle known already from the H atom or the H2 molecule. In the following we have to transform the possible changes of the relative momentum p, as a consequence of the collision, into experimentally observables in the laboratory system. In the context of these kinematic considerations we are only interested in the asymptotic behaviour before and after the collision (i.e. for R → ∞, V (R) → 0), which we mark by un-dashed and dashed quantities, respectively. Energy conservation then implies for the relative kinetic energy in the CM system: 1 1 ¯ 2 = M¯ u2 = Mu ∓ W = TCM ∓ W. (6.31) TCM 2 2 This includes the possibility of inelastic and super-elastic processes, which due to changes of internal energy ±W of the participating particles A and/or B may lead to a corresponding change of the relative velocity. For reactive processes even the reduced mass M¯ → M¯ may change. In the lab-frame the total kinetic energy is then 1 1 1 2 + MB vB2 = MV 2 + TCM . (6.32) TLab = MA vA 2 2 2 By replacing in this expressions all quantities by dashed ones (v → v ), one obtains the lab energies after the collision process. With (6.32) it is clear that of the total kinetic energy which the collision partners have in the laboratory system, only a fraction is available for the collision process, that is TCM . Very clear and helpful for visualizing the kinematics is the so called N EWTON diagram of the scattering process, which we introduce in Fig. 6.13. It represents the

398

6


Fig. 6.13 Example of a N EWTON diagram prior to a collision process – here for a mass ratio MA :MB = 3:2. The centre of mass (CM) divides the relative velocity u at a ratio MB :MA

vA u

MB uA = u − M CM V MA uB = - u – M

O

vB

laboratory velocities v A and v B together with the relative velocity u and the velocity of the centre of mass V . We rewrite the definition (6.27) for the momentum of the centre of mass as M A v A + M B v A − MB v A + MB v B , M MB MA u and uB = − u, so that with uA = M M MB MA V = vA − u = v A − uA = v B + u = v B − uB M M

V=

finally: v A = V + uA and v B = V + uB .

or (6.33)

The N EWTON diagram Fig. 6.13 represents the last equation graphically. The centre of mass, CM, is thus represented by a point which divides the relative velocity u = v A − v B proportional to the mass MB and MA , respectively. One big advantage of this diagram is that one may use it equally well after the collision process, to transform the kinematics from the CM system back into the lab-frame. Since the centre of mass velocity V remains constant while the relative velocity u after the collision may be different from that before, u, one obtains all potentially possible laboratory velocities after the collision from the relations v A = V + uA

and v B = V + uB .

(6.34)

For the graphical construction one simply has to draw circles around the CM with a radius u MA /M and u MB /M for particle B and A, respectively. On these circles all kinematically allowed laboratory velocity vectors v A and v B will be found (their origin being at O). The magnitude of u is derived from the energy balance (6.31) for the process studied. In the case of reactive processes the masses may also change, i.e. MA and MB may differ from MA and MB . Figure 6.14 illustrates the situation for elastic, inelastic and super-elastic scattering processes without reactions. The examples shown assume in all three cases a centre of mass scattering angle θCM = 45◦ . The physical quantity one wants to determine is the differential cross section I (θCM ) in the CM system, i.e. the probability to observe (for a specific collision process studied) a certain CM scattering angle θCM . This is the quantity which reveals the details about the interaction dynamics. In contrast to the latter, N EWTON

6.2


Fig. 6.14 N EWTON diagrams for non-reactive processes before and after the collision. (a) elastic, (b) inelastic, (c) super-elastic. Full and dashed lines give the velocities before and after the collision, respectively. In all three cases the scattering angle in the CM-frame is assumed to be θCM = 45◦ . It leads to different laboratory scattering angles θBlab of particle B in each of these cases. The dotted circles indicate all velocities of particle B which in principle could occur after the collision

399

(a)

uA

vA

uA'

u B'

CM V

vA' θ Blab

θ CM vB

O

vB' uB

uA

(b) vA'

vA

CM V

vB' θ CM

θ Blab

vB

uB

O

(c)

vA

uA

uA' vA'

u B'

CM V

vB' θ CM

θ Blab O

vB

uB

diagrams describe what one calls kinematics, i.e. the transformation from the CM system into the laboratory system and vice versa. We have to point out that the conversion of data measured in the lab-frame into the CM-frame is not always unambiguously possible – if only the scattering angle is determined. If the circles with radius uA or uB , respectively, do not fully enclose the CM velocity V , then there exist laboratory scattering angles which belong to two different CM scattering angles, as seen in Fig. 6.14(a) and (b). If one knows the type of process studied, a velocity analysis may help. However, the situation becomes very complicated if all three process types illustrated in Fig. 6.14(a)–(c) occur simultaneously. Another complication arises when transforming the scattering rates measured in the lab into differential cross sections dσCM /dΩCM in the CM system: the detector acceptance angle changes from dΩlab → dΩCM , and one has to apply an appropriate transformation factor. Corresponding considerations hold for the energy scale. These transformations are achieved by applying the corresponding Jacobian determinant. Finally, at low and possibly thermal energies one has to account for the fact that different initial velocities lead to uncertainties in determining the CM system. The de-convolution of all these distributions can be rather complicated (PAULY and T OENNIES 1965). Thus, in the following we just present for illustration a partic-

400

6


ularly elegant example which shows how one may even exploit the cumbersome kinematic relations with advantage to solve specific problems.

6.2.3

Mass Selection of Atomic Clusters

The investigation of atomic and molecular clusters – i.e. of more or less loosely bound systems of many atoms and/or molecules – is an important field of modern research. It has developed essentially out of atomic and molecular physics and is closely related with today’s very fashionable ‘nano’ physics. To generate such clusters one may use supersonic, cold molecular beams. By adiabatic expansion of gases at high pressure (typically 1–100 bar) through a thin nozzle into vacuum, the atoms or molecules are efficiently cooled over a short distance of flight, and partially condense, thus forming clusters. Their size N may range from 2 to many ten thousands – usually in a more or less broad distribution. When studying them it is of course desirable to know their size. If one treats ionic clusters, N may be fixed by mass spectrometric methods. But neutral clusters have to be ionized prior to detection, and one usually has to accept that the ionization process may change the cluster distribution. B UCK and M EYER (1984) have developed an elegant, albeit rather elaborate, method to select even neutral clusters according to their size: by using the very same collision kinematics which we have just discussed. The idea is to deflect the different cluster sizes N by collisions, and use the kinematics to select them. Figure 6.15(a) explains the principle using again the N EWTON diagram, here for the scattering of ArN clusters by He atoms – as applied in the first, pioneering experiment. Since the mass of the ArN clusters is large compared to the light collision partner He, the relevant part of the N EWTON diagram is localized around the initial velocity of the ArN clusters. This region is sketched enlarged in Fig. 6.15(a). The relative velocity u (red, intermitted arrow) prior to the collision defines the CM system. The red circles mark the possible positions of velocity vectors after a collision for clusters with 2 ≤ N ≤ 5. As seen, for each laboratory angle θlab there are two scattering angles in the centre of mass system which lead to different lab velocities (black arrows). One may distinguish these by measuring the velocity of the scattered clusters. The experimental realization is illustrated schematically in Fig. 6.15(b). ArN cluster beam and He beam cross each other at right angle. The scattered clusters obtain a temporal structure by passing the chopper (C). They are collimated and detected behind a quadrupole mass spectrometer. Their time of flight allows to determine their velocity. The measured signal is shown in Fig. 6.15(c) for three different lab scattering angles θlab . As predicted by the N EWTON diagram (a) one finds at θlab = 14◦ only two peaks, corresponding to Ar2 clusters at the two different possible scattering angles in the CM system. At θlab = 10◦ one already detects also Ar3 , while at θlab = 8◦ in addition two more peaks from Ar4 are seen. Overall, these results show a rather impressive experimental confirmation of the kinematic considerations explained in the previous subsection. One may now use the thus documented angular and velocity distribution of the scattered clusters for spectroscopic studies with mass selected neutral clusters. One


(a)

401

u

10 N=2

≈

14˚

N=2

≈

10˚ 8˚

N=3 N=4 N

=5

≈

θLab

ArN velocity vA He beam

(b)

rt ape

ure

s e det

C

cto

r

skimmer

ArN source

He source

θ lab= 8°

4

5

3

0 10 N=2

3

2

θ lab=10° 3

5

2

0 10 N = 2

5

θ lab skimmer

(c) 3 4

100 m/s normalized scattering intensity

He velocity v B

6.2

θ lab=14°

2

ArN beam 0 0.6

0.8 1.0 1.2 time of flight / ms

Fig. 6.15 Cluster size selection by elastic scattering according to B UCK and M EYER (1984). (a) N EWTON diagram for scattering ArN by He atoms at velocities vA = 570 m /s and vB = 1790 m /s. The red circles give the positions of all possible final velocity vectors for elastically scattered clusters (note the breaks in the velocity arrows). (b) Experimental setup schematic, featuring a pseudo-statistical chopper C for the time of flight analysis of the scattered clusters. (c) Measured time of flight spectra after ArN He collisions at three different laboratory angles, corresponding to the N EWTON diagram (a)

may, e.g. localize a laser beam at a fixed position in space behind the scattering centre, irradiating the cluster perpendicularly to the scattering plane of the clusters indicated here. Buck and collaborators have developed and used this method as a very powerful tool for spectroscopy and for studying fragmentation dynamics of atomic and molecular clusters (see e.g. P OTERYA et al. 2009; B ONHOMMEAU et al. 2007; S TEINBACH et al. 2006; FARNIK et al. 2004). Section summary

• Differential cross sections dσ/dΩ (DCS) describe the angular distribution of scattered particles and give deep insights into the interaction dynamics. They may be measured in well defined beam-gas or in crossed beam experiments. Both have their specific problems and advantages. • Collision kinematics – as opposed to collision dynamics – is merely defined by momentum and energy conservation. The so called N EWTON diagrams (see examples Figs. 6.13–6.15) provide very useful tools for visualization and quantitative evaluation of the kinematic constraints.

402

6.3

6


Elastic Scattering and Classical Theory

Before turning to the quantum mechanical treatment of scattering processes, we briefly recall in this section the basics of classical scattering theory – which the readers may perhaps remember from classical theoretical mechanics. We shall use this opportunity to bring its limits to mind. In the spirit of the previous section, all problems will be treated in the centre of mass (CM) system: all velocities, energies distances refer to it unless otherwise mentioned. This is equivalent to treating the scattering problem as if a particle A with mass M¯ = MA MB /M interacted with a particle B whose mass is infinite. Even though, finally, the interaction of atomic particles has to be treated quantum mechanically, very often classical mechanics turns out to give a rather good first approach to understand the scattering dynamics. At any rate, the concept of a classical trajectory is found to have rather far reaching significance for a number of scattering problems – as long as the typical atomic dimensions are large compared to the DE B ROGLIE wavelength λdB of the interacting particle. In atomic units the latter is written 2π λdB , = a0 ¯ (2M/me )(T /Eh )

(6.35)

with T being the relative kinetic energy of the particles and M¯ their reduced mass. ¯ For electron scattering (M/m e 1) a classical treatment may only be envisaged for rather high kinetic energies (on the order of keV). In contrast, for heavy particle collisions usually λdB /a0 1 holds – even in the thermal energy range. Since a0 also characterizes the typical range of atomic interactions, a classical description of heavy particle collisions is usually an appropriate first step.

6.3.1

The Differential Cross Section

We treat now the elastic scattering of particle A in the isotropic potential V (R) of particle B and interpret the differential cross section (6.21) geometrically as sketched in Fig. 6.16: Let A approach B with a range of impact parameters b to b + db. After the interaction A is scattered into an angle between θ and θ + dθ . Taking advantage of the axial symmetry, the elastic scattering rate for the whole conical ring with dΩ = 2π sin θ dθ becomes proportional to dσ = 2πbdb. Particle A must approach B within this cross section in order to be scattered into scattering angles θ Fig. 6.16 Definition of impact parameter b, scattering angle θ and classical scattering cross section dσ = 2πbdb

dθ A

db

θ

b B


Fig. 6.17 Classical light scattering at a water droplet with radius R for wavelengths λ R, schematic. The classical rainbow arises for trajectories of type p = 2. Further reflections are possible and lead to (weaker) secondary rainbows (not shown here)

403 p=0

light from the sun

impact parameter b

6.3

α

β

R b α

p=1 β

Θ2 α

α

β

p=2

to θ + dθ . Hence, the classical, differential cross section becomes: I (θ ) =

b(Θ) 1 d(b2 ) bdb = = . sin θ dθ 2 sin θ dθ sin θ |dΘ/db|

(6.36)

The dynamical problem is now simply, to determine the classical deflection function Θ = Θ(b). Note: we distinguish here between scattering angle θ , which is always taken as positive, and the deflection function Θ, which may assume positive as well as negative values, depending on whether the overall interaction acts as attractive or repulsive, respectively (the deflection sketched in Fig. 6.16 is negative). It is important to realize that this classical cross section diverges if the deflection function goes through a maximum or a minimum, i.e. when |dΘ/db| = 0 holds. This leads to the so called classical rainbow. This interesting phenomenon is indeed closely related to the well known and popular optical rainbow, which arises from scattering of sunlight by small water droplets.

6.3.2

The Optical Rainbow

So let us make a little detour into the physics of optical rainbows, which will allow us to understand quite intuitively how the phenomena arise which we have just indicated. Figure 6.17 shows a light beam arriving from the sun, at the interface between air and water droplet, where it is refracted and/or reflected. Light may be reflected at the surface (p = 0) or after entering the droplet and refraction may undergo several additional reflections before finally leaving the droplet after back-refraction (p = 1, 2, 3, . . . ).6 The famous optical rainbow is formed for p = 2, possibly followed by one or more secondary rainbows p > 2. 6 As long as for the radius of the droplet R λ holds (with λ being the wavelength of the light) – and this is usually the case for standard rainbows – we may treat the nearly parallel light arriving from the sun as a bundle of light beams and apply geometric optics. Light scattering at smaller objects shows very pronounced interference structures and is described by the M IE theory – an application of classical electrodynamics. It is treated in detail in standard textbooks on optics (see e.g. B ORN and W OLF 2006, p. 759ff).

404

(a)

6 180˚

Basics of Atomic Collision Physics: Elastic Processes 0˚

(b)

p=0

- 137˚ rainbow scattering angle θ

scattering angle θ

90˚ p=1

0˚ -90˚

p=2

-180˚ -270˚ -360˚

p=3 0

0.2

p=4

0.4 0.6 x = b /R

0.8

-140˚ p=2 -150˚ -160˚ -170˚ -180˚

1.0

0

0.2

0.4 0.6 x = b/R

0.8

1.0

Fig. 6.18 Classical deflection function Θ(b) for the scattering of light at a water droplet (radius R) as a function of the ratio x = b/R of impact parameter to radius. (a) Different orders p for an average value of the index of refraction. (b) Enlarged scale for p = 2 for red (red line) and blue wavelength (grey line). The clearly visible maximum at deflection angles −137◦ and −139◦ , respectively, lead to the rainbow phenomenon

We discuss the classical deflection function Θp (b) as a function of impact parameter and wavelength, considering a cut through an equatorial plane of the droplet, in which the beam propagates. The trajectory may be read directly from Fig. 6.17: obviously α < β and the deflection of the beam when entering the droplet is α − β (we count deflection towards the beam axis as negative). Each reflection within the droplet leads to an additional deflection −2β. On the exit one more deflection α − β is added. With S NELL’s law of refraction in the form sin(π/2 − β) cos β = = 1/n sin(π/2 − α) cos α and the ratio x = b/R = cos α of impact parameter b to droplet radius R the ray deflection becomes: Θp = 2α − 2pβ = 2 arccos(x) − 2p arccos(x/n).

(6.37)

Hence, the deflection function Θp (b) depends only on the ratio of impact parameter to droplet radius x = b/R and on the index of refraction n. Notably it does not depend on the size of the droplet as such! For water, the index of refraction for blue light is 1.3427 and for red light 1.3282. With these values (6.37) leads to the deflection function as a function of b/R shown in Fig. 6.18. The deflection function for p = 2 has a maximum at about −137.2◦ and −139.3◦ for red and blue light, respectively. This implies that many light rays with different b lead to nearly the same deflection angle and thus contribute to a maximum of intensity at these angles. In (6.36) this is expressed as a singularity. For different wavelength this rainbow angle is slightly different – that is what we experience as

6.3


405

Fig. 6.19 Schematic illustration of how the observed inclination angles of the rainbows arise, for the main rainbow and one secondary rainbow. Note that the ordering of the colours is inverted for the secondary rainbow

ligh the t from sun 49˚ rain (rad drople t ius R)

p = 3: secondary rainbow

obs

erve

43˚

r

p = 2 : main rainbow

the wonderful colours of an optical rainbow – red light appearing to come from a slightly higher origin (or spanning a larger circle) than blue light. Sometimes a so called secondary rainbow (p = 3) may be observed in addition to the main rainbow (p = 2). It arises from one more reflection within the droplet. Since the intensity of this scattered light is of course much weaker the secondary rainbow is much more pale. As illustrated in Fig. 6.19 the observer sees the main rainbow at an inclination 180◦ − |Θ| (relative to the direction of the incoming sunlight) i.e. at ca. 41◦ and 43◦ for blue and red, respectively. For the secondary rainbow one finds according to (6.37) an inclination of 49.6◦ and 54.4◦ for red and blue, respectively. The order of the colours is thus inverted in the secondary rainbow as compared to the main rainbow. According to Fig. 6.19 this is an immediate consequence of the reflection and refraction geometry.

6.3.3

The Classical Deflection Function

Let us get back to particle scattering! We consider a classical trajectory in the centre of mass system as illustrated in Fig. 6.20(a). ¯ as well as During elastic scattering the magnitude of linearmomentum p = Mu ¯ the angular momentum || = pb is conserved (u = 2T /M is the magnitude of the relative velocity). Hence, the impact parameter too is a constant of motion. With the

(a) b

Θ = π – Á(t = ∞)

Á

(b) R b

b

Rc

R

Á

ψ

ψ Rc

b

2 = d/

Θ = π – 2ψ

Fig. 6.20 Classical trajectories in cylinder coordinates R, φ with the classical turning point Rc (distance of closest approach). The deflection angle Θ is a function of the impact parameter b. (a) General case, (b) elastic collision of point like particle with a solid hard sphere of diameter d

406

6


¯ 2 of the relative motion one may express || by the angular moment of inertia MR ˙ velocity φ: ¯ ¯ 2 φ˙ = const. || = Mub = MR

(6.38)

We have used these relations already in Chap. 2, Vol. 1 when formulating the H AMILTON operator for bound states. With the effective potential Veff (R) = V (R) +

||2 b2 = V (R) + T ¯ 2 R2 2MR

(6.39)

and the total energy T the H AMILTON function of the relative motion is written as: H=

M¯ ˙ 2 R + Veff (R) = T . 2

(6.40)

With (6.38)–(6.40) one finds φ˙ dφ b = = ˙ dR R R 2 1 −

,

(6.41)

Veff (R) T

from which by integration we obtain the classical deflection function: Θ(b, T ) = π − φ(R = ∞) (6.42) ∞ ∞ dR dR = π − 2b = π − 2b . Rc R 2 1 − Veff (R) Rc R 2 1 − V (R) − b2 T T R2 Rc is the classical turning point (distance of closest approach of the particles) in the effective potential Veff .

The Limit of a Hard Sphere The most simple case is elastic scattering of a point like object by a hard sphere of diameter d as indicated in Fig. 6.20(b). Here Rc = d/2 always holds, and with b/(d/2) = sin ψ = cos(Θ/2) the deflection function becomes (independent of energy) Θ(b) = π − 2ψ = 2 arccos

2b d

(6.43)

as long as b ≤ d/2, while Θ(b) = 0 for b > d/2. With a little bit of algebra we derive the differential cross section (6.36) to be I (θ ) = d 2 /16,

(6.44)

independent of the scattering angle. We thus find a completely isotropic scattering distribution!

6.3


407

R UTHERFORD Cross Section An prominent example is RUTHERFORD scattering, i.e. the interaction of two charges qA e and qB e in a pure C OULOMB potential V (R) = qA qB /R (written in atomic units). The integral (6.42) may be solved by substitution of x 2 = 1/R 2 and correspondingly rewritten limits of integration. One finds θ qA qB = arctan 2 2T b

and b =

qA qB θ cot . 2T 2

Inserted into (6.36) this leads to the well known RUTHERFORD cross section: I (θ, T ) =

dσ = dΩ

qA qB T

2

1 16 sin4

θ 2

.

(6.45)

Here we have used atomic units, i.e. T is given in units of Eh , the differential cross section is measured in units a02 sr−1 .

The Limit of Small Scattering Angles A very important limit is that of small scattering angles. The scattering process probes in this case the long range attractive part of the interaction potential which may always be written as V (R) = −C/R s (typically with s = 1, 2, . . . , 6 as discussed in Sect. 8.3, Vol. 1). One may then expand the integral (6.42) and obtains (here without proof) √ (s − 1)f (s) C π Γ ((s − 1)/2) V (b) θ= with f (s) = . (6.46) ∝ T bs T 2 Γ (s/2) For the scattering of two neutral atoms in their isotropic ground state (L ENNARD J ONES potential) with s = 6 the prefactor becomes (s − 1)f (s) = 15π/16. For the differential cross section (6.36) in general one obtains 1 (s − 1)f (s)C 2/s −2(s+1)/s I (θ ) = θ and specifically s T 1/3 C = 0.2389 θ −7/3 for s = 6. T

(6.47) (6.48)

Reduced Scattering Angle and Cross Section For small angle scattering (θ sin θ ) one often uses the reduced scattering angle τ (b, T ) = θ T ,

(6.49)

and instead of the differential cross section (6.36) the reduced cross section τ db2 . (6.50) ρ(b, T ) = θ sin θ I (θ, T ) = 2 dτ

408

6


With these definitions, for large impact parameters according to (6.46) τ (b, T ) = θ (b, T ) × T ∝ V (b)

(6.51)

holds. Hence, to 1st order, τ depends only on the impact parameter b and not on the kinetic energy T . More generally, one may show (see e.g. S MITH et al. 1966), that τ can be expanded in powers of T −n : θ τ (b, T ) = τ0 (b) + T −1 τ1 (b) + · · · = τ0 (b) + τ1 (b) + · · · . τ

(6.52)

Correspondingly, the reduced cross section may be written as θ ρ(b, T ) = ρ0 (b) + T −1 ρ1 (b) + · · · = ρ0 (b) + ρ1 (b) + · · · . τ

(6.53)

The approximation holds for high energies and arbitrary scattering angles – or alternatively for arbitrary energies and small scattering angles (large impact parameters).

Integral Elastic Cross Section At this point, we have to come back to the integral cross section once more. With (6.22) and (6.36) one may write it as 0 π 2 σ = 2π I (θ ) sin θ dθ = 2π bdb = πbmax . (6.54) 0

bmax

For collisions with hard spheres the situation is unambiguous: for b > d no scattering occurs, hence the maximum impact parameter is bmax = d/2. Alternatively one may also insert (6.44) into the left integral and obtain the same result, σ = πd 2 /4, corresponding to the geometric visualization. However, if we study a potential with infinite interaction range, on first sight it looks as if the integral elastic cross section diverges: however large one chooses bmax , classically there remains always some deflection, albeit very small. Hence, the question arises, which is the maximum impact parameter bmax that leads to a still measurable deflection. This question can only be answered quantum mechanically, i.e. in last consequence by the uncertainty relation. From this perspective only such scattering angles Θ(b, T ) may be considered reasonable for which the product of ¯ and impact parameter b is larger than h. Hence, there lateral momentum Θ(b, T )Mu exists always a maximum impact parameter bmax for which this is just fulfilled: ¯ × bmax h. Θ(bmax , T )Mu

(6.55)

For large b the interaction potential is given by V (b) = −C/R s and the classical deflection function is given by (6.51). Hence, with (6.55) Θ

h C ∝ s ¯ max bmax Mu ¯ 2 Mub

=⇒ bmax ∝

C hu

1/(s−1) .

6.3


409

Finally, the integral elastic cross section becomes approximately 2/(s−1) C σ =K . hu

(6.56)

Thus, the integral cross section remains finite – except for s = 1, the C OULOMB potential (which is always a special case). Clearly, the numerical prefactor K requires an in depth quantum mechanical derivation (see e.g. F LUENDY et al. 1967). However, we note that (for high enough energies) the integral elastic cross section σ depends only on the relative velocity u and on the form and magnitude of the potential – and not explicitly on energy or momentum of the colliding particles. We have seen this e.g. already in Fig. 6.4 where for large relative velocity u the cross sections for H + He and D + He became equal. And Fig. 6.6(a) gives an example for a linear trend (on average) of the cross section in a log-log plot, corresponding to the power law (6.56).

6.3.4

Rainbows and Other Remarkable Oscillations

We now want to give a qualitative overview on the behaviour of classical trajectories and the resulting phenomena which one observes in low energy elastic scattering of atoms or ions by atoms and similarly by molecules (an excellent, more extended introduction, still timely today, is found in PAULY and T OENNIES 1965). Schematically, Fig. 6.21 summarizes the situation. Typically, atomic interaction potentials have a minimum as shown in Fig. 6.21(a). Even if the collision partners are chemically not bonding, such a minimum is the consequence of the superposition of the long range attractive VAN DER WAALS (R −6 ) and the short range repulsive potential between the atomic cores. Depending on impact parameter b, according to Fig. 6.21(b) trajectories pass mainly the repulsive part of the potential (small b = b1 and b1 ), or the attractive part (larger b = b3 ) or both regions (b = b2 ). As overall outcome the trajectories experience an effectively repulsive (Θ > 0) or attractive (Θ < 0) interaction. For b = bg attractive and repulsive forces are just compensated and as a result no overall deflection occurs, and we expect forward scattering. The (schematic) deflection function Fig. 6.21(c) summarizes these considerations and represents a special kind of image of the potential Fig. 6.21(a). If – as illustrated here – several classical trajectories with different impact parameters bj lead to the same scattering angle they must of course all be accounted for. In the classical limit, (6.36) has to be summed over all bj for which θ = |Θ(bj )| holds: dbj 1 . (6.57) I (θ, T ) = bj (Θ) sin θ dΘ j

As illustrated in Fig. 6.21(d) the differential cross section has a singularity, I (θ, T ) → ∞, for br where dΘ/db goes through zero. The same holds for bg , where sin θ disappears. Classical scattering theory thus claims a remarkable singularity in

410

6


Fig. 6.21 Schematic illustration of classical, elastic scattering: (a) atom-atom interaction potential, (b) some characteristic trajectories at different impact parameters b, (c) classical deflection function, (d) classical, differential cross section weighted with sin θ ; note the rainbow structure at θr

V(R)

(a) R0

-Θ

b1'

- De

R

(b)

bg

Θ

π

b1

b2 b 3

b→∞

(c) 0 -θr br

sinθ × I(θ)

(d) 0˚

θr

180˚

the differential cross section at the rainbow angle θr . Just as in the case of the optical rainbow, the phenomenon arises since a whole range of impact parameters around br leads to the same scattering angle θr – just there where Θ(b) goes through a minimum. Of course, nature will smooth out such singularities. In a quantum mechanical or semiclassical treatment instead of the sum over scattering cross sections (6.57), the square of the sum of corresponding scattering amplitudes f (θ ) will have to be used: 2 fj (θ ) . (6.58) I (θ, T ) = j

This will lead to characteristic interference phenomena. A very nice example of a classical rainbow with interference structures is provided by the elastic Cs-Hg scattering, documented in Fig. 6.22 for thermal energies. The differential cross section shows a pronounced maximum just below the classical rainbow angle (arrow) and then decreases very rapidly for θ > θr since there only one trajectory contributes to the cross section (6.58). In addition, one recognizes several less pronounced maxima for θ < θr . These supernumerary rainbows may be compared to those which we have seen for the optical rainbow. However, we obviously reach the limits of a classical interpretation when trying to understand the


Fig. 6.22 Experimentally determined differential cross section for the system Cs-Hg according to B UCK et al. (1972). (a) I (θ) weighted with sin θ , (b) blow up of I (θ) up to the rainbow angle θr , weighted with θ 7/3 (see Eq. (6.48))

411 sinθ × I(θ) 1000

(b)

100 0° 10

1

θr

0°

8

θ 7/3 × I(θ) / arb. un.

Fig. 6.23 Differential elastic cross section I (θ) for Li-Hg scattering according B UCK et al. (1974), weighted with θ 7/3 . Note the very high angular resolution: one recognizes several supernumerary rainbows, and on top of them, very clearly, the rapid oscillations

θ 7/3 × I(θ)

6.3

10° 20° 30° 40°

(a)

60° 120° scattering angle θ

180°

Li-Hg

6 4 2 0

0°

10° 20° 30° scattering angle θ

40°

various oscillations. These interference phenomena are a consequence of the wave nature of the colliding atoms. Also clearly seen is the forward maximum as expected according to (6.48). This is borne out most clearly in Fig. 6.22(b) where the differential cross section has been scaled with θ 7/3 so that the average remains constant. The oscillations are even more pronounced in the differential cross section for Li-Hg shown in Fig. 6.23 which has been measured with high angular resolution (again scaled with θ 7/3 ). How do all these different oscillatory structures in the DCS arise? A closer analysis shows that they may be rationalized by interferences between (the amplitudes for) trajectories with different impact parameters which lead to the same scattering angle, as indicated in Fig. 6.21 by b1 , b2 , b3 . Just as in light optics the particle waves are superposed and, due to phase differences varying with scattering angle, characteristic interference oscillations emerge. Trajectories which proceed on the same side of the scattering centre (e.g. with b2 and b3 ) lead to small phase differences and correspondingly slow oscillations (supernumerary rainbows). In contrast, trajectories passing the target on different sides (b1 and b2 or b1 and b3 ) collect large phase differences and give rise to rapid oscillations.

412

6

(a)

b3 supernumerary b2 rainbows

(b)

rapid oscillations


(d)

orbiting resonance

(e)

shadow scattering (diffraction) θ ~ λ dB / a

(f )

symmetryoscillations

b 2, b 3

(c)

glory oscillations

b1'

bg

Fig. 6.24 Schematic illustration of interfering trajectories (see Fig. 6.21) as origin of different types of oscillations in the low energy elastic atom-atom scattering

Figure 6.24 summarizes schematically the type of trajectories responsible for the different anomalies. Figures 6.24(a) and (b) illustrate the origin of the just documented slow (supernumerary rainbows) and rapid oscillations (in Fig. 6.23 recognizable on the rainbow and on the supernumerary rainbows). Related are the so called glory oscillations according to Fig. 6.24(c). They arise by interference between trajectories with very large and small impact parameters bg , both leading to forward scattering – the latter by compensation of attractive and repulsive parts of the potential (see also Fig. 6.21(c)). Experimentally one observes them most clearly in the integral cross section as already shown in Fig. 6.6. The phenomenon indicated in Fig. 6.24(d) is called an orbiting resonance. For a short time a trajectory is – so to speak – resonantly captured into a quasi-bound state of the interaction potential (collision complex). A particularly nice example according to T OENNIES (2007) gives the elastic, integral cross section for H + Xe scattering shown in Fig. 6.25. Quantum mechanically such short-lived resonances may e.g. be formed behind a rotational barrier which arises by the superposition of an attractive potential and the repulsive centrifugal potential, as schematically illustrated in the inset of Fig. 6.25. This may lead to resonances for very low kinetic energies. Experimentally the observation is a big challenge. In the example shown here the very low kinetic energies have been achieved by crossing two well cooled and velocity selected beams at an angle of 45◦ (instead at the usual 90◦ ). We note in passing, that similar resonances are also known in molecular spectroscopy, there called predissociation, which we have already mentioned in Sect. 3.6.7. They are formed e.g. when a molecule is excited into a quasi-bound state, shortly above the nominal dissociation energy, just as sketched in Fig. 6.25 (so called shape resonances). Another type of resonance are the so called F ES HBACH resonances which occur shortly below bound states, e.g. by rotational or vibrational excitation of molecules or in heavy particle collisions or in electronic excitation of atoms by electron impact. We shall come back to these in Sect. 6.5.

6.3


413

integral elastic cross section σ / 10-16 cm 2

CM collision energy T / meV 0.5

1

2

5

10

20

50

100

200

500

500 Veff

ℓ =7 ℓ =6 ℓ =5 R

ℓ =7 ℓ =6 100

ℓ =5

H + Xe

50 200

500

1000 2000 relative velocity u / m s-1

5000

10000

Fig. 6.25 Orbiting resonances in the integral elastic cross section for H-Xe scattering from two different experiments (experimental points with error bars) in comparison with quantum mechanical model calculations (lines) according to T OENNIES (2007). The inset (top right) illustrates schematically the origin of these resonances in the effective potential Veff for angular momenta > 0

Closely related to the interference phenomena just discussed is the so called shadow scattering, which in a certain sense marks the diffraction limit in collision physics, as symbolized in Fig. 6.24(e). Shadow scattering is the wave mechanical analogue to optical diffraction by a small object at very small scattering angles. For thermal heavy particle scattering it cannot be distinguished from the types of oscillations already discussed. However, for small DE B ROGLIE wave¯ i.e. for electron and ion scattering at kinetic energies lengths λdB = 2π/k = h/Mu, in the keV range it is of interest: one expects for the particle waves in full analogy to (1.67) a typical F RAUNHOFER diffraction pattern for the differential cross section: ∞ 2 I (θ ) ∝ bJ0 (kbθ )T (b)db . (6.59) 0

Here T (b) is a transmission function – in general complex – for the collision process studied. It also accounts for the range of the interaction potential. For a hard sphere of radius a by which a point like projectile is scattered, the integral leads to the same result (1.68) as in the optical case, with an opening angle of the central diffraction disc on the order of θ1 = 0.61λ/a. In the past, experimental observations of this phenomenon have occasionally been reported, mostly, however, accompanied by some controversial discussion (see e.g. G EIGER and M ORÓN -L EÓN 1979; B ONHAM 1985). The experimental requirements on angular resolution are just extreme.

414

6 6 laser beams (Na trap)

Li detector

Li+

Basics of Atomic Collision Physics: Elastic Processes Na+ detector (position sensitive)

+ _

u = 0.15 a.u.

y x

(a)

deflection coils

Li + be

z

pz / a.u. 2.5 0 - 2.5

(b) am -2.5 0 2.5

MOT coils

(c)

0.40 a.u.

-2.5 0 2.5

pz / a.u.

0.20 a.u. u = 0.15 a.u.

0

0.135 a.u.

0 0.04

0.0 04 θz / de eg

0

0 -0.04

θy / deg

0 0.04

0

0 -0.04

0.40 a.u.

0.28 a.u.

Fig. 6.26 F RAUNHOFER diffraction patterns in very small angle scattering for the charge exchange process 6 Li+ + 23 Na(3s) → Li(3s) + Na+ according to VAN DER P OEL et al. (2002). (a) Experimental setup with MOT (the heavy red arrows mark the laser beams that cool the Na atoms and position them in the trap), (b) experimental raw data, (c) reconstructed differential cross sections for small scattering angles θ for different velocities in a.u. (energies 2.4 keV to 40 keV)

For the charge exchange 6 Li+ + 23 Na(3s) → Li(3s) + Na+ such F RAUNHOFER diffraction rings have been reported by VAN DER P OEL et al. (2002) for collision energies from 2.4 keV to 40 keV. If the typical interaction radius is assumed to be a 5 to 10a0 , diffraction angles θ1 in the range 0.01◦ –0.005◦ are expected. To be able to resolve such structures cutting edge measurement technology is required. Figure 6.26(a) shows the scheme of the experimental setup used. With the help of a so called reaction microscope a complete momentum analysis has been achieved, detecting both collision partners in coincidence with position and temporal analysis after the collision.7 To achieve this analysis with sufficient precision, one has to use a well collimated Li+ projectile beam (the angular resolution was <0.3◦ ), and also the Na target gas has to be extremely cold, so that the initial state is overall very well defined. In this experiment a so called magneto optical trap (MOT) has been used, which achieved 7 For

details about this extremely powerful detection method, originally introduced as COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy), we refer the interested reader to the fundamental review by U LLRICH et al. (2003). See also Appendix B.4.

6.3


415

temperatures <1 mK. The combination of all these techniques finally provided the necessary momentum and angular resolution. In Fig. 6.26(b) we show the raw data for two velocities. A three dimensional reconstruction of the angular distribution on the basis of the experimental data is shown in Fig. 6.26(c). The impressive, characteristic diffraction patterns are indeed very similar to optical diffraction. They allow very critical tests of the relevant quantum mechanical computations of the charge exchange process. As an important final topic we want to properly recognize the significance of the so called symmetry oscillations. Schematically their origin is illustrated in Fig. 6.24(f). They occur due to the superposition of forward and backward scattering for systems with two identical particles which cannot be distinguished quantum mechanically. Since only the relative velocity of the two particles can be measured and both particles leave the collision in opposite directions, we do not know which particle goes forward and which backwards. In fact, the magnitude of the scattering amplitude for forward and backward scattering is identical and there is no way to distinguish which particle is scattered forward and which backward. Hence, corresponding to (6.58) one expects that the differential cross section becomes 2 I (θ, T ) = fj (θ ) ± f (π − θ ) .

(6.60)

According to the general symmetry rules, the positive sign holds for bosons, the negative sign for fermions. Since the phase changes with the scattering angle this gives again rise to characteristic oscillations which of course may also be observed in the integral elastic cross section. The pioneering experiment on this fascinating phenomenon has been carried out by F ELTGEN et al. (1982). Unfortunately it has, over the years, remained without much attention by main stream atomic physics. Figure 6.27 illustrates the experimental setup to study the integrated elastic cross section σel for He + He scattering. As one easily recognizes: to carry out such experiment requires heroic efforts. It requires among other sophisticated technology, two low temperature cryostats and affords highest professional skills and frustration tolerance of the experimenters. The target gas should be as cold as possible, and good velocity selection of the projectile beam is required (as indicated by M, mechanical selectors are used, based on the concept used by F IZEAU to measure the velocity of light: the molecular beam is chopped with fast rotating slit discs). High angular resolution must be achieved to limit the kinematic uncertainties and to allow for a clear observation of the cross section as a function of relative velocity8 u. The experimental results are summarized in Fig. 6.28. 4 He as well as 3 He have been studied in the various possible combinations. The symmetry oscillations are clearly visible as a function of the velocity for the symmetric systems 4 He + 4 He 8 Actually,

F ELTGEN et al. (1982) give the lab velocity of the primary beam, which is essentially identical to the relative velocity u since the target is very cold, i.e. at rest.

416

6

(a)

GA

Basics of Atomic Collision Physics: Elastic Processes GB

CB RS V C2 C3

C1

CA RS

BB

D ( ( ((( (((

S M

(b)

S

BC C1 Ø6

4×3 384

SC

RS

C2 C 3

SC Ø3

SEM

4.5×4.5

Ø5

178 12 656 265

Fig. 6.27 Experiment for low energy elastic He–He scattering designed by F ELTGEN et al. (1982) as a characteristic example for glory oscillations according to the scheme Fig. 6.24(f). (a) Details of the apparatus GA , GB : gas inlet for A and B, respectively; CA , CB : cryostat; S: primary source for the projectile A; M: mechanical velocity selector (see text); SC: scattering chamber; D: detector (ionization and deflectors); SEM: secondary electron multiplier; RS: radiation shield; BC: beam chopper; BB: beam blocker; C1 –C3 : collimating apertures; V: isolating valve. (b) Dimensioning of the apparatus in mm

σel / arb. un.

η 0= 0

−N= 2

1 0.5

100

1.5

0.5

3

4 5 6

2.5 1.5

1

4He

+ 4He

3He

+ 3He

3He

+ 4He

2.5 2

3

4

5

η 1= 0 1.6 K 10

100

1000

u / ms-1

Fig. 6.28 Integral elastic cross section σel for He–He scattering according to F ELTGEN et al. (1982) as a function of the primary beam velocity (i.e. in this case the initial relative velocity u of the colliding particles). The so called g − u oscillations are very pronounced for the symmetric pairs 4 He + 4 He and 3 He + 3 He, but completely disappear for 3 He + 4 He – in spite of completely identical interaction potential in all three cases

as well as 3 He + 3 He – albeit with a pronounced shift of maxima and minima. In contrast, no such oscillations are observed for the system 4 He + 3 He. As F ELTGEN et al. (1982) explain, “physically, the backward glory oscillations of 4 He2 and 3 He2 originate from the indistinguishability of the He atoms via zero-angle in-

6.3


417

terference between primary particles and secondary particles which are scattered backward by the repulsive part of the potential.” To fully appreciate this astonishing result, we remind the readers that the atomic charge clouds of the interacting particles – which are finally responsible for the interaction potential – are identical in all three cases. The decisive difference is, that 4 He is a boson (nuclear spin 0) and 3 He a fermion (nuclear spin 1/2), i.e. 4 He2 and 3 He2 obey pure B OSE E INSTEIN or F ERMI -D IRAC statistics, respectively, and the scattering amplitudes have to be added according to (6.60) with a plus or minus sign, respectively. In contrast, 4 He+ 3 He are distinguishable particles and squared amplitudes must be added. For more detail, such as the velocities for which s and p wave phase shifts (η0 and η1 ), respectively, go through zero, or for the meaning of the oscillation index −N , the readers are referred to the original publication. Undoubtedly these experimental observations touch upon the most subtle and basic aspects of quantum mechanics: none of the known interactions communicates among the interacting particles their bosonic or fermionic character: apart from being bosons or fermions, the ingredients of the interaction potential are identical in all three cases. And nevertheless the collision partners ‘know’ from the very beginning of the collision (i.e. essentially at infinite distance) how they have to arrange their wave function: symmetric or antisymmetric or not at all. Of course the experimental observations may be predicted precisely according to the rules of quantum mechanics. One just needs to apply the proper recipe: that for bosons or fermions or distinguishable particles. But why the particles do, what they do, is not a question to be asked. Clearly, the physics at work here is fundamentally the same as that needed to construct the periodic system of elements: only the fact that electrons are fermions makes atomic matter possible in the form as we know it. However, the present experiment may even be seen somewhat more crucial. Here we do not discuss bound states (i.e. particles which are essentially closely spaced at all times). We discuss collision processes, i.e. continuum states which extend essentially over infinite space. Section summary

• Classical scattering theory provides a good first approach towards understanding many important phenomena in elastic, low energy heavy particle collision physics. The differential elastic cross section is derived in a straight forward manner through (6.36), simply by evaluating (6.42) for the classical trajectory Θ(b, T ) as a function of kinetic energy T and impact parameter b. Interesting singularities (rainbow scattering) are predicted (and experimentally observed) for |dΘ/db| = 0. • In the limit of small scattering angles the classical differential elastic cross section I (θ, T ) may be evaluated explicitly to give (6.47). It turns out to be useful to introduce a reduced scattering angle τ (b, T ) = θ T and a reduced cross section ρ(b, T ) = θ sin θ I (θ, T ) which (at high enough T ) both depend only on the impact parameter b.

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6


• The classical interpretation of scattering processes is, however, limited. For example one finds (in contrast to classical expectation) the integral elastic cross section (6.56) to be finite for long range potentials −CR −s if they decay faster than the C OULOMB potential with s = 1. And to understand experimentally observed interference phenomena – such as supernumerary rainbows, glory oscillations, rapid oscillation, symmetry oscillations and orbiting resonances – one has to superpose quantum mechanical amplitudes coherently with their phases, rather than to add cross sections.

6.4

Quantum Theory of Elastic Scattering

As we have just seen, classical theory does not suffice to describe all scattering processes and the broad variety of experimentally observed phenomena fully. For electron scattering this is already evident according to (6.35) as a consequence of the large DE B ROGLIE wavelengths λdB . In heavy particle scattering interference effects force us to use a quantum mechanical interpretation. In principle, quantum ¯ is comparable with characmechanics has to be used when λdB = 2π/k = h/(Mu) teristic dimensions of the interaction. In this section we shall continue to focus on elastic scattering and introduce the basic quantum mechanical concepts of scattering amplitudes and partial wave analysis, and discuss the behaviour for scattering phases in some characteristic cases.

6.4.1

General Formalism

The L IPPMANN -S CHWINGER Equation Elastic scattering leaves the internal wave functions of the interacting particles (electronic, vibrational, rotational) unchanged. Thus, we simply have to translate (6.30) in the usual manner, to obtain a stationary wave function ψ(R) for the relative motion – now at positive energies: 2 2 2 k 2 ψ(R) = 0. (6.61) − ∇ + V (R) − 2M¯ 2M¯ Here k = p/ is the wave vector of the relative particle motion, T = 2 k 2 /2M¯ the initial kinetic energy in the CM system. In this stationary picture one describes the projectile beam (more precisely the relative motion of the particles with reduced ¯ as a plane wave propagating into direction k i , and the CM is the origin of mass M) an outgoing spherical wave. We thus want to find asymptotic solutions for (6.61) of the type ψ(R) −→ ψ (ki ) (R) + R→∞

eikf R f (θ, ϕ) R

with ψ (k) (R) = eik·R .

(6.62)

6.4


419

This normalization of ψ (k) (R) follows the most common practice in scattering physics (see e.g. B URKE 2006).9 As well known and easy to verify, ψ (k) is a solution of the (homogeneous) free particle wave equation ∇ 2 ψ (k) (R) + k 2 ψ (k) (R) = 0, and one rewrites (6.61) for the scattering problem as ∇ 2 ψ(R) + k 2 ψ(R) =

2M¯ V (R)ψ(R). 2

(6.63)

To find a special solution one uses the corresponding G REEN’s function G(R) G(R) = −

eikR , 4πR

a solution of ∇ 2 G(R) + k 2 G(R) = δ(R),

and obtains as a formal, general solution for the scattering problem: 2M¯ (k i ) ψ(R) = ψ0 (R) + 2 G R − R V R ψ R d3 R .

(6.64)

(6.65)

This is the so called L IPPMANN -S CHWINGER equation in coordinate representation. By inserting G(R), the full solution ψ (+) is then derived from ikf |R−R | 1 M¯ e V R ψ (+) R d3 R . (6.66) ψ (+) (R) = eiki ·R − 2 2π |R − R | The scattering signal is detected at very large distances R, for which R − R −→ R 1 − eR · R + · · · , R→∞

with eR = R/R. By identifying the direction of the scatted particle with k f = kf eR (in the elastic case kf = ki = k), we can pull an outgoing spherical wave in front of the integral in (6.66) and obtain M¯ eikf R (6.67) ψ (+) (R) −→ eiki ·R − e−ik f ·R V R ψ (+) R d3 R . 2 R→∞ 2π R This has exactly the form desired by (6.62), with the so called scattering amplitude M¯ f (θ, ϕ) = f (k f , k i ) = − (6.68) ψ (kf )∗ (R)V (R)ψ (+) (R)d3 R. 2π2 9 For

plane wave normalization see also Appendix J.2 in Vol. 1. We note that the present, standard scattering theory normalization of the plane wave differs by a factor (2π)3/2 from normalization in k scale. For determining cross sections this is irrelevant, since particle fluxes are finally normalized to each other. However, normalization in k scale is also used quite often in the literature (see e.g. B RAUNER et al. 1989) and leads to slightly different prefactors in the expressions for scattering amplitudes, T-matrix elements and cross sections. We shall occasionally point this out where appropriate.

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6


With 2 /me = Eh a02 and ψ (+) defined dimensionless, while the interaction potential V has the dimension Enrg, one finds that the scattering amplitude is of dimension L. Note that (6.68) for the scattering amplitude is still exact if ψ (+) is an exact solution of the S CHRÖDINGER equation. All high energy approximations start at this point by substituting reasonable approximations for ψ (+) (R).

Scattering Amplitude, T-matrix and Cross Section The scattering amplitude f (θ, ϕ) describes the dependence of the scattered spherical wave on polar and azimuthal angle, while k i and k f are the wave vectors of the relative particle motion before and after the collision, respectively. In compact form it may be written f (θ, ϕ) = −

M¯ M¯ (+) k |V |ψ = − k f |T|k i . f 2π2 2π2

(6.69)

We have introduced here the transition matrix or transition operator T (short: Tmatrix) which is formally defined by V ψ (+) = T|k i ,

(6.70)

were |k i represents the initial plane wave and |ψ (+) the full solution of the scattering problem. The T-matrix is a useful tool in formal scattering theory, as we shall see in Sect. 6.4.6. Its elements have the dimension Enrg × L3 . The second term in (6.62) represents the scattered particle. The flux which reaches the detector is10 Φf =

k f |f (θ, ϕ)|2 , 2 M¯ Rdet

so that

(6.71)

2 N˙ = Φ f · Adet = |Φf |Rdet Ω

(6.72)

scattered particles N˙ per unit of time and per scattering atom reach the detector 2 Ω is the detector in a distance Rdet from the scattering centre. Here Adet = Rdet area for an acceptance angle Ω. We finally normalize to the initial particle flux Φ i = k i /M¯ and obtain from the definitions (6.20) and (6.21) the DCS: I (θ, ϕ) =

2 kf dσ N˙ = = f (θ, ϕ) . dΩ Φi Ω ki

one obtains it by application of the quantum mechanical expression Φ = /2mi[ψ ∗ ∇ψ − (∇ψ ∗ )ψ] onto the asymptotic wave function (6.62). Note, however, that with ψ being defined here dimensionless, the thus derived flux has the dimension LT−1 , i.e. it is given per time (T), per area (L2 ) and per volume in k space (L−3 ). Final normalization to the initial flux allows us to ignore these dimensions.

10 Formally

6.4

Quantum Theory of Elastic Scattering incident particle beam

421 p = ħk

b R y (col) ℓ e i kz

x (col)

z (col) +Θ classical trajectory

Fig. 6.29 Standard collision frame, with the x (col) z(col) scattering plane defined by the position coordinate R and the momentum p = k prior to collision; note that y (col) points here perpendicularly into the scattering plane. With the impact parameter b the magnitude of the angular momentum is || = = bp

In summary, differential (DCS) and integral (ICS) cross sections become 2 ¯ 2 2 kf dσf i (M/m e ) kf = ff i (θ, ϕ) = Tf i (k f , k i ) 2 4 2 dΩ ki (2π) Eh a0 ki 2 kf ff i (θ, ϕ) dΩ. and σf i = ki

(6.73) (6.74)

For general reference, we have included the indices i and f referring to initial and final states of the whole interacting system, respectively.11 Specifically, for elastic scattering, the magnitudes of the wave vectors are kf = ki = k.

6.4.2

Angular Momentum and Impact Parameter

The L IPPMANN -S CHWINGER equation (6.65) may be solved iteratively by suitable approximations – a procedure preferentially used for high energies (and thus short interaction time). At low energies one may try to express the quantum mechanical wave function in terms of angular momenta, following the scheme used for bound states. The key difference is, however, that typically many angular momenta contribute to the wave function in the continuum: the scattering amplitude just discussed can simply not be expressed by a single spherical harmonics, in contrast to discrete, bound states. One has to expand the problem into a series of (so called) partial waves. This may be rationalized in the classical, standard collision frame “col” defined by scattering plane and impact parameter as illustrated in Fig. 6.29. 11 With

plane wave normalization in k scale the right side of (6.73) is written in a.u. dσf i kf (k−scale) 2 T = M¯ 2 (2π)4 dΩ ki f i

(k−scale) (see e.g. OVCHINNIKOV et al. 2004), with Tf i = (2π)−3 Tf i .

422

6


The angular momentum is conserved during the collision. We have derived it classically in Sect. 6.3.3 from (relative) momentum p and position vector R in the CM system. In vectorial form it is = R × p = R × k

(6.75)

and defines also the collision plane to which it is perpendicular. For spatially isotropic potentials one usually chooses the direction of the incident particle beam as quantization axis z(col) and to be perpendicular to the x (col) z(col) collision plane, i.e. parallel to y (col) . Angular momentum conservation leads to an important conclusion: elastic collisions occur exclusively in the plane defined by linear momentum and scattering centre. The final momentum also lies in this plane. This still holds for inelastic collisions – if the interaction potential is isotropic prior to collision. The scattering plane may change, however, if angular momentum is transferred onto the collision partner. For the magnitude of the angular momentum ¯ || = = bp = bk = b 2MT holds, (6.76) and thus b = /k.

(6.77)

Hence, collisions with impact parameters between 0 ≤ b < 1/2k may be viewed as dominated by s waves, for 1/2k ≤ b < 3/2k by p waves and so on. Adapting this to quantum mechanics, one sets bk = ( + 1) ¯ = ( + 1/2) (6.78) with the latter being an excellent approximation for large .

6.4.3

Partial Wave Expansion

One thus solves the S CHRÖDINGER equation (6.61) by the usual separation ansatz ψ(R) =

u (R) Ym (θ, ϕ) R

(6.79)

with the spherical harmonics Ym , where and m are as usual the quantum numbers for magnitude and orientation of the angular momentum in space.12 As in the case of bound states, the radial equation is written as 2 d 2M¯ ( + 1) 2 u (R) = 0. + k − 2 V (R) − (6.80) dR 2 R2 12 Here and in the following we use the notation for the angular momentum of the relative nuclear motion and call its quantum number – as usual done in this context – and we have to accept the little inconsistency with respect to denoting the molecular angular momentum by N and its quantum number by N , according to Chap. 3.

6.4

(a)


423

η>0

(b)

u(R )

η<0 sin(kR+η)

u(R )

sin(kR+η) sin(kR)

V(R ) T

sin(kR)

V(R ) T

R

R potential well (attractive)

potential barrier (repulsive)

Fig. 6.30 Illustration of the scattering phase η for a (hypothetical) one dimensional model problem: (a) attractive and (b) repulsive potential (heavy black lines). The full red lines represent the scattered waves, the dashed red lines the unperturbed waves

For simplicity we assume an isotropic potential V (R), so that f (θ, ϕ) = f (θ ). To get acquainted with these partial waves u (R) and with scattering phase shifts, let us first visualize them for a one dimensional model. Figure 6.30 illustrates the asymptotic behaviour of the wave function u(R) = sin(kR + η) which has passed a scattering potential V (R). The scattered wave functions differ from the freely propagating wave sin(kR) just by the phase shift η of the wave trains. The sign of η is positive or negative for attractive and repulsive potentials, respectively. Let us go back to the full three dimensional radial equation. To solve (6.80) we choose the coordinate system sketched in Fig. 6.29. Prior to collision we have m = 0, and since angular momentum is conserved this holds also after the collision. In contrast to the bound states in atomic physics, in a collision problem always many angular momenta have to be superposed – simply because it is impossible to prepare the scattering system prior to collision with a well defined collision parameter or angular momentum. This makes the solution of the scattering problem substantially more complicated than the search for bound states. One writes the total wave function (6.62) as a so called partial wave expansion ∞

ψ

(+)

1 (R) = A u (R)P (cos θ ) kR

(6.81)

=0

√ with the L EGENDRE polynomials P (cos θ ) = 4π/( + 1)Y0 (θ, 0). Correspondingly, we also expand the plane wave of the asymptotic initial state in partial waves, following (J.13)–(J.18), Vol. 1: ψ (ki ) (R) = eiki Z =

∞ (2 + 1)i j (kR)P (cos θ ).

(6.82)

=0

We have written the solutions of (6.80) for vanishing potential V (R) → 0 as spherical B ESSEL functions j (kR) = u0 (kR)/kR.

424

6


For the evaluation of the scattering processes we are simply interested in the asymptotic behaviour for R → ∞ (see Appendix J, Vol. 1). For vanishing potential the radial wave function is given by: π (0) u (R) ∝ sin kR − . (6.83) R→∞ 2 Among the general solutions u (R) of (6.80) we now have to select those which describe the scattering function (6.62) correctly as a partial wave expansion (6.81). Asymptotically we expect π (6.84) u (R) ∝ sin kR − + η R→∞ 2 π π ∝ sin kR − + tan η cos kR − . R→∞ 2 2 This differs from the free, plane wave (6.83) simply by the scattering phase shift η , which depend of course on . The partial wave expansion of the scattered wave is obtained from the definition (6.62) by subtracting the plane wave (6.82) from the full wave function (6.81): ψ (+) (R) − ψ (ki ) (R) → −

eikR f (θ ). R

(6.85)

With the asymptotic behaviour (6.84), (6.83) and with some algebra we obtain finally the elastic scattering amplitude for an isotropic potential f (θ ) =

∞ 1 (2 + 1) e2iη − 1 P (cos θ ). 2ik

(6.86)

=0

With (6.73) follows from this the differential and with (6.74) the integral cross section. We have thus derived a direct relation between measurable observables and the S CHRÖDINGER equation (6.61). Summarizing, the task of the quantum theory of elastic scattering consists in determining the phase shifts η between the partial waves of angular momentum with and without potential. The square of the scattering amplitude f (θ ) according to (6.86) provides the differential cross section. Integration over all scattering angles leads us to the integral elastic cross section: σel =

∞ 4π (2 + 1) sin2 η . k2

(6.87)

=0

Finally, we note that for θ = 0 and P (1) = 1 the scattering amplitude (6.86) becomes ∞ 1 f (0) = (2 + 1) sin η eiη . (6.88) k =0

6.4


425

Inserted into (6.87) this leads to the so called optical theorem: σel =

6.4.4

4π Im f (θ = 0). k

(6.89)

Semiclassical Approximation

As just discussed, to determine the elastic cross sections we have to solve the radial equation (6.80) and to determine the scattering phases η from the asymptotic solutions (6.84). For elastic scattering of electrons with a few eV kinetic energy this is not too difficult (if the scattering potential is sufficiently well known or can be calculated to some reasonable approximation). According to (6.76) only a few partial waves will be relevant. However, if the angular momenta can be large – and for heavy particle scattering this almost always the case – then a fully quantum mechanical computation of all scattering phases becomes an extremely cumbersome task. Thus, appropriate approximations are required. Semiclassical procedures are the method of choice if the DE B ROGLIE wavelength λdB = 2π/k is small compared to typical dimensions over which the potential changes and at the same time dλdB

1 dR

(6.90)

holds during the collision process. Then, classical theory which computes for each impact parameter (6.77) b = /k a trajectory is a good first approximation. To include interference effects, in the so called eikonal approximation one computes the phases along all these trajectories in the effective potential Veff = V (R) +

2 ( + 1) b2 2 ( + 1/2)2 = V (R) + T 2 V (R) + , ¯ 2 ¯ 2 R 2MR 2MR

(6.91)

where we have applied (6.78). The effective kinetic energy Teff = T − Veff (R) changes slowly along the trajectory, and the wavenumbers for the free spherical wave k and the partial wave k in the scattering potential V (R) are k (R) =

k2 −

( + 1/2)2 R2

k2 −

2M¯ ( + 1/2)2 V (R) − , 2 R2

k (R) =

and

(6.92)

respectively.

(6.93)

426

6


The semiclassical scattering phase shift13 is thus obtained by ∞ ∞ SC η (T , ) = k (R)dR − k (R)dR, or R

ηSC (T , b) = k

∞

Rc

R˜

b2 V (R) − 2 dR − k 1− T R

∞ b

(6.94)

1−

b2 dR, R2

(6.95)

with the classical turning points R = Rc given by the root of (6.92) and by R˜ = b = ( + 1/2)/k. In practice, due to the singularities at the classical turning point, the computation of the JWKB phases is not trivial (see e.g C OHEN 1978, for efficient computational methods). For high kinetic energies T V (R) one may expand the first term of (6.95) in powers of V (R)/T and obtains the J EFFREYS -B ORN phase ∞ V (R) k JB η (T , b) = − dR. (6.96) 2T b 1 − b2 /R 2 To see the connection between quantum mechanics and classical trajectory one exploits the asymptotic expansion of the L EGENDRE polynomials,

1 π 2 →∞ cos + θ− P (cos θ ) −→ π sin θ 2 4 for large , and obtains the scattering amplitude (6.86) conveniently by integration, rather than by summation: ∞√ 1 f (θ ) = √ (6.97) eiΦ+ () + eiΦ− () d ik 2π sin θ 0 π (6.98) with Φ± () = 2η ± θ ∓ . 4 If one expands the phase Φ± () around a well defined value of = 0

dη (0) Φ± () = Φ± + 2 ± θ ( − 0 ) + · · · , d 0 one sees that the integral in (6.97) usually averages out to zero, due to fast oscillations as a function of . Only for such 0 for which dΦ± dη =2 ±θ ≡0 (6.99) d d 0 13 Usually called Wenzel-Kramers-Brillouin (WKB) or Jeffreys-Wenzel-Kramers-Brillouin (JWKB) phase shift.

6.4


π Θ ηℓ

427

Θ bg

br

θ -θ - θr

0 b, ℓ ηℓ b'1 b 2

b3

Φ+ Φ-

Φ+

0 Φ-

b, ℓ

Fig. 6.31 Classical deflection function Θ(b) (red) and scattering phase η (black) as a function of impact parameter b and angular momentum = bk, respectively. Below, the corresponding phases Φ+ () (dotted) and Φ− () (dashed) according to (6.98) are shown. Marked are the impact parameters and scattering angles for glory scattering bg and for the rainbow br at the maximum and turning point of the scattering phase ηl , respectively. Also indicated are the three points of stationary phase at b1 , b2 and b3 which contribute to the scattering amplitude for a given scattering angle θ . They can be related directly to the corresponding classical trajectories (schematically) shown in Fig. 6.21

holds, the integral in (6.97) remains finite. These values of Φ± (0 ) are called stationary phases. This remarkable result allows us, in one further step, to draw a direct connection between partial wave expansion (6.97) and classical trajectory: we evaluate the slope of the phase shift explicitly in JWKB approximation (6.94) (with ¯ = + 1/2):

∞ ¯ ¯ dR dR − 2 ¯ b Rc R 2 k 2 − 2M¯ V (R) − R 2 k 2 − R 2 2 ∞ π dR Θ(b) = −b . = √ 2 1 − V (R)/T 2 2 R Rc eff

dη = d

∞

¯2 R2

(6.100)

(6.101)

The last step follows directly from comparison with the classical deflection function Θ(b) according to (6.42). Inserting this into (6.99) one recognizes: stationary phase conditions are realized exactly when the scattering angle θ is identical to the classical deflection function ±Θ(b). This is a rather nice result: only such partial waves (or impact parameters) contribute to the scattering amplitude f (θ ) according to (6.97) which are close to the corresponding classical trajectory. Schematically this is illustrated in Fig. 6.31 for one scattering angle θ – arbitrarily chosen here close to the rainbow angle θr . The figure is so to say the semiclassical equivalent to Fig. 6.21. The thin, vertical, red lines indicate the impact parameters

428

6


for which the phases Φ+ or Φ− have a maximum or minimum (i.e. are “stationary”) and hence contribute to the scattering amplitude. It should be pointed out that these semiclassical methods are very powerful and allow one to describe elastic heavy particle collisions (as discussed in Sect. 6.2), as accurately as demanded by the experiments.

6.4.5

Scattering Phase Shifts at Low Kinetic Energies

The semiclassical approximation is excellent as long as the DE B ROGLIE wavelength is small compared to the characteristic dimensions of the interaction potential. For electron scattering this is, however, only the case for rather high kinetic energies (at least ≥100 eV). As a rule, one has to solve the radial equation (6.80) explicitly for each in order to derive the scattering phases. This also holds for the heavy particle scattering in the very low energy range, e.g. for the orbiting resonances in the sub-thermal energy range, already discussed in Sect. 6.3.4. Interestingly enough, collision processes with “ultracold atoms” – a topic of considerable interest during the last years – have revived the interest in the respective method of scattering theory dating back to second half of the past century. In principle, scattering phase shifts may be obtained today without problems by solving (6.80) numerically if the scattering potential V (R) is known. Conversely, one may also try to reconstruct the latter by partial wave analysis from precisely measured integral or differential cross sections.

Potential Well and Barrier Nevertheless, it is very useful to know some general trends and simple approximations beyond the black box “computer code”. We thus want to give a survey about the typical behaviour of scattering phase shifts in the low energy regime and illustrate it with classical examples. We start by investigating the phase shifts for a (hypothetical) potential well (or a potential barrier) V (R) of the depth (or height) ¯ One may then rewrite (6.80) in dimensionless form, measurV0 = ∓(2 k0 )2 /2M. ing distances R in units of the well (barrier) width a and energies in units of V0 . This is a standard problem solved in standard textbooks of quantum mechanics and scattering theory (see e.g. B RANSDEN and J OACHAIN 2003). On finds that (6.80) is solved exactly by a combination of spherical B ESSEL and N EUMANN functions, j (kR) and n (kR), respectively, and the scattering phase shifts are given by ˜ = tan η (κ)

κj ˜ (κ)j ˜ (κ) − κj (κ)j (κ) ˜ . κn ˜ (κ)j ˜ (κ) − κj (κ)n (κ) ˜

(6.102)

Here κ = (ka)2 + (k0 a)2 and κ˜ = ka are the magnitudes of the wave vectors in dimensionless form within and outside the potential well (barrier).14 Since the scat14 When

evaluating (6.102) one has to be careful with the arctan function which is multiple valued.

6.4


Fig. 6.32 Low energy behaviour of the s, p and d scattering phase shifts η for the elastic scattering at a potential well (barrier) of depth (height) V0 = ∓2 k02 /2M¯ and radius a. The characteristic dimensionless potential parameter is k0 a. Left η (ka) are shown for an attractive potential (well), right for a repulsive one (barrier). The small oscillation seen for larger k0 a originate from the sharp boundary of the potentials and have no practical relevance

429 attractive

repulsive

η / rad 2π

η / rad

π

k 0a = 6

-π

ka

0 ηs 5 ηp 0

k 0a = 4.4 π -─ 2 ka

0

ηs η p 5

k 0a = 1.5

5

10

ka

10 ka

0

ηp 0

ηp 5

10

ηs

─ 4

ηs

0

10

ηd 5

π

10 ka

ηp

π

5

0

ηs

10 ka

ηp

π

-─ 5

ηs

tering phase shifts are in principle only meaningful modulo 2π one defines: η −→ 0. k→∞

(6.103)

Figure 6.32 illustrates some general trends of the scattering phases ηs , ηp and ηd ( = 0, 1, 2) at low energies for different attractive (left) and repulsive (right) potential well depths and barrier heights, respectively. As discussed already in Sect. 6.4.3 and illustrated in Fig. 6.30 the phase shift is positive (negative) in an attractive (repulsive) potential; we have to add that this holds at moderate energies and not too high . The absolute value of the phase shift grows usually rather rapidly with k (or with energy), reaches a maximum and finally falls to zero as defined by (6.103). There are, however, a few remarkable exceptions in the case of an attractive potential if that is sufficiently deep: while for small well depths (k0 a = 1.5) all phases start at η (ka = 0) = 0, for k0 a = 4.4 the s phase as well as the p phase has the limit π for vanishing k. And for the deepest potential well shown here (k0 a = 6) the s phase shift at energy zero corresponds already to a full wavelengths (ηs = 2π ). The shifts for p and d waves start in this case at ηp (0) = ηd (0) = π . In contrast, the repulsive potential (right in Fig. 6.32) shows no such peculiarities.

L EVINSON Theorem, Scattering Length and R AMSAUER Effect A closer investigation of the mathematics behind these observations shows that the number N of bound states with angular momentum that can exist in a given potential determine the behaviour of the scattering phase shifts at vanishing kinetic energy. One may show that for an arbitrary potential the so called L EVINSON theo-

430

6


rem holds: η (k) −→ N π.

(6.104)

k→0

Hence, Fig. 6.32 shows us that one bound s and one bound p state can exist in a potential well with k0 a = 4.4. For k0 a = 6 even two bound s states as well as one p and one d state exist. Very interesting is also the behaviour of the d phase shift in the middle panel left in Fig. 6.32: for k0 a = 4.4 obviously a bound d state does just not yet exist – in the limit η2 (0) = 0. However, the d phase shift rises very rapidly at about ka 0.7 from 0 to nearly π . One interprets this as due to a quasi-bound state in the continuum. We have seen such resonance phenomena already, e.g. in Fig. 6.25 as orbiting resonances in the heavy particle scattering, and we shall treat scattering resonances in more detail in a moment. Here we simply note that such rapid changes of a scattering phase (by nearly π ) must lead according to (6.86) and (6.87) to pronounced structures in the DCS and ICS. The dominance of low values at low energies, as illustrated in Fig. 6.32 for the one dimensional case, suggests that a corresponding series expansion might be useful. Such expansions are known as effective range expansion valid under certain conditions in the general case (see e.g. OM ALLEY et al. 1961, and further references therein):15 k 2+1 cot η = −

1 + k 2 r + O k 4 . a

(6.105)

Here, a and r are constants. For the s phase shift they are called scattering length (as ) and effective range (rs ), respectively, and the scattering length is positive for repulsive and negative for (not too strongly) attractive potentials. In the limit of vanishing kinetic energy T → 0 one may combine (6.104) and (6.105) to: η −→ −k 2+1 a + N π. k→0

(6.106)

Thus, for very low energies (e.g for collisions of ultra-cold atoms at temperatures below µK, i.e. for kinetic energies T < 10−11 eV!) practically only s scattering is relevant. Then ηs → −kas and the scattering amplitude (6.86) becomes f (θ, k) −→ k→0

ηs 1 −2iηs −as e −1 2ik k

so that I (θ ) → as2 .

(6.107)

Hence, elastic DCS and ICS according to (6.73) become dσel −→ a 2 dθ k→0 s 15 Today,

and σel −→ 4πas2 . k→0

(6.108)

owing to the ever increasing experimental precision and large data collections improved concepts for effective range expansions exist (see e.g. G ULLEY et al. 1994; R AKITYANSKY and E LANDER 2009).

6.4

Quantum Theory of Elastic Scattering ηs

π 0

as<0

RAMSAUER minimum

as > 0 as<0 as > 0

431

attractive I II

III repulsive

IV k

Fig. 6.33 Four different possibilities for the behaviour of the s scattering phase shift ηs at low energies. According to the effective range expansion (6.105) the scattering lengths as is derived from the slope of ηs (k) as a function of k in the limit (6.106) for very small k

The integral cross section at these low kinetic energies is thus finite! Interestingly, we recall that for elastic scattering of a point-like particle by a hard sphere of diameter d the scattering length is just as = d/2 (the phase shift of a spherical wave originating from the centre of the sphere is −η = kd/2 at the surface of the sphere). Hence, the effective elastic ICS of a sphere is at low energies according to (6.108) σel = πd 2 , i.e. four times its geometrical cross section (6.54) which is predicted by classical theory. By the way, this also holds for light waves diffracted by a small disc. In both cases, interference effects are responsible. With increasing energy the phases change. For s phases Fig. 6.33 sketches schematically four different types of possible behaviour for ηs (k) at low energies. Here terms up to k 3 are considered. Different magnitudes and signs of as and rs and a different number of bound states is assumed (one for I and II each, none for III and IV). In all cases the integral elastic cross section according to (6.108) is finite for k → 0. One particularly interesting case is displayed in curve I: at a particular value of k the s phase shift ηs passes through π , so that sin ηs = 0. Hence, the integral elastic cross section will assume there a minimum, it may even disappear nearly completely since all other phases in (6.87) may still be very small due to the k 2+1 behaviour. This effect has first been discovered by R AMSAUER in 1921 and we have seen impressive examples for it already in the introduction to the present chapter (see Fig. 6.8). We can now understand this quite remarkable phenomenon: as a transition of the scattering phase shift through N × π (usually the s phase).

Examples of Partial Wave Analysis: e− Scattering by He and Ne Scattering phases at very low collision energies may be determined experimentally from precision measurements of differential cross sections by a so called partial wave analysis – as long as only a few partial waves contribute significantly to (6.86). Scattering phase shifts η are then optimized as parameters to obtain best fits to the experimental data. A state-of-the-art documentation on electron scattering by rare gases is found in A DIBZADEH and T HEODOSIOU (2005). Some typical results for the elastic electron scattering by He and Ne are summarized in Fig. 6.34. On the left, examples of experimentally determined DCS are shown, on the right the phase shifts derived from such data are presented as a function of kinetic energy T of the scattered electrons. To each measured experimental angular distributions of the

432

6

I(θ ) / Å2 sr -1 0.6

e- + Ne 0.4 10 eV

2 eV

(a)

(c)

5.8 5.6

2.5

12 eV

3.0 0.06

10 -1

(b)

0.04 0.1

(d)

10 -2

d

(f )

0 0º

90º

180º θ

0º

90º

180º

p

(h)

p

0.2

0.4

s

(g)

3.1

0.3

50 eV

1

e- + Ne

s

(e)

2.0

0

0.6

0

6.2 6.0

e- + He

3.0

0.2

0

0.2

ηℓ / rad

ηℓ / rad

I(θ ) / Å2 sr -1

e- + He

0.4 0.2


0

10

d

(i)

0.02 0 20 0 T /eV

f 2

4

6

Fig. 6.34 Partial wave analysis for e− − He scattering according to A NDRICK and B ITSCH (1975) and e− − Ne according to G ULLEY et al. (1994) and A DIBZADEH and T HEODOSIOU (2005). (a)–(d): measured differential cross sections I (θ) (for He the error bars are on the order of the linewidths); (e)–(i) phase shifts η derived from a corresponding partial wave analysis for the s, p, and d waves (for Ne also f )

DCS one set of phase shifts may be fitted. For e− − He scattering one finds that in the energy range from 0 to about 15 eV excellent agreement is obtained by just including the s, p and d phase shifts as documented by Fig. 6.34(e) and (f), while for e− − Ne scattering, f waves have to be included at even lower energies. As a side remark we note: the s wave phase shift for e− − He of π at zero energy shown in (e) does not allow the conclusion that a He anion exists! While the L EVINSON theorem (6.104) in principle supports an additional bound s state (with n = 1), helium already contains two bound (1s)2 electrons, i.e. the 1s shell is fully occupied, and according to the PAULI principle no additional (low energy) electron can be attached. A similar argument holds for Ne. The limiting s and p phases are 2π and π (Fig. 6.34(g) and (h)), respectively, but due to the PAULI principle no extra electron can be added to the full Ne (2s)2 or (2p)6 shells: thus no stable neon anion exists.

6.4.6

Scattering Matrices for Pedestrians

We shall now introduce a somewhat formal, but very important concept of scattering theory. In the spirit of this book we shall try to keep the mathematics simple and approach the subject somewhat heuristically by discussing Fig. 6.35 which schematically illustrates the key elements of a scattering experiment.

6.4


Fig. 6.35 Scheme of the scattering formalism. The T-matrix acts on the incoming plane wave |k i . The thus constructed scattered wave T|k i is projected onto the plane wave |k f which the detector can register

collimating aperture for projectile beam

433 detector aperture

detektor for plane |k f 〉 scattered wave

θ incoming ^ T unscattered wave plane source wave |k 〉 collision 0 ^ scattered wave T |k 0 〉

The incoming projectile beam, well collimated by apertures, can be described as a plane wave |k i = |ψ (k i ) (R) which propagates into direction k i . The main part of this wave passes the scattering centre without any disturbance – a very small part is scattered. The scattered wave is obtained quite formally by letting an operator S act onto |k i : |k i → S|k i = |k i + T|k i .

(6.109)

The right part of this equation explicitly expresses the (small) scattered wave by S the scattering matrix or scattering operator, while the T-matrix T|k i . One calls (6.69) and the T operator (6.70) have already been introduced above. Equation (6.109) is so to say the mathematical description of Fig. 6.35. The detector too, positioned far away from the scattering centre, detects a well collimated beam which also may be described by a plane wave |k f propagating into direction k f . In this conceptual framework, the scattering amplitude (6.69) ff i (θ, ϕ) = −

¯ M¯ |k i = − M/me k f |T|k i k | T f 2π2 2πEh a02

(6.110)

is simply the probability amplitude for finding |k f in the scattered wave T|k i , i.e. one projects former onto the latter. The differential cross section is then given by (6.73), with ki = kf = k in the elastic case. We emphasize again, that the above Tmatrix element in (6.110) has the dimension L2 Enrg so that ff i has the dimension L. A partial wave expansion of the T-matrix is found by expanding the plane waves before and after the scattering process. Exploiting the orthogonality relations of the spherical harmonics Ym one obtains the scattering amplitude:16 2π ff i (θ, ϕ) = i kf ki

m m

∗ i− T m m Y m (θ, ϕ)Ym (θi , ϕi ).

(6.111)

T m ,m (k) are the matrix elements of T, while θ, ϕ and θi , ϕi characterize the direction of the scattered plane wave |k f (as seen by the detector) and the incoming wave |k i , respectively. The latter is usually chosen to be parallel to the z-axis, so that θi = ϕi = 0 and m = 0 holds. It is instructive to have a look at the dimensions in 16 With plane wave normalization in k scale, the T-matrix elements have to be multiplied by a factor −2πi.

434

6


(6.111). Since the wave vectors lead to a dimension L which also is the dimension of the scattering amplitude, the T-matrix elements obviously must be dimensionless – in contrast to the one in (6.110) (the dimension of k f |T|k i is Enrg L3 ). This is due to the fact that these matrix elements are obtained from radial wave functions which are normalized in energy scale according to (J.11) in Vol. 1. Integration over dR and the dimension of T (which is Enrg) cancel the dimensions of this normalization. For elastic scattering by an isotropic potential we may compare this with the partial wave expansion (6.86) which is diagonal in . Hence for elastic scattering T m ,m = δ δm m T ,

(6.112)

and by comparing directly (6.111) with (6.86) one finds T = e2iη − 1.

(6.113)

In deriving this relation we have used the addition theorem of the spherical harmonics (C.22), Vol. 1. Alternative to the T-matrix one may also use the S-matrix, for which holds in the isotropic, elastic case S = e2iη = 1 + T .

(6.114)

So far, S and T matrices simply offer a trivial possibility to rewrite the partial wave expansion (6.86) for the scattering amplitude: f (θ ) =

1 (2 + 1)P (cos θ ) S (k) − 1 2ik

(6.115)

1 (2 + 1)P (cos θ )T (k). 2ik

(6.116)

=

The integral cross section (6.87) is then written as σel =

∞ π (2 + 1)|T |2 . k2

(6.117)

=0

However, for the general case of non-isotropic potentials and/or inelastic scattering the angular momenta of partial waves are coupled to the eigenstates of projectile and target. Hence, T and S are genuine matrices with nonvanishing off-diagonal terms. They allow a transparent representation of the scattering process by (6.111). In that case one defines17 1. S = 1 + T with S S† =

(6.118)

plane waves normalized in k scale, the relation between S- and T-matrix is S = 1 − 2π T. The prefactor (2π)−1 in the scattering amplitude becomes (2π)2 . 17 With

6.4


435

The scattering matrix S is thus defined as a unitary operator. In view of (6.109) this unitary relation expresses nothing else but the conservation of particle flux during the scattering process. The S and T operators allow to formulate the symmetry of a scattering process with respect to incoming and outgoing plane waves in a clear manner: we may invert the scattering process by simply inverting the beam directions. In Fig. 6.35 the source would become detector and vice versa. We then have to replace (6.109) by the time inverse equation: S † |k f . |k f → |k f + T† |k f =

(6.119)

Inverse to (6.110) is thus the scattering amplitude: M¯ k i |T† |k f 2π2 M¯ = k f |T|k i ∗ = ff∗←i (θ, ϕ). 2π2

fi←f (θ, ϕ) =

(6.120)

For later use we finally note an alternative representation of the asymptotic solutions for the partial waves (6.84) as superposition of in and outgoing waves: π u (R) → sin kR − + η = a∗ e−ikR + a e+ikR 2 (6.121) 1 π −ikR+i π2 +ikR−i π2 ikR−i π2 − T e ∝e − S e ∝ sin kR − 2 2 π 1 a (6.122) with a = exp −i + iη and S = − ∗ eiπ = e2iη . 2i 2 a

Section summary

• Quantum mechanical scattering theory has to find solutions of the S CHRÖDIN GER equation which asymptotically are a superposition (6.62) of an incoming plane wave and outgoing spherical wave. The amplitude f (θ, ϕ) of the latter is called scattering amplitude. The DCS is (kf /ki )|f (θ, ϕ)|2 – with (kf /ki ) = 1 for elastic scattering. • Solutions for the scattering problem can be obtained either from the L IPP MANN -S CHWINGER equation (6.65) or by a partial wave expansion into a series of spherical harmonics Ym (θ, ϕ). Angular momenta may become large. The asymptotic phase shifts η between the partial waves in the scattering potential and the free outgoing spherical wave determine the scattering amplitude (6.86). • Classical and quantum picture of a collision process are related by impact parameter √ b, wavenumber k and angular momentum quantum number through bk = ( + 1) ( + 1/2).

436

6


• For heavy particle scattering, semiclassical approximations of phase shifts are very useful, such as the WKB approximation (6.94). They also allow to establish a direct relation between classical trajectories and partial wave expansion for so called stationary phases (6.99). • Phase shifts are (for not too deep potentials) positive for attractive and negative for repulsive potentials. For deeply attractive potentials, the L EVINSON theorem (6.104) relates the number of potentially bound states to the scattering phase. At low energies, phases can be described well by effective range theories (6.105) and (6.106). The R AMSAUER effect, a minimum in the integral cross section at low energies, is understood in terms of the s phase passing through a multiple of π . In the limit of vanishing energy, the s phase dominates the cross section and f (θ ) → −as , with as being the so called scattering length. • With (6.109) we have introduced the (unitary) scattering matrix S and the transition matrix T with S = 1 + T. The scattering amplitude is given by (6.110). While for elastic scattering this concept just implies some trivial reformulation of the partial wave expansion, for inelastic scattering it will be an indispensable instrument of book-keeping for different contributions to scattering cross sections.

6.5

Resonances

6.5.1

Types and Phenomena

Resonances belong to the most fascinating phenomena of physics in practically each of its topical areas. Resonances show up as pronounced structures in certain observables when studied as a function of frequency or energy. They originate as interference phenomena due to the existence of quasi-bound states imbedded into a continuum, into which they may decay and with which they interfere. In Vol. 1 we have already discussed such resonances on several occasions, e.g. in Sect. 7.6, Vol. 1 in the context of autoionization of He. There, quasi-stable configurations above the ionization potential have been investigated. In molecular physics too such phenomena are known (beyond autoionization), e.g. as predissociation: above the dissociation limit of a molecular system, rotational or even vibrational states may exist, which are quasi-bound by the centrifugal barrier. Such states may also be observed in the vicinity of avoided crossings, as we have got to know them in Sects. 3.6 and 3.7. When studying electronic, atomic and molecular collisions, quasi-bound states are frequently encountered as well. For example if potential barriers are to be considered. The only difference here is that the states in question (or their wave function) are originally formed at large internuclear distances in the continuum of states of the interacting particles. The colliding pair of particles may then be caught for a short time into the quasi-bound state, it its relative kinetic energy is the same as the resonance energy of this state.

6.5

Resonances

Fig. 6.36 Characteristic potentials (very schematically) for the formation of (a) a shape resonance and (b) a F ESHBACH resonance at an energy Wres

437 Veff (R)

V(R) excited state centrifugal barrier Wres

Wres

R

(a)

shape resonance

R

(b)

FESHBACH resonance

This is illustrated schematically in Fig. 6.36 for two most common resonance types. Figure 6.36(a) shows the characteristic potential for the formation of a so called shape resonance: for angular momenta > 0 the centrifugal barrier may be high enough to allow for one or more quasi-bound states in the effective potential Veff (R) – above the asymptotic zero energy. Clearly, these states are not stable: their finite lifetime τ = /Γ is determined by the tunnelling probability Γ through the centrifugal barrier. Nevertheless, they may influence the scattered wave significantly if the relative kinetic energy of the system is nearly resonant with the energy Wres of the quasi bound state. The centrifugal barrier may be tunnelled (into either direction), so that the quasi-bound state may be occupied temporarily. When the kinetic energy passes such a resonance the scattering phase jumps by nearly π – as we have already discussed in Sect. 6.4.5 for a potential well. And we have already identified there in Fig. 6.32 such a phase jump, e.g. for k0 a = 4.4 in the d partial wave. A somewhat different situation is sketched in Fig. 6.36(b). This so called F ESH BACH resonance manifests itself as a quasi-bound state in the potential of an excited state which is embedded into the dissociation or ionization continuum of the initial state. The resonance would thus form a stable, bound state of the excited scattering system – if there were no coupling between this state and the collision continuum. In this situation too a phase jump is encountered by π within a small range of kinetic energies. It leads to pronounced interference structures in the DCS as well as in the ICS. A little warning at this point appears in order when discussing such concepts for electron scattering: The two ‘potential images’ shown in Fig. 6.36 have to be used with care, since the potential which an electron experiences when scattered (e.g. by an atom) involves also all electrons of the target. In such a situation – due to the similar velocities of projectile electron and the electrons in the target atom – no such thing as a B ORN -O PPENHEIMER approximation is strictly valid. At best, one may resort to the independent particle model which has proved very valuable when computing the wave functions of complex atoms (see Sect. 10.1, Vol. 1) – i.e. one might average the interaction potential over the wave functions of all atomic electrons. Such pseudopotentials are indeed used in the theory of electron scattering at intermediate energies and often turn out to be quite successful. In the general case, however, especially for excitation processes, one has to treat the problem with more rigorous methods which we shall outline in Sect. 7.3.

438

6.5.2

6


Formalism

Both resonance types just discussed are characterized by an end state (i.e. a scattered wave) of well defined energy, which may be reached by a direct process as well as by a temporary capture into a quasi-bound, intermediate states. We shall now describe these two processes by two scattering amplitudes fdir and fres . We have discussed a very similar situation already in Sect. 7.6, Vol. 1 for autoionization. There, direct ionization and double excitation were interfering. The following considerations are adapted to the scattering problem. Here too, both channels ABres A+B

fres ,τ

fdir

A+B

can – in principle – not be distinguished. Thus, the amplitudes have to be added coherently in order to obtain the elastic DCS: dσ = I (θ ) = |fdir + fres |2 . dΩ

(6.123)

This leads to typical interference structures as a function of the phase difference between the two amplitudes – just as in YOUNG’s double slit experiment. Here it is the resonance scattering amplitude which changes rapidly over a small range of energies, while the direct amplitude remains essentially constant. Since the resonance has a finite lifetime τ , one may describe this quasi-bound state formally by attributres which accounts for the decay. We have already ing to it a complex energy W employed that method earlier, e.g. in Sect. 5.1.1, Vol. 1 to describe the spontaneous decay after optical excitation in a semiclassical scheme. We thus set for the resonance energy res = Wres − iΓ /2, W where Wres would be the energy of a bound state without decay, and Γ = /τ gives its width due to the decay. With the concepts developed in Sect. 6.4.6 we may now characterize resonance scattering as follows: a resonance occurs in a particular partial wave, say for = ζ . It has purely outgoing character, hence, the incoming partial res . This implies that the correspondwave must disappear for the complex energy W res ) → 0 at resing partial wave coefficient according to (6.121) must become aζ∗ (W onance. For scattering energies T Wres we may expand aζ∗ around the resonance energy: ∗ iΓ daζ ∗ aζ = T − Wres + . (6.124) 2 dT Wres

6.5

Resonances

439

With this the scattering matrix (6.122) for the partial wave ζ becomes Sζ = −

aζ iζ π T − Wres − e = aζ∗ T − Wres +

iΓ 2 iΓ 2

eiζ π

daζ /dT |Wres . daζ∗ /dT |Wres

Of course |Sζ | ≡ 1 holds (unitary matrix). Also the magnitude of the last fraction in this expression is ≡ 1, so that we may abbreviate eiζ π

0 daζ /dT |Wres = e2iηζ . ∗ (daζ /dT |Wres )

The scattering matrix may now be rewritten as: 0 res 2i 2iηζ0 1− = e2iηζ +2iηζ . Sζ = e "+i

(6.125)

Here " is the relative collision energy, related to the width of the resonance, "=

T − Wres , Γ /2

and ηζres = arctan

−1 "

(6.126)

is the resonance phase.18 In contrast, within the present approximation ηζ0 represents a constant background phase shift. Hence, we see that the resonant part of the scattering phase changes rapidly by π when the energy passes the resonance energy and assumes an odd multiple of π/2 for Wres − T . For the special case that all scattering phases except the resonance phase vanish, the integral cross section (6.90) becomes σ=

4π (2ζ + 1) sin2 ηζres , k2

which leads with (6.126) to a L ORENTZ profile of the resonance19 σ=

4π (Γ /2)2 (2ζ + 1) , k2 (Wres − W )2 + (Γ /2)2

(6.127)

which we know from optical excitation of atomic levels. In electronic and atomic scattering, however, usually many partial waves contribute. They generate a nonresonant, slowly varying background, while the rapid changes due to a resonance usually occur only in one partial wave. We have seen this already in Fig. 6.32 (left, middle) for the d wave scattering phase in a potential well: there a consequence of a corresponding shape resonance. easily verifies this from (6.125) using the trigonometric relation tan 2η = 2 tan η/(1 − tan2 η).

18 One

19 Often

this is called B REIT-W IGNER distribution since B REIT and W IGNER (1936) have applied this formula for the first time onto resonance scattering of neutrons.

440

6


We are now able to derive expression (6.123), heuristically introduced, from the partial wave expansion (6.86), simply by inserting there (6.125). For a resonance in the partial wave = ζ we obtain: + * ∞ 1 2iηζ0 +2iηζres 2iη0 (2 + 1)e P (cos θ ) + (2ζ + 1)e Pζ (cos θ ) f (θ ) = 2ik =0 =ζ

*∞ 1 0 = (2 + 1)e2iη P (cos θ ) 2ik =0 + (2ζ + 1)e

2iηζ0 2iηζres

e

+ − 1 Pζ (cos θ ) .

(6.128)

Hence, we are indeed able to separate two interfering amplitudes f (θ ) = fdir (θ ) + fres (θ, "),

(6.129)

of which the direct scattering amplitude fdir (θ ) is obtained from the usual partial wave expansion with phase shifts η0 , slowly varying with energy. The resonant amplitude in partial wave = ζ is given by fres (θ, ") =

0 res 1 (2ζ + 1)e2iηζ e2iηζ − 1 Pζ (cos θ ). 2ik

(6.130)

It depends in the resonance region with ηζres (") according to (6.126) strongly on the kinetic energy. Alternatively, the decisive energy depending factor 2iηres e ζ − 1 ∝ fres (")

may also be written

res 2 2i 2"i − 2 . =√ eiηζ = 2 2 " + 1 " +1 " +1

(6.131) (6.132)

This allows us to directly compare the corresponding formulas for autoionization (7.74) developed in Vol. 1. They just differ by an arbitrary phase and normalization factor 1/2i. We see again that the purely resonant cross section ∝ |fres |2 shows the typical B REIT-W IGNER energy dependence (6.127), while the direct scattering amplitude remains nearly constant over the resonance. Both fdir (θ ) as well as fres (θ, ") depend on the scattering angle. However, according to (6.130), the angular dependence of the resonant contribution determined by Pζ (cos θ ) depends only on one partial wave = ζ . Depending on the relative phase and magnitude of direct and resonant amplitude, fdir and fres , respectively, the interference may be constructive or destructive or assume various dispersion type forms: just as in the case of autoionization as we have discussed it in Vol. 1 and illustrated in Fig. 7.10, Vol. 1. Of course, one may

6.5

Resonances

441

use here too a parametrization according to the corresponding FANO lineshape σ (T ) = σres

(" + q)2 + σdir 1 + "2

(6.133)

with the asymmetry parameter q as we have done it for autoionization. The partial wave analysis which we have presented here is, however, much more powerful for analyzing the differential scattering cross section, since (6.128) allows in principle a consistent description for all scattering angles with very few free parameters.

6.5.3

An Example: Electron Helium Scattering

One finds scattering resonances in heavy particle collisions (at very low energies, see Fig. 6.25), as well as and rather frequently in electron scattering by atoms and molecules. They occur in a wide energy range and have been studied over the past decades very thoroughly and comprehensively. A good survey on the state-of-theart of theory and experiment for e− atom scattering gives B UCKMAN and C LARK (1994), and the review of H OTOP et al. (2003) introduces into a broad variety of phenomena observed in the low energy scattering of electrons by molecules and clusters. A ‘bench mark’ case is the He− (1s2s 2 2 S1/2 ) F ESHBACH resonance – a resonance in s wave scattering. Historically it gave the first experimental evidence of resonance structures in electron atom scattering of the type discussed here, and has been discovered by S CHULZ (1963) in the total e− − He cross section. The first angularly resolved measurements of this and similar resonances have been reported by A NDRICK and E HRHARDT (1966). The energy resolution was at that time about 50 meV,20 which also illuminates the limits of spectroscopic investigations in the scattering continuum: the price to be paid for the high depth of information by energy and angular dependence, when studying collision dynamics, is a significantly reduced energy resolution as compared to optical spectroscopy. Since these first experiments many groups have worked hard on an improvement of this situation. The so to say ‘ultimate’ experiment, developed by H OTOP and collaborators (see e.g. G OPALAN et al. 2003), has achieved an energy resolution down to 4 meV FWHM (B OMMELS et al. 2005). It has been employed for a number detailed, very informative investigations of resonances and the threshold behaviour in the elastic and inelastic electron atom and electron molecule scattering. Figure 6.37 illustrates schematically the most important aspects of this sophisticated experimental setup. It is fair to say that here the preceding 40 years of methodical and experimental development of scattering physics culminate. 20 In a traditional experiment one generates the electron beams by thermal emission from a cathode. From its rather broad energy distribution one selects a more or less broad fraction by an electrostatic monochromator, limits the emission angle and accelerates it to the desired energy T . After the collision, one usually has to analyze the scattered electrons correspondingly.

442

6

Basics of Atomic Collision Physics: Elastic Processes towards detector for metastables

z

(a)

potassium beam

target beam x

electron extraction

SC electron beam

SV PIEQ

electron lenses and deflection

FARADAY cup

155 mm Kontinuum continuum

(c)

(b) electron beam e-

beam focussing

SC

FARADAY cup

T0

423< λ2< 478 nm

39K(4p)

42P3/2 1730.4 GHz

≈ F MHz 3 21.0 2 1 0 9.3 3.2 2

42P1/2

λ1= 766.7 nm

y x

39K(4s)

target beam

42S1/2

1

57.7

≈ ≈ 2 461.7

retarding field detector(s) 1

Fig. 6.37 ‘Ultimate’ electron scattering experiment with ultra high energy resolution according to G OPALAN et al. (2003). Experimental setup: cuts along the (a) zx, and (b) yx plane; the most important components are the photoionization electron source (PIES) with the source volume (SV) and the scattering chamber (SC). (c) Resonant two-photon ionization scheme for the generation of the scattering electrons with an initial kinetic energy of T0 < 1 meV

The heart of the apparatus is the photoionization electron source (in Fig. 6.37(a) marked as PIEQ), in which a potassium beam (−z-direction) is ionized by two narrow band laser beams (z-direction). By this device electrons with a well defined initial energy are created in a very small start volume (SV). Crucial for this concept is that the laser defined initial kinetic energy T0 of the photoelectrons is not broadened significantly by the space charge of the K+ ions from which they are emerging. As detailed simulations show, one has to keep the ionization energy just above the ionization potential and restrict the extracted current to some 10 pA to 100 pA. Hence, the intricate, resonant two-photon ionization scheme via the K(4 2 P3/2 ) intermediate state, which is excited by a CW T I :S APPH laser (λ1 = 766.7 nm) (about optical pumping see also Appendix D). The two hyperfine levels F = 2 and 3 are excited si-

6.5

Resonances

443

multaneously, as sketched in Fig. 6.37(c), the laser being stabilized correspondingly and modulated electro-optically. To obtain enough intensity for ionization (the cross sections are rather low) the intermediate state is ionized intra cavity by a narrow band dye laser (Stilben 3). The wave length (λ2 = 455 nm) is chosen such that the initial kinetic energy of the photoelectrons is T0 < 1 meV. The electrons are then extracted by a very weak extraction field (E = 10 V /m), accelerated with a specially designed electron lens system. Finally the electrons are deflected by a special electron optics into the scattering chamber (SC in Fig. 6.37(a) right). Electrons hit the target atoms in a well collimated supersonic beam (zdirection) with an angular divergence below 1◦ . Electron detection occurs, as indicated in Fig. 6.37(b), by up to five electron detectors with channeltrons (see Appendix B.1) positioned in the scattering plane, each of them equipped with an electric retarding field to distinguish elastic from inelastic processes. Metastable He(23 S1 ) atoms which may possibly be excited by electron collision are deflected at threshold by 11.5◦ due to momentum transfer. They may be recorded by an additional channeltron and used e.g. for energy calibration. With the standards set by this experiment, the He atom can no longer be assumed as being at rest. Rather, the kinetic energy T of the electron has to be referred to the centre of mass system (see Sect. 6.2.2). Pronounced attention is devoted in this experiment to the calibration of the voltage supplies and to screening of stray electric and magnetic fields. Figure 6.38 shows some experimental and theoretical results for the above mentioned He− resonance according to G OPALAN et al. (2003) at three different scattering angles. The experimental points and their partial wave analysis (essentially performed with the procedure explained in the preceding subsection) document the exceptional quality of the experimental data. The comparison with the so called RMPS theory of BARTSCHAT (1998), an ambitious modified R-matrix theory (see Sect. 7.3) with pseudo-states witnesses the theoretical standards reached today for a precise ab initio description of such processes. Energetic position and linewidth of the He− (1s2s 2 2 S1/2 ) resonance have been determined in this experiment with highest accuracy to be Wres = 19.365(1) eV and Γ = 11.2(5) meV, respectively. We emphasize that the energy dependence of this resonance, shown in Fig. 6.38 – which at first sight looks rather complex – is fully and exactly described by (6.128)– (6.132) for all scattering angles from the known, smoothly varying direct scattering amplitude and just two additional parameters – Wres and Γ ! And we recall that the non-resonant, direct amplitude for e− + He scattering is excellently parameterized with only three scattering phase shifts for the s, p and d waves, as documented in Fig. 6.34. Section summary

• Scattering resonances are interesting and often encountered phenomena. They occur due to quasi-bound states in the collision continuum which open two indistinguishable channels. These interfere quantum mechanically and lead to characteristic structures in the cross sections.

Fig. 6.38 He− (1s2s 2 2 S1/2 ) F ESHBACH resonance in the elastic differential scattering cross section of electrons by atomic He according to G OPALAN et al. (2003). The experimental data points are marked red, the red lines present a consistent partial wave fit with an energy resolution of 7.4(5) meV FWHM and a resonance width of Γ = 11.2(5) meV. The black, dashed lines (hardly distinguishable from the red lines) represent the results of the RMPS theory, convoluted with the energy resolution of 7.4 meV; the vertical, dash-dotted line marks the resonance energy Wres = 19.365 eV

Basics of Atomic Collision Physics: Elastic Processes 4 2.0 2 22°

1.0 0.0

0 4

1.5 1.0

2 45°

0.5 0.0 0.8

0

I(θ) / a02 sr -1

6

count rate / 1000 per 30 s

444

0.8 0.4

90°

0

0.4 0.0

19.30 19.40 electron energy T / eV

• We specifically have identified shape resonances which involve quasi-bound states behind the centrifugal barrier in a partial wave > 0, and F ESHBACH resonances which correspond to bound excited states embedded into the collision continuum. • A formal treatment allows to identify the two interfering amplitudes in a partial wave expansion, and to relate it with the famous FANO profile for resonances, e.g. observed in the spectroscopy of autoionization. • We explicitly present the He− (1s2s 2 2 S1/2 ) F ESHBACH resonance, as investigated in a high resolution, state-of-the-art benchmark experiment, and its theoretical interpretation in the framework of the formalism discussed above.

6.6

B ORN Approximation

At higher initial kinetic energies partial wave solutions of the S CHRÖDINGER equation become increasingly expensive. Approximative methods to solve the L IPPMANN -S CHWINGER equation (6.65) are a possible alternative. We remember that we have used B ORN approximation already in the context of photoionization (Sect. 5.5.2 in Vol. 1) and shall resume this discussion in Chap. 8. B ORN (1926a,b) introduced this approximation in the early days of quantum mechanics as a first successful attempt to tackle the continuum problem – and it is still used today. It yields useful results for high kinetic energies and small momentum transfer, which in the classical picture is equivalent to large impact parameters (small scattering angles) where the interaction between projectile and target is week

6.6

B ORN Approximation

445

on average. The main advantage of the B ORN approximation is its simplicity, in particular at high energies where high values of limit the applicability of a rigorous ¯ 2 )V (R) so that the partial wave expansion. At high energies we have k 2 (2M/ right side of (6.63) may be treated as a small perturbation, so that first order B ORN approximation (FBA) is derived from ∇ 2 ψ (1) + k 2 ψ (1) =

2M¯ V (R)ψ (0) . 2

(6.134)

The plane wave ψ (k) (R) = exp(ik · R) = ψ (0) is taken as 0th order solution. Second order B ORN approximation (SBA) follows from ∇ 2 ψ (2) + k 2 ψ (2) =

2M¯ V (R)ψ (1) 2

(6.135)

and so on. SBA and higher terms of this B ORN series are actually used successfully in modern research for a number of problems, e.g. for rather accurate solutions of electron impact ionization. Presently, we treat first order B ORN approximation (FBA) applied to elastic scattering.

6.6.1

Scattering Amplitude and Cross Section in FBA

Starting from the general expression for the scattering amplitude (6.68), all we have to do is to substitute the incoming plane wave ψ (ki ) in place of the exact solution ψ (+) to obtain the elastic scattering amplitude in FBA: f (FBA) (θ, ϕ) = − =−

M¯ ψ (k f )∗ R V R ψ (ki ) R d3 R 2 2π

(6.136)

M¯ k f |V R |k i . 2 2π

(6.137)

And the T-matrix (6.69) in FBA becomes (FBA) Tf i = k f |V R |k i = e−ik f R V R eiki R d3 R

(6.138)

2π2 (FBA) eiKR V R d3 R = − (θ, ϕ), (6.139) f M¯ f i √ ¯ / in the elastic case, and the momentum transwith |k i | = |k f | = k = 2MT fer K = k i − k f . Obviously the B ORN scattering amplitude is proportional to the F OURIER transform of the potential (see Appendix I.5 in Vol. 1). The integral (6.139) is rather easy to handle. We have already encountered it in similar form in the context of photoionization in Sect. 5.5, Vol. 1. For H atoms it may be evaluated explicitly as documented by (5.75), Vol. 1. =

446

6


For spherically symmetric potentials (6.138) may further be simplified: = Tf(FBA) i

2π 0

π 0

∞

dϕ sin θ dθ eiK·R V R R 2 dR

(6.140)

0

∞

=

RV (R) sin(KR)dR

(6.141)

0

θ with K = |K| = 2k sin . 2

(6.142)

In the second line we have chosen the z-axis parallel to K and integrated over sin θ dθ = −d(cos θ ). Differential and integral cross sections are again given by (6.73) and (6.74), respectively. Some general properties of the B ORN approximation are emphasized: • With (6.138) the T operator in FBA is identical to the potential V (R ). • According to (6.139) the FBA has only one symmetry axis, parallel to the momentum transfer K = k i − k f . • The scattering amplitude (6.136) or (6.141) is a real function of the momentum transfer K and depends only through K on scattering angle θ and collision en¯ ergy T = 2 k 2 /2M.

6.6.2

R UTHERFORD Scattering

A special case is RUTHERFORD scattering, i.e. scattering by the C OULOMB potential VC (R) = qA qB e2 /(4π"0 R) of two charges qA e and qB e. The explicit evaluation of (6.141) is done most conveniently21 for the more general case of a Y UKAWA potential V (R) = VC (R) exp(−R/R0 ). For clearness we introduce again a.u. (a0 , Eh and me ). Then (6.141) simply becomes f (FBA) (θ ) = =

2M¯ qA qB me Ka0

∞

exp(−R/R0 ) sin(KR)dR 0

2M¯ qA qB a0 . me (a0 /R0 )2 + (Ka0 )2

(6.143)

Thus, for a Y UKAWA potential the DCS is 2 2 2 ¯ 2 4(M/m dσ e ) (qA qB ) a0 = f (FBA) = . 2 dΩ [(a0 /R0 ) + (Ka0 )2 ]2

(6.144)

notice by trial and error that direct insertion of a potential ∝ 1/R leads to problems which are related to the long range of the C OULOMB potential.

21 The attentive reader may

6.6

B ORN Approximation

447

For a pure C OULOMB potential, i.e. for RUTHERFORD scattering, R0 → ∞ and the DCS becomes ¯ 2 ¯ a02 M 4(qA qB )2 2 M qA qB 2 dσ = a = . (6.145) 0 dΩ me me T /Eh 16 sin4 (θ/2) (Ka0 )4 Here we have inserted (6.142) and used the identity (ka0 )2 = 2T /Eh . Interestingly, in this particular case FBA leads to the same result as the classical trajectory approximation (6.45). We also mention here – without proof – that the exact quantum mechanical treatment again leads to the same result. This is of course a lucky mathematical incident – owing to the special properties of the C OULOMB potential. Often the DCS is given with respect to the quantity WK = (K)2 /2me = Eh (Ka0 )2 /2, somewhat loosely called energy transfer.22 With the momentum transfer K according to (6.142), dΩ = 2π sin θ dθ = πdK 2 /k 2 , and the kinetic energy T = Eh (ka0 )2 /2 we have to replace dΩ = πdWK /T , and obtain dσ = dWK

6.6.3

¯ 2 ¯ 2 4π(qA qB )2 2 π(qA qB )2 2 M M a = a . me me T (WK /Eh )2 0 T (Ka0 )4 0

(6.146)

B ORN Approximation for Phase Shifts

The scattering amplitude in B ORN approximation (6.137) may, of course also be expanded into spherical harmonics and then be compared to the partial wave expansion (6.86). Without going through the algebra in detail we just point out that one has to expand the two plane waves in (6.137) according to (J.13)–(J.18), Vol. 1 so that the result for the B ORN phase shift appears plausible: tan ηB

2M¯ −k 2

∞

2 V (R) j (kR) R 2 dR.

(6.147)

0

For large impact parameters kb = ( + 1/2) ka and high kinetic energies (ηB 1) one may show that this expression approaches the J EFFREYS -B ORN phase (6.96).23 For not too deep potentials (ηB < π ) the sign of the B ORN scattering phase shift (6.147) is obviously positive or negative for attractive and repulsive potentials, respectively – just as in the low energy case discussed in Sect. 6.4.5. Using B ORN a matter of fact, the energy transfer from projectile A onto a target B at rest is (p)2 /2MB = ¯ B = 2T (M/M B )(1 − cos θ) (see Fig. 6.11 and the text associated with it) – which for electron impact on heavy particles is very small.

22 As

(K)2 /2M 23 The

readers may verify this by applying the following approximation for large : 2 −1/2 1 . k 2 R 2 j (kR) → 1 − ( + 1/2)2 /k 2 R 2 2

448

6


(6.147) or J EFFREYS -B ORN (6.96) phase shifts – instead of the more difficult to compute WKB phase shift (6.94) – may be helpful even when the B ORN approximation as such cannot be applied: B ORN phases are always a good approximation if the interaction remains weak on average, i.e. for large angular momenta (impact parameters). For many problems one thus obtains very reasonable results by solving the exact radial equation (6.80) for a few, low values of and using (6.147) for all higher angular momenta . For inverse power potentials of the type V (R) = ∓CR −ν , that is essentially for all potentials and large (or impact parameters) at sufficiently high energies, one may integrate (6.147) in closed form and finds: ηB ∝ ±k ν−2 1−ν

for 1.

(6.148)

This expression is something like a high angular momentum counterpart to the effective range formula (6.106), which is applicable only to low kinetic energies. Section summary

• B ORN approximation is the most simple quantum mechanical approach towards the scattering problem. Historically important, it is still used today for its straight forward derivation of scattering amplitudes, as introduced here for elastic scattering. • B ORN approximation uses a perturbation approach, identifying the transition operator with the interaction potential. From its matrix elements between incoming and outgoing plane wave the scattering amplitude is obtained in a straight forward manner. Typically, FBA results are rather accurate at high energies and small scattering angles, while the low energy cross sections are usually overestimated. • For C OULOMB (RUTHERFORD) scattering amplitude derived in FBA is identical to the classical and quantum result. • The partial wave expansion of FBA allows one to derive B ORN (6.147) and J EFFREYS -B ORN (6.96) phase shifts. These may be used with advantage for large in combination with exact quantum calculations for a few low angular momentum phase shifts.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2 in Vol. 1. CM: ‘Centre of mass’, coordinate system (or frame) (see Sect. 6.2.2). CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. DCS: ‘Differential cross section’, see Sect. 6.2.1. FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects. 6.6 and 5.5.2, Vol. 1, respectively).

References

449

FWHM: ‘Full width at half maximum’. ICS: ‘Integral cross section’, see Chaps. 6 to 8. JWKB: ‘J EFFREYS -W ENTZEL -K RAMERS -B RILLOUIN’, semiclassical method to determine scattering phases. MOT: ‘Magneto optical trap’, for a typical setup see e.g. Fig. 6.26. RMPS: ‘R-matrix with pseudo-states method’, advanced quantum mechanical theory for electron scattering. SBA: ‘Second order B ORN approximation’, second order term in the B ORN series (see Sect. 6.6). SEM: ‘Secondary electron multiplier’, see Appendix B.1. Ti:Sapph: ‘Titanium-sapphire laser’, the ‘workhorse’ of ultra fast laser science. WKB: ‘W ENTZEL, K RAMERS, and B RILLOUIN’, semiclassical method to determine the evolution of the quantum mechanical phase of a wave function as a function of time; basically an approximative method to solve the S CHRÖDINGER equation, specifically for the motion of heavy particles.

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B UCK , U., K. A. KOHLER and H. PAULY: 1971. ‘Measurements of glory scattering of Na-Hg’. Z. Phys., 244, 180. B UCK , U. and H. M EYER: 1984. ‘Scattering analysis of cluster beams – formation and fragmentation of small Arn clusters’. Phys. Rev. Lett., 52, 109–112. B UCKMAN , S. J. and C. W. C LARK: 1994. ‘Atomic negative-ion resonances’. Rev. Mod. Phys., 66, 539–655. B URKE , P.: 2006. ‘Electron-atom, electron-ion and electron-molecule collisions’. In: G. W. F. D RAKE, ed., ‘Handbook of Atomic, Molecular and Optical Physics’, 705–729. Heidelberg, New York: Springer. C OHEN , J. S.: 1978. ‘Rapid accurate calculation of JWKB phase-shifts’. J. Chem. Phys., 68, 1841– 1843. FARNIK , M., C. S TEINBACH, M. W EIMANN, U. B UCK, N. B ORHO and M. A. S UHM: 2004. ‘Size-selective vibrational spectroscopy of methyl glycolate clusters: comparison with ragoutjet FTIR spectroscopy’. Phys. Chem. Chem. Phys., 6, 4614–4620. F ELTGEN , R., H. K IRST, K. A. KOHLER, H. PAULY and F. T ORELLO: 1982. ‘Unique determination of the He2 ground-state potential from experiment by use of a reliable potential model’. J. Chem. Phys., 76, 2360–2378. F LUENDY , M. A. D., R. M. M ARTIN, E. E. M USCHLITZ J R . and D. R. H ERSCHBACH: 1967. ‘Hydrogen atom scattering – velocity dependence of total cross section for scattering from rare gases hydrogen and hydrocarbons’. J. Chem. Phys., 46, 2172–2181. G EIGER , J. and D. M ORÓN -L EÓN: 1979. ‘Electron-atom shadow scattering’. Phys. Rev. Lett., 42, 1336–1339. G ENGENBACH , R., C. H AHN and J. P. T OENNIES: 1973. ‘Determination of H-He potential from molecular-beam experiments’. Phys. Rev. A, 7, 98–103. G OPALAN , A., J. B OMMELS, S. G OTTE, A. L ANDWEHR, K. F RANZ, M. W. RUF, H. H OTOP and K. BARTSCHAT: 2003. ‘A novel electron scattering apparatus combining a laser photoelectron source and a triply differentially pumped supersonic beam target: characterization and results for the He− (1s 2s2 ) resonance’. Eur. Phys. J. D, 22, 17–29. G ULLEY , R. J., D. T. A LLE, M. J. B RENNAN, M. J. B RUNGER and S. J. B UCKMAN: 1994. ‘Differential and total electron-scattering from neon at low incident energies’. J. Phys. B, At. Mol. Phys., 27, 2593–2611. H OFSTÄDTER , R.: 1961. ‘The N OBEL prize in physics: for his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1961/. H OTOP , H., M. W. RUF, M. A LLAN and I. FABRIKANT: 2003. ‘Resonance and threshold phenomena in low-energy electron collisions with molecules and clusters’. In: ‘Advances in Atomic Molecular, and Optical Physics’, vol. 49, 85–216. Amsterdam: Elsevier, Academic Press. L EVINE , R. D.: 2005. Molecular Reaction Dynamics. Cambridge: Cambridge University Press, 554 pages. OM ALLEY , T. F., L. S PRUCH and L. ROSENBERG: 1961. ‘Modification of effective range theory in presence of a long-range (r−4 ) potential’. J. Math. Phys., 2, 491–498. OVCHINNIKOV , S. Y., G. N. O GURTSOV, J. H. M ACEK and Y. S. G ORDEEV: 2004. ‘Dynamics of ionization in atomic collisions’. Phys. Rep., 389, 119–159. PAULY , H. and J. P. T OENNIES: 1965. ‘The study of intermolecular potentials with molecular beams at thermal energies’. In: ‘Adv. Atom. Mol. Phys.’, vol. 1, 195–344. New York: Academic Press. VAN DER P OEL , M., C. V. N IELSEN , M. RYBALTOVER , S. E. N IELSEN , M. M ACHHOLM and N. A NDERSEN: 2002. ‘Atomic scattering in the diffraction limit: electron transfer in keV Li+ Na(3s, 3p) collisions’. J. Phys. B, At. Mol. Phys., 35, 4491–4505. P OTERYA , V., M. FARNIK, U. B UCK, D. B ONHOMMEAU and N. H ALBERSTADT: 2009. ‘Fragmentation of size-selected Xe clusters: Why does the monomer ion channel dominate the Xen and Krn ionization?’ Int. J. Mass Spectrom., 280, 78–84. R AKITYANSKY , S. A. and N. E LANDER: 2009. ‘Generalized effective-range expansion’. J. Phys. A, Math. Gen., 42, 225302.

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S CHULZ , G. J.: 1963. ‘Resonance in elastic scattering of electrons in helium’. Phys. Rev. Lett., 10, 104–105. S MITH , F. T., R. P. M ARCHI and K. G. D EDRICK: 1966. ‘Impact expansions in classical and semiclassical scattering’. Phys. Rev., 150, 79–92. S TEINBACH , C., M. FARNIK, U. B UCK, C. A. B RINDLE and K. C. JANDA: 2006. ‘Electron impact fragmentation of size-selected krypton clusters’. J. Phys. Chem. A, 110, 9108–9115. S ZMYTKOWSKI , C., K. M ACIAG and G. K ARWASZ: 1996. ‘Absolute electron-scattering total cross section measurements for noble gas atoms and diatomic molecules’. Phys. Scr., 54, 271– 280. T OENNIES , J. P.: 2007. ‘Molecular low energy collisions: past, present and future’. Phys. Scr., 76, C15–C20. T OENNIES , J. P., W. W ELZ and G. W OLF: 1976. ‘Determination of H-He potential well depth from low-energy elastic-scattering’. Chem. Phys. Lett., 44, 5–7. U LLRICH , J., R. M OSHAMMER, A. D ORN, R. D ÖRNER, L. P. H. S CHMIDT and H. S CHMIDTB ÖCKING: 2003. ‘Recoil-ion and electron momentum spectroscopy: reaction-microscopes’. Rep. Prog. Phys., 66, 1463–1545.

7

Inelastic Collisions – A First Overview

In the previous chapter we have introduced potential scattering. Even though the concepts discussed there describe elastic heavy particle scattering very well (and in some cases even elastic electron scattering), we had to exclude so far completely the very important field of atomic and molecular excitation by collisions, as well as reactions: quite generally, atomic collisions are many body problems, and whenever changes of the internal states of the collision partners are possible, one has to account for these degrees of freedom.

Overview

We introduce some characteristic questions about inelastic and reactive collisions and approaches to answer them for several important examples. We start in Sect. 7.1 with very simple models. The general trends for excitation processes as a function of the relative kinetic energy are presented in Sect. 7.2. Specifically, in Sect. 7.2.7 we focus on the threshold region. In Sect. 7.3 we introduce multichannel theory, and discuss the alternative adiabatic and diabatic viewpoints. In Sect. 7.4 we extend the semiclassical methods already employed in the elastic case. In Sect. 7.5 we make a short excursions into the world of collision processes with highly charged ions. Finally, we address reactive scattering processes in Sect. 7.6.

7.1

Simple Models

We begin with some remarkably simple, but rather instructive models, which will allow us first guesses on the energy dependence of inelastic and reactive processes.

7.1.1

Reactions Without Threshold Energy

Starting from the effective potential Veff (R) according to (6.39) one may already make some predictions on the order of magnitude of cross sections. Let us first have a look at exothermic reactions without threshold energy and let us assume the © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_7

453

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7

Fig. 7.1 Schematic illustration of an exothermic reaction: the role of the centrifugal barrier during interaction of an ion with an atom or molecule. The pure interaction potential V (R) is drawn as full line, the effective potential Veff as dashed line. The inset on the bottom right sketches typical trajectories as a function of impact parameter b


potential V(R), kinetic energy T T = Veff (Rm)

Veff (R)

Rm

V(R)

b bm Rm

interaction potential V (R) is purely attractive. Typical examples are ion-atom and ion-molecule reactions according to the scheme Aq+ + B → ABq+ → C + Dq+ .

(7.1)

In the most simple case this may represent a charge exchange process. Since we consider an exothermic reaction it will always happen if the approaching particles with their initial kinetic energy T can overcome the centrifugal barrier. This is illustrated in Fig. 7.1 where the interaction potential V (R) and the effective potential Veff (R) according to (6.39) are shown. The small inset (bottom right) shows characteristic trajectories as a function of the impact parameter b. A trajectory for a given kinetic energy T always leads to a reaction if the impact parameter b is small enough to reach the critical radius Rm , i.e. if at the maximum of the potential barrier (max)

Veff

= V (Rm ) + T

b2 ≤T 2 Rm

(7.2)

holds. Let us assume, reaction (7.1) to be determined by a polarization potential V (R) = −αq 2 /(2R 4 ) (given in atomic units), the maximum (7.2) is found 2 = αq 2 /T b2 . The equal sign in (7.2) gives the maximum impact parameter at Rm bm = (2αq 2 /T )1/4 which still leads to a reaction – only for b ≤ bm the reaction may occur. From this consideration the so called L ANGEVIN cross section for ionmolecule reactions is derived: 2 (7.3) = πq 2α/T . σL = πbm For other interaction potentials the expression has to be modified correspondingly. For example, for molecules with high permanent dipole moment (H2 O, CO etc.) the potential is ∝ −1/R 2 . For these one derives in the same manner an energy dependence σ ∝ 1/T .

7.1

Simple Models

455

Fig. 7.2 Energy relations for an endothermic reaction with threshold energy Wth . The model of an absorbing sphere with radius Rth is shown. Full lines give the pure potentials, the dashed lines represent the effective potentials – black are those in the entrance channel, red those in the reactive channel to be reached

Veff (R ) T = W th + T b 2/R th2

W th R Rth

reaction cross section

Fig. 7.3 Typical energy dependence of reactive cross sections for exothermic and endothermic reactions according to the L ANGEVIN model and according to the absorbing sphere model, respectively

LANGEVIN cross section

absorbing sphere model

W th

7.1.2

kinetic energy T

The Absorbing Sphere Model

For endothermic reactions, which occur only above an energetic threshold Wth , the initial kinetic energy T changes to T − Wth during the process, and we have to modify our considerations with respect to the critical distance Rth and the maximum impact parameter bm . Figure 7.2 illustrates the relations for such an inelastic reaction process in the absorbing sphere model. The reaction may only occur if the trajectories reach the crossing of the potential curve of the ground state (black) with the excited state (red), that is to say if the energy at the crossing radius Rth is sufficient to overcome the threshold energy Wth . In this case we have to replace (7.2) by Veff (Rth ) = Wth + T

b2 ≤ T, 2 Rth

(7.4)

from which with Veff (Rth ) = T again the maximum impact parameter for the reac2 = R 2 (1 − W /T ) and the reaction cross tion is obtained. Here it is given by bm th th section becomes 2 2 = πRth (1 − Wth /T ) σr = πbm

for T > Wth .

The energy dependence predicted by these models is sketched in Fig. 7.3.

(7.5)

7 reaction cross sectin σ / 10-16 cm 2

456


HD+(v)+Ne →NeH++D HD+(v)+Ne →NeD++H

5 4

v=4

3 2

absorbing sphere

v=1 LANGEVIN

1 0

0 Wth

1

2 3 4 kinetic energy T / eV

5

Fig. 7.4 Reaction cross sections for the formation of NeH+ and NeD+ ions (full and open experimental points, respectively) formed in the reaction HD+ + Ne as a function of initial kinetic energy T (in the CM system) according to D RESSLER et al. (2006). Both reactions are exothermic if the initial vibrational quantum number is v = 4, and both are endothermic if v = 1; they are compared with the respective simplified models according to Fig. 7.3

7.1.3

An Example: Charge Exchange

Reality, however, is usually somewhat more complicated. We illustrate this for the example of charge exchange in a still fairly simple ion molecule reaction HD+ (v) + Ne → NeH+ + D + HD → NeD+ + H + HH ,

(7.6) (7.7)

for which experiments with vibrationally selected HD+ molecular ions have been performed by D RESSLER et al. (2006). Figure 7.4 shows the cross sections for two initial vibrational states of HD+ . In the case of v = 1 the reactions (7.6) and (7.7) are endothermic, with threshold energies of Wth = 0.29 and 0.25 eV, respectively, in the case of v = 4 both are exothermic. One recognizes in Fig. 7.4 the typical behaviour, essentially predicted by the models discussed above: exothermic reactions show reaction cross sections which monotonously decrease with energy, while the endothermic cross sections first rise rapidly beyond the threshold, reach a maximum and fall again slowly with further increasing energy (that the signals at very low energies are nearly constant for v = 4 is an experimental artifact due to the finite width of the energy distribution – the same holds for the small residual signal below threshold Wth in the v = 1 case). The dashed lines in Fig. 7.4 indicate our attempts to fit the experimental data by the models discussed above – by a L ANGEVIN cross section for the exothermic case v = 4 and by the absorbing sphere model for the endothermic case v = 1. As documented, this is only partially successful. While the decay of the cross section for v = 4 may be predicted, and the threshold behaviour for v = 1 is verified, the strong decay with increasing energy is not reproduced by these simple models. The

7.1

Simple Models

457

most obvious explanation is the purely geometric nature of these models. They imply that the reaction probability is 100 % if only the classical trajectory reaches the critical radius Rm and Rth , respectively. In reality this is clearly not the case: we expect a reduction of the reaction probability with decreasing interaction time, i.e. with increasing kinetic energy – as observed in the experiment.

7.1.4

M ASSEY Criterium for Inelastic Collisions

We end these introductory considerations on modelling inelastic collision processes with a dynamical view point – it may be considered complementary to the geometrical approach adopted above. Let us look at the inelastic process A + B(a) + T → A + B(b) + (T − Wba )

(7.8)

with a kinetic energy T of the collision partner before and T − Wba after the collision (both with respect to the CM system). In this process, B is initially in a state |a of energy Wa , and is excited by the collision into the state |b with energy Wb > Wa . During the collision, the relative kinetic energy is reduced by the excitation energy Wba = Wb − Wa . The interaction time of this process may roughly be estimated ¯ from a characteristic range a of the potential and the relative velocity u = 2T /M: a tcol = = u

M¯ a. 2T

(7.9)

During this short collision time tcol , the internal energy of the system is defined only within the limits of the uncertainty relation: W tcol ≥ .

(7.10)

The shorter the collision time, the larger the uncertainty W of the internal energy. For electron atom (or molecule) impact tcol is on the same order of magnitude as the electronic transition time te = /Wba for the inelastic process. More generally, with (7.9) we may rewrite (7.10) as M¯ Wba a 2 T ≥ , Wba me 2Eh a02 with the atomic units Eh = 27.2 eV, a0 0.053 nm and the electron mass me (2 = me Eh a02 ). In words: if the excitation energy Wba is on the order of Eh (as typical for electronically excited states), and the range of the potential is on the order of a0 , ¯ we expect maximum excitation cross sections for kinetic energies T /Wba ≥ M/m e. ¯ For electrons (M/m 1) this holds just above threshold, while for heavy particle e ¯ collisions (M/m e 1) kinetic energies substantially above threshold are required for optimal excitation!

458

7

(a)

(b)

x u

-2 -1

b R(t) z,t

Inelastic Collisions – A First Overview |cba (∞)|2

V(t) 1

(c)

2 t / t col

-1/R 4 2 2 e- t / 2t col -1/R

V(R) t=0

0

1

2 1/ ξ ∝ u

3

√ Fig. 7.5 Illustrating the M ASSEY parameter ξ : (a) Trajectory R(t) = z2 + b2 with z = ut , (b) time dependent potentials V (t), which lead to the (c) transition probability |cab (∞)|2 which depends on the relative velocity u ∝ 1/ξ

In the spirit of the B ORN -O PPENHEIMER approximation (Sect. 3.2) one expects a strictly non-adiabatic behaviour of the electronic wave function during e− − atom interactions. In contrast, in heavy particle collisions at kinetic energies on the order of Wba , the collision time is tcol te . As a consequence, the electronic wave function of the atoms may follow the interaction potentials adiabatically throughout the whole scattering process. Thus, in atom-atom (molecule) collisions, only at very high kinetic energies inelastic processes are expected (when tcol and te become comparable). For a quantitative interpretation, we consider the interaction potential V (R) as time dependent along a classical trajectory R(t). As indicated in Fig. 7.5(a), for simplicity we assume the relative motion along a straight line trajectory, parallel to √ the z-axis. With relative velocity u and impact parameter b we obtain 2 2 R(t) = b + z = b2 + (ut)2 . For sufficiently large b, the potential is attractive, −C/R −s , and thus may be written as: −s/2 V (t) = −C 1 + (ut/b)2 . Figure 7.5(b) shows the characteristic time dependence for such a potential for the pure C OULOMB case ∝ −1/R and for a polarization potential ∝ −1/R 4 (for this display we have chosen b = utcol ). As we are interested here only in general trends, we represent these potentials by a G AUSS function √ (7.11) V (t) −U0 exp −t 2 /( 2tcol )2 , as shown in Fig. 7.5(b). With its finite range, it keeps the following computation simple and the results clear. We identify the collision time according to (7.9) through a characteristic average range a of the potential. We now recall from Chap. 4 in Vol. 1, that time dependent perturbation theory turned out to be very successful for treating optically induced transitions. It should

7.1

Simple Models

459

also work here as long as the interaction potentials remain weak, i.e. for U0 /Wba

1. With (4.47), Vol. 1 we had derived in the optical case that the probability amplitude cba (∞) for the transition from state |a into state |b is nothing but the F OURIER transform of the perturbation potential at frequency ωba of the atomic transition. In the present case we thus have to compute: √ i ∞ i ∞ 2 2 −iωba t cba (∞) = − V (t)e dt U0 e−t /( 2tcol ) −iωba t dt. −∞ −∞ This integration can be done in closed form and leads to a probability for finding the target B after the collision in the excited state |b: 2 U0 U0 2 2 ω2 2 2 cba (∞)2 = 2πe−tcol ab tcol = 2πξ 2 e−ξ . (7.12) Wba Here we have introduced the so called M ASSEY parameter |Wba |a |Wba |a M¯ ξ = ωba tcol = = u 2T

(7.13)

(also called “adiabaticity parameter”). We specifically emphasize that it depends on the reduced mass M¯ of the system and on the relative kinetic energy T . Figure 7.5(c) illustrates the dependence of the excitation probability (7.12) on the relative collision velocity u ∝ 1/ξ . Obviously the excitation probability reaches indeed its maximum for ξ 1 (so called M ASSEY criterium). This corresponds to the equal sign in (7.10). Thus, the M ASSEY parameter characterizes an adiabatic energy range (ξ 1), for which inelastic processes are without relevance, and a diabatic range (ξ ≤ 1), where transitions in atoms or molecules may be induced by collisions. It underlines again the fundamental difference between electron scattering and heavy particle scattering: for the former the diabatic region ξ ≤ 1 typically begins at kinetic energies above threshold T > |Wba |. In contrast, for all heavy particle collision processes which we have discussed in Chap. 6 the M ASSEY parameter is on the order of ξ ≈ 102 to 103 , thus inelastic processes do not occur. They usually begin to play a role only for initial kinetic energies in the range of some keV to MeV where ξ 1. Only if the energy difference Wba (which has to be surmounted during collision) is very small, e.g. in rotational or vibrational excitation of molecules, thermal or hyperthermal kinetic energies may be effective in achieving the transition. Clearly, (7.12) gives only a qualitative description of the velocity dependence for inelastic processes. And the choice of a ‘characteristic average range’ a is always somewhat arbitrary. However, practice shows that the observations are usually quite well described by (7.12). In particular, the M ASSEY criterium ξ 1 gives a rather good indication for the maxima of excitation functions1 for many processes – as we shall document in the next section. 1 Excitation

cross sections as a function of collision energy are called excitation functions.

460

7


Section summary

• For exothermic reactions occurring in an attractive potential the cross sections are essentially determined by the trajectory (relative kinetic energy T ) which must lead over the centrifugal barrier. Specifically for charge exchange between an ion (charge qe) √ and an atom (polarizability α) the L ANGEVIN cross section (in a.u.) is πq 2α/T . • For endothermic reactions with a threshold Wth = Wba which is reached at a critical radius Rth the absorbing sphere model assumes that all trajectories which reach the sphere of radius Rth contribute, so that the cross section be2 (1 − W /T ). Both models ignore the general trend that cross comes πRth th sections decrease at higher energies. • A semi-quantitative dynamical model is derived by a simple time dependent approximation. The crucial parameter is the collision time, tcol = a/u, where a is an effective interaction range and u the relative velocity. The M ASSEY parameter, ξ = (|Wba |/)(a/u), distinguishes the adiabatic (ξ 1) from diabatic energy regime (ξ ≤ 1) where inelastic processes can occur. An astonishing good first order guess for the maxima of inelastic processes gives ξ = 1.

7.2

Excitation Functions

7.2.1

Impact Excitation by Electrons and Protons

We illustrate this first for the excitation of He atoms from the ground state. Figure 7.6 shows the excitation functions for the processes e− + He 1s 2 1 S0 + T → e− + He 1s2p 1 Po1 + T − 21.22 eV (b) p+ + He 1s 2 1 S0 + T → p+ + He 1s2p 1 Po1 + T − 21.22 eV

(a)

as a function of the relative velocity u in the CM system. They are compared with each other and with several theoretical approximations (CC represents “close coupling”, CCC “convergent close coupling” – approximations which we shall get acquainted with in Sect. 8.1). First B ORN approximation, FBA, gives at low energies obviously only a very rough guess, but it is astonishingly realistic at higher energies (dashed lines in Fig. 7.6). If we assume that the “effective, average interaction range” a of the potential is given by the VAN DER WAALS radius for He (0.14 nm), the velocity u in the CM system in Fig. 7.6 may be expressed by the M ASSEY parameter ξ according to (7.13), u = Wba /(aξ ). The experimentally observed trend and the theoretical modelling with ambitious methods confirm the simple considerations of the preceding section. The relevant literature documents for a multitude of cases that the positions of the maxima of


Fig. 7.6 Experimental cross sections for (a) electron and (b) proton impact excitation of He from the ground state into the 21 P state after M ERABET et al. (2001). The experimental data ( and ) are compared with several theories: (a) - - - - improved R-matrix with FBA, CCC; pseudo-states, and (b) - - - - FBA, two different CC calculations

461 e- + He(1s 2 1S) → e- + He(1s2p 1P)

12 exctitation cross section / 10 -18 cm2

7.2

(a)

8 4 0

ξ =1 0 uth

2

4

6

8

p+ + He(1s 2 1S) → p+ + He(1s 2p 1P)

20 15 10

(b)

5 0

ξ =1 0

uth 2

4

6

8

relative velocity u / a.u.

the excitation functions are estimated rather well by the M ASSEY criterium ξ 1. This holds, at least qualitatively, in electron, atom and ion collisions for so called ‘direct’ impact excitation. Clearly, the model cannot make any predictions about many important and interesting details. And for heavy particle scattering, we shall encounter in Sects. 7.3 and 7.4 a variety of processes which occur in the vicinity of potential curve crossings – for which the M ASSEY criterium hast to be modified dramatically.

7.2.2

Electron Impact Excitation of He

We shall now give an introduction into the typical behaviour of excitation functions for electron impact with atoms, by way of example. Rare gases are excellent prototypes. Probably the best studied case is e− + He(11 S0 ) excitation into various He(n 2S+1 LJ ) states. Numerous experiments have been compared with theoretical approaches. Today, elaborate computational methods and programmes allow to determine cross sections with high reliability even when experiments are not available. For the summary of excitation functions shown in Fig. 7.7 we have made use of the comprehensive data base of NIFS and ORNL (2007). Such data banks represent the cumulated ‘know-how’ of several decades. In detail, it may be somewhat delicate to extract exact cross sections for specific processes. But trends and orders of magnitude for many systems are certainly represented correctly. In addition to electron impact excitation of He(1s 2 1 S0 ) into the short-lived 1s2p 1 P1 state, a transition which we have already discussed above and which is

462

7

40

10

He(21P

1)

He(23P)

2

He+ + e-

20

5 σ inel

σ inel

1

0 σ / 10-18 cm2


0

0 He(23S1)

He(21S0) 2

2

σ inel

0 10

102

103

0 104 10

He (total) 200

4

1

σ ion

σ inel

102

σ tot 0 10

102

103

104

electron energy T / eV

Fig. 7.7 Excitation function for several final states in e− + He(1s 2 1 S0 ) collisions. Summarized are data derived from NIFS and ORNL (2007), using also the results of G OPALAN et al. (2003) for the 23 S1 state. For comparison, also shown is the total cross section (including elastic scattering) according to V INODKUMAR et al. (2007). Note the different scaling of σ for different processes

also allowed for optical E1 processes, Fig. 7.7 shows the cross sections σinel for several optically forbidden transitions into metastable states, 1s2s 1 S0 , 1s2p 3 P and 1s2s 3 S1 . Immediately evident is the much sharper rise of these cross sections above threshold and their much faster decay at higher energies – in comparison to the optically allowed 2 1 P1 excitation. In summary, the excitation function for optically forbidden transitions are significantly narrower on the energy scale, and the maximum of their cross section is typically a factor of five lower than the corresponding optically allowed transition. Looking back to our preliminary considerations in Sect. 7.1.4 it is not surprising that optically allowed as well as optically forbidden transitions occur in collisions over a rather broad energy range: the electron passing the atom rapidly generates a time dependent interaction potential with a broad spectrum of frequencies, which can in principle induce these transitions as soon as they are energetically possible (T ≥ Wba ). And since in and outgoing electron waves may contain in principle all angular momenta , there are no selection rules which would prohibit certain processes – in contrast to optically induced E1 transitions, where one photon of angular momentum is absorbed. The particular rapid rise of the cross section above threshold for the 2S excitations is obviously due to a dominance of the s scattering wave – as we have seen it already in elastic e− + He(1s 2 1 S) scattering (see e.g. Fig. 6.34). Clearly, at threshold,

7.2


463

where the kinetic energy T − Wba of the outgoing electron disappears, the s wave dominates in the outgoing channel (see also Eq. (6.106)). In the 1s 2 S → 1s2sS excitation process for the scattered waves = 0 must hold, i.e. the incoming s wave is responsible for threshold excitation. In contrast, for the 1s 2 S → 1s2pP transition = −1 describes the partial waves involved and at threshold only the incoming p wave can induce the transition. And since its phase shift ηp is significantly smaller the cross section rises much slower than in the former case. However, with increasing kinetic energy T of the electrons the forbidden processes obviously decrease much more rapidly than those optically allowed. We thus note (and we shall justify this statement on strict theoretical grounds in Sect. 8.2): With respect to excitation of atoms and molecules, fast electrons behave essentially like white light! Also characteristic is the excitation function for the 2 3 S and 2 3 P triplet states: they are even more localized to a narrow energy range above threshold – as compared to the excitation of the (also optically forbidden) singlet state 2 1 S0 . In the case of the triplet states the total spin of the He changes from S = 0 to S = 1. As we have learned in Vol. 1, spin orbit coupling is very weak for the He atom – as for all light atoms. Thus, even during a collision the spin of an atomic electron will not flip! The only possibility to induce a singlet-triplet transition by electron impact is exchange of the projectile electron with one of the atomic electrons: in such an exchange process with excitation the chance is 50 % that the total atomic spin changes according to the scheme e(↑) + He 1s ↑ 1s ↓ 1 S0 → He 1s ↑ n ↑ 3 LJ + e(↓).

(7.14)

However, such exchange processes are only probable at very low relative velocity. This explains why the cross sections for exciting 2 3 P and 2 3 S1 are of a similar order of magnitude as for 2 1 S0 and 2 1 P excitation, and why they decrease rapidly with increasing energy. Higher excited states of the He atom may also be excited by electron impact and show similar trends, albeit with significantly smaller cross sections as n increases. As for electromagnetically induced transitions this is explained by the lesser overlap between the respective wave functions and that of the ground state. For comparison we also show in Fig. 7.7 the integral cross section σion for electron impact ionization over a broad energy range. We shall come back to this in Sect. 8.4. The cross section shows clear similarities with the 21 P excitation function, however, it is about a factor of four larger. We note, that this is completely different to photoinduced processes where excitation is typically orders of magnitude larger than ionization. Finally, Fig. 7.7 shows the total electron cross section σtot for He, which is dominated by σion and the elastic cross section σel for which the low energy behaviour has already been reported in Fig. 6.7. The contribution of excitation processes to σtot is relatively small.

464

7

integral 2 3S1

(a) σ / 10-18 cm 2

6

Inelastic Collisions – A First Overview integral 2 3S1

(b)

total

1.5

2 3 S1

4 20.616 eV

2

21S0 2 3P

(c)

2.0

CCC

CCC

1.0

RMPS

0.5 0.0

0 20

21 20 electron energy T / eV

30

40

0

100

200

Fig. 7.8 Excitation function for e− + He(1s 2 1 S0 ) collisions in the metastable 2S and 2P states. (a) Threshold region according to G OPALAN et al. (2003) (taken with the apparatus shown in Fig. 6.37); on top: highly resolved part between 20.55 and 20.70 eV (2 1 S0 threshold at 20.616 eV). Calculations: total cross section —, individual processes · · · (state-of-the-art R-matrix theory with pseudo-states, RMPS); (b) — similar to RMPS according to BARTSCHAT (1998); (c) higher energies: CCC theory according to F URSA and B RAY (1995) - - -; older measurements ,

7.2.3

Finer Details in e− + He Impact Excitation

We now want to have a look at some selected examples to familiarize ourselves also with some finer details of the cross sections. First, as a relatively straight forward case study, we discuss the excitation of the metastable states in He. G OPALAN et al. (2003) have studied the processes e + He 1s 2 1 S0 → e + He 1s2s 3 S1 − 19.81 eV (7.15) and → e + He 1s2s 1 S0 − 20.62 eV just above the excitation threshold, both experimentally and theoretically with very high electron energy resolution. The metastable, excited atoms are registered with a special L ANGMUIR -TAYLOR detector (see Sect. 1.9.3 in Vol. 1). Figure 7.8(a) shows the total cross section for excitation of all metastable states as a function of electron energy. Just above the 2 3 S1 excitation threshold (at ca. 19.8 eV) one observes on this fine energy scale not simply a smooth rise of the integral cross section but rather pronounced structures which are reproduced by theory with astonishing precision. They are, obviously, related with the threshold (20.616 eV) for the 2 1 S0 excitation. An even better resolved measurement in the vicinity of this threshold (blow up on top of Fig. 7.8(a)) is not completely matched by theory which only shows a sharp indent (a so called “cusp” due to the unitarity of the wave function also at this threshold). The physics behind this feature is probably a very sharp resonance. For the pure 2 3 S1 excitation, theory predicts in the intermediate energy range between 22 and 25 eV some more, sharp resonances as shown in Fig. 7.8(b). Unfortunately, only in the higher energetic region some experimental data points are available in this case. The decrease of the cross section extends up to 200 eV as documented in Fig. 7.8(c).

excitatino cross section / 10-16 cm2

7.2


465

3

10

σ el

2 He

1

σ ion

Ar

5

σ tot

0

0 4

30

20 Ne

2

20 10

0 0

1000


Kr

2000 0

Xe

10 0

0

1000

2000

0

1000

2000

Fig. 7.9 Integral cross sections for electron scattering by different rare gases according to V IN ODKUMAR et al. (2007): σel elastic, σion ionization, σtot total

7.2.4

Electron Collisions with Rare Gases

To illustrate that such investigations are not restricted to the most simple atoms we now briefly mention other rare gases which are experimentally accessible with relative ease. Quantitative computations can rely for comparison on sufficient experimental data. Again we first give a general overview on trends and orders of magnitude. Elastic scattering has already been discussed briefly in Sect. 6.1.3. We continue here and show in Fig. 7.9 the integral elastic cross sections σel , ionization cross sections σion as well as total cross sections σtot for the rare gases He to Xe over a broad energy range. The cross sections shown are based on an detailed evaluation of many sets of experimental data and a careful comparison with most recent ab initio calculations by V INODKUMAR et al. (2007). Only the theoretical computational results are presented here. They set a bench mark for electron scattering of rare gases over a broad energy range. The behaviour of these cross sections as such is quite unspectacular and shows the expected rise and decay with energy, as we already know it from helium. The elastic cross section provides in all cases the major contribution. The ionization cross section is the next important ingredient for σtot while inelastic processes play only a minor role – just as in the He case. The rare gas atoms differ essentially by the order of magnitude of the cross sections which increases from He to Xe by about a factor of 10 to 15. This reflects very clearly the corresponding polarizabilities of the rare gas atoms, which dominate the interaction potential with the scattered electron at large distances: they are α 1.4, 2.6, 11, 17 and 27 a.u. for He, Ne, Ar, Kr and Xe, respectively. The fact that such cross sections can be computed for a large variety of targets is of great fundamental, but also of high practical importance. Modern computational methods for electron scattering – in Sect. 8.1.2 we shall get to know some

7

σ / 10-18cm 2

466


4 e- + Ne(2p 6) → e- + Ne(2p 5 3s) 3 2 1 0 17

18

19

T / eV

Fig. 7.10 Excitation function for e− + Ne(2p 6 ) → e− + Ne(2p 5 3s) collisions close to the excitation threshold. Compared are experiments with high energy resolution by B OMMELS et al. (2005) and B UCKMAN et al. (1983) with R-matrix calculations of Z ATSARINNY and BARTSCHAT (2004) . The experimental data are not calibrated but scaled to the theoretical data

of them in more detail – are very powerful and are improved continuously. Reliable experimental data, preferably with much structure, are crucial for testing and advancing these methods. As we shall discuss later, special, highly selective parameters, depending on polarization and/or spin of the collision system, may provide such crucial tests for theory. Similarly, pronounced structures, such as scattering resonances in the low energetic region, may serve the same purpose – apart from being interesting phenomena as such. A special example, according to B UCKMAN and S ULLIVAN (2006) a “bench mark I”, is low energy electron excitation of neon atoms. Figure 7.10 shows a small energy range above threshold. We see a series of sharp resonances, corresponding to short-lived negative ions. Without going into details we emphasize the excellent agreement between theory and experiment, documenting the capabilities of modern scattering theory very nicely. More recently, such studies have been extended successfully also to krypton (H OFFMANN et al. 2010). We finally mention just two very recent developments. The high energy resolution achievable with the laser photoelectron source used in these experiments is now also applied to study electron collisions with rare gases at very low energies (see e.g. K ITAJIMA et al. 2012). New perspectives for such studies may arise from recently proposed monochromatic high flux electron sources based on field ionization of RYDBERG states excited in laser-cooled atom beams (K IME et al. 2013).

7.2.5

Electron Impact at Atomic Mercury – The F RANCK-H ERTZ Experiment

We now address electron impact excitation of atomic mercury, Hg, as another, already rather complex example. The system is not only of practical importance, e.g. for understanding the plasma in Hg discharge lamps. It also has a remarkable historical perspective: the famous experiment of F RANCK and H ERTZ (1914), an essential

7.2


467 4

6 3P1 (4.89eV)

σ / 10-16cm2

3 2

6 3P2 (5.46eV)

1 0

6 1P1 (6.68eV)

5 6 7 6 3P0 (4.67eV) T / eV

8

9

Fig. 7.11 Excitation function for e− + Hg(6 1 S0 ) → e− + Hg(6 3,1 P) collision, i.e. for the cross sections populating the lowest 4 excited states of Hg just above threshold; red metastable states, black short-lived states. The data have been sketched (somewhat schematically) after S IGENEGER et al. (2003) and H ANNE (1988), also accounting for KOCH et al. (1984) and N EWMAN et al. (1985). Note the sharp resonances just above the thresholds for 6 3 P0 and 6 3 P1

proof for the quantization of atomic states in the early development of quantum mechanics, was based on electron impact excitation of Hg. As it turns out, the study of mercury, for this purpose, was a particularly happy choice. Would the electron impact excitation functions in Hg for the first excited states have the ‘typical’, rather broad energy width as we have seen them e.g. for e− + He(1s 2 1 S) → e− + He(1s2p 1 P) in Fig. 7.6(a) this classical experiment would not have shown any significant structures – and who knows what consequences this might have had on the further evolution of quantum physics. In reality, the excitation functions look roughly as sketched (schematically) in Fig. 7.11. The available experimental data for this case are, even today, still not fully satisfying. Most significant and well confirmed are, however, the extremely sharp resonances with high cross sections shortly above the thresholds for 6 3 P0 and 6 3 P1 excitation. It is the latter resonance at 4.89 eV to which the F RANCK -H ERTZ experiment owes its historical dimension (N OBEL prize in 1925). This very resonance is responsible for a significant jump of the excitation probability whenever the nominal electron kinetic energy T would be a multiple of 4.9 eV. This in turn leads to an energy loss of the electrons – which is finally recognized by a reduction of the current reaching the anode. A detailed modelling of the F RANCK -H ERTZ was and still is today a substantial intellectual challenge (R APIOR et al. 2006; S IGENEGER et al. 2003; H ANNE 1988). Today excitation functions for metastable states are known over a large energy range. They document the variety of possible phenomena quite impressively. Figure 7.12 shows a survey for the sum of all metastable processes of the type e− + Hg 5d 10 6s 2 1 S0 → e + Hg 5d 10 6s6p 3 Po0,2 and → e + Hg 5d 10 6s7p 3 Po2 etc.

468

7 6d 3D

6p 3Po1

5


σ / arb. un.

×30 6p 1Po1

0

6 4 6p 3Po0 6p 3Po2

8

7p 3Po2

10 11d 3D

12

14

16

T / eV

Fig. 7.12 Excitation function for e− + Hg(6 1 S0 ) into metastable states. Vertical lines —— (grey) mark the thresholds, - - - (grey) indicate potential parent states for resonances. Summarized from data by N EWMAN et al. (1985) and KOCH et al. (1984)

from threshold for the first excited state up to and beyond the thresholds for HgII(2 S1/2 ), HgII(2 D3/2 ), and HgII(2 Do5/2 ) ionization. The data presented here are collected from several different experiments, and the relative calibration is somewhat uncertain. Note also, that the sum of cross sections for excitation of all metastable states has been measured: one may assume that the energetic region above 5.46 eV is dominated by excitation of the 6 3 Po2 state, and above 8.83 eV by the 7 3 Po2 excitation. These excitation functions are obviously characterized by a number of resonances, i.e. by temporarily formed Hg− anions, for which N EWMAN et al. (1985) give plausible configurations. Present theory appears to be just on the verge of a quantitative interpretation of these interesting data. Clearly, in this case spin-orbit coupling cannot be treated as small perturbation, but recent relativistic convergent close-coupling calculations (RCCC) reproduce already several of the resonances shown here (B OSTOCK et al. 2010; B OSTOCK 2011). They even allow a convincing comparison with experimental data for differential cross sections (not shown here).

7.2.6

Molecular Excitation by Electron Impact

The excitation of molecules by electron impact (as well as purely elastic scattering) is a wide field of intensive research in which during the last decades a host of exciting material has been collected. We can give here only a slight taste. Everything that makes collisions of electrons with atoms interesting, and sometimes complicated, is also found when studying molecules. However, the electronic structure of molecules is significantly more complex, as we have seen in Chaps. 3 and 4. In addition, vibrational and rotational structure of molecules comes on top – and with it a further degree of difficulty arises. We know this already from spectroscopy and may summarize it by the keyword “F RANCK -C ONDON (FC) factors”. Dissociatative processes, including dissociative attachment of electrons, or for larger molecules the

Excitation Functions 15

e- + H2

10 5 30

total cross section σ tot

Fig. 7.13 Total excitation functions for electron impact with some important diatomic molecules e− + AB after S ZMYTKOWSKI et al. (1996); data from several experimental and theoretical sources are compared with the authors own experiments (red)

469

/ 10-16 cm2

7.2

e- + N2

20 10 40 30 20 10

e- + CO

e- + NO

12 10 8 6 10 8 6 4 0.5

e- + O2 1

5 10 50 100 electron energy T / eV

rearrangement of components within the molecules, constitute a further, broad field for interesting experiments and sophisticated theoretical calculations. All these processes are of substantial fundamental but also practical relevance – not the least as a basis for an elementary understanding of many chemical reactions as well as for radiation chemistry. We show in Fig. 7.13 just excitation functions for some of the most important diatomic molecules according to S ZMYTKOWSKI et al. (1996). They may communicate a feeling how complex and fascinating the study of such collision processes can be. The pronounced structures shown in these data may – with some effort – be attributed to the electronic, vibrational and rotational structures of these molecules which have been discussed in Sect. 3.6. Most of the observed features reflect FC factors for the transition from the ground state potential into the various excited states, including resonant transitions into the respective negative ions as short-lived intermediates. We emphasize in particular the data for NO in the energy interval from 0.5–2 eV, where a clear vibrational structure can be recognized. It is caused by temporarily exciting the NO− anion, whose vibrational states fall into this energy range as one verifies by comparison with Fig. 3.53. Such resonances are observed and intensively studied for many small and larger molecules, both in the elastic as well as in inelastic electron scattering cross sections.

470

7.2.7

7


Threshold Laws for Excitation and Ionization

Tacitly, one key question has already been addressed in much of the above discussion: how do integral cross sections behave just above threshold as a function of kinetic energy? Ever since the early days of collision physics this has been a challenging theme, and – albeit in principle solved (but just in principle) – remains of interest until today. The fundamental theoretical concepts have been derived by W IGNER (1948), and a summary on the present state-of-the-art is found e.g. in S ADEGHPOUR et al. (2000) or H OTOP et al. (2003). The threshold laws are derived from phase space arguments and the behaviour of the wave function of the outgoing particle(s). We recall: a photon (angular frequency ω) can excite an atom or molecule only if the transition is resonant, i.e. if ω = Wba holds. The excitation function in this case is thus a delta function (apart from the natural linewidth which may be neglected in the present context). In contrast, an electron can excite whenever its kinetic energy T is larger than the threshold energy Wth of the process studied (for atoms, usually the threshold energy and the excitation energy are identical: Wth = Wba ). In the case of excitation of a neutral atom or molecule by an electron according to W IGNER the excitation cross section above threshold is σ ∝ kf2+1 ∝ (T − Wba )+1/2 ,

(7.16)

for an outgoing electron wave with wavenumber kf , energy T − Wba and angular momentum . Since the √ latter usually is = 0 (s wave), this typically leads to an energy dependence σ ∝ T − Wba . With respect to ionization too, photons and electrons behave differently. According to W IGNER, for photoionization where one electron leaves the system in an attractive C OULOMB potential, at threshold σ ∝1

(7.17)

holds, i.e. the photoionization cross section starts at threshold (ω = WI ) with a finite value. We have seen characteristic examples for this behaviour already in Sect. 5.5, Vol. 1. For electron impact ionization the basic concepts have been developed by WANNIER (1953) and R AU (1971). The situation here is significantly more complex, since in this case two electrons leave the atomic ion, and the excess energy T − WI = WA + WB is shared among these two. For generating a Z fold charged ion by electron impact, the threshold behaviour of the energy and angle integrated cross section is according to WANNIER and R AU 1 100Z − 9 (ion) μ/2−1/4 σ ∝ (T − WI ) with μ = 2 4Z − 1 (7.18) hence for Z = 1 :

σ (ion) ∝ (T − WI )1.127 .

Thus, when a neutral atom or molecule is ionized by an electron, the threshold exponent of the energy becomes 1.127.

7.2


471

cross section

Fig. 7.14 Threshold laws for excitation (a, c) and ionization (b, d) of a neutral particle by a photon (a, b) or an electron (c, d). The cross section depends on the excess energy W = T − Wth , where the threshold energy Wth corresponds to the excitation energy Wba or the ionization potential WI , respectively. The range of validity of such threshold laws may, however, be very limited

ħω +A → A*

ħω +A → A++ e -

(a)

(b) Wba

e - +A

→

WI A*+

e-

e - +A

W 0.5

(c)

→ A++2e W 1.27

(d) Wba

WI

energy T of the exciting particle

excitation function / arb. un.

Figure 7.14 summarizes the above discussion. Of course, threshold laws are valid only for energies just above threshold. The key question as to the range of validity of such behaviour cannot be answered universally – and has kept generations of experimentalists and theoreticians intensively busy. All cross sections for excitation, reaction and ionization pass through a maximum and decrease at high energies more or less rapidly to zero. As a general rule, the range of validity for threshold laws is very narrow, typically some tenth or even only a hundredth of an eV. Figure 7.15 illustrates this for a process already well known to us: for electron impact excitation of helium into the 2 3 S1 state. This threshold study has again been performed with the very high resolution apparatus of G OPALAN et al. (2003) described in Sect. 6.5.3. The excellent agreement between experiment and state-of-the-art R-matrix theory is truly remarkable. But as one clearly sees, W IGNER’s square root threshold law holds only for a very small energy range. For somewhat larger energies experiment and theory may

e - + He(11S0) → e - + He(23S1) 2 W 0.5

W 0.391

1

0 19.80 Wth 19.84 electron energy T / eV

19.88

Fig. 7.15 Excitation function for the 2 3 S1 state in e− + He(2 1 S0 ) collisions according to G OPALAN et al. (2003). The data points are compared with R-matrix theory, which has √ been . W IGNER’s threshold law σ ∝ W = convoluted with the experimental energy width √ holds only for less then 10 meV above the threshold for excitation at Wth = Wba = T − Wth 19.820 eV

472

7


be approximated by a W 0.391 behaviour, which is attributed to the high polarizability of the excited state. We shall come back to the threshold behaviour of electron impact ionization in Sect. 8.4.3. Section summary

• Characteristic examples of excitation functions for ion and electron impact have been discussed. For electron and proton excitation of He(1s2p 1 P) the maximum of the excitation cross section is predicted very well by the M ASSEY criterium. • If finer details are overlooked, cross sections for electron induced (optically allowed) excitation and ionization rise smoothly from threshold up to the M ASSEY maximum, and then decrease more or less rapidly to zero with increasing energy. • For optically forbidden transitions, the rise of the cross section is much steeper and the overall energetic width of the excitation function is much narrower – particularly narrow for spin exchange processes (e.g. triplet excitation in He). • State-of-the-art experiments with high resolution and recent results from scattering theory reveal a wealth of finer structures in the excitation cross sections due to resonances and cusps. • Particularly structured are electron impact excitation functions for atomic mercury. The famous F RANCK -H ERTZ experiment owes its historic dimension to one of these extremely sharp resonances. • Electron molecule scattering shows even richer structures due to the complexity of molecules and the formation of short-lived, excited states of molecular anions. • Threshold laws for photons and electrons differ. For excitation, with photons strict resonance is required (delta √ function) while with electrons the cross section is predicted to follow σ ∝ W (with W = T − Wba ). For ionization with photons, the cross section starts with a finite value at the ionization threshold, while for electrons the WANNIER law with σ ∝ W 1.127 should hold. The range of validity for these threshold laws is typically very narrow (less than tenth of an eV).

7.3

Scattering Theory for the Multichannel Problem

7.3.1

General Formulation of the Problem

After the discussion above, the relevance of a flexible and efficient, fully quantum mechanical scattering theory including inelastic processes is evident. Today, a variety of very powerful computer codes are available fulfilling these requirements. The underlying concepts can be explained most clearly for inelastic heavy particle collisions. The basic terminology, however, is also applicable to electron scattering theory. A major part of Chap. 8 is devoted to this topic.

7.3

Scattering Theory for the Multichannel Problem y (col) prior to collision ℓy |a〉 B ri(B) R ΘCM ri(A) A P z (mol)

ℓy' B'

ri(B)

x (col)

z (col)

x (col)

y (col) after collision

473

R'

A' ri(A)

P' ' ΘCM

z' (mol) z (col)

|b〉

Fig. 7.16 Scheme of an inelastic scattering process A + B → A + B with excitation |a → |b. The internuclear distance is R, the relative momentum of the nuclei P . The inner coordinates r i of the system A–B are indexed here in addition by (A) and (B) to emphasize that these inner coordinates refer either to collision parter A or B. In the standard collision frame the coordinates x (col) , y (col) , z(col) are space fixed. However, the molecular axis z(mol) rotates during the collision. To keep these schematic simple, we have positioned the centre of mass in this picture into the centre of B (as if the mass of B was very large compared to that of A)

Here we shall focus for simplicity onto collisions of the type (7.8), involving two atoms A and B as illustrated in Fig. 7.16 (B may also be an atomic ion). Both collision partners may also be internally excited. Let the total state of the system A–B prior to collision be characterized by |a, after the collision by |b. It is important to note, that in such an inelastic collision the angular momentum of the nuclear motion y , perpendicular to the scattering plane, may change – while for elastic collisions it was a conserved quantity. The relative motion of the colliding particles A and B is described by the internuclear distance R, the inner degrees of freedom by r i representing the position coordinates of i = 1 . . . N electrons (mass me ) of the colliding partners. The S CHRÖDINGER equation (6.61) may then be written − W )Ψ (r 1 . . . r i . . . r N , R) = 0, (H

(7.19)

and the H AMILTON operator is = T(R) + T(r) + V (AB) (r 1 . . . r i . . . r N , R). H

(7.20)

The overall potential V (AB) (r 1 . . . r i . . . r N , R) comprises the various repulsive and attractive C OULOMB terms within A and B, as well as those between them.2 The 2 Often the computation can be dramatically simplified by distinguishing between active valence electrons and the passive core electrons and by using correspondingly pseudopotentials rather than summing over all N . The generalization to molecules as targets is for this ansatz without problems, however, when solving the problem in detail, much more complicated.

474

7


operators for the kinetic energy of the relative motion A–B and those of all electrons are 2 2 T(R) = − ∇R = 2M¯

N

2 1 ∇ 2r n , P · uˆ and T(r) = − 2 2m e n

(7.21)

respectively. Here M¯ is the reduced mass of the system A–B, while its operators ¯ respec = /M, for the relative momentum and relative velocity are P k and uˆ = P tively. We may write the total energy of the system as W = T + Wa =

2 kb2 2 ka2 + Wa = + Wb = T + Wb , 2M¯ 2M¯

(7.22)

with the relative kinetic energies T and T , the corresponding wave vectors k a and k b , the internal energies Wa and Wb of the collision system, before and after the collision, respectively. The various possible excited states |b of the system are called channels. A channel b is called open, if the corresponding excited state |b may in principle be excited during the collision, i.e. if T ≥ Wb − Wa . In the spirit of the B ORN -O PPENHEIMER approximation (Sect. 3.2) a suitable general ansatz for the solution of (7.19) is Ψ (r 1 . . . r i . . . r N , R) =

φj (r 1 . . . r i . . . r N ; R)ψj (R).

(7.23)

j

Here φj (r 1 . . . r i . . . r N ; R) represents internal states of the system A–B, including the initial and the final state of interest, j = a and b, respectively. In the following we shall abbreviate r 1 . . . r i . . . r N ≡ r. Basically, the φj (r; R) are functions of the inner coordinates r. They do, however, also depend implicitly on R, just as in the case of a molecule. For larger distances between A and B, i.e. before and after the collision, they become independent of R: φj (r; R) −→ φj (r). R→±∞

Finally, one has to ensure orthonormality φj |φj = δjj . Asymptotically, φj may be described as a product wave function of the isolated particles A and B. The index j represents all relevant quantum numbers. The wave function ψj (R) describes the relative motion of A and B; it contains the scattering dynamics for excitation of the states |j . For the elastic channel one looks for a solution with incoming plane wave and outgoing spherical wave in the form of (6.62). For the inelastic case with a = b, only outgoing spherical waves are expected. Hence, asymptotically the full scattering wave function is described by Ψ (r, R) −→ eik a R φa (r) + R→∞

1 fba (θ, ϕ)eikb R φb (r). R b

(7.24)

7.3


475

To be precise: the summations in (7.23) and (7.24) have to be carried out in principle over a complete set of basis states, including the respective continua. The fine art of scattering theory consists mainly in finding appropriate basis sets: these must describe the scattering process of interest as completely as possible, while on the other hand rapid convergence of the series expansions is desired. From such a solution one obtains a whole set of scattering amplitudes fba (θ, ϕ) for the transitions from an initial state |a into the final states |b. In the most general case they depend on the polar as well as on the azimuthal scattering angle, and of course on the relative initial kinetic energy T in the CM system. For a fully quantum mechanical solution of the problem one will again expand the scattering wave functions into partial waves. The asymptotic solutions of the corresponding radial equations – complementary to the elastic case (6.84) – may be written as: π π Γ uΓba (R) ∝ δba sin kb R − b + Kba (7.25) cos kb R − b R→∞ 2 2 for open channels and uΓba (R) = 0 R→∞

for closed channels.

The K-matrix introduced here is real and symmetric, and Γ represents a set of quantum numbers which are conserved during the collision. The S- and T -matrices already introduced in Sect. 6.4.6 are related to the K-matrix: Γ 1 + iK SΓ = Γ 1 − iK

and TΓ = SΓ − 1=

Per definition (6.118) the S-matrix is unitary, ∗ Sij Sj†k = Sij Skj = δik and specifically j

j

Γ 2iK . Γ Eˆ − iK

|Sij |2 = 1,

(7.26)

(7.27)

j

so that for open channels always |Sij | < 1

(7.28)

holds. In contrast to purely elastic scattering the S-matrix elements may no longer be expressed by just one real scattering phase according to (6.114). Instead, one may represent inelasticity of the process by a complex scattering phase ηba + iμba : Sba = e2i(ηba +iμba ) .

(7.29)

Quite formally, the scattering process may now be written in analogy to (6.109) S|k a a = |k a a + T|k a a. |k a a →

(7.30)

Just as in Sect. 6.4.1 the scattering amplitude is found by projection onto the final state, in complete analogy to (6.110):

476

7

fba (θ, ϕ) = − =−


¯ M/m e k b b|T|k a a 2πEh a02 ¯ M/m e ∗ k a a|T† |k b b∗ = fab (θ, ϕ). 2πEh a02

(7.31)

The second line describes the time inverse process: de-excitation of the atom from the excited state |b into the ground state |a. This simply follows from the fact that T is a Hermitian operator. In terms of the partial wave expansion the scattering amplitude may now be written in analogy to (6.115) fba (θ ) =

1 (Γ ) (2 + 1)P (cos θ )Tba (k), √ 2i ka kb

(7.32)

assuming for the moment cylindrical symmetry of the problem. We shall see in Sect. 8.1, that it may be necessary to extend the summation if different possibilities exist to realize the set Γ of conserved quantum numbers. The differential elastic (a = b) and inelastic (a = b) cross sections (DCS) for scattering into the solid angle dΩ at θ, ϕ and into a specific channel b is, according to (6.73): Iba (θ, ϕ) =

2 kb dσba = fba (θ, ϕ) . dΩ ka

(7.33)

The respective integral cross sections are found by integrating over all scattering angles; the total cross section is obtained by summation (and possibly integration) over all open channels. These two different quantities are unfortunately not always clearly distinguished in the literature. With (7.33) and (7.31), the relation between the differential cross sections for excitation and de-excitation may be written ka2 Iba (θ, ϕ) = kb2 Iab (θ, ϕ)

or T Iba (θ, ϕ) = T Iab (θ, ϕ),

(7.34)

¯ a2 and T = (2 /2M)k ¯ 2 = T − Wba . For the corresponding inwith T = (2 /2M)k b tegral cross sections we have ka2 σba ka2 = kb2 σab kb2 or

T σba (T ) = T σab T .

(7.35)

This is the quantum mechanical derivation of micro-reversibility which we have introduced heuristically at the start of Chap. 6. The two equations (7.35) and (6.19) are completely equivalent: As we describe here state specific transitions, the degeneracy factors are ga = gb = 1. In experimental verifications one has to account for the fact that the T-matrix elements depend on the kinetic energies before and after the collision, respectively. One has to compare the equivalent processes, for b ← a

7.3


477

at an initial kinetic energy T , for process a ← b at an initial kinetic energy T . In addition, for a differential experiment one has to make sure with respect to the scattering angles θ, ϕ that indeed exactly the inverse experiments are carried out. We come back to this in Sect. 9.2.2.

7.3.2

Potential Matrix and Coupling Elements

The ansatz (7.23) is still fairly general,3 and we shall now specialize it for heavy particle scattering. Corresponding to B ORN -O PPENHEIMER approximation the full Hamiltonian (7.20) will be split into (el) = T(R) + H H

(7.36)

(el) = T(r) + V (AB) (r 1 , r 2 . . . r N , R). with H

(7.37)

The choice of an appropriate basis for the φj (r; R) will be a key theme in the following sections. In any case, a set of coupled equations for the relative motion in R is obtained by insertion of (7.23) into (7.19), multiplication from the left with φj (r, R)| and integration over all inner coordinates r. One obtains 2 2 (R) (R) + Tjj + Ujj − (7.38) kj ψj (R) T 2M¯ P jj · uˆ + Tjj(R) =− + Ujj ψj (R). j =j

The potential matrix used here is given by (el) φj (r; R) , Ujj (R) = φj (r; R)H

(7.39)

and the so called non-adiabatic coupling elements are determined according to the definition (7.21) by jj · uˆ = − φj |∇ R |φj ∇ R P M¯ 2

2 (R) Tjj = − φj |∇ 2R |φj . 2M¯

and

(7.40) (7.41)

The potential matrix Ujj as well as the matrix elements of the momentum term jj · uˆ and of the kinetic energy Tjj are functions of R and depend on the choice P of the basis. A key point here is the skillful choice of this basis. In the following 3 The antisymmetrization necessary in the case of electron scattering is, however, not yet included and will have to be added in Sect. 8.1.

478

7


we shall discuss two alternative representations and their implications, as first investigated systematically by S MITH (1969) with respect to the treatment of inelastic heavy particle collisions.

7.3.3

The Adiabatic Representation

The most obvious approach towards a theoretical treatment of inelastic heavy particle collisions is to follow the spirit of the B ORN -O PPENHEIMER approximation outlined in Sect. 3.2 – which is particularly well suited for not too high kinetic ener(el) gies. One uses a basis set φj (r; R), for which the potential matrix (7.39), i.e. H becomes diagonal (Ujj = Ujj δjj ). In the case of atom-atom or atom-ion scattering the thus defined wave functions φj correspond exactly the electronic eigenfunctions of the molecule AB, (el) φj (r; R) = Ujj (R)φj (r; R) H

(7.42)

with φj |φj = δjj for all R, orthonormalized with respect to all internal coordinates r. In the asymptotic limit of large R the matrix elements are identical to the electronic energies of the A + B system, Ujj (±∞) = Wj . The big advantage of this adiabatic approach is, that the non jj · uˆ and T(R) adiabatic coupling elements P jj may become relevant only in a limited range of R values. For most internuclear distances they may be neglected: with respect to computing molecular energies and wave functions, the essence of B ORN O PPENHEIMER approximation is precisely this complete neglect of non-adiabatic coupling terms. In that case, (7.38) would reduce to the standard S CHRÖDINGER equation (6.61) for elastic scattering which we have already treated in Sect. 6.4. However, these very coupling matrix elements are responsible for inducing the inelastic processes of interest here. To develop a feeling for the order of magnitude of the terms (7.39)–(7.41), we make a rough estimate: The potential energy of the system (electronic energy) is on the order of an atomic energy unit, Ujj 2 /(2me a02 ) = 0.5Eh = 13.6 eV. The gradients of the internal wave functions change significantly over distances of an atomic unit of length and their magnitude is thus on the order of |∇R φj (r; R)| |φj (r; R)|/a0 . For this estimate, the scattering wave may be approximated by a plane wave, so that |∇R ψj (R)| |ψ|/λ k|ψ|. Extending the discussion in Sect. 3.2.3 we estimate for the orders of magnitude: # " 2 2 2 2 1 = Eh ∇ 2R k = T and Ujj 2 2 2M¯ 2M¯ 2me a0 me 2 T me k jj · u ˆ P Ujj T = 2 a0 M¯ M¯ 2me a M¯ 0

(R) Tjj

2

¯ 2 2Ma 0

=

2

me me a02 Ujj . ¯ M 2me M¯

7.3


479

Table 7.1 Order of magnitude of energies and coupling matrix elements for molecular bonding and inelastic collisions as relevant in B ORN -O PPENHEIMER approximation Application Dominant in BO Neglected in BO

Bound molecule T Ujj Eh /2 e T m Ujj M¯ jj ·u ˆ P T Tjj

e 4 m M¯

(R)

Order of magnitude

T

Inelastic scattering T ≥ Ujj Ujj jj ·u ˆ P Ujj Tjj

(R)

me M¯

Ujj

me T M¯ Ujj

=

uR ur

me M¯

The brackets indicate here integration over all R, i.e. over the whole collision process. We now have to distinguish two cases if one wants to compare the terms in the S CHRÖDINGER equation (7.38): 1. Bound molecular states: the kinetic energy in this case is identified with the vibrational energy of the molecule AB, T = Wv . The diagonal potential matrix element Ujj = We is its electronic energy. According to (3.6) we had estimated: ¯ jj . T ∼ me /MU 2. The (electronically) inelastic collision problem where excitation is only possible if the kinetic energy is sufficiently large, at least T > Ujj . To first order, B ORN -O PPENHEIMER approximation may be applied if 2 2 jj · u ˆ + Tjj(R) ∇ R /2 + Ujj P . Table 7.1 gives an overview. For the molecular bonding problem – as well as for very low collision energies, i.e. forelastic scattering at thermal energies – the kinetic energy of the system T jj · u ˆ and Tjj(R) Ujj me /M¯ has to be compared with P . In heavy particle ˆ Ujj T . The collisions inelastic processes will not occur as long as P jj · u (R) kinetic operator matrix elements Tjj can always be neglected in atom-atom (ion) collisions, since the ratio of electron mass me to (reduced) heavy particle mass M¯ is always very small. jj · uˆ is determined by The relative magnitude of the non-adiabatic coupling P the ratio of heavy particle to electron velocity, uR to ur , respectively. As long as uR /ur is small, the (adiabatic) B ORN -O PPENHEIMER approximation holds and inelastic processes have a very small probability only. This obviously corresponds to the M ASSEY criterium treated in Sect. 7.1.4. Inelastic processes may occur either at much higher energies, or if the energy difference Ubb (R) − Uaa (R) to overcome gets small, e.g. in the vicinity of potential curve crossings. With these assumptions, the following set of linear ODEs emerges from (7.38), describing the inelastic heavy particle scattering problem in the adiabatic represen-

480

7


tation: 2 kj2 2 2 jj · uψ ˆ j (R) − ψj (R) = − P ∇ + Ujj (R) − 2M¯ 2M¯ j or more explicitly

=

2 j

M¯

(7.43)

φj |∇ R |φj ∇ R ψj (R).

These ODEs thus replace (6.61), valid for elastic scattering, and the asymptotic behaviour of their solutions defines the scattering amplitudes fba (θ, ϕ) according to (7.24). With (7.22) the wavenumbers kj for each channel are derived from the initial kinetic energy T and the respective asymptotic inner energies Ujj (±∞) = Wj (in particular Wa and Wb ). We emphasize that the non-adiabatic coupling terms on the right hand side of (7.43) are scalar products of two vectors. One may view this in full analogy to the situation with optically induced dipole transitions (Sect. 4.3, Vol. 1), where the scalar product of the dipole transition matrix element and electric field or (more specifically) its polarization vector is responsible for the transitions and determines jj · uˆ determines which transitions are posthe selection rules. In the present case P sible and which cannot be induced by collisions. Since uˆ = −i∇ R /M¯ and acts on R as well as on ΘCM and ΦCM , one distinguishes radial and rotational coupling. As we shall explain in detail in Sect. 7.4.2, radial coupling can only induce a change in the main quantum number (without change of angular momentum), while rotational coupling can be expressed as a linear combination of the orbital angular momentum components (see e.g. S MITH 1969). Hence, changes of the atomic angular momenta are possible as a consequence of collisions.

7.3.4

The Diabatic Representation

While the adiabatic procedure just outlined can be used fairly general for inelastic heavy particle scattering, it does not necessarily converge rapidly, in particularly not for high kinetic energies. But even at low kinetic energies, as an alternative to the adiabatic basis a so called diabatic basis may be convenient. One tries to choose jj (R) and T(R) this basis in such a manner, that the matrix elements P jj of the nonadiabatic coupling disappear, i.e. ! φj (r; R)∇ R φj (r; R) ≡ 0

(7.44)

should hold. It is not always possible to achieve this in an unambiguous manner, and the price to be paid for it is in any case a non-diagonal potential matrix Ujj (R). We shall not go into the details of how to find such a diabatic representation. Rather, we shall later on discuss some characteristic examples. In Sect. 7.4.5 we shall illustrate, e.g., that the characterization of a basis as adiabatic or diabatic may even depend on the type of coupling.


potential matrix and coupling elements

7.3

trajectory Wba(Rx)

Ubb Uaa

481 adiabatic Ubb Uaa

uPab trajectory Wba(Rx)

Fa Fb

diabatic (D)

Ubb

4a (D)

Uaa

(D)

4×Uba

Rx

R

(D)

Fig. 7.17 Adiabatic Ujj and diabatic Ujj representation of the potentials (upper and lower panel, respectively), describing inelastic processes in the vicinity of an (avoided) crossing according to (D) (D) S MITH (1969). In the diabatic case, one may expand Ujj (R) = Ujj (Rc ) + Fj (R − Rc ) linearly (· · · · · ·). In the adiabatic representation, transitions between states |a and |b are induced by the non-adiabatic coupling element uP ab (- - -), while in the diabatic representation the non-diagonal (D) potential matrix element Uab (- - -) is responsible for the transition

Figure 7.17 compares the two different perspectives for the important case of a localized (avoided) crossing of two potentials at an internuclear distance Rx . We have already encountered such crossings previously, e.g. for the alkali halides in Sect. 3.7.4. They are found rather often in diatomic systems A–B, in particular for molecules with ionic bonding. In the adiabatic representation, upper panel of Fig. 7.17, we denote the potentials with Uaa and Ubb and the corresponding non-adiabatic coupling element with Pab (here we show very schematically only the radial component). Note, that the character of the adiabatic states which belong to these adiabatic potentials changes usually very rapidly in the vicinity of the crossing, as indicated in Fig. 7.17 by the line colour. At the crossing point, the energetic distance Wba (Rx ) of the potentials is of course much smaller than its asymptotic value Wba (∞) = Ubb (∞) − Uaa (∞). Hence, transitions between the two states |a and |b may occur already at rather moderate kinetic impact energies T (at some tenth eV to a few keV), rather than at much higher energies expected according to the M ASSEY criterium ξ ≤ 1 for Wba (see Sect. 7.1.4). In collisions of A with B these avoided crossings are typically reached by trajectories from a rather narrow range of impact parameters b Rx . They may, however, be identified easily via the reduced scattering angle T θ = τ (b). Thus, when treating such transitions it is usually sufficient to consider a two state model.

482

7


Complementary to this adiabatic point of view – but completely equivalent – in (D) the lower panel of Fig. 7.17 the corresponding potential matrix Ujj for the diabatic basis is illustrated. The diabatic states which are connected with these potentials do not change their character at the crossing point – in contrast to the adiabatic states discussed above. We have indicated this again by coloured lines. For large R the (D) (D) potentials in both representations merge, Uaa → Uaa and Ubb → Ubb , while for (D) small R the association is inverted. The off-diagonal element Uab is shown here to scale. The splitting of the adiabatic potentials shown in the upper panel follows from the diabatic representation, as already detailed in Sect. 8.1.6, Vol. 1. According to (8.36), Vol. 1 in present terminology we have Uaa

(D) (D) (D) 2 Uaa + Ubb 1 (D) (D) 2 ± Uaa − Ubb or Ubb = + 4Uab , 2 2

(7.45)

where the plus and minus sign holds for Uaa and Ubb , respectively. At the (diabatic) (D) (D) crossing Uaa (Rx ) = Ubb (Rx ), the splitting of the adiabatic potentials is thus given by (D) (7.46) Wba (Rx ) = Uaa (Rx ) − Ubb (Rx ) = 2Uab (Rx ). Equivalently, if the splitting is known, the diabatic coupling potential at the crossing point is obtained from: (D)

Uba (Rx ) =

1 Uaa (Rx ) − Ubb (Rx ) . 2

(7.47)

In Sect. 7.4 we shall explain, in the context of the semiclassical approximation, somewhat more detailed how such excitation processes are quantitatively evaluated in one or the other representation. Here we have to mention one more, so to say trivial, possibility to define diabatic states. It will always be an option if B ORN -O PPENHEIMER approximation is even in 0th order not a useful starting point. This is specifically the case for ion atom scattering at very high energies, and a fortiori for electron atom and electron molecule scattering, i.e. for the so called direct excitation processes. They are characterized by very small M ASSEY parameters (7.13) with respect to the asymptotic energy difference Wba . The basis φj of choice for describing the states according to (7.23) is in that situation simply the product wave function of the separated particles A and B, completely independent of the distance R of the two collision partners: φj (r; R) = φjAA (r A )φjBB (r B ).

(7.48)

Here, r A and r B stand for the respective inner coordinates of A and B, jA and jB for their quantum numbers. Since φj (r, R) is now independent of R, per definition (R) (7.44) holds. Also, all Tjj disappear. The components of the Hamiltonian (7.20)

7.3


483

may be associated correspondingly: (A) H

(B) H

= T(R) + V (R, r) + T(r A ) + V (A) (r A ) + T(r B ) + V (B) (r B ) . H

(el) H

Now, the non-diagonal terms of the potential matrix Ujj (R) according to (7.39) do not vanish, and may induce transitions. The further calculation becomes more transparent if the inner structure of one of the collision partners does not play a role. For example, A may be an ion with a closed rare gas shell which cannot be excited; or A may be an electron at high kinetic energy so that electron exchange with the target B can be excluded. In this case (7.48) simplifies to φj (r; R) = φjB (r), i.e. our AB basis is now simply given by the eigenfunctions of the target. Instead of (7.36) and (7.37) the Hamiltonian is now = T(R) + V (R, r) + H (B) . H

(7.49)

V (R, r) is the interaction of the N electrons and the nucleus (charge Ze) of the (B) is the N electron Hamiltonian of the target target B with the projectile A, and H atom B. Its eigenvalues Wj and wave functions φj (r) ≡ φj (r 1 , r 2 . . . r N ) are computed as we have discussed in detail in Sect. 10.1, Vol. 1. In electron atom scattering the potential is V (R, r) =

N n=1

e2 Ze2 − , 4πε0 |R − r n | 4πε0 R

(7.50)

and the potential matrix elements (7.39) are written as Ujj = Vjj + Wj δjj

with Vjj (R) = φj (r)V (R, r)φj (r) .

(7.51)

Instead of (7.43) the S CHRÖDINGER equation becomes finally

2 kj2 2 2 ψj (R) = − − Vjj (R)ψj (R). ∇ − Wj − ¯ ¯ 2M 2M j

(7.52)

Obviously, the structure of this system of coupled ODEs is independent of the specific nature of the interaction potential V (R, r) between projectile A and target B. They are exact as long as the basis φj used is complete and electron exchange does not play an essential role. In practice, however, the calculation will always be limited to a finite basis and exchange can only be neglected for truly high kinetic energies. The implications have to be judged in each case individually (see e.g. K IMURA and L ANE 1989). We shall come back to this aspect in Chap. 8.

484

7


Section summary

• The quantum mechanical treatment of multichannel heavy particle scattering starts with an ansatz (7.19)–(7.23) in the spirit of the B ORN -O PPENHEIMER approximation – trying to separate the internal coordinates (r of the colliding partners A and B from their relative motion R by a sum j φj (r; R)ψj (R) over all relevant channels j . • Generalizing the formalism used for elastic collisions, the asymptotic solution (7.24) for the inelastic case is composed of (i) the product of an incident plane wave with the initial internal wave function φa (r; R), and (ii) the sum over products of outgoing spherical waves with φj (r; R) for all relevant final channels j . The respective amplitudes are the scattering amplitudes fj a (θ, ϕ) from which the inelastic DCS (7.33) are obtained. • The dynamics of inelastic collisions is described by the wave functions ψj (R) for each channel j . They may be derived from a set of coupled ODEs (7.38). In principle, inelastic processes can be induced by non-adiabatic coupling elˆ or by off-diagonal elements of the interaction ements, dominantly by P jj · u, potential Ujj . • Two representations of the internal wave functions φj (r; R) are discussed: in the adiabatic basis (standard BO approximation) the potential matrix is diagˆ In the diabatic onal, Ujj = δjj Ujj , and transitions are induced by P jj · u. basis the adiabatic coupling vanishes and the off-diagonal potential matrix elements induce transitions. While both approaches may differ with respect to rapid convergence, they are in principle equivalent as discussed in some detail for an (avoided) curve crossing (Fig. 7.17).

7.4


7.4.1

Time Dependent S CHRÖDINGER Equation

As we have seen, inelastic heavy particle collisions typically require high kinetic energies. Partial wave expansion – for electron scattering the method of choice – will thus converge extremely slowly since high angular momenta are involved – even at moderate kinetic energies. On the other hand, even at relatively low energies the DE B ROGLIE wavelengths in heavy particle collisions are small compared to typical distances over which the interaction potential changes significantly. Hence, semiclassical methods are ideally suited for these scattering problems. In Chap. 6, in particular in Sect. 6.4.4, we have already documented and discussed their efficiency for elastic scattering. For inelastic heavy particle collisions semiclassical theory is beyond any doubt the most commonly used approximation. With appropriate modifications it is used whenever transitions from one potential surface onto another are to be described, if they are induced by nuclear motion. Specifically, this also holds for the dynamics in isolated, polyatomic molecules, where electronic transitions, internal rearrangement or dissociation may occur as a consequence of an initial photoexcitation process.

7.4


485

The basic idea is to compute a classical trajectory R(t) for the relative motion of the collision partners – if appropriate a trajectory for all the constituent atoms or ions if molecules are involved. This trajectory is used to derive time dependent in jj (R(t)). For complex teraction potentials Ujj (R(t)) and/or coupling elements P systems at moderate kinetic energies present state-of-the-art computations integrate the classical motion of all heavy particles involved in the problem as completely as possible. For such molecular dynamics (MD) calculations powerful commercial programmes exist. They even allow one to predict the relevant potentials and forces in the vicinity of the classical trajectory, ‘on the fly’ is the standard term, by means of suitable quantum chemical structure programmes. When treating polyatomic systems, this allows one to limit the computational efforts of complex potential hypersurfaces to those geometrical regions which are really required for the dynamical study. For more simple atom-atom or atom-ion collisions it is usually sufficient to determine trajectories on an potential which has been suitably averaged between initial and final state. Even the assumption of a straight line trajectory R(t) = ut + b often leads to satisfactory results – specifically so at very high energies. The electronic transitions are then determined from the time dependent interaction potentials or non-adiabatic coupling elements, depending on whether a diabatic or an adiabatic basis is used. The time dependent S CHRÖDINGER equation ∂ (el) Ψ (t, r) = 0 (7.53) H − i ∂t (el) according to (7.37), together with for the electronic wave function, with H the classical trajectory R(t), now replaces the stationary S CHRÖDINGER equation (7.19) for the whole scattering system. Instead of the ‘ansatz’ (7.23) a time dependent series is now used, (7.54) cj (t)e−iϕj (t) φj r; R(t) , Ψ (t, r) = j

with the semiclassical phases derived from 1 R Ujj (R(t)) 1 t dR. Ujj R t dt = ϕj (t) = −∞ ∞ uR (R(t))

(7.55)

The potential Ujj is defined by (7.39) or possibly by (7.51), and uR = dR/dt is the relative radial velocity. With (7.54) and (7.55) the usual manipulations of (7.53) finally leads to a set of coupled, time dependent, linear ODEs:4 c˙j (t) = −

1 Gjj (t)ei(ϕj −ϕj ) cj (t).

(7.56)

j =j

4 One finds slightly different notations in the literature, which differ by i or i/ in the definition of the coupling element Gjj .

486

7

7.4.2


Coupling Elements

As discussed in Sects. 7.3.3 and 7.3.4, it depends on the individual case whether one uses an adiabatic or diabatic representations for a practical computation of the transition probabilities according to (7.56). There are no general rules and often both approaches are tested. If one chooses the adiabatic basis according to (7.42), the non-adiabatic coupling elements for j = j are now time dependent and can be derived from (7.53): ∂ Gjj (t) = φj r; R(t) φj r; R(t) . ∂t

(7.57)

As illustrated in Fig. 7.16, the classical trajectory is given by the internuclear distance R(t) and the angle ΘCM (t) in the centre of mass system.5 The collision coordinates are usually defined such that the z(col) -axis is parallel to the relative momentum P and to the relative velocity u of the colliding particles, prior to collision. The x (col) -axis lies in the collision plane defined by P and P and points into the direction into which the particle is scattered. With these definitions, the angular velocity Θ˙ CM has a positive or negative sign depending at which side the particles pass each other: positive and negative sign of ΘCM (∞) correspond to an effectively attractive or repulsive potential (for the trajectory shown in Fig. 7.16 Θ˙ CM > 0 holds and ΘCM (∞) > 0). The nuclear angular momentum6 in y (col) -directions is ¯ ¯ 2 Θ˙ CM is obtained from im= MR y = R × P and its magnitude |y | = = Mbu pact parameter b and relative velocity u. With the reduced mass M¯ and the time dependent internuclear distance R(t) the angular velocity of the relative motion becomes: Θ˙ CM = ±

bu =± . ¯ 2 R2 MR

(7.58)

With this we may write the time derivative of the wave function, which enters into the coupling matrix element (7.57), explicitly as ∂φj (r; R(t)) dR ∂φj ∂φj dΘCM ∂φj = + ± = uR iL φ . ¯ 2 y j ∂t dt ∂R dt ∂ΘCM ∂R MR

(7.59)

Here we have inserted the radial velocity uR = dR/dt as well as the electronic any = −i∂/∂ΘCM . With (7.59) the coupling element gular momentum of the atom L

5 As

in the elastic case, the classical deflection angle Θ has a well defined sign, in contrast to the scattering angle θ . 6 We

recall that we use (a vector) and (a number) for the nuclear angular momentum and its quantum number, respectively (to be distinguished from the electronic orbital angular momentum L).

7.4


487

(7.57) finally becomes Gjj (t) = uR φj |

∂ y |φj . φj |iL |φj ± ¯ ∂R MR 2

radial coupling

(7.60)

rotational coupling

The two components of Gjj (t) represent radial and rotational coupling, respectively – which we have already mentioned above. The radial coupling element arises from changes of the internal wave function of the collision system A–B with internuclear distance R. It can only couple states with equal angular symmetry, i.e. Σ with Σ and Π with Π states. In contrast, rotational coupling changes the symmetry of the molecular states. y changes the electronic angular momentum with respect to the The operator L (col) y -axis by ±, so that it can induce Σ → Π or Π → Σ or Π → transitions: we have to note here that the symmetry of the molecular states refers to the internuclear axis z(mol) between A and B (see Fig. 7.16). The latter rotates during the collision around y (col) with the angular velocity Θ˙ CM . The electronic charge cloud would stay space fixed parallel to z(col) – except for the molecular interaction which tries to keep the orbitals aligned along z(mol) . It is this very fact that the nuclear angular velocity translates directly into electronic angular momentum. Typically one y |φj is nearly independent of R, so that overall the rotational coufinds that φj |L pling (7.60) which is ∝ 1/R 2 dominates for small internuclear distances. Alternatively one may of course choose a diabatic basis according to (7.44). In Sect. 8.1 we shall come back to this again in the context of electron scattering. For (D) heavy particle scattering one will try to make the diagonal matrix elements Ujj of the potential as equal as possible to the adiabatic energies of the system – except in the vicinity of crossings. In any case, the diabatic coupling element (7.57) is obtained from (el) φj r; R(t) . Gjj (t) = Ujj R(t) = φj r; R(t) H (7.61)

7.4.3

Solution of the Coupled Differential Equations

If one finally knows all potential matrix elements (7.60) or coupling terms (7.61) as well as the trajectories, the coupled ODEs (7.56) can be integrated in principle by standard methods. One has to evaluate the excitation amplitudes cb (t) for t → ∞. Specifically, for exciting state |b from state |a the initial conditions ca (−∞) = 1 and cb (−∞) = 0 for j = a have to be applied. The transition probability (for one trajectory at a given impact parameter) is then |cb (∞)|2 . More generally, the transition amplitudes cj (t) are brought into the form (7.25) by relating the impact parameter b = /kj to the corresponding angular momentum and the outgoing wavenumber kj . Then one applies (7.26) to obtain the Sand T-matrix elements, from which the scattering amplitudes (7.32) are derived, and finally the differential cross section (7.33). Alternatively, for sufficiently high ener-

488

7


gies one may – as for elastic scattering according to Sect. 6.4.4 – also use the eikonal method. In the limit of small scattering angles and high energies one obtains (see e.g. D UBOIS et al. 1993) the scattering amplitude fba (θ, ϕ) =

¯ Mu (−i)1+|Mb −Ma | e−i(Mb −Ma )ϕ ∞ × bdbJ|Mb −Ma | (Kb) cba (b, ∞) − δba

(7.62)

0

(cf. Eq. (6.59) for shadow scattering). This scattering amplitude describes the transition between states |aMa and |bMb with projection quantum numbers Ma and Mb of the electronic angular momentum of the target with respect to the z-axis. The latter is taken here parallel to the relative velocity before collision. J|Mb −Ma | (x) is a ¯ sin(θ/2) the change of the wave vector during B ESSEL function and K = 2(Mu/) the collision (∝ to the momentum transfer). A few general characteristic of the solution may be discussed. The special significance of the exponential phase factors exp[i(ϕj − ϕj )] is already recognized in 1st order perturbation theory, where for a transition b ← a cba (∞)2 = 1 2

+∞

−∞

Gba (t)e

i(ϕb (t)−ϕa (t))

2 dt .

(7.63)

One sees that – independent of the strength of the coupling Gba – excitation is only possible in those regions of the trajectory where exp[i(ϕb (t) − ϕa (t))] does not vary too fast – otherwise the oscillations would lead to cancellation of positive and negative contributions. Two cases can be distinguished: 1. For direct excitation the potentials of the two states a and b are separated over the whole trajectory and their distance is approximately given by the asymptotic conditions. With (7.55) the phase difference is then approximately ϕ(t) = ϕb (t) − ϕa (t)

(Wb − Wa ) R (Wb − Wa ) t, ∼ uR

(7.64)

and the exponential factor in the integrand of (7.63) oscillates with time. Depending on whether this oscillation is slow (high velocity uR ) or fast (small uR ) along a typical interaction range, the transition probability will be significant or disappear. This consideration confirms quantitatively and generally M ASSEY’s adiabaticity criterium: the phase difference ϕ(tcol ) for the total interaction time tcol is according to (7.64) indeed identical with the M ASSEY parameter (7.13). 2. Complementary to this, non-adiabatic transitions may also occur at a crossing of the potential curves for state |a and |b, as illustrated in Fig. 7.17. This may happen already at low velocities and between channels with asymptotically very different internal energies. In this case, the phase difference ϕ(t) remains sufficiently small in the vicinity of the crossing at Rx , at least as long as (R − Rx ) ≤ uR /(Ubb − Uaa ) uR /Wab (Rx ) holds, so that the coupling

7.4


489

can induce the transition. In this case one may focus onto the two relevant states, and has to solve simply a system of two coupled, linear differential equations according to (7.56). In addition to the direct numerical solution of the coupled ODEs (so to say by ‘brute force’) a number of more or less elegant approximations exist which also lead to the desired result. In the following we shall discuss the important method developed by L ANDAU (1932) and Z ENER (1932), which even today is still often used as a simple but reasonable approach for a quantitative understanding of transitions at (avoided) crossings.

7.4.4

L ANDAU -Z ENER Formula

We consider the transition from state |a to |b through a crossing localized at Rx , following the diabatic picture as represented in the lower panel of Fig. 7.17. We calculate the transition probability between the diabatic states |a (D) and |b(D) . According to (7.56) with (7.61) one has to solve the following coupled differential equations for the initial conditions caa (−∞) = 1 and cba (−∞) = 0 (the second index refers to the initial state): i (D) c˙aa (t) = − Uab (t)eiϕab cba (t) i (D) c˙ba (t) = − Uba (t)e−iϕab caa (t) 1 t (D) (D) with ϕab (t) = Uaa t − Ubb t dt .

(7.65) (7.66) (7.67)

(D) (D) In the vicinity of the crossing at Rx , where Uaa (Rx ) = Ubb (Rx ), one expands the 7 diabatic potentials for small distances R = R − Rx :

(D) ∂ (D) (D) (D) Uaa − Ubb R Ubb − Uaa = R x ∂R

(7.68)

= (Fa − Fb )R = Fab R.

(7.69)

The constant Fab is the difference of the slopes of both potentials at the crossing (i.e. the difference of the respective forces onto the trajectory), and R = uR t, if (D) time zero is set when the trajectory passes the crossing. The coupling element Uab is assumed to be constant over the crossing region: (D) U (R) = U (D) (R) = U (D) . (7.70) ab ba ab 7 For

simplicity of the derivation we consider here only radial coupling, for which the L ANDAU Z ENER model is typically used. A LLAN and KORSCH (1985) have shown, however, that the formalism may also be applied to rotational coupling.

490

7


With these definitions and (7.68) the phase difference (7.67) is π ϕab (t) = α t 2 2

with α =

Fab uR , π

(7.71)

and the coupled ODEs (7.65) and (7.66) simply become: i (D) π 2 c˙aa (t) = − Uab eiα 2 t cba (t) π 2 i (D) c˙ba (t) = − Uba e−iα 2 t caa (t).

(7.72)

It is instructive to first have a look at the solution in 1st order perturbation theory. We thus insert on the right of (7.72) caa (−∞) = 1 and cba (t) = cba (−∞) = 0, √ substitute x = αt, integrate from −∞ to ∞, and obtain: i (D) cba (∞) = − √ Uab α

∞

−∞

e−i 2 x dx = π

2

signum(α) − i (D) Uab . √ |α|

For exp(−iπx 2 /2)dx = cos(−πx 2 /2)dx + i sin(−πx 2 /2)dx we have inserted here the well known limits of the F RESNEL integrals. Thus, in 1st order perturbation theory the probability for the transition |a (D) → |b(D) during one transit of the crossing is: (D)

(D)

wba

(D)

2|Uab |2 2π|Uab |2 = = 2πξ. |Fab |uR 2 |α|

(7.73)

Of course this approximation only holds for 2πξ 1. As already shown by Z ENER (1932), the system (7.72) of coupled ODEs can be solved exactly. An attractive modern derivation is found e.g. in W ITTIG (2005). Accordingly, the probability for a transition between the adiabatic states |a → |b is wba = e−2πξ ,

(7.74)

which is identical to the probability for remaining on one of the diabatic potentials. Inversely, the probability for a transition from one to the other diabatic state |a (D) → |b(D) (i.e. to stay on the adiabatic potential) is wba = 1 − e−2πξ , (D)

(7.75)

which in the limit of small ξ reproduces the perturbation result (7.73). It is worthwhile to have a closer look at the key parameter (D)

|Uab |2 ξ= uR |Fab |

with Fab =

∂ (D) (D) U . − Uaa ∂R bb

(7.76)

7.4


Fig. 7.18 Trajectory with twofold curve crossing (‘in’ and ‘out’) at Rx . Rc is the classical turning point

491 uR

in

u b

Rx

out

- θ CM

Rc z (col) x (col)

(D)

For pure radial coupling, according to (7.46) the coupling potential Uab = Wab /2 is related to the splitting Wab of the adiabatic potentials. Hence, ξ=

1 |Wab |2 a |Wba |a |Wab |2 = = = ωba tcol , 4uR |Fab | uR |Fab |4a uR

(7.77)

and we recognize ξ again as the well known M ASSEY adiabaticity parameter (7.13) – if we interpret the effective interaction range a suitably, so that 4aFab = Wab : at a distance (R − Rx ) = 4a from the crossing point the diabatic splitting is identical to the adiabatic splitting Wab at the crossing point. We have marked this in the lower panel of Fig. 7.17. Clearly, the M ASSEY criterium has now to be reinterpreted appropriately: it is restricted to the crossing. There, the energetic splitting of the potentials the Wba is the key quantity which determines the transition probability. The larger the splitting, the smaller the probability (7.74) for a jump from or to the other adiabatic state. Conversely, the transition probability increases with the radial velocity uR as well as with the difference of the slopes. For u → 0 (and thus uR → 0) the transition probability goes to zero, wba → 0. Finally, we have to consider the overall result of such a collision. As indicated in Fig. 7.18, each trajectory passes the crossing point at Rx twice (for sufficiently high energy and small enough impact parameter): (i) on the way ‘in’ towards the classical turning point Rc and (ii) on the way ‘out’ as marked by grey circles in the figure. During each passage of the crossing point transitions may occur or not. There are two different trajectories which lead to a non-adiabatic transition |b ← |a: 1. For a trajectory, which has achieved the jump |b ← |a on the way in with a probability wba , on the way out the probability to remain in this state |b is given by (1 − wba ). The total transition probability from one to the other adiabatic state is thus wba (1 − wba ). 2. On the other hand, a trajectory which has not made the jump on the way in (probability 1 − wba ), may make it on the way out with a probability wba . Again, the overall probability for a jump is (1 − wba )wba . Both trajectories contribute to transitions, and the total probability for the transition |b ← |a at a given impact parameter is8 the probability to remain in the initial state during the overall process is wba wba + (1 − tot . wba )(1 − wba ) = 1 − wba

8 Inversely,

492

7 tot wba = 2wba (1 − wba )


with wba = exp(−2πξ ).

(7.78)

This is the L ANDAU -Z ENER probability for a non-adiabatic transition at a localized crossing, ξ being the modified M ASSEY parameter (7.77). tot on the adiabaticity parameter ξ is similar to that sketched The dependence of wba tot = 1/2: in Fig. 7.5(c) on page 458. It has, however, at ξ 0.11 a maximum of wba overall, at most 50 % of all collision processes can lead to a non-adiabatic transition. While the L ANDAU -Z ENER model is still rather qualitative, it has proven to be very useful for the discussion of non-adiabatic transitions in atomic collisions as well as for intramolecular dynamics. It is still used today on many occasions – even though for quantitative investigations, obviously, an exact solution of the coupled differential equations (7.56) is required.

7.4.5

A Simple Example: Na+ + Na(3p)

As a still rather straight forward example we discuss the collision of an ion with an excited atom, during which two pronounced, non-elastic processes can occur: a “super-elastic” (exothermic) de-excitation and an excitation process: * 2 → Na+ + Na(3s 2 S) + (T + 2.10 eV) + Na + Na 3p P3/2 + T (7.79) → Na+ + Na(3d 2 D) + (T − 1.48 eV). Figure 7.19 shows the relevant interaction potentials of the system Na+ 2 . Marked by circles are three crossings of which we consider here only (C) for the transition |3p → |3s and (B) for the transition |3p → |3s.9 In both cases we have a crossing between molecular |nΣu and |n Πu states. Their potentials are obtained as solutions of the electronic S CHRÖDINGER equation (7.42) for the system Na+ 2. In the usual static picture, used in Chap. 3 for the presentation of molecular potentials as a function of R, these are genuine crossings. Even though the radial parts of the wave function depend on R, due to the different symmetries of their angular parts the transition matrix element disappears nΣu |∂/∂R|n Πu ≡ 0, and the states do not split along the R coordinate. On the other hand, according ¯ 2 )nΣu |iL y |n Πu does not disappear to (7.60) the rotational coupling (/MR and may induce transitions. Quantitatively one finds that the coupling matrix eley |3sΣu and 3pΣu |Lˆ y |3dΠu are about (A LLAN and KORSCH ments 3pΠu |L 1985). The terminology “diabatic” and “adiabatic” is here somewhat confusing, rather one should speak about crossing and non-crossing potentials, respectively. When discussing the L ANDAU -Z ENER model in Sect. 7.4.4, we have denoted (D) (D) the potentials of the states which cross as a function of R by Uaa and Ubb . In the present case (with strong rotational coupling) these potential are the adiabatic eigen9 Crossing (D) is important for the dependence of the process on polarization – which we cannot describe here.

Semiclassical Approximation 6

B

6

B

5

5

4

3d 2u

3

C

5

3p 2u

2

7

4

5

6

Na(3d 2D)+ Na+ 3p 2 Σ g

3p 2 Σ u

2 1

6

C

3

≈

Fig. 7.19 Potentials for Na+ 2 from data of M AGNIER and M ASNOU -S EEUWS (1996). The relevant potentials are drawn in red. For the observed inelastic ion impact processes the crossings (B) and (C) are responsible. In the insets these crossings are magnified. The other potentials have no influence on the processes under discussion

493

potential energy / eV

7.4

D

Na(3p 2P)+ Na+

3p 2 Σ g

3s 2 Σ u 0

Na(3s 2S1/2)+ Na+

3s 2 Σ g -1 0

10

20 R / a0

30

40

values of the electronic Hamiltonian. However, as a function of the deflection angle ΘR (t) they split.10 The coupling element (7.60) which enters in the coupling ODEs (7.56) is in the present case pure rotational coupling. The probability to remain on the respective initial (crossing) potentials is according to the L ANDAU -Z ENER model y |n Πu |2 2 |nΣu |L . (7.80) wab = exp(−2πξ ) with ξ = 2 ¯ uR |Fab | MR For very small relative velocities u → 0 ( → 0 and uR → 0) becomes wab → 1 – which is equivalent to vanishing transition probability (1 − wab ) for the transition |nΣu → |n Πu . However, at intermediate and higher kinetic energies such transitions occur indeed at the crossings, and the transition probability depends of ¯ course on the impact parameter b. It enters into the computation via = Mub and uR = u[1 − (b/R)2 ]1/2 (the latter is read directly from Fig. 7.18). A reliable computation of the differential cross section in the framework of the semiclassical approximation requires a calculation of the classical deflection function Θ(b) on a suitably adapted potential: it has to accounts for the jumps between the potentials. One then has to solve the coupled differential equations (7.56) either exact or according to an appropriate model (for a finite number of states involved). Finally, the scattering phase shifts ηa,b have to be computed as a function of , e.g. 10 This

is a consequence of the rotation of the internuclear axis during the collision (rotational coupling). We note here in passing, that this splitting corresponds directly to lambda-type doubling in molecular spectroscopy (Sect. 3.6.6), a splitting of energy levels into Λ+ and Λ− states for higher rotational quantum numbers due to the coupling of the electronic angular momentum Λ with the nuclear rotation N .

494

7 |S12|2 1.0

0

1

2

3


4

b / a0 5 6

0.8 0.6 LZ 0.4 0.2 AKo85 0.0 0

400

800

1200 ℓ

1600

Fig. 7.20 Calculated transition probability for the transitions |3p → |3s in collisions of Na(3p) with Na+ at an initial kinetic energy 47.5 eV – here as a function of the relative nuclear angular momentum or of the impact parameter b, respectively. Shown are the semiclassically exact values of |S12 |2 according to A LLAN and KORSCH (1985) ( AKo85), compared with the simple ) L ANDAU -Z ENER (LZ) transition probability 4 exp(−2πξ )[1 − exp(−2πξ )] (

as JWKB phases according to (6.94). Without entering into details (see e.g. A L LAN and K ORSCH 1985; BANDRAUK and C HILD 1970), we also communicate the resulting S-matrix: () = e−2πξ + 1 − e−2πξ e−2iδ e2iηa Saa () Sbb = e−2πξ + 1 − e−2πξ e2iδ e2iηb (7.81) () () Sab = Sba = 2ie−πξ 1 − e−2πξ sin δei(ηa +ηb ) . S is of course uniHere, ξ , ηa , ηb and δ depend on (or b, respectively), and tary. The additional phase δ() introduced here accounts for the possible alternative pathways in the crossing region which have been discussed in Sect. 7.4.4: along these trajectories different phases may be collected. In the S-matrix this leads to characteristic interferences, so called S TÜCKELBERG oscillations. The scattering amplitudes are finally obtained by inserting (7.81) in (7.32), from which again the differential cross section I (θ ) follows according to (7.33). Alternatively, for very short wavelengths (high kinetic energies, large masses) I (θ ) may be computed as in the case of shadow scattering as diffraction integral (6.59). The complex transmission function T (b) is then given by the semiclassical transition amplitude cba (∞, b) which one calculates as accurately as possible from the coupled ODEs, including the real phase shifts. Figure 7.20 shows, for the first of the two processes in (7.79), the semiclassically calculated transition probabilities A LLAN and KORSCH (1985) induced by rotational coupling – see crossing at (C) in Fig. 7.19. For comparison the L ANDAU -Z ENER transition probability is also shown – computed for identical conditions by (7.80). Classically, the maximum active impact

(a)

fluorescence detector

Na-oven

θ Lab

ion detecto

r

atom bea

495

ion source

m shutter beam expansion λ /4 plate polarizer dye laser

( ) dσ / dΩ CM / 100 a20 sr -1

Semiclassical Approximation laser beam

7.4

3

3 d ← 3p 3s ←3p

2

1

0 0º

2º 4º θ CM

6º

8º

10º

Fig. 7.21 Na+ + Na(3p) → Na+ + Na(3s, 3d) scattering experiment: (a) experimental setup, schematic. (b) Differential cross section as a function of the scattering angle θCM in the CM system at a kinetic energy TCM = 47.5 eV: measured data •, with line to guide the eye - - - according to according to A LLAN and KORSCH (1985) BÄHRING et al. (1984); semiclassical computation (slightly rescaled angle, see text)

parameter b = Rx 4.9a0 leads to a nearly straight line trajectory. As documented by Fig. 7.20, only the quantum mechanical treatment can produce interference structures and transitions occur also for values of b somewhat larger than Rx . These processes have been studied by BÄHRING et al. (1984) in great detail (see also the review of C AMPBELL et al. 1988). Figure 7.21(a) gives only a very schematic overview of the ion beam apparatus used. The Na+ ions have been generated in a hot, Na impregnated metal surface. They were then passed through an electrostatic 180◦ spherical capacitor (see Appendix B.3), selecting their energy with a bandwidth of ca. 150 meV. The ion optics finally formed a well collimated ion beam. At laboratory energies of 40 eV to 100 eV the latter crossed a sodium atomic beam at right angle. The Na+ ions scattered at a laboratory angle θLab have been analyzed with an electrostatic energy analyzer (again a 180◦ spherical capacitor) and were finally detected by a particle multiplier. It was also possible to vary the relative azimuthal angle of the detector plane. The Na target atoms in the scattering centre were irradiated by linearly polarized light, incident perpendicularly to the scattering plane, from a CW laser. The Na atoms were excited into the 3 2 P3/2 , F = 3 state (see Appendix D for optical pumping). For small scattering angle the kinematics of the system is readily transformed from the laboratory into the CM system (see Sect. 6.2.2): TCM TLab /2 and ΘCM 2θLab . Hence, the reduced scattering angle – as in the elastic case a function of the impact parameter b, is given by τ = TCM ΘCM = TLab θLab . Figure 7.21(b) compares the experimentally determined differential cross sections for both reactions (7.79) with semiclassical calculations according to A LLAN and KORSCH (1985). Since the position of the crossing points Rx are somewhat larger (as calculated by M AGNIER and M ASNOU -S EEUWS 1996) than those used

496

7

Fig. 7.22 Scattering angle for maximum transition probability in excitation and de-excitation processes during Na+ + Na(3p) collisions (7.79) as a function of initial kinetic energy, according to BÄHRING et al. (1984). These reduced scattering angles τ = T θ for the respective maxima of the two processes are nearly independent of the initial kinetic energy T


θ CM 10º 8º

3d← 3p

τ = 258 eV º 3s← 3 p

6º

τ = 185 eV º

4º 2º 0º 0.0

0.01 0.02 0.03 0.04 0.05 -1

TCM / eV -1

in these calculations, we have rescaled the scattering angle ΘCM ∝ 1/Rx slightly in each case. This is reasonable, since according to (6.46) the scattering angle is approximately ΘCM ∝ 1/b (in the relevant range of distances we deal with a pure repulsive C OULOMB potential). This leads to a better agreement between theory and experiment which is now surprisingly good, considering the simplicity of the theoretical model. Even the relative cross sections of both processes are computed correctly (theory was used for absolute calibration of the cross sections). Finally, in Fig. 7.22 the measurements at different collision energies are summarized. Obviously, the reduced scattering τ = T θ for which the differential cross section reaches its maximum is indeed nearly constant, as expected in first order approximation according to (6.52).

7.4.6

S TÜCKELBERG Oscillations

We have already mentioned above, that different phases may be accumulated on different trajectories in a collisional excitation process via curve crossing (see Fig. 7.18 where the excitation may occur either at the crossing marked in or out). Thus, the () (7.81) for inelastic processes may lead to oscillascattering matrix elements Sab tions as a function of (or b) and consequently as a function of scattering angle (as well as of kinetic energy). The phenomenon has already been treated by S TÜCKEL BERG (1932) in the heydays of early quantum mechanics. The calculation shown in Fig. 7.20 for the process just discussed exemplifies such behaviour. Unfortunately, after conversion into differential cross sections, these oscillations are not significant enough to be observed experimentally in this case as documented by Fig. 7.21(b). There are, however, numerous examples where such interferences can be observed. We discuss here charge exchange between the quadruply charged C4+ ion and a neutral He atom. This example is interesting for several reasons and quite popular in scattering physics. Experimentally, the system is easy to access (highly charged C ions are obtained from standard ion sources), and a whole series of exothermic electron capture processes into doubly or triply ionized Cq+ ions may be studied. These can easily be identified by their different energy gain W for different final

7.4


497

C3+(3p)+He2+ [11] U(R) / eV

(a)

C2+(2s3p)+He2+ [10] C4+ + He [9]

U(R) / eV

0

(b)

20 C3+(3s)+He2+ [8] C2+(2s3s)+He2+ [7]

-10

0

C2+(2p 2 1S)+He2+ [6]

[2]

-20

C2+(2s2p)+He2+ [4]

2

6

8

[9] [2] [3]

- 20

[1]

R / a0

≈ 20

40

60

80

12

CHe 4+

(c)

0

C3+(2s)+He+ [1] 0

10

20

C2+(2s 2)+He2+ [2] -40

[3] 4

40

C3+(2p)+He+ [3]

-30

[1]

-20

C2+(2p 2 1D)+He2+ [5]

[7] [5]

[6] [4]

∞

0

2

4

6

Fig. 7.23 Potentials and energetics for the system CHe4+ according to P ICHL et al. (2006). State [9] represents the entrance channel, dashed grey lines - - - indicate 2e− capture, full grey lines − —– 1e capture. (a) Overview; (b) critical crossing region; (c) simplified four state model according to BARAT et al. (1990) with · · · · · · indicating a diabatic basis. Red circles mark the relevant crossings

channels: ⎧ 3+ 2 2 + ⎪ ⎨C (1s n L) + He (1s) + W C4+ 1s 2 1 S + He 1s 2 1 S → C2+ (1s 2 2sn 1 L) + He2+ + W ⎪ ⎩ 2+ 2 C (1s 2pn 1 L) + He2+ + W.

(7.82)

From a theoretical point of view, this is a case of pure radial coupling, in contrast to the previous example, and both collision partners have initially identical electronic structures (1s 2 1 S0 ). In Fig. 7.23 we show the potential curves for all exothermic processes according to P ICHL et al. (2006) (for simplicity, configurations are given without the 1s electrons). The potentials are dominated by C OULOMB repulsion in the exit channels, the initial channel corresponds to a nearly horizontal line, interrupted by avoided crossings at which the charge exchange (7.82) occurs. The enlarged presentation (b) shows these avoided crossings more clearly (red circle). Here too, a classical trajectory can take (several) different paths towards the classical turning point on the way in and out – just as we have discussed it in Sect. 7.4.4. In this case, the resulting phase differences lead to pronounced oscillations in the differential cross sections which have been studied by BARAT et al. (1990) at high kinetic energies. Figure 7.24 gives some examples for the one and two electron

498 dσ __ 2 / a 0 rad -1 dθ

7 2e- capture 9.6 keV

1600 1200


500

1e- capture 9.6 keV

400 300

800

200

400

100

0

0 0

2

4

6

8 10 12

0

2

4

6

8 10 12 τ / keV deg

Fig. 7.24 Differential cross section for C4+ + He as a function of reduced scattering angle τ according to BARAT et al. (1990) at 9.6 keV. The measured data points are compared with semifor the four state model shown in Fig. 7.23(c) classical calculations

capture process at 9.6 keV. Shown is the differential cross section summed over different final states as a function of the reduced scattering angle τ = T θ . The experimental data are compared with computations from a semiclassical calculation using the models discussed above. BARAT et al. (1990) have used the diabatic version of a simplified four state model with the potentials shown in Fig. 7.23(c). The agreement between theory and experiment is astonishingly good, considering the simplicity of the model. Note that for single electron capture (potential curves [1] and [3] in Fig. 7.23(c)) the relevant crossings are localized at very small internuclear distances. Due to the dominant C OULOMB repulsion these can be reached only at rather high kinetic energies. More recently, two electron capture has been studied by H OSHINO et al. (2007) at significantly lower energies. Figure 7.25 shows a 2D representation of the doubledifferential cross section as a function of the energy gain (W = T − T ) and the laboratory scattering angle θLab . The dominant process in this case is obviously the 2e− capture into the ground state of C2+ (1s 2 2s 2 ). Here too, S TÜCKELBERG oscillations are very clearly resolved as a function of the scattering angle. Section summary

• Semiclassical methods for inelastic heavy particle collisions follow the general concepts developed for elastic processes (see Sect. 6.4.4) and combine these with time dependent coupled differential equations for multichannel transitions, summarized by (7.56). • For inelastic processes which occur at localized potential curve crossings, two types of coupling are distinguished according to (7.60): Radial coupling and rotational coupling. The latter is closely related to lambda type doubling of molecular electronic states, and may be rationalized due to rotation of the

Collision Processes with Highly Charged Ions (HCI)

Fig. 7.25 Doubledifferential cross section for the 2e− capture in C4+ − He collisions at low kinetic energy as a function of the laboratory scattering angle θLab and the measured energy gain W according to H OSHINO et al. (2007). The smooth and the dashed curves represent W for the C2+ ion as expected from the kinematics of two double electron capture processes with nominal energy gain Wab = 33.4 and Wab = 20.7 eV, respectively

499

30 20 10 0

W / eV

7.5

-10 30 20 10 0 -10 30 20 10 C2+(1s 22s 2 1S) C2+(1s 22s 2p 1P)

0 -10 -8˚

-4˚

0˚

θ Lab

4˚

8˚

molecular symmetry axis of the collision system with respect to the space fixed laboratory system. • In the semiclassical picture, at these localized curve crossings transitions occur along a trajectory with a probability wba on the way in (towards the classical turning point) as well as on the way out. The L ANDAU -Z ENER model tot = 2w (1 − w ) for a non-adiabatic tranpredicts an overall probability wba ba ba sition, with wba = exp(−2πξ ) where ξ is essentially the M ASSEY parameter – modified for a localized crossing according to (7.76) and (7.77). • As an example, we have discussed excitation and de-excitation in Na+ + Na(3p 2 P3/2 ) collisions. The differential cross sections are found to peak strongly, with reduced scattering angles τ = ΘCM T nearly independent of kinetic energies – corresponding to a narrow range of impact parameters. • S TÜCKELBERG oscillations may occur in such inelastic heavy particle collisions as a consequence of different phases accumulated by different trajectories leading to the same scattering angle. An impressive example is electron capture in C4+ + He collisions.

7.5


In the context of L AMB shift we have already mentioned in Sect. 6.5, Vol. 1 that today quite fundamental experiments are carried out with highly charged ions (HCI). For example, H like ions of heavy elements (up to 91 times ionized uranium with a nuclear charge Z = 92) allow to perform very sensitive tests of quantum electro-

500

7 106

Xe q +

105 Wpot (q) / eV

Fig. 7.26 Potential Wpot of highly charged atomic ions as a function of their charge q according to W INTER and AUMAYR (1999). In principle Wpot (q) can be released when q electrons are captured


Thq +

Ar q +

104 103 102 10

0

20

40

60

80

100

q

dynamics. As QED related contributions to atomic energies are typically expanded into a series of powers of the fine structure constant (α 1/137) times effective nuclear charge q, perturbation theory becomes increasingly problematic as the latter increases. With promising perspectives about new facilities for cutting edge research with HCI (G RIESER et al. 2012), the field is expected to remain attractive and productive. Extending our above discussion on charge transfer with C4+ to higher charged states we want to focus now onto some specific problems in collisions with HCI. This interesting special area of collision physics has developed in the last two or three decades very productively (M ORGENSTERN and S CHMIDT-B ÖCKING 2009). One very special property of highly charged ions is their extremely high potential energy – when interacting with neutrals – which may lead to a variety of violent reactions. In Fig. 7.26 these energies Wpot as a function of the charge number q are collected for three examples.11 One realizes that e.g. the naked argon nucleus (Ar18+ ) carries already a potential energy of nearly 16 000 eV, and complete recombination of Th90+ with all its electrons liberates nearly 1 MeV. Aside of the fundamental academic interest in such processes, highly charged ions have a number of practical implications. HCI are observed relatively often in space and may be used for diagnostics, e.g. – just to mention a somewhat exotic application – to determine the temperature of solar winds in collisions with the atmospheres of comets. Also in fusion plasmas, HCI play a role when by ion bombardment heavy elements may be knocked out of the walls. And in the context of generating nanostructures on surfaces or in biomedical objects, the high energy density and excellent focusability of HCI beams should be emphasized. Today, highly charged ions are generated by various methods. Originally, accelerator based methods were used, exploiting the fact that high energy ions, when shot through thin foils, loose their electrons. They are now replaced by so called electron cyclotron resonance (ECR) sources, special designs of which are called electron beam ion source (EBIS) or . . . trap (EBIT). A lot of progress has been made during the past decade and such sources are provided in several specialized 11 W pot

is the sum of all ionization potentials WI (q ) for q ≤ q.

7.5


501

laboratories worldwide. They are all based essentially on the same physical principles, but are quite different in their specific construction. In all three cases HCI are generated by multiple collisions with electrons and stored in an electric field, while strong magnetic fields (in some cases generated by superconducting magnets) keep the electrons focussed together. In the EBIS design, electrons play an important role in ion storage. Addition of several low Z gases allows an efficient cooling by small angle scattering (so called evaporative cooling). For the study of collision processes, the HCI are then extracted from the source, accelerated, selected according to mass and energy and finally focussed onto a target or inserted into a storage ring. In HCI collisions a manifold of processes may occur, so that efficient detection and measuring methods for the various reaction products is crucial for the success of such experiments. Typically, time and position sensitive methods are used, often with coincident determination of the momentum components for several reaction products (e.g. by COLTRIMS, which is briefly explained in Appendix B.4).

7.5.1

Above-Barrier Model

Charge exchange processes are of particular importance. For simplicity, we restrict the following discussion to single electron capture (SEC) such as Aq+ + B → A(q−1)+ (n) + B+ .

(7.83)

Let the binding energy (<0) of electron transferred be Wb after the collision and Wb prior to it. For high charge q, the energy defect of the process Q = Wb − Wb

(7.84)

will typically be negative, since the captured electron is bound much stronger in A(q−1)+ than before in B. Thus, (7.83) describes exothermic processes which – as discussed in Sect. 7.1.1 – may have high cross sections. Due to its high potential energy, the HCI (Aq+ ) acts in a collision with the neutral target (B), so to say, as a ‘vacuum cleaner’ which sucks up the electrons from the target as soon as it comes close enough. As sketched in Fig. 7.27, the potential barrier between projectile and target is reduced during this process dramatically. This classical, so called “over-the-barrier” model has already been developed during the late 80s of the past century (see N IEHAUS 1986, and further work quoted there), but still enjoys great popularity for qualitative discussions and makes predictions of astonishing accuracy. We discuss here the basic ideas. When the projectile Aq+ approaches the target B, the valence electrons (coordinate r) of B ‘see’ an effective potential V (r; R) which changes as sketched in the scenarios Fig. 7.27(a–h) with internuclear distances R. At a critical distance R = Rth the barrier decreases below the local binding energy of the target electrons as indicated in Fig. 7.27(c). Projectile and target then form – for a very short time – a quasi-molecule and the HOMO of the target B and high lying, unoccupied n states of Aq+ come very close, energetically.

502

7 0

20

- 20

0

20

≈

- 20

≈

0


R→ ∞

-2

t

(a)

-4

R→ ∞

Aq+

B

(h)

-2

n=8

-4

A(q-1)+

B+

≈

≈

0

0

(b)

-2 V(r;R ) / E h

0

(g)

-4

R = 50

0

(c)

-2 -4 0

n=9 8 7 R th = 12.4a 0

-2

n=8 R = 50

(f )

0 n=9 8 7 R th = 12.4 a 0

R= 8 - 20

0

20

-4

R= 8 40 60 - 20 0 electron coordinate r / a 0

-4

-2

-2 -4

-2

0

(e)

(d )

-4

20

40

60

Fig. 7.27 Classical over-the-barrier model for a single electron capture process for the example He + Ar18+ (we follow here a presentation of M ORGENSTERN 2009). Plotted are the potentials V (r; R) which the electrons (coordinate r) experience in the system for different internuclear distances R. The heavy red arrows indicate the time sequence. Red, horizontal lines correspond to energy levels in the isolated collision partners, and for R ≤ Rth in the quasi molecule, respectively (full line: occupied, dashed: unoccupied). For energetic consistency the two He electrons are drawn on different levels – even if they are of course indistinguishable in the first ionization step

Thus, the valence electron with the smallest binding energy may change its place and will occupy these n states with high probability. It will stay there on the way out, as indicated in Fig. 7.27(f): the probability for recapture from the n states of A(q−1)+ to the ionized target B+ is very low, since phase space (degeneracy) is much higher in these high lying states than in the typically light target. Mostly, processes with high kinetic energies are studied, so that – in contrast to Sect. 7.1 – one may assume approximately straight line trajectories. The critical distance Rth may thus be identified with the maximum impact parameter bm for a charge exchange process 2 may become rather to happen. Thus, the resulting capture cross sections σ πRth large. Quantitatively, one assumes a simplified potential q t . V (r; R) = − − r R−r Here and in the following we use again atomic units (a.u.), Eh (energy) and a0 (length). The screening of the nuclear charge of B by its inner electrons is modelled by an effective charge t – quite similar as done in Sect. 7.1.2 (Vol. 1) for bound state

7.5


503

calculations. With simple calculus one √ finds the maximum of the barrier for a given internuclear distance R at rM = R/( q/t + 1), its value being √ √ VM (R) = −( q + t)2 /R. (7.85) We write the binding energy of the HOMO electron at infinite distance Wb (∞, HOMO) = −WIB (with the ionization potential WIB ). As the highly charged ion approaches, this will be lowered. In the above-barrier model one assumes that at the critical distance Rth this decrease is given simply by the potential of the HCI: (B)

Wb (Rth , HOMO) = −WI

− q/Rth .

(7.86)

With (7.85) this critical distance is derived from (B)

VM (Rth ) = −WI − q/Rth √ √ (B) ( q + t)2 /Rth = WI + q/Rth , so that it becomes Rth =

√ t + 2 qt (B)

(7.87)

.

WI

And in analogy to (7.86) the energies of the n final states of the projectile ion are lowered at Rth : Wb (Rth , n) = Wb (∞, n) − t/Rth .

(7.88)

Electrons captured from B will preferentially populate such levels n for which the energy at Rth matches best with the energy of the target electron according to (7.86): −WIB −

q t Wb (∞, n) − , Rth Rth

Wb (∞, n) −WIB −

√ q + 2 qt q −t = −WIB √ , Rth t + 2 qt

(7.89)

where the latter result is obtain by using (7.87). Hence, the energy defect (7.84) becomes: Q = Wb (∞, n) − Wb (∞, HOMO) = −WI

(B)

(q − t) √ . t + 2 qt

(7.90)

If we now write the binding energy of the captured electron in the projectile ion Wb (∞, n) = −q 2 /(2n∗2 ), we finally find with (7.89) the effective quantum number of the state predominantly populated: √ (B) −1/2 t + 2 qt 1/2 . n∗ q 2WI √ q + 2 qt

(7.91)

504

7.5.2

7


An Experiment on Electron Exchange

As a genuine example we shall now discuss the reaction Arq+ + He → Ar(q−1)+ (n) + He+ .

(7.92)

It has been studied by K NOOP et al. (2008) for q ≥ 15 at projectile energies of 14 keV /q in a COLTRIMS experiment, and was compared with state-of-the-art close-coupling calculations.12 One of the He electrons is captured into a RYDBERG state n in this process, and the experiment measures the surplus energy Q(n) by determining the momentum p of the back scattered He+ ion. One detects the latter in coincidence with the respective projectile ion after the collision and thus ensures that only signals arising from the SEC process (7.92) are analyzed. Let us have a look at the kinematics on which the analysis of this process is based. At these high kinetic energies one may assume the trajectories to be nearly straight lines and the scattering angle θ to be very small. Hence, p is nearly parallel to the momentum pA of the projectile. From the perpendicular component p⊥ = p sin θ one determines the scattering angle, from the parallel component |p | p the energy defect. The mass of the projectile Aq+ is MA and (MA + me ) before and after the collision, respectively (with me being the electrons mass), v = pA /MA is its velocity. B is assumed at rest prior to the collision, its mass is MB and (MB − me ) prior and after the collision, respectively. For simplicity we consider only such target recoil ions which are scattered along the projectile axis (forward or backward). The momentum of the projectile after the collision (with the electron captured) is then = p A − p . pA

(7.93)

2 /2M before and p 2 /2(M + m ) after the collision, the With kinetic energy pA A A e A energy defect (7.84) follows from the energy balance as 2

Q=

p (pA − p )2 (pA )2 − . − 2MA 2(MA + me ) 2(MB − me )

(7.94)

By expanding the second and third denominator into a series for me /MA and me /MB , respectively, up to the linear term, and using pA = MA v, one obtains p2 MB me v 2 1+ , + p v − Q= 2 2MB MA

(7.95)

when neglecting me /MA and me /MB 1 in the final expressions (much less than 1/1000 of the last two terms). In the relevant literature one usually finds only the first scenarios sketched in Fig. 7.27 correspond to just this reaction for q = 18 (i.e. for the naked Ar nucleus). Not completely correct but illustrative we have shown in these schematics also the reduction of the states at R > Rth – according to (7.86) and (7.88).

12 The

ion yield / counts

7.5


1500

8

(a)

600

7

500 n =6

10

9 10

Ar15+

n =6

8

200 0

0

Ar18+

n =6 n =7 n =8

(c)

0.1 0.01 0.0

7

(b)

400

0 -40 -120 -80 Q value / eV

dσ/dθ /10-12cm2

800

Ar18+

1000

1

505

-120

-80

-40 0 Q value / eV

10 1

Ar15+

(d)

0.1

0.1

0.2 0.3 θ / mrad

0.4

0.01 0.0

0.1

0.2 0.3 θ / mrad

0.4

Fig. 7.28 One electron capture according to K NOOP et al. (2008) in the reactions He − Ar18+ → He+ − Ar17+ (1sn) (a, c) and in He + Ar15+ (1s 2 2s) → He+ + Ar14+ (1s 2 2sn) (b, d); the measured Q value spectra (a, b) are fitted (red lines) assuming occupation of the n states 6 to 10; in (c) and (d) the experimentally determined differential cross sections as a function of scattering angle θ are compared with theory

two terms (see e.g. U LLRICH et al. 2003; D EPAOLA et al. 2008, where, however, the signs are defined differently). For high energies (in the experiment discussed here >200 keV) the third term may indeed be neglected too, as one easily verifies. Also, me v 2 /2 is usually much smaller than |p v|. Hence, negative Q (exothermic reaction) leads to p < 0, i.e. to backscattering of the B+ . In Fig. 7.28 some of the results from K NOOP et al. (2008) are shown. They show very clearly that electron capture into the projectile ion occurs indeed very selectively to a few states n as illustrated in Fig. 7.28(a, b). For q = 18 preferentially n = 8 and 7 are populated, for q = 15 we see n = 7 and 6 to dominate. A look onto Fig. 7.27(c) shows that this essentially agrees with the predictions of the overthe-barrier model. Quantitatively according to (7.91) the values n∗ = 8.5 and 7.3, respectively, are expected – slightly above the experimental observations – but in view of the simplifications of the model this is an extremely surprising agreement. The observed differential cross sections shown in Fig. 7.28(c, d) are strongly peaked for small scattering angles, as expected. The agreement with theory is impressive, considering the complexity of the processes. The computed integral cross section by which the experiment has been calibrated are very large – owing to the large values of the critical distance Rth . Summed over all occupied states n∗ one finds σ (q) to be between 24 and 26.9 × 10−16 cm2 for q = 15 to 18.

506

7.5.3

7


HCI Collisions and Ultrafast Dynamics

The attentive reader will have noticed that the over-the-barrier model used here is very similar to the concepts which we have used in Sect. 8.5, Vol. 1 for describing the interaction of atoms and molecules with intense femtosecond laser pulses. Indeed, the electric field strengths which are active in the interaction with highly charged ions are very similar to those in an intensive laser field. Thus, it is not surprising that the ionization and fragmentation patterns observed in the two cases often show astonishing similarities. Also the interaction time in such HCI collisions may be extremely short. By increasing the interaction energy or by focussing onto smaller regions of interaction this time may even be reduced by an order of magnitude or so. Thus, we are discussing here attosecond physics – also a very ‘hot topic’ in state-of-the-art laser matter interaction. However, the interaction pulses induced by HCI are strictly monodirectional – a property of which laser physicists often dream. Nevertheless, one has to state in all fairness that collision physics with HCI does (in contrast to attosecond laser science) not know how to do pump-probe experiments – which are a basic requirement for real time observations of dynamical processes in physics and chemistry. Section summary

• Collision experiments with highly charged ions are an area of particular current interest. The high potential energy inherent in highly charge ions leads to a rich variety of interesting interaction processes. • Electron capture processes are visualized and semi-quantitatively modelled by the classical above-barrier model. In the course of the collision, the strong C OULOMB field of the HCI modifies the potential which the valence electrons the neutral target ‘see’. At a critical interaction distance the C OULOMB barrier is suppressed such that for a short time a quasi-molecule is formed. The n states of the HCI ion which are energetically resonant at that moment to the target states will be occupied. • The physics of these processes closely resembles the dynamics of atoms and molecules in intense, ultrafast laser pulses.

7.6

Surface Hopping, Conical Intersections and Reactions

So far, our introduction into inelastic heavy particle scattering was restricted to relatively small colliding systems (electrons, atoms, ions, small molecules), i.e. to a very small section of relevant bimolecular interactions. Today one is, however, able to investigate by experimental and theoretical methods also reactions of complex, polyatomic systems. Instead of the potential curve crossings and L ANDAU -Z ENER type transitions discussed above, one has to deal in this case with multidimensional potential hypersurfaces and one speaks about “conical intersections” and “surface

7.6

Surface Hopping, Conical Intersections and Reactions

507

hopping”. Such processes do not only occur in (bimolecular) scattering processes but also in isolated, larger molecules and clusters. This intra- and intermolecular dynamics may be studied today for many interesting examples by time resolving methods. For the theoretical description of such processes it has become clear in the past years that the above mentioned conical intersections play a key role for the dynamics of large molecular systems: they describe points (more precisely: hypersurfaces of reduced dimensionality), on which different states are energetically degenerate. For diatomic molecules we have learned that crossings of states with equal symmetry are forbidden. In the multidimensional case, however, such crossings may well occur: on surfaces of reduced dimension – and are even a rather typical phenomenon. For understanding photoinduced reaction dynamics on an molecular level it is indeed crucial to understand transitions between different potential surfaces. Typical chemical reactions may, however, already occur on a single, albeit multidimensional, potential hypersurface. The desire to follow the mechanism of chemical processes on an atomic level has been – from the very beginning of scattering physics – one of the key motives for performing robust and conclusive experiments with crossed molecular beams. Between 1960 and 1990 pioneering work towards this goal has been carried out worldwide in many laboratories, most prominently in the groups of Dudley R. H ERSCHBACH, Yuan T. L EE and John C. P OLANYI and their students. Their work was honoured in (1986) by the N OBEL prize in chemistry. Today a broad variety of experimental and theoretical methods are available to study and describe such processes. Already in his N OBEL lecture, H ERSCHBACH (1987) mentioned more than 500 reviews and estimated the number of research papers on molecular reaction dynamics to exceed 5 000. Ever since then, a steady flow of new concepts, methods and detailed insights has continued to emerge from this field – reaching another culmination with the N OBEL prize for Ahmed Z EWAIL (1999) for his pioneering work on femtochemistry. At this point, we want to conclude our discussion on heavy particle collision physics with just a glimpse on some recent developments of experimental techniques in reaction dynamics. While in earlier times one had to rely for the study of reactive collisions on rather large and mechanically demanding molecular beam equipment, today improved preparation and detection techniques allow to study many questions with small, handy apparatus. Figure 7.29 shows an example for this type of experiments, using a velocity map imaging (VMI) technique, optimized here for the experimental tasks at hand. Key ingredients are a pulsed source for low energetic ions and a molecular target beam, also pulsed. Slow, thermic ions are generated in a very short pulsed neutral atom beam by electron bombardment. Neutral beam and ion beam have well defined kinetic energies and cross at the reaction centre. In this manner both, the relative kinetic energy TCM prior to collision as well as the time at which a collision may occur are well defined, so that a precise time of flight analysis of the reaction products becomes possible. Ion extraction is achieved with the help of (again) pulsed high voltage supplies, and detection occurs in a well defined time window. The scattered ions or reaction products are imaged by a carefully designed and calibrated lens system onto a micro channel plate. The position at

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7


to micro channel plates and CCD e-- gun imaging ion optics pressure gauge extra xtra ction xtrac erepeller Cl− ion source

CH3I supersonic beam

Fig. 7.29 Imaging spectrometer according to M IKOSCH et al. (2006). The key components are a pulsed projectile beam (red dashed), an also pulsed target gas beam (white dashed) and the imaging ion lens system which images the reaction products (pink line) onto a multi channel plate. A special pressure gauge and an electron gun are used to characterize the gas beam and to calibrate the detector

which the ions hit the MCP are registered, position sensitive, via a phosphorescence screen by a computer controlled CCD camera. The time of flight is determined by a pulsed voltage switch, triggering the detector system. From the position at which the ions hit the detector and their time of flight one can derive the velocity components of the scattered ions and reaction products unambiguously. With this apparatus one may investigate, e.g. vibrational and rotational excitation of small molecules in low energy collisions, or reactions of small molecules of the type X− + RY → XR + Y− .

(7.96)

In this case, one ion is exchanged by another ion. One measures the angular and energy resolved differential cross section by the imaging method. As an example for a characteristic result Fig. 7.30(a) shows a 2D plot of reaction probabilities for a so called nucleophilic substitution reaction CH3 I + Cl− → CH3 Cl + I− as experimentally determined by M IKOSCH et al. (2008). The colour shades (red: very high, white: average, black: vanishing) represent the scattering intensity as a function of the x- and y-components of the relative velocity of the scattered I− ions after the reaction in the CM system. At the kinetic energy used here (1.9 eV) the reaction obviously is most efficient when the product ion I− is emitted into the direction of the incoming Cl− ion, which is equivalent to saying that the reaction product CH3 Cl is scattered into the backward direction: reaction (7.96) is found to be a direct process. This may be explained in a suggestive manner by a reaction mechanism as illustrated in Fig. 7.30(b): the Cl− ion attacks the target molecule CH3 I on the side opposite to the I atom. Shown is the potential (red line), calculated ab initio (on the MP2 level), along the so called reaction coordinate. The reaction coordinate characterizes the path from the educts (initial molecules) to the

Surface Hopping, Conical Intersections and Reactions Cl -

0 Cl -

-4.0

I V / eV -0.4 -0.8

uy / m s -1

500 0

CH3 -11.6

-21.6

(a)

-10 0 500 ux / m s -1

1000

(b)

CH3 -12.2

-500 -1000 -1000 -500

I-

Cl

0

-10 kcal mol -1

I-

θ

-20

1000 CH I 3

509

0

7.6

10

R C-I - R C-Cl / 10 -8 cm

Fig. 7.30 Nucleophilic reaction CH3 I + Cl− → CH3 Cl + I− according to M IKOSCH et al. (2008). (a) ‘Velocity map’ for TCM = 1.9 eV measured as a function of the relative velocity in x- and y-direction in the CM system. The white N EWTON rings mark positions of equal energy in the CM system. The largest circle corresponds the maximum kinetic energy after the collision. (b) ab initio calculation of the potential along the ‘reaction coordinate’ RC−I − RC−Cl ; characteristic maxima and minima are given in kcal /mol

final products along a trajectory with locally minimal changes of potential energy (the minimum energy path). One may visualize this concept by a sledge ride which leads you downhill as much as possible, occasionally also uphill but always with minimal energetic effort. In the present case, the system has all together 12 internal degrees of freedom – and the reaction coordinate is determined by the difference between the distances RC−I − RC−Cl . As the potential curve in Fig. 7.30(b) shows, the reaction proceeds through a number of local maxima and minima on the potential hypersurface (so called transition states). Clearly, the geometry sketched here is only one out of many possibilities: different collision parameters may also lead to a successful reaction, and the orientation of CH3 I relative to the approaching Cl− ion is statistically distributed. The preference for I− emission along the (positive) CM axis with essentially maximum kinetic energy possible – as experimentally observed by M IKOSCH et al. (2008) at 1.9 eV – simply says that the orientation indicated in Fig. 7.30(b) is the most efficient geometry for a reaction to take place. We cannot enter here into further details, but mention that this behaviour depends strongly on the initial kinetic energy. Section summary

• In this short section we have discussed the nucleophilic substitution reaction CH3 I + Cl− → CH3 Cl + I− as an example for a detailed study of a reaction process. Advanced experimental methods, with miniaturized molecular beams, VMI methods, and state-of-the-art ab initio calculations of the interaction potential along the reaction path can give deep insights into complex reaction mechanisms.

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7


Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2 in Vol. 1. BO: ‘B ORN O PPENHEIMER’, approximation, the basis when solving the S CHRÖ DINGER equation for molecules (see Sect. 3.2). CC: ‘Close-coupling’, calculations, computation of scattering cross sections by solving (part of) the coupled integro-differential equations (see Sect. 8.1.1). CCC: ‘Convergent close-coupling’, calculations, special solutions of the coupled integro-differential equations for collisions (see Sect. 8.1). CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). CM: ‘Centre of mass’, coordinate system (or frame) (see Sect. 6.2.2). COLTRIMS: ‘Cold target recoil ion momentum spectroscopy’, see Appendix B.4. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. DCS: ‘Differential cross section’, see Sect. 6.2.1. E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. EBIS: ‘Electron beam ion source’, source for highly charged ion beams see Sect. 7.5. EBIT: ‘Electron beam ion trap’, source for highly charged ion beams see Sect. 7.5. ECR: ‘Electron cyclotron resonance’, used e.g. in sources for highly charged ion beams see Sect. 7.5. FC: ‘F RANCK -C ONDON’, introduced an important approximation for optical transition between electronic states (see Sect. 5.4.1). FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects. 6.6 and 5.5.2, Vol. 1, respectively). HCI: ‘Highly charged ions’, see Sect. 7.5. HF: ‘H ARTREE -F OCK’, method (approximation) for solving a multi-electron S CHRÖDINGER equation, including exchange interaction. HOMO: ‘Highest occupied molecular orbital’. JWKB: ‘J EFFREYS -W ENTZEL -K RAMERS -B RILLOUIN’, semiclassical method to determine scattering phases. MCP: ‘Multi channel plate’, electron multiplier with many amplifying elements. MD: ‘Molecular dynamics’, classical trajectory computations for molecular systems. MP2: ‘M ØLLER -P LESSET correction of 2nd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials. ODE: ‘Ordinary differential equation’. QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. RCCC: ‘Relativistic convergent close-coupling’, relativistic version of CCC calculations (including spin orbit interaction). RMPS: ‘R-matrix with pseudo-states method’, advanced quantum mechanical theory for electron scattering.

References

511

SEC: ‘Single electron capture’, see Sect. 7.5.1. VMI: ‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B).

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8

Electron Impact Excitation and Ionization

In the preceding chapter we have given a survey on inelastic processes in general, we have introduced into the theory of excitation by heavy particle collisions and illustrated this by various examples. Now we dare to make a further step onto somewhat more difficult ground, and try to develop a profound understanding for electron impact excitation and ionization. In particular the latter aspect is not only an intellectual challenge but also of great practical importance.

Overview

In Sect. 8.1.1 we develop – on a somewhat abstract level – the general formalism of close-coupling theory (CC). We then return in Sect. 8.2 once again to B ORN approximation as the most simple theoretical approach to electron impact excitation. We present – complementary to optical excitation – the concept of the generalized oscillator strength for e− -atom collisions. Section 8.4 treats electron impact ionization, beginning with integral cross sections which are of particular importance for practical applications. Singly and doubly differential cross sections follow, while finally, triply differential cross sections contain the maximum information about any (e, 2e) process. This is further elaborated in Sect. 8.4.6 with a brief excursion into (e, 2e) spectroscopy, which may be understood as complementary to photoelectron spectroscopy (see Sect. 5.8) in the VUV and XUV spectral region. Finally, in Sect. 8.5 we discuss an example for electron ion recombination – the inverse process to photoionization.

8.1

Formal Scattering Theory and Applications

Readers who are not interested in formal details may well skip this, somewhat ambitious section without concern for understanding the following text.


515

516

8.1.1

8


Close-Coupling Equations

The theory for electron impact excitation, according to the scheme e− + B(i) + T → e− + B(f ) + (T − Wf i )

(8.1)

was first introduced by P ERCIVAL and S EATON (1957) (rather comprehensive reviews on the present state-of-the-art are B URKE 2006; B URKE et al. 2007). Electron impact differs fundamentally from heavy particle collisions in that the scattered electron has velocities which are comparable to those of the electrons inside the target. In contrast, the adiabatic potential approach (used throughout the last chapter) is based on the assumption that the collision proceeds slowly in respect of the internal electron dynamics. Thus, for electrons it may only be used at very low kinetic energies T (if at all). As a general rule the diabatic representation (7.48) is the method of choice for the multichannel electron scattering problem (7.19). This leads to a system of ODEs of the type (7.52) which has to be solved. An additional complication arises from the fact that the scattered electron, let its position coordinate be R, has an electron spin. Thus, in full analogy to the computation of bound atomic states in Sect. 10.2, Vol. 1, the total wave function (7.23) ms the asymptotic wave function must be anti-symmetrized. With the spin function χ1/2 (7.24) is then written as: msi Ψ (R, r) eik i R χ1/2 φi (r) +

ff i (θ, ϕ)

f

eikf R msf χ φf (r). R 1/2

(8.2)

In the framework of RUSSEL -S AUNDERS coupling (i.e. for light atoms) a set of (good) quantum numbers Γ ≡ LM L SM S Ptot

(8.3)

for the total system scattered electron and target remains conserved through the collision. Here L and S are the quantum numbers for total orbital angular momentum and total spin of the system, respectively, while ML and MS are their components with regard to a given z-axis, and Ptot is the parity of the system. The states |Γ of the whole collision system are constructed from the states |Lj MLj Sj MSj of the target atom1 and the scattered electron |j mj s = 12 msj – in the usual notation for the total orbital angular momenta of the target atom and of the scattered electron, respectively – the additional index j refers to the different channels which are considered. The standard rules for angular momentum coupling apply (using the relevant C LEBSCH -G ORDAN coefficients). The partial wave expansion is now somewhat more complicated than (6.82). Instead of the radial equation (6.80), due to the necessary antisymmetrization of the overall electron wave function one obtains a system of coupled integro-differential 1 This implies already recoupling from the standard (LS)J M

also footnote 3.

J

coupling scheme for light atoms; see

8.1


517

equations for the radial wave functions uΓf i (r) of the scattered electron:

f (f + 1) d2 2 Γ − + k f uf i (R) dR 2 R2 ∞ 2μ Γ Γ R, R uΓji R dR . Kfj = 2 Vfj (R)uΓji (R) + 0

(8.4)

j

Here j is the angular momentum of the scattered electron in channel j and kj is its wave vector. On the right hand side of (8.4) one has to sum over all quantum numbers and states j of the chosen set of basis states. The local, so called direct potential VjΓ (R) of the type (7.51) accounts for the C OULOMB attractions and repulsions, averaged over all electrons, while KjΓ (R, R ) represents the nonlocal exchange potentials which are described by rather complicated expressions. Restricted to a finite basis set, equations (8.4) are called close-coupling (CC) equations. In practice, they often are complemented by a suitable set of functions which aim at describing electron correlation. Building on (7.24)–(7.33) the inelastic scattering amplitude ff i (θ, ϕ) for electron scattering from a neutral atom may be expressed by T-matrix elements TΓf i , to be derived from asymptotic solutions of (8.4) corresponding to (7.25). In the LS coupling scheme we have # " 1 2π i −f ff i (θ, ϕ) = i i Li MLi i mi |LM L Si MSi msi SM S 2 kf ki LSPtot i f

# " 1 × Lf MLf f mf |LM L Sf MSf msf SM S 2 × TΓf i Yf mf (θf , ϕf )Y∗i mi (θi , ϕi )

(8.5)

for a transition from the initial state |i = |γi Li MLi Si MSi msi into the final state |f = |γf Lf MLf Sf MSf msf . The projectile electron is characterized by i mi and f mf (orbital angular momenta) and the spin orientation quantum numbers msi and msf , each prior and after the collision, respectively. Even if (8.5) looks somewhat complicated, it just introduces into (7.32) the coupling of the orbital angular momenta (Li,f and i,f ) as well as for the spins (Si,f and 1/2) of atomic and scattered electron respectively. Each again prior and after the collision, i and f , respectively. They form the overall total orbital angular momentum L and the overall total spin S. Both are conserved during the collision. The T-matrix elements TΓf i = SΓf i − δf i describe the transition amplitudes between different partial waves i and f of the scattered electron for each set of conserved quantum numbers Γ . In a partial wave expansion for the elastic case (without electron exchange), as we have discussed it in Sect. 6.4.6, each orbital angular momentum corresponded to just a single value of Γ = , and TΓf i became diagonal according to (6.112).

518

8


The direction of the k i vector of the incoming electron in this notation is given by θi , ϕi with respect to the z-axis. In the standard collision frame, as sketched in Fig. 7.16, the reference axis z points into the direction k i of the incoming electron, while x lies in the scattering plane and points to the side into which the electron is scattered.2 In this coordinate frame θi = ϕi = ϕ ≡ 0 and the projectile electron is characterized prior √ to collision by mi = 0 so that (8.5) is further simplified with Y∗i mi (θi , ϕi ) → (2i + 1)/(4π). Consequently, in this frame ML = MLi so that mf = MLi − MLf and Yf −mf (θ, 0) = (−1)mf Yf mf (θ, 0). Thus, with angular momentum and parity conservation we obtain from (8.5) an important symmetry relation for the scattering amplitude: fMLf MLi = (−1)MLi −MLf f−MLf −MLi .

(8.6)

The differential cross sections are obtained again from (7.33). Note, however, that now the azimuthal angle ϕ enters explicitly into the scattering amplitude, and thus conservation of the scattering plane is no longer compulsive, in contrast to what has been found in the classical treatment of elastic scattering. Whenever the target is prepared or detected in a non-isotropic state, the azimuthal dependence of the cross section is relevant. To measure it requires, however, additional selectivity of the experiment. The individual spin quantum numbers of atom and scattered electron (Sj MSj and 1 m 2 sj , with j = i, f ) may well change during the collision. If spin orbit coupling may be neglected – which is a good approximation for all light atoms – the overall total spin is conserved during collision – with respect to magnitude and direction (S and MS ). However, the components MSi,f and msi,f may well change due to exchange – as we have seen in Sect. 7.2 for excitation of a triplet state out of a singlet, or when the outer electron is exchanged during electron impact with a 2 S1/2 atom.3 One may also study changes of spin direction of the scattered electron directly – in experiments with spin polarized electrons. Such experiments are an interesting and active field of atomic collision physics (the interested reader is referred to K ESSLER 1985; A NDERSEN et al. 1997; A NDERSEN and BARTSCHAT 2003). 2 Alternatively, many good reasons speak for a z-axis perpendicular to the scattering plane, as we shall explain in Sect. 9.3.2 (see also A NDERSEN et al. 1988). 3 We

mention at this point, that the description of the electron scattering process given here – LS coupling of atomic and scattered electrons – implies (tacitly) the so called hypothesis of P ERCIVAL and S EATON (1958) which is an excellent approximation for electron collisions where usually the collision time is short compared to h/WFS – with WFS being the fine structure splitting. Prior to the collision the initial atomic state is (for light atoms) best described in a (Li Si )Ji MJi coupling scheme. This is projected onto an uncoupled basis (Li MLi )(Si MSi ) which in turn is then coupled together with the scattered electron in an (Li i )L(Si s = 1/2)S coupling scheme. After the collision this is recoupled to the (Lf MLf )(Sf MSf ) basis. Finally, this has to be recoupled again into the (Lf Sf )Jf MJf scheme in order to obtain the scattering amplitudes for the final atomic states, e.g. to distinguish singlet and triplet excitation in (7.14) or (7.15).

8.1


Table 8.1 Amplitudes for e− scattering processes at a quasi-one-electron system, considering the electron spin. The spin orientation ±1/2 is indicated by ↑ and ↓, respectively

519

Spins before after collision

Amplitude

msa

MSa

msb

MSb

↑

↑

↑

↑

f1 =f −g

↑

↓

↓

↑

−g = (f 1 − f 0 )/2

↑

↓

↑

↓

f = (f 1 + f 0 )/2

↓

↑

↑

↓

−g = (f 1 − f 0 )/2

↓

↑

↓

↑

f = (f 1 + f 0 )/2

↓

↓

↓

↓

f1 =f −g

Here we can only sketch some basics, exemplified for the still relatively simple case of electron scattering by a (quasi) one electron system with one active electron (Si,f ≡ 1/2). We focus on the relation between the experimentally accessible exchange cross section Iex (θ ) and the scattering amplitudes (8.5). In this case, the C LEBSCH -G ORDAN coefficients describing the spin coupling of atomic and scattered electron, may be pulled out of the summation, and (8.5) is split into a singlet (S = 0) and a triplet amplitude (S = 1), f 0 and f 1 , respectively: #" # " 1 1 1 msf msi 0 1 (8.7) MSi msi 00 ff i = ff i MSi msf 00 2 2 2 2 " #" # 1 1 1 1 1 + ff1i MSf msf 1MS MSi msi 1MS . 2 2 2 2 MS =−1

This expression allows one to compare the results of scattering calculations with quantities that can be measured experimentally. One may, e.g. determine the probability for changing the spin orientation (spin flip) of the scattered electron in a collision with an unpolarized target. Alternatively to the singlet (f 0 ) and triplet (f 1 ) scattering amplitudes one often defines a direct scattering amplitude (f ) and an exchange amplitude (g). For a quasi one electron system the different possible processes and the respective amplitudes are summarized in Table 8.1. These relations follow directly by entering the respective C LEBSCH -G ORDAN coefficients into (8.7). As long as the spin may only change due to electron exchange, these relations are still rather straight forward. However, for electron scattering by heavy atoms the situation gets much more complex: We have seen this in Sect. 7.2 already for the example of mercury. In the radial equation (8.4) one has to include then spin-orbit interaction terms, i.e. the interaction potential now explicitly depends on MS and ML , and also L and S are no longer independent good quantum numbers. The spin of the electron may now change by flipping (and not only by electron exchange). Correspondingly the T-matrix will depend on further parameters and the number of independent scattering amplitudes emerging from (8.5) increases dramatically. Scattering amplitudes may in principle obtain a left-right asymmetry, i.e. depending on spin orientation the amplitudes can differ for ϕ = 0 and ϕ = π .

520

8.1.2

8


Theoretical Methods and Experimental Examples

Elastic, inelastic and ionizing interaction processes of electrons with atoms, molecules and their ions play a key role in many areas of application – and this also holds for photoionization, a related process in regard to the theoretical methods and the experimental detection schemes used. It is thus important to understand these processes in much detail, and to obtain quantitatively reliable cross sections which may also be compared with theory. The significance of such data reaches from astrophysics (gaseous interstellar matter, atmospheres of stars and planets etc.) via the physics and chemistry of our earth atmosphere and its modelling (not the least in the context of the present challenges due to global warming), via plasma physics (e.g. in the context of controlled fusion or intelligent design of energy saving gas discharge lamps) right to the understanding of biologically relevant radiation damage or the interaction of molecules at catalytic surfaces. To think about a complete experimental acquisition of the necessary data is completely unrealistic, considering the multitudes of relevant targets and broad range of energies of interest (from sub thermal to MeV). Thus – in a joint, worldwide effort of a number of strong theoretical groups – over the past decades powerful, quantitative methods have been developed for solving the scattering problem (8.1) in a general way, and efficient computer codes have been implemented. The necessary mathematical and numerical effort is substantial. Collisional problems will probably never be ‘solvable’ in a streamlined manner comparable to today’s state-of-the-art for bound states of atoms and molecules. As discussed in previous chapters, bound state problems, even for large molecular systems, may nowadays be solved on the basis of commercially available quantum mechanical ab initio programmes, routinely and with high precision. In contrast, continuum states as basis for the treatment of scattering problems, or collisional ionization and photoionization are – since the early days of quantum mechanics – a particular challenge which was attacked relatively late – namely by B ORN (1926a,b) with its famous approximation. The special boundary conditions, the large numbers of angular momenta involved, and the – principally – infinite extension of the basis wave functions afford costly approximation methods. For a number of examples we have already illustrated in Sect. 7.2 that modern methods and advanced computer codes have reached a remarkable accuracy and broad applicability. In this development it was of utmost importance that theoretical approaches and results could be tested by selected, special examples which were experimentally relatively easy to access and provided an extensive testing ground for advanced theory. Particular challenges for theory were provided not only by exact measurements of integral and differential cross sections, but also by very specific observables which may today be studied for some selected examples with sophisticated experiments. They allow to test and continually improve modern methods of computation. We cannot enter into details at this point, but we want to introduce some of the few often used terminology in order to give the reader some first orientation on the way into the specialized literature. In each case one has to solve the close-coupling


521

(a)

(b)

exact target spectrum

approximated spectrum pseudo states

continuum states

8.1

finite nuber of bound states

all bound states

ionization threshold

Fig. 8.1 (a) Exact spectrum of an atom or ion and (b) its representation by pseudo-states according to B URKE (2006)

equations (8.4) in a suitable approximation for the boundary conditions (8.2). Towards this aim one writes the wave function (7.23) of the system scattered electron with atom, molecule or ion typically (see e.g. BARTSCHAT 1998) as ΨkΓ (1 . . . N , N + 1) aijΓ k ψjΓ (1 . . . N )R −1 uij (R) + bjΓk χjΓ (1 . . . N + 1), = Aˆ ij

(8.8)

j

with the anti-symmetrization operator Aˆ and the numbers 1 . . . N , N + 1 for the position and spin coordinates of the target and scattered electron, and R stands for the distance of the scattered electron from the atomic nucleus. The atomic orbitals ψjΓ (1 . . . N ) may e.g. be derived from H ARTREE -F OCK orbitals of the target wave functions. The functions χjΓ (1 . . . N + 1) account for correlations which are not contained in the first sum. The radial function of the scattered electron uij (R) in the different channels are the solutions of the close-coupling equations. Their asymptotic behaviour leads, according to (7.25) and (7.26), to the K-, T- or S-matrix. And from the latter one derives scattering amplitudes and cross sections or other, experimentally accessible parameters. The summation over a complete basis set which – in principle – is necessary to obtain the desired parameters, has of course to be limited to a finite number of channels, as indicated in Fig. 8.1. The real art of theoretical atomic scattering physics is to find a suitable, rapidly converging set of basis states. In the simplest case only a small number of states is accounted for (in addition to initial and final states, those for which one assumes the strongest overlap with these). The thus obtained set of close-coupling equations has to be solved at all energies of interest. This procedure will, however, only lead to reasonable results for initial kinetic energies of the

522

8


electron which are not much larger than the excitation energy of the channels enclosed. More sophisticated calculations include in additions the other states by so called pseudo-states, whose characters are supposed to represent as many bound and unbound states as possible, as Fig. 8.1 suggests. By choosing the pseudo-states wisely one may describe, in addition to inelastic processes, also collisional ionization and photoionization quite well – for not too high energies. Depending on the method used to select the pseudo-states and on the method of integration, one speaks of pseudo-state method, of convergent close-coupling calculations (CCC) or of Rmatrix theory. The latter enjoys continuously increasing popularity during the past years as a very efficient tool for the successful computation of a variety of processes. In this approach one divides the whole space into two (more precisely into three) regions. The inner region rn , R ≤ a is selected such that only here electron exchange must be accounted for. At this ‘boundary’ the target wave functions have typically decreased to about 0.1 % of their maximum value. The radial equations of the projectile are solved such that they form a basis which is (as good as possible) orthogonal to the target states. These solutions must obey the R-matrix boundary conditions: a duij uij (0) = 0 and = b. (8.9) u (a) dR ij

R=a

Thus, one uses a constant b for the logarithmic derivation of the radial function on the boundary surface which can be freely chosen (e.g. =0). Inside this boundary the computation has to be performed only once. From this follows the R-matrix, which connects the projectile function inside and outside. Even though outside this boundary one has to solve the radial equations for each( energy of interest, one has to deal here only with long range potentials of the type ∝ Cs R −s ; electron exchange is no longer relevant. Finally, from these solutions one obtains the asymptotic behaviour (third region) uij (R)|R→∞ which leads to the K-, T- and S-matrix. Another approximative method to account for the continuum – applicable in particular for high collision energies – is the introduction of local or non-local polarization potentials. Finally, the so called distorted-wave (DW) approximation derives the T-matrix from the close-coupling equations by using a simple first order approximations for the scattering wave. This may be seen as intelligent expansion (DWB) of the first order B ORN approximation, applied to inelastic electron scattering. The latter will be described in the next section. Several examples for the ability of these modern methods of scattering theory have already been presented at the beginning of Chap. 7. A wealth of data are found in the literature (see e.g. B UCKMAN and S ULLIVAN 2006). We want to end this excursion with two further ‘benchmark’ type systems. A still relatively simple case with a quasi one electron target is electron impact excitation of Na atoms as shown in Fig. 8.2. The resonance doublet 3p 2 P1/2,3/2 is excited out of the 3s 2 S1/2 ground state. For several kinetic energies T of the exciting electron and for a wide range of scattering angles (0◦ to 140◦ ) we may compare here the experimental data for the DCS from different sources directly with theory. The grey dotted line is a CC calculation, into which only the 3 2 S, 3 2 P and 3 2 D


523

10 4

I (θ) / Å2 sr -1

Fig. 8.2 DCS for inelastic electron scattering e− + Na(3 2 S) → e− + Na(3 2 P) according to B RAY et al. (1991). For several initial kinetic energies T experimental data from different authors ( ) are compared with close-coupling calculations (3CC · · · ), also including a non-local polarization potential (3CCO, )

DCS

8.1

10 2

T = 20 eV

T = 54.4 eV

1 10 -2 10 4 10 2

3CC

3CCO

T = 10 eV

T = 100 eV

45° 90° 135° θ

0° 45° 90° 135° 180°

1 10 -2 0°

states enter as H ARTREE -F OCK target orbitals. The red line accounts generally also for all other states (including the continuum) by a non-local polarization potential: one recognizes the improvement. Nevertheless significant differences from the experimental data remain. Considering the fact, that the differential cross sections are shown here on a logarithmic scale, it becomes obvious that, even for this seemingly rather simple case of e− + Na scattering, significant improvement is still desirable. In view of the different experimental results (the error bars are typically given much smaller than the differences between different authors) we tend here to trust the theoretical results. Figure 8.3 documents that this kind of calculations can also provide reliable results for electron molecule collisions, here for the example of elastic and inelastic e− + N2 scattering. Shown is a blow up for a small, particularly interesting section of the excitation function which we have already introduced in Fig. 7.13. We had already mentioned in Sect. 3.6.9 that a short-lived anion N− 2 exists (see Fig. 3.43). Figure 8.3(a) and (b) provides the proof for this by a very impressive resonance structure: each time when the electron energy corresponds to the energy of a vibrational state of this instable anion, one observes pronounced, albeit broad maxima. The FWHM of these structures give an indication of the lifetimes of these states, which amounts to some fs. The scattering process may schematically be described as follows: 2 1 + − e− + N2 X 1 Σg+ v = 0 → N− 2 X Πg v → e + N 2 X Σ g v . (8.10) As discussed already in Sect. 6.5, in addition to resonance scattering a direct process may occur – both together give rise to interference structures. The slightly different energetic positions of the maxima in elastic (a) and vibrationally inelastic (b) are due to the vibrational dynamics in the resonance state. In Fig. 8.3(c) and (d) the angular dependence of the differential cross section is shown. Even without a detailed partial wave analysis the angular distribution is clearly recognized as being essentially of pz type: a consequence of the = 1 wave in the partial wave expansion, that reflects the Π character of the X 2 Πg ground state in the N− 2 anion which is formed

524

8

(a)

1.6

4 3

1.2

I (θ) / Å2 sr -1

(c)

5

1.4

DCS


v=0→0

1.0

v=0→0

2 1

0.8

0 1

2

(b) 0.20

3

0°

4 1.0

v=0→1

45°

90°

135° 180°

(d)

0.8

v=0→1

0.6 0.4

0.10

0.2

0.00

0 1

2 3 electron kinetic energy T / eV

4

0°

45° 90° 135° 180° scattering angle θ

Fig. 8.3 DCS for the elastic (a, c) and inelastic (b, d) electron scattering by N2 in the energetic region of the N− 2 resonance according to S UN et al. (1995). (a, b) energy dependence of the cross sections at θ = 60◦ , (c, d) I (θ) as a function of the scattering angle indicated at the energy indicated in (a, b) by arrows. Experimental data ( ) are compared with CCC calculations — including 15 vibrational channels 2 for a short time. N− 2 (X Πg ) may only be formed by attachment of a pπ type electron onto the ground state X 1 Σg+ of the neutral N2 molecule: what is shown here is a shape resonance. The experiments are compared again with CCC calculations for the electronic ground state of N2 . The CC equations have been solved with 15 vibrational states in the wave function. Rotation is considered to be very slow (adiabatic) in these calculations. Of course, the non-isotropic scattering potential of the molecule complicates these calculations in respect of atoms. In this case, however, it obviously suffices to average statistically over all molecular alignments. Given all these difficulties, the agreement between experiment and theory in this case is indeed impressive!4

Section summary

• The theoretical treatment of low energy elastic and inelastic electron scattering requires a partial wave expansion of the radial wave functions in terms of a suitable set of good quantum number (8.3) for the total system (scattered elec4 More

recent experimental and theoretical data (see e.g. T ELEGA and G IANTURCO 2006, and references given there) show an even slightly better agreement. For the sake of clarity we omit here a comparison.

8.2

B ORN Approximation for Inelastic Collisions

525

tron plus target atom or molecule). This leads to a system of coupled integrodifferential equations (8.4). A judicious choice of states or pseudo-states to be included in these so called closed coupling (CC) equations is the key to their successful solution, predicting differential cross sections and various, even more specific observables. • We have illustrated what today may be achieved with such calculations by comparing experimental and theoretical differential cross sections and their energy dependence for the examples e− + Na and N2 . The agreement is impressive, including in the latter case pronounced low energy scattering resonances.

8.2


8.2.1

FBA Scattering Amplitude

In Sect. 6.6 we have introduced first B ORN approximation (FBA) for elastic electron atom scattering. It can be expanded in straight forward manner to inelastic channels.5 FBA gives a quick, first overview onto the orders of magnitude of inelastic cross sections, while for higher energies it often provides even rather reliable data: In a classical picture high energies correspond to a short interaction time during which the state of the target cannot change very much. In that case one may also neglect exchange processes. Starting point is the fully diabatic expansion of the S CHRÖDINGER equation (7.52) – the close-coupling equations without exchange – which we write 2 2M¯ Vjj (R) + Wj δjj ψj (R), ∇ + kj2 ψj (R) = 2

(8.11)

j

B ORN approximation assumes as 0th order a plane wave in the initial channel |i and zero for all other channels |j : (k)

ψi (R) = ψi (R) = eiki R

and ψj (R) ≡ 0 for all j = 0.

(8.12)

Inserting this into the right hand side of (8.11) only one term j = j remains and one obtains for each final, inelastic channel |f 2 2μ ∇ + kf2 ψj (R) = 2 Vf i (R)eik i R . 5 For

(8.13)

a detailed description we refer to the excellent review by I NOKUTI (1971), which even today after more than 40 years has not lost any of its validity and generality.

526

8


Using G REEN’s function one follows essentially the same procedure as outlined for the elastic case by (6.64)–(6.68). Substituting in (6.68) the plane wave exp(ik i · R) (+) for ψi (R) we obtain, in analogy to (6.136), the inelastic scattering amplitude in FBA for exciting the state |f out of the state |i (FBA)

ff i

(θ, ϕ) = −

¯ M¯ (FBA) M/m e (FBA) = − , T T f i 2π2 2πEh a02 f i

(8.14)

with the reduced mass M¯ of the system and the transition matrix element (FBA) Tf i = k f |Vf i R |k i = e−ik f R Vf i R eiki R d3 R = eiKR Vf i R d3 R .

(8.15)

As in the elastic case, B ORN approximation has again axial symmetry – in contrast to the general solution, for which the wave vectors of the electron before and after collision, k i and k f , respectively, define a symmetry plane. Thus, the inelastic scattering amplitude in FBA is again characterized by the momentum transfer K from the scattered electron to the target, which depends on the scattering angle θ : K = ki − kf

with K =

and kf2 − ki2 =

kf2 + ki2 − 2kf ki cos θ ,

2μ 2μ (Wf − Wi ) = 2 Wf i . 2

(8.16) (8.17)

Obviously, (8.15) and (8.16) are only slightly more complicated than (6.140) and (6.142) for the elastic case. The potential matrix element Vf i (R) replaces now the classical potential V (R). We may further simplify the inelastic T-matrix element (8.15) by inserting (7.51) with (7.50) for the interaction potential: e2 = Tf(FBA) i 4πε0

eiK·R

(8.18)

N × φf (r 1 . . . r N )

n=1

Z 1 − φi (r 1 . . . r N ) d3 R . |R − r n | R

Due to the orthogonality of the atomic wave functions φj the interaction Z/R with the atomic nucleus does not contribute. For the remaining double integral we invert according to B ETHE the sequence of integration over scattering coordinate R and internal coordinates r i and use the so called B ETHE integral (here without proof)

eiK·r 3 4π d r = 2. r K

(8.19)

8.2


527

With this we may rewrite

eiK·R d3 R = eiK·r n |R − r n |

eiK·(R −r n ) 3 4π iK·r n d R = 2e . |R − r n | K

Inserting atomic units, e2 /(4πε0 ) = Eh a0 , into (8.18) we finally obtain the FBA T-matrix for the transition from |i to |f as a function of momentum transfer6 (FBA) Tf i (K) =

4πEh a03 4πEh a03 iK·r n φ | e |φ = Ff i (K) f i (Ka0 )2 (Ka0 )2 n

where we have abbreviated the above matrix element Ff i (K) = φf | eiK·r n |φi =

(8.20)

(8.21)

n

φf∗ (r 1 . . . r N )

eiK·r n φi (r 1 . . . r N )d3 r 1 . . . d3 r N .

n

This matrix element Ff i (K) is called inelastic atomic form factor. For f = i it is identical to the atomic form factor (1.97) in Vol. 1 which we have encountered (FBA) there in the context of X-ray diffraction. Ff i (K) is dimensionless, while Tf i (FBA)

and ff i have the dimension Enrg × L3 and L, respectively. As we shall see in Sect. 8.3.2, Ff i (K)/K is directly comparable to what we have called transition matrix element in a radiation field, specified in Appendix H.1.6, Vol. 1: its dimension is also L, and we shall see that eK = K/K effectively assumes the role of the unit linear polarization vector elin .

8.2.2

Cross Sections

∼ ¯ In the following, we restrict the discussion to electron impact excitation (M/m e= 1), but note that one may also treat high energy ion impact in essentially the same manner. We also assume that all quantities are given in a.u. to further simplify the writing. With (6.73) the DCS is then in compact form (see also footnote 11 in Chap. 6): (FBA)

dσf i

(θ, ϕ)

dΩR 6 For

=

2 4kf 1 1 kf (FBA) Ff i (K)2 . (8.22) Tf i (k f , k i ) = 2 4 ki K (2π) ki

larger atoms and molecules with closed shells, the summation in the matrix element ( φf | n eiK·r n /K|φi needs to be carried out only over the active electrons. The interaction with the core potential can be summarized by an effective core potential V (core) (R), which in FBA drops out due to the orthogonality of the φj .

528

8


For an isotropic target the differential cross section does not depend on ϕ and the scattering angle θ is determined exclusively by the magnitude of the momentum transfer K according to (8.16). The integral inelastic cross sections σf i is obtained as usual by integration . . . d(cos θ ). As in the elastic case (Sect. 6.6) we change variables using (8.16); we note that dK 2 = 2KdK = ki kf (2π sin θ dθ )/π = ki kf dΩR /π and write (θ ) dσf(FBA) i dK 2 (FBA)

dσf i

dW

(θ )

=

2 4π 1 2 1 (FBA) Tf i (K) = 2 4 Ff i (K) 2 4πki ki K

=

2 π 1 Ff i (K) . T W2

or

(8.23)

(8.24)

We have inserted T = ki2 /2 and the so called energy transfer WK = K 2 /2 (see footnote 22 in Chap. 6). When integrating over dW the integration limits change to WKmax

2 Wf i Wf i 1 Wf i and WKmin 1+ , 4T 1 − 2T 4 T 2T

(8.25)

respectively (obtained from expanding (8.16) for Wf i T ). With (8.16) one finally obtains 2 π Wmax 1 (FBA) Ff i (K) dW. = (8.26) σf i 2 T Wmin WK The above considerations are based essentially on B ETHE (1930), who also evaluated the integral Ff i (K) explicitly for several excited states of the H-atom. A high energy approximation in general form is given by the so called B ETHE formula, (now reintroducing a.u. explicitly): σf(Bethe) i

8.2.3

2πa02 T A ln +B . = T /Eh Eh /2

(8.27)

B ORN Approximation and R UTHERFORD Scattering

We add here a note on a peculiarity of the differential cross section in B ORN approximation according to (8.22) and (8.24). These relations are obviously the product of (i) a pure C OULOMB part according to (6.145) and (6.146), respectively, which simply describes RUTHERFORD scattering of the projectile electron by a free electron (q1 = q2 = 1), and (ii) the square of the absolute value of an inelastic atomic form factor Ff 0 (K), as one realizes by comparing (8.20) with (1.97) in Vol. 1. This inelastic atomic form factor thus modifies the pure C OULOMB repulsion of two free electrons corresponding to the electronic density distribution in the initial and final state of the target. Without this complication by initial and final state one might

8.2

B ORN Approximation for Inelastic Collisions σint / Å2

FBA

529 e- + Na (3 2P←3 2S) BE scaled

30

CCC

20 R matrix

10

3 2D←3 2S 0 1

W th

10

100

1000

T / eV

Fig. 8.4 Integral cross section for electron impact excitation e + Na(3 2 S) → e + Na(3 2 P) and → e + Na(3 2 D) according to L IN and B OFFARD (2005). Compared are as a function of collision energy T experimental data from different sources (◦, •) with CCC ( ), R-matrix theory ( ), B ORN approximation (– – –) and BE scaled (see text) B ORN approximation (—). At high energies the B ETHE formula (pink) is also displayed – nearly identical with FBA

even think of a method for determining the momentum distribution of the electrons in the target. We shall, however, come back to this idea in the context of collisional ionization in Sect. 8.4.6.

8.2.4

An Example

The B ETHE formula (8.27) and suitable modifications are used very successfully, e.g. in radiation chemistry to determine the stopping power of electrons, protons, and alpha particles (see e.g. B ERGER et al. 2005). Quite generally one may say that the B ORN approximation – albeit strictly applicable only for high energies T Wf i = (Wf − Wi ) – often gives a reasonable idea about the general behaviour of excitation functions, and allows for simple estimates of processes without exchange. If one knows the atomic or molecular states sufficiently well (today many reliable programme packages provide such data), one may evaluate (8.22) with moderate effort. For high energies, integral excitation cross sections found in this way approach the B ETHE formula. To obtain an impression of the effectiveness of FBA, we have again a look at electron scattering by atomic sodium. Figure 8.4 shows the experimentally determined integral cross section for 3P ← 3S and 3D ← 3S excitation. The experimental data (somewhat older and not in full agreement with each other) are compared with B ORN approximation, convergent close-coupling (CCC) and R-matrix calculations. As seen, FBA reflects the general trends and agrees with experimental data for high kinetic energies where the more sophisticated theories can no longer be applied. Interestingly, the optically forbidden 3D excitation – which is about a factor of 10 weaker – is reasonably well reproduced by FBA even close to the excitation maximum (probably due to the fact that the perturbation is much smaller than for

530

8


3P excitation). We note that the B ETHE formula (8.27) agrees in this case very well with FBA over the whole high energy region. According to K IM (2007) there are good reasons to re-scale the B ORN approximation for optically allowed transitions by (BE)

σf i

=

T (FBA) σ . T + Wf i + WI f i

(8.28)

Here WI is the ionization energy (= −binding energy) of the active electrons. This so called BE scaled approximation reduces the overshooting maximum of the B ORN approximation without changing the high energy behaviour; drawn in Fig. 8.4 as full black line. It matches the experimental data astonishingly well. This finding has been confirmed for a number of cases. Section summary

• B ORN approximation (more precisely FBA, see Sect. 6.6) for elastic scattering is readily extended to electron impact excitation (inelastic collisions). Using again a plane wave as 0th order approximation for the free electron, one obtains from the diabatic representation of the S CHRÖDINGER equation (without exchange) the rather plausible expression (8.15) for the transition matrix elements. • The differential excitation cross section (8.22) for a transition f ← i in FBA turns out to depend (apart from a flux factor) only on the momentum transfer

K = |k f − k i | = kf2 + ki2 − 2kf ki cos θ . • The integral cross sections in FBA represent the trends with initial kinetic energy reasonably well: they rise from threshold to a maximum at a few times the excitation energy, and then decay slowly, following essentially the B ETHE formula (8.27), a simplified version of FBA. While the maxima of the cross sections are typically over-estimated for optically allowed excitations, one obtains rather good agreement with experiment for weak, optically forbidden transitions. • Further improvement is obtained by judiciously rescaling of the B ORN cross sections.

8.3

Generalized Oscillator Strength

8.3.1

Definition

The B ORN approximation facilitates a direct comparison of collisional and optical excitation: We note that the operator exp(iK · r) active in the FBA scattering amplitude (8.21) resembles the transition operator for the electric field (see Eq. (H.22) in Vol. 1). The key difference is the missing oscillatory prefactor which leads to exclusively resonant excitation by irradiation with an oscillating electromagnetic field. Inelastic electron scattering thus acts similar to broadband electromagnetic radia-

8.3

Generalized Oscillator Strength

531

tion (‘white light’). This is born out in mathematical terms by defining the generalized oscillator strength (GOS) – in full analogy to the optical oscillator strength (5.27), Vol. 1. It was first introduced by B ETHE (1930) (for details see e.g. I NOKUTI 1971): ff(GOS) (K) = i or

=

2 2Wf i Ff i (K) 2 2me Wf i Ff i (K) = Eh Ka0 2 K 2

(8.29)

2 2Wf i K 2 (FBA) (K) T f i (4π)2

(8.30)

in a.u.

The GOS is again dimensionless, and determined by the excitation energy Wf i = Wf − Wi and the matrix element Ff i (K) according to (8.21). The differential, inelastic excitation cross section (8.22) may then be written as (FBA) (GOS) (GOS) dσf i (θ, ϕ) 2a02 Eh kf ff i 2a02 Eh T − Wf i ff i = = . (8.31) dΩ Wf i ki (Ka0 )2 Wf i T (Ka0 )2 We see again that in first B ORN approximation collisional dynamics does not depend directly on kinetic energy T or scattering angle θ : both enter only through the momentum transfer K into the final result (apart from the flux factor kf /ki which at high energies differs little from one). B ETHE (1930) showed that a generalized sum rule holds for the GOS, (GOS) ff i =N b

– in close analogy to the T HOMAS -R EICHE -K UHN sum rule (5.28), Vol. 1.

8.3.2

Expansion for Small Momentum Transfer

To show explicitly the close relation between the thus defined GOS and the standard optical dipole oscillator strengths according to (H.34) in Vol. 1, we expand in (8.29) the matrix element Ff i (K) from its definition (8.20) into a power series of K · r (for small momentum transfer Ka0 1). For simplicity and clarity we consider here only one electron systems (such as H, Na, K etc.) and express again all physical quantities in a.u. (i.e. energies in Eh , lengths in a0 , wave vectors in a0−1 ): (GOS)

ff i

2 2Wf i φf (r)1 + iK · r − |K · r|2 /2 + · · · φi (r) 2 K 2 K 2 = 2Wf i φf (r) eK · r + i |eK · r| − · · · φi (r) . 2

(K) =

(8.32)

In the second line we have exploited the orthogonality of the wave functions φj . The first term of this sum may directly be compared to the standard (optical) oscillator strength (H.34), Vol. 1 for one specified transition. The unit momentum transfer vec-

532

8


tor eK = K/K obviously plays now the role of the polarization vector e in optical excitation. However, in contrast to optical excitation where the higher order terms of the expansion (H.24), Vol. 1 can be neglected as a consequence of the large optical wave length (k · r 1), the magnitude of the momentum transfer in electron impact excitation is of the same order of magnitude as atomic dimensions and |K · r| cannot be neglected a priori in the series expansion (8.32). To evaluate the matrix elements f |(K · r)n |i, the quantization axis z is conveniently chosen parallel to K, so that K · r = Kz = Kr cos γ .7 This leads to a selection rule M = 0 for electron impact excitation in FBA – in full analogy to the corresponding optical transitions with linearly polarized light. We may rewrite (8.32): 2 2 (GOS) ff i = 2Wf i f |z|i + K 2 f |z2 |i + 2f |z|if |z3 |i · · · (opt) = ff i − f2 K 2 + f4 K 4 + O K 6 . (8.33) Obviously this is some kind of multipole expansion into a power series of K 2 , with (opt) the first term being the optical dipole oscillator strength ff i for linearly polarized light with polarization vector e = eK . The second term contains a contribution from quadrupole transition moments and a product of dipole and octupole moments, while the forth term is mainly determined by the octupole transition moment. We thus find a nearly perfect analogy to the electromagnetically induced excitations. The key difference is the broad band character of the excitation already mentioned and the fact that the momentum transfer vector K, which replaces here the wave vector of the photon cannot be neglected. Rather, according to (8.16) it may become substantial. Hence, with electron impact one may observe nearly arbitrary types of transitions which may be strictly forbidden for optical excitation, such as e.g. octupole transitions (see e.g. H ERTEL and ROSS 1969). For high collision energies T differential cross sections typically are strongly forward peaked, as one reads from their dependence on 1/K 2 according to (8.31). Thus, in many practical situations very few terms of the power series (8.33) are sufficient to describe small angle scattering rather well. In practice on determines usually the effective GOS from the measured differential cross sections by inverting (8.31). One may extrapolate it then towards vanishing momentum transfer K and compare with the usually well known optical oscillator strengths. Note, however, that according to (8.25) the limiting value K = 0 is experimentally not accessible for inelastic collision. Electron energy loss spectroscopy (EELS) is a powerful technique to study atoms, molecules and solid-state systems. As mentioned above, a monochromatic electron beam may spectroscopically be used as a white light source. Electron beams with sufficiently narrow energy spread and careful energy analysis of the scattered electron allow one to determine the energy loss Wf i in (8.33), and hence to obtain 7γ

is the polar angle of the position vector r of the target electron with respect to K – not to be confused with the scattering angle θ contained in (8.16) defining K.

8.3

Generalized Oscillator Strength (GOS)

ffi

1.0 0.8 0.6

533

e- + Na(3s) → e- + Na(3p) 100eV 40eV 20eV

(opt)

10eV

0.4

ffi - f2K 2 +f4K 4 FBA

0.2 0

10-3

0.01

0.1 1 (Ka 0) 2

10

Fig. 8.5 Generalized oscillator strength for the e− + Na(3s) → e− + Na(3p) excitation as a function of the square of the momentum transfer, K 2 . The 3CCO differential cross sections (Fig. 8.2) of B RAY et al. (1991) have been transformed into generalized oscillator strengths ( ) and are compared with FBA — and the series expansion of FBA for very small K 2 - - -

spectra which are essentially equivalent to optical absorption spectra – their quantitative analysis being based on (8.1). The foundations for EELS have been laid by L ASSETTRE et al. (1968) and others, and a recent overview on the present status of experiment and theory is found in TAIOLI et al. (2010). Even though optical spectroscopy is superior in the VIS and near UV spectral range with respect to spectral resolution, electron spectroscopy (including EELS, PES and AUGER electron spectroscopy) becomes compatible in the high energy regime. In particular in the VUV, XUV and X-ray region where laser sources are not readily available, EELS offers a relatively inexpensive alternative to synchrotron radiation. One interesting feature is its sensitivity for optically forbidden transitions: while their cross sections disappear for K 2 → 0, they may become quite significant for larger momentum transfer. From a careful study of the generalized oscillator strengths as a function of K 2 one may directly infer the type of transition studied.

8.3.3

Explicit Evaluation of GOS for an Example

Figure 8.5 presents as an example electron impact excitation of the 3s 2 S → 3p 2 P transition in the Na atom. The limits of B ORN approximation become obvious here in comparison to the (essentially) exact scattering calculations. For comparison with (8.29), the theoretical data of the so called coupled-channel optical method (3CCO) from B RAY et al. (1991) have been recalculated according to (8.31) into GOS. Alternatively we show three terms of the series (8.33), as used typically in electron spectroscopy. The evaluation of (8.20) with realistic wave functions is described in Appendix A as a realistic example. Figure 8.5 shows again that the B ORN approximation overestimates the GOS (i.e. the scattering cross sections) for low energies, but nevertheless reproduces the trend as a function of the scattering angle roughly correct. (It is unclear how reliable the

534

8


3CCO calculations really are for very small K 2 . It may well be that in the limit K → 0 the numerical uncertainties of the Na wave functions used distort the results somewhat.)

8.3.4

Integral Inelastic Cross Sections

The computation of integral inelastic cross section too, may now be pushed one step further. If one inserts into (8.26) the generalized oscillator strengths according to (8.29), one obtains (FBA)

σf i

=

πa02 (Wf i /Eh )(T /Eh )

Kmax a0

Kmin a0

1 (GOS) f d(Ka0 ). Ka0 f i

(8.34)

(GOS)

Expanding ff 0 for small momentum transfer with (8.33) and inserting the limits of integration according to (8.25), one obtains a specialized form of the B ETHE formula (8.27) for practical applications: (Bethe) σf i

πa02 T (opt) + B(T ) . f = ln (Wf i /Eh )(T /Eh ) f i Eh /2

(8.35)

Section summary

• One introduces a generalized oscillator strength (8.29) for electron impact excitation. This concepts allows a direct comparison with optically induced transitions. • For not too large momentum transfer K, the GOS may be expanded into a series of powers K 2n , the prefactors being essentially the squared dipole, quadrupole, octupole etc. moments. Thus, for larger K i.e. scattering angles, optically forbidden transitions may be accessed. • Based on this concept, EELS has been developed to a powerful spectroscopic method for atoms, molecules, clusters and solid-state materials.

8.4

Electron Impact Ionization

One of the most challenging problems of AMO collision physics is impact ionization of an atomic or molecular target Tg by electron impact, short (e, 2e), more specific Tg(e, 2e)Tg+ process,8 and in some more detail: 8 We focus again on electron impact only, for clarity and also because the richest experimental and theoretical data are available for these processes. We note, however, that fast ion collisions may be treated in a very similar manner, as already mentioned for excitation processes. The kinematics, of course, is quite a bit more complicated.

8.4


(a)

e-

535 kB

(b)

WA

z x

ħω

e-

e-

T

y

qI = k 0 - k B - k A

- φB θB θA

Wba WI

θK

k0 kA

K

0 =k

-k

A

Fig. 8.6 (a) Energetics and (b) kinematics of the (e, 2e) process: energies and wave vectors (momenta) prior (T and k i ) and after ionization for the primary (WA and k A ) as well as the secondary electron (" and k B ) according to (8.37) and (8.39); for comparison in (a) on the left, the energetics for a photoionization process is also indicated (photon energy ω)

e− (T , k i ) + Tg → e− (WA , k A ) + e− (WB , k B ) + Tg+ (γj, q I )

(8.36)

with T ∼ = WI + WA + WB

(8.37)

or WB + WI ≡ " + WI ∼ = T − WA

(8.38)

and k i = k A + k B + q I .

(8.39)

In comparison to the photoionization process, discussed in Sect. 5.5.1, Vol. 1, the complexity of impact ionization is obvious.9 Already in the most simple case, the hydrogen atom, we have to treat a genuine three body problem, which in general cannot be solved in closed form – neither in classical nor in quantum mechanics. We denote the momentum (wave vector)10 and energy of the projectile electron prior to collision with k i and T , respectively, while the momenta and energies of the − two electrons e− A (and eB ) after the ionization process are k A (and k B ) and WA (and WB ≡ "), respectively. In general, the ionization energy WI (=−Wn , the binding energy of the electron in the initial state) depends also on the specific electron j which is ejected from the atom (or molecule) as well as on the state γ in which the ion Tg+ remains after the process. Figure 8.6 illustrates energetics and kinematics – not a completely trivial situation. Only a complete analysis of all parameters for both free electrons after the ionization process can lead to a full understanding on an atomic level. Usually one distinguishes pragmatically between a scattered primary electron (energy WA ≥ ", momentum k A ) and the emitted secondary electron (energy ", momentum k B ) – but we have to keep in mind that, from a strictly quantum mechanical view point, both electrons are indistinguishable. They are both free after the ionization process and share the excess energy T − WI according to (8.37). The energy comparability with photoionization, in the following we shall mostly use the letter " ≡ WB to denote the energy of the ejected electron.

9 For 10 In

the following we shall often use wave vector k and momentum k as synonyms.

536

8


partitioning is one of the key parameters determined by the dynamics of the ionization process. The sign in (8.37) indicates that we have neglected the recoil energy which is transferred onto the target ion Tg+ – which is finite, albeit very small in comparison to that of the electrons due to the large mass ratio MTg /me 1 between target and electron. Still the momentum balance (8.39) can always be satisfied by the recoil momentum q I = k i − k A − k B which is transferred to the Tg+ ion. The total momentum transfer K = k i − k A from the projectile to the target atom + is shared between the secondary electron e− B and the remaining ion Tg . As shown − in Fig. 8.6, for each scattering angle θA of the electron eA the secondary electron e− B may exit the interaction region in principle at arbitrary polar and azimuthal angles θB , ϕB . The problem poses also a substantial challenge for the experiment. Not only the integral ionization cross section has to be measured – which for applications is very important – one also has to determine the probability of final angles and kinetic energies for both electrons. Depending on the degree of detail detected, one distinguishes the single-differential ionization cross section (SDCS), double-differential (DDCS) and triple (TDCS) differential cross section, SDCS =

dσ , dW

DDCS =

and TDCS =

d2 σ , dW dΩ

d3 σ , dWA dΩA dΩB

(8.40) (8.41)

while the integral ionization cross section is related to the SDCS by σ (ion) = 0

T −WI

dσ dW. dW

(8.42)

The SDCS only specifies W , the energy of one scattered electron after the ionization process; the missing index indicates that we do not know which of the two electrons (with energies WA or WB = ") is detected. The DDCS specifies in addition the scattering angle Ω of the detected electron – again without recording anything about the second electron. Finally, for determining the TDCS one has to register the − scattered electron e− A in coincidence with the ejected electron eB , thereby recording both scattering angles. In addition, the energy WA of one of these electrons is determined (and thus also that of the other free electron WB = T − WI − WA ). A full theoretical description of this complex process has to follow the trajectories of both electrons over large distances – if one takes a classical point of view. Quantum mechanically the asymptotic wave functions of both receding electrons must be computed in the long range C OULOMB field of the finally remaining ion. The impact ionization problem was already attacked by theory in the first half of the past century, notably by B ETHE (1930). But only in the its last three decades a comprehensive picture has emerged, based on concentrated efforts with highly

8.4


537

selective experiments and the development of powerful theoretical methods. Today one understands the dynamics of collisional ionization in its essential aspects quite well, and may also use this knowledge for important applications. Again, we can only illuminate some key aspects and refer the interested reader to the rich literature on the subject (good access is provided e.g. by the reviews of C OPLAN et al. 1994; M C C ARTHY and W EIGOLD 1991; B YRON and J OACHAIN 1989; E HRHARDT et al. 1986; I NOKUTI 1971, and the references given there).

8.4.1

Integral Cross Sections and the LOTZ Formula

In a first step we want to obtain an overview about integral cross sections for impact ionization – integrated over all scattering angles and energy distributions of the pri(ion) mary and secondary electron. They are called total, σtot , if they are summarized also over all electrons of the target (which is usually the case if no further selection occurs). For modelling plasmas of various kinds, these very total cross sections as a function of initial kinetic energy are of utmost importance. It is important to know them for many atomic and ionic species. Well kept and documented data collections are found e.g. at NIFS and ORNL (2007). To use them efficiently (e.g. for plasma modelling) in the past years various quasi-empirical relations have been developed. As a rule, starting point is the B ETHE formula (8.35) for the Nj electrons of all atomic or ionic subshells j of a target with the respective ionization potentials WIj , which replace the excitation energies Wf i in (8.35): * (ion) σj

=0 ln(T /W ) ∝ WIj TIj

for T < WIj for T ≥ WIj .

(8.43)

We recall here that the B ETHE formula, which in turn is based on the B ORN approximation,is distinctively a high energy approximation. If one wants to improve it for low energies T one has to add further empirical parameters fitted onto the available experimental material. A generally used standard is still the formula, developed by L OTZ (1967, 1968, 1970): (ion)

σtot

=

N j =1

a j Nj

ln(T /WIj ) 1 − bi exp −ci (T /WIj − 1) . T WIj

(8.44)

Corresponding to (8.43) each term j contributes only for T ≥ WIj . One easily verifies that the L OTZ formula reproduces a linear behaviour ∝ (T − WIj ) in the threshold region (T WIj ), while at higher energies it approaches the B ETHE formula (8.43). The parameters aj are typically between 2.6 and 4.5 × 10−14 cm2 eV2 and are for each atom and each shell individual constants (the same holds for bj and cj ). The parameter values, tabulated by L OTZ for many atoms and ions, are however

(ion)

σ tot / 10-17 cm2

5 4 3 2 1 0

10

13.6 eV


e- + H → 2e- + H+

25

1s: a = 3.9 b = 0.56 c = 0.45 WI =13.6 eV

20

medium WT

6

low WT

8

threshold region

538

high WT > 10 WI

100

1000

15 10 5 0 10

e- + Ar → 2e- + Ar+ 3s: 2a = 7.2 b = 0.69 c = 0.0 WI = 29.2 eV 3p: 6a = 28 b = 0.61 c = 0.16 WI =15.76 eV 100 15.8 eV

1000


Fig. 8.7 Total ionization cross section as a function of kinetic energy T . (a) e− + H: experimental data ( ) from S HAH et al. (1987) and CCC calculations ( ) according to BARTSCHAT and B RAY (1996). (b) e− + Ar: experimental data from K RISHNAKUMAR and S RIVASTAVA (1988) ) according to ( ) and S OROKIN et al. (2000) ( ), respectively, as well as SCOP calculations ( V INODKUMAR et al. (2007). The experimental data are approximated by the L OTZ formula (—–) with fit parameters a, b, c for the contributing shells as noted in the legend

based on older experiments and have to be refitted if necessary to the most recent experimental data. We show in Fig. 8.7 as examples the total ionization cross sections for atomic hydrogen and argon together with more recent measurements and calculations which we have fitted by the L OTZ formula.11 The typical dependence of ionization cross sections on kinetic energy T is rather similar to that for electron impact excitation, except for the fact that in the threshold region it depends nearly linearly on energy, cf. (7.16) and (7.18). Beyond the maximum (at typically 4 to 5 times threshold energy) the cross sections fall essentially ∝ ln(T /WIj )/T as predicted by the B ETHE formula. The agreement between theory, experiment and L OTZ formula is very good for the H atom. Helium is considered a benchmark case for (e, 2e) processes. With WI = 24.6 eV the integral ionization cross section peaks at ca. 120 eV (not shown here). Very accurate CCC(469) calculations (supported by several sets of experimental data) are believed to be correct to within a few percent (B RAY and F URSA 2011). We shall refer to He again in the context of SDCS, DDCS, and TDCS. For argon some further clarification is still needed: while measurements and calculations agree rather well above ca. 200 eV, at low energies the experimental data are higher than the optimal fit by the L OTZ formula, while a quite recent theory of V INODKUMAR et al. (2007) with a spherical, complex optical potential, SCOP, are too low. Obviously the ionization probability for the weaker bound 3p 6 electrons (WI = 15.8 eV) is underestimated. 11 These experimental data do, however, not allow to test the theoretical prediction for the threshold

law ∝ (T − WI )1.127 , which anyhow holds only in a very narrow energy range above threshold (see Sect. 8.4.3).

8.4


8.4.2

539

SDCS: Energy Partitioning Between the Electrons

The next question concerns the energy balance of the ionization process (8.36): how is the excess energy T − WI = WA + WB shared among primary and secondary electron? One electron is detected after ionization and its kinetic energy W is measured – which is either WA or WB = ". According to M OTT (see e.g. RUDD 1991; K IM 1975a), a first understanding may be gleaned from the RUTHERFORD cross section (6.146) for the interaction of projectile and ejected electron (qA = qB = 1). What is there called energy transfer WK corresponds here to W + WI . Thus, the SDCS will be ∝ (W + WI )−2 . However, since both free electrons are in principle indistinguishable this relation has to be symmetrized and one derives with (8.38) from (6.146) the so called M OTT cross section:

1 1 1 dσ (M) πa02 Eh2 = . + − dW T (W + WI )2 (T − W )2 (W + WI )(T − W )

(8.45)

The last term accounts for interference between direct (RUTHERFORD) and exchange amplitude. It turns out that already this somewhat hand waving guess explains the experimentally observed trends qualitatively rather well. RUDD (1991) has modified the M OTT formula and parameterized it in a semiempirical manner, so that the behaviour of the (angle integrated) single-differential cross sections (SDCS) for electron impact ionization of H and He is well described at higher energies: dσ (R) S = F (t)f1 (w, t) dw WI / eV 2 A1 ln t + A2 + A3 /t 2 Eh and , F (t) = with S = qI π(a0 ) WI t f1 (w, t) =

(8.46)

1 1 1 + − . (w + 1)n (t − w)n (w + 1)n/2 (t − w)n/2

Reduced energies w = W/WI and t = T /WI have been introduced here. For the e− +He →2e− +He+ RUDD (1991) uses the values n = 2.4, A1 = 0.85, A2 = 0.36, and A3 = −0.1. As documented by Fig. 8.8(d) this formula fits nicely the somewhat older data of O DA (1975) for the SDCS at 500 eV. Using the same parametrization, (8.46) agrees remarkably well with the more recent experimental data from S CHOW et al. (2005) close to the threshold region and above. The CCC calculations show significantly more variation than the simple M OTT formula (8.45); nevertheless, the experimental errors do not permit a clear decision about their validity. Characteristic for these energy distributions is the symmetry with respect to W = (T − WI )/2. For very low excess energies (T − WI ) WI the distribution is nearly constant, while at very high energies the distributions become essentially bimodal: for T WI one may indeed speak about a primary electron (with nearly maximum energy) and one (very low) energy secondary electron. More recent experimental

540

8 5×10-2

(a)

5×10-2

26.3 eV

1×10-2 SDCS / Å2 eV -1


(c)

40.7 eV

1×10-2 0.0

5×10-2

0.02

(b)

0.04

0.06 0.0691

0.0 10-1

28.3 eV

0.2

0.4

(d)

500 eV

5

10

0.6 0.6545

10-2 10-3 1×10-2

10-4 0.0

0.05

0.10

0.15

0

15

19.32

W / WI

Fig. 8.8 dσ/dw (SDCS) for the He(e, 2e)He+ process (log-lin scales) as a function of reduced electron energy w = W/WI after ionization. Experimental data (◦) and CCC calculations ( ) from S CHOW et al. (2005) at low primary kinetic energy and at 500 eV from O DA (1975) are compared with the modified M OTT /RUDD formula (8.46) ( )

data confirm this behaviour also for intermediate initial kinetic energies (see e.g. B RAY et al. 2003). Recent measurements for other atoms at intermediate energies indicate a somewhat more complex behaviour (YATES and K HAKOO 2011). Advanced theoretical efforts by Z ATSARINNY and BARTSCHAT (2012) show even for He significant structures and good agreement with experimental data as illustrated in Fig. 8.9. The RUDD formula is still surprisingly accurate – but of course it does not reproduce the wiggly structure of the sophisticated BSR theory.

8.4.3

Behaviour at the Ionization Threshold

Hyperspherical Coordinates The Hamiltonian for the fundamental three body C OULOMB problem, i.e. for H(e, 2e)H+ but also for H− and H+ 2 , is given in a.u. by = − 1 − 2 − 1 − 1 + 1 . H 2 2 r1 r2 r12

(8.47)


SDCS / 10-18 cm 2 eV - 1

8.4

3

0

541

10

2

W / eV 20 30 40 50 experiment BSR theory Rudd formula

60

70

1

0

0

0.5

1 1.5 W / WI

2

2.5

3

Fig. 8.9 dσ/dw (SDCS) for the He(e, 2e)He+ process (linear scale) as a function of reduced electron energy w = W/WI after ionization at 100 eV; experimental data () from M ÜLLER -F IEDLER et al. (1986) and state-of-the-art BSR calculations (—) from Z ATSARINNY and BARTSCHAT (2012), Fig. 2. Comparison with the RUDD formula ( ) shows a surprising agreement – only the finer details (wiggles) are missing

According to M ACEK (1967) one may express this with advantage in hyperspherical coordinates, R, α and θ12 : with 0 ≤ R ≤ +∞ R = r12 + r22 cos α = r1 /R, r1 · r2 cos θ12 = r1 r2

sin α = r2 /R

with 0 ≤ α ≤ π/2

(8.48)

with 0 ≤ θ12 ≤ π.

In these coordinates the S CHRÖDINGER equation is written as 5/2 1 d2 Λ2 + 15/4 ζ (α, θ12 ) + W R Ψ = 0, − − 2 2 2 dR R 2R

(8.49)

with the so called C ASIMIR operator: 2 2 ˆ 1 ˆ 2 d 2 2 sin α cos α + + . Λ =− 2 dα cos2 α sin2 α sin α cos2 α 2

1

(8.50)

Here, ˆ 1 and ˆ 2 are the usual orbital angular momentum operators for the two electrons, and the potential ζ (α, θ12 )/R is given by: ζ (α, θ12 ) = −

1 1 R R R 1 − +√ − + =− . r1 r1 r12 cos α sin α 1 − sin 2α cos θ12

(8.51)

The Potential Surface We cannot go into the mathematical details of solving the S CHRÖDINGER equation in the form (8.49) (see e.g. M ACEK 1967; L IN 1974; D EB and C ROTHERS 2002, and further references there). However, already a closer look at the potential hypersur-

542

8


Fig. 8.10 Potential function ζ (α, θ12 ) in hyperspherical coordinates (see e.g. L IN 1974) for the system e + e + H+ to illustrate the three body problem. Equipotential lines are coloured in white, the black dot-dashed line indicates the WANNIER ridge

90˚

60˚

α

ζ

30˚

0˚ 4 2 0 -2 -4 -6

90˚

60˚ α

1 30˚

0˚ -1

0 cosθ12

face is instructive and leads to some kind of ‘visual’ understanding of the correlated dynamics between the two electrons. Figure 8.10 gives a 3D representation of ζ (α, θ12 ), i.e. the potential energy is plotted as a function of the two correlation angles α and θ12 which describe the position of the two electrons with respect to each other. In one further dimension one has to imagine the decrease of the potential ∝ 1/R with the hyper-radius. Intuitively, one visualizes quite easily that in the region of the ridge of this potential surface (α = 45◦ ), the so called WANNIER ridge (black dash-dotted in Fig. 8.10) stable forms of motion at low energies are possible. Now, α = 45◦ implies r1 = r2 even for large R – which corresponds to ionization. This motion will tend to lead towards cos θ12 = −1, to the minimum of the ridge, where both electrons recede from the ion with θ12 = 180◦ in opposite direction. The more α deviates from α = 45◦ , while the three particles are still interacting strongly, the higher will be the chance that the trajectory ‘roles’ into the valley, i.e. towards α → 90◦ or → 0◦ (in particular this is expected for cos θ12 > 0 where the electrons are close). According to (8.48) α = 90◦ or α = 0◦ implies that one of the two electrons does not leave the atom, i.e. that no ionization occurs. In Sect. 8.4.5 we shall discuss that our conjectures are actually not too bad when compared to reality.

The WANNIER Threshold Law WANNIER (1953) and R AU (1971) have treated the three body problem in hyperspherical coordinates, using classical and quantum mechanical theory, respectively. They find in both cases a threshold law σ (ion) ∝ (T − WI )1.127 for the integral ionization cross section – as mentioned already in Sect. 7.2.7. The very fact that classical and quantum theory arrive at exactly the same result is quite remarkable, in particular as this concerns very low kinetic energy of the outgoing electrons. Many attempts have been made to prove the threshold law also experimentally. However, due to the small deviation of the slope from the power 1, this is not trivial. Quite convincing is the first experimental verification by C VEJANOV and R EAD (1974b,a). They have used an ingenious trick: instead of measuring the integral cross

8.4


543

He(1s 4 ℓ )

low energy electron signal ∝ SDCS at W =0

1s5ℓ

1s6ℓ

He+(1s) + e-

7ℓ ∝ ( T - WI ) 0.127

23.5

24.0

24.5 WI

25.0

25.5

26.0

T / eV

Fig. 8.11 Yield of slow electrons from the ionization process He(e, 2e)He+ in the threshold region (ionization potential WI ) as function of the initial kinetic energy of the ionizing electron T is a best fit through the as determined by C VEJANOV and R EAD (1974b). The full red line experimental data according to (8.52)

section as a function of collision energy T , the SDCS for threshold electrons was studied. For detection they used a method which is sensitive only for electrons of nearly vanishing energy W (emitted into all scattering angles) – one may see this as early precursor of ZEKE spectroscopy (see Sect. 5.8.3) for electron collisions. As we have seen in Sect. 8.4.2, for a given primary energy T close to threshold, the SDCS, dσ/dW (integrated over all scattering angles) is practically independent of the energy W of the outgoing electrons and depends only on T (we recall: 0 ≤ W ≤ T − WI ). Hence, the integral ionization cross section (8.42) is approximately T −WI dσ dσ (ion) σ = dW (T − WI ). dW dW 0 One thus expects the SDCS to rise as σ (ion) dσ ∝ (T − WI )0.127 dW T − WI

(8.52)

if the threshold law (7.18) of WANNIER and R AU holds. By an experimental determination of dσ/dW at W 0 one may thus test the deviation of the threshold law from a slope of 1 with high sensitivity. Figure 8.11 shows for e− + He → 2e− + He+ the yield of zero kinetic energy electrons as measured by C VEJANOV and R EAD (1974b) close to threshold, plotted as a function of the kinetic energy T of the projectile electron. The dependence ∝ (T − WI )0.127 for T above the ionization threshold WI is clearly documented by the red line, fitted to the data. In this case, the threshold law (7.18) appears to be valid in an energy range of at least 1.5 eV above threshold.

544

8


It is interesting to note that also below threshold a similar excitation law seems to hold (a mirror image) for high RYDBERG states. The high lying RYDBERG states were detected in this experiment by field ionization – by a similar method as in the (much later) ZEKE experiments (see Sect. 5.8.5). In view of the potential hypersurface Fig. 8.10 this trend is not too surprising: the genuine electron dynamics should be similar slightly below and above threshold. From this viewpoint, RYDBERG excitation corresponds to trajectories which just ‘do not make it’ into the continuum. Rather, they ‘role into the valley’ at α = 0◦ or 90◦ . The threshold region (WANNIER region) up 1.5 eV above threshold was and still is an object of sophisticated experimental and theoretical work. Of specific interest is the angle θ12 between both electrons. An educated glance at Fig. 8.10 suggests, that at low energies the two electrons will preferentially separate at θ12 = 180◦ – if ionization occurs at all. This has indeed been confirmed experimentally (not shown here, see C VEJANOV and R EAD 1974b, Fig. 2) as well as by state-of-the-art quantum mechanical calculations (see e.g. BARTLETT and S TELBOVICS 2004b). Of course one observes a distribution of angles, the width (FWHM) of which increases with excess energy above ionization threshold. For the system e− + H → 2e− + H+ one finds (π − θ12 )FWHM 3(T − WI )1/4 . Recent results will be discussed in Sect. 8.4.5 (see in particular Fig. 8.21).

8.4.4

DDCS: Double-Differential Cross Section and the B ORN -B ETHE Approximation

Still deeper insight into ionization dynamics is gleaned from double-differential cross sections (DDCS), i.e. from the relative cross section per energy and angular intervals, dW and dΩ, respectively. For sufficiently high kinetic energy T WI and small momentum transfer K < a0−1 , B ORN approximation is again a good first approach (see e.g. the classical papers of K IM 1975c,a,b; I NOKUTI 1971). One has to be aware that the B ORN approximation tacitely implies that the two electrons can be distinguished: The projectile electron is represented by a plane wave which is supposed to change little. For high energies this is a reasonable approximation, since the kinetic energy T of the projectile electron before and after the ionization process (WA ) differ little – both being large compared to the ionization potential WI WA T . As we shall see, the distinction between a “scattered” (A) and an “ejected” particle (B) remains a helpful model when discussing the experimental results. Since, however, a DDCS measurement does detect only one electron, we drop for the moment the indices A and B and refer to the energy of the detected electron as W while the ‘other’ electrons energy is ". We now recall the energy and momentum balance shown in Fig. 8.6 for ionization. In order to apply FBA this situation we have to translate (8.22) which holds for excitation: we replace the excitation energy Wf i there (for a bound-bound transition) for ionization here by Wf i → " + WI = T − W

(8.53)

8.4


545

and obtain the DDCS per energy interval d" = dW and solid angle dΩ into which the detected electron is scattered after ionization: (FBA) 2 a02 1 4a02 (k f , k i ) 2 dσ"i(FBA) (θ, ϕ) W W T"i F = (K) = "i 1/2 d"dΩ Eh (2π)2 T (Ka0 )4 T Eh a03 √ with (Ka0 )2 = 2(T + W − 2 T W cos θ )/Eh . (8.54) √ We have expressed the flux factor kf /ki = W/T in terms of the energy W of the detected electron and the initial kinetic energy T . The matrix element F"i (K) is formally still given by (8.21). But it describes now a transition of the (ejected) target electron between a bound, discrete initial state |i and a free, continuum final state "|. The latter is normalized in energy scale (dimension L−3/2 Enrg−1/2 ) as described in Appendix J.1 of Vol. 1, so that (FBA) F"i (K) and T"i have now the dimension Enrg−1/2 and Enrg1/2 L3 , respectively, and the DDCS has the dimension L2 Enrg−1 per solid angle. We may compare the step from bound-bound to bound-free transitions in electron impact with that from photoexcitation (Sect. 5.2.2 in Vol. 1) to photoionization (Sect. 5.5 in Vol. 1). The energy balance (5.59), Vol. 1 for photoionization (opt) was ω = " + WI and the OOSD df"i /d" (dimension Enrg−1 ) was defined by (5.62), Vol. 1. Correspondingly one extends the concept of the (generalized) oscillator strength to the continuum (in the framework of FBA). With the substitution (8.53) for Wf i we write in analogy to (8.31) for the DDCS (FBA) (GOS) 2a02 Eh W (θ, ϕ) (K) d2 σ"i 1 df"i = . (8.55) 2 d"dΩ T − W T (Ka0 ) d" By comparison with (8.54) the generalized oscillator strength density, GOSD, introduced here may be computed (in a.u.) from (GOS)

df"i

d"

(K)

(opt) Ff i (K) 2 df"i −→ = 2(T − W ) K K→0 d"

(8.56)

with the matrix element (8.21). In the limit K → 0 the GOSD approaches the optical oscillator strength density – just as for the excitation process according to (8.33). The GOSD has the dimension Enrg−1 , and as in the case of photoionization it strongly depends on the excitation energy " + WI = T − W of the electron raised into the continuum. For the H atom the GOSD may be evaluated in closed form, as already done by B ETHE. Following I NOKUTI (1971) this somewhat complicated formula, the so called B ETHE surface, is plotted in Fig. 8.12 as a function of the excitation energy " + WI and the logarithm of K 2 , again in a.u. In the optical limit12 K → 0 GOSD and OOSD decay very rapidly with " + WI . Quite remarkable is the occurrence of a a collision process the limit K → 0 may be approached arbitrarily well, but can never be reached completely, as obvious from (8.54).

12 In

546 Fig. 8.12 B ETHE surface of the generalized oscillator strength density (GOSD) for electron impact ionization of atomic hydrogen. The GOSD per atomic energy unit (Eh ) is plotted for values of the excitation energy " + WI from threshold at 13.6 eV up to 270 eV and for momentum transfer Ka0 from 0.01 to 10 (in log K 2 scaling). The B ETHE ridge is indicated by the black dash-dotted line

8


10-2

10-4

1

102 GOSD / E h

1.5

1.0

0.5 0 WI ( + W

I) /

2 E

h

4 10 -4

10 -2

1

102 (Ka0) 2

maximum for the GOSD at higher K and the shift of this maximum as the excitation energy " + WI rises. As a closer analysis shows (see e.g. C OPLAN et al. 1994), in the region of this so called B ETHE ridge the ionization process is dominated by binary interaction between the scattered and the ejected electron. In this situation the remaining ion participates in the process more or less as a spectator only. We now illustrate these considerations for the reaction He(e, 2e)He+ – a benchmark system for electron impact ionization. Of course no analytical formula for the GOSD is available here. However, a number of experimental and theoretical studies have been performed, also for somewhat higher initial kinetic energies. Figure 8.13 shows for T = 100 eV and 500 eV the DDCS, i.e. the angular distribution of electrons detected at a number of selected energies W . For He the ionization energy is WI = 24.6 eV, thus, in Fig. 8.13, the energy " of the other, not detected electron ranges from practically 0 eV up to 20 eV (and up to 40 eV at T = 500 eV). The strongly forward peaked distribution is quite remarkable (most dramatically at 500 eV and low energy " of the ejected electron). With increasing excitation energy " + WI = T − W the cross section decreases (as already expected from Fig. 8.8(d), and the angular distribution gets flatter. The experimental data of M ÜLLER -F IEDLER et al. (1986) are compared to several state-of-the-art computational results, including very recent BSR calculations of Z ATSARINNY and BARTSCHAT (2012). The theoretical models reproduce the experimentally observed trends quite well. As a comparison between the data of M ÜLLER -F IEDLER et al. (1986) and AVALDI et al. (1987) at 500 eV shows, there are some uncertainties also among the experiments.13 13 We must note here, that without the re-calibration (suggested quite convincingly by

1996) the agreement with the BSR calculations at 100 eV would even be better.

S AENZ et al.

DDCS / 10 -23 m 2 eV -1 sr -1

8.4


(a)

10 2

547

T = 100eV

(b) T = 500eV

10 2

W = 73.4 eV W = 71.4 eV

101

W = 471.4 eV

101

W = 55.4 eV 100

W = 435.4 eV

10-1 0°

20°

Avaldi 435.4

100

BSR DWB GA

DWB GA

10-1 40°

60°

80°

0°

20°

40°

scattering angle θ

Fig. 8.13 Double-differential cross section (DDCS) for electron impact ionization of He at a primary energy T = 100 and 500 eV for several energies of the “scattered” electron W as a function of scattering angle. Experimental data from M ÜLLER -F IEDLER et al. (1986), one data set from AVALDI et al. (1987). The data were re-calibrated by S AENZ et al. (1996). Theory (lines): G LAUBER approximation, GA (R AY et al. 1991), distorted wave B ORN, DWB (M C C ARTHY and Z HANG 1989), and BSR (Z ATSARINNY and BARTSCHAT 2012)

Even though, from a fundamental point of view, one cannot really distinguish the scattered electron from the (secondary) ejected electron, the pronounced forward scattering for small secondary energies " indicates that there is little exchange at these relatively high initial kinetic energies (the energy distribution discussed in Sect. 8.4.2 also supports this finding). Thus, the intuitive model of a scattered electron (with little energy loss, deflected only little) and an ejected electron (at rather low energy) is quite reasonable in this range of primary energies. This becomes even more evident, when the angular distribution is studied for electrons with low energies as shown for a few examples in Fig. 8.14: identifying this low energy electron as ‘ejected’ (i.e. originally bound) it is plausible that it has no pronounced forward peak. On the contrary, for the lowest energies of 2 eV and 4 eV we even recognize a distinctive backward tendency, complementary to the sharp forward peak of the high energy counterpart at 75.4 eV and 73.5 eV in Fig. 8.13(a): this appears to correspond rather plausibly with a trajectory around cos θ12 = −1 on the WANNIER surface Fig. 8.10. As the detected energy gets larger, the distribution becomes more isotropic – albeit still structured. The agreement with theoretical models is quite satisfactory, especially so for CCC from B RAY and F URSA (1996). It is instructive to express the measured DDCS for the (primary) scattered electron in terms of the generalized oscillator strength density (GOSD). This is displayed for the He(e, 2e)He+ process in Fig. 8.15. Using (8.55) and (8.56), the experimental cross section data from M ÜLLER -F IEDLER et al. (1986) (re-calibrated according to S AENZ et al. 1996) are thus compiled into one single plot, for six different initial kinetic energies T and three different values of the excitation energy " + WI . The result is quite convincing: while there are – as expected – some deviations in the GOSD at larger momentum transfer Ka0 , the overall convergence

8

DDCS / 10 -23 m 2 eV -1 sr -1

548

4

BSR CCC

4 W = 2eV

2

2

0

W = 4eV

0

2

2

W = 10eV

1 0


W = 20eV

1

0

90º

0 180º 0 scattering angle θ

90º

180º

Fig. 8.14 Double-differential cross section (DDCS) for electron impact ionization of He at 100 eV for different energies W of the detected (“secondary”) electron as a function of its scattering angle θ . Note the different scales for the DDCS in the upper and lower panels. Experimental data from M ÜLLER -F IEDLER et al. (1986) are compared with recent BSR calculations (Z ATSARINNY and BARTSCHAT 2012) and CCC (B RAY and F URSA 1996)

Theorie Experiment bei T = 100 eV 200 eV 300 eV 400 eV 500 eV 600 eV

1.5 1.0 0.5 0.0 1.0

+WI = 24.6 eV

0.5 GOSD / Eh-1

Fig. 8.15 Generalized oscillator strength density GOSD per atomic energy unit Eh for electron impact ionization of He as a function of the momentum transfer in units of a0−1 . For three different values of the energy transfer " + WI = T − W the graph compares FBA calculations of S AENZ et al. (1996) and effective values, determined from the experimental data of M ÜLLER -F IEDLER et al. (1986) at different primary kinetic energies T . Note the indication of a “B ETHE ridge” at " + WI = 44.6 eV and (Ka0 )2 1.5

+WI = 34.6eV 0.0

0

0.5

+WI = 44.6eV 0.0 0.1

0.5

1

5

(Ka0)2

of the data towards Ka0 → 0 is impressive. B ORN approximation reproduces the behaviour rather well, even at relatively high momentum transfer – at least the trend is displayed very nicely, considering the fact that this is the most simple approximation for understanding a process as complex as impact ionization. Note that for " + WI = 44.6 eV 1.6Eh one recognizes at (Ka0 )2 1.5 a slight maximum, and

8.4


549

one may envisage easily a more pronounced “B ETHE ridge” for higher excitation energy, similar as in the H(e, 2e)H+ case according to Fig. 8.12. A more differentiated picture emerges from recent experimental and theoretical work at lower kinetic energies, in particular in the threshold region for ionization. The observations may often be explained by using the hyperspherical potential Fig. 8.10 in a quite intuitive manner. For details the readers are referred to the original literature (see e.g. S CHOW et al. 2005; B RAY et al. 2003).

8.4.5

TDCS: Triple-Differential Cross Sections

Triple-differential cross sections (TDCS) provide maximum information about the fundamental three body break up process initiated by electron impact ionization of atoms. State-of-the-art particle imaging methods (VMI) and advanced reaction microscopes (COLTRIMS) allow today an efficient, coincident registration of the final momenta of all three charged particles (and thus the determination of their energies and emission angles), so that cross sections are measured with the kinematics fully determined as sketched in Fig. 8.6. Indeed, during the past few years an impressive wealth of information has been (and continues to be) accumulated, and beautiful images have emerged which illustrate the three body quantum mechanics. At the same time substantial progress has been made in theoretical methods and computational power, so that one may claim today that for small atoms (He and H) electron impact ionization is fully understood. We can only provide a glimpse of these developments. The first pioneering experiments were performed by E HRHARDT et al. (1969) and A MALDI et al. (1969) – each with different goals. At that time, such measurements were extremely time consuming and the necessary technical efforts were considerable, as may be gleaned from Fig. 8.16 with the historical setup of the E HRHARDT group. The figure is essentially self-explaining and shows the key ingredients of any such experiment: an energy selected projectile electron beam, crossing an atomic target beam, and two, energy selective detectors which register the two free electrons after the ionization process in coincidence. One recognizes the geometrical constraints of these early experiments, a limited range of scattering angles accessible for the two detectors and a specific detection geometry: Here the two electrons are detected coplanar with respect to the plane defined by the direction k i of the projectile electron before, and k A after the interaction with the target atom. Over the years a large body of data has been accumulated, mostly for electron impact ionization of He and H atoms. Clearly, the data for the He(e, 2e)He+ process are experimentally much more easy to access and the material is thus more detailed and probably more reliable than for the system e + H(e, 2e) + H+ . The problems with using atomic hydrogen should not be underestimated, even though from a theoretical point of view one would prefer the H system as a test case for obvious reasons: it is the simplest and the only genuine three particle break up system.

550

8

Electron Impact Excitation and Ionization multiplier 2

5 cm

collector B 235º

electron gun

70º

lenses 2 field free energy selector

190º

multiplier 1

eatomic beam ┴ plane

collimation

lenses 1

repeller

lenses 5

cathode

lenses 4 energy analysor

125º

collector A

lenses 3

Fig. 8.16 Pioneering experimental setup by E HRHARDT et al. (1969) for the determination of triple differential cross sections (TDCS) for the He(e, 2e)He+ process with coincident detection of the scattered and ejected electron

Both cases have been studied worldwide intensively – mostly in close collaborations between theoretical and experimental groups. Today the agreement between experiment and theory is quite impressive. Relatively clear and easy to understand intuitively is the situation for intermediate and high energies (beyond the M ASSEY maximum of integral cross sections) while at initial kinetic energies below that maximum and in particular in the threshold region a very rich and complex dynamics is observed.

B ORN Approximation for the TDCS Before we discuss the experimental data and compare it with theory, let us have a brief look at the B ORN approximation which – even though reliable results can be expected only for high kinetic energies (T 1 keV) and small momentum transfer (Ka0 1) – may provide a reasonable first interpretation of what is observed experimentally. We now have to include the emission angle θB for the second electron into (8.54), which in a.u. is thus rewritten as14 (FBA) 2 dσ"i (θ, ϕ) 1 kA (FBA) , = (k , k ) T f i "i d3 k B dΩA (2π)2 ki

the plane waves used to determine T"i(FBA) are normalized in k scale (see Appendix J.2.2 in Vol. 1), as e.g. in OVCHINNIKOV et al. (2004), the factor (2π)−2 has to be replaced by (2π)4 .

14 If

8.4


551

where d3 k B refers to the ejected electron. Following OVCHINNIKOV et al. (2004), this “is best expressed in terms of ionization states normalized on the energy scale”.15 Using d3 kB = kB d"dΩB we obtain in a.u. (FBA) 2 4kB kA (θ, ϕ) dσ"i 1 kB kA (FBA) F"i (K)2 , (K) = T"i = 2 4 dΩA dΩB d" (2π) ki ki K

(8.57)

where F"i (K) is formally still given by (8.21). However, while there the integration is assumed to be over all N electrons of the target, it is now carried out only over N − 1 electrons, while the momentum k B of the ejected electron is kept as a parameter which determines energy and angles of the ejected electron. As in im(FBA) (K) as well as pact excitation, in FBA the form factor F"i (K) – and hence T"i the triple differential cross – are symmetric around the momentum transfer K. Explicit expressions (available for atomic H) contain characteristic terms of the type K · k B = K · kB cos θK where θK is the angle between the ejected electron and the momentum transfer K. According to (8.56), in the limit of really very small momentum transfer the GOSD approaches the OOSD, and one even expects ultimately an angular distribution of the ejected photoelectron as in the case of photoionization: (opt) 1 df" 1 WA d3 σ (θ ) 1 + βP2 (cos θB ) . (8.58) −→ 2 dΩA dΩB d" K→0 2π T K (WI + ") d" Here β is the anisotropy parameter as introduced by (5.80), Vol. 1. Integration over all emission angles θB leads again to (8.55), independent of β. In photoionization, we have β = 2 if the initial state |i is an s state, i.e. the outgoing will be a p electron described by the characteristic dumbbell shape. Clearly, electron impact ionization is substantially more complicated as no optical selection rule confines the outgoing electrons to p continuum states. Nevertheless, the momentum transfer K plays effectively the role of the photon polarization vector e and defines the symmetry axis of the B electron distribution – as far as FBA is applicable.

TDCS for Electron Impact Ionization of He What does the experiment say to these predictions? We concentrate our discussion on the e− + He →2e− + He+ and three characteristic situations at T = 250 eV which are summarized in Fig. 8.17.16 The experimental data of S CHLEMMER et al. (1991) are compared with a quite expensive complete close-coupling calculation CCC(101), including 101 states and pseudo-states, by B RAY and F URSA (1996) – 15 In view of this definition, in the theoretical literature the TDCS is often referred to as “FDCS” since, including the azimuthal angles, strictly speaking it refers to five variables in the denominator. 16 As

illustrated in the kinematics, Fig. 8.6, we use an xz scattering plane defined by k i and k A , the scattering angle θA being positive by definition, and this also holds for the momentum transfer angle 0 ≤ θK ≤ 360◦ . In the experiment discussed here the secondary electron is detected in the scattering plane (ϕ = 0), its emission angle being 0 ≤ θB ≤ 360◦ . In the literature one finds a variety of slightly different notations.

552

8

(a) B

kA

0° k

Electron Impact Excitation and Ionization 0° k

(b) B

315°

45°

kA

45°

315°

5.0 2.5

1.0 binary

0.5 270°

binary 270°

90°

recoil 135° A = 4° B = 2.5

135°

225° eV,

180° = 311°,

= 0.38 0-1

(c)

B=

180° 5 eV, = 295°,

A = 10° = 0.77 0-1

0° k

B

45°

225°

kA

315°

0.5 0.25 270°

90°

135° A = 14° B = 10

eV,

225° 180° = 204°, = 1.05 0-1

Fig. 8.17 TDCS for e− + He → 2e− + He+ at T = 250 eV. Polar diagram as a function of the emission angle θB of the ejected electron (energy WB ), at fixed scattering angle θA of the projectile electron and fixed momentum transfer K (its length is not drawn to scale here). The magnitude of the TDCS is proportional the distance from origin (values at the circles are in 10−22 m2 sr−2 eV−1 ). Experimental data from S CHLEMMER et al. (1991) and as communicated by B RAY and F URSA (1996) who also performed the CCC(101) calculations ( ); for (a) at small momentum transfer also FBA ( ) from B YRON et al. (1986); the experimental data in (b) where recalibrated to match the binary peak of FBA

and we refrain from discussing a large variety of further theoretical approaches, all reproducing the experimental results rather satisfactorily. For the smallest momentum transfer (Fig. 8.17(a)) we may compare the data with FBA (dashed red line), and see that even at K = 0.38/a0 we are far from the

8.4


553

expected dumbbell shape in the limit K → 0. However, experiment, FBA, and the nearly exact CCC(101) theory display a characteristic double lobe structure. The forward lobe, pointing more or less into the direction of the momentum transfer +K, is called binary peak and is understood to arise essentially from a binary interaction between the projectile electron and the ejected atomic electron – with the atomic nucleus remaining a spectator. In contrast, the recoil peak, emitted more or less in the opposite direction, is considered to arise from backscattered electrons undergoing multiple interactions with the nucleus. It is interesting to note – and quite typical – that the FBA cross section overestimates the binary peak and underestimates the recoil peak. Closer inspection also shows, that the symmetry predicted by FBA does not completely match the observations – even at very low K. Rather, the maximum of the binary peak is slightly shifted towards smaller emission angles (typically 270◦ ≤ θB < θK ) while the recoil maximum is found at slightly larger angles (θK − 180◦ < θB < 180◦ ). In contrast to FBA, the CCC(101) results reproduce the experimental data at the smallest scattering angle θA = 4◦ almost perfectly (Fig. 8.17(a)). At larger scattering angles and higher momentum transfer, displayed in Fig. 8.17(b) and (c), the overall cross section decreases as expected, and K still provides an approximate symmetry axis. However, while the binary peak dominates now the process, the recoil peak tends almost to vanish. This is a general observation at higher kinetic energies and momentum transfer approaching that of the B ETHE ridge where ionization occurs essentially by binary encounter of projectile and target electron. Over the years, a variety of theoretical approaches has been tested for reproducing these and other experimental data, among others the B ORN series, modified B ORN approximations e.g. with outgoing C OULOMB waves and various other “distorted wave” methods as well as the so called G LAUBER approximation (an eikonal approximation similar to that sketched in Sect. 7.4 as semiclassical approach to heavy particle scattering). Nearly all these approaches predict similar overall trends, even B ORN approximation (except that in FBA K is a strict symmetry axis, in all other theories it is not). However, it is not trivial to make exact predictions, especially for the ratio between binary and recoil peak. Only CCC(101) and similarly expensive calculations match the presently available experimental data almost perfectly, as illustrated by the few examples shown in Fig. 8.17.17 Most of the earlier experiments were restricted to two particle coincidence in the scattering plane – so some important information was missing. Recently, R EN et al. (2011) performed an impressive 3D benchmark experiment using a state-of-theart reaction microscope, testing CCC and TDCC theory (at relatively low kinetic energy, T = 70.6 eV, for several scattering angles θA and different energy sharing between the two electrons). The momenta of all three charged particles were determined and full 3D images were generated of the probability distributions for the ejected second electron. 17 The

experimental data points in Fig. 8.17(b) have been rescaled to match the binary peak maximum from CCC. This appears justified as the experiment was calibrated by extrapolation to the OOSD according to (8.58) – which is questionable at these high K values.

554

8

(a) experiment


(b) CCC theory

ki

kA θA

kA

binary

I II

recoil

Fig. 8.18 TDCS for the He(e, 2e)He+ process at T = 76 eV according to R EN et al. (2011). Plotted is the cross section for a fixed scattering angle θA = 20◦ as a function of the emission angle θB , ϕB of an electron with energy WB = 5 eV (a) experiment with an advanced reaction microscope, (b) TDCS calculated with state-of-the-art CCC; note that here the length of K (K = 0.878a0−1 ) is to scale with respect to k i and k A ; the maxima in experiment and CCC theory are found at θB 286◦ (binary) and θB 151◦ (recoil) – for comparison θK = 317◦ and θK − 180◦ = 137◦

One sample from these beautiful studies is shown in Fig. 8.18.18 At this particular emission angle and energy partitioning the observed bimodal lobe still resembles the p shaped dumbbell. However, kinked downward at both ends, it is far from being symmetric with respect to the momentum transfer K. We specifically emphasize the pronounced, non-zero waist in the perpendicular plane – in contrast to a p dumbbell – and recall that total momentum conservation between k i and any two finally detected electrons (with energy conserving k A , k B ) may be ensured by recoil q I to the target ion, as sketched in Fig. 8.6. A detailed comparison of experiment and theory by R EN et al. (2011) documents excellent agreement between CCC and the experimental results (with very small error bars) for all geometries studied (not shown here), while TDCC tends to overestimate the recoil peak. In general, at these relatively low energies the recoil peak (i.e. electron interaction with the atomic core) dominates at small momentum transfer, while the binary peak (i.e. electron-electron interaction) dominates for larger momentum transfer. For equal energy partitioning between the two outgoing electrons and larger momentum transfer, the recoil peak disappears almost completely.

The e− + H System From the above discussion, the solution of the tree body problem appears to require ‘brute force’ computational methods to obtain good agreement between theory and experiment, in particularly so since the e− − He system is not a true three body 18 We are grateful to the authors for providing the specially adapted version of their 3D images shown here.

8.4

Electron Impact Ionization 12

555

(a) FBA SBA TBW

8

θ A= 3° T=250eV W B= 5 eV K = 0.38a 0-1 θ K = 321°

TDCS / 10 -22 m 2 eV -1 sr -2

4 0 2

recoil

(b) TDCC ECS CCC

1

binary

θ AB= 180° T =17.6eV W B=W A=2eV

0

0.10

(c)

θ AB= 90° T=17.6eV W B=W A=2eV

0.05

0 0°

90°

180°

270°

360°

θB

Fig. 8.19 TDCS for the H(e, 2e)H+ process, comparison of experiment and theory for detection of the second electron in the scattering plane. Plotted is the cross section as a function of the emission angle θB of an electron with well defined energy WB . (a) High energy regime (T = 250 eV), the scattered electron is detected at a fixed angle θA = 3◦ ; the experimental data from E HRHARDT et al. (1986) (recalibrated ×0.88) are compared with FBA and SBA from DAL C APPELLO et al. (2011), and (almost identical to SBA) with TBW from B RAUNER et al. (1989); (b) initial kinetic energy very close to threshold (T = 17.6 eV), the excess energy of 4 eV is equally shared between the two electrons, the angle between them is now kept fixed at θAB = 180◦ ; the experimental data RÖDER et al. (2003) are compared with ECS (BAERTSCHY et al. 2001), CCC (B RAY 2002), and – added here – TDCC (C OLGAN and P INDZOLA 2006) calculations; (c) same as (b) but now the angle between the two electrons is kept at θAB = 90◦

problem: the influence of the second atomic electron certainly adds even more complications. Irrespective of the excellent understanding which has been achieved for the He(e, 2e)He+ system, the genuine three body problem, H(e, 2e)p+ , remains a unique challenge for theory and experiment alike. Figure 8.19 compares experiment and theory for some selected geometries at two different energies. Figure 8.19(a) documents the characteristic quantitative failure of FBA. We note, however, that already the second term in the B ORN series, SBA, in the special variety reported by

556 Fig. 8.20 Toroidal, multidimensional analyzer setup for coincident registration of two electrons with angular and energy analysis (perpendicular geometry) according to VAN B OEYEN and W ILLIAMS (2005)

8


electron gun electron trajectories

entrance lenses FARADAY cup

toroidal sector fields

position sensitive detectors

exit lenses

DAL C APPELLO et al. (2011) gives excellent agreement with experiment at an intermediate energy (T = 250 eV) and competes well with the calculations of B RAUNER et al. (1989) based on an asymptotically exact form of the three-body C OULOMB wave function. Figure 8.19(b) and (c) document the situation for very low initial kinetic energy – just 4 eV above the ionization threshold. In this case the detection angle θAB between the two electrons (equal final energy) has been kept constant. Obviously, two electrons emission into opposite directions (b), θAB = 180◦ , is by a factor of 20 more probable than emission at right angles (c). This nicely verifies the classical intuition of preferred motion on the WANNIER ridge as discussed in Sect. 8.4.3. For this geometry it is obviously most probable that one electron just follows the direction of the projectile while the other electron is ejected in the opposite direction as displayed in Fig. 8.19(b). The (less probable) θAB = 90◦ geometry obviously involves more complex dynamics. For obvious reasons, there is much less experimental material available then for He (it is just not trivial to generate a stable, well controlled beam of atomic hydrogen). Thus, efficient data collection is very important. An interesting alternative to the reaction microscope is the toroidal energy analyzer shown in Fig. 8.20, devised by VAN B OEYEN and W ILLIAMS (2005). The two final electrons are simultaneously selected according to their energy and emission angle and detected in coincidence. So called wedge and strips anodes are used, which allow simultaneous temporal and spatial registration of particles (at the bottom of Fig. 8.20 shaded in light red). In contrast to the traditional planar geometry used in the early experiments by the E HRHARDT group in Kaiserslautern, here the perpendicular geometry is used where scattered and ejected electron are both detected perpendicular to the electron beam. In our terminology displayed in Fig. 8.6, this implies θA = θB = 90◦ . Both electrons are detected in coincidence at azimuthal angles 0◦ ≤ ϕB ≤ 360◦ – just by simultaneous registration of the respective positions on the wedge and strips anodes behind

Electron Impact Ionization TDCS / 10 -20 cm 2 sr -2 eV -1

8.4

557

20

T-WI = 0.5eV W B /WA=1:10 15 W B =WA

T -WI = 6.8eV W B /WA=1:10 W B =WA

10 ×5 5 0 0

90°

φB

180°

270°

360°

Fig. 8.21 TDCS for e + H → 2e + H+ in perpendicular geometry (θA = θB = 90◦ ) as a function of the azimuthal angle ϕB for two different collision energies T at two different ratios for energy sharing between the two electrons. Experimental data points (determined with the setup shown in Fig. 8.20) and exact quantum mechanical computations (lines, see text) according to W ILLIAMS et al. (2006)

the toroidal analyzer. Even though the TDCS is very small in this configuration, it is particularly sensitive for critical tests. Figure 8.21 shows the results of W ILLIAMS et al. (2006) for e− + H in the energy range just above threshold, T WI . For excess energies 0.5 eV and 6.8 eV each, two types of energy sharing have been studied: one very asymmetric and a fully symmetric version with WA = " = (T − WI )/2. It is interesting to note that the TDCS changes only little with energy partitioning. However, the ϕB dependence varies dramatically with the primary energy T . While at T − WI = 0.5 eV the behaviour predicted by the WANNIER model – the electrons emerge at opposite direction (ϕB = 180◦ ) is very pronounced, it has disappeared already at 6.8 eV above the ionization threshold and yields a structured angular distribution. The full lines in Fig. 8.21 correspond to exact quantum mechanical calculations with the so called propagating exterior complex scaling method of BARTLETT and S TELBOVICS (2004a): on a grid of the overall size Rmax = 180a0 the S CHRÖDINGER equation is solved numerically for partial waves up to = 5! The fantastic agreement between this theory and experiment documents nicely the stateof-the-art for this fundamental three body problem.

Recent Developments, Non-isotropic Targets Experiments with state-of-the-art imaging techniques together with efficient theoretical methods and computer codes provide presently a wealth of new data and insights, also into more complex ionization problems. We can only mention some recent publications and refer the reader to these and references quoted there. In the He and H case discussed above, the ejected electron is originally in an s state. If the electron to be ionized is initially in a p state, as e.g. in the larger rare gases, the situation changes. This already holds for photoionization where in principle all values of the anisotropy parameter −1 ≤ β ≤ 2 are possible. R EN et al. (2012a) have studied the TDCS for electron impact ionization of argon atoms, using

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8


the efficient imaging techniques described above. The experiments show clearly, that the simple double lobe structure seen in the latter case is no longer observed for Ar(e, 2e)Ar+ . The 3D images observed experimentally and computed by BSR theory show a rather complex structure – and do agree very well. As yet another direction of present and future research in this field, molecules come into view (see e.g. A L -H AGAN et al. 2010; S ENFTLEBEN et al. 2012, and further references there). The Heidelberg group (R EN et al. 2012b) reported again a new type of electron impact ionization study for H2 where the TDCS is even correlated with a specific alignment of the molecular axis – determined by registering the momentum of a proton emitted by dissociative ionization.

8.4.6

Electron Momentum Spectroscopy (EMS)

Before ending this section an important spectroscopic aspect of the (e, 2e) experiments has to be mentioned. It has been developed and exploited very successfully by several groups over the past decades. While all our above discussion focussed on the dynamics of the ionization process, we shall now show that the TDCS depends directly on the momentum distribution of the target electrons. Under certain conditions, the coincidence technique may thus be used to determine the latter for arbitrary atomic or molecular targets. The basic idea of this electron momentum spectroscopy (EMS) starts by assuming that the momentum k B of the ejected (secondary) electron is determined by the sum of the momentum transfer K = k i − k A and its own momentum q at the time of the collision, i.e. k B = K + q.

(8.59)

The angular distribution of the ejected electron, that is the idea, should then allow direct conclusions about the momentum density distribution of the ejected electron prior to the collision. And the latter is the F OURIER transform of the electron density distribution in position space. In this manner one may thus obtain a direct image of the atomic or molecular wave function. Let us have a look again at the kinematics of the (e, 2e) process according to Fig. 8.6. We assume the target atom or molecule to be initially at rest (the slight thermal motion may usually be neglected). The target as a whole has initially no net momentum. One may recast this trivial fact into the statement, that the initial momentum of the target electron (averaged over the whole momentum density distribution) is zero. After the ionizing collision, the target ion has in general a nonvanishing momentum q I = ki − kB − kA = K − kB, so that momentum conservation is satisfied. With (8.59) this leads to q I = −q: the electron momentum q prior to collision is thus equal in magnitude to the ion recoil momentum q I and directed opposite to it. From this point of view the momentum

8.4


559

balance (8.39) may be written k i = k A + k B − q.

(8.60)

In contrast, the respective recoil energy q 2I /2Mion of the ion is very small and may be neglected. Since the measurement of the TDCS implies according to (8.39) a determination of q I one thus determines – according to this concept – also q = −q I . However, to really obtain a genuine momentum distribution of the target electron one has to make sure that the process can be described as a truly binary interaction between scattered electron and ejected target electron. Effectively, neither the scattered projectile electron nor the ejected electron must directly interact with the remaining target ion. At high energies, where T as well as WA and " are large compared to the binding energy WIj of the electron studied, one may hope that this assumption is justified under certain conditions. One uses the so called plane-wave impulse approximation (PWIA), by representing the projectile electron before and after as a plane wave, which leads to (8.61). In addition, one now further assumes that also the ejected secondary electron is approximated by a plane wave. This further simplifies the atomic (or molecular) form factor defined in (8.21) and leads to (in a.u.) 2 d3 σ (θ ) 4 kA kB −iq·r B (N −1) (N ) e = 4 φγ B (r) φi (r, r B ) dΩA dΩB d" K ki

(8.61)

with q = k A + k B − k i = k B − K. According to (8.60) q should be equivalent to the momentum of electron (B) immediately prior to the interaction with electron (A). Here r B and r represent the electron to be ionized and the remaining N − 1 (N ) electrons, respectively, while |φi (r, r B ) describes the target state with N elec(N −1) trons prior to the collision and |φγ B (r) the ionic state (electron B missing) with quantum numbers γ . Note that even though (8.61) looks similar to (8.57), the wave function of the outgoing electron has now been approximated by a plane wave, hence K is replaced by q. Hence, the TDCS (8.61) in PWIA is nothing more than the product of the binary cross section for RUTHERFORD scattering ∝ K −4 which we have discussed several times, a flux factor and the F OURIER transform of the overlap integral between initial state and final ionic state at a value q which can be determined experimentally together with the TDCS. One often introduces so called DYSON orbitals (typically as spin-orbitals, see P ICKUP 1977) √ (N −1) (N ) (8.62) gγ (r B ) = N φγ B (r)φi (r, r B )dr. They effectively represent the electron hole created in the cationic wave function and can be computed by standard quantum chemical programmes. The DYSON orbital gγ (r B ) reduces to the initial MO of the electron studied if the most simple

560

8


approximation is used: assuming that target and ionic wave functions may be factorized in MOs, and that the ionic core remains unchanged when the secondary electron emerges from the target (so called frozen-core or sudden approximation). Using DYSON orbitals the TDCS (8.61) is in any case written simply as 2 d3 σ (θ ) 4 kA kB gγ B (p) , = 4 dΩA dΩB d" K ki

(8.63)

where gγ B (p) is the F OURIER transform of the DYSON orbital. Thus, the TDCS is indeed proportional to the density in momentum space of the target electron studied. Here . . . is the overlap integral of the spectator orbitals in the ion and target, respectively, and the last integral is just the F OURIER transform of the target orbitals φB (q), i.e. the wave function in momentum space. At which experimental conditions these approximations are valid and how reliable the density maps obtained are, has been subject to detailed experimental and theoretical studies in the 1990ies. We cannot discuss the details and refer the interested reader again to the relevant literature (e.g. W EIGOLD and M C C ARTHY 1999; C OPLAN et al. 1994; M C C ARTHY and W EIGOLD 1991). In the meantime (e, 2e) EMS has become a well established, powerful method in molecular spectroscopy – often called ‘synchrotron of the poor’ as one may obtain similar (typically more expensive) information from PES using XUV and X-ray radiation from synchrotron sources. High electron kinetic energies (T > 1 keV) are required for a clean interpretation of the data. One typically does not use the coplanar setup, which has provided most information about the dynamics. Rather, one often uses symmetric geometries (θA = θB and WA = "), keeps the momentum transfer K constant, and varies instead the azimuthal angle ϕB and thus q. Since the experimental signals are low in this geometry, one uses today parallelized measuring techniques with position sensitive detectors for registering the electron coincidence, and electron energy analyzers of the type shown in Fig. 8.20. This allows to collect data with sufficiently good statistics in acceptable measuring time. In the past years impressive data have been collected, determining experimentally the momentum density distributions for a variety of atoms and molecules. We illustrate this by a more recent experiment with the H2 O molecule from N ING et al. (2008). The kinetic energy was T = 1200 eV, with good energy and angular resolution (W = 0.68 eV, ϕB = ±0.84◦ , θ = ±0.53◦ ) under fully symmetric conditions with WA = " = (T − WIj )/2 and θA = θB = 45◦ . Here WIj is again the ionization potential (= −binding energy) of the orbitals studied. The coincidence rate, that is the TDCS, is measured as a function of the azimuthal angle ϕB with a position sensitive detector behind a double toroidal analyzer through which both electrons pass. From the energy balance (8.36) the detected signal can be attributed uniquely to a certain binding energy, −WIj , which allows unique identification of the orbital from which the detected electron originates. In this geometry the recoil momentum q I = −q is a unique function of the azimuthal angle ϕB . Thus, assuming the PWIA to be valid (with frozen ion core), a direct determination of the momentum density distribution prior to the ionization process is possible for all orbitals

8.4


Fig. 8.22 Electron momentum spectroscopy (EMS) for the valence electrons of H2 O according to N ING et al. (2008): (a) experimentally determined momentum density distributions plotted as a function of binding energy and azimuthal angle ϕB . The recoil momentum qI = −q is a unique function of ϕB . Note the nodal planes in the 1b1 , 3a1 and 1b2 orbitals, in contrast to the 2a1 orbital. (b) Energy spectra of the valence orbitals obtained by integrating the density plots over all ϕB , projected onto the axis of binding energies (compare to Fig. 5.45(a))

561 30º 1b1 3a 1 φB

2a 1

1b 2

20º

(a)

10º 0º -10º -20º -30º 10 1b1

20

30

40 eV

(b)

3a 1 2a 1

1b 2 ×5 10

20

30

40 WI / eV

studied. To obtain a visual image of this momentum distribution (and thus of the wave function in the different orbitals) one plots the TDCS in a 2D presentation as a function of the binding energy and the azimuthal angle. For the valence electrons of H2 O this is displayed in Fig. 8.22(a). The projection of these density distributions onto the axis of the binding energies Fig. 8.22(b) reproduces nicely the spectra known from photoelectron spectroscopy (see Fig. 5.45(a)). One recognizes very clearly in Fig. 8.22(a) the different symmetries of the orbitals which are directly reflected in the momentum density distributions. We recall the discussion of the H2 O orbitals in Chap. 4. As schematically sketched in Fig. 4.22 we verify that 1b1 , 3a1 and 1b2 each have one nodal plane, while 2a1 does not have one. This is directly mimicked in the momentum density distribution which is the F OURIER transform of the spatial distribution. Nodes in the wave functions are reflected as minima in Fig. 8.22(a). In contrast, the 2a1 distribution shows no such minimum. When discussing these images one has, however, to bear in mind that the spatial orientation of gaseous free water molecules (as studied here) is statistically distributed. Thus, we cannot expect any information about this orientation from the momentum density distributions. A quantitative comparison between experiment and modern quantum chemical methods for the computation of these orbitals thus requires averaging over the orientation angle as well as a convolution over experimental resolutions. Carefully analyzed, EMS experiments may thus provide an important, critical test to molecular orbital calculations beyond the standard spectroscopy – which usually ‘only’ measures energies, albeit with high precision. EMS, in addition, contains information on the spatial structure of the wave functions which otherwise is not trivial to obtain.

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8


Section summary

• Electron impact ionization of atoms and molecules is both, of fundamental physical interest and of great practical relevance. The simplest process, H(e, 2e)H+ , represents a three body C OULOMB break up which cannot be solved exactly in analytic form – in spite of its apparent simplicity. In principle, the probability distributions of all final momenta can be determined, leading to information on four levels of detail: • An overview is obtained from the integral cross section as a function of initial kinetic energy T above ionization threshold WI . For small excess energies (few eV) the WANNIER /R AU threshold law σ (ion) ∝ (T − WI )1.127 is now well established. Beyond that, σ (ion) rises more or less linearly and reaches a maximum at about three to six times threshold. It then decreases slowly, essentially ∝ ln(T )/T as predicted by the B ORN -B ETHE approximation. Excellent semi-empirical fits are obtained with the L OTZ formula (8.44), very useful for practical applications. • The SDCS, dσ/dW , describes the energy partitioning between electron A and B in the final channel, with T − WI = WA + WB , and W the energy of the detected electron – both being in principle indistinguishable. The M OTT /RUDD formula (8.46) describes dσ/dW quite well, with symmetry between T − WI ≥ W ≥ 0 and a minimum at W = (T − WI )/2. • The DDCS dσ/dW dΩ specifies in addition also the emission angle of the detected electron. At high kinetic energies T and low momentum transfer K = k i − k A the B ORN -B ETHE approximation agrees reasonably well with experiment. It tacitely implies a model – valid for high T and negligible exchange – which distinguishes between the forward scattered electron (high kinetic energy) and the ejected electron (essentially scattered backward with low kinetic energy). For the former the GOSD is introduced by (8.55), in analogy to photoionization, which for very small K approaches the optical limit. • Finally, the TDCS, dσ/d"dΩA dΩB , also refers to the emission angle of the second electron. It contains maximum information on the ionization dynamics experimentally accessible. It depends in principle on four observables, the energy W = T − " − WI of one electron, its scattering angle θA (" being the energy of the “other” electron), and the polar and azimuthal scattering angles θB and ϕB of the latter. Characteristic for s states is a bimodal distribution of electron B, as beautifully imaged in Fig. 8.18: the binary peak arises essentially from interaction of the two electrons, the ionic core being just a spectator, while the recoil peak involves multiple interactions of the ejected electron with the latter. • Today, one may consider the (e, 2e) process as well understood for H and He atoms, based on sophisticated experiments and advanced theory. For more complex atoms and molecules it is still subject to current research. • EMS is an efficient application of the (e, 2e) process, providing detailed insights into molecular structure – complementary to standard spectroscopy.

8.5

Recombination

563

8.5

Recombination

8.5.1

Direct and Dielectronic Recombination

Finally we briefly discuss recombination, the process inverse to ionization. Direct inversion of the reaction scheme (8.36) would imply three particle interaction and is thus extremely improbable – except in very dense plasmas. However, the process inverse to photoionization, whereby an electron is captured by an ion, is frequently observed. It is, as such, an interesting process, and again also important – e.g. for a quantitative description of plasmas. The excess energy which becomes available in such processes is directly or indirectly released as a photon. Particularly well studied is recombination with highly charged ions (HCI) for which cross sections are large (the interested reader will enjoy the informative review of M ÜLLER 2008, which covers the general field of electron-ion scattering). We have encountered HCIs already in Sect. 7.5 as interesting collision partners. The most simple case is the direct or radiative recombination Aq+ + e− (slow) → A(q−1)+ (f ) + hν,

(8.64)

where the photon is emitted immediately during the transition of the captured electron from the continuum into a bound state |f of the target. The cross sections are usually rather small – essentially due to the limited phase space of the final state, i.e. the density of states is much larger in the continuum than in bound states. The cross sections do, however, increase with decreasing electron energy and may become significant. An altogether different situation is encountered with the so called dielectronic recombination, which is kind of inverse AUGER process (see Sect. 10.5.1, Vol. 1). Schematically one may view this process as being composed of three phases, sketched in Fig. 8.23. Fig. 8.23 Schematic sequence of a dielectronic recombination process – from left to right: (a) a continuum electron is captured and an electron from an inner shell is simultaneously excited; (b) the excited electron releases its energy by photoemission; (c) the captured electron remains in the highly excited RYDBERG state whose lifetime is very long

(a)

(b)

(c)

T( n ) 0 Wn

|n〉

Wf

|f 〉 WI hν

Wf i | i〉

Wi -W

Aq+

[A(q -1)+]**

[A(q -1)+]*

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8


The critical first step (a) is only possible if energy is conserved, i.e. if the kinetic energy of the electron T (n) together with the (negative) binding energy Wn of the state |n just corresponds to the excitation energy Wf i = Wf − Wi (one may call this exchange of a virtual photon): T (n) − Wn = Wf i .

(8.65)

The short-lived state |f looses its energy by spontaneous emission of a photon hν. However, the electron is typically captured into long-lived, high lying RYDBERG states. Thus, the state |n remains occupied while the inner electron is typically found in its initial state. We note that dielectronic recombination – in contrast to direct radiative recombination – is a resonant process which occurs only at well defined energies of the free electron to be captured: 2 Eh qeff T (n) = Wf i − . (8.66) 2 n2 We have here rewritten (8.65) and assumed that only highly excited states |n of the ion A(q−1)+ capture an electron. For these we may apply the RYDBERG formula for hydrogen like atoms, if necessary considering incomplete core screening by qeff .

8.5.2

The Merged-Beams Method

Electron-ion collisions (see M ÜLLER 2008) at low kinetic energies are studied with advantage in so called merged-beams. They are used to investigate elastic scattering, impact excitation and multiple ionization of ions, and of course electron-ion recombination. Figure 8.24 shows (very schematically) a typical setup, this one has been used by B ÖHM et al. (2002) at the Stockholm heavy ion storage ring CRYRING. The electron beam originates from a thermionic cathode, is accelerated to some 100 eV, focussed and guided by toroidal magnetic fields to merge with the target Aq+ ion beam (3.3 MeV to 9.4 MeV). After interaction the two beams are separated in the same manner. Recombination is detected with the help of a dipole magnet which separates the parent ion Aq+ and products A(q−1)+ . One finally measures the A(q−1)+ ion yield as a function of electron energy (and thus as a function of available relative kinetic energy T in the CM system). The experiment is by no means trivial. For important details we refer to the original publication of B ÖHM et al. (2002). Of general interest are the advantages of the parallel trajectories of ions and electrons in this merged-beams setup. Not only Fig. 8.24 Example of a typical merged-beams experiment (very schematic); this specific setup was used by B ÖHM et al. (2002) to study dielectronic recombination

thermionic cathode electron beam ion beam Aq+

Aq+

toroidal correcting dipole magnets magnets magnet

A(q-1)+ recombination detector

8.5

Recombination

565

does it allow for a sufficiently long interaction region, the adjustment of the relative energy in this setup is also advantageous. For kinetic energies (masses) Te (me ) and TI (mI ) of electron and ion, respectively, the relative kinetic energy is given by19 T=

M¯ 2 M¯ vrel = (ve − vI )2 2 2

(8.67)

with the velocities ve,I = 2Te,I /me,I of electron or ion, and the reduced mass is M¯ = me mI /(me + mI ). At a given ion energy, say of 7 MeV for oxygen, the required relative kinetic energies of 2 eV ≤ T ≤ 15 eV may be obtained by laboratory electron energies between 200 eV and 375 eV, which are much easier to generate and control than in a genuine, low energy electron beam. At the same time the energy width of the beam is reduced by a factor between 10 and about 5. Thus, the large energy width of electrons from a thermionic cathode is significantly decreased without dedicated energy selection.

8.5.3

Some Results

As a characteristic example we discuss electron recombination with fivefold ionized oxygen atom O5+ studied by B ÖHM et al. (2002). The process investigated in this experiment may be viewed as (8.68) O5+ 1s 2 2s + e− → O4+ 1s 2 2pn → O4+ 1s 2 2sn + hν. With a transition energy W2p←2s = 11.95 and 12.02 eV for the 2 P1/2 and 2 P3/2 levels of O4+ , respectively, and with qeff = 5 according to (8.66), the lowest accessible RYDBERG state that can be populated will have a principle quantum number n = 6, corresponding to T (n) 2.5 eV. The experimental data shown in Fig. 8.25 even allows to recognize 6 states to different orbital angular momenta . The results are published as rate constants σ vrel , i.e. as product of cross section and relative velocity, averaged over the energy distribution of electrons and ions. The population of higher, unresolved RYDBERG states up to the continuum can be recognized clearly. A significant fraction of the total recombination strengths is accumulated there. The fact that the signal decreases strongly at the ionization threshold is attributed to quenching of high lying RYDBERG states by the electric and magnetic fields used in the optics for guiding ions and electrons, i.e. it is considered an experimental artifact (B ÖHM et al. 2002). Profound understanding of the underlying physics has been achieved in this field of research, not the least owing to sophisticated experimental methods and ambi19 At

the velocities used here of less than 0.05c one does not yet need to compute these numbers relativistically – even though this is possible without problems (see Eq. (45) in M ÜLLER 2008). The results are identical within the experimental energy resolution.

Fig. 8.25 Dielectronic recombination rate for O5+ + e− → O4+ (n) according to B ÖHM et al. (2002). The rate is plotted as a function of relative electron kinetic energy in the CM system. Note the good energy resolution which allows to identify the resonant population even of high lying RYDBERG levels

8 rate cofficient / 10 -10 cm 3 s -1

566


1.5 O5+ n=6

7

8

9 10 12 14

∞

1.0

0.5

6d 6p 6s

0 2 4 6 8 10 12 electron-ion collision energy T / eV

tious theory. Beyond the intellectual interest a variety of applications are based on this knowledge, in particular in plasma physics. Section summary

• Recombination of a free electron with an ion may occur in a direct, nonresonant process where all excess energy is released by emission of a photon. Alternatively and more efficiently the so called dielectronic recombination occurs resonantly: the electron populates a particular RYDBERG state and the excess energy is used to excite another electron. • Such processes may be studied in merged-beams experiments which allow to efficiently control very small relative kinetic energies between ions and electrons with good energy resolution. • Dielectronic recombination is particularly efficient with HCIs. Results for e− + O5+ are discussed as an illustrative example.

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. a.u.: ‘atomic units’, see Sect. 2.6.2 in Vol. 1. BSR: ‘B-spline atomic R-matrix codes’, a general program (beyond CCC) to calculate atomic continuum processes, including electron-atom and electron ion scattering and radiative processes (Z ATSARINNY 2006). CC: ‘Close-coupling’, calculations, computation of scattering cross sections by solving (part of) the coupled integro-differential equations (see Sect. 8.1.1). CCC: ‘Convergent close-coupling’, calculations, special solutions of the coupled integro-differential equations for collisions (see Sect. 8.1). CM: ‘Centre of mass’, coordinate system (or frame) (see Sect. 6.2.2). COLTRIMS: ‘Cold target recoil ion momentum spectroscopy’, see Appendix B.4. DCS: ‘Differential cross section’, see Sect. 6.2.1. DDCS: ‘Double-differential cross section’, in e, 2e ionization processes (see Sect. 8.4).


567

DW: ‘Distorted wave’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.1.2). DWB: ‘Distorted wave B ORN approximation’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.1.2). ECS: ‘Exterior complex scaling’, method for solving the close-coupling equations for scattering problems. EELS: ‘Electron energy loss spectroscopy’, see Sect. 8.3.2. EMS: ‘Electron momentum spectroscopy’, method to determine the momentum distribution of electrons in atoms and molecules exploiting e, 2e processes (see e.g. M C C ARTHY and W EIGOLD 1991). FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects. 6.6 and 5.5.2, Vol. 1, respectively). FWHM: ‘Full width at half maximum’. GA: ‘G LAUBER approximation’, method for approximate solution of the close coupling equations in electron scattering (see Sect. 8.4.5). good quantum number ‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the H AMILTON operator (see Sect. 2.6.5 in Vol. 1)’. GOS: ‘Generalized oscillator strength’, characterizes the strength of electron impact excitation in analogy to the optical oscillator strength see Sect. 8.3.2. GOSD: ‘Generalized oscillator strength density’, characterizes the strength of electron impact ionization per energy interval in analogy to the optical oscillator strength density (see Sect. 8.4.4). HCI: ‘Highly charged ions’, see Sect. 7.5. MO: ‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation. ODE: ‘Ordinary differential equation’. OOSD: ‘Optical oscillator strength density’, characterizes the strength of photoionization per energy interval (see Sect. 5.5.1 in Vol. 1). PES: ‘Photoelectron spectroscopy’, see Sect. 5.8. PWIA: ‘Plane wave impulse approximation’, basic assumption for EMS (see Sect. 8.4.6). SBA: ‘Second order B ORN approximation’, second order term in the B ORN series (see Sect. 6.6). SDCS: ‘Single-differential cross section’, in e, 2e ionization processes (see Sect. 8.4). TBW: ‘Three body approximate scattering wave function’, accurate method for calculating triple-differential cross section for ionization (B RAUNER et al. 1989). TDCC: ‘Time dependent close-coupling calculations’, a method, in principle accurate, for solving the S CHRÖDINGER equation for scattering problems (C OLGAN and P INDZOLA 2006). TDCS: ‘Triple-differential cross section’, in e, 2e ionization processes (see Sect. 8.4). UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007).

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VIS: ‘Visible’, spectral range of electromagnetic radiation. Wavelengths between 380 nm and 760 nm according to ISO 21348 (2007). VMI: ‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B). VUV: ‘Vacuum ultraviolet’, spectral range of electromagnetic radiation, part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007). XUV: ‘Soft x-ray (sometimes also extreme UV)’, spectral wavelength range between 0.1 nm and 10 nm according to ISO 21348 (2007), sometimes up to 40 nm. ZEKE: ‘Zero kinetic energy’, photoelectron spectroscopy (see Sect. 5.8.3).

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9

The Density Matrix – A First Approach

Quite a few readers will associate with “density matrix” a rather dry concept of advanced lectures and textbooks in quantum mechanics, to be avoided if possible. However, this concept is indispensable for the interpretation of measurable observables as soon as a real quantum system cannot be fully described by a single set of quantum numbers – and that is, unfortunately, the most commonly encountered situation in experimental physics. Thus we try here a heuristic, pragmatic way to access this important tool, and to illustrate its usefulness by some simple, practical examples. We thus hope to communicate that the density matrix is indeed a very useful tool for everyday’s research which, in principle, is easy to handle.

Overview

Those who want to benefit from reading the next chapter and other advanced text in AMO physics and are not yet familiar with the density matrix, should study the summary given in this chapter with some care. After some introductory remarks we shall start in Sect. 9.1.1 by defining the concept of pure and mixed states. In Sect. 9.1.2 we formally introduce the density operator and in Sect. 9.1.3 its matrix representation which will be illustrated by some simple examples. A general formalism for describing a physical measurement is derived in Sect. 9.2 and applied in Sect. 9.3 to two typical examples. Finally, in Sect. 9.4 we present an introduction to the general theory of radiation from quantum systems in non-isotropic and oriented excited states, based on the formalism developed by FANO and M ACEK (1973). Up to now, we have usually assumed that the quantum systems discussed can be characterized by states or wave functions with a set of well defined quantum numbers – implying that it is completely known prior to an interaction to be discussed, i.e. at time t → −∞. The evolution under the influence of a H AMILTON operator describing the whole system occurs then according to the time dependent S CHRÖDINGER equation. At a later time t → +∞ the whole system is still fully characterized by a wave function, typically a coherent superposition of several


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states. This is strictly true for any completely isolated quantum system, as long as the whole system is considered. In the real world, however, the basic assumption of pure states is hardly ever a realistic description of the situation in an actual experiment. We often do not know the initial state completely, but even if we do the final outcome of an experiment must not necessarily be a pure state either. Consider e.g. the thermal population of rotational and vibrational states of a molecule and their alignment and orientation is space. It may only be described by a probability distribution. Also, we often deal with quite complex quantum systems which frequently consist of several subsystems (photons, electrons, atoms, molecules) whose individual fate cannot be followed in details – and may not be of interest. Thus, at the end of an experiment as a rule only some characteristic observables of a quantum system are measured. In the language of quantum mechanics: we project the state of the whole system onto a certain subset of states which are experimentally accessible. Very often the system of interest interacts with a dissipative environment – a so called “bath” which is a very large quantum system whose state cannot be described in quantum mechanical detail. There, our nice, coherent states are flushed away, so to say. Hence, real experiments always average over the unobserved quantum numbers, and this has to be accounted for when comparing with theoretical predictions. The density matrix provides a convenient method of book-keeping for this purpose. We mention that there are indeed approaches to completely avoid the concept of the density matrix or density operator. One may instead use expectation values of certain tensor operators (multipole moments) which are constructed from angular momentum operators (see Appendix C in Vol. 1). As it turns out, these quantities are equivalent to irreducible components of the density matrix. – But even if one finds the density matrix “sometimes frightening” (Z ARE 1988) it remains dubious whether these irreducible tensor operators are more comforting to the ordinary reader. On the contrary, we hope to show that the density matrix offers in many cases a much more direct access to standard physical intuition. Nevertheless – as we shall briefly outline in Sect. 9.4 – the representation of certain experiments in terms of such multipole moments allows a very convincing disentanglement of dynamical parameters (characterizing a physical interaction process) from purely geometric factors (which characterize a specific experimental setup). Also, for irreducible tensor moments it is often much simpler to perform rotations in space and recoupling of angular momenta (if necessary) – than for the density matrix itself. Hence, this elegant ansatz has been used by FANO and M ACEK (1973) to describe the radiation characteristics and the polarization of light emitted by atoms after collisional excitation. M ACEK and H ERTEL (1974) extended this concept for collisions with laser excited atoms, and G REENE and Z ARE (1983) further developed it for the analysis of laser induced fluorescence from aligned molecules. The density matrix and its irreducible moments are treated in several textbooks quite comprehensively (A NDERSEN and BARTSCHAT 2003; M UKAMEL 1999; K LEIMAN et al. 1998; B LUM 2012; B RINK and S ATCHLER 1994; W EISSBLUTH 1989; Z ARE 1988). We just try here to give a compact version of some important

9.1

Some Terminology

575

state-of-the-art tools for the conception and interpretation of many advanced experiments in laser spectroscopy and collision physics. In this presentations – so to say for pedestrians – we shall refrain (as elsewhere in these books) from presenting lengthy mathematical excursions and detailed proofs and built on the willingness of the readers to simply accept certain results and concepts – and find out that they are useful.

9.1

Some Terminology

9.1.1

Pure and Mixed States

To have something concrete in mind, Fig. 9.1 shows – very schematically – three typical experimental arrangements for studying atomic and molecular interaction processes with state selection and/or analysis. Figure 9.1(a) represents what may be called an ‘ideal’ interaction experiment between two subsystems (A, B) – we may think of a scattering experiment or a chemical reaction – where the quantum states |γA |γB of both partners before the interaction process are assumed to be well characterized. One may achieve this by suitable state selectors (sel A, B), the physical details of which is of no interest to us at this point. The total system prior to interaction may be described by a wave function of the type r A r B R | γA γB k = eikR φ (A) (r A )φ (B) (r B ),

(9.1)

with r A and r B being the coordinates, γA and γB the quantum numbers of all interior degrees of freedom for A and B, respectively (including spin variables). The relative momentum1 between A and B is characterized by k, while R describes their relative coordinates. The fact that the total wave function can be factorized expresses the independence of the subsystems prior to the interaction. During the interaction process the state of the system changes. A formal description for a scattering process has been presented in Sects. 6.4.6 and 8.1, using the transition operator T (T-matrix) of the process. It transfers |γA γB k i asymptotically (t → ∞) into T|γA γB k i . In principle, both subsystems (A, B) may undergo changes, they may e.g. be excited, ionized, rearranged internally or even form new products (C, D). The final state T|γA γB k i and the respective wave function may thus describe a rather complex situation. In principle one can no longer assume that final state may still be written as a product state φC (r C )φD (r D ) of the subsystems (C, D) after the process. To use a fashionable word: as a rule one deals with entangled states of the subsystems after the interaction. (See also Appendix E.3 in Vol. 1.) The constituents ‘know’ of each other even at time t → ∞. In an ‘ideal’ interaction experiment one will try to obtain as much information about the final states of the subsystems after the interaction, by using suitable state 1 We

use here again, somewhat loosely, wave vector (k) synonym for momentum (k).

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(a) A

sel A

(b) C

anl C A

B

D

sel B


anl D

sel A

anl C anl D

(c) A bath B

Fig. 9.1 Schematics of prototypical interaction experiments: one tries to characterize the quantum state of the system prior and after the interaction (T) as good a possible by using state selectors (sel) and analyzers (anl), respectively; (a) reaction or collision experiment between partners A and B; (b) photoexcitation process in an isolated system; (c) photoexcitation of A in a “bath” B

analyzers (anl C, D) to determine the quantum states |γC and |γD of the separated reaction products C and D. And this should be done in a correlated fashion, i.e. by detecting C and D in (if necessary delayed) coincidence. Using quantum mechanical terms, this detection process projects the final state T|γA γB k i of the total system onto product states |γC γD k f of the subsystems, with k f representing the magnitude and direction of the final relative momentum of C and D. Thus one obtains a scattering amplitude γC γD k f |T|γA γB k i and a cross section corresponding to (7.31) and (7.33). This kind of measurement implies automatically a significant reduction of the full reality – which is described by T|γA γB k i . In addition, only in exceptional cases it will be possible to determine all relevant quantum numbers of all subsystems after the interaction process. In a quite similar manner the experiment indicated in Fig. 9.1(b) may be described: one or more photons interact with a well prepared atom or molecule A. In this process A may e.g. be excited, subsequently undergo internal rearrangement, re-emit photons and finally may even fragment. Here too, new end products will be created (C, D). And the T operator is in this case the interaction operator for the atom in an electromagnetic field which we have encountered already in Chap. 4, Vol. 1 – as far as necessary supplemented by the description of further reaction steps. Still somewhat more complicated is the experiment sketched in Fig. 9.1(c). Formally, however, it will be described in very similar terms. In this case, the T operator has to describe also the interaction with the bath B, which in the individual situation may be rather complicated – even if one assumes that prior to photo-absorption A was very well characterized and did not interact with the bath.

9.1

Some Terminology

577 z (sel)

angular collimation by apertures source A

x (sel) velocity selector

m state selector

x (col) z (col) y (col)

Fig. 9.2 State selection in a particle beam

For one particularly simple, special case we shall now specify this general description in order to illustrate the concept. Be A an atom which initially is found in a γ 2 S1/2 state. It may assume two sets of quantum numbers: {γ 2 S1/2 , m = +1/2} or {γ 2 S1/2 , m = −1/2}, i.e. the electron spin may point up or down. These two states (let us call them |+ and |− for briefness) correspond then to the initial state |γA γB k i in (9.1). Figure 9.2 shows a schematic of state selection prior to an interaction experiment. Let us have a somewhat closer look at it. After collimation (by apertures) and velocity selection, the atomic beam may be described approximately as a “plane wave”. It passes an m state selector (e.g. a S TERN -G ERLACH magnet or a circularly polarized laser beam employed for optical pumping as described in Appendix D) which prepares one of the states |+ or |−. We emphasize that such a selection characterized by + or − always refers to a well defined z(sel) -axis of the m selector, i.e. to the direction of the magnetic field B in a S TERN -G ERLACH magnet or to the direction of light propagation in the case of optical pumping. Note that the interaction experiment is often described with advantage in a different coordinate system, e.g. in the standard collision frame as introduced in Sects. 6.4.2 and 7.3.1, denoted by x (col) y (col) z(col) .

Case (a): Pure State in the Selector System Let us first assume the most simple situation, i.e. let both coordinate systems be the same, and let the selector transmit a beam in which to the choice of the experimenter 100 % of the atoms are exactly in one of the states, |+ or |−. We thus deal with an eigenstate (9.2) |γ = |φ0 = γ 2 S1/2 ± 1/2 = |± 2 of the angular momentum operators J and Jz . In general, an ideal state selector prepares a quantum system in an eigenstate |γ of some physical operator, representing a set of physical observables. These are then characterized by the set of quantum numbers {γ }.

Case (b): Pure State in the Collision System However, the thus prepared states |+ or |−, given by (9.2) with respect to the selector system (sel), will typically have to be transformed into a coordinate system in which the interaction process is described most conveniently. This may e.g. be the collision frame (col), here with z(sel) x (col) . As explicated in Appendix E.2, Vol. 1 one has to rotate the latter into the selector system (sel) through the E ULER angles

578

9


(0, π/2, π) so that the z(col) axis points into z(sel) direction. One obtains (E.20), Vol. 1 which we write here ∓1 |φ0 (sel) = |± = √ |+1/2 ± |−1/2 . 2

(9.3)

The states |±1/2 are characterized by the projection quantum numbers m = ±1/2 with respect to the z(col) -axis.2 We emphasize that (9.3) is a linear (or coherent) superposition of eigenstates. More generally, any so called pure state may be written as a linear superposition of basis states |γ |α = aγ(α) |γ (9.4) γ (α) aγ .

with the probability amplitudes One has to sum over all quantum numbers which characterize the system studied. With the eigenfunctions φγ (r) which belong to these quantum numbers γ we may write |α also as a wave function: φ0 (r) = r | α = aγ(α) φγ (r). (9.5) γ

Case (c): Imperfect Selector – Description in the Selector System Let us first assume now that we do not have a state selector for m states at all. Instead of what has been said in case (a), we can now only describe the state of the atomic beam by saying that the system is either in state |γ 2 S1/2 + 1/2 or in state |γ 2 S1/2 − 1/2 with respect to the given coordinate system. There is simply no further information about the system. But in the real world, even a state selector will never select a 100 % pure basis state. Rather, we expect that the atoms after passing a real selector are found with the probabilities p+

in state |+ and with

p−

in state |−.

(9.6)

Thus, reality confronts us with the situation that the initial state can no longer be described by a pure state (or a wave function) – not even by any kind of linear combination of pure states. Such a state of the system is called a mixed state or an incoherent superposition of pure states. Of course the probabilities p± are normalized such that still p+ + p − = 1

(9.7)

holds. The quality of the selector (or of the selected atomic beam) is described by its polarization Eq. (E.20), Vol. 1 these spin states |±1/2 are called |α and |β, respectively, not to be confused with the state designation in the following text.

2 In

9.1

Some Terminology

579

P = p+ − p − =

I+ − I− , I+ + I−

(9.8)

where I + and I − stands for the measurable beam intensities of atoms with spin up or spin down, respectively.

Coherent or Incoherent? At this point a warning is appropriate against too naive usage of the terms coherent and incoherent. For example, the question about the probability to find the states |+1/2 or |−1/2 with respect to the collision coordinates (col) in case (b) would correctly be answered with p+1/2 = p−1/2 = 1/2. But clearly, these probabilities do not fully characterize the pure states (9.3): we deal here with a coherent superposition of two states. According to (9.2) the same two states written in selector frame (sel) are |+ and |−. And the respective probabilities in this coordinate system are {p+ = 1, p− = 0} and {p+ = 0, p− = 1}, respectively, depending on the choice of selector transmission. It is evident that the probability to find a specific basis state depends on the coordinate system. With reference to Fig. 9.2: only when considering the state of the atoms A before the beam enters the selector we find a completely incoherent mixture of states {p+ = p− = 1/2} which does not change upon changing the coordinate system {p+1/2 = p−1/2 = 1/2}. This may be generalized to the case of N possible basis states |γ = |1, |2 . . . |N with probabilities p1 , p2 . . . pN .

(9.9)

Again, in general such probability distribution refers to a particular coordinate system and is normalized: N

pγ = 1.

(9.10)

γ =1

Thus, if we do not know more than this probability distribution we cannot distinguish whether we deal with a pure or mixed state. A quantum system is said to be in a pure state if and only if a linear combination (9.4) of basis states exists, which fully describes the system. Conversely, a quantum system with quantum numbers γ is said to be in a fully incoherent state if and only if each of the basis states |γ is found with equal probability. As a consequence of (9.10) this implies pγ = 1/N for each basis states |γ with respect to any coordinate system. To check this quality in our case (in a “Gedanken experiment”), one would have to analyze the fully or partially selected beam emerging from the selector with a

580

9


second (if possible ideal) selector, the reference axis of which could be aligned into an arbitrary direction in space. If the first selector would have prepared a pure state, then the second selector would transmit the beam to 100 % only if aligned into one specific direction in space. If the state prepared by the first selector is a mixed state, always less than 100 % will pass the second selector irrespective of its alignment.

Case (d): Imperfect Selector – Description in the Collision System As in case (b) we now change again our perspective and describe the incompletely selected beam in the collision system. We assume that in the selector system (sel) the states |+ and |− were prepared with probabilities p+ and p− , respectively. With reference to the collision system (col) we must characterize the system as being in a mixed state with the probabilities −1 for |+ = √ |1/2 + |−1/2 and 2 1 for |− = √ |1/2 − |−1/2 . 2

p+ p−

(9.11)

Again, there is no possibility to transform this description by a linear transformation of basis states into a single pure state. This is not surprising since we still discuss the same incoherent superposition of states as in (9.6), only described in a different coordinate system. We see, however, that the coordinate transformation may allow to see a certain degree of coherence even in the mixed state – it describes the presence of two states with different probabilities which as such are coherent superpositions of basis states |1/2 and |−1/2. Thus, one has to be very careful when using the terms coherence and incoherence. Hence, it is desirable to obtain a quantitative description of terms such as “mixed state”, “degree of coherence” and “degree of polarization”. This becomes even more evident in the general case of mixed states which are described by an incoherent superposition of pure states |α – i.e. by a set of probabilities with respect to a reference coordinate system: p1 for state |1 = aγ(1) |γ γ

for state |2 =

p2

aγ(2) |γ

γ

.. . pN

for state |N =

γ

again with

N α=1

pα = 1.

aγ(N ) |γ

(9.12)

9.1

Some Terminology

581

Obviously, this kind of description of an initial state (not to speak of the final state after an interaction) is rather clumsy – in particular in comparison with pure eigenstates |γ , which we have assumed so far tacitly as starting point for our considerations (also in Vol. 1). When including the detection process into the description, the situation becomes increasingly complex (see Fig. 9.1), since the analyzers too are usually far from perfect.

9.1.2

Density Operator and Density Matrix

We thus need an efficient and clear method of bookkeeping. For this purpose one defines a density matrix of the states to be described. In the most trivial case of a pure state |α according to (9.4) its elements are given by γ |ρˆ (α) |γ = ργ γ = aγ aγ∗(α) . (α)

(α)

(9.13)

In compact form the density operator of a pure state |α is written as ρˆ (α) = |αα| =

γ γ

ργ γ |γ γ |. (α)

(9.14)

(α)

In the case of mixed states according to (9.12), with amplitudes aγ of individual states α = 1 . . . N , the density operator it is simply given as the weighted sum of its components: ρˆ =

N α=1

pα ρˆ (α) =

N

pα |αα| =

γ γ

α=1

pα aγ aγ∗(α) |γ γ |. (α)

(9.15)

α

A compact form of the density operator is ρˆ =

ργ γ |γ γ |,

(9.16)

γ γ

with its matrix elements extracted from (9.15) ργ γ = γ |ρ|γ ˆ =

N α=1

(α)

pα ρ γ γ =

N

(α) pα aγ aγ∗(α) = aγ aγ∗ ,

(9.17)

α=1

aγ aγ∗ being an abbreviation for the averaging. One verifies easily that the density matrix is Hermitian: ργ γ = ργ∗ γ .

(9.18)

582

9


In the following, the trace of the density operator will play an important role, also in combination with other operators. For normalized, pure states a (α) 2 =α | α= 1 ργ(α) (9.19) Tr ρˆ (α) = γ = γ γ

γ

holds, and the same is true for the trace of any density operator: Tr ρˆ =

ργ γ =

γ

pα ργ(α) γ =

αγ

pα = 1.

(9.20)

α

The density matrix3 is thus a convenient abbreviation for lengthy expressions required for a realistic description of an averaging processes which typically occurs in a measurement. of an arbitrary observable O in a Let us work out the expectation value O mixed state according to (9.12). We first recall from (2.42), Vol. 1 that for a pure state ∗(α) = α|O|α = aγ(α) . O aγ γ |O|γ (9.21) γ γ

For a mixed state, in addition one has to average over all pure states |α involved: = O

= pα α|O|α

N ∗(α) γ |O|γ pα aγ aγ(α) . γ γ

α

(9.22)

α=1

With the definition (9.17) this may be written = O

ργ γ Oγ γ = Tr(ρˆ O),

(9.23)

γ γ

in the basis {|γ }. We shall where Oγ γ are the matrix elements of the observable O meet such kind of relations in several contexts as useful abbreviations for measurable quantities. All these definitions may appear still somewhat abstract, even artificial. They turn out, however, to be very useful if one tries to formulate the theory of measurement for specific examples in some detail. Before doing so we need to familiarizes ourselves somewhat better with the density matrix.

9.1.3

Matrix Representation for Selected Examples

It is often useful to write down the elements of a density matrix explicitly. We do this for the four cases discussed in Sect. 9.1.1, where we have described an atomic 3 One uses the terms density operator and density matrix more or less synonymous, the latter emphasizing the concrete representation by a matrix.

9.1

Some Terminology

583

beam with γ 2 S1/2 atoms which were (partially) selected according to their quantum numbers.

Case (a): Pure State Observed in the Selector System In selector coordinates the density matrices for the pure |+ and |− states are

0 1 0 ρˆ = = and 0 0 0 0 0 0 0 (−) ρˆ = = , respectively. 0 ρ 0 1 (+)

ρ++ 0

(9.24) (9.25)

Case (b): Pure State Observed in the Collision System With respect to collision coordinates as illustrated in Fig. 9.2 we obtain with the definitions (9.3) and (9.13) for the very same, pure initial states ρ1/2 −1/2 ρ1/2 1/2 with ρ−1/2 1/2 ρ−1/2 −1/2 1 1 1 1 1 −1 (+) (−) ρˆ = and ρˆ = , 2 1 1 2 −1 1

ρˆ (±) =

(9.26) respectively.

(9.27)

We recognize now the characteristic off-diagonal terms ρ−1/2 1/2 , which imply coherence among the substates |+1/2 and |−1/2. They ensure that these density matrices represent pure states, while the diagonal matrix elements |a1/2 |2 = |a−1/2 |2 = 1/2 just give the probability to find states |+1/2 and |−1/2, respectively, with reference to the collision coordinates (col). This may, e.g., be verified with a second selector which is aligned along the z(col) -axis.

Case (c): Imperfect Selector Observed in the Selector System We consider a real experiment with a partially selected atomic beam in which the states |+ and |− are populated with the probability p+ and p− , respectively. By definition (9.17), we obtain the density matrix for this mixed state in respect to the selector coordinates (sel) from (9.24) and (9.25): ρˆ = p+ ρˆ (+) + p− ρˆ (−) =

p+ 0

0 p−

=

1 2

1+P 0

0 1−P

.

(9.28)

In the last equality we have used the polarization P as defined in (9.8). A completely unpolarized, mixed state (P = 0) is represented by ρˆ =

1 2

1 0 0 1

1 1. = 2

(9.29)

584

9


This matrix would describe e.g. the initial atomic beam prior to entering the selector. Note that the zero off-diagonal matrix elements in (9.28) and (9.29) indicate a complete lack of coherence between the states |+ and |−.

Case (d): Imperfect Selector Observed in the Collision System Let us describe the same situation as in case (c), now with respect to the collision system (col). According to (9.17) we obtain now with (9.26) and (9.27) p− 1 −1 p+ 1 1 + (9.30) ρˆ = 2 1 1 2 −1 1 1 1 P 1 1 p+ − p− = . = 1 2 p+ − p − 2 P 1 Now the density matrix has a nonvanishing off-diagonal matrix element which documents (partial) coherence between the basis states |1/2 and |−1/2. We thus clearly recognize that any quantitative expression of coherence among basis states depends on the choice of the basis system: (9.28) and (9.30) describe exactly the same mixed state but have different off-diagonal matrix elements. Mere rotation of the coordinate system may thus ‘generate’ or ‘destroy’ coherence terms (here P)! This should be born in mind if one reads or writes about the observation of coherence in an experiment. Only for a completely unpolarized state the density matrix is diagonal independent of the coordinate system – as one verifies by inserting P = 0 into (9.28) and (9.30).

The General Case We may now summarize the above considerations and generalize the formalism to a system which consists of N basis states. Using the notation (9.16) and (9.17) for the density matrix of a mixed state, we may write it as ⎛

ρ11

⎜ ⎜ ρˆ = ⎜ ⎝ c.c. c.c.

ρ12 ρ22

··· ··· .. .

⎞

⎛

a1 a1∗ a1 a2∗ ⎟ ⎜ a2 a2∗ ⎟ ⎜ ⎟=⎜ ⎠ ⎝ c.c. ρN N c.c. ρ1N ρ2N .. .

··· ··· .. .

∗ a1 aN ∗ a2 aN .. .

∗ aN aN

⎞ ⎟ ⎟ ⎟ , (9.31) ⎠

while for a pure state the averaging ai ak∗ over the amplitudes is suppressed. This general form of the density matrix may look somewhat complicated, and in general one does not recognize immediately what kind of state is described by it – a pure or a mixed state. All we can say is that the probability to find the states |1, |2 . . . |N is given by ρ11 , ρ22 . . . ρN N and that a certain coherence between the state |i and |k exists if ρik = 0. The picture becomes much clearer by diagonalizing (9.31). This may be achieved using the rules of linear algebra by a suitable unitary transformation

9.1

Some Terminology

585

{|γ } = {|γ (d) } of the basis states |γ . One thus obtains a density matrix in diagU onal form: ⎞ ⎛ (d) ρ11 0 0 ··· 0 ⎟ ⎜ (d) 0 ··· 0 ⎟ ρ22 ⎜ ⎜ .. ⎟ .. ⎟ ρˆ U † = ⎜ . (9.32) ρˆ (d) = U ⎜ . ⎟. ⎜ .. ⎟ .. ⎟ ⎜ . ⎝ . ⎠ (d) 0 ρN N Such a transformation removes the coherence terms which depend on the choice of basis states. It allows one to express the density matrix exclusively by the probabilities ρ (d) for finding the states |γ (d) in this particular choice of basis states. The magnitude of these probabilities gives a clear measure for the degree of purity or mixture of states under consideration. There are two extreme cases: for a pure state one and only one diagonal element (=1) remains, all others disappear: ⎛ ⎞ 0 0 ⎜ ⎟ 0 ⎜ ⎟ (d) ⎜ ⎟. 1 (9.33) ρˆpure = ⎜ ⎟ ⎝ ⎠ 0 0 0 In contrast, the extreme opposite is a mixed (unpolarized) state where all basis states are found with equal probability p = 1/N : ⎞ ⎛ 1/N 0 0 ··· ··· 0 ⎜ .. ⎟ ⎜ . ⎟ 1/N 0 ⎟ ⎜ 1 ⎜ (d) .. ⎟ = ρûnpol = ⎜ . (9.34) . ⎟ 1/N ⎟ N ⎜ ⎜ .. ⎟ ⎝ . ⎠ 1/N Note that the matrix representation of this particular state is completely independent of the choice of the basis: any unitary transformation of the identity matrix contained in (9.34) reproduces simply the identity matrix. We have seen this already for the 2 × 2 matrices according to case (c) and (d), if P = 0 is inserted into (9.28) and (9.30), respectively.

9.1.4

Coherence and Degree of Polarization

We shall now try to formulate the above discussion quantitatively without recurring to the sketched diagonalization procedure, which can be rather tiresome. What would be desirable is a single, easy to compute parameter which allows us to quantify the degree of mixture or purity in between the two extremes just defined.

586

9


We first recall that the trace of a density matrix is always Tr ρˆ = 1 per definition (9.20) – for a pure as well as for any mixed state. In contrast, the trace of the square of a density matrix is unity Tr ρˆ 2 = 1 for a pure state, as obvious from (9.33). However, for a completely unpolarized state Tr ρˆ 2 = N × 1/N 2 = 1/N holds, as evident from the diagonal form (9.34). For any other mixed state one finds values in between the two extremes. This is again evident for the two diagonal representations of ρ. ˆ And since the trace of any matrix is independent of any unitary transformation, the general relation also holds for any density matrix: 1 ≥ Tr ρˆ 2 ≥ 1/N .

(9.35)

The limits 1 and 1/N refer to a pure and a completely unpolarized state, respectively. Let us familiarize ourselves with these terms for the example of a 2 × 2 density matrix ρ11 ρ12 . (9.36) ρˆ = ρ21 ρ22 The diagonal matrix elements are real (they represent probabilities) and Tr ρˆ = ρ11 + ρ22 = 1, while the off-diagonal matrix elements may be complex with ∗ . Hence, three real parameters suffice to fully characterize the most general ρ21 = ρ12 2 × 2 density matrix, let us say ρ11 , |ρ12 | and arg ρ12 . Thus 2 ρ11 ρ12 + ρ12 ρ22 ρ11 + ρ12 ρ21 , ρˆ 2 = 2 +ρ ρ ρ11 ρ21 + ρ21 ρ22 ρ22 12 21 and with (ρ11 + ρ22 )2 = 1 and ρ12 ρ21 = |ρ12 |2 we obtain for (9.35) 1 ≥ Tr ρˆ 2 = 1 + 2 |ρ12 |2 − ρ11 ρ22 ≥ 1/2.

(9.37)

For a pure state (left limit) the condition is thus √ |ρ12 | ≡ ρ11 ρ22 = ρ11 (1 − ρ11 ),

(9.38)

while for a fully unpolarized (completely incoherent) state |ρ12 | ≡ 0 and ρ11 ≡ ρ22 = 1/2 holds. On the other hand, the matrix may be diagonalized by det ρˆ − ρ (d) 1 =0

(9.39)

(9.40)

which leads to two eigenvalues for the diagonalized density matrix ρˆ (d) ρ (d) =

12 1± 2

3 1 + 4 |ρ12 |2 − ρ11 ρ22 .

(9.41)

9.1

Some Terminology

587

With (9.37) one may write the difference between these eigenvalues as (d)

(d)

ρ11 − ρ22 =

1 + 4 |ρ12 |2 − ρ11 ρ22 = 2 Tr ρˆ 2 − 1.

(9.42)

For the pure state (Tr ρ 2 = 1) this gives ±1, for the unpolarized state (Tr ρ 2 = 1/2) (d) (d) we obtain ρ11 − ρ22 = 0. (d) (d) This difference ρ11 − ρ22 corresponds to the polarization P of the two state system defined by (9.8), as we see by direct comparison with the diagonal form (9.28). Since Tr ρˆ 2 is independent of the choice of the coordinate system, (9.42) offers a good starting point for the sought-after definition of a degree of polarization. For the two state system we define it (omitting the sign in order to be truly independent of the coordinate system) as: (d) (d) |P| = ρ11 − ρ22 = + 2 Tr ρˆ 2 − 1 with 1 ≥ |P| ≥ 0.

(9.43) (9.44)

The left and right limits correspond to the pure and unpolarized states, respectively. The thus defined degree of polarization is more tangible than the somewhat abstract expression Tr ρˆ 2 . We generalize (9.43) for an N × N density matrix ρˆ and define the degree of polarization of an arbitrary (pure or mixed) state: (d) +1 (d) 2 P (N ) = √ or ρjj − ρkk 2(N − 1) j,k P (N ) = √

+1 N −1

N Tr ρˆ 2 − 1.

(9.45)

(9.46)

While (9.45) corresponds to the geometric mean value of all degrees of polarization of any two states of the diagonalized density matrix ρˆ (d) , the definition (9.46) is independent of the coordinate system. The identity of the two expressions may be verified with a little bit of algebra, using the normalization according to (9.20). The √ normalization constant ∝ 1/ N − 1 is chosen such that 1 ≥ P (N ) ≥ 0,

(9.47)

again with the left and right limits representing fully coherent and unpolarized (fully incoherent) states, respectively. One may also be interested in the coherence (or purity) of a subset of N basis states. The degree of polarization of such a subset is obtained in full analogy to (9.46), where instead of ρˆ the corresponding N × N submatrix ρˆ and instead of N → N is to be used.

588

9


Section summary

• The density operator ρˆ defined by (9.16) and the density matrix ργ γ according to (9.17) has been introduced as a useful and flexible tool for book keeping when describing experiments with mixed (not fully coherent) states of quantum systems. • Pure states are represented by a linear superposition of basis states. In contrast, mixed states cannot be described in that way. Rather they are characterized by the probabilities for finding different pure states. • Expectation values of observables are obtained from the trace of the their product with the density matrix according to (9.22). • We have familiarized ourselves with the concept by simple 2 × 2 matrices, describing pure as well as mixed states of a system with two basis states. It is very important to realize that the density matrix depends crucially on the choice of the coordinate system – and so does the observation of coherence among states. • Each density matrix may be brought into diagonal form by suitable unitary transformation. Pure states are characterized by a single unit element in this diagonal form, while the fully unpolarized (incoherent) state has equal probability 1/N for all its component states. • A unique measure for the degree of purity of a state is obtained from 1 ≥ Tr ρˆ 2 ≥ 1/N . Based on this we define a general degree of polarization (9.45) for which 1 ≥ P (N ) ≥ 0. The upper and lower limits refer to pure and completely unpolarized (mixed) states, respectively.

9.2

Theory of Measurement

9.2.1

State Selector and Analyzer

We shall now develop the tools described above into a handy formalism to describe the signal detected in real experiments. With experiments in mind as sketched in Fig. 9.1 we have to account for the fact that neither the state selectors nor the analyzers are perfect. Up to now we have used the density matrix (or the density operator) to describe a quantum system in a mixed state of several pure states |α. We now change the perspective slightly and use ρ(α) ˆ = ρˆ (sel) to describe a state selector acting as a filter which transmits (from an initially completely unpolarized state) a mixed state described by ρˆ (sel) – which is constructed according to (9.31). In complete analogy we use the density matrix ρ(β) ˆ = ρˆ (anl) to describe an an(anl) out of a completely unpolarized alyzer. Again, it filters the state described by ρˆ ensemble. We may now describe a preliminary experiment, sketched in Fig. 9.3: the key parameter is the transmission probability p(β, α) through a state analyzer – characterized by ρˆ (anl) – for a quantum ensemble prepared by a state selector – characterized by ρˆ (sel) .

9.2


589

source

>

state selector ^ ρ(α) state analyzer ^ ρ(β)

{|q } ^ ρ(α)

^ ρ(β)

towards detector

detector

Fig. 9.3 Schematic of a preliminary experiment: quantum states prepared by a state selector are detected after passing an analyzer. Indicated by the large circle on the bottom left is the total set of all basis states {|q}. From these, the selector would transmit the subset denoted by ρ(α), ˆ the analyzer would transmit ρ(β). ˆ The intersection of these subsets of states reaches the detector after passing through both, selector and analyzer – as indicated by the pink coloured area

Let us first consider an ideal state selector which prepares a pure state |α, i.e. we assume its density operator ρ(α) ˆ = |αα| to project just this state out of the initially unpolarized ensemble. And similarly, we assume the state analyzer to be characterized by a density operator ρ(β) ˆ = |ββ| which projects only the state |β out of whatever enters into it (in our case the state |α). Thus, the state passing through the analyzer is given by ρ(β)|α ˆ = |ββ|α,

(9.48)

and the squared absolute value of the probability amplitude β|α is just the probability to detect a signal (in state |β) after passing the analyzer: 2 p(β, α) = β|α . (9.49) With β|α∗ = α|β, using (9.4) we may rewrite this with respect to the basis states {|γ } as ∗(β) (α) p(β, α) = β|αα|β = aγ aγ aγ∗(α) aγ(β) . (9.50) γ γ

Using the definition (9.13) of the density matrix elements for the pure states |α and |β the signal expected in the experiment will be proportional to (β) (α) p(β, α) = ργ γ ργ γ = Tr ρ(β) ˆ ρ(α) ˆ . (9.51) γ γ

The generalization is obvious: let us now consider an imperfect state selector which prepares a mixture of states |α, each with a probability pα , and an imperfect analyzer which transmits several states |β with the probabilities pβ . We thus have to sum in (9.50) over all prepared and transmitted signals, weighted with the probabilities pα and pβ characterizing selector (sel) and analyzer (anl), respectively. The

590

9


overall signal which reaches the detector is then proportional to ∗(β) (α) pα pβ aγ aγ aγ∗(α) aγ(β) . S(sel, anl) = p(β, α) =

(9.52)

α,β γ γ

By reorganization this sum we recover the density matrix elements (9.17) for state selector and analyzer and obtain the measured signal: S(sel, anl) =

γ γ

(sel) (anl) ργ γ ργ γ = Tr ρˆ (sel) ρˆ (anl) = Tr ρˆ (anl) ρˆ (sel) .

(9.53)

This is the basis for all further considerations about measurable signals with state selection and/or analysis. We thus emphasize again the significance of the operators for selector and analyzer, ρˆ (sel) and ρˆ (anl) , respectively: they describe the mixed state which the selector and analyzer would filter out from a completely unpolarized source.

9.2.2

Interaction Experiment with State Selection

We are now prepared to exploit the book keeping function of the density matrix for the evaluation of experiments with (partial) state selection prior to an interaction experiment and (partial) state analysis after it – i.e. we use the density matrix for a realistic description of real experiments. We have to derive an expression for the signal detected after an interaction process with an experimental scheme as sketched in Fig. 9.1. For compact writing we abbreviate the quantum numbers of the system prior to the collision with {γ } = {γA , γB }, including if relevant also k i . The basis states are correspondingly |γ . The operator of the state selector is then ρˆ (sel) =

γ γ

ργ γ |γ γ |. (sel)

(9.54)

Correspondingly, we characterize the analyzer by ρˆ (anl) =

ε ε

ρε ε |ε ε|. (anl)

(9.55)

The different indices γ , γ in (9.54) and ε , ε in (9.55), respectively, indicate that initial and final state may belong to different subsets of basis states of the system, {|γ } and {|ε}, respectively. As briefly discussed in Sect. 9.1.1, the interaction process may formally be described using the transition operator (T-matrix) formalism, i.e. by replacing each basis state |γ with T|γ . Thus, the quantum system which leaves the selector, characterized by ρˆ (sel) according to (9.54), is transformed during the interaction process

9.2


591

into T

ρˆ (sel) −→

γ γ

(sel) ργ γ T|γ γ |T† = Tρˆ (sel) T† .

(9.56)

The strength of the signal detected in the interaction experiment is obtained by replacing ρˆ (sel) in (9.53) with (9.56):4 S(sel, anl) = Tr Tρˆ (sel) T† ρˆ (anl) = Tr ρˆ (sel) T† ρˆ (anl) T .

(9.57)

Note that state selector and analyzer enter into this expression in fully equivalent manner. We may thus use the same theoretical tools to describe the time inverse experiment, by just interchanging source and detector in Fig. 9.3: one obtains the same signal if the selector is described by ρˆ (anl) and the analyzer by ρˆ (sel) . We point out that the important relation (9.57) may also be written in terms of matrix elements for the transition operator T and density matrix elements. One simply has to insert the quantum mechanical unit operators (2.43), Vol. 1 in between the operators in (9.56). We do not want to discuss the resulting, somewhat lengthy expressions for the general case, but focus on two specialized examples for illustration.

Example 1: Scattering Experiment with Unpolarized Beams and State Analysis After the Interaction We describe the case sketched in Fig. 9.1(a), but assume that there is no initial state selection in the A–B system. Hence, the state selector is represented by a diagonal density matrix for a mixed state, consisting of a set {|ε} of basis states statistically populated with probabilities pε according to (9.34) (completely unpolarized)5 with ( pε = 1. We also consider explicitly the relative momenta k i prior and k f after the interaction process. For simplicity we assume that A and B encounter each other in well collimated, velocity selected beams. After the interaction the emerging particles are assumed to be analyzed equally well. This is described by (sel) i γ ,k i γ

ρk

= pγ δγ γ δ k i − k i ,

(9.58)

(anl) = ρεε δ k f − k f

(9.59)

prior, and correspondingly by (anl) f ε,k f ε

ρk

4 Strictly speaking one has to multiply the following expression with normalizing factors which account for the fact that selector and analyzer operators are not necessarily normalized to 1. For clarity of writing we suppress this detail here, knowing that in each experiment anyhow several further transmission and detection factors have to be accounted for when evaluation a measured signal. 5 In

a next step of sophistication pγ˜ could e.g. follow a B OLTZMANN distribution.

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9


after the process. Using the left equality of (9.57) the measured signal is S(sel, anl) = dk f pγ ε k f |T|γ k i γ k i |T† |εk f εk f |ρˆ (anl) |ε k f ε γ ε

=

εε

(anl) ∗ pγ Tε γ (k f , k i )Tεγ (k f , k i ) ρεε .

(9.60)

γ

Obviously, the term in square brackets [. . . ] describes the total system directly after the collision, it defines what may be called the density matrix of the collision process: − → (col) ∗ pγ Tε γ (k f , k i )Tεγ (k f , k i ). (9.61) ρε ε = CT−1 γ

The summation over γ may be rather elaborate and comprises in principle a full partial wave expansion as given in (8.5). Alternatively we may exploit the proportionality between the scattering amplitude f and the T-matrix according to (7.31) and write − → (col)

ρε ε = Cf−1

∗ pγ fε γ (θ, ϕ)fεγ (θ, ϕ).

(9.62)

γ

The normalizing factors CT and Cf , respectively, have to be chosen such that − →

Tr ρˆ (col) = 1.

(9.63)

With this notation we may rewrite (9.60) in the general form (9.53). We obtain the differential cross section for such an experiment by inserting (9.62) into (9.60) and accounting for the flux factors (7.31):

(anl) −→ dσ (k f , k i ) I (θ, ϕ) = = I0 (θ ) Tr ρˆ (col) ρˆ (anl) . dΩ

(9.64)

The prefactor I0 denotes the standard inelastic differential cross section – not state specific, averaged over all initial and summed over all final states:

(av) 2 kf kf dσ (k f , k i ) I0 (θ ) = = Cf = pγ fεγ (θ, ϕ) . (9.65) dΩ ki ki εγ Equation (9.64), with the definitions (9.62) and (9.65), is the basis for the evaluation of all scattering experiments with state analysis after collision.6 In full analogy one includes of course the experiment without any state analysis. In that case ρˆ (anl) = 1/Nβ and Tr(ρˆ (col) ) = 1. Thus I (θ, ϕ) = I0 (θ) which is given by the standard formula (9.65) for the differential cross section. 6 This

9.2


593

may evaluate optical excitation and detection, such as the experiments sketched in Fig. 9.1(b, c), with T representing in that case the electromagnetic interaction operator D. − →

At this point it is important to ensure that ρˆ (col) and ρˆ (anl) used in (9.64) refer to the same coordinate system. As we have seen in Sect. 9.1.1, in practice this is often − →

not the case. Typically, it is convenient to describe ρˆ (col) in a particular coordinate system – e.g. in the standard collision frame (col), while ρˆ (anl) is best described with respect to other reference axes (anl) as dictated by the experimental setup. Hence, one of these coordinate systems has to be transferred into the other reference frame. This implies rotation by D(αβγ ) through the E ULER angles (αβγ ) as described in Appendix E, Vol. 1. Introducing such rotation into (9.64) leads to → −→ † (− −→ ρˆ (anl) = Tr ρˆ (col) D ρˆ (anl) D † . ρˆ col) D Tr ρˆ (col) ρˆ (anl) → Tr D

(9.66)

Explicitly one has to insert unit operators in between all these operators, so that this formalism obviously results in somewhat clumsy expressions. Alternatively one may rewrite it in terms of expectation values of the irreducible tensor operators into which the density operator may be expanded. This allows a much more elegant formulation of transformation problem. This procedure has long been established in nuclear physics in the context of so called “perturbed angular correlations”. In atomic physics it was introduced by FANO and M ACEK (1973) to describe the angular distribution and polarization of light emitted from atoms excited in atomic collisions obtained in coincidence experiments. The essentials are derived in Sect. 9.4. This theory of measurement may well be extended to other types of state selection or analysis, e.g. to experiments with electron spin polarization (BARTSCHAT et al. 1981) or to collisions with laser excited atoms (M ACEK and H ERTEL 1974).

Example 2: Scattering Experiment with State Selection Prior to Collision but Without State Analysis after Conversely, we may perform a state analysis prior to the collision, e.g. by laser optical pumping, by spin analyzers, special pump-probe techniques etc. For simplicity, we detect instead the signal without state analysis after the interaction. In analogy to (9.58) and (9.59) we now have (sel) (sel) ρk ε,k ε = ρεε δ k i − k i and (9.67) i

i

(anl) ρk γ ,k γ f f

= pγ δγ γ δ k f − k f .

(9.68)

(Often one has pγ 1 for a particular group of states, and = 0 for all others, which may be suppressed e.g. by the energy analyzer.)7 Note the slightly different use of 7 This

shows again the necessity to introduce a normalization constant for detector and state analyzer (which we have omitted here for simplicity). If a quantitative signal evaluation is aimed for one always has to ensure that density matrices are normalized to Tr ρˆ = 1.

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9


the indices (for reasons which will become obvious at the end of this section): we still denote the group of not analyzed states by γ and γ , the selected group by ε and ε (as in the previous example). However, the former now refer to the states after the interaction process, the latter to those before. We now evaluate the right side of the detection equation (9.57) in full analogy to the preceding example (9.60). Introducing unit operators and (9.68) gives ε k i |ρˆ (sel) |εk i εk i |T† |γ k f pγ γ k f |T|ε k i S(sel, anl) = dk i εε γ

=

εε

(sel)

ρε ε

pγ Tγ ε (k f , k i )Tγ∗ε (k f , k i ) ,

(9.69)

γ

where in the second line we have used (9.67). With the notation ← −

(col) ρεε = CT−1

pγ Tγ∗ε (k f , k i )Tγ ε (k f , k i )

(9.70)

pγ fγ∗ε (θ, ϕ)fγ ε (θ, ϕ)

(9.71)

γ

= Cf−1

γ

and

← −

Tr ρˆ (col) = 1

(9.72)

we may express the scattering intensity in analogy to (9.61)–(9.65) by ← − ←− I (θ, ϕ) = I0 (θ ) Tr ρˆ (sel) ρˆ (col) = I0 Tr ρˆ (col) ρˆ (sel) 2 kf kf with I0 = Cf = pγ fγ ε (θ, ϕ) . ki ki εγ

(9.73) (9.74)

Since now we perform state selection prior to the collision, and since the T-matrix now describes a scattering process from the states |ε to |γ (while before T was referring to transitions from |γ to |ε) a subtle difference between both examples becomes apparent: the density matrix of the scattering process (9.70) is constructed such that one does not only replace ε by γ (and vice versa); one also has to apply the complex conjugate definition to (9.61). In addition, we have so sum over all detection probabilities pγ for the individual states, while in example 1 the quantity pγ refers to the probability of finding the state |γ ) prior to the collision process. A somewhat more symmetric representation of the present result is obtained when we view example 1 and example 2 as being the time inverse of each other, i.e. as processes |γ → |ε and |ε → |γ , respectively. In this scheme we also have to replace k i by −k f and k f by −k i so that ∗ Tγ ε (k f , k i ) = Tεγ (−k i , −k f ).

(9.75)

9.2


595

Fig. 9.4 Schematic sketch of two inelastic collision experiments discussed in the text: (a) detection of the collision products with state analysis after the collision; (b) detection of the time inverse process by state selection prior to the collisions

A

(a)

coincidence B

collision

B

exciting laser A

(b)

B

collisio B

Finally, we may write (9.70) ← − (col)

ρεε (k f ← k i ) = CT−1

pγ Tεγ (−k i , −k f )Tε∗ γ (−k i , −k f )

(9.76)

γ

= Cf−1

− → ∗(col)

pγ fεγ (θ˜ , ϕ)f ˜ ε∗ γ (θ˜ , ϕ) ˜ = ρεε

(−k i , −k f ), (9.77)

γ

where Cf is the flux factor according to (9.65), and the angles θ˜ , ϕ˜ indicate that the scattering amplitudes have now to be described in a coordinate system which is defined by −k i and −k f . Expressions (9.76)–(9.77) are fully equivalent to (9.61)– (9.62) in the previous example – except that we have replaced k i by −k f and k f by −k i . Thus, a scattering experiment inducing transitions ε → γ with state selected particles and no final state analysis, provides identical information as an experiment for the inverse process γ → ε without initial state selection but with final state analysis. All measurable quantities are contained in the respective collision density matrix ρˆ (col) . Figure 9.4 illustrates the two time inverse experiments schematically. One example for such experiments has already been treated in Sect. 7.4.5. We emphasize, however, that the formalism described here is by no means limited to scattering processes. We have presented here a rather general approach to the theory of measurement with state analysis. In completely similar manner one may e.g. describe an optical excitation process which is analyzed by a suitable detection scheme – e.g. by measuring the polarization of the fluorescence. And analogue considerations are to be made e.g. for pump-probe studies of the temporal evolution in molecules, clusters or liquids. A comprehensive description is found in the standard textbook by M UKAMEL (1999) on this particular subject.

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9


Section summary

• We have made use of the density matrix to introduce a general theory of measurement involving mixed (or partially polarized) states. To do so we have extended the concept of the density matrix to describe also state selectors and analyzers. These are represented by a matrix, equivalent to the density matrix of the state which they filter out from a completely unpolarized state. • The signal in such experiments is described essentially by S ∝ Tr[ρˆ (anl) ρˆ (sel) ] = Tr[ρˆ (sel) ρˆ (anl) ], where ρˆ (sel) denotes the state prepared either by a selector or in a collision process or by any other dynamics, while ρˆ (anl) refers to a state analyzer. A scattering process with an unpolarized projectile and an unpolarized target is characterized by ρˆ (sel) = ρˆ (col) = TT† , where T is the standard transition matrix (T-matrix) defined in collision physics.

9.3

Selected Examples of the Density Matrix

9.3.1

Polarization Matrix and S TOKES Parameters

Partially polarized light and the S TOKES parameters P1 , P2 and P3 have already been introduced in Sect. 1.3.3 – somewhat heuristically. The S TOKES parameters allow us to fully characterize light which is not necessarily fully coherent – i.e. partially polarized light. The following, more rigorous derivation of these parameters and their properties is used as a practical example for exploring the use of density matrices.

The Coherent, Fully Polarized Case Polarization of electromagnetic waves has been defined in Sect. 1.3 essentially on the basis of the unit vector eel for elliptically polarized light (1.85). And in Chap. 2 we have described the coherence properties of quasi-monochromatic light in a quantitative manner. We have introduced state vectors |N for photons, while G LAUBER states (2.115) where discussed as a quantum mechanical representation of fully coherent light. In the following we shall now use these concepts – suppressing, however, for ease of writing the photon number N . We assume fairly intensive, quasimonochromatic light (e.g. a laser beam) with a high degree of temporal and spatial (lateral) coherence. A pure state of elliptically polarized light is then described by |eel = a1 |1 + a0 |0 + a−1 |−1

(9.78)

with a1 = e−iδ cos β, a0 = 0 and a−1 = −eiδ sin β. We refer here again to the standard, spherical unit basis vectors e+1 and e−1 , and denote the corresponding states for left and right hand polarized light with |1 and |−1, respectively, assuming that the light propagates in the +z-direction (helicity basis). Due to the transversality of electromagnetic radiation the amplitude for the |0 component is zero. We also recall: the alignment of the polarization ellipse with

9.3


597

Table 9.1 Pure polarization states of light (apart from an overall phase factor) and the corresponding density matrices in the helicity basis; we may consider these expressions as ρˆ (anl) to describe the respective polarization filter; these are actually all 3 × 3 matrices, but for compact writing we have omitted all zero components Row

Polarization

β

Basis state |eel

1

left circular (LHC)

π 2

|1

2

right circular (RHC)

0

|−1

3

linear at δ

π 4

√1 [e−iδ |1 − eiδ |−1] 2

4

δ = 0◦

π 4

√1 [|1) − |−1)] 2

5

δ = 90◦

π 4

−i √ [|1) + |−1)] 2

6

δ = 45◦

π 4

1 2 [(1 − i)|1 − (1 + i)|−1]

7

δ = 135◦

π 4

− 12 [(1 + i)|1 − (1 − i)|−1]

Density matrix ρêl 4 5 ρˆLHC = 10 00 4 5 ρˆRHC = 00 01 4 5 1 −e−2iδ ρˆδ = 12 2iδ −e 1 4 5 1 −1 1 ρˆ0◦ = 2 −1 1 4 5 ρˆ90◦ = 12 1 1 11 4 5 1 i 1 ρˆ45◦ = 2 −i 1 4 5 ρˆ135◦ = 12 1 −i i 1

respect to ex is specified by the polarization angle δ, while ellipticity angle β characterizes the ellipticity of the light: β = 0 and π/2 refer to LHC and RHC light, respectively, while β = π/4 specifies linearly polarized light parallel to the x-axis. We describe this pure polarization state (9.78) by its density matrix: ⎛

|a1 |2 (pol) ⎝ ρˆ = 0 a1∗ a−1

⎞ ⎛ 2 ∗ 0 a1 a−1 cos β 0 0 ⎠=⎝ 0 0 |a−1 |2 c.c.

0 0 0

⎞ − 12 sin 2βe−2iδ ⎠ (9.79) 0 2 sin β

with ρik = ai ak∗ and Tr ρˆ (pol) = ρ11 + ρ−1−1 = 1. 2 = 1 (full coherence), One easily verifies for this effective 2 × 2 matrix that Tr ρˆpol and that the degree of polarization, introduced in (9.46), is P = 1 (remember we have assumed a pure state in (9.78). Table 9.1 presents some special cases of practical importance, to be compared with (4.7)–(4.9), Vol. 1. Each pair of polarization states, {|1, |−1}, {|e0◦ , |e90◦ } as well as {|e45◦ , |e135◦ }, represents a complete orthonormal basis set for photons propagating into z-direction. Each of these pairs may in principle be used to construct a density matrix. In the literature the basis {|0◦ , |90◦ } or {|x, |y} is often used. However, we recommend and use the helicity basis {|1, |−1} as particularly convenient (in agreement with many other authors, e.g. B LUM 2012). We note one important advantage of the helicity basis: the effective 2 × 2 matrix is a submatrix of the 3 × 3 matrix which describes any system of angular momentum with J = 1 and its |J, M states. We may interpret J as photon spin, and M as its projection onto the z-axis of the detector coordinates – with zero amplitude for the photon state when M = 0.

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9


Of special interest are the three pairs (1, 2), (4, 5) and (6, 7) of photon states according to Table 9.1. Their differences may be written as:

0 −1 with Tr(ρˆ0◦ − ρˆ90◦ ) = 0 ρˆ − ρˆ = −1 0 0 i ρˆ45◦ − ρˆ135◦ = with Tr(ρˆ45◦ − ρˆ135◦ ) = 0 −i 0 1 0 with Tr(ρˆLHC − ρˆRHC ) = 0. ρˆLHC − ρˆRHC = 0 −1 0◦

90◦

(9.80) (9.81) (9.82)

Interestingly, these expressions are identical with the PAULI spin matrices according to (2.101) in Vol. 1 (apart from an overall phase factor −1 for the first two). They are unitary and the following ‘orthogonality’ relations hold: 1 ρˆLHC + ρˆRHC = ρˆ0◦ + ρˆ90◦ = ρˆ45◦ + ρˆ135◦ =

(9.83)

Tr(ρˆLHC + ρˆRHC ) = Tr(ρˆ0◦ + ρˆ90◦ ) = Tr(ρˆ45◦ + ρˆ135◦ ) = 2 Tr (ρˆLHC − ρˆRHC )(ρˆ0◦ − ρˆ90◦ ) = 0 Tr (ρˆ0◦ − ρˆ90◦ )(ρˆ45◦ − ρˆ135◦ ) = 0 Tr (ρˆ45◦ − ρˆ135◦ )(ρˆLHC − ρˆRHC ) = 0

(9.84)

Tr(ρˆLHC − ρˆRHC ) = Tr(ρˆ0◦ − ρˆ90◦ ) = Tr(ρˆ45◦ − ρˆ135◦ ) = 2.

(9.88)

2

2

2

(9.85) (9.86) (9.87)

As we shall see in a moment, these relations allow a convenient evaluation of the transmission of light through a polarization filter.

Incompletely Polarized Light Before deriving explicit expressions for the evaluation of experimental polarization studies, we have to apply the density matrix formalism to light which is only partially polarized. In physical reality quasi-monochromatic light beams are used which are an incoherent mixture of the pure states just discussed. At best, pure states can be approximated. The physical background to this partial coherence has already been discussed in Sect. 2.1.3: what we observe are light trains (wave packets), the phase of which is approximately constant over an average coherence time τ0 . This also holds for the phase differences between light trains with orthogonal polarization vectors, e.g. for σ + and σ − light. Replacing the time averages by ensemble averages (ergodicity) as discussed in Sect. 2.1.2, the pure photon polarization state (9.78) must be replaced by a mixed state (an ensemble) {α} of pure polarization states (α) (α) |ep (α) with the amplitudes a1 and a−1 . These individual quasi-monochromatic photons are the constituents of the light beam, each present with a probability pα , so ( that pα = 1. With the definitions (9.17) and (9.79) we obtain the density matrix

9.3


599

for a partially polarized light beam: ⎞ ⎞ ⎛ ( ( (α) 2 (α) (α)∗ ∗ p |a | 0 p a a |a1 |2 0 a1 a−1 α α α α 1 1 −1 ⎟ ⎜ 0 0 0 ρˆ (pol) = ⎝ 0 0 0 ⎠=⎝ ⎠ ( ( (α) (α)∗ (α) 2 a1∗ a−1 0 |a−1 |2 p a a 0 p |a | α α −1 1 α α −1 ⎛ ⎞ ( ( 2 0 − α pα sin βα cos βα e−2iδα α pα sin βα ⎠. =⎝ ( 0 0 0 ( +2iδ 2 α − α pα sin βα cos βα e 0 α pα cos βα (9.89) ⎛

Of course for each one of the wave trains |ep (α) in the mixed ensemble the basis vectors are correlated with a fixed phase difference 2δα and an ellipticity angle βα : each of them represents pure elliptic polarization over a time scale τ0 . However, averaged over a whole ensemble this may nevertheless lead to an unpolarized state: e.g. in the case that the phases δα are distributed statistically, and hence the offdiagonal matrix elements ρ+− average out (i.e. the coherence term disappears). In the general case the light described by (9.89) is neither fully polarized nor completely unpolarized. The matrix, i.e. the polarization state of the light, may be ∗ described by three real parameters, since ρ++ + ρ− − = 1 and ρ+− = ρ−+ is a complex quantity. The relations (9.85)–(9.87) suggest to attribute these three parameters to the three differences of polarization matrices (9.80)–(9.82) and to add an unpolarized background (identity matrix): ρˆ (pol) =

1 1 + P1 (ρˆ0◦ − ρˆ90◦ ) + P2 (ρˆ45◦ − ρˆ135◦ ) + P3 (ρˆRHC − ρˆLHC ) . (9.90) 2

This defines a convenient parametrization of the most general polarization matrix (9.89) for light travelling into z-direction. Written as 2 × 2 matrix it is 1 ρ1−1 −P1 + iP2 1 − P3 ρ11 (pol) = . (9.91) ρˆ = ρ−1+1 ρ−1−1 1 + P3 2 −P1 − iP2 Up to now, P1 , P2 and P3 are still free parameters. We shall derive their physical significance in the following and identify them as S TOKES parameters which we have introduced in Sect. 1.3.3. For reference we also communicate the polarization matrix in the {|ex , |ey } basis which is often used alternatively in the literature: 1 ρxx ρxy P2 + iP3 1 + P1 ρˆ (pol) (x, y) = = . (9.92) ∗ ρyx ρyy 1 − P1 2 P2 − iP3 This is readily derived from (9.91) using the relations (4.4)–(4.6), Vol. 1 between the basis states. We finally mention that for linearly polarized light it is sometimes convenient to use an alternative coordinate frame, with its z-axis parallel to the polarization vector

600

9


i.e. perpendicular to the wave vector, z ⊥ k. In this coordinate system the polarization vector is simply elin = e0 while all other components vanish. The respective polarization matrix is given by ⎛ ⎞ 0 0 0 ρˆ (pol) = ⎝ 0 1 0 ⎠ . (9.93) 0 0 0

Experimental Determination of S TOKES Parameters We now apply the theory of measurement as derived in Sect. 9.2. We recall: the density matrix may be used to describe a state selector or state analyzer. From this perspective the density matrices collected in Table 9.1 represent polarization filters which prepare or analyze pure polarization states. A light beam, characterized by a polarization matrix parameterized as (9.91), may be analyzed by passing it through different polarization filters described by ρˆ (anl) according to Table 9.1. With (9.53) the signal behind the analyzer is I (pol) = I0 Tr ρˆ (pol) ρˆ (anl) , (9.94) where the normalization constant I0 turns out to be the total intensity in the beam; this is verified by inserting (9.91) and the analyzer matrices from Table 9.1, and exploiting (9.83): I (0◦ ) + I (90◦ ) = I (45◦ ) + I (135◦ ) = I (RHC) + I (LHC) 1 = I0 . ≡ I0 Tr ρˆ (pol)

(9.95)

And for the relative intensity differences one finds from (9.92) with the ‘orthogonality’ relations (9.85)–(9.88) 1 I (0◦ ) − I (90◦ ) = Tr ρˆ (pol) (ρˆ0◦ − ρˆ90◦ ) = P1 Tr(ρˆ0◦ − ρˆ90◦ )2 I (0◦ ) + I (90◦ ) 2 = P1 , I (45◦ ) − I (135◦ ) I (45◦ ) + I (135◦ )

= P2

(9.96) and

I (RHC) − I (LHC) = P3 . I (RHC) + I (LHC)

(9.97) (9.98)

We thus have expressed the parameters used in (9.90) and (9.91) in terms of quantities which are directly accessible to the experiment. And these turn out to be indeed the S TOKES parameters defined in Sect. 1.3.3. A suggestive abbreviation is the S TOKES vector of the light beam: P = (P1 , P2 , P3 ).

(9.99)

9.3


601

Measuring Polarization in the General Case The matrices ρˆ (anl) used above describe perfect analyzers, transmitting one type of polarization to 100 % while the orthogonal polarization is not transmitted at all. According to Sect. 9.1.4 they are characterized by the coherence relation Tr(ρˆ (anl) )2 = 1, or equivalently with (9.43) by a degree of polarization |P| = 1. A more general, not necessarily perfect analyzer, is described by a set of S TOKES (anl) (anl) (anl) parameters (P1 , P2 , P3 ) = P (anl) , which define an analyzer matrix ρˆ (anl) corresponding to (9.91) – for which 1 ≥ Tr(ρˆ (anl) )2 ≥ 1/2, according to (9.35). Inserting this ρˆ (anl) into (9.94) one obtains after a brief calculation the signal behind the analyzer: I0 (anl) (anl) (anl) 1 + P1 P1 + P2 P2 + P3 P3 2 I0 1 + P · P (anl) . = 2

I (pol) =

(9.100)

We have already anticipated this expression with (1.102). The specific relations (9.96) and (9.97) may also be recovered by alternatively setting one of the three S TOKES parameters of the analyzer = ±1, the other two = 0.

Degree of Coherence, Degree of Polarization The coherence properties of the polarization matrices correspond to those derived in Sect. 9.1.4 for 2 × 2 matrices. For full coherence (P = 1) ρ11 ρ1−1 = |ρ1−1 |2

and ρxx ρyy = |ρxy |2

(9.101)

holds, while for completely incoherent, unpolarized light (P = 0) we have 1 ρ1−1 = ρxy ≡ 0 and ρ11 = ρ−1−1 = ρxx = ρyy ≡ . 2

(9.102)

This may suggest the definition of a so called degree of coherence: √ |μ| = |ρxy |/ ρxx ρyy .

(9.103)

However, while this quantity is used in the literature quite often, its definition is somewhat unfortunate as it depends on the choice of the basis and the coordinate system used: this is immediately seen by (9.92), where P2 depends of course on the choice of the x-axis, while P3 is independent of this selection. Much more robust is the degree of polarization as defined by (9.43). If one inserts ρˆ (pol) according to (9.91) or alternatively according to (9.92), it is determined by the S TOKES parameters completely independent of the coordinate system

|P| = + P12 + P22 + P32

with 1 ≥ |P| ≥ 0.

(9.104)

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9


The limiting cases indicate completely polarized and unpolarized light, respectively. In summary, we note that the three S TOKES parameters are easy to determine experimentally, and they contain the complete information on the polarization state of the light beam. For the experimental characterization of a light beam, described by (9.91) in the helicity basis, by means of a linear polarizer (as ideal as possible) one has to apply (9.94). With the analyzer matrix ρˆδ from Table 9.1 (row 3) one obtains the transmitted signal as a function of the analyzer angle δ: I (δ) = I0 Tr ρˆ (pol) ρˆδ . We leave it to the reader as an easy exercise to show that this leads to the formulas already communicated in (1.103)–(1.105).

9.3.2

Atom in an Isolated 1 P State

General Discussion We continue our exploration of the concept and use of density matrices by a look at specific atomic states. We want to describe the simple case of an atom in a p state with its three substates |m = ±1, |m = 0, pertaining to important atomic model systems. For the sake of simplicity we ignore for the moment the electron spin and the nuclear spin and simply focus on the electronic orbitals, assuming a 1 P np state (e.g. in He). We further assume that the p level considered is energetically well separated from other energy levels of the system. The general 3 × 3 density matrix (for any system with angular momentum j = 1), ⎞ ⎛ ρ11 ρ10 ρ1−1 (9.105) ρˆ = ⎝ c.c. ρ00 ρ0−1 ⎠ , c.c. c.c. ρ−1−1 defines in principle 8 independent, real parameters: three complex off-diagonal terms and with Tr ρˆ = 1 = ρ11 + ρ00 + ρ−1−1 two real diagonal terms. Before discussing how such an atom is prepared by an optical excitation process or by collisional excitation, let us have a look at the general properties of the thus described p atoms and find the symmetries which occur in most applications. These will reduce the number of independent parameters further. We first assume the p atom to be in a pure state: |p = b1 |m = 1 + b0 |m = 0 + b−1 |m = −1.

(9.106)

The angular component (solid angle ) of the wave function for this pure state is given by |p = b1 Y11 (θ, ϕ) + b0 Y10 (θ, ϕ) + b−1 Y1−1 (θ, ϕ).

(9.107)

9.3


603

The probability distribution of a thus characterized charge cloud with respect to is understood in the most direct manner by simply inserting the spherical harmonics explicitly: |p2 = 3 b0 b∗ cos2 θ + 1 b1 b∗ + b−1 b∗ − 2 Re b1 b∗ e2iϕ sin2 θ 0 1 −1 −1 4π 2 ∗ + terms proportional to b0 b±1 cos θ sin θ.

(9.108)

In the general case of mixed states, several different contributions of similar type ∗ in (9.108) must then constitute the charge cloud: the amplitude products bm bm ∗ be replaced by density matrix elements ρm m = bm bm . To simplify the situation somewhat we restrict the discussion in the following to a situation (often found in experiments) where at least one symmetry plane exists with respect to which the charge density – proportional to (9.108) – is symmetric. For convenient writing we choose the xy plane to be this symmetry plane, and call this reference system the atomic coordinate system,8 x (at) y (at) z(at) . The last terms in (9.108) contain products of the type sin θ cos θ which in the upper and lower hemisphere, z > 0 and z < 0, respectively, have different sign and are not mirror symmetric with respect to the x (at) y (at) plane. We drop them completely in (9.108) so that the corresponding off-diagonal terms disappear: ρ0±1 = ρ±10 ≡ 0. The most general distribution of an electron charge cloud for a p state with xy reflection symmetry in the “atomic” coordinate system is thus 2 3 1 ρ00 cos2 θ + ρ11 + ρ−1−1 I (θ, ϕ) = |p = 4π 2 + 2|ρ1−1 | cos 2(ϕ − γ ) sin2 θ ,

(9.109)

with ρ1−1 = |ρ1−1 |ei arg(ρ1−1 ) = −|ρ1−1 |e−2iγ , as plotted in Fig. 9.5. The corresponding density matrix is ⎛ ⎞ ρ11 0 −|ρ1−1 |e−2iγ (at) ⎠. 0 ρ00 0 ρˆ = ⎝ 2iγ −|ρ1−1 |e 0 ρ−1−1

(9.110)

With the usual normalization ρ11 + ρ−1−1 = 1 − ρ00

(9.111)

only four independent real parameters remain. They completely characterize a p charge distribution which is symmetric with respect to reflection at the xy plane. 8 In the literature related to collisional alignment and orientation studies this is usually called the “natural” coordinate system (see e.g. A NDERSEN et al. 1988).

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9

Fig. 9.5 Example for a p state charge cloud with its characteristic parameters: length l ∝ Imax , width w ∝ Imin , height h ∝ Iz , alignment angle γ and z in angular momentum L z-direction; plotted is the probability per solid angle according to (9.109)

The Density Matrix – A First Approach z (at) ^

< Lz >

ki

+

–

l

y (at)

w

kf

γ

θcol x (at) h

The restriction to such charge distributions – characteristic for quite a number of experiments – leads to the particularly clear structure of the density matrix (9.110). Obviously ρˆ (at) is composed of a 2 × 2 sub-matrix ρˆ + and a single element submatrix ρˆ − ; they describe the population of the m = ±1 states and the m = 0 state, which have positive and negative reflection symmetry in respect of the xy plane, respectively:9 positive: ρˆ + =

ρ11 −|ρ1−1 |e−2iγ c.c. ρ−1−1

(9.112)

−

negative: ρˆ = ρ00 . As the structure of this density matrix shows, a fully coherent state implies that one of the two sub-matrices disappears: only then (9.110) can be transformed into the form (9.33) by diagonalization. If ρ00 = 0 one determines from (9.38) whether ρˆ + represents a pure or a mixed state. We also note that this special choice of the coordinate system allows a direct physical interpretation of the density matrix elements of ρˆ (at) (which otherwise are somewhat abstract quantities). We first recall that the diagonal matrix elements ρ11 , ρ00 , ρ−1−1 give the relative probability for finding the substates m = 1, 0 and −1. From this a particular important observable is deduced: the expectation value of the angular momentum10 in z-direction: z ) = ρ11 − ρ−1−1 . z = Tr(ρˆ L L

(9.113)

9 Note

that this xy reflection symmetry must not be confused with the xz reflection symmetry used to define the real angular momentum states and real tensors introduced in Appendix D.3, Vol. 1. For a graphical illustration see Fig. D.1 in Vol. 1.

10 For

simplicity of writing we use here again atomic units, i.e. angular momenta are measured in z |m = m|m. units and L

9.3


605

Even the off-diagonal matrix element, the so called coherence term, obtains in these coordinates a direct physical meaning which may be read directly from the charge distribution (9.109). A graphic illustration is given in Fig. 9.5. We first note that for |ρ1−1 | ≡ 0 the charge distribution becomes a rotational ellipsoid with z being the symmetry axis. For ρ00 = 1/3 this degenerates to a completely isotropic distribution with ρ11 + ρ−1−1 = 2/3. But even in this case the angular momentum (9.113) may still remain finite: a fully spherical charge distribution may still contain an inherent asymmetry. In general, ρ00 = 1/3 and |ρ1−1 | will have a finite value so that the charge cloud has three symmetry axes as illustrated in Fig. 9.5: the z-axis and axes of maximal and minimal probability in the xy plane. The latter point into the direction ϕ = γ and ϕ = γ + π/2, respectively. The alignment angle γ is obtained according to (9.109) directly from the phase arg(ρ1−1 ) of the off-diagonal matrix element: 1 γ = − arg(ρ1−1 ) ≥ π/2. 2

(9.114)

For symmetry reasons γ is defined only modulo π . The absolute value |ρ1−1 | characterizes the polarization of the ρˆ + submatrix, which we shall call linear polarization of the charge cloud:11 Plin = (Imax − Imin )/(Imax + Imin ) = 2|ρ1−1 |.

(9.115)

According to (9.109), Imax = I (π/2, γ ) and Imin = I (π/2, γ + π/2) correspond to the maximum and minimum values of the charge density in the xy plane. Figure 9.5 summarizes these findings. z , The p state charge cloud is fully characterized by the four parameters L γ , Plin and h (height of the charge density in the z-direction Iz = I (0, ϕ)). The latter is proportional to ρ00 : Iz Iz + Imin + Imax

= ρ00 ∝ h.

(9.116)

For the relative length l of the charge cloud we find 1 − ρ00 Imax + |ρ1−1 | ∝ l, = Iz + Imin + Imax 2

(9.117)

and its width w is given by 1 − ρ00 Imin − |ρ1−1 | ∝ w. = Iz + Imin + Imax 2 11 Note

(9.118)

that the corresponding, explicit expressions (9.8) and (9.43) refer to the diagonal form of a 2 × 2 matrix, which in the present case can be obtained by rotation through −γ around the z(at) -axis.

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9


Sometimes one also reports the relative thickness in x-direction: λ= =

I (π/2, ϕ = 0) 1 − ρ00 = − Re(ρ1−1 ) Iz + Imin + Imax 2

(9.119)

1 − ρ00 + |ρ1−1 | cos 2γ . 2

Incoherence Induced by Collisional Excitation We now want to describe the preparation of such a charge cloud. Of particular interest is the question what kind of processes lead to incoherence. One type of mechanisms to be discussed are inelastic collisions between the atom A of interest and another particle B: A(i) + B(i) → A(np) + B(f ).

(9.120)

Let us assume that the initial states of A and B are fully known and can be described (A) (B) by wave functions φi (r A ) and φi (r B ), respectively. In the following we shall analyze two situations of different complexity.

The Most Simple Situation Particle B cannot change its state during the collision process while atom A is excited into a level |n with substates |nm. Writing the combined internal wave functions of A and B as (A)

(B)

φi (r) = φi (r A )φi (r B )

(A)

(B)

and φf (r) = φnm (r A )φi (r B ),

(9.121)

and following the derivations in Sects. 7.3 and 7.3.1, we write the asymptotic wave function of the whole quantum system as Ψ (R, r) = exp(ik i R)φi (r) +

1 ff i (θ, ϕ) exp(ikf R)φf (r). R

(9.122)

f

The wave vectors k i and k f characterize again the relative motion before and after the collision process, respectively, and ff i (θ, ϕ) is the corresponding scattering amplitude. The wave function of the total system prior to the interaction (i.e. for t → −∞, R → ∞) can thus be written as a product function: (A)

(B)

Ψ (R, r) = exp(ik i R)φi (r A )φi (r B ).

(9.123)

Note that this expression represents a pure state, and the question is: how can a collision process ever lead to the observation of incoherent ensembles of states? The sum in (9.122) must in principle be carried out over all possible final states φf (r) of the total system. Now, let the scattering experiment be designed such that after the interaction process only one specific set of states of the excited atom A is selected by a state analyzer – let us say it detects all substates |npm in one level. This may e.g. be achieved by measuring the relative kinetic energy after the collision

9.3


607

or by a spectral analysis of the fluorescence from atom A after the collision. The thus prepared state of the total system is then described by Ψanl (R, r) =

+1 1 (B) (A) exp(iknp R)φi (r B ) fnpm φnpm (r A ), R

(9.124)

m=−1

with the abbreviation fnpm = ff i (θ, ϕ). Note that the sum in this expression describes the wave function of the atom excited to the np level by the collision. We emphasize that the total wave function Ψanl (R, r) still represents a pure state: being written as product of the wave functions of the collision partners A and B and their relative momentum. Thus, the coherence of the quantum system is conserved during and after this particular interaction process – and the selection of the specified states. This still holds if we ask the question in which state particle A is found after the collision process: the simple answer is that its wave function after the interaction ( (A) (r A ). and after analysis describes a pure, coherent state: fnpm φnpm Before discussing the next case, we also must recall that the scattering amplitudes obey certain symmetry relations, such as (8.6), which refer there to scattering amplitudes in the standard collision frame.12 One finds (here without proof) that these symmetry relations translate into the atomic frame as: symmetry with respect to the x (at) y (at) plane (now the scattering plane) is conserved in the collision process. Thus, if we start with an xy positive state, e.g. in an n1 s → n2 p transition, only the m = ±1 states are excited. In an n3 d → n2 p transition md = ±1 → mp = 0 are possible, as well as md = ±2 → mp = ±1 and md = 0 → mp = ±1.

A More General Situation We now allow the quantum numbers of B to also change during the process. To keep it still simple, we assume that B is originally in state |1 and can change into the states |j = |1 or |2 during collision. We may e.g. think of an electron with its (B) (B) two spin states. These states are described by wave functions φ1 (r B ) and φ2 (r B ), respectively. The scattered and analyzed wave function of the total system is then Ψanl (R, r) =

1 exp(iknp R) R 6 7 +1 +1 (B) (B) (1) (A) (2) (A) × φ1 (r B ) fnpm φnpm (r A ) + φ2 (r B ) fnpm φnpm (r A ) . m=−1

m=−1

Clearly, the total quantum system, consisting of particle A and particle B, is still in a coherent state as given by the two particle wave function in the square brackets. However, if we now ask how to describe atom A as an isolated particle after the interaction, it is no longer possible to factor out the wave function of particle B – 12 To

compare the standard collision frame introduced in Fig. 7.16 with the “atomic frame” (at) used here (also called the “natural frame”) the latter has to be rotated through the E ULER angles (−π/2, −π/2, 0).

608

9


which was the critical step applied to (9.124) in the previous situation: No longer can we describe A after the collision by one well defined wave function – and the same holds for particle B! The total system consisting of particle A and particle B has to be described by an entangled state. If we insist on looking only at the state of particle A after the collision – and that is what a standard scattering experiment usually does – A has to be characterized (j ) now by a density matrix. With the scattering amplitudes fm , which are the probability amplitudes for simultaneously finding B in state |j and A in state |npm, we obtain ⎛ (1) (1)∗ ⎞ (1)∗ f1 f1 0 f1(1) f−1 ⎜ ⎟ (1) (1)∗ ρˆ (at) = p (1) ⎝ 0 f0 f0 0 ⎠ (1) (1)∗ c.c. 0 f−1 f−1 ⎛ (2) (2)∗ ⎞ (2) (2)∗ f1 f1 0 f1 f−1 ⎜ ⎟ (2) (2)∗ (9.125) + p (2) ⎝ 0 f0 f0 0 ⎠. (2) (2)∗ c.c. 0 f−1 f−1 This description (in the atomic frame) allows also to describe processes which change the reflection symmetry in respect to the x (at) y (at) plane. The general shape of the resulting p state charge cloud is illustrated in Fig. 9.5. The probabilities p (j ) are obtained from (B) 2 3 ( (j ) 2 |φj | d r B m |fm | (j ) × . p = (B) ( (B) (1) 2 (2) 2 (|φ1 |2 + |φ2 |2 )d3 r B (|f | + |f | ) m m m As a typical example we treat the excitation of a hydrogen atom (A = H) by an electron (B = e− ) from the ground state into the first excited p state: e− + H(1s) → e− + H(2p). S e remains constant during the collision, The total spin of the system S = SH + since spin-orbit interaction can be neglected for this very light atom. Two possible total spin quantum numbers are possible, S = 0 or 1, i.e. the total system is found to be either in a singlet state (S) with the statistical weight p (S) = 1/4, or in a triplet state (T) with the weight p T = 3/4. Both spin states may have rather different scattering amplitudes. Thus, the density matrix elements of the excited hydrogen state (2p) in its 2p state are given by ρmm = C1 ( 34 fmT fmT∗ + 14 fmS fmS∗ ), with the normalization constant being 1 3 T 2 1 S 2 . (9.126) fm + fm C= 4 4 m=−1

One easily verifies that in general this does not lead to a coherent state. Trivially, a mixed state is obtained when both wave functions participate, those with positive reflection symmetry (m = ±1) as well as antisymmetric ones (m = 0). In the present

9.3


609

case, the latter processes would require spin flip processes – expected only for atoms with large Z. However, even if only states with positive reflection symmetry are excited one has to distinguish by (9.38) whether a pure or a mixed state is observed. Only a strictly linear relation between singlet and triplet amplitudes would allow a simplification of (9.125) such that coherence is maintained. As a rule this will not be the case and the excited state will be incoherent.

Optical Excitation Alternatively we discuss optical excitation of an atom into an np state. For simplicity we restrict ourselves here to a situation where optical pumping can be neglected (see however Appendix D). We thus consider a 1 P1 state |b without hyperfine structure, excited by single photon absorption from an initial 1 S0 state |a. As discussed in Chap. 4, Vol. 1 the density formalism is not needed as long as the excitation is achieved with fully polarized, coherent light: in this case the excited atom is described by a wave function as in (9.107). To warm up to our task we nevertheless write down the density matrix for this case, recalling the results from Sect. 4.7 in Vol. 1. As far as possible the polarization states are described in a coordinate system with its +z-axis parallel to the direction of light propagation (wave vector kz). The well known selection rules m = +1 and −1 hold for left (σ + , LHC) and right (σ − , RHC) circularly polarized light, respectively. This light thus excites the |1 Pm = 1 and |1 Pm = −1, respectively, the excitation amplitudes are b1 = 1 and b−1 = 0 (alternatively b1 = 0 and b−1 = 1). In contrast, if one uses linearly polarized light propagating also into kz-direction (its electric field vector E(r, t) lying in the xy plane aligned at an angle γ with respect to the x-axis, so called σ light), then a linear combination of√both states is excited as described in (4.136), Vol. 1. The amplitudes are √ b1 = +(1/ 2)e−iγ and b−1 = −(1/ 2)e+iγ . The respective density matrices are ⎛ ⎞ 1 0 0 for σ + light with kz: ρˆLHC = ⎝ 0 0 0 ⎠ , (9.127) 0 0 0 ⎛ ⎞ 0 0 0 (9.128) for σ − light with kz: ρˆRHC = ⎝ 0 0 0 ⎠ and 0 0 1 ⎞ ⎛ 1 0 −e−2iδ 1⎝ 0 0 ⎠. for σ light with E ⊥ z: ρˆlin = (9.129) 2 2iδ −e 1 We note here two interesting aspects: 1. Of all the states, potentially excited, according to (9.110), only those with positive reflection symmetry can be excited: as long as the exciting light propagates into z-direction and the E vector lies in the xy plane, we have ρ00 = 0. 2. According to (9.127)–(9.129), the sub-matrices of ρˆ + are completely identical to those of the exciting light (see Table 9.1).

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9


The generalization to excitation by incompletely polarized light which propagates into z-direction is obvious: we just have to weight the matrices (9.127)–(9.129) with the probabilities to find the respective polarization. Most conveniently this is again done by using the S TOKES parameters as in Sect. 9.3.1. Excitation by light propagating into kz-direction with S TOKES parameters P1 , P2 , P3 leads to a density matrix of the excited 1 P1 state in complete analogy to (9.91): ⎞ ⎛ 1 − P3 0 −P1 + iP2 1⎝ ⎠. 0 0 0 ρˆ = (9.130) 2 −P − iP 0 1+P 1

2

3

In contrast, if we want to optically excited the state |m = 0 with negative xy reflection symmetry, we have to use light which does not propagate into z-direction. The most transparent situation is encountered when the light propagates in the xy plane with its electric field vector E being parallel to the z-axis. The density matrix for this so called ⎛ ⎞ 0 0 0 π light with Ez: ρˆπ = ⎝ 0 1 0 ⎠ . (9.131) 0 0 0 In practice a light beam has always some divergence (not all components of it propagate exactly into one direction) and we have to average over the contributing directions of incidence. If the laser beam is slightly divergent and its central axis is assumed to be parallel to the z-axis, not all components with negative reflection symmetry average out. The density matrix of a 1 P1 state excited with such a light beam will contain a small contribution ρ00 : ⎛ ⎞ 1 − P3 0 −P1 + iP2 1 ⎝ ⎠. 0 ρ00 0 (9.132) ρˆ = 2 + ρ00 −P − iP 0 1+P 1

2

3

Again, four real parameters P1 , P2 , P3 , ρ00 , describe such a 1 P1 state. They may again be related to the physical quantities discussed above: the angular momentum in z-direction according to (9.113) is Lz = ρ11 − ρ−1−1 = −P3 ,

(9.133)

with (9.114) the alignment angle of the charge cloud is13 γ=

1 arg(P1 + iP2 ) 2

so that

tan 2γ = P2 /P1 .

(9.134) (9.135)

practice one has to be somewhat cautions when automatically extracting γ from P1 and P2 . Since the standard function arctan is not univalued one has always to keep the physical geometry in mind.

13 In

9.4

Angular Distribution and Polarization of Radiation

611

The linear polarization of the charge cloud (9.115) is equal to the linear polarization of the exciting light: 1/2 Plin = 2|ρ1−1 | = P12 + P22 . (9.136) Finally, we may introduce a fourth S TOKES parameter of the charge cloud which measures the relative difference between its length in x-direction and its height. With (9.116), (9.119) and −2 Re ρ1−1 = 2|ρ1−1 | cos(2γ ) = Plin cos(2γ ) this quantity becomes P4 =

I ( π2 , 0) − Iz λ − ρ00 1 − 3ρ00 − 2 Re ρ1−1 = = . I ( π2 , 0) + Iz λ + ρ00 1 + ρ00 − 2 Re ρ1−1

(9.137)

Section summary

• Partially polarized light may be described with advantage by a density matrix ρ (pol) according to (9.91). Convenient parametrization is provided by the three S TOKES parameters, summarized as S TOKES vector P = (P1 , P2 , P3 ). • A polarization analyzer can also be described by such a density matrix ρ (anl) or by a S TOKES vector P (anl) . The signal from a partially polarized source which passes through this analyzer is given by I0 Tr(ρ (pol) ρ (anl) ) = (I0 /2)(1 + P · P (anl) ). • The density matrix of an isolated 1 P state has a quite similar 3 × 3 structure. It becomes most transparent if an “atomic” coordinate system is chosen with reflection symmetry of the charge distribution in respect to its x (at) y (at) plane. The density matrix then decomposes into a 2 × 2 submatrix corresponding to wave functions with positive reflection symmetry and a single element ρ00 with negative wave function symmetry. The state is fully characterized by 4 parameters: in addition to ρ00 which reflects the height of the charge cloud, we have its alignment angle γ = arg(ρ1−1 ), its linear polarization Plin = 2|ρ1−1 | and its inherent angular momentum L⊥ = ρ11 − ρ−1−1 . • Incoherence in an excited atom A∗ can be generated in a collision processes with a partner B which may change its quantum state: After such a process the state of the whole system is entangled, and the isolated atom A can no longer be described as a coherent wave function – it requires a density matrix description.

9.4


9.4.1

Formulation of the Problem

As a last topic in this chapter we give a brief introduction to a general theory of radiation emitted from excited quantum systems – with special emphasis on systems that cannot be described as pure, coherent states. What is the angular distribution and polarization of radiation from systems characterized by a density matrix? What

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9


information can be gleaned about the excited system by a judicious choice of the experimental observation geometry? In the terminology used above: How do we have to design the state analyzer in order to disentangle mere geometry of the experimental setup from the information about the state of the system studied? How can we measure the density matrix of an excited atom or molecule that has been prepared e.g. in a binary collision, by photoexcitation, photo-dissociation or internal conversion? Such knowledge can provide much deeper insight into the preceding dynamics than average cross sections or probabilities. The subject has different names in different areas of physics. In nuclear physics one speaks about perturbed angular correlations, in atomic collision physics about coherence and correlation studies. Elegant theoretical concepts have been introduced to atomic physics, based on a key publication by FANO and M ACEK (1973).14 However, this original work is somewhat difficult to read for the unexperienced reader. We shall therefore try to draw out the key considerations, following essentially the step by step derivation presented in the review of A NDERSEN et al. (1988), Appendix C. We have to describe the radiation from a set of excited states |b which decay into a set of final states |a. According to (4.53) and (4.56), Vol. 1 the relevant dipole transition operator is for photon emission D† = r · e∗ and its adjoint is D = r · e. As outlined in Chap. 4, Vol. 1 it describes the essentials of the radiation process. We use again the helicity basis for the position vector r of the atomic electron and recall (4.76), Vol. 115 r =r C1q (θ, ϕ)e∗q = −r C1−q (θ, ϕ)eq . (9.138) The polarization vector e∗ of the emitted photon is e∗ =

1 q=−1

aq∗ e∗q

while e =

1

aq e q .

(9.139)

q=−1

For the moment we assume that atom and photon are described in the same coordinate system, characterized by the spherical unit basis vectors eq introduced in D contains all relevant information Sect. 4.1, Vol. 1. The dipole transition operator on the geometry of the experiment – including the detected polarization determined by the polarization amplitudes aq . The emission probability (4.67), Vol. 1 from one 14 For more details and illustrative examples we refer the interested reader to the specialized literature (see e.g. A NDERSEN and BARTSCHAT 2003; A NDERSEN et al. 1997a,b, 1988; B LUM 2012; Z ARE 1988; H ERTEL and S TOLL 1978; M ACEK and H ERTEL 1974). 15 The

minus sign can be pulled out of the sum as only terms with q = ±1 participate.

9.4


613

initially excited state |b = |γj m to one final state |a = |γ¯ j¯m ¯ is simply propor† tional to the squared matrix element | Dab |2 . With (9.138) the dipole transition matrix elements for emission are γ¯ j¯m| ¯ D† |γj m = r ab · e∗ = −γ¯ |r|γ

1

j¯m|C ¯ 1−q |j maq∗ .

(9.140)

q=−1

In Chap. 4, Vol. 1 as well as in Sect. 2.3.6 we have discussed only transitions between pure states. Now, we want to generalize this to transitions from a set of excited states into a set of final states of a lower level. As a first step we rewrite the ( spontaneous transition probability (2.153) for a coherent superposition |b = bj m |γj m of excited states to all accessible final substates |a = |γ¯ j¯m. ¯ The emission rate per solid angle is obtained by summing coherently over all initial and incoherently over all final states: 2 dR (spont) ¯ =C b j m| ¯ D |j m m dΩ j¯m ¯ jm

or Re (θ, ϕ) = C

bj m bj∗m

j m j m

with C =

3 3 αωba e2 ωba = 2πc2 8π 2 ε0 c3

j¯m| ¯ D† |j mj m | D|j¯m. ¯

(9.141)

j¯m ¯

Note that the emission rate dR (spont) /dΩ = Re (θ, ϕ), short Re , is given here for specified polarization e and emission angles θ, ϕ (see e.g. in Fig. 4.3, Vol. 1). Re (θ, ϕ) is a probability16 per atom, per solid angle and per unit of time and has the dimension T−1 . We can easily generalize (9.141) for an incoherent superposi( (β) tion of initial states bj m |j m, each present with a probability pβ : Re = C

j m j m

bj m bj∗m

j¯m| ¯ D† |j mj m | D|j¯m. ¯

(9.142)

m ¯

With (9.17) we identify

(β) (β)∗ bj m bj∗m = pβ bj m bj m = ρˆj m j m

(9.143)

β

as elements of the density matrix of the excited atom. Hence (9.142) may be written as (det) † = C Tr (9.144) Re = C Tr ρˆ D D = C Tr ρˆ ρˆ Dρˆ D† . 16 Integration over the excited state lifetime and 4π solid angle should give 1 (if Tr ρ = 1): each excited atom eventually emits one photon (its average lifetime being ∝ 1/C). In the literature, the 4 /(2πc3 ) – which would reflect the corresponding photon energy constant C is often given as e2 ωba emitted, the electron charge e being measured in esu!

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9


ρˆ (det) = D† D,

(9.145)

We have identified here a detector matrix

which in the present experiment acts in the same manner as the analyzer matrix introduced in Sect. 9.2. In principle, (9.144) may be evaluated by inserting (9.143) and (9.140). By suitably chosen polarization, represented by the amplitudes aq , one can hope to extract the density matrix elements ρˆj m ,j m of the atom under investigation. The atom may have been prepared e.g. by optical excitation, internal rearrangement, or collision – all of these processes may be studied in some detail by analyzing the polarization of the emitted light. The key question is, however, which physical information can be extracted by which specific experiment and how can the geometry of the experiment be disentangled from the dynamics of the preceding preparation process. This requires a transparent formulation of (9.144). Also, we have to account for the fact that the density matrix ρˆ of the system and that for the emitted fluorescence, ρˆ (det) , are often described conveniently in different coordinate systems. Thus, we have to rotate either ρˆ or ρˆ (det) according to (9.66). In terms of matrix elements, the detected signal becomes (j )∗ (j ) (det) Dmm˜ ρˆm˜ m˜ Dm˜ m ρˆm m , (9.146) Re = C mm ˜m ˜ m

(j )

with the rotation matrix elements Dmm˜ (0, θ, ϕ) as detailed in Appendix E, Vol. 1. This is obviously a rather clumsy expression, with the additional complication that the matrix elements of D† D describing the polarized light emission are also a nontrivial construct. Hence, this approach is not suited for a direct disentanglement of geometrical and dynamical parameters. Consequently – like it or not – we switch to an irreducible representation of the density and detector matrix in terms of state multipoles. Details are explicated in Appendix C. For the present discussion we just need to know that the density matrix can be expanded according to (C.3) † tˆ j j KQ tˆ j j KQ (9.147) ρˆ = KQ

in terms of the so called state multipoles † (−)j −m (2K + 1)1/2 tˆ j j KQ = mm

j m

j K ρ . −m −Q j m ,j m

(9.148)

Some characteristic examples are summarized in Table C.1 for 0 ≤ j = j ≤ 2. One of the great advantages of this irreducible representation of the density matrix is its transformation under frame rotation in the same simple way as angular momentum states, described by (C.13). This greatly simplifies the procedure indicated by (9.146). Also, recoupling of angular momenta is often necessary in such experiments, which is greatly facilitated when using this expansion.

9.4


615

Thus we insert (9.147) and (9.140) into (9.144) and apply the summations necessary according to (9.141): † Tr tˆ j j KQ Dtˆ j j KQ D† Re = C KQ

=C

1 † tˆ j j KQ γ |r|γ¯ γ¯ |r|γ × aq aq∗

×

¯ j¯j j mmm

(9.149)

q,q =−1

KQ

∗ j¯m|C ¯ 1−q |j m j m |tˆ j j KQ |j mj m|C1−q |j¯m. ¯

Here we have made one more generalization: as indicated by aq aq∗ we average over the products of amplitudes according to (9.139) which describe the detected polarization. As explained in Sect. 9.3.1, the polarization amplitudes define the polarization matrix (pol) (9.150) ρˆq q = aq aq∗ of the light transmitted by the polarization analyzer. Note that this polarization matrix ρ (pol) has to be distinguished from the detector matrix ρ (det) (9.145) introduced above. Their multipole moments differ, however, only by recoupling factors, as we shall see in a moment. A perfect polarization analyzer would be represented by (9.79), a realistic one by (9.89). We use again the convenient parametrization (9.91) in the helicity basis, (det) (det) (det) with the three S TOKES parameters P1 , P2 , P3 . Now we insert the matrix elements (C.5) of the statistical tensor operator into ∗ = (−1)−q C1q , and apply the W IGNER -E CKART theorem (9.149), replace C1−q (C.8), Vol. 1 to the matrix elements of the renormalized spherical harmonics Ckq . The reduced matrix elements, γ¯ j¯rγj = γ¯ |r|γ j¯C1 j and γj rγ¯ j¯, can be pulled out. One obtains a sum over a triple product of 3j symbols to which the contraction formula (B.69), Vol. 1 can be applied. Careful evaluation of all phase factors17 finally leads to (see also B LUM 2012, Chap. 5) Re =

C˜ j¯j j

with

3

sˆ (11)†KQ =

¯

(−1)1+j +j

(−1)K+Q

KQ 1 q,q =−1 ¯

1−q

(−1)

1 j

1 j

√ 2K + 1

K † tˆ j j KQ sˆ (11)†KQ (9.151) ¯ j

1 q

1 −q

K (pol) ρ . (9.152) −Q q q

also the factor (−1)j −j according to (C.52), Vol. 1 when inverting one of the reduced matrix elements.

17 Note

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9


Equation (9.151) is the key relation for the analysis of radiation from anisotropically populated excited atoms.18 C˜ contains all numerical factors such as C and the product of the reduced matrix elements, i.e. |γ¯ j¯rγj |2 for j = j . By comparison with (9.148) we identify the parameters ˆs (11)†KQ as state multipoles of the polarization analyzer matrix ρˆq q , while the state multipoles tˆ(j j )†KQ characterize the excited atom. In this (pol)

terminology, the state multipoles ˆs (11)†KQ of the polarization matrix ρˆ (pol) differ only by the recoupling factors from those of the detector matrix ρˆ (det) . Compared to (9.146) we have clearly achieved our goal of disentangling the characteristic of the atom studied from the geometry of the experiment. After this detailed derivation, we may now appreciate the more general perspective given by FANO and M ACEK (1973): the operators D and D† are of rank k1 = k2 = 1. Their matrix elements are j m |r|j¯m ¯ and j¯m|r|j ¯ m, displayed in the original equation (9.142). They couple to tensors of rank K. This coupling scheme of four angular momenta, as read from the matrix elements, may schematically be indicated by [(j j¯)1(j¯j )1](K) . However, we want to obtain the irreducible representation of the detector matrix ρˆ (det) defined by its matrix elements between j m | and |j m. This corresponds to a coupling scheme [(j j )K(j¯j¯)0](K) (rank 0 since we sum incoherently over the final states j¯m). ¯ The recoupling of the former scheme into the latter is achieved by a 9j symbol (recoupling coefficient for four angular momenta). Fortunately, it simplifies to a 6j symbol since one of the j ’s is 0. From (B.78), Vol. 1 our result can be retrieved. Alternatively, we may apply (C.50), Vol. 1 which gives the reduced matrix elements for products of two tensor operators acting on the same system.

9.4.2

General Discussion

Before illustrating this important result for specific experimental situations, we have to realize and discuss a few essential points. 1. The polarization analyzer is characterized by state multipoles ˆs (11)†KQ of rank K = 0, 1 and 2, since (9.152) as well as the 6j symbol in (9.151) requires the triangular relation δ(11K) = 1 to hold – a consequence of the photon spin 1 and single photon detection. Hence, such an experiment can only probe state multipoles tˆ(j j )†KQ of the atom up to rank K = 2. We recall that K = 2 reflects the shape of the atomic charge cloud, the so called alignment, while K = 1 measures the expectation values of angular momentum components, called orientation. Finally, K = 0 represents the isotropic part of the atom and the emitted light. that in the helicity basis Q drops out of the recoupling factors (−1)K {. . . } since it is even, as we shall see below.

18 Note

9.4


617

2. According to (B.71), Vol. 1 the 6j symbol is for K = 0

1 j

1 j

0 j¯ = j¯ 0

j 1

¯

δjj (−1)1+j +j 1 = √ . j 3(2j + 1)

(9.153)

¯

Hence, the phase factor (−1)1+j +j is cancelled and the isotropic part of the sum (9.151) is indeed positive as required. 3. On the other hand, the atomic system is characterized by angular moments j, j from which multipole moments up to rank K = j + j may be constructed. They all may, in principle, be excited e.g. in atomic collisions. However, if j + j > 2, a full analysis of all anisotropy parameters is not possible with single photon detection. This problem can be overcome by experiments with laser excited atoms. Instead of analyzing the emitted radiation, in a time inverse experiment the atom is prepared by optical pumping as sketched in Fig. 9.4 (see also Appendix D). Since the optical pumping process involves many photons and occurs in the hyperfine coupling scheme with angular momentum F , state multipoles up to K = 2F are prepared by this “state selector”. The same formalism can be used to evaluate such experiments. One simply has to replace (−1)K+Q {6j }ˆs (11)†KQ in (9.151) by ˆs (F F )†KQ . Details are described in H ERTEL and S TOLL (1978) and A N DERSEN et al. (1988), Appendix D. 4. The angular momenta j and j of the excited atom studied may represent either the hyperfine quantum number F , or the total electronic angular momentum J , or even the orbital angular momentum L – depending on the preparation process. According to the so called hypothesis of P ERCIVAL and S EATON (1958) the relevant coupling scheme depends essentially on the interaction time tint during which the atom is excited. If tint /FS (with FS being the fine structure splitting) spin and orbit are decoupled during the process and the orbital angular momentum L = j is the relevant quantity. This is typically the case for electron or fast ion impact excitation. If on the other hand /FS tint /HFS , where HFS is the hyperfine splitting, one has to describe the process in terms of the total electronic angular momentum J = j . Typical cases are fine structure transitions induced in thermal atom-atom collisions. Finally, if /HFS tint as in the optical pumping case just described, the hyperfine quantum number F = j is relevant for describing the atom. 5. This brings us to the temporal behaviour of the emission process, which so far we have completely neglected. However, if the excitation time is experimentally recorded, e.g. in a coincidence experiment between exciting laser pulse (or impacting particle) and emitted photon one has to account for the temporal evolution of the excited state. If different excitation energies are involved the time dependence of the emission amplitudes may lead to quantum beats as already introduced in Sect. 4.7.2, Vol. 1. Even in the absence of electric or magnetic fields, excited state levels j and j may be split by Ej − Ej = ωj j due to fine and/or hyperfine structure. This may be accounted for by factors (see e.g. B LUM 2012,

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9


Chap. 5) ∝

1 − exp[−iωj j − (Aj + Aj )/2] −iωj j − (Aj + Aj )/2

by which in principle all tˆ(j j )†KQ with j = j have to be multiplied. Here Aj are the inverse spontaneous lifetimes of the excited levels involved. It is interesting to note, that multipole moments higher than K = 0 are required to observe such FS or HFS quantum beats at all, since for K = 0 the δjj factor in (9.153) cancels all terms with j = j . 6. Often the experiment integrates over the whole decay time of the excited atom and the quantum beats average out. As a consequence only terms with j = j remain in (9.151). This may, however, lead to depolarization of the radiation. Appropriate mathematical treatment is achieved by exploiting the recoupling formalism for the state multipoles – as sketched in Appendix C.3. In effect, depolarization as well as recoupling is accounted for by multiplying the state multipoles by suitable depolarization and recoupling factors GK . The reader interested in the details is referred to A NDERSEN et al. (1988), Appendices A–C. In the following we shall further discuss only such experiments where j = j . For ease of writing we also shall assume that only one final state j¯ is involved. 7. In (9.151) both state multipoles, tˆ(j j )†KQ and ˆs (11)†KQ describing the atom and the detector, respectively, refer to the same coordinate system. Often experiments may indeed be arranged in such manner as we shall see below. Alternatively one has to rotate one or the other according to (C.13) or (C.14) – whichever is appropriate. For an explicit expression of the fluorescence intensity as a function of emission angles we refer again to A NDERSEN et al. (1988), Appendix C. In the relevant literature different authors use different representations of the multipole moments. In the tradition of FANO and M ACEK (1973) we prefer the real multipole moments constructed from angular momenta as detailed in Appendix C.2 – partially because the numerical factors involved look somewhat simpler, while e.g. B LUM (2012) and A NDERSEN and BARTSCHAT (2003) use the state multipoles. State multipoles and multipole moments are proportional to each other as described by (C.16)–(C.20). For compact writing we denote the real multipole moments of the atom by TKQp and those describing the polarization analyzer PKQp . We rewrite (9.151) in this notation for j = j after pulling the 6j symbol (9.153) for K = 0 out the sum. This makes the result quite transparent, noting that T00+ = P00+ = 1 and T00− = P00− = 0 (see also Table C.1): Re (θ, ϕ) =

C gK (j, j¯)GK TKQp PKQp 3

(9.154)

KQp

1 1+j +j¯+K ¯ with gK (j, j ) = (−1) 3(2j + 1) j

1 j

K j¯

v(K, j )v(K, 1) v(0, j )v(0, 1)

9.4


619

with v(K, j ) given by (C.16). With this definition g0 = 1 holds. The depolarization factors GK (with G0 = 1) can be derived using the recoupling concepts sketched in Appendix C.3 and appropriate time averaging over the decay process. Explicit expressions for GK are documented in A NDERSEN et al. (1988), together with a compact treatment of the closely related quantum beats. Also given there is an introduction to the theory for the time inverse experiment – scattering from laser excited atoms. The reader may also be interested in the main part of that review and its companions A NDERSEN et al. (1997a,b), which give a comprehensive and unified summary of pioneering experiments on alignment and orientation in atomic collisions.

9.4.3

Details of the Evaluation

Determining the S TOKES Parameters In this final subsection we briefly discuss some typical geometries without going into details. To have something concrete and simple in mind, we may consider a transition between pure orbital angular momentum singlet states, 1 P1 → 1 S0 or 1 D → 1 P , the respective values for g (j, j¯) being 2 1 K g0 (j, j¯) = 1s 3 2 1 g2 (1, 0) = 2

g1 (1, 0) =

3 4 1 g2 (2, 1) = . 12 g1 (2, 1) =

(9.155)

Let us study an aligned and/or oriented excited atom as illustrated in Fig. 9.5 (the geometry is the “atomic frame”, called “natural frame” in collision experiments). We first assume that the experiment detects fluorescent light propagating into +z(at) -direction. The analyzer is characterized in the helicity basis by the polarization matrix (9.91). The values of the real multipole moments PKQp for the detector are derived from this polarization matrix ρˆ (pol) as outlined in Appendix C.2. With (C.10) and (C.17)–(C.20), or from the explicit values read from Table C.1 for J = 1, we obtain: P00+ = 1 (pol)

P10+ = −P3 P20+ = 1

√ (pol) P22+ = − 3P1

(9.156) √ (pol) P22− = − 3P2 .

All other multipole moments of the analyzer are zero. Note in particular, that all PKQ± are zero if Q is odd – this is a consequence of the special structure of ρˆ (pol) ,

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9


expressing the transversality of light (see also footnote 18). Inserting these parameters into (9.154) and abbreviating gK = gK (j, j¯) gives Re =

√ (pol) √ (pol) C 1 + g2 T20+ − g2 T22+ 3P1 − g2 T22− 3P2 3 (pol) . − g1 T10+ P3

(9.157)

We have assumed here that no hyperfine splitting is involved and all GK = 1. This restriction can easily be removed by replacing gK → gK GK . Clearly, Re (θ, ϕ) ≥ 0 as one may verify for specific situation with the explicit expressions for the multipole moments given in Table C.1. As a consequence, the extremely simple structure of (9.157) for the signal allows a straight forward evaluation: Specific multipole moments may be measured by a specific choice of the analyzer S TOKES parameters. The total intensity emitted into +z(at) = z(ph) -direction is formally obtained by adding the signal for two orthogonal we obtain: polarizer settings. Omitting the overall scaling factor C, (pol) (pol) 2 = 1 + Re P i = −1 = {1 + g2 T20+ } Rz = Re Pi 3

(9.158)

(pol)

may be any of the three S TOKES parameters). The S TOKES parameters Pi (Pi of the detected light are derived from the corresponding differences of the signal: (pol) (pol) 2 Rz P1 = Re P1 = 1 − Re P1 = −1 = − √ g2 T22+ 3 (pol) (pol) 2 Rz P2 = Re P2 = 1 − Re P2 = −1 = − √ g2 T22− 3 (pol) (pol) 2 Rz P3 = Re P3 = 1 − Re P3 = −1 = − g1 T10+ . 3

(9.159) (9.160) (9.161)

Alternatively, when observing linearly polarized light emitted in the x (at) y (at) plane, with linear polarization perpendicular to it, one may with advantage use a “photon frame” (ph) with k ⊥ z elin , i.e. the z-axis is this time perpendicular to the wave vector k and parallel to the polarization. The polarization matrix (9.93) in this case has only one non-zero component, ρ00 . From Table C.1 one finds the detector multipole moments: ⊥ ⊥ P00+ = 1 and P20+ = −2.

(9.162)

All others disappear and the signal detected with such polarization – parallel to the z(at) , propagating in the x (at) y (at) plane – is thus R⊥ =

1 1 ⊥ 1 + g2 T20+ P20+ = [1 − 2g2 T20+ ]. 3 3

(9.163)

9.4


621

Angular Distribution It is important to note at this point that in all above discussion the polarization (pol) parameters Pq as well as the atomic multipole moments TKQ± were referring to the same coordinate frame. In the above discussion we have called it “atomic frame” (at) x (at) y (at) z(at) which was identical to the photon frame. Detection always occurs in the photon frame. Thus, we now mark the multipole moments in (9.157) by the (ph) superscript TKQ± to remind us of this fact. In the most general case the detector points in an arbitrary direction θk , ϕk , thus defining the x (ph) y (ph) z(ph) photon frame (with the wave vector k z(ph) ). We now allow this frame to be different from the atomic frame (for an illustration see Fig. 4.3 in Vol. 1). Thus, to eventually extract (at) multipole moments TKQ+ which describe the atom in a most convenient “at” frame, we have to rotate the coordinate system by (C.13). For the total emission rate this is relatively easy to achieve, since (9.158) in the (ph) photon frame contains only the zero component TK0+ of the atomic alignment tensor. And that relates to the multipole moments of interest (in the atomic frame) corresponding to (C.14): (ph)

TK0+ =

K

(at)

TKqp CKq± (θk , ϕk ).

q=0,p=±

Inserting this into (9.158) and using the real spherical harmonics CKq± as tabulated in Table D.1, Vol. 1, we obtain 6 2 2 (ph) R(θk , ϕk ) = 1 + g2 T20+ = 1 + g2 3 3 =

2

7 (at) C2q± (θk , ϕk )T2qp

q=0,p=±

(at) 2 1 + g2 3 cos2 θ − 1 T20+ 3 3 √ (at) (at) + 3 sin2 θ cos 2ϕT22+ + sin 2ϕT22− √ (at) (at) + 2 3 sin θ cos θ cos ϕT21+ + sin ϕT21− .

(9.164)

This is a general expression for the angular distribution of emitted radiation (without polarization analysis). In principle, it can be applied to radiation from any excited atoms or molecule (if depolarization is relevant one has to replace g2 → g2 G2 ). Specifically in a collision experiment, with the x (at) y (at) plane be(at) (at) ing defined as collision plane, T21+ and T21− are zero, and the charge cloud in the “at” frame corresponds to that illustrated in Fig. 9.5. One may also transform the polarization dependent signal (9.157) into an arbitrary photon frame direction. The procedure is straight forward, but somewhat tedious. We thus end our discussion here with one simple, but important case. (pol) Let us consider pure linear polarization along the x (at) -axis with P1 = 1 (and (pol) (pol) = P3 = 0). From Fig. 4.3 (Vol. 1) it is intuitively evident that this signal P2

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9


is independent of the polar angle θk as long as it is emitted with k ⊥ x (at) and polarization elin x (at) . Thus, emission of this polarization within the x (at) y (at) plane (pol) will also be described by (9.157) with P1 = 1:19 R =

√ 1 (at) (at) 1 + g2 T20+ − g2 T22+ 3 . 3

(9.165)

With (9.163) and (9.165) one may define an extra S TOKES parameter for the linear polarization detected in the x (at) y (at) plane (light propagating in the y (at) -direction). (at) (at) √ 3g2 T20+ + g2 T22+ 3 R − R⊥ (9.166) = P4 = √ . R + R ⊥ 2 − g2 T (at) − g2 T (at) 3 20+ 22+ Using this in conjunction with (9.158) and (9.159) we derive: (at)

T20+ =

1 2P4 + P4 P1 − P1 . g2 3 + P 1 + P 4 − P 1 P 4

(9.167)

With this and Rz from (9.158) we can derive from (9.159) and (9.160) similar expressions for the other two alignment parameters: P1 P2 (at) (at) (at) (at) T22+ = − √ 1 + g2 T20+ and T22− = − √ 1 + g2 T20+ . 3g2 3g2

(9.168)

We thus can reduce the measurements necessary for a full determination of the three non-zero alignment parameters to just three relative quantities. These may easily be recorded as S TOKES parameters P1 and P2 for light travelling in the +z(at) direction, and the parameter P4 which characterizes the linear polarization emitted into the +y (at) -direction. (at) Finally, the remaining orientation parameter T10+ can be directly derived from a measurement with circular polarization. From (9.161) we obtain (at)

T10+ = −

P3 (at) 1 + g2 T20+ . g1

(9.169)

As one easily sees from the tabulated values Table C.1 of the multipole moments, (at) T10+ is simply the expectation value of the atomic angular momentum with respect (at)

to the z(at) -axis. With reference to Fig. 9.5, in contrast T20+ (which for a pure L = 1 state is 1 − 3ρ00 ) reflects essentially the height of the atomic charge cloud, the latter (at) (at) being ∝ ρ00 . Finally, from the alignment parameters T22+ and T22− one recovers the width and the alignment angle of the charge cloud. We leave it to the interested reader as a simple exercise to derive the parameters of the density matrix of the atom 19 Of course this can be verified explicitly with some careful effort by rotating (E.25)–(E.28), Vol. 1

through the E ULER angles α = ϕk = −π/2 and βk = π/2, using the expressions given in (E.13), 2 2 Vol. 1 for the d±20 and d±2±2 matrices.


623 (at)

(e.g. for a pure L = 1 state) by comparing the TKQ± via Table C.1 with (9.132). Note, that the S TOKES parameters used there to describe the atom relate to the directly measurable quantities Pi discussed here by Pi /(gK GK ). Section summary

• Based on the concepts of FANO and M ACEK (1973), we have derived the general theory for the angular and polarization distributions Re (θ, ϕ) of light emitted from anisotropically and incoherently populated excited states. D† = r · e∗ for emission of a • Starting from the dipole transition operators D† D. The photon with polarization e we define a detector matrix ρˆ (det) = (det) ), where ρˆ describes expected signal is then given by Re (θ, ϕ) = Tr(ρˆ ρˆ the excited system. • In order to disentangle geometrical and dynamical parameters in such an experiment both density matrices are expanded into series of state multipoles of rank K, with 0 ≤ K ≤ 2 for ρˆ (det) in single photon detection. • The detector matrix ρˆ (det) , originally described in a [(j j¯)1(j¯j )1](K) coupling scheme has to be recoupled into a more transparent and convenient [(j j )K(j¯j¯)0](K) coupling scheme for the polarization analyzer ρˆ (pol) . • The final result (9.154) for Re (θ, ϕ) is expressed in terms of multipole moments (constructed from angular momenta) for both, the atom and the polarization analyzer. • Detailed evaluation in terms of S TOKES parameters and angular distribution of the total intensity leads to a clear disentanglement of experimental geometry from the atomic alignment and orientation parameters to be determined.

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. c.c.: ‘complex conjugate’. esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3 in Vol. 1). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6 in Vol. 1). HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9 in Vol. 1). LHC: ‘Left hand circularly’, polarized light, also σ + light. RHC: ‘Right hand circularly’, polarized light, also σ − light.

References A NDERSEN , N. and K. BARTSCHAT: 2003. Polarization, Alignment and Orientation in Atomic Collisions. Berlin, Heidelberg: Springer.

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A NDERSEN , N., K. BARTSCHAT, J. T. B ROAD and I. V. H ERTEL: 1997a. ‘Collisional alignment and orientation of atomic outer shells: 3. Spin-resolved excitation’. Phys. Rep., 279, 252–396. A NDERSEN , N., J. T. B ROAD, E. E. B. C AMPBELL, J. W. G ALLAGHER and I. V. H ERTEL: 1997b. ‘Collisional alignment and orientation of atomic outer shells: 2. Quasi-molecular excitation, and beyond’. Phys. Rep., 278, 108–289. A NDERSEN , N., J. W. G ALLAGHER and I. V. H ERTEL: 1988. ‘Collisional alignment and orientation of atomic outer shells: 1. Direct excitation by electron and atom impact’. Phys. Rep., 165, 1–188. BARTSCHAT , K., K. B LUM, G. F. H ANNE and J. K ESSLER: 1981. ‘Electron-photon coincidences with polarized electrons’. J. Phys. B, At. Mol. Phys., 14, 3761–3776. B LUM , K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics 64. Berlin, Heidelberg: Springer, 3rd edn., 343 pages. B RINK , D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3 edn., 182 pages. FANO , U. and J. H. M ACEK: 1973. ‘Impact excitation and polarization of emitted light’. Rev. Mod. Phys., 45, 553–573. G REENE , C. H. and R. N. Z ARE: 1983. ‘Determination of product population and alignment using laser-induced fluorescence’. J. Chem. Phys., 78, 6741–6753. H ERTEL , I. V. and W. S TOLL: 1978. ‘Collision experiments with laser excited atoms in crossed beams’. In: ‘Adv. Atom. Mol. Phys.’, vol. 13, 113–228. New York: Academic Press. K LEIMAN , V., H. PARK, R. J. G ORDON and R. N. Z ARE: 1998. Companion to Angular Momentum. New York: Wiley, 208 pages. M ACEK , J. and I. V. H ERTEL: 1974. ‘Theory of electron-scattering from laser-excited atoms’. J. Phys. B, At. Mol. Phys., 7, 2173–2188. M UKAMEL , S.: 1999. Principles of Nonlinear Optical Spectroscopy. Oxford: Oxford University Press, 576 pages. P ERCIVAL , I. C. and M. J. S EATON: 1958. ‘The polarization of atomic line radiation excited by electron impact’. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci., 251, 113–138. W EISSBLUTH , M.: 1989. Photon-Atom Interactions. New York, London, Toronto, Sydney, San Francisco: Academic Press, 407 pages. Z ARE , R. N.: 1988. Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics. New York: Wiley, 368 pages.

10

Optical B LOCH Equations

Up to now we have treated optically induced processes exclusively in the framework of perturbation theory. If, however, the probability densities of the excited states become comparable to those of the initial states this is no longer sufficient. Also, perturbation theory does not answer the question about the type of radiation which is re-emitted in such a case. By spontaneous emission, inevitably, mixed states are created. To include these into a formal description we have to apply the concepts developed in Chap. 9. These and related questions are at the heart of quantum optics and subject to the present chapter.

Overview

To set the stage, in Sect. 10.1 we take a look onto an experiment from modern quantum optics. In Sect. 10.2 the important “dressed state” model is introduced to analyze the two level system in a quasi-monochromatic light. Section 10.3 presents several characteristic experiments which may be explained effortless with this model. Section 10.4 derives the theoretical framework for treating such systems quantitatively, starting with the fundamental L IOUVILLE - VON -N EUMANN equation from which the “Optical B LOCH equations” are derived. In Sect. 10.5 we apply these to a number of important questions which in earlier chapters could only be discussed by hand waving arguments. On these grounds, Sect. 10.6 develops some basics of short pulse spectroscopy. Finally, Sect. 10.7 introduces a somewhat more complex application, the STIRAP method, today of increasing interest also in the context of “quantum information”.

10.1

Open Questions

Perturbation theory can only be applied for a quantitative description of light induced excitation if the changes in the quantum system studied are very small. It fails if the exciting electromagnetic field (e.g. a laser field) is high enough to substantially modify the population of the states involved – and with laser intensities © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5_10

625

626

10


available today this is a situation often encountered, in particular if a transition is induced with nearly resonant radiation. In addition, a number of profound questions about the re-emitted radiation cannot be answered by simple perturbation theory. If, for example, a narrow band laser is tuned slightly off resonance to an atomic absorption line: what is the frequency of the resonance fluorescence? Or more general, how precisely is the spectrum of the resonance fluorescence structured if an atom or molecule is excited by an extremely narrow band laser, whose bandwidth is smaller than the natural linewidth of the excited level studied? Does one observe the natural linewidth corresponding to the usual spontaneous emission, or does the spectral distribution of the re-emitted light rather reflect the incident radiation? How does nature secure energy conservation in such a case? There are plenty of such questions which have fuelled the turbulent development of quantum optics, which started in the 60ies and 70ies of the past century and continues until today. The theoretical background for the B LOCH equations which are at the heart of the present chapter has already been developed even earlier – in the context of EPR and NMR spectroscopy. (The standard textbook by A LLEN and E BERLY 1975, has established these concepts in optical physics.) Presently, quantum information science appears to rise and expand with no less ardour – much of it based on single atoms interacting with specifically tailored light (see also T ICHY et al. 2011; W EISSBLUTH 1989). To set the stage for the following discussion we refer to some relatively recent work of W EBER et al. (2006) (see also VOLZ et al. 2007; P IRO et al. 2011). These experiments are part of the quest for systems suitable to quantum computing and quantum information storage. Here one studies the fluorescence which is emitted from a single, laser cooled rubidium atom, 87 Rb, stored in a so called FORT trap (far-off-resonance optical dipole trap), an optical dipole trap (partially overlapping with the MOT) in which the atom is clipped to the focus (3.5 µm) of a laser beam, which is detuned from resonance by 62 nm. The setup is illustrated schematically in Fig. 10.1. First, the atoms are laser cooled in a magneto-optical trap (MOT) and then filled into the FORT. One atom only is stored at a time in the trap for each measurement. One recognizes this from the detected signal (top right inset in Fig. 10.1): the atom (continuously re-excited after decay into the ground state) emits photons for a few seconds before falling out of the trap and being replaced by a new atom. The average time span of emission is ca. 4 s. The excitation scheme is illustrated in Fig. 10.2. The cooling laser (CL) is tuned to the 5 2 P3/2 F = 3 ← 5 2 S1/2 F = 2 hyperfine transition in 87 Rb, slightly off resonance by CL . To avoid optical pumping into the 5 2 S1/2 F = 1 ground state one further, so called “repump laser” (RL) is tuned to the 5 2 P3/2 F = 2 ← 5 2 S1/2 F = 1 transition. One may now determine e.g. the second order degree of coherence g (2) (δ) according to (2.36). As sketched in Fig. 10.3(a), a setup of HBT type (see Sect. 2.1.6) is used. The fluorescence light is split into two parts (remember, the light originates always from the same, isolated atom). The detector probes the probability for receiving a second photon after a first one, temporally delayed by δ. The observed

Open Questions

627 photon count rate / ms

10.1

to vacuum pump Rb-dispenser objective (NA = 0.38)

MOT coils

dichroic mirror

4 2 0

40 10 20 30 measuring time / s Si avalanche diode

laser beams for cooling

single mode glas fibre

from dipole trap laser

Fig. 10.1 Experimental setup according to W EBER et al. (2006) for measuring the 2-photon correlation function for a single atom in a MOT trap

Fig. 10.2 Hyperfine pump scheme for a single 87 Rb atom according to W EBER et al. (2006)

5 2P3/2 F' = 3 F' = 2

Δ CL CL RL 5 2S1/2 F = 2

F =1

(a)

t1 ST

(b)

3.0 2.0

t2

D2

from detection optics

g (2)(δ )

Fig. 10.3 Intensity correlation function g (2) (τ ) of the fluorescence from a single Rb atom in a FORT trap according to W EBER et al. (2006) (see also VOLZ et al. 2007). One clearly sees the anti bunching at a delay time δ = 0, as well as the damped R ABI oscillations; CL = cooling laser, RL = resonant repump laser

D1

correlation detector

1.0 dark current 0

-50

0

50

100 δ / ns

628

D2

from detection optics single atom fluorescence / arb. un.

Fig. 10.4 Fluorescence spectrum at very low excitation intensity. One measures (black line and data points) a bandwidth much below the natural linewidth (6 MHz). The fluorescence spectrum is only slightly broadened with respect to the exciting laser line (red). The difference originates from D OPPLER broadening of the cold atoms at a temperature of 105 µK(!) (data adapted from W EBER et al. 2006; VOLZ et al. 2007)

10

Optical B LOCH Equations FPI D1

ST 1.0

scanning piezo

0.8 0.6 0.4 0.2 0.0 - 1.0

- 0.5

0.0 0.5 Δν / MHz

1.0

signal shown in Fig. 10.3(b) gives clear evidence of anti-bunching: if one photon has been emitted, the stored atom is found in the ground state and it takes some time until it has a high probability for being found in the excited state again. Obviously, this probability oscillates between a maximum and a minimum. Later in this chapter, we shall develop a quantitative understanding of these damped oscillations, the so called R ABI oscillations. In Fig. 10.4 the corresponding experimental setup (top) for the spectral analysis of the fluorescence (bottom) is shown. One uses an FPI, scanned by a piezo. For comparison the laser linewidth is studied simultaneously (red line). Both profiles are much narrower than the natural linewidth of the atom (6 MHz). At the very low intensity used in this particular experiment one simply observes R AYLEIGH scattering of the incident light. As only one atom is captured in the trap at a given time, many individual measurements have to be carried out to obtain these nice data. The thus calibrated atoms, albeit very cold, still have a finite temperature which leads to a measurable D OPPLER broadening of the fluorescence line. The authors derive from the linewidth an atomic temperature of 105 µK. In view of experiments discussed below, we have to point out that the extremely narrow line profile shown in Fig. 10.4 is only observed at extremely low laser intensities. As soon as the level shift due to the dynamic S TARK effect (see Sect. 8.4.1 in Vol. 1) gets into the order of the natural linewidth, the picture changes completely – as we shall see below. Section summary

• Perturbation theory can only describe optical excitation at low intensities. It fails, if the excited state probability becomes substantial. Also, it cannot address questions about the spectral distribution of light emitted from a system excited with narrow band lasers.

10.2

Two Level System in Quasi-Monochromatic Light

629

• Significant improvements of radiation theory are needed, as e.g. demonstrated by fluorescence from a single, isolated atom in a trap: R ABI oscillations are observed and (at very low radiation intensity) a linewidths far below the natural linewidth of the atom.

10.2


The two level system is an extremely simple and useful approximation for understanding the temporal evolution of quantum systems in intense, quasimonochromatic, nearly resonant radiation fields. We have already used this concept in Sect. 2.3.5 (see Fig. 2.18). Here we shall demonstrate that it provides qualitative and quantitative insight into the key processes. The following questions will have to be addressed – without recurrence to perturbation theory: How does the population and coherence of a quantum system with two levels develop in time if it is exposed to a strong, nearly resonant electromagnetic radiation field? How does it depend on the detuning from resonance? And which radiation is re-emitted?

10.2.1 Dressed States From Sect. 2.3.5 we recall some basics about the fully quantized description of transitions induced by a single mode electromagnetic field. We distinguish the H AMIL A of the atom, H F of the field and the interaction U . With the energy TON operator H ωγ of the unperturbed atomic state |γ and the energy N ω of a photon state |N for N photons in the mode, A |γ = ωγ |γ and H F |N = N ω|N H the total energy of the system without interaction is obtained from 0 |γ N = Wγ N |γ N H 0 = H A + H F and Wγ N = ωγ + N ω. with H Neglecting for a moment the interaction of the field with the atom, we have for a system with two levels γ = a, b 0 b(N − 1) = ωb + (N − 1)ω b(N − 1) H 0 |aN = [ωa + N ω]|aN . H In Fig. 10.5 the energy levels with different photon numbers N are sketched for nearly resonant excitation ω ω, the detuning being defined as ω = ωba − ω

with ωba = ωb − ωa .

(10.1)

630

10

Fig. 10.5 Energy levels of the uncoupled two level system, “dressed” with photons


|b, N +2〉 |a, N+3〉

ħΔω

~ ~ ħωba

|b, N +1〉 |a, N+2〉 ~ ~

ħω

~ ~

ħΔω

ħωba ħω

|b, N 〉 |a, N+1〉

~ ~ ħωba

|b, N-1〉 |a, N 〉

~ ~

ħΔω

ħω ħΔω

The states are grouped in pairs, slightly split as a consequence of the detuning. In principle this ladder may be extended below and above the levels shown. For clarity and without loss of generality we choose the energy of the state |aN as zero energy. But as we shall see, only two such pairs of levels need to be considered for describing the two level system in a nearly resonant, oscillating field. With Fig. 10.5 in mind, the term “dressed states” has been coined – evoking the image of a quantum system dressed by photons.

10.2.2 R ABI Frequency ( = er · E = i ke eCk [ We recall now the interaction operator U Daˆ k − D† aˆ k+ ] for E1 transitions according to (2.132) and its non-zero matrix elements according to (2.135a)–(2.135d). For high intensities (N 1) according to (2.126) the matrix elements do not explicitly depend on the photon number, but on the field amplitude Dba = r ba · e: E0 , and we obtain, again with the abbreviation |aN = i e bN − 1|U Dba E0 2 |bN = i e aN − 1|U Dab E0 2 |bN = − i e aN + 1|U D∗ E 0 2 ba |aN = − i e bN + 1|U D∗ E 0 . 2 ab

(10.2a) (10.2b) (10.2c) (10.2d)

10.2


631

In a genuine two level system, only one lower substate |a = |γa ja ma and one upper substate |b = |γb jb mb must be involved. The transitions are thus induced by radiation with a basis polarization vector e = eq with q = mb − ma . The relevant transition matrix element is then r ba · e = rq (ma mb ) as defined in (4.77), Vol. 1, which in this particular case (two level system) is also identical to |r ba | according to (4.79), Vol. 1.1 We now introduce the (resonant) R ABI frequency (we mention in this context the N OBEL prize for R ABI 1944) ΩR =

Dba E0 e e|r ba |E0 = ,

(10.3)

proportional to the dipole transition matrix element |r ba | between the two levels and to the electric field strength E0 . For quantitative comparisons it is useful to relate |r ba | to the E INSTEIN coefficients Aab and Bba , (2.151): Aab =

4α 4h 1 |r ba |2 ω3 = 3 Bba = . 2 T1 3c 3λ

(10.4)

Here we call the natural lifetime of the excited state T1 . We must emphasize that these relations hold in this special form only for a genuine two level system. R ABI frequency and intensity I of the laser field are related to each other by ΩR2 =

e2 |r ba |2 E02 2e2 |r ba |2 I 2I 3λ3 Aab = = Bba = I. 2 2 πc 2πhc ε0 c

(10.5)

10.2.3 Rotating Wave Approximation For the two level system we shall now apply the rotating wave approximation (RWA) which we have already introduced in Sect. 2.3. It simplifies the problem significantly: only the near energy resonant processes (10.2a) and (10.2c) are taken into account, i.e. induced emission and absorption of a photon. As we have seen in the framework of perturbation theory this neglects the fast oscillating terms which average out more or less perfectly when the time dependent S CHRÖDINGER equation is integrated.2 While deriving in the following a non-perturbative solution, it is a two level system may be realized for jb = ja + 1 with maximum or minimum projection quantum numbers by the states |γa ja ±ja and |γb jb ±jb with circularly polarized light, q = ±1, for excitation. A classical example is the 3 2 S1/2 F = 2 MF = 2 ↔ 3 2 P3/2 F = 3 MF = 3 transition, populated by optical pumping (see Appendix D).

1 Typically,

2 However, when studying processes of higher order (e.g. multi-photon processes), the application of the RWA is no longer trivial.

632

10


useful to keep in mind that inserting the electric field strength into the matrix elements – as done in (10.2a)–(10.2d) – implies a strictly single mode, monochromatic radiation field (in contrast to the previous treatment in Sect. 2.3.6 where we have assumed a broad-band, multi-mode radiation field). This is certainly permissible if the bandwidth ω1/2 of the radiation field used is small enough, i.e. much narrower than the natural linewidth 1/T1 ω1/2 . One may approach the remaining problem in two different ways: 1. One tries to solve the time dependent S CHRÖDINGER equation i

∂|ψ(t) A + H F + U )ψ(t) = (H ∂t

(10.6)

by solutions of the type ψ(t) = cb (t)b(N − 1) e−i(ωb +(N −1)ω)t + ca (t)|aN e−i(ωa +N ω)t . With this ansatz one obtains, in analogy to (2.142) and (2.143), a set of differential equations (independent of N ) 1 c˙b = + ΩR ca eiωt 2 1 c˙a = − ΩR cb e−iωt 2

(10.7)

where ω is the detuning according to (10.1). The solutions ca (t) and cb (t) are the time dependent probability amplitudes of the upper, |b(N − 1), and lower, |aN states, respectively. One finds that they are exponential functions which contain the fast temporal oscillations corresponding to ωb + (N − 1)ω and ωa + N ω, respectively, and in addition some side bands. 2. Somewhat more compact one takes the “dressed state” picture, Fig. 10.5, literally F + U with the interaction in matrix =H A + H and writes the Hamiltonian H form. The state |aN defines the energy zero: 0 −iΩR . (10.8) H= 2 iΩR 2ω With this not only the lowest pair of states is described. Rather, for all other pairs of states the same energy operator acts, raised by ω for each additional level pair. In between these states no coupling exists if only single photon processes are considered.3 The time dependent S CHRÖDINGER equation reads in the rotating wave approximation dc i c, =− H dt 3 This

will be different as soon as another photon is involved which is absorbed or emitted.

(10.9)

10.2


633

where c(t) stands for a vector constructed from the time dependent probability amplitudes c1 (t), c2 (t) . . . . In this representation the fast oscillations drop out and only the physically relevant frequency differences are displayed. The differential equations are now 1 c˙b = + ΩR ca − iωcb 2 1 c˙a = − ΩR cb . 2

(10.10)

They are – apart from the fast oscillations of which only iωcb reminds us – fully equivalent to (10.7).

10.2.4 The Coupled System Before entering in Sects. 10.4 and 10.5 into details of the temporal dynamics, using the so called optical B LOCH equations, we want to have a somewhat closer look at the energetics. The H AMILTON operator (10.8) with the interaction term represented by the R ABI frequency ΩR , may readily be diagonalized to explore the energetic shift due to the interaction with the electromagnetic field. As usual, the secular determinant must disappear: −W − W det(H 1) = iΩR 2

R − iΩ 2 = 0. ω − W

(10.11)

From this relation one obtains the new energy eigenvalues W± = (ω ± Ω ) 2

(10.12)

with the so called non-resonant R ABI frequency Ω =

(ω)2 + ΩR2 .

(10.13)

The thus identified energy shifts hold for each of the level pairs sketched in Fig. 10.5. In Fig. 10.6 this is illustrated for the level pairs |bN , |aN + 1 and |bN − 1, |aN . Thus, both level pairs split in the field (beyond ω), the total slitting being Ω . One occasionally calls this splitting in the field R ABI splitting, which is proportional to the field for resonant excitation (ω = 0). The additional splitting is a consequence of the coherent superposition of the two initial states in each pair |bN , |aN + 1 and |bN − 1, |aN , respectively. The new basis states in the diagonalized

634

10

|+'〉

|b, N 〉

(ΩΔ+Δω)/2

Δω

|a, N +1〉

~ ~ ~ ~ ω

ΩΔ

(Ω Δ- Δω)/2

~ ~

~ ~

ω ωba

|b, N-1〉

~ ~ ~ ~

|-'〉

ω

ω −ΩΔ

ω+ΩΔ

|+ 〉 (ΩΔ+Δω)/2

Δω

|a, N 〉


(Ω Δ- Δω)/2

ΩΔ

|-〉

Fig. 10.6 “Dressed states” – for illustration of the four energy levels arising due to R ABI splitting from the unperturbed states |a N , |b(N − 1), |a(N + 1) and |bN of the two level system. Excitation occurs with a frequency ω which is detuned by ω = ωba − ω in respect to the resoABI frequency ΩR and detuning ω add geometrically to the nance frequency ωba . The resonant R non-resonant R ABI frequency Ω = ω2 + ΩR2 which determines the splitting. The emitted radiation forms a so called M OLLOW triplet, as indicated by red arrows, with the angular frequencies ω − Ω , ω and ω + Ω

c = W± c and are4 system are obtained from (10.8) with (10.12) by H |+ = −i sin Θ|aN + cos Θ b(N − 1) |− = cos Θ|aN − i sin Θ b(N − 1)

(10.14) (10.15)

with cos Θ ΩR = . sin Θ Ω − ω

(10.16)

In the limit of vanishing laser field the amplitudes are cos Θ → 1 and sin Θ → 0, while for very high fields (high √ R ABI frequencies) or exact resonance the limiting values are cos Θ = sin Θ = 1/ 2, i.e. both states contribute with equal amplitude to the new basis states. The original “ground states” |aN or |a(N + 1) then become |− and |− , respectively, while |b(N − 1) and |bN are transferred into the linear combinations |+ and |+ , respectively, as indicated in Fig. 10.6. From this energy splitting, illustrated in Fig. 10.6, one may already understand the characteristic spectral profiles which such a “dressed atom” emits. This is indicated in the figure by red marked double arrows and the dashed single arrows (side bands): one expects from this picture that the fluorescence of a near resonantly phase factors −i can be traced back to our choice of the phase in the definition (2.87) of the electric field.

4 The

10.3

Experiments

635

excited atom consists of three components: a resonant part (R AYLEIGH scattering) with the frequency ω of the laser field, and two side bands which are shifted by ±Ω with respect to the former. This so called M OLLOW triplet is indeed observed as we shall see in the next section. The splitting is given by the non-resonant Ω according to (10.13). Thus, it depends on the laser intensity – via (10.5) – and on the detuning ω. Section summary

• The “dressed state” model Fig. 10.5 pictures the energies of a two level quantum system in a quasi-monochromatic, nearly resonant electromagnetic radiation field as a ladder of levels separated by the photon energy ω. Each level is split into a doublet which – in the limit of low intensity – is split by the detuning ω = ωba − ω of the radiation from resonance ωba . • The interaction (10.2a)–(10.2d) of the field with the atom is characterized by the resonant R ABI frequency ΩR = e|r ba |E0 /. Eigenvalues (10.12) and eigenstates (10.14)–(10.15) in the dressed state model are derived in the rotating wave approximation RWA. • This leads to quantitative predictions about the emitted fluorescence radiation. One expects a so called M OLLOW triplet as illustrated in Fig. 10.6, arising from the ground and excited state doublets, each with a splitting due to Ω = ω2 + ΩR2 , the non-resonant R ABI frequency.

10.3

Experiments

10.3.1 M OLLOW Triplet Pioneering experiments have been performed by Herbert WALTHER and his collaborates 1974 and 1976, followed by G ROVE et al. (1977). They studied the prototypical 3 2 P3/2 F = 3 ↔ 3 2 S1/2 F = 2 transition in Na atoms, which may be considered a genuine quasi two level system. Its preparation is described in Appendix D in some detail. A typical experimental setup is illustrated schematically in Fig. 10.7. Key Fig. 10.7 Experiment for studying the M OLLOW triplet according to H ARTIG et al. (1976)

vacuum apparatus atomic beam FABRY -

A1

A2

PEROT

laser from back

photomultiplier

636

(a)

10 fluorescence signal

Ω

Ω/γ

P / mW


(b) 8

35 26

6

17 10

4

7.5 4.0

2

0.5 -100

- 50

0

50

0

∆ν / MHz

0

10

20

30

P / mW

Fig. 10.8 M OLLOW triplet in resonant excitation at different power P of the exciting laser according to H ARTIG et al. (1976). (a) Experimentally observed spectra (base line are shifted vertically for each measurement); (b) splitting Ω of the triplet in units of the natural linewidths γ as function of P

elements are a well collimated Na atom beam, a stable, narrow band CW dye laser, and an FPI for analyzing the spectrum of the fluorescence. Figure 10.8(a) shows the observed spectrum of the resonance fluorescence for exactly resonant irradiation (ω = 0) at different laser intensities. M OLLOW (1969) has predicted this triplet, named after him, and provided a quantitative theory – first semiclassically, later on fully quantum mechanically. Both calculations essentially lead to the same result (M OLLOW 1975). Quantitatively the splitting Ω as a function of the laser power P is shown in Fig. 10.8(b). From our above discussion one expects that Ω equals the R ABI frequency in the resonant limit and should be

theory for Ω/(2π) = 78 MHz, γ /(2π) = 10 MHz

(a)

(b)

Δν = - 50 MHz

Δν = 0 MHz

convolution of theory with experimental profile

Δν = 50 MHz

experiment

-100 - 50 0 50 100 - 100 - 50 0 50 100 -100 - 50 0 fluorescence shift with respect to laser frequency (νem - ν) / MHz

50

100

Fig. 10.9 M OLLOW triplet according to G ROVE et al. (1977) for three different values of detuning ν – the intensity was I = 640 mW cm−2 in all three cases. (a) Theory with pronounced elastic (R AYLEIGH) scattering in the case of significant detuning. (b) Comparison of experiment (black) and the theory (red), the latter has been convoluted with the experimental profile

10.3

Experiments

637

√ √ proportional to I ∝ P as stated (10.5). And this is exactly what is observed experimentally as documented in Fig. 10.8(b). H ARTIG et al. (1976) have also studied already the non-resonant case. While the theory predicts full symmetry of the fluorescence around the irradiating laser frequency, the first experiments showed some asymmetric profiles. However, G ROVE et al. (1977) were able to show that a very clean preparation of the two level system and accounting for geometrical effects does lead indeed to fully symmetric M OLLOW triplets, both in the resonant as well as in the non-resonant case. In these experiments the laser frequency has been stabilized with the Na resonance fluorescence as reference. As documented in Fig. 10.9 one observes excellent agreement between the measured spectrum of the fluorescence and the theory – which has been properly convoluted with the experimental profiles.

10.3.2 AUTLER -TOWNES Effect The observation of the M OLLOW triplet confirms nicely the predictions about the resonance fluorescence of a two level system for quasi resonant excitation which we have gleaned from Fig. 10.6. Nevertheless, this is not yet a direct proof of a splitting of the originally non-degenerate levels into two terms each. However, the so called AUTLER -T OWNES effect gives direct evidence for it, by using two lasers and three levels. The concept is sketched in Fig. 10.10. An intense pump laser “A” irradiates and atom with a fixed frequency which is nearly in resonance with the transition between states |a and |b. It thus generates the level splitting (e.g. between the two hyperfine levels in the case of the 3 2 S1/2 and 3 2 P3/2 levels in Na discussed above). The weaker probe laser “B” is tuned such that it may induce a transition from the upper state |b to a third, higher lying level |c. The population of the intermediate levels |b is determined by the fluorescence detected from level |c (in the present example the F = 4 level in the 3 2 D5/2 state of Na). G RAY and S TROUD (1978) have indeed measured this splitting of the |b state as documented in Fig. 10.11. Figure 10.11(a) shows the measured excitation probability |b of the intermediate level as a function of the detuning of the probe lases with respect to the transition |c ← |b in a weak laser field “A”. One sees a clear splitting of the state |b, which grows with the intensity of the pump laser “A”. Fig. 10.10 Scheme for measuring the AUTLER T OWNES effect in Na

3 2D5/2 F'' = 4, M'' = 4 laser B (probe)

σ+

detection by fluorescence 3 2P3/2 F' = 3, M' = 3

σ+

>

|c

|b

>

|a

>

laser A (pump, splitting)

3 2S1/2 F = 2, M' = 2

10

IA = 5.3

(a) level splitting / MHz

excitation probability

638

IA = 86 IA = 470


(b)

60 40 20 0

-50

0 Δν B / MHz

50

0

5

10

15

20

25

IA1/2 / (mW cm-2)1/2

Fig. 10.11 Experimentally observed AUTLER -T OWNES effect (see scheme Fig. 10.10) according to G RAY and S TROUD (1978). (a) Fluorescence spectra with resonant irradiation at different intensities IA /(mW cm−2 ) as a function of the detuning of√the probe laser νB (intensity IB = 15 mW cm−2 ); (b) measured line distance as a function of IA Fig. 10.12 AUTLER T OWNES splitting according to G RAY and S TROUD (1978), recorded as a function of the probe laser detuning νB at different values νA of the pump laser detuning; the intensities of pump and probe laser are kept constant at IA = 780 mW cm−2 and IB = 3 mW cm−2 , respectively

ΔνA = 25 MHz 20 15 10 5 0 -5 -10 -15 - 20

ΔνB / MHz: - 60 0 60

Figure 10.11(b) summarizes the measurements of the splitting (AUTLER -T OWNES effect) as a function of the pump laser intensity IA , and compares them with theory which in turn is proportional to the square root of IA . The observation that the splitting already disappears at finite intensity is a consequence of the finite natural linewidth – as we shall see quantitatively in the following section. Finally, one may study the dependence of the effect on the detuning of the pump laser. This is illustrated in Fig. 10.12: with this detuning, νA , the excitation profile of the transition |c ← |b becomes asymmetric and leads in the limit to one single line as characteristic for a resonant two-photon excitation |c ← |a. Section summary

• Experimental observations of the resonance fluorescence in an intense, nearly resonant, quasi-monochromatic laser field confirm the predictions of the dressed state model – both for resonant and non-resonant excitation.

10.4

Quantum Systems in Strong Electromagnetic Fields

639

• A direct experimental proof of the level splitting into a doubled – due to interaction with the electromagnetic radiation field – is obtained in a three level scheme, with a pump and a probe laser as illustrated in Fig. 10.10. When tuning the probe laser two resonances are observed in the fluorescence from the third level: this observation is called AUTLER -T OWNES effect.

10.4


For a quantitative understanding of such experiments one has to account correctly for spontaneous emission. In order to do so, a multitude of initially unoccupied field modes must be included in the treatment so that the solution of the problem can no longer be given in terms of a wave function. Rather, we have to use the density matrix formalism to which we have familiarized ourselves in Chap. 9. Up to now we have, however, not yet discussed the temporal evolution of the density matrix.

10.4.1 Temporal Evolution of the Density Matrix † we write the ∗ = H Accounting for the Hermiticity of the H AMILTON operator H time dependent S CHRÖDINGER equation i

∂|ψ(t) ψ(t) =H ∂t

or

− i

∂ψ(t)| . = ψ(t)H ∂t

(10.17)

With this, we derive the time dependence of the density operator (9.15) as i

∂ ∂ ρˆ pα α(t) α(t) = i ∂t ∂t ∂|α(t) ∂α(t)| = i α(t) + α(t) pα ∂t ∂t α =H ρˆ − ρˆ H , α(t) α(t) − α(t) α(t)H = pα H

(10.18)

α

which will be the basis of all following considerations. This so called L IOUVILLE - VON -N EUMANN equation is written in compact form as i

∂ ρˆ , ρ]. = [H ˆ ∂t

(10.19)

640

10


10.4.2 Optical B LOCH Equations for a Two State System We apply now again the RWA. For a two level system we use the R ABI frequency ΩR according to (10.3) to characterize the strengths of the coupling. We just need to insert the Hamiltonian (10.8) into the L IOUVILLE - VON -N EUMANN equation (10.19) to obtain without problems the optical B LOCH equations for the two level system with a detuning ω = ωba − ω, for the moment still without relaxation: ΩR (ρba + ρab ) 2 ΩR ρ˙bb = (ρab + ρba ) 2 ΩR (ρbb − ρaa ) − iωρba . ρ˙ba = − 2

ρ˙aa = −

(10.20) (10.21) (10.22)

Only the last of the three equations is complex. Normalization of the density matrix, ∗ this finally ρbb + ρaa = 1, makes one of these equations redundant. With ρab = ρba leads to a system of three independent, coupled, linear ODEs. First, however, we have to add two generalizations: 1. Often the electric field amplitude is not constant, e.g. when the system is excited by a laser pulse such as (1.110). However, as long as the amplitude varies slowly with time (we recall the SVE approximation) one still may use the equations just derived and simply replace E0 → E0 h(t) and correspondingly ΩR → ΩR (t) = ΩR h(t). Note that the envelope function h(t) is dimensionless, usually with h(0) = 1. 2. We have to account for relaxation processes of various origins. For example, if spontaneous emission is the cause of relaxation we would have to sum incoherently over all modes of the vacuum field, since all empty modes contribute statistically to spontaneous emission. Both, the excitation probability ρbb as well as the coherence term ρba are affected. With quite some effort this may be done to derive a clean theory of relaxation. For simplicity we choose a much simpler, phenomenological approach. We first remember that the excited state decays with a natural lifetime say T1 = 1/Aab = 1/Γab . Without external electric field the probability to find the excited state thus decays as (0)

ρbb (t) = ρbb exp(−Γab t) (0)

(10.23)

from its original population ρbb at t = 0. In differential form this reads ρ˙bb = −Γab ρbb . Thus, to account for this decay in our density matrix equations one adds −Γab ρbb to (10.20) and (10.21). In analogy one adds −Γbb ρba to (10.22) for a phenomenological description of phase relaxation. With these additions we write the Optical B LOCH equations with relaxation:

10.4


ρaa = 1 − ρbb

641

(10.24)

1 ρ˙bb = h(t)ΩR (ρab + ρba ) − Γab ρbb 2 1 ρ˙ba = − h(t)ΩR (ρbb − ρaa ) − iωρba − Γbb ρba , 2

(10.25) (10.26)

where the decay of excitation is characterized by Γab = 1/T1 ,

(10.27)

Γbb = 1/T2 .

(10.28)

and the decay of coherence by With ρI = Im ρba and ρR = Re ρba we write (10.25) and (10.26) in real form: ρ˙bb = h(t)ΩR ρR − Γab ρbb

(10.29a)

ρ˙R = −h(t)ΩR (ρbb − 1/2) + ωρI − Γbb ρR

(10.29b)

ρ˙I = −ωρR − Γbb ρI .

(10.29c)

For practical calculations it is often useful to write these equations in dimensionless form. One multiplies both sides of the equation by some characteristic time τ (e.g. by the duration of the laser pulse) and replaces t/τ → ϑ (i.e. all times are now measured in units of τ ): ρbb dρbb = h(ϑ)τ ΩR ρR − dϑ T1 /τ

(10.30a)

dρR ρR = −h(ϑ)τ ΩR (ρbb − 1/2) + τ ωρI − dϑ T2 /τ

(10.30b)

dρI ρI = −τ ωρR − . dϑ T2 /τ

(10.30c)

Typically, the envelope of the field could be a Gaussian: h(ϑ) = exp −ϑ 2 /2 .

(10.31)

For purely optical decay, the relation between Aab = Γab and Γbb may be gleaned from the field free limit at very low population density in the excited state. We recall the origin of the density matrix elements from amplitudes. In the coherent case we had ρik = ci ck∗ , and for weak excitation ca 1 holds, so that ρba = cb ca∗ cb . Thus dcb = −Γbb ρba = −Γbb cb =⇒ cb (t) = cb0 exp(−Γbb t) dt 2 =⇒ ρbb (t) = cb (t) = |cb0 |2 exp(−2Γbb t).

ρ˙ba =

642

10


If we compare this with (10.23) we find: Γbb = 1/T2 = Γab /2 = Aab /2 = 1/2T1

or

T2 = 2T1 .

(10.32)

Note that this only holds, if other relaxation processes can be neglected. In the general case other mechanisms may be important, such as collision processes within the system studied, internal relaxation or interaction with an environment (so called bath). In such a case T2 and T1 may well be independent of each other. For example, collision processes in the gas phase may lead to shorter T2 (so called dephasing) but not necessarily to a decay of the excitation. On the other hand, loss processes (e.g. into a third state not coupled to the field) may dominantly lead to a shortening of T1 without substantial dephasing of the population remaining in states a and b. Finally, we end these general considerations by noting for later use that the RWA Hamiltonian (10.8) of the two level system in a near resonant electromagnetic field with relaxation reads now 0 −iΩR (t)/2 + iΓab (t) = . (10.33) H iΩR (t)/2 + iΓab iΓbb + ω into the L IOUVILLE equation (10.19) and This is easily verified by inserting H comparing the result with the optical B LOCH equations (10.29a)–(10.29c). Section summary

• In the general case of mixed states, the temporal evolution of the population and coherence is described by the L IOUVILLE equation (10.19) for the density matrix ρˆ of a quantum system. Diagonal terms ρii describe the population, the off-diagonal terms ρij represent the coherence among the states. • Both can change under the influence of a near resonant electromagnetic radiation field. Inserting the coupling strength (described by the R ABI frequency ΩR ) into the L IOUVILLE equation leads – after some manipulations and after introducing relaxation terms – to the optical B LOCH equations (10.24)–(10.26) for a two level system. • In the case of pure optical decay, the relaxation times for population decay, T1 , and for decay of coherence, T2 , are related by T2 = 2T1 .

10.5

Excitation with Continuous Wave (cw) Light

We now consider several special examples. For some of them, the optical B LOCH equations may be solved in closed form, while for others a numerical treatment is required. In the present section we set h(t) ≡ 1 for t ≥ 0, and thus study excitation with continuous light, which is switched on at time t = 0, just as in the semiclassical and quantized perturbational treatment.

10.5


643

10.5.1 Relaxed Steady State If the damping is finite, we just need to wait sufficiently long after switching on the field (t T1 , T2 ), and all time derivatives disappear, so that (10.29a)–(10.29c) becomes Γab ρbb = ΩR ρR

(10.34a)

Γbb ρR = −ΩR (ρbb − 1/2) + ωρI

(10.34b)

Γbb ρI = −ωρR .

(10.34c)

The probability to find the system in the excited state is then: ρbb =

ΩR2 1 . 2 Γab Γbb + (Γab /Γbb )ω2 + ΩR2

(10.35)

In the limit of very high intensities (R ABI frequency ΩR ω, Γab , Γbb ) this leads to ρbb → 1/2, i.e. to equal population of ground and excited state – as expected. This is called saturation as already predicted by phenomenological semiclassical considerations. We mention that in this case the off-diagonal matrix element disappears, since it becomes ρba = ρR + iρI ∝ 1/ΩR . We also note that for large detuning ω ΩR the excitation probability decreases as ρbb ∝ 1/ω2 , the typical Lorentzian behaviour.

10.5.2 Saturation Broadening For purely radiative decay into the ground state with Γbb = Γab /2 = A/2 the population density (10.35) becomes ρbb =

ΩR2 /4 (A/2)2 + ΩR2 /2 + ω2

=

ΩR2 /4 . Ωs2 /4 + ω2

(10.36)

Thus, we have derived an expression for the power broadening (or saturation broadening) of spectral lines. Obviously (10.36) describes a L ORENTZ profile with a FWHM (10.37) Ωs = A2 + 2ΩR2 . Figure 10.14 illustrates the saturation profile for various values of ΩR (and thus intensities). We recall (10.5), ΩR2 = (3λ3 A)I /(2πhc), the relation between R ABI frequency ΩR and intensity I , and introduce a saturation intensity (see e.g. A SHKIN 1978) Is =

2πhcA . λ3

(10.38)

644

10

Fig. 10.13 Population probability ρbb of the upper state for resonant irradiation (ω = 0) as a function of laser intensity I – measured in units of the saturation intensity √ Is , at which ΩR = 3A


ρbb 0.50

0.25

0.00

0

1

2

3

4

5

I / Is

√ For I = Is the R ABI frequency is 3 times the natural linewidth A. By writing ΩR2 = 3(I /Is )A2 the average population density (10.36) of the upper state becomes: ρbb =

6(I /Is ) 1 . 2 1 + 6(I /Is ) + (2ω/A)2

(10.39)

For exact resonant irradiation (ω = 0) this excitation probability is plotted as a function of the intensity in Fig. 10.13. It approaches the maximum value of 50 % rapidly with increasing intensity, and is already 43 % for I = Is . This saturation of the excitation probability is the genuine origin of the power broadening illustrated in Fig. 10.14: in the steady state, due to spontaneous emission, one can never excite more than 50 % of the atoms – independent of how high the intensity is, hence the term saturation broadening. As a numerical example we have again a look at the ‘Drosophila’ of atomic physics: for the 3 2 P3/2 ← 3 2 S1/2 transition in sodium we have at λ = 589 nm a natural lifetime of T1 = 1/Aab 16 ns. Thus, with (10.38) the saturation intensity becomes Is 38 mW cm−2 and one obtains saturation already at very low intensities if the radiation is exactly resonant. We may also obtain an estimate on how tightly the laser beam was focussed for the measurement of the M OLLOW triplet shown in Fig. 10.8. At P = 30 mW one reads there a splitting of ΩR 7Aab ; with (10.5) this corresponds to I 627 mW cm−2 , which according to (1.60) is achieved with a G AUSS radius of a 1.2 mm. Fig. 10.14 Line broadening in an intense laser field (so called power broadening). Plotted is the excitation probability as a function of the detuning ω (measured in units of the natural linewidth Γab = A) for several different R ABI frequencies ΩR

ρbb 0.5 ΩR =10A

ΩR = 0.5A ΩR = 0.1A

ΩR= 5A ΩR= 1A

-20

-10

0 Δω / A

10

20

10.5


645

It is worthwhile noting that the exact (non-perturbative) stationary solution (10.36) or (10.39) is identical to that gleaned in Sect. 5.1.1, Vol. 1 from semiclassical considerations based on perturbation theory – even though the latter is strictly not valid at such high intensities.

10.5.3 Broad Band and Narrow Band Excitation Up to now we have assumed in this chapter that excitation occurs with strictly monochromatic light (bandwidth ω1/2 A). We discuss – now again for low intensities ΩR2 A2 – the general case of excitation with quasi-monochromatic laser light (average angular frequency ωc ). For simplicity we assume the spectral distribution of the radiation to be a L ORENTZ distribution (2.20) with a FWHM ω1/2 and a total intensity I . We have to convolute (10.36) with this distribution in order to obtain the average excitation probability. We recall from Appendix G.5, Vol. 1 that a convolution of one L ORENTZ distribution of FWHM Γ1 with another one of FWHM Γ2 has a convoluted linewidth Γ = Γ1 + Γ2 . Thus, we obtain ρbb =

3λ3ab 1 (A/2 + ω1/2 /2) I . 2πh π (A/2 + ω1/2 /2)2 + ω2 c

(10.40)

We consider two limiting cases: 1. Excitation with a very broad band source (ω1/2 A) whose maximum is tuned into resonance ωc = ωba . One obtains from (10.40) and (2.20) for the population density: ρbb =

3λ3ab 3λ3 I˜(ωba ) I . = ab 2πh πcω1/2 4h c

(10.41)

Here I˜(ωba )/c = u(ω ˜ ba ) = 2I /(πcω1/2 ) is the spectral radiation density of the light source at ωba . Note that only the wavelength is specific for the transition studied and no other atomic properties enter into the relative population of the excited state. This may appear surprising at first sight. However, it is simply a consequence of the fact that at stationary conditions spontaneous emission and absorption balance each other. While the time to reach the stationary state depends on |r ba | – as we shall see in Sect. 10.5.6 – the stationary state itself does not. 2. The other extreme is a very narrow band laser ω1/2 A. Again we excite in the line centre (ω = 0) and find in that case: ρbb =

3λ3ab 2I . 2πh πcA

(10.42)

The excitation probability under stationary conditions is obviously inversely proportional to the natural linewidth A of the excited state – and of course proportional to the intensity. One may rationalize this by considering that excitation occurs only in the centre of the line, spontaneous emission, however, which counter

646

10


acts excitation, occurs over the whole natural linewidth A. One could also say that the irradiation into a spectral region smaller than the natural linewidth is equivalent to a source which has a bandwidth equal to A, such radiation corresponds in this respect to a spectral radiation density u(ω ˜ ba ) = I˜(ωba )/c = 2I /(πcA).

10.5.4 Rate Equations So far we have completely ignored the temporal evolution of the density matrix elements – assuming that the quantum system studied was exposed to the radiation field already for a long time (t → −∞) so that transient responses have already settled. A reasonable, first approach to attack the time dependence of the population densities is to assume that coherences have soon reached their steady state (e.g. when exciting with a broadband, low intensity source, or when collisions play a major role). We then set in (10.26) simply ρ˙ba = 0, switch on the light source at t = 0 and let it have constant intensity thereafter, i.e. h(t) ≡ 1. We then obtain from (10.26): ρba + ρab =

−ΩR Γbb (ρbb − ρaa ). 2 ω2 + Γbb

Inserting this into (10.25) and expressing again ΩR2 according to (10.5) by the linewidth A and the laser intensity I one finally finds: I ˜ ρ˙bb = −B(ω) (ρbb − ρaa ) − Aρbb c I ˜ ρ˙aa = B(ω) (ρbb − ρaa ) + Aρbb c ˜ with B(ω) =

Γbb A2 /4 3λ3 A 3λ3 = . 2 2 2 4πhc Γbb + ω 2πh A /4 + ω2

(10.43) (10.44) (10.45)

In the last step we have assumed purely radiative processes with Γbb = Γab /2 = A/2.5 Often it is appropriate to have an equation for half the inversion probability ρD = (ρbb − ρaa )/2: ˜ /cρD + A(ρD + 1/2). ρ˙D = B(ω)I

(10.46)

Thus, we have now derived the well known rate equations, which have been introduced for the first time heuristically by E INSTEIN in the context of deriving the P LANCK radiation law (see Sect. 4.2.5 in Vol. 1). Expression (10.45) is, however, 5 If the intensity is sufficiently low, so that optical pumping can be neglected, these expressions may

also be applied to transitions with several sublevels involved. One just has to replace the factor 3 in (10.45) by gb /ga (see footnote 2 in Chap. 5, Vol. 1).

10.5


647

more detailed in as far as it includes now the dependence on ω, the detuning with respect to resonance for finite absorption linewidth. We could say that we have truly derived the frequency dependence of the E INSTEIN B coefficient. In Sect. 1.1.4 we had already used these relations, based essentially on guess work. For direct comparison of (10.43) and (10.44) with the classical rate equation – where a broad band source is assumed – we have to integrate (10.45) over all frequencies and replace 3λ3 A I˜(ωba ) I ˜ → = B u(ω ˜ ba ). B(ω) c 4h c As mentioned on several occasions the relation between B and A differs from others, often found in the literature, by a factor 3/2π , since we discuses light beams instead of isotropic sources (factor 3) and b) relate the spectral radiation intensity u(ω) ˜ to the angular frequency and not to the frequency (factor 1/2π ). Such rate equations are often used and describe the behaviour of quantum systems in the electromagnetic radiation field usually rather well – the basic assumption being that rapidly oscillating coherence terms have reached their equilibrium. They even allow to introduce additional terms beyond the two state system, and we have made use of this possibility in previous chapters occasionally. Clearly, rate equations cannot be used to describe coherence effects which are observed for narrow band excitation and are of importance when studying ultrafast processes.

10.5.5 Continuous Excitation Without Relaxation We now take a step into the opposite direction and assume that there is no relaxation = Γbb = 0). With the resonant and non-resonant R ABI frequencies, ΩR at all (Γab and Ω =

ΩR2 + ω2 , respectively, the optical B LOCH equations (10.24)–(10.26)

may be solved for initial conditions6 ρbb = 0 and ρab = 0 in analytical form (we use SWP 5.5 2005):

Ω t with ρaa = 1 − ρbb 2 2 Ω 2 2 Ω t Ω t ΩR Ω ρR = − sin cos 2 2 2 Ω 2 ΩR ω 2 Ω t ρI = − sin . 2 2 Ω

ρbb =

6 These

ΩR2

sin2

and

initial conditions imply again that the radiation field is switched on at t = 0.

(10.47)

648 Fig. 10.15 Temporal evolution of the population density ρbb of the excited state for different detuning ω and R ABI frequencies ΩR (i.e. for different intensities) – neglecting all relaxation (Γab = Γbb = 0)

10


ρbb(t ) ∆ω/ΩR = 0

0.8 0.6

∆ω/ΩR = 1

0.4 0.2 0

∆ω/ΩR = 2.5 π

ΩRt

2π

3π

In Fig. 10.15 we have plotted the population density ρbb (t) of the excited state as a function of time (in units of 1/ΩR ) for three different values of the detuning ω/ΩR . One sees that the population oscillates between ground and excited state. These are the so called R ABI oscillations. For exact resonance ω/ΩR = 0 the whole population which was originally in the ground state is after a time t = π/ΩR found in the excited state – and returns at t = 2π/ΩR again completely back into the ground state. If one uses a square wave pulse, switching the laser field on at t = 0 and off at t = π/ΩR , the system remains to 100 % in the excited state (as long as we can neglect relaxation, as assumed here). Such a laser pulse with ΩR t = π is called a π pulse. Correspondingly, a π/2 pulse generates equal population in the ground and excited state, while a 2π pulse brings the hole population completely back into the ground state. Of course, the same may be achieved by two π pulses following each other with some arbitrary time delay. With increased detuning the R ABI oscillations get faster, as determined by (10.37) while the maximum excitation probability is reduced. To obtain a feeling for the time tπ which such a π pulse takes, we choose again the excitation of Na atoms as an example. Let us assume an intensity of I 630 mW cm−2 (see Sect. 10.5.2). In this case ΩR 7 × 2π × 10 MHz and hence tπ = 7 ns. However, since the natural lifetime of the excited state is only 16 ns, we cannot really describe the situation well without accounting for relaxation. Nevertheless, equations (10.47) may be used as a realistic description for very short times t T1 , T2 . At very high intensities as obtained with state-of-the-art short pulse lasers these equations without relaxation are quite adequate to describe the R ABI oscillations.

10.5.6 Continuous Excitation with Relaxation In the general case one usually has to solve the optical B LOCH equations numerically. Only in the special case of exact resonance, ω = 0, the solutions can be

10.6

B LOCH Equations and Short Pulse Spectroscopy

Fig. 10.16 Resonant excitation: temporal evolution of the population density ρbb of the excited state, taking account of relaxation (spontaneous linewidth A) at two different intensities or R ABI frequencies ΩR , respectively

ρbb(t ) 0.8

649

∆ω = 0 ΩR /A = 10

0.6 0.4

ΩR /A = 2

0.2 0

ΩR t 0

5π

10π

15 π

given in closed form, again for the initial conditions ρbb = 0 and ρab = 0:

ΩR2 1 3A 3tA ρbb = 1 − cos Ω t + sin Ω t exp − x x 2 /2 + Ω 2 2 Γab 4Ωx 4 R A2 with Ωx = ΩR2 + . 8 In Fig. 10.16 this solution is shown for ΩR /A = 2 (strong damping) and =10 (weak damping). It is possible to detect such oscillations experimentally, e.g. by determining the second order degree of coherence as documented for a single atom in a trap quite impressively at the beginning of the present chapter in Fig. 10.3. Section summary

• Using the stationary limit of the optical B LOCH equations (excitation with a narrow band CW laser) we have derived power (or saturation) broadening of the natural linewidth A with increasing find a radiation intensity I . We √ 2 2 L ORENTZ profile with a linewidth Ωs = A + 2ΩR , where ΩR ∝ I is the R ABI frequency. • For the temporal evolution of the excited state density after switching on a narrow band CW laser we find an oscillatory behaviour. Without relaxation, ground and excited state population oscillate with half the non-resonant R ABI (angular) frequency, Ω /2 between 0 and 1. • On resonance, a square wave pulse of duration t = π/ΩR (so called π pulse) leads to complete population inversion.

10.6


10.6.1 Excitation with Ultrafast Laser Pulses When exciting an atom or molecule with an ultrashort laser pulse usually the optical B LOCH equations cannot be solved in closed form. However, numerical integration is straight forward. We start the discussion with some preliminary remarks.

650

10


According to (10.5), the R ABI frequency ΩR is proportional to the square root of the intensity; for pulses we have to use the temporal maximum I0 . To be specific, we assume a Gaussian pulse in time and position according to (1.124) for which the envelope of the field amplitude is h(t) = exp[−(t/τG )2 ]. For simplicity we only consider the local maximum of the intensity I0 = I0 (z = 0). The fluence F0 (in J/cm2 ) of the laser beam on √ the beam axis (ρ = 0) is related to its intensity according to (1.125) by I0 = 2/π F0 /τG , and can be obtained from the total energy √ Wtot of the pulse as F0 = Wtot /(πa 2 ) with the G AUSS beam radius a = w0 / 2 (at 1/e intensity). Laser and excitation parameter which determine the optical B LOCH equations (10.30a)–(10.30c) can be summarized in the dimensionless phase angle ΩR τG . With (10.5) it becomes 3λ3 A 3λ3 AF0 I0 τG = τ ∝ |r ba |F0 τG . (10.48) ΩR τ G = G 2πhc 21/2 hcπ 3/2 τG This relation is quite remarkable: it says that for quasi-resonant excitation with short laser pulses the laser intensity is not the key parameter; rather, it is the product of fluence (i.e. laser pulse energy per area) and pulse duration which determines the process. To obtain some feeling for numbers, we compute ΩR τG for the excitation of the Na resonance at λ = 589 nm with a laser pulse of 50 fs. A typical pulse energy might be 0.3 mJ. Focused on a (1/e) diameter of 100 µm this gives a fluence of F0 4 J cm−2 (at a maximum intensity I0 = 6.3 × 1013 W cm−2 ). With A = 2π × 10 MHz we obtain a phase angle ΩR τG = 222. This is a large multiple of π , and we have to question the whole approximation at such intensity. This becomes quite evident when computing the R ABI frequency ΩR = 222/(50 fs) = 4.44 × 1015 s−1 , which is already larger than the angular frequency of the transition studied, which is 2πc/λ 3.2 × 1015 s−1 . Thus, strictly speaking, we should not use the RWA and even the SVE approximation becomes questionable – it only holds if the field amplitude changes little during one optical cycle. With today’s short pulse lasers one easily enters into an intensity region where even the optical B LOCH equations may no longer be a good approximation (note, however, that the example discussed here has an exceptionally high oscillator strength). In the following considerations we shall therefore assume a significantly smaller fluence. As long as only radiative relaxation plays a role (nano second time scale), one may neglect relaxation terms altogether when working with femtosecond laser pulses. We concentrate for the moment on the nearly resonant case τG ω 1. Thus, we need only consider the first term on the right hand side of (10.30a) and (10.30b), and have to solve the following coupled equations: dρbb = h(ϑ)ΩR τG ρR dϑ dρR = −h(ϑ)ΩR τG (ρbb − 1/2). dϑ

(10.49) (10.50)

10.6

B LOCH Equations and Short Pulse Spectroscopy ρbb 1.0 0.8

651

Á ba = π ∞

GAUSS I(t)

Á∞ba = 1.5π

0.6 0.4

Á∞ba = 2π

0.2 0.0 -2

-1

0

1

2

3

t /τ

Fig. 10.17 Population of the excited state in a two level system for resonant excitation with a short laser pulse. The numerical solution of the optical B LOCH equations has been carried out for ∞ . The time scale is calibrated in different laser pulse fluence, characterized by the phase angle φba units of the FWHM of the exciting G AUSS pulse (dashed red) and set to zero at its maximum

For this undamped resonant case an analytical solution exists, ϑ h ϑ dϑ ρbb (ϑ) = 1 − cos φba (ϑ) /2 with φba (ϑ) = ΩR τG −∞

(10.51)

with the integrated, time dependent phase angle φba (ϑ). For a quantitative comparison of the excitation by different pulse forms (including square pulses as discussed in Sect. 10.5.5), one has to compare the effective phase angle by integrating over the full pulse duration. For a G AUSS pulse, h(ϑ) = exp(−ϑ 2 ), we obtain: ∞ √ ∞ φba = φba (∞) = ΩR τG h(ϑ)dϑ = π ΩR τG . (10.52) −∞

√ Thus an effective Gaussian π pulse is obtained for ΩR τG = π – where the R ABI frequency ΩR refers to the maximum of the intensity. The results are shown for ∞ in Fig. 10.17. Clearly, with φ ∞ = π the whole population is some values of φba ba pumped from the ground state |a into the excited state |b – just as for the square ∞ = 1.5π we see already pulse – except that there a π pulse implies ΩR τ = π . For φba the decrease of population so that at longer times only 50 % of the system remains ∞ = 2π the system passes during the pules a in the excited state. Finally, for φba whole cycle of excitation and de-excitation (2π pulse). Excitation with such pulses or pulse sequences plays a key role when studying so called photon echoes. They are standard tools in NMR and EPR spectroscopy. Typically, experiments in short pulse spectroscopy are performed by pump-probe methods at sufficiently low laser intensities to avoid such effects: what is needed is the preparation (pump) of an excited state density in atoms, molecules or condensed matter which is high enough to be probed conveniently with a second (probe) laser. Usually one operates far away from saturation, ρbb 1/2. In Fig. 10.18 the population is plotted at very large times after the pump pulse (relaxation is neglected) as a ∞ accumulated during the pulse. Note that the popfunction of the effective phase φba ulation in the excited state increases for small phase angles, but decreases again for

652

10


ρbb (t→ ∞) 1.0

0.5

0.0

0

0.5 π

π

Á∞ ba

Fig. 10.18 Population ρbb of the excited state in a two level scheme for nearly resonant excitation with a short laser pulse at negligible relaxation. Shown is the population density at times t long compared to √ the laser pulse duration as a function of the effective phase angle ∞ = Ω τ ∞ h(ϑ)dϑ ∝ F τ φba R −∞ ∞ ≥ π after the system has undergone half a R ABI cycle. For significantly higher φba fluence several maxima and minima can be reached.

10.6.2 Ultrafast Spectroscopy For free atoms and molecules with isolated excited states the dissipation constants Γab = A and Γbb = A/2 describe the (spontaneous) radiative decay completely, typically on the ns time range. Additional relaxation mechanisms may arise from the interaction with a bath (gas, surface, liquid). In larger molecules and clusters a multitude of additional, internal relaxation mechanisms (electronic or nuclear) may occur. It is a key topic in short pulse spectroscopy to follow and understand this kind of dynamics. Femtosecond laser pulses are particular useful to study these processes in pump-probe experiments. One creates two synchronized laser pulses which can be delayed with respect to each other by using an optical delay line (typically an interferometer of the M ICHELSON or M ACH -Z ENDER type) as sketched in Sect. 1.5. The first pulse excites the system (pump) the second induces emission or leads to ionization which allows to probe the process as a function of the variable delay time t. The optical B LOCH equations turn out to be very useful when analyzing the experimentally observed dynamics of quantum systems in the field of short laser pulses. In such pump-probe experiments one uses the parameters T1 = 1/Γab and T2 = 1/Γbb to model an experimentally determined transient signal of the system – with the hope that these parameters allow one to glean information on the characteristic temporal behaviour of the inherent processes in the system, i.e. non-adiabatic or adiabatic transitions, dissociation processes, internal ro-vibrational redistribution etc. Even though, as a rule, such phenomenological numbers on lifetimes may be a somewhat course approach towards the many underlying processes, the information is very valuable. In the following we want to discuss a few special cases, with emphasis on the use of the optical B LOCH equations and finally present and discuss some experimental results. For a more detailed introduction into the methods and

10.6


653

results of short pulse spectroscopy in the gas phase we refer the interested readers to a review by H ERTEL and R ADLOFF (2006).

10.6.3 Rate Equations and Optical B LOCH Equations Quite generally, an extension of the two level scheme for which the optical B LOCH equations in the form (10.24)–(10.26) are originally designed is in principle possible without problems. If several sublevels (or even other electronic states) are involved, one just has to add for each interacting pair of sublevels an extra set of equations – and one must account for all the ‘relaxation channels’ which can empty or fill such sublevels individually. This is even more straight forward if only rate equations are treated, as we shall show in the following. Typical applications of short pulse spectroscopy focus on the fs or ps time domain so that one may safely ignore radiative decay which occurs on the ns time scale. Often the dephasing, i.e. the loss of coherence, is much faster than population or depopulation of the excited state, i.e. Γbb Γab = 1/T1 . One also has to be aware that the processes studied are often rather complex and proceed in several steps, which cannot be resolved individually. Nevertheless, one may account for these sub-processes phenomenologically by additional rate equations which supplement the optical B LOCH equations for the two level system. These extra rate equations describe the population and depopulation of different ‘channels’ (see e.g. L IPPERT et al. 2003; F REUDENBERG et al. 1996). In addition to the equations describing the excitation in the sense of (10.30a)–(10.30c) one adds equations of the type ρ˙kk = ρii Γki − ρkk Γj k , j

( which have to be solved simultaneously. Note that j Γj( k = 1/τk is then the total decay rate of a channel (k) with the lifetime τk and Γik / j Γj k characterizes the branching ratio for a transition from channel k to channel i. The observed signals (ions, electrons, fluorescence) are considered to be proportional to the population densities of the states involved, ρbb (t), ρii (t), ρkk (t), as they evolve with time. Finally, the thus modelled signals have to be weighted with the temporal profile of the probe pulse and must be added (possibly with different specific weights) to the total signal which is finally observed. These weights, decay rates and branching ratios are then used to obtain a fit to the experimental data which provides optimal agreement. Even if such models cannot serve to obtain an accurate picture of all processes which occur in a molecule, they allow nevertheless a good comparison among different species and processes, they allow to determine appropriate time scales for the dynamics within the system, and they provide measurements which can possibly be compared to theoretical predictions derived from more detailed models.

Limiting Cases For a genuine pump-probe experiment one wants to avoid saturation phenomena ∞ ∼ Ω τ 1, i.e. close to zero and thus works with low laser intensity, i.e. with φba R

654

10


in Fig. 10.18. In this case one may even integrate the optical B LOCH equations (10.30a)–(10.30c) by a perturbative approach in closed form. One obtains for the depopulation of the ground state (again with ϑ = t/τ ) 1 − ρaa (t) =

τ 2 ΩR2 4

t/τ

−∞

dϑ h ϑ e−τ Γbb ϑ +iτ ω·ϑ

ϑ

dϑ h ϑ e+τ Γbb ϑ −iτ ω·ϑ

−∞

+ c.c.,

(10.53)

which for a long-lived excited state is identical to ρbb (ϑ). Clearly, contributions to this integral arise only during the time of the pulse duration τ . For exact resonant excitation ω = 0 one may evaluate the double integral (10.53) even further by partial integration (F REUDENBERG et al. 1996) and can now distinguish two limiting cases: • τ Γbb 1: the dephasing time is much larger than the pulse duration. The exponential expression in (10.53) becomes then 1 and we have as coherent limit τ 2 ΩR2 t/τ 2 dϑ h ϑ (10.54) 4 −∞ √ 2 t→∞ π 1 = πτ 2 ΩR2 erf t/(τ 2) + 1 −→ τ 2 ΩR2 , 8 2 √ where the error integral erf(t/(τ 2)) determines the temporal behaviour of the signal. • τ Γbb 1: Conversely, if the dephasing time is much shorter than the pulse duration, (10.53) leads to the incoherent limit 1 − ρaa (t) =

τ ΩR2 1 − ρaa (t) = 2Γbb

t/τ

−∞

2 dϑ h ϑ

(10.55)

√ t→∞ 1 π 2 2 1 √ τ 2 ΩR2 τ ΩR , = erf(t/τ ) + 1 −→ π 4 τ Γbb τ Γbb 2

which differs significantly from the coherent case. • For large detuning τ ω ≥ 1 the situation is more complicated. Numerical simulations show that detuning in combination with a fast decay of the excited state (Γab 1/τ ) makes the process incoherent and in the limit τ ω 1 leads to maximum excitation at t = 0 (i.e. at the pulse maximum) which is not so in the fully coherent case.

Excitation of a Quasi-Continuum Up to now we always have assumed that the excited state |b which is prepared by the pump pulse is completely isolated as indicated in Fig. 10.19(a). However, as a rule in a large molecule or cluster one has to account for a multitude of excited vibrational and rotational states, especially so in higher electronic levels. To a

10.6


Fig. 10.19 Schematic illustration (a) of a two level system with decay rate Γ12 and dephasing rate Γ22 of the excited state and (b) of a quasi-continuum as a model for deriving the excitation probability in dense excited states

655

(a)

(b) Γ22

∆ω

Wb

|b 〉

Wb

|b 〉 ħω

δW

Γ12

ħω |a〉

Wa

Γ12 |a〉

Wa

Fig. 10.20 Simulated pump probe signal for a very long-lived excited state |b with its population density ρbb (t) according to (10.54) and (10.55), excited with a G AUSS pulse (grey dashed) of a FWHM t1/2 . For comparison the signals are normalized to each other at t → ∞. The incoherent signal reaches the half maximum at a time delay t = 0; when exciting coherently this point is shifted by 0.327t1/2

simulated pump-probe signal ρbb

first approximation one may model this by an infinite number of states with different detuning, as sketched in Fig. 10.19(b). With some mathematical effort (see e.g. H ERTEL and R ADLOFF 2006) one may apply the above formalism just derived also for the situation sketched in Fig. 10.19(b). One obtains an expression equivalent to (10.55) and may thus draw the following important conclusion: the coherent excitation of a quasi-continuum of states may be treated in the same way as the incoherent limit of a pure two level system. This is a very practical result for pump-probe experiments in very large molecular systems. It justifies indeed the rate equations which typically are used to evaluate such experiments. Experimentally the cases may well be distinguished if the zero of the delay time is well known (this is, however, not a trivial requirement). The density of the exited state is plotted in Fig. 10.20 according to (10.54) and (10.55) for the case of a longlived excited state |b with Γab → 0 and ρbb (t) = 1 − ρaa (t) for a pump pulse with a temporal Gaussian profile. Note that in the incoherent case the half maximum of the signal is reached at time delay zero (with respect to the pulse maximum). In contrast, in the coherent case this point is reached for 0.385τG = t1/2 , with t1/2 being again the FWHM of the probe pulse. In life experiments this offset is crucial when one wants to determine relaxation times T1 below the width of the laser pulse. An illustrative example are experiments with ammonia molecules and clusters reported by F REUDENBERG et al. (1996). Figure 10.21 shows the transient ion sig˜ = 4) state in NH3 . The pump pulse nals after excitation of the fast decaying A(v 1.0 incoherent limit

GAUSS pulse profile

coherent limit half maximum

0.5

0.327 -1.0

0.0

1.0

2.0 t / Δt1/2

656

10 in resonance

NH3+ resonant

(a)

offset ca. 70 fs NH3+ res. (204nm)

(b)

ND3+ res.


Bz+ 204 nm NH3+ non res.

-0.5

(NH3)3NH4+

0

0.5

1

Δt / ps

Bz+ 200nm NH3+ non resonant (200 nm) -0.5

0

0.5

1

Fig. 10.21 (a) Calibration of the delay in a pump probe experiment with the help of resonant ˜ = 4) state. Note (204 nm) and non-resonant excitation (200 nm) of NH3 in the fast decaying A(v the offset towards positive delay times in the resonant case. The offset is even somewhat larger for the longer living ND3 molecule as well as for ammonia clusters. The fit curves have been obtained by numerical integration of the optical B LOCH equations (10.34a)–(10.34c). For these fits the experimentally determined laser pulse width of 160 fs FWHM was used. Optimal fits are obtained for a lifetime of 40 fs for NH3 and 180 fs for ND3 . Shown for comparison are cluster ion signals, resonantly excited by two different wavelengths (204 nm and 200 nm) (data according to F REUDENBERG et al. 1996). (b) Comparison with benzene (Bz)

has been tuned alternatively to exact resonance (∼204 nm) or in between two resonances at (∼200 nm). The following ionization in this experiment has been achieved with a probe pulse of 267 nm. The pulse duration in this experiment was ∼150 fs to 170 fs for both, pump and probe pulses. Shown are the experimental ion signals. The fit curves shown have been derived numerically from the optical B LOCH equations (10.34a)–(10.34b). With a decay time of 40 fs, a laser pulse duration 160 fs and a detuning of 1.5 nm in the non-resonant case one obtains excellent agreement for the NH+ 3 signal. This may be used to calibrate the time zero, if one assumes that this experimental signal is essentially symmetric around time zero, as predicted by the calculations. In the case of resonant excitation the signal maximum is shifted by 70 fs, which is excellently reproduced by the B LOCH fit. Simultaneously fitting the resonant and non-resonant transients allows to determine the lifetime of excited ˜ = 4) with an accuracy of about ±10 fs, even though the half width of the NH3 A(v laser pulse is much larger. The result agrees well with estimates derived from the measured linewidth in NH3 A˜ (Z IEGLER 1985). For ND3 , which has been excited with the same wavelength, an effective lifetime of ∼180 fs has been determined. It should be pointed out that such extremely short excited lifetimes can of course not be explained by radiative processes. Rather, they must be attributed to fast internal conversion and dissociation of the system. For comparison we also show in Fig. 10.21 the transients in benzene and ammonia clusters. The fit curves correspond to the solution of the B LOCH equations for a resonantly excited two level system. Obviously the density of excited states is in this case high enough to allow resonant excitation, but not high enough to generate incoherence according to (10.55).

10.7

STIRAP

657

Section summary

• After CW excitation treated in the preceding section, we have now applied the optical B LOCH equations to excitation with short pulses. It turns out that the key parameter for the strength of the excitation is the time integrated phase (10.52), i.e. √ essentially the product of R ABI frequency and pulse duration ΩR τG ∝ |r ba |F0 τG . Note that here the product of fluence (∝ total laser pulse energy) and pulse duration enters, rather than the intensity! • The population in the excited state increases for small phase angles, but de∞ ≥ π after the system has undergone half a R ABI cycle creases again for φba as shown in Fig. 10.18. • If coherence decays very fast one may supplement or even fully replace the optical B LOCH equations by rate equations. • As illustrated in Fig. 10.20 incoherent excitation is slightly faster then coherent excitation. For a long-lived excited state, half maximum population is reached (a) for incoherent excitation simultaneously with the laser pulse maximum, (b) for coherent excitation at a time delay of 0.327t1/2 (the latter being the FWHM pulse duration).

10.7

STIRAP

10.7.1 Three Level System in Two Laser Fields Up to now we have discussed exclusively two level systems in one nearly resonant laser field – even though we have included loss channels by rate equations. In the following we show how this limitation can be overcome. The B LOCH equations have shown us that a nearly resonant laser field generates a coherent superposition of lower state |a and excited state |b. This implies a periodic population change between the two levels as illustrated in Fig. 10.15. For exactly resonant radiation the system may temporally be found to nearly 100 % in the excited state. Relaxation processes (e.g. spontaneous emission) destroy coherence and damp out the population oscillations as illustrated in Fig. 10.16. For still longer times finally the rate equations as proposed by E INSTEIN lead to the well known equilibrium populations of lower and upper states with ρaa ≥ ρbb – which in the limit of very high intensity can lead at most to equal population. It is now interesting to consider for a multilevel system, how one may transfer the initially very high population of the excited state into a third level – by a judicious sequence of several laser pulses. One possible scheme is the so called lambda configuration (Λ) as sketched in Fig. 10.22. (Alternatively, in a so called ladder configuration, the third state lies above the second.) A typical example would be the population transfer between vibration-rotation states of a molecule. Assume |1 to be a low lying and |3 a higher lying vibrationrotation state in the electronic ground, while |2 is an electronically excited state. The goal is now to shift as much population as possible into level |3, using level

658

10

Fig. 10.22 Three level scheme (lambda configuration) for the STIRAP process with pump (ωP ) and S TOKES angular frequency (ωS ) and the detuning ωP and ωS with respect to state |2; the goal of the experiment is to achieve a population transfer from |1 to |3 as complete as possible


|2 〉 ΔωP

ΔωS spontaneous emission

ωS

ωP |3 〉 |1 〉

|2 as intermediate. The underlying molecular potentials may be visualized as e.g. shown in Fig. 5.13. Let us assume the states |1 and |2 are coupled by a pump pulse (P) which generates as time dependent population in state |2 similar to that shown in Fig. 10.17. Intuitively one would now try to apply a second pulse, the so called S TOKES pulse (S), slightly delayed so that it hits the maximum population of |2 in order to transfer it to the final state |3. This method, the so called stimulated emission pumping, has been and still is successfully used in many spectroscopic applications. However, one finds that in this manner only a relatively low percentage, typically up to 25 %, may be transferred into |3. Surprisingly, however, it is possible to transfer nearly 100 % of the population from |1 to |3 if the S TOKES laser acts prior to the pump laser onto the quantum system. This method, the so called stimulated R AMAN adiabatic passage STIRAP, has been realized for the first time by B ERGMANN and collaborators for scattering experiments with rotationally excited Na2 molecules (for a review see e.g. B ERGMANN et al. 1998). Today it is widely used for many applications, e.g. for the study of atomic and molecular interaction processes, in chemical dynamics, in quantum optics and more recently also for the preparation of ultracold atoms and molecules (see e.g. S TELLMER et al. 2012; W INKLER et al. 2007). To understand this scheme one has to describe the three level system in analogy to the two level system. Applying again the RWA – now for two, nearly resonant radiation fields – the corresponding Hamiltonian reads in analogy to (10.8):7 ⎞ ⎛ 0 −iΩP (t) 0 (t) = ⎝ iΩP (t) 2ωP −iΩS (t) ⎠ . (10.56) H 2 2ω 0 iΩ (t) S

3

The result is very plausible, so we refrain from deriving it explicitly. For clarity we have not introduced here any damping terms. The R ABI frequencies for the pump and the S TOKES laser are now ΩP (t) = 7 Note

e|r 21 |E0 hP (t)

and ΩS (t) =

e|r 32 |E0 hS (t)

(10.57)

that due to our definition (1.35) for the electric field of the radiation the off diagonal matrix elements differ from those of B ERGMANN et al. (1998) by an (insignificant) phase factor ±i.

10.7

STIRAP

659

with the envelopes hP,S (t) of the respective field amplitudes. Note that for the STIRAP process the time dependence of the R ABI frequencies plays a key role. In Λ configuration one defines the detuning ωP = ωP − (W2 − W1 )/ ωS = ωS − |W3 − W2 |/

(10.58)

ω3 = ωP − ωS with the corresponding angular frequencies ωP and ωS . In the ladder configuration, i.e. if W3 > W2 , one has to set ω3 = ωP + ωS . Depending on the tuning of the two laser frequencies one distinguishes between two-photon resonance ω3 = 0 and one-photon resonance, more precisely pump resonance ωP = 0 or S TOKES resonance ωS = 0.

10.7.2 Energy Splitting and State Evolution If damping plays a decisive role – and this is often the case in practice – one has to add the corresponding damping terms to the Hamiltonian (10.56), in analogy to (10.33). Using the L IOUVILLE equations (10.19), one easily derives the coupled linear differential equations for the temporal evolution of the three level system. It allows one to describe the dynamics as well as the losses during the population transfer. We discuss here only the case without damping. It is easier to analyze by solving the coupled equations (10.9). For details we refer to F EWELL et al. (1997). Here we only discuss the most important results and one example. In analogy to the two level system, one has to treat now a coherent superposition of the three levels |1, |2 and |3. If one follows B ERGMANN et al. (1998) and assumes exact two-photon resonance (i.e. ωP = ωS ) one may verify by diagonalization of the Hamiltonian (10.56) that the (time dependent) eigenvalues of the three dressed states are given by (10.59) ω+ = ωP + ωP2 + ΩP2 + ΩS2 , ω0 = 0 and ω− = ωP − ωP2 + ΩP2 + ΩS2 . For the respective eigenstates one finds + a = sin Θ sin Φ|1 + cos Φ|2 + cos Θ sin Φ|3 0 a = cos Θ|1 − sin Θ|3 − a = sin Θ cos Φ|1 − sin Φ|2 + cos Θ cos Φ|3,

(10.60)

with the mixing angle Θ from tan Θ =

ΩP (t) , ΩS (t)

(10.61)

660

10


– |3〉

|a-〉

|a+ 〉 Θ

|a0 〉 |1〉

| ψ〉

|2〉

losses

Fig. 10.23 Schematic representation of the states for the three level system in H ILBERT space: |1, |2 and |3 correspond to the basis vectors in the unperturbed system, |a + , |a 0 and |a − those in the laser fields; |ψ is the state vector of the total system. By variation of the mixing angle Θ one may in principle transfer |1 → |a 0 and |a 0 → |3 without involving state |2 which is subject to losses

and Φ being known function of detuning and R ABI frequencies, here without further interest. The three new basis states |a + , |a 0 and |a + of the system with laser field are – as the original basis – again orthogonal. Of particular interest is the state |a 0 which obviously may be changed continuously from |1 to |3 by choosing the appropriate mixing angle. The latter can be adjusted to values between Θ = 0 and Θ = π/2 depending on the intensity of the two laser fields and their temporal evolution. This is illustrated schematically in Fig. 10.23. The optimal temporal sequence of the laser pulses for this population transfer and the corresponding temporal evolution of the mixing angle (10.61) is sketched in Fig. 10.24(a) and (b), respectively. The temporal evolution of the energies of the three basis states in the field (for one and two-photon resonance) is shown in Fig. 10.24(c). As already mentioned, amazingly the realization of state |a 0 requires that at the beginning of the process (region I) first the S TOKES pulse must rise and consequently also ΩS (t) – it couples states |2 and |3! Conversely, at a later time (region III) the S TOKES pulse must decrease before the pump pules. During the time in between (region II) where the three states are clearly split due to the laser interaction both pulses must overlap. In short: the interaction in the system must start with the S TOKES laser shortly before the pump intensity rises, however, it must end before the pump laser. Albeit completely counterintuitive, this scheme is highly effective. The reason is, that for these conditions state |a 0 consists initially to 100 % of |1 and at the end to 100 % of |3, as indicated in Fig. 10.24(d). So far, this is just a conceptual idea. Whether in reality the physical state of the system (in Fig. 10.23 indicated as |ψ(t)) is identical with |a 0 (more precisely: whether the |ψ(t) follows the basis vector |a 0 adiabatically) requires a detailed, time dependent calculation and, of course, experimental proof.

STIRAP

661

Fig. 10.24 (a): Optimal temporal sequence of S TOKES (S) and probe laser pulse (P) for the population transfer from |1 to |3; (b) the corresponding mixing angle Θ; (c) temporal evolution of the energy splitting between state |a + , |a 0 and |a − ; (d) corresponding content of states |1 and |3 in the state |a 0 as a function of time

(a)

ΩS,P

S

P

0

π/2 Θ

(b) 0

(c) energy

10.7

|1〉

ħω +

population

1

|3〉

ħω 0

ħω |1〉

|3〉

(d) 0 I

II

III

time

10.7.3 Experimental Realization Let us first have a look at the experiment. The temporal variation of the interaction may be detected by two different methods: B ERGMANN and collaborators typically have used a molecular beam setup as sketched in Fig. 10.25. Here a well collimated target beam passes two slightly displaced CW laser beams with a Gaussian spatial profile. The target atoms or molecules ‘see’ effectively a temporal sequence of two pulses G AUSS pulses. Focussing the laser beams to about 100 µm up to 3 mm (typical laser powers are some mW to W) one generates an effective “pulse width” in a

preparation if required

D1

D2

signal

target beam pump laser STOKES laser

probe laser

Fig. 10.25 Scheme of a STIRAP experiment with a molecular beam and spatially slightly displaced, continuous S TOKES and pump laser beams. The displacement realizes the temporal pulse sequence. The STIRAP process occurs in the region of overlap; detector D1 registers the fluorescence emitted there. Further downstream the final state is excited with a probe laser and detected with D2 (again by fluorescence)

662 J=1

S

probe 2p 5 3s

|1〉 ~ ~

~ ~ 1S

0

VUV

135

nm

P

633

nm

140

nm

145

0

J=2

Ne K =1/2 K= 1/2 K =1/2 K =3/2 |2〉 K= 3/2 |4〉 K =5/2 2p 5 3p

616

150


588

wavenumber / 103 cm-1

J=0

losses

Fig. 10.26 Part of the term scheme of neon with the first excited states (valence electron in the M shell). The STIRAP states used are denoted by |1, |2 and |3 (P indicates the pump and S the S TOKES laser); state |4 is used for probing the population transfer

10

~ ~

|3〉 1s2 2s2 2p6

range of 100 ns to µs. The tunable dye lasers used are highly stabilized and thus have a very narrow bandwidth (1 MHz) – and correspondingly high coherence. The interaction is monitored via the fluorescence of the short-lived intermediate state |2. The resulting population transfer into the (long-lived) final state |3 is probed down stream with a further laser tuned to the optically allowed transition between |3 and a further, short-lived state |4. Alternatively one may use pulsed ns lasers (in this case fully overlapping) with pulse energies in the Joule region. The demand for stability is in this case particularly critical and the pulse duration must be F OURIER limited. Experiments with ps lasers are in principle also possible, while ultrashort pulses are not appropriate, in particular due to their broad spectral bandwidth. Over all, experience shows that the conditions required for the adiabatic passage between the states involved is most conveniently achieved with CW lasers as described above. A nice example (B ERGMANN et al. 1998) is STIRAP with excited, metastable Ne atoms. The term scheme is sketched in Fig. 10.26 (see also Sect. 10.4.2 in Vol. 1, specifically Fig. 10.9). We remember: the ground state of Ne, the completely closed rare gas shell, is characterized by 1s 2 2s 2 2p 6 1 S0 , the first excited states have the configuration 1s 2 2s 2 2p 5 3s and 3p and have an excitation energy of 16.6 to 19 eV. They correspond to transitions in the VUV. The total angular momentum of the core j couples to the angular momentum l of the excited electron, which leads to an angular momentum K, which in turn couples with the electron spin to J = K ± 1/2. They are thus characterized by (2S+1 Lj )nl 2S+1 [K]J . For optical dipole transitions we have the usual selection rule J = 0, ±1, with 0 0 forbidden. Two of the 2p 5 3s states are metastable and are used for the STIRAP process as initial and final states:8 |1 = |(2 P◦1/2 )3s 2 [1/2]◦0 and |3 = |(2 P◦3/2 )3s 2 [3/2]◦2 . One generates the metastable Ne in a gas discharge and depopulates state |3 by optical pumping, i.e. by excitation of |4 = |(2 P◦3/2 )3p 2 [5/2]◦2 with 633 nm, followed by decay into the two lowest lying J = 1 states, which in turn decay into the ground 8 We

use here, different from B ERGMANN et al. (1998), the standard terminology according to NIST.

STIRAP

Fig. 10.27 Fluorescence signals from the STIRAP process with metastable Ne atoms as a function of the detuning of the pump laser at fixed S TOKES laser frequency according to B ERGMANN et al. (1998). One detects the VUV emission into the 1 S0 ground state. (a) Fluorescence from detector D1 above the interaction region, documenting the dark resonance; (b) population of the final state as seen by detector D2, following excitation with a probe laser

663 signal / 103 counts s-1 losses/ 103 counts s-1

10.7

16

(a)

12 8 4 0 60

(b)

40 20 0 - 800

0 - 400 ΔωP / MHz

400

800

state. The thus prepared atomic Ne beam in state |1 enters after collimation (to limit D OPPLER broadening) the STIRAP setup. As intermediate state one chooses |2 = |(2 P◦ 1/2 )3p 2 [1/2]◦ 1 which decays spontaneously (J = 1) into the two lowest 3s states with J = 1. In principle, this opens an efficient loss channel. The interaction region is observed with a channeltron which is a simple and efficient detector for the VUV emission into the ground state. Downstream, long after the interaction, one finally probes by laser induced fluorescence the population of the final state |3. Again the transition |3 → |4 at 633 nm is used and the following VUV emission from the two lowest lying J = 1 states into the 1 S0 ground state. In Fig. 10.27 the experimentally observed VUV signal from both detection regions is shown as a function of the detuning of the pump laser. The S TOKES laser frequency, slightly off resonance, remains fixed in this measurement. Figure 10.27(a) shows the fluorescence which monitors the population of the intermediate level |2: as long as the two-photon resonance is not realized exactly one observes large fluorescence losses. The spectral width of the laser induced fluorescence is larger than the D OPPLER width and reflects the power broadening of the transition |1 → |2. However, if two-photon resonance conditions are reached (ωP = ωS ) the sharp and pronounced minimum in the loss signal documents that exclusively the ‘dark’ state |a 0 is populated. In the experiment less then 0.5 % of the signal without S TOKES laser is observed! This disappearance is all the more remarkable as the atoms spent more than 20 times the natural lifetime of state |2 in the interaction region. Under these conditions the final state |3 is populated by adiabatic passage as documented impressively by the signal shown in Fig. 10.27(b). The broad background observed corresponds to a certain fraction which decays spontaneously from |2 to |3 while the intermediate state is populated (ωP = ωS ). A quantitative determination of the population transfer is reproduced in Fig. 10.28. The measurement shows convincingly that nearly complete population transfer occurs when the S TOKES laser is active distinctively before the probe laser,

664

10

S

P

S P

P S

P

S

100 transfer efficiency / %

Fig. 10.28 Efficiency of the population transfer in the STIRAP process with metastable Ne, as a function of the spatial distances between S TOKES (S) an pump laser beam (P) as reported by B ERGMANN et al. (1998). The upper part of the figure illustrates schematically the offset between the two lasers. Optimal transfer is obtained only if (S) interacts before (P) with the target atom


50

0 -1000

-500

0

500

beam displacement / μm

but overlaps with it. The transfer efficiency decreases to about 25 % if both beams fully overlap. The experiments thus proof the effectiveness of the stimulated R AMAN process and the realization of the adiabatic passage. In view of Fig. 10.24(c) and Fig. 10.23 one thus finds that the state vector |ψ(t) of the system (with |ψ(−∞) = |1), originally parallel to state |a 0 , follows this state indeed during the whole process so that at the end of the interaction the population has been transferred completely and coherently from |1 → |3. This is not trivial. One may well imagine that during the process transitions can be induced into the STIRAP states |a + and |a − according to (10.60) and thus state |2 would be excited – as seen for the case ωP = ωS . The necessary conditions for true adiabatic passage have been intensively analyzed both experimentally as well as theoretically. Good overlap between the two pulses as well as sufficiently strong coupling appears mandatory. B ERGMANN et al. (1998) give as a general criterium

Ωeff τ > 10,

(10.62)

where Ωeff = ΩP2 + ΩS2 is an effective mean R ABI frequency and τ the overlap time between the two pulses. Only if this condition is fulfilled one may assume that during the whole process |ψ(t) ∝ |a 0 and that transitions to the other states with losses can be neglected. The STIRAP method has been successfully exploited for a variety of atomic and molecular processes. More recently it has been used even in solid state physics and – as already mentioned – appears to be a promising road to create ultra cold molecules. After this brief introduction we shall leave it to the reader to explore this fascinating subject further. Section summary

• STIRAP is a very efficient scheme to transfer population from one state into another in a three level system, using two nearly resonant laser fields, a pump and a S TOKES laser as illustrated in Fig. 10.22.


665

• Contrary to intuition, for a most efficient population transfer the quantum system must interact first with the S TOKES laser, while the pump laser follows and overlaps with it. • The scheme may quantitatively be understood by a generalization of the optical B LOCH equations. The properties of the three dressed states (10.60) are schematically illustrated in Fig. 10.24. • The key to efficient population transfer by “adiabatic passage” is to keep the system during the whole process in state |a 0 .

Acronyms and Terminology c.c.: ‘complex conjugate’. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. EPR: ‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2 in Vol. 1). FORT ‘Far-off-resonance optical dipole trap’, for trapping single atoms; a typical setup is shown in Fig. 10.1. FPI: ‘FABRY-P ÉROT interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2 in Vol. 1). FWHM: ‘Full width at half maximum’. HBT: ‘Hanbury B ROWN and T WISS’, experiment, to determine the lateral correlation of light by a second-order interferometric measurement (see Sect. 2.1.6). MOT: ‘Magneto optical trap’, for a typical setup see e.g. Fig. 6.26. NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. NMR: ‘Nuclear magnetic resonance’, spectroscopy, a rather universal spectroscopic method for identifying molecules (see Sect. 9.5.3 in Vol. 1). ODE: ‘Ordinary differential equation’. RWA: ‘Rotating wave approximation’, allows to solve the coupled equations for a two level system in a strong electromagnetic field in closed analytical form (see Sect. 10.2.3). STIRAP: ‘Stimulated R AMAN adiabatic passage’, special type of optical pumping, see Sect. 10.7. SVE: ‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38)). UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). VUV: ‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

666

10


References A LLEN , L. and J. H. E BERLY: 1975. Optical Resonance and Two-Level Atoms. New York: Dover, 2nd edn., 233 pages. A SHKIN , A.: 1978. ‘Trapping of atoms by resonance radiation pressure’. Phys. Rev. Lett., 40, 729–732. B ERGMANN , K., H. T HEUER and B. W. S HORE: 1998. ‘Coherent population transfer among quantum states of atoms and molecules’. Rev. Mod. Phys., 70, 1003–1025. F EWELL , M. P., B. W. S HORE and K. B ERGMANN: 1997. ‘Coherent population transfer among three states: Full algebraic solutions and the relevance of non adiabatic processes to transfer by delayed pulses’. Aust. J. Phys., 50, 281–308. F REUDENBERG , T., W. R ADLOFF, H. H. R ITZE, V. S TERT, K. W EYERS, F. N OACK and I. V. H ERTEL: 1996. ‘Ultrafast fragmentation and ionisation dynamics of ammonia clusters’. Z. Phys. D, 36, 349–364. G RAY , H. R. and C. R. S TROUD: 1978. ‘Autler-Townes effect in double optical resonance’. Opt. Commun., 25, 359–362. G ROVE , R. E., F. Y. W U and S. E ZEKIEL: 1977. ‘Measurement of spectrum of resonance fluorescence from a 2-level atom in an intense monochromatic-field’. Phys. Rev. A, 15, 227–233. H ARTIG , W., W. R ASMUSSEN, R. S CHIEDER and H. WALTHER: 1976. ‘Study of frequencydistribution of fluorescent light-induced by monochromatic radiation’. Z. Phys. A, 278, 205– 210. H ERTEL , I. V. and W. R ADLOFF: 2006. ‘Ultrafast dynamics in isolated molecules and molecular clusters’. Rep. Prog. Phys., 69, 1897–2003. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. L IPPERT , H., V. S TERT, L. H ESSE, C. P. S CHULZ, I. V. H ERTEL and W. R ADLOFF: 2003. ‘Analysis of hydrogen atom transfer in photoexcited indole(NH3 )n clusters by femtosecond timeresolved photoelectron spectroscopy’. J. Phys. Chem. A, 107, 8239–8250. M OLLOW , B. R.: 1975. ‘Pure-state analysis of resonant light-scattering – radiative damping, saturation, and multi-photon effects’. Phys. Rev. A, 12, 1919–1943. M OLLOW , R. B.: 1969. ‘Power spectrum of light scattered by two-level systems’. Phys. Rev., 188, 1969–1975. P IRO , N. et al.: 2011. ‘Heralded single-photon absorption by a single atom’. Nat. Phys., 7, 17–20. R ABI , I. I.: 1944. ‘The N OBEL prize in physics: for his resonance method for recording the magnetic properties of atomic nuclei’, Stockholm. http://nobelprize.org/nobel_prizes/physics/ laureates/1944/. S CHUDA , F., C. R. S TROUD and M. H ERCHER: 1974. ‘Observation of resonant stark effect at optical frequencies’. J. Phys. B, At. Mol. Phys., 7, L198–L202. S TELLMER , S., B. PASQUIOU, R. G RIMM and F. S CHRECK: 2012. ‘Creation of ultracold Sr2 molecules in the electronic ground state’. Phys. Rev. Lett., 109, 115302. SWP 5.5: 2005. ‘Scientific work place’, Poulsbo, WA 98370-7370, USA: MacKichan Software, Inc. http://www.mackichan.com/, accessed: 9 Jan 2014. T ICHY , M. C., F. M INTERT and A. B UCHLEITNER: 2011. ‘Essential entanglement for atomic and molecular physics’. J. Phys. B, At. Mol. Phys., 44, 192001. VOLZ , J., M. W EBER, D. S CHLENK, W. ROSENFELD, C. K URTSIEFER and H. W EINFURTER: 2007. ‘An atom and a photon’. Laser Phys., 17, 1007–1016. W EBER , M., J. VOLZ, K. S AUCKE, C. K URTSIEFER and H. W EINFURTER: 2006. ‘Analysis of a single-atom dipole trap’. Phys. Rev. A, 73, 043406. W EISSBLUTH , M.: 1989. Photon-Atom Interactions. New York, London, Toronto, Sydney, San Francisco: Academic Press, 407 pages. W INKLER , K., F. L ANG, G. T HALHAMMER, P. VON DER S TRATEN, R. G RIMM and J. H. D EN SCHLAG : 2007. ‘Coherent optical transfer of Feshbach molecules to a lower vibrational state’. Phys. Rev. Lett., 98, 043201. Z IEGLER , L. D.: 1985. ‘Rovibronic absorption analysis of the A˜ ← X˜ transition of ammonia’. J. Chem. Phys., 82, 664–669.

Appendices

Overview

These appendices contain supplementing material for those readers who want to obtain some more in depth information on a few selected topics. Except for Appendix C they do not contain essential tools for the main text. Appendix A Gives an example for the explicit evaluation of the first B ORN approximation in the case of inelastic electron scattering. Appendix B acquaints the reader with important devices for the detection, manipulation and energy selection of electrons and ions which are used in many experiments discussed throughout the main text. Appendix C introduces the so called state multipole moments as irreducible representation of the density matrix, a concept which is intensely used in Chap. 9, it compares different types of multipole expansions and introduces some useful tools for working with them. Appendix D, finally, illustrates the concept of optical pumping by way of example for the very often used hyperfine pumping of Na atoms.

A

First B ORN Approximation for e + Na(3s) → e + Na(3p)

A.1

Evaluation of the Generalized Oscillator Strength

We explicate here for a simple example the computation of scattering amplitudes in first B ORN approximation (FBA) for the inelastic electron scattering. According to (8.29) the generalized oscillator strength (GOS) is (GOS)

ff i

(K) =

2 2Wf i Ff i (K) 2 K

(A.1)

(Wf i and K are given in a.u.) with the matrix element Ff i (K) = φf | eiK·r n |φi n

as defined in (8.21). The integration has to be carried out over all active electrons with the coordinates r n . The key quantity in this approximation is the momentum transfer vector K = k i − k f with K = kf2 + ki2 − 2kf ki cos θ , (A.2) which depends on the electron scattering angle θ , the transition energy Wf i = Wf − Wi and the initial kinetic energy T , with ki2 = 2T and kf2 = 2(T − Wf i ). As a particularly simple example we consider the quasi one electron system Na and investigate the electron impact excitation of the first resonance transition 3p 2 P ← 3s 2 S. The integration has to be carried out just for the valence electron. We neglect in good approximation any spin-orbit and exchange interaction, since fine structure is usually not resolved in electron collisions and exchange can be neglected at higher energies (T Wf i ) for which B ORN approximation can be applied. We thus have only to average over all initial quantum numbers and sum over all final states. The problem may then be described completely in the uncoupled atomic basis |nm. The wave function is given by φnm (r) = Rn (r)Ym (θB , ϕB ) =

unl (r) Ym (θB , ϕB ) r

© Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5

669

A First B ORN Approximation for e + Na(3s) → e + Na(3p)

670

with the spherical harmonics according to (B.20)–(B.22), Vol. 1 (−1)+m (2 + 1)( − m)! Ym (θB , ϕB ) = 2 ! 4π( + m)! × (sin θB )m

(A.3)

d+m (sin θB )2 exp(imϕB ), d(cos θB )+m

and the radial wave functions Rn (r) for the active orbital. For unl (R) we use the Na orbitals introduced in Sect. 3.2.5, Vol. 1 (S CHUMACHER 2011). Specifically for a p ← s transition we have to evaluate the matrix element iK·r 2 3pmf |e |3smi = drr R3s (r)R3p (r) ×

dϕB

sin θB dθB Y0mi (θB , ϕB )eiKz Y1mf (θB , ϕB ) .

The most reasonable coordinate system has its z-axis parallel to K, and z = r cos θB . Here θB is the polar angle of the position vector r of the target electron with respect to K – not to be confused with the scattering angle θ , which according to (A.2) is built into K. As a consequence of the symmetry with respect to this z-axis only transitions with m = 0 may occur. Thus, the only nonvanishing matrix element to be evaluated is iK·r 3p0|e |3s0 = 2π dr u3s (r)u3p (r) ×

Y10 (θB , ϕB )eiKz Y00 (θB , ϕB ) sin θB dθB .

√ √ If we insert Y10 (cos θB ) = 3/4π cos θB and Y00 (cos θB ) = 1/ 4π the angular integration can be carried out in closed form: √ √ π 3 3 iKr cos θB iKr cos θB . sin θB dθB = e (iKr cos θ − 1) cos θB e B 2 2 2 2K r 0 The whole matrix element thus becomes √ ∞ Kr cos(Kr) − sin(Kr) dru3s (r)u3p (r) × . 3p0|eiK·r |3s0 = −i 3 K 2r 2 0 The next integration step can be carried out numerically without problems. The quality of the result depends on the quality of the radial wave functions u3s (r) and u3p (r) used. Finally, we obtain for the generalized oscillator strengths (A.1): 6W3p3s ∞ Kr cos(Kr) − sin(Kr) 2 (GOS) dru (r)u (r) (A.4) f3p3s = 3s 3p . K2 0 K 2r 2

A.1

Evaluation of the Generalized Oscillator Strength

671

It is instructive to expand the fraction in the integrand for small K in powers of Kr. One then obtains matrix elements of powers of r: ∞ n r = u3s (r)r n u3p (r)dr. (A.5) 0

This leads to a power series for the generalized oscillator strength. Explicitly one finds in the present case for the GOS up to the 6th power in K: rr 5 r 3 2 2 1 (GOS) f3p3s = W3p3s r2 − K 2 r r 3 + K 4 + 3 5 140 100 7 3 5 rr r r + + O K8 . − K6 7560 1400

(A.6)

We see here, that the step from the series expansion (8.33) in the main text to computable radial matrix elements is not completely trivial. Alternatively and valid for any single electron system one may obtain this result also by expressing the powers of z which appear in (8.33) in terms of spherical harmonics Y0 . Thus one obtains a genuine multipole expansion (GOS) ff i

2Wf i = gi =

...

... 2 K 2 s γf Jf Mf |r Y0 |γi Ji Mi

(A.7)

Ji Mi Jf Mf =0

f K 2 ,

=0

which one can extend as far as the experimental accuracy requires it. The matrix elements may be rewritten with the help of the W IGNER -E CKART theorem into reduced matrix elements while the M dependence averages out (as long as the initial state is populated isotropically). One finally obtains expressions of the type (A.6) with coefficients f which can be determined experimentally or theoretically. The advantage here is that this procedure can be used in principle for arbitrary transitions and coupling schemes. The coefficients may be compared to the corresponding expressions for optical transitions (dipole, quadrupole etc. transitions). For a comparison with theory one has of course to compute the necessary matrix elements of powers of r by integration over the wave functions. In any case, it is recommended to check in each individual case whether a complete integration of (A.1) is possibly more convenient. The results presented it in Sect. 8.3.3 for the p ← s transition in Na originate from such full integration. In our case the evaluation of the matrix elements gives r = −4.2687, so that the first, constant term in the series (A.6) assumes the value 0.939. This limiting value should for K → 0 be identical to the optical oscillator strength. The experimentally determined literature value is f (opt) = 0.960. It gives us a feeling for the quality or deficits of the wave functions used. According to K IM (2007) it is advisable to

A First B ORN Approximation for e + Na(3s) → e + Na(3p)

672

rescale the thus computed oscillator strengths as well as the differential cross section as (GOS)

(K) =

ff

f (opt) (GOS) (K) f f(GOS) (0)

(f )

and

dσif (θ, φ) dΩ

=

Born f (opt) dσif (θ, φ) . dΩ f(GOS) (0)

(GOS)

We have done this in Fig. 8.5 so that ff optical limit with K → 0.

A.2

(A.8)

shown there indeed approaches the

Integration of the Differential Cross Section

The integral inelastic cross section is obtained according to (8.34) from the thus computed generalized oscillator strength (all quantities again in a.u.): σ=

π × T Wf i

Kmax f(GOS) (K) f

Kmin

K

dK.

(A.9)

(GOS)

This integration may be simplified by approximating ff (K) by a function which can be integrated in closed form. In the present case one finds that (GOS) ff (A.10) (K) = A exp −(K/w)2 1 + c1 K + c2 K 2 + c3 K 3 with A = 0.95935, w = 0.42351, c1 = 0.00322, c2 = −1.64602 and c3 = 1.70711 gives an excellent fit. The integral may then be expressed with the help of the error integral and the error function. We have made use of this in Fig. 8.4.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2 in Vol. 1. FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sects. 6.6 and 5.5.2, Vol. 1, respectively). GOS: ‘Generalized oscillator strength’, characterizes the strength of electron impact excitation in analogy to the optical oscillator strength see Sect. 8.3.2.

References K IM, Y. K.: 2007. ‘Scaled Born cross sections for excitations of H2 by electron impact’. J. Chem. Phys., 126. S CHUMACHER, E.: 2011. ‘FDAlin programme, computation of atomic orbitals (Windows and Linux)’, Chemsoft, Bern. http://www.chemsoft.ch/qc/fda.htm, accessed: 5 Jan 2014.

B

Guiding, Detecting and Energy Analysis of Electrons and Ions

In the main text we refer on various occasions to methods for the manipulation, detection or energy analysis of electron and ion beams which today are standard experimental tools far beyond modern atomic, molecular and optical physics. Electron and ion optics and the detailed layout of energy selectors are nowadays designed with the help of commercial programmes (see e.g. SIMION 2012). Nevertheless it is useful to have some elementary knowledge about principles and typical components. One essential basis of all these methods are the forces which act on these particles in electric or magnetic fields which lead to characteristic deflections of particle beams as described already in Chap. 1, Vol. 1. The following short introduction focusses onto low energy particle beams for which today usually electrostatic guiding and focussing is used. In the last section we shall, however, also show an important application of magnetic fields. We do not discuss here space charge effects since they usually do not play any role for the detection and analysis of particles (very low currents).

B.1

SEM, Channeltron, Microchannel Plate

We start with a summary on secondary electron multipliers (SEM) and related devices which are most commonly used in low energy atomic and molecular physics for direct detection of ions, electrons, VUV and XUV photons. By repeated secondary electron emission they provide a typical amplification of ca. 108 . Each single particle thus generates a pulse of some ns duration with currents in the range of mA. These may then be further amplified with conventional electronics, be discriminated against noise and are finally counted. This allows to count single electrons, ions or photons with high efficiency. The classical standard arrangement of an SEM is sketched in Fig. B.1(a). The electron to be detected hits the first dynode and ejects there one, two or even more secondary electrons of low kinetic energy. These are accelerated towards the next dynode where the secondary electron emission is repeated for each impacting electron. The whole process is repeated many times so that an avalanche of electrons is © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5

673

Fig. B.1 Secondary electron multiplier: (a) classical prototype SEM scheme with individual dynodes, (b) principle of a channeltron, (c) typical realization of a channeltron

B


(a)

dynodes

anode pulse

674

e-

or ion

to amplifier C

_

high voltage (HV)

+

(c)

layer of semiconductor secondary electrons

e- or ion

glass wall _ HV +

input funnel

electron exit

(b)

ca.15mm

built up which is finally captured on an anode. From there it is coupled into an electronic circuit for further processing. For particle detection one typically uses 14–18 dynodes, for photomultipliers usually less. An overall voltage (HV in Fig. B.1(a)) of 2–4 kV is applied and distributed to the dynodes via a voltage divider chain as shown in Fig. B.1(a). In front of each dynode the electrons have a kinetic energies of 100 eV to 200 eV, which allows for efficient secondary electron emission. For the detection of photons (also in the visible) standard photomultipliers (PM) are still built according to this classical scheme. In this case a photocathode is mounted in front of the SEM where the photons are efficiently converted by photoelectron emission into electrons. The whole assembly – photocathode and SEM – are mounted in a compact, evacuated tube, the voltage for the dynodes is fed into this tube by connector pins in the bottom of the tube. This is today a very well established technology, and a broad range of very efficient devices for various applications is commercially offered: low dark currents and high sensitivity cover today a broad spectral range for single photon counting. For the detection of electrons, ions and fast neutral particles one uses today almost exclusively channel electron multiplier (short channeltrons or CEM). Their principle is illustrated in Fig. B.1(b). Instead of discrete dynodes in an SEM (Fig. B.1(a)) a channeltron uses the whole inner wall of a thin (some mm diameter) glass tube continuously as a kind of extended dynode for secondary electron emission. This inner wall is coated with a thin semiconducting layer of some 100 nm thickness with a high resistance (in the 100 M region). On top of it a several 10 µm thick layer of SiO2 is applied for passivation and improving the secondary electron emission. At typical voltages of 3000 V (current ca. 30 µA) the potential rises linearly over the total length of the tube (some cm), so that along the whole length secondary electrons are ejected from the walls and accelerated continuously. A variety of specific shapes for channeltrons are used. Figure B.1(c) shows an example

(a)


secondarelektronen

ca.

103-104

microchannel plate

electical e or ion contact entrance material

_ +

_ HV ca. 1000 V

e-

on exit

(d)

675

(b)

single microchannel

channel wall

B.1

2 microchannel plates in chevron configuration

+ glass structure e- exit

primary particle

microchannels

channels _

MCP1 MCP2

+_ +

electron avalanche on exit

anode

pulse exit

Fig. B.2 Microchannel plate (MCP) schematic: (a) function of a single channel, (b) cut through a single MCP, (c) two MCP’s in chevron configuration to enhance amplification with typical coupling to the electronics

with an entrance funnel (important for “hitting” the channeltron) and an amplifying channel in spiral shape. This particular shape is supposed to improve the emission geometry and to minimize echoes from ions moving into the back direction which may be ejected from the walls. Alternatively, compact channeltrons embedded into ceramics in sinusoidal shape (Ceratron) and other shapes are offered. Amplifications of more then 108 are possible, which decreases, however, rapidly at higher count rates above 104 s−1 (saturation). Microchannel plates (MCP) may be seen as a consequent further development of the channeltron, schematically illustrated in Fig. B.2. An MCP consists of many microchannels as shown in Fig. B.2(a) with a diameter of 6 µm to 10 µm, each of them acting just like a channeltron. They are combined to plates of some tenth mm up to 2 mm thickness as seen in Fig. B.2(b). Typically the channels are tilted by a small angle (ca. 8◦ ) with respect to normal in order to avoid direct transmission of the particles to be detected and to reduce the ion back passages. The amplification of a single MCP is, however, moderate, only about 103 –104 . Thus, as indicated in Fig. B.2(c), one uses two plates in tandem, either in the so called “Chevron” or in a V configuration (one plate is turned around the normal by 180◦ ). With such a setup one reaches a gain of 106 to 107 , which as a rule is sufficient for particle counting; if necessary one may even post three plates behind each other (Z configuration). The main advantage of microchannel plates is their large area: diameters

Fig. B.3 Detection probability for electrons by secondary emission as a function of their kinetic energy T at the entrance into the channeltron

B

Guiding, Detecting and Energy Analysis of Electrons and Ions 100 detection probability P / %

676

80 60 40 20 0 10

100

1000

104

T / eV

of 8 cm (or even 12 cm) are available today without problems (corresponding to about 107 channels). For simple applications the large area enables detection of extended signal currents without focussing. However, the most genuine features of microchannel plates are exploited in connection with position sensitive detection. They allow to record a combination of time, energy, momentum and angular distribution of the detected particles – so to say in “one shot”. A number of sophisticated schemes for such velocity map imaging (VMI) methods have been established, based on MCP detection. In the most simple case one uses stripe-anodes which allow a 1D detection. 2D methods work e.g. with arrays of such anodes, with crossed wires (2N wires for N 2 positions), with resistive anodes or – most common today – with time of flight methods where the amplified electron pulse hits two or more crossed or meandering delay lines, arranged behind each other. When the delay time for each of theses signals is measured, in principle the x- and y-positions can be computed. A third wire in some setups may help to avoid ambiguities. Finally, also the direct optical detection enjoys great popularity: the electrons emerging from the MCP hit a fluorescent screen which allows to detect and record the spatial distribution directly with a CCD camera. At the end of this section a few words are appropriate on the detection probability for electrons and ions by secondary electron emission – as basis of all the configurations discussed here. Most clear is the situation for electrons. In Fig. B.3 the typical behaviour of the secondary electron emission coefficient and thus the detection probability P is plotted as a function of the impact energy T of the primary electron. As seen, P is nearly 100 % for electrons at about T 200 eV. This is also the optimal voltage between the dynodes of an SEM shown in Fig. B.1(a). Consequently a channeltron according to Fig. B.1(b) will be constructed such that typical pathways of the electrons between hitting the walls in the channel correspond to a few 100 eV voltage difference on the semiconductor layer. For ions detection the situation is significantly different. Ions eject secondary electrons with a much lower probability when hitting the wall. For a (semi-)quantitative treatment one has to account for the fact that we use here a statistical process. The detection probability results from a P OISON distribution for the probabilities to

B.1


677

emit N secondary electrons: Pe (N ) =

γeN exp(−γe ). N!

(B.1)

The quantity γe is called secondary electron emission coefficient. The probability that no electron is emitted is Pe (0) = exp(−γe ). Thus, the detection probability, which we look for, is given by the probability to emit one or more electrons: P = 1 − exp(−γe ).

(B.2)

In principle γe and thus P depends on the velocity v as well as on the mass M of the ions studied. In general one expect the detection efficiency to rise with the velocity, and for the same velocity probably also with mass. During the past decades there have been a number of attempts to determine γe experimentally, in particular for larger masses, unfortunately with somewhat uncertain outcome (see the discussion by F RASER 2002). We communicate two relatively recent results for MCPs. In both cases a calibration was attempted with alternative ion detection methods, assumed to be quantitative. Different masses with molar weights of some 100 u up to some 1000 u were investigated. W ESTMACOTT et al. (2000) find by comparison with superconducting tunnel contacts an empirical formula for the “reduced” secondary electron emission coefficient (per unit of mass): γe /u = Av B .

(B.3)

The authors report a value B = 4.3 ± 0.4, while A = 5.6748 × 10−24 can be gleaned from their data. Here v is measured in m s−1 . In practice, usually of interest is the detection probability as a function of kinetic energy T (in keV) and mass M (in u) of the detected ion. We find: B P (M, T ) = 1 − exp −M × A × 4.4 × 105 T /M (B.4) = 1 − exp −10.5M −1.15 T 2.15 . In contrast T WERENBOLD et al. (2001) have derived by calibration with cryodetectors 3.5 v γe = . (B.5) 53000 This secondary electron emission coefficient depends only on the particle velocity and leads to a detection probability P (M, T ) = 1 − exp −1639.2(T /M)1.75 .

(B.6)

Both results are compared in Fig. B.4 with each other for two different masses. As seen, for typical extraction voltages 3–4 kV the detection probability for ions of

B

Fig. B.4 Detection probability P for ions by secondary electron emission as a function of their kinetic energy T when hitting an MCP. They grey lines show relation (B.4) derived from W ESTMACOTT et al. (2000), while the red lines illustrate relation (B.6) after T WERENBOLD et al. (2001). Full lines refer to mass M = 720 u (C60 ), dashed lines to M = 120 u Fig. B.5 About the definition of the index of refraction for particle beams in an electrostatic field

Guiding, Detecting and Energy Analysis of Electrons and Ions 100 80 P/ %

678

60 40 20 0 0

5

10 T / keV

15

y

T1

U1 < v1

20

T2 U2

vy θ 1

v2

v θ2 y z

the higher mass is below 10 %. Special efforts are needed to obtain maximum detection efficiency for large masses (e.g. for protein analysis). Either one has to place the MCP’s at very high negative potential – far above the operation voltage for the plates. Alternatively, one may generate the ions at a very high positive potential – which is technically not easy to realize since usually one wants to have the source on ground potential. Unfortunately, the calibration data reported above from the literature do not agree. We would tend to use the data of T WERENBOLD et al. (2001), i.e. relation (B.6) illustrated in Fig. B.4 by red lines; their calibration by cryodetectors appears relatively straight forward (see, however, F RASER 2002).

B.2

Index of Refraction, Lenses and Directional Intensity

The deflection and focussing of charged particle beams may be treated in a very similar manner as done for light beams in geometrical optics, hence the field is called electron and ion optics. Refraction of a particle beam occurs when it passes through different electric potentials. In contrast to light optics there usually are no sharp boundaries, rather the change of direction occurs continuously corresponding to the respective local kinetic energy, as sketched in Fig. B.5. Here we show the example of a homogeneous electrostatic field between two transparent plane, parallel metal grids in a distance d at the potential U1 and U2 , respectively. When the particle beam (particle mass m, charge qe) passes through such a field its kinetic energy changes from T√ 1 = qeU1 to T2 = √qeU2 . The magnitude of the velocity thus changes from v1 = 2T1 /m to v2 = 2T2 /m. Perpendicular to the optical axis (z-axis, normal to the surface) the velocity component vy remains con-

B.2

Index of Refraction, Lenses and Directional Intensity

(a)

(b) eU1U1

679

(c)

(d) e-

U1 U2 U 1

U1 >U2

U 1 U2

Fig. B.6 Examples for electron lenses

stant. For entrance and exit angles (with reference to the z-axis) θ1 and θ2 , respectively, we have vy = v1 sin θ1 and vy = v2 sin θ2 . From this follows immediately the refraction law for particle beams sin θ1 v2 = = sin θ2 v1

T2 n 2 = , T1 n 1

(B.7)

in full analogy to S NELLIUS’ law in geometrical optics. The index of refraction for particle √ beams is thus proportional to the square root of the local kinetic energy n ∝ T . Note that this relation does not depend on the magnitude of the angle and for the geometry sketched in Fig. B.5 is not limited to small entrance or exit angles. Electron and ion lenses may thus be constructed in analogy to light optics. Some characteristic examples are summarized in Fig. B.6. The red lines in (a) and (c) indicate typical electron trajectories in collecting lenses. One distinguishes einzel lenses (a, b) and immersion lenses (c, d). The former consist of three elements (typically apertures or cylinders), of which the two outer ones are fixed to the same potential, so that the energy of the particles is conserved. In contrast, immersion lenses are made of two elements at different potential. In addition to focussing or defocussing the charged particles, they also change their energy. The focal lengths and other imaging properties cannot be written in terms of simple formulas as in light optics. Rather, a detailed calculation or measurement is needed (a survey on the classical literature is given by M ULVEY and WALLINGTON 1973). Today one uses commercially available programmes, already mentioned above, to simulate particle trajectories. These programmes allows one to design much more sophisticated and better optimized geometries than implied by the simple lens patterns introduced above. The so called H ELMHOLTZ -L AGRANGE relation (also known from light optics as A BBE sine condition), is an other important relation for the propagation of particle beams: n1 y1 sin θ1 = n2 y2 sin θ2 or T1 sin θ1 = β T2 sin θ2 .

(B.8)

It describes the relation between lateral magnification β = y2 /y1 and the respective divergence angles θ1 and θ1 for a particle beam which images an object of the size y1 onto y2 as sketched in Fig. B.7. It is valid only for near-axial rays (small

680

B


Fig. B.7 On the H ELMHOLTZ -L AGRANGE relation (B.8)

y y1

imaging optic

Δθ1

z y2

Δθ2

divergence angles) – in contrast to the S NELLIUS law of refraction (B.7), with which it must not be confused. In two dimensional perspective, i.e. with reference to the differential areas dA1 and dA2 of a particle beam at two different positions along the beam path one obtains the usually discussed form of the H ELMHOLTZ -L AGRANGE relation T1 dA1 dΩ1 = T2 dA2 dΩ2 ,

(B.9)

where dΩ1 and dΩ2 are the respective differential solid angles. One defines now the so called directional intensity1 of a particle beam: R=

dI , dAdΩ

(B.10)

with the current dI which passes through the area dA along the beam axis. The directional intensity thus characterizes the current flux dI /dA of the beam with respect to its divergence angle dΩ. With (B.9) for the directional intensity the following conservation law holds for two different positions in one beam: R1 dI1 dI2 R2 = = = . T1 T1 dA1 dΩ1 T2 dA2 dΩ2 T2

(B.11)

For each well collimated particle beam the ratio of directional intensity to kinetic energy, R/T , is a conserved quantity – of course only if no particles are lost and if no energy dispersive elements are active on the beam path. This relation is of quite fundamental importance for the design of electron and ion optics. If one wants to obtain as much as possible current through a small cross section – which often is the standard requirement – one has to make sure that already at the beginning, i.e. when the ions or electrons are generated, the directional intensity is as high as possible (which means we have to design the cathode of an electron source well, or we have to focus the light source properly if we are talking about photoelectrons etc.). High energies are also helpful, they may, however, not be what is otherwise required for the experiment.

B.3

Hemispherical Energy Selector

For energy analysis in photoionization spectroscopy, today one typically uses electrostatic methods if continuous radiation sources are involved (CW lasers, syn1 Also

denoted as “brightness of the beam”, or by the original German term “Richtstrahlwert”.

B.3

Hemispherical Energy Selector

Fig. B.8 Hemispherical energy analyzer (here for the example of detecting photoelectrons from a solid state surface) with imaging optics and two dimensional registration of the measured signal

681 U0 x1 e-

U

A α R1 R0

focussing optics

R2

detector MCP's

w max

B x2

chrotron sources), while for pulsed sources (short pulse lasers, isolated SR pulses) time of flight methods are appropriate (they will briefly be discussed in the next section). In atomic, molecular and cluster science electrons and/or ions are detected. And in photoemission spectroscopy from solid surfaces electrons are detected. The basic concept of all electrostatic energy selectors is a combination of (i) spatial separation of particles with different kinetic energy by deflection in the electric field with (ii) geometric focussing of different angles at the entrance slit of the selector. In this way one tries to collect as much signal as possible to pass through the dispersive element. This concept succeeds with different perfection depending on the geometry used (cylindrical, hemispherical, toroidal, trochoidal etc.). Such electrostatic energy monochromators can be used for both, energy analysis of charged particles as well as for the generation of monoenergetic electron and ion beams. Here we discuss as an example the hemispherical analyzer, the advantages of which are well proven in practice. A number of commercial realizations are marketed with great success. This analyzer consists of two concentric hemispheres as sketched in Fig. B.8. This setup has first been described by P URCELL (1938) and was introduced to low energy photoelectron spectroscopy by K UYATT and S IMP SON (1967). The following description is based on these papers and includes modern types of realization. We assume an idealized field distribution which is not disturbed by boundary effects. One may achieve this according to H ERZOG (1935) by suitable limiting apertures which we consider to be realized in Fig. B.8 by an optimally positioned entrance slit and the first detector plate at the exit of the analyzer, respectively. The electrostatic potential between the hemispheres in the geometry of Fig. B.8 is according to the laws of electrostatics C1 /R + C2 . The magnitude of the electric field thus becomes E(R) = U R1 R2 /[(R2 − R1 )R 2 ], with U being the overall poten-

682

B


tial difference between the hemispheres. The field is directed radially and we expect circular (more generally elliptical) orbits just as for the K EPLER problem. Let us first discuss electrons which start at point A on the radius R0 = (R2 + R1 )/2 (i.e. exactly in the middle between the spheres) and enter perpendicular to the connecting axis between points AB (dash dotted line). Their orbit is exactly a circle if the centrifugal force and the electric field compensate each other, i.e. if qeE(R0 ) = −mv02 /R0 . Hence, to guide an electron of kinetic energy T0 = eU0 = mv02 /2 on the nominal circular orbit with R0 , the voltage difference between inner and outer semi-sphere must be U = U0 (R2 /R1 − R1 /R2 ),

(B.12)

while U1 = U0 [3 − 2(R0 /R1 )] and U2 = U0 [3 − 2(R0 /R2 )] are the potentials on the inner and outer hemisphere, respectively, with the middle potential being U0 . This implies that charged particles, generated at ground potential with zero kinetic energy, which are accelerated up to the potential of the entrance slit U0 just pass the analyzer on the nominal orbit. The symmetry of the sphere ensures that under these conditions all electrons, which move on a grand circle with radius R0 and enter at the point A tangentially to the equipotential areas of the field, also exit at point B. These are particles which move on any plane through the connecting axis between AB, which enter perpendicular to this axis at point A: the geometry of the setup thus ensures perfect angular focussing in a direction perpendicular to the plane displayed in Fig. B.8. Now, what about the focussing in this sectional plane, i.e. focussing with respect to angular divergences denoted as α in Fig. B.8? And how large is the dispersion (respectively the energy resolving power) of the hemispherical capacitor? The classical problem rather straight forward (P URCELL 1938). One solves the equations of motion by linearizing them for small deviations from the nominal values. We discuss here just the results. Let x2 be the radial deviation of the electron from R0 when it leaves the analyzer near B, and let x1 be the corresponding deviation at point A. Further, let T = T − T0 be the deviation of the electron kinetic energy T from the nominal energy T0 and α be the entrance angle with respect to the nominal trajectory in the sectional plane. The calculation then gives x2 /R0 = −x1 /R0 + 2(T /T ) − 2α 2 .

(B.13)

The fact that the angular divergence only enters in quadratic form means that the spherical analyzer does focus different angles to first order in the sectional plane shown in Fig. B.8. To derive the energy resolution of such a system one has to account for the width of the entrance and exit apertures. In case of slits with equal width w and neglecting the α 2 term, one expects a triangular transmission function with a FWHM w T0 . (B.14) T1/2 = 2R0

B.4

Magnetic Bottle and Other Time of Flight Methods

683

For best energy resolution T1/2 one thus works advantageously at low transmission energies which can be obtained by suitable particle optics (immersion lenses) – also for high initial kinetic energies. However, according to (B.11) this can only be achieved at the expense of a larger divergence angle. The most useful geometry and electron optics has thus to be assessed specifically for each individual experiment. In the setup sketched in Fig. B.8 the entrance aperture is considered to be a vertical slit. At the exit a multichannel plate with 2D resolution is used – the exit slit is thus realized by the spatial resolution of the plate. In the situation shown here an extended area of a target is illuminated with photons. This is imaged onto the entrance slit of the analyzer. In the sectional plane shown, one thus collects all electrons which are emitted from a finite part of the target in this plane, emitted perpendicular to the target. Correspondingly electrons which are emitted with angles differing from normal to the target plane are imaged onto the entrance slit below or above A. The analyzer images these below or above the sectional plane onto the detector MCP – again selected for energies in the x2 direction. In summary, this allows a two dimensional recording of the photoelectron emission: according to energy (horizontal direction) and emission angle (vertical direction).

B.4

Magnetic Bottle and Other Time of Flight Methods

As mentioned above, if the light source is pulsed and thus marks sharply the time of the photoionization process, time of flight (TOF) methods are appropriate for energy analysis. An often used setup is the so called magnetic bottle, whose main merit is a high collection efficiency, specifically for low energy electrons. It has been used by K RUIT and R EAD (1983) for low energy electron spectroscopy for the first time. For many applications it is still the method of choice. If one is interested in the kinetic energy of electrons from weak sources it helps if one can register – if possible – all electrons emitted into one half full solid angle 2π . With simple time of flight methods which allow the electrons to drift without guiding field, one typically looses a substantial part of the signal which is emitted with lateral velocity components. If, on the other hand, one first extracts the electrons into the direction of the drift tube, different electron emission angles lead to different lifetimes for the same initial kinetic energy: without special precautions one just measures the velocity component parallel to the extraction and detection region. The magnetic bottle is the ideal remedy to this problem: one first parallelizes the momenta of the emitted electrons during an initial, short time of flight. This is done in a strongly inhomogeneous magnetic field without changing the total kinetic energy. The essence of this method is schematically illustrated by Fig. B.9. A strong magnetic field B i (some Tesla, typically generated by permanent magnets) is maintained at the interaction centre, where the photoelectrons are created. In cylindrical symmetry the field is smoothly changed along the z-axis into a weak field B f that

684

B


Fig. B.9 Essentials of a magnetic bottle (Bi Bf ) illustrated by two characteristic trajectories

vf

vi

θf

θi

ez e-

Bi

Bf

extends along the actual, electric field free, drift distance of the TOF. Typically B f is some mT, generated by a solenoid which in turn is shielded by µ-metal from external magnetic fields. Let an electron (mass me , charge −e) originally be generated in the lower part of the setup with a velocity v i and be emitted at an angle θi with reference to the z-axis. In the strong field B i it will spiral with the cyclotron frequency ωi =

eBi . me

(B.15)

The initial radius of the orbit ri is defined by the radial component vi sin θi of the velocity: ri =

vi sin θi me vi sin θi = . ωi eBi

(B.16)

This circular movement has an angular momentum i = Θωi = me ri2 ωi

= me

vi sin θi ωi

2 ωi = me

vi2 sin2 θi m2 v 2 sin2 θi = e i . (B.17) ωi eBi

We remember: the electron kinetic energy me vi2 /2 remains constant in a purely magnetic field. The magnitude of the velocities is thus constant, |v|i = |v|f = v. The magnetic field is assumed to change “adiabatically” in z-direction, i.e. the change is negligibly small during one cyclotron cycle. In this case one may show that also the angular moment remains constant so that sin θi = sin θf

Bi Bf

1/2 .

(B.18)

Thus, the transverse component of the velocity decreases strongly while the longitudinal component in z-direction which determines the time of flight grows from initially v cos θi to 2 vzf = v cos θf = v 1 − sin θf = v 1 − (Bf /Bi ) sin2 θi .


685

The trajectories thus become indeed parallel. For a magnetic field ratio of Bf /Bi = 1:1000 the z-component of the velocity vzf in the TOF region differs from the overall magnitude v only by 0.5 h. With (B.16) one may write (B.18) as ri = rf

Bf Bi

1/2 .

(B.19)

This means that the magnetic flux Bπr 2 through one circular orbit of the electron is a constant of motion. Aside from this particular variety of a TOF electron spectrometer, optimized for maximum collection efficiency, during the past years a variety of powerful imaging techniques has been developed for photoelectrons and photoions. In addition to energy measurement these allow a complete determination of the particle momentum by direction and magnitude (velocity map imaging VMI, also energy imaging, momentum mapping etc.) which we have mentioned already several times (e.g. in Sect. 5.5.4, Vol. 1). With the help of modern electronics one may detect with such methods several particles simultaneously (electrons and one or more fragment ions). These methods make a whole new class of experiments accessible for modern atomic and molecular physics in collision processes or laser matter interaction – with hitherto unknown high efficiency and detail. Particularly successful in this context is the so called reaction microscope, originally introduced by S CHMIDTB ÖCKING and collaborators as COLTRIMS (cold target recoil ion momentum spectroscopy). Recent developments of it are MOTRIMS (magneto-optical trap recoil ion momentum spectroscopy). They all are extremely powerful but by no means trivial to handle and to evaluate. Among other ingredients the evaluation requires a detailed analysis of the particle trajectories in the magnetic guiding fields which are used in these devices, as well as sophisticated position and time resolved coincidence detection techniques. In the MOTRIMS variety one needs in addition expertise with cooling and storing of particles by “state-of-the-art” laser traps. A good entrance for the interested readers give the reviews of U LLRICH et al. (2003) and D EPAOLA et al. (2008) (while optical traps are reviewed by G RIMM et al. 2000).

Acronyms and Terminology CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). CEM: ‘Channel electron multiplier’, see Appendix B.1. COLTRIMS: ‘Cold target recoil ion momentum spectroscopy’, see Appendix B.4. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. FWHM: ‘Full width at half maximum’. MCP: ‘Multi channel plate’, electron multiplier with many amplifying elements. MOTRIMS: ‘magneto-optical trap recoil ion momentum spectroscopy’, see Appendix B.4.

686

B


PM: ‘Photomultiplier’, see Appendix B.1. SEM: ‘Secondary electron multiplier’, see Appendix B.1. SR: ‘Synchrotron radiation’, electronmagnetic radiation in a broad range of wavelengths, generated by relativistic electrons on circular orbits. TOF: ‘Time of flight’, measurement to determine velocities of charged particles, and consequently their energies (if the mass to charge ratio is known) or their mass to charge ratio (if their energy is known). UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). VMI: ‘Velocity map imaging’, experimental method for registration (and visualization) of particle velocities as a function of their angular distribution (see Appendix B). VUV: ‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007). XUV: ‘Soft x-ray (sometimes also extreme UV)’, spectral wavelength range between 0.1 nm and 10 nm according to ISO 21348 (2007), sometimes up to 40 nm.

References D EPAOLA, B. D., R. M ORGENSTERN and N. A NDERSEN: 2008. ‘Motrims: magneto-optical trap recoil ion momentum spectroscopy’. In: ‘Adv. At. Mol. Opt. Phys.’, vol. 55, 139–189. Amsterdam: Elsevier. F RASER, G. W.: 2002. ‘The ion detection efficiency of microchannel plates (MCPs)’. Int. J. Mass Spectrom., 215, 13–30. G RIMM, R., M. W EIDEMÜLLER and Y. B. OVCHINNIKOV: 2000. ‘Optical dipole traps for neutral atoms’. Adv. At. Mol. Opt. Phys., 42, 95–170. H ERZOG, R.: 1935. ‘Berechnung des Streufeldes eines Kondensators, dessen Feld durch eine Blende begrenzt ist’. Arch. Elektrotech., 29, 790–802. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. K RUIT, P. and F. H. R EAD: 1983. ‘Magnetic-field parallelizer for 2π electronspectrometer and electron-image magnifier’. J. Phys., E J. Sci. Instrum., 16, 313–324. K UYATT, C. E. and J. A. S IMPSON: 1967. ‘Electron monochromator design’. Rev. Sci. Instrum., 38, 103–111. M ULVEY, T. and M. J. WALLINGTON: 1973. ‘Electron lenses’. Rep. Prog. Phys., 36, 347–421. P URCELL, E. M.: 1938. ‘The focusing of charged particles by a spherical condenser’. Phys. Rev., 54, 818–826. SIMION: 2012. ‘Industry standard charged particle optics simulation software’, Scientific Instrument Services, Inc., Ringoes, NJ, USA. http://simion.com/, accessed: 5 Jan 2014. T WERENBOLD, D., D. G ERBER, D. G RITTI, Y. G ONIN, A. N ETUSCHILL, F. ROSSEL, D. S CHEN KER and J. L. V UILLEUMIER : 2001. ‘Single molecule detector for mass spectrometry with mass independent detection efficiency’. Proteomics, 1, 66–69. U LLRICH, J., R. M OSHAMMER, A. D ORN, R. D ÖRNER, L. P. H. S CHMIDT and H. S CHMIDTB ÖCKING: 2003. ‘Recoil-ion and electron momentum spectroscopy: reaction-microscopes’. Rep. Prog. Phys., 66, 1463–1545. W ESTMACOTT, G., M. F RANK, S. E. L ABOV and W. H. B ENNER: 2000. ‘Using a superconducting tunnel junction detector to measure the secondary electron emission efficiency for a microchannel plate detector bombarded by large molecular ions’. Rapid Commun. Mass Spectrom., 14, 1854–1861.

C

Statistical Tensor and State Multipoles

C.1

Multipole Expansion of the Density Matrix

As explicated in Chap. 9, the density matrix describes mixed states in concise form. It is the key tool for evaluating experiments with imperfect state selection prior to and/or incomplete state analysis after an interaction process – i.e. essentially for any experiment in the real world. In practice, the final expressions involving the density matrix may become rather clumsy, in particular when state selection, interaction and analysis are best described in different coordinate systems and in different angular momentum coupling schemes. In this context, a multipole expansion of the density matrix often facilitates such evaluations and allows to disentangle the dynamics of the processes studied from the geometry of a specific experiment. We give here a short introduction to the concept of state multipoles as first introduced by FANO (1953). They provide an irreducible representation of the density matrix. Alternatively, the multipole moments introduced in Appendix F.3.2, Vol. 1 may be used. As we shall see below, these different representations are related to each other by simple numerical factors derived from the W IGNER -E CKART theorem Appendix C.1.2, Vol. 1. Slightly different definitions are used in the literature. We essentially follow the notation of B LUM (2012), who gives a rigorous derivation of the concept. In A NDERSEN et al. (1988) both notations – state multipoles and multipole moments are used. The reader interested in details finds there also extensive tabulation of the relevant conversion relations and parameters. We start with the density operator (9.15)–(9.18) and specify the states |γ of the system as standard angular momentum states |j m: ρˆ =

ρj m ,j m |j m j m|.

(C.1)

j m j m

The density matrix is Hermitian ρj m ,j m = ρj∗m,j m © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5

(C.2) 687

688

C Statistical Tensor and State Multipoles

and characterizes a quantum system (or a specified subset) completely. One may rewrite (C.1) as a multipole expansion of the type ρˆ =

† tˆ j j KQ tˆ j j KQ ,

(C.3)

KQ

with tˆ(j j )KQ representing appropriate irreducible tensor operators. At this point the quantities tˆ(j j )†KQ are introduced as suitable expansion coefficients for a given quantum system. (We use here the lower case letter tˆKQ to avoid confusion with the multipole moment operators Tkq introduced in Appendix F.3.2, Vol. 1.) In view of (C.1) the so called statistical tensor operators are constructed by coupling angular momentum states |j m and j m | in essentially the same manner (see J2 = K as one couples angular momenta in a vector coupling scheme J1 + Eq. (B.31), Vol. 1), except that here the outer product |j m j m| of these states has to be used: (−1)j −m j m j − m|KQ|j m j m| tˆ j j KQ = m m

j −m 1/2 j = (−1) (2K + 1) m mm

j −m

(C.4)

K |j m j m| −Q

with j − j ≤ K ≤ j + j and −K ≤ Q ≤ K.

(C.5)

The matrix elements of this operator are obviously given by1 j m |tˆ j j KQ |j m = (−1)j −m j m j − m|KQ or √ j K j (C.6) j m |tˆ j j KQ |j m = (−1)j −m 2K + 1 m −m −Q j K j j −m 2j + 1 j tK j . = (−1) m −m −Q The latter equality is derived from the W IGNER -E CKART theorem (C.8), Vol. 1. Hence, the reduced matrix elements of the statistical tensor operators are2

j tK j =

1 Interestingly, 2 in

2K + 1 . 2j + 1

(C.7)

this very relation was used by FANO (1953) to define the “state multipoles”.

the notation of B RINK and S ATCHLER (1994), while according to E DMONDS (1996), A NDER √ SEN et al. (1988), B LUM (2012) j tK j (Ed) = 2K + 1.

C.1

Multipole Expansion of the Density Matrix

689

Exploiting Hermiticity of the operator and the symmetry of the C LEBSCH -G ORDAN coefficients † j m|tˆ j j KQ |j m = j m |tˆ j j KQ |j m∗ = (−1)j −m j m j − m|KQ

(C.8)

= (−1)j −j +Q (−1)j −m j mj − m |K − Q = (−1)j −j +Q j m|tˆ jj K−Q |j m , one obtains the adjoint operators3 † tˆ j j KQ = (−1)j −j +Q tˆ jj K−Q .

(C.9)

Their expectation values – the so called state multipoles – are according to (9.23) † † † tˆ j j KQ = Tr ρˆ tˆ j j KQ = ρj m ,j m j m|tˆ j j KQ |j m =

m m

m m

ρj m ,j m j m |tˆ j j KQ |j m

† tˆ j j KQ = (−1)j −m (2K + 1)1/2 mm

j m

or j K ρ . −m −Q j m ,j m (C.10)

Inserting this and the statistical tensor operator (C.5) into (C.3) and exploiting the orthogonality relation (B.40), Vol. 1 for 3j symbols, we indeed recover (C.1). Sim ilarly, (C.10) may be inverted by multiplying both sides with (−1)j −m (2K + 1)1/2 × C LEBSCH -G ORDAN coefficient and summing over K and Q. One obtains the density matrix elements: ρj m ,j m =

j (−)j −m (2K + 1)1/2 m KQ

j −m

K † tˆ j j KQ . (C.11) −Q

If all tˆ(j j )†KQ are known, the density matrix elements may be derived from them. With (C.3), (C.4), (C.10), and (C.11) we have thus fully achieved our initial goal: to reformulate the density matrix as a multipole expansion. Note that state multipoles up to rank K = j + j may be constructed from the angular momenta j and j of the set of states to be described. The simplest situation is the completely incoherent, equal population of all sublevels with ρm ,m = δm m /gj with the degeneracy gj = 2j + 1. The density matrix 3 The star indicating complex conjugate is actually redundant in this case since the matrix elements are real.

690


is then ρˆ =

1 , 2j + 1

in which case only the zero rank state multipole moment is nonvanishing:

1 tˆ(jj )†00 = √ . 2j + 1

The state multipoles are as such also irreducible tensor operators of rank K and provide an alternative possibility to characterize a given quantum system. In general, they are complex quantities, and we record here some of their properties – easily derived from the above definitions, exploiting the Hermiticity of the density matrix (C.2): † ∗ † tˆ j j KQ = tˆ j j KQ = (−1)j −j +Q tˆ jj K−Q , and (C.12) † ∗ tˆ j j KQ = tˆ j j KQ = (−1)j −j +Q tˆ jj K−Q . This is in agreement with the relations (D.5), Vol. 1 communicated earlier for irreducible tensor operators. Under rotation through E ULER angles (ϕk θk 0) – see e.g. Fig. 4.3 in Vol. 1 – the state multipoles transform as described by (E.15), Vol. 1: † # † k tˆ j j KQ = tˆ j j Kq DqQ (ϕk θk 0)∗ .

(C.13)

q

Specifically, the zero components tˆ(j j )†K0 # of a quantum system in respect to a detector or analyzer system are obtained from the state multipoles in a different coordinate system by using (E.3), Vol. 1: K † # † tˆ j j K0 = tˆ j j Kq CKq (θk , ϕk ).

(C.14)

q=−K

Note that these expressions are dramatically more simple and transparent than (9.146) which describes rotation of the density matrix itself. This is one of the key advantages for using the state multipoles.

C.2

State Multipoles and Expectation Values of Multipole Tensor Operators

While the definition and evaluation of state multipoles is rather straight forward, they are nevertheless somewhat abstract quantities, and the numerical factors involved (3j symbols) may be inconvenient. In contrast, the multipole tensor operators, TKQ , which we have introduced in Appendix F.3.2, Vol. 1 are constructed from

C.2

State Multipoles and Expectation Values of Multipole Tensor Operators

691

angular momentum operators. Supposedly, their expectation values TKQ , short multipole moments, provide a more direct ‘physical feeling’ for atomic anisotropies and are numerically simpler to handle. Fortunately, the W IGNER -E CKART theorem gives a one to one relation between the matrix elements of any irreducible tensor operator of rank K for a set of states – say for the state multipoles tˆ(j j )KQ in a |j m basis – to any other irreducible tensor of the same rank in that basis – say the TKQ . With (C.12), Vol. 1 this relation is simply given by j TK j j m |tˆ j j KQ |j m, j m |TKQ |j m = j tK j

(C.15)

and also holds for the respective expectation values. It allows us by comparison with (F.32), Vol. 1 (see the examples given in Table F.1, Vol. 1) to associate the better ‘physical intuition’ with the state multipoles. We point out here, however, that for evaluation of experiments it is irrelevant which kind of multipole moments are used, as long as their definition is clearly stated.4 Of special interest is the case with ‘sharp’ angular momentum j = j . The ratio of the reduced density matrix elements is then given by (see e.g. M ACEK and H ERTEL 1974, and references quoted there):

1/2 j tK j 2K (2K + 1)1/2 (2j − K)! v(K, j ) = = . j TK j K! (2j + K + 1)! Hence, the multipole moments are related to the state multipoles by T(j )KQ = tˆ(j )†KQ /v(K, j ).

(C.16)

(C.17)

In standard literature both types of multipole expansions are used, e.g. for describing the angular distribution and polarization of radiation from anisotropically populated excited atoms. Often it is convenient to rewrite them as real parameters which directly express important symmetry properties as outlined in Appendix D.3, Vol. 1. According to (D.5), Vol. 1 and using (C.15) and (C.16) one constructs them from their complex counterparts for 0 < Q ≤ K by √ (C.18) T(j )KQ+ = (−1)Q 2 Re T(j )KQ and √ (C.19) T(j )KQ− = −(−1)Q 2 Im T(j )KQ , while for Q = 0 T (j )K0+ = T (j )K0 and TK0− = 0, (C.20) and correspondingly for the state multipoles. 4 Atomic

anisotropies and orientation may uniquely be represented by a set of expectation values of tensor operators. However, as illustrated in Sect. 9.3, we find the density matrix usually a more instructive representation – provided an appropriate coordinate system is chosen. In this spirit, multipole moments of any kind are seen mainly as an intermediate step for designing and evaluating experiments.

692


Table C.1 Complex multipole moments T(J )KQ i.e. expectation values of multipole tensor operators, up to rank K = 2 for sharp angular momentum values up to J = 2, as a function of the respective density matrix elements; real multipole moments are obtained from (C.18)–(C.20) and state multipoles by multiplication with v(K, Q) J

K

Q

T(J )KQ

v(K, Q)

any

0

0

1

1

0∗

1

1

(ρ 1 , 1 − ρ− 1 ,− 1 )/2 2 2 2 2 √ −ρ 1 ,− 1 / 2

1

0

ρ1,1 − ρ−1,−1

1

1

−(ρ0,−1 + ρ1,0 )

2

0a

2

1

2

2

1 − 3ρ0,0 √ 3(ρ0,−1 + ρ1,0 ) √ 6ρ1,−1

1

0

1

1

(J + 1)−1/2 √ 2 √ 2 √ 1/ 2 √ 1/ 2 √ 1/ 6 √ 1/ 6 √ 1/ 6 √ 1/ 5 √ 1/ 5

2

0

2

1

2

2

1

0

1

1

1/2

1

3/2

2

2

2

+ 12 ρ 1 , 1 − 12 ρ− 1 ,− 1 − 32 ρ− 3 ,− 3 2 2 2 2 2 2 √ √ √ −( 3/2ρ 3 , 1 + 2ρ 1 ,− 1 + 3/2ρ− 1 ,− 3 ) 3 2 ρ 32 , 32

2 2

2

2

2

2

3(ρ 3 , 3 − ρ 1 , 1 − ρ− 1 ,− 1 + ρ− 3 ,− 3 ) 2 2 2 2 2 2 2 2 √ 3 2(−ρ 3 , 1 + ρ− 1 ,− 3 ) 2 2 2 2 √ 3 2(ρ 3 ,− 1 + ρ 1 ,− 3 ) 2

2

2

2

2ρ2,2 + ρ1,1 − ρ−1,−1 − 2ρ−2,−2 √ √ √ √ − 2ρ2,1 − 3ρ1,0 − 3ρ0,−1 − 2ρ−1,−2

6[2(ρ2,2 + ρ−2,−2 ) + 12 (ρ1,1 + ρ−1,−1 ) − 1] √ √ 2 1 3[ 6ρ−1,−2 + ρ0,−1 − ρ1,0 − 6ρ2,1 ] √ 2 2 6[ρ2,0 + 3/2ρ1,−1 + ρ0,−2 ] ( a Here we have made use of ρM,M = 1 2

0a

1/6 1/6 1/6 √ 1/ 10 √ 1/ 10 √ 1/(3 14) √ 1/(3 14) √ 1/(3 14)

For some examples of T(j )KQ , constructed from angular momenta with quantum numbers 0 ≤ j ≤ 2, explicit expressions are given in Table C.1 (see also Table 3 in A NDERSEN et al. 1988). Multipole moments up to rank K = 2j are possible. We communicate these here up to rank K = 2 only, since higher rank multipoles cannot be determined in experiments with single photon emission (see Sect. 9.4).

C.3

Recoupling

When evaluation experiments which are sensitive to anisotropies, often an additional problem arises since the density matrix ρˆ of the system studied and that of the state analyzer σˆ (sometimes also state selector may be involved) must be described in different angular momentum coupling schemes. This requires recoupling

C.3

Recoupling

693

of the respective angular momentum states. Again, this is best done in by using an irreducible representation of the density matrix. To have something concrete in mind, let us consider the coupling scheme (J I )F of electronic angular momentum J and nuclear spin I to the total angular momentum F in an atomic system. According to (C.19), Vol. 1 one may construct tensor operators of rank K from direct products of state multipole operators tˆ(J J )kJ qJ and tˆ(I I )kI qI , describing the electronic and nuclear component of the system, respectively:

t J J k ⊗ t I I k KQ = J I

tˆ J J k

J qJ

tˆ I I k

I qI

kJ qJ kI qI |KQ.

qJ qI

Note that tˆ(J J )kJ qJ is generated from the coupling scheme (J J )kJ while tˆ(I I )kI qI corresponds to the scheme (I I )kI . The above direct product thus corresponds to an overall coupling scheme |(J J )kJ , (I I )kI ; K. In contrast, in the statistical tensor operator tˆ(F F )KQ in the coupled system (J I )F must be constructed according to (C.4) from |F mF states, i.e. in a coupling scheme |(J I )F, (J I )F ; K. Thus, to rewrite tˆ(F F )KQ in terms of multipole moments of its subsystems we have to recouple the four angular momenta involved correspondingly: tˆ F F KQ =

(J I )F, J I F ; K J J kJ , I I kI ; K

J J I I kJ kI

× t J J k ⊗ t I I k KQ . J

I

And since the state multipoles (i.e. the tensor operators averaged over the population densities of the states described) are also irreducible tensor operators, the same relation holds also for these. Using above relations and (B.77), Vol. 1 we may relate the state multipoles in the coupled and uncoupled scheme by † 1/2 (2F + 1) 2F + 1 (2kJ + 1)(2kI + 1) tˆ F F KQ = J J I I kJ kI

⎧ ⎨J × J ⎩ kJ

I I kI

(C.21)

⎫ F ⎬ † † F tˆ J J k q tˆ I I k q kJ qJ kI qI |KQ. J J I I ⎭ K qJ qI

Now, one is often confronted with the situation that one subsystem is completely isotropically populated while the other is not. For example, in an atomic collision processes the nuclear spin states are typically without relevance, and remain statistically populated and the density matrix elements of the nuclear spin subsystem are simply ρMI ,MI = δMI MI /(2I + 1). With (C.10) only state multipoles of rank zero exist, hence, kI = 0 so that K = kJ and the whole summation reduces to a single term, while the 9j symbol reduces to a 6j symbol according to (B.78), Vol. 1. With

694


√ tˆ(I )†00 = 1/ 2I + 1 one finally obtains: † † tˆ F F KQ = V (K, F ) × tˆ J J KQ with V (K, F ) =

[(2F + 1)(2F + 1)]1/2 2I + 1

(C.22)

F J

F J

K . I

In the case that the electron spin S too is statistically distributed and all information on an atomic anisotropy is contained in the orbital part L of the density matrix, we may just repeat the above derivation for the coupling scheme (LS)J . The state multipole moment in the hyperfine coupling scheme are then related to that in the orbital angular momentum subsystem by † † tˆ F F KQ = V (K, F )V (K, J ) × tˆ LL KQ [(2J + 1)(2J + 1)]1/2 J J with V (K, J ) = L L 2S + 1

(C.23) K . S

The same recoupling relations also hold for the TKQ± . Thus, if recoupling is necessary, in the transformation relations (C.18)–(C.20) the factor 1/v(K, F ) has to be replaced by V (K, F )V (K, J )/v(K, F ). For further details we refer the interested reader to the original literature, specifically to B LUM (2012), A NDERSEN et al. (1988), Appendices A and B, and to H ERTEL and S TOLL (1978), Chap. IV.

References A NDERSEN, N., J. W. G ALLAGHER and I. V. H ERTEL: 1988. ‘Collisional alignment and orientation of atomic outer shells: 1. Direct excitation by electron and atom impact’. Phys. Rep., 165, 1–188. B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics 64. Berlin, Heidelberg: Springer, 3rd edn., 343 pages. B RINK, D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3rd edn., 182 pages. E DMONDS, A. R.: 1996. Angular Momentum in Quantum Mechanics. Princeton: Princeton University Press, 154 pages. FANO, U.: 1953. ‘Geometrical characterization of nuclear states and the theory of angular correlations’. Phys. Rev., 90, 577–579. H ERTEL, I. V. and W. S TOLL: 1978. ‘Collision experiments with laser excited atoms in crossed beams’. In: ‘Adv. Atom. Mol. Phys.’, vol. 13, 113–228. New York: Academic Press. M ACEK, J. and I. V. H ERTEL: 1974. ‘Theory of electron-scattering from laser-excited atoms’. J. Phys., B At. Mol. Phys., 7, 2173–2188.

D

Optical Pumping

The concept of optical pumping implies the repeated, nearly resonant absorption and reemission of photons, due to which the state population of a quantum system is modified. Dramatic changes of the thermal equilibrium population of atoms, molecules or solids may thus be achieved. The origin of optical pumping goes back into the thirties of the past century – long before the laser was invented – and are connected to names like H ANLE, B ERNHEIM, B ROSSEL and B LOOM. The related atomic spectroscopy reached a first summit with the N OBEL prize for Alfred K ASTLER (1966). It entered into a second, very fruitful period in the beginning of the seventies with the development of tunable dye lasers. They allowed first the first time to exploit the full potential of this method. As an example, our group succeeded in 1973 by laser optical pumping to study electron scattering processes from exited states in an atomic beam (H ERTEL and S TOLL 1974a). Today optical pumping belongs, so to say, to the standard tools of working with light, in laser physics, spectroscopy, collision physics or in creating ultracold atoms and molecules. Also in the present textbook several experiments have been discussed where optical pumping is of importance to prepare atoms and molecules with a non-thermal population. We thus want to briefly illustrate the method by way of one, very often used example.

D.1

A Standard Case: Na(3 2 S1/2 ↔ 3 2 P3/2 )

As discussed in Chap. 10 one often wants to prepare a pure two level system. In nature, such a system exists only in very special cases, e.g. as a pure spin 1/2 system. Thus, one tries to populate two states which may be transferred by absorption or emission of radiation only into each other. Such a nearly ideal two state situation can be achieved for a well defined hyperfine transition (HFS) of the Na D2 Line by optical pumping. Figure D.1 illustrates two alternative optical pumping processes in this system. The HFS term scheme Fig. D.1(a) and the allowed optical transitions have already been explicated in Fig. 9.5, Vol. 1. © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 2, Graduate Texts in Physics, DOI 10.1007/978-3-642-54313-5

695

696 F' =3 2 1 0

D 32P3/2

59.8 MHz 35.5 MHz 5.155 THz

3 2P

1/2

188.9 MHz 0.508 PHz

F= 2 F= 1

32S1/2 1771.6 MHz

(a)

Optical Pumping

MF' = -3 -2 -1 0 1 2 3

MF' = -3 -2 -1 0 1 2 3

σ + light pumps ∆ MF = +1 z

π light

x

pumps ∆ MF = 0

y

k

k E (σ +)

x

MF = -2 -1 0 1 2

(b)

z

E (π) y MF =-2 -1 0 1 2

(c)

Fig. D.1 Hyperfine optical pumping with quasi-monochromatic laser light (red double arrows) of Na between the 32 S1/2 (F = 2) and 32 P3/2 (F = 3) levels. (a) Energy levels (not to scale) with HFS: F, F and FS: 3 2 P3/2,1/2 ; (b) MF = +1 excitation with σ + light for preparation of a pure two level system; (c) MF = 0 excitation with π light. In both cases spontaneous emissions with MF = ±1, 0 lead to the redistribution of population

If the laser is tuned into resonance with the 3 2 S1/2 F = 2 ↔ 3 2 P3/2 F = 3 transition, spontaneous emissions occur only between 3 2 P3/2 F = 3 → 3 2 S1/2 F = 2, owing to the selection rule F = 0, ±1. These two HFS levels consist of 7 and 5 projection levels, with MF and MF , respectively. If one uses, however, circularly polarized σ + light (alternatively σ − light) as indicated in Fig. D.1(b) for excitation one enforces for the excitation process the selection rule MF = +1 (respectively MF = −1) with reference to z k, where k is the wave vector of the light. In contrast, for spontaneous emission all transitions with MF = ±1, 0 are allowed. In summary this leads to a positive transfer of angular momentum onto the ensemble of atoms. For each repetition of the process, again positive angular momentum is transferred on average. With a lifetime of 16 ns in the excited state the atoms undergo typically several 100 pump cycles when passing through the laser beam (at a velocity of ca. 1000 m s−1 in an atomic beam). Already after a few cycles one finds practically all atoms only in one ground state |3 2 S1/2 F = 2MF = 2, and – coupled by the laser light – in the corresponding excited state |3 2 P3/2 F = 3MF = 3. Transitions into other hyperfine sublevels MF are no longer possible. Thus, one has prepared a pure two state system. The situation is somewhat different in the case of linearly polarized π light as indicated in Fig. D.1(b). With reference to a coordinate system with z E, the electric field vector of the pumping light, the selection rule is now MF = 0 and the electric field couples the states F = 2, MF in the electronic ground state 3 2 S1/2 only with F = 3, MF in the excited 3 2 P3/2 state. However, spontaneous emission nevertheless leads to changes in the population density among the MF states. This is summarized in Fig. D.2. To model this quantitatively, we first note that the coordinate systems chosen above lead to a diagonal density matrix (9.17), here a (7 + 5) × (7 + 5) matrix. Thus, we simply have to evaluate the population probabilities for ground and excited state sublevels, w(MF , t) and w(MF , t). To follow the optical pumping process with

D.1

A Standard Case: Na(3 2 S1/2 ↔ 3 2 P3/2 )

697

π light pumping

σ + light pumping

(a)

during the first pumping cycle

(c)

during the first pumping cycle

(b)

stationary

(d)

stationary

MF' = - 3

-2

-1

0

1

2

3

MF' =- 3

-2

-1

0

1

2

3

Fig. D.2 Population of the sublevels −3 ≤ MF ≤ 3 during HFS pumping of Na between 3 2 S1/2 (F = 2) and 3 2 P3/2 (F = 3) level (see e.g. H ERTEL and S TOLL 1974b). (a, b) linearly, (c, d) circularly polarized pumping light; (a, c) during the first pumping cycle, (b, d) in the stationary limit

time t in detail one has to setup 12 − 1 rate equations and to solve them. Following (10.43)–(10.44) they can be read from the term schemes Fig. D.1(b) and (c) for π and σ + pumping, respectively. The required transition probabilities are according to (4.113) and (4.123) in Vol. 1: A F MF , F MF , q ∝ ωF3 F 2F + 1

F MF

1 q

F MF

2

F C1 F 2

3λ3 B F MF , F MF , q = B F MF , F MF , q = ba A F MF ; F MF . 4h Since C1 acts only onto the spatial part of the state one may recouple the reduced matrix element with (C.46) in Vol. 1 and obtains for the hyperfine transitions within a fine structure transition: 2 1 F F . (D.1) B F MF , F MF , q ∝ 2F + 1 (2F + 1) −MF q MF In our case (F = 3, F = 2) this gives for the initial values of MF = −2, −1, 0, 1, 2 when exciting with circularly polarized light: B(MF , q = 1) = 1/3, 1, 2, 10/3, 5 and for linear polarization B(MF , q = 0) = 5/3, 8/3, 3, 8/3, 5/3. The rate equations are solved numerically. Two limiting cases are easily evaluated: during the first excitation cycle the probability of finding excited states will be proportional to the transition probabilities, we thus expect w(MF , t = +0) ∝ B(MF + q, q). This is displayed in Fig. D.2(a) and (c). With little more effort one may derive the stationary limit of induced processes and spontaneous emission which is illustrated in Fig. D.2(b) and (d) for the two pumping conditions. Optical pumping has been developed to great perfection in modern laboratories. A survey about the early developments of this method in atomic collision physics is found in H ERTEL and S TOLL (1978).

698

D

D.2

Optical Pumping

Multipole Moments and Their Experimental Detection

We have just described how by optical pumping one may generate a non-equilibrium population of the states involved. One creates an orientation and/or and alignment. Orientation implies that the average angular momentum in the pumped ensemble of atoms is finite. Alignment characterizes in addition an anisotropic population of M states – in the case of unequally populated orbitals this corresponds to an electric quadrupole moment. For a quantitative description one uses multipole moments as introduced in Appendix C, Vol. 1. More details are found in Appendix C. For describing the optical pumping process in question we use the expectation values of the multipole tensor operators (F.32), Vol. 1 (short: multipole moments). Their relations to the respective density matrix elements are described in Appendix C, and explicitly given for some cases in Table C.1. Of interest here are only orientation o0 (J ) = T10 (J ) = Jz and (D.2) 2 J . alignment a0 (J ) = T20 (J ) = 3Jz2 −

(D.3)

With we emphasize the averaging over the angular momentum components with J being the relevant angular momentum quantum number. Explicitly one determines these quantities in the HFS coupled scheme for the excited state F from the density matrix elements ρM M˜ according to (9.22): F

F

ρM M˜ F˜ M˜ F |Tkq± (F )|F MF . Tkq± (F ) = MF M˜ F

F

F

(D.4)

In our case, with a diagonal density matrix, the calculation is simplified to ; +F MF w(MF ) w(MF )

z = o0 (F ) = F

+F MF

=−F

a0 (F ) = T20 (F ) =

and

(D.5)

M=−F

; +F +F 2 3MF − F F + 1 w(MF ) w(MF ). MF =−F

M=−F

(D.6) The orientation parameter is simply the expectation value of the angular momentum with respect to the z-axis, −F ≤ o0 (F ) ≤ +F . For the alignment parameter one verifies easily that it disappears for equal population of the M states, w(M) = 1/(2F + 1), and must take a value between the minimum and maximum −F (F + 1) ≤ a0 (F ) ≤ F (2F − 1). In the present case with F = 3 we thus have −12 ≤ a0 (F ) ≤ +15. These limiting values are, however, only reached for circular polarized pumping in the stationary limit – as one can guess from Fig. D.2. We have introduced the multipole moments Tkq± (F ) as expectation values of certain combinations of the angular momentum operator J , here of the total angular in the HFS coupling scheme (J I )F . Often this coupling scheme is, momentum F

D.2

Multipole Moments and Their Experimental Detection

699

however, without relevance for the physical process studied, e.g. since the interaction with the nuclear spin is negligible – the nuclear spin behaves, so to say just as a spectator. But also the angular momentum J is a composite quantity, in RUSSEL S AUNDERS coupling (LS)J , again composed of the orbital angular momentum L and the electron spin S. The latter often is also a spectator only. The key interaction – e.g. in a collision process – may just be determined by the electron charge distribution and its orbital angular momentum, in summary it is specific to L. Often one is thus only interested in multipole moments which are defined by the components of L. As described in Appendix C.2, one may derive these from the Tkq (F ) by reduction with the help of the W IGNER -E CKART theorem. With the respective reduced matrix elements one obtains: LTk L Tkq± (F ) . Tkq± (L) = F Tk F

(D.7)

We also may write the parameters of interest here as: z |F M = L M˜ |L o0 (L) = F M˜ F |L F L z |L ML = T10 (L) = T10 (F ) /3 2 2 2z − 2z − a0 (L) = F M˜ F |3L L |F MF = L M˜ L |3L L |L ML = T20 (L) = T20 (F ) /15.

(D.8) (D.9) (D.10) (D.11)

The second equality in both, (D.8) and (D.10), holds since the choice of the basis cannot influence the result of the averaging procedure. The numerical factors in (D.9) and (D.11) are derived from detailed evaluation of the reduced matrix elements (D.7) as outlined in Appendix C.3 (see also F ISCHER and H ERTEL 1982). As described in Sect. 9.4 one may experimentally determine the a0 (F ) and o0 (F ) parameters by measuring the fluorescence emitted from the excited Na(3 2 P3/2 ) – in several geometries. Figure D.3 shows the results of such a measurement from an Na atomic beam excited by the pump process as characterized in Figs. D.1 and D.2 (F = 3). The parameters a0 (L) and o0 (L) (renormalized for the L basis) are shown together with the excitation density Nex as a function of the pump laser intensity. The figure documents that indeed, by optical hyperfine pumping one may achieve considerable electronic alignment and orientation. Excitation, alignment and orientation is obtained with particularly high efficiency when pumping with circularly polarized light: one obtains at sufficiently high intensity nearly the maximum values of o0 (L) = 1 and a0 (L) = 1 – the latter corresponds to the oblate shaped charge distribution, which we have already mentioned often here, symmetric around the wave vector of the σ + light (note the footnote 4 in Appendix F, Vol. 1). Even at low pump intensity substantial values are reached (the red numbers at the scales indicate the values expected for o0 (L) and a0 (L) during the first pumping cycle as illustrated in Fig. D.2(d). Clearly less efficient is pumping with linearly polarized light as documented by Fig. D.2(b): the minimal value (at L = 1) of a0 (L) = −2 cannot be reached by HFS

700

D

max

1.0 10

(a) o1(L)

Nex / ar.un.

1.0

π pump

0.1

0.9

norm.

σ+ pump

Optical Pumping

σ+ pump

(c)

0.8 0.7

- 0.4

0.667 1.0

π pump

- 0.5

0.8

(b)

a0(L)

a0(L)

- 0.480

max σ+ pump

- 0.6

0.6

(d)

0.4 10-3

10-2

10-1

-0.667

0.240

10-3

10-2

10-1

1

I pump / Wcm-2

Fig. D.3 Experimentally determined data points for the 3 2 P3/2 F = 3 ↔ 3 2 S1/2 F = 2 pump process in Na atoms. (a) Excitation probability, (b, d) alignment a0 (L), and (c) orientation parameter o1 (L) as a function of the laser intensity; grey: linearly polarized light (π pumping), red: circularly polarized light (σ + pumping). The data are taken from F ISCHER and H ERTEL (1982), the calculation (dashed red line) corresponds to the saturation profile according to (10.39) with Is = 0.038 W cm−2

pumping due to the coupling of nuclear and electron spin. With the MF distributions according to Fig. D.2(b) one expects a minimum of a0 (L) = −0.667, which is approached at sufficiently high intensity. Even the excitation density of 50 % cannot be reached with linearly polarized light (albeit this cannot be seen directly in Fig. D.2(a) since all experimental data have been normalized to the model at one point as indicated). Clearly, there are deviations from the pure two level model when pumping with linearly polarized light, which leads, as we shall see in a moment, to substantial losses: D OPPLER broadening and a partial overlap of the F = 3 levels with F ≤ 2 leads to population of the F = 1 ground state which cannot be excited again by the one pump laser used in this experiment.

D.3

Optical Pumping with Two Frequencies

Obviously the initially assumed model of coupling the two preferential hyperfine levels 3 2 P3/2 F = 3 ↔ 3 2 S1/2 F = 2 with just one laser frequency does not correspond completely to reality. The HFS levels have a finite linewidth and – most importantly – power broadening at higher intensities leads to population of other

D.3

Optical Pumping with Two Frequencies

F' =3 2 1 0 hν1

32P3/2

F= 2 F= 1

F' = 3 2 1 0 hνhν 11

(a)

32S1/2 1771.6 MHz

701

(b) hν2

F= 2

N

a

b

m ea

F= 1

N

hν2

hν1

1mm

1mm

a

be

am

hν1

Fig. D.4 Population density in a Na beam optically pumped with (a) one frequency on the F = 2 → 3 transition and (b) with an additional laser tuned to F = 1 → 2 transition. The fluorescence signal from the excitation region has been recorded with a CCD camera and is proportional to the excitation density. The data have been adapted from C AMPBELL et al. (1990)

excited HFS levels with F < 3. Hence, optical pumping via spontaneous emission into the 3 2 S1/2 F = 1 ground state leads to substantial losses. The evident solution – we have mentioned it already in the main text (e.g. in Sect. 6.5.3) – is to use two laser frequencies hν1 and hν2 for pumping, to empty just this particular sink 3 2 S1/2 F = 1 again. Figure D.4 show the dramatic differences in both cases. Plotted is the spatially (in 2D) resolved fluorescence from the excited Na(3 2 P3/2 ) state. The signal images the excitation density for excitation (a) with one and (b) with two laser frequencies. The pump scheme is sketched in the respective insets. In Fig. D.4(a) one observes a slightly asymmetric, so to say sluggish excitation profile. The asymmetry is mainly due to the D OPPLER broadening in the wings of the atomic beam due to which the ‘false’ HFS levels are excited, which relax immediately into the 3 2 S1/2 F = 1 ground state. Irradiation by the second laser frequency, here propagating into opposite direction, leads to a massive enhancement of the excitation density, which in addition is quite symmetric. Obviously the second laser field manages efficiently to empty the sink in the ground state, thus supporting a high excitation density. Clearly, two separately tunable, well stabilized dye lasers imply a significant experimental effort, which one certainly would like to reduce. C AMPBELL et al. (1990) thus have developed a two mode laser specifically for this type of excitation. It exploits neatly the spatial variation of the gain profile in the liquid jet of a dye laser: the gain profile explained in Sect. 1.1.7 shows of course also a spatial variation along the z-axis since laser modes are standing waves. In nodes of these standing waves a second mode can be built up. By suitable dimensions of the laser resonator and positioning of the gain medium one may indeed generate simultaneously two laser modes which have a frequency distance corresponding to the HFS splitting of the ground state in Na.

702

D

Optical Pumping

With such a setup and one additional laser it was possible to excite Na in sufficient density even in the 4D state and to study charge exchange processes with K+ ions. Alternatively one may modulate the frequency of a single mode laser with tunable, acousto-optical modulators, which oscillate at the frequency of the HFS splitting of the ground state: the thus induced light modulation leads to corresponding side bands impressed onto the carrier frequency. This again leads to effectively two stable optical frequencies which can be tuned jointly as well as with respect to each other. This method is used today preferentially since it is robust and stable. We have quoted an example in Sect. 6.5.3.

Acronyms and Terminology CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6 in Vol. 1). HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9 in Vol. 1).

References C AMPBELL, E. E. B., H. H ÜLSER, R. W ITTE and I. V. H ERTEL: 1990. ‘Near resonant charge transfer in Na(4D) + K+ → Na+ + K∗ : optical pumping of the Na(4D) state and energy dependence of rank 4 alignment’. Z. Phys. D, 16, 21–33. F ISCHER, A. and I. V. H ERTEL: 1982. ‘Alignment and orientation of the hyperfine levels for laser excited Na-atom beam I. The 3 2 S1/2 F = 2 ↔ 3 2 P3/2 F = 3 transition’. Z. Phys. A, 304, 103– 117. H ERTEL, I. V. and W. S TOLL: 1974a. ‘A crossed beam experiment for the inelastic scattering slow electrons by excited sodium atoms’. J. Phys. B, At. Mol. Phys., 7, 583–592. H ERTEL, I. V. and W. S TOLL: 1974b. ‘Principles and theoretical interpretation of electronscattering by laser-excited atoms’. J. Phys. B, At. Mol. Phys., 7, 570–582. H ERTEL, I. V. and W. S TOLL: 1978. ‘Collision experiments with laser excited atoms in crossed beams’. In: ‘Adv. Atom. Mol. Phys.’, vol. 13, 113–228. New York: Academic Press. K ASTLER, A.: 1966. ‘The N OBEL prize in physics: for the discovery and development of optical methods for studying Hertzian resonances in atoms’, Stockholm. http://nobelprize.org/nobel_ prizes/physics/laureates/1966/.

Index of Volume 2

Symbols 12, 6 potential, 149 A A BBE sine law, 679 Adiabatic representation, 478–480 A IRY diffraction pattern, 25 A IRY disc, 26 Alignment, 596, 604, 616, 619, 698–700 angle, 36, 44, 605, 610, 611 parameter, 622, 623, 698 Alkali halide potentials, 220–224 Ammonia maser, 1, 252 Amplification profile, 15 Amplified spontaneous emission, 11 Anharmonicity constant, 162 Anti-S TOKES lines, 334 Anti-symmetrization, 521 Antibonding orbital, 183 Ar2 TPES spectrum, 369 Arn clusters, 400 Atomic form factor inelastic, 527 AUGER electron, 533 AUTLER -T OWNES effect, 637–639 Autocorrelation function, 53–60 Avoided crossing, 220–224, 263 B Baseline in interferometry, 87, 91 Beam divergence, 21, 22 Beam parameter, 17–21 product, 26, 34, 35 Beam radius, 17–35, 650 Beam waist, 17–21 and coherence volume, 87 and lateral coherence radius, 91

Benzene D6h point group, 262 H ÜCKEL orbitals, 277–285 β (anisotropy) parameter, 356, 360, 551, 557 B ETHE formula, 528–530, 534, 537, 538, 546 ionization, 537 B ETHE integral, 526 B ETHE ridge, 546, 548, 549, 553 B ETHE surface, 545 Binary peak in (e, 2e) process, 553 Birefringence, 38, 40, 41 B IRGE -S PONER plot, 177 Black body radiation, 105 B OLTZMANN distribution, 155, 173, 335, 389, 591 statistics, 158 Bond order, 195 Bonding orbital, 183 B ORN approximation first order, elastic, 444–448 generalized oscillator strength (GOS), 530–534 inelastic collisions, 460, 461, 522, 525–530 integral inelastic cross sections, 534 B ORN phase shift, 447 B ORN series, 553 B ORN -O PPENHEIMER approximation, 139–151 collision processes, 458, 474 BPP, see Beam parameter product Branching ratio, 653 C C6 H + 4 absorption spectra, 328 C2 DFWN spectrum, 354 Carrier envelope, 46 phase, 46, 55, 188


703

704 Carrier frequency, 45 C ASIMIR operator, 541 C AUCHY-S CHWARZ inequality, 83 Cavity quantum electrodynamics, 127–130 Cavity ring down spectroscopy, 327, 328 CCD camera, 315, 508, 676, 701 Centre of mass system, 389, 395, 396 Ceratron, 675 CH4 , electronic states, 273 Channeltron, 674 Character tables, 257–262 C2v , 258, 259 Cs , 259 D2h , 260 D3h , 322 D6h , 278, 279 Oh , 260–262 Td , 273 Characteristic equation, 180 Charge exchange H+ 2 system, 184–190 H+ + H, 185–188 highly charged ions, 504, 505 over-the-barrier model, 501–503 Chemical shift, 358, 360 Chemical-potential, 106 Classical rainbow, 403 Classical trajectory, 402 Classical turning point, 406 Close-coupling convergent calculations (CCC), 522, 524, 529, 538, 539, 547, 548, 551, 553 theory, 460, 461, 516–519, 521, 523 Clusters mass selection, 400–402 supersonic molecular beams, 321 CO nuclear spin statistics, 156, 157 CO2 infrared spectrum, 245 laser, 245 normal modes, 244, 245 Coherence, 72–100 and incoherence, 580 angle, 90 area, 91 length, 76, 78–82 spatial, 86–91 time, 43, 76, 77, 79, 82, 84, 598 volume, 91 Coherent anti-S TOKES R AMAN spectroscopy (CARS), 351 Coherent S TOKES R AMAN spectroscopy (CSRS), 351

Index of Volume 2 Collision channels, 474 Collision frame, 421, 422, 473 Collision process elastic, 383–451 highly charged ions, 499–506 and ultrafast dynamics, 506 inelastic, 453–513 introduction, 383–393 kinematics, 396–400 COLTRIMS, 414, 501, 504, 685 Complex beam parameter Gaussian beams, 19 Confocal parameter in Gaussian beam, 21, 30 Conical intersection, 263, 312, 323, 332, 507 Conical skimmer, 321 Convergent close-coupling, 460, 464 Correlation diagram, 192–194, 217 Correlation function, 52–60 1st order, 72, 74 higher order, 55–58 interferometric measurement, 54–58 examples, see also Convolution C OULOMB integral, 182 Coupling elements non-adiabatic, 486, 487 Cross section differential, 393–402 integral, 394 momentum transfer, 395 total, 385–387 Cusp in the e + He excitation, 464 Cytosine photoelectron spectroscopy, 360 D D E B ROGLIE wavelength, 402 Decay of coherence (T2 ), 641 Decay of excitation (T1 ), 641 Degenerate four wave mixing (DFWM), 351 Degree of coherence, 601, 602 1st order, 72 2nd order, 84 N th order, 82 Degree of polarization, 601, 602 coherence, 585–588 linear, 43 Degree of temporal coherence 2nd order, 82 Delay line, 52 Density functional theory, 211 Density matrix, 573–624 1 P state, 602–611

Index of Volume 2 optical excitation, 609–623 transformation, 585 Density of states photons, 105 Density operator, 581, 582 Depletion spectroscopy, 321 Detailed balance, 387–389 Diabatic representation, 480–483 Diatomic molecules, 135–228, see also Molecules, diatomic, 229 Dielectronic recombination, 563, 564 Differential cross section, 393–402 Diffraction F RAUNHOFER, 23–26, 89, 413, 414 F RESNEL factor, 25 F RESNEL number, 8, 20, 25 H UYGEN -F RESNEL principle, 23 Dipole moment of a diatomic molecule, 167 Dipole transition, 166–178 matrix element, 305, 480 diatomic molecule, 167 polyatomic molecules, 242 operator, 119, 612 Direct excitation process, 482 Directional intensity, 680 Dissociative ionization, 363, 366, 558 Distorted wave approximation, 547, 553 Double resonance spectroscopy, 295, 330 Dressed states, 629, 630 coupled systems, 633–635 three, 659 D UNHAM coefficients, 165, 166 DYSON orbital, 559 E e− rare gases integral elastic cross section, 392 (e, 2e) process, 534 Effective potential, 152, 163, 406, 413, 453–455 Effective range expansion, 430 Eikonal approximation, 425 Elastic scattering, 383–451 classical theory, 402–418 Electric field and photon number, 116, 117 Electron and ion optics, 673–685 Electron configuration, 141, 195, 205, 217, 226, 259, 261, 263, 264, 267 Electron impact excitation GOS (for e− Na), 533, 534 of He, 461–463 fine details, 464

705 of Hg, 466–468 of molecules, 468, 469 of rare gases, 465, 466 Electron impact ionization, 534–562 at threshold, 540–544 double-differential cross section, 544–549 GOSD, 545 integral cross section, 537, 538 L OTZ formula, 537 single-differential cross section, 539, 540 triple-differential cross section, 549–558 WANNIER threshold law, 542–544 Electron jump in ion molecule reactions, 221 Electron momentum spectroscopy, EMS, 558–561 Electron photo-detachment, 364 Electron scattering theory, 515–525 Electronic spectra of molecules, 305–317 classical spectroscopy, 314–317 laser induced fluorescence (LIF), 317–320 laser spectroscopy biomolecules, 328–333 REMPI laser spectroscopy, 320–327 rotational transitions, 312, 313 selection rules, 309–311 Electronic states NH3 , 274, 275 of conjugated organic molecules, 277–285 triatomic molecules, 266–277 Ellipsoid of inertia, 232 Ellipticity angle, 36, 44, 597 Emission in a narrow band laser field, 635–639 Energy defect charge exchange, 501 Energy imaging, see Velocity map imaging Energy selector hemispherical, 680–683 magnetic bottle, 683–685 time of flight method, 683–685 Entanglement, 575, 608 Ergodicity, 75, 598 ESCA, 355–383 Ethylenfluoroacetat, 361, 362 Exchange amplitude, 519 cross section, 519 integral, 182 symmetry, 156 electrons, 203 Excitation continuous, with relaxation, 648, 649 continuous, without relaxation, 647, 648

706 narrow band vs. broad band, 645, 646 with short laser pulses, 649–652 Excitation function, 459, 460–472 F FABRY-P ÉROT resonator, 4–6, 50 FANO lineshape, 441 FANO -M ACEK theory, 611–623 Fast and slow axis, 38 Femtochemistry, 507 Femtosecond spectroscopy, 224 F ERMI resonance, 244 F ESHBACH resonance, 437 Field envelope phase of, see Carrier envelope phase Field quantization and transitions, 110–131 Fine structure in H UND’s case (a), 200 Fluence, 49 FORT trap, 626 F ORTRAT diagram, 314 F OURIER transform limited pulses, 46, 47 spectroscopy, 170, 293 F OURIER transform spectroscopy, 298–302, 343 of H2 O in the visible, 247 Fragment spectroscopy KETOF, 364 MATI, 364 F RANCK -C ONDON factor, 305–312 principle, 306–309 F RANCK -H ERTZ experiment, 467 F RAUNHOFER diffraction, see Diffraction Frequency comb, 49–52 F RESNEL rhomb, 40 Fringe spatial frequency stellar interferometer, 91 Frozen-core approximation, 560 FTIR, see F OURIER transform spectroscopy G Gain narrowing, 11 Gas kinetic cross section, 389 G AUSS radius, 22 Gaussian beam, 17–35 beam waist, 19–32 complex beam parameter, 19 divergence angle, 22 intensity and power, 21–23 nonlinear processes in, 61–67 radius of curvature of the wave front, 19, 21, 30, 31

Index of Volume 2 R AYLEIGH length, 19 Gaussian profile, 20, 59, 83, 644, 651 Generalized oscillator strength density (GOSD), 545–549 (GOS), 530–534 Genetic algorithm, 332 G LAUBER approximation, 553 G LAUBER states, 114–117, 596 Glory oscillations, 391, 412 Glow bar, 297 G OUY phase, 20 H H + He, integral elastic cross section, 390 H2 MO ansatz, 206–210 nuclear spin statistics, 156, 157 ortho and para hydrogen, 156 potentials, 208, 209 reflection symmetry, 203, 204 valence bond theory, 210, 211 H+ 2 MOs, 181–184 H2 O absorption spectrum, 269 C2v group, 258 electronic states, 266–270 orbitals (EMS), 561 photoelectron spectroscopy, 358–360 rotational levels, 237, 238 structure, 237 vibrational spectrum, 245–247 H∗2 , predissociation, 207 Half wave plate, 40 H AMILTON operator diatomic molecules, 137, 138 heavy particle collision, 396 relative motion, 397 Hanbury B ROWN -T WISS, 72 experiment, 84–86 Harmonic oscillator, 143–145 He-He potential, 149 Helicity basis, 36, 37, 42, 596, 597, 602 polarization vectors, 101 H ELMHOLTZ -L AGRANGE relation, 679 Hemispherical capacitor, 680–683 H ERMITE functions, 144 H ERMITE polynomials, 144 Hessian matrix, 240 Hindered pseudorotation, 324 Hole burning, 14–16 with IR and UV, 330 HOMO, 195 HOOO IR action spectrum, 302

Index of Volume 2 microwave spectrum, 295 H ÜCKEL method, 279–285 H UND’s coupling cases, 199–201 Hybrid orbitals, 219 double bonding, 275, 276 LiH, 219 σ bonding, 273, 274 sp 3 orbitals, 270–272 triple bonding, 276, 277 Hyperspherical coordinates, 540–542 I I2 studied by LIF, 318, 319 Imaging methods, 357, 365, 367, 369, 370 Impact parameter, 402, 404, 405, 408–412, 421, 422 Incoherence by collisional excitation, 606–609 Index of refraction particle beams, 678–680 Inertia tensor, 232 Infrared spectroscopy, 296–305 action spectroscopy (IAS), 302–304 Integral cross section, 408, 409 Intensity correlation function, 626 Intensity interferometry, 98 Interaction experiment state selective, 590–596 Intercombination lines, 309 Interference experiment, 78, 79, 87 spatial, 87 YOUNG’s, 78 Interferometer M ACH -Z EHNDER, 52 M ICHELSON, 52, 59, 78 stellar, M ICHELSON, 91–95 Internal conversion, 312 Inversion frequency, 250 Inversion symmetry, 183, 202, 203 Inversion vibration in NH3 , 247–252 p toluidine, 294 Ion imaging, 66, 67 Ion velocity map, see Velocity map imaging Irreducible representation density matrix, 614, 687, 693 dipole operator, 311 of point groups, 257 J JABLONSKY diagram, 335 JAHN -T ELLER effect, 237, 262–265, 321–323 theorem, 262 vibronic coupling, 265

707 J EFFREYS -B ORN phase shift, 426, 448 JWKB phase shift, 426, 427, 494 K K type doubling, 237 K-matrix, 475, 521, 522 L Laboratory frame, 396 Lambda-doubling, 205, 206 Lambda-half plate, 40 Lambda-quarter plate, 38–40 L AMBERT-B EER law, 386 L ANDAU -Z ENER formula, 489–492 L ANGEVIN cross section, 454–457 Laser, 1–17 amplifier medium, 9–11 basic principle, 3, 4 diffraction losses, 8 history, 1, 2 longitudinal modes, 5, 16 population inversion, 14–16 rate equations, 12–14 stability diagram, 7 threshold condition, 11, 12 transverse modes, 6–8 Laser beam diameter, 22 M2 factor, 34, 35 profile measurement, 34 Laser pulse, 45–48 frequency spectrum, 45–48 Gaussian temporal profile, 46 highest intensities, 3 measurement of ultrashort, 52–60 mode coupled, 50 spatial and temporal profile, 49 time dependence of sech2 , 46 ultrafast, 59 Laser spectroscopy, 317–334 LCAO, 179, 180 L ENNARD -J ONES potential, 149 L EVINSON theorem, 429, 432 Li3 high resolution spectroscopy, 321–327 potential surface, 323 Lifetime excited molecular states, 306, 307, 312 H2 O, 270 internal conversion, 656 natural, 631, 640, 644 photons in a resonator, 4–6, 8, 12, 14, 327, 328 rate equations, 653

708 resonance scattering, 523 RYDBERG state, 563, 564 Light beam, 17 LiH potentials, 218–220 Line broadening saturation, 643–645 Linewidth homogeneous and inhomogeneous, 14–16 natural, 9, 127–130 L IOUVILLE equation, 639, 640 L IPPMANN -S CHWINGER equation, 418–420 Lone pair (of electrons), 223, 267 ammonia, 274 Longitudinal coherence, 80 L ORENTZ profile, 4, 9–11, 76, 77, 81, 91, 107, 127, 300, 301, 643, 645 resonance scattering, 439 L OTZ formula electron impact ionization, 537, 538 LUMO, 195 M Magic angle, 357 Main axis of symmetry, 254 MALDI, 330 M ALUS’s law, 44 Maser microwave amplification by stimulated emission of radiation, 1 NH3 , 252 M ASSEY criterium, 457, 459, 479, 481, 488, 491 modifed, 461, 491 M ASSEY parameter, see M ASSEY criterium Matrix isolation spectroscopy, 333 Mean free path length, 386 Measurement state analyzer, 588–590 state selector, 588–590 theory of, 588–596 Merged-beams experiment, 564, 565 M ICHELSON interferometer, 298 Micro reversibility, 388 Microchannel plate, 675 Microwave spectroscopy, 292–296 Mode density, 105 Mode locking, 50 passive, 328 Mode synchronization, 50 Modes, see Laser Modes of the radiation field, 103 Molecular beam seeded, 321 Molecular orbitals

Index of Volume 2 diatomic, heteronuclear, 215–218 diatomic, homonuclear, 179–197 Molecular potential, 140 Molecular spectroscopy, 289–381 Molecules diatomic, 135–229 electron spin, 197, 198 electronic energy, 138 equilibrium distance, 136 heteronuclear, 215–226 rotational energy, 139 total angular momentum states, 197–214 total energy diagram, 291 vibration, 160–163 vibrational and rotational constants, 154 polyatomic, 231–288 RYDBERG states, 224 valence states, 224 M OLLOW triplet, 635–637 Momentum conservation, 554, 558 Momentum imaging methods, see Velocity map imaging M ORSE potential, 145–147, 163 MOs, 179, 180, see Molecular orbitals MOTRIMS, 508, 685 Multi-mode states, 117, 118 Multi-photon ionization, 67 Multiplicity, 197, 205, 214 Multipole moment, 690–692, 698–700 Multipole tensor operator, 690–692 N N2 nuclear spin statistics for bosons, 347 potentials, 211, 212 R AMAN spectrum, 343, 344 N2 + e− shape resonance, 524 Na atom hyperfine transition, 695–697 Na+ + Na(3p), inelastic and super-elastic processes, 492–496 Na + Hg integral elastic cross section, 391 Na+ 2 potentials, 493 Na3 REMPI spectroscopy, 321 NaCl potentials, 221, 222 NaI potentials, 222–224 Natural lifetime, 127 N EWTON diagram, 397, 398, 400, 401 NH3 , umbrella mode, 248–252 (NH3 )n , FEICO spectra, 371 NIST data bank, 147, 162, 168–170 NO potentials, 224–226

Index of Volume 2 N OBEL prize in physics F RANCK and H ERTZ (1925), 467 Chandrasekhara V. R AMAN (1930), 334 Isidor I. R ABI (1944), 630 Robert H OFSTÄDTER (1961), 385 B LOEMBERGEN, S HAWLOW, S IEGBAHN (1981), 2, 317, 355 R AMSEY, D EHMELT, PAUL (1989), 695 G LAUBER, H ALL, H ÄNSCH (2005), 49, 52, 72 H AROCHE and W INELAND (2012), 128 N OBEL prize in chemistry Robert S. M ULLIKAN (1966), 258 Gerhard H ERZBERG (1971), 314 H ERSCHBACH, L EE, P OLANYI (1986), 221, 507 C URL, K ROTO, S MALLEY (1996), 242 Ahmed H. Z EWAIL (1999), 224, 507 F ENN, TANAKA, W ÜTHRICH (2002), 330 Non-adiabatic coupling, 477 Non-crossing rule, 194 Nonlinear spectroscopy, 348–355 basics, 349–353 BOXCARS, 353 four wave mixing processes, 352 Normal modes, 239–243 asymmetric stretch, 243 bending vibrations, 244 symmetric stretch, 243 transitions between, 242, 243 triatomic molecule, 245–247 Nuclear spin statistics, 157, 343–348 population of rotational levels, 155–157 Nuclear wave function, 142, 143, 151–161 Number operator, 111 O O2 H ERZBERG bands, 315, 316 nuclear statistics (bosons), 348 paramagnetism, 214 potentials, 211–213 R AMAN spectrum, 344, 345 reflection symmetry, 204, 205 O5+ + e− dielectronic recombination, 563 Oblate, 234–236, 699 Odd and even molecular orbitals, 183 Optical B LOCH equations, 625–665 and short pulse spectroscopy, 649–657 Optical multi channel analyzers, 316 Optical pumping, 626, 695–702 with two frequencies, 700–702 Optical theorem, 425

709 Optical-optical double resonance in Li3 , 325 Orientation, 616, 619, 698–700 Orientation parameter, 698 Oscillations glory, 391 rainbow, 411 shadow scattering, 412–415 S TÜCKELBERG, 494, 496–498 symmetry, 415–417 Oscillator strength density, 545 Overlap integral, 181 P P toluidine, microwave spectrum, 294 Partial wave analysis, 391, 392, 418, 427, 428, 433, 434, 475, 476, 484, 516, 517, 523, 592 e− − He and e− − Ne, 431, 432 Partial wave expansion, 422–425, see Partial wave analysis Partial waves, 423 Partition function, 156, 173 PAUL trap, 333 P ENNING trap, 333 P ERCIVAL -S EATON hypothesis, 617 Periodic boundary conditions, 103 Periodic system diatomic molecules, 194–197 Phase matching, 352 Phase shift scattering, 423 Phosphorescence, 311 Photo-dissociation of H+ 2 , 188–190 Photo-fragment spectroscopy, 333 Photoelectron spectroscopy (PES), 355–372 anions, clusters, 364–366 basics, 355–358 PEPICO, TPEPICO, 366–371 PFI, 363 TPES, 363 ZEKE, 363 Photoionization anisotropy parameter, 356, 360 magic angle, 357 Photomultiplier, 674 Photon introduction, 100–102 modes of the radiation field, 102–105 number per mode, 106–108 photon states, 100–108, 110 Photon annihilation operator, 113

710 Photon bunching, 84 Photon creation operator, 113 Photon number states, 110–114 Photon spin, 101 π pulse, 648 P LANCK’s radiation law, 106 Plane wave impulse approximation, 559 PN emission spectrum, 314 P OCKELS cell, 41 P OISSON distribution, 116 Polarizability, 337 Polarization, 35–45, 611 analyzer for linear, 44 circular, 38, 40, 44, 101, 155, 596, 597, 609 degree of, 42, 43, 356, 580 density matrix for, 596–602 elliptical, 36 field induced, 349 incomplete, 41–44 lambda-half plate, 40 lambda-quarter plate, 38–40 linear, 36, 38, 102, 154, 155, 189, 356, 495, 609, 695 linear, elliptic, circular, 37 measuring the degree of, 43, 44 nonlinear, 350 rotating the plane of linearly polarized light, 40 selection rule in electron impact excitation, 532 S TOKES parameter, 600 S TOKES vector, 43, 600 time dependence of intensity by, 36–38 Polyatomic molecules vibration, 239–253 Population inversion, 9, 14, 16, 252 Potential anharmonic, 147 diatomic molecules, 141 hypersurface, see Potential surface L ENNARD -J ONES, 149 surface, 140, 141, 262, 263, 265, 269, 506–509 H atom as three body problem|(, 541, 542 ‘Mexican hat’, 323 VAN DER WAALS , 141, 148–150 Potential hypersurface, see Potential surface Predissociation of H∗2 , 207 Principle moments of inertia, 232 Prolate, 234–236, 699 Pseudo-states, 522 Pseudopotential, 437 Pseudorotation, 321–327

Index of Volume 2 Pulse train, 50 Pure state, 577 Pyrazine ZEKE and MATI spectra, 365 Q Q factor of a resonator, 4 QED cavity, 127–130 Quality factor of a resonator, 4 Quantum beats, 617 Quantum optics, 72–134, 626 Quantum system in electromagnetic field, 639–642 temporal evolution, 639 Quarter wave plate, 38–40 Quasi-monochromatic light, 43, 45, 75–77 R R-matrix theory, 443, 464, 472, 522, 529 R ABI frequency, 630, 631 non-resonant, 633 resonant, 631 R ABI oscillation, 627, 648 Radial coupling collision induced transitions, 480 Radiationless transitions, 311, 312 Radiative corrections, 128 Radiative recombination, 563 Rainbow heavy particle collisions, 409–417 optical, 403–405 rapid oscillations, 411 supernumerary, 410 R AMAN active transitions, 338, 341 R AMAN scattering differential cross section, 341 graph, 339 R AMAN spectroscopy, 334–348 classic interpretation, 337, 338 experimental aspects, 342, 343 principle, 334–337 quantum mechanical theory, 338–342 R AMSAUER minimum, 391, 431 Rate constant, 386 Rate equations, 386, 646, 647, 653–657 Ray tracing, 26 Ray transfer matrix, 26–29 R AYLEIGH criterium resolution of optical instruments, 89 R AYLEIGH length, 19, 29, 30, 32, 63, 65 R AYLEIGH line, 334 Rb atom, hyperfine transition, 626 Reaction coordinate, 508, 509

Index of Volume 2 Reaction microscope, 414, 508, 685 Reactions absorbing sphere, 455 without threshold, 453, 454 Recoil peak in (e, 2e) process, 553 Recombination, 563–566 Reduced cross section, 407, 408 Reduced scattering angle, 407, 408 Reflection symmetry, 203 Refraction of particle beams, 678 Relative velocity, 187, 386, 387, 390, 397, 398, 400, 409, 509 Resolving power FABRY-P ÉROT interferometer, 5 R AYLEIGH criterium, 26 Resonances, 436–444 autoionization, 436 electron scattering by molecules, 523–525 He− , 441–443 F ESHBACH, 412 formalism, 438–443 in electron scattering H− 2 , 207 N− 2 , 211 O− 2 , 214 orbiting, 412 predissociation, 412, 436 shape, 412 types, 436, 437 Resonant capture, see Resonances, orbiting Resonator mode, 16 Resonator Q factor, see FABRY-P ÉROT resonator Resonator quality factor, see FABRY-P ÉROT resonator Resonator turnaround time, 6 Richtstrahlwert, 680 Room temperature, 148 Rotating wave approximation, 123, 631–633 Rotational constant, 155 diatomic molecules, 153 Rotational coupling collision induced transitions, 480 Rotational spectrum CO, 168 Rotational temperature, 155 Rotational transitions, 167–170 Rotor, diatomic non-rigid, 163–165 rigid, 152–157, 160–162, 200 Rotor, polyatomic, 231

711 asymmetric (non-rigid), 295, 303 asymmetric (rigid), 236–239 spherical (rigid), 234 symmetric top (rigid), 234–236 Ruby laser, 2 RUTHERFORD scattering, 407, 446, 447, 528, 529, 539, 559 RYDBERG states in molecules, 224 RYDBERG -K LEIN -R EES method, 177, 178 S S-matrix, 434, 475, 487, 494, 521, 522 S-triazine, R AMAN spectrum, 345 Saturation broadening, see Line width Saturation intensity, 643 Scattering, elastic, 383–451 Scattering amplitude, 410, 418–420, 424, 426, 433, 434, 488, 494, 576, 592, 606, 669 direct, 519 first B ORN approximation, 525–527 inelastic, 475, 476, 517, 518 semiclassical, inelastic, 487, 488 Scattering cross section differential beam-gas experiment, 394 crossed beam experiment, 395 elastic, 385 elastic, integral, 389–392 inelastic, 385 ionizing, 385 reactive, 385 total absorption experiment, 386 Scattering kinematics, 396–400 Scattering length, 429–431 Scattering matrix, 432–435 Scattering phase, 418 Scattering theory classical, 405–409 multi channel problem, 472–484 phase shifts, 428–435 quantum mechanical, 418–436 semiclassical, elastic, 425–428 semiclassical, inelastic, 487–489 semiclassical approximation, 484–499 S CHRÖDINGER equation molecules, 137 sech2 function, 47 Secondary electron multiplier, 673–678 Seeded molecular beam, 321

712 SF6 Oh group, 261 Shape resonance, 437 Short pulse generation, 49–52 σ ± light, see Polarization, circular Single atom in a MOT trap, 627 Single electron capture, 501 Slowly varying envelope approximation, 18, 42 S OLEIL -BABINET compensator, 40 sp 2 orbital double bonding, 275, 276 Spatial coherence degree of, 89 Spatial filter, 33 Specific heat capacity, 158, 160 Spectrum electromagnetic radiation for molecular spectroscopy, 290–292 Spontaneous line broadening, see Line broadening, natural Stability diagram laser oscillation, 7 Standard deviation P OISSON distribution, 116 S TARK effect, 171, 172 State multipole, 614, 687–694 State of a quantum system coherent and incoherent, 579 pure and mixed, 575–581 State selection, 577 Stationary phase, 426–428 Statistical tensor, 687–694 operator, 688 Statistics B OLTZMANN, 389 exponential distribution, 76 of coincidence methods, 560 of measurement, 368 quantum, 72 Stellar interferometer Hanbury B ROWN -T WISS, 95–98 Stimulated emission pumping, 658 STIRAP, 657–665 energy splitting and evolution of states, 659, 660 experiments, 661–665 three level system, two laser fields, 657–659 S TOKES lines, 334 S TOKES parameters, 41, 42, 596, 598–600, 602 experimental determination, 600, 601 S TOKES shift, 308 S TOKES vector, 42, 600

Index of Volume 2 Sudden approximation, 560 Surface hopping, 312, 506–509 SVE approximation, see Slowly varying envelope approximation Symmetry character tables, see Character tables cylinder, 190 g, u, 191 molecular physics, 253–265 point groups, 254–257 reflection, 201–205 T T-matrix, 420, 433, 434, 475, 476, 487, 517, 519, 521, 522, 592, 594 inelastic scattering, 517 Telescope systems K EPLER and G ALILEI, 32, 33 Temporal coherence, 80 Tensor operator statistical, 688 Tetrahedral angle, 272 Three body problem, 535, 541, 542, 555, 557 Threshold amplification, 14 Threshold inversion, 14, 15 Threshold laws, 470, 472 Tight binding method, 285 Time-bandwidth product, 48 Toroidal energy analyzer, 556 Transition field quantization, 110–131 L ANDAU -Z ENER, 489–492 non-adiabatic, 477, 478 perpendicular, 245 Triatomic molecules linear, 243–245 nonlinear, 245–247 Tunnelling in predissociation, 207 level splitting in NH3 , 247–252 Two level system, 119, 629–635 excitation with CW light, 642–649 U Ultrafast laser pulses measurement, interferometric, 60 Ultrashort light pulse, see also Correlation function Unimolecular dissociation, 321 V Valence states in molecules, 224 Van C ITTERT-Z ERNICKE theorem, 89 VAN DER WAALS

Index of Volume 2 contact distance, 149 equation, 148 potential, 148–150, 409 radius, 150 Variational method, 179, 180 R ITZ, 208 Velocity of electrons and atomic nuclei, 136 Velocity map imaging (VMI), 66, 67, 188, 357, 384, 507, 508, 676–685 Vertical binding energy, 358 Vertical ionization potential, 358 Vibration-rotation spectra, 174–177 CO, 174 PQR branches, 175 Vibrational quantum number, 152 Vibrational transitions diatomic molecules, 172, 173 in polyatomic molecules, 242

713 polyatomic molecules, 253 Vibronic coupling, 262, 265, 321–327 Visibility, 80 VMI, see Velocity map imaging W WANNIER ridge, 542, 556 WANNIER threshold law, 470, 542–544 Wave equation, 17 general case, 350 Wave-packets, 45–52, 82 Whole burning spectroscopy, 330 W IGNER -E CKART theorem, 615, 671, 687, 688, 691, 699 WKB phase shift, 426, 448 Z Zero point energy, 112, 145 C60 , 242

Index of Volume 1

Symbols 1, identity matrix, 99 3D F OURIER transform, 655–657 3j symbols, 300, 564–568 orthogonality, 565, 566 special cases, 567, 568 symmetries, 565, 566 6j symbols, 311, 568–572 orthogonality, 569, 570 special cases, 571, 572 symmetries, 569, 570 9j symbols, 572 A Above-barrier ionization, 436, 437 Above-threshold ionization (ATI), 266, 441 of Ar, 441 of C60 , 441, 442 Absorption, 193–196 coefficient, 18, 241, 422 molar, 18 cross section, 18, 240–242 broad band light, 242 monochromatic light, 241 definitions, units, 180 edge, 521, 525 E INSTEIN coefficient, 180 inner shells, 520–524 introduction, 17, 18, 178–180 probability, 190–192 rate, 179, 191 X-ray, 520–525, 530 C OMPTON scattering, 526 pair production, 526 photoionization, 526 T HOMSON scattering, 526 Addition theorem for Ckq , 579

Adjoint operator, 97, 576 ADK theory, 437 Air mass coefficient (AM), 36 A IRY function, 280 Al atom, G ROTRIAN diagram, 518 Alignment, 173, 588, 620 angle, 173 parameter, 620 Alkali atoms, 144, 146, 151, 165 comparison with H atom, 146 overview about term energies, 146 quantum defect, 146, 147 spectroscopy, 145, 146 Alkaline earth metals fine structure, 361, 362 G ROTRIAN diagram, 371 Angular dispersion, 275 Angular momentum, 5, 65, 66, 72, 107–117 algebra, 575–592 commutation rules, 559 conservation, 81 E1-transitions, 196 coupling, 297, 298 definition, 109, 559–573 eigenstates, 561 intrinsic, 78 matrix elements, 561 operators, 109, 111 scalar product, 298, 299, 585, 586 projection theorem, 578 real and helicity basis, 560 Anomalous magnetic moment of the electron, 331 Anomalous Z EEMAN effect, 380 Anti-symmetrization, 346, 357, 358, 388 Areas of physics, 2 Atom model, 63


715

716 Atom radius, 121 Atomic beam, 73–75, 282, 317 Atomic core, 148 Atomic form factor, 41, 430, 431 Atomic hydrogen eigenfunctions, 123–126 Atomic orbitals, 118, 122, 125, 126 Atomic radius, 29, 68, 143, 144 Atomic size, 64 Atomic units, 67, 68, 118, 119 Atoms in a magnetic field, see Z EEMAN effect Atoms in an electric field, see S TARK effect Atoms in intense laser fields, 432, 442 Attometer, 6 Attosecond, 8 Attosecond laser pulses, 440 Aufbau principle, 138, 139 AUGER electron, 522 Autocorrelation function, 623, 626, 628, 646, 648 Autoionization, 260, 366–370 Avoided crossing, 392–394 S TARK effect in RYDBERG states, 411 B Band structure, 28 electronic, in a solid, 132 Barions, 48, 49 Barn, 524 Basis spin function, 115 Basis states: real and complex, 595–599 Basis vectors, 95 G ROTRIAN diagram, 372 BESSY, 42, 532 β (anisotropy) parameter, 260, 263, 264, 269, 529 B EUTLER -FANO profile, 368 Black body radiation, 31–34 Blazed grating, blaze angle, 277 B LOCH wave function, 132 B OHR, 2, 64 magneton, 378, 449 model of the atom, 64–70 comparison with quantum mechanics, 127, 128 limits, 69, 70 orbital radius, 67, 124 B OLTZMANN constant, 19, 33 distribution, 20, 24, 25, 235, 624, 627 factor, 23 statistics, 21–23 B ORN approximation first, applied to photoionization, 257 first, for X-ray photoionization, 524

Index of Volume 1 B OSE -E INSTEIN condensate, 8 B OSE -E INSTEIN statistics, 23 Boson, 20, 22, 139 Boundary condition, 104 B RAGG reflection, 40, 41 B REIT-R ABI formula, 393, 468 Z EEMAN effect in hyperfine structure, 467–471 Bremsstrahlung (X-ray), 530 B RILLOUIN zone, 41, 133 C C atom, 22 G ROTRIAN diagram, 514 Candela, 38 CCD camera, 59, 269, 277 Central field approximation, 496 Centre of mass system, 119 Centrifugal potential, 117, 120, 123 Centrosymmetric problems, 108 Cesium fountain atomic clock, 320 Characteristic X-ray radiation, 530 Chemical shift NMR spectroscopy, 488 Chemical-potential, 23, 24, 105–107 Classical forced oscillator model for photon absorption cross section, 240 C LAUSIUS -M OSSOTI formula, 421 C LEBSCH -G ORDAN coefficients, 300, 564–568 orthogonality, 565, 566 symmetries, 565, 566 CODATA, 9, 321, 329, 330, 334–336, see also Fundamental physical constants data bank, 551 Coherently excited states, 217–220 Collisional line broadening, 233, 234 Colour temperature, 39 Commutation rules, 114, 115, 634 Commutator, 96, 101 Complex spectra, 512–519 C OMPTON effect, 2, 28, 29 C OMPTON wavelength, 29 Configuration interaction (CI), 363–366, 508, 509 Continuum, 659–663 normalization, 659–661 of eigenvalues, 102 wave function, 262 Convolution, 623–629 Gaussian profile, 626, 627 Hyperbolic secant, 627, 628 L ORENTZ profile, 628, 654

Index of Volume 1 theorem for F OURIER transform, 645 VOIGT profile, 628, 629 C OOPER minimum, 260, 529 Coordinate rotation, 575, 605–611 exercise, 207 Coordinate system atomic vs. photon, 174 cartesic and polar, 108 electron vs. nucleus, 295 Core electron, 148, 151 Core potential, 149 alkali atoms, 165 Correlation function 1st order, 625 Correspondence principle, 127 C OULOMB gauge, 631 C OULOMB integral, 356, 359, 365, 506 C OULOMB law, 50, 59 C OULOMB potential, 119, 121, 123 H atom, 121 screened, 148, 149 C OULOMB wave, 262 Coupling jj , 301, 354, 360 LS, 301, 307, 354, 360, 361 LS vs. jj , 512–514 RUSSEL -S AUNDERS, 301, 354, 360, 361, 508 break down, 362 spin-orbit, see Spin-orbit Crystal lattice, 40 C URIE constant, 396 C URIE’s law, 396 Cyclotron frequency, 54, 332 electron, 79 Cylindrical capacitor, 52 D Damping constant, 229 DARWIN term, 305 D E B ROGLIE, 57, 89 wavelength, 57–59 Decay constant, 15 Decay rate, 15 Degeneracy, 21, 22 , 123 m, 111 , removal of, 137–146 m, removal of, 130, 131 Degree of coherence 1st order, 625 Degrees of freedom, 19 Density distribution, 125 Density functional theory, 510–512

717 Density of states, 21–24, 238, 240 particle in a box, 105, 106 Detailed balance principle of, 184 Detuning, 229 Diamagnetism, 396–398 Dielectric function, 57 Diffraction D EBYE-S CHERRER, 58, 59 experiment, 60 He scattering, 60 image, 42 LEED, 59 Dimensional analysis, 119, 192, 554, 556, 661 Dipol vector, 183 Dipole approximation, 189, 631–640 electric, 635 magnetic, 250–254 Dipole excitation, linear combination of states, 217–225 Dipole matrix element, 193 length approximation, 193, 636 velocity approximation, 193, 636 Dipole operator, 594 magnetic field, 386 multi-electron system, 264, 363 Dipole oscillator, classical, 182 Dipole radiation angular characteristic, 203–212 Dipole transition amplitude, 193 E1 transitions in the H atom, 201–203 in He, 362–365 length approximation, 636 matrix element, 190, 635 operator, 190, 193, 263 selection rules for E1 transitions, 196–203 selection rules for E1-transition, 202 velocity approximation, 636 D IRAC delta function, 644 D IRAC equation, 79, 93, 296, 303, 333 Direct product, 578 Dispersion, 421 anomalous, 422 close to resonance, 232, 233 normal, 420, 422 Dispersion relation, 131–134 matter waves, 94 Displaced terms alkaline earth term schemes, 372 C atom, 515 Ne atom, 518

718 D OPPLER free spectroscopy, see Spectroscopy, D OPPLER free D OPPLER broadening, 234, 236, 285 D OPPLER effect, 285, 317 classical, 14 quadratic, 13 relativistic, 13 D OPPLER narrowing, 283 D OPPLER profile, 287 Double slit experiment, YOUNG’s, 88, 89 Doubly excited states, 348, 365–367 Drehimpuls, 81 Dressed states, 418 D RUDE frequency, 57 Duality wave-particle, 2, 4 Duality, wave-particle, 58 E E1-transitions, 635 Echelle spectrometer, 277 Effective mass of an electron, 132 Effective nuclear charge, 159 Effective potential, 117 Eigenenergy, 91 Eigenfunction, 91, 99 nodes of the, 103 of momentum, 102 Eigenstate, 98, 99 Eigenstates of angular momentum, 109–113 Eigenvalue, 97 Eigenvalue equation, 97, 99 energy, 129 z , 110 of L of momentum, 102 Eigenvalue problem, 102 Eigenvector, 97, 99 E INSTEIN E = mc2 , 3, 10 photoelectric effect, 31 E INSTEIN A and B coefficients, 184, 185, 193, 194, 212, 215, 216, 228, 230, 241, 314, 489 E INSTEIN’s paradigm on speed of light, 425 E INSTEIN - DE -H AAS effect, 79, 81 Electric dipole (E1) transitions, 588 Electric dipole moment of the electron-nucleus pair, 189 Electric quadrupole (E2) transition, 250–254, 588 Electric quadrupole moment atomic nuclei, 449 Electromagnetic spectrum, 31 Electromagnetic waves, 170–176

Index of Volume 1 Electron, 49–51, 114 angular momentum, 50 classical electron radius, 50 C OMPTON wavelength, 29 elementary charge, 50, 51 g factor, 79 mass, 68 M ILLIKAN experiment, 50 orbital magnetic moment, 129 point like, 49 spin, 50, 70, 78, 112, 114, 128 Electron bunches, 532 Electron configuration, 138, 142, 144, 151, 302, 500, 508, 509 Electron diffraction D EBYE-S CHERRER, 58, 59 LEED, 59 Electron hole, 521 Electron magnetic moment, 50, 79, 81, 294, 387, 449 anomaly, 331, 336 Electron shell, 139, 140 Electron spin, 114–116 resonance spectroscopy (ESR), see Electron paramagnetic resonance (EPR) Electron storage ring, 53 synchrotron radiation, 531 Electronvolt, 51 Electrostatic potential, expansion, 614–616 Electroweak interaction, 44 Ellipticity angle, 172, 174 Emission, 193–196 inner shells, 520–524 spectrum, 130 Emittance, 532 Energy analyzer, 52, see also Energy selector Energy balance, 120 Energy conservation operator form, 91 relativistic, 11, 28 Energy levels fine structure splitting, 294 Energy packet, 27 Energy quantization, 26 Energy scales, 7 Energy terms H atom, 122 Energy zero H atom, 118 Entanglement, 354, 609, 610 EPR spectroscopy, 484–487 high B field, 487 X band, 486

Index of Volume 1 Equivalence of mass and energy, 10 Ergodicity, 626 Error function complementary, 629 complex, 629 E ULER angles, 605 Exchange boson, 44 integral, 356, 357, 359, 365 interaction, 343 spin orientation, 358, 360 operator, 351–353 Expectation value, 98, 99, 126 observable, 98 r k for the H atom, 126, 127 spin component, 115 the momentum in a 1D box, 103 Experiment of B ETH, 176 Extinction coefficient, 241, 421 molar, 18 F FABRY-P ÉROT interferometer, 279–281 finesse, 280 finesse coefficient, 280 FADEEVA function, 629 FANO lineshape(, 366 FANO lineshape, 369 Fast light, 422–427 F ERMI contact term, 456, 458, 459 F ERMI energy, 25, 106 F ERMI level, 107 F ERMI’s golden rule, 238 F ERMI -D IRAC statistics, 23, 106 Fermion, 20, 22, 44, 138 Ferromagnetism, 80, 358 F EYNMAN diagrams, 324–326 energy conservation, 326 for ge − 2, 335 L AMB shift, 326 neutron decay, 49 pair annihilation, 325 pair generation, 325 propagator, 326 self-energy of the electron, 325 vacuum polarization, 325 vertices, 326 Fine structure, 293–316 alkali atoms, 307, 309 alkaline earth metals, 362, 372 and electron spin, 293 BALMER Hα line, 288 D IRAC theory, 303 DARWIN term, 305

719 relativistic correction, 304, 305 spin-orbit term, 306 H atom, 293, 296, 297, 306, 307 He and He like ions, 360, 361, 362 interaction, see Spin-orbit interaction interval rule (L ANDÉ), 308, 361, 362, 372 Hg atom, 372 Na D doublet, 293 normal ordering of terms, 308 quantum defect, 308, 309 splitting, 293, 308 theory and experiment, 303–310 transitions branching ratios, 315 multiplet, 312, 315 transitions and selection rules, 310–316 Fine structure constant, 9, 69, 293, 296 electromagnetic coupling, 326 high precision measurement, 335 Finesse, see FABRY-P ÉROT interferometer Four vector (momentum), 11 F OURIER transform, 643–658 analysis, 54 exponential distribution, 653–655 Gaussian, 650, 651 inverse, 643 L ORENTZ, 653–655 rectangle, 652 sech, 651, 652 spectroscopy, 643 Free electron gas, 105–107 Free electron laser (FEL), 542, 543 Free spectral range, 279 FT-ICR, 54, 55 Fundamental interactions the four, 43–51 Fundamental physical constants, 67, 551, 553 G g − 2, see Electron magnetic moment, anomaly Galaxy, 6 Gas kinetic cross section, 20 Gaussian profile, 235 convolution, 626, 627 Generalized cross section for multi-photon processes, 245 Geonium atom, 333–335 gJ factor definition, 77 quantum mechanical derivation, 380, 381 vector model, 381, 382 Gratings, 274–279 G ROTRIAN diagram, 514–518, 520 Al atom, 518

720 alkaline earth atoms, 371 Be atom, 372 C atom, 514 H atom, 202 He atom, 343 Hg atom, 372 Li atom, 145 N atom, 515 Ne atom, 517 O atom, 516 Ground state H atom, 64 Group in periodic system of elements, 140 Group index, 424 Group velocity, 422–424 Gyromagnetic ratio, 71 H H atom, 69, 117–128, 296, 302, 305, 306, 316 1S–2S transition, 290, 291 BALMER series, 126 density plots, 124–126 energy levels, 122 expectation values of r k , 126, 127 fine structure, 287 fine structure transition, 310 H AMILTON operator, 117 in a magnetic field, 129 L AMB shift, 317 LYMAN series, 126 orbitals, 125 PASCHEN series, 126 radial wave function, 120, 121, 123, 124 spectrum, 65, 69, 126 wave functions (2D plot), 125 Hadrons, 49 H AMILTON operator, 91–94, 99, 101, 117, 121, 130 He atom, 345 magnetic fields, 129 H ANKEL transform spherical, 657 H ARTREE equations, 498–500 H ARTREE -F OCK, 503–510 equations, 506, 508 restricted, 504 unrestricted, 504 He atom, 22, 59, 341–375 0th order approximation, 346–348 diffraction, 59 electron exchange, 351–355 excited states, 351–360 G ROTRIAN diagram, 343

Index of Volume 1 ground state, 348 H AMILTON operator, 345 He like ions ground state, 349 H EISENBERG representation, 100 H EISENBERG uncertainty relation, 100, 101 Helicity, 196 Helicity basis, 171, 172, 174, 267, 560 angular momentum, 560 transition amplitudes, 198–200 Hemispherical capacitor, 52 Hermitian operator and conjugate, 97, 576 HFS, see Hyperfine structure Hg atom, 372–374 G ROTRIAN diagram, 372 High harmonic generation, 439, 440 plateau, 440 H ILBERT space, 100 History of physics, 2, 3 Hole burning, 284, 285 H UND’s rules, 357 H UYGENS -F RESNEL principle, 89 Hydrogen anion, 350 Hydrogen like ions, 68 Hydrogen maser, 459 H YLLERAAS wave function, 350 Hyperbolic secant convolution, 627, 628 Hyperfine structure, 287, 447–493 coupling constant, 452 coupling tensor, 456, 457, 480 deuterium, 454, 460 E1 multiplet transitions, 460 H atom, 453, 459 intervall rule, 481 isotope shift and electrostatic interaction, 471–482 L ANDÉ’s interval rule, 452 magnetic dipole and quadrupole, 481 magnetic dipole interaction, 452 mass effect, 473, 474 Na atom, 282, 287, 290, 454, 460 nuclear quadrupole moment, 477–481 quadupole interaction, 481 total angular momentum, 449 vector diagram, 451 volume shift, 475, 477 I Identity matrix, 1, 99 Independent particle model, 345, 355, 496, 497 Index of refraction, 57, 62, 400, 420–422 Induced transitions dipole approximation, 189, 190 probability, 215, 216

Index of Volume 1 Insertion device, 540 Intensity, 34, 632, 633 cycle averaged, 632 Intensity spectrum, 648, 649 Intercombination lines forbidden in He, 363 Interference, 89, 90 Interferometer, 278–281 FABRY-P ÉROT, 279–281 free spectral range, 279 opitcal path difference, 279 optical path difference, 278 resolving power, 278 Interval rule, see Fine structure Invariant mass, 10 Ion beam spectroscopy, see Spectroscopy Ion cyclotron resonance Spectrometer, 54 Ionization above-barrier, 436 non-sequential, double, 438 Ionization potential, 27, 75, 142, 146 of alkali atoms, 147 IR spectral range, 37 Irradiance, 34 Irreducible representation rotation group, 560, 575, 606 tensor operator, 560 J jj coupling, see Coupling, 513 K K shell, 139, 151 K ELDYSH parameter, 434, 436 Kinematic correction, 68, 119, 321, 323, 474 Kinetic gas theory, 18–20 K IRCHHOFF’s diffraction theory, 60, 89 K LEIN -G ORDON equation, 93 KOHN -S HAM equations, potential, orbitals, 510 KOOPMAN’s theorem, 509 L L shell, 139, 151 L AGUERRE polynomials, 121 L AMB dip, 286 L AMB shift, 316 1st order perturbation theory, 328 BALMER Hα, 316 experiment of L AMB and R ETHERFORD, 317, 318 highly charged ions, 322, 324 optical precision spectroscopy, 319, 322 theory, 326, 331

721 L AMBERT-B EER law, 18, 178, 524 L ANDÉ g factor, see gJ factor L ANDÉ’s interval rule fine structure, 308 hyperfine structure, 452 L ANGEVIN function, 396 L ANGMUIR -TAYLOR detector, 75, 317 L APLACE expansion, 613 L ARMOR frequency, 72, 77, 79, 331, 485 Laser based X-ray sources, 543, 544 Lattice plane, 40, 42, 59 L EGENDRE polynomial, 563 associated, 111 Leuchtelektron (valence electron), 144 Level splitting, 130 Li atom, 146 G ROTRIAN diagram, 145 Light quantum, 27 Light scattering, 427–432 coherent, 430 C OMPTON, 430, 431 from relativistic electrons, 431 incoherent, 431 M IE, 427 R AYLEIGH, 428, 429 T HOMSON, 429 Light storage, 281 Light year, 6 Limits of classical physics, 87 Line broadening, 227–238 by finite measuring time, 288 homogeneous, natural, 231, 232 inhomogeneous, 236 Lorentzian linewidth, 234 Line spectra, 2 Line strength, 212, 636, 637 Line triplet, 130 ‘normal’ Z EEMAN effect, 131 Liquid drop model for nuclear radius, 476 Long range potentials, 414 induced dipole – induced dipole, 417, 418 monopole – induced dipole, 416 monopole – monopole, 414 monopole – permanent dipole, 415 monopole – quadrupole, 415 permanent dipole – induced dipole, 416 permanent dipole – permanent dipole, 415 quadrupole – quadrupole, 416 L ORENTZ factor, 10 L ORENTZ force, 55, 295 L ORENTZ profile, 229 convolution, 628 numerical examples, 231, 232 LS coupling, see also Coupling, 512

722 LS interaction, see Spin-orbit interaction Lumen, 38 Luminous efficacy, 38 Luminous efficiency, 38 photopic, 37 Luminous flux, 38 Luminous intensity, 38 M Magic angle, see Photoionization Magic angle spinning (MAS), see NMR Magnet poles, 73 Magnetic dipole (M1) transition, 250–254, 484, 488, 589, 590 Magnetic field of the electron cloud, 453, 457 Magnetic moment, 251 and angular momentum, 70, 71 atomic nuclei, 447, 450 in a magnetic field, 71, 72, 294, 295 of the electron, 331 precessing in a magnetic field, 386, 388 Magnetic resonance spectroscopy, 482–491 Magnetic susceptibility, 395 diamagnetism, 397 paramagnetism, 396 Magnetization, 395 Magneton, 71 B OHR, 79, 129, 294 Magnetron frequency, 332 Main group periodic system of elements, 141 Mass absorption coefficient, 525 X-ray, 524 Mass correction relativistic, 54 Mass polarization, 474 Mass selection, 53 Mass spectrometer, 54–56 double focussing, 55 quadrupole, 55 time of flight, 55 Matrix eigenvalue equation, 164 Matrix element, 100, 116, 575, 592 angular momentum components, 586 operator, 97 reduction, 582–587 spherical harmonics LS-coupling, 583, 585 Matrix representation, 116 Matter wave, 87–94 plane, 58 M AXWELL’s equations, 92 M AXWELL -B OLTZMANN velocity distribution, 21, 72

Index of Volume 1 Mean free path length, 18 Mesons, 49 Metastable states of rare gases, 517 M ICHELSON interferometer, 278 Microscope resolution according to Abbé, 62 M IE scattering, 427 M ILLER indices, 41 Molar susceptibility, 395 Molecular beam, 72, 282 resonance spectroscopy, 482–484 M ØLLER -P LESSET perturbation theory, 498, 508 Momentum conservation, 429, 430 F EYMAN graphs, 326 relativistic, 11, 30 Momentum eigenfunction, 102 Momentum operator, 91, 92 M OSLEY diagram, 160,161 Na-like ions, 160 X-ray absorption edges, 523 Multi-electron atom, 344, 495–547 H AMILTON operator, 496–498 H ARTREE method, 500 self-consistent field method, 500, 501 with one valence electron, 144 Multi-electron photoionization, 524–530 Multi-photon ionization, 244, 265–269 angular distribution of electrons, 266–269 kinetic energy of the electrons, 265 saturation, 434–436 Multi-photon processes, 244–250 Multiple beam interference, 40 Multiplicity, 76, 300, 302, 357, 363 Multipole expansion, 613–622 Multipole moment, 613–622 Multipole tensor operator, 616–621 general, 619–621 N N atom, G ROTRIAN diagram, 515 Na atom, 149 electron density distribution, 150–152 radial electron density, 151 radial wave function, 149 Natural lifetime, 229 Natural linewidth, 227–229 Natural unit of energy, 67 Ne atom, G ROTRIAN diagram, 517 Neon shell, 151 Neutron, 48 Neutron diffraction, 59 NIST data bank, 50, 118, 140, 146, 147, 157, 159, 160, 166, 203, 259, 260, 309,

Index of Volume 1 319, 342, 347, 349, 351, 371, 373, 431, 514, 522, 525, 529, 551, 629 NMR spectroscopy, 487–491 apparatus, 488 CW spectrum of ethanol, 488 magic angle spinning (MAS), 490 occupation probability of levels, 489 N OBEL prize in chemistry Richard R. E RNST (1991), 490 KOHN and P OPLE (1998), 510 F ENN, TANAKA, W ÜTHRICH (2002), 490 N OBEL prize in physics Wilhelm C. RÖNTGEN (1901), 530 L ORENTZ and Z EEMAN (1902), 377 Joseph J. T HOMSON (1906), 50 Albert A. M ICHELSON (1907), 278 Max K. E. L. P LANCK (1918), 31 Johannes S TARK (1919), 399 Albert E INSTEIN (1921), 27, 31 Niels H. D. B OHR (1922), 64 C ORNELL, K ETTERLE, W IEMAN (1925), 25 Arthur H. C OMPTON (1927), 28 Louis DE B ROGLIE (1929), 57 Werner K. H EISENBERG (1932), 100 S CHRÖDINGER and D IRAC (1933), 90 Otto S TERN (1943), 70 Isidor I. R ABI (1944), 482 Wolfgang PAULI (1945), 22, 138 Max B ORN (1954), 88, 89 L AMB and K USCH (1955), 317 T OMONAGA, S CHWINGER, F EYNMAN (1965), 79, 324, 534 R AMSEY, D EHMELT, PAUL (1989), 55, 225, 288, 332, 482 G LAUBER, H ALL, H ÄNSCH (2005), 248, 319 M ATHER, S MOOT (2006), 6, 7 E NGLERT and H IGGS (2013), 46 N OBEL prize in physiology or medicine L AUTERBUR, M ANSFIELD (2003), 490 Non-crossing rule, 391, 394 Non-local potential, 507 Non-stationary problems dipole excitation (E1), 169–225 light matter interaction, 227–270 Non-stationary states, 186, 187 ‘normal’ Z EEMAN effect, 128–131, 382–386 Nuclear gN factor, 447, 449 Nuclear magnetic moment, 447, 449, 485, 487, 488 Nuclear magnetic resonance, see NMR Nuclear mass, 119 energy correction, 68, 69

723 Nuclear quadrupole moment, 447, 449, 616 oblate or prolate, 478 Nuclear radius, 29, 121 liquid drop model, 476 Nuclear spin, 449 eigenvalue equations, 449 Nuclear spin resonance, see NMR Nucleons, 48 O O atom, G ROTRIAN diagram, 516 Oblate, 477, 478, 619 nuclear shape, 478 Observable, 97–99, 118 commuting, 101 non-commuting, 101 simultaneous measurement, 100, 101 One electron cyclotron oscillator, 334 One particle problem, 117–134 One sided exponential distribution, 653 One-loop QED effects, 323, 324 Operator, 96, 97, 100 energy, 101 momentum, 101 position in space, 101 simultaneous measurement, 101 Optical path difference, 40 Orbital angular momentum, 71, 295, 299, 302 components, 110, 111 eigenfunctions, 109–113 square, 111, 112 Orbital energies, 509 Orders of magnitude, 5–9 energy scales, 7 length scales, 5 time scales, 7 Orthonormality relation, 96 Oscillator strength, 238, 239, 636–640 density, 256 sum rule, 239, 639, 640 P Pair production, 526 Paramagnetism, 394–396 Parity, 593, 594 conservation in E1 transitions, 202 multi-electron systems, 594–603 Parity violation, 249 Particle detection, 75 Particle diffraction, 58–61 C60 , 60, 61 He atoms, 59, 60 Particles and waves, 57, 64

724 Partition function, 21 PASCHEN -BACK effect, see Z EEMAN effect, high field PAUL trap, 332 PAULI spin matrices, 116 PAULI principle, 22, 138, 139, 351–355, 503–506 P ENNING trap, 332 Periodic system, 137–144, 168 table of elements, 140–144 Perturbation hierarchy with electric field, 402 Perturbation theory, 129 1st order, 162, 163 2nd order, 163, 164 alkali atoms, 165 degenerate states, 164, 165 stationary, 161–167 time dependent, 186–196 1st order, 190 Phase diagram, 88 Phase difference FPI interferometer, 280 Phase index, see Index of refraction Phase shift in QDT, 159 Phase space, 88 Phase velocity, 422–424 Photo-absorption cross section, 42 aluminum, 525 lead, 525 Photo-detachment, 255 angular distribution of electrons, 264 Photoelectric effect, 26–28 Photoelectron spectroscopy (PES), 28, 254, 268, 269 imaging spectrometers (EIS), 268 Photographic plate, 59, 73, 74 Photoionization, 254–269 angular dependence, 260, 261 anisotropy parameter, 260, 264 Ar atom, 527 B ORN approximation, 256–260 cross section, 255–258 energy dependence, 259, 527–530 magic angle, 261 theory and experiment, 261–264 with X-ray, 520, 524, 530 Photometry, 37–40 Photon, 4, 26–43, 62, 88–90, 92 angular momentum, 30, 175–198 elastic scattering, 527 flux, 179, 245 momentum, 29

Index of Volume 1 P LANCHEREL’s theorem, 644 P LANCK constant, 4, 33 P LANCK energy, 9 P LANCK length, 5 P LANCK time, 7 P LANCK’s radiation law, 31–34 E INSTEIN’s derivation, 185 Plane wave, 94 partial wave expansion, 661–663 Plasma frequency, 56, 57 Plasma oscillations, 56 Plasmon resonances, 57 Pointing vector, 632 Polar coordinates, 107–110 H atom, 117 Polarizability, 144, 150, 411–413 Polarization circular, 172, 173 dielectric, 411–413 induced, 412 orientation, 412 elliptical, 172–174 linear, 171, 173, 174 vector, 170–176 basis, 171–174 Polarization ellipse, 174 Polarization potential, 416 Ponderomotive potential, 432–434, 634, 635 Positron emission tomography (PET), 30 Potential box one dimensional, 103, 104 three dimensional, 104–107 Potential well model, 27 Power broadening, 230 Precession of angular momentum in a magnetic field, 72 Principle quantum number, 67 Probability amplitude, 87–90 dependence on time, 92 matter waves, 89, 90 photon, 88, 89 time dependent, 187 Probability distribution, 61, 89, 151 energy, 20 position, 123 Probability interpretation, 89 Product ansatz, 93, 345 Projection theorem for angular momenta, 578 Prolate, 477, 478, 619 nuclear shape, 478 Proton, 48 Proton radius, 7, 29, 324, 329

Index of Volume 1 Q Quadrupole coupling constant, 481 Quadrupole field, 55, 332 Quadrupole moment, 617–619 electric, 253 intrinsic, 479 spectroscopic, 479 Quadrupole tensor atomic nucleus, 478 electric, 253 Quantization, 2–5 Quantization of the electromagnetic field, 325 Quantum beats, 220–224 Quantum defect, 146–148 fine structure, 308, 309 He atom, 343 theory, 152–159 Quantum electrodynamics (QED), 181, 324–326 Quantum jumps, 224, 225 Quantum mechanics axioms, 95–99 definitions, 95–104 introduction, 87–134 representations, 99, 100 Quantum number, 127 angular momentum, 77, 112, 138 good, 121, 299, 595, 597, 599 in a box, 104 principle, 122, 123, 138, 139, 147 projection, 76, 138 spin, 78, 114, 138 spin projection, 78 Quantum state, 95, 96 Quasi-one-electron system, 144–161 Quasi-two-electron system, 371–374 Quiver motion high, oscillating field, 433 R Radial electron density computed with DFT, 511 Radial matrix elements, 590, 592 Radial wave function, 166 Radian, 556 Radiance, 533 spectral, 533 Radiant flux, 34 Radiation spectral density, 33 spectral distribution, 31, 34 Radio frequency spectrum lithium iodide, 483 R AMSEY fringes, 288, 289, 320, 484

725 Rare gas, 142, 517 no anions, 142 radii, 144 Rare gas configuration, 144 Rare gas shell, 139 Rate equations, 184 R AYLEIGH criterium, 276 R AYLEIGH SCATTERING, 428, 429 Real solid harmonics renormalized, 597 Real spherical harmonics, 596 renormalized, 596 Reciprocal lattice vector, 40 Recollision, 438, 439 Reduced mass, 119 Reduced matrix element, 577 Reflection operator, 597, 617 Reflection symmetry, 595–603 Relativity, see Special theory of relativity Removal of degeneracy, 137–144 Removal of m degeneracy, 130, 131 REMPI H atom, 1S–2S, 291 Resolving power FABRY-P ÉROT interferometer, 280 interferometer, 277 Resonance denominator, 164 Rest mass, 10 Rotation group, 575 irreducible representation, 560, 606 Rotation matrix, 606 Rule of D UANE -H UNT, 531 RUNGE -K UTTA method, 150 RUSSEL -S AUNDERS coupling, 512 RUTHERFORD, 65, 326 RYDBERG, 2 atoms, 253 atoms and diamagnetic interaction, 398 atoms in electric fields, 409–411 constant, 67, 69 constant, precision measurement, 321 states, 529 RYDBERG -R ITZ formula, 69 S Saturation broadening, 230 Saturation spectroscopy, see Spectroscopy, D OPPLER free Scalar product of states, 95 of tensor operators, 578 Scattering cross section, 326 S CHRÖDINGER equation, 90–92 alkali atoms, 149

726 H atom, 117 stationary, 91 time dependent, 92–94 S CHRÖDINGER representation, 100, 107 Screening He atom, 343 of nuclear charge, 138 Screening parameter, 153 alkali like atoms, 159 Na atom, 159 sech2 function, 651, 652 convolution, 627 Selection rules, 194 Self adjoint operator, 97, see also Hermitian operator Self consistent field method, 498 Self-energy, 323 S ELLMEIER equation, 420 Separation ansatz, 117 Shell closure, 142 Shell structure of atoms, 137–144 Sinc function, 652 Singlet function, 353 Singlet states, 301 Singlet system, 342 Singlet transitions, 203 S LATER determinant, 358, 503–508 Slow light, 422–427 Solid harmonics, 616 Space quantization, 73–78 Special theory of relativity, 10–14 and fine structure, 69 L ORENTZ contraction, 13 rest energy, 58 rest frame, 13 time dilation, 13 twin paradox, 13 Spectral brilliance, 534 of various X-ray sources, 534 Spectral intensity distribution, 180 Spectral radiance, 34 Spectral radiation density, 180 Spectrometer echelle, 277 Spectroscopy absorption, 177 D OPPLER free ion beams, 283, 284 microwave and RF transitions, 317 molecular beams, 282, 283 saturation, 285–288 two-photon, 289–291 emission, 177 fluorescence spectroscopy, 177

Index of Volume 1 general concepts, 177, 178 high resolution, 274–293 Spectrum He atom, 342 Hg atom, 372 visible, 34 Spherical harmonics, 112, 113 matrixelements, 580–582 products, 579 Spin angular momentum, 99 components, 115 function, 301 projection, 99, 140 Spin orientation and exchange interaction, 358 Spin-orbit coupling, 568 coupling parameter, 296, 307, 378 interaction, 293–303, 513 splitting D IRAC theory, 306 Spin-orbital, 503–508 Spinor equation, 93 Splitting of energy levels due to magnetic field, 130 Spontaneous decay rate, 229 Spontaneous emission and QED, 186 E INSTEIN A coefficient, 183 frequency dependence, 185 introduction, 181–183 Spontaneous transition probability, 213, 214 Standard deviation, 15, 627 B OLTZMANN distribution, 235 Standard model, 4 of elementary particle physics, 46–48 Standard phase convention, 563 S TARK states, 408 S TARK effect, 399–411 dipole states, 408 dynamic, 418, 420 H(2s, 2p) states, 408 interaction potential, 400, 401 linear, 407, 408 matrix elements, 402 high field, 403, 404 low field, 404 perturbation series, 405 quadratic, 405, 406 RYDBERG atoms, 409–411 significance, 399, 400 State of a quantum system, 95, 96 State vector, 95, 129 States of a quantum system, 95

Index of Volume 1 Stationary states, 176 Statistics classical, 20–26 elementary, 14–26 quantum, 20–26 S TEFAN -B OLTZMANN constant, 34 law, 34 Steradian, 554, 556 S TERN -G ERLACH experiment, 70–78, 99, 114, 115 interpretation, 75, 76 setup, 72, 73 Stimulated emission introduction, 180, 181 Strength of dipole transitions, 212–217 Strong force, 59 Structure analysis, 40–43 Subgroup periodic system of elements, 141 Subshell, 142 Superluminal light propagation, 424–427 Superposition principle, 89, 92 Susceptibility, 413, 414, 420–422 Synchrotron radiation, 42, 53, 531–539 angle and energy dependence, 539 critical wavelength, 538 generation schematically, 537 T Tensor operator, 575–578 irreducible representation, 560 products, 578–582 real, 595 Term levels, see G ROTRIAN diagram Term scheme, see G ROTRIAN diagram Terminology of atomic structure, 301 Theory of special relativity rest energy, 67 time dilation, 13 T HOMAS -F ERMI equation, 502 T HOMAS -F ERMI potential, 501, 502 T HOMSON cross section, 527 T HOMSON parabolas, 56 T HOMSON scattering, 429 nonlinear, relativistic, 543 Time dependent density functional theory, 511 Time-bandwidth product, 651 Total angular momentum eigenstates, 299–301 of the atomic charge could, 449 Total wave function, 138, 139

727 Trajectory classical, 87 Transition amplitude perturbation ansatz, 187, 189 spherical basis, 198–200 Transition matrix element radial, 312 Transition operator dipole approximation, 190, 635, 636 Transition rate absorption, 191 Transition rates in the continuum, 238 Transmission grating, 60 Trembling motion of an electron (so called Zitterbewegung), 327 Triangular relation, 197, 201, 298, 565, 577 Triplet functions, 353 Triplet states, 301 Triplet system, 342 Tunnel ionization, 436, 437 Tunnelling effect, 75 21 cm line, 460 Two electron system, 341–375 Hamiltonian, 344, 345 probabilities, 345 quantum mechanics, 344–351 Two level system thermodynamic equilibrium, 184 Two particle wave function, 345, 346 Two particle problem, 119 Two sided exponential distribution, 654 Two wire field, 73 Two-photon emission, 248, 249, 317 Two-photon excitation, 245–248, 289, 320 H atom, 291 U Ultrafast physics, 8 Uncertainty relation Gedanken-experiment, 61 H EISENBERG, 61, 63 Undulator, 532 Undulator and wiggler, 540–542 Unit operator quantum mechanics, 98, 99, 577, 583 Unit vector of polarization, 170 V Vacuum field, 181 interaction of an electron with the, 324 Vacuum polarization, 323 Vacuum-ultraviolet, 31

728 Valence electron, 75, 144, 146, 148 Na atom, 151 VAN DER WAALS potential, 417, 418 radius, 142, 143 Variance, 15, 235, 626 Variational method, 350, 351 Vector boson, 44 Vector diagram, 78, 112, 129 Vector model, 298, 381, 385 Vector operator, 91, 114, 576, 577 Vector potential, 396, 631–634 Velocity distribution, 285 Virial theorem, 66 VIS, visible spectral range, 31 VOIGT profile, 236, 237 convolution, 628, 629 Volume term in HFS, 473 W Wave function, 88–96, 100, 104 symmetric and antisymmetric, 353 two particles, 345, 346 Wave nature of matter, 58–63 Wave vector, 4, 13, 57, 94 two-photon excitation, 289 Wave-packets, 88 Wave-particle duality, 58, 61 Wavenumber, 31, 67 SI units, 31 W EHNELT cylinder, 530 W IEN filter, 56 W IEN’s displacement law, 34 Wiggler, 532, 541, 542 W IGNER -E CKART theorem, 576–578, 580, 582, 586, 617, 620, 621, 637 W IGNER -S EITZ radius, 143, 144 X X-ray absorption edge, 520, 521, 523

Index of Volume 1 bremsstrahlung, 530 characteristic emission line, 521 diffraction, 40–43 sources, 530–544 spectroscopy, 519–524 AUGER electron, 522 characteristic lines, 522–524 M OSLEY formula, 522–524 X-ray tube, 530, 531 Y YOUNG’s double slit experiment, 88 Z Z EEMAN effect, 9 ‘normal’, 128–131 anomalous, 78 fine structure, 377–399 ‘normal’, 382–386 anomalous, 379 avoided crossings, 391 classical triplet in a high field, 386 examples, 382–384 high B field, 384–386 interaction Hamiltonian, 377–380 intermediate field, 388–392 limiting cases, 379, 392 line strengths, 384 low B field, 380–384 selection rule for transitions, 386 hyperfine structure, 461–471 B REIT-R ABI formula, 467–471 ground state of 6 Li, 470 high B field, 464–466 low B field, 462, 464 Na D lines, 464 transition to very high fields, 469 Zero point energy, 181 Zero range potential, 436 Zitterbewegung (trembling motion), 327

Atomic Molecular Optical Physics by Hertel C Schulz, Volume 2

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