Atomic Molecular Optical Physics by Hertel C Schulz , Volume 1

Graduate Texts in Physics

Ingolf V. Hertel Claus-Peter Schulz

Atoms, Molecules and Optical Physics 1 Atoms and Spectroscopy

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.

Series Editors Professor Richard Needs Cavendish Laboratory JJ Thomson Avenue Cambridge CB3 0HE, UK [email protected]

Professor William T. Rhodes Department of Computer and Electrical Engineering and Computer Science Imaging Science and Technology Center Florida Atlantic University 777 Glades Road SE, Room 456 Boca Raton, FL 33431, USA [email protected]

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

Professor H. Eugene Stanley Center for Polymer Studies Department of Physics Boston University 590 Commonwealth Avenue, Room 204B Boston, MA 02215, USA [email protected]

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 1 Atoms and Spectroscopy

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 978-3-642-54321-0 ISBN 978-3-642-54322-7 (eBook) DOI 10.1007/978-3-642-54322-7 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2014952813 © 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

Atomic, Molecular and Optical physics – short AMO physics – is one of the canonical fields of physics, a profound knowledge of which is essential for understanding almost any other area of modern physics. And while its roots reach back over a century and are closely connected with the early days of modern physics, current research in AMO physics is still highly productive in respect of both, cutting edge applications and fundamental insights – as several N OBEL prizes in recent years have documented convincingly. Looking back at the technical development of modern industrial society – which is closely connected with modern physics – one may refer (COSE, 1998) to the 20th century as that of the electron while the 21st is the century of the photon. This interesting particle, the essential ingredient of modern optics and quantum optics, surprises humankind since N EWTON with its wave-particle dualism. It does not only play a key role in today’s information technology but is, from a general point of view, also the primary carrier of any information which can be obtained about the constituents of matter and materials. Even collisions of particles with mass under the influence of the C OULOMB force may be viewed as exchange of virtual photons. The textbooks presented here try to give a fairly comprehensive overview on the whole field. They cover state of the art experimental methods, and combine this with preparing the basis for a serious, theory based understanding of key aspects in modern AMO research. The two volumes, originally written in German language (H ERTEL and S CHULZ, 2008), are a genuine authors translation – not just an English mirror image of the original. We have rewritten much of the text, extended it wherever appropriate, and updated a number of aspects to catch up with recent progress in the field. On the one hand we address advanced students of physics, chemistry and other neighbouring fields, typically at the end of their undergraduate studies, or during their doctoral work. On the other hand we also wish to reach young postdocs or even mature scientists, who feel it is time they connect freshly with the topics addressed here. We consider the basics of classical geometrical optics and wave optics as well as electrodynamics to be well known by our readers. We also expect a cervii

viii

Preface

tain basic knowledge and understanding of atomistic concepts in physics, as well as of elementary quantum mechanics. We do, however, provide in Chaps. 1 and 2 of this Vol. 1 a brief repetition of these topics – essentially an extended list of keywords focussed on basic understanding and knowledge. In the main part we cover the standard scope of atomic physics, touch some modern aspects of spectroscopy, and try to lead the reader up to stateof-the-art research in some main areas of the field – wherever possible and as far as space permits. The sequence of chapters follows essentially the logics of perturbation theory. The strongest perturbation is treated first. Thus, after the introductory chapters where pure C OULOMB interaction and the H atom have been discussed, in Chap. 3 we allow for coarse deviations from the 1/r potential and focus on quasione-electron systems. This, and some common sense, allows us already to introduce the periodic system of elements. Next, in Chap. 4, we have to treat optically induced and spontaneous transitions: they are a central theme in AMO physics. This requires a brief introduction to time dependent perturbation theory, a topic which is indispensible in AMO physics, but which is often neglected in undergaduate quantum mechanics. To allow the reader a step by step approach towards the more demanding topics, we implement at this point ‘only’ the semiclassical approach – by which 95 % of standard atomic physics may be treated (resorting occasionally to somewhat hand waving arguments) – and postpone field quantization to Vol. 2. Chapter 5 further extends this knowledge, treating shapes and widths of spectral lines and introducing multiphoton processes as well as transitions into the continuum. We are now ready to understand in Chap. 6 a next step of complication, fine structure (FS) interaction. In order to allow the reader to appreciate the experimental efforts, we also give a brief introduction to high resolution and precision laser spectroscopy. This leads us automatically to the L AMB shift and calls for a short side step into the basics of quantum electrodynamics (QED). In Chap. 7 two electron systems are treated, mainly the He atom and He like ions. Exchange interaction may be smaller or larger than FS, depending on the system, but the step to multielectron systems adds a new degree of complexity and sets the stage for a quantitative treatment of the PAULI exclusion principle. The next finer step in the hierarchy of perturbations is treated in Chap. 8, including interactions between atomic electrons and external magnetic and electric fields, leading to Z EEMAN and S TARK effect, respectively. At this point, a small detour into the world of interaction between atoms and very intense laser fields is appropriate, as the theoretical formalism used is essentially an extension of the so called dynamical S TARK effect. As a last refinement we include in Chap. 9 hyperfine interactions between the atomic nucleus and electrons. These lead to very small but highly significant splittings of atomic energy levels (HFS) and offer a wealth of practical applications. In the last Chap. 10 of Vol. 1 we are finally ready to treat genuine multi-electron systems with a large number of electrons. We discuss the appropriate theoretical tools (such as HF equations, CI methods, and DFT), and present some relevant methods of X-ray spectroscopy and sources for generating X-ray radiation. As a rule, we try to avoid extensive mathematical derivations. Rather, in the ‘spirit of these books’ we prefer to give the reader some general guidance on how to reach

Preface

ix

the final, physically important results – which we discuss and illustrate usually in some detail. In addition, we provide several appendices for the reader interested in more detail. We have e.g. collected a toolbox for angular momentum algebra in atomic and molecular physics – without any claim for full mathematical consistency, but quite compact and possibly useful in practice. Some words about formats, notation, units, typography appear in order: • Each chapter begins with a brief “motto” setting the tune of the chapter, followed by short abstract guiding the reader through the text. At the end of each section a short summary recalls what the readers should have learned from the preceding text. All chapters build upon each other, but may be read by advanced readers also individually: this is facilitated by intensive cross referencing of formulas and figures, extended indices covering both volumes, a list of acronyms and important terminology as well as references at the end of each chapter. • For clarity and homogeneity we do not reproduce original drawings or other material from the literature. Rather, all published data have been redrawn (after digitalization if necessary), are presented in a standard format, and all sources used in the figures and text are properly quoted. • We consequently use the SI-System for all measurable quantities, and we emphasize the pedagogical and practical value of a “dimensional analysis” for complex physical formulas.1 On the other hand, atomic units (a.u.) facilitate the writing of many relations in atomic and molecular physics dramatically. Hence, we use them intensively – considering, however, Eh , a0 and t0 etc. simply as abbreviations for quantities with dimensions. Phrasings such as “we set , e, me , c equal to unity” are avoided, since they are highly misleading. • The finite number of letters in the Latin and Greek alphabets makes some inconsistencies or unusual designations unavoidable: we mention specifically, that in order to allow the use of E for the electric field strength (an important quantity in AMO) we use W (with appropriate indices) for energies of various types (with the exception of the atomic unit of energy which is internationally defined as Eh ). Occasionally we use the letter T for kinetic energy and try to avoid the neighbourhood of time and temperature which are often also designated by T . Vectors are written as r or k, unit vectors in these directions are er and ek , respectively. , vector-operators as We write operators as H p and tensors of rank k as Ck . For the unit operator and unit matrix we use 1. For integer numbers we mostly use calligraphic letters such as N , while number densities are simply N to distinguish them from the index of refraction n which is also an often used quantity throughout this text. Oscillations and other periodic processes are mostly characterized by their angular frequencies ω (sometimes also by their frequencies ν) and the corresponding energies are ω (or hν). 1 We make, however, use of allowed prefixes (NIST, 2000a), such as cm−1

as unit of wavenumbers (which appears ineradicable in the literature). We also use “accepted” units outside the SI (NIST, 2000b), such as the enormously practical energy unit eV (electronvolt), or b (barn) as unit for cross sections.

x

Preface

Finally, we hope that these books will become a continuing source of reference for the fastidious reader, working in or just needing to use AMO physics in her or his special field. We ask all of you to kindly provide us with the necessary feedback. We shall try to react to useful suggestion promptly. At the home page of the books, http://www.mbi-berlin.de/AMO/book-homepage, we shall continuously report on the status, list errata and possibly present additions. For additional reading and cross referencing we have collected a few related textbooks and monographs in the reference list below, just as typical examples without any claim for completeness. Berlin Adlershof, Germany January 2014

Ingolf V. Hertel Claus-Peter Schulz

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. a.u.: ‘atomic units’, see Sect. 2.6.2. CI: ‘Configuration interaction’, mixing of states with different electronic configurations in atomic and molecular structure calculations, using linear superposition of S LATER determinants (see Sect. 10.2.3). DFT: ‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6). HF: ‘H ARTREE -F OCK’, method (approximation) for solving a multi-electron S CHRÖDINGER equation, including exchange interaction. HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9). NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction.

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: Walter der Gruyter, 902 pages. B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics 64. Berlin, Heidelberg: Springer Verlag, 3rd edn., 343 pages. B ORN, M. and E. W OLF: 2006. Principles of Optics. Cambridge University Press, 7th (expanded) edn.

Preface

xi

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. COSE (Committee Optical Science and Engineering): 1998. Harnessing Light: Optical Science and Engineering for the 21st Century. Washington, D.C: National Academy Press, 360 pages. D EMTRÖDER, W.: 2010. Atoms, Molecules and Photons. Berlin, Heidelberg, New York: Springer, 2nd edn. 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, NJ, USA: Princeton University Press, 154 pages. H ERTEL, I. V. and C. P. S CHULZ: 2008. Atome, Moleküle und optische Physik 1; Atomphysik und Grundlagen der Spektroskopie. Springer-Lehrbuch. Berlin, Heidelberg: Springer-Verlag, 1st edn., 511 pages. H ERTEL, I. V. and C. P. S CHULZ: 2010. Atome, Moleküle und optische Physik 2; Moleküle und Photonen - Spektroskopie und Streuphysik, vol. 2 of Springer-Lehrbuch. Berlin, Heidelberg: Springer-Verlag, 1st edn., 639 pages. NIST: 2000a. ‘Reference on constants, units, and uncertainties: SI prefixes’, NIST. http://physics. nist.gov/cuu/Units/prefixes.html, accessed: 8 Jan 2014. NIST: 2000b. ‘Reference on constants, units, and uncertainties: Units outside the SI’, NIST. http://physics.nist.gov/cuu/Units/outside.html, accessed: 8 Jan 2014. S TEINFELD, J. I.: 2005. Molecules and Radiation, An Introduction to Modern Molecular Spectroscopy. Mineola, NY: Dover Edition, 2nd edn. W EISSBLUTH, M.: 1978. Atoms and Molecules. Student Edition. New York, London, Toronto, Syndey, 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, all other sources are also documented there which we have used for information and which have provided the data 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.

xiii

Contents of Volume 1

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 xv

xvi

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 S TERN-G ERLACH 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

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

xvii

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

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

xviii


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

. . . .

. . . .

. . . .

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


xix

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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 . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . . . . . . . . . . . . . . .

xx


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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

310 310

. . . . . . . . . . . . .

. . . . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .


xxi

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . .

. . . . .

. . . . .

. . . . .

447 447 452 452 453

. . . . .

. . . . .

. . . . .

. . . . .

xxii


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 . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

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 . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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


xxiii

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 . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

536 540 542 543 543 544 546

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

559 559 559 562

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

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 . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

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 . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

xxiv


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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

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 . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

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

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . . . . .


xxv

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

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .


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 . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 6 9 11 12 14 17 17 23 26 29 34 34 35 36 38 41 45 45 49 49 52 52 53 54 59 61 61 xxvii

xxviii


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

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

63 65 66 68 69

2

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 72 72 75 78 82 84 86 91 95 98 100 100 102 105 106 109 110 110 113 114 117 118 121 127 132 133

3

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

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

135 136 137 138 138 139 139 139 141 142 143

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . . . . . . . .


4

xxix

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 148 151 151 152 155 158 160 163 165 166 167 170 171 172 174 177 179 179 181 184 190 197 197 197 198 199 201 205 206 210 211 215 215 216 218 220 224 226 228

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

. . . .

. . . .

. . . .

231 231 231 234

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

xxx


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

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

234 236 239 239 242 243 245 247 253 253 254 257 262 266 266 270 274 275 276 277 285 286

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 . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

289 289 292 296 296 298 302 305 305 309 311 312 314 317 317 320 327 328 333 334 334 337


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

xxxi

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

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 . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

xxxii

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 . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

453 453 453 455 456 457 460 460 461 464 465

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . .

. . . . . . . .

. . . . . . . .

515 515 516 520 525 525 527 528


xxxiii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

529 530 530 531 533 534 534 537 539 540

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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 . . . . . .

. . . . . .

. . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

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

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

625 625 629 629 630 631

xxxiv


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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

687 687

. . . . . .

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 . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxv

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

695 695 698 700 702 702

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

703

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

715

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.

xxxvii

1

Basics

Here we provide a compact summary of the most important concepts, experiments, observations, phenomena and models that are essential in understanding matter on an atomic and sub-atomic length-scale. These are taught typically to second year undergraduates in physics. Readers who feel sufficiently familiar with these basics of today’s quantum view of matter may safely skip the chapter.

Overview

Section 1.1 gives a brief survey of the canonical subject areas in physics, of physics history and the quantum nature of atomic phenomena. An introduction to orders of magnitude of length, time and energy follows. Section 1.2 summarizes some essentials of special relativity. Section 1.3 introduces some elements of statistical mechanics and thermodynamics. The photon, key particle in this text book, enters the scene in Sect. 1.4. Section 1.5 makes a very short excursion into the nature of the “four fundamental interactions” and to the “standard model” of elementary particles. Section 1.6 deals with the mundane subject of how free, charged particles move under the influence of an external electromagnetic field. Particles and waves (Sect. 1.7) and the B OHR model of the H atom (Sect. 1.8) lead us to the foundations of modern physics. Section 1.9 introduces one of the key concepts of quantum mechanics: space quantization as discovered in the famous S TERN -G ERLACH experiment – which in turn led, more or less directly, to the discovery of the electron spin, treated in Sect. 1.10.

1.1

Overview, History and Magnitudes

Table 1.1 gives a compact overview of the “canonical subject areas of modern physics” and connects them to the content of these textbooks: marked in italics are those themes that are – at least partially – treated here. Otherwise, the table is more or less self explaining.

© Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7_1

1

2

1 Basics

Table 1.1 Standard subject areas of physics Theory

Canonical themes of modern physics

Applications and special subjects

Classical mechanics and special relativity

Atomic physics

Meteorology

Thermodynamics and statistics

Molecular physics

Metrologya Chemical physics

Electrodynamics and optics

Scattering physics

Quantum mechanics

Quantum optics Nonlinear optics

Quantum electrodynamics (QED)

Lasers Ultrafast physics Cluster physics Solid state physics (condensed matter)

Surface physics Semiconductor physics Medical physics Biophysics

Quantum field theory Quantum chromodynamics

Nuclear physics

General relativity

Elementary particle physics

“Grand unification”

Astrophysics

Quantum geometrodynamics

Astro particle physics

a Scientific

Reactor physics Plasma physics

standards (units) and measuring techniques

At the beginning of the past century, the history of atomic, molecular and optical (AMO) physics was almost identical to the history of physics at large. Without any claim for completeness, we collect in Table 1.2 some milestones in the topical fields of this textbook, and Table 1.3 highlights some key developments in theoretical particle physics which we shall touch very briefly in Sect. 1.5. Of course, a host of fascinating details during the development of modern physics cannot even be mentioned here.

1.1.1

Quantum Nature of Matter

Based on experiments, many fundamental observations in physics can only be understood in terms of quantum mechanics. Some examples are • • • •

photoelectric effect (E INSTEIN 1905), C OMPTON effect (C OMPTON 1922), frequency distribution of blackbody radiation (P LANCK 1900), diffraction and interference of particle beams – i.e. the wave-particle duality (DE B ROGLIE wavelength 1923), • specific heat capacity at low temperatures (E INSTEIN, D EBYE 1906), • line spectra of atoms (RYDBERG, B OHR 1913).

1.1


3

Table 1.2 Highlights of science history from the idea of the atom to modern physics of atoms, molecules and quantum optics (incomplete list) 400 BC 1808 AD 1811 1814 1834 1868 1869 1886 1895 1896 1897 1898 1898 1900 1903 1905 1913 1913 1921–1922 1925

ατ oμoς (indivisible) Multiple proportions Molecular theory of gases First useful spectrometer Electrolysis (FARADAY constant) Periodic table of elements Cathode rays Channel rays X-rays Radioactivity e/m for electrons Polonium, radium e/m for ions E = hν Atomic nuclei E = mc2 Atom model e-determination Space quantization Fundamental research in quantum mechanics 1926 S CHRÖDINGER Wave equation 1927 H EISENBERG Uncertainty relation 1947 L AMB and R ETHERFORD L AMB shift for excited H 1958–1966 S CHAWLOW, T OWNES, Basov, P ROKHOROV, Maser, laser and spectroscopy M AIMAN, JAVAN, K ASTLER 1986 N OBEL prize Dudley R. H ERSCHBACH, Dynamics of chemical elementary Yuan T. L EE and John C. P OLANYI processesa 1989 N OBEL prize Norman G. R AMSEY, R AMSEY fringes, atomic clocksb , ion Hans D EHMELT and Wolfgang PAUL trapsa 1996 N OBEL prize R. F. C URL Jr., H. K ROTO, Discovery of fullerenesa . . . C60 etc. R. E. S MALLEY 1997 N OBEL prize S. C HU, Methods to cool and trap atoms with laser C. C OHEN -TANNOUDJI, W. D. P HILLIPS light 1999 N OBEL prize Ahmed Z EWAIL Femto(second) chemistrya 2001 N OBEL prize Eric A. C ORNELL, Cold atoms and Wolfgang K ETTERLE, Carl E. W IEMAN B OSE -E INSTEIN condensationa 2002 N OBEL prize John F ENN, Koichi TANAKA Electro spray, molecular beamsa , MALDI mass spectroscopya 2005 N OBEL prize Roy G LAUBER, Theory of optical coherencec and John H ALL and Theodor H ÄNSCH laser precision spectroscopya 2007 N OBEL prize Gerhard E RTL Chemical processes at surfaces a Work

D EMOCRITOS DALTON AVOGADRO F RAUNHOFER FARADAY M ENDELEEV H ITTORF G OLDSTEIN RÖNTGEN B ECQUEREL J.J. T HOMSON Marie & Pierre C URIE W IEN P LANCK RUTHERFORD E INSTEIN B OHR M ILLIKAN S TERN & G ERLACH Max B ORN (N OBEL prize 1954)

from several preceding years

b Work

from the 1950s

c Work

from the 1960s

4

1 Basics

Table 1.3 Theory on the way from electrodynamics to the standard model of the fundamental interactions (the dates of N OBEL prizes given here refer to discoveries and developments which typically happened much earlier) ca. 1850

James Clerk M AXWELL

Electrodynamics

1918 N OBEL prize Max P LANCK

Energy quanta

1921 N OBEL prize Albert E INSTEIN

Law of the photoelectric effect

1932 N OBEL prize Werner H EISENBERG

Creation of quantum mechanics

1933 N OBEL prize Erwin S CHRÖDINGER, Paul D IRAC

Wave equations for matter

1949 N OBEL prize Hideki Y UKAWA

Prediction of mesons

1954 N OBEL prize Max B ORN

Statistical interpretation of quantum mechanics

1963 N OBEL prize W IGNER, G OEPERT-M AYER, J ENSEN

Structure of the nucleus

1965 N OBEL prize T OMONAGA, S CHWINGER, F EYNMAN Quantum electrodynamics 1967 N OBEL prize B ETHE

Theory of nuclear reactions

1969 N OBEL prize G ELL -M ANN

Quark model

1979 N OBEL prize G LASHOW, W EINBERG, S ALAM

Theory of weak interaction

1982 N OBEL prize W ILSON

Re-normalization, critical phenomena

1999 N OBEL prize ’ T H OOFT, V ELTMAN

Quantum structure of electro-weak interaction

2004 N OBEL prize G ROSS, P OLITZER, W ILCZECK

Asymptotic freedom of quarks

We encounter the phenomenon of quantization that will be a steady companion during all volumes of this textbook. Of central importance is the relation between the momentum p of a particle and its wavelength λ (or wave vector k) that quantifies the so called “wave-particle dualism” by p = h/λ

and p = k

with k = 2π/λ,

(1.1)

with the fundamental P LANCK constant (see also Sect. 1.4.6): h = 6.62606896(33) × 10−34 J s = 4.13566733(10) × 10−15 eV s.

(1.2)

Closely related and supplementing the above we shall also have to discuss the quantization of energy. As examples we mention the • energy of a photon (see Sect. 1.4.1): Wph = hν = ω

(1.3)

• energies of the H atom (see Sect. 1.8): Wn = Eh /2n2

with n = 1, 2, 3, . . . .

(1.4)

1.1


5

Equally important is the quantization of angular momentum and its direction in space (so called space quantization) – as we shall explicate later on various levels of abstraction (see e.g. Sect. 1.9). At this point it may suffice to say that angular momenta are measured in either integer or half integer multiples of . The following equations summarize this: • absolute value of angular momentum: J = j (j + 1)2

(1.5)

angular momentum quantum numbers for bosons j = 0, 1, 2, . . .

(1.6)

j = 1/2, 3/2, . . .

(1.7)

for fermions • (2j + 1) angular momentum projections: Jz = mj

(1.8)

mj = −j, −j + 1, . . . , j.

(1.9)

with projection quantum numbers

Here J is the absolute value of the angular momentum, Jz its projection on a given axis (here z) in space. One distinguishes orbital angular momentum (integer quantum number) and intrinsic angular momentum or spin (integer for so called bosons, half integer for so called fermions, see Sect. 1.5). To characterize these properties of a given orbit or particle we shall refer to its angular momentum (or spin) as being 0, 1/2, 1, etc.

1.1.2

Orders of Magnitude

Before going into details, we want to give an orientation about orders of magnitude for some relevant physical observables with which we shall have to deal. Orders of magnitudes are always important in physics and the student is well advised to roughly memorize some of these data. This will turn out to be very helpful when trying to find her or his bearings – later on when indulging into real, own measurements. (A list of the most important fundamental constants of nature is provided in Appendix A.)

Length Scales from Atomic Physics to Astrophysics An overview about the whole length scale on which relevant physics occurs gives Fig. 1.1. The smallest possible length unit one may construct from the known fun damental constants is the so called P LANCK length P = /(mP c) = G/c3 =

6

1 Basics

10−35 10−15 dominant forces:

10−10

strong interaction

10−5

100 105 length / m

1010

1015

universe

diameter of our galaxy

next star

distance to the sun

cells

human

bacteria insects

distance to the moon

diameter of the earth

nano world

virus

molecules atoms

e− (Coulomb law) atomic nuclei

Planck length

see enlarged scale

1020

gravitation (among quasi neutral objects)

electroweak interaction electromagnetic interaction

Fig. 1.1 Length scales in the universe and the range of dominance of the four fundamental forces: strong, electro-weak, electromagnetic and gravitational interaction (see Sect. 1.5)

1.616252 × 10−35 m, which may √ be viewed as describing the ‘granularity’ of space (with the P LANCK mass mP = c/G = 2.176437375 × 10−8 kg. Both, P and mP , are constructed from the reduced P LANCK constant , the gravitational constant G and the velocity of light c (see Appendix A). At the other end of the length scale are cosmic objects such as our galaxy (milky way) with a diameter of ca. 100 000 light years (1 light year 9.46 × 1015 m) and finally the known universe. Its age is currently estimated to be about 13.7 × 109 a (see M ATHER and S MOOT 2006), from which one has to assume an upper limit of its extension of 1.30 × 1026 m. In contrast, atomic and molecular physics focus on objects in a length range from 0.5 × 10−10 m–10−9 m. In this context it is useful to visualize typical dimensions (radii) of the building blocks of matter as done very schematically in Fig. 1.2. “Elementary particles” in the strict sense are only quarks and electrons, their extension being (if any) definitively below 1 am. While such length scales are in general of little relevance to phenomena observed in atoms and molecules, it should be Fig. 1.2 Blow up of the length scale Fig. 1.1 for the size range of building blocks of matter – on the left side given in m, right in attometers (1 am = 10−18 m). The ‘images’ of elementary particles are schematics only (see also Sect. 1.5)

size in am

size 10−10 m

atom

100 000 000 am

10−14 m

nucleus

10 000 am

10−15 m

proton, neutron (nucleons)

1 000 am

≤ 10−18 m

≤1 am

quark ?

electron ? (lepton)

see enlarged scale

10-44

10-16

10-12

10-8

10-4 100 time / s

104

108

1012

universe earth

homo sapiens

human life

neutron lifetime

one year

7

lifetime


π 0 meson

Planck time

1.1

1016

Fig. 1.3 Time scales in the universe. The range marked with the fat double arrow is shown in Fig. 1.4 on an enlarged scale

pointed out that the finite extension of the atomic nuclei play a role (proton radius ca. 0.88×10−15 m) for ultra-high precision spectroscopy (see Chap. 9). On the other end of the atomic length scale we have to be aware that the wavelength of electromagnetic radiation (spectral range of the visible, VIS, from 380 nm to 760 nm, see Sect. 1.4.5) is of high relevance in all spectroscopic investigations.

Time Scales from Atomic Physics to Astrophysics Typical time scales are communicated in Fig. 1.3. Again one may define a shortest time (granularity of time), the P LANCK time as tP = P /c = G/c5 = 5.39124(27) × 10−44 s. For comparison, the mean lifetime of a π0 meson is 0.84 × 10−16 s, atomic excited states live on the order of 10−9 s (see Chap. 4), and the neutron’s mean lifetime is 886 s (see Sect. 1.5.3). On the other end of the scale we find the age of our earth: it exists now for about 4.55 × 109 a. The universe that was created 13.7 × 109 a ago (according to our present understanding, see M ATHER and S MOOT 2006). Out of these more than 60 decades, the time scale of interest for AMO physics as well as for technology, chemistry, biology and medicine is illustrated in Fig. 1.4. Energy Scales in Physics Besides space and time, energy plays a key role in AMO physics. An orientation gives Fig. 1.5 where the typical energy content of physical objects and excitation energies of quantum systems are summarized. In our context mainly energies between some μeV and some ten’s of keV are significant. It is useful to characterize atomic and molecular phenomena according to their characteristic interaction energies. All of them are in some way based on electromagnetic forces. In the following chapters we shall first treat the strongest interactions and then refine our considerations step by step towards smaller perturbations of the dominant effects. The quantitative treatment occurs then in the spirit of perturbation theory: one first formulates and solves the most simple problem and improves the calculations by adding one perturbation after the other, ideally each some orders of magnitude smaller than the former. The changes may then be calculated

8

1 Basics

electron motion in atoms and molecules

molecular vibration / rotation

primary processes of photosynthesis

chemical reaction (explosion)

fast digital electronics life time of excited atomic states

stop clock

fast camera shutter

ultrafast physics 10−18

10−15 femto

atto

10−12 pico

10−9 nano

10−6 micro

10−3 milli

1 s

characteristic time scales / seconds (s)

Fig. 1.4 Enlarged time section from seconds to attoseconds (10−18 s) from Fig. 1.3. A presently very active area of research in AMO is ultrafast and attosecond physics

1 μK

Bose-Einstein condensates

1 mK

1K CMBR

1eV

1 keV

1 MeV

1000 MK 1000 K 1 MK room teminterior fusion perature of stars reactor

W & Z boson

1 GeV

1 TeV 1 PeV LHC lead

1 nK

1meV

LHC protons

1μeV

rest mass p+

molecular rotation

electronic excitation in atoms, molecules, molecular solids vibrations

rest mass of e−

with excellent success by approximative methods. This hierarchy of perturbations is summarized quantitatively in Table 1.4. Corresponding to different spectroscopic precision one often uses different units for characterizing the relevant interaction. Most often in spectroscopy the unit wavenumber or cm−1 is used (see Eq. (1.79)). Alternatively, energies in atoms and molecules are measured in eV or in atomic units (a.u., see Sect. 1.8.3) of energy, Eh = 27.211 eV. The upper end of the energy scale for AMO physics may be seen as related to the rest mass energy of the electron me c2 0.511 MeV (with me being the mass of the electron and c the speed of light in vacuum, see Sect. 1.2).

Fig. 1.5 Energy and equivalent temperature of quantum systems: the scale starts today at one or several 100 pK, referring to the coldest B OSE -E INSTEIN condensates (BEC) and cosmic microwave background radiation (CMB(R)) at 2.725 K on the one end of the scale. On the other end we find collisions artificially generated at the large hadron collider (LHC) with 14 TeV for protons and more than 1000 TeV for lead nuclei. AMO is interested mainly in the red marked energy range

1.1


9

Table 1.4 Orders of magnitude of atomic interactions – here for typical examples such as H, alkali metal atoms and He; the structure of volume 1 of these textbooks essentially follows this energetic scheme Interaction Pure C OULOMB ∝ Z/r

Order of magnitude

See

cm−1

eV

kHz

K

30000

4

1015

43000

Chap. 2

0.12 to 0.7

3 × 1010

1400 to 8600

Chap. 7

to 1.8×1011

1014

4300

Chap. 3

3 × 1010 to 3 × 1013

1.4 to 1400

Chap. 6

3 × 1010

1.4

Chap. 8

3 × 107 to 3 × 1010

1.4 × 10−3 to 1.4

Chap. 9

Exchange (in He n = 2)

1000 to 6000

C OULOMB screening

3000

0.4

Fine structure (FS)

1 to 1000

10−4

External el. mag. fields

1

10−4

Hyperfine structure

10−3

10−7

to 1

to 0.1

to

10−4

The ratio of these two energies defines a dimensionless fundamental constant, the so called fine structure constant α=

1 Eh e2 , = me c2 4πε0 c 137

(1.10)

which should be memorized. We shall often come across this very important constant, typically indicating some influence of, or connection with special relativity. The currently best value of α, measured with very high accuracy, is reported in Appendix A, based on 2010 CODATA (NIST 2010). There, one also finds precise conversion factors between different units of energy. We finally mention that one may, from fundamental constants, also construct a so called P LANCK energy WP = c2 (c/G)1/2 = 1.221 × 1019 GeV. This is an energy of cosmic magnitude which may be related to the first moments after the “big bang”. There are indications for unification of the fundamental forces at such energy, i.e. at least three coupling constants might become equal at this energy. Section summary

• The history of AMO physics is identical to the early history of physics at large. We have recalled some basic observations documenting the quantum nature of submicroscopic matter. • The essence of (any) physics is to obtain a quantitative description of nature. Numbers and orders of magnitude are thus essential, and a good feeling for the scales of lengths, time and energy is important for working in physics. Section 1.1.2 gives a summary. • Specifically, we memorize P LANCK’s constant h 6.63 × 10−34 J s and the dimensionless fine structure constant α 1/137 which will be steady companions on our voyage through AMO physics.

10

1 Basics

1.2

Special Theory of Relativity in a Nutshell

1.2.1

Kinematics and Dynamics

We cannot give here a serious introduction to the theory of relativity and assume the reader to be somewhat familiar with E INSTEIN’s special theory of relativity. Generally speaking, throughout this textbook relativistic effects will be treated as a kind of afterthought where necessary (e.g. in Chap. 6). It is, however, useful to recall here some formulas for later use. We first remember that N EWTON’s equations, specifically his second axiom, remain fully valid under relativistic conditions. Thus, dp =F dt

(1.11)

at high velocities v, has just to be applied to the relativistic momentum: mv p= = γ mv and p = γ mβc = mc γ 2 − 1. 1 − β2

(1.12)

Here m is the rest mass of a particle (its intrinsic mass in its rest frame, also called invariant mass) and γ=

1 1 − β2

with β =

v c

(1.13)

is the so called L ORENTZ factor. One may also rewrite (1.13) as γ 2 − γ 2β 2 = 1

(1.14)

which is invariant under L ORENTZ transformation (it is a constant). For highly relativistic particles β 1 one obtains 1 − β 1/ 2γ 2 . (1.15) Next we recall the equivalence of mass and energy. In the particle’s rest frame the famous E INSTEIN relation reads Wrest = mc2 .

(1.16)

2 we obtain If we multiply (1.14) on both sides by Wrest

γ 2 m2 c4 − γ 2 β 2 m2 c4 = m2 c4 , and with (1.12) follows γ 2 m2 c4 − p 2 c2 = m2 c4 W 2 − p 2 c2 = m2 c4 .

(1.17)

1.2


11

Here W is interpreted as total relativistic energy of the moving particle:1 W = γ mc = 2

m2 c4 + p 2 c2

or γ = W/mc2 .

(1.19)

For the particle at rest (p = 0 and γ = 1) it becomes identical to the E INSTEIN relation (1.16). Obviously with (1.13) the fundamental relation β = v/c < 1 must hold if divergence of energy is to be avoided: the velocity of a particle with finite rest mass is always less than the speed of light. In contrast, particles without rest mass, such as the photon, exist only moving at the speed of light. We emphasize that for several interacting particles in a given frame of reference relativistic energy conservation

Wi = const

holds – with energies Wi according to (1.19) to which potential energies may have to be added due to internal or external fields. In addition relativistic momentum conservation p i = const holds, with momenta p i according to (1.12). This may be summarized in most compact form by saying that the so called momentum four-vector P is conserved in a closed system: P=

P i = const

with P i = (Wi /c, p i ).

i

Note that P 1 · P 2 = W1 W2 /c2 − p1 · p 2 is the inner product of two four vectors P 1 and P 2 . The length of a four-vector is invariant under L ORENTZ transformation (M INKOWSKI Norm): 1 One

may abbreviate m mrel = = γm 1 − β2

(1.18)

and write the relativistic momentum and energy as p = mrel v

and

W = mrel c2 ,

respectively. In modern theoretical literature the introduction of this “relativistic mass” is, however, usually omitted to avoid confusion: it is the energy that changes with velocity, while the rest mass m is L ORENTZ invariant.

12

1 Basics

P2 =

W2 − p2 c2

W 2 − p 2 c2 = W 2 − p 2 c2 = const.

(1.20)

This expression describes a system of particles with total energy and total momentum W , p in one frame, and W , p in an other frame of reference, respectively. This is completely equivalent to (1.17), which was referring to a single particle and its rest frame on the right hand of the equation. As an example, for two (or more) particles the transformation from the laboratory system (lab) to the centre of mass system (cm) would read 2 2 2 2 Wlab − plab c = Wcm ,

(1.21)

where in the laboratory frame Wlab and p lab are the sums of relativistic energies and momenta of all particles, respectively, while Wcm is the sum of all particle energies in the centre of mass system – in which per definition the sum of all momenta is p cm = 0. The kinetic energy T of a relativistic particle is defined by T + mc2 = W = γ mc2 .

(1.22)

The L ORENTZ factor may thus be written as γ = 1 + wk

with wk =

T . mc2

(1.23)

Often it is useful to know the velocity of a particle as a function of its kinetic energy. With (1.23) and the definition (1.13) one finds √

3 1 + wk /2 β = 2wk 2wk 1 − wk + · · · . (1.24) 1 + wk 4 The approximate equality is an expansion for small kinetic energies. The relation between the magnitude of the momentum p and the kinetic energy T is derived by inserting (1.22) into (1.19) and squaring, so that p 2 c2 = T 2 + 2T mc2 from which follows

√ √ T T p = 2mT 1 + 2mT 1 + − ··· 2mc2 4mc2

wk − ··· . p = mc 2wk 1 + wk /2 mc 2wk 1 + 4

(1.25)

For small kinetic energies one recovers the classic relation T = p 2 /2m. A nonrelativistic treatment is only possible if T mc2 . For an electron this is a rather limited range of energies since Wrest e− = me c2 = 0.511 MeV. (1.26)

1.2


13

Fig. 1.6 Relativistic Doppler shift moving source

1.2.2

θ

observer in lab-system v

Time Dilation and LORENTZ Contraction

We want to transform times and positions that are known in one frame of reference, say the rest frame of a particle with coordinates x , y , z , t , moving at a velocity v in respect of another frame, say the laboratory system with coordinates x, y, z, t. A distance x in the moving system is seen from the laboratory system at rest as much shorter (L ORENTZ contraction):

x = x /γ .

(1.27)

Conversely, the time difference t of two events measured in the moving system will appear to be stretched in a measurement in the laboratory system (so called time dilation):

t = γ t .

(1.28)

We recall the so called twin paradox: the twin brother travelling in a spacecraft at nearly the speed of light returns only a little bit older, while his twin who remained on earth has already become an old man. Closely related is the relativistic D OPPLER shift. Assume, as indicated in Fig. 1.6, a source moving with velocity v to emit radiation of angular frequency ω (wave vector k with k = ω /c, wavelength λ = c/ν ). In the laboratory system at rest one registers an angular frequency ω (wave vector k) at an angle θ with respect to v: ω 1 1 ν k λ = = = = = . ω ν k λ γ (1 − β cos θ ) γ − γ 2 − 1 cos θ

(1.29)

For absorption one just has to replace −β by +β. For perpendicular observation (θ = π/2) this leads in both cases to the non-classical, so called quadratic D OPPLER effect ν/ν = 1/γ = 1 − β 2 . In forward and backward direction, θ = 0 and π , respectively, (1.29) simplifies (again with opposite signs for absorption): ω ν k 1 = γ (1 ± β) = = = = ω ν k γ (1 ∓ β)

1±β =γ ± 1∓β

γ 2 − 1.

(1.30)

In the limit of highly relativistic energies, with γ 1, radiation is emitted essentially into forward direction at frequencies ω = 2γ ω .

(1.31)

14

1 Basics

We shall come back to this remarkable fact in the context of synchrotron radiation in Sect. 10.6.2. In the limit of very small velocities v one expands (1.29) in powers of β = v/c to recover the classical D OPPLER shift:

ν/ν → β cos θ = (v/c) cos θ v→0

or

ω = k · v.

(1.32) (1.33)

For later use we also give an expansion in terms of the kinetic energy. For wk = T /(mc2 ) = γ − 1 1 we obtain from (1.30) (at θ = 0 and π ):

wk

ν 2 = wk ± 2wk + wk ± 2wk 1 ± + ··· . (1.34) ν 2 Section summary

• Most of the relations from special relativity communicated here will be used quite often throughout this textbook. The reader may want to memorize at least (1.12), (1.13), (1.17), (1.20), (1.22), and (1.26)–(1.28).

1.3

Some Elementary Statistics and Applications

Thermodynamics and statistics are together a big and important theme in physics and physical chemistry. Many substantial textbooks exist on the subject and a variety of important aspects may be found well presented by online scripts in the Internet. Here we present only a collection of topics and formulas from this wide field with particular relevance to atomic and molecular physics. We start with some rather elementary remarks about exponential probability distributions. They will be exemplified for spontaneous decay of unstable (excited) states of quantum systems and for absorption of radiation. We then present a collection of formulas from kinetic gas theory and end with probability distributions for classical particles, fermions and bosons. But before going into specific examples we define some general terminology. A probability distribution w(x) describes the probability w(x)dx to find a random variable between the values x and x + dx. Properly normalized the probability to find the system with any value of x must be unity: ∞ w(x)dx = 1. (1.35) 0

The average value of any observable f (x) which depends on x is ∞ f = f (x)w(x)dx. 0

(1.36)

1.3


15

Specifically, the mean or expectation value of the variable is given by ∞ x = xw(x)dx,

(1.37)

0

and the so called variance is ∞ 2 σ2 = x − x w(x)dx = 0

∞

x 2 − 2x x + x 2 w(x)dx

0

= x 2 − 2 x 2 + x 2 = x 2 − x 2 . The square root of the variance is called standard deviation, σ = gives a measure of the width of the distribution.

1.3.1

(1.38)

x 2 − x 2 , and

Spontaneous Decay and Mean Lifetime

Exponential decay probabilities play an important role throughout quantum physics in general, be it in AMO, nuclear or condensed matter physics. We have already used notions like “unstable” particles or states, β decay and half lifetime, tacitly assuming that the reader is somewhat familiar with these concepts, and we shall have to use them again quite often. Quantum mechanics does not give us any certainty for finding quantum objects at a given time, at a particular position in space or in a specific state: it only makes predictions about probabilities. Thus, if a particle or a state of a quantum system is not stable it decays into one or more other particles or states with a certain decay constant A (also called decay rate) which is measured in units [A] = s−1 . In many cases it may be calculated by quantum mechanics (or possibly by QED or QCD). We note explicitly two characteristic of such probabilities which hold for most of the processes we are interested in: 1. The decay constant A does not depend on the number of particles that are incidentally present in the experiment but only on the properties of the object(s) studied. 2. It does also not depend on the time at which the particle or state decays: the decay occurs at some arbitrary, a priori unknown time and we cannot predict what that time will be; we only know the probability Adt that a decay may happen within a time interval dt. No predictions can be made about the destiny of a specific particle or state. – However a quite accurate prediction can be made for a large ensemble of, say N , particles or objects in unstable states.2 Here and in the following N gives the number of particles in a volume V of interest while N = N /V refers to the number density (or particle density) measured in units [N] = m−3 .

2 Note:

16

1 Basics

In the process N decreases during the time dt by dN = −N Adt.

(1.39)

The minus sign here indicates reduction. If we start at time t = 0 with N0 particles in one particular initial state, the number N (t) of particles that at time t are still found in their initial state is obtained by integration: t dN =− Adt ⇒ ln N (t) − ln N0 = −At N 0 N (t) = N0 e−At = N0 e−t/τ = N0 e−t ln 2/τ1/2

(1.40)

where we have introduced the mean lifetime τ = 1/A and the so called half lifetime τ1/2 = ln 2/A = τ ln 2 = 0.692τ . The latter is the time during which half of the initial particles have decayed. Often one is interested in the number of decay processes per unit time in the sample (at time t):3 A=−

N (t) ln 2 dN N0 −t/τ = AN (t) = = e N (t) = . dt τ τ1/2 τ

(1.41)

The exponential decay law is of fundamental nature and describes the statistical (also called spontaneous or natural) decay of excited atomic or molecular states or of electron hole pairs (excitons) in a solid as well as the decay of radioactive nuclei (and thus the decrease of radioactive radiation with time) or of elementary particles (barions, mesons, etc.) such as the β decay of the neutron which we shall describe in Sect. 1.5.3. The exponential distribution or decay law (1.40) is displayed in Fig. 1.7. One directly recognizes the significance of the half lifetime τ1/2 . Alternatively to a linear display shown in Fig. 1.7(a) one often uses a logarithmic scale for the probability. As seen in Fig. 1.7(b) the exponential decay then gives a straight line. This is often used to recognize exponential decays and to estimate the half lifetime. Finally, we may recast (1.40) into a probability w(t)dt for any of the initial N0 particles to decay between a time t and t + dt: w(t) = −

1 1 dN = e−t/τ . N0 dt τ

(1.42)

We note that this probability distribution is normalized according to (1.35): any given particle will definitely decay at some time between t = 0 and ∞. Conversely, the probability that a particle has not yet decayed before a time t is exp(−t/τ ). And with (1.37) we verify indeed that the mean lifetime (also mean or average decay time) is t = τ = 1/A as introduced in (1.40). The variance (1.38) for the exponential decay function is τ 2 . 3 In nuclear physics this is called activity – which must not be confused with the decay constant (or rate) A for which in nuclear physics often the letter λ is used.

1.3


(t) /

17

(t) /

0

(a)

1.0

0

in log. scaling

(b)

1.0 1/2

0.8

1/4 0.6 1/2

1/8

0.1

0.4 1/4 0.2

1/8

0.0 0

1

2

3

4

5

6 t / 1/2

0.01 0

1

2

3

4

5

6 t / 1/2

Fig. 1.7 Exponential decay law (a) in linear, (b) in logarithmic display. Note that the time axis is scaled here in units of the half lifetime τ1/2 so that for t = 1 and t = 2 and t = 3τ1/2 the probability decreases to 1/2, 1/4 and 1/8, respectively as indicated by the dashed lines

1.3.2

Absorption, LAMBERT-BEER Law

Exponential distribution functions of the kind just discussed do not only describe probabilities as a function of time. Another important application is the absorption of fast moving particles or electromagnetic radiation (i.e. photons, including visible light as well as γ -rays) when passing through matter. We discuss prototypically the absorption of visible light of intensity I (z) when passing over a distance z through a medium. The intensity is defined as the total energy transported in the light beam per unit time and area. It is measured in [I ] = J s−2 m−2 = W m−2 . Alternatively, in the particle picture, we may describe the photon flux Φ(z) = I (z)/(hν) with [Φ] = particles s−1 m−2 as a function of position z (each photon having an energy hν). As indicated in Fig. 1.8, light with intensity I (z) at position z is absorbed on the way through a medium. In analogy to (1.39) and (1.40), the intensity dI absorbed over a short distances dz is proportional to that distance and

Fig. 1.8 Absorption of light schematic – for deriving of the L AMBERT-B EER law

I(z)

I(z) -│dI│ dz z

18

1 Basics

the incoming radiation dI = −μI (z)dz. In a homogeneous medium which extends from zero to infinity, we find that the initial intensity I0 (at z = 0) is reduced to4 I (z) = I0 exp(−μz)

(1.43)

at position z. Thus, the result is again an exponential decay law – here it is the light intensity which decays with the optical path z travelled. The proportionality constant is called absorption coefficient μ and is measured in units [μ] = m−1 . It may be understood on an atomistic, statistical basis: the absorber medium may e.g. be an atomic gas in its electronic ground state, it may also be glass or a liquid. In any case it consists of many absorbing particles (atoms, molecules, defects in a solid), each of which has a characteristic absorption cross section σ ([σ ] = m2 ) for the photons which pass through the medium. The absorption coefficient μ is then the product of cross section and particle density (of the absorbers), μ = σ N . For a medium of finite thickness d the transmitted intensity is thus given by the so called L AMBERT-B EER law I (d) = I0 exp(−μd) = I0 exp(−σ N d).

(1.44)

and the decrease of the photon flux with z is given by Φ(z) = Φ0 exp(−μz) = Φ0 exp(−σ N z).

(1.45)

The probability w(z)dz that any of the photons arriving in the initial flux Φ0 at z = 0 is absorbed between z and z + dz is given by the probability distribution w(z) =

1 1 dΦ = e−z/ l Φ0 dz l

with l = 1/(σ N ).

(1.46)

In analogy to the notation used in Sect. 1.3.1, here l is called mean free path length of the photon. It is the distance a photon can – on average – move freely in the medium without being absorbed. And exp(−z/ l) is the probability that a photon has survived up to distance z.

1.3.3

Kinetic Gas Theory

The statistical interpretation of the properties of ideal and real gases by the kinetic theory of gases has played an important role in the history of atomic and molecular physics. Here we just communicate some basic terminology and results without derivation. the chemical literature one often writes μ = ln 10εC 2.303εC, with C being the concentration of the absorbent (e.g. in dilute liquids or gases), measured in [C] = mol L−1 . The molar absorption (or extinction) coefficient ε thus has the dimension [ε] = L mol−1 cm−1 = 1000 mol−1 cm2 . The so called absorption (or extinction) is then log(I0 /I (x)) = εCx. 4 In

1.3


19

One “mol” of a gas (an SI unit) is defined as the amount of a substance that contains as many elementary entities (e.g. atoms, molecules, ions, electrons) as there are atoms in 12 g of the carbon isotope 12 C. The corresponding number of particles is the AVOGADRO constant: NA = 6.02214179(30) × 1023 mol−1 .

(1.47)

The relative atomic (or molecular) mass Mr , formerly called standard atomic (molecular) weight, of an isotope (substance) X is defined as Mr (X) = m(X)/ [m(12 C)/12] where m(X) is the mass of the corresponding atom and m(12 C) that of 12 C. Note, that one typically finds Mr given in [Mr ] = g mol−1 so that the mass of one atom (molecule) of this substance is m = Mr /NA . Formally, however, Mr is a dimensionless number and m = (Mr /NA ) g mol−1 . The classical equipartition theorem states that in a system at thermodynamic equilibrium each degree of freedom that enters quadratically into the total energy of the system contributes equally to it with kB T /2. The total internal energy of the complete system is then given by5 U = f × NA

RT kB T =f × 2 2

(1.48)

where f is the number of degrees of freedom per particle, T the absolute temperature of the gas, kB the B OLTZMANN constant (1.83) and R the molar gas constant R = N A kB . In an ideal (atomic) gas only the kinetic energy is of relevance and f = 3. Temperature then corresponds to the average internal kinetic energy u of the particles according to 3 1 u = mv 2 = kB T . 2 2

(1.49)

p = N mv 2 /3 = N kB T .

(1.50)

√ These particles move in the gas with an average velocity v 2 , their average momen√ 2 tum being m v . From this, one may calculate the pressure p of an ideal gas as the momentum transfer per unit time and area by elastic collisions and back reflections with and from the walls containing the gas. Since 1/6 of all atoms move into one direction we obtains with the particle density N

For νNA particles, i.e. for ν mol in a volume V we have N = νNA /V and obtain from (1.50) the well known ideal gas law for a macroscopic system: pV = νNA kB T = νRT 5 We

use the traditional letter U for this energy/ mol and u for its average per particle.

(1.51)

20

1 Basics

In a real gas particles do collide – not only with the walls but also with each other. With the particle velocity v, the particle flux (i.e. the number of particles passing through a unit area per unit time) is vN . If σ is the gas kinetic cross section, measured in [σ ] = m2 , the time tcol and the distance l a particle moves on average freely between two collisions are tcol =

1 σ vN

and l = v tcol ,

(1.52)

respectively. The bracket indicates that one has to average over the velocity distribution N(v) of the gas particles. Somewhat more precisely, in the denominator v is the relative velocity between the colliding particles. If only one particle species √ is of interest their average relative velocity is 2 v . Thus, the so called mean free path becomes 1 , (1.53) l=√ 2σ N where σ is an average gas kinetic cross section. For typical elastic collisions between atoms and molecules it is on the order of 10−19 m2 . Relations similar to (1.53) may also be written for the absorption of ions and nucleons or of light, X- or γ -rays, see (1.46). Finally, we come back to the equipartition theorem. For molecules one has to add 1 to the number of degrees of freedom f for each accessible rotation, and 2 for each vibrational mode (for kinetic and potential energy). Thus, in a solid f = 6 per atom, in a diatomic molecular gas f = 7 per molecule (three translational coordinates, two rotational axes, one vibrational mode), while for linear and bent triatomic molecules we have f = 9 and 10, respectively – and so on. However, in these cases one has to account for quantization of rotational and vibrational energies: they become inaccessible (‘frozen’) at very low temperature. Thus, the equipartition theorem in its simple form (1.48) holds only at sufficiently high temperature.

1.3.4

Classical and Quantum Statistics – Fermions and Bosons

When speaking of mean energy, velocity, lifetime or free pathway of particles (photons, atoms, molecules, ions, electrons, nuclei etc.) one implies that these quantities are described by a statistical distribution. Classically, B OLTZMANN statistics provides the statistical distribution of energies in all fields of physics. It refers to the probability for finding a certain energy u per particle in an ensemble. This energy may be kinetic energy or internal excitation of the particles (e.g. electronic, vibrational and rotational excitation in a molecule). Quantum physics requires certain modifications, but for low enough particle densities and sufficiently high temperatures these are very small, as we shall see in a moment. In view of the possible quantization of energy we have to distinguish discrete and continuous energy states. Energies of the system may be realized by different quantum mechanical states. Different states i with identical energies ui are denoted

1.3


21

as degenerate and the number of possible realizations of one energy ui is called degeneracy gi . In the case of a continuum of energies u a density of states g(u) characterizes the number g(u)du of states (here per unit volume) in an energy interval between u and u + du. The B OLTZMANN distribution is derived from classical statistical mechanics. The number density Ni (or dN , respectively) of particles with energy ui (or between u and u + du, respectively) may be written as

or

Ni gi = exp(−ui /kB T ) N Z(T )

(1.54)

dN ∝ g(u) exp(−u/kB T )du

(1.55)

with the total number density N of the particles and the so called partition function Z(T ) = gi exp(−ui /kB T ) which ensures that summation over all states i on the right hand side of (1.54) gives 1. Normalization in the continuum case needs some more detailed consideration. We exemplify this by the distribution of velocities vx , vy , vz in an ideal gas. With the particle mass m the kinetic energy is u = m(vx2 + vx2 + vx2 )/2 and g(vx , vy , vz ) = const (since for −∞ < vx,y,z < ∞ no velocity vector is a priori more probable than another) one may write the velocity distribution dN = N

m 2πkB T

3/2

m(vx2 + vx2 + vx2 ) exp − dvx dvy dvz , 2kB T

(1.56)

which is normalized such that integration over all velocities gives 1. If, on the other hand, one is interested in the probability of finding a particle with a certain magnitude of velocity in a range v and v + dv one has to integrate over all angles so that dvx dvy dvz = 4πv 2 dv. This leads to the well known M AXWELL -B OLTZMANN velocity distribution

m 3/2 2 2 mv 2 dN = dv, (1.57) v exp − N π kB T 2kB T which is again properly normalized so that√the integral over all velocities 0 ≤ v < ∞ is 1. The most probable velocity is vm = 2kB T /m. One may rewrite (1.57) as an energy distribution by substituting u = mv 2 /2: 2 dN =√ N π

1 kB T

3/2

√ u u exp − du. kB T

(1.58)

Comparing with (1.55) √ we see that the density of states in the continuum of kinetic energies is g(u) ∝ u. The mean energy is given by 2 u = √ π

1 kB T

3/2 0

∞

√ 3 u du = kB T u u exp − kB T 2

(1.59)

22

1 Basics state

1

2

3

Boltzmann

1

2

3

Bose-Einstein

1

2

3

Fermi-Dirac

Fig. 1.9 How two particles may be distributed onto three states; this illustrates the key difference between the statistics according to B OLTZMANN (distinguishable classical particles, here red and black), B OSE -E INSTEIN (indistinguishable particles, no further limitations), and F ERMI -D IRAC (indistinguishable particles, PAULI principle)

and thus fully recovers the classical equipartition theorem for three degrees of freedom in the It also confirms the expression (1.49) for the averaged square form (1.49). √ √ velocity v 2 = 3kB T /m, while v = 8T kB /πm. So much about classical statistics. When reconsidering this from a quantum mechanical point of view one has to account for phase space quantization as well as for the indistinguishability of identical particles. The former aspect implies that even the continuum is not completely continuous. Rather, the 6 dimensional phase space (3 position and 3 momentum coordinates) has a finite cell size h3 . From this and with the degeneracy ge = 2s + 1 due to the spin s of the particles under consideration (see Eq. (1.8)) follows the density of states for a gas of non-interacting particles in the continuum: √ gs (2m)3/2 √ 4 2πm3/2 √ g(u) = u = gs u. (1.60) 4π 2 3 h3 Its dimension is Enrg−1 L−3 . We shall give a derivation of g(u) for the model of a free electron gas in Sect. 2.4.3 and discuss the quantization of electromagnetic radiation in Sect. 2.2.2, Vol. 2. Here we refrain from elaborating on the derivation of the statistical distributions, refer to the standard textbooks on statistical thermodynamics and quantum statistics, and just summarize some key results. In respect of indistinguishability, quantum mechanics knows two kinds of particles that behave differently: Bosons and fermions which we have introduced already in Sect. 1.1.1 as particles with integer and half integer spin s, respectively. For fermions (e.g. e− , e+ , p, 3 He, etc.) the PAULI exclusion principle (N OBEL prize 1945) holds so that each (discrete or continuum) quantum state can only be occupied by one particle at most. In contrast, bosons (e.g. photons, 2 H = D, 4 He, 12 C, etc.) are not restricted by the PAULI principle, i.e. each state may be occupied by many particles. But again, identical bosons are indistinguishable in contrast to classical theory. Figure 1.9 illustrates the fundamental differences between the three

1.3


23

statistics for the most simple example: how can two particles be distributed onto three states? Detailed considerations on the probabilities for populating (many) energy levels for many particles lead to the different statistics: the B OLTZMANN distribution for classical particles, the B OSE -E INSTEIN distribution6 for bosons, and the F ERMI D IRAC distribution7 for fermions. To compare all three statistics (concentrating on a quasi continuum of states) we also rewrite the B OLTZMANN distribution (1.55) suitably: B OLTZMANN

dN =

1 × g(u)du exp[(u − μ)/(kB T )]

(1.61)

F ERMI -D IRAC

dN =

1 × g(u)du exp[(u − μ)/(kB T )] + 1

(1.62)

B OSE -E INSTEIN

dN =

1 × g(u)du. exp[(u − μ)/(kB T )] − 1

(1.63)

We have introduced the so called chemical-potential8 μ which here allows us to properly normalize the distributions. Note that the dimensions of these equations are m−3 since with the density of states according to (1.60), g(u)du is a number per unit of volume, and so is N . In the case of discrete states, one simply has to replace dN by Ni /N and g(u) du by gi /Z(T ). The fractions displayed in the middle of these equations are the B OLTZMANN factor, the F ERMI function and the B OSE -E INSTEIN function, respectively (not distribution, as they are sometimes called colloquially). Proper normalization requires that the integration over all energies 0 ≤ u < ∞ is carried out using the whole right hand expression including the density of states. The result must be N , the number density of the gas under consideration. In this way one determines the chemical-potential μ. As we shall see in a moment, it depends on T , N , m, and ge and on the type of statistics to be applied. It is interesting to note that the three statistics differ only by an additive constant 0, 1 or −1 in the denominator – and of course by the specific value of μ. As it turns out, for sufficiently high temperature T and/or not too high density N they are virtually indistinguishable. However, they differ substantially at low temperature: at T = 0, bosons are all in the lowest state and the total energy of the system is zero, while fermions populate a band of energies up to a maximum value F , called Fermi energy (see Sect. 2.4.3). 6 B OSE

first applied this to photons while E INSTEIN generalized it to any bosons.

7 F ERMI

and D IRAC developed it independently from each other in 1926, F ERMI somewhat earlier than D IRAC.

thermodynamics, the chemical potential is defined as the partial derivative ∂G/∂ N of the free enthalpy G (G IBBS potential) with respect to the particle number N at constant temperature and pressure. Thus, μ gives the amount of energy that is necessary to change the number of particles of a system (by 1) without disturbing the equilibrium of the system.

8 In

24

1 Basics

For a gas of free, noninteracting particles, we may insert the density of states g(u) from (1.60) explicitly, introduce a quantity √ 4 2πm3/2 A = gs (1.64) N h3 which emphasizes the quantum nature of these statistics, and obtain: √ dN udu =A× B OLTZMANN N exp( u−μ kB T ) √ dN udu F ERMI -D IRAC =A× u−μ N exp( kB T ) + 1 √ dN udu =A× B OSE -E INSTEIN . u−μ N exp( kB T ) − 1

(1.65)

(1.66)

(1.67)

For normalization we have to integrate the right hand sides of (1.65)–(1.67). By substituting x = u/(kB T ) and ξ = μ/(kB T ) the normalization condition becomes √ ∞ xdx ! 3/2 =1 (1.68) A(kB T ) exp(x − ξ ) + δ 0 with δ = 0 and ±1 for the B OLTZMANN, F ERMI -D IRAC and B OSE -E INSTEIN probability distributions, respectively. For the√B OLTZMANN distribution the integral can be evaluated in closed form and gives π /2 exp(ξ ) so that one finds for the chemical-potential μ: exp(−ξ ) = or

ξ=

gs (2πmkB T )2/3 N h3

(1.69)

μ gs (2πmkB T )3/2 = − ln . kB T N h3

(1.70)

At high temperatures and not too high densities the chemical-potential is thus negative. Just to obtain some feeling for typical values of this quantity we note that at normal conditions N = NL = 2.687 × 1025 m−3 and 273 K for He with m 4 u, s = 0 and gs = 1 one finds exp(−ξ ) 252106 – compared to which the additive constant δ = ±1 in the denominators of (1.66) and (1.67), respectively, is irrelevant. This is characteristic for gases under standard conditions where no difference among the three statistics is observable, and μ −0.293 eV is virtually identical for all three statistics. However, at low temperatures and/or high particle densities significant differences exist. For F ERMI -D IRAC and B OSE -E INSTEIN statistics μ has to be determined from a numerical integration of (1.68) with δ = ±1, respectively. It is important to note, that for B OSE -E INSTEIN gases μ must not be positive to avoid singularities in (1.67) (see, however, Sect. 2.2.2, Vol. 2). For μ = 0 the integral on the left side of (1.68) reaches its maximum and can be evaluated in closed form.


√u exp (u – μ) + δ

1.3

0

1

25

(a)

(b)

He 100 mbar at 2.5 K Tc = 0.18K

He 100 mbar at 1K Tc = 0.32K

2

3

4

5 6 u / k BT

0

1

2

3

4

5

6

Fig. 1.10 Comparison of the three statistical energy distributions for an atomic mass 4 u at 100 mbar and two different temperatures (a) 2.5 K and (b) 1 K. The red lines refer to B OSE -E INSTEIN statistics as applicable to He atoms, the grey line illustrates the classical M AXWELL-B OLTZMANN distribution and the black line represents the energy distribution obtained from F ERMI -D IRAC statistics

By comparing it to (1.64) one derives the so called critical temperature −2/3 2 2 −2/3 2/3 = 3.31 gs N Tc = 2πN 2/3 ζ (3/2)gs mkB mkB

(1.71)

using the value ζ (3/2) = 2.612 from the R IEMANN zeta function. At this critical temperature the much celebrated B OSE -E INSTEIN condensation (BEC) occurs (the pioneering work of C ORNELL, K ETTERLE, and W IEMAN was honoured with the N OBEL prize in C ORNELL et al. (2011)). In Fig. 1.10 we show two examples of the three statistical distributions – with T somewhat above the critical temperature Tc . The energy distributions shown have been calculated for a gas of mass 4 u at a pressure of 100 mbar for two different temperatures (a) T = 2.5 K and (b) 1 K. Clearly, in nature He gas would be described by the red curve corresponding to B OSE -E INSTEIN statistics, while the other two curves are just drawn for comparison. We recognize the differences for the tree statistics, but it is also clear by extrapolating the trends seen in this figure that at only several times the critical temperature, not the slightest difference will be recognizable. In contrast to B OSE -E INSTEIN statistics where the chemical-potential has to be μ ≤ 0, in the F ERMI -D IRAC case μ can assume also positive values (no singularities occur in Eq. (1.66)). One particularly important application is the model of a free electron gas for electrons in a metal. There the number density of electrons is very high and temperatures may be low. In that case, μ is called F ERMI energy, which can assume rather high positive values (in units of kB T ). We shall discuss and illustrate this in Sect. 2.4.3 and shall find that the energy distribution looks very different from a M AXWELL -B OLTZMANN distribution.

26

1 Basics

Section summary

• Statistical distributions are important in many areas of classical and quantum physics. They describe the probability to find an observable at a certain position in space or time or with a particular energy, frequency etc. Characteristic are the mean value (1.37) and the variance (1.38). • Most common is the exponential distribution which we have introduced in the context of spontaneous decay of excited states (1.40). It may be characterized by the half-lifetime of the excited states (the time after which half of the excited states have decayed) which is t1/2 = ln 2/A = τ ln 2, with τ the mean lifetime and A the transition probability. Analogue relations hold for the absorption of radiation though matter, described by the L AMBERT-B EER absorption law (1.44). • We have reviewed a few basic concepts from kinetic gas theory: the average free energy per atom and degree of freedom √ is kB T /2; the mean free path between two collisions in a gas is l = 1/( 2σ N ), with σ 10−15 m2 the gas kinetic cross section and N the particle density. • We have also discussed the three relevant statistics (1.61)–(1.63) for energy distribution in an ensemble of particles: the classical B OLTZMANN statistics, F ERMI statistics (valid for fermions, i.e. particles with half integer spin) and the B OSE -E INSTEIN statistics (valid for bosons, particles with integer spin quantum number). At temperatures a few degrees above the critical temperature (1.71) for B OSE -E INSTEIN condensation all three distributions are almost identical.

1.4

The Photon

From classical wave optics we know that light can be described as electromagnetic waves: diffraction and interference are the experimental observations onto which this viewpoint is based. In terms of geometrical optics, light propagation may even be described simply by so called light rays or beams – a notion which can be based on wave optics as a special case (more in Chap. 1, Vol. 2). However, light has also particle properties. The key observations documenting this aspect are summarized in this section. Quantum mechanics, to be addressed in some detail in the next chapter, reconciles both points of view – or rather: it provides a set of rules for a consistent interpretation of the experimental observations.

1.4.1

Photoelectric Effect and Quantization of Energy

One of the fundamental observations on the quantum nature of light is the photoelectric effect. With light of a wavelength λ (frequency ν = c/λ) one illuminates a metal surface from which electrons emerge. One measures the kinetic energy T of

The Photon

27 potential energy

1.4

e− hν electron sea

T WA x

Fig. 1.11 Simple potential well model to explain the photoelectric effect. The photon of energy hν rises an electron • from the ‘electron sea’ in the metal (bound) into the continuum (free). A hole is left in the sea

these emitted electrons and makes some quite remarkable observations (astonishing at least at the time early in the 20th century when this was discovered): • in contrast to the classical expectation the energy of the photoelectrons is independent of the intensity of the irradiating light: the latter only determines the number of emitted electrons • the observed kinetic energy T of the photoelectrons has a maximum value T (max) = hν − WA ,

(1.72)

where h is the P LANCK constant (1.2) and WA the so called work function or electron affinity of the metal surface from which electrons emerge (for experiments in the gas phase WA has to be replaced by the ionization potential WI of the atoms or molecules studied). E INSTEIN (1905) in his ‘annus mirabilis’ presented the interpretation for the photoelectric effect – one of the key steps in the early days of modern physics for which he received the N OBEL prize in physics9 in 1921: Light energy exists only in well defined energy packets of Wph = hν = ω.

(1.73)

This energy packet is the elementary quantum of light, called photon. Light obviously has both: wave and particle character. To get some numerical feeling, let us consider yellow light (from the sun or from a sodium street lamp) at a wavelength of λ = 589 nm. With c = λν we have ν = 5.09 × 1014 Hz. Thus, Wph = hν = 3.37 × 10−19 J = 2.10 eV is the energy of the photon! One may visualize the photoelectric effect by a very simple potential well model for quasi free electrons in a metal. The energetic relations between T , WA and hν are illustrated in Fig. 1.11. If the photon has an energy hν > WA it may eject an electron from the ‘electron sea’ (where it is bound but may freely move within the metal) into the continuum (where it is unbound). If the electron originates from the 9 Note:

not for his at least equally important theory of special relativity.

28

1 Basics scattered radiation

(a)

collimating apertures incoming

detector

(b)

scattered photon

p'

θ

γ-radiation metal target

incoming photon p=h/λ W = hν

free electron pe = 0 We = m e c 2

p' = h/ λ' W' = hν'

θ p'e + p' = p p'e

electron scattered electron We' = √c2p'e2 + me2 c 4

Fig. 1.12 (a) Experimental scheme to study the C OMPTON effect. (b) Kinematics of the scattering process. Prior to collision energy (momentum) are W (p) and We (pe ) for photon and electron, respectively; the dashed quantities refer to the situation after the process

surface, it will have a kinetic energy T = T (max) as given by (1.72). An electron hole is left in the sea. The photoelectric effect is the basis for modern photoelectron spectroscopy (PES) as we shall elaborate on in some detail in Sect. 5.8 in Vol. 2. One may easily visualize from Fig. 1.11 that a precise determination of the spectrum of kinetic energies T of the emitted electrons may serve as a sensitive tool to determine the electronic structure of the object studied. The potential well is of course only a very rough approximation of reality, which more correctly would be described by the band structure of a solid surface. Note: The photoelectric effect as described above refers to low photon and electron fluxes (linear regime). If one uses very intense laser pulses which can readily be generated with state-of-the-art laser techniques, the situation changes. With increasing light intensity the process becomes no longer linear, i.e. more than one photon will be involved in the emission process of one electron (see Sect. 8.5.1). Then the phenomena observed may even approach the classical expectation, which so surprisingly was not met in the original photoelectron emission experiments at low intensity.

1.4.2

COMPTON Effect and Momentum of the Photon

The C OMPTON effect (N OBEL prize in 1927) may be observed with an experimental setup as sketched schematically in Fig. 1.12(a). Highly energetic photons (γ -radiation) are scatted from quasi free metal electrons. The momentum of the photon is p = k

and p = h/λ = ω/c.

(1.74)

It enters into the kinematics of the experiment as illustrated in Fig. 1.12(b). Both, momentum p = p + p e and energy W + We = W + We have to be conserved during the scattering process, using the relativistic expression (1.19) for the electron energy. With some algebra one finds that the wavelength λ of the scat-

1.4

The Photon

θ = 0º

29

incoming radiation (dashed)

θ = 45º

θ = 90º λ

intensity of the scattered γ - radiation

λ θ = 135º λ

λ λ = 0.710 Å

λ = 0.710 Å

λ' = 0.751 Å

Fig. 1.13 Wavelengths of γ radiation after C OMPTON scattering

tered γ -radiation is shifted in respect of the incoming λ. The shift depends on the scattering angle θ : λ − λ = λC (1 − cos θ ).

(1.75)

Experimentally one observes the predicted shift of wavelength in the scattering signal as a function of the scattering angle θ illustrated in Fig. 1.13. This shift is a direct consequence of the momentum of the photon. Thus, along with the photoelectric effectthe C OMPTON effect constitutes an important proof for the particle properties of the photon. The parameter in (1.75) is the so called C OMPTON wavelength of the electron (with the electron mass me ):10 λC =

h = 2παa0 = 2.4262 × 10−12 m. me c

(1.76)

It determines the overall magnitude of the wavelength shift and is independent of the irradiating wavelength λ. We compare several characteristic lengths to convey some feeling for the relevant orders of magnitude. The C OMPTON wavelength of a particle corresponds to the wavelength of a photon whose energy is equal to the rest mass energy mc2 of that particle. For the electron it is in between atomic and nuclear radius: atomic radius (H atom 1s )

a0 = 0.529 × 10−10 m

C OMPTON wavelength of e−

λC = 2.4262 × 10−12 m

proton radius

Rp = 0.875 × 10−15 m.

the reduced C OMPTON wavelength /me c = αa0 = 3.8110 × 10−12 m is used. In relativistic quantum mechanics one typically measures lengths in units of the reduced C OMPTON wavelength /me c, and atomic energies in units of me c2 . 10 Often

30

1.4.3

1 Basics

Pair Production

The photoelectric effect and the C OMPTON effect are two major mechanisms by which high energy photons interact with matter. For completeness we also mention pair production: in the vicinity of a nucleus a photon can be converted into an electron and a positron. This process is symbolically written as γ → e− +e+ and has the energy balance hνγ = 2me c2 + Te− + Te+ ,

(1.77)

so that this process becomes possible if and only if the energy of the photon hνγ > 2me c2 1.022 MeV, the rest mass of one electron and one positron. The excess energy is converted (essentially) into kinetic energy Te+ +Te− of the two emerging particles. For momentum conservation (at threshold the electron and positron momenta are very small) the process can only occur in the presence of a nucleus which – by C OULOMB interaction – carries away the surplus momentum of the photon. Pair production may also be viewed as excitation of an electron into the world from the “D IRAC sea” in which a whole is created (the positron). All three processes, photoelectric effect, C OMPTON effect and pair production, are the key mechanisms for absorption of high energy photons in by atoms (specifically in the solid state). We shall come back to this in Sect. 10.5.3. For completeness we mention that the exactly inverse process is extremely improbable due to phase space considerations. In contrast, the generation of two photons by positron-electron annihilation e− +e+ → 2γ is a well known process, in which energy and momentum conservation is easily realized. The two photons are emitted in exactly opposite direction. This process is exploited in positron emission tomography (PET), today a widely used medical technique for precise tumour imaging. The positron in this case originates from an artificial isotope attached to a drug which is accumulated specifically in tumour cells. Detecting the two photons after e− e+ annihilation in coincidence allows one to localize their origin in the human body.

1.4.4

Angular Momentum and Mass of the Photon

For completeness we mention already here that the particle “photon” also has an intrinsic angular momentum , called the photon spin with a quantum number s = 1. We shall learn more about the experimental evidence in Sect. 4.1.4. The photon spin will play an important role in various contexts, and a quantum mechanical description of the photon will be presented in Sect. 2.2, Vol. 2. We may even attribute a mass to the photon, the equivalent of its energy: mP h = hν/c2 .

(1.78)

Note, however, that the rest mass of the photon is zero: it exists only as a particle moving with the speed of light. We shall later on discuss that this has serious consequences for space quantization of its angular momentum. Briefly, the massless parti-

1.4

The Photon

31

cle photon has only two substates with sz = jz = ±, while according to (1.9) a particle with rest mass and angular momentum j = s = 1 has three possible substates.

1.4.5

Electromagnetic Spectrum

Electromagnetic radiation is the key for most spectroscopic studies in atomic and molecular physics. The relevant radiation ranges from radio frequency photons whose energy is in the μeV energy range up to the hard X-ray region with energies up to MeV. Figure 1.14 gives an extended overview on the whole electromagnetic spectrum of relevance. Note that slightly different definitions are used for the spectral ranges shown in Fig. 1.14, depending on the field of application. We follow the specifications of ISO 21348 (2007).11 Different units are used in different spectral regions: Frequencies ν in the very low energy range, wavelengths λ in the infrared (IR), in the visible (VIS), ultraviolet (UV) and vacuum-ultraviolet (VUV) spectral range.12 For still shorter wavelengths, i.e. in the extreme ultraviolet (XUV), in the soft and hard X-ray region as well as for γ -rays one uses energy units (ω in eV, keV, MeV). In spectroscopy a convenient measure is the reciprocal wavelength, the so called wavenumber ν¯ = 1/λ

(1.79)

that is proportional to the photon energy Wph = hcν¯ = ω = hν = hc/λ

(1.80)

= ν¯ × 1.239841875(31) × 10−4 eV cm. The SI unit of wavenumbers is m−1 but commonly one still uses [¯ν ] = cm−1 , often literally called “wavenumber”. Up to date energy conversion factors are found e.g. at NIST (2011).

1.4.6

PLANCK’s Radiation Law

Quantum mechanics, one may say, was triggered by understanding the photoelectric effect due to E INSTEIN (1905) (N OBEL prize 1921). But before that (1900), P LANCK’s law had already revolutionized the world of physics (N OBEL prize 1918) – giving an accurate interpretation of black body radiation for which the dependence on wavelengths λ and absolute temperature T had been measured with very high precision. The characteristic behaviour is illustrated for several temperatures in Fig. 1.15. The interpretation of this fundamental distribution of radiation forced 11 Except 12 Called

for RF and MW where we follow the technical literature.

“vacuum-ultraviolet” since absorption in air forces one to work in vacuum with these wavelengths.

109 Hz

1m

1 meV microwave oven

EPR

clystron, magnetron

radar

1 eV

1 m

IR-FEL

TiSa

1 keV

HHG

FEL

1018 Hz

1 nm

laser plasma

XFEL

X-ray tube

medical diagnostics

nuclear reactions

1 MeV

1021 Hz

1pm

Ȗ-ray

1 GeV

1 fm

cosmic Ȗ-radiation nuclear reactions

hard X-ray

Cu K-edge

soft X-ray XUV

between K-edge of C and O

H2O window

EUV for lithography (13.5 nm)

1015 Hz

synchrotron radiation

light bulb

EUV

A BC VUV

visible UV

MIR NIR

optical fibre communication

IR

FIR

H Lyman-Į

maximum of solar radiation

various lasers and NLO

1012 Hz

1 mm

THz

CMBR

microwave

Cs atomic clock

cellular phones

1 ȝH9

NMR

radio, TV

electronic oscillators and antennas

1 neV

106 Hz

1 km

UHF

9192631770 Hz

Fig. 1.14 Spectrum of electromagnetic waves (for acronyms see text and p. 81ff.). (a) Specific sources and properties, (b) terminology, (c) wavelength scale, (d) frequency scale, (e) energy scale, (f) examples for applications, (g) examples for methods of generation. Note the narrow range of visible radiation. Only a few special sources of radiation are specifically emphasized such as CMB(R) and H Lyman-α. The ultraviolet (UV) regions UVA, UVB, and UVC are abbreviated here by A, B, and C

(g)

(f)

(d) (e)

(c)

RF

HF

VHF

MF

(b)

LF

21cm H-line

(a)

32 1 Basics

1.4

The Photon

33 1.0 5772 K (sun)

1.0

1073 K

u( ) / J m−3 m-1

3400 K ×10 2856 K ×10

0.5

×4000

0.5

0

5

288 K (earth) ×106

10

15 μm

1873 K ×100 0.0 0

1

2

3

4

5 wavelength

visible spectrum (390 to 750) nm

/ m

Fig. 1.15 P LANCK’s law at different temperatures: 5772 K effective black body temperature of our sun, 3400 K special purpose, short-lived incandescent bulbs, 2856 K CIE standard “illuminant A” (about 100 W tungsten incandescent bulb), 1873 K blast furnace at the discharge aperture, 1073 K dark red glow (kitchen stove, grill fire), 288 K earth surface temperature (average)

Max P LANCK to introduce, at the beginning very reluctantly, a new fundamental constant of action h (units [h] = J s). P LANCK was well aware that this manifested the breakdown of classical physics! Today, the P LANCK constant is a fundamental physical constant, known with very high precision (see Eq. (1.2)). Here without derivation (it will follow in Sect. 2.2.3, Vol. 2) we communicate P LANCK’s law for the spectral energy density distribution of the black body radiator as u(ν)dν ˜ =

dν 8πhν 3 , 3 exp(hν/kB T ) − 1 c

(1.81)

per unit of frequency with [u(ν)] ˜ = J m−3 Hz−1 , or as u(ω)dω ˜ =

dω ω3 π 2 c3 exp(ω/kB T ) − 1

(1.82)

per unit of angular frequency, [u(ω)] ˜ = J m−3 s, with the speed of light c and the B OLTZMANNconstant kB = 1.3806504(24) × 10−23 J K−1 .

(1.83)

Alternatively one plots – as done in Fig. 1.15 – the distribution as a function of the wavelength λ so that [u(λ)] ˜ = J m−4 = 10−3 mJ m−3 μm−1 and u(λ)dλ ˜ =

dλ 8πhc . 5 λ exp(hc/kB λT ) − 1

(1.84)

34

1 Basics

Equations (1.81) and (1.84) are the standard form of P LANCK’s law for the black body radiation density – as originally published by P LANCK (1900). Often one is also interested in the intensity at a given point, direction and wavelength per unit of projected area, solid angle and frequency interval with [Lν ] = W m−2 sr−1 Hz−1 (or [Lλ ] . . . per wavelength interval), also called spectral radiance. Since black body radiation is intrinsically isotropic, the corresponding expressions are obtained by simply multiplying (1.81) and (1.84) with c/4π , i.e. the prefactor 8πhc is replaced by 2hc2 . Typically measured is the spectral distribution of the intensity,13 i.e. of the radiation power emitted (per area) into the forward hemisphere, with [I˜(λ)] = W m−2 nm−1 . Integration over the cos θ angular distribution (projection of the surface area onto the direction of emission) gives a factor of π so that dλ 2πhc2 . I˜(λ)dλ = 5 exp(hc/kB λT ) − 1 λ

(1.85)

The wavelength λmax at which this spectral distribution of radiation reaches its maximum, decreases with temperature as seen in Fig. 1.15. Explicitly one finds (from dI˜(λ)/dλ = 0 for the maximum) the so called W IEN wavelength displacement law: λmax T = b

with b = 2.8977721(26) × 106 nm K.

(1.86)

Finally, the total power emitted per unit area from the surface of a black body is obtained by integrating (1.85) over all wavelengths. This gives the important S TEFAN -B OLTZMANN law for the intensity of black body radiation: I (T ) = 0

∞

2 π 5 kB4 4 T = σB T 4 . I˜(λ)dλ = 15 h3 c2

(1.87)

Thus, the (spectrally integrated) intensity depends on the fourth (!) power of the absolute temperature T . The proportionality factor σB = 5.670373(21) × 10−8 W m−2 K−4 is called S TEFAN -B OLTZMANN constant.

1.4.7

Solar Radiation on the Earth

At this point, a few words are in order on the radiation which we receive every day from our sun. Some relevant parameters are summarized in Table 1.5. The spectral distribution of the solar radiation at the top of the earth atmosphere is shown in Fig. 1.16 (wiggly red line). Since about 2004 it is constantly monitored by satellite. The solar constant S is the integral over this spectrum. The daily results can be obtained online from SORCE (2012) (the data reported there are renormalized to 13 In radiometry one uses the terms irradiance or radiant flux (see also Sect. 1.4.8). For consistency

with the general custom in AMO physics we usually call this quantity intensity of the radiation, measured in [I ] = W m−2 .

1.4

The Photon

35

Table 1.5 Some properties of sun and earth in the context of solar radiation (data from SSE 2012, unless otherwise specified) Mean radius sun

R

6.9551 × 105 km

Mean radius earth

RE

6371.0 km

RSE

149.60 × 106 km

(from KOPP and L EAN 2011)

Mean distance

sun-eartha

1 uab

S

1360.8(5) W m−2

Radiation power emitted

by sun

384 × 109 PW

(into 4π sr)

Radiation power received

by earthd

173.5 PW

(from the sun)

Effective temperature

TS

5772 K

(for the above value of S)

Sun surface temperature

Tph

4400 K to 6600 K

(top to bottom photosphere)

Albedo (B OND)

a

0.306

(fraction of radiation reflected)

Earth temperaturee

TE

254 K

(effective black body)

TEa

288 K

(average at surface)

Solar

constantc

a Due

to the ellipticity of the orbit the distance varies between parhelion (minimum) and aphelion (maximum) by about 6.9 % between 4th of January and 4th of July b Per definition the astronomical unit of length is 1 ua = 149597870700 m; 1 ua is almost identical to the average distance RSE of the earth from the sun c Defined d Above eT E

as mean irradiance at 1 ua from the sun

atmosphere

2 , emitting 4πR 2 = [S × (1 − a)/(4σB )]1/4 from (1.87); receiving surface πRE E

solar spectrum at the top of the earth atmosphere

2.0

RAYLEIGH scattering

~ I (λ) / W m−2 nm-1

O2 PLANCK at 5772 K

O2

0.05

1.0 H 2O

O3 0.0

solar spectrum at sea level

0

H2O H2O H2O

1000 UV

vis

×20

IR

2000

H2O, CO2

×20

3000

4000

wavelength λ / nm

as measured by SORCE Fig. 1.16 Spectral intensity distribution I˜(λ) of the solar radiation: (AM1.5 (2012) above the earth’s atmosphere (for λ > 2400 nm from ASTM 2008); at sea level black body radiator at 5772 K. Also indicated global tilt spectrum from ASTM 2008, see text); are the main absorbing molecules in the earth’s atmosphere

36

1 Basics

Fig. 1.17 Definition of air mass coefficient AM for solar radiation standards. Table on the right: solar radiation (integrated from 280 to 4000 nm) at normal incidence for different zenith angles ζ

a the astronomical unit of length 1 ua, see Table 1.5). The spectral distribution is remarkably stable over time, the average changes being less than the width of the red line in Fig. 1.16. As illustrated, the distribution is approximated reasonably well by a P LANCK distribution from a black body at 5772 K (smooth black line). The latter is obtained from (1.85) by multiplying it with (R /1 ua)2 , where R is the solar radius. No further scaling is needed to obtain this fit. As we see, the fit is not perfect, but considering the gigantic nuclear fusion reactor which our sun actually is, and considering its complicated photosphere, the relative similarity with a black body radiator is quite remarkable – and so is the stability of the distribution. A variety of values for the ‘sun temperature’ may be found in the literature. From the recent, highly accurate measurement of the solar constant, S = 1360.8 W m−2 , by KOPP and L EAN (2011) (see also SORCE 2012) and with appropriate scaling of the S TEFAN -B OLTZMANN law (1.87), we obtain the effective black body temperature of the sun as TS = (1 ua /R )1/2 (S/σB )1/4 = 5772 K. Also shown in Fig. 1.16 is the spectral distribution of radiation which reaches the ground level after partial absorption and scattering of the incoming solar radiation by atmospheric gases. As sketched in Fig. 1.17, this obviously depends on the zenith angle ζ of the sun (ζ = 90◦ – latitude angle). For not too large ζ the optical path length through the atmosphere is given by hx = h0 / cos ζ , where h0 7.7 km is the effective vertical height of the earth’s atmosphere (i.e. the height where the pressure has dropped to 1/e of its value on ground). The ratio AM =

hx 1 h0 cos ζ

(1.88)

is called air mass coefficient. An approximate empirical formula (see e.g. H ONS BERG and B OWDEN 2012) for the intensity reaching the earth surface is I = 1.1 I0 × 0.7AM

0.678

,

(1.89)

1.4

The Photon

37

where I0 is the radiation arriving on top of the atmosphere (essentially the solar constant S, given in Table 1.5). The prefactor 1.1 accounts for radiation scattered from the air and retroreflected from the ground, the exponentials reflect somehow the L AMBERT-B EER absorption law. With reference to Fig. 1.17 we also note that the intensity depends in addition on the angle of incidence (ζ − θ ) onto the receiving surface, Isur = I × cos(ζ − θ ), with (1.89) referring to normal incidence. As a standard value, AM = 1.5 (short AM1.5) has been adopted, considered representative for most industrialized countries at noon time. Based on measurements and modelling, one also has defined two standards spectral distributions arriving on the earth surface, both for AM1.5: (i) direct normal incidence of radiation and (ii) hemispherical (global) incidence on a 37◦ degree tilted surface, which also includes scattered and retroreflected radiation (ASTM 2008, G173-3).14 The latter spectrum is shown in Fig. 1.16 and referenced the table with Fig. 1.17. We note at this point that this spectrum has its maximum at ca. 500 nm, i.e. near the maximum of the spectral sensitivity of the human eye at 555 nm. The evolutionary context behind this ‘coincidence’ is evident. In contrast, the surface of our earth with an average temperature of ca. 288 K emits in the IR spectral range with a maximum at about 10 μm where the “greenhouse gas” CO2 absorbs (we shall come back to this in Sect. 5.3.1, Vol. 2) – fortunately we have to say: without it, the temperature on the earth surface would correspond to the effective black body temperature of the earth TE = 254 K and life would not be possible.

1.4.8

Photometry – Luminous Efficiency and Efficacy

Black body radiators are very inefficient when used to generate visible light. Even the sun emits only a fraction of its total radiation energy into the visible (VIS) spectral region: only 46 % of the total radiation power are emitted between 380 and 760 nm, as one finds from integrating (1.87). Incandescent light bulbs at typical temperatures convert much less energy into the visible spectral range (e.g. ∼6.5 % for a 100 W bulb with its tungsten wire at 2856 K). Worldwide it is understood that intense efforts are needed to ‘save energy’. Primary energy resources must be used in a most efficient way and energy efficient lighting is an import potential. As a consequence, the good old light bulb, which humankind has gotten so used to since over 100 years, is approaching its last days very rapidly. The overall efficiency in creating luminosity as registered by the human eye is even much worse, since u(λ) ˜ has to be multiplied by the physiological sensitivity V (λ) of the eye. This so called photopic luminous efficiency function peaks at 555 nm,15 where the human eye has its maximum sensitivity. It is standardized by 14 See also ISO 60904-3 (2008) or DIN EN 60904-3. Note that for AM1.5 normal incidence and incidence on a 37◦ degree tilted surface differs only by 2 %. 15 Vision

in bright light is called photopic, in contrast to scotopic at low light levels and maximum sensitivity at ca. 498 nm.

38

1 Basics

Table 1.6 Relation between photometric and radiometric quantities after O HNO (2010) Photometric

Unit

Relation to lm

Radiometric

Unit

Luminous flux

lm (lumen)

Radiant flux

W (Watt)

Luminous intensity

cd (candela)

lm sr−1

Radiant intensity

W−1 sr−1

Illuminance

lx (lux)

lm m−2

Irradiance

W m−2

Luminance

cd m−2

lm sr−1 m−2

Radiance

W sr−1 m−2

Luminous exitance

lm m−2

Radiant exitance

W m−2

Luminous exposure

lx s

Radiant exposure

J m−2

Luminous energy

lm s

Radiant energy

J (joule)

Color temperature

K (Kelvin)

Radiance temperature

K

the Commission international de l’éclairage (CIE) and forms the basis for all luminosity determinations. For back of an envelope calculation it may roughly be approximated by a Gaussian, centred at 560 nm with a FWHM 104 nm (from a least square fit). But for precise work tabulated values from the literature have to be used, e.g. from CIE (or as plotted by D ICK LYON 2008, with detailed references). For a brief and concise introduction into photometry we refer to O HNO (2010) from whom we also have adopted Table 1.6. It compares (physiologically weighted) photometric quantities with (directly energy related, physical) radiometric quantities.16 The relevant photometric SI unit is the candela, cd, defined as “. . . the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency17 540 × 1012 Hz and that has a radiant intensity in that direction of (1/683) W per sr.” However, the physically more fundamental unit is the lumen (lm), the unit for the luminous flux. It measures the photometric equivalent of the total radiation power emitted by a given light source. Thus, the hypothetical, ideal light source driven by 1 W electric power emits 683 lm at 555 nm. It is said to have an overall luminous efficacy of 683 lm W−1 and its luminous efficiency is defined as 100 % – it emits the maximum possible amount of visible light. All other light sources have (much) lower luminous efficacy and efficiency. One may compute the ∞ V (λ)I˜(λ)dλ luminous efficiency = 0 ∞ (1.90) I˜(λ)dλ 0

from the spectral radiation intensity I˜(λ) = 4u(λ)/c ˜ of the source and the photopic luminosity function V (λ). For a black body radiator I˜(λ) = cu(λ)/4 ˜ is obtained from (1.84) and the denominator in (1.90) is σB T 4 , the total radiation power emit16 Note that this radiometric terminology somewhat confusingly differs from the standards used elsewhere in optical and laser physics. A laser beam e.g. is typically characterized by its intensity measured in units [I ] = W m−2 , while here in radiometry the corresponding quantities in Table 1.6 are called irradiance or radiant exitance. 17 That

is at a wavelength of 555 nm where the human eye has its maximum sensitivity.

1.4

The Photon

39

ted per unit area. This luminous efficiency is a function of T and has a maximum of about 14.5 % at T = 7000 K. For our sun it is ca. 14 %, while for the above mentioned CIE standard “Illuminant A” the efficiency is only about 2.5 %, corresponding to a luminous efficacy of ca. 17 lm W−1 . A typical standard 100 W incandescent bulb emits a total luminous flux of 1360 lm and thus its efficacy is 13.6 lm W−1 . Of all the electric energy used to heat the tungsten wire of the bulb, 98 % is lost as IR radiation and eventually dissipated as heat! So called halogen lamps are somewhat more efficient since a special chemical process reduces evaporation of the cathode material and the temperature of the tungsten wire can be significantly higher. Still, the general efficiency problem of incandescent lamps remains. Thus, the challenge is, to exploit more efficient ways for converting electrical energy into visible light. One direction that currently is followed uses miniaturized fluorescent lamps. Typically, in a mercury gas discharge UV light is generated at 253.7 and 185 nm. It is then converted into visible light of different wavelengths by fluorescing materials, so called phosphors. Typically three to five different phosphors with reasonable efficiency are used, whose fluorescence bands by additive colour mixing appear more or less as ‘white’ light. Much current development is focused onto these materials. By judicious choice of the fluorescent bands one may obtain a more or less continuous coverage of the visible range – thus imitating the visible part of the black body radiation without loosing energy into other spectral regions. The colour temperature of such a source is defined as the temperature of a black body radiator that best matches that spectrum in the visible. Presently, the luminous efficacy of compact fluorescent lamps (commercial energy saving lamps) is on the order of (50 to 60) lm W−1 (7.3 % to 8.7 % efficiency) but up to more than 100 lm W−1 (14.6 % efficiency) can be achieved with long tubular fluorescent lamps. For street lighting also high and low pressure sodium discharge lamps are used, easily recognized by their yellow-orange light (around 590 nm). Low pressure Na discharges have the highest luminous efficacy in state-of-the-art lighting technology of up to 200 lm W−1 (29 % efficiency). Light emitting diodes (LED) could, in principle, be able to supersede these values: they convert electric current – so to say – more or less directly into light. Again, additive colour mixing of at least three LED’s is required for generating white light in practical devices. The luminous efficiency is presently of a similar order of magnitude as for fluorescent lamps, somewhere between 15 and 25 %. However, strategies are discussed to achieve much higher efficiencies (e.g. B RETSCHNEIDER 2007), and the price for such high-tech products typically decreases exponentially with years of experience and mass production. Occasionally the question comes up about a theoretical maximum of luminous efficiency. It is difficult to answer: consider a (very hypothetical) light source converting electric power to a 100 % into a spectrum, which in the visible (380 and 760 nm) corresponds exactly to the ideal white light of our sun, and is zero elsewhere. According to (1.90) this would correspond to a luminous efficiency of 38 %. Hence, conversion of 100 % electric energy into visible light is – per definition – a completely unrealistic goal: it would mean totally green illumination. Thus, any

40

1 Basics

Fig. 1.18 B RAGG reflection at two lattice planes of a crystal with a distance d. Note that the B RAGG angle δ is defined complementary to the angle of incidence commonly used in reflection optics

G X-r

θ

ay

l sta e cry rfac su

k'

in k

n sig

al o

ut

d lattice planes

d d

s

claims for a luminous efficiency higher than 34–38 % have to make massive compromises on the effective colour temperature. We shall certainly witness an exciting development in the lighting industry over the coming years.

1.4.9

X-Ray Diffraction and Structural Analysis

Electromagnetic radiation in all spectral ranges – light in the widest sense – is today one of the most important tools to reveal the structure and dynamics of matter. In later chapters we shall learn a lot about the various spectroscopic techniques that are used in this context. At this point we want to mention, at least briefly, one of the key methods for structural analysis: X-ray diffraction, i.e. scattering and interference of electromagnetic radiation of very short wavelength from crystalline matter. It is treated systematically in textbooks and monographs on solid state physics (see also Sect. 1.7.2). The basis for a variety of such methods is multiple beam interference from the crystal lattice structure studied. As sketched in Fig. 1.18 a multitude of parallel, so called lattice planes may be thought to reflect the X-rays. According to Fig. 1.18 the optical path difference between two rays reflected from neighbouring planes is 2s = 2d sin θ , where d is the distance of two lattice planes in the crystal and θ the so called B RAGG angle. Thus, the reflected X-ray beams (shown in Fig. 1.18 are only two of them) interfere constructively if and only if B RAGG’s law (also called B RAGG condition) 2d sin θ = zλ

with z = 0, 1, 2 . . .

(1.91)

holds. Here λ is the wavelength of the scatted X-ray radiation and z is an integer. For reference we also report some further quantitative relations relevant to X-ray scattering. One defines a reciprocal lattice vector G = hg 1 + kg 2 + lg 3

(1.92)

1.4

The Photon

41

constructed with the basis vectors of the unit cell g 1 , g 2 , g 3 in the reciprocal lattice, using the so called M ILLER indices h, k, l that characterize the lattice planes in the crystal. Without entering into details of crystal lattice theory we just note that these basis vectors in the reciprocal lattice relate to the ordinary basis vectors a 1 , a 2 , a 3 in the crystal lattice by g i · a j = 2πδij ,

(1.93)

and in Fig. 1.18 we identify d ≡ 2π/|G|. With wave vectors k and k of incoming and outgoing X-ray radiation B RAGG’s law (1.91) is now written

k = k − k = G.

(1.94)

Normally one investigates only elastic scattering where |k| = |k | and one may rewrite (1.94) as (k − G)2 = k 2 or G2 − 2kG = 0. We may finally write the condition for diffraction: 2kG = G

2

or

G k = 2

G 2

2 .

(1.95)

In the latter form the diffracted wave vectors describe a plane bisecting the reciprocal lattice vector G and being perpendicular to it. One may construct such planes for all elementary reciprocal lattice vectors and combine them to a closed surface in reciprocal lattice space. They form the so called B RILLOUIN zone (BZ) that represents all wave vectors of radiation that can be B RAGG reflected by the crystal. BZs are a very important concept in solid state physics, specifically relevant for the theory of band structure as we shall briefly discuss in Sect. 2.8. Since several different lattice vectors with different alignment in space exist (typically more than 3, see Eq. (1.92)), BZs may be rather complex surfaces. Corresponding surfaces may also be constructed from 2G, 3G etc. and one distinguishes the 1st, 2nd, 3rd, etc. B RILLOUIN zone. The intensity of the diffracted X-ray radiation depends on the (electron) charge density distribution (r) in the unit cell of the crystal and is proportional to the absolute square of the so called structure factor d3 r(r) exp(iG · r) (1.96) F(hkl) = cell

=

Fj (G) exp i2π(xj h + yj k + zj l) .

j

The summation has to be carried out over all atoms in the unit cell of the crystal, each of which is characterized by its atomic form factor18 Fj (q) = d3 rN (j ) (r) exp(iq · r). (1.97) atom 18 The

correct quantum mechanical equivalent (8.21) will be discussed in Vol. 2.

42

(a)

1 Basics

(b)

Fig. 1.19 (a) X-ray diffraction image from a large single crystal (80 × 100 × 50 μm) of the human enzyme prolidase, recorded with hν = 13.05 keV at the beamline BL14.1 of the Free University Berlin at BESSY.The maximum resolution of the diffraction image corresponds a lattice plane distance of the crystal lattice of 0.25 nm. In the squares magnified sections of the image are shown. (b) Secondary structure mode of the enzyme in dimer form with unbound Mn2+ (red spheres). With kind permission from M UELLER et al. (2007) and private communications

These form factors have to be determined for each atom by integration over the particle density N (j ) (r) of all its electrons (the dimension of N being L−3 ). Without entering into the details we note here that the imaginary part of the atomic form factor is related to the photo-absorption cross section, while the real part characterizes the elastic (coherent) photon scattering. For radially symmetric charge distributions it is ∞ (j ) N (r) sin(qr) 2 Re Fj (q) = 4π r dr, (1.98) qr 0 with the momentum transfer q = 2k sin(θ/2) = 4π sin(θ/2)/λ where θ = 2δ is the light scattering angle. Today these atomic form factors are well known and tabulated (see e,g. C HANTLER et al. 2005). We cannot go into details of different experimental methods for obtaining X-ray diffraction data. But we mention that in addition to laboratory X-ray sources, synchrotron radiation (SR, see Sect. 10.6.2) plays a key role in structural analysis, in particular for large biological molecules. In order to illustrate the astonishing performance of state-of-the-art SR diffraction techniques, we show in Fig. 1.19 one particularly impressive example: an X-ray diffraction pattern from a human enzyme which has been obtained with synchrotron radiation by rotational exposure. One illuminates the object with rather monochromatic X-ray light (W/ W ∼ 5000 to 10 000) and rotates the crystal with a certain angular increment, in the present case through 0.5◦ . During this rotation the multitude of reflexes shown in Fig. 1.19 becomes visible.

1.5

The Four Fundamental Interactions

43

Section summary

• Photons have (i) a well defined energy W = hν = ω (with ν = c/λ and ω = ck) as determined by the photoelectric effect, (ii) a momentum p = k documented by the C OMPTON effect, and (iii) an angular momentum . They have no rest mass and exist only moving with the speed of light. • The spectrum of electromagnetic radiation (we call it “light” in a general sense) ranges from radio frequencies (λ ∼ km, hν ∼ 10−9 eV) to γ -rays (10−5 nm, hν ∼ 108 eV). The visible spectrum is only a very small part of it (the wavelength range from 380 and 760 nm). • P LANCK’s radiation law (1.81) was a corner stone in the development of quantum physics. It describes the spectrum of a “black body” such as our sun or incandescent light bulbs. The maximum of the spectrum shifts with temperature according to W IEN’s displacement law λmax T 2.9 × 106 nm K, the total intensity (per unit area) follows the S TEFAN -B OLTZMANN law I (T ) = σB T 4 . • A hypothetical, ideal light source with a luminous efficiency of 100 %, driven by 1 W electric power emits 683 lm (lumen) at 555 nm. Incandescent light bulbs have a luminous efficiency of only 2–3 %. • X-ray radiation is a very powerful tool for structural analysis. Constructive interference occurs at angles θ of incidence (in respect of the lattice plane) according to the B RAGG law 2d sin θ = zλ. Other formulations of the B RAGG law make use of the reciprocal lattice vector (1.92) and M ILLER indices. The structure factor (1.96) and the atomic form factor (1.97) describe the intensities in the diffraction pattern.

1.5


We cannot give here an even brief introduction into particle physics and the underlying theory. The so called standard model (SM) of quantum chromodynamics (QCD) is a sophisticated theory which cannot be treated ‘in passing’. However, it is important to know a few basics when discussing the interactions relevant in atomic, molecular and optical physics. The four fundamental interactions 1. 2. 3. 4.

Gravitation Electromagnetic interaction Weak interaction Strong interaction

may be seen as “what holds the world together in its innermost folds” – as far as we understand it today. In our daily life we are confronted almost exclusively with the first two of these forces – in particular gravitation plays a key role in our everyday experience, while electromagnetic interactions are perceived commonly in a more indirect manner: through the action of various machines and equipment, via lighting gears (or occa-

44

1 Basics

Table 1.7 The four fundamental interactions, exchange bosons and coupling constants Interaction

Fermion (e.g.)

Exchange boson

Mass / GeV c−2

Couplingb

Range /m

Dependence on distance

Gravitation

e± , p, n

Gravitona

0

5.9 × 10−39

∞

1/r 2

Electromagn.

e± , p

Photon

0

7.30 × 10−3

∞

1/r 2

Weak

e± ,

W± -boson

10−18

1/r 5

Strong a Hypothetical, b Values

80.4

10−5

Z0 -boson

91.2

10−7

p, n

π-meson

135.139

1

Quarks

Gluons

0

0.119

ν

1/r 7 10−15

1/r 7

not yet found

for gravitation and electromagnetic interaction refer to a pair of protons

sionally electrostatic discharges), magnetic attraction (e.g. by compass needles) or even via natural phenomena such as lightning. In contrast, the whole physics and chemistry of atoms, molecules, condensed matter and quantum optics is essentially determined by electromagnetic interaction – while the extremely weak gravitational force is of relevance only in exceptional cases – and so are the weak and strong forces which can be neglected almost completely in AMO and condensed matter physics (we shall mention some exceptions in later chapters). Weak and strong interactions play, however, a crucial role when we are interested in the structure of nuclei, in their stability and decay – and of course if we are interested in the genesis and development of the universe. The unified description of electromagnetic and weak interaction is called electroweak interaction. In Table 1.7 the most important characteristic of the four fundamental interactions are summarized: The second column gives some examples of building blocks of matter (here fermions, i.e. particles with spin quantum number s = 1/2) which are subject to these forces. These fermions (and their composites) interact with each other by exchanging characteristic virtual particles, the so called exchange bosons or vector bosons (with spin quantum number s = 1, except for the graviton which should have a spin s = 2 and is as yet unobserved). They are listed in the third column and their respective mass is given in column four of Table 1.7. An order of magnitude of the strengths of these forces is estimated in column five of Table 1.7 – a direct comparison being, however, impossible due to the different dependence on distance and symmetry properties of these forces. Only electromagnetic interaction (C OULOMB’s law) and gravitation decrease universally with 1/r 2 (if r refers to the distance of two particles), the range being given as ∞ in contrast to the weak and strong forces that act only on close encounter.

1.5.1

COULOMB and Gravitational Interaction

The interaction of two particles depends in general in a complex manner from their distance r. Only the well known force laws for gravitation

1.5


45

F g = −G

m1 m2 r r2 r

(1.99)

and electromagnetic interaction, i.e. C OULOMB’s law Fe =

1 q1 q2 r 4πε0 r 2 r

(1.100)

can be written in this simple, closed form, with masses m1 , m2 and charges q1 , q2 of the interacting particles, respectively. They may be derived from a scalar potential V (r) ∝ 1/r by F = − grad V (r).

(1.101)

While the gravitational constant G has to be measured as one of the fundamental constants experimentally, the electric constant ε0 is related to the speed of light c = ε0 μ0 which in SI units is a defined quantity as well as the magnetic constant μ0 . The potential energy of an electron at a distance a0 (first B OHR orbit, see atomic units in Sect. 1.8) from a positive charge e is V (a0 ) = e2 /(4πε0 a0 ) = α 2 me c2 . The fine structure constant α according to (1.10) is thus the coupling constant of C OULOMB’s law in units of me c2 .19 Exact values for the coupling constants G and 1/(4πε0 ), as well as α, are documented in Appendix A. The electromagnetic force between two protons is 1.2×1036 stronger than gravitation among these particles. Thus, electromagnetic interaction plays a dominant role on the atomic and molecular length scale (and is still relevant at subatomic distances). For the structure and properties of atoms, molecules and condensed matter it is more or less exclusively responsible. It determines e.g. the orbits of electrons around atomic nuclei (or more precisely: the probability distributions of the electrons). In contrast, macroscopic objects of our daily life as well as planets and stars in the universe are essentially uncharged when seen from the outside: they consist of an almost identical number of protons and electrons (quasi-neutrality). Thus, macroscopic objects interact with each other only via gravitation that determines their motion (in spite of the extremely small coupling constant G) – as long as these macroscopic objects are not driven by other mechanical, chemical or electromechanical means. Even then gravitation usually plays a crucial role, if we think e.g. of driving a car, flying an airplane, steering a rocket, or if we consider the tidal motions of the oceans. The whole complex structure of today’s universe has developed almost exclusively under the influence of graviton for now about 13.75 billion years – since about 1 million years after the big bang.

19 We

note here that for very high energies the electromagnetic coupling constant αem changes (essentially ∝ ln W ); at energies 90 GeV (roughly corresponding to the mass of the Z boson) experiments have determined 1/αem 1/128.

46

1 Basics

1.5.2

The Standard Model of Fundamental Interaction

The other two forces, strong interaction and weak interaction, play an important role on a subatomic length scale only: as indicated in Table 1.7, their range of interaction is finite – in complete contrast to C OULOMB’s law and gravitation. Strong interaction is mainly responsible for the cohesion of nuclear matter, i.e. it takes care of the stability of nuclei which otherwise under the influence of electrostatic repulsion would simply explode. Weak interaction on the other hand plays a decisive role in β-decay, i.e. when an electron is emitted from an atomic nucleus. Electromagnetic, strong and weak interaction are described today consistently and convincingly by the standard model of QCD – only gravitation resists so far a unified interpretation. At the core of the theory is a set of objects that may be considered point like20 (<10−18 m) real elementary particles without any internal structure – but with some well defined properties in terms of mass, charge and spin (intrinsic angular momentum). They are summarized in Fig. 1.20. The standard model distinguishes two varieties of elementary particles: 1. Matter particles • 6 quarks are subject to strong interaction and carry one of three colour charges red, green or blue; they also carry a charge – +2/3e or −1/3e – and are subject to electromagnetic as well as to weak interaction. • 6 leptons are all subject to weak but not to strong interaction – i.e. they are colourless. The electron, muon and tauon carry also an electric charge −e and are subject to electromagnetic interaction. In contrast, the corresponding neutrinos do not have an electric charge. The have, however, a (very small) mass. All of these are fermions with spin 1/2. They are complemented (not shown in Fig. 1.20) by the corresponding antiquarks (in the colours antired, antigreen, antiblue) and antileptons, their charges are opposite. 2. Exchange particles (or gauge bosons) are considered to be exchanged between the matter particles during interaction. They are all bosons with spin quantum number s = 1 and may be seen as mediators of the forces. They are formed for a very short time only, i.e. ‘virtually’, and are unobservable directly.21 experts: e− − p collision experiments at HERA, with momentum transfer up to Q2 = 40 000 GeV2 , have not shown any deviations of the observed scattering signal from that expected theoretically for a point-like electron and quark structure. This corresponds to a smallest resolved size <10−3 fm, i.e. less than 1/1000 of the proton radius.

20 For

21 The

“virtual” exchange of particles may be seen to support our phantasy in understanding the interaction at a distance: a virtual particle of energy W or mass m = W/c2 can be created for a very short time given by the limits of the energy-time uncertainty relation to be discussed in Sect. 1.7.3.

1.5


47

Leptons

Neutrinos colourless

Mass1) Charge Spin Name

Quarks: each may have one of the colours red green blue

The three generations of matter particles (fermions)

1) 1MeV/c 2 2) each

Gauge or exchange particles (bosons)

2.4 MeV 2/3 1/2 Up

u

1.27 GeV 2/3 1/2 Charm

171.2 GeV 2/3 1/2 Top

t

0 1 Photon

4.8 MeV -1/3 1/2 Down

104 MeV -1/3 1/2 Strange

4.2 GeV -1/3 1/2 Bottom

0 0 1 Gluon2)

d

c s

b

γ

g

electromagnetic interaction strong interaction

< 2.2 eV 0 1/2 El. neutrino

<0.17 MeV 0 1/2 Myon neutr.

νμ

<15.5 MeV 0 1/2 Tau neutr.

91.2 GeV 0 1 Z Boson

Z0

electroweak interaction

0.511MeV -1 1/2 Electron

105.7MeV -1 1/2 Myon

1.777 GeV -1 1/2 Tauon

80.4 GeV ±1 1 W Boson

electroweak interaction

νe

e

μ

= 1.073 ×10-3 u = 1.957 a.u. (me)

1 colour + 1 anti colour (8 combinations)

ντ

τ

W±

HIGGS Boson*)

Fig. 1.20 Elementary particles according to the standard model: matter particles (fermions, spin 1/2) and exchange particles (gauge bosons, spin 1). The masses and charges are indicated: masses are given in terms of their energy equivalent Wm = mc2 . *) According to CERN (2013) “On 4 July 2012, the ATLAS and CMS experiments at CERN’s Large Hadron Collider announced they had each observed a new particle in the mass region around 126 GeV” /c2 “. . . consistent with the H IGGS boson but it will take further work . . . ”. This is based on theoretical work for which E NGLERT and H IGGS (2013) received the N OBEL prize

We have discussed already in Sect. 1.4 the photon. It is the exchange particle for electromagnetic interaction by which all charged particles interact (in addition to weak or strong interaction and to gravitation). Strong interaction is characteristic for quarks as well as for antiquarks (not shown in Fig. 1.20) with charge −2/3e and +1/3e. They are never observed as free particles and occur only as compound systems (confinement). On the other hand, at very close distance their interaction tends to zero (so called asymptotic freedom). In analogy to electric charge one attributes to the quarks a colour charge by which they interact with each other. The colour comes in three varieties red, green or blue (in our graphs coloured red, grey or black). In each composite particle the colours have to add in such a manner that their sum leads to white (in analogy to the colours observed from the visible spectrum of light). Antiquarks are coloured with complementary colours antired, antigreen or antiblue. The exchange bosons of the strong interaction are called gluons (reminding us of ‘glue’). They come in 8 different varieties encompassing one colour and one anticolour each (while the

48

1 Basics Proton u u d

Charge = 1e0 Spin = 1/2 stable

Mass mp = 1836.15 me = 1.6726×10−27 kg = 938.27231 MeV/c 2 = 1.007 276 47 u

Neutron u d d

Charge = 0 mn = 1838.68 me = 1.6749 ×10−27 kg Spin = 1/2 = 939.5656 MeV/c 2 unstable = 1.008664 92 u Δt1/2 = 10 min 14s

Fig. 1.21 Structure of the two nucleons: proton and neutron. They are built from up and down quarks (u and d, respectively). The special choice of colours is here without direct relevance and is supposed to just illustrate the principle of colours in quantum chromodynamics: each of the three fundamental colours (commonly called red, green, blue) has to be present exactly once. The wiggly lines indicate exchange of gluons (strong interaction)

photon does not contain any charge). Both, gluons and the photon are massless particles while the three exchange bosons of weak interaction Z0 , W+ and W− are again colourless but have each a (different) mass.

1.5.3

Hadrons

Barions and Mesons All further particles observed in nature or generated by particle accelerators and storage rings are composed of the elementary particles summarized in Fig. 1.20. Of special importance are the two nucleons, proton p and neutron n. Their composition by three quarks each is sketched in Fig. 1.21. Note that the charge of the proton (q = 1e) and neutron (q = 0) is obtained by simply adding the charges of the quarks involved (2/3, +2/3, −1/3) and (2/3, −1/3, −1/3), respectively. The three colour charges red, green and blue of the three quarks add to white in each case. The spin of proton and neutron has to be constructed from the three constituents, each of which has spin 1/2. According to the rules of angular momentum algebra a spin of 1/2 or 3/2 could in principle result. Apparently 1/2 for both, neutron and proton is realized. In contrast, the mass is not simply additive. While neutron and proton have masses about 1840 times that of an electron, the mass of up and down quarks are only somewhat larger than the mass me of the electron – according to Fig. 1.20 a factor 5 and 10, respectively. Obviously the binding energy Wb = mc2 contributes most to the masses of proton and neutron. In addition to these two long-lived nucleons, which are special cases of baryons (heavy particles), about a hundred others are known or predicted. They are constructed from three quarks in a similar manner. All of them are rather short-lived and differ in their inner structure, energy (and thus mass), spin and charge.

1.5


n

49

u d d

u d u

W−

p

_ νe e−

Fig. 1.22 F EYNMAN diagram for the neutron decay by weak interaction (half lifetime τ1/2 = 10 min +14 s): by emission of an electron e− and an antielectron neutrino ν¯ e a down quark “d” (charge −1/3e) is changed into an up quark “u” (charge +2/3e). The interaction occurs via a “virtual” W− boson

In contrast to the barions the so called mesons (medium heavy particles) are built from a quark and an antiquark with complementary colours (colour and anticolour), which makes them also white. Barions and mesons together are called hadrons. Hadrons are all subject to strong interaction, to weak interaction and – as far as electrically charged also subject to electromagnetic interaction.

Decay of the Neutron One typical example of such an interaction is β-decay of the neutron through weak interaction. The isolated neutron is not stable outside the atomic nuclei, albeit rather long-lived (half lifetime about 10 min 14 s, mean lifetime 885.7 ± 0.8 s). Schematically this decay is illustrated in Fig. 1.22 by a so called F EYNMAN diagram (more about these useful diagrams in Sect. 6.5.6). The three constituents of the neutron move as indicated by the arrows through time. A down quark emits a virtual W− boson and thus the neutron becomes a proton (see Fig. 1.21). The W− boson finally decays into a newly generated lepton pair ν¯ e + e− . From energy and momentum conservation it is clear that at least two particles have to be created, since these two fundamental conservation laws could otherwise not be fulfilled.

1.5.4

The Electron

The leptons electron, muon and tauon and their respective neutrinos are, in contrast to the hadrons genuine elementary particles, at least as far as we know today. They interact among each other as well as with quarks, protons and other barions and atomic nuclei by electro-weak and (as far as charged) by electromagnetic interaction. In atomic, molecular and optical physics we are almost exclusively concerned about electromagnetic interactions and by far the most important particle is the electron. Let us have a somewhat closer look at it. Its rest mass is me = 9.10938291(40) × 10−31 kg = 5.4857990946(22) × 10−4 u.

50

1 Basics

Fig. 1.23 Principle of the setup for determining the elementary charge according to M ILLIKAN

d

+ U ~ 5000 V –

As an elementary particle, the electron is considered as point like, more precisely: the C OULOMB law is valid at least down to r ≤ 10−18 m. Thus, the electron is at least three orders of magnitudes smaller than the proton. The electron has a mass me and an inherent angular momentum, the so called electron spin. The latter is characterized by a spin quantum number s = 1/2 and has a measurable value of angular momentum in respect of any given direction in space, say z, of sz = ±/2. Connected with the spin is its magnetic moment −ge μB s, as we shall discuss in more detail in Sect. 1.9. Note: the electron has all these properties irrespective of its vanishing spatial extension. From a classical viewpoint this is a very strange object! The so called “classical electron radius” re = e2 / 4πε0 me c2 = α 2 a0 = 2.82 × 10−6 nm

(1.102)

is to be considered a purely numerical quantity equating the C OULOMB energy at a distance re to the rest mass energy me c2 of the electron, with α being the fine structure constant (1.10). Historically, re was thought to be related to the radius of a homogeneously charged sphere whose C OULOMB energy is equal to me c2 . Today we know that the electron is as point-like a particle as we can measure it (see footnote 20). Nevertheless, re often appears as a useful abbreviation in physical relations. From Sect. 1.4.2 we recall in this context the reduced C OMPTON wavelength of the electron λ¯ C = λC /2π = αa0 – another length scale related to the electron. Attempts to measure the electron charge have already been undertaken in 1899 by J.J. T HOMSON, who determined e/me from the deflection of a moving electron in a magnetic field (Nobel prize for T HOMSON 1906). The first accurate determination of e goes back to M ILLIKAN (1913) who received the N OBEL prize in physics 1923 for this work. As sketched in Fig. 1.23 he determined the charge q of small oil droplets with mass m by letting them fall or rise in an electric field. The velocity v of the particles is determined by the gravitational force mg, the electric force ±qE and the frictional force 6πηrv = mg ± qE with E = U/d. The charge q = ± N e is a (small) multiple of the elementary charge e and can be determined by variation of the field if one knows the radius of the particles r and the viscosity η of the carrier gas. The elementary charge is of course a very important fundamental constant that may be determined today with great accuracy: e = 1.602176487(40) × 10−19 C .

(1.103)

Its precise value is constant subject of improvement and the interested reader may want to consult NIST (2010).

1.6

Particles in Electric and Magnetic Fields

51

Generally, free and isolated particles are only observed with an integer multiple of e (positive, zero or negative). We point out again, however, that the bound quarks and antiquarks have charges ±2/3e or ∓1/3e. Directly related to the elementary charge is the energy unit electronvolt 1 eV = 1.602176487(40) × 10−19 J [per particle].

(1.104)

It corresponds to the change in kinetic energy T = (mv 2 /2) = eU of a charge e after passing a voltage difference U = 1 V. The unit eV is conveniently and commonly used in AMO physics. Section summary

• The four fundamental interaction responsible for “keeping the world at its innermost together” are gravitation, electromagnetic interaction, weak interaction, and strong interaction. Gravitation and electromagnetic interaction energies are ∝ 1/r, i.e. have in principle infinite range, and are dominant in macroscopic physics and atomic physics, respectively. Weak and strong interaction are of short range on the order 10−18 and 10−15 m, respectively, and – from an atomic physics point of view – refer to forces inside the nuclei. • The standard model of elementary particle physics describes electromagnetic, weak and strong interaction in a unified scheme. It comprises 6 quarks (matter particles, fermions), 6 leptons (including the electron, all fermions, subject to weak and electromagnetic interaction, if charged) and 4 types of exchange bosons, including the photon which mediates electromagnetic interaction. The quarks have charges ±2e/3, −e/3, spin 1/2 and come with three kinds of “colour charges”. They are subject to weak, strong and electromagnetic forces. They are building blocks of all other massive particles, including the nucleons (neutron and proton) which in turn are constituents of atomic nuclei. • Leptons are a different kind of elementary particles, subject only to weak and electromagnetic interaction (if charged). Of particular interest in atomic and molecular physics is the electron. We memorize its electric charge e 1.6 × 10−19 C. To move it through an electric field requires energy or sets energy free. The atomic unit of energy is thus 1 eV 1.6 × 10−19 J.

1.6


We now have a look at characteristic motions and frequencies of charged particles in electric (E) and magnetic fields (B) as useful background for later discussion. One aspect is mass spectroscopyand energy analysis or energy selection for which the properties of trajectories can be exploited. In the following we shall explore the possibilities and limits of such methods.

52

1 Basics

A particle of momentum p, velocity v and charge q experiences in an electromagnetic field the L ORENTZ force: F=

dp = q(E + v × B). dt

(1.105)

In this form the equation is correct even in the relativistic limit. However, for simplicity we shall restrict the following discussion to the nonrelativistic limit unless pointed out differently.

1.6.1

Charge in an Electric Field

Consider a particle beam with velocity v moving in +x-direction. It enters a purely electric field E = U/b at x = 0. We choose a geometry such that mv˙x = 0 and mv˙y = qU/b and assume the field to be strictly limited to the lengths of the capacitor as sketched in Fig. 1.24. This can actually be achieved by placing the aperture plates (vertical, grey lines) at potential U/2 in a specific distance to the capacitor plates as investigated in the early days of particle optics by H ERZOG (1935). For the lateral deflection d of the beam we obtain from integrating N EWTON’s second law of motion d=

q U 2 q U t = 2m b 2m b

2 qU 2 . = v 4T b

(1.106)

Geometry (b and ) as well as the voltage U applied are known. The ratio (and only the ratio) of kinetic energy T to charge q may be determined according to (1.106) by the electric deflection method! This is valid in general, completely independent of how complicated the setup may be. In practical applications one often uses special capacitor designs e.g. cylindrical or hemispherical capacitors (the latter is described in some detail in Appendix B.3, Vol. 2) as well as segments of these for determining the particle energy if its charge is known. In addition to a measurement of the energy, for extended, divergent particle beams such setups allow to focus the trajectories from the entrance aperture onto the exit aperture of an energy analyzer.

Fig. 1.24 Trajectory (red line) of a charged particle in the electric field of a plane capacitor (heavy black lines) which is limited by entrance apertures (heavy grey lines)

+ U x

b q, v

d – 0 y

ℓ

1.6


1.6.2

53

Charge in a Magnetic Field

To also determine the mass of a particle one needs an additional measurable quantity. In a purely magnetic field B the movement of the charge q is determined by F=

dp = qv × B dt

with the force F being perpendicular to v and B (see Fig. 1.25). The change in energy dW = vF = qv · v × B ≡ 0 dt is thus identical to zero, and v = const. This holds also relativistically so that γ˙ = 0 and γ = const. The relativistic equation of motion thus reads dv dp = γ m = qv × B, dt dt with the rest mass m of the particle and p = γ mv. The particle moves on a circle that is determined by balancing the centrifugal force Fc and the magnetic force F = qvB: 2 2 dp = Fc = mv = γ mv = qvB, (1.107) dt r r p = rB (1.108) so that γ mv = p = qBr or q γ mv p and r = = . (1.109) qB qB If one wants to use this orbital radius r for mass selection, again only one parameter can be extracted: with a magnetic field the ratio of momentum and charge p/q can be measured. The expressions given above also hold in the nonrelativistic case where γ → 1. For highly relativistic energies as they are e.g. realized in electron storage rings built for the generation of synchrotron radiation (see Sect. 10.6.2) one has more or less Fig. 1.25 Moving charge (red line) in a magnetic field B grey pointing into the drawing plane

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ B ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

q, m v

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ r ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

×

v

54

1 Basics

exactly v c and the relativistic energy (1.19) becomes γ mr v 2 γ mc2 = W . Thus, (1.107) can be written dp W c or β˙⊥ = (1.110) dt r r where β˙⊥ is the radial acceleration a⊥ related to c. The latter relation follows from (1.107) with a⊥ = Fc /(γ m) = v 2 /r c2 /r.

1.6.3

Cyclotron Frequency and ICR Spectrometers

According to (1.109) in a sufficiently extended magnetic field a charged particle with rest mass m, charge e and kinetic energy T moves on a circular orbit. Its angular frequency is called cyclotron frequency: ωc =

eB 1 v eB 1 = = . r m γ m 1 + T /(mc2 )

(1.111)

The turnaround time is Tc = 2π/ωc = 2πγ m/Be. At low kinetic energies (T mc2 ), cyclotron frequency and turnaround time are independent of radius and velocity of the particle studied and simply depend on the ratio of mass to charge. At high velocities a relativistic correction is required. Ion cyclotron resonance (ICR) mass spectrometers are today among the most important and accurate mass spectrometers for chemical and biological analysis. One stores ions with different m/q in an ion trap and lets them interact with high frequency electromagnetic radiation (RF from MF to HF, see Fig. 1.14). If the frequency of this radiation is in resonance with ωc for a particular m/q in the trap, these ions (and only these) are accelerated. The thus induced image currents can readily be detected. In practice one may either tune the RF field over the bandwidth of interest, or alternatively use a very broad band (short pulse) and F OURIER analyze the resulting signal (FT-ICR). This kind of technique is also used for optical spectroscopy and discussed in some detail in Sect. 5.3.2, Vol. 2. Today’s commercial devices allow to analyze m/q up to several 1000 u (where q can be rather high so that even large biomolecules may be studied) and achieve mass resolutions (FWHM) up to m/ m 100 000.

1.6.4

Other Mass Spectrometers

The so called magnetic mass spectrometers are based in fact on a combination of electric and magnetic fields. As we have seen the electric field selects: the magnetic field selects:

T = f (U, geometry) q p = rB. q

1.6


55

If one combines the two types of fields judiciously one may achieve with T = p 2 /2m a determination of m/q: m 1 p 2 /q 2 = . q 2 T /q

(1.112)

In the classical, so called double focussing mass spectrometers, ions pass successively electric and magnetic fields. Such devices are still used today but have lost much of their importance in relation to the FT-ICR mass spectrometers discussed in Sect. 1.6.3. Another type of mass spectrometers, today widely used, is the so called quadrupole mass spectrometer (QMS). It exploits the mass dependence of the dynamic stability regions for ion trajectories in a rapidly oscillating electric quadrupole field (see e.g. D EHMELT and PAUL 1989): for specific ratios of DC to AC fields only ions with a specific mass to charge ratio m/q may pass on a stable trajectory through the setup while all other ions undergo increasing oscillations and do not reach the exit slit. The interesting mathematics follows M ATHIEUS’s differential equation (see e.g. W IKIPEDIA CONTRIBUTORS 2013). In time of flight mass spectrometers one accelerates ion in electric fields to select m/q through their time of flight to the detector. In state-of-the-art setups of this type one exploits a variety of tricks to focus ions of the same mass but from different starting positions at the ion source (W ILEY and M C L AREN 1955) as well as those with different initial kinetic energies (M AMYRIN 1994) at the same time on the detector. With present commercial solutions high mass to charge ratios m/q (up to 4000 u) may be reached and high mass resolution (m/ m 60 000) can be achieved. For special applications two other, classical setups are still useful and deserve to be mentioned. We already mentioned J.J. T HOMSON’s method with parallel electric (E) and magnetic (B) fields which nowadays is used to analyze particle emission with high kinetic energies T . Assume the fields to point into +y-direction, the ions enter the field perpendicular in +z-direction (velocity vz ) and pass through them on a length l. In this case the L ORENTZ force (1.105) leads to acceleration both in yas well as in x-direction. After a time t1 = l/vz the particles have passed the fields and acquired velocities vy =

qE qEl t1 = m mvz

and vx =

qBvz qBl t1 = . m m

(1.113)

Behind the fields the ions are detected on a position sensitive detector: J.J. T HOM SON used a photographic plate, today sophisticated electronic devices are available as we shall describe in Appendix B.1, Vol. 2. If the detector is positioned at z = s far behind the field, the ions hit the detector at a time t2 = s/vz and their deflections are y = v y t2 =

qEls qEls = 2 2Wkin mvz

and x = vx t2 =

qBls qBls =√ . mvz 2mWkin

(1.114)

56

1 Basics

Fig. 1.26 Charges in a solid, displaced by an external electric field

electrons ––––––– ++++++ ––––––– ++++++ ––––––– ++++++ ions x

Eliminating vz leads to the so called T HOMSON parabolas y=

m E 2 x . q B 2 ls

(1.115)

They describe the positions where the ions hit the detector and allow to determine m/q and for known q. In addition, with (1.114) one may also determine the kinetic energy. In contrast to the T HOMSON setup, the so called W IEN filter uses crossed electric and magnetic fields to select charged particles according to their velocity. For particles that hit the crossed E and B fields perpendicularly one may set the ratio of the fields such that the resulting forces (1.105) for a given velocity v just compensate each other, i.e. q(E − vB) = 0. For v = E/B the charged particles just fly on straight line trajectories. If the mass is known one may thus determine the momentum and hence the kinetic energy.

1.6.5

Plasma Frequency

Electrons in plasmas but also in clusters and condensed matter (in particular quasi free electrons in metals and semiconductors) may carry out collective oscillations, so called plasma oscillations – with a characteristic frequency that plays an important role in many areas of physics. The simplest model starts with is a quasi neutral plasma of electrons e− and ions with a density N of quasi free electrons, equal to that of the ions. Assume we displace all electrons in respect of the ions by a distance x as shown in Fig. 1.26. This leads to a surplus of surface charge density σ = −eN x that causes an electric field between electrons and ions E = σ/εr ε0 . The equation of motion for each electron in this field is thus me x¨ = eE = eσ/εr ε0 = e2 N/εr ε0 x. This differential equation is simply that of a harmonic oscillator with a force constant kP = e2 N/εr ε0 . The angular frequency of the oscillations kP N e2 = (1.116) ωp = me me εr ε0

1.7

Particles and Waves

57

is called plasma frequency, sometimes also L ANGMUIR or D RUDE frequency. It plays a key role when describing electrical and optical properties of electron gases in metals, semiconductors or plasmas. For reference but without going into details we mention that the plasma frequency is, however, not an oscillation frequency in the standard sense, and that the dielectric function of the plasma is given by εr (ω) = 1 −

ωp2 ω2

,

(1.117)

which for ω < ωp becomes negative. To interpret this result, we consider a plane wave exp[i(kx − ωt)] propagating in +x-direction, with k = ωn/c being the magnitude of the wave vector in √ the medium and n = εr the index of refraction. According to (1.117), below the plasma frequency n becomes imaginary and we write k = iκ. Correspondingly the wave, ∝ exp(−κx) exp(−iωt), is damped. Thus, the medium absorbs at all frequencies ω < ωp and is transparent for ω > ωp . We finally note that in isolated particles, the situation is somewhat more complicated, e.g. so called plasmon resonance phenomena will be observed at ωp . Section summary

• In an electromagnetic field E, B particles with a charge q are subject to the L ORENTZ force F = q(E + v × B), where v is the particle’s velocity. • This can be used to manipulate and guide the charged particles (typically electrons and ions) and to select their energy and/or mass/charge ratio. • We note two interesting frequencies: (1) In a homogeneous magnetic field B a free electron moves on a circle, perpendicular to the field (or spirales) with the cyclotron frequency ωc = v/r = eB/(γ me ). (2) The plasma frequency 2 = /(m Ne ωp e εr ε0 ) is characteristic for oscillations of quasi free electrons in clusters and metals (particle density N ).

1.7

Particles and Waves

1.7.1

DE BROGLIE Wavelength

In 1923, Louis DE B ROGLIE (N OBEL prize in physics 1929), a French aristocrat, argued in his PhD thesis that particles should also have wave properties, just as electromagnetic waves have particle properties – as demonstrated by the photoelectric and the C OMPTON effect (see Sect. 1.4.1 and Sect. 1.4.2). His postulate, now well proven, states that in full analogy to light (1.74) the momentum p of a particle is related to its wave vector k with k = 2π/λdB and wavelength λdB by p = k

and p = h/λdB = ω/c,

(1.118)

58

1 Basics

(a)

diffraction patterns

(b) target

cathode

screen

e-beam +U aperture

Wkin low

Wkin high

Fig. 1.27 D EBYE -S CHERRER diffraction schematically: (a) setup and generation of the diffraction cones; (b) typical diffraction patterns for a poly-crystalline target: top view onto the screen (a) as seen from the right

with λdB commonly called DE B ROGLIE wavelength. The postulate of DE B ROGLIE gave birth to the famous concept of wave-particle duality. In its most simple form a matter wave may be written as ψ(r) = C exp(ik · r),

(1.119)

i.e. as a plane wave with C being a suitably chosen normalization constant. In the nonrelativistic limit a slow electron with kinetic energy T has a wavelength 1.23 nm h =√ . λdB = √ T / eV 2me T

(1.120)

At a kinetic energy 100 eV the wavelength of an electron is thus on the order of the size of an atom 0.1 nm. Note, however, that (1.118) also holds in the relativistic case, with p given by (1.25). Specifically, for electrons that have been accelerated by a voltage U one has to use p=

2me eU 1 +

eU . 2me c2

(1.121)

With the rest energy me c2 = 0.511 MeV for electrons this leads already at a moderate kinetic energy of 50 keV to a reduction of the wavelength of 2.5 % in comparison to (1.120). In a precision experiment this clearly has to be accounted for.

1.7.2

Experimental Evidence

DEBYE-SCHERRER Diffraction of Electrons We remind the reader here of a some applications that illustrate the wave nature of matter in a particularly obvious manner. The D EBYE -S CHERRER method, which is also well established in X-ray diffraction, is one of the most prominent examples. It is used for the structural analysis of poly-crystalline material which may e.g. be deposited on a thin carbon film. The setup is sketched in Fig. 1.27(a).

1.7

(a)

Particles and Waves fluorescence screen

59 electron energy selecting grids

(b)

electron gun target electron beam elastically diffracted electrons

Fig. 1.28 Low energy electron diffraction (LEED): (a) experimental setup; (b) typical diffraction pattern from a single crystalline ordered surface

The diffraction structures are generated by many single reflexes from the micro-crystals and appear according to (1.91) under the B RAGG angles 2θ = arcsin(zλ/2d) with z = 0, 1, 2 . . . The various distances d of the crystal’s lattice planes contain information about the crystal structure. Since the crystals are aligned at random into all directions of space, all electrons diffracted into an angle 2θ will be emitted into a cone around the incoming electron beam. For each m and each d such a cone exists. For each cone a circle is formed at its intersection with the plane of observation (photographic plate, or today more likely a CCD camera). In Fig. 1.27(b) such diffraction patterns for electrons of low and higher energy are sketched schematically.

Low Energy Electron Diffraction (LEED) Electron diffraction is used in a variety of methods for structure determination of matter. In contrast to the D EBYE -S CHERRER method, diffraction from single crystals leads to point patterns. Low energy electrons may be used with advantage for structure determination of surfaces in a setup illustrated in Fig. 1.28. The method is known as LEED (low energy electron diffraction) and commercially available. Diffraction of Neutrons, Atoms and Molecules from a Grating Low energy neutrons have a DE B ROGLIE wavelengths on the order of atomic diameters (1 eV = 0.029 nm). In contrast to charged particles that interact by the C OULOMB force with the electron shells of the atoms, neutrons interact only at short distance with the atomic nuclei through the strong force (see Sect. 1.5). At the same time they penetrate very deep into the matter studied. Thus, thermal neutrons are an excellent tool for structural analysis. The special form of atomic shell does not play any role in neutron scattering which makes a detailed evaluation of diffraction patterns much more simple than those from charged particles. Such experiments are today an indispensable method in structure determination – from materials research to biology – and a number of dedicated nuclear reactors are available for this purpose worldwide. In future, alternatively so called spallation sources are planned or under construction, e.g. the European Spallation Source (ESS) is currently being built in Lund, Sweden. First neutrons are expected for 2019. Thermal and supra-thermal neutral He atoms are also used for structural investigations. Since they cannot enter into the material they are particularly well suited for surface analysis. Figure 1.29 documents impressively that slow He atoms may

60

1 Basics

Fig. 1.29 Scattering of a neutral helium atom beam through a transmission diffraction grating with 100 nm slit distance as reported by S CHÖLLKOPF and T OENNIES (1996)

10 5

0. 1. order

intensity / particle s-1

10 4 3. 10 3 2. 10 2 10

–6

–4

–2

0

2

5. 4. 7. 6. 8. 9.

4

6

scattering angle / mrad

even be diffracted by quasi macroscopic optical elements, and they interfere. For this nice experiment a transmission grating was manufactured by state-of-the-art nano-technology. The diffraction pattern may be completely understood in terms of K IRCHHOFF’s diffraction theory which has been developed more than 150 years ago. One may of course ask how far one can push such kind of wave optics with particles and particle beams. This is an interesting theme which may even be potentially relevant for technical applications. Atom lithography and atom optics with very cold atoms is subject to current research. We have already mentioned briefly in Sect. 1.3.4 research on B OSE -E INSTEIN condensation (BEC). One particular fascinating aspect is that the wavelength λ of the matter waves increasing with decreasing temperature according to (1.118). It eventually reaches values comparable to the average distance between atoms (particle density N λ−3 ). With (1.118) and (1.50) this happens when T

h2 N 2/3 . 3mkB

(1.122)

Comparison with (1.71) shows that this estimate agrees with the critical temperature for BEC – apart from a numerical prefactor and the missing spin degeneracy. This is a nice, plausible result. One may also push the wave concepts into the opposite direction and ask about the macroscopic limits of the wave-particle √ duality: at constant kinetic energy the DE B ROGLIE wavelength decreases ∝ m, i.e. with the square root of the mass. Diffraction experiments with larger molecules are thus increasingly more difficult and much more challenging. Z EILINGER and his associates (see e.g. A RNDT et al. 1999) were able to show that even such big objects as Fullerene molecules, C60 , are subject to wave mechanics. In a single slit diffraction experiment one observes quite ‘normal’ diffraction patterns, as documented in Fig. 1.30 – and as one actually would expect. The ultimate conundrum in this context is about really large objects – as sketched in the cartoon Fig. 1.31 the question is, what will happen? We leave this as a little mental

1.7

Particles and Waves

61 signal pulses / (50 s)

Fig. 1.30 Diffraction of C60 molecules by a slit. Experiment according to A RNDT et al. (1999). The grey background indicates the structure of C60

1200 1000 800 600 400 200 – 100 – 50 0 50 detector position / μm

Fig. 1.31 Wave-particle duality for macroscopic objects? Cartoon after Wolfram v. Oertzen

100

?

〈ψ ⎜ψ〉

exercise for the reader who may want to estimate orders of magnitude for a pertinent experiment.

1.7.3

Uncertainty Relation and Measurement

The H EISENBERG uncertainty relation

px x ≥

(1.123)

belongs today well-nigh to the canonical intellectual inventory of an educated member of society (as far as she or he shows any interest for sciences at all). Expressed in words the uncertainty relation (1.123) states that space and momentum – or somewhat more general – canonical conjugate quantities cannot be measured simultaneously with accuracy.22 The classical Gedankenexperiment devised by H EISENBERG and illustrated in Fig. 1.32 analyzes the attempt to study an electron with an optical microscope. As it turns out, all endeavours to localize the electron with precision, are limited by its wave nature. If we want to observe the electron by light, its wavelength λhν has to 22 Note that measurements of microscopic observables are based on probability distributions. Thus, the exact value on the right hand side of the uncertainty relation (, h/2, h etc.) depends on the nature of that distribution and on the exact definition of uncertainty, e.g. FWHM, foot width, position of the first minimum, etc.

62

1 Basics

Fig. 1.32 Classical “Gedankenexperiment” according to H EISENBERG why an electron may not be localized by an optical microscope better than “allowed” by the uncertainty relation

∆ px

∆x

Fig. 1.33 Trying to localize an electron beam by an aperture leads to diffraction and hence to uncertainties in the determination of the electrons lateral momentum

e-

p

θ

x

be small and the opening angle θ of the microscope has to be kept large, in order to obtain high resolution. According to A BBÉ the smallest structure x one may resolve in an optical microscope is determined by its numerical aperture n sin θ and the wavelength λhν , (1.124) n sin θ with n = 1 for the optical index of refraction (in vacuum). Small λ and large sin θ imply, however, an intrusion into the experimental situation, since the photon will thereby lose or change its momentum p by

x =

px = p sin θ =

h sin θ, λhν

and transfer it to the electron. Together with the A BBÉ relation (1.124) we obtain

px x ∼ h – and thus have “derived” the uncertainty relation (1.123). Any other method trying to localize the electron leads eventually to the same result. Assume e.g. we try to localize an electron beam by a slit this leads to diffraction as indicated in Fig. 1.33. To obtain an estimate for the uncertainty in the xcomponent of the electron momentum px = p · sin θ we may use the angle θmin at which the first minimum of the diffraction pattern occurs. The latter is well known from the corresponding formula in wave optics: sin θmin =

λ .

x

Using this one reads from Fig. 1.33

px = p · sin θ = p

h/p λ =p

x

x

=⇒

px x = h,

which again is equivalent to the uncertainty relation (1.123).

1.7

Particles and Waves

63

We note at this point that an analogues uncertainty relation between energy W and time t also holds:

W t ≥ .

1.7.4

(1.125)

Stability of the Atomic Ground State

The big dilemma of physics at the beginning of the 20th century was the inability to understand stable electron orbits around the atomic nuclei. Electrons on circular orbits are accelerated charged particles – which according to classical physics should continuously radiate electromagnetic waves and hence loose energy. Thus, they should get slower and slower with time and – being negatively charged – should eventually fall into the positively charged nucleus! Alas! Why do the electrons remain on stable orbits? As we shall see in Sect. 1.8, B OHR simply postulated that stable electron orbits have angular momenta equal to integer multiples of . Among theses stationary states – according to B OHR – there are also excited ones which decay spontaneously by emitting radiation according to an exponential decay law exp(−At). But still, the ground state is completely stable even in the B OHR model! A quantitative solution of this problem is given by quantum mechanics. Interestingly, the few basics about the wave nature of the electron which we have discussed so far, already allow us to answer this puzzle in principle. We simply use the H EISENBERG uncertainty relation to obtain an estimate for the minimum energy without adopting a specific model for the atom: Let a be the mean radius of the atom. With a high probability electrons will be found within that radius. This immediately leads to an estimate of the uncertainty p in a determination of the electron’s momentum by writing (1.123) now a · p ≥

and p ≥ p ≥ /a.

(1.126)

Consequently the kinetic energy of the electron becomes T=

p2 2 ≥ . 2me 2me a 2

Together with the potential energy V =−

e2 4πε0 a

this gives a total energy W ≥ T + V . Assuming the equality to hold one obtains W=

2 e2 . − 2me a 2 4πε0 a

(1.127)

64

1 Basics

W obviously depends on the atomic diameter a. So let us now find the lowest possible energy! We just apply the standard rule of calculus to find this minimum: dW 1 42 πε0 − e2 me a ! = 0, =− da 4 me a 3 πε0 from which we obtain for the radius of the atomic ground state: a0 =

42 πε0 ε0 h2 = . e2 me e2 πme

(1.128)

Putting this back into (1.127) the total energy of the atomic ground state becomes Wmin = −

1 me e4 me e4 =− 2 . 2 2 32 π 2 ε0 8ε0 h2

(1.129)

This is exactly the value of the B OHR model for the ground state of the hydrogen atom! Big surprise with such a simple estimate? – Well, the idea is obviously right. One should, however, not overestimate the quantitative assertion: the correct numerical result is a direct consequence of the specific form of (1.126) with on the right hand side, rather than h/2 or something else (see footnote 22). And from this we have derived the specific estimate for the minimum product of position and momentum uncertainties. Section summary

• We have discussed here some elements of quantum mechanics. While the photon was recognized as the particle associated with electromagnetic waves, we now associate “matter waves” with the well known particles. The DE B ROGLIE wavelength of particles λdB = h/p also holds for relativistic velocities if the correct expression for p is used. √ In the nonrelativistic limit, for slow electrons, we memorize λdB 1.2 nm / T / eV. Convincing experimental evidence comes from diffraction experiments with particles. • The uncertainty relation may be said to express the particle wave duality in terms of minimal uncertainties in a measurement of two canonic conjugate variable, e.g. for momentum and position p x ≥ or for energy and time

W t ≥ . The precise value of the lower limit (, h/2 or the like) depends on the specific problem and on the accuracy which appears tolerable. • The stability of the atomic ground state can be viewed as a direct consequence of the uncertainty relation. (Why do negative electrons not fall into the positive nucleus?)

1.8

BOHR Model of the Atom

In 1913, during the big time of quantum mechanical discoveries, Niels B OHR (N OBEL prize in physics 1922) worked as a young postdoc with Ernest RUTHER -

1.8


65

FORD in England. RUTHERFORD had developed an atomic model based on his scattering experiments of alpha particles by atoms according to which “. . . the atoms consist of a positively charged nucleus surrounded by a system of electrons kept together by attractive forces from the nucleus; the total negative charge of the electrons is equal to the positive charge of the nucleus. Further, the nucleus is assumed to be the seat of the essential part of the mass of the atom, and to have linear dimensions exceedingly small compared with the linear dimensions of the whole atom . . . ”. J.J. T HOMSON expanded this by assuming that “. . . the atom consists of a sphere of uniform positive electrification, inside which the electrons move in circular orbits” (both quotes from B OHR 1913). As discussed above, the big challenge in physics was then to explain the highly problematic fact that these atoms do not collapse: why do electrons circle on stable atomic orbits without loosing energy? – Remember, this was 10 years before DE B ROGLIE postulated his matter waves as discussed in Sect. 1.7.1! B OHR , so to say, just cut the Gordian knot and developed the fundamental ideas for a theory of atomic structure. He knew from BALMER’s spectroscopic work, that the energy levels of atoms follow the phenomenological equation W ∝ n−2 with integer numbers n. These numbers became B OHR’s quantum numbers. Since according to classical mechanics the energy of a particle on a circular orbit scales inversely to the square of its orbital angular momentum, B OHR suggested that the angular momenta of stable orbits be directly proportional to these quantum numbers. And since the P LANCK constant h, then already known from P LANCK’s radiation law and E INSTEIN’s explanation of the photoelectric effect, has the unit of an angular momentum, he postulated that angular momenta be quantized in units of h/2π = . From a classical view point, this quantization could neither be proven nor even be justified, but B OHR demonstrated that with these assumptions the experimentally observed spectrum of hydrogen could be explained with unique precision.

1.8.1

Basic Assumptions

Many versions and numbers of “B OHR’s postulates” may be found in the literature and in a multitude of Web pages on the subject. In his original paper B OHR (1913) used the following two principle assumptions: 1. The dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis. 2. The latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by P LANCK’s theory. B OHR, in principle, considered elliptical orbits but (wisely) focused on circular ones for which he assumed a “simple condition: that the angular momentum of the

66

1 Basics

electron round the nucleus in a stationary state of the system is equal to an entire multiple of a universal value” which he identified as h/2π , today written as . In concise form, the angular momentum L = r × p is quantized and its magnitude on a stationary circular orbit is L = rme ve = n

h = n, 2π

(1.130)

with the velocity ve , the electron mass me and the radius r of the orbit. The second of B OHR’s principle assumptions is about the photon energies emitted in a transition from one stationary orbital n to another one n . In mathematical form it reads: hν = Wn − Wn .

(1.131)

In classical mechanics, an electron (charge = −e) may be on any circular orbit around the nucleus (charge = +Ze) for which the centrifugal force is balanced by the C OULOMB force: me ve2 Ze2 = r 4πε0 r 2

=⇒

r=

Ze2 . 4πε0 me ve2

(1.132)

Thus, we have derived a classical expression for the radius, while the kinetic energy may be written in terms of the potential V (r)

T=

me ve2 Ze2 1 = = − V (r). 2 8πε0 r 2

(Note that the latter equality is directly given by the C OULOMB law but also corresponds to the classical virial theorem.) Thus, the total energy of the system is

W = T + V = −T = −

Ze2 < 0, 8πε0 r

(1.133)

with the minus sign characterizing bound states: one defines zero total energy W = 0 for an electron that is just no longer bound (no kinetic energy at r → ∞). Finally, the quantization condition (1.130) is introduced by rewriting the kinetic energy, again using classical mechanics:

W = −T = −

L2 (n)2 = − . 2me r 2 2me r 2

(1.134)

1.8


1.8.2

67

Radii and Energies

One may now obtain independent expressions r(n) = an and W (n) = Wn for any given n = 1, 2, . . . < ∞ by solving the set of Eqs. (1.133) and (1.134) accordingly. One finds B OHR’s orbital radii an =

n2 ε0 h2 n2 = a0 Z e2 πme Z

(1.135)

for the stationary states of hydrogen (Z = 1) and hydrogen like ions (Z > 1), with n called principle quantum number and a0 B OHR’s radius: a0 =

ε0 h2 = 5.2918 × 10−11 m. e2 πme

(1.136)

The latter is used as unit of length in atomic physics (we memorize a0 0.05 nm). The total energy (or term energy) of the stationary states is Wn = −

Z 2 me e4 Z2 Z2 = E = − R∞ hc. h n2 8ε02 h2 2n2 n2

(1.137)

The quantity Eh is used as atomic energy unit23 and amounts to Eh = 2R∞ hc =

me e4 e2 = = 4.359 × 10−18 J = 27.211 eV, 4ε02 h2 4πε0 a0

(1.138)

while the RYDBERG constant R∞ = Eh /(2hc) = 10 973 731.568527(73) m−1 is used in spectroscopic context, often given in “wavenumbers”, cm−1 . Today it is the most precisely measured fundamental constant (see also Sect. 6.5.4).

1.8.3

Atomic Units (a.u.)

It is often convenient to measure observable properties of quantum systems in so called atomic units (a.u.), i.e. write energies as W/Eh with the atomic unit of energy Eh according to (1.138), lengths in r/a0 with B OHR’s Radius a0 according to (1.136), and times in t/t0 . With the atomic velocity v0 = Eh /me = Eh /me c2 c = /me a0 = αc, (1.139) (with the fine structure constant α, Eq. (1.10)) one defines t0 =

2ε 2 h3 a2 a0 = 2πme 0 = 0 4 = 2.4189 × 10−17 s. v0 h πme e

(1.140)

to be confused with the rest energy of the electron Wrest = me c2 , also called natural unit of energy (see e.g. NIST 2010).

23 Not

68

1 Basics

Fig. 1.34 Orbital radii rn = a0 n2 /Z and energies Wn = Eh Z 2 /(2n2 ) for H (Z = 1, red) and He+ (Z = 2, black) according to the B OHR model

The atomic unit of mass is the rest mass of the free electron: 1 a.u. ≡ 1me = 9.10938291(40) × 10−31 kg.

(1.141)

Note, however, that this must not be confused with the unified atomic mass unit 1 u = 1.660538921(73) × 10−27 kg,

(1.142)

which is defined as 1/12 of the mass of an unbound atom of the nuclide 12 C – at rest and in its ground state. It is a unit outside the SI but widely used and recommended by the Comité International des Poids et Mesures.

1.8.4

Energies of Hydrogen Like Ions

Ions with only one electron, i.e. He+ , Li++ , . . . , U91+ are called hydrogen like. For the examples H and He+ Fig. 1.34 illustrates the magnitudes of orbital radii rn = n2 a0 /Z and energies Wn = −(Z 2 /2n2 )Eh .

1.8.5

Correction for Finite Nuclear Mass

So far we have always assumed that the nuclear mass is at rest (mn = ∞). However, correctly we should describe the electron motion in the centre of mass system. For a two particle system this kinematic correction is done by simply replacing ¯ e: the electron mass me with the reduced mass m me → m ¯e =

me mn . me + mn

(1.143)

1.8


69

B OHR’s radius has thus to be replaced by a0 → a¯ 0 = a0

me , m ¯e

(1.144)

and the term energies become Wn = −

Z2 ¯ Z2 = − RH hc, E h 2n2 n2

(1.145)

with Eh and the RYDBERG constant R∞ (1.138) being replaced by m ¯e E¯ h = Eh me

1.8.6

and RH =

m ¯e R∞ , me

respectively.

(1.146)

Spectra of Hydrogen and Hydrogen Like Ions

From the term energies (1.137) one derives with (1.131) the famous RYDBERG (RYDBERG -R ITZ) formula for the spectra of hydrogen and hydrogen like ions H, D, He+ , Li++ , Be+++ , . . . , U91+ :

1 1 2 (1.147) Wn1 n2 = ω = hν = Wn1 − Wn2 = Z R∞ 2 − 2 n1 n2

1 1 1 2 ν¯ = = Z R∞ 2 − 2 in wavenumbers (1.148) λ n1 n2

1 1 ν = Z 2 R∞ c 2 − 2 as transition frequency. (1.149) n1 n2 State-of-the-art spectroscopy of the hydrogen atom contributes today decisively to the most precise measurements of fundamental constants. We shall discuss their sophisticated methods and their amazing precision in Sect. 6.5.4. In order to compare these experimental results with various theoretical predictions one has to correct them of course according to (1.146) with the different kinematic correction factors for each nuclear mass.

1.8.7

Limits of the BOHR Model

The B OHR model works surprisingly well for H and H like ions. B OHR’s term energies (1.137) in these cases are identical to those derived from nonrelativistic quantum mechanics. Unfortunately, the model fails for all other atoms in spite of some serious efforts made in the early days of quantum mechanics. It also fails in the relativistic case when ve /c = (e2 /2ε0 hc)(Z/n) = Zα/n. Here the fine structure constant α 1/137 according to (1.10) appears again. Relativistic effects obviously become important for large Z and low n. We may neglect them as long as Zα/n 1.

70

1 Basics

So far we have also completely ignored the spin of the electron which is also of relativistic origin. The electron spin leads to an extension of the concept of angular momentum and one of its practical consequences is fine structure splitting (see Sect. 6), which the B OHR model of course does not consider. We have to be aware that the concept of electron orbitals with well defined radii will have to be reconsidered very critically: strictly speaking, from today’s understanding of quantum mechanics, the B OHR model is simply not correct – as we shall discuss in detail in Sect. 2.6.11. Still, it serves in many cases as a helpful first step towards formulating and understanding the correct quantum mechanical theory. Section summary

• The B OHR model of hydrogen (and H-like atoms), historically one of the cornerstones in the development of modern physics, makes surprisingly correct predictions for energy levels, Wn = −Eh Z/(2n2 ) and transition frequencies hν = Wn − Wn . . . though just based on heuristic postulates. Even the B OHR radii rn = a0 n2 /Z allow a reasonable interpretation as we shall see in the next chapter. Thus, the B OHR model remains a reference for quick cross checks and estimates. Also, its educational value should not be underestimated. • For higher precision, some small correction for finite mass of the nucleus are needed as summarized in Sect. 1.8.5. • Using the results of the B OHR model, we have introduced the system of atomic units (a.u.) with energies measured in Eh (twice the binding energy of the H atom’s ground state), lengths in a0 (the radius of the first B OHR orbit), time in t0 (the time an electrons needs to circle once around that orbit), and the mass in me (the rest mass of the electron).

1.9

STERN-GERLACH Experiment and Space Quantization

One of the key experiments for quantum mechanics has been performed in 1922 by S TERN (N OBEL prize 1943) and G ERLACH. It showed with then unsurpassed clarity that classical mechanics and electrodynamics are unable to explain the observations on the level of atomic dimensions.

1.9.1

Magnetic Moment and Angular Momentum

Before trying to understand this experiment and its consequences, we have to recall some basics from mechanics and electrodynamics. According to the B OHR model the electron circles around the atomic nucleus, a motion that in a classical picture may be viewed as a spinning top. As indicated in Fig. 1.35 this orbit is connected with an electric current I=

e ev = . t 2πr

1.9

S TERN -G ERLACH Experiment

71

Fig. 1.35 Magnetic moment of an electron on a circular orbit

ML

e0 r

v

+I L

of This current, enclosing an area A (A = πr 2 ), generates a magnetic moment M the magnitude M = IA =

ev eL evr πr 2 = = . 2πr 2 2me

It is proportional to the magnitude of L and oriented into the opposite direction. The magnetic moment of an orbital angular momentum is thus M=−

e L L = −μB . 2me

(1.150)

Note: the gyromagnetic ratio derived here, M/L = −e/2me = −ge μB is independent of the specific geometry of the motion. It does not depend on whether a point charge or an extended charge distribution orbits around the nucleus. However, this above classical derivation is valid only for orbital angular momenta. It will have to be modified in view of the results of the S TERN -G ERLACH experiment. In the B OHR model angular momenta (1.130) are quantized and appear only in units of . Thus, L/ is an integer number in B OHR’s theory, and the unit of magnetic moment is the so called B OHR magneton μB =

1.9.2

e −1 = 927.400915(23) × 10−26 J T . 2me

(1.151)

Magnetic Moment in a Magnetic Field

In a magnetic field B – as a consequence of the L ORENTZ force – a magnetic moment M experiences a torque T = M × B. Its potential energy

(1.152)

T · dθ is thus given by VB = −M · B = −MB cos θ,

(1.153)

where θ is the angle between M and B and zero energy has been assumed for M ⊥ B. Minimum potential energy corresponds to M parallel to B, and L antiparallel to it.

72

1 Basics

Fig. 1.36 Precession of the angular momentum L in a magnetic field B

T → ML

B ωL φ

dL L

φ e0

The force on the magnetic dipole M as whole in this potential is F = − grad V (r). Since M is independent of r, one obtains F = M · ∇B = Mx

∂B ∂B ∂B + My + Mz , ∂x ∂y ∂z

(1.154)

(see e.g. S TERN 1921, rewritten in SI units). Thus, in a homogeneous magnetic field no net force acts on a magnetic dipole, while in an inhomogeneous field it experiences a force proportional to its magnetic moment. However, the torque (1.152) acting on the dipole moment leads to a change of the angular momentum of the spinning top. With dL = Ldϕ the equation of motion is dϕ dL =L = LωL . (1.155) T= dt dt In the special geometry shown in Fig. 1.36 with L ⊥ B, this becomes L LωL = MB = −μB B. The torque T forces the angular momentum L to precess around B (gyroscopic motion) with an angular frequency, called L ARMOR frequency: ωL =

1.9.3

μB e M B= B= B. L 2me

(1.156)

The Experiment

Otto S TERN had invented the so called molecular beam method24 and already applied it to determine the M AXWELL -B OLTZMANN velocity distribution in gases (see Sect. 1.3.4). In his famous paper “Ein Weg zur experimentellen Prüfung der 24 In this and later context, an atomic or molecular beam source may be thought to just consists of a reservoir with the species to be investigated (possibly heated, or cooled, possibly mixed with an inert “carrier gas”). Through a small orifice in it, the atoms or molecules diffuse or stream into the vacuum, and their divergence angle is collimated by one or more apertures along the beam axis.

1.9


73

z atom oven

z

(a) magnet S

B on B off

N magnet

x B on photo plate

slit

(b) S y N

profile

Fig. 1.37 Schematic of the S TERN -G ERLACH experiment. (a) The atomic beam from the “oven” is collimated through a slit, may then be deflected in an inhomogeneous magnetic field, and will finally hit a photographic plate. (b) Side view of magnet poles and atom beam

(a)

z y

z

(b)

y

Fig. 1.38 Two wire field: it corresponds approximately to the field created by two wires through which a current flows. The corresponding magnetic field lines are illustrated in (a). In an experiment the field is realized by a pair of suitably shaped permanent magnets as indicated in (b). The red line parallel to the y-direction indicates the position of the atomic beam entering the space between the magnets from the back. It will be deflected by the inhomogeneous field in ±z-direction

Richtungsquantelung im Magnetfeld” he proposed (S TERN 1921) an experiment to determine the magnetic moment of an atom, based on the deflection of an atomic beam in an inhomogeneous magnetic field.

The Original Atomic Beam Experiment The experiment as it was actually carried out in 1922 by S TERN and G ERLACH with silver atoms (Ag) is illustrated schematically in Fig. 1.37. With (1.154) and B = (0, 0, B) at y = 0 the symmetry of the setup leads there to Fx = 0 and Fy = 0. Since the components Mx and My will precess around the z-axis, the force due to these components averages out. Thus, we are left with the z-component of the force (1.154): z Fz = M

∂B . ∂z

(1.157)

A small but important detail is the physical realization of an inhomogeneous field. In the original S TERN -G ERLACH experiment a so called two wire field was used which is illustrated in Fig. 1.38.

74

1 Basics z

(a)

x z

(b)

x z

(c)

x

Fig. 1.39 Classical expectation about the S TERN -G ERLACH experiment: (a) View onto the atomic beam coming out of the yz plane without magnetic field. (b) Alignment of the magnetic moments prior to entering the magnet, distributed at random. (c) Atomic beam profile after deflection by the magnetic field. The distribution would arise from ∂B/∂z being strongest for x = 0 and decreasing for |x| > 0

What Do We Expect? Let us assume the profile of the atomic beam without magnetic field arriving in the detector plane is shaped as indicated in Fig. 1.39(a). Classically, the magnetic moments M are expected to be randomly distributed in all directions as indicated in z onto the +z-direction ranging from −μB L/ to Fig. 1.39(b), their projections M +μB L/. The atomic beam enters the inhomogeneous magnetic field and its atoms z . Behind the magnet the reare deflected corresponding to their distribution of M sulting spatial distribution is registered by a suitable detector. With the magnetic field switched on, the classical expectation is to observe a broadening of the beam in z-direction proportional to the field and the average magnetic moments – reflecting the randomly distributed Mz components. The deflection on the z-axis (x = 0) will be largest since there the field gradient ∂B/∂z is highest. And what Is Observed in the Experiment? In their experiment with Ag atoms S TERN and G ERLACH used a photographic plate as detector. The traces of the silver beam condensed on it. The highly remarkable result of these efforts is a pattern schematically sketched in Fig. 1.40 – in complete contrast to the classical expectation: obviously there are only two directions into which the atomic magnetic moments are oriented. The particular (elliptical) shape is due to the vanishing field gradient at both sides of the beam.25 More clearly one recognizes the result by tracing the measured signal along the z-axis in Fig. 1.40 and plotting the absorbance due to the silver deposited as a function of z. This is schematically shown in Fig. 1.41. The comparison with and without 25 A

quaint note aside: the silver traces only became visible after Otto S TERN blew the smoke of his (sulfur containing) cigar onto the photographic plate – an early, unintended contribution to the photochemistry and catalysis of the photographic development process.

1.9


75

Fig. 1.40 Schematic sketch of the pattern observed in the S TERN -G ERLACH experiment: the beam is split into two components

Fig. 1.41 Result of the S TERN -G ERLACH experiment (cut along the z-axis in Fig. 1.40): Splitting in two discrete states when the magnet is switched on and comparison with the classical expectation

z

y

z

classical expectation B on

B=0

B on

atom signal

magnetic field bears out the dramatic splitting into two components. This observation, which is also made for many other atoms, is a direct consequence of space quantization as we shall discuss in a moment.

L ANGMUIR -TAYLOR Detector We note here in passing that today many very efficient methods for particle detection exist. Today’s reproductions of the S TERN -G ERLACH experiment (but also many modern atomic beam experiments) use detectors exploiting the so called L ANGMUIR -TAYLOR effect – which in itself is interesting. It is based on the quantum tunnelling effect, to which the valence electron of an atom with low ionization potential WI (e.g. K) is exposed when hitting a metal surface with high work function WA (e.g. W). Figure 1.42 gives an essentially self explaining schematic of such a detector with the relevant potential energy diagram (see also the potential well model introduced in Sect. 1.4.1). The electrons tunnel into the metal, the K+ ions remain and can be detected as ion current after field extraction and amplification, possibly with a particle multiplier (see Appendix B.1, Vol. 2).

1.9.4

Interpretation of the S TERN-G ERLACH Experiment

How can the results of the S TERN -G ERLACH experiment be interpreted? We remember, the B OHR model postulates that angular momenta are quantized according to (1.130): Thus, we expect L = with to be a positive integer number or zero. Let us assume now, that a similar relation holds for the components of angular momenta, say for Lz in z-direction. Then Lz would assume 2 + 1 integer multiples of : Lz = m

with m = − , − + 1, . . . , .

(1.158)

76

1 Basics tungsten surface

(a) W WA electron lake

(c)

V(x )

electron lake

W

vacuum x

x

WI

V(x )

WA (tungsten) > WI (potassium)

K atom on a tungsten surface

electron tunnells

K atom

(b)

vacuum

x after e– tunnelling a K+ ion remains on the surface

detector setup schematically

(d) K atoms

electronics for ion current amplificaton + 50V

hot tungsten wire in a metal cylinder

Fig. 1.42 Schematic of a L ANGMUIR -TAYLOR detector: (a) Potential for an electron at an isolated tungsten surface, (b) potential and energy of the valence electron in a K atom, (c) potential felt by a K atom when hitting a tungsten surface, (d) simple detector setup

This quantization is called space quantization (in German language somewhat more precisely: Richtungsquantisierung). The number m is often called “magnetic quantum number” since it was originally observed in a magnetic field. We prefer, however, the more general term projection quantum number. In contrast to a classical statistical distribution of angular momenta, as illustrated in Fig. 1.39(b), there are only 2 + 1 allowed projections of L = onto the zaxis. The number 2 + 1 is called multiplicity of the state (in Sect. 1.3.4 we have introduced it as “degeneracy” – a term used in the context of energies). According to (1.150) the magnetic moment is M = −μB L/, thus space quantization of L also implies quantization of M. The component Mz may thus assume values from |M| to −|M| which correspond to the components −|L| to |L| of the orbital angular momentum. A closer quantum mechanical analysis (see Chap. 2) shows that the conjectures made above are nearly correct – except for the fact that the magnitude of the angular momentum is given by (1.159) L = ( + 1), which for large values of gives again L ∼ . But even this heuristic introduction of space quantization, guided by B OHR’s postulates – which even turns out to be quantum mechanically correct for orbital angular momenta – does not provide a full explanation of the S TERN -G ERLACH experiment: the observed splitting in two components (see Fig. 1.41), i.e. the observation of a multiplicity 2, is obviously contrary to the multiplicity predicted for the lowest nonvanishing orbital angular momentum ( = 1): one would expect a multiplicity 2 + 1 = 3, i.e. triple splitting!

1.9


1.9.5

77

Consequences of the STERN-GERLACH Experiment

The S TERN -G ERLACH experiment revealed three dramatic non-classical facts: 1. Space quantization – completely unexpected from a classical point of view but plausible in the framework of B OHR’s quantum theory 2. The observed multiplicity does not correspond to the expected whole numbered angular momenta according to the B OHR model. The twofold splitting 2 = 2j + 1 only allows the conclusion that the atom studied (Ag) has an angular momentum quantum number j = 1/2. This holds for silver atoms but also for the alkali metal atoms such as Na, K, . . . . We thus have to extend the B OHR model and assume that also half integer angular momenta exist. All experimental observations of quantum systems confirm the hypothesis: Angular momenta J only exist as integer or half integer multiples of . We have already anticipated the relevant quantum mechanical relations in (1.5)–(1.9). 3. Finally, a quantitative evaluation of the S TERN -G ERLACH experiment shows that also the magnitude of the classically predicted magnetic moment (1.150) does not lead to a correct estimate of the observed deflections. One has to generalize (1.150) and defines the magnetic moment for J : MJ = −gJ μB

J

(1.160)

and its projection onto a given axis, say z, is given by MJ z = gj μB mj

with mj = −j, −j + 1, . . . , j.

(1.161)

The so called L ANDÉ g-factor is gL = 1 for pure orbital angular momenta and ∼ =2 for electron spin states. We shall derive general expressions gj in Chap. 8. A multitude experiments confirms that the magnetic moment of atoms and molecules is indeed proportional to the total angular momentum J and oriented opposite to it. Equation (1.156) has also to be modified: L ARMOR frequency for J :

ω j = gj

e B = gj ω L . 2me

(1.162)

Section summary

• The S TERN -G ERLACH experiment has been another corner stones in the development of quantum mechanics. It proofs what we call space quantization: angular momenta J are oriented in space so that their projection on a given axis is Jz = mj with mj = −j, −j + 1, . . . , j where j is the angular momentum quantum number. • Orbital angular momenta have integer quantum numbers, intrinsic quantum numbers of particles may be half integer (fermions) or integer (bosons).

78

1 Basics

Fig. 1.43 Vector diagram for electron spin (heavy black arrow): It precesses on a cone around the z-axis with two possible orientations (dotted circles with arrow), so that its projection onto the z-axis (indicated by the two red, dashed lines) is either +/2 or −/2

z S √3 S =-ħ 2

x S

ħ Sz =+ 2

y ħ Sz = - 2

• With each angular momentum J a magnetic moment M = −gj μB J / is −1 associated, with the B OHR magneton μB = e/2me 927.4 × 10−26 J T . The L ANDÉ gj factor is 1 for orbital angular momenta and 2 for the electron spin. • In an external magnetic field B magnetic moments have a potential energy VB = −M · B = −MB cos(M, B), they precess under the influence of the torque T = M × B with the L ARMOR frequency ωj = gj Be/2me . Only in an inhomogeneous field an overall force (1.154) acts on them.

1.10

Electron Spin

A conclusive explanation for the ground-breaking observations of S TERN and G ER LACH was presented not until 1925 by G OUDSMIT and U HLENBECK in the context of unusual atomic line splitting in a magnetic field (anomalous Z EEMAN effect, see Sect. 8.1.2): The electron has an intrinsic magnetic moment MS which is associated with an intrinsic angular momentum S, the electron spin. Identifying in (1.5)–(1.9) J with S and setting the spin quantum number s = 1/2 explains the multiplicity of 2 observed in the S TERN -G ERLACH experiment. Two possible orientations for √ the spin exist with projection quantum numbers ms = ±1/2, so that S = |S| = 3/2 and Sz = ±/2. One may visualize these relations in a so called vector diagram, shown in Fig. 1.43. In summary, the electron spin is characterized as follows: √ √ 3 magnitude |S| = s(s + 1) = 0.88 2 quantum number s = 1/2 (1.163) multiplicity 2s + 1 = 2 z component

Sz = ms with ms = ±1/2.

1.10

Electron Spin

79

1.10.1 Magnetic Moment of the Electron Quantitative evaluation of the deflection observed in the S TERN -G ERLACH experiment for a range of atoms such as Ag, K, Na, . . . shows that the projection of the magnetic moment onto a given axis is Mz = ∓μB , with the B OHR magneton (1.151). If we were to apply relation (1.150) M = −(e/2me ) L = −μB L/, which is valid for orbital angular momenta, to the z-component of the spin, Sz / = ±1/2, we would obtain Mz = ∓μB /2 – in contrast to the experimentally observed value Mz = ∓μB . Thus, we must apply the generalized expression (1.160) for the magnetic moment of the spin. Specifically for an electron without angular momentum, i.e. for = 0 (as it is the case with H, Na, K, Ag in the ground state) the magnetic moment of the atom is determined exclusively by the intrinsic magnetic moment of the electron. From this follow the magnetic moment of the electron MS = −ge μB the g factor of the electron

ge 2

S

and

(1.164) (1.165)

We emphasize that this value of the electron g factor cannot be explained by any kind of classical charge distribution on an orbit, for which always (1.150) holds, i.e. gL = 1. In contrast, the relativistically correct D IRAC theory leads exactly to ge = 2 for the electron spin. Thus, according to (1.161) the component of the magnetic moment of the electron in respect of a given axis, say z, becomes indeed Mz = ∓ge μB /2 = ∓μB as experimentally observed. It is interesting to note that the L ARMOR frequency (1.162) for an electron with spin s and ge = 2 becomes ω s = ge ω L = ge

e e B B, 2me me

(1.166)

and is, according to (1.111), (nearly) identical to the cyclotron frequency ωc = (e/me )B of an electron in a magnetic field (in the nonrelativistic limit). We point out, however, that high precision measurements show a small, but significant difference between ωL and ωc , i.e. document a deviation from (1.165), in fact one finds ge 2.0023 . . . . A theoretical understanding of this so called anomalous magnetic moment of the electron is provided by quantum electrodynamics (QED) for which T OMONAGA, S CHWINGER and F EYNMAN received the N OBEL prize in 1965 – as we shall explicate in Sect. 6.6.

1.10.2 EINSTEIN-DE-HAAS Effect The so called E INSTEIN - DE -H AAS effect gives one further impressive piece of evidence for the non-classical nature of the magnetic moment of the electron – based

80

1 Basics

Fig. 1.44 Principle of the E INSTEIN - DE -H AAS experiment

spring wire laser beam setup for measuring mirror the twisting of the wire C

+ -

solenoid ferromagnetic iron core

on a macroscopic measurement. In this experiment one uses the fact (not to be discussed here in detail) that ferromagnetism is caused by many electron spins which are oriented in parallel. One determines the torque exerted onto a (soft) ferromagnetic iron core (cylinder) due to a change of magnetization: the magnetic field of the solenoid tends to orient the magnetic moments of the electron spins in parallel to this magnetic field – and hence there will also be a preferential orientation of the electron spins. These spins all together constitute a total inherent angular momentum of the cylinder. In order to change it, a torque has to act on this cylinder which in the setup shown in Fig. 1.44 arises from the external magnetic field. The torque is measured by the twisting angle ϕ of a thin spring wire (silica). In a quantitative experiment one starts with a completely demagnetized probe and allows a well defined current pulse to flow through the solenoid. This leads to a changing magnetic field Hsolenoid (t) as sketched in Fig. 1.45. During this pulse a torque acts on the system which was originally at rest. After the magnetic field has decayed, the remanent magnetization Mrem is assumed to be N times the z-component of the magnetic moment Mz of the individual electrons. With (1.161) one obtains Mz = ge μB ms = −ge

e Sz 2me

⇒

Mrem = Brem = −N ge

e Sz , 2me

(1.167)

where N is the number of electrons that after the pulse remain oriented (magnetized) in the direction of the applied magnetic field (remanence). In turn, Brem may easily Hsolenoid

Bmagnet Brem t

Fig. 1.45 Temporal evolution of the magnetic field generated by the solenoid in Hsolenoid after starting to discharge the capacitor C – along with the magnetic induction Bmagnet resulting from it in the ferromagnetic cylinder leading to a remanent part Brem

Acronyms and Terminology

81

be determined e.g. by measuring a voltage induced by the remanent magnetic field in a probe coil. Due to angular momentum conservation we have N Sz = Irod ωrod ,

(1.168)

with Irod = mR 2 /2, the moment of inertia of the ferromagnetic cylinder, m being its mass and R its radius. The angular frequency of the cylinder ωrod after application of the magnetic field may be determined from the maximum twist angle ϕmax of the silica wire onto which the cylinder is mounted: the initial kinetic energy has then been transformed into potential energy: Irod 2 kr 2 ω = ϕmax . 2 rod 2

(1.169)

From the ratio of Brem according to (1.167) and the angular momentum Irod ωrod according to (1.168) we obtain e Brem N ge S z e = = ge . Irod ωrod N Sz 2me 2me This finally allows one to determine ge . A quantitative evaluation of such experiments confirms ge 2. Section summary

• The spin related properties of the electron have again be summarized in (1.163)–(1.165). • The E INSTEIN - DE H AAS experiment demonstrates the electron spin on a macroscopic level using in a ferromagnetic rod. When its magnetization is reversed in external magnetic field, the magnetic moments of N the electrons have to reversed from +/2 to −/2. Thus, the total angular momentum of all electrons in the rod changes by N . Since N is a large number, this induces a macroscopic rotation of the rod, which can be measured, confirming the microscopic measurements of ge .

Acronyms and Terminology AC: ‘Alternating current’, oscillating electric voltage and current. AMO: ‘Atomic, molecular and optical’, physics. a.u.: ‘atomic units’, see Sect. 2.6.2. BEC: ‘B OSE -E INSTEIN condensation’. BESSY: ‘Berlin Electron Strorage ring for Synchrotron Radiation’, Germany’s third generation synchrotron radiation source in Berlin-Adlerhof. BZ: ‘B RILLOUIN zone’, represents all wave vectors of radiation which can be B RAGG-reflected by a crystal lattice. Important concept in solid state physics.

82

1 Basics

CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). 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. CIE: ‘Commission international de l’éclairage’, International Commission on Illumination, provides e.g. colorometric tables (http://files.cie.co.at/204.xls). CMB(R): ‘Cosmic microwave background’, radiation at 2.725 K from the origin of the universe. DC: ‘Direct current’, unidirectional electric voltage and current. ESS: ‘European Spallation Source’, large scale facility for generating neutrons for structural research, see http://europeanspallationsource.se/. EUV: ‘Extreme ultraviolet’, part of the UV spectral range. Wavelengths between 10 nm and 121 nm according to ISO 21348 (2007). FIR: ‘Far infrared’, spectral range of electromagnetic radiation. Wavelengths between 3 μm and 1 mm according to ISO 21348 (2007). FT: ‘F OURIER transform’, see Appendix I. FWHM: ‘Full width at half maximum’. HERA: ‘Hadron-Elektron-Ring-Anlage’, for collision experiments between electrons of 30 GeV and protons of 820 GeV – operated at DESY very sucessfully until 2007. HF: ‘High frequency’, part of the RF spectrum. Wavelengths from 10 m to 100 m or frequencies from 3 MHz to 30 MHz according to ISO 21348 (2007). ICR: ‘Ion cyclotron resonance’, spectroscopy (specifically mass spectroscopy) based on irradiation with a radio frequency in resonance with the cyclotron frequency of an ion in magnetic field (see Sect. 1.6.3). IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). LED: ‘Light emitting diode’. LEED: ‘Low energy electron diffraction’, see Sect. 1.7.2. LF: ‘Low frequency’, part of the RF spectrum from 30 kHz up to 300 kHz. LHC: ‘Large hadron collider (not to be confused with left hand circularly polarized light)’, high energy physics facility at CERN (Geneva) providing particles with collision energies up to 14 TeV for protons and up to 1 PeV for heavy ions. MF: ‘Medium frequency’, part of the RF spectrum from 300 kHz up to 3 MHz. MIR: ‘Middle infrared’, spectral range of electromagnetic radiation. Wavelengths between 1.4 μm and 3 μm according to ISO 21348 (2007). 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 and 300 GHz; ISO 21348 (2007) defines it as the wavelength range between 1 mm and 15 mm. 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.


83

PES: ‘Photoelectron spectroscopy’, see Sect. 5.8, Vol. 2. PET: ‘Positron emission tomography’, medical exploitation of positron-electron annihilation (see Sect. 1.4.3). QCD: ‘Quantum chromodynamics’, the theory of strong interaction (color force). A fundamental force describing the interactions of quarks and gluons, the constituents of all hadrons. QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. QMS: ‘Quadrupole mass spectrometer’, a brief explanation is found in Sect. 1.6.4. 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. 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. SM: ‘Standard model’, of elementary particle physics. The basis of today’s understanding of matter. SR: ‘Synchrotron radiation’, electronmagnetic radiation in a broad range of wavelengths, generated by relativistic electrons on circular orbits. THz: ‘Tera-Hertz’, spectral region of electromagnetic radiation. Wavelengths range covering parts of MW and IR. UHF: ‘Ultra high frequency’, part of the RF spectrum. Wavelengths from 10 cm to 1 m or frequencies from 3 GHz to 300 MHz according to ISO 21348 (2007). UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007). UVA: ‘Ultraviolet a’, part of the UV spectral range. Wavelengths between 315 nm and 400 nm according to ISO 21348 (2007). UVB: ‘Ultraviolet b’, part of the UV spectral range. Wavelengths between 280 nm and 315 nm according to ISO 21348 (2007). UVC: ‘Ultraviolet c’, part of the UV spectral range. Wavelengths between 100 nm and 280 nm according to ISO 21348 (2007). VHF: ‘Very high frequency’, part of the RF spectrum. Wavelengths from 1 m to 10 m or frequencies from 300 MHz to 30 MHz 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 electromagentic 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.

84

1 Basics

References A RNDT , M., O. NAIRZ, J. VOS -A NDREAE, C. K ELLER, G. VAN DER Z OUW and A. Z EILINGER: 1999. ‘Wave-particle duality of C60 molecules’. Nature, 401, 680–682. ASTM: 2008. ‘G173-03 Reference Spectra Derived from SMARTS v. 2.9.2’, American Society for Testing and Materials (ASTM). http://rredc.nrel.gov/solar/spectra/am1.5/ASTMG173/ ASTMG173.html, accessed: 7 Jan 2014. B OHR , N.: 1913. ‘On the constitution of atoms and molecules’. Philos. Mag., 6, 26, 1–25. B OHR , N. H. D.: 1922. ‘The N OBEL prize in physics: for his services in the investigation of the structure of atoms and of the radiation emanating from them’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/1922/. B RETSCHNEIDER , E.: 2007. ‘Efficacy limits for solid-state white light sources’, Laurin Publishing Co., Inc., Pittsfield MA, USA. http://www.photonics.com/Article.aspx?AID=28677, accessed: 7 Jan 2014. DE B ROGLIE , L.: 1929. ‘The N OBEL prize in physics: for his discovery of the wave nature of electrons’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1929/. CERN: 2013. ‘The search for the H IGGS boson’, Geneva. http://home.web.cern.ch/about/ physics/search-higgs-boson, accessed: 8 Jan 2013. C HANTLER , C. T., K. O LSEN, R. A. D RAGOSET, J. C HANG, A. R. K ISHORE, S. A. KO TOCHIGOVA and D. S. Z UCKER : 2005. ‘X-ray form factor, attenuation, and scattering tables (version 2.1)’, NIST. http://physics.nist.gov/ffast, accessed: 7 Jan 2014. C OMPTON , A. H.: 1927. ‘The N OBEL prize in physics: for his discovery of the effect named after him’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1927/. C ORNELL , E. A., W. K ETTERLE and C. E. W IEMAN: 2001. ‘The N OBEL prize in physics: for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/2001/. D EHMELT , H. G. and W. PAUL: 1989. ‘The N OBEL prize in physics: for the development of the ion trap technique’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1989/. D ICK LYON: 2008. ‘Luminosity function’, Wikimedia Commons. http://commons.wikimedia. org/wiki/File:Luminosity.png, accessed: 7 Jan 2014. E INSTEIN , A.: 1905. ‘Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt’. Ann. Phys., 17, 132. E INSTEIN , A.: 1921. ‘The N OBEL prize in physics: for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect’, Stockholm. http://nobelprize. org/nobel_prizes/physics/laureates/1921/. E NGLERT , F. and P. W. H IGGS: 2013. ‘The N OBEL prize in physics: for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s large hadron collider’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/2013/. H ERZOG , R.: 1935. ‘Berechnung des Streufeldes eines Kondensators, dessen Feld durch eine Blende begrenzt ist’. Arch. Elektrotech., 29, 790–802. H ONSBERG , C. and S. B OWDEN: 2012. ‘PVCDROM – Air Mass’, UNSW and Solar Power Labs at ASU, Australia. http://www.pveducation.org/pvcdrom/properties-of-sunlight/air-mass, accessed: 7 Jan 2014. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. KOPP , G. and J. L. L EAN: 2011. ‘A new, lower value of total solar irradiance: evidence and climate significance’. Geophys. Res. Lett., 38, L01706. M AMYRIN , B. A.: 1994. ‘Laser-assisted reflectron time-of-flight mass-spectrometry’. Int. J. Mass Spectrom. Ion Process., 131, 1–19.

References

85

M ATHER , J. C. and G. F. S MOOT: 2006. ‘The N OBEL prize in physics: for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/2006/. M ILLIKAN , R. A.: 1923. ‘The N OBEL prize in physics: for his work on the elementary charge of electricity and on the photoelectric effect’, Stockholm. http://nobelprize.org/nobel_prizes/ physics/laureates/1923/. M UELLER , U., F. H. N IESEN, Y. ROSKE, F. G OETZ, J. B EHLKE, K. B UESSOW and U. H EINE MANN : 2007. ‘Crystal structure of human prolidase: the molecular basis of PD disease’, Hinxton, UK: PDB entry 2okn. The European Molecular Biology Laboratory (EMBL-EBI). http://www.ebi.ac.uk/pdbe-srv/view/entry/2okn/summary.html, accessed: 7 Jan 2014. NIST: 2010. ‘Reference on constants, units, and uncertainties’, NIST. http://physics.nist.gov/cuu/ Constants/, accessed: 7 Jan 2014. NIST: 2011. ‘Conversion factors for energy equivalents’, NIST. http://physics.nist.gov/cuu/ Constants/energy.html, accessed: 7 Jan 2014. O HNO , Y.: 2010. ‘Radiometry and photometry for vision optics’. In: M. BASS, ed., ’Handbook of Optics’, vol. II, 37.1. New York: McGraw-Hill. PAULI , W.: 1945. ‘The N OBEL prize in physics: for the discovery of the exclusion principle, also called the Pauli principle’, Stockholm. http://nobelprize.org/nobel_prizes/physics/ laureates/1945/. P LANCK , M.: 1900. ‘Zur Theorie des Gesetzes der Energieverteilung im Normalenspektrum’. Verh. Dtsch. Phys. Ges., 2, 235–245. P LANCK , M. K. E. L.: 1918. ‘The N OBEL prize in physics: in recognition of the services he rendered to the advancement of physics by his discovery of energy quanta’, Stockholm. http://www.nobelprize.org/nobel_prizes/physics/laureates/1918/. S CHÖLLKOPF , W. and J. P. T OENNIES: 1996. ‘The nondestructive detection of the helium dimer and trimer’. J. Chem. Phys., 104, 1155–1158. SORCE: 2012. ‘SORCE Solar Spectral Irradiance’, Boulder, Co.: Laboratory for Atmospheric and Space Physics, University of Colorado and NASA. http://lasp.colorado.edu/ lisird/sorce/sorce_ssi/, accessed: 7 Jan 2014. SSE: 2012. ‘Solar System Exploration – Our Solar System’, NASA. http://solarsystem.nasa.gov/ planets/profile.cfm?Display=Facts&Object=Sun, accessed: 7 Jan 2014. S TERN , O.: 1921. ‘Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld’. Z. Phys., VII, 249–253. Nachdruck: Z. Phys. D, Atoms, Molecules and Clusters 10, 111– 116 (1988). S TERN , O.: 1943. ‘The N OBEL prize in physics: for his contribution to the development of the molecular ray method and his discovery of the magnetic moment of the proton’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1943/. T HOMSON , J. J.: 1906. ‘The N OBEL prize in physics: in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases’, Stockholm. http://www.nobelprize.org/nobel_prizes/physics/laureates/1906/. T OMONAGA , S.-I., J. S CHWINGER and R. P. F EYNMAN: 1965. ‘The N OBEL prize in physics: for fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles’, Stockholm. http://nobelprize.org/nobel_prizes/physics/ laureates/1965/. W IKIPEDIA CONTRIBUTORS: 2013. ‘Mathieu function’, Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/wiki/Mathieu_function, accessed: 7 Jan 2014. W ILEY , W. C. and I. H. M C L AREN: 1955. ‘Time-of-flight mass spectrometer with improved resolution’. Rev. Sci. Instrum., 26, 1150–1157.

2

Elements of Quantum Mechanics and the H Atom

Quantum mechanics provides the tools for a quantitative understanding of atoms and molecules. The reader is expected to be familiar at least with the main concepts. Here we want to repeat and refurbish the most important notions and methods so that we can work with them directly in the following chapters.

Overview

If the reader is already familiar with quantum mechanics he may just want to browse this chapter and return later if necessary. However, readers who have experienced quantum mechanics up to now only as compulsory mathematical exercise may perhaps read this chapter with advantage and find it helpful to approach the indispensable instruments without big formal hurdles. In Sects. 2.1–2.3 we summarize a minimum of formalism. Section 2.4 treats as a first example the well known particle in a box and the free electron gas which in atomic and solid state physics is an important elementary model. Section 2.5 gives an overview of how to treat angular momenta, needed in all following text, specified in Sect. 2.5.4 for the electron spin s = 1/2. Section 2.6 offers a ‘crash course’ in nonrelativistic quantum mechanics of the H atom – essential knowledge for all the following chapters. We refrain here from formal derivations in favour of a plausible, possibly somewhat hand waving introduction. Finally, Sect. 2.7 presents a first, elementary approach to interactions of atomic electrons with external fields, which will be extended and deepened in Chap. 8. Clearly, such a brief introduction into quantum mechanics cannot substitute a profound study of the formal theory. It should, however, facilitate the access to it and make the reader ‘fit’ for the following chapters.

2.1

Matter Waves

2.1.1

Limits of Classical Theory

The classical picture of a well defined trajectory with known coordinate x(t) and momentum p(t) looses its validity in quantum mechanics, as sketched in the phase © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7_2

87

88

2


diagram Fig. 2.1: position and momentum cannot be measured simultaneously. They may only be determined with an accuracy in accordance with the uncertainty relation pi xi ≥ h/2π . Quantum mechanics only makes statements about probability amplitudes Ψ (r, t). These may assume the form of so called Wave-packets which localize particles in space and time as well as the uncertainty relation allows it. One finds a particle at position r and time t with the probability 2 w(r, t) = Ψ (r, t) .

(2.1)

This is the key assumption of the statistical interpretation of quantum mechanics as formulated by Max B ORN (1927) in the early days of quantum mechanics, for which he received the N OBEL prize in physics in 1954. Quantum mechanics allows to describe the evolution of a wave packet quantitatively. If formed at time t = 0 with widths in momentum and space, pi (0) and xi (0), respectively, one finds for larger times t > 0 always pi (t) xi (t) > pi (0) xi (0): the wave packed diverges as indicated in Fig. 2.1.

2.1.2

Probability Amplitudes in Optics

One may easily visualize the concept of a probability amplitude for the example of photons in a YOUNG’s double slit experiment. The probability to find a photon at position r and time t is proportional to the intensity I (r, t) of the light, and the latter is proportional to the square of the field amplitude. Let us consider only one component of polarization, say Ex . The dependence of the intensity on position may then be written as 2 2 I (r) ∝ Ex (r) = ψ(r) = w(r). The last two equalities can help us to get used to quantum mechanical terminology: we call the quantity ψ(r) now the (position dependent) probability amplitude or wave function. In the case of light, it is simply represented by the electric field component Ex . One determines it according to the laws of optics as solution of the corresponding wave equation

ψ(r) + k 2 ψ(r) = 0 Fig. 2.1 Classical trajectory (red line) and quantum mechanical probability (grey shaded) in phase space. Note: also at the beginning of a quantum mechanical ‘trajectory’, i.e. a wave-packet, its position and momentum are not precisely defined – corresponding to the uncertainty relation

(2.2)

p

ics

han

ec mm

ntu

qua

classical x

2.1

Matter Waves

89

(with = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z2 ) for given boundary conditions. For optical applications a variety of approximations may be used such as H UYGENS -F RESNEL principle or the diffractions theory of K IRCHHOFF. As the wave function is derived from a linear partial differential equation (PDE), we may use the linear superposition principle to describe the interference of waves. For diffraction by two slits we thus have ψ(r) = ψ1 (r) + ψ2 (r),

(2.3)

if ψ1 and ψ2 each describe the wave from one or the other slit. With these amplitudes, the probability to find an atom at the detector positions r is given by: 2 2 w = ψ(r) = ψ1 (r) + ψ2 (r) . This expression contains interference terms ψ1∗ ψ2 and is not simply a superposition of probabilities but rather (we omit here the r): w = |ψ1 |2 + |ψ2 |2 + 2 Re ψ1∗ ψ2 . (2.4) We may see this expression as a reinterpretation of the classical YOUNG double slit experiment in terms of quantum mechanical probability amplitudes. In an actual experiment one may now reduce the light intensity such that only one single photon at a time is near the double slits and participates to the observed interference pattern. One may easily verify such a setup with a particle counter detecting single photons – one may even ‘hear the individual photons click’.1 If a sufficient number of such single photon events are recorded and added up, one finds – quite counter intuitively – that the classical diffraction pattern known from optics is recovered! Thus, the probability distribution of each individual photon behind the double slit is determined by the wave amplitudes ψ = ψ1 + ψ2 and it is impossible to say through which slit the particle has passed (see Fig. 1.31). One also says that one photon always interferes only with itself. We shall discuss this statement in some detail in Chap. 2, Vol. 2 and quantify it statistically.

2.1.3

Probability Amplitudes and Matter Waves

From the perspective just discussed let us have a look at the matter waves introduced in Sect. 1.7.1. We have already familiarized ourselves with the relation between momentum and wavelength as postulated by DE B ROGLIE. Matter waves are 1 For the experts: One may invest a lot of thought and substantial experimental efforts to make absolutely sure that only isolated photons reach the double slit setup (for a recent review on single photon sources see E ISAMAN et al. 2011). In the present context we shall be content if the average time tav between each interference event (photon counted) is long compared to the coherence time of the photon source τc = 1/ ω tav , with ω being the bandwidth of the source (see Sect. 2.1.4, Vol. 2).

90

2


also characterized by an amplitude ψ(r), determining the probability for finding a particle at position r in a volume element d3 r: 2 dw(r) = w(r)d3 r = ψ(r) d3 r.

(2.5)

And as in optics, diffraction and interference occurs, e.g. at a double slit for which (2.4) describes the experimentally observable signal of particles. However, in contrast to electromagnetic radiation, where ψ was identified with the electric (or magnetic) field, for matter waves ψ(r) cannot be attributed to any directly measurable physical quantity: the observable physics is described by the probability w(r). So we call ψ(r) simply the probability amplitude for finding a particle at position r. Otherwise, analogue considerations are valid for photons and particles of matter. If e.g. we try to localize them on their pathways, we loose the interference patterns! One important general rule holds: interference phenomena are observed if different but indistinguishable pathways exist on which the particles may proceed toward the detector. In contrast, no interference is observed if the two pathways may in principle be distinguished – even when no distinction is made in the actual experiment. Section summary

• In classical mechanics well defined trajectories in position r(t) and momentum p(t) space describe the motion of particles. In contrast, quantum mechanics describes probability amplitudes ψ(r) and probabilities dw(r) = |ψ(r)|2 d3 r for finding a particle at a certain position r in a volume element d3 r. • Diffraction and interference of matter waves can be described by concepts in analogy to those used in wave optics. The big difference is that for photons the “probability amplitude” is also a directly measurable quantity (electric or magnetic field) while for matter waves ψ(r) cannot be measured directly – only probabilities |ψ(r)|2 refer to the real world.

2.2

SCHRÖDINGER Equation

Wave equations for matter waves were derived independently by Erwin S CHRÖDINGER and PAUL D IRAC who in 1933 jointly received the N OBEL prize in physics for their ground breaking work. The nonrelativistic S CHRÖDINGER equation is most commonly used in AMO physics and will briefly be introduced here. As we shall see in Chap. 6, the relativistically correct D IRAC equation, valid for fermions, is significantly more complex and describes inherently also the electron spin. Its key results may, however, also be introduced as perturbations into the S CHRÖDINGER equation which then leads to sufficiently accurate results in the non relativistic energy regime.

2.2


2.2.1

91

Stationary SCHRÖDINGER Equation

In contrast to photons, particles of matter may be exposed to external forces which change their momentum and hence their wavelength. One may start from the wave equation (2.2) and try to guess, how to modify it for matter waves. Considering a particle of mass m with a total energy W moving in a conservative potential V (r), we simply use the (nonrelativistic) energy conservation law of classical mechanics W = Wkin + V to determine its momentum from the kinetic energy Wkin : p 2 = 2mWkin = 2m W − V (r) . With W being a constant of motion we ‘derive’ from this and (1.118) the absolute value of the wave vector k = p/ and insert it into (2.2):

ψ(r) +

2m(W − V (r)) p2 ψ(r) = ψ(r) + ψ(r) = 0. 2 2

Rewritten, this is already the stationary S CHRÖDINGER equation −

2

ψ(r) + V (r)ψ(r) = W ψ(r), 2m

(2.6)

or somewhat more compact ψ(r) = W ψ(r) H

(2.7)

with the eigenfunction ψ(r) and the eigenenergy W . The H AMILTON operator H represents the total energy (briefly just Hamiltonian): = − + V (r). H 2m 2

(2.8)

In the often encountered one-dimensional case the S CHRÖDINGER equation (2.6) is further simplified: −

2.2.2

2 d2 ψ(x) + V (x)ψ(x) = W ψ(x). 2m dx 2

(2.9)

HAMILTON and Momentum Operators

We may write the H AMILTON operator (2.8) even more suggestive: 2 p2 = − ∇ 2 + V (r) = + V (r). H 2m 2m

Here we have introduced the momentum operator (a vector operator)

(2.10)

92

2


⎛ ⎞ ∂/∂x p = −i∇ = −i ⎝∂/∂y ⎠ ∂/∂z

(2.11)

so that p2 = p · p = −2 ∇ 2 = −2 = −2

∂2 ∂2 ∂2 . + + ∂x 2 ∂y 2 ∂z2

(2.12)

Thus, (2.10) is the operator form of the classical energy conservation law W = Wkin + V =

2.2.3

p2 + V (r). 2m

Time Dependent SCHRÖDINGER Equation

So far we have discussed probability amplitudes only as a function of r, the particle’s position in space. For many problems in AMO physics such a description of quantum systems by stationary states is fully sufficient. However, often time dependence is equally important. In respect of photons we know, that electromagnetic waves follow a time dependent wave equation, which is a 2nd order partial differential equation (PDE) in space and time derived from the M AXWELL equations. In contrast, for matter waves the time dependent S CHRÖDINGER equation holds: Ψ (r, t) = i ∂Ψ (r, t) H ∂t or more explicitly: −

(2.13)

2 ∂Ψ (r, t)

Ψ (r, t) + V (r)Ψ (r, t) = i . 2m ∂t

It cannot be derived, and we just communicate it here as it was ‘found’ by Erwin S CHRÖDINGER in the beginning of 1926 – by the way: during a winter ski holiday in the Swiss Alps. We point out some key aspects: • The S CHRÖDINGER equation is a linear PDE of 2nd order in space and 1st order in time! As a consequence, time dependence of stationary states is truly complex. • The linear superposition principle may be applied to the solutions. • The statistical interpretation of quantum mechanics understands solutions Ψ (r, t) of this PDE as probability amplitudes for finding a particle at a position r in space at time t according to (2.1). • Although the S CHRÖDINGER equation cannot be ‘derived’, it is able to describe a wealth of atomic phenomena and to predict observables quantitatively (under nonrelativistic conditions) with excellent accuracy: we emphasize that is this very fact which defines the “validity” of a physical theory. • Equivalently, formal (algebraic) quantum mechanics can be deduced from a consistent set of axioms, which are, however, also heuristically assumed.

2.2


93

• Consistent alternatives of wave equations for matter waves are the D IRAC equation for fermions, a multi component spinor equation and the K LEIN -G ORDON equation, a single component PDE of 2nd order in time, which turns out to be valid for bosons – both are relativistically correct. (r, t) = H (r), the time depenIf the Hamiltonian itself does not depend on time, H dence of the wave function may be factored by a product ansatz: Ψ (r, t) = ψ(r)ϕ(t) Ψ (r, t) = i ∂Ψ (r, t) H ∂t

⇒

(2.14)

ψ(r)ϕ(t) = i ∂ψ(r)ϕ(t) H ∂t H ψ(r) i ∂ϕ(t) = ≡ W. ψ(r) ϕ(t) ∂t

The last identity (with the constant W which has to be determined) must hold so that the former equality can be valid for all values of r and t. One thus has to ψ(r) = W ψ(r). In this case, the time dependent solve idϕ(t)/dt = W ϕ(t) and H equation leads to the trivial solution

W ϕ(t) ∝ exp −i t .

(2.15)

The position dependent part is nothing but the stationary S CHRÖDINGER equation (2.6) and the parameter W introduced is the total energy of the system. The overall wave function is given by

W Ψ (r, t) = ψ(r) exp −i t .

(2.16)

Note: The time dependence is truly complex and the imaginary unit i in the prefactor = H (t) the time is needed for the solution! In the present case, however, with H dependence is trivial in the sense that only 2 2 w(r, t) = Ψ (r, t) = ψ(r)

(2.17)

can be measured, which contains only information about the stationary state. In order to find these stationary atomic states for a given potential V (r) – which are the equivalent to B OHR’s stationary orbits – one has to solve the stationary S CHRÖDINGER equation (2.6) under suitably chosen boundary conditions. For bound states this typically leads to a whole series of discrete total energies Wγ < 0 with a set of quantum numbers γ characterizing the states. The corresponding series of wave functions – for individual electrons called orbitals – describe the probability of finding the particles at a given position in space.

94

2

2.2.4


Freely Moving Particle – The Most Simple Example

As the most simple example we consider now a freely moving particle with mass m, (kinetic) energy W and momentum p. The relevant stationary S CHRÖDINGER equation (2.6) reads −

2

ψ(r) = W ψ(r) 2m

and has solutions ψ(r) = C · exp(ikr).

(2.18)

As one verifies by inserting the solution with wave vector k = p/ into the equation, the total energy is given by W = 2 k 2 /(2m) = p 2 /(2m) = Wkin . With the time dependence (2.16) the whole probability amplitude for this free particle is simply a plane wave:

2 k t − kr (2.19) Ψ (r, t) = C · exp −i(ωt − kr) = C · exp −i 2m

pr W t− . = C · exp −i With this we obtain the important dispersion relation for free matter waves: ω=

W p2 k 2 = = = ω(k) 2m 2m

or W =

2 2 k . 2m

(2.20)

Note: The probability to find this particle, w(r, t) = |Ψ (r, t)|2 = |C|2 , is independent of position and time – as expected for an infinitely extended plane wave. In other words, a particle with well defined momentum may not be localized at all – as expected according to the uncertainty relation (1.123). Section summary

• The stationary S CHRÖDINGER equation (2.6) may be gleaned from the classical wave equation combined with a free interpretation of the DE B ROGLIE wavelength in a conservative potential. It is validated by the excellent agreement of its predictions with experimentally observed data in the submicroscopic world at nonrelativistic energies. ψ = W ψ, where the H AMILTON operator • In its most compact form it reads H (2.10), also called Hamiltonian, is constructed in full analogy to its classical counterpart, just replacing the momentum by the quantum mechanical momentum operator p = −i∇. • The time dependence of the wave function is described by the time dependent S CHRÖDINGER equation (2.13). In the case that the Hamiltonian itself is not time dependent, the (stationary) solutions are given by the product of the solutions ψ(r) of the stationary equation (2.6) and a simple exponential function ∝ exp(−i(W/)t). • The most simple solution of the time dependent S CHRÖDINGER equation is a plane wave exp[−i((W/)t − kr)], with the energy W = k 2 /(2m).

2.3

Basics and Definitions of Quantum Mechanics

95

2.3


2.3.1

Axioms, Terminology and Rules

Here we summarize briefly the fundamental axioms of quantum mechanics and recall some terminology and rules which we shall use later on:

Quantum States and Wave Functions States of quantum systems (in the world, in atomic physics . . . ) are described by state vectors for which we introduce here the “bra” and “ket” notation of D IRAC, ψ| and |ψ , respectively. Typically, states may be expressed in terms of basis states, or basis vectors, say |f1 , |f2 , |f3 , . . . , |fn , . . . . We speak about a complete basis set, if each state of a system can be written as ket vector |ψ =

∞

ci |fi ,

alternatively as bra vector ψ| =

i=1

∞

ci∗ fi |.

(2.21)

i=1

Note: the sum includes all bound (discrete) and free (continuum) states. One calls a basis orthonormal if fk |fi = δki ,

(2.22)

where φ|ψ is the scalar product of two state vectors |ψ and |φ . With this, the projection of the state |ψ onto the basis vector |fk is fk |ψ =

∞

ci fi |fk = ck .

(2.23)

i=1

Wave functions, the most commonly used representation of states, are formally obtained by expanding |ψ in a continuous “position basis” {|r } where r extends over all points in 3D position space. We write (2.21) as ket |ψ =

d r ψ r r 3

or as bra ψ| =

d 3 r ψ ∗ r r .

(2.24)

Per definition we have, in analogy to (2.22), r r = δ r − r ,

(2.25)

and obtain from (2.24) the definition of a wave function in position space, r|ψ = ψ|r ∗ =

d3 r ψ r r r = ψ(r),

(2.26)

96

2


and ψ ∗ (r) = ψ|r . In practice, wave functions are determined by solving the stationary S CHRÖDINGER equation. With (2.24) and (2.25) we derive (after one 3D integration) the scalar product in terms of wave functions:2 ψ|φ = ψ ∗ (r)φ(r)d3 r = φ|ψ ∗ . (2.27) For the eigenstates |ψk of the S CHRÖDINGER equation the orthonormality relation (2.22) thus reads: ψi∗ (r)ψk (r)d3 r = δik . (2.28) ψi |ψk = We finally note that, equivalent to the state expansion in position space, one may define wave functions ψ(p) in momentum space by |ψ = d3 p ψ p p and p|ψ =

d3 p ψ p pp = ψ(p).

Operators Linear operators play a key role in quantum mechanics: in general an operator changes an object onto which it acts (e.g. a state vector, a wave function, another and let it act on a ket operator) into something different. Let us call the operator A vector |ψ : ˜ A|ψ = |ψ . Linearity implies that for a superposition of states |ψ = c1 |ψ1 + c2 |ψ2 + · · · c1 |ψ1 + c2 |ψ2 + · · · = c1 A|ψ 1 + c2 A|ψ 2 + · · · . A B is defined by The product of two operators A B)|ψ B|ψ (A =A .

(2.29)

B C = (A B) C = A( B C) but not necesOperator multiplication is distributive, i.e. A sarily commutative. In general AB = B A and one defines a commutator B] =A B − B A [A,

(2.30)

2 We shall use the bra| and |ket symbols rather loosely. In particular, we shall often write wave functions simply as |ψ , |ψk , etc.

2.3


97

which only in special cases may vanish (see Sect. 2.3.3). With |ψ and |ϕ being between two state vectors or wave functions one defines as matrix element of A states |ϕ and |ψ (which may or may not be basis states of an operator A: = ψ ∗ (Aφ)d 3 r. (2.31) Aψφ = ψ|Aφ Without going into details we define the so called adjoint (or Hermitian conjugate) † by operator A † † ψ = ψ|Aφ ψ φ = ψ|Aφ ∗ or φ A A (2.32) † ∗ 3 ψ φd r = ψ ∗ (Aφ)d 3 r. or A (2.33) Of particular importance are the so called Hermitian operators, let us call them O. They are self-adjoint: † ≡ O. O

(2.34)

By this definition and with (2.31)–(2.33) the matrix elements = (Oψ) ∗ φd3 r = ψ ∗ (Oφ)d 3r Oψ|φ = ψ|Oφ

(2.35)

= φ|Oψ ∗ = ψ|Oφ

(2.36)

are all equivalent for Hermitian operators.3

Observables Observables are all physical quantities which can in principle be measured (observed). Every quantity which is observable in classical physics is represented quan A quantum systum mechanically by a linear Hermitian operator, let us call it O. tems can be characterized by a set of eigenstates (eigenvectors) |fk of an observable which it may in principle assume. From the eigenvalue equation O k = ωk |fk O|f

(2.37)

for the eigenvectors |fk . In one determines the eigenvalues ωk of the observable O only one of its eigenvalues any individual physical measurement of the observable O ωk can be observed. From (2.37) and the orthonormality of the |fk basis one sees that the matrix in the basis of its eigenstates between |fi and |fk are elements of O k = ωk δik , Oik = fi |Of

(2.38)

verify these relations one simply expands |ψ and |φ in a basis of eigenvectors (eigenfunc tions) of O.

3 To

98

2


ik is diagonal and because of (2.36) the eigenvalues ωk must be real i.e. the matrix O – as one has to demand for measurable quantities!

Superposition and Expectation Values In general, a quantum systems to be investigated may be found in a state |ψ which Let the state |ψ be a linear superposition of is not an eigenstate of the operator O. eigenstates |fi of the operator O with eigenvalues ωi according to (2.37): |ψ =

ci |fi

with the expansion coefficients ci = fi |ψ .

(2.39)

i

The latter relation follows directly with (2.22). If we measure now the observable many times (as one does in a real experiment), the result of each individual meaO The probability to detect this particular surement is one of the eigenvalues ωi of O. eigenvalue ωi is determined by the probability amplitude ci : O|ψ =O

ci |fi =

ci ωi |fi .

(2.40)

The average value measured for this observable, i.e. the result of many individual measurements applied to the same state |ψ , is called expectation value of the observable: ≡ O |ci |2 ωi = ψ|Oψ . (2.41) The latter identity follows from ψ|O|ψ =

k ci fi O ck f k = ck Of ci f i

i

=

i

k

ωk ci∗ ck fi |fk =

k

i

ωk ci∗ ck δik =

k

ωi |ci |2 .

i

(ψ) Somewhat more general, if the state of the system is |ψ = i ci |gi , i.e. it is its expectation given in an arbitrary basis {|gi } and |gi are not eigenstates of O, value is = ψ|O|ψ O =

∗(ψ)

ci

gi |O|

i

(ψ)

ck |gk =

k

∗(ψ)

ci

k c(ψ) . (2.42) gi |O|g k

ik

Unit Operator We note in passing a nice mathematical trick by rewriting (2.39) as |ψ =

i

|fi ci =

i

|fi fi |ψ =

i

|fi fi | |ψ .

2.3


99

From this we deduce the often very useful fact that the quantum mechanical unit operator4 (which does not change the operand) may be written as 1= |fi fi |, (2.43) i

as long as the states |fi represent a complete orthogonal basis and the summation goes over all states of this basis.

Quantization is determined in an individual measurement Experimentally, when an observable O one always finds one of its eigenvalues. With the measurement one prepares also One the corresponding eigenfunction (eigenstate, eigenvector) of the observable O. may say that by a measurement one projects the eigenvector out of the state |ψ under investigation. Example: HAMILTON Operator are the eigenenergies Wn of a sysThe eigenvalues of the H AMILTON operator H tem. Thus, the Hamiltonian is a particularly important example of an observable. Its eigenvectors |ψn and eigenenergies are determined by solving the S CHRÖDINGER equation (2.7). We may also write it in algebraic form: |ψn = Wn |ψn . H

Example: Spin Projection onto the z-Axis As a further example we mention the projection of the spin of a particle onto a given axis in 3D space. Typically, but not necessarily, one measures the Sz component of the spin. We have already discussed such a measurement in Sect. 1.10, the S TERN G ERLACH experiment. The eigenvalues are here ms and we write the eigenstates quite formally as |sms . The eigenvalue equation is then Sz |sms = ms |sms .

2.3.2

Representations

We have already made use of different ways to describe quantum systems and their changes. The standard terminology speaks about representations of states and operators. 4 In

matrix representation this corresponds to the identity matrix ⎛ ⎞ 1 0 ··· 0 ⎜0 1 ··· 0⎟ ⎜ ⎟ 1=⎜. . . ⎟. . . ... ⎠ ⎝ .. .. 0 0 ··· 1

100

2


SCHRÖDINGER Representation In the S CHRÖDINGER representation (or picture) the operators are differential operators. The states are represented by wave functions. The scalar product is an integral according to (2.27) and the orthogonality of basis states is defined by (2.28). The in respect of an arbitrary basis set |fk , |fi , . . . matrix elements of an operator A are (in general not eigenfunctions of A) † ∗ fi = A†∗ , k = fi∗ (r)Af k (r)d3 r = fk A Aik ≡ fi |Af (2.44) ki where we have used the definition of adjoint operators (2.32). If the operator is Hermitian, i.e. represents and observable, this relation reads k (r) d3 r Oik ≡ fi |Ofk = fi∗ (r) Of (2.45) =

k∗ (r) ∗ fi (r)d3 r Of

∗

∗ i ∗ = Oki = fk |Of .

(2.46)

HEISENBERG Representation In the H EISENBERG representation (or picture) the operators (we mention in this are matrices, which are decontext the N OBEL prize for H EISENBERG 1932) A termined by their matrix elements Aik . The states, say |ψ or |φ , are vectors in H ILBERT space5 which we write |ψ = ψ = b1 f 1 + b2 f 2 + b3 f 3 + · · · |φ = φ = c1 f 1 + c2 f 2 + c3 f 3 + · · · with their components bi f i or ci f i , respectively. The scalar product is here (2.47) ψ|φ = bi∗ ci . Both representations are physically as well as mathematically fully equivalent.

2.3.3

Simultaneous Measurement of Two Observables

The following is a generalization of the H EISENBERG uncertainty relation which states that two canonically conjugate coordinates, such as position and momentum, cannot be measured simultaneously. and B can be measured simultaneously if and only if the Two observables A are also eigenstates of B, i.e. if the following holds: eigenstates of A i = αi |ϕi and B|ϕ i = βi |ϕi . A|ϕ 5A

H ILBERT space is an extension of the 3D vector space to an arbitrary or infinite number of dimensions – in quantum mechanics to an infinite dimensional function space.

2.3


101

Thus, also the following relations must be valid for simultaneous measurability of and B: the operators A B|ϕ i = Aβ i |ϕi = βi A|ϕ A|ϕ i . i = βi αi |ϕi = B A B =! B A. Equivalently: Hence, the operators must commute, A and B is possible if and only Simultaneous measurement of two observables A if their commutator vanishes: B − B A = [A, B] = 0. A

2.3.4

(2.48)

Operators for Space, Momentum and Energy

From the above follow some simple recipes of how to translate classical quantities into quantum mechanical operators. One simply has to substitute: r −→ r

and pi −→ −i

∂ =p i ∂xi

∂ ∂ ∂ = −i∇ = p. or p −→ −i , −i , −i ∂x ∂y ∂z

(2.49)

All other substitutions are derived from these rules. In particular, the classical Hamiltonian total energy Hclass =

p2 + V (r) = Wkin + V 2m

with p 2 = p · p

becomes the H AMILTON operator: 2 = 1 (−i∇) · (−i∇) + V (r) = − + V (r). H 2m 2m

Position in space x and its canonically conjugate momentum p x are the prime example for non-commuting observables. As seen from comparing

∂ ∂ xϕ(x) = −i x ϕ(x) + ϕ(x) p x xϕ(x) = −i ∂x ∂x with x p x ϕ(x) = −ix

∂ ϕ(x) ∂x

the commuted operators p x x and x p x generate completely different results – and hence, the observables x and p x cannot be measured simultaneously. This is, so to say, a formal endorsement of the H EISENBERG uncertainty relation.

102

2.3.5

2


Eigenfunctions of the Momentum Operator p

We shall now find the eigenfunctions and eigenvalues of the momentum in the S CHRÖDINGER picture, starting with the one dimensional case: p x ϕ(x) = px ϕ(x)

⇒

−i

dϕ(x) = px ϕ(x). dx

One easily verifies that ϕ(x) = exp(ipx x/) = exp(ikx x) are solutions of this eigenvalue problem. Each value of px (with −∞ < px < ∞) is an eigenvalue of the momentum operator p x in x-direction: the eigenfunction is thus a plane wave with a continuum of eigenvalues. This is easily extended into 3D space. The eigenvalue equation p ϕ(r) = −i∇ϕ(r) = kϕ(r) a plane wave

is solved by

ϕ(r) = C exp(ik · r).

The unit directional vector for an arbitrary direction in space is e = ax ex +ay ey + az ez with ax2 + ay2 + az2 = 1. If we are interested in the magnitude of the momentum p in any given direction e, the operator providing this information is:

∂ ∂ ∂ + ay + az . (2.50) p = ax p x + ay p y + az p z = −i ax p e = e · ∂x ∂y ∂y Any plane wave exp(ik · r) in arbitrary direction k and with arbitrary magnitude k is an eigenfunction of p e , with eigenvalue p cos γ , as one easily verifies p e exp(ik · r) = (ax kx + ay ky + az kz ) exp(ik · r) = e · k exp(ik · r) = k cos γ exp(ik · r) = p cos γ exp(ik · r),

(2.51)

with p = k and the angle γ between e and k – just as one would guess. Section summary

• In the S CHRÖDINGER picture, states (bra |ψ and ket φ|) of quantum systems are represented by wave functions ψ(r) and ψ ∗ (r), respectively. The H EISENBERG picture uses state vectors in H ILBERT space. • Quantum states may be expressed as a linear superposition of states (2.21) from a complete, orthonormal basis – with fi |fk = δik . The unit operator may be written i |fi fi |. • The S CHRÖDINGER representation uses differential operators. Classical theory is translated by replacing r → r and p → p = −i∇. k = f ∗ (r)Af k (r)d3 r are Aik ≡ fi |Af • Matrix elements of an operator A i in the S CHRÖDINGER picture. In the Heisenberg representation operators are defined by the corresponding matrices. is defined by A † ψ|φ = ψ|Aφ . Hermitian † of an operator A • The adjoint A † operators are self-adjoint A = A, and Aik = A∗ki holds.

2.4

Particles in a Box – And the Free Electron Gas

103

k = ωk |fk • Observables are represented by Hermitian operators. With O|f their eigenvalues ωk are the only values of that observable which can be observed experimentally. • Such an experiment projects the eigenstate |fk of the observable out of a and B, state |ψ under investigation. If the operators for two observables, A B] = 0) they can be measured simultaneously. commute (commutator [A • The average value of an observable in a state |ψ is called its expectation 3 r. = ψ|Oψ = ψ ∗ (Oψ)d value: O • Eigenstates of the momentum operator p are plane waves exp(ik · r), with p = k.

2.4


2.4.1

One Dimensional Potential Box

We first consider the one dimensional problem of a particle in a potential box, i.e. between two infinitively high walls at a distance L. In between the walls, the 1D S CHRÖDINGER equation (2.9) simply reads −

2 d2 ψn (x) = Wn ψn (x). 2m dx 2

Solutions can in principle be ψ(x) = sin(kx) or cos(kx). However, since the wave function cannot penetrate into the wall, it must vanish on the walls which at x = 0 is only possible for the sin(kx) solution. For continuity on the other wall sin(kL) = 0 and thus k = nπ/L must hold, with n = 1, 2, 3, . . . . In summary, the solutions (eigenfunctions) are standing waves with nodes on both walls of the box. The eigenenergies assume discrete values Wn : nπx h2 n2 2 2 k 2 sin with Wn = = ψn (x) = . (2.52) L L 2m 8mL2 The expectation value of the momentum p x is derived from px = ψn p x ψn = ψn∗ (x) px ψn (x)dx

d sin nπx nπx L −i dx L dx 0 nπx −i2nπ L nπx cos dx ≡ 0. = sin L L L2 0

=

2 L

L

sin

This corresponds to the fact that the particle in the box moves with equal probability back and forth. In contrast, the square of the momentum,

2 2 L nπx 2 d2 sin nπx L dx x2 ψn = ψn∗ p x2 ψn dx = sin p x = ψn p L 0 L dx 2

104

2

=

2(nπ)2 L3

L

sin2

0


1 h2 2 nπx dx = n , L 4 L2

= is not zero. With this and H px2 /2m we verify Wn in (2.52).

2.4.2

Three Dimensional Potential Box

A next step towards reality is the extension into the 3D space. The movement of a particle is now restricted to a large but finite 3-dimensional box, with rigid walls beyond which the probability to find a particle is zero. For simplicity we assume the box to be a cube with edge length L as illustrated in Fig. 2.2(a). Inside the box particles move freely. Stationary solutions are plane waves (2.18) which we write in the box as real functions – now as product in three dimensions: ψ(x, y, z) = sin(kx x) sin(ky y) sin(kz z).

(2.53)

To be continuous, the wave function must vanish on the walls of the cube: sin(kj L) = 0

⇒

kj = n j

π L

for j = x, y, z.

(2.54)

With these boundary conditions, and in analogy to the 1D case (2.52), the energy of the particle in a box becomes W=

2 k 2 2 π 2 2 2 2 kx + ky2 + kz2 = = n , 2m 2m 2mL2

(2.55)

now with three integer quantum numbers nx , ny , nz and n2 = n2x + n2y + n2z . One may view this in k or n space as indicated in Fig. 2.2(b). Equation (2.54) says that precisely one solution exists for each lattice point with integer nx , ny and nz . Thus, one reads from the figure that the total number of states with quantum numbers 1 to nx , 1 to ny and 1 to nz , i.e. the number of states with n ≤ n2x + n2y + n2z is given by NZ (n) = 1/8 × 4π/3 n3 . Expressing n by the energy W , the total number of states with energies ≤W is NZ (W ) = Fig. 2.2 Boundary condition for a particle in a potential box (cube of edge length L); (a) the nodes of the wave function are on the walls of the box in 3D position space; (b) one counts the number of states n in the n or k space

(a)

1 (2mW )3/2 3 L . 6π 2 3 z

(b)

n z ∝ kz

L n

L L x

y

ny ∝ ky nx ∝ kx

2.4


105

If the particle has a spin s, we also have to account for the energy degenerate ge = 2s + 1 possible orientations in space. Dividing by the volume of the box L3 , we obtain the total number of available states per unit volume: NZ (W ) =

ge (2mW )3/2 . 6π 2 3

(2.56)

From this we derive the number of states in an energy interval from W to W + dW , called density of states (DOS), here per unit volume: √ ge (2m)3/2 √ 4 2πm3/2 √ dNZ (W ) = W = ge W. (2.57) g(W ) = dW 4π 2 3 h3 We note in passing, that assuming the phase space to be quantized in unit cells of size h3 leads to exactly the same result.6 For later use we also give the density of states in respect of a specific element of solid angle dΩ and express the energy by k, the magnitude of the wave vector: dg =

2.4.3

ge mk d2 NZ (W ) dΩ = dΩ. dW 4π (2π)3 2

(2.58)

The Free Electron Gas

An important application are electrons (mass me , s = 1/2, ge = 2). In the context of the photoelectric effect (Sect. 1.4.1) we have already introduced the free electron gas model, where one assumes the electrons in a metal to move freely in an ‘electron sea’. Now, as a 1st order approximation, we describe this situation quantitatively by electrons in a 3D box. It turns out that such a model serves well in many areas of physics, as a first step e.g. when introducing electron bands in solid state physics, but also in atomic physics for statistical models of electrons in a large atom (see Sect. 10.1.5). Due to the high particle density Ne (electrons per unit volume), F ERMI -D IRAC statistics of electrons in the bulk is quite different from that in gases treated in Sect. 1.3.4. We estimate Ne = ν × NA ρ/Mr g mol−1 , with ν the number of valence electrons per atom, NA the AVOGADRO number, ρ the (mass) density of the material and Mr the relative atomic mass. It is typically on the order of 1028 –1029 m−3 . In contrast to the situation treated in gases, the chemical-potential μ is now positive and μ kB T . At absolute zero temperature, T = 0 (or rather at very low values of T ), each available state (with any given spatial wave function according to Eqs. (2.53) and (2.54)) will be filled by two electrons (with spin up and spin down, ge = 2). The potential box will thus be ‘filled’ up to an energy F . With (2.56) the 6 The

size of phase space with momenta up to p is (4π/3)p 3 L3 . Expressing p in terms of kinetic energy W , and dividing by h3 and L3 gives the number of phase space cells per unit volume: NZ = (4π/3)(2mW )3/2 / h3 . Differentiation in respect of W yields (2.57).

106

2

(a)

w(W ) / eV -1

∝ √W

0.2

w (W ) / eV -1

∝ √W

0.2

0.1

0


0.1

Ne 0

2

4

6

8 F

0

T = 0K T = 50 K T = 500K T = 293 K

Ne 6.8

W / eV

(b)

6.9

7.0

7.1

7.2

F

Fig. 2.3 F ERMI -D IRAC probability distribution according to (2.61) as a function of energy for electrons in a metal with a F ERMI energy F = 7 eV at different temperatures T : (a) energy range from 0 eV to 9 eV, (b) expanded scale around F ; at T = 0 the electron density Ne extends up to F √(grey shading up to the dashed vertical red line); the full red line indicates the density of states ∝ W

total number of electrons per volume with kinetic energies between 0 and F is thus

1 2me F 3/2 Ne = 2NZ (F ) = . (2.59) 3π 2 2 We may invert this to obtain the so called F ERMI energy F . At absolute zero temperature F is identical to the chemical-potential μ introduced in Sect. 1.3.4, and corresponds to the maximum electron energy. One state after the other is filled, each by two of electrons, up to the F ERMI energy: F =

2 2 2/3 . 3π Ne 2me

(2.60)

Typical F ERMI energies for metals range from 1.6 eV (Cs) to 14.3 eV (Be). Expressing the prefactor A given by (1.64) with ge = 2 in terms of the F ERMI energy F one may write the probability distribution for finding a given energy W in the electron gas in a box: √ dNe 3 1 3/2 W w(W )dw = dW = Ne 2 F exp[(W − μ)/(kB T )] + 1

(2.61)

with μF F as long as kB T F . This is illustrated in Fig. 2.3 for a characteristic example with F =√7 eV (about the value for Cu). The full red line indicates the density of states ∝ W according (2.57). The grey shaded area under this curve indicates for T = 0 K the filling with electrons up to F (dashed vertical red line). For temperatures T > 0 K the states will be occupied according to the F ERMI -D IRAC statistics (1.66). As clearly seen on an enlarged energy scale in Fig. 2.3(b) the boundary between occupied and

2.5

Angular Momentum

107

unoccupied states broadens as temperature increases (width 2kB T ) – the reduction of the probability for W < F is approximately compensated ∞by the increase at W > F , and the F ERMI distribution remains normalized, i.e. 0 w(W )dw 1. For metals at room temperature one typically finds kB T /F < 1/100 and the normalization condition holds very well. However, at higher temperatures, kB T /F 0.1, or for different densities of states (e.g. in semiconductors), the chemical-potential μ must be readjusted in order to maintain the normalization of w(W ). In solid state physics it is often called F ERMI level WF = μ, which, strictly speaking, is identical to the F ERMI energy F only at T = 0. Section summary

• The particle in a box model allows a most simple description of electrons moving freely in a metal. The wave functions (2.53) in the 3D box (volume L3 ) have nodes at theboundaries of the box. The energies are W = 2 π 2 n2 /(2mL2 ) with n = n2x + n2y + n2z and ni representing positive values in 3D integer number space. • From this√ the DOS (2.57) is derived, which according to (2.57) is ∝ ge m3/2 W , where m is the particle’s mass and ge = 2s + 1 the degeneracy due to its spin. • In the case of electrons (Fermions) each state can be filled with up to 2 electrons. Then, at T = 0 K the highest energy with occupied states is the F ERMI energy F = 2 (3π 2 Ne )2/3 /(2me ). • At temperatures T > 0 the boundary between occupied and unoccupied states smears out according to (2.61). The width of the boundary layer is on the order of kB T .

2.5

Angular Momentum

Angular momenta play a central role in atomic and molecular physics and Appendix B gives a summary on the essentials: their abstract definitions, properties, combinations and the relevant algebra. Here we introduce the S CHRÖDINGER picture of orbital angular momenta as used in the quantum mechanics of the H atom. The orbital picture emerging from this concept is directly accessible to physical imagination and visualization. At the end of this section we shall generalize the basic concepts, including electron spin.

2.5.1

Polar Coordinates

Quantum mechanical problems may be treated with advantage in polar rather than in Cartesian coordinates if they have a symmetry centre, e.g. if the potential depends only on the distance r from origin V (r) = V (r) as in the C OULOMB case. The

108

2


Fig. 2.4 Cartesian (x, y, z) and polar coordinates (r, θ, ϕ)

z r

θ

y

φ x

transformations between Cartesian (x, y, z) and polar coordinates (r, θ, ϕ) are read from Fig. 2.4, x = r sin θ cos ϕ y = r sin θ sin ϕ

(2.62)

z = r cos θ, and the volume element transforms as dxdydz → r 2 sin θ dθ dϕdr.

(2.63)

Conversion of the relevant observables such as the kinetic energy

2 2 p2 2 ∂ 2 ∂2 ∂2 =− ∇ =− + + 2m 2m 2m ∂x 2 ∂y 2 ∂z2 requires some – in principle trivial – partial differentiations and mathematical rearrangements which we shall not explicate here. The result is

2 2 2 L p2 2 1 ∂ 2 ∂ =− ∇ =− r + with 2m 2m 2m r 2 ∂r ∂r 2mr 2

1 ∂ ∂ 1 ∂2 2 2 . sin θ + 2 L = − sin θ ∂θ ∂θ sin θ ∂ϕ 2

(2.64) (2.65)

In full analogy to classical mechanics, the form of (2.64) suggests two components of kinetic energy:

2 1 ∂ 2 ∂ radial energy Hr = − r and (2.66) 2m r 2 ∂r ∂r rotational energy

2 rot = L . H 2mr 2

(2.67)

This suggestive way of writing the energy already implies that L might be the quantum mechanical equivalent to the classical angular momentum. In the following, a more formal approach to derive an expression for the angular momentum will be sketched. 2

2.5

Angular Momentum

2.5.2

109

Definition of Orbital Angular Momentum

In classical mechanics angular momentum is defined as L = r × p. The quantum L which accordmechanical equivalent is the orbital7 angular momentum operator ing to the recipe (2.49) is obtained by substituting p → p: L = r × p.

(2.68)

This has to be expressed in polar coordinates. We just show this for one coordinate by way of example:

z = −i x ∂ − y ∂ . (2.69) L ∂y ∂x With (2.62) one transforms the expression in brackets into polar coordinates: ∂ ∂x ∂ ∂y ∂ ∂z ∂ ∂ ∂ = + + = −r sin θ sin ϕ + r sin θ cos ϕ +0 ∂ϕ ∂ϕ ∂x ∂ϕ ∂y ∂ϕ ∂z ∂x ∂y = −y

∂ ∂ ∂ ∂ +x =x −y . ∂x ∂y ∂y ∂x

Thus, the operator for the z component of orbital angular momentum is z = −i ∂ . L ∂ϕ

(2.70)

Obviously this is constructed in complete analogy to the linear momentum p x = −i∂/∂x, with the pair of canonically conjugate coordinates ( px , x) replaced by z , ϕ). (L y is slightly more complicated. In summary one x and L The transformation of L finds, here without proof, that the operator of the squared orbital angular momen2 2x + L 2y + L 2z is indeed given by (2.65). Some details are explained in tum L =L Appendix B.

2.5.3

Eigenvalues and Eigenfunctions

We are now ready to discuss eigenvalues and eigenstates of angular momentum operators. They will be used in more or less all of the following chapters. In the spirit of this textbook, again we only sketch the basic concepts, present a collection of necessary tools, and refer the reader to the standard textbooks on quantum mechanics and angular momentum algebra for details. 7 “Orbital”

to distinguish it e.g. from the spin angular momentum.

110

2


z of the Orbital Angular Momentum Component L In polar coordinates the z-axis represents a preferred direction, since the polar angle z is an ODE θ is defined in respect of it. With (2.70), the eigenvalue equation of L of 1st order z Φ(ϕ) = z Φ(ϕ) L −i

∂ Φ = z Φ(ϕ) ∂ϕ

(2.71)

and may be integrated directly. The solution is

z Φ = C exp i ϕ with a normalization constant C. We now have to apply some physics: which values of z are really meaningful? Obviously Φ(ϕ) has to be unique:

z ! ! Φ(0) = Φ(2π) or equivalently exp(0) = exp i 2π . (2.72) This is only possible for integer values of z / = m with m = 0, ±1, ±2, . . . . Then

z exp i 2π = exp(im2π) = 1, and Φm (ϕ) = Cm exp(imϕ) are the eigenfunctions solving (2.71). We call m the projection quantum number (in the literature often also – somewhat misleadingly – called magnetic quantum number). These wave functions are orthonormal: 2π ! ∗ δmm = Φm |Φm = Cm Cm exp(−imϕ) exp im ϕ dϕ (2.73) √ with Cm = 1/ 2π.

0

The standard phase convention uses real normalization constants Cm . In summary, for the projection of the angular momentum onto the z-axis the eigenvalue equation is

z Φm = mΦm , L

with eigenfunctions

1 Φm = √ exp(imϕ), 2π

and eigenvalues

m

(2.74)

where m = 0, ±1, ±2, . . . .

Components in x- and y-Direction x and L y components the calculation is more complicated but in principle For the L x , L y , and L z cannot be measured simultrivial. We communicate without proof: L taneously. Each pair of them does not commute. On the contrary, one may show that the following commutation rules for angular momenta hold:

2.5

Angular Momentum

111

x , L y ] = L x L y L z , y − L x = iL [L z ] = iL x y , L [L

z , L x ] = iL y . and [L

(2.75)

i are represented by different functions of ϕ Corresponding to this all components L and θ (not shown here).

Square of the Orbital Angular Momentum 2 With (2.65) we write the eigenvalue equation for L as 2 L Y (θ, ϕ) = L2 Y (θ, ϕ)

(2.76)

with the eigenvalue L2 and use a the product ansatz towards its solution: Y (θ, ϕ) = Θ(θ )Φ(ϕ).

(2.77)

z and substitute Θ(θ )Φm (ϕ) As azimuthal part we try the eigenfunctions (2.74) of L into (2.65). This leads to −

2

1 ∂ ∂Θ(θ ) 1 ∂2 sin θ Φm (ϕ) + 2 Θ(θ ) 2 Φm (ϕ) sin θ ∂θ ∂θ ∂ϕ sin θ

= L2 Θ(θ )Φm (ϕ) = L2 Y (θ, ϕ)

1 ∂ ∂Θ m2 ⇒ −2 sin θ − 2 Θ = L2 Θ. sin θ ∂θ ∂θ sin θ

(2.78)

Thus, only one ODE remains to be solved. Several procedures lead to the correct solutions. One may directly use the associated L EGENDRE polynomials, known from the mathematics of ODEs, or (perhaps more elegantly) by exploiting the properties of the angular momentum operators to find appropriate recursion formulas. In any case, one has to demand physically reasonable boundary conditions. In analogy to (2.72) for the z-component, the wave functions must be finite and unique for 0 ≤ θ ≤ π . Without proof we communicate here that such physically meaningful solutions exist for which the following relations hold (for completeness we include again the z-component): eigenvalue equation for L

2 L Y m (θ, ϕ) = ( + 1)2 Y m (θ, ϕ) (2.79)

eigenvalues of L

L2 = ( + 1)2

(2.80)

eigenvalue equation for Lz

z Y m (θ, ϕ) = m Y m (θ, ϕ) L

(2.81)

quantum numbers

= 0, 1, 2, . . .

degeneracy

2 + 1

2

2

and m = 0, ±1, . . . ±, (2.82) (2.83)

112

2


Fig. 2.5 Vector diagram illustrating the 2 + 1 possible orientations of the angular momentum in space (shown for = 2 as an example)

z

Lz / ħ 2

L L

1 0 y -1

x

-2

z only acts The validity of (2.81) follows directly from (2.74) and (2.77), since L onto the ϕ component of Y m (θ, ϕ). This implies 2 2 z L L L z=L

or

2 z = 0. L ,L

(2.84)

2 2 z can be measured simultaneously. This also holds for Equivalently: L and L L 2 x as well as for y – not, however, for the components L x , L y and L z and L L and L which according to (2.75) do not commute with each other.

Vector Diagram With (2.79) one may write the magnitude of the angular momentum as: | L| = ( + 1).

(2.85)

For a given set of quantum numbers m the exact direction of the total angular momentum is undefined. Precisely defined is only the magnitude and component Lz = m in respect of the z-axis. One visualizes these relations with the help of a vector diagram shown in Fig. 2.5. of√the electron spin (see Fig. 1.43). Figure 2.5 z / = m = −2, −1, 0, 1, 2. illustrates the example = 2, | L|/ = 6 2.45 with L One may consider the vector arrows statistically distributed around the z-axis, i.e. √ on cones of height m with a side length ( + 1).

Spherical Harmonics 2 z are called spherical harmonics Y m (θ, ϕ). General The eigenfunctions of L and L formulas and properties are summarized in Appendix B.1.2, and specific expressions up to = 3 are tabulated in Table B.1. A graphical illustration of the angular dependence is shown in Fig. 2.6. The Y m (θ, ϕ) are orthonormalized:

2π

π

dϕ 0

0

∗ Y m (θ, ϕ)Y m (θ, ϕ) sin θ dθ = δ δm m .

(2.86)

2.5

Angular Momentum

113

z

x

y

|Y00|2

|Y10|2

|Y1±1|2

|Y20|2

|Y2±1|2

|Y2±2|2

Fig. 2.6 3D-plot of the s, p and d spherical harmonics; plotted are the squared moduli as a function of angles, the shading colours indicate the sign of Y m (θ, 0). For an alternative representation in the real basis see Appendix D.3 and in particular Fig. D.1

The complex conjugate is given by ∗ Y m (θ, ϕ) = (−1)m Y −m (θ, ϕ),

(2.87)

and inversion at the origin (r → −r) leads to Y m (π − θ, π + ϕ) = (−1) Y −m (θ, ϕ),

(2.88)

which describes (see detailed discussion in Appendix D) so called positive or negative parity depending on whether is even or odd, respectively. At this point, we introduce an important, commonly used notation: orbital angular momenta = 0, 1, 2, 3, 4, . . . are labelled s, p, d, f, g, . . . . In the following text we shall, for compactness, usually write the spherical harmonics in “bra” and “ket” form, substituting Y m (θ, ϕ) → | m

∗ and Y m (θ, ϕ) → m|.

(2.89)

In this notion, the orthogonality relations (2.86) and the matrix elements of an operator Aˆ are written as m m = δ δmm

m . and A m, m = m|A

(2.90)

Several useful relations are summarized in Appendixes B and C. It has to be pointed out, that the complex form of the spherical harmonics Y m (θ, ϕ) illustrated in Fig. 2.6 is well adapted for many problems in atomic physics – but it is by no means the only possible representation of the angular part of atomic orbitals. An alternative frequently used in molecular physics (see Chaps. 3 and 4, Vol. 2) and always in chemistry, are real combinations of spherical harmonics. They are described in some detail in Appendix D.

114

2.5.4

2


Electron Spin

As shown by the S TERN -G ERLACH experiment, the electron has in addition to mass and charge one further property which we have identified in Sect. 1.10 as an intrinsic angular momentum, called spin. The√ spin is characterized by the spin quantum number s = 1/2. Its magnitude is |S| = s(s + 1), and two orientations with angular momentum components /2 and −/2 are possible. Closely related to the spin, the electron has also a magnetic moment with a g factor – as defined by (1.162) – close to ge 2. One simply transfers the formal rules which we have introduced in Sects. 2.5.2– 2.5.3 onto the properties of the spin. Clearly, the spin may not be imaged in position space. However, we may define quite formally in analogy to the orbital angular 2 and a momentum L a new vector operator S with a square magnitude operator S component Sz in z-direction for which the general commutation rules for angular momenta (2.75) and (2.84) are valid: 2 S , Sz = 0. (2.91) They imply, as in the general case, that the components of the spin cannot be measured simultaneously. However, it is possible to determine its magnitude together with one of its components is (e.g. the z-component Sz ). We now introduce spin states |sms . In analogy to (2.79)–(2.83) (for the orbital angular momentum) [ Sx , Sy ] = i Sz ,

[ Sy , Sz ] = i Sx ,

2 |sms = s(s + 1)2 |sms S

[ Sz , Sx ] = i Sy ,

and Sz |sms = ms |sms

(2.92)

with ms = −s, −s + 1, . . . , s holds. Specifically, for the electron spin with s = 1/2 and ms = ±1/2 only two basis states exist, 1 1 1 1 − = |β = |− , (2.93) 2 2 = |α = |+ and 2 2 for which the spin points into +z- and −z-direction, respectively. We have introduced here three equivalent notations which are commonly used for compactness. One also finds in the literature the notation spin function α and β without bra or ket. Alternatively, one speaks about “spin up” (↑) and “spin down” (↓) states. In any case, the relations 2 |α = 3 2 |α S 4 Sz |α = |α 2

2 |β = 3 2 |β S 4 Sz |β = − |β 2

(2.94)

hold, together with the orthonormality relations α|β = β|α = 0 and α|α = β|β = 1.

(2.95)

2.5

Angular Momentum

115

From the commutation rules (2.91) one may derive how the other components of S act onto the basis (here without proof): Sx |α = |β 2

Sx |β = |α 2

i Sy |α = |β 2

i Sy |β = − |α . (2.96) 2

In this basis each arbitrary spin state of an electron may be expressed as |χ = χ+ |α + χ− |β ,

(2.97)

with the probability amplitudes χ+ and χ− normalized to unity: χ|χ = |χ+ |2 + |χ− |2 = 1. The probabilities for finding in this state the α and β component (or spin up ↑ and spin down ↓) is given by |χ+ |2 and |χ− |2 , respectively. The expectation values of the spin components Sk (with k = x, y or z) in the spin state (2.97) are readily obtained from Sk = χ| Sk |χ using (2.94) and (2.96). Note that although the three components of the spin cannot be measured simultaneously, magnitude and phase of the (complex) amplitudes χ+ and χ− determine the orientation of the spin in three dimensional space uniquely. Assume we start with a pure basis state – say |α prepared with an ideal S TERN -G ERLACH experiment, pointing into an arbitrary direction of space. In a given coordinate system with x-, y-, and z-direction (differing from that S TERN -G ERLACH coordinate frame) this state |α will again be a superposition of the type (2.97). The mathematics to do such a coordinate rotation in general is summarized in Appendix E. Experimentally one may determine the expectation values Sk in respect of the x-, y-, and z-coordinates by using a second, rotatable S TERN -G ERLACH setup. With this one may perform experiments in each of the corresponding directions – and repeat them many times to obtain expectation values. From these (one needs at least two of them) the probability amplitudes χ+ and χ− in the new system may easily be derived. It is convenient to write these amplitudes with (E.16) and γ = 0 as

ϕ ϕ θ θ χ+ = cos exp −i and χ− = sin exp i , (2.98) 2 2 2 2 which are normalized by definition. As an exercise, the reader may show with the aid of (2.94) and (2.96), that the parameters θ and ϕ give the polar and azimuthal angles, respectively, at which the so defined spin state is oriented in space. The above provides a toolbox which is fully sufficient to describe the properties of the spin states. Nevertheless one often writes – perhaps for historical reasons or better visualization – operators in the form of matrices and eigenstates as vectors, the so called spinors χ+ χ= and χ † = χ+∗ χ−∗ (2.99) χ− 1 0 with the basis α = and β = . 0 1

116

2


The matrix elements of the operators Sx , Sy and Sz are obtained from (2.96). They may be summarized as spin operators in matrix form, S= σ, 2

(2.100)

with σ being a vector operator, the so called PAULI vector, which is built from the PAULI matrices:

0 −i 1 0 σy = and σz = σx = i 0 0 −1

3 1 0 2 . Sx2 + Sy2 + Sz2 = 2 S = 0 1 4 0 1 1 0

(2.101) (2.102)

For later use we note here that the PAULI matrices anti-commutate σi σj + σj σi = 2δij ,

(2.103)

1 and σx σy = −σy σx = iσz . σx2 = σy2 = σz2 =

(2.104)

and thus

Another useful relation is obtained by applying (2.103):

3 2 3 2 2 1 0 2 2 2 = r 2 3(S · r) = (σx x + σy y + σz z) = r S . 0 1 4 4

(2.105)

Section summary

• Orbital angular momentum operators may be derived from L=r × p , with p = −i∇, or be constructed from the commutation rules (2.75). 2 z according to • From the spatial representations of the operators L and L (2.65) and (2.70), respectively, one obtains eigenvalues and wave functions as summarized in (2.79)–(2.83), and schematically illustrated in the vector diagram Fig. 2.5. • The shape of the orbitals with lowest angular momentum – s, p and d – shown in Fig. 2.6 should be memorized. General formulas and properties are summarized in Appendix B.1.2, specific expressions are tabulated in Table B.1. • The electron spin obeys the same commutation rules as orbital angular momenta. Its intrinsic angular momentum, the spin, is however s = 1/2 and the projection quantum number ms = ±1/2. A frequently used representation of the spin operators are the PAULI matrices (2.101) and (2.102) which act on the so called spinors – two component representations of the spin eigenfunctions.

2.6

One Electron Systems and the Hydrogen Atom

2.6

117


We are now fully prepared to treat a particle in a centrosymmetric potential V (r), specifically the one electron problem. Much of the following will be based on what we have learned about angular momenta in Sect. 2.5. We keep the discussion general as long as possible and specialize to the H atom, i.e. to the pure, attractive C OULOMB potential V (r) < 0 in Sect. 2.6.5.

2.6.1

Quantum Mechanics of the One Particle System

We start by formulating the S CHRÖDINGER equation precisely. With (2.64)–(2.67) we may write the Hamiltonian (2.10) as: p2 r + L + V (r) =− + V (r) = H H 2me 2me r 2

2 1 ∂ 2 ∂ r , with Hr = − 2me r 2 ∂r ∂r 2

(2.106) (2.107)

and obtain the one particle S CHRÖDINGER equation in polar coordinates:

r + H

2 L + V (r) ψn m (r, θ, ϕ) = Wn m ψn m (r, θ, ϕ). 2me r 2

(2.108)

In Sect. 2.5.3 we have discussed in detail the eigenfunctions Y m (θ, ϕ) and eigen2 L . To solve (2.108) we make the separation ansatz values 2 ( + 1) of ψn m (r, θ, ϕ) = Rn (r)Y m (θ, ϕ), so that (2.109)

2 r + L + V (r) Rn (r)Y m (θ, ϕ) = W Rn (r)Y m (θ, ϕ). H 2me r 2 2 r and V (r) act only on the radial and Since H L only on the angular part we obtain with (2.107) and (2.79):

2 ( + 1) 2 1 d 2 d r + − + V (r) Rn (r) = W Rn (r). 2me r 2 dr dr 2me r 2

(2.110)

The sum of centrifugal potential 2 ( + 1)/(2me r 2 ) and C OULOMB potential ∝ −1/r is called effective potential:

118

2


Veff (r) = V (r) +

2 ( + 1) . 2me r 2

(2.111)

With this and the substitution Rn (r) = un (r)/r

(2.112)

one obtains a relatively simple, one dimensional, ordinary differential equation (ODE) which may be integrated without problems: 2 d2 un + Wn − Veff (r) un (r) = 0. 2 2me dr

(2.113)

Note: The total energy does not depend on the projection quantum number m and is thus be written W = Wn . Zero energy is usually set for the completely unbound system, i.e. for electron and nucleus at infinite distance with no kinetic energy. Bound electrons have negative energies Wn < 0, while free electrons have total energies W > 0. Extending (2.89) to the full electron wave function one often abbreviates Rn (r)Y m (θ, ϕ) → |n m .

(2.114)

More specifically, one even writes these atomic orbitals shorthand as n = 1s, 2s, 2p, 3s, 3p, 3d, etc. in the notation introduced in Sect. 2.5.3.

2.6.2

Atomic Units

We recall here the concept of atomic units (a.u.), introduced in Sect. 1.8.3: energy

Eh = me e4 ε0−2 h−2 /4

length

√ a0 = ε0 h2 e−2 m−1 e /π = / me Eh

time

t0 = 2ε02 h3 e−4 m−1 e /π.

(2.115)

Numerical values of these quantities are given in Appendix A, and the most recent, accurate updates can be found at NIST (2010). We use these definitions here to 2 rewrite the radial S CHRÖDINGER equation √ (2.113). We multiply (2.113) by me / 2 and a0 , and apply the identity a0 = / me Eh to obtain in dimensionless form:

1 d2 un ( + 1) V (r/a0 ) un (r) = 0. + W /E − + n h 2 d(r/a0 )2 Eh 2(r/a0 )2

(2.116)

For simplicity, one may just substitute r/a0 → r and Wn / Eh → Wn as well as V (r)/Eh → V (r). This just implies that all observables are measured in a.u. and the radial S CHRÖDINGER equation (2.113) reads now

2.6


119

1 d2 un + Wn − Veff (r) un (r) = 0 (2.117) 2 2 dr ( + 1) with Veff (r) = V (r) + and for the H atom V (r) = −Z/r. 2r 2 Following this scheme one may rewrite all atomic equations in a dimensionless, rather clean looking form. Theoreticians, in particular, like this kind of equation very much and even give the recipe to just set = e = me = 1 – which really oversimplifies what has to be done. The procedure has one serious disadvantage: a dimensional analysis is no longer possible – which is often highly commendable to check complex calculations. Thus, we typically try to use equations in a form indicated by (2.116), and carry the a.u. a0 , Eh , and t0 explicitly along. Sometimes one may even be able to combine elementaryconstants to truly dimensionless quantities, such as the fine structure constant α = Eh /me c2 according to (1.10).

2.6.3

Centre of Mass Motion and Reduced Mass

Up to now we have treated the problem as if the electron would orbit around a space fixed centre. As the nuclear mass is much larger than the electron mass me – in the case of the H atom with a proton as nucleus mp 1840me – the centre of mass is indeed very close to r = 0. For more demanding precision one has, however, to correct for the difference. As in classical mechanics (see corrections to the B OHR model described in Sect. 1.8.5), in quantum mechanics too, one transforms the two body problem into an effective one particle problem by replacing the electron mass ¯ e (1.143) of the system. All a.u. have, in principle, to me with the reduced mass m be replaced correspondingly (kinematic correction): a0 → a¯ 0 = a0

me , m ¯e

Eh → E¯ h = Eh

m ¯e , me

and t0 → t¯0 = t0

m ¯e . me

(2.118)

For simplicity we shall, however, in the following text continue to use me and the units a0 , Eh and t0 , and refer to the exact calculations if relevant.

2.6.4

Qualitative Considerations

While we know already the angular part of the hydrogenic wave functions, we are still left with the task to find physically meaningful solutions to the radial S CHRÖDINGER equation (2.117). They have to behave reasonably at r = 0 (see below) and must not diverge for r → ∞. From this follows necessarily that only a particular set of discrete total energies Wn < 0 leads to such solutions. To find these is the task at hand. Before applying mathematics we want to obtain a qualitative picture to support our physical intuition. Figure 2.7 illustrates this for the = 0 case in a C OULOMB potential V (r) = −Z/r. We derive the kinetic energy of the electron at different

120

2


Wn ℓ

r

W,V(r )

V Wkin

0

rcl

λ large classically for-

un ℓ (r )

bidden region: exponential decrease of

λ = h /p small

wavefunction

r

Fig. 2.7 Bound state radial wave functions for s states, schematic. Top: C OULOMB potential V (r) ∝ −1/r (red) and total energy Wn < 0 (black) determine the classical turning point rcl (onset of the classically forbidden region). Bottom: The characteristic behaviour of radial wave functions un is explained by changes of the kinetic energy Wkin in different regions of the potential (see text)

positions in the potential from Wkin = Wn − V (r) and take the corresponding DE B ROGLIE wavelength λ = h/p = h/(2me Wkin )1/2 as an indication for changes in the radial wave function un (r). It obviously will change more rapidly for small r (large Wkin ) than in the neighbourhood of the classical turning point rcl , where Wkin = 0. In the classically forbidden area with r > rk (negative kinetic energies) we expect exponential damping of the wave function, as illustrated in Fig. 2.7. In a next step we explore the limits for very large and very small r. For the limiting case r → ∞ we may neglect the potential altogether and (2.117) becomes a simple oscillator equation: 1 d2 un + Wn un (r) = 0. 2 dr 2 √ The classical solution is un (r) ∝ exp(±i 2Wn r). Since for bound states Wn < 0, we note for large r lim un (r) ∝ r n exp − 2|Wn |r .

r→∞

(2.119)

In the opposite limit r → 0 the centrifugal term ( +1)/2r 2 dominates the potential in (2.117) and 1 d2 un ( + 1) − un (r) = 0 2 dr 2 2r 2 r→0

has the solution un (r) = Ar +1 as one easily verifies. Thus, we note for small r lim Rn (r) = lim un (r)/r ∝ r .

r→0

r→0

(2.120)

2.6


2.6.5

121

Exact Solution for the H Atom

We now specialize to the H atom. In the present chapter we neglect the size of the atomic nucleus (positively charged with +Ze) since nuclear radii rnuc are much smaller than atomic radii, typically ratom ≈ 105 rnuc . Thus, we are dealing with a pure C OULOMB potential V (r) = −

1 Ze2 , 4πε0 r

(2.121)

apart from small and very small effects which will be treated in Chaps. 6 and 9, respectively. The general solution of the radial S CHRÖDINGER equation (2.110) is found by using a power series of the type ... Ak r k , Rn (r) = exp − 2|Wn |r k=

which includes the limiting cases just discussed. Well known results from mathematics are used and we summarize here the results, again without proof. For hydrogen like systems – i.e. for one electron in the C OULOMB potentialof a Z fold charged nucleus – the radial function is 2 +1 (ρ) Rn (r) = unl (r)/r = An e−ρ/2 ρ Ln+

where ρ =

3/2 2 Z 2Z r/a0 and An = − n a0 n2

(2.122)

(n − − 1)! [(n + )!]3

with the so called associated L AGUERRE polynomials: 2 +1 (ρ) = Ln+

n− −1

(−1)k+1

k=0

ρk [(n + )!]2 . (n − − 1 − k)!(2 + 1 + k)! k!

(2.123)

With An these radial functions are orthonormalized:

∞

Rn (r)Rn (r)r 2 dr = δnn δ .

(2.124)

0

We introduce here finally the often used term good quantum numbers: They characterize the eigenvalues of such observables that may be measured simultaneously with the H AMILTON operator, i.e. their operators commutate with the Hamiltonian. We already know and m as typical examples: they are part of the set of quantum 2 z are simultaneously numbers characterizing the total energy of the system; L and L measurable with H .

122

2

Wn ℓ / eV 0

n ∞

- 0.85

8 6 4

- 1.51

3

- 3.40

2


ℓ=0

ℓ=1

ℓ=2

ℓ=3

ℓ=4

5s 4s

5p 4p

5d 4d

5f 4f

5g

3s

3p

3d

2s

2p

Z2 Wnℓ = – —— E h 2n 2 independent of ℓ

-13.6

1

1s

Fig. 2.8 Term energies of the hydrogen atom (Z = 1) for different n and Table 2.1 The lowest atomic levels, their energies in the H atom and the degeneracy of the states (with Eh = 27.2 eV) Shell

Orbital

n

m

Wn

Degeneracy Without spin With spin sum in shell sum in shell

K

1s

1

0

0

−Eh /2

1

L

2s

2

0

0

−Eh /8

1

2p

2

1

0, ±1

3s

3

0

0

1

2

3p

3

1

0, ±1

3

6

3d

3

2

0, ±1, ±2

5

4s, p, d, f

4

M

N

2.6.6

3 −Eh /18

−Eh /32

1

2

2

2 4

9 16

6

10

8

18 32

Energy Levels

These solutions of the S CHRÖDINGER equation are the quantum mechanical equivalent to B OHR’s stationary orbits. Substituting unl (r) according to (2.122) into the radial equation (2.113) one finds the eigenenergies Wn for the H atom. Remarkably, they are identical to the energies (1.137) from the B OHR model.8 These results are summarized in Fig. 2.8 and Table 2.1. We recall that the states are characterized by the principle quantum number n = 1, 2, 3, . . . , the angular momentum quantum number (0 ≤ ≤ n − 1) and the projection quantum number m (− ≤ m ≤ ). Each set n m of quantum numbers refers to a different wave function (atomic orbital). In addition, we also have to consider the spin of the electron, with projections ms = ±1/2. 8 The

corrections for finite mass of the atomic nucleus (Sect. 1.8.5) also apply.

2.6

One Electron Systems and the Hydrogen Atom Veff (r ) / E h ℓ =1 0.2

ℓ =2 5

0

123

10

15

n=2

20 n=3 r /a 0

n =1

- 0.5

COULOMB potential = Veff for ℓ = 0

Fig. 2.9 Illustration of degeneracy: Shown are the pure C OULOMB potential (red line) and the effective potentials (black) for the H atom. The term energies Wn are indicated by horizontal lines in their respective effective potentials: for = 0 (dotted red), for = 1 (dashed grey) and for = 2 (heavy, full red). As indicated, the C OULOMB potential leads to a characteristic degeneracy of states with equal n but different

However, for the H atom the energies Wn depend only on the principle quantum number n. Thus, the energy levels are degenerate, the total degeneracy in a shell n being gn = 2n2 . This is summarized in Table 2.1 for n = 1 to 4, also showing the assignment of orbitals with equal principle quantum number n (i.e. with equal energies and similar orbital radii) to specific shells (K, L, M, N . . . corresponding to n = 1, 2, 3, 4, . . .). It is important to note that degeneracy is a special property of the pure C OULOMB potential, while m degeneracy occurs for all atoms if no external field is present. Figure 2.9 illustrates degeneracy in the potential energy diagram, also showing the effective potentials (2.111) for = 1 and = 2 (for = 0 effective and C OULOMB potential are identical).

2.6.7

Radial Functions

The radial wave functions have very specific shapes, which may be understood from the properties of the effective potentials. Figure 2.10 illustrates this schematically by way of example for the n = 3 level and orbital angular momenta = 0 and 1. Shown are C OULOMB potential, centrifugal potential and effective potential for = 1. The classically forbidden areas (Wkin < 0) are grey shaded, the classical turning points. While the radial functions for = 0 start with a finite value at r = 0, for = 1 the probability there is = 0 (since ( + 1)/(2r 2 ) → ∞). Oscillations of the radial functions are expected only in between the classically turning points. Table 2.2 presents the radial wave functions Rn (r) for the six energetically lowest states (n ≤ 3) of atomic hydrogen (Z = 1) and H-like atoms (Z > 1) in closed form. From the radial wave functions one derives the probability distributions 2 w(r)dr = Rn (r) r 2 dr

(2.125)

124

2

Fig. 2.10 Schematic illustration of the relation between potential (top) and corresponding wave function (bottom) for different , exemplified for the n = 3 level with = 0 and 1. Grey shaded is the classically forbidden region, red-black circles indicate classical turning points

Table 2.2 Radial wave functions for the energetically lowest states of H and H-like atoms

Elements of Quantum Mechanics and the H Atom V, W

0 W30

V, W

ℓ=0

Veff

0

–1/r

ℓ=1

ℓ (ℓ +1)/ 2r 2 r

r W 31

–1/r Rnℓ (r) 0

0

r

n

Rn (r) with ρ = 2Zr/(na¯ 0 ) and a¯ 0 = a0 me /m ¯e

1

0

R10 (r) = 2( a¯Z0 )3/2 e−ρ/2

2

0

R20 (r) =

1 √ ( Z )3/2 (2 − ρ)e−ρ/2 2 2 a¯ 0

1

R21 (r) =

1 √ ( Z )3/2 ρe−ρ/2 2 6 a¯ 0

0

R30 (r) =

1 √ ( Z )3/2 (6 − 6ρ 9 3 a¯ 0

1

R31 (r) =

1 √ ( Z )3/2 ρ(4 − ρ)e−ρ/2 9 6 a¯ 0

2

R32 (r) =

√1 ( Z )3/2 ρ 2 e−ρ/2 9 30 a¯ 0

3

r

+ ρ 2 )e−ρ/2

for the electron to be found between r and r + dr. For comparison with the classical picture of an orbiting electron one has to consult these probability distributions as a function of distance from the nucleus. A graphical illustration of the radial wave functions Rn (r) and the radial probability distributions w(r) is shown in Fig. 2.11. Closer inspection of the probability distributions shows, that their maxima match exactly the radii of the respective B OHR orbits for the largest = n − 1 at any given n!

2.6.8

Density Plots

The complete solutions (atomic orbitals) ψn m (r, θ, ϕ) of the S CHRÖDINGER equation (2.108) consist of radial and angular part. With the spherical harmonics Sect. 2.5.3 and the just discussed radial functions the original ansatz (2.109) ψn m (r, θ, ϕ) = Rn (r)Y m (θ, ϕ) is validated. With (2.86) and (2.124) these wave functions are already orthonormalized: ∗ ψn m ψn m r 2 dr sin θ dθ dϕ = δnn δ δmm . (2.126) n mn m = The square of the wave function |ψn m (r, θ, ϕ)|2 represents the probability distribution for finding electrons per volume element, or equivalently the electron density within the atom as a function of position in space. Since these atomic orbitals form

2.6


125

R n l (r) /a 0-3/2

r 2 [R n l (r) ] 2 /a 0-1

0.1

0.4 3s

0.1

0.1

0.04 3p

3d

3s

0

3p

2s

0.1

0 2.0

0

0.1

2p

1.0

0.1

2s

0

10 20

10 20

blow up 1s

0.4 0.2

0

r /a0

10 20

2p

1s

0.2

0

0

0 10 20

0.4

1s

3d

0.2

0.2

0.5

0.1

10 20

0

2

4

2 (r) for Fig. 2.11 Radial wave functions of the H atoms Rn (r) and probability distributions r 2 Rn the K, L and M shell. The dashed, vertical lines in the probability plots for the highest at a given n indicate the corresponding B OHR radii. Note the drawing for the 1s orbital on a blown up r-scale

z

3

1s 0

5

0

-3 10

-3

0

-5

-10

-10

-5

0

-10

-10 10

10 0

10

0

3s0

0

0

-10

10

0

3

3d 1

2p 1

0

5

3d 2

-10

0

10

20 4 s 0

10

0

10

0 -40

-20 0

-10 40 5s 0

0

-10

3p 1

-20

0

20

x

-40

0

40

Fig. 2.12 Density plots for some characteristic H atom wave functions ψn m . Plotted are equidensity lines (red high, grey low probability density). The distances are given in a0

a base for solving many key problems in atomic and molecular physics, we recommend our readers to visualize and memorize their general shape intensively. One finds a host of instructive Internet-sites, e.g. with Java applets to generate the H orbitals in a variety of displays. Thus, we present in Fig. 2.12 only a small selection of cuts through the 3D density distributions. Plotted are the contour lines of the density |ψn m (x, y = 0, z)|2 = |Rn (r)Y m (θ, ϕ)|2 in the zx plane on a linear scale (in contrast to many presentations in the WWW, where one finds the density plotted on a log-scale). For clearness we have indicated the highest densities by red areas.

126

2


We finally note again, that only the ns orbitals have a finite density at the origin, |ψns (0)| > 0, while √ ψn m (0, θ, ϕ) ≡ 0 for > 0. Specifically, from (2.122) and with Y00 (θ, ϕ) = 1/ 4π we record for later use: ψn00 (0)2 = Z 3 / πa 3 n3 . (2.127) 0

2.6.9

Spectra of the H Atom

Just as in the B OHR model, the transition energies (wavenumbers, frequencies) in atomic hydrogen are given by the RYDBERG formula (1.147)–(1.149). Traditionally, one characterizes spectral lines series for transitions n ↔ n

1 ¯e 1 Eh m 2 ω = hν = Wn − Wn = Z − 2 (2.128) 2 2 me n n according to n (with n < n). Clearly, due to the m and (for the H atom) degeneracy, the angular momentum quantum numbers do not influence the position of the spectral lines – as one also verifies by a glance at Fig. 2.8. This holds at least in 1st order approximation. Finer effects will be discussed in Chap. 6. A lot of scientific detective work of the early pioneers went into discovering the connection between spectra and term energies, and the series are named after those who originally discovered them. Most prominent are the LYMAN (n = 1 in the VUV), BALMER (n = 2 in the VIS and near UV) and PASCHEN series (n = 3 in the near IR). Within the series, the BALMER lines are historically referred to as “Halpha”, “H-beta”, “H-gamma” etc., more generally the lines of the different series are designated as Ly-α, and Ly-β, Ly-γ , . . . , Ba-α, Ba-β, Ba-γ , . . . and so on, with α = n + 2, β = n + 3, etc.

2.6.10 Expectation Values of r k One often needs to know the expectation values of the electron distance from origin r to a certain power k. In principle, they could be determined by a large number of individual measurements of this value in an suitably designed experiment. Quantum mechanics provides: ∞ ∞ k 2 r = n |r k |n = Rn (r)r k Rn (r)r 2 dr = Rn (r)r 2+k dr (2.129) with 0

0

∞

0

2 Rn (r)r 2 dr = 1.

These integrations are trivial but somewhat tedious. Using (2.122) and (2.123) one obtains (for later reference):9 9 Here

too, for high precision measurements one has to replace a0 → a¯ 0 = a0 me /m ¯ e.

2.6


1 ( + 1) n2 1+ 1− Z 2 n2 "

# 2 ( + 1) − 1/3 3 n4 r n = a02 2 1 + 1 − 2 Z n2 1 1 Z = r n a0 n2 1 Z2 1 = r 2 n a02 n3 ( + 1/2) Z3 1 1 . = r 3 n a03 n3 ( + 1/2)( + 1)

127

r n = a0

(2.130)

2.6.11 Comparison with the BOHR Model One often hears the verdict that the B OHR model is basically wrong – albeit predicting the correct term energies Wn and spectra (1.131) for the H atom in agreement with quantum mechanics – and thus should be abolished when teaching modern atomic physics. We do not adhere to such dogmatic view. Of course, B OHR’s orbits have to be replaced by atomic orbitals, i.e. by density distributions of the electrons within the atoms, and B OHR’s quantization condition (1.130), L = n, has to be confronted with its quantum mechanical analogue, z Φ(ϕ) = mΦ(ϕ). Obviously, m ≤ ≤ n − 1 is not equal to n as postulated in the L B OHR model – but it is a good first guess. On the other hand, the concept of angular momentum quantization, of stationary states, and of radiation emitted upon transition between them with hν = Wn was a brilliant, daring and instrumental step on the way to understand the quantum nature of microscopic phenomena – even though the fact that states without orbital angular momentum ( = 0) do exist, does not fit at all into the B OHR model. However, the higher the angular momenta, the closer the atomic orbitals correspond to B OHR’s orbits. And in general, the correspondence principle holds: • quantum mechanical and classical values of observables approach each other for very large quantum numbers. A quantitative comparison of atomic orbitals with B OHR’s orbits in hydrogen like atoms shows that the maxima of the electron radial distribution (see Fig. 2.11) for max = n − 1 are indeed found at r = a0 n2 /Z, i.e. they correspond exactly to the B OHR radii according to (1.135). And more specifically, electron orbitals with m = ± may to some extend be associated with electrons moving on a circle – a notion which holds again particularly well for large . Such “circular states” are actually an interesting subject studied in highly excited RYDBERG atoms.

128

2


However, the maxima of the probability distributions are of course not directly observable; rather one may compare expectation values such as n |r|n = r . For the largest angular momenta = n − 1 one obtains from (2.130)

1 a0 2 a0 = n2 , r = lim n + n n 1 2 Z Z

(2.131)

which obviously agrees in the limit with B OHR’s prediction (1.135). Conversely, for the smallest values of = 0, the average radius r is distinctively larger than predicted by the B OHR model, namely (3/2)n2 a0 /Z. In summary, we do not feel that the B OHR model should be completely forgotten. Apart from its outstanding historical impact, we shall come across a number of specific aspects in modern AMO physics for which useful, simple models or concepts have been stimulated by images of electrons moving on classical trajectories – combined with appropriate quantization rules. We may think of B ORN -O PPENHEIMER approximation for molecular physics and of semiclassical trajectory calculations in atomic scattering theory, to mention just two important, and very successful examples – or, as we shall see in the following section, the derivation of the magnetic moment of an electron associated with its orbital angular momentum. Section summary

• The eigenenergies of the H atom, Wn = Z 2 /(2n2 )Eh , depend only on the principle quantum number n . . . a specific consequence of the pure C OULOMB potential. For precision measurements this has to be corrected by m ¯ e /me where m ¯ e is the reduced mass of the electron. • Electron wave functions for the H atom can be expressed in analytical form ψn m (r, θ, ϕ) = Rn (r)Y m (θ, ϕ) with the spherical harmonics Y m (θ, ϕ) and the radial function Rn (r) being proportional to the L AGUERRE polynomials. • The asymptotic behaviour √ of the wave functions should be memorized: limr→∞ Rn (r) ∝ exp(− 2|Wn |r) and limr→0 Rn (r) ∝ r . • good quantum numbers characterize the eigenvalues of such observables that may be measured simultaneously with the H AMILTON operator.

2.7

Normal ZEEMAN Effect

The so called normal Z EEMAN effect is actually not normal at all. It is observed only in special cases. It concerns atoms in an external magnetic field. The word ‘normal’ simply alludes to the classical interpretation of such a situation – ignoring the electron spin. We shall treat atoms in external fields in great detail in Chap. 8. Thus, the subject is touched here just briefly, and only since it confronts us for the first time with the removal of a specific energy degeneracy, the m degeneracy in this case.

2.7

Normal ZEEMAN Effect

129

Fig. 2.13 Electron circling in a magnetic field B and its orbital magnetic moment M

B M

z Mz = - μB L z / ħ v

er

I

2.7.1

Angular Momentum in an External B-Field

As discussed in Sect. 1.9, an orbital angular momentum has a magnetic moment (1.150) which we rewrite in operator form L =− e M L = −μB , 2me

(2.132)

with the B OHR magneton μB . Its potential energy (1.153) in an external magnetic field B is · B = μB L · B. B = −M (2.133) V B = +μB (L z /B), as indicated in the vector diagram We choose z B so that V B , Fig. 2.13. The Hamiltonian contains now an additional term V =H 0 + V B = H 0 + μB Lz B, H

(2.134)

assuming the unperturbed S CHRÖDINGER equation to be 0 ψn m = W (0) ψn m . H 0 may e.g. be the H AMILTON operator (2.106) for the H atom. Thus, in the presH ence of an external field the S CHRÖDINGER equation to be solved reads 0 + μB (L z /)B |n m = W (0) + W |n m , H (2.135) with W being the change of total energy in respect of the unperturbed state. Here and in the following we use state vectors instead of wave functions ψn m → |n m for compact writing. The present problem is a particular simple case for the so called perturbation theory to which we shall come later in some detail. At this point we just remember that according to (2.79) and (2.81) the eigenfunctions ψn m (r, θ, ϕ) = Rn (r)Yn (θ, ϕ) of the H atom, or their eigenstates |n m , respectively, are also eigenfunctions 2 z with (eigenstates) of L and L z |n m = m|n m . L

(2.136)

130

2


∆Wm

Fig. 2.14 The ‘normal’ Z EEMAN effect

m=1

m=0

B

m =−1

0 are also eigenstates Thus, the eigenstates |n m of the unperturbed Hamiltonian H so that we may insert (2.136) into (2.135) to obtain: of the full H 0 + μB mB)|n m = W (0) + Wm |n m (H with Wm = μB mB.

2.7.2

(2.137)

Removal of m Degeneracy

The latter relation obviously implies that in a magnetic field the energy degeneracy for different m is removed. The originally identical energies of the 2 + 1 states |n m which correspond to a given value of n and , now split into 2 + 1 different energy sub-levels. According to (2.137) the splitting is proportional to m and B. The origin of this splitting is that the magnetic field breaks the spherical symmetry characteristic for the unperturbed H atom. We illustrate this for the example of an np state |npm with = 1 and sublevels m = −1, 0, and 1. Figure 2.14 shows the energy changes Wm as a function of the magnetic field B. One may observe this splitting e.g. in optical emission spectra. Figure 2.15 compares the emerging spectra for (a) a p → s and (b) a d → p transition. The individual transitions are indicated by black, downward arrows. The selection rules applied in this plot are = ±1 and m = 0, ±1. They will be derived and discussed in Sect. 4.4. As the degeneracy and hence the number of split levels is 2 + 1, a manifold of transitions may occur if states with > 1 are involved. This is illustrated in Fig. 2.15(b) where the upper levels correspond to a d state. Since, however, for the normal Z EEMAN effect the splitting between neighbouring levels is always μB B according to (2.137), independent of and m, and since for all transitions

m = 0, ±1, one nevertheless sees only a line triplet in all cases. As mentioned at the beginning of this section: in reality this kind of Z EEMAN effect is seen only in special situations (see Sect. 8.1.2), since usually the spin of electrons plays an important role and complicates the observations. We note here an important message from a situation which one typically encounters in quantum systems with two or more degenerate states |1 , |2 , |3 : The degeneracy is removed as soon as some additional, perturbing interaction V1 has to be considered for which the matrix elements i|V1 |j between some of these states do not vanish.

2.8

Dispersion Relations

(a)

131

p

m=+2 +1 0 −1 B B −2

(b)

+1 BB 0 −1

d

BB

s

p

∆m = −1 0 +1

∆m = −1

0

+1

+1 0 −1

Fig. 2.15 ‘Normal’ Z EEMAN effect for (a) p → s and (b) d → p transitions. In case of equal splitting in the excited and ground states one observes in each case a line triplet – in spite of the 5-fold splitting in the d state

Section summary

• A magnetic field B removes the central symmetry and hence the m degeneracy. For the so called ‘normal’ Z EEMAN effect, theory predicts a level splitting

Wm = μB mB. • The selection rules for optical transitions are = ±1 and m = 0, ±1.

2.8


We make a brief detour here, illuminating the borderline between atomic, molecular, optical and solid state physics. Traditionally, dispersion relations are used in optics and characterize an important material property: the dependence of the wavelength λ (or wavenumber k = 2π/λ) of electromagnetic radiation on its frequency ω. From a quantum mechanical point of view one may generalize this to describe the relation between the energy of a photon W = ω (or in fact the energy of any other particle) and its wave vector k. For the massless particle photon in vacuo, with c = νλ = ω/k, the dispersion relation W = ω = c|k|

(2.138)

is obviously a linear relation between energy and wave vector. In contrast, the energy of a freely moving, nonrelativistic electron (whose mass is me ) is W (k) = WP +

me 2 p2 2 k 2 v = WP + = WP + , 2 2me 2me

(2.139)

where we have used the DE B ROGLIE relation (1.118). WP allows for arbitrary energy calibration and may e.g. account for a potential energy or the rest mass energy. Thus, in this case the dispersion relation is quadratic. Both cases are illustrated graphically in Fig. 2.16.

132

2


Fig. 2.16 Dispersion relations (a) linear for a massless particle (e.g. a photon) and (b) quadratic for a particle with mass (e.g. an electron), moving freely in 3D space

(a)

W

(b)

k

W

k

It should be noted that the quadratic relation (2.139) for particles with mass is also in accord with the quantum mechanical calculation for a free particle in a 1Dor 3D-box according to (2.52) or (2.55). Providing the box is large enough, the energies may be considered continuous, i.e. they generally follow (2.139). However, this is only the most simple model for the electronic band structure in a solid. The particle in a box model does not account for the fact that the electrons in a solid do not really move freely: rather, the electrons move in the lattice of atomic ions and experience a periodic potential with strong attraction close to the ionic cores. Elsewhere the potential is partially screened by bound and other free electrons in the solid. We mention here two crucial consequences arising from this situation. First, the dispersion relation (2.139) will have to be modified, which in principle requires a serious band structure calculation. Many phenomena may, however, be explained by the so called parabolic approximation. It parameterizes the band energy by introducing an effective mass m∗e of the electron which may even depend on the direction into which the electron moves: 2

ky2 kz2 kx 2 . (2.140) + + W (k) = WP ± 2m∗x 2m∗y 2m∗z The ± sign allows to extend the concept also to electron holes (positive charges in the latice) and both these “quasi-particles” may have different masses. This mass enters into all further calculations on the dynamics and statistics of electrons and holes in a solid, e.g. into the F ERMI -D IRAC statistics discussed in Sect. 2.4.3. Second, one has to account for the periodicity of the motion in the lattice. The solutions for this problem are so called B LOCH waves ψ(r) ∝ exp(ikr)uk (r)

(2.141)

which are the product of a plane wave exp(ikr) and a periodic function uk (r). The latter has to obey the periodic boundary condition uk (r + T ) = uk (r),

(2.142)

where T is any translation from one elementary cell of the lattice into another. Overall we still expect the dispersion relation (2.139) for electron energies to be more or less valid. As detailed studies show, one may have to replace the electron mass me by a (different) effective mass meff but the quadratic dependence of the energy W

2.8


133

on the wave vector typically holds. The electron motion in general averages over the periodic potential. However, the energies and eigenfunctions will be influenced strongly by the lattice potential if the electrons are particularly close to the ions – or particularly far away from them. This is specifically relevant when the electron wave vector k = k BZ is at (or close to) the B RILLOUIN zone (BZ), i.e. if the corresponding wave functions are constructively interfering (see Sect. 1.4.9). Such electrons ‘feel’ the periodic potential strongly, all others experience only an average. We cannot go into detail of these concepts which are fundamental for the theory of band structure in solid states. We just emphasize some aspects which are also of importance in molecular physics. Thus, let us discuss the particularly simple situation of the 1D case, which e.g. describes a chain of atoms with a distance a from each other (also, with slight modifications, a ring like molecule). According to (1.95) and (1.93) the B RAGG condition for the ntn BZ then reads π k = kBZ = n . a

(2.143)

The corresponding wave functions are essentially exp(ikBZ x) and exp(−ikBZ x), and for free travelling electrons the energy in that situation would be 2 k 2BZ /2me in both cases. We have two energetically degenerate solutions. However, the most general solutions are linear superpositions of both, i.e. standing waves, with the two physically reasonable cases: ψ(x) ∝ exp(ikBZ x) ± exp(−ikBZ x).

(2.144)

These correspond to cos kBZ x and sin kBZ x. The probability |ψ(x)|2 to find the electron close to the lattice ions is maximal in the first case, minimal in the latter. The consequence of the perturbing periodic potential is removal of the degeneracy, similar to Z EEMAN splitting discussed in the last section. And since the perturbing potential is highly attractive, we expect the energy to be lowered in the first case, and to be risen in the second case. In effect, at the boundary of the BZ we expect the energies to split into two, and a gap to arise between two bands of the otherwise continuous 2 k 2 /2me distribution. This is illustrated in Fig. 2.17(a). Since the periodicity of the system does not favour any particular origin in the reciprocal lattice, one projects all the possible energies onto the first BZ as shown in Fig. 2.17(b). Electrons in the system may assume all energies W in the grey shaded “energy bands”. The energy regions marked “band gap” are energetically forbidden. In summary, in the solid state case continuous energy bands, with gaps in between them, replace the discrete energy levels which we have discussed for the atomic case (specifically for the H atom in the present chapter). One must, however, be aware of the fact that Fig. 2.17 shows only a particularly simple situation (1D case, one valence electron only). In general, the band structure of solid states is much more complicated, and depending on how many electrons are available to fill the bands, it provides the basis for such different materials as metals, isolators or semiconductors.

134

2


W

(a) 3rd

2nd

1st

W 2nd

1st

3rd

(b)

band gap

band gap band gap 2 -– - 3— a -— a a

0

– a

2 3 — a — a

k

a–

k

Fig. 2.17 Emergence of energy bands in a periodic system (lattice constant a) and band gaps. ) with distortions at the boundaries of the 1st, 2nd, and 3rd B RILLOUIN (a) Allowed energies ( zone in comparison with the free particle (· · · · · ·) as a function of k. (b) Projection onto the first B RILLOUIN zone

Section summary

• Dispersion relations describe how the energy W of the system depends on the wave vector k. The most simple cases are (a) the photon (a massless particle) W = ck, and (b) an electron (a particle with mass) moving freely W = 2 k 2 /(2me ). • The influence of an average potential can be accounted for by an effective mass, replacing me – which even may depend on the direction of the electrons momentum. • B LOCH waves, ψ(r) ∝ exp(ikr)uk (r), are constructed to include the periodicity of the lattice in the function uk (r) = uk (r + T ).

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. a.u.: ‘atomic units’, see Sect. 2.6.2. BZ: ‘B RILLOUIN zone’, represents all wave vectors of radiation which can be B RAGG-reflected by a crystal lattice. Important concept in solid state physics. 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. DOS: ‘Density of states’, number of states for a specified observable per unit of that observable. Most frequently the observable is the energy of a particle in a system. Typically it is given per unit volume of the system studied. 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)’. IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007).

References

135

NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. ODE: ‘Ordinary differential equation’. PDE: ‘Partial differential equation’. 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 electromagentic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References B ORN , M.: 1927. ‘Das Adiabatenprinzip in der Quantenmechanik. Z. Phys., 40, 167–192. B ORN , M.: 1954. ‘The N OBEL prize in physics: for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wave function’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1954/. E ISAMAN , M. D., J. FAN, A. M IGDALL and S. V. P OLYAKOV: 2011. ‘Invited review article: single-photon sources and detectors’. Rev. Sci. Instrum., 82, 071101. H EISENBERG , W. K.: 1932. ‘The N OBEL prize in physics: in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases’, Stockholm. http://www.nobelprize.org/nobel_prizes/physics/laureates/1932/. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. NIST: 2010. ‘Reference on constants, units, and uncertainties’, NIST. http://physics.nist.gov/cuu/ Constants/, accessed: 7 Jan 2014. S CHRÖDINGER , E. and P. A. M. D IRAC: 1933. ‘The N OBEL prize in physics: for the discovery of new productive forms of atomic theory’, Stockholm. http://nobelprize.org/nobel_prizes/ physics/laureates/1933/.

Periodic System and Removal of Degeneracy

The S CHRÖDINGER equation for the hydrogen atom has been solved in a fully analytic manner. This was possible due to the special mathematical properties of the 1/r C OULOMB potential. We introduce now step by step deviations from this particular simple case, and aim at describing more and more subtle effects – and later on also more complex problems as they are observed in spectroscopic and dynamic studies of atoms, molecules and clusters.

Overview

This is a quite compact and important chapter. The attentive reader should be able to work through it rather quickly after having refreshed her or his basic knowledge in the previous two chapters. We shall make here the first steps to generalize the methods successfully applied for the H atom, and allow for an interaction potential which is no longer strictly proportional to 1/r. This forms the basis for understanding the physics behind the periodic system of the elements which is summarized in Sect. 3.1. The most simple multi-electron systems are the alkali atoms. Their energy levels are discussed in Sect. 3.2 phenomenologically and analyzed qualitatively and quantitatively, briefly explaining quantum defect theory. In Sect. 3.3 we introduce time independent perturbation theory – a tool that will be used later on quite frequently – and illustrate it by way of example for the energies of alkali atoms.

3.1

Shell Structure of Atoms and the Periodic System

The periodic system provides the structural basis for our understanding of atoms and molecules. Thus, albeit fairly well known, it appears worthwhile to summarize here the underlying concepts, observations and definitions.


137

3

138

3

3.1.1


Electron Configuration

The theory of the hydrogen atom as a prototype atom contains already all ingredients needed for understanding of how the more complex atoms are built up. The periodic system of the elements follows naturally from the so called Aufbau principle (from German “Aufbau” = build-up): to a first order approximation one treats the N electrons of an atom (with a nuclear charge Z) as independent of each other and assumes that their respective wave functions look very similar to those of the electron in a hydrogen atom. It turns out that this approach serves astonishingly well for a first guess. Of course one has to modify the potential that each individual electron ‘sees’ – with all the other electrons around it clearly cannot simply be ∝ −Z/r: the nuclear charge will partially be screened off by (N − 1) electrons. We shall discuss this screening and its consequences in detail later. Here we simply note, that each electron (numbered as i = 1, 2, . . . , N ) is characterized by a set of quantum numbers: Quantum numbers

(ni i mi msi )

with the principle quantum number

ni = 1, 2, . . . , ∞

angular momentum quantum number

i = 0, 1, 2, . . . , ni − 1

the projection quantum number

mi = − i , − i + 1, . . . , i

and the spin quantum number

msi = ±1/2.

(3.1)

They correspond to the quantum numbers of the electron in an H atom. The entirety of quantum numbers for all electrons of an atom in a particular state is called its configuration

{n1 1 m1 ms1 , n2 2 m2 ms2 , . . . , nN N mN ms N },

(3.2)

or somewhat more precisely the electron configuration of the atom.

3.1.2

PAULI Principle

In its best known form the PAULI exclusion principle (short PAULI principle, N OBEL prize 1945) states that no two electrons in the same atom can have the same four quantum numbers, briefly (na a ma msa ) = (nb b mb msb )

if a = b.

(3.3)

More generally, no two identical fermions (i.e. particles with half integer spin s = 1/2, 3/2, etc.) can be in the same quantum state. This is an empirically confirmed property of fermions. In Sect. 7.3.1 we shall discuss and use the quantum mechanical formulation: The total wave function of identical fermionsis antisymmetric in respect of the exchange of two particles. We shall see there that both formulations are completely equivalent.

3.1


Table 3.1 Electron shells

Shell

States

139 n

Number of states

K

1s

1

2

L

2s, 2p

2

8

M

3s, 3p, 3d

3

18

N

4s, 4p, 4d, 4f

4

32

The PAULI principle has profound consequences on the structure of matter. It accounts for the properties of atoms, for the order of the periodic system, for the rules that determine how molecules are formed and condensed matter is built up. It is completely mind-boggling to think about it in philosophical terms, using the concepts and experience of everyday life. – One might ask: how do fermions communicate with each other, how do they know which quantum numbers their equals occupy? Is the PAULI exclusion principle something like a force? A nonlocal force? And so on. . . . The strict, formally correct answer is of course: these are simply the wrong questions and we just have to accept it as a mathematical constraint. – In any case, the PAULI principle is certainly among the most profound concepts in physics and part of the “Miracle of Existence” (M ARGENAU 1984). The counterpart of the PAULI principle states for bosons, i.e. for particles with zero or integer spin (e.g. for photons or 4 He atoms), that the total wave function of identical bosons is symmetric in respect of the exchange of two particles – which as such is amazing enough. But there is no constraint with respect to filling theses states: each respective quantum state may be filled with any number of identical bosons.

3.1.3

How the Shells are Filled

One defines so called electron shells which are occupied by all electrons of an atom with the same principle quantum number n. The shells are denoted by the letters K, L, M, N, . . . , as summarized in Table 3.1. As already noted in Sect. 2.5.3, the letters s, p, d, f, g stand for the angular momentum quantum numbers = 0, 1, 2, 3, 4. The number of states in a shell with the principle quantum number n is 2n2 (including spin states with ms = ±1/2) – all of which are degenerate in the H atom case. According to the PAULI principle, each state may be occupied with no more than one electron. The electron configuration of an atom is written in compact form as illustrated in Table 3.2 for the ground state of some light atoms. For larger atoms (e.g. Na) one summarizes the inner shells by the symbol of the next smaller rare gas atom in brackets [ ]. The Aufbau principle of the periodic system assumes that these shells are subsequently filled as the number of electrons increases. Figure 3.1 illustrates the shell filling scheme up to neon graphically.

140

3 s

p

L K

s


p

L H

K

s

He

L

L

L

K

K

K

Li

Be

L

L

L

K

K

K

N

O

p

s

p

L B

K

C

L F

K

Ne

Fig. 3.1 How the K and L shell of the periodic system of elements are filled with electrons. Arrows indicate the spin projection (±1/2) of the electrons Table 3.2 Ground state electron configuration of some light atoms

3.1.4

Z

Atom

Ground state configuration

Shell

1

H

1s

K

2

He

1s 2

3

Li

1s 2 2s

4

Be

1s 2 2s 2

5

B

1s 2 2s 2 2p

...

...

...

10

Ne

1s 2 2s 2 2p 6

11

Na

1s 2 2s 2 2p 6 3s = [Ne]3s

L

M

The Periodic System of Elements

Table 3.3 gives a complete overview of the periodic table of elements, indicating by colour shadings how the electron shells are filled. The elements within one group (vertical columns) have equivalent outer shell electrons (differing only by n) and typically show corresponding similarities in their chemical and physical properties. Each period (horizontal row) corresponds to the filling of one particular shell and comprises elements with different electron configuration and usually different properties. Exceptions are the d block of transition elements, the Lanthanides and the Actinides, for which deferred inner electron shell filling occurs. The radioactive elements of the 7th period have been (except for Fr and Ra) generated artificially at heavy ion storage rings (in very low concentration). They typically have mean lifetimes of only seconds to minutes, and little is known about their electron configuration or even about their physical and chemical properties. Nevertheless, they are interesting objects for fundamental studies. We refer the reader to numerous, well linked presentations in the internet, such as W IKIPEDIA CONTRIBUTORS (2014). The source par excellence for further information is, however, NIST (2011) from where a host of tabulated properties of the elements including all available spectroscopic data can be retrieved. Quite instructive is also the animation from the U NIVERSITY OF C OLORADO (2000).

d electrons leads to the subgroups in the middle of the periodic table that contain all transition metals. The deferred addition of 4f and 5f electrons occurs for Lanthanides and Actinides

Table 3.3 Periodic system of the elements. Left on the top of the elements the atomic number is given, below each element the configuration of the last built in electron is shown. The filling of different shell is marked by colour shadings: s electrons and p electrons determine the main groups, adding

3.1 Shell Structure of Atoms and the Periodic System 141

142

3 Be

Mg


Zn

Yb

Cd

Hg [Xe]4f 145d 106s 2

[Xe]4f 146s 2

[Kr]4d 105s 25p 6

[Kr]4d 105s 2

3d 104s 24p 6

15

[Ar]3d 104s 2

3s 2

WEA and WI / eV

20

3s 23p 6

25

10 5 0

Z=

0

20 He

Ne

Ar

40 Kr

60 Xe

80 Rn

Fig. 3.2 Ionization potentials WI (red) and electron affinities WEA (grey) of the atoms as a function of nuclear charge Z. The full vertical lines indicate shell closures, the dashed lines the closure of subshells, corresponding to the electron configuration given

The essential basis for the experimentally well confirmed shell structure of the elements is (beyond the PAULI principle) the astonishingly good model of quasiindependent electrons in the so called central field approximation. We shall explain this in detail in Chap. 10 where a profound understanding of the periodic system will emerge.

3.1.5

Some Experimental Facts

The ionization potentials (IP) WI for all natural elements of the periodic system are presented in Fig. 3.2. The diagram illustrates impressively the shell nature of atomic structure: The IPs assume very pronounced maxima for the rare gas atoms (He, Ne, Ar, Kr, Xe, Rn) that are characterized by closed shell. They are chemically particularly inert. Smaller maxima are also in between, whenever a subshell for one particular is completely filled. A complementary behaviour is seen in the electron affinities WEA (grey line), i.e. the negative of the binding energies, which may be gained by the formation of a negative ion (anions) energies. They correspond to the negative binding energies of the electrons in the anion (WB = −WEA ) and are particularly high, if the corresponding neutral atom has just one vacancy in the outer shell for the electron to be accepted – which holds for the H atom and the halogens (F, Cl, . . .) – and becomes zero for the rare gases with their closed shells: no stable rare gas anions are known to exist (see also Sect. 6.4.5, Vol. 2). The atomic radii also support the shell structure of atoms very clearly. They are plotted in Figs. 3.3 and 3.4 as functions of nuclear charge Z. Of course, the term


Fig. 3.3 Atomic radii as a function of nuclear charge – determined by different methods: W IGNER -S EITZ radii (red) and VAN DER WAALS radii (black)

H Li 0.3

143 Na

K

Rb

van der Waals

Cs

Wigner-Seitz

r / nm

3.1

0.2

0.1 He

Ne

Ar

Kr

Xe

Z H Li

Na

K

Rb

Cs

0.3

r / nm

Fig. 3.4 Calculated atomic radii (red), covalent radii (black star) and averages from various binding lengths (black circle)

0.2

0.1

0 He

Ne

Ar

Kr

Xe

Z

“atomic radius” is not very well defined – as we have learned in Chap. 2 the ‘size’ of an atom is characterized by the probability to find its electrons in a certain distance of the nucleus, and a limit cannot be defined uniquely. One may e.g. use the so called W IGNER -S EITZ radius, rWS . That is the radius of a sphere of the same volume that the atom occupies on average in the condensed phase – if such a phase exists for that particular atom. From particle density N or mass density ρ, relative atomic mass Mr and AVOGADRO number NA one finds 3 3Mr 3 = 3 . (3.4) rWS = 4πN 4πNA ρ A similar quantity is the VAN DER WAALS radius, that gives the distance up to which chemically not bound atoms can approach each other (for a more precise definition see Sect. 3.2.6, Vol. 2). Both quantities are shown in Fig. 3.3 for the elements H to Ba. Alternatively, Fig. 3.4 shows calculated atomic radii. They are extracted e.g. from quantum mechanical expectation values, as discussed in Sect. 2.6.11 – relying on good computations of the respective atomic wave functions. An empirical determination is based on so called covalent radii that are derived from a set of well known binding lengths of molecules, preferably diameters. By comparison with

144

3


other molecules one may improve these estimates without big efforts as also shown in Fig. 3.4 (asterisks). In spite of all ambiguities in defining the term “atomic radius” one recognizes very clearly the general trends as a function of atomic number. Rare gases typically have the smallest radii, alkali metals the largest: electrons in closed shells all ‘see’ essentially the same, high charge, while in the outermost shell of an alkali atom this charge is strongly screened for the one valence electron – due to the core electrons. We shall discuss this in detail in the following sections of the present chapter. Before doing so we mention one more little difference between Fig. 3.3 and Fig. 3.4: while the minima of the atomic radii for rare gases are quite pronounced in the latter case, such clear cut minima cannot be identified for the VAN DER WAALS and W IGNER -S EITZ radii. Obviously, in addition to the electron density the influence of polarizability plays here an important role. Section summary

• According to the PAULI principle each identical fermion in a quantum system must differ by at least one quantum number. • The periodic system of the elements is based on the PAULI principle. The PAULI principle prevents electrons from falling all into the energetically lowest level and thus ensures the variety of the chemical elements. One may say that it is behind the “mystery of life”. • The atomic shells are denominated by K, L, M, N, . . . according to their principle quantum number n = 1, 2, 3, 4, . . . , respectively. Each shell may contain up to 2n2 electrons. The Aufbau principle (building up principle) of the periodic system says that electrons fill states with quantum numbers n mms one after the other (essentially) in numerical sequence. • Chemical and physical properties of the elements depend crucially on the number of electrons in the outermost shell. Completely filled shells correspond to the rare gases; within each main period they have the highest ionization potential and the smallest radius (as represented e.g. by the W IGNER S EIZ radius Eq. (3.4)).

3.2

Quasi-One-Electron System

The most simple cases of multi-electron systems are those where one or several inner shells are completely filled with electrons (closed shells) and only one electron is found in the outermost shell. A look at the periodic system in Table 3.3 identifies these as the elements of the 1st group, the alkali atoms and the alkali like ions. Their electronic ground state configuration is {[Rg]ns}, where [Rg] stands for a rare gas configuration of the atomic core, e.g. Li: {[He]2s} Na: {[Ne]3s} K: {[Ar]4s}, etc. In the following we shall focus on the ‘active’ electron in the respective new shell, filled only with this one electron of particular interest. It is called valence electron (in the original German literature also Leuchtelektron = lighting electron).

Quasi-One-Electron System WI = 5.392

+ ion

5 5s

3d

5f 4f .7 69 18

460 .2 610 .3

3p

3s

2p

67

1

0

3.2.1

5d 4d

0.8

2

5p 4p

2.6 81

3

4s

323. 2

4

.2 497

Fig. 3.5 G ROTRIAN diagram for the lithium atom: for the ns, np, nd, and nf configuration of the valence electron the energy terms are marked by horizontal lines. Some optically allowed transitions (double arrows) are indicated with wavelengths given in nm. The diagram has been generated from the NIST data bank. The term energies are related to the respective (negative) binding energies Wn of the valence electron by Wn − W2s , with WI = −W2s being the IP of the system

145

term energy ( Wnℓ - W2s ) / eV

3.2

2s

angular momentum ℓ

Spectroscopic Findings for the Alkali Atoms

The most detailed and precise information comes again from spectroscopy. The spectra and hence the term schemes of the alkalis turn out to be similar to those of the H atom with its 1s ground state electron – with two important modifications: (i) the active valence electron is now initially in an ns state with n > 1 and the inner shells are filled (thus no VUV radiation is observed) and (ii) the degeneracy is now removed (which makes the spectra richer, i.e. more complicated to interpret, than that of the H atom). As a characteristic example Fig. 3.5 shows the term diagram of Li(1s)2 n as derived from a host of spectroscopic information collected by generations of spectroscopists. This type of term scheme indicating the possible transitions is usually called G ROTRIAN diagram. The general relation between the energies Wn of stationary states and spectral lines ν¯ observed (in wavenumbers) is again 1 1 ν¯ n ←→ n = = (Wn − Wn ), λ hc

(3.5)

now, however, with different energies for each .1 The basics of emission, absorption and fluorescence spectroscopy will be introduced briefly in Sect. 4.2.2 and a variety of more refined modern methods will be described later, e.g. in Sect. 6.1. Here we only give a survey over the collected re1 Strictly speaking, one should even introduce an additional quantum number j for the total angular momentum, which becomes increasingly relevant as Z gets larger. We have already familiarized ourselves with it in the context of the S TERN -G ERLACH experiment in Sect. 1.9.5 – and shall come back to it in detail in Sect. 6.2.5.

146

3

Wnℓ / E h

H


Li

Na

K

Rb

ionization continuum 0 n=8

- 0.020 - 0.031 - 0.056 - 0.125

n=4 4s 4p 4d 4f n=3 3s 3p 3d n=2

4d 4f 4s 4p 3p 3d 3s

2s 2p

n=1

4s

5p 4d 4f 4p 3d

6s

3s

6p 4f 4d 5p

5s 4p

3p

2p 2s

- 0.500

5s

4s

7s 7p 5d 6p 6s 5p

5f

5s

1s

Fig. 3.6 Overview on the term energies of the alkali atoms in comparison to the H atom. Characteristic is the removal of degeneracy and the decrease of the energies in respect of the H atom for small angular momenta

sults for the alkalis and refer the interested reader to the spectroscopic data bank of K RAMIDA et al. (2013) which we have already mentioned. A comparison of the energies of the H atom with those of all alkali atoms (valence electron) shows Fig. 3.6. The characteristic removal of degeneracy lets the term energies of the alkali valence electrons always lie below those of the H atom, Wn < −Eh /(2n2 ), owing to the higher nuclear charge Z. However, this lowering of energy gets smaller as the orbital angular momentum gets larger. Figure 3.6 clearly documents that Wns < Wnp < Wnd < Wnf . For the nf terms the energies of the alkali atoms are practically identical to those in atomic hydrogen. We shall understand in a moment why that is so.

3.2.2

Quantum Defect

We start by summarizing the experimental findings in a compact form: the just described similarity with the H atom suggests to write the energies of the alkali atoms as Wn = −

Eh 2n∗2

with n∗ = n − μ.

(3.6)

The parameter μ is called quantum defect and n∗ the effective quantum number. For the moment this is simply a parameter, determined experimentally, which allows one to order the spectroscopic data in a systematic manner. A comparison of experimental data shows that μ strongly depends on the orbital angular momentum – and upon closer inspection also slightly on the principle


Fig. 3.7 Quantum defects μ(n, ) of the Na atom as a function of binding energy Wn for orbital angular momenta = 0, 1, 2, 3 and 4. Full black circles are experimental data taken from K RAMIDA et al. (2013), open circles are denoted there as “determined by interpolation or extrapolation . . . or by semi-empirical calculations. . . ”. Full lines represent linear least square fits to the measured data. Note the break in energy scale for the ns and np series, and the different scale for the nd, nf and ng series

147

μ(n,ℓ )

3.2

1.37

Na

ns

1.36 1.35 0.89 0.87 0.85

np - 4 - 2 - 0.8

- 0.4

0

0.014 nd

0.012 0.010 0.0015

nf

0.0010 0.0004 0.0000 - 1.6

ng - 1.2

- 0.8

- 0.4 0 Wnℓ / eV

quantum number n. Figure 3.7 illustrates this for Na as example. Obviously, for large n the quantum defect approaches a constant value that depends only on . We shall come back to the theoretical interpretation of μ in the framework of the so called quantum defect theory (QDT) in Sect. 3.2.6. For a quantitative comparison one fits the experimentally determined term energies, i.e. the excitation energies in respect of the ground state, by an extended RYDBERG -R ITZ formula (see e.g. W EBER and S ANSONETTI 1987): term energy Wn = Wn0 0 −

Eh 2(n − μ(n, ))2

(3.7)

with μ(n, ) = μ + B/(n − μ )2 + C/(n − μ )4 + · · ·

(3.8)

or μ(n, ) μ + DWn .

(3.9)

Here Wn0 0 is the ground state energy, which is related to the IP by WI = −Wn0 0 . The quality of present day spectroscopic data (K RAMIDA et al. 2013) allows a very precise determination of the parameters. The thus derived quantum defects μ of all alkali atoms for large n are summarized in Table 3.4 and Fig. 3.8. One sees very clearly that the quantum defect increases strongly with the nuclear charge Z and decreases with increasing orbital angular momentum. We also want to emphasize that for large n (small |Wn |) μ decreases indeed linearly with Wn (not shown here) according to (3.9) which allows for (i) a determination of WI with spectroscopic accuracy and (ii) analytic continuation of μ(n, ) → μ(W, ) into the continuum where W > 0. In the following we shall try to understand these findings

148

3


Table 3.4 Quantum defect μ of the alkali atoms for large n Valence electron Atom H

0

1

2

3

4

ns

np

nd

nf

ng

0

0

0

0

0

Z 1

Li

3

0.40

0.05

0.002

0.00

0.00

Na

11

1.3478

0.8551

0.01489

0.0016

0.00025

K

19

2.180

1.7115

0.2577

0.0013

0.0017

Rb

37

3.1223

2.6535

1.3355

0.017

0.0025

Cs

55

4.0494

3.5916

2.4663

0.03341

0.007

H

Li

Na

K

Rb

4 quantum defect μℓ

Fig. 3.8 Quantum defect of the alkali atoms for high n as a function of . Note the scale change in the ordinate at 0.05. In particular the f orbitals and even more so the g orbitals are so far away from the atomic core that the quantum defect is close to zero

Cs ns

3 Z=3 11

19 np

2 nd 1 Z=37

0.04 0.02 0.00

55 nf ng

1

Fig. 3.9 Model of an alkali atom as quasi-one-electron system

10

20

30

40

50

Z r

+Ze0

1e0

(Z-1)e0

qualitatively and quantitatively by looking at the potentials and wave functions, and finally we shall introduce QDT which explains that the quantum defect μ is not just some kind of ‘fudge’ parameter.

3.2.3

Screened COULOMB Potential

A quite reasonable model for the alkali atoms assumes that the Z − 1 core electrons (which fill the inner shells of the core ion completely) do influence the energy terms of the one outermost electron (the valence electron) only by screening the pure C OULOMB potential of the nucleus. As illustrated in Fig. 3.9, we consider an N electron problem (nuclear charge Z = N ). At large distances, the valence elec-

3.2


149 2

VS (r ) / E h

0

-1/r

-2

4

VS (r)

6 r / a0

-4 - Z/r

-6 -8 -10

Fig. 3.10 Effective potential VS (r) in which the valence electron of an alkali atom moves (red), compared to a fully screened C OULOMB potential −1/r and the unscreened potential −Z/r that the electron experiences close to the nuclear charge Z (here with Z = 11 representing the Na atom). The grey shaded area indicates the radius of the ionic core

tron ‘sees’ only the screened nuclear charge 1e and experiences something about the nucleus only if it dives into the ionic core. One may treat this simplified problem with nearly the same methods as the H atom – except that we no longer have a pure −1/r potential. Rather we have to solve the problem for a screened −Z/r potential (in a.u.) ⎧ r →0 ⎨ −Z/r VS (r) = −1/r + VC (r) in between (3.10) ⎩ −1/r r →∞ where the ionic core is accounted for by a suitable, smooth potential, in the simplest case e.g. VC (r) = −(Z − 1) exp(−r/rS )/r

(3.11)

as illustrated schematically in Fig. 3.10.

3.2.4

Radial Wave Functions

Thus, the S CHRÖDINGER equation for this quasi-one-electron model is 2

p + VS (r) ψn m (r) = Wn ψn m (r). H ψn m (r) = 2me

(3.12)

Its solution depends on the spatial coordinates of one electron only. Tacitly we have also assumed the core potential to be spherical symmetric. Consequently, we may derive a radial equation in the same manner as done for the H atom, using the ansatz ψn m (r) = Rn (r)Y m (θ, ϕ). Again we substitute un (r) = rRn (r) and obtain in complete analogy to (2.113) the radial differential equation (in a.u.)

150

3


d2 un ( + 1) un (r) = 0 + 2 W − V (r) + n S dr 2 2r 2

(3.13)

with the important difference that VS (r) is no longer the pure C OULOMB potential but rather a suitably screened one as described by (3.10). The task at hand now is to identify from the infinite manifold of solutions for (3.13) those which are physically meaningful. They have to reproduce the correct asymptotic behaviour both for large and small distances r. Since we have derived the corresponding relations in the case of the H atoms without any reference to the interaction potential, (2.119) and (2.120) must be valid in the present case too. Thus, for √ r → ∞ the radial wave function has to be damped exponentially, un ∝ exp(− 2|Wn |r), and Wn is the sought-after energy of the stationary states of the system. On the other hand, for r → 0 the radial function has to follow un ∝ r +1 . This very behaviour allows us to explain the experimentally observed dependence of the quantum defect (Table 3.4): according to (2.125) the probability to find 2 (r) = 4πu2 (r). an electron at a distance r from the nucleus is w(r) = 4πr 2 Rn n 2 +2 Specifically for small distances it is thus ∝ r . Hence, the higher its angular momentum, the less the electron is influenced by the ionic core. For very large the wave function will be essentially hydrogen like, and the quantum defect correspondingly small, just as documented in Sect. 3.2.2. For a truly quantitative analysis of wave functions and eigenenergies we have to integrate (3.13) explicitly. Even though the radial S CHRÖDINGER equation for a screened potential of the type shown in Fig. 3.10 cannot be solved analytically, numerical integration of (3.13) may be done today on any PC without problems. A number of robust and simple procedures are available, e.g. the often used RUNGE K UTTA method. Typically one integrates for a given trial energy Wn from the inside outwards as well as from the outside inwards, starting with the asymptotic forms just discussed. One accepts only such solutions that can be matched continuously and differentiable – and by variation of Wn one thus obtains the discrete eigenstates and eigenenergies of the system. Of course, the result of such calculations can only be as good as the potential VS (r) – which so far we have only described qualitatively. There are a number of useful approximations to estimate VS (r). In the most simple approach one guesses a parametric form which is fitted such that the experimentally observed energies of a few states are reproduced exactly. With the thus determined empirical core potential one may derive the wave functions for any state of the system and compute other quantities, such as the energy terms, optical transition probabilities, polarizabilities and so on.

3.2.5

Precise Calculations for Na as an Example

Today, the quasi-one-electron model discussed above is, generally speaking, of mainly pedagogical value and helps us to understand the physical origin of the ex-

3.2


(a) -1 2

(b)

w(r ) 1

4 r 2Rnℓ (r) / a 0

151

2p

K-shell L- shell of the ion core / 40

0.3

3s

0 1

0.2

2s

0

3p

0.1

5

3d

1s 0.0

0 0

1

2

0

5

10

15

r / a0

Fig. 3.11 Radial electron probability distributions w(R) in Na for (a) the core electron orbitals and (b) orbitals of the valence electron (alternatively in the 3s ground state and in the 3p and 3d excited states). Also shown is the cumulated radial electron probability in the Na+ core (Z = 11, closed [Ne] shell), down-scaled by a factor of 40. Grey shaded is the ‘radius’ of the ion core (1.8a0 ). The distributions shown by red lines have been calculated ab initio using a finite difference atomic structure code, FDAlin, conveniently provided by S CHUMACHER (2011). The black dashed and dash dotted lines are derived from QDT and are not expected to be valid inside the ionic core (see Sect. 3.2.6). Note in particular the excellent agreement for 3s

perimental observations. With efficient, compact computing programmes and computers available, one may calculate energy eigenvalues, wave functions, transition probabilities and other properties of the alkali atoms with nearly unlimited precision.2 There is no need to restrict such calculations to the valence electron, and the wave functions of all filled orbitals for the alkalis may be obtained readily. Some details of appropriate methods will be discussed in Chap. 10 in the context of genuine multi-electron systems. However, the general character of the orbitals for each individual electron remains very similar to that of the corresponding electron orbital in atomic hydrogen. Figure 3.11 shows by way of example the probability distributions for the electrons in Na (electron ground state configuration 1s 2 2s 2 2p 6 3s). They have been computed with a simple DFT programm that generates these data on any PC within seconds (the principles of DFT will be introduced in Sect. 10.3). Shown in Fig. 3.11(a) are the radial probability distributions (integrated over all angles) for the orbitals of the core electrons (1s, 2s, 2p). The grey shaded areas indicates again the literature value for the ion core ‘radius’ (0.95 nm = 1.8a0 ). Figure 3.11(b) shows the corresponding probability density for the valence electron in the 3s ground state (alternatively 3p or 3d). In addition, the cumulated radial 2 (r) in units [w] = 1/a is given electron probability w(r) = core Nn × 4πr 2 Rn 0 for the ion core with Nn being the number of electrons in the respective n states, i.e. 2 for the K shell (1s 2 ), and 2 + 6 for the L shell (2s 2 2p 6 ). This cumulated distribution for the ionic core clearly exhibits the different shells and underlines again 2 This holds at least for the smaller alkali atoms where relativistic effects play a minor role and spin-orbit interaction can be treated as a small perturbation.

152

3


quantitatively that the valence electron stays predominantly outside the core. This holds a fortiori for excited states, here very clearly seen for the 3d state the orbital: its radial wave function (R3d (r) ∝ r 2 for small r) leads to very little probability for finding the electron inside the core as Fig. 3.11 illustrates. This explains again convincingly why the quantum defect decreases so rapidly with as illustrated in Fig. 3.8. It also clarifies the overall striking similarity of the alkali spectra with those of the hydrogen atom which we have seen in Fig. 3.6. The nf electrons, for which at the origin R4f (r) ∝ r 3 and thus w(r) ∝ r 8 , practically never come close to the nucleus and the quantum defect can essentially be neglected – as for all higher values of (see Fig. 3.8). We recall again: degeneracy observed in H and H like ions is just a very specific consequence of the pure C OULOMB potential. Deviations from the C OULOMB potential lead to different energies for different , i.e. to a removal of this degeneracy. The deviations are the larger the more the electron ‘feels’ of the ionic core. For very large the situation is practically identical to the pure −1/r potential in the H atom case.

3.2.6

Quantum Defect Theory

Even though in principle the full quantum mechanical problem of bound states in small alkali atoms may be treated numerically, it is useful to return once more to the model of a single electron in an effective potential. So far, we have not yet understood why the extended RYDBERG -R ITZ formula works so well – with essentially constant quantum defect (3.8) for large n as documented by Fig. 3.7. This remarkable observation hints at some genuine physical background, even though the effective quantum number n∗ and the quantum defect μ were introduced in (3.6) just as useful empiric parameters. It turns out that understanding this physical origin of the quantum defect allows one to extract useful concepts and further quantitative information. The roots of the so called quantum defect theory (QDT) go back to the early days of quantum mechanics and H ARTREE (1928), while between 1950 and 1990 S EATON (1983), FANO (1986), J UNGEN (1996) and their students have developed it into a powerful theoretical framework and extended it to multichannel problems (MQDT). It has been applied successfully to calculate oscillator strengths for atomic and molecular transitions including high lying RYDBERG states, to determine accurate ionization potentials and photoionization cross sections (see also Sect. 5.5), to understand autoionizing series and perturbations in multi-electron spectra, as well as in electron ion scattering theory and is used even in solid state structure theory. The key to QDT is that for large r – far outside the atomic core – electronic wave functions evolve in a pure C OULOMB potential and may be derived as analytic functions also for non-integer quantum numbers – with the correct damping at large r (in a.u.)

3.2


∗ ∗ un (r) ∝ r n exp − 2|Wn |r = r n exp −r/n∗ , r→∞

153

(3.14)

Wn being the binding energy and n∗ the effective quantum number.3 Effective range theories for collisions (which will briefly be covered in Sect. 6.4.5, Vol. 2) may be viewed as a generalization of QDT for the non-C OULOMB case. Recently, QDT has attracted renewed interest in the context of collisions and reactions with ultra-cold atoms and molecules (e.g. O SPELKAUS et al. 2010; I DZIASZEK and J ULIENNE 2010, and references there). As an introduction we take a quantitative look at the radial wave functions for large principle quantum numbers n where μ(n, ) is essentially independent of n. We solve (3.13) numerically to obtain un (r) = rRn (r), using the S CHRÖDINGER applet from S CHMIDT and L EE (1998). As specific examples we choose the 18s and 20s states and use the screened potential VS (r) according to (3.10) shown in Fig. 3.10.4 The screening parameter was chosen rS = 0.4190a0 to yield the experimentally determined quantum defect μ = 1.348. In Fig. 3.12(a) we show for comparison the exact radial wave functions for atomic hydrogen, in (b) the computed wave functions of n = 18 and 20 in Na are plotted. For r > 0 the wave functions u18s (r) and u20s (r) in both cases (H and Na) have of course the same number of nodes (crossings through zero), n − 1 = 17 and 19 for 18s and 20s, respectively. The probability distributions of these wave functions lies almost entirely outside the ionic core (Na+ core radius is 0.095 nm = 1.8a0 ). At intermediate r they show a pronounced oscillatory character √ √resembling effectively sin(kr r) or cos(kr r) functions, with kr = 2π/λr ∝ T = |VS | − |Wn | corresponding essentially to the DE B ROGLIE wavelength λr for the local kinetic energy T . We also see (in the intermediate r range) that a change from n → n − 2 corresponds approximately to a shift by 1 oscillation (counting from the last maximum) downward – or a phase shift of 2π when extrapolated into the continuum. There, for large r one might essentially expect a sin kr or cos kr like behaviour (see, however, the more precise discussion below). Most important in the present context: the wave functions for Na are ‘phase shifted’ towards smaller r in respect of the strictly C OULOMB wave functions for H – clearly by more than 1/2 of the oscillatory period as indicated by the red dotted lines and the red arrow. Extrapolated into the continuum this translates into a phase shift of δ = πμ as we shall see in a moment. The physical origin of this “phase shift” is recognized in the blow ups Fig. 3.12(c) for H and (d) for Na: Even though the electron comes only very rarely close to the atomic core (grey shaded region), due to the strong attraction of the nucleus in Na (Z = 11) its wave function oscillates there much more rapidly than for the H atom – one may attribute this simply to the much shorter DE B ROGLIE wavelength λr in the 3 The r n∗

factor used here in contrast to (2.119) can improve the convergence.

4 This is a rather crude choice. It leads, however, to qualitatively correct wave functions. The radius

of the ionic core for Na+ is typically given in the literature as 0.095 nm = 1.8a0 . At this distance this screened potential VS (r) is about −1.1/r.

154

3

(a)

0.6

(c)

0.2

H 18s

0

0

H 20s -0.2 H 18s

50

0

H 20s

-1.0

(b) 0.6

0.2

Na+ core

r Rn ℓ / arb. un.


10

(d)

20

Na 18s Na 20s

0

0

-0.2 50

Na 18s

Na 20s

0

-1.0 500

0

1000

10

20 r / a0

Fig. 3.12 Illustration of the quantum defect μ as “phase shift” δ = πμ for Na in the 18s and 20s states: (a) The radial wave functions rRn (r) for a pure C OULOMB potential (H atom) are compared (d) to those in an Na+ pseudopotential VS (r) (see Fig. 3.10). All wave functions are normalized to their minimum, which is set to −1. The blow ups of the r scale (c) and (d), respectively, illustrate the strong influence of the Na+ ionic core (grey shaded area) onto the Na wave functions. It leads to a shift of the rRn maxima in respect of the H atom; this shift is directly related to the quantum defect μ(n, )

deeper attractive potential VS . While the Na 18s and 20s wave functions are nearly identical in this small r range (apart from their magnitude owing to normalization), they differ dramatically from their counterparts in the H atom. Outside the core the potential is purely C OULOMBic for both Na and H – hence the two pairs of wave function differ essentially only by this “phase shift” δ. QDT treats the outside wave functions as a linear superposition of two pure C OULOMB functions. The mathematics involved is not completely trivial and we can only outline here just a few basics, following the excellent review of S EATON (1983) and his concise mathematical summary from 2002. The radial S CHRÖDINGER equation (3.13) is solved in two parts: " u(r) =

FI (r) FII (r)

for 0 ≤ r < r0 with VS (r) for r0 ≤ r → ∞ with −1/r.

(3.15)

(If alkali like ions are discussed with a remaining charge ZC of the ionic core, the −1/r potential has to be replaced by −ZC /r.)

3.2


155

Both functions must be joined at a suitable distance r0 with continuous logarithmic derivatives. The first part FI (r) requires either some experimental input or a numerical solution, but FI is insensitive to the total energy W as long as |W | |VS (r)|; Fig. 3.12(d) may serve as illustration. The second part of (3.15) can be solved fully analytically. One uses a scaled energy and redefines a scaled radial distance r: = 2(W/Eh )/ZC2

and r := ZC r/a0 .

(3.16)

With this (3.13) is rewritten as "

# d2 ( + 1) 2 − + + u = 0. r dr 2 r2

(3.17)

Since this is an ODE of 2nd order it has two linearly independent sets of C OULOMB functions as solutions for which one finds various forms in the literature. For QDT S EATON (1983) defines s(, ; r) as the regular solution, and c(, ; r) as the irregular solution. When treating the H atom in Sect. 2.6.5 we have considered only the regular solution un ∝ s(n , ; r) for n = −1/n2 (n being an integer); asymptotically they are ∝ r +1 for r → 0 and ∝ r n exp(−r/n) for r → ∞. It is important to note that for all other values of the regular solution s(n , ; r) diverges at large r – hence the n give the eigenvalues of the H problem. The irregular solutions c(, ; r) are ∝ r − at the origin so that any radial wave function R(r) ∝ rc(, ; r) would diverge – hence they were considered nonphysical for the H problem. It is interesting to note, however, that for large r and n = −1/(n + 1/2)2 (and only for these) they are damped ∝ r n exp(−r/n). Hence, it is plausible – as shown already by H ARTREE (1928) – that at arbitrary energies = −1/n∗2 a physically meaningful solution of the problem for r > r0 is: FII (r) = − cos πn∗ s(, ; r) + sin πn∗ c(, ; r) = −(−1)n cos(πμ)s(, ; r) + sin(πμ)c(, ; r) .

(3.18)

Here n∗ is the effective quantum number and μ = n − n∗ is the quantum defect as defined by (3.6) and more specifically by (3.7). These superpositions of regular and irregular C OULOMB function for a given quantum defect μ are used in QDT for r > r0 . In general they cover most of the interesting part of the wave function as documented in Fig. 3.12. For not too low n they are fully sufficient to calculate matrix elements of r – and hence transition probabilities, as we shall discuss in Chap. 4. Typically, in the Na case one would choose r0 1.8a0 , corresponding to the Na+ ionic core radius. Generally speaking, C OULOMB wave functions are special cases of confluent hypergeometric functions which are derived from (3.17), possibly recast into various other standard mathematical forms.

156

3


Bound States Specifically for bound states = −1/(n∗ )2 < 0, S EATON shows

∗ n∗3 1/2 sin(πn∗ ) ξ − cos πn Kθ 2 (2ν)1/2 πK

∗ ∗ n∗3 1/2 cos(πn ) ξ + sin πn Kθ c(, ; r) = (−1) 2 (2ν)1/2 πK s(, ; r) = (−1)

with the normalization factor −1/2 K n∗ , = n∗2 Γ n∗ + + 1 Γ n∗ − where Γ (x) is the Gamma function (with Γ (n − 1) = n!). 2ρ θ n∗ , ; r = Wn∗ , +1/2 ∗ n

(3.19)

(3.20)

(3.21)

is the a so called W HITTAKER W function, while ξ(n∗ , ; r) is a linear combination of two such functions. The asymptotic behaviour for large r is ξ ∝

r→∞

r n∗

−n∗

n∗

r r r and θ ∝ exp ∗ exp − ∗ . r→∞ n∗ n n

For integer values n∗ = n the regular functions are orthonormalized: ∞ s(n , ; r)s(n , ; r)dr = n3 /2 δnn .

(3.22)

(3.23)

0

Since δn = (n+1 − n ) 2/n3 in the limit of large n, the bound state functions s(n , ; r) are said to be normalized per unit energy in this limit. Since the ξ component of the solutions (3.19) diverges for large r it is not suitable for the description of any realistic physical wave function. And indeed, when inserting (3.19) with (3.21) into the general solution (3.18) the ξ terms cancel and one obtains for r > r0 simply: ∗3 1/2 n 2r K n∗ , Wn∗ , +1/2 ∗ . FII n∗ , ; r = (−1) 2 n

(3.24)

This is the sought-after general wave function in a pure C OULOMB potential outside the atomic core for arbitrary energy! The W HITTAKER functions are included in advanced mathematical programmes, e.g. in Mathematica, and we have computed these functions for the 18s and 20s case of Na with n∗ = n − 1.3848. The result is completely identical to the wave functions shown in Fig. 3.12(b) and (d) within the limits one could see in the graphs – except inside the ionic core where it is not expected to be valid.

3.2


157

As a second example, illustrating the power as well as the limitations of QDT, we have computed the 3s, 3p and 3d functions of Na in the same manner, using the precise (experimentally determined) values μ(3s) = 1.37289, μ(3p) = 0.88283 and μ(3d) = 0.01023 derived from the K RAMIDA et al. (2013) data. The results for the electron densities FII2 (r) are plotted in Fig. 3.11(b) as dashed and dash dotted black lines and can be compared with the ‘exact’ results from the DFT calculations. The agreement is astonishingly good (and we cannot discuss here potential shortcomings of the DFT method). Clearly, the FII (r) wave functions are not expected to be relevant inside the ionic core, where 3p and 3d are seen to diverge. But with a suitable cutoff one may confidently use such type of wave functions (even for low n), e.g. for calculating transition probabilities, polarizabilities, and other properties of the alkali atoms: QDT appears to lead to wave functions that represent most of the significant r range very well. One final brief and potentially useful remark (see e.g. F REEMAN and K LEPPNER 1976): A small change in the quantum defect (say by μ n∗ ) changes the term energies according to (3.7) by

1

μ −1 1 −1 1− ∗3 . (3.25)

Wn = ∗2 + = 2n 2(n∗ + μ)2 2n∗2 (1 − μ/n∗ )2 n Conversely, any perturbation Wn known to scale (e.g. for the H atom) with the principal quantum number as 1/n3 may be accounted for by a corresponding addition to the quantum defect. As we shall see in later chapters, a variety of important perturbations that originate from interactions inside the ionic core do indeed scale like 1/n3 , e.g. fine and hyperfine structure splitting, certain effects from interactions with an electric fields, polarization and so on. They may all be treated as additive to the quantum defect as long as μ n∗ . Hence, also the corresponding changes of the wave functions may be described by QDT as explained above.

Continuum States We have already indicated that the behaviour of continuum wave functions may be gleaned from analytic continuation of the quantum defect μ(n, ) → μ(, ) for = k 2 > 0. What we have vaguely addressed as “phase shift” between the oscillations of the bound state wave functions will then become indeed a genuine phase shift δ () = πμ(, )

(3.26)

between a pure outgoing C OULOMB wave and the continuum wave function distorted by the screened Z/r potential of the ionic core. Here too, (3.18) applies for the outside part of the wave function, FII , and at a suitable r0 one has to join FI and FII . Again various forms for the regular and irregular pure C OULOMB continuum wave functions are used. According to S EATON (2002), Eqs. (86)–(90) and (113)–(118) ( ' s(, ; r) ∝ Im exp i(σ − π/2) Wi/km +1/2 (−2ikr) (3.27) ' ( c(, ; r) ∝ Re exp i(σ − π/2) Wi/km +1/2 (−2ikr)

158

3


s (ϵ, ℓ;r)

0.0

c(ϵ , ℓ;r)

0.4

-0.4 0

r

5

10

Fig. 3.13 C OULOMB continuum s waves ( = 0) at ε = 4 for attractive potential: regular solution (red line, zero at origin), irregular (black line, finite at origin) adapted from S EATON (2002). The asymptotic behaviour (3.30) (pink and grey dashed lines, respectively) gives a nearly perfect match to the exact solutions already at r 3

where Wi/km +1/2 (−2ikr) is again a W HITTAKER W function, this time to a complex argument, and for the presently considered attractive C OULOMB potential we have r > 0. The C OULOMB phase shift σ = σ (k, ) = arg Γ (1 + − i/k)

(3.28)

decreases rapidly with k and corresponds to the phase difference of the C OULOMB in respect of a free spherical wave (not to be confused with the phase shift δ () due to the ionic core potential VS being different from purely C OULOMB). We have suppressed here a discussion on the important question of normalization and simply note that ∞ s(1 , ; r)s(2 , ; r)dr = δ(1 − 2 ), (3.29) 0

i.e. s(1 , ; r) is normalized to scale (see Appendix J). Again, the regular solution goes to 0 at the origin while the irregular solution stays finite or diverges. For large r these wave functions are essentially outgoing spherical waves – apart from a slowly varying logarithmic phase shift characteristic for the C OULOMB potential (see (J.8)) – i.e. they behave essentially like sine and cosine functions: 2 2 lim s(, ; r) = sin ζ and lim c(, ; r) = cos ζ (3.30) r→∞ r→∞ πk πk 1 with ζ = kr + ln(2kr) − π/2 + σ (k, ). (3.31) k Figure 3.13 illustrates these continuum functions for = 0 at a moderate energy = 4 or k = 2 where σ0 0.078 × 10−2 π . The exact solutions (adapted from S EATON (2002), Fig. 1) are apparently matched rather perfectly by their asymptotic form (3.30) already at surprisingly small values of r 3. We now come back to the general QDT theme of constructing a wave function for r > r0 , this time in the continuum, using the analytic continuation μ () derived from bound state quantum defects. Being interested in large r we insert the

3.2


159

asymptotic functions (3.31) into (3.18) and obtain for r > r0 2 sin ζ + δ () FII (, ; r) −→ r→∞ πk

(3.32)

with the phase shift δ () according to (3.26) that just describes the shift of the maxima between the regular continuum C OULOMB wave and the wave function of the system with quantum defect – very similar to the bound state situation illustrated in Fig. 3.11, except that now the phase shift is a genuine phase between the sin[ζ ] in a clean −1/r potential and sin[ζ + δ ()] influenced by the screened ionic core. We shall return to these types of outgoing spherical waves in the context of photoionization in Sect. 5.5.4 and collision theory in Chaps. 6 and 7, Vol. 2.

3.2.7

MOSLEY Diagrams

It is often advantageous to use an alternative empirical description of the energy terms that supports the comparison of different elements within a row of the periodic system. Instead of using an effective quantum number n∗ one introduces an effective nuclear charge Z ∗ , thus accounting for the fact that electrons ‘see’ only a fraction of the nuclear charge. The quantity qs = Z − Z ∗ is then called screening parameter. The binding energies of the electrons thus become Wn = −

Z ∗2 (Z − qs )2 Eh = − Eh , 2 2n 2n2

(3.33)

or in wavenumbers (Z − qs )2 (Z − qs )2 E /2 = R h ∞ hcn2 n2 2|Wn | ν¯ n Z ∗ Z − qs = . = = Eh R∞ n n ν¯ n =

(3.34)

Plots of the square roots of the energies as a function Z are called M OSLEY diagram and should give according to (3.34) a straight line. The screening parameter √ qs is then obtained from the axis intercept of 2|Wn |/Eh where Z − qs = 0. Full screening of the C OULOMB potential would imply qs Z − 1. By way of example we discuss the energy terms for Na and Na-like ions with data taken from the K RAMIDA et al. (2013) data bank. The series starts with Z = 11 (Na) and continues with Mg, Al, Si, P, S, Cl, etc. Figure 3.34 presents the data for the ns, np and nd terms with n = 3, 4 and 5. The graph documents that indeed the square root of the energies follows to a very good approximation a straight line as a function of the nuclear charge Z as predicted by (3.34). Admittedly we still need two parameters, qs and n → n∗ to fit a whole series well. As expected, for states with the highest orbital angular momentum the

160

3


4

Z-10.03 2.91

√2Wnℓ / E h

3

3s

3d

4s

2

4p Z-10.02 3.90

1

3p

5s

4d

5p 5d Z-10.02 4.97

0 10 11 12 13 14 15 16 17 18 19 20 Z Na I Al III P V Cl VII Ca X

Fig. 3.14 M OSLEY diagram for Na like ions and different n states. Following spectroscopic traditions we designate the degree of ionization with roman numbers: I for neutral atoms, II for singly ionized, III for doubly ionized atoms, etc. Symbols correspond to the experimental data according to K RAMIDA et al. (2013), the lines are fits following essentially (3.34). Nearly perfect screening (qs = 10) is shown for the d states

energies agree best with the straight lines as validated in Fig. 3.14 particularly well for the 3d, 4d and 5d series. The screening in all three cases is nearly perfect (qs = 10) and die slope of the lines is almost ideally 1/n. For the p and s states the screening is obviously not perfect: in particular for the 3s states the screening is only just above qs = 9, i.e. the electron still ‘sees’ effectively nearly two charges of the nucleus. In any case, the surprisingly good agreement of experimental data with the prediction (3.34) supports very impressively the model of a quasi-one-electron system over a whole isoelectronic series. For higher Z the lines typically fall into the X-ray region. M OSLEY diagrams are therefore also used successfully to characterize X-ray spectra from inner shells (see Sect. 10.5.2). Section summary

• The spectra of alkali atoms resemble those of the H atom in the VIS – except that now degeneracy is removed and consequently several series of lines are observed for each principle quantum number n. • Thus, a good concept to understand these spectra is a quasi-one-electron model where all but one active electron (the valence electron) constitute the closed shell of the ionic core. They (partially) screen the charge of the nucleus but otherwise do not participate in standard spectra. • The energy terms of the alkali atoms (and alkali like ions) depend thus on n and of the valence electron and may be described by Wn = −Eh /[2(n − μ)2 ]. The quantum defect μ(n, ) depends strongly on and weakly on n. For large n it becomes nearly independent it. The higher , the smaller is μ, i.e. the closer the energy levels are to those of the H atom (and H like ions, respectively).

3.3

Perturbation Theory for Stationary Problems

161

• These findings are explained by the low probability of the valance electron to be found close to the nucleus, being ∝ r 2 +2 for r → 0. • Quantum defect theory (QDT) gives a very useful analytic foundation to the observed behaviour of quantum defects: it turns out that the quantum defect μ relates to the phase shift δ = πμ between a pure C OULOMB wave function (H atom) and the wave function for the alkalies. • M OSLEY diagrams summarize the spectral lines of a whole series of ions with equal electron configuration as a function of the nuclear charge Z. Plotting √ 2|Wn |/Eh vs. Z leads to essentially straight lines.

3.3


3.3.1

Perturbation Ansatz for the Non-degenerate Case

As we have seen, numerical integration of the S CHRÖDINGER equation is straight forward – at least for a quasi-one-electron problem. Nevertheless it is important not to rely completely on the ‘black box’ of powerful computer programs. As we have just seen from QDT by way of example, alternative approaches may give profound new insight into the underlying physics based on experimental observations. An even more general approach, which in many cases helps to develop a genuine understanding, is to view an additional interaction as a perturbation on top of a problem that has already been solved. We thus use the opportunity to supplement the above discussion by introducing an important quantum mechanical tool that will be used frequently in the following chapters: time independent perturbation theory. Instead of solving a problem by direct integration of the S CHRÖDINGER equation as described in Sect. 3.2.5, one may try to estimate the changes expected when slightly changing the interactions in a given problem for which the solution is known. The basic idea is to consider the already solved problem (in the present example the H atom wave functions and energies) as 0th order solution for the modified problem (here the system with a potential changed from −1/r to a screened −Z/r potential). If that perturbation is not too large one may hope that the new solution may not be too much different from the 0th order solution and can be derived from the latter. We summarize here the procedures in recipe like manner and refer the readers to standard textbooks on quantum mechanics for a rigorous presentation of the underlying perturbation theory. So, let the Hamiltonian for the original problem (here 0 , the eigenstates ψ (0) (a complete basis), and the corresponding the H atom) be H k (0) eigenenergies Wk . The 0th order S CHRÖDINGER equation 0 ψ (0) = W (0) ψ (0) H k k k

(3.35)

is assumed to be solved. Let the Hamiltonian for the new problem be given by (r, p =H 0 + U ), H

(3.36)

162

3


(in the most simple case just a scalar potential). with the perturbation operator U For perturbation theory to be applicable we have to assume that for the averaged per | | H 0 | holds. We thus write the S CHRÖDINGER equation (3.12) turbation | U in the form 0 + U )ψk = Wk ψk , (H

(3.37)

which has to be solved by the perturbation approach. One then expands the energies and wave functions into a series in terms of small quantities Wk and ψk – is also small on the order of . The formally correct procedure is assuming that U to compare quantities of the same power in the smallness parameter and thus to obtain correction terms of the order , 2 , etc. These are the corrections in 1st, 2nd, etc. order perturbation theory. The key quantities characterizing the perturbation are the matrix elements: (0) (0) (r, p Uj k = ψj U ψk = ψj(0)∗ (r)U )ψk(0) (r)d3 r. (3.38)

3.3.2

Perturbation Theory in 1st Order

We abbreviate the procedure just outlined somewhat. Since the eigenfunctions ψi(0) 0 , derived from (3.35), form a complete, orthonorof the unperturbed Hamiltonian H mal basis set, any function may be constructed as linear superposition from them – including the solution of (3.37). Thus, we use the perturbation ansatz (0) ai ψi with |ak | 1 and |ai | 1 for i = k. (3.39) ψk = i

The conditions |ak | 1 and |ai | 1 are crucial; they establish the essence of the perturbative approach. Introducing (3.39) into the S CHRÖDINGER equation (3.37) we obtain 0 + U − Wk ] ai ψ (0) = 0 [H i 0 H

i

i

(0) ai ψ (0) − Wk ai ψ (0) = 0 ai ψi + U i i i

i

(0) (0) ai ψ (0) = 0. ai W i − W k ψi + U i i

(3.40)

i

In the last step we have made use of 0th order solution (3.35). Next, (3.40) is multi(0)∗ (0) (0) plied from the left with ψk and integrated over all space. With ψk |ψi = δki one obtains (0) (0) (0) ai ψk ak Wk(0) − Wk + ak ψk(0) U ψ k + U ψi = 0. i=k

3.3


163

Since |ai | 1 for i = k, and since the matrix elements of the perturbation operator are small too, one can neglect to 1st order the whole sum over i = k in this U expression. With this ak may be factored out, and one arrives immediately at the 1st order correction for the energy (0) (0) (0)

W = Wk − Wk = Ukk = ψk U ψ k ,

(3.41)

determined by the diagonal matrix element of the perturbation operator. In order to also derive the correction for the wave functions, one multiplies (3.40) (0)∗ from the left with ψj for j = k and integrates: (0) (0) (0) 0 = aj W j − W k + ai ψj U ψ i . i

Inserting the 1st order solution (3.41) for the energy correction this becomes (0) (0) (0) (0) (0) (0) 0 = aj Wj − Wk − ψk + ai ψj U ψ k U ψ i . i

If again one neglects all terms that are quadratically small, the third term can be dropped and of the sum only the term with i = k remains (since ak 1), so that (0) (0) (0) (0) U ψ k , 0 = aj Wj − Wk + ψj from which finally aj =

|ψ ψj |U k (0)

(0)

(0)

(0)

Wk − Wj

for j = k

follows. Thus, the wave function in 1st order perturbation theory is |ψ (0) ψj(0) |U k

(0)

ψk = ψk +

3.3.3

(0) j =k Wk

(0) − Wj

(0)

(0)

ψj = ψk +

Uj k

(0) j =k Wk

(0)

(0)

− Wj

ψj . (3.42)

Perturbation Theory in 2nd Order

For the next step we insert the results of 1st order perturbation theory again into the S CHRÖDINGER equation (3.37) and repeat the whole procedure. The 2nd order correction for the energy is thus derived as follows: Wk =

0 ψk(0) H

|ψ (0) ψj(0) |U k (0) ψ (0) + +U ψ j k (0) (0) j =k Wk − Wj

164

3

= ⇒

(0) Wk

+

(0) (0) ψk U ψk +

(0)

Wk = Wk + Ukk +


|ψ (0) ψj(0) |U k (0) j =k Wk

(0) − Wj

|Uj k |2

(0) j =k Wk

(0)

− Wj

(0) (0) ψk U ψj

(3.43)

.

|ψ (0) are the matrix elements of the perturbation. The proAgain, Uj k = ψj(0) |U k cedure may in principle be repeated as often as necessary. A few points need to be considered: 1. We see that many states (in principle an infinite number) may contribute to the determination of wave functions as well as of energies. Practical considerations will limit the efforts, and one has to choose judiciously which states are included in a perturbation expansion. 2. The contribution of individual states j depends both on the magnitude of the (0) (0) matrix elements |Uj k | and on the resonance denominators 1/(Wk − Wj ). The closer a the energy of a perturbing state j to that of state k which is perturbed, the stronger the modification. (0) (0) (0) (0) 3. If several states ψ1 , ψ2 , ψ3 are degenerate, i.e. have the same energy W1 = (0) (0) W2 = W2 , one has to be careful with the perturbation approach because of the resonance denominators. Only if the non-diagonal matrix elements of the perturbation disappear, i.e. only if U12 = U13 = U23 ≡ 0, one may follow the perturbation procedure just described.

3.3.4

Perturbation Theory for Degenerate States

In the more general case with g fold degeneracy and nonvanishing off-diagonal matrix elements, the problem has to be approached in a more general manner. For clearness we write the Hamiltonian in matrix form with 0 + U |ψk = W (0) δj k + Uj k j k = ψj |H H k ⎛ (0) W1 + U11 U12 U13 ⎜ (0) U W + U U23 ⎜ 21 22 2 (0) =⎜ H ⎜ U31 U21 W3 + U33 ⎜ ⎝ ... ... ... Ug2 Ug3 Ug1

... ... ... ... ...

⎞ U1g ⎟ U2g ⎟ ⎟ ⎟ . (3.44) U3g ⎟ ⎠ ... (0) Wg + Ugg

ψk = Wk ψk thus becomes a matrix eigenvalue The S CHRÖDINGER equation H equation, − W (H 1)|ψ = 0,

(3.45)

with the g × g identity matrix 1, W an eigenvalue, and |ψ is now an eigenstate vector with g components (which may refer to the basis of the unperturbed system).

3.3


165

In other words, we have now to solve a set of g linear algebraic equations, which can lead to a set of g eigenvalues W and state vectors. Generally speaking, one has to diagonalize the Hamiltonian matrix by a unitary transformation. The matrix elements of the diagonalized matrix are the eigenvalues of the system. As well known, a solution of (3.45) only exists if the determinant of the matrix vanishes. Thus, one first has to find the solutions of − W det(H 1) = 0.

(3.46)

In general this leads to a nonlinear equation of degree g in W with up to g solutions Wk , the eigenenergies of the perturbed system. With these one can go back into (3.45) and solve the equations for each value Wk to find the eigenstates of the system. We note that the procedure described here does not require the perturbation to be small. It does not even need the 0th order states to be degenerate (which would imply (0) (0) (0) W1 = W2 = · · · = Wg ). The procedure may ultimately be used universally – as long as the perturbation matrix Uij is sufficiently well known. The accuracy of the solution finally depends only on the number and artful choice of the basis states included in the computation. We shall encounter many applications in later chapters.

3.3.5

Application of Perturbation Theory to Alkali Atoms

To illustrate how to use 1st order perturbation theory, we briefly return to the alkali atoms as a simple example. We emphasize the pedagogical aspect of this subsection that is not meant to generate accurate numbers for energy levels. With the interaction potential (3.10) discussed in Sect. 3.2.3 the Hamiltonian of the perturbed problem (in a.u.) is written = − 1 2 + VS (r) = H 0 + VC (r) H 2

with VS (r) = −1/r + VC (r),

(3.47)

and we recognize 0 = − 2 /2 − 1/r H being the unperturbed Hamiltonian of the H atom. The spherically symmetric perturbation potential U (r) = VC (r) originates from the interaction of the valence electron with the full nuclear charge Z at small r and screening at larger distances. For demonstration we use again the particularly simple model potential VC introduced in (3.11). Figure 3.15 illustrates this potential for Na (Z = 11). The screening radius has now been calibrated, as described below, to fit the experimentally determined ground state binding energy W3s . The matrix elements of the perturbation (3.11) are computed with the hydrogen (0) eigenfunctions Rn (r)Y m (θ, ϕ) (0th order solutions), which have been treated in detail in Sect. 2.6.1. Since the perturbation potential is spherically symmetric and

166

3


Fig. 3.15 Core screening in Na with rS = 0.98a0

W / Eh 1

0

2

3

4

5

W3s

r / a0

-2 -4

VC (r) =

-6 – -8

Table 3.5 Binding energies for some levels of the s, p, and d series in Na. Experimental data (exp.) are compared with 1st order perturbation theory (1st order). WH are the corresponding energies for the H atom. For details see text

(Z– 1) -r/rS e r

Levels

Binding energies / eV exp.

1st order

WH

3s

−5.13907

−5.139

−1.5117

3p

−3.0357

−2.70

−1.5117

3d

−1.5221

−1.63

−1.5117

4s

−1.9477

−1.87

−0.8503

4p

−1.3864

−1.30

−0.8503

5s

−1.0227

−0.92

−0.54422

5p

−0.7951

−0.74

−0.54422

6s

−0.6294

−0.55

−0.37794

acts only on the radial part of the wave function, the matrix elements simply are ∞ (0) (0) VC n m,n m = δ δmm VC (r)Rn (r)Rn (r)r 2 dr = VC n ,n . (3.48) 0

In 1st order perturbation theory only the diagonal matrix elements are needed. We (0) make explicit use of the radial wave functions Rn (r) of the H atom as given in (2.122), using the series expansion (2.123) for the L AGUERRE polynomials. The numerical integration of (3.48) can be done with standard desktop computing programmes within seconds. (0) Inserting the binding energies Wn = −1/2n2 of the hydrogen atom into (3.41) and noticing that with VC (r) < 0 the matrix elements are also negative, the energies of the alkali atoms in 1st order perturbation theory finally are (in a.u.) (0) + VC n ,n = − Wn = Wn

1 − |VC n ,n |. 2n2

(3.49)

All energy terms are indeed lowered in respect of the H atom – as experimentally observed. In Table 3.5 we communicate some numbers that we have derived in this manner for Na and compare them with the experimental data from the NIST data bank (K RAMIDA et al. 2013). For comparison, H atom binding energies are also shown.


167

The data illustrate quite clearly what may be achieved with such a simple approximation. Of course, the results can only be as good as the perturbation potential that describes the additional interaction and one cannot expect miracles from (3.11) with only one free parameter – a rather crude guess of the perturbation potential. An additional problem is, that the wave functions experience a strong phase shift in the ionic core, as we have seen in Sect. 3.2.6. This is of course not taken into account in 1st order perturbation theory where the energy is computed from 0th order wave functions for the H atom. In view of these difficulties, the results shown in Table 3.5 may be considered ‘reasonable’. But it becomes also obvious that 1st order perturbation theory should only be used for a first orientation in such problems with several competing influences. If precision is demanded one has to resort at least to the methods outlined in Sect. 3.2. Section summary

• Perturbation theory can often give important qualitative and semi- quantitative insights into the physics expected from exposing atoms or molecules to specific interactions – as long as the averaged perturbation is small compared to | | H 0 |. the Hamiltonian defining the basic interaction in the system, | U • In 1st order perturbation theory the change of energy (of non-degenerated (0) (0) |ψk , while the 1st order wave function is states) is W = Ukk = ψk |U given by (3.42). • Higher order approaches are needed if first order (diagonal matrix elements) vanishes. • For degenerate states a different approach has to be taken. One expresses the perturbed Hamiltonian in matrix form (3.44), using as many and as good basis functions as appropriate. One then has to diagonalize this matrix, using the standard procedures of linear algebra. The procedure can be extended to quite general problems, providing suitable basis functions are available.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2. DFT: ‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3). IP: ‘Ionization potential’, of free atoms or molecules (in solid state physics the equivalent is called “workfunction”). MQDT: ‘Multichannel quantum defect theory’, advanced form of QDT for the interpretation of complex atomic and molecular spectra, especially of highly excited RYDBERG states (see Sect. 3.2.6). NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. ODE: ‘Ordinary differential equation’.

168

3


QDT: ‘Quantum defect theory’, interprets experimental spectra by phase shifts in the radial wave functions and makes predictions for scattering processes (see Sect. 3.2.6). 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 electromagentic radiation. Part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References FANO , U. and A. R. P. R AU: 1986. Atomic Collisions and Spectra. Orlando: Academic Press Inc., 409 pages. F REEMAN , R. R. and D. K LEPPNER: 1976. ‘Core polarization and quantum defects in high angular-momentum states of alkali atoms’. Phys. Rev. A, 14, 1614–1619. H ARTREE , D. R.: 1928. ‘The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods’. Proc. Camb. Phil. Soc., 24, 89–110. I DZIASZEK , Z. and P. S. J ULIENNE: 2010. ‘Universal rate constants for reactive collisions of ultracold molecules’. Phys. Rev. Lett., 104, 113202. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. J UNGEN , C.: 1996. Molecular Applications of Quantum Defect Theory. New York, London: Taylor & Francis, 664 pages. K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. M ARGENAU , H.: 1984. The Miracle of Existence. Woodbridge, CT, USA: Ox Bow Press. NIST: 2011. ‘NIST physics laboratory holdings by element’, NIST. http://physics.nist.gov/ PhysRefData/Elements/per_noframes.html, accessed: 7 Jan 2014. O SPELKAUS , S. et al.: 2010. ‘Quantum-state controlled chemical reactions of ultracold potassiumrubidium molecules’. Science, 327, 853–857. PAULI , W.: 1945. ‘The N OBEL prize in physics: for the discovery of the exclusion principle, also called the pauli principle’, Stockholm. http://nobelprize.org/nobel_prizes/physics/ laureates/1945/. S CHMIDT , K. and M. A. L EE: 1998. ‘Visual Schrödinger: A visualizer-solver’. http://fermi.la.asu. edu/Schroedinger/, accessed: 7 Jan 2014. 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. S EATON , M. J.: 1983. ‘Quantum defect theory’. Rep. Prog. Phys., 46, 167–257. S EATON , M. J.: 2002. ‘Coulomb functions for attractive and repulsive potentials and for positive and negative energies’. Comput. Phys. Commun., 146, 225–249. U NIVERSITY OF C OLORADO: 2000. ‘David’s wizzy periodic table’, Physics 2000. http://www. colorado.edu/physics/2000/applets/a2.html, accessed: 7 Jan 2014. W EBER , K. H. and C. J. S ANSONETTI: 1987. ‘Accurate energies of ns, np, nd, nf, and ng levels of neutral cesium’. Phys. Rev. A, 35, 4650–4660. W IKIPEDIA CONTRIBUTORS: 2014. ‘Periodic table’, Wikipedia, The Free Encyclopedia. http://en. wikipedia.org/wiki/Periodic_table, accessed: 7 Jan 2014.

4

Non-stationary Problems: Dipole Excitation with One Photon

A quantum system, such as an atom, may only be observed by changing it. Electromagnetic waves can induce transitions between stationary states and are the basis of spectroscopy – one of the most important methods for studying quantum systems in general. In this chapter we want to recapitulate the quantum mechanical tools needed and then treat in detail the rules and phenomena which underly light induced, electric dipole (E1) transitions.

Overview

The present chapter concentrates on electric dipole transitions (E1), while Sect. 5.4 will also treat electric quadrupole (E2) and magnetic dipole (M1) transitions. After some basics and terminology on electromagnetic radiation, polarization, and photon spin (Sect. 4.1), the essentials of spectroscopy are introduced in Sect. 4.2, the E INSTEIN A and B coefficients are defined, and the classical model of a radiating oscillator is reviewed. The advanced reader may jump over this section and ignore also Sect. 4.3.1–4.3.5, where the elements of time dependent perturbation theory are summarized. However, Sect. 4.3.6 with terminology and some key results as well Sect. 4.4 with essentials on selection rules for dipole (E1) transitions are needed in the following sections and should be read carefully. The same holds for Sect. 4.5 where the angular characteristics of dipole radiation are presented. Section 4.6 may be used by the expert reader just as a source of reference with details on the evaluation of matrix elements and E INSTEIN coefficients. In Sect. 4.7 photoinduced linear combinations of states are discussed – a theme of broad relevance. In this context we also introduce quantum beats and indicate some spectroscopic perspectives. Finally, we ask the very fundamental, almost philosophical question whether electrons may really ‘jump’ from one stationary state into another – and present experiments illuminating this profoundly.


169

170

4.1

4


Electromagnetic Waves: Electric Field, Intensity, Polarization and Photon Spin

Before discussing radiation induced transitions we define the terminology for describing electromagnetic waves used here and in most of the following text. For brevity, we shall speak about “electromagnetic waves”, “electromagnetic radiation” and “light” more or less synonymously, although the latter is often used more specifically for the visible part of the spectrum.

4.1.1

Electric Field and Intensity

For the present purpose it is sufficient to consider only plane, monochromatic waves extending over all space, and to concentrate on the electric field component of the wave which is responsible for E1 transitions.1 The electric field vector E(r, t) is an observable in the real world depending on position vector r and time t. We thus write it as2 i E(r, t) = E0 eei(kr−ωt) − e∗ e−i(kr−ωt) 2

(4.1)

with the (real) field amplitude E0 , the unit polarization vector e, and the wave vector k with |k| = 2π/λ = ω/c. Even though it is sometimes convenient to use a complex representation (and take its real part as observable after all calculations are done), we emphasize strongly that it is important to write E(r, t) as a real quantity in order to be able to describe all observed physical phenomena as done here – neither of the two summands in (4.1) can be ignored as we shall see shortly! From classical electrodynamics we know that the field amplitude E0 is related to the (time averaged) intensity I of the electromagnetic wave by √ E0 = 2I /(ε0 c) = 2I Z0 = 27.45 I "1/2 (4.2) with the electric constant ε0 , the speed of light c and Z0 the characteristic vacuum impedance. For practical purposes we communicate a handy numerical expression: I E0 = 2745 V m−1 . (4.3) W cm−2 1 We find this approach (leading to correct results for E1 transitions) conceptually more accessible than the general, rigorous treatment of all transition types based on the vector potential. The latter is outlined in Appendix H, while in Chaps. 1 and 2, Vol. 2, we shall generalize (4.1) and learn how to treat spatial distributions and quasi-monochromasy of real light beams adequately. 2 The

overall phase angle φ0 does not play a role in the present discussion and will generally be ignored. However, we shall have to come back to it in Chaps. 1 and 2, Vol. 2.

4.1

Electromagnetic Waves: Electric Field, Intensity, Polarization

171

In reality, one usually has to deal with light of a certain bandwidth, i.e. with an intensity distribution I˜(ω) (intensity per unit angular frequency). Then one has to replace I → I˜(ω)dω and to integrate expressions derived for transition probabilities over the whole available spectral range.

4.1.2

Basis Vectors of Polarization

In (4.1) the vector nature of the field is expressed by e, the unit polarization vector – which may be complex since it is always accompanied by its conjugate complex e∗ . Using a (real) Cartesian basis ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 ex = ⎝ 0 ⎠ , ey = ⎝ 1 ⎠ , ez = ⎝ 0 ⎠ , (4.4) 0 0 1 and assuming for simplicity that the light propagates in +z-direction (z k) each arbitrary polarization vector may be expressed as a suitable linear combination of the basis pair (ex , ey ). This is due to the fact that freely propagating light is transversely polarized and the ez component has no relevance. The basis vectors are clearly orthonormal ei · ej = δij . Alternatively, one may write e in a (complex) spherical basis, also called helicity basis with basis vectors eq ⎛ ⎞ ⎛ ⎞ 1 0 −1 −1 ⎝ ⎠ ∗ ⎝ i = −e−1 , e+1 = √ (ex + iey ) = √ e0 = ez = 0 ⎠ 2 2 0 1 ⎛ ⎞ (4.5) 1 1 1 ⎝ ⎠ −i = −e∗+1 . and e−1 = √ (ex − iey ) = √ 2 2 0 In this basis, the polarization of light which propagates in the +z-direction (e0 ) may again be expressed by only two basis vectors (e+1 , e−1 ). One easily verifies again the orthonormality eq · e∗q = δqq

with e∗q = (−1)q e−q .

(4.6)

For atomic problems the helicity basis is often better adapted, since atoms too are usually described in a spherical coordinate system. For reference, we express the Cartesian basis in terms of the helicity basis: −1 ex = √ (e+1 − e−1 ) 2

i and ey = √ (e+1 + e−1 ). 2

(4.7)

For later use we also give the unit polarization vectors for linearly polarized light in 45◦ and 135◦ direction in respect of the x-axis: 1 e 45◦ = √ (ex + ey ) 2

−1 and e 135◦ = √ (ex − ey ). 2

(4.8)

172

4


In the spherical basis this is written as: 1 e 45◦ = (i − 1)e+1 + (i + 1)e−1 2 1 e 135◦ = (i + 1)e+1 + (i − 1)e−1 . 2

(4.9)

Again, for light propagating in +z-direction this pairs of polarization vectors constitutes an orthonormal basis set. We now give a few explicit examples for the wave fields. If e = ex , i.e. for light polarized linearly in x direction, (4.1) becomes simply E x (r, t) = −E0 sin(kr − ωt)ex ,

(4.10)

while linear polarization in y direction is described by E y (r, t) = −E0 sin(kr − ωt)ey .

(4.11)

The unit vector e = e+1 describes left hand circularly polarized light (LHC), also called σ + light. Inserting (4.5) into (4.1) gives 1 E +1 (r, t) = √ E0 sin(kr − ωt)ex + cos(kr − ωt)ey , 2

(4.12)

and e−1 stands for right hand circularly polarized light (RHC) or σ − light: 1 E −1 (r, t) = √ E0 sin(kr − ωt)ex − cos(kr − ωt)ey . 2

(4.13)

An illustration of σ + light gives Fig. 4.1. Shown is the E vector at a fixed time t = π/2ω along the z-axis. As indicated E rotates – at a fixed position – clockwise around the wave vector k of the light, i.e. with positive helicity. One verifies this most directly in (4.12) with kr = 0, or correspondingly in Fig. 4.1, if one considers the time propagating.3 The most general unit polarization vector for light propagating into an arbitrary direction can be written in either Cartesian or spherical basis: e = ax e x + ay e y + az e z =

1 q=−1

aq e q

with

1

|aq |2 = 1.

(4.14)

q=−1

In the spherical basis it may be specialized for any elliptically polarized light propagating parallel to the +z axis (i.e. into e0 direction) by eel = a+ e+1 + a− e−1 = e−iδ cos βe+1 − eiδ sin βe−1 , 3 Methods

for generating and detecting polarized light will be discussed in Chap. 1, Vol. 2.

(4.15)

4.1


173 z

σ+ light (LHC)

k x

+t

+t

k

0 +t

y

Fig. 4.1 Schematic illustration of left hand circularly polarized light σ + (LHC). The grey arrows indicate the direction of the electric field perpendicular to the z k axis at a fixed time, corresponding to ωt = π/2. For larger times the sense of the rotation is indicated. The somewhat surprising definition for left hand circular polarized light is of historical origin: in times prior to the laser, people used to look into the oncoming light beams, hence they considered σ + light as rotating anti-clockwise, i.e. LHC polarized

with the component a0 = 0 in this coordinate system). The ellipticity angle β describes the degree of ellipticity, the alignment4 angle δ gives the direction of the ellipse in respect of ex . Converting (4.15) into Cartesian coordinates with (4.5) and inserting into (4.1) gives an explicit, parameterized expression for the elliptic field vector in terms of β and δ: √ E el (r, t) = (E0 / 2) ' × cos β sin(kr − ωt − δ) + sin β sin(kr − ωt + δ) ex (4.16) ( + cos β cos(kr − ωt − δ) − sin β cos(kr − ωt + δ) ey . This describes the standard form of an ellipse on which the E vector rotates if (for fixed r) the phase kr − ωt + δ varies between 0 and 2π . As sketched in Fig. 4.2, this ellipse is inclined by an angle δ in respect of the x-axis, and for 0 ≤ β ≤ π/2 its major and minor axes a and b, respectively, are: √ a = E0 (cos β + sin β)/ 2 = E0 sin(β + π/4) √ b = E0 (cos β − sin β)/ 2 = E0 cos(β + π/4).

(4.17)

4 Unfortunately the terms “alignment” and “orientation” are mixed up time and again in the literature: Alignment refers to the direction of a polar vector (e.g. the E vector in the case of linearly polarized light). In contrast, orientation specifies the sense of rotation of an axial vector (e.g. the angular momentum of left and right circularly polarized light).

174

4


Fig. 4.2 Polarization ellipse seen for right hand elliptically polarized light, seen into the +z-direction (k points into the plane). As indicated, the light is right hand polarized (negative helicity). The parameters are δ ∼ 60◦ , β ∼ −64◦ or ∼ 0.34

x

δ a b

y

Note that a 2 + b2 = E02 . In the literature instead of the ellipticity angle β often a (somewhat ambiguously defined) so called ellipticity is used: = b/a = cot(β + π/4).

(4.18)

Three special cases of are of particular significance according to (4.15): 1. Left hand circularly polarized light (σ + ) corresponds to an ellipticity angle β = 0◦ , while 2. right hand circularly polarized light (σ − ) is characterized by β = +π/2. √ 3. Linearly polarized light corresponds to β = π/4, i.e. sin β = cos β = 1/ 2. From (4.16) one finds that in this case: E(r, t) = E0 e(δ) sin(kr − ωt) = E0 (cos δex + sin δey ) sin(kr − ωt).

4.1.3

(4.19)

Coordinate Systems

The main part of this chapter will be concerned with absorption and emission of electromagnetic radiation from atoms. In general, one has to distinguish two different coordinate systems: one in which the atom is best described – we call it the “atomic frame” (at) – while the photon may possibly be better described in another frame – we call it the “photon frame” (ph). These coordinate systems may differ from each other. In the spirit of Sect. 4.1.2 a convenient choice for the photon frame is illustrated in Fig. 4.3. We shall use this henceforth unless otherwise stated. The atomic frame may e.g. be defined by an external electric or magnetic field in respect of which the projection quantum numbers m are defined. The photon frame is defined as shown in Fig. 4.3 by the direction of light propagation (wave vector k), with the axes z(ph) k and y (ph) ⊥ z(at) . (at) The unit vectors eq in the atomic frame (helicity basis) indicated in Fig. 4.3 by heavy red arrows may in a classical picture be seen to represent three different classical dipole oscillators. This will be discussed in Sect. 4.5.1.

4.1


Fig. 4.3 Coordinate system for the radiating atom (red, (at)) and for the photon (black, ph). The three classical oscillators which correspond to the basis (at) (at) vectors e(at) 1 , e0 and e−1 are indicated by thick, red arrows

z (at)

z ( ph) θk

e 0(at )

y (ph) k

x ( ph)

Δm= 0 (at) e -1

y (at)

Δ m= +1 x (at)

4.1.4

175

(at) e +1

Δm = -1

φk

Angular Momentum of the Photon

At this point we want to extend our knowledge about the properties of photons summarized in Sect. 1.4. In addition to momentum and energy the photon also possesses an intrinsic angular momentum, the so called photon spin with a quantum number sph = 1. Projected onto the wave vector k LHC and RHC polarized light have angular momenta ms with ms = 1 and −1, respectively. Owing to the transverse nature of electromagnetic waves the third component with ms = 0 does not exist. In a particularly beautiful and fundamental experiment this has been observed for the first time by B ETH (1936). The setup and the results are shown schematically in Fig. 4.4. LHC photons are prepared by passing linearly polarized light through a λ/4 plate. These photons then pass through a λ/2 plate where they are converted into RHC. This implies an angular momentum change of −2 per photon which is transferred to the λ/2 plate in order to conserve the total angular momentum of the system. For more efficiency the light beam is retroreflected behind another λ/4 and quartz fibre

mirror and light beam for measuring the twist

x

/4 plate coated on top

y

LHC

experimental result: light torque M (measured by the twist of the quartz fibre) as a function of circular polarization defined by the angle

LHC

/2 plate RHC

RHC

x

y

/4 plate

xy-polarization plane of linear polarization linearly polarized light in

M / 10-9 dyn×cm

5

0

-5 = -135° -90° -45°

0°

45°

Fig. 4.4 Experiment of B ETH (1936) on “mechanical detection and measurement of angular momentum of light”

176

4


passes the λ/2 plate again. As a consequence a measurable torque M = dL/dt = −2 × (2I A/ω) twists the plate which is suspended on a quartz fibre, I being the intensity of the light beam, A its effective area and ω the energy of one photon. The experiment confirms quantitatively that the photon has indeed a spin projection ± for LHC and RHC light, respectively. The photon is a boson! In contrast to electrons (which are fermions) several photons may occupy identical states – a prerequisite for a laser, as we shall see in Chap. 1, Vol. 2. Section summary

• We describe electromagnetic radiation by a monochromatic plane wave. As a physical observable it is represented by a real valued function (4.1). Both exponential terms will turn out to be relevant when describing the interaction of the electromagnetic field with quantum systems. • The helicity basis (4.5) is convenient for describing orthonormal (4.6) unit basis vectors for problems in atomic physics. The most general unit polarization vector for light propagating into +z-direction is given by eel = e−iδ cos βe+1 − eiδ sin βe−1 . Ellipticity angles β = 0 and π/2 describe LHC (σ + ) and RHC (σ − ), respectively, β = π/4 corresponds to linear polarization (π ). The alignment angle δ gives the direction of the major axis of the polarization ellipse (for linear polarization the E vector) with respect to the x-axis. • The photon is a particle with angular momentum (spin) sph = 1, i.e. it is a boson. As experimentally verified by B ETH (1936), the spin projection onto the z-axis of propagation is ± for σ ± light, respectively.

4.2

Introduction to Absorption and Emission

4.2.1

Stationary States

Figure 4.5 schematically shows a typical system of stationary states of an atom which converge towards the ionization limit. As we have seen for the H atom and the alkali atoms the term energies are found by solving the S CHRÖDINGER equation |n = Wj |n H at physically ‘reasonable’ boundary conditions (in the following we shall often abbreviate all quantum numbers by one letter, e.g. n j = j, a or b). Energy zero is defined such that for bound states Wj < 0 while the free electron has a continuous spectrum of energies Wkin = W ≥ 0. As discussed in the previous two chapters, the asymptotic behaviour of the radial wave function of the bound states is given by lim Rj (r) ∝ exp − 2|Wj |r r→∞

while the continuum states are described essentially by C OULOMB wave functions.

4.2


Fig. 4.5 Stationary states of an atom with IP WI

177 W 0

ionization continuum |j〉

bound | 4 〉 stationary states | 3〉

4.2.2

W2

| 2〉

W1

| 1〉

WI

Optical Spectroscopy – General Concepts

According to B OHR, transitions |b ←→ |a between two stationary states are accompanied by emission (or are induced by absorption) of electromagnetic radiation with a frequency νba given by: hνba = ωba = hcν¯ ba = |Wb − Wa |.

(4.20)

Here ωba = 2πνba is the angular frequency of the radiation, λba = c/νba its vacuum wavelength and ν¯ ba = 1/λba = νba /c its wavenumber. Figure 4.6 illustrates three basic types of setups for spectroscopic investigations. Specialties and refinements will be discussed in detail later, but essentially all spectroscopic measurements follow in principle one of these three concepts: 1. Emission spectroscopy: A hot gas or a plasma (e.g. a spectral lamp) emits by spontaneous decay energy in the form of photons. The emitted light is analyzed according to its wavelength by some type of spectrometer (in Fig. 4.6 symbolized by a prism). As indicated in Fig. 4.6 (right) many excited states |b are involved, each of them may decay into several states |a of lower energy. Thus, the light emitted originates from a variety of different combinations of upper and lower states. Hence, emission spectra are typically rich of lines and difficult to analyze. 2. Absorption spectroscopy: Only light of the characteristic transition frequencies according to (4.20) is absorbed, all other light just passes through the target. Ideally, only one initial (ground) state |a is populated in the cold target, and the number of lines observed is significantly smaller as compared to emission spectroscopy. One typically uses white light (e.g. from a synchrotron radiation source in the VIS, UV, VUV or X-ray spectral region, or from a glow bar in the IR). After passage through the target the light is analyzed, today usually by optical multichannel analyzers (OMA) which allow for simultaneous registration of the signal over a whole spectral range. The alternative, to make the light monochromatic before sending it through the target and to scan its wavelength over the region of interest is usually much less efficient. 3. Fluorescence spectroscopy: In a first step, the atoms or molecules to be studied are excited very selectively into one specific state |b , typically by a laser (laser

178

4


hot gas, plasma

emission spectroscopy

cold gas white light lamp

absorption spectroscopy

las

fluorescence spectroscopy, laser induced fluorescence

er

exciting laser line

specifically excited atoms

Fig. 4.6 3 basic methods of spectroscopy very schematic. Left: experimental setups, middle: type of observed spectra, right: term schemes and transitions

induced fluorescence, LIF). All observed emissions |a ← |b then start from this one level, leading to very clear spectra. Since the exciting wavelength may also be varied, the method is very powerful in identifying the nature of unknown transitions. If the emission occurs delayed (on the ms to s time scale) the process is called phosphorescence.

4.2.3

Induced Processes

Transitions which occur under the influence of an external electromagnetic radiation field are called induced. Such transitions |b ←→ |a may occur in both directions: we distinguish absorption and stimulated emission.

Absorption As derived in Sect. 1.3.2 the L AMBERT-B EER absorption law (1.44) dI = −Na σ I dz

⇒

I (z) = I0 e−σ Na z = I0 e−μz

(4.21)

describes the exponential reduction of the intensity I (z) of electromagnetic radiation when passing through an absorbing medium of particle density Na with [Na ] = number of absorbing particles m−3 . Here Na is assumed to be constant over distance z and time t. We now want to understand the process on a microscopic level and derive quantitative expressions for the absorption cross section σ ([σ ] = m2 ) and the absorption coefficient μ = σ Na ([μ] = m−1 ). Since the energy of a photon is ω we can

4.2


Fig. 4.7 Atomistic view on the absorption of electromagnetic radiation

179 |b〉

Nb

Nph(0) nħω

Nph(t) |a〉

(n-1)ħω Na

translate the intensity of the radiation field I ([I ] = W m−2 ) into its photon density Nph ([Nph ] = number of photons m−3 ) or energy density u = ωNph ([u] = J m−3 ) by I = cωNph = cu.

(4.22)

As schematically indicated in Fig. 4.7, on an atomistic level the absorption process corresponds to a loss of photons from the field, i.e. a reduction of Nph by dNph – which is identical to a loss dNa of particle density Na in the initial state |a and gain dNb of density Nb in the final excited state |b . In contrast to the situation described by (4.21), we are now interested in the change of N with time t at fixed position z for a constant intensity I . Since a change over a distance dz corresponds to a change over a time dt = dz/c we may rewrite (4.21) with (4.22) as −Na σ

dNph dNa I dI dI dNb = = = = =− . ω ωdz cωdt dt dt dt

(4.23)

With this one defines an absorption (or excitation) rate5 I 1 dNa = −σba = σba Φ R¯ ba = − Na dt ω

(4.24)

for the transition into the state |b from state |a . We have introduced here also the photon flux Φ = I /ω. Note that the rate R¯ ba has the dimension T−1 ; it is the probability for absorption of photons by an atom per unit of time. To obtain the total number of events per unit time one has to multiply it by the number of atoms in the volume under observation. In the above discussion we have tacitly assumed that the radiation to be absorbed is strictly monochromatic and tuned into resonance with the absorption line. In reality, however, the intensity has a spectral distribution, which we indicate by I˜(ω) with [I˜(ω)] = W m−2 s−1 . Thus, only that fraction of intensity can be absorbed which is in resonance with the angular frequency ωba of the transition. Usually one refers the transition probability to the spectral radiation density u(ω), ˜ i.e. the energy density clarity, throughout this chapter we shall use R¯ ba , B¯ ba etc. for rates and coefficients averaged over initial and summed over final substates, since in general we have to account for degenerate energy levels. In contrast, Rab (e) refers to transitions between specific substates b and a with polarization e. 5 For

180

4


Table 4.1 Definitions in the context of absorption and emission of electromagnetic radiation Symbol

Term

Unit

Remarks

σ

absorption cross section

m2

effective absorbing area

N

particle density

m−3

absorption coefficient

m−1

I = I0 exp(−μx) if μ < 0

μ

Eq.

=N ·σ

γ

= −μ

amplification coefficient

m−1

I

= c · Nph · ω

light intensity

W m−2

Nph

= I /(c ω)

photon density

m−3

I

= E02 /(2Z0 )

intensity

Z0

= 1/(cε0 )

impendence in free space

376.73Ω

also “vacuum impedance”

Nph ω

=

energy density

J m−3

radiation field

spectral radiation density

J m−3 Hz−1

u(ω) ˜

= =

εr ε0 2 2 E0 Nph ω

ω energy Vol× ω

=

I c ω

=

E0 field amplitude

ε0 E02 2 ω

=

I (ω) 2c

W s2 m−3

ν

bandwidth (frequency)

Hz

ω

. . . (angular frequency)

s−1

u(ν) ˜ = 2πu(ω)

du per spectral range dω, which is related to the spectral intensity distribution I˜(ω) by dν du u(ν) ˜ I˜(ω) du = = = , (4.25) c dω dω dν 2π with the spectral radiation density given alternatively per unit angular frequency ω = 2πν or per unit frequency ν of the electromagnetic radiation. (We shall use the former unless mentioned otherwise.) Relevant for the transition probability is the intensity at the transition frequency. Consequently, we now rewrite (4.24) as: u(ω) ˜ =

I˜(ωba ) 1 dNa R¯ ba = − = B¯ ba = B¯ ba u(ω ˜ ba ). Na dt c

(4.26)

The thus defined constant B¯ ba is called E INSTEIN coefficient for absorption with [Bba ] = m3 s−2 J−1 . Note that rates, transition probabilities and matrix elements from state |a to state |b are typically indexed from right to left. Table 4.1 summarizes the terms and definitions most frequently used for the quantitative description of radiation induced dipole transitions. Most of the discussion to follow in this chapter will focus on the E INSTEIN coefficients. They contain the essence of the transitions in quantum systems. Polarization and frequency dependence play an important role.

Stimulated Emission Up to now we have assumed that only one dominantly populated ground state level contributes to the absorption process. And the absorption as such was thought to be weak enough so that it does not change the population significantly. We now con-

4.2


181

Fig. 4.8 Stimulated emission

|b〉

Nb

Nph(0) nħω

Nph(t) |a〉

(n+1)ħω Na

sider the inverse process, the so called stimulated emission which we have schematically illustrated in Fig. 4.8. For this to happen, the excited state must be populated. For simplicity we assume for the moment that all atoms are initially found in the excited state |b as indicated in Fig. 4.8. In complete analogy to (4.24) and (4.26) the corresponding emission rate is I˜(ωba ) 1 dNb R¯ ab = − = B¯ ab = B¯ ab u(ω ˜ ba ). Nb dt c

(4.27)

We thus find that more photons may come out of the system than going into it – if the excited state has a higher population than the ground state so that this process dominates. B¯ ab is called E INSTEIN coefficient for stimulated emission. In general, both processes – absorption and stimulated emission – happen in a quantum system. According to (1.54) the ratio of the population densities are given by Nb /Na = (gb /ga ) exp[−(Wb − Wa )/(kB T )] in thermodynamic equilibrium. At room temperature kB T 25 meV is small, while electronic excitation in atoms typically requires some eV; hence Wb − Wa kB T and Nb Na . In contrast, if the energy gaps are much smaller, as it is the case e.g. for molecular vibrational and rotational states, one has to consider induced emission also in standard spectroscopic absorption experiments. A completely different situation arises when skillful schemes are applied to achieve a significant population of some specific excited state |b such that Nb > Na (for some state |a , not necessarily the ground state). Then, stimulated emission can be stronger than absorption and laser action becomes possible. We shall discuss this in some detail in Chap. 1, Vol. 2.

4.2.4

Spontaneous Emission – Classical Interpretation

From experimental experience we know that excited atoms may also decay spontaneously. Quantum electrodynamics (QED) explains this by the interaction of quantum systems with the vacuum field: In a fully quantized description the electromagnetic field is represented by harmonic oscillators, their energy being (Nj + 1/2)ωj . Here Nj is a (typically very large) integer number proportional to the intensity of the radiation at the frequency ωj , and ωj /2 is the so called zero-point energy. The latter represents the eigenenergy of the ground states of the field oscillators at zero intensity of the radiation. This “vacuum field” is essentially a consequence of the uncertainty relation. One may say, somewhat figuratively, that it is this (isotropic)

182

4


zero point oscillation of the quantized radiation field, which forces an excited quantum system to decay spontaneously. We shall present a quantitative description of spontaneous emission based on this concept in Chap. 2, Vol. 2. Here we consider a heuristic, classical approach in which the atomic electron is treated as an accelerated charge with a dipole moment D(t) = −er(t). This classical picture does not allow to extract quantitative results. It leads, however, to an intuitive understanding. According to classical electrodynamics such a dipole emits electromagnetic waves. In the nonrelativistic limit the electric field at large distances R from the source is given by E(R, t) =

k k 1 ¨ t = er¨ ⊥ (t ) , × × D 2 2 k 4πε0 c R k 4πε0 c R

(4.28)

with t = t − R/c being the retarded time, and r¨ ⊥ representing the components of the dipole acceleration perpendicular to the wave vector k. We identify this with projecting r¨ onto the x (ph) y (ph) plane of the photon frame introduced in Fig. 4.3. For a harmonic dipole oscillator r(t) = r (at) exp(−iωba t) with amplitude r (at) in the atomic frame and angular frequency ωba the field amplitudes are given by Ex (R, t) =

2 eωba (ph) r (at) · ex ei(kR−ωba t) 4πε0 c2 R

(4.29)

Ey (R, t) =

2 eωba (ph) r (at) · ey ei(kR−ωba t) . 4πε0 c2 R

(4.30)

In Sect. 4.5.1 these expressions will be used for a semiclassical derivation of angular and polarization characteristic of atomic dipole radiation. Here we are just interested in the overall power emitted. From (4.28) we obtain with (4.2) the intensity I = ε0 c|E(R, t)|2 , and on average6 an energy 2 ¨ dW |D(t)| dΩ = I R 2 dΩ = sin2 θk dΩk dt (4π)2 ε0 c3

(4.31)

is emitted per unit time into a solid angle dΩk = sin θk dθk dϕk . Here θk is the angle ¨ and k. Integration of sin2 θk over all solid angles gives a factor 8π/3 so between D that the total radiation power emitted is P=

2 dp 1 dW e2 ¨ 2= = | D| 3 2 3 dt 6πε0 c 6πε0 me c dt

(4.32)

with [P ] = J s−1 . For later use we have introduced also the momentum p = me r˙ of the oscillating electron. 6 The

bar on top of quantities indicates temporal averaging.

4.2


183

¨ 2 = |ω2 D|2 = e2 |r (at) /2|2 ω4 For the harmonic dipole oscillator we have |D| ba ba and hence the average energy radiated by one atom per unit time is P=

4 dW 1 e2 |r (at) /2|2 ωba . = dt 6π ε0 c3

(4.33)

Due to this radiation the excited state of the atom decays. In a classical picture the oscillation amplitude (i.e. the orbital radius of the electron) continuously decreases – contrary to the spectroscopic observation of discrete, stationary states: the atom (or any other quantum system) is either in the excited or in the ground state. Thus, quantum mechanically (4.33) is interpreted as a probability statement: The probability that one excited atom emits a photon of energy ω during the time interval dt is 3 1 e2 |r (at) /2|2 ωba dW (spont) R¯ ab = dt = dt = A¯ ab dt 3 ω 3π ε0 c

(4.34)

(spont) with the spontaneous decay rate R¯ ab for transitions |a ← |b between two energy levels. The decay constant ([Aab ] = s−1 ) derived from this classical model is

A¯ ab =

3 |er (at) /2|2 ωba 4α 3 (at) 2 32απ 3 c |r (at) /2|2 = ω r /2 = , 3 3πε0 c3 3c2 ba λ3

(4.35)

with α 1/137 being the fine structure constant (1.10). The exponential decay law ensuing form (4.34) has already been discussed in Sect. 1.3.1. For the excited state density Nb we obtain according to (1.39) and (1.40) in the absence of other processes −

1 dNb 1 = A¯ ab = Nb dt τab

⇒

¯

Nb (t) = Nb0 e−Aab t = Nb0 e−t/τab

with the average excited state lifetime τab = 1/A¯ ab , the half lifetime τ1/2 = (ln 2)τab , and Nb0 the excited state density at time t = 0. We note at this point, that (4.35) is almost identical to the exact expression for the so called E INSTEIN A-coefficient for spontaneous emission, which will be discussed in Sect. 4.6.2. The specific properties of the emitting atom, here represented by |r (at) /2|, require of course a stringent quantum mechanical interpretation. For a prominent case, the 3p ↔ 3s transition in sodium (the so called Na D line at λ = 589 nm), we try an intelligent guess by setting r (at) 0.190 nm (the atomic radius, see Fig. 3.4). From (4.35) we obtain A¯ 3s3p 3.2 × 107 s−1 or τ = 1/A¯ ab 31 ns – which is at least on the same order of magnitude as the correct, experimentally determined value 16.2 ns. Of course, we do not expect exact results from this classical model.

184

4


absorption |b〉

Nb

|a〉

Na

nħω

(n -1)ħω

initial I

characteristics of spectra

stimulated emission nħω

final

initial

spontaneous emission

(n+1)ħω final

I

ωab

ω

ωab

ħω initial

final

I

ωab

ω

ω

Fig. 4.9 Absorption, induced and spontaneous emission schematically. In the top row the atomistic view point is indicated, below typical spectroscopic patterns are illustrated: the detected intensity as a function of angular frequency of the radiation

Fig. 4.10 Two level system and E INSTEIN coefficients; for deriving the principle of detailed balance

|b 〉

Nb =

Bab u(ωab)

= Aab

Na

4.2.5

= Bba u(ωab) |a 〉

The EINSTEIN A and B Coefficients

Figure 4.9 summarizes the above considerations and findings. In a real experiment one has to consider all three processes simultaneously for describing an atomic system in the presence of an electromagnetic field correctly. A special case is a system of atoms, molecules or condensed matter oscillators in thermodynamic equilibrium with its own radiation field. E INSTEIN has taken this thermodynamic equilibrium as a starting point for a very elegant and simple derivation of P LANCK’s radiation law Fig. 1.15. It is based on the so called principle of detailed balance: under stationary conditions each process must be in equilibrium with its inverse. One treats the problem as a representative two level system sketched in Fig. 4.10. The kinetics of this system is described by so called rate equations. The population of excited and ground state may change due to the three processes as follows: dNa dNb = −A¯ ab Nb − Nb B¯ ab u(ω . ˜ ba ) + Na B¯ ba u(ω ˜ ba ) = − dt dt

(4.36)

Stationarity means that the particle densities do not change: dNb,a ˜ ba ) + Na B¯ ba u(ω ˜ ba ) = 0 = −A¯ ab Nb − Nb B¯ ab u(ω dt A¯ ab Nb A¯ ab ⇒ = u(ω ˜ ba ). =1 ⇒ (Na B¯ ba − Nb B¯ ab )u(ω ˜ ba ) B¯ ba (Na /Nb ) − B¯ ab

4.2


185

On the other hand, according to (1.54) thermodynamic equilibrium implies Nb gb − W gb − ωba = e kB T = e kB T Na ga ga

(4.37)

with the degeneracies gb and ga of ground |a and excited states |b , respectively. Thus the spectral radiation density becomes u(ω ˜ ba ) =

A¯ ab A¯ ab = . ¯ ¯ ¯ Bba (Na /Nb ) − Bab Bba (ga /gb )eωba /(kB T ) − B¯ ab

By comparing this with P LANCK’s radiation law (1.81) u(ω) ˜ =

1 ω3 , 2 3 ω/(k BT ) − 1 π c e

the following relations between the A and B coefficients emerge: ω3 (spont) = 2 3 B¯ ab A¯ ab = R¯ ab π c

and gb B¯ ab = ga B¯ ba .

(4.38)

Note the ω3 factor – interestingly, it agrees with the classical prediction (4.35). We point out that the B coefficients used here refer to spectral radiation densities u(ω) ˜ per unit angular frequency.7 Section summary

• We have introduced three prototype methods for obtaining information from quantum systems by interaction with an electromagnetic field: emission, absorption and fluorescence spectroscopy. • Induced processes (absorption and induced emission) are introduced phenomenological. The rates for both processes are proportional to the spectral radiation intensity u(ω ˜ ba ) = I˜(ωba )/c at the transition frequency studied. • A classical interpretation of spontaneous emission encounters severe problems – which have led to B OHR’s second theorem and to the probabilistic interpretation of the emission process. Nevertheless, it provides an informative first guess on the spontaneous emission probability and useful information (4.32) for later use. • E INSTEIN’s has derived P LANCK’s radiation law from rate equations for a system of quantum oscillators in thermal equilibrium with the radiation field. This leads immediately to quantitative relations between the E INSTEIN coefficients for spontaneous and induced emission. • The readers should memorize the famous factor ω3 (or ν 3 ) between them: A¯ ab ∝ ω3 B¯ ab = ω3 (ga /gb )Bba – it will play an important role on several occasions. 7 In

the literature often u(ν) ˜ = 2π u(ω) ˜ per unit frequency is used. Then one has to replace B¯ ba → = B¯ ba /2π in all relevant equations.

(ν) B¯ ba

186

4.3

4


Time Dependent Perturbation Theory

After this phenomenological introduction into absorption and emission of photons and equipped with a working description of electromagnetic waves, we are now ready to develop a quantum mechanical understanding of dipole transitions. In order to quantitatively describe the temporal change of atoms under the influence of electromagnetic radiation, a semiclassical, perturbative approach will be applied at this point: the atom is treated fully quantum mechanically. The electromagnetic field, (r, t), which however, is considered as a classical time dependent perturbation U leaves the initial state of the atom essentially intact – apart from the small transition amplitudes to be derived. Spontaneous emission will be treated as a kind of ‘afterthought’ by making use of the A/B ratio (4.38) for the E INSTEIN coefficients just discussed. As already mentioned, a rigorous derivation of spontaneous emission requires QED and the quantization of the electromagnetic field. Since this concept is significantly less transparent than the semiclassical approach we defer its introduction to Chap. 2 in Vol. 2. All spectroscopic relations derived in the following will, however, remain valid – as long as the intensity of the radiation field is high enough so that its statistical properties are irrelevant, and on the other hand not too high so that the problem may be treated with sufficient accuracy by perturbation theory. These constraints will be released in Chaps. 2 and 10 in Vol. 2, respectively.

4.3.1

General Approach

We briefly recall from quantum mechanics how non-stationary states are treated by solving the time dependent S CHRÖDINGER equation (2.13) approximatively. We communicate the necessary, basic tools from time dependent perturbation theory for deriving probabilities for transitions to state |b from state |a . The reader who is sufficiently familiar with these tools may continue directly at Sect. 4.3.6. One starts with the known 0th order solution for an unperturbed problem: 0 ψj (r) = Wj ψj (r). H

(4.39)

0 = H 0 (t) has a ‘trivial’ The time dependent S CHRÖDINGER equation (2.13) for H time dependence (2.16) which we write here with ωj = Wj /

Wj (0) t = ψj (r) exp(−iωj t). Ψj (r, t) = ψj (r) exp −i ( Adding a time dependent perturbation U p , r, t) the Hamiltonian becomes (t) = H 0 + U ( H p , r, t).

(4.40)

Abbreviating Ψ (r, t) → |Ψ (t) and ψj (r) → |j we now make the ansatz ∞ Ψ (t) = cj (t)e−iωj t |j j =0

(4.41)

4.3


187

where cj (t) is the time dependent probability amplitude of state ψj (r) and the probability to find a certain final state |j at time t is given by 2 wj (t) = cj (t) . (4.42) If the summation (4.41) is carried out over sufficiently many states and the basis |j is complete, it represents a solution as close to exact as desired. Using this ansatz and the time dependent Hamiltonian (4.40), the time dependent S CHRÖDINGER equation (2.13) is written as ∂|Ψ (t) ( 0 + U p , r, t) Ψ (t) = i H ∂t ∂[cj (t)e−iωj t ]|j 0 + U ( . cj (t)e−iωj t H p , r, t) |j = i ∂t

j

(4.43)

j

0 |j according to (4.39) on the left hand side of this equation, and using Inserting H the rules of product differentiation on the right hand side leads to ( cj (t)e−iωj t Wj + U p , r, t) |j j

=i

cj (t)(−iωj )e

j

−iωj t

+ e

−iωj t

dcj (t) |j . dt

With i(−iωj ) = Wj the first terms in the sums on either side cancel. One multiplies from the left with b|eiωb t and applies b|j = δbj . With Wb = ωb and the time dependent perturbation matrix elements ( p , r, t)|j = ψi∗ (r)U p , r, t)ψj (r)d3 r (4.44) Uij (t) = i|U ( a system of linear differential equations is obtained for the probability amplitudes cb (t) – with the (angular) transition frequency ωbj = (Wb − Wj )/: i dcb (t) =− cj (t)Ubj (t)eiωbj t . dt

(4.45)

j

Note that this is a general and still exact formulation for atoms or molecules exposed to a time dependent interaction. In particular, it is not restricted to the dipole approximation nor even to monochromatic plane waves.

4.3.2

Perturbation Ansatz for Transition Amplitudes

In practice one can only sum over a finite number of terms. A perturbation approach ( 0 |, i.e. if the averaged perturbation is small comis possible if | U p , r, t) | | H pared to the averaged Hamiltonian of the unperturbed system (H atom, alkali atom

188

4


etc.). Then the initial conditions will change only slightly with time. To obtain the probability amplitude cb (t) for the final state b in 1st order time dependent perturbation theory one starts in 0th order by setting on the right hand side of (4.45) ca(0) (t) ≡ 1 (initial state a) and cj(0) (t) ≡ 0 for all j = a. From this one obtains for b = a in 1st order: dcb (t) i (4.46) = − Uba (t)eiωba t . dt If the interaction is switched on at time t = 0, integration over time leads to the time dependent transition amplitude in 1st order perturbation theory: cb (t) = −

i

t

Uba t eiωba t dt .

(4.47)

0

The full solution in 1st order is then given by Ψ (t) ≈ |a e−iωa t + cj (t)|j e−iωj t .

(4.48)

j =a

One may iterate the procedure by inserting (4.47) into (4.45) to obtain the 2nd order solution and so on. Often one is interested only in how the probability amplitude cb (t) evolves after many oscillation of the field, i.e. for t 1/ωba . In the limit t → ∞ the 1st order probability amplitude (4.47) i ∞ )ba (ωba ) cb (∞) = − Uba t ei(ωba )t dt = U (4.49) 0 is – apart from numerical prefactors – nothing but the F OURIER transform (I.2) of the perturbation potential at the frequency ωba of the transition. In conclusion atomic or molecular transitions can only be excited by an interaction potential )ba (ω) which are in resonance with the tranwhich contains F OURIER components U sition frequency ωba = ω.

4.3.3

Transitions in a Monochromatic Plane Wave

We now specialize to an electromagnetic monochromatic plane wave of frequency ω and intensity I . Guided by the mathematical form of the electromagnetic field (4.1), we introduce now – still quite general – a perturbation amplitude eE0 and a transition operator D= D( p , r):8 somewhat heuristic derivation of eE0 and D( p, r) for electric dipole transitions will be presented below in Sect. 4.3.4, while a more rigorous, general derivation and specialization is found in Appendix H.

8A

4.3


189

−iωt i ( Uba (t) = b|U p , r, t)|a = b| eE0 De − D† eiωt |a 2 i † = eE0 with Dba e−iωt − Dba eiωt Dba = b| D|a . 2 The transition amplitude (4.47) now becomes eE0 t † Dba ei(ωba −ω)t − Dba ei(ωba +ω)t dt cb (t) = 2 0

† D ei(ωba +ω)t eE0 Dba ei(ωba −ω)t − ba . = 2 i(ωba − ω) i(ωba + ω)

(4.50)

(4.51) (4.52)

We note here that both terms, exp(−iωt) and exp(+iωt), are relevant. And one sees already in (4.51) that significant contributions are expected only for stationary phases, i.e. for ωba ∓ ω = 0, respectively. Otherwise the contributions to the integral oscillate rapidly and disappear in the limit t → ∞. With the angular frequency of the radiation ω > 0 (per definition), and a referring to the initial, b to the final state, the first term in (4.52) describes absorption, the second induced emission: absorption b ← a: ωba > 0 emission

a ← b: ωba < 0

⇒

D exp(−iωt)

term relevant

⇒

D† exp(+iωt)

term relevant.

(4.53)

Obviously, both exponential functions – originating from writing the field of an electromagnetic wave (4.1) as a real quantity – are indispensable.

4.3.4

Dipole Approximation

We now apply this general formalism to the so called electric dipole transitions (E1 transitions) and present here a slightly heuristic introduction of the interaction potential. It leads to the same result as the more stringent (but also less evident) derivation given in Appendix H.1.6. In both cases the key simplification exploits the fact that the wavelengths of interest are typically large compared to atomic dimensions (λ a0 ). Hence, the electromagnetic field E(r, t) described by (4.1) may be expanded in powers of r/λ or k · r 1. To 1st order exp(±ik · r) reduces to 1, and only the temporal change of the electric field E(t) needs to be considered. The force on an electron from this oscillating field is −eE(t), and for the electron-nucleus pair the electric dipole moment is D = −er,

(4.54)

the interaction energy in semiclassical dipole approximation is i U (r, t) = −D · E(t) = er · E(t) = E0 er · ee−iωt − e∗ eiωt . 2

(4.55)

190

4


It depends on the electron position r and on time t. The field amplitude E0 is related to the radiation intensity I via (4.2), and e describes the polarization of the field. Hence, by comparing (4.55) with (4.50) one finds that the dipole transition operator for absorption is given by D = r · e = −D · e/e

and eE0 = e 2I /(cε0 ),

(4.56)

D† = r · e∗ . The relevant and since r = r † its adjoint, the operator for emission, is dipole transition matrix elements are

for absorption Dba = r ba · e,

† for emission Dab = D∗ba = r ab · e∗ , (4.57)

with r ab = r ∗ba . Note that E0 is measured here in V m−1 and is not a F OURIER component (per unit ω) of the field, so that U (r, t) is indeed an energy. Of course it is not mandatory to treat only strictly monochromatic waves, and we shall extend the description correspondingly in the next subsection. The electric dipole approximation is a very reasonable, far reaching approximation. For many spectroscopic applications treating it in 1st order perturbation theory is sufficient. However, only single photon transitions can be described in 1st order. Multi-photon processes require higher order perturbation theory and will be addressed in Sect. 5.3. They are very important in modern laser spectroscopy where often intense radiation fields are used. For two level systems, one may even find essentially exact solutions – still within the framework of the electric dipole approximation. This will be discussed in Chap. 10, Vol. 2. On the other hand, it is important to realize at this point, that the 1st order perturbation formalism summarized in (4.46)–(4.53) is not restricted to E1 transitions. It may also be applied to higher order interactions arising from expanding exp(±ik · r) in (4.1) beyond the first term – as long as only one photon is involved. In order to treat such processes accurately one has to use the quantum mechanically correct per( turbation operator U p , r, t) derived from the vector potential A(r, t) as elaborated in Appendix H.1.6. In Sect. 5.4 we shall treat the most important cases, magnetic dipole and electric quadrupole transitions (M1 and E2 transitions, respectively). They may become important if E1 transitions between two levels of interest are forbidden.

4.3.5

Absorption Probabilities

In the following we evaluate explicitly the probability amplitude for absorption, i.e. we focus for the moment only on the first term in (4.52), with ωba > 0 (Wa < Wb ). If the electromagnetic field is switched on at t = 0, we have cb (t) ≡ 0 for t < 0 while for t > 0 the transition amplitude evolves as cb (t) =

ei(ωba −ω)t − 1 eE0 Dba . 2 i(ωba − ω)

(4.58)

4.3


191

Fig. 4.11 g(ω) = sin2 [(ωba −ω)t/2]/(ωba −ω)2 as a function of ωba − ω in units of 2π/t . For t → ∞ it becomes proportional to the delta function

g( ) / (t/2 )

-3

-2

1

-1 0 1 ( ab - ) / (2 /t)

2

3

Hence, the probability to find state |b populated at time t > 0 is given by (abs) wba (t)

i(ω −ω)t ba 2 e2 E02 − 1 2 2 e = cb (t) = 2 |Dba | 2i(ωba − ω) =

2 ωba −ω e2 E02 D02 πt 2 sin ( 2 t) g(ω). | D | = | Dba |2 ba 2 2 2 2 (ωba − ω)

(4.59)

Figure 4.11 illustrates the characteristic frequency dependence g(ω). For large times t 1/(ωba − ω) it becomes arbitrarily narrow, and at the same time arbitrarily high ∞ (t/2π at ω = ωba ). This line profile is normalized such that −∞ g(ω)dω = 1, and is given by g(ω) =

ω −ω 2 sin2 ( ba2 t) t→∞ −→ δ(ωba − ω). πt (ωba − ω)2

(4.60)

Thus, g(ω) becomes a representation of the D IRAC delta function in the limit t → ∞. (abs) According to (4.59) wba (t) grows linearly with time. The approximation is of course only valid for not too large fields and not too large times so that the general assumption of perturbation time |cb (t)|2 1 is fulfilled for all b = a. At the same time we are interested in the stationary state, i.e. in times t 1/(ωba − ω).9 (abs) (abs) We now divide wba (t) by time t to obtain the transition rate Rba = wba (t)/t, i.e. the transition probability per unit time which in this approximation obviously becomes independent of time. This is indeed the quantity one determines in an absorption experiment. Finally, we have to account for the fact that in practice strictly monochromatic electromagnetic waves do not exist. Each radiation has a certain bandwidth ω = 2π ν. We introduce this frequency dependence by replacing the intensity I in a spectral range from ω to ω + dω by I˜(ω)dω, where I˜(ω) = cu(ω) ˜ is the spectral 9 We

shall see in Chap. 5 that these somewhat contradictory requirements can already be overcome by accounting for a finite lifetime of excited states, and a full treatment will be given in Chap. 10, Vol. 2 in the framework of the optical B LOCH equations.

192

4


intensity distribution. Thus, in (4.59) the square of the perturbation amplitude e2 E02 according to (4.56) has to be replaced by e2 E02 → e2

2 ˜ I (ω)dω. cε0

(4.61)

(abs)

The absorption rate dRba = wba (t)dω/t for radiation of angular frequency between ω and ω + dω is then given by dRba =

πe2 ˜ I˜(ω) Dba |2 g(ω)dω = 4π 2 α Dba |2 g(ω)dω, I (ω)| | 2 ε0 c

(4.62)

which is identical to the excitation probability of state |b from |a per unit time, with the fine structure constant α 1/137 according to (1.10). So far we have tacitly assumed that the absorption occurs between two isolated states |a and |b with sharp energies Wb and Wa , i.e. at a well defined sharp line of angular frequency ωba = (Wb −Wa )/. The total transition rate Rba is then obtained by integrating (4.62) over all frequencies: Rba =

∞

−∞

dRba =

∞

−∞

4π 2 α

I˜(ω) 4π 2 α Dba |2 g(ω)dω = Dba |2 I˜(ωba ). (4.63) | |

A brief dimensional analysis [Rba ] = −1 r 2 [I˜] = Enrg−1 T−1 L2 Enrg L−2 T−1 1/T−1 = T−1 shows that Rba is indeed a rate, i.e. gives the probability for an absorption process per atom per unit of time. In the present perturbative approach we have assumed that Rba tobs 1 during the relevant observation time tobs . We see now explicitly that the excitation probability is proportional to the ˜ = I˜(ω)/c at the tranF OURIER component of the spectral radiation density u(ω) sition frequency ωba . The integration (4.63) is of course based on the assumption that I˜(ω) is constant over the absorption line. For typical classical radiation sources this is trivially a correct assumption. In laser spectroscopy, however, this is not necessarily true – as we shall explain in Sect. 5.2.3 and more profoundly in Chap. 10, Vol. 2. For the integration in (4.63) we have assumed that the electromagnetic wave interacts for a sufficiently long time with the system, so that the limit 1/ωba t → ∞ in (4.60) is well approximated. With typical periods of the light field on the order of femtoseconds (fs) this approximation may even be applied to excitation by ns pulses often used in laser spectroscopy. But (4.62) is still valid for much shorter interaction times: according to (4.60) this does, however, imply a broadening of the line. We shall come back to this aspect in Sect. 6.1.7.

4.3


4.3.6

193

Absorption and Emission: A First Summary

To derive (4.63) for the absorption process we have evaluated the first exponent in (4.51). The second becomes only relevant if ωba < 0, i.e. for induced emission (Wa > Wb ). This leads in complete analogy to the rate Rab ∝ | Dab |2 = |r ab · e∗ |2 for induced emission, with identical prefactors. Because of hermicity r ab = r ∗ab holds and thus Rba = Rab . The rates for induced emission and absorption for a specific transition between the states |b ←→ |a are identical. In contrast to the general discussion in Sects. 4.2.3–4.2.5 we refer here to individual substates described by a set of quantum numbers e.g. γj m and not to energy levels which may consist of several such (degenerate) sublevels. The key quantity which characterizes an electric dipole transition is the dipole transition operator, according to (4.56) Dab = r ba · e, i.e. essentially the scalar product between dipole matrix element D ba = −er ba and polarization vector e with the dipole transition amplitude r ba = b|r|a = ψb∗ (r)rψa (r)d3 r = a|r|b ∗ = r ∗ab . (4.64) Note that r ba is a vector, specific for each system and each transition,10 and the transition probabilities also depend critically on the polarization e of the radiation. Before evaluating this in detail, we briefly summarize the results for electric dipole transitions obtained so far. We rewrite the probability (4.63) for inducing E1 transitions per unit time between the substates |a = |ja ma and |b = |jb mb with polarization e: Rba = =

4π 2 α ˜ I (ωba )|r ba · e|2 = Rab

(4.65)

4π 2 αc |r ba · e|2 u(ω ˜ ba ) = B(jb mb ; ja ma ; e)u(ω ˜ ba ).

In full analogy to (4.26) and (4.27) we have introduced the substate and polarization specific E INSTEIN B coefficients: B(jb mb ; ja ma ; e) =

4π 2 αc |r ba · e|2 = B(ja ma ; jb mb ; e).

(4.66)

Again, these B coefficients are defined in respect of a spectral intensity distribution I˜(ωba ) = cu(ω ˜ ba ) per unit angular frequency ω at the transition frequency ωba = |Wb − Wa |/. 10 More precisely, er ba is called dipole length matrix element. According to (H.25) the corresponding dipole velocity matrix element is

e b| p|a = iωba me e b|r|a . Both formulations lead to identical results if exact wave functions are used. For approximate solutions (i.e. quite generally) significant differences may occur.

194

4


At this point the wisdom of the semiclassical approximation ends. Spontaneous emission cannot be understood in this manner. However, we may apply some hand waving arguments based on the E INSTEIN relation (4.38) with A/B = ω3 /(π 2 c3 ). It was derived from P LANCK’s radiation law and detailed balance between two isolated levels, assuming a completely isotropic, unpolarized radiation field. In the present discussion, absorption and induced emission between the substates |γa ja ma and |γb jb mb are the result of a plane wave with well defined direction k and polarization e. In contrast, spontaneous radiation can in principle be emitted into 4π with two orthogonal polarization vectors. Thus, we divide (4.38) by 4π × 2, multiply it by the solid angle element, and insert (4.66) for B. Thus, we may glean for the probability of spontaneous emission with well defined polarization per unit time into a solid angle dΩ: (spont)

dRab

= B(ja ma ; jb mb ; e) 2 = C r ab · e∗ dΩ

ω3 dΩ = C|r ba · e|2 dΩ π 2 c3 8π 3 3 αωba e2 ωba with C = = . 2πc2 8π 2 ε0 c3

(4.67)

A discussion of the corresponding E INSTEIN A coefficients will be postponed until we have derived the angular distributions of the radiation patterns in Sect. 4.5. A clean derivation of (4.67) will be given in Chap. 2, Vol. 2. In summary, absorption and emission of electromagnetic radiation due to E1 transitions is characterized by (spont)

(abs) (ind) Rba = Rab ∝ dRab

(4.68)

† 2 2 Dba |2 = |r ba · e|2 = Dab = r ab · e∗ . ∝ |

According to (4.57) the matrix elements for transitions |ja ma ↔ |jb mb are Dba = † ∗ ∗ r ba · e and Dab = r ab · e = Dba for absorption and emission, respectively. They determine • whether the transition may occur at all (so called selection rules), • its dependence on polarization and angular distribution, • and its overall strength. In the following sections these aspects will be discussed in detail, in particular the key quantity r ba · e in (4.68) will be evaluated for different polarizations and geometries. As we shall see, this may be somewhat involved since in general, r ba and e are described most conveniently in different coordinate frames, e.g. in the atomic (at) and photon (ph) frame, respectively, as introduced in Sect. 4.1.3. However, we do not want to end this section without treating at least one particularly simple example. We choose excitation (and induced emission) by linearly polarized light and exploit the fact that r ba = (xba , yba , zba ) is a well defined vector which may be represented by its Cartesian components. Let θ (r) be the angle

4.3


195

between r ba and the polarization vector e of the absorbed electromagnetic radiation – not to be confused with the polar angle of e in respect of the z-axis or with the azimuthal angle θk of z(ph) introduced in Fig. 4.3. For a transition between well defined initial and final states |ja ma ←→ |jb mb one may now write (4.65) simply as Rba = Rab = 4π 2 α

I˜(ωba ) |r ba |2 cos2 θ (r)

(4.69)

2 + y 2 + z2 ). This expression may be somewhat misleading in as with |r ba |2 = (xba ba ba far as depending on the alignment of the polarization vector not all components of r ba contribute equally; e.g. for the simplest geometries where the linear polarization is chosen parallel to any of the axes (4.69) becomes

⎧ ⎨ |xba |2 ˜ ) I (ω ba 2 Rba = Rab = 4π α × |yba |2 ⎩ |zba |2

for e x for e y for e z.

(4.70)

In a typical absorption experiment with a given polarization vector, say e z, one sums over all final states b and averages over all initial states a. For an isotropically populated target one may show that the absorption rate then becomes independent of polarization (see e.g. Sect. 4.6.3 or Appendix H.2.1). Section summary

• The semiclassical approach describes radiation induced processes by a classical, time dependent electromagnetic field and treats the interacting atomic or molecular system quantum mechanically. • Time dependent perturbation theory is usually a good approximation for low enough radiation intensities. Time dependent transition amplitudes (4.47) are (t). computed from the matrix elements of the perturbation operator U • In the limit t → ∞, the transition amplitude (4.49) between initial (a) and final state (b) is essentially the F OURIER component of Uba (t) at the transition frequency ωba . • The dipole approximation describes so called E1 transitions and is usually an excellent approximation for a quantitative description of transition probabilities as long as the wavelength of the radiation is λ a0 . It neglects the change of the field with position over atomic distances. In the dipole length formulation the interaction operator is ∝ E0 D where E0 is the field amplitude and Dba = r ba · e, with e being the unit polarization vector of the radiation. • Both exponential terms of the real field amplitude (4.1) matter: the D exp(−iωt) term is responsible for absorption, the D† exp(iωt) term for induced emission. While the semiclassical treatment gives quantitatively correct results for absorption and induced emission, spontaneous decay requires, strictly speaking, the use of QED to be discussed in Chap. 2, Vol. 2.

196

4


4.4

Selection Rules for Dipole Transitions

4.4.1

Angular Momentum and Selection Rules

As we have learned in Sect. 4.1.4 the photon has a spin sph = 1. But only two projections ms ≡ q = ±1 in respect of the direction of propagation, |sph q = |1 +1 and |1 −1 are realized, describing LHC and RHC polarized light, respectively. One speaks of positive and negative helicity. The fact that the third component q = 0 is missing, reflects the transverse nature of free electromagnetic waves. According to relativistic quantum field theory this is a consequence of the photon being a massless particle. Quite down-to-earth, the angular momentum properties of the photon allow us to derive selection rules for E1 transitions without any computational efforts: we simply apply the classical conservation law for angular momenta. For the system atom + photon the total angular momentum and its projection onto a particular direction in space is conserved during absorption or emission of a photon. Thus, transitions may only be observed if the total angular momentum of the whole system J is conserved. Denoting the total angular momentum operators of the atom with Ja and J b in the lower and the upper state, respectively, the following rules in vectorial form hold: for absorption J = Ja + S ph ⇒ Jb for emission J = J b ⇒ J a + S ph .

(4.71)

J b with the atomic angular momentum quanFor general use we associate J a and tum numbers ja and jb , the corresponding projection quantum numbers being ma and mb . They may be integer or half integer as discussed already in Sect. 1.1.1 and 1.9.5. For the model of a spinless electron (which has been assumed so far) one identifies j → . In the following chapters, the general rules developed below will be extended to examples of increasing complexity. With a photon spin quantum number sph = 1 and conservation of the total angular momentum of the complete system, jb can only assume three values: ja + 1, ja and ja − 1. The vector diagram Fig. 4.12 illustrates this graphically. However, if ja = 0 the transition is only allowed if jb = 1 – and vice versa if finally jb = 0 initially only ja = 1 is possible. No transitions are allowed between states with ja = jb = 0. Finally, for ja = 1/2, a dipole transition may only lead to final states with jb = 1/2 or 3/2. One summarizes these selection rules in compact form as so Fig. 4.12 Triangular relation between angular momenta of initial and final atomic states J a and J b , respectively and the photon spin S ph

Sph Jb Ja

Ja

Sph Ja Jb

Sph Jb

Fig. 4.13 Absorption (top, Dba = r ba · e) and emission † Dab = r ab · e∗ ) of (bottom, circularly polarized light (left: LHC σ + ) and (right: RHC σ − ). The selection rules m = ±1 are summarized in (4.74). Here the atomic at and photon (ph) coordinate frames are identical, with the photon wave vector k z(at)

197 mb=ma +1

z (at)

z (at)

mb=ma - 1

ma k || z x (at)

ma y (at)

y (at)

k || z

σ +(LHC) light

out

|a 〉

x (at) in

in

k

|b 〉

absorption


σ -(RHC) light

mb

k

ma = m b - 1

out

mb ma=mb +1

y (at) x (at)

|b 〉 |a 〉

emission

4.4

y (at) x (at)

called triangular relation:

j = 0, ±1 with ja , jb ≥ 0, but 0 0 abbreviated: δ(ja jb 1) = 1,

(4.72)

while δ(ja jb 1) = 0 indicates that the triangular relation does not hold. Clearly, the sum of the projections of the angular momenta before and after the process have to be conserved as well. Hence, for dipole transitions the corresponding selection rule is q = m = mb − ma = 0, ±1.

(4.73)

Different q (photon spin projection quantum number) correspond to different polarizations of the light (and possibly also to different coordinate systems). In respect of the z(at) axis of the atom we have: type of radiation σ + (LHC) σ − (RHC) π (lin. pol.)

z(at) k z(at) k z(at) E(r, t)

q=

m 1 −1 0

absorption ja ma → jb mb mb = ma + 1 mb = ma − 1 mb = ma

emission jb mb → ja ma ma = mb − 1 ma = mb + 1 ma = mb

(4.74)

Schematically this is illustrated in Fig. 4.13 for absorption and emission of pure circularly polarized σ + (LHC) and σ − light (RHC), propagating in +z(at) direction. Figure 4.14 shows the situation for linearly polarized light. Note that due to the transverse nature of electromagnetic waves the selection rule

m = 0 refers to a coordinate system perpendicular to that in the two other cases: for linearly polarized light it is convenient to choose the photon frame (ph) in respect

198

4

Fig. 4.14 Emission and absorption of linearly polarized light (π transitions) with m = 0. The coordinate system is chosen such that the (linear) polarization vector e z(at)

Non-stationary Problems: Dipole Excitation with One Photon mb= ma

z (at)

ma

k

linearly polarized light

|a 〉 y (at)

φ

x (at)

|b 〉

e 0 || z (at)

mb

z (at)

ma= mb

absorption

|b 〉 |a 〉 y (at)

x (at)

φ

emission e 0 ||

z (at)

k

of the atomic coordinate frame (at) with the electric field vector E z(ph) z(at) and the light travelling in the x (at) y (at) plane, as illustrated in Fig. 4.14.11

4.4.2

Transition Amplitudes in the Helicity Basis

The selection rules just outlined are necessary but not sufficient conditions for E1 transitions. To obtain quantitative expressions for the transition probabilities summarized in (4.68) one has to evaluate | Dba |2 = |r ba · e|2 explicitly. To do so, a specific coordinate system has to be chosen. For the unit polarization vector e we have discussed suitable basis sets in Sect. 4.1.2. In principle, one may use Cartesian (xyz) or polar coordinates (rθ ϕ). Depending on the geometry of the experiment one or the other choice may be advantageous. For linear polarization we have shown by way of example that a Cartesian basis with e parallel to one of the axes leads directly to the particularly simple result (4.70). However, since the angular part of atomic wave functions is usually presented in polar coordinates, for a general description the corresponding helicity basis (introduced in Sect. 4.1.2) is more appropriate for expressing the polarization vector. We introduce spherical components rq of the position vector r. With x = r sin θ cos ϕ, y = r sin θ sin ϕ and Table B.1 one easily verifies

11 This

is possible as long as no other preferential direction is enforced by a particular experiment. More generally, linearly polarized light propagating in z(at) direction √ may be written with (4.15) as a linear combination of σ + and σ − light (cos β = sin β = 1/ 2). Absorption or emission of this light involves superpositions of m = ±1 transitions (sometimes called σ ) and leads to the generation of linear combinations of states – as we shall explain in more detail in Sect. 4.5.2 and Sect. 4.7.1.

4.4


199

√ 4π Y1±1 (θ, ϕ) = rC1±1 (θ, ϕ) r±1 = ∓(x ± iy)/ 2 = r 3 4π and r0 = z = r Y10 (θ, ϕ) = rC10 (θ, ϕ). 3

(4.75)

With (4.5)–(4.7) the position operator r (a real valued vector) becomes: ∗

r = r = r = xex + yey + zez = †

1

rq e∗q

q=−1

≡r

1

∗ C1q (θ, ϕ)eq = r

q=−1

1

=r

1

C1q (θ, ϕ)e∗q

(4.76)

q=−1

(−1)q C1−q (θ, ϕ)eq =

q=−1

1

(−1)q r−q eq = r † .

q=−1

We write the transition amplitudes between angular momentum substates |γb jb mb ← |γa ja ma for a specific polarization q – with mb |rq |ma as abbreviation if no misunderstanding is possible – γb jb mb |rq |γa ja ma = γb |r|γa jb mb |C1q |ja ma := mb |rq |ma

1 ja jb jb −mb jb C1 ja , 2jb + 1 = rba (−1) −mb q ma (4.77) where the radial transition matrix element is ∞ ∗ Rb (r)Ra (r)r 3 dr. γb |r|γa = rba = rab =

(4.78)

0

Explicit evaluation of these matrix elements is greatly facilitated by the W IGNER E CKART theorem (C.7)–(C.9), which we have used in the second line of (4.77). It allows us to factorize the overall transition probabilities, not only into radial and angular part, but also into a contribution from geometrical effects (dependence on projection quantum numbers m, q), and from total angular momenta.12 We may now evaluate the dipole transition matrix elements (4.57). For absorption of light with polarization e we find r ba · e = b|r|a · e = γb jb mb |r|γa ja ma · e =

1

mb |rq |ma e∗q · e

q=−1

= γb |r|γa

1

jb mb |C1q |ja ma e∗q · e.

q=−1

12 More

details for evaluating these matrix elements are made available in Appendix C.

(4.79)

200

4


Note that emission differs from absorption as described by (4.57). The polarization vector e∗ (creation of a photon) replaces e (annihilation of a photon): r †ab · e∗ = rab

1

∗ ja ma |C1q |jb mb eq · e∗ = (r ba · e)∗

q=−1

= rab

1

(−1)q ja ma |C1−q |jb mb eq · e∗ .

(4.80)

q=−1 † † However, since Dab = r ab · e∗ = (r ba · e)∗ = D∗ba , and since transition rates depend only on the absolute squares of the matrix elements, the probabilities for absorption and emission between well defined substates |γb jb mb ←→ |γa ja ma are the same if the photon polarization q is the same. We come back to this in Sect. 4.4.3. The magnitude of the radial matrix element rba determines the overall strength of a dipole allowed transition. Explicit analytical formulas are available only for atomic hydrogen and H like ions (see Appendix C.5). In general rba has to be evaluated numerically from the radial wave functions Ra (r) and Rb (r). The latter are usually obtained from numerical integration of the radial S CHRÖDINGER equation as detailed in Sect. 3.2. The angular part jb mb |C1q |ja ma of the transition amplitude (4.77) is of a more general nature. It depends only on the respective angular momentum quantum numbers of the system studied and determines the selection rules for E1 transitions. In Sect. 4.6 some general rules and expressions will be given for the transition probabilities (A and B coefficients). In the following we shall explicitly recover and quantify the selection rules (4.72) and (4.73), which were derived there from the concept angular momentum conservation.

4.4.3

Transition Matrix Elements and Selection Rules

We now discuss the simplest case, assuming that the light polarization is given by a basis vector in the atomic frame, e := eq . All transition rates (4.68) are now proportional to | Dba |2 = |r ba · eq |2 . With (4.79) this implies (abs)

Rba

(ind)

= Rab

(spont)

∝ dRab

2 2 ∝ γb |r|γa jb mb |C1q |ja ma .

(4.81)

The selection rules for any E1 transition are essentially contained in the angular part of the matrix elements. With (4.77) the angular part of the transition probability (4.81) is jb mb |C1q |ja ma 2 = (2jb + 1)

jb −mb

1 q

ja ma

2 jb C1 ja 2 .

(4.82)

4.4


201

Whether a dipole transition is possible at all between two levels with total angular momenta ja and jb depends on reduced matrix element jb C1 ja . It does not vanish if and only if the triangular relation (4.72) holds, as already gleaned from angular momentum conservation. Whether transitions can occur between specific sublevels depends on the 3j symbol in (4.82). It can only be finite if the sum of its projection quantum numbers in the second row vanishes (see Appendix B.2). Hence, (4.73) is confirmed as additional selection rule. The underlying conservation of total angular momentum has been illustrated in Fig. 4.13.

Note: We point out again the differences between absorption and emission as a

consequence of replacing e in (4.79) by e∗ to obtain (4.80) – with r † = r (real valued vector). E.g., as spelled out in (4.74), σ + light absorption increases the projection quantum number in the final state by q = 1: mb = ma + 1; in contrast, emission of σ + decreases of m in the final state: ma = mb − 1. But in both cases the se† Dab = r ab · e∗ = lection rule (4.73) holds per definition. This is a consequence of ∗ ∗ Dba : the transition probabilities depend only on the absolute squares of (r ba · e) = the matrix elements. A corresponding consideration holds for σ − light with q = −1.

For linearly polarized light, according to (4.74) and Fig. 4.14, the unit polarization vector becomes e = e0 so that with (4.81) and (4.82) the transition rate is simply proportional to

2 1 ja jb for E z(at) ⇒ mb = ma . (4.83) −mb 0 ma Thus, with such linearly polarized light one observes transitions with m = 0, often called π transitions. In contrast to (4.74) the E vector of the photon and not its wave vector k points into the z(at) direction. The propagation angle ϕ within the x (at) y (at) plane does not play a role in this case. For reference we also note that |r ba |2 =

1 mb |rq |ma 2 = |xba |2 + |yba |2 + |zba |2 .

(4.84)

q=−1

The explicit evaluation of the matrix elements in (4.81) may become quite complicated, depending on the angular momentum coupling schemes in which the states |ja ma and |jb mb are described. In later chapters we shall have to discuss this in more detail for various examples.

4.4.4

An Example for E1 Transitions: The H Atom

For the moment we just evaluate the most simple case for single electron transitions between uncoupled, pure orbital angular momentum states, |na a ma and |nb b mb , ignoring the electron spin. Inserting (C.28) into (4.81) one obtains

202

4

Wnℓ / eV 0 -0.85 -1.51

n ∞ 8 6 4 3

-3.40

2

-13.6

1


ℓ=0

ℓ=1

ℓ=2

ℓ=3

ℓ=4

5s 4s 3s

5p 4p 3p

5d 4d 3d

5f 4f

5g

2s

2p

1s

Fig. 4.15 G ROTRIAN diagram for the H atom illustrating the selection rule = ±1 in E1 transitions (shown are only emission lines, the electron spin is neglected)

2 2 Rba ∝ na |r|nb a ma |C1q | b mb 2 = na |r|nb (2 a + 1)(2 b + 1)

2 1 b a a × δma mb +q δ( a 1 b ) −ma q mb 0

1 0

b 0

2 .

(4.85)

The first 3j symbol confirms again the selection rules (4.72) and (4.73). The second 3j symbol, however, adds an additional selection rule: since its projection quantum numbers all vanish, a + 1 + b must be even, lest it disappears according to (B.49). Thus with the constraint δ( a 1 b ) = 1 a new selection rule holds: a = b ± 1.

(4.86)

The physical origin of this rule is parity conservation which is discussed in more detail in Appendix D. Briefly, the overall parity of the system has to be conserved, i.e. its symmetry in respect of inversion at the origin has to be the same before and after absorption or emission of the photon. The angular part of the wave functions for such pure states is completely described by spherical harmonics, their parity being (−1) . The photon which is represented by C1 has negative parity. The parity of the whole system before absorption of the photon is thus (−1) a +1 while it is (−1) b after the photon has been absorbed. Thus, for parity conservation a + 1 + b must indeed be even. Up to now, we have used pure states for the hydrogen and alkali atoms. As an example, Fig. 4.15 shows the term scheme of the H atom with some E1 transitions. Specifically, for an s −p transition three processes are allowed: |0ms ↔ |1mp with ms = 0√and mp = 0 or ±1. According to (C.44) all three angular matrix elements are 1/ 3, independent of the substates, so that for an = 0 ←→ = 1 transition: Rab ∝

| ns |r|np |2 . 3

(4.87)

4.5

Angular Dependence of Dipole Radiation

203

The electron spin, so fare neglected, will modify details of the term scheme Fig. 4.15. The corresponding selection rules will be discussed in Chap. 6. In Appendix C one finds compact formulas for the evaluation of (4.82) in the general case. In Chap. 7 we shall see, however, that transitions between states with angular momentum quantum numbers 0 and 1 are characteristic for closed shell multi-electron systems. They occur e.g. among so called singlet states where the spins of several electrons compensate each other. E1 transitions may then indeed be described by the orbital angular momenta of the optically active electron, e.g. p ←→ s. In the remainder of the present chapter we shall identify jb = 1 ←→ ja = 0 processes as 1 P ←→ 1 S transitions within a singlet system, indicated by the superscript, while 1 0 the subscript stands for j . To actually calculate the transition probabilities, in addition to the angular part of the matrix elements, which we have so far discussed in detail, one also has to compute the radial transition matrix elements. For the H atom Appendix C.5 summarizes the necessary formulas. A rather comprehensive collection of spontaneous transition probabilities for many atoms of the periodic system is found at K RAMIDA et al. (2013). Section summary

• Selection rules for light induced E1 transitions may directly be inferred from angular momentum conservation. • For the total angular moments the triangular relation δ(ja jb 1) = 1 must hold, the photon having a spin 1. • For the projection quantum numbers q = m = mb − ma = 0, ±1 reflects the photon polarization q = ±1 (with respect th z(at) k) and q = 0 (with respect th z(at) E). • Quantitatively, the transition probabilities are proportional to

2 (2jb rab

4.5

jb + 1) −mb

1 q

ja ma

2 jb C1 ja 2 .


We now turn to the angular characteristics for absorption and emission of electromagnetic radiation in dipole transitions. We make explicit use of the coordinate frames introduced in Sect. 4.1.3: the atomic frame (at) for describing the atom and the photon frame (ph) in which the light is best described. The latter is characterized by the polar (θk ) and azimuthal angle (ϕk ) of the wave vector k as shown in Fig. 4.3. Section 4.5.1 presents a first – somewhat heuristical – semiclassical approach by interpreting the classical oscillator as dipole transition matrix element (at) r ab . In Sect. 4.5.2 we shall show that this may be cast into an exact quantum mechanical formulation with essentially identical results, now including all necessary numerical prefactors.

204

4.5.1

4


Semiclassical Picture

To express the classical electron oscillator r(t) = r (at) exp(−iωba t) described in Sect. 4.2.4 in quantum mechanical terms we assume its amplitude r (at) to be propor(at) tional to the dipole transition matrix element r ab and reintroduce the time depen(at) dence of the stationary initial and final states (2.16). Multiplying r ab from the left with eiωa t and from the right with e−iωb t , and using the representation (4.79) in the helicity basis we obtain the quantum mechanical equivalent to a classical oscillator: ∗ (at) (at) r(t) ∝ eiωa t r ab e−iωb t = ma |r0 |mb e0 e−iωba t (4.88) (at) −iω t ∗ + ma |r−1 |mb e−1 e ba ∗ (at) + ma |r1 |mb e+1 e−iωba t . The three components directly reflect the three types of classical oscillators illustrated graphically in Fig. 4.3. They correspond to the three types of transitions which we have already summarized in (4.74): • For q = 0 (π light) the electron oscillates linearly in z(at) direction with an amplitude ma |r0 |mb . From the classical picture we expect zero radiation in ±z(at) direction and maximum intensity emitted into the x (at) y (at) plane with linear polarization z(at) . • The terms with q = +1 and q = −1 represent an electron on a circular orbit in the x (at) y (at) plane. The emitted σ light may propagate into the whole solid angle 4π . Maximum intensity and fully circularly polarized σ ± light is intuitively expected for propagation into +z(at) direction, while σ light detected in the x (at) y (at) plane will be linearly polarized ⊥ z(at) axis. More quantitatively, one may glean electric field amplitudes from (4.29) and (4.30) (at) by projecting r ab onto the x (ph) y (ph) plane. For π light emitted from a linear oscil(at)∗ lator with amplitude ma |rq |mb e0 one reads from Fig. 4.3 (ph)

Ey

(ph)

= 0 and Ex

∝ ma |r0 |mb sin θk

(4.89)

independent of ϕk . Hence, π light is always linearly polarized, with e x (ph) . The emitted intensity distribution (ph) 2 (ph) 2 2 I (θk ) ∝ Ey + Ex ∝ ma |r0 |mb sin2 θk (4.90) is of doughnut type as illustrated in Fig. 4.16. (at)∗ By the same arguments, the circular dipoles ma |r±1 |mb e±1 oscillate in the x (at) y (at) plane. With (4.7) and Fig. 4.3 the projections onto the x (ph) y (ph) plane are (ph)

Ey

∝i

ma |r±1 |mb √ 2

(ph)

and Ex

∝±

ma |r±1 |mb cos θk . √ 2

(4.91)

4.5


205

z (at)

k

θk= 90˚ θk< 90˚ E, π

φk x (at)

y (at) E, π

k

Fig. 4.16 Angular distribution of dipole radiation from a linear oscillator (π light) I (θk , ϕ) ∝ sin2 θk . For two different directions of the wave vector k the alignment of the polarization vector is indicated (red, full and dashed vector arrows, respectively)

k

z (at)

E, σ+ θk

x (at)

φk

y (at ) E, σ k

Fig. 4.17 Angular distribution of dipole radiation from a circular oscillator (σ light) I (θk , ϕ) ∝ (1 + cos2 θk )/2. The polarization vector is indicated for two different directions of observation: at 0 < θk < π/2 (full red ellipse with vector arrow), corresponding to elliptically polarized light, and for θk = π/2 (dashed red arrow) corresponding to linear polarization

Hence, the angular distribution for σ light emitted in q = ±1 transitions is 2 (1 + cos2 θk ) (ph) 2 (ph) 2 I (θk ) ∝ Ey , + Ex ∝ ma |r±1 |mb 2

(4.92)

again independent of ϕk , as illustrated in Fig. 4.17. In contrast to π light, the polarization of σ light changes with θk . The field amplitudes (4.91) imply that full circular polarization σ ± is only observed for θk = 0 and π , i.e. in ±z(at) direction. (ph) The cos θk factor reduces the Ex component, and in general elliptically polarized light is observed at angles in between these limits as indicated in Fig. 4.17. One (ph) special case is θk = π/2 (half maximum intensity) where Ex = 0 so that σ light (at) becomes linearly polarized with its e vector aligned in the x y (at) plane ⊥ z(at) . In Fig. 4.18 the standard geometry for observation of the different transition types is shown, as e.g. realized for observing the so called normal Z EEMAN effect (see Sect. 2.7). The three cases q = 0, ±1 may in principle be observed independently since the corresponding atomic lines are split in the magnetic field, with frequencies ν0 and ν±1 = ν0 ± ωL /2π for transitions with m = q = 0 and ±1, respectively.

206

4

(a)


(b)

+

-

(c)

-

z (at )

+

1P

1

-1

0

1

spectrum

+1 0

mb = -1 -1

0

1

Bex field 1S

0

(d)

mg = 0

spectrum magnet -1

0

1

Fig. 4.18 ‘Normal’ Z EEMAN effect as realized in a 1 S0 ↔ 1 P1 transition; (a) energy levels in an external magnetic field B ex z(at) , (b) experimental setup, (c) spectrum from σ ± light emitted at θk = 0, (d) spectrum from σ and π light emitted at θk = π/2

We emphasize again that the level scheme Fig. 4.18(a) should not be misinterpreted: the arrows indicating transitions with ν±1 and ν0 refer to different coordinate systems. The external magnetic field defines here the direction of z(at) B ex ; emission parallel to that axis (i.e. k z(at) B ex ) implies transitions with m = q = ±1 (σ ± light) only; in contrast, m = q = 0 transitions imply π light, linearly polarized with the electric field E B ex , emitted into any direction except in ±z(at) direction.

4.5.2

Angular Distributions from Quantum Mechanics

What has been deduced above from a more or less classical picture also follows from a rigorous quantum mechanical treatment. The angular and polarization characteristic of emission and absorption rates are obtained by inserting (4.80) and (4.79), respectively, into the general expression (4.68). Typically, the matrix element r ba and the polarization vector e are described in different coordinate frames, most conveniently in the atomic (at) and photon frame (ph) as illustrated in Fig. 4.3. The (spont) (ind) (abs) transition rates dRab , Rab , and Rba between two angular momentum basis states |jb mb ←→ |ja ma are then proportional to 2 1 (at) (ph) 2 (at) (ph)∗ 2 r · e = r · e = mb |rq |ma eq(at)∗ · e(ph) ba ab q=−1

2 1 (at)∗ (ph) = γb |r|γa jb mb |C1q |ja ma eq · e . q=−1

(4.93)

4.5


207

We emphasize that the unit vector e(ph) represents the polarization of the • detected radiation for a spontaneous transition between |jb mb → |ja ma , • inducing radiation for an induced transition between |jb mb → |ja ma , • absorbed radiation for absorption between |jb mb ← |ja ma . Note that – irrespective of the direction of the above arrows – (4.93) holds as written (see also the note on p. 201). To obtain practical relations for the angular dis(at) tributions, such as (4.90) and (4.92), we have to express eq and e(ph) in the same coordinate system. Which of these frames is more convenient depends on the experimental situation. For illustration we carry this out explicitly as a kind of exercise on coordinate rotation, using the rules explained in Appendix E.

Angular Distribution of Spontaneous Emission We first consider spontaneous emission. We are interested in the angular distribution and polarization of radiation emitted from a well defined transition in the (at) (at) system. Thus we have to express eq in the photon frame. We rotate the photon system (old) through the E ULER angles αβγ into the atomic system (new). No initial rotation around z(ph) is required as seen in Fig. 4.3, i.e. α = 0. Rotation around y (ph) through β = −θk makes z z(at) . Finally, we have to rotate the frame around z(at) through γ = −ϕk so that x is transferred into x (ph) . Thus, the atomic basis vectors (ph) are expressed in terms of basis vectors eq according to (E.14): iqϕk e(at) q =e

q

(ph)

dq1 q (−θk )eq .

(4.94)

For the three standard transitions m = q = 0, ±1 we insert the rotation matrices dq1 q (θk ) according to (E.12) into (4.94) and obtain explicit expression for the atomic basis in the photon system according to Fig. 4.3 with z(ph) k: • m = q = 0 (π component) (at)⊥

e0

sin θk (ph) (ph) (ph) = − √ e1 − e−1 = sin θk ex . 2

(4.95)

• m = q = 1 (σ + component) e1(at)⊥

=e

iϕk

θk (ph) cos2 e1 2

θk (ph) + sin2 e−1 2

.

• m = q = −1 (σ − component)

θk (ph) θk (ph) (at)⊥ e−1 = e−iϕk sin2 e1 + cos2 e−1 . 2 2

(4.96)

(4.97)

208

4


Note that in the photon frame only the q = ±1 basis vectors are considered, with polarization perpendicular to the photon wave vector k. Only these are of physical (ph) (at)∗ relevance in the scalar products (4.93) eq · e(ph) . The e0 component is dropped since it would indicate (un-physical) linear polarization in z(ph) direction. Hence, (at)⊥ eq is no longer a unit vector and its magnitude depends on θk . We may, however, express it in terms of an angle dependent normalization factor fq (θk , ϕk ) and the (ph) general unit vector (4.15) for elliptically polarized light in the photon frame eel : (ph)

= fq (θk , ϕk )eel , so that e(at)⊥ q (ph)∗ (at) r ab · e(ph) = ma |rq |mb fq (θk , ϕk )eel · e(ph) .

(4.98) (4.99)

q

Assuming we can detect one specific transition q = m only, and register all light (ph) emitted from this transition without further discrimination, i.e. e(ph) := eel , the emission rate becomes proportional to (at) (ph) 2 r · e = ma |rq |mb 2 fq (θk , ϕk )2 . (4.100) ab el Specifically, according to (4.95) for q = 0 (π component) the emitted light is (ph) (ph) always linearly polarized x (ph) so that eel := ex and f0 (θk , ϕk ) = sin θk . Thus, with (4.100) the rate (4.67) for spontaneous emission into a solid angle dΩ due to a transition |ja ma ← |jb mb is (spont)

dRπ

=

3 αωba mb |r0 |ma 2 sin2 θk dΩ. 2 2πc

(4.101)

One easily verifies that this probability has indeed the dimension T−1 . The intensity of radiation emitted per atom seen at a detector of an area A in a distance R is obtained by multiplying (4.101) with the photon energy ωba , dividing it by A, and identifying the solid angle with dΩ = A/R 2 : (spont)

Iπ

=

4 α ωba mb |r0 |ma 2 sin2 θk . 2 2 2πc R

(4.102)

This angular distribution corresponds exactly to the previously derived semiclassical pattern (4.90) shown in Fig. 4.16. The angular part jb mb |C1q |ja ma of the matrix element mb |r0 |ma according to (4.77) is typically on the order of 1, while the radial matrix element γb |r|γa may vary over a wide range depending on the size of the atomic orbitals and the overlap and symmetry of the wave functions. We now turn to the circular components of the emitted radiation pattern (q = ±1). The atomic basis vectors projected onto the photon frame are (ph) (at)⊥ e±1 = f±1 eel = f±1 (θk , ϕk ) e−iδ cos βe+1 − eiδ sin βe−1 . (4.103) (ph)

The parameters δ and β of the unit elliptic basis vector eel (4.15) are found from a comparison of (4.103) with (4.96) and (4.97). We see immediately that δ = −π/2

4.5


209

(the main axis of the polarization ellipse is y (ph) ); with cos2 β +sin2 β = 1 and with the trigonometric identity 2(cos4 (θk /2) + sin4 (θk /2)) = (1 + cos2 θk ) we derive f±1 (θk , ϕk ) = −ie±iϕk

(1 + cos2 θk )1/2 √ 2

(4.104)

as normalizing prefactor.13 The ellipticity angle β is given by: √ " 2 2 cos (θk /2) cos β = × sin2 (θk /2) (1 + cos2 θk )1/2

if q = 1 if q = −1.

(4.105)

For emission at θk = 0, i.e. along the z(at) axis we have cos β = 1 or = 0 and sin β = 0 or = 1 for q = +1 or −1, respectively. As discussed already, this implies emission of pure circularly polarized σ +√(LHC) or σ − (RHC) light. In contrast, at θk = π/2 one finds cos β = sin β = 1/ 2, corresponding to light emission with linear polarization vector in the x (at) y (at) plane (see Fig. 4.3). The angular distributions for m = q = +1 and m = q = −1 follow naturally by inserting f±1 (θk , ϕk ) from (4.104) into (4.100) and this in turn into (4.67). Again, (ph) the overall intensity of the emitted light is found by setting e(ph) := eel . We finally obtain for q = 1 transitions (spont)

dRσ ±

=

3 2 αωba mb |r±1 |ma 2 1 + cos θk dΩ, 2 2πc2

(4.106)

and the intensity per atom at a distance R is (spont)

Iσ ±

=

4 2 α ωba mb |r±1 |ma 2 1 + cos θk . 2 2πc2 R 2

(4.107)

This confirms again the angular characteristics (4.92) of the radiation found by the semiclassical model as illustrated Fig. 4.17. We may integrate the radiation characteristic (4.101) and (4.106) over the full solid angle and thus derive the overall spontaneous emission probability for one specific transition. Both, the sin2 θk distribution (4.103) for the π -components as well as the (1 + cos2 θk )/2 distribution (4.107) for the σ transitions lead to the same factor 8π/3: π 1 8π 2 3 = 1 + cos2 θk dΩ. (4.108) sin θk dΩ = 2π sin θk dθk = 3 2 4π 4π 0 Hence, from (4.101) as well as from (4.106) the integral spontaneous emission probability for a specific transition with ma + q = mb becomes: overall phase factor −i exp(±iϕk ) is here of no significance for measurable quantities, since these are proportional to the absolute squares of the matrix elements.

13 The

210

4 (spont)


= A(ja ma ; jb mb ; q) =

Rq

≡

3 4 αωba mb |rq |ma 2 2 3 c

3 4 αωba γb |r|γa 2 jb mb |C1q |ja ma 2 . 2 3 c

(4.109)

Equations (4.95)–(4.109) describe spontaneous emission processes between well defined states |ja ma ← |jb mb . This situation is realized experimentally if a specific initial state is prepared and only one specific final state is detected. Then the angular distributions (4.102) and (4.107) are observed. One method (by far not the only one) to prepare and detect such specific transitions is the Z EEMAN effect as illustrated in Fig. 4.18. We finally mention that a more general theory of radiation emitted from excited atoms will be presented in Sect. 9.4, Vol. 2 as an interesting example for applications of the density matrix.

Angular Dependence for Absorption and Induced Emission In contrast to spontaneous emission where the polarization and angular distribution was seen as the result of a specific transition, one may also be interested in excitation (or induced emission) by a specific polarization e(ph) . In which case we have to express e(ph) in the atomic frame. To do so, one rotates the atomic system (at) into the photon system (ph). One reads from Fig. 4.3 that first the atomic frame has to be rotated around its z(at) axis through α = ϕk and then around the new y axis through β = θk , while a final rotation around the new z(at) axis is not necessary so that γ = 0. Inserting these E ULER angles into (E.14) we obtain relations inverse to (4.94): (ph) (at) e−iqϕk dq1 q (θk )eq . (4.110) eq = q

Explicit, general expressions similar to (4.95)–(4.97) will be presented in Sect. 4.7. We shall derive there the general case in which absorption or induced emission from a well defined initial substate, say |ja ma , leads to a coherent superposition of final states |jb mb . Presently we specialize to the angular dependence of induced transition rates for absorption (and induced emission) between specific substates |jb mb ↔ |ja ma (with q = mb − ma ) by a well defined polarization q = 0 or ±1 of the light. In this case we have to evaluate Rba = 4π 2 α

2 I˜(ωba ) (ph) 2 mb |rq |ma eq(at)∗ · eq = Rab .

(4.111)

Without going into details of the derivation we just report a few examples: 1. Excitation of a m = q = +1 transition with LHC or RHC light (σ ± ) implies (ph) e(ph) := e±1 . We obtain Rba = 4π 2 α

" 4 2 I˜(ωba ) cos (θk /2) for σ + light mb |r+1 |ma × sin4 (θk /2) for σ − light.

4.5


211

2. Similarly, the excitation probability for a m = q = −1 transition induced with σ + or σ − light as a function of polar angle θk is given by Rba = 4π 2 α

" 4 2 I˜(ωba ) sin (θk /2) for σ + light mb |r−1 |ma × cos4 (θk /2) for σ − light.

3. And for excitation of a m = q = 0 transition with circularly polarized light the angular dependence is Rba = 4π 2 α

2 sin2 (θk ) I˜(ωba ) mb |r0 |ma × . 2

4. Finally, if the same transition can also be induced by linearly polarized light (ph) x (ph) , i.e. if e(ph) := ex . We find Rba = 4π 2 α

2 I˜(ωba ) ma |r0 |mb × sin2 (θk ),

which obviously is a more efficient process. The pattern is the same as shown in Fig. 4.16. We must emphasize, however, that it is nontrivial to study such specific transitions: the initial substate would have to be prepared and the final substate to be detected selectively. While in principle possible, standard spectroscopic experiments average over all initially populated substates and sum over all accessible final substates – as discussed in the following section. Section summary

• We have derived the angular distributions of radiation emitted for q = 0 (q = mb − ma ) transitions and for q = ±1 from a semiclassical picture as well as from quantum mechanical rates for E1 transitions. • We should memorize these distributions according to Figs. 4.16 and 4.17. • The polarization depends on the emission angle θk with respect to the z(at) axis. For q = 0 transitions, only linearly polarized light (π light) is emitted, with its polarization vector parallel to z(at) for θk = π/2. For q = ±1 transitions, at θk = 0 pure circularly polarized light (σ ± light) is emitted, while at θk = π/2 one finds linearly polarized σ light with polarization vector perpendicular to z(at) . For other emission angles the light is elliptically polarized. • The three components of radiation may be separated experimentally, e.g. by exploiting the Z EEMAN effect as illustrated for a 1 S0 ← 1 P1 transition in Fig. 4.18. • For induced transitions one may derive the angular dependence of rates for transitions induced between specific sublevels by a specific polarization. The general case is more complex – however, as we shall see in a moment, it may be greatly simplified for an initially isotropic population of all substates.

212

4


4.6

Strength of Dipole Transitions

4.6.1

Line Strength

According to (4.68) the key parameter for all absorption and emission processes is |r ba · e|2 . It contains the necessary information on polarization and angular characteristics for E1 transitions between specific substates. In contrast, to characterize the overall strength of a dipole transition between levels ja and jb with several degenerate ma and mb substates one introduces (see also Appendix H.2) a symmetrically defined line strength with the dimension L2 : |r ba |2 S(jb ja ) = mb ma

=

γb jb mb |rq |γa ja ma 2

mb ma q

2 = (2jb + 1) γb |r|γa jb C1 ja 2 2 = (2ja + 1) γb |r|γa ja C1 jb 2 ≡ S(ja jb ).

(4.112)

Here we have used (4.84) with (4.77), (4.82) and the 3j orthogonality relation (B.41). The radial matrix element γb |r|γa is given by (4.78). We recall that γb and γa represent all quantum numbers needed to characterize the radial wave function. We also note that ja and jb may refer to various kinds of the angular momentum quantum numbers, representative for the states under discussion – in the simplest case these may be orbital angular momenta or L, but could also be total angular momenta as we shall discuss in Chaps. 6 and 9. The line strength is a useful reference quantity. However, when describing specific cases one has to distinguish between types of radiation (polarized or unpolarized, unidirectional or averaged over all solid angles) as well as between types of transitions (selected substates, sums over final states, averages over initial states). This will be discussed below in terms of E INSTEIN coefficients. Figure 4.19 defines these for transitions between the substates |jb mb and |ja ma with polarizations characterized by q = mb − ma . jb

mb B( ja ma ; jb mb ;q) A( ja ma ; jb mb ;q)

B( jb mb ; ja ma ;q) ja

ma

Fig. 4.19 E INSTEIN coefficients for absorption, B(jb mb ; ja ma ; q), for induced emission, B(ja ma ; jb mb ; q), and for spontaneous emission, A(ja ma ; jb mb ; q), between substates of an upper and a lower level characterized by the angular momentum quantum numbers jb and ja , respectively

4.6


4.6.2

213

Spontaneous Transition Probabilities

The spontaneous transition probability (4.109) for one specific transition |ja ma ← |jb mb and polarization q is now rewritten with (4.82), (4.77) and (4.112) A(ja ma ; jb mb ; q)

2 3 2 4αωba jb 1 ja γb |r|γa (2jb + 1) = jb C1 ja 2 −mb q ma 3c2

2 3 3 4αωba 1 ja jb mb |rq |ma 2 = 4αωba = S(jb ja ). −mb q ma 3c2 3c2

(4.113)

It has the dimension T−1 . The 3j symbol accounts for all dependencies on angular momenta and polarization while all specific properties of the atom (or molecule) are included in S(jb ja ). Total decay probability and spontaneous lifetime of the upper level are determined by summing over all final states and polarizations: A¯ ab =

1 τ ja jb

=

A(ja ma ; jb mb ; q)

ma q

3 jb 4αωba = S(jb ja ) −mb 3c2 m q a

1 q

ja ma

2

3 4αωba S(jb ja ) . = 3c2 (2jb + 1)

(4.114)

Here we have again used the orthogonality relation (B.41) for the 3j symbols. It is important to note that the spontaneous lifetime τja jb is independent of the quantum number mb of the sublevel in the excited state! Explicitly, the prefactor in the above equations is given by 3 2 4αωba 32π 3 αc 7.235 × 108 m −19 3 s . = = 1.083 × 10 ω = ba 2 s 3c2 m 3λ3ba λ3ba

(4.115)

For practical use we insert the line strength (4.112) into (4.114) and introduce atomic units a0 and Eh : 2 4α 3 2 A¯ ab = 2 3 Wba a0 γb |r/a0 |γa jb C1 ja 2 3c 2 4 α 5 me c2 (Wba /Eh )3 γb |r/a0 |γa jb C1 ja 2 = (4.116) 3 2 2.1420 × 1010 (Wba /Eh )3 γb |r/a0 |γa jb C1 ja 2 . = s The reduced matrix element jb C1 ja is typically on the order of 1. For the simplest case, j = , it is given by (C.30), in the general case it may be evaluated as explicated in Appendix C.3. The radial matrix element expresses the genuine atomic

214

4


physics: it is determined by the overlap between initial and final state wave functions, weighted with the radius. The dependence of A on the nuclear charge Z is particular interesting. For hydrogen like ions we have Wba ∝ Z 2 and r ∝ 1/Z, so that the spontaneous lifetime τja jb = 1/A¯ ab ∝ Z −4 decreases with the 4th power of the nuclear charge, a result which is important for atomic X-ray physics. With (4.114) one may derive the line strength directly from the experimentally accessible natural lifetime of the excited state: S(jb ja ) =

3c2 (2jb + 1) 1 . 3 4α τ ja jb ωba

(4.117)

On the other hand, with (4.113) the individual transition probabilities between individual substates may be written as

2 1 1 ja jb A(ja ma ; jb mb ; q) = (2jb + 1) , (4.118) −mb q ma τ ja jb and (4.77), the transition amplitude between substates, may be written

1 ja jb mb |rq |ma = (−1)jb −mb S(jb ja ). −mb q ma

(4.119)

We also recall at this point, that – even though according to (4.114) the excited state lifetime is identical for all excited substates – the angular radiation characteristic and the polarization of the emitted light depends on mb and q as described by (4.101)– (4.109). Experimentally one often detects the fluorescence of an atom without polarization and final state analysis. The respective transition probability is obtained by summing over all polarizations q according to (4.101)–(4.106) and over all final states: α (spont) dRja ←jb mb (θ ) = dΩ ω3 S(jb ja ) (4.120) 2πc2 ba *+

2

2 , jb 1 + cos2 θk 1 ja 1 ja jb × + −mb 1 ma −mb −1 ma 2 ma

2 1 ja jb 2 sin θk . + −mb 0 ma Usually, this still needs to be averaged over all populated initial states mb and describes in general a non-isotropic angular distribution. Only if all initial excited states |jb mb are equally populated (∝ (2jb + 1)−1 ) the 3j orthogonality relation (B.41) may be applied and finally leads to an isotropic distribution: (spont)

dRtot

=

1 dΩ . τjb ja 4π

(4.121)

4.6


215

The general case of radiation from an anisotropically populated atom will be treated in Sect. 9.4, Vol. 2.

4.6.3

Induced Transitions

We now consider induced processes, i.e. absorption and stimulated emission. For transitions between specific substates induced by a polarization q (in respect of the atomic frame) we rewrite (4.66) using (4.79): B(jb mb ; ja ma ; q) =

2 2 4π 2 αc γb |r|γa jb mb |C1q |ja ma

(4.122)

= B(ja ma ; jb mb ; q). Again B refers to a spectral radiation intensity distribution I˜(ω) = cu(ω) ˜ per unit angular frequency. The ratio of spontaneous (4.113) to induced transition probability (4.122) for a specific set of two substates is given by 3 ωba 4 h A(ja ma ; jb mb ; q) = = . B(ja ma ; jb mb ; q) 3π 2 c3 3 λ3ba

(4.123)

Note that this ratio differs by a factor 1/3 from (4.38) where isotropic radiation and summation over all final states is assumed. In contrast, presently a transition between specific substates is considered. The spontaneous rate has been integrated over all emission angles to obtain A, while B has been determined for a specific polarization and direction of the inducing light. In full analogy to (4.113) we may write the coefficients for stimulated emission and absorption between specific sublevels as B(ja ma ; jb mb ; q) = B(jb mb ; ja ma ; q) =

4π 2 αc

1 jb −mb q

2 ja ma

3π 2 c3 (2jb + 1) 1 jb = 3 −m q τ b ωba jb ja

2 3λ3 1 ja jb (2jb + 1) = A¯ ab , −mb q ma 4h

ja ma

2 S(jb ja ) (4.124)

using the identities A¯ ab = τj−1 and λab = 2πc/ωba . b ja In a typical absorption or induced emission experiment (even with a polarized laser beam) one measures the absorption coefficient averaged over all initial and summed over all final states. Assuming isotropic initial population we obtain for the averaged E INSTEIN absorption coefficient (independent of q):

216

4

B¯ ba =

1 B(jb mb ; ja ma ; q) 2ja + 1 m m a

b

= =


1 2ja + 1

4π 2 αc

S(jb ja )

jb −mb

mb ma

1 q

ja ma

2 (4.125)

λ3ab 2jb + 1 4π 2 αc S(jb ja ) π 2 c3 2jb + 1 1 = = × A¯ ab . 3 2j + 1 τ 3 (2ja + 1) ωba 4h 2ja + 1 a ja jb

And equivalently the averaged induced emission coefficient is λ3ab π 2 c3 1 4π 2 αc S(jb ja ) = × A¯ ab . = B¯ ab = 3 τ 3 (2jb + 1) ωba 4h ja jb

(4.126)

The relations (4.125) and (4.126) just derived express a very important property of absorption and induced emission without which quantitative spectroscopy would be much more difficult (see also Appendix H.2): The averaged B¯ ba and B¯ ab coefficients do not depend on the polarization q. Finally, as a consequence of this independence of polarization of induced processes, we recover the standard E INSTEIN relation for the averaged coefficients of absorption, induced, and spontaneous emission by comparing (4.125), (4.126), and (4.114). With g = 2j + 1 we obtain: ga B¯ ba = gb B¯ ab

λ3ab π 2 c3 ¯ × A¯ ab . and B¯ ab = = A ab 3 4h ωba

(4.127)

The agreement with (4.38) eventually justifies the hand waving introduction of numerical factors for spontaneous emission in (4.67). However, we emphasize again, that this is not a derivation of the spontaneous emission rate which we postpone to Chap. 2 in Vol. 2. Section summary

• The line strength S(jb ja ) introduced in (4.112) characterizes the overall strength of an atomic transition between energy levels jb and ja . • Various relations among the E INSTEIN A and B coefficients have been derived which are useful for practical computations. • The spontaneous lifetime of substates in a given excited level jb is independent of the initial projection quantum number mb . • Absorption as well as stimulated emission in an isotropically populated target is independent of the polarization of the light used!

4.7

Superposition of States, Quantum Beats and Jumps

217

4.7


4.7.1

Coherent Population by Optical Transitions

So far we have discussed only transitions between basis states in the atomic frame. But of course, initial and final states may also be linear superpositions of these. Even if the initial state is a basis state, absorption as well as emission of light with polarization e(ph) may create a linear superposition of states |ψ(b) and |ψ(a) , respectively: ψ(b) ∝ mb |rq |ma eq(at)∗ · e(ph) |jb mb (4.128) mb q

ψ(a) ∝ ma |rq |mb ∗ e(at) · e(ph)∗ |ja ma . q

(4.129)

ma q

For illustration we begin by discussing the absorption of elliptically polarized (ph) light characterized by eel as defined by (4.15). We assume it to propagate parallel (ph) to the z(at) direction, i.e. k z(ph) = z(at) . The amplitudes of eel are e−iδ cos β and (ph) (ph) −eiδ sin β for e+1 and e−1 , respectively. With (4.128) absorption of this light by a pure basis state |ja ma leads to an coherent superposition of excited substates: ψ(b) ∝ ma + 1|r1 |ma ∗ eiδ cos β|jb ma + 1 (4.130) − ma − 1|r−1 |ma ∗ e−iδ sin β|jb ma − 1 . Often one cannot choose the atomic frame arbitrarily, since the experimental setup already defines a preferential (at) frame, e.g. if an external magnetic or electric field is applied, or if one wants to analyze a scattering experiment where a well defined collision plane exists. The atom has to be described then in an appropriately chosen coordinate system, which may differ from the (ph) frame which refers e.g. to the position of a photon detector or to a laser beam for exciting the target. We choose atomic (at) and photon system (ph) again according to Fig. 4.3 and shall (ph) into the atomic frame as have to transform either e(at) q into the photon frame or e discussed in Sect. 4.5.2. There, our discussion was focussed on the characteristic of emission and absorption for well defined transitions between basis states. Here we want to resume the discussion about transitions induced by absorption or stimulated emission of electromagnetic radiation with well defined polarization (characterized by e(ph) ). By using (4.110) we now expand e(ph) explicitly into an atomic basis. Referring to the angles defined in Fig. 4.3 we obtain: θk (at) sin θk (at) e − √ e0 + e−iϕk sin2 2 −1 2 θk (at) sin θk (at) (ph) e+1 = eiϕk sin2 e−1 + √ e0 + e−iϕk cos2 2 2 (ph)

e−1 = eiϕk cos2

θk (at) e 2 +1

(4.131)

θk (at) e . 2 +1

(4.132)

218

4


Fig. 4.20 Excitation of a coherent superposition of substates m b with b = 1 form a = 0 by linearly polarized light (1 P1 ←− 1 S1 transition). The electric field vector E is orthogonal to the z(at) axis

k, z ( ph)

z ( at )

θk E

y ( ph )

E - δ ( at ) x ( at )

φk k

θk

y ( at )

x (ph )

Elliptically polarized light propagating into an arbitrary direction is described (ph) (ph) by a superposition of e+1 and e−1 according to (4.15). The resulting expressions are somewhat √ complicated in the general case. For linearly polarized light (cos β = sin β = 1/ 2) which propagates into an arbitrary direction θk , ϕk with arbitrary polarization angle δ we find (ph)

eδ

=

i sin δ − cos θk cos δ iϕk (at) (at) e e−1 + sin θk cos δe0 √ 2 i sin δ − cos θk cos δ −iϕk (at) e − e+1 , √ 2

(4.133)

which for δ = 0 and −π/2, respectively, simplifies to cos θk (at) cos θk (at) (at) = −eiϕk √ e−1 + sin θk e0 + e−iϕk √ e+1 2 2 i (ph) (at) (at) ey = √ eiϕk e−1 + e−iϕk e+1 , respectively. 2 (ph)

ex

(4.134) (4.135)

A full description of the excited atomic state |ψ(b) after absorption of such photons is obtained by inserting (4.133) into (4.128). We exemplify this for two specific situations. Let us assume first, as sketched in Fig. 4.20, excitation from an initial state |ja ma with linearly polarized light, its electric field vector E being aligned perpendicular (δ = −π/2) to the z(at) axis (σ light). Thus, the polarization vector is (ph) ey as described by (4.135), independent of the propagation angle θk . Inserting (4.135) into (4.129) leads to the excited state: ψ(b) ∝ √i eiϕk ma − 1|r−1 |ma |jb ma − 1 2 + e−iϕk ma + 1|r1 |ma |jb ma + 1 .

(4.136)

4.7


219

Specifically, for a transition jb = b = 1 ←− ja = a = 0 (1 P1 ←− 1 S0 ). In this case ma = 0, so that from (4.119) with the√3j symbols (B.48) the matrix elements ma − 1|r−1 |ma = ma + 1|r1 |ma = rba / 3 are identical. Hence, the angular part of the excited state wave function simply becomes i ψ(Ω) = Ω ψ(b) ∝ mb |r1 |ma √ eiϕk Y1−1 (θ, ϕ) + e−iϕk Y11 (θ, ϕ) 2 rba 1 sin θ sin(ϕ − ϕk ). (4.137) = 2 π The angular dependence of the charge distribution may be expressed as |ψ(Ω)|2 , with θ and ϕ referring to the atomic coordinate frame. In Fig. 4.20 this dumbbell shaped charge cloud is imaged, with its main axis aligned along −δ (at) = ϕk − π/2 in the x (at) y (at) plane (parallel to the electric field vector E) and independent of the propagation angle θk as seen from (4.135). Another, trivial case has already been discussed in the context of Fig. 4.14 where the E vector points into the z(at) direction: in the present language we would de(ph) (at) scribe this by θk = π/2 with ex = −e0 according to (4.134). Thus we expect only m = 0 transitions. The excited state dumbbell now points into the z(at) direction. As a slightly more complex, but instructive example we discuss excitation by circularly polarized light, which propagates in the x (at) y (at) plane. Inserting (4.132) with θk = π/2 into (4.129), one obtains for the upper state, excited from the initial |ja ma state: ψ(b) ∝ 1 eiϕk mb |r−1 |ma |jb ma − 1 + e−iϕk mb |r1 |ma |jb ma + 1 2 1 + √ mb |r0 |ma |jb ma . (4.138) 2 If we specialize again to an b = 1 ←− a = 0 transition (1 P1 ←− 1 S0 ), we √ have mb |r1 |ma = mb |r−1 |ma = mb |r0 |ma = rba / 3, and obtain for the angular part of the wave function rba rba ψ(Ω) ∝ √ eiϕk Y1−1 (θ, ϕ) + e−iϕk Y11 (θ, ϕ) + √ Y10 (θ, ϕ) 2 3 6 1 rba −i sin θ sin(ϕ − ϕk ) + cos θ . (4.139) = 2 2π Apart from the phase factor i the first term of (4.138) and (4.139) [in square brackets] is obviously identical to the wave function generated by linearly polarized light as described by (4.136) and (4.137) and imaged in Fig. 4.20. The second term corresponds to a dumbbell in z(at) direction.

220

4


Fig. 4.21 Illustration of a 1 P1 ← 1 S0 transition, induced by circularly polarized light which propagates in y (at) direction

We remember that LHC polarized light corresponds to a pure m = 1 transition (ph) in respect of the (ph) with z(ph) k, represented by a toroidal Y11 state. However, in respect of the atomic frame it is described by a linear superposition of states as illustrated in Fig. 4.21 for the special case k y (at) . In the general case of elliptic polarization (not spelled out here mathematically) the charge distribution will be sort of a squeezed doughnut instead of the dumbbell shown in Fig. 4.20, lying in the x (ph) y (ph) plane, also aligned along −δ (at) .

4.7.2

Time Dependence of Optically Excited States – Quantum Beats

An interesting aspect of coherent, linear superposition of states is the time dependence which may be generated by optical excitation. While degenerate states within one atomic level just exhibit the trivial exp(iωba t) time dependence of wave functions derived from the S CHRÖDINGER equation, this may dramatically change if the atomic (or molecular) sublevels studied are no longer degenerate. Such situations arise e.g. in external magnetic fields where the energies of the |j m levels are Wj m = Wj 0 + mgj μB B = Wj 0 + mωj .

(4.140)

Wj 0 is here the energy of the level without magnetic field, gj the L ANDÉ g factor of the level, μB the B OHR magneton, B the magnitude of the external field, and ωj = gj ωL the L ARMOR frequency for J (1.162). If we neglect spontaneous decay the temporal evolution of the magnetic substates will be given by ψj m (t) = |j m e−i(ωj 0 +mωj )t , (4.141) with ωj 0 + mωj = Wj m /. The atomic frame is chosen such that z(at) B, and in all relevant equations of the preceding section the basis states |j m have to be replaced by (4.141). A common trivial time dependence exp(−iωj 0 t) may be pulled out of the expressions and contributes to an overall non-measurable phase. However, the

4.7


221

frequency shift mgj ωL = mgj 8.79 × 1010 s−1 T−1 B is different for each magnetic substate and leads to changing amplitudes. We first have a look at the simplest case, excitation by linearly polarized light described by (4.136) and (4.137). The E vector of the light is assumed to be perpendicular to the magnetic field. Inserting the states (4.141) into (4.136) instead of |j m and identifying ϕk = ϕk (0) with the phase angle at time t = 0 we obtain the angular part of the wave functions for a 1 P1 state: irba ψ(θ, ϕ, t) ∝ √ e−iω10 t 6 i(ϕ (0)+g ω t) −i(ϕk (0)−gj ωL t) j L Y × e k Y11 (θ, ϕ) 1−1 (θ, ϕ) + e rba 1 −iω10 t e = sin θ sin ϕ − ϕk (t) 2 π with ϕk (t) = ϕk (0) + gj ωL t.

(4.142) (4.143) (4.144)

The charge distribution |ψ(θ, ϕ, t)|2 contains now a time dependent azimuthal angle ϕk (t). In order to observe this dynamics, the time origin has to be known with sufficient accuracy – in any case better than 1/(2gj ωL ). For a magnetic field on the order of 1 T it should be around 0.1 ns. Assume then the atom is excited by a short light pulse. The dumbbell shown in Fig. 4.20 will then rotate around the z(at) axis (ϕk (t) decreases with time) with an angular frequency gj ωL which may be observed as quantum beat. In contrast, if the atom is excited by a continuous wave (CW) light beam the L ARMOR precession will tend to generate a disc like charge cloud. The other example discussed above, excitation by circularly polarized light propagating perpendicular to the magnetic field according to (4.138), may also lead to quantum beats. Since m = 0 states do not show an energy shift in the magnetic field the corresponding component Y10 (θ, ϕ) of the wave function remains unchanged. In contrast, the Y1±1 components describe again a dumbbell rotating in the x (at) y (at) plane as before. Thus, pulsed excitation leads to counter clockwise rotation around the z(at) axis of the toroidal charge distribution |+1 (ph) shown in Fig. 4.21. For CW excitation a completely isotropic distribution is expected. No such dynamics will emerge if the exciting circularly polarized light is propagating parallel to the B field (z(ph) B): the toroidal charge cloud will just rotate around its symmetry axis without observable change of shape or alignment. The same holds for linear polarization with e B. In summary, the dynamics in a linear superposition of non-degenerate states can only be observed if the substates are populated non-isotropically and if a suitably geometry is chosen. How can one actually measure such dynamics – or wave-packets – of the charge cloud distribution? The so called quantum beat spectroscopy exploits these oscillations and is used today in several varieties as very efficient method to measure small energy differences in excited states. Two (or more) neighbouring energy levels in the excited state are needed as indicated in Fig. 4.22. The initial ground state |a is excited coherently into the two states |b1 and |b2 by a short light pulse. For

222

4

Non-stationary Problems: Dipole Excitation with One Photon |b 2 |b1

ħ

21

|f |a

Fig. 4.22 Schematic of a typical four level scheme for the observation of quantum beats. Note that the excited levels must be populated coherently. Quantum beats are generated by interference of the emission lines from different excited states decaying into the same final state

the moment,14 the term “coherent” simply expresses a well defined phase relation among the excited states as a consequence from a sufficiently short duration of the exciting light pulse τ 1/ω21 (with ω21 being the excited state level splitting). Equivalently, this implies a F OURIER limited bandwidth, broad enough to simultaneously excite both levels b1 and b2 . At time t = 0 the excited state is then given by ψ(0) = c1 |b1 + c2 |b2 , (4.145) where cj are e.g. the coefficients used in (4.142). However, the following considerations are valid much more generally, for two or more states which are excited coherently and have slightly different energies Wj . Quantum beats are not restricted to j mj sublevels which are split in a magnetic field, as just discussed. Other examples are highly excited RYDBERG states in atoms, molecular rotational and vibrational states etc. The excited states evolve in time according to cj |bj exp(−iWj t/) with Wj / = ωj . Let us now assume that these states decay by spontaneous emission into a lower lying, final state |f . The lifetime of the excited states τj = 1/Aj is accounted for by a damping term exp(−Aj t/2) for the amplitude, i.e. by exp(−Aj t) = exp(−t/τj ) for the probability: ψ(t) = c1 |b1 e−(iω1 +A1 /2)t + c2 |b2 e−(iω2 +A2 /2)t .

(4.146)

The characteristic frequency for the dynamics clearly is ω21 = ω2 − ω1 since the trivial time dependence exp(−iω1 t) may be pulled out. The intensity of the emitted radiation according to (4.93) will in this case be 2 If b ∝ |r f b · ed |2 = f |r f b · ed ψ(t) ,

14 A

detailed discussion of coherence will be given in Chap. 2, Vol. 2.

(4.147)


(a)

(a’)

J=1 M

(b)

+1 Re (FT)

0 -1

intensity

223

intensity

4.7

0

2

t/ s

4

6

Re (FT)

(b’)

0

4

8 12 0 t/ s

1 2

3 4 / MHz

0

10

20 / MHz

30

40

Fig. 4.23 Quantum beats in molecules according to C ARTER and H UBER (2000): (a) fluorescence of the Z EEMAN levels for a R(0) line of the transition 17U in CS2 (magnetic field B 1.5 mT). The laser polarization was perpendicular to B and enables coherence between the M = ±1 sublevels; (b) fluorescence after coherent excitation of the 000 −101 rotational line in the vibrational band (3,0) 231+ 0 I(0,0) of the S1 ← S0 transition in acetone; quantum beats arise from a mixture between singlet and triplet states; (a ) and (b ) give the real parts of the F OURIER transformed signals (a) and (b), respectively

with ed being the polarization registered in the detector system. Obviously, the transition amplitudes for both processes |f ← |b1 and |f ← |b2 interfere. Inserting (4.146) into (4.147) one obtains Iab ∝ |r f b1 · ed |2 |c1 |2 e−A1 t + |r f b2 · ed |2 |c2 |2 e−A2 t (4.148) −(A +A )t/2 cos (ω21 + φ)t . + (r ab1 · ed )(r ab2 · ed )c1 c2 e 1 2 Without exploring the details of the polarization dependence we recall, however, that the coefficients cj also depend on expressions of the type r bj a · ee with ee being the polarization vector of the exciting radiation. The phase between excitation and emission amplitudes is summarized in φ. Clearly, a typical interference pattern is recognized in (4.148): the coherent superposition of radiation from both states leads to periodic increase and decrease of the observed intensity, the so called quantum beats, with a period 2π/ω21 and damped as exp[−(A1 + A2 )t/2]. If both states have the same lifetime 1/A and their excitation matrix elements are identical (4.148) simplifies to ' ( Iab ∝ 1 + cos (ω21 + φ)t e−At . (4.149) One may interpret the scheme Fig. 4.22 as a YOUNG’s double slit experiment: since a priori it is undetermined which path the light will take in the overall absorptionemission process, i.e via state |b1 or |b2 in analogy to the two slits in YOUNG’s experiment, and since on the other hand the final state is well defined, the amplitudes have to be added coherently and the typical interference pattern is generated.

224

4


Quantum beats have already been observed in the 60s of the past century. However, only after the introduction of tuneable, pulsed lasers it was possible to study and use them in full beauty. Figure 4.23 presents two nice examples from molecular physics: (a) a rather clear spectrum is shown for a rotational transition of free CS2 molecules in a magnetic field where exactly two Z EEMAN levels participate. The F OURIER transform of the temporal evolution in (a ) shows essentially one difference frequency. A somewhat more complicated molecule is acetone for which a quantum beat spectrum is reproduced in (b). It originates from contributions of several, close lying electronic states. The temporal spectrum thus looks somewhat intricate, and the F OURIER transform (b ) documents that more than four transitions participate in the beats. The fact that the signals are not fully modulated from zero to maximum is attributed to additional, incoherent contributions from other levels.

4.7.3

Quantum Jumps

We want to conclude this chapter with a short consideration of quantum jumps – in some sense the opposite of quantum beats. Studies of the type reported here are of fundamental relevance for our basic understanding of stationary states and transitions between them. We remember: stationary states in a quantum system (atom, molecule, etc.) have a discrete, well defined energy, notwithstanding that we may create quantum systems in coherent superpositions of states, as just exemplified. As we shall see in Chap. 10, Vol. 2, even superpositions of ground and excited states may be generated. However, if we try to determine experimentally in which of the states the system at a specific instant of time resides, we obtain a unique answer: either in the excited or in the ground state. Quantum mechanics only allows us to determine a probability amplitude for one or the other state. The system is never ‘in between’ – just as the famous “S CHRÖDINGER’s cat”, which is either dead or alive. An experiment which documents this in a particularly clear manner has been carried out by S AUTER et al. (1988) and is illustrated in Fig. 4.24. One single(!) Ba+ ion is trapped in an ion trap (see e.g. D EHMELT and PAUL 1989) and is excited continuously with a laser tuned into resonance between ground state |g and a short-

exc on

ng

itati

nce

esce fluor |g〉

τm >> τe

rki

|e〉

(b)

fluorescence / 103 pulses s-1

τe ≅ 10 -8 s

pa

(a)

|m〉

6 3 0 0

100

200

300

400

time / s

Fig. 4.24 Quantum jumps as observed by S AUTER et al. (1988). (a) Part of the term scheme for Ba+ and the transitions involved. (b) One observes the fluorescence of the state |e as a function of time. Grey shaded are times where the fluorescence signal vanishes while the ion is ‘parked’ in state |m


225

lived excited state |e . One observes the fluorescence signal emitted from state |e . At the same time one also induces (with a much smaller probability) a transition to the metastable state |m from where the ion can decay only with a very low probability back into the ground state (the average lifetime |m is several seconds). As documented in Fig. 4.24(b) the almost continuous fluorescence signal emitted from |e is interrupted in irregular intervals (grey shaded) – a clear indication that the ion is not available for the excitation-fluorescence cycle. Remember, there is always one but only one ion in the trap! We thus have to conclude that during the dark periods the ion is ‘parked’ in the excited, metastable state |m . Obviously, most of the time one observes a fluorescence signal, i.e. the ion spends most of its time in the ground state |g from where it can easily be excited for 10−8 s into the state |e and thus be monitored. The intermissions by excitation into |m are rather infrequent (intervals of up to more than 100 s) and have different duration times. A statistical analysis of these intermissions (not shown here) simply reproduces the exponential lifetime distribution of the state |m . This experiment thus documents very explicitly that the ion exists always in a very well defined state: except for very short times (on average for some 10−8 s) in the fluorescing state |e the ion is found either in the ground state |g and may be detected by excitation-fluorescence cycles, or it is found in the (dark) meta stable state |m and cannot be detected by fluorescence. There is nothing in between and a transition from one into the other state occurs (arbitrarily fast) by a quantum jump. It is only the probability for such a quantum jump which is described by the exponential decay function for the state |m . Section summary

• Absorption and induced emission with fully polarized light (linear, circular or elliptic polarization) leads in general to a coherent superposition of basis states. We have derived closed expressions for these states as a function of polarization. • If at least two states b1 and b2 with slightly different energies ωb1 and ωb2 are exited coherently and with an anisotropic overall population, one may observe quantum beats if they radiate to the same final states. Coherent excitation in this context implies short pulses with a duration t ≤ 1/ ωb2 b1 . • Single ion storage allows one to observe quantum jumps in an atomic system which is excited with a certain, low probability: the atom is found either in the excited or in the ground state. The observed change of population occurs suddenly and in statistically determined time intervals.

Acronyms and Terminology 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.

226

4


E2: ‘Electric quadrupole’, transitions induced by the interaction of a quadrupolar charge distribution with the electromagnetic radiation field. 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). LHC: ‘Left hand cicularly’, polarized light, also σ + light. LIF: ‘Laser induced fluorescence’, radiation emitted from a quantum system after excitation by laser radiation (see Sect. 5.5.1, Vol. 2). M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. OMA: ‘Optical multichannel analyzer’, spectrometer which allows simultaneous registration of a whole spectrum. QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. RHC: ‘Right hand cicularly’, polarized light, also σ − light. 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 electromagentic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References B ETH , R. A.: 1936. ‘Mechanical detection and measurement of the angular momentum of light’. Phys. Rev., 50, 115–125. C ARTER , R. T. and J. R. H UBER: 2000. ‘Quantum beat spectroscopy in chemistry’. Chem. Soc. Rev., 29, 305–314. D EHMELT , H. G. and W. PAUL: 1989. ‘The N OBEL prize in physics: for the development of the ion trap technique’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1989/. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. S AUTER , T., H. G ILHAUS, I. S IEMERS, R. B LATT, W. N EUHAUSER and P. E T OSCHEK: 1988. ‘On the photo-dynamics of single ions in a trap’. Z. Phys. D, 10, 153–163.

5

Linewidths, Photoionization, and More

What we have learned in the preceding chapter still requires some consolidation, quantification and extension. Linewidth, dispersion, oscillator strength and cross sections have to be defined and understood. Multi-photon processes, M1 and E2 transitions will be introduced. Finally, we shall address in some detail photoionization processes, i.e. various types of photoinduced transitions from discrete, bound states into the continuum of unbound states.

Overview

Section 5.1 introduces realistic finite linewidths into the description of optical transitions. It should be easy to read and is of central importance for the entire spectroscopy. In Sect. 5.2 – closely related to this topic – cross sections for excitation involving lines of finite width are discussed. The concept of optical oscillator strength is introduced (for which Appendix H.2 provides additional background). Section 5.3 offers a brief introduction into multi-photon processes without which much of modern spectroscopy would not be possible. Likewise, E2 and in particular M1 processes are important in many areas of spectroscopy; the reader may, nevertheless, consider the somewhat mathematical Sect. 5.4 as being mainly for later reference – without loosing the present context. This holds to some extend also for Sect. 5.5 where photoionization is treated which plays a key role in many areas of modern physics. We shall make use of it later, e.g. in Sect. 7.6.2 and Chap. 10.

5.1

Line Broadening

5.1.1

Natural Linewidth

Even if we are still missing some key tools for treating spontaneous emission it is important to develop a pragmatic way to work with it correctly in the context of spectroscopy. One essential consequence is the finite lifetime τ = t of “stationary” states and the finite width Γ = W of atomic levels, both connected through the uncertainty relation (1.125). In Sect. 4.7.2 we have already introduced in passing a © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7_5

227

228

5


Fig. 5.1 Broadening of an excited state by spontaneous decay – with L ORENTZ distribution

Γ

|b〉

Wba |a〉

somewhat heuristical method to account quantum mechanically for finite lifetimes of excited states. We now want to have a closer look at this approach – without claiming formal mathematical stringency. One describes the temporal evolution of atomic states by time dependent probability amplitudes cj (t) as previously done in the perturbation ansatz (4.41). Now we account, however, for the finite lifetime τ = 1/A of excited states |j which has been derived from 1st order perturbation theory – more precisely: from the E INSTEIN relation (4.38) between A and B coefficients. An atom which is prepared in this excited state at time t = 0 will decay with time t according to the probability distribution 2 wj (t) = cj (t) = e−Aj t = e−t/τj . In the spirit of perturbation theory the former 0th order approximation cj (t) = 1 now becomes cj (t) = exp(−t/2τj ). In the ansatz (4.41) we thus replace ψj (r)e−iωj t

by

ψj (r)e−t/2τj e−iωj t = ψj (r)e−i(ωj −i/2τj )t .

(5.1)

One may read this equation as if excited states were now characterized by a complex eigenfrequency or eigenenergy, substituting Aj i = ωj − i or 2τ 2 Γj Wj → Wj − i with Γj = Aj = . 2 τj ωj → ωj −

(5.2a) (5.2b)

Schematically this is illustrated in Fig. 5.1. With (5.2a) the system of ODEs (4.45) for the time dependent probability amplitudes becomes i dcb (t) =− cj (t)Ubj (t)ei(ωbj t−iAj t/2) , dt j

(5.3)

5.1

Line Broadening

229

and instead of (4.58) the final result in 1st order perturbation theory is cb (t) =

ei((ωba −ω)t−iAt/2) − 1 eE0 Dba 2 i(ωba − ω − iA/2)

(5.4)

Dba in dipole approxi(we drop here the index of Aj ) with E0 given by (4.56) and mation by (4.57). For times t 1/A (steady state) the probability amplitude for |b becomes independent of time, we call it cr :

cr = cb (t → ∞) ∝

i . ωba − ω − iA/2

(5.5)

cr is a typical resonance amplitude, well known from classical physics where it describes a “forced oscillator”. It depends on the detuning ωba − ω and on the damping constant A.

LORENTZ Profile Somewhat more precisely, the decay constant A used here has to be identified with A¯ ab given in (4.114)–(4.117) – as long as only transitions between two isolated levels are considered. From (5.5) follows the probability to find an atom in the excited state if CW light of angular frequency ω is absorbed: A2 /4 cb (ω)2 ∝ gL (ω) = 2 × . πA (ω − ωba )2 + A2 /4

(5.6)

This is the well known L ORENTZ profile (L ORENTZ distribution), describing an absorption line with a natural linewidth. The distribution is given here per unit angular frequency, i.e. it has the dimension T. Its FWHM is ωnat = A ≡ A¯ ab . In respect of energy the L ORENTZ profile reads gL (W ) =

(Γ /2)2 2 Γ /2π × = . 2 2 πΓ (W − Wba ) + (Γ /2) (W − Wba )2 + (Γ /2)2

(5.7)

ωnat is related to the frequency bandwidth νnat , the energy width Γ of the state, its natural lifetime τnat and its spontaneous decay rate A:

ωnat = 2π νnat = A =

1 τnat

=

Γ .

Both profiles, gL (ω) and gL (W ), are normalized to unity: ∞ ∞ gL (ω)dω = gL (W )dW = 1. −∞

−∞

(5.8)

(5.9)

Their maxima at ω = ωba are 2/π ωnat and 2/πΓ , respectively. Frequency integrated excitation and absorption rates have been treated in the preceding chapter. With (4.62) and I˜(ω) (intensity distribution per unit angular fre-

230

5


quency, dimension Enrg T−1 L−2 T = MT2 ) the absorption rate in a frequency interval ω to ω + dω may be written dRba = BgL (ω)

I˜(ω) dω. c

(5.10)

B is the E INSTEIN coefficient according to (4.124) or (4.126), depending on experimental conditions.1 The rate for spontaneous emission of a photon with angular frequency ω is given by (spont)

dRab

= AgL (ω)dω,

with the spontaneous decay rate A ≡ A¯ ab , equal to the inverse lifetime of any of the substates of level b in respect of decay into all substates of level a. We may now introduce a frequency dependent absorption coefficient by multiplying B¯ ba according to (4.126) with the normalized line shape (5.6): A¯ 2ab /4 gb λ3ab B˜ ba (ω) = B¯ ba gL (ω) = . ga 2πh (ω − ωba )2 + A¯ 2ab /4

(5.12)

B¯ ba is the averaged absorption probability from the lower levels a to the upper levels b for a given polarization.2 The attentive reader may find the above ‘derivation’ of the frequency dependent absorption coefficient somewhat heuristic. We shall come back to this in Sect. 10.5, Vol. 2 where a more systematic deduction will be presented – which turns out to lead to identical results as the one above.

Population Density and Saturation Broadening It is instructive at this point to learn something about the population density in the excited state from (5.12). We recall the rate equation (4.36) and apply it for stationary conditions, now at nearly resonant excitation by quasi-monochromatic CW radiation. We consider an isolated ensemble of atoms (no thermalization) in which only the ground state (density Na ) and the excited state b (density Nb ) are populated. To keep things simple we assume a pure two level system,3 which may easily be realized, e.g. if jb = ja + 1 by mb = jb , ma = ja and q = 1. In this case we insert (5.12) instead of Bba into (4.36), we set gb /ga = 3, Bab (ω) = Bba (ω) and the 1 If

instead of individual transitions between substates, averaged probabilities are studied, we have to replace in (4.68) the squared transition matrix element 1 1 | Dab |2 = |r ab · e|2 by | Dab |2 = |r ab · e|2 , (5.11) ga m m ga m m a

b

a

b

2 If one is interested in the absorption profile for one specific transition between substates jb mb ←→ ja ma , one would have to replace in (5.12) the prefactor (gb /ga ) → 3 and the linewidth A → A(jb mb ja ma ; q) according to (4.123). 3 If several excited or lower state sublevels are involved one has, in addition, to account for optical pumping (see Appendix D in Vol. 2).

5.1

Line Broadening

231

decay constant A = Aab for this particular transition (see footnote 2). We replace the spectral radiation density u(ω) ˜ → I /c – with I0 being the intensity of the (very narrow band) laser radiation. Then we obtain from (4.36) Nb /Na =

B˜ ba (ω)I /c , Aab + B˜ ab (ω)I /c

from which we extract the relative population of the excited state Nb (ΩR /2)2 = Nb + Na (Aab /2)2 + ΩR2 /2 + (ω − ωba )2

(5.13)

with the abbreviation ΩR2 = (3λ3 Aab )I /(2πhc). At very low intensity (ΩR Aab ) the L ORENTZ distribution is recovered with a FWHM= Aab . The maximum excitation probability is proportional to the laser intensity I . At high intensity (ΩR > Aab ) the distri line profile is still a L ORENTZ √ 2 2 bution, however, the FWHM is now Ωs = Aab + 2ΩR and increases ∝ I – we speak of power broadening or saturation broadening: even for extremely high intensity the maximum population (stationary limit) in the excited state at resonance ω = ωba is always Nb /(Nb + Na ) < 1/2, i.e. the transition becomes saturated. By excitation with CW radiation one can never excite more than 50 % of the atoms! We emphasize again, that the above ‘derivation’ is to some extent guess work: perturbation theory is strictly valid only for population densities Nb /(Nb + Na ) 1/2. However, as we shall see in Sect. 10.5.2, Vol. 2, the non-perturbative treatment confirms (5.13).

Homogeneous Line Broadening Closely related to the above, we introduce here the concept of homogeneous line broadening – as opposed to inhomogeneous broadening which will be discussed in Sect. 5.1.4. If the absorption (or emission) profile as a function of frequency is identical for each individual quantum system (atom), the broadening is called homogeneous. In contrast, if each individual quantum absorber/emitter has a different frequency profile, the broadening is called inhomogeneous. The L ORENTZ profile (natural line profile) describes the prototype of homogeneous line broadening: absorption of a photon – irrespective of its frequency – always excites the upper state as a whole. All atoms of a kind (say H atoms) are identical in this respect and cannot be distinguished by their absorption frequency. Conversely, if excited to a specific state they all emit the same spectrum with a probability described by gL (ω). The concept of homogeneous vs. inhomogeneous line broadening plays an important role in many spectroscopic problems and methods – and is of particular relevance for understanding amplification in laser systems (see Sect. 1.1 in Vol. 2). Some Numbers To develop some feeling for the orders of magnitude of natural linewidths we communicate two typical values:

232

5


• For the LYMAN alpha line (1s ← 2p) in hydrogen like atoms we use (4.117) with (C.56) and Table C.2. With these values4 we obtain A(1s2p) = 6.2658 × 108 Z 4 s−1 . Specifically, for the 2p state of the H atom (Hα line at λ = 121.57 nm) the lifetime becomes τnat = 1.596 ns. From this follows a linewidth

νnat 99 MHz. • For other atoms one has to compute the radial wave functions and from these the radial matrix elements numerically, as described by way of example for alkali atoms in Chap. 3. One of the strongest atomic transitions is the Na D line (3s ← 3p) at λ = 589 nm, with a lifetime τ ∼ 16.2 ns, ωnat = 1/τnat = 6.15 × 107 s−1 or νnat 9.8 MHz. These linewidths have to be compared to the transition frequencies νba = c/λ, being 24.66 and 5.089 × 1014 Hz for H and Na, respectively. Natural linewidths are thus extremely small, with νnat /νba = λ/λ 4 × 10−8 and 2 × 10−8 , respectively. For the Na D line the bandwidth in wavenumbers is νnat /c = 0.00033 cm−1 and in wavelengths units λ = −c νnat /ν 2 ∼ 1.06794 × 10−3 nm. Hence, extremely good spectral resolution is required to record such line profiles experimentally. We note here that the natural linewidth is a lower boundary for the width of any spectral line. It is directly related to the lifetime of the excited states by the H EISEN BERG uncertainty relation. A variety of influences and experimental conditions may lead to further broadening of spectral lines – as we shall see later.

5.1.2

Dispersion

For later reference we note that the complex resonance amplitude (5.5) with resonance energy Wba and linewidth Γ = /τ (FWHM) may be written: cr (W ) = |cr | exp(iφ) with

Γ 2 /4 (Wba − W )2 + Γ 2 /4

Γ /2 and phase φ = arctan + (π for W > Wab ). Wba − W

magnitude |cr | =

(5.14a) (5.14b)

Here we have normalized |cr (W )| at resonance to |cr (Wba )| = 1. Alternatively, we may express cb (W ) by its real and imaginary parts:

4 In

Re(cr ) = |cr | cos φ =

(Wba − W ) Γ /2 (Wba − W )2 + Γ 2 /4

(5.15a)

Im(cr ) = |cr | sin φ =

Γ 2 /4 . (Wba − W )2 + Γ 2 /4

(5.15b)

the particular case of s ←→ p transitions A is even independent of the electron spin.

5.1

Line Broadening

233

1

(a)

1.0

(b) phase

dispersion

absorption

-8

-4

0

Г

0.5

Г

0.5

absorption

4

-8

-4

0

8

- 0.5

8 (W-Wba) / Γ

Fig. 5.2 Alternative representations of the resonance amplitude (5.14a)–(5.15b) as a function of photon energy W = ω. (a) Magnitude and phase, (b) imaginary part (absorption) and real part (dispersion)

We note that the imaginary part reproduces (apart from the normalization) the L ORENTZ distribution (5.7) and thus corresponds to the absorption probability, while the real part describes a dispersion type line shape. Both profiles are illustrated in Fig. 5.2. We shall come to these again in Sect. 7.6.2 where so called FANO resonances are treated, as well as in Sect. 8.4.3 in the context of dispersion of the refractive index as a function of wavelength.

5.1.3

Collisional Line Broadening

At higher pressures atoms and molecules in the gas phase collide frequently with each other. One may rationalize the influence of such collisions on the linewidth by assuming that the emission (or absorption) process is not completely disrupted, but looses all its phase memory. One assumes that the interactions occur statistically distributed (exponential probability distribution), with an average time interval tcol . As will be discussed in detail in Sect. 2.1.3, Vol. 2 these considerations lead again to a L ORENTZ distribution of the frequency spectrum with a bandwidth inversely proportional to the collision time. The average time tcol between two collisions is derived with (1.52) from the particle density N of the perturbing gas, the collision cross section σ and the relative velocity v between radiating atoms and perturbing gas particles (being not necessarily the same). Thus, collisional line broadening (or pressure broadening) is characterized by a Lorentzian line shape with a FWHM ¯ BT )

ωcol = 1/tcol = N σ v = σ v p/kB T σp 8/(π Mk (5.16) ¯ B T )/2π, (5.17)

νcol σp 8/(π Mk where indicates averaging over the relative velocity distribution in the gas studied. The particle density is related to the pressure p by (1.50) at temperature T . The

234

5


k

Fig. 5.3 On the origin of the D OPPLER profile for an absorption line: A red or blue shift occurs, depending on the direction of the atomic velocity (red and black marked atoms, respectively). No shift occurs for atoms at rest or moving ⊥ k (grey)

¯ is derived from the M AXWELL average (relative) velocity v 8kB T /(π M) B OLTZMANN distribution (1.57). The reduced mass M¯ of the colliding pairs has to be used here, since it is their relative velocity that counts. Typical gas kinetic cross sections are σ 10−15 cm2 (mostly elastic). For our standard example, Na atoms, say in a cell with 100 mbar Ar buffer gas (M¯ = 14.6 u) at T = 554 K, the Na vapour pressure is 1 Pa = 0.01 mbar, sufficiently high for nice spectroscopy. At these conditions the frequency bandwidth becomes about νcol = 1.84 × 108 Hz, about an order of magnitude larger than the natural linewidth. It increases linearly with pressure and may become very important for dense gases. The collisional broadening too is a homogeneous line broadening since it is of statistical nature and concerns all atoms in a gas in the same way. If this additional broadening is of similar magnitude as the natural linewidth one has to convolute both profiles. Specifically, the convolution of two Lorentzian profiles (see Appendix G.5) leads again to a Lorentzian line profile with additive linewidths (FWHM):

ω1/2 = ωcol + A =

5.1.4

1 tcol

+

1 τnat

.

(5.18)

DOPPLER Broadening

While collision broadening may often be avoided in spectroscopic studies of isolated atoms and molecules, simply by reducing the target density, a much more cumbersome and important broadening mechanism is due to the thermal motion of atoms and molecules in the gas phase. Figure 5.3 illustrates the origin of this D OPPLER broadening of spectral lines. The D OPPLER shift for a particle with a velocity component vx in k direction is, in the non-relativistic limit, according to (1.32)

ωba = ωba

2π vx = vx , c λba

(5.19)

where ωba = 2πνba is the angular transition frequency in the rest frame of the atom, λba the corresponding wavelength, and ωba = 2π νba the shift of absorbed (or emitted) angular frequency in respect of ωba .

5.1

Line Broadening

235

Δω1/2

-4

-2

0 ω- ωb a / Δω1/2

2

4

Fig. 5.4 Comparison of a L ORENTZ profile (black line), typical for the natural line profile and collision broadening, and a Gaussian distribution (red line), arising from thermal line broadening. Compared are here profiles of the same FWHM ω1/2 , renormalized to equal maxima for comparison

According to (1.56), the B OLTZMANN distribution taken for one velocity component vx (in the direction of light propagation k) is w(vx )dvx =

1 2πv02

exp −vx2 / 2v02 dvx

kB T with v02 = v 2 = m

(5.20)

being the variance of the distribution, v0 = kB T /m its standard deviation, m the atomic mass, T the temperature, and kB the B OLTZMANN constant. Thus, with (5.19) each atom (molecule) absorbs at a different frequency ν according to its velocity component vx . Inserting vx from (5.19) into (5.20) leads to a Gaussian line profile

kB T 1 1 ω − ωba 2 dω, with ωD = ωba exp − (5.21) gD (ω)dω = √ 2 ω mc2 2πωD D being the standard deviation of the Gaussian. The FWHM is √ √ kB T kB T 2π √ .

ωD = 8 ln 2ωD = ωba 8 ln 2 = 8 ln 2 λba m mc2

(5.22)

Since dν/νD = dω/ωD , the D OPPLER line profile in frequency space is ob2 = ω /2π = ν tained from (5.21) by just √ replacing ω → ν, with ν D D ba kB T /mc 2 2 and a FWHM νD = νba 8 ln2 kB T /mc 2.4νba kB T /mc . As usual, the D OPPLER profile is normalized: gD (ω)dω = gD (ν)dν = 1. Figure 5.4 illustrates that the wings of a G AUSS distribution are strongly suppressed in comparison to a L ORENTZ distribution of the same FWHM. We note here, that D OPPLER shift satisfies of course momentum and energy conservation. We prove this in the non-relativistic limit for emission of a photon with energy ω into direction n. Energy conservation requires 1 1 ω + ωa + mv 2a = ωb + mv 2b , 2 2

(5.23)

236

5


where v b and v a are the atomic velocities, ωb and ωa the electronic energies in the upper and lower state, respectively. Momentum conservation implies mv b = mv a +

ω n. c

(5.24)

Inserting v a from (5.24) into (5.23), abbreviating as usual ωba = ωb − ωa , and neglecting terms ω/mc2 vx /c, one easily verifies

vx vx ω − ωba ωba or νba = ω= ωba 1 + = νba . 1 − vx /c c 2π c Here vx = v b · n is the projection of the initial atomic velocity v b onto the direction of propagation of the photon. Thus we have recovered the non-relativistic D OPPLER shift according to (5.19), q.e.d. In practice, D OPPLER broadening in the gas phase is usually much larger than the natural linewidth or collision broadening at not too high pressures. As an example, we refer once again to the Na D lines. At 554 K with m = 23 u the D OPPLER width becomes νD 1.8 × 109 Hz at ν = 5.09 × 1014 Hz – as compared to νnat = 9.79 × 106 Hz corresponding to a natural lifetime τnat = 16.2 ns. The D OPPLER width in this case is thus two orders of magnitude larger than the natural linewidth and one order of magnitude larger than the collision broadened line at 100 mbar. Spectroscopy of emission or absorption lines in the gas phase – the most important source of our knowledge about the structure of atoms and molecules – has to fight massively against D OPPLER broadening. In later chapters we get to know a variety of interesting methods which deal artfully with this problem. In view of Fig. 5.3 one easily recognizes that D OPPLER broadening – in contrast to natural line broadening and collisional broadening – is inhomogeneous: each atom emits its specific wavelength according to its instantaneous velocity. In an absorption process each particular frequency ν excites only a specific group of particles in a velocity interval from vx to vx + vx related through (5.19) to the natural linewidth by vx = λba νnat .

5.1.5

VOIGT Profile

In the most general case (i.e. at high temperatures and high pressures) one has to account for both effects, D OPPLER broadening and collision broadening. One has to convolute then a Lorentzian with a Gaussian profile. This is not completely trivial, but can be done as briefly sketched in Appendix G.6. The result, a so called VOIGT profile, is illustrated in Fig. 5.5 for the specific example of a L ORENTZ and D OPPLER profile with equal width. A rather accurate approximate formula for the linewidth of a VOIGT profile is (G.25). Such delicate studies and evaluations of spectral lines were done rather extensively in the second half of the past century, partially in order to better understand

5.1

Line Broadening

Fig. 5.5 Comparison of G AUSS (black), L ORENTZ (grey), and VOIGT (red) line profiles – the latter being a convolution of the former two. All three profiles are to normalized ∞ −∞ g(ω)dω = 1. The arrows indicate the FWHM of the distributions

237 1.0 0.5

ΔωD ΔωL ΔωV

- 2.0

0 ω - ω ba / Δω1/2

2.0

collision processes. Modern spectroscopy and collision physics have advanced over the past decades substantially. Thus it may be said that collisional and D OPPLER broadening have more or less been overcome by a toolbox of sophisticated spectroscopic methods. Nevertheless, a detailed evaluation of line profiles still plays an important role for remote spectroscopy of dense gases, high pressure gas discharges, and plasmas in general – in laboratory studies as well as in astrophysics. In addition to the effects mentioned here, several other mechanisms such as S TARK broadening in plasmas contribute to the line shapes. In otherwise inaccessible media systematic profile measurements of spectral lines often provide the only viable experimental access to important physical properties – such as particle densities and temperatures. Section summary

• The natural line profile arises from the finite lifetime τab of the excited state. It is described by a L ORENTZ distribution (5.12) with a FWHM ωnat = A¯ ab = 1/τab . • Collisional line broadening is also represented by a L ORENTZ profile. It is relevant for studies in gases at high pressure p, its FWHM being ωcol ¯ B T ) at a temperature T . σp 8/(π Mk • D OPPLER broadening arises from transition frequencies shifted due to different velocities of freely moving atoms or molecules in a gas. The B OLTZMANN velocity distribution √ leads√to a Gaussian line profile (5.21) with a FWHM of

ωD (2π/λba ) 8 ln 2 kB T /m. • Typically ωnat ωcol and ωnat ωD . • The VOIGT profile is a convolution of L ORENTZ and Gaussian profile and describes a situation where both, D OPPLER broadening and collisional broadening, are important, i.e. at high pressure and high temperature. • We distinguish homogeneous and inhomogeneous line broadening, if each absorber or emitter has the same line profile or a different resonant frequency, respectively. The natural line profile and collision broadening are typical for the former, D OPPLER broadening for the latter.

238

5


5.2

Oscillator Strength and Cross Section

5.2.1

Transition Rates Generalized

In Sect. 5.1.1 we have started to extend the concept of transition rates between discrete stationary states to levels broadened by spontaneous emission. We may consider this as a first step to make the transition into the continuum. The considerations presented there need not be restricted to the natural linewidth or to homogeneous line broadening. They can also be applied to transitions broadened by collisions or by the D OPPLER effect as just discussed. We simply have to replace gL (ω) in (5.10) by the corresponding profiles. In this context one usually normalizes to energy scale. With W = ω and I˜(ω) = dI /dω = dI /dW the transition rate (4.62) is rewritten as dRba = 4π 2 α I˜(W )| Dba |2 g(W ), (5.25) dW with | Dba |2 according to (4.57) in dipole approximation (see also footnote 1). As introduced in Sect. 1.3.4, g(W ) is now an appropriate density of states which describes the probability to encounter and excite the corresponding final states. The area under this distribution function (whatever its overall width) must of course be properly normalized according to (5.9). In this reading dRba /dW is a transition rate (or a rate of absorption from the photon point of view) per unit energy interval (dimension T−1 Enrg−1 ), and the density of states g(W ) may in the most general case describe an extended continuum or quasi-continuum of very closely spaced final states: it specifies how many states exist per unit energy. Replacing in (5.25) the specific interaction for electromagnetic radiation with as introduced in Sect. 4.3.1, this expresmatter by the general transition operator U sion turns out to be just a special application of the famous F ERMI’s golden rule:

2π 2 dRba = |Uba | g(W ), dW

(5.26)

which in the literature is usually written without explicit reference to its differential nature per unit energy interval.

5.2.2

Oscillator Strength

In Sect. 4.6.1 we have introduced the line strength S(jb ja ), which has the dimension L2 . For ease of comparing dipole transitions between different transitions and atoms, one also defines a dimensionless quantity, called oscillator strength (opt) fba

2 Dba 2Wba S(jb ja ) 2Wba 2Wba zba 2 = = = . Eh 3ga a02 E h a0 E h a0

(5.27)

5.2


239

Briefly, the line strengths S(jb ja ) is the sum over the squares of all transition matrix (opt) elements between sublevels of a and b, while fba characterizes absorption (or induced emission) from one substate |ja ma of the initial level a to the final level b – averaged over all initial substates and summed over all final states of that level.5 The degeneracy factor ga = 2ja + 1 compensates the summation over all initial states in the definition of S(jb ja ) according to (4.112), the factor 1/3 compensates (opt) summation over all polarizations. We emphasize that fba is independent of the polarization used! With the transition energy Wba = ωba being related as usual to the term energies by Wba = (Wb − Wa ) and with the definition (5.27) we have Wba = −Wab

(opt)

and ga fba

(opt)

= −gb fab .

(opt)

Thus fba > 0 for absorption and <0 for induced emission. As shown in Appendix H.2 the important (opt) fba = N , (5.28) T HOMAS -R EICHE -K UHN sum rule b

strictly holds in dipole approximation, with N being the number of active electrons. The summation (and possibly integration) has to be carried out over a complete basis set of states describing the system, and potentially includes positive as well as (opt) negative values of fba . The term “oscillator strength” is best understood in the context of the classical (opt) picture of one oscillating electron: an oscillator strength fba = 1 for a particular transition implies that the whole ability of an active electron to absorb electromagnetic radiation is concentrated in that one particular transition. A nearly perfect example provides Na. As discussed in detail in Sect. 3.2 its one valence electron is perfectly described by a quasi-one-electron system. The oscillator strength for its yel(opt) low Na–D line (which we have already mentioned several times) is f3p←3s ∼ 0.98, i.e. it contains ca. 98 % of all available oscillator strength. In contrast, for the H (opt) (opt) (opt) (opt) atom f2p←1s = 0.416, f3p←1s = 0.073, f4p←1s = 0.029, ∞ n=5 fnp←1s = 0.041. In addition one has to consider ionization for which integration over all continuum (opt) states gives dfba = 0.435. When applying the T HOMAS -R EICHE -K UHN sum rule to larger systems one has, however, to be aware that the sum (5.28) is to be taken literally and may include transitions to fully occupied levels – which are forbidden transitions as a consequence of the PAULI principle. (opt) A very useful numerical relation between oscillator strength fba (for excitation b ← a) and spontaneous transition probability Aab (for the decay process a ← b) is derived from (5.27) and (4.114): 5 In the second and third equality of (5.27) we have omitted these summations for simplicity. Precise

details and alternative expressions are summarized in Appendix H.2. The last expression refers to linear polarization, parallel to the z-axis.

240

5 (opt)

ga fba

= Cgb Aab λ2

with C =

5.2.3


(5.29)

ε0 me c = 1.4992 × 10−14 nm−2 s. 2πe2

Absorption Cross Section

If one describes the absorption process so to say from the view point of the photon, the question arises, which cross section does the photon ‘see’ during its interaction with an atom. With the photon energy W = ω we may rewrite (5.25) for the number of photons absorbed per unit of time and energy interval as (see also footnote 1) I˜(W ) dRba = 4π 2 α W | Dba |2 g(W ). dW ω

(5.30)

Now, I˜(W )/(ω) = Φ˜ is the number of photons per unit time per unit area and per spectral energy range, i.e. photon flux per unit energy. Thus, generalizing (4.24) we obtain the absorption cross section for a transition from |a to |b as a function of the energy of the absorbed photons: Dba |2 g(W ) = 2π 2 α σba = 4π αW | 2

2

Dba 2W g(W )Eh a 2 . 0 E h a0

(5.31)

On the right hand side we have rewritten this expression using atomic units a0 and Eh , so that σba is given essentially as a product of dimensionless factors: Since Dba |2 /a02 is dimensionless, Eh is an energy, and the density of states g(W ) has the | dimension Enrg−1 , σba has indeed the dimension L2 of a cross section (here given in units a02 ). If we refer here to the averaged cross section for transitions between levels a and b (see footnote 1) the term in square brackets is the oscillator strength as defined in (5.27) and (H.37). Thus we may write in compact form (opt)

(opt)

σba = 2π 2 αfba g(W )Eh a02 = 2π 2 re cfba g(W ).

(5.32)

Inserting numerical values for the atomic constants we obtain6 σba = 4.034 × 10−18 cm2 Eh fba g(W ) = 1.098 × 10−16 cm2 eV fba g(W ). (opt)

(opt)

The classical electron radius re = e2 /(4πε0 me c2 ) in (5.32) recognizes the fact that this equation can also be obtained from the classical forced oscillator model – except for the oscillator strength which can only be determined quantum mechani(opt) cally (for a one electron system 0 ≤ fba ≤ 1). 6 Note

that different authors use different units. Thus, one finds different prefactors in different publications. The value used here corresponds to FANO and C OOPER (1968), while e.g. C OOPER (1988) uses 8.067 × 10−18 cm2 and measures energies in RYDBERG = Eh /2.

5.2


241

For excitation of a resonance level, which is determined only by the natural lifetime, the cross section (5.32) may be rewritten in an even more transparent manner. (opt) With (5.29) we express fba by Aab and identify g(W ) as the L ORENTZ distribution (5.7) with a FWHM = Γ = ωnat , normalized according to (5.9). By inserting the atomic constants the photo-absorption cross section becomes σba (W ) =

(Γ /2)2 gb λ 2 . ga 2π (W − Wba )2 + (Γ /2)2

(5.33)

Its maximum value σA reached for W = Wba (corresponding to ω = ωba ) is universally given by σA = σba (Wba ) =

gb λ2ba , ga 2π

(5.34)

and depends only on the wavelength λba at resonance and on the degeneracies. To give a numerical value we refer again to our standard example, the Na D transition at λba = 589.6 nm from which we obtain σA = 1.6 · 10−9 cm2 (gb /ga = 3). This is indeed an extremely large absorption cross section. The relation between absorbed light intensity I , particle density Na,b and E IN STEIN coefficient Bba has already been discussed in Sect. 4.6.3. There I expressed the total intensity absorbed over the distance x by the whole spectral line considered arbitrarily narrow. With σba (W ) we may write this now frequency or energy dependent. The intensity I˜(W ) absorbed in an energy interval W is now simply

I˜(W ) = −Na σba (W )I˜(W ) x,

(5.35)

and the corresponding energy dependent absorption coefficient (in the chemical literature called extinction coefficient) becomes μ(ω) = Na σba (ω). However, when deriving (5.34) from (5.33) we have tacitly assumed essentially monochromatic light with ωlight ωnat . This may be achieved – but not in standard absorption experiments. Rather, one usually excites with light for which the opposite is true, ωlight ωnat . In that case the experiment averages over the whole spectral line and one measures an averaged cross section ∞ σba (Wc ) =

˜

−∞ I (W − Wdet )σba (W )dW ∞ ˜ −∞ I (W )dW

.

(5.36)

Here we have accounted for some detuning Wdet of the light source in respect of the center of the spectral line. This convolution (see Appendix G) thus allows one to describe a realistic spectroscopic experiment where the laser is tuned over the spectral lines. We assume now for simplicity that the spectral distribution of the light source (e.g. a tuneable dye laser) is also characterized by a Lorentzian profile, its FWHM being ωlight . Using (G.21) the convolution integral (5.36) can be evaluated, leading

242

5


again to a (properly normalized) L ORENTZ profile gL (ω; FWHM) with a FWHM=

ωnat + ωlight . One finds that σba (ω) =

λ 2 gb π ωnat gL (ω; ωnat + ωlight ) 2π ga

(5.37)

with ω − ωba being the detuning of the light source from the maximum ωba of the spectral line. On resonance (ω = ωba ) the averaged absorption cross section becomes ___ σA =

ωnat λ2 gb . 2π ga ωnat + ωlight

(5.38)

Thus, in practice the huge maximum absorption cross section (5.34) is reduced to an effectively much smaller value if the bandwidth of the light source is large compared to the natural linewidth. The total absorbed intensity (W cm−1 ) is then given by ___

I = −Na σA I x. For other line profiles similar considerations may be applied to derive averaged cross sections. For instance, to describe a mainly collision broadened line, one would have to replace ωnat = Γ / by ωcol according to (5.16) and reduce the cross section (5.33) by ωnat / ωcol . Note, however, that the linear averaging described above is only applicable in the linear absorption regime, i.e. as long as I˜(W )/I˜(W ) 1, and the relative absorption may be computed directly from (5.35). Otherwise one has to apply its integrated form, the L AMBERT-B EER law (1.45) at each incident frequency, and finally average over the line shape of the radiation.

5.2.4

Different Notations – Radiative-Transfer in Gases

In general, this is not a trivial task, in particular if molecular spectra with many lines are considered. They may even arise from several different absorbing gas molecules – as it is the case in radiative-transfer of the incoming solar radiation through our earth’s atmosphere. In the spectroscopic literature and in data banks (see e.g. ROTH MAN et al. 2009) one often rewrites (5.33) as a function of the wavenumber ν¯ = 1/λ σba (¯ν ) =

γ /π gb γ λ 2 × = Sba × L(¯ν , ν¯ ba , γ ), ga 2 (¯ν − ν¯ ba )2 + γ 2

(5.39)

where the L ORENTZ line shape L(¯ν , ν¯ ba , γ ) is given as a function of wavenumbers, and γ is the half width of the spectral line profile at half maximum (HWHM) in wavenumbers [γ ] = cm−1 . Note that L(¯ν , ν¯ ba , γ ) is measured in units [L] = cm and normalized so that L(¯ν , ν¯ ba , γ )d¯ν = 1. The parameter Sba is usually also called line strength (not to be confused with the line strength S(jb ja ) which we have used elsewhere, with the dimension L2 ). It is measured in units [S] = cm−1 /(molecule cm−2 ) – to be identified as wavenumber per column density. In radiative-transfer calculations one has also

5.2


243

to account for changes in the density Na (x) of absorbing molecules with position x in space and defines the column density as CD = Na (x) × x, measured in units [CD ] = molecule cm−2 . Thus, (5.35) is rewritten for a given spectral line, centred at ν¯ ba , as dI˜(¯ν ) = −I˜(¯ν )Sba L(¯ν , ν¯ ba , γ )dCD .

(5.40)

If only one single absorbing line is considered, the relative intensity transmitted over a spectral range ¯ν is determined by

ν¯ ba + ¯ν /2 I γ /π 1 d¯ν , = exp −CD Sba I0 ¯ν ν¯ ba − ¯ν /2 (¯νba − ν¯ )2 + γ 2 where we have assumed that the incident spectral intensity is constant over ¯ν and centred at ν¯ ba . For the general multiline problem (also accounting for D OPPLER broadening) one has to replace Sj L(¯ν , ν¯ j , γ ) → j Sj V (¯ν ; ν¯ j ) in (5.40), with VOIGT line profiles (Sect. 5.1.5) centred at each relevant spectral line ν¯ j . By integrating (5.40) over CD one derives the relative transmission I˜(¯ν )/I˜0 (¯ν ) for the spectral intensity at each ν¯ of interest. To model a genuine molecular spectrum one also has to account for the thermal population of the initial levels, and include induced emission in addition to absorption. Finally, this spectrum I˜(¯ν ) thus obtained has to be convoluted with the resolution of the spectrometer. For atmospheric modelling, reemission of radiation and scattering from the atmosphere has also to be included, as well as absorption and scattering from aerosols and small droplets. Thus, the problem gets rather involved. References to a number of codes available for atmospheric radiative-transfer are reported by W IKIPEDIA CONTRIBUTORS (2013). They all rely on information from molecular spectroscopy as documented in databanks such as HITRAN, and allow to model solar spectra on the surface of the earth as presented in Sect. 1.4.7. Section summary

• We have related the rate for optically induced E1 transitions to the famous F ERMI golden rule (5.26). (opt) • The optical oscillator strength fba is a useful, dimensionless quantity which characterizes the strength of an E1 transition. For a (quasi)single electron system, in the classical limit of a harmonic electron oscillator it would be equal to 1, in quantum mechanical reality it obeys (5.28), the T HOMAS -R EICHE (opt) K UHN sum rule, b fba = 1 and is always <1. • The photo-absorption cross section (5.32) for an infinitely narrow bandwidth laser beam is very large, specifically σA = λ2 /(2π)(gb /ga ) at the maximum of a L ORENTZ distribution if only natural line broadening (FWHM ωnat = 1/τab ) is relevant. The measured cross section depends, however, strongly on the line profile of the absorbed light (FWHM ωlight ). If the latter is large ___ compared to the former, the photo-absorption cross section becomes σA σA ωnat / ωlight .

244

5.3

5


Multi-photon Processes

Up to now we have determined transition probabilities in 1st order perturbation theory. Thus, we have included only processes where a single photon is absorbed or emitted. This is in order as long as the light intensities used are low enough so that the excitation probabilities are small and changes of the wave function of the target are negligible. However, with today’s powerful laser sources, this is by no means a trivial assumption. On the contrary: multi-photon processes are at the heart of today’s laser spectroscopy – on the one side they may be something to avoid carefully if one aims at strictly linear spectroscopic data. On the other hand, they may be used artfully for accessing a broad range of objects, states and transition types which would otherwise be unaccessible. Not only in physics and physical chemistry are they of high spectroscopic relevance, they also have become indispensable tools in biology and medicine. A quick Internet search for “two-photon” will convince you that a host of applications relates to confocal microscopy, fluorescence microscopy and biological imaging. Hence, we are well advised to approach this growing field of trans-disciplinary research by trying to understand at least the basic physics involved – applying tools we have already acquired. In the following we shall define the terminology, derive some quantitative expressions for the cross section of two-photon excitation, and mention some experimental aspects (more examples will be met on many occasions in later chapters). We shall then discuss two-photon decay processes for simple atoms with very long-lived excited states, which can only decay by two-photon emission – the dipole selection rules prevent any other decay channels. Multi-photon ionization will be the subject in Sect. 5.5.5 and – from the perspective of strong laser fields – in Sect. 8.5. We consider the excitation of a target atom or molecule Tg into a state |b with energy Wb from an initial state |a with energy Wa by several, say N photons of angular frequency ω: Tg(a) + N ω → Tg(b).

(5.41)

Of course, energy must be conserved in such processes, so that N ω = Wb − Wa = Wba = ωba .

(5.42)

While in (4.46) perturbation theory was applied only in 1st order, one now has to carry it out consequently at least up to N th order to determine the transition probability for this particular type of process. While in 1st order the transition rate (4.63) was proportional to the intensity I of the laser field, in N th order it becomes proportional to the N th power of it. Thus, the N photon transition rate will be (N )

(N )

Rba = σba Φ N ∝ I N

with Φ = I /ω,

(5.43)

5.3


245

(N )

where σba is a so called generalized cross section. The photon flux Φ is given in units [Φ] = Photons m−2 s−1 and the unit of the generalized cross section [σ (N ) ] = m2N sN −1 is thus a little bit unused but meaningful. In the following we shall talk about processes which are induced by an intense laser field which is, however, not yet strong enough to completely deform the subject of investigation – or even to destroy it completely.

5.3.1

Two-Photon Excitation

The question whether two photons may be absorbed simultaneously was first studied theoretically by a student of Max B ORN, Maria G ÖPPERT-M AYER (1931), who later in 1963 won the N OBEL prize (together with W IGNER and J ENSEN) for her work on the nuclear shell structure. In her PhD thesis “Über Elementarakte mit zwei Quantensprüngen” (On elementary acts with two quantum jumps) she laid the foundations of this field – which, however, only started to flourish with the advent of powerful lasers, long after her pioneering work. We have already prepared in the last chapter the tools for understanding what happens, so that now it will take only a few steps to sketch the basic theory necessary for catching up with state-of-the-art research in this field. In Sect. 5.5.5 we shall return to it. We simply have to reintroduce the transition amplitude (4.52) (obtained in 1st order perturbation theory) on the right hand side of the original ODE (4.45), use (4.50) and to integrate. This leads to the probability amplitude in 2nd order perturbation theory: cb (t) =

−ie2 E02 42

t 0

dt

Dbγ Dγ a ei(ωba −2ω)t

γ

(ωγ a − ω)

+

† D Dγ a eiωba t

bγ

(ωγ a − ω)

(5.44)

+ emission terms . The expression sums (in principle) over all basis states |γ of the system.7 Characteristic for all absorption terms of this sum (first line) are the resonance denominators (ωγ a − ω) where ωγ a = Wγ − Wa gives the energy difference between initial state |a and an intermediate state |γ , each with exponentials in the nominator resembling those in the 1st order approximation (4.51). We see now clearly how two photon absorption comes into play: the first exponent i(ωba − 2ω)t in (5.44) – common to all terms in the sum – simply replaces i(ωba − ω)t in (4.51): the double angular frequency 2ω of the electromagnetic wave assumes the role of ω. These terms, characteristic to 2nd order, oscillate rapidly and average out in the integration – unless 2ω = ωba – in analogy to terms with ωba − ω 7 We have written out here only the absorption terms; the ‘emission terms’ are identical, except that

−ω is replaced by ω and D by D† (and vice versa).

246

5


in 1st order. Terms with exp(iωba t ) vanish in any case when the integration is carried out. Hence, for absorption only the first terms of the sum over γ in (5.44) contribute to the integral, and exp[i(ωba − 2ω)t ] is pulled out of the sum. (The corresponding ‘emission terms’ in (5.44) contain exp[i(ωba + 2ω)t ] and describe two-photon emission in the case that ωba < 0.) All further steps correspond to those in 1st order: one carries out the integration from t = 0 to t and makes the transition from probability amplitudes to transition probabilities in the limit t → ∞. Dividing that by t gives a transition rate proportional to E04 (hence to I 2 ) and to the D IRAC delta function – this time of the argument ωba − 2ω. Thus, the resonance frequency for a transition from |a to |b by simultaneous absorption of two photons (both of ω energy) becomes 2ω = ωba as expected. The necessary extension to levels of finite width or for transitions into the continuum follows the scheme used in the one-photon case. Specifically, for a L ORENTZ broadened line one replaces δ(ωba − 2ω) → gL (2ω),

(5.45)

where the line profile (5.7) of the excited state |b is now a function of twice the photon energy. From the thus derived rate one pulls out the square of the photon flux (I /ω)2 to obtain the generalized cross section for two-photon excitation in 2nd order perturbation theory: 2 b| D|γ γ | D|a (2) gL (2ω). σba = (2π)3 α 2 (ω)2 (5.46) Wγ a − ω γ

As indicated in this expression, in principle one has to sum not only over infinitely many discrete basis states |γ but also to integrate over the continuum. This can lead to rather elaborate calculations. Fortunately, however, with γ differing more and more from a, the increasing denominators as well as the decreasing magnitude of the matrix elements Dγ a = γ | D|a usually leads to satisfactory results with only a finite number of terms. One verifies easily that this generalized two-photon cross section is derived in (2) units [σba ] = m4 s since the line profile is given in [g(2ω)] = eV−1 and the ma(2) trix elements are measured in units of length. Note that σba as defined by (5.43) refers in principle to a strictly monochromatic electromagnetic wave (e.g. a highly stabilized laser source) which is much narrower than the atomic linewidth given by gL (2ω). For a radiation source of broader bandwidth only the fraction of the radiation spectrum I˜(ω) overlapping with the atomic linewidth is relevant – just as in the one-photon case. For a spectrum with finite bandwidth, in the present case of two-photon absorption one has first to convolute I˜(ω) with itself, i.e. with I˜(ω ), and the result in turn with gL (2ω) to finally obtain an expression analogous to (5.38). Several extensions of the procedure described above are obvious. Excitation processes with two photons of different frequency and/or polarization may be considered by adding to the interaction potential (4.55) a second electric field vector. The

5.3


247

expressions obtained by carrying this consequently through to 2nd order perturbation theory are correspondingly more complicated and contain interesting cross terms of the two frequencies. Also, one may include more than two photons into the calculation by applying time and again the procedure described above to the original ODE (4.45) – as often as required, which leads to accordingly more complex final expressions for the generalized cross section. Clearly, (5.46) is also valid for multi-electron systems. One ‘simply’ has to reD by a sum over all electrons according to (H.29) as place the matrix elements of exemplified for the two electron case and single photon absorption in Sect. 7.5. In detail the evaluation will not always be trivial to do. As mentioned above, although in principle understood since 1931, multi-photon transitions became experimentally accessible only after 1960 when lasers were introduced as efficient light sources for spectroscopy: generalized multi-photon cross sections are extremely small – but according to (5.43) this may be compensated by sufficiently high light intensity. To give one example: the two-photon ionization cross section for Xe is σ (2) = 1.16 × 10−49 cm4 s at a wavelength of λ = 193 nm (L AMBROPOULUS 1985). From a typical classical light source one expects at such wavelengths a few W cm−2 at most, which would lead to an ionization rate of ca. 10−13 s−1 per atom, i.e. on average one expects an atom to be ionized in 105 years! In contrast, a typical excimer laser at that wavelength (pulse duration of 10 ns, pulse energy 100 mJ) may be focussed onto an area of 0.1 mm2 . This corresponds to 1010 W cm−2 and would lead to an ionization rate of about 107 s−1 per atom. In other words, during 10 ns, the duration of one laser pulse, there is already a 10 % chance for such a process to happen, and thus the two-photon process can be detected conveniently. Consequently, excitation and ionization by two or more photon absorption plays an important role in today’s laser spectroscopy. One important advantage of multi-photon processes is that they allow to overcome some of the limitations imposed by the rather rigid dipole selection rules for single photon spectroscopy. Two-photon selection rules can be read directly from D|γ γ | D|a implies that selection (5.46). The product of the matrix elements b| rules are multiplied – so to say. Each of the two matrix elements corresponds to a set of specific dipole selection rules. The product thus has the same effect as if the two processes were carried out one after the other – although the absorption of the two photons is simultaneous. In place of the selection rules for one-photon transitions summarized in Sect. 4.4.3, for two photon transitions on obtains for the total angular momentum j = 0, ±2, the parity selection rule becomes = 0, ±2, and for the projection quantum number m = 2q holds (if two photons of polarization q = 0, ±1 are absorbed). Using different polarizations for the two photons allows for additional flexibility. Important examples for two-photon absorption transitions between s states as illustrated in Fig. 5.6 for the 1s 2 S → 2s 2 S transition in the H atom. The intermediate states |γ over which we have to sum here are all the bound |np states as well as the corresponding states |p in the ionization continuum – only to these the matrix Dbγ and Dγ a do not vanish. We point out that these states are energetielements cally not positioned ‘in between’ initial and final state and thus, are far from being

248

5

Fig. 5.6 Two-photon transition 1s → 2s in atomic hydrogen (full red arrows). The red dashed arrows indicate the coupling from initial state |1s to the ‘intermediate’ |p states and from them to the final |2s states by Dp1s and D2sp , respectively


W / Eh

єs

єp

0.0 3s - 0.125 2s

3p 2p

|γ〉

ħω ħω - 0.500

1s

resonant. Nevertheless such processes can be studied with high efficiency and have played a key role in the ultra high precision spectroscopy of atomic hydrogen which led to the N OBEL prize for Ted H ÄNSCH (2005). As we shall discuss in Sect. 6.1.8, crucial for this kind of precision spectroscopy is the possibility to study two-photon absorption in the gas phase completely free of D OPPLER broadening. As a somewhat exotic but interesting field of atomic physics we finally mention spectroscopic studies relating to parity violation. They rely on such high precision spectroscopy. In this context, e.g. so called nuclear anapole moments have been observed in 2 photon excitation of heavy alkali atoms (see e.g. B OUCHIAT 2007).

5.3.2

Two-Photon Emission

The inverse process, induced two-photon emission, is of course equally possible. But even spontaneous two-photon emission has been observed in some cases where one-photon transitions are strictly forbidden. Since in effect it is the vacuum field which induces such transitions, these are extremely weak processes which accompany the decay of very long-lived excited states. Particularly well studied examples are 2s 2 S states8 of hydrogen like ions (much easier to manipulate and to detect than neutral species). First experiments for two-photon decay were reported by L IPELES et al. (1965) and N OVICK (1972). Metastable, hydrogen like He+ ions in the 2s 2 S state were investigated – with at that time still rather elaborate photonphoton-coincidence counting techniques. For He+ the 2s 2 S state the lifetime was determined to be 2 ms. This is in very good agreement with theoretical predictions of a two-photon decay rate R1s←2s = 8.228 Z 6 s−1 , i.e. for Z = 2 ca. 1.9 ms. The fact that the decay rate increases ∝ Z 6 suggests to study such processes for highly charged ions which are available in state-of-the-art ion storage rings. Figure 5.7 illustrates such an experiment with He like 58 Ni26+ performed by S CHÄF FER et al. (1999). Figure 5.7(a) shows the term scheme of the ion. In addition to the 2 1 S0 state of interest in this experiment several other states are excited in this experiment which may decay by one or the other mechanism into the ground state. 8 The

designations 2 S, 2 P used here and 1 S0 , 3 P0 , etc. used in Fig. 5.7 will be explained in detail in Chap. 6.


W- W11S / keV

(a)

0

7.80

1.65fs

21P

1

71ps

E1 21S0 ħω1

154ps

M1

0

58Ni26+

2

V / ke

11S0

4

0

6

E1 ħ ω2

ħω 1

7.70

8

M2

2E1

2 3 P2 13fs 23P1 3 2.5ns 2 P0 E1E1 2 3 S1 2.3ns

(b)

10

7.75

249

coincident poton rate

5.3

0

2

6 4 / keV 2 ω ħ

8

10

Fig. 5.7 Two-photon decay of the 1s2s 1 S0 state in He like 58 Ni26+ . (a) Term scheme with several neighbouring levels and transitions. (b) Two-photon coincidences as a function of the emitted photon energies ω1 and ω2 . The red dotted line emphasizes the transition of interest 2 1 S0 → 1 1 S0 . Figure adapted from S CHÄFFER et al. (1999)

Lifetimes and process types are noted in Fig. 5.7(a).9 Figure 5.7(b) shows the measured coincidence rate for detecting simultaneously two photons as a function of their respective energies, ω1 and ω2 . Accidental coincidences with some of the other processes make the results somewhat intransparent – hence the red dotted line indicates the 2E1 decay process of interest. Energy conservation requires that ω1 + ω2 = W2 1 S0 − W1 1 S0 . One sees clearly that energy sharing among the two photons leads to a broad distribution with a flat maximum for similar energies while the probability that one of the photons carries all energy is zero (barely recognizable in Fig. 5.7 due to accidental coincidences). Precision measurements of such decay processes are of considerable interest as an important testing ground for relativistic multi-electron quantum mechanical computations. Interestingly, special two-photon transitions (3 P0 → 1 S0 otherwise strictly forbidden for 2E1 processes) are discussed in the context of investigating parity violation in atomic physics (B OUCHIAT and B OUCHIAT 1997). Section summary

• Multi-photon processes can be induced by intense electromagnetic radiation. (N ) (N ) The rate for absorption of N photons is Rba = σba (I /ω)N (at not too (N ) high intensities I ), the generalized cross section σba being measured in m2N sN −1 . Resonance conditions are now N ω = ωba . 9 We cannot enter into a discussion of the interesting details here, which require knowledge of the underlying atomic physics still to be explored in later chapters. For the advanced reader it may be particularly interesting that the high nuclear charge Z implies strong spin-orbit interaction and leads to configuration mixing. Thus, the transition 2 3 P1 → 1 1 S0 is dominated by an E1 process, while transitions 2 3 P2,0 → 1 1 S0 with J = 2 and 0, respectively, are strictly E1 forbidden due to angular momentum conservation.

250

5


• N photon cross sections may be computed in N th order perturbation theory. This involves – in principle – summation (and integration) over all intermediate discrete (and continuum) states as exemplified for two-photon excitation in (5.46). Corresponding expressions hold for induced multi-photon emission processes. • In addition, (very weak) spontaneous two-photon (2E1) emission processes are also observed. Their probability rises with the nuclear charge ∝ Z 6 and has been studied for highly charged ions.

5.4

Magnetic Dipole and Electric Quadrupole Transitions

If electric dipole transitions (E1) are forbidden, higher order single photon processes may become effective (in addition or alternatively to the multi-photon processes just discussed). We have already noticed this in passing while discussing two-photon decay in Sect. 5.3.2. The present section describes how magnetic dipole (M1) and electric quadrupole transitions (E2) arise when treating the full perturbation operator in 1st order perturbation theory. Albeit much weaker than E1 processes, such transitions are observable in a variety of interesting cases. Of particular spectroscopic and practical relevance are M1 transitions, being the basis for all magnetic resonance spectroscopy (see Sects. 9.5.2 and 9.5.3 for more details about EPR and NMR, respectively). So far, we have treated the interaction potential of the electromagnetic wave with the atomic (or molecular) electrons only approximatively. The full expression (H.21) −iωt (r, t) = eE0 (5.47) De − D∗ e+iωt U 2 2I (5.48) with (H.23) eE0 = eωA0 = ecB0 = e cε0 and (H.22) D=

eik·r −ieik·r e · p= e·∇ ωme ωme

(5.49)

contains the term exp(ik · r) = 1 + ik · r + · · · , which in dipole approximation was just replaced by 1. We shall now treat the second term of this expansion: i D(2) = (k · r)(e · p ). ωme We first rewrite10 this by trivial addition and subtraction of an extra term: i i D(2) = [k · r e · p −e ·r k · p] + [k · r e · p +e ·r k · p ]. 2ωme 2ωme 10 Above we have indicated once the precedence of vector multiplication by brackets. In all follow-

ing expressions we omit this notation: vector products always have precedence over scalar products while these again have precedence over simple scalar multiplications.

5.4


251

We exploit now some rules for multiple products from vector algebra, using the fact that k ⊥ e ⊥ B. The first bracket is identical to the scalar product of two vector products (making sure not to permutate r and p): D(2) =

i i k ×e ·r × p+ [k · r e · p +e ·r k · p ]. 2ωme 2ωme

Per definition r × p= L is the angular momentum of the electron and k × e B 0 = −iωt iωt (B 0 /2)(e − e ), see also (H.7). In addition, we note that with k ⊥ e the projection of the position vector onto the polarization vector, e · r, and the projection of the momentum vector onto the wave vector, k · p , represent perpendicular components of the r and p operator, respectively. Hence, they commute and we may write D(2) =

ik B 0 i ·L+ [k · r e · p +k · p e · r]. 2ωme B0 2ωme

] Next we apply the commutation rule (H.17), according to which p=−i(me /)[r, H and obtain: D(2) =

ik 1 ]. [H k · r e · r − k · r e · r H B0 · L− 2ωme B0 2ω

Multiplication with eE0 according to (5.48) and substituting in the first term k/ω = 1/c finally gives: ecB0 D(2) = i

e ecB0 ]. [H k · r e · r − k · r e · r H L · B0 − 2me 2ω

(5.50)

We have thus managed to expand the operator (5.47) describing the interaction of electromagnetic waves with matter beyond the simplest dipole approximation in a suggestive manner. Obviously, (5.50) expresses the second most important term as two different types of interactions due to the electromagnetic field: as we shall see in a moment, the first one is responsible for magnetic dipole (M1), the second one for electric quadrupole transitions (E2). Both have the same time dependence as specified in (5.47) and may be treated completely analogous to electric dipole transitions in 1st order perturbation theory. We thus simply need to determine the transition matrix elements between |a and |b of ecB0 D(2) for the two components. The magnetic dipole term may then be written as 1 L · B0 (ecB0 D)M1 = μB · B 0 = −M i

(5.51)

where according to (1.150) we have inserted explicitly B OHR’s magneton μB to express the characteristic nature of this interaction. The magnetic dipole interac · B(t) induces M1 transitions – in complete analogy to the electric tion energy M dipole interaction energy D · E(t) which is responsible for E1 transitions. Let us now obtain an estimate for the relative order of magnitude for these two types of

252

5


transitions. Both interactions enter the transition probability quadratically. We assume L/ 1 and D = erab ea0 /Z we obtain (exploiting E0 = cB0 ) as an order of magnitude · B 0 |2 Z 2 2 (Zα)2 |M 1.33 × 10−5 Z 2 . = 4 |D · E 0 |2 4m2e a02 c2

(5.52)

The probability to induce an M1 transition is thus (for light atoms) about five orders of magnitude smaller than for an E1 transition – at a given intensity of the radiation. The same holds for spontaneous decay probabilities. Sure enough, for high charge states M1 or even M2 may become appreciable, as we have already seen in Sect. 5.3.2. The M1 transitions are crucial for all magnetic resonance spectroscopy where transitions between different magnetic substates |j mb ↔ |j ma within the same electronic level are studied (see Sects. 9.5.2 and 9.5.3) – strictly forbidden for electric dipole transitions. For a quantitative treatment of M1 processes we also have to account for the electron spin. As mentioned already in Chap. 1, the magnetic moment associated with L = μB the electron spin is ∼ L/ = 2μB . This implies that the magnetic moment M = μB ( L + 2 S)/. Since of the orbital angular momentum has to be replaced by M M1 transitions are single photon processes, selection rules and line strength for M1 transitions thus follow from the corresponding transition matrix elements. For the determination of transition rates and cross sections one simply has to replace in all relevant expressions of Sects. 4.3–5.2 the electric dipole transition matrix element eE0 Dba by the corresponding one for the magnetic dipole interaction: B μB B 1 μB (M1) (ecB0 b|L + 2 S|a · = b|J + S|a · . D)ba = i ec B ec B

(5.53)

The components of total angular momentum and spin are thus projected onto the magnetic field direction of the electromagnetic wave with B ⊥ E ⊥ k. Tools for explicit evaluation of the matrix elements are summarized in Appendix C.4.3. Details will be discussed in several of the following chapters. Let us come back now to the second part in (5.50), the electric quadrupole term, from which the transition matrix element follows as eE0 (Wb − Wa ) b|k · r e · r|a 2ω eE0 π =− b|ek · r e · r|a . λba

a|eE0 D|b (E2) = −

(5.54)

With k = ω/c we have introduced here the unit vector ek = k/k in the direction of the wave propagation and the transition wavelength λba . Alternatively with (5.47)– (5.49) and (4.1) the Hamiltonian for the electric quadrupole interaction may also be written as (r, t)(E2) = e ωba E(t) · r (k · r). U (5.55) 2 ω

5.4


253

Since k and e are always perpendicular to each other, the two projections of r onto these directions correspond to orthogonal components of r. Thus, depending on the choice of the (at) and (ph)√frame (see Fig. 4.3), the relevant matrix ele(propagation in x-direction, polarization in ments are of the type xy = Q22− / 3 √ in z-direction, polarization y-direction or vice versa), xz = Q21+ / 3 (propagation √ in x-direction or vice versa) or yz = Q21− / 3. Light propagation under 45◦ in the xy plane with a polarization angle vector also aligned within √ the xy plane (azimuthal angle −45◦ ) is described by (x 2 − y 2 )/2 = Q22+ / 3. The terminology used here for the real components of the so called quadrupole tensor (an irreducible tensor operator) is summarized in Appendix F.3. Any other geometry may be expressed as a linear combination of the components of this tensor operator, typically by rotation of this operator with the help of the rotation matrix of rank 2 according to Appendix E. Quantitatively the transition probabilities and cross sections for E2 transitions are obtained by replacing the electric dipole transition matrix elements Dba throughout Sect. 4.3 with eE0 b|eE0 D|a (E2) =

(−1)q+1 π b|Q2q± |a . √ 3λba

(5.56)

Here q± characterizes the geometry of absorption or emission as just outlined. It is advantageous that angular momentum algebra allows to derive the selection rules in a relatively straight forward manner. For multi-electron atoms one has to account for the respective angular momentum coupling schemes which will be discussed in later chapters. A brief survey on selection rules for E2 transitions is given in Appendix C.4.2. We finally estimate the relative probability for such transitions. The E2 matrix element (5.54) is ∝ eE0 xi xj /λba , xi and xj being two position coordinates, while the E1 matrix element is ∝ eE0 xi . For a rough estimate we derive the averaged position coordinates from (2.130) with xi xj ∝ r 2 n n4 a02 /Z 2 and xi ∝ r n n2 a0 /Z. Thus, the ratio of the transition probabilities at identical intensities is on the order of 2 2 D|b (E2) |2 n a0 | a|eE0 . (5.57) 2 Zλba |D · E 0 | This ratio is thus determined by the square of the ratio of atomic dimension to wavelength. For a typical transition in the VIS or UV spectral range for small nuclear charges Z and low principle quantum numbers n, one estimates a ratio of (0.1 nm /300 nm)2 10−7 . For high Z, the probability for E2 processes even decreases with Z −2 – in contrast to M1 transitions which increase with Z 2 as discussed above. Thus, E2 processes have not played a major role in classical atomic, molecular or solid state spectroscopy – except in the X-ray region where they may be considerable. However, E2 transitions are of increasing interest as states with ever higher principle quantum numbers n are studied, since orbital radii increase with n2 . A so called RYDBERG atom with n = 100, say, has a diameter of ca. 500 nm, and thus

254

5


a size comparable to optical wavelengths. For a recent experimental example, also illustrating the details of evaluating (5.54), see e.g. T ONG et al. (2009) who studied RYDBERG states in ultra-cold rubidium for n = 27–59. Quite generally speaking, RYDBERG atoms, these nearly macroscopic quantum objects, often surprise with a lot of interesting and ‘rich physics’.11 Section summary

• By expanding the spatial dependence exp(ik · r) of the electromagnetic field beyond the first, constant term one obtains magnetic dipole and electric quadrupole interaction terms, proportional to μB b| J + S|a and to b|Q2q± |a /λba , respectively. • The rates for the resulting M1 and E2 transition can be evaluated in 1st order perturbation theory – in full analogy to those for E1 transitions. In comparison to the latter M1 transitions are typically less probable by a factor of 10−5 Z 2 , E2 transitions by about n4 (a0 /λba )2 Z −2 .

5.5

Photoionization

The discovery and interpretation of the “photoelectric effect” at the beginning of the 20th century was one of the fundamental mile stones on the way to modern physics – as we have outlined in Sect. 1.4.1. Today one classifies the corresponding processes as photoionization and addresses with this term a major part of the continuum physics of atoms and molecules. While in the past chapters we have essentially discussed transitions between discrete, bound states which are induced by electromagnetic fields, we now want to give at least a brief introduction into, so to say, this other half of atomic and molecular physics. The fundamental concepts were already explored in the 1930s, while in the 1960s and 1970s they have been brought into full fruition, when extensive experimental studies on photoionization have been carried out: The availability of synchrotron radiation (M ADDEN and C ODLING 1963) as intensive light source in the VUV and X-ray region gave way to rich harvesting of precise data and profound physical insight. Modern laser physics too has made an important contribution to the development of photoelectron spectroscopy as a key tool for studying the electronic and ro-vibronic structure of atoms, molecules and solid states. Today’s efforts are mainly concentrated on molecules and clusters. In surface physics too these methods, original developed in atomic physics, are now widely used. Photoelectron spectroscopy has matured to an indispensable tool in many areas of physics, chemistry and materials research. The following introduction outlines some basic theoretical considerations and illustrates these by a few experimental examples. Characteristic findings for multielectron systems will be presented in Chap. 10, and a more detailed discussion of 11 This

expression is often used like a synonym for ‘up to now only poorly understood’.

5.5

Photoionization

255

methods and results is given in the context of molecular photoelectron spectroscopy in Sect. 5.8, Vol. 2. Readers interested in the historical development of this important field are referred to the original literature and a number of good review articles (see e.g. B ETHE and S ALPETER 1957; B URGESS and S EATON 1960; C OOPER 1962; C OOPER and Z ARE 1968a; M ANSON and S TARACE 1982; S AHA 1989; S CHMIDT 1992, and references there).

5.5.1

Process and Cross Section

We want to obtain a quantitative description of the ionization process for an atomic (or molecular) target, denoted in the following as “Tg”, by interaction with an electromagnetic field. We assume that the field induces a transition from a bound state |a = |n into an unbound continuum state |b = | during which one photon of energy ω is absorbed. For simplicity we ignore again the electron spin (as a rule the spin does not change its orientation for low Z atoms) and also concentrate on effective one electron systems. While bound states are characterized by a principle quantum number n, an orbital angular momentum quantum number , and a binding energy Wn < 0, continuum states describe the free electron with an asymptotic kinetic energy > 0 and its angular momentum . The continuum energy is related to the wavenumber ke of the electron at infinite distance from the nucleus by = 2 ke2 /(2me ). The photoionization process may be written schematically as Tg(n ) + ω → Tg+ + e− with ω − WI = ,

(5.58)

(5.59)

with the ionization potential WI which is related to the binding energy Wn = −WI < 0 of the initial target state Tg(n ). A closely related process is the removal of an electron from an anion, the so called photo-detachment: Tg− (n ) + ω → Tg + e− . (5.60) It may be treated in a similar manner as photoionization, the ionization potential WI being replaced by the electron affinity WEA (see Sect. 3.1.5). The main difference between the two processes is the rapid decrease of the interaction potential with r between the remaining neutral atom and the emerging electron, in contrast to C OULOMB interaction in the ionization case, which leads to a quite different asymptotic behaviour of the continuum wave functions. We shall discuss this in Sect. 5.5.4. Just as for optical excitation, the photoionization cross section is given by (5.31). Da = r a ·e in E1 approximation according However, the transition matrix element, to (4.57), is now determined between the discrete initial state |a and the final continuum state | . Usually one normalizes the latter in energy scale (see Appendix J.1)

256

5


so that its dimension is L−3/2 Enrg−1/2 . In this normalization the density of states (number of states per energy interval), g(W ) in (5.31), is already built into the wave functions.12 The photoionization cross section thus becomes Da |2 . σa (ω) = 4π 2 αω|

(5.61)

Note that the squared matrix element has the dimension L2 Enrg−1 so that σa is still a genuine cross section, with dimension L2 , measured e.g. in [σa ] = b (see Appendix A.2). Extending now the definition (H.34) to the continuum, one introduces the optical oscillator strength density (OOSD) (opt)

dfa d

=2

ω + WI | Da |2 = 2 |za |2 . 2 E h a0 Eh a02

(5.62)

The photoionization cross section may thus be written as (opt)

σa (ω) = 2π 2 αEh a02

dfa . d

(5.63)

The numerical prefactor is the same as for (5.32) while g(ω) is now included in the OOSD. It has the dimension Enrg−1 – in contrast to excitation of discrete states (opt) where the oscillator strength fba for a specific transition is a dimensionless number. In the literature one also finds different notations of (5.63). They may, however, be transferred into each other by (H.34)–(H.44). For linear polarization, often the field vector is assumed to point into z-direction, without loss of generality. With (4.77) one finds (still neglecting the electron spin) za = |z|a = r|n m C10 | m . (5.64) Various other choices of geometry are possible – in analogy to the treatment of transitions between bound states which we have discussed earlier.

5.5.2

BORN Approximation for Photoionization

Following B ETHE and S ALPETER (1957), for high but not relativistically high photon energies WI ω me c2 12 Alternatively

(5.65)

and completely equivalent one may employ continuum wave functions which are determined as an orthogonal basis set for a large but finite volume L3 . In this case a density of states corresponding to (5.31) has to be used in (5.61).

5.5

Photoionization

257

Fig. 5.8 Coordinates and angles for describing photoionization

z θ K k e

γ

ke y

φ

x

one may approximate the continuum wave function of a free electron by a plane wave (1st order B ORN approximation, FBA): me ke ike ·r r|k e = e . (5.66) (2π)3 2 The normalization factor under the square root is the density of states in an element of solid angle dΩ per unit volume (dimension L−3 ) and energy according to (2.58). The geometry for describing the photoionization process is sketched in Fig. 5.8: the ionizing light propagates into z-direction with a wave vector k. We assume it to be linearly polarized, its polarization vector e pointing in x-direction. After ionization the electron is emitted into the direction k e . At these high energies we can no longer use the dipole approximation. Rather the full expression (H.22) for the transition operator has to be employed: D=

1 ik·r 1 ik·r e e · p= e p E . ωme ωme

(5.67)

We denote the component of p in the direction of the polarization vector e as p e – as introduced in (2.50). The transition matrix element for photoionization of the state |a is thus written Da = k e | D|a = k e |

1 ik·r 1 ik·r e p e |a = pe k e | e |a . ωme ωme

(5.68)

The latter identity holds since p e e ⊥ k so that p e and eik·r commute, and we can apply (2.35). We recall now that according to (2.51) each plane wave is an eigenvector of p e with the eigenvalue p cos γ = ke cos γ , where γ is the angle between polarization vector e and the direction of electron emission k e . Thus, applying (5.66) the matrix element is written in integral form: Da = k e | D|a =

3/2

ke √ cos γ (2π)−3/2 ω me

With K = k e − k we may abbreviate −3/2

ψa (K) = (2π)

ei(k−k e )·r ψa (r)d3 r.

e−iK·r ψa (r)d3 r

(5.69)

(5.70)

258

5


which is just the F OURIER transform of the initial wave function ψa (r), i.e. its repDa refers to electrons which resentation in momentum space. The matrix element are emitted into a specific solid angle. Hence, inserting (5.69) into (5.61) leads to the differential cross section for photoionization in FBA: dσa (ω) = 4π 2 α

2 1 ke3 cos2 γ ψa (k e − k) dΩ. ω me

(5.71)

The function ψa (k e − k) may be derived either by (5.70) from the wave function in space or directly from the S CHRÖDINGER equation transformed into momentum space (see e.g. B ETHE and S ALPETER 1957, Chap. 8).

Specialization to ns Initial States and H Like Atoms The general form of the F OURIER transform is ψa (p) = Fn (p) Y m (θ, ϕ). For ns initial states with radial wave functions Rns (r) it may be written as: π ∞ 1 2 Rns (r)r dr e−iK·r sin θ dθ. (5.72) ψa (K) = √ 2π 0 0 For the angular integration we choose z K so that e−iK·r = e−iKr cos θ and ∞ 2 drRns (r)r sin Kr (5.73) ψa (K) = √ K 2π 0 which can be readily integrated. For large K = |k e − k| and hydrogen like initial ns states it may be approximated by √ 3/2 a0 2 2Z 5/2 ψns (k e − k) = πn3/2 a04 |k e − k|4

(5.74)

(B ETHE and S ALPETER 1957, Eq. (70.4), in a.u.). Introducing this into (5.71) leads to a differential photoionization cross section for H like ns electrons: dσa (ω) = 4π 2 α = 32α

a3 1 2 ke3 8Z 5 cos2 γ 2 3 8 0 dΩ ω me π n a0 |k e − k|8

cos2 γ Z5 1 dΩ. me n3 ω (a0 ke )5 [1 − (v/c) cos θ ]4

(5.75)

The expression in brackets [ ] in the second row is obtained by exploiting (5.65) 2 2 ∼ so that with (5.58) √ and (5.59) ω = = ke /2me ; with the velocity of the ejected electron v = 2/me and the photon momentum k = ω/c one obtains 2k/ke = v/c 1 and neglects (k/ke )2 . Consequently, further expansion in powers of v/c and neglecting all nonlinear terms finally gives dσa = 64α

Z 5 a02 cos2 γ [1 + 4(v/c) cos θ ] dΩ. n3 (2/Eh )7/2

(5.76)

5.5

Photoionization

259

Angular distributions will be discussed in the next subsection. Here we just integrate over the full solid angle. From Fig. 5.8 we read cos γ = sin θ cos ϕ and integration 4π 2π π over . . . dΩ = 0 dϕ 0 . . . sin θ dθ simply gives a factor 4π/3. Thus, for H like atoms in ns states the integral photoionization cross section in FBA is given by:13 4π a02 dσa 256π Z 5 σns = . (5.77) dΩ = α 3 dΩ 3 n (2ω/Eh )7/2 0 Even if this result of the B ORN approximation is only valid at high (but not relativistically high) energies, it describes the general trend for H like orbitals reasonably well: a dramatic decrease of the cross section with photon energy ∝ (ω)−7/2 , and strong dependence on nuclear charge ∝ Z 5 which leads to substantial photoionization cross sections for the heavier atoms, as well as the decrease for higher orbitals ∝ n−3 . As we shall see in Chap. 10 the inner shells of large atoms may very well be described by H like orbitals so that the FBA leads to quite realistic first estimates for absorption of X- and γ -rays (see Sect. 10.5.3). A comprehensive data source for such cross sections is C HANTLER et al. (2005). Specifically for photoionization of the H 1s orbital (5.77) gives in numbers: σ1s / cm2 = 1.609 × 10−23 (ω/ keV)−7/2 .

(5.78)

Even though FBA is a high energy approximation, it is instructive to show it over the whole energy range. One has to replace the approximation (5.74) by the exact F OURIER transform of the wave function and obtains σ1s / cm2 = 1.609 × 10−23 (ω − WI )3/2 (ω/ keV)−5 .

(5.79)

Thus, FBA predicts the photoionization cross section to disappear at threshold ω = WI = 13.6 eV, to rise quickly to a maximum at around 20 eV and then to decrease ∝ (ω)−7/2 . As illustrated in Fig. 5.9 this is not even close to reality at threshold: Generally one finds (also from theoretical arguments) that photoionization cross sections are finite at threshold (see also Sect. 7.2.7 in Vol. 2). They decrease rapidly with increasing energy but convergence of FBA and the ‘exact’ cross section from the NIST data base (C HANTLER et al. 2005) is reached only when ω is several times WI . At ω = 10WI the cross section is already below 1 % – a rather typical behaviour. The NIST data shown in Fig. 5.9 are based on a careful selection of state-of-the-art theory. The experimental data in Fig. 5.9 are from one of the very few quantitative data sets for photoionization of atomic hydrogen. They have been determined from a H plasma in a shock tube (PALENIUS et al. 1976). Considering the experimental difficulties one may speak about satisfactory agreement between theory and experiment. We point out, however, that the extremely structureless behaviour seen in Fig. 5.9 is a specialty of the H atom. Already for He (Chap. 7) we shall see between the 13 Since

the integral is independent of ϕ the same expression holds for unpolarized light.

5 experiment (shock tube) 4

(b)

10-20

exact calc. (NIST)

2


10-18

(a)

BORN approximation

σє1s / cm2

σє1s / 10-18 cm2

260

10-22 10-24 10-26 10-28 10-30

0 20

40 60 ħω / eV

80

100

10-32 0.01

0.1

1 10 log( ħω / keV )

100

Fig. 5.9 Cross section for photoionization of atomic hydrogen as a function of photon energy ω. Exact calculations (red lines) according to C HANTLER et al. (2005) are compared with the B ORN approximation (black lines) computed by (5.79), and with experimental data (grey squares) from PALENIUS et al. (1976). (a) Linear plot for low ω, (b) logarithmic plot for energies from 13.7 eV to 400 keV. The dashed line indicates the decrease ∝ W −7/2

first and the second ionization threshold (He+ and He++ , respectively) a rich spectrum of so called autoionizing lines, superposed on the general trend of the photoionization cross section. For larger atoms, the photoionization cross section may even pass at low energies through a so called C OOPER minimum as will be discussed in Chap. 10. Nevertheless, in the limit of very high photon energies one always finds the dramatic decrease of the photoionization cross section discussed here.

5.5.3

Angular Distribution of Photoelectrons

The angular distribution of the emitted photoelectrons for polarized light is characterized according to (5.76) mainly by cos2 γ = sin2 θ cos2 ϕ. At high photon energies the term 4(v/c) cos θ leads to a shift of the angular distribution into the direction of the ionizing light, but may in general be neglected for non-relativistic electrons. Without derivation we note here the result for the general, low energy case. Strict application of the dipole approximation (E1) leads to an angular dependence of the differential photoionization cross section dσa σa = 1 + βP2 (cos γ ) dΩ 4π

where −1 ≤ β ≤ 2,

(5.80)

which is characterized by the so called β parameter (also anisotropy parameter), with the L EGENDRE polynomial P2 (cos γ ) = (3 cos2 γ − 1)/2. For ionization of pure ns states as in H(1s) case the just discussed, (5.76) is reproduced with β = 2 – neglecting the high energy term which is included in the B ORN approximation in contrast to the simplified dipole approximation. The maximum of the electron signal for ns ionization is thus found at γ = 0, i.e. in the direction of the polarization vector e (see Fig. 5.8) while perpendicular to it, i.e. parallel to the wave vector k of the ionizing light no electrons are emitted.

5.5

Photoionization

261

One may say, the angular distribution of the state generated by linear polarized light corresponds to a |px orbital. Its angular probability distribution is represented by the angular distribution of the photoionization cross section. For not too high photon energies the continuum states generated by photoionization are thus equivalent to those obtained in dipole excitation (see Sect. 4.7). For unpolarized or circularly polarized light one simply has to average the differential cross sections for polarization in x- and y-direction. According to Fig. 5.8 this implies averaging over the corresponding polarization angles with cos2 γ = sin2 θ cos2 ϕ and = sin2 θ sin2 ϕ, respectively and leads to

dσa β σa 1 − P2 (cos θ ) . = dΩ 4π 2

(5.81)

For β = 2 this characterizes a torus shaped (‘doughnut’ like) distribution around the z-axis. Both expressions are generally valid for not too high ω in photoionization of atoms and molecules initially prepared in an isotropic population of states. One finds, however, that β usually depends on photon energy. As we shall see, β is an important quantity which describes the contributions of different partial waves in single photon ionization processes. We finally note that the photoelectron yield emitted at the so called magic angle γmag = 54.736◦ – for which P2 (cos γmag ) = 0 – does not depend on β. Systematic studies of the integral photoionization cross section σa are preferably done at this emission angle (or at θ = γmag , respectively).

5.5.4

Cross Sections in Theory and Experiment

The B ORN approximation is a high energy approximation. Its advantage is its simplicity and the fact that it includes all multipole moments of the incoming plane wave up to infinite order. We have seen, however, that it does not give reliable results at energies ω below about 5 to 10 × WI . Yet it is this very energy range that contains most of the continuum oscillator strengths, and provides the background for many interesting atomic and molecular phenomena. Fortunately, one may in principle apply here the electric dipole (E1) approximation which according to (5.62) and (5.64) just requires an accurate computation of the relevant radial matrix elements. Thus, one ‘only’ has to determine sufficiently exact wave functions for the initial bound and the final continuum states. As discussed in detail in Chaps. 2 and 3, the initial state |a of the system is described by a wave function with well defined quantum numbers n m: |a = Rn (r)Y m (θ, ϕ) =

un Y m (θ, ϕ). r

(5.82)

262

5


The asymptotic behaviour of the radial functions for the bound states is according to (2.119) and (2.120) (5.83) un (r) −→ r +1 and un (r) −→ exp − 2|Wn |r . r→∞

r→0

In contrast, the continuum wave functions | for a given energy of the free electron may in principle contain all orbital angular momenta :

| =

∞ =0 m =−

a

u (r) Y m (θ, ϕ). r

(5.84)

The expansion coefficients a characterize the respective boundary and ionization conditions, while the radial functions u (r) = rR (r) have to be calculated from the radial S CHRÖDINGER equation (2.117). There, the binding energy Wn has to be replaced by the continuum energy = k 2 /2 (in atomic units) and the potential has to be adapted. While stable solutions for bound states exist only for well defined, discrete binding energies, to any positive energy a meaningful wave function may be determined. As already discussed in Sect. 3.2.6, wave functions in the continuum have to be normalized on the energy scale as described in Appendix J. In the case of photoelectron emission (5.84) has to represent a radial outgoing electron flux and the functions u (r)/r have essentially the character of spherical waves. Their asymptotic behaviour (in atomic units) is given by u (r) −→ r +1 r→0

u (r) −→

r→∞

and

π ZC 2 cos kr − + ln(2kr) + δ + σ πk 2 k

(5.85) (5.86)

√ with k = 2, the magnitude of the wave vector,14 and the remaining charge ZC of the atom after the ionization process. For photoionization the outgoing wave in the field of the remaining ion is characterized the standard phase shift δ of the wave in respect of a free C OULOMB wave in a −ZC /r potential, by σ , the C OULOMB phase shift (3.28) which rapidly decreases with k and by its characteristic logarithmic phase.15 In the case of √ photo-detachment (5.60) from anions (ZC = 0) the asymptotic radial function is 2/πk cos(kr − π/2 + δ ). In practice, it may often suffice to determine the radial functions un (r) and u (r) by solving the simple radial, one particle S CHRÖDINGER equation (2.117). However, depending on the complexity of the system studied and on the desired quality of the computation it may become nec14 In

the literature the energy is often given in RYDBERG units so that the factor 2 under the root is omitted.

15 For

the ionization of H like atoms only a naked ion remains so that for the outgoing wave in the pure C OULOMB potential δ = 0. For all other cases δ is related with the quantum defect μ according to (3.26).

5.5

Photoionization

263

essary to use more sophisticated methods, such as multi configuration H ARTREE F OCK (MCHF) to be discussed in Chap. 10. Finally, for multi-electron systems one has to sum over the ionization cross sections for all electrons in the atom or molecule studied which can be ionized at a given photon energy. With the thus characterized wave functions of the initial bound state (5.82) and the final free continuum state (5.84) the integral cross section can be calculated. Without loss of generality as in (5.64) we again assume the polarization vector D = ie · r = to point into z-direction, i.e. we set the dipole transition operator irC10 (θ, ϕ). Then (5.61) gives after averaging over all initial states and summation over all accessible final states: 2 4π 2 αω C10 | m . σa (ω) = m (5.87) r|n 2 + 1 mm

While the radial matrix elements |r|n are determined by the specific properties of the atom and the photon energy, the matrix element m |C10 | m (see Appendix C.2.2) may be evaluated in close analogy to Sect. 4.4 where we have described optical excitation processes. This leads to corresponding dipole selection rules for photoionization: again = ±1 must hold, since a photon with angular momentum is absorbed and the transition = 0 is forbidden as a consequence of parity conservation. We note, however, that in the continuum both final states, with = ± 1 may be superposed at each photon energy ω above the photoionization threshold. Also the selection rule m = m holds again for light linearly polarized in z-direction. Evaluation of the sum and the matrix elements in (5.87) finally leads to σa (ω) =

2 4π 2 2 αω r, −1 + ( + 1)r, +1 3

with the corresponding radial matrix elements ∞ ru ±1 (r)un (r)dr. r, ±1 =

(5.88)

(5.89)

0

Hence, unless the initial state is an s state with = 0 the photoionization cross section contains contributions from two matrix elements involving continuum functions with = + 1 and − 1. With some additional effort the continuum function (5.84) may be determined in such a way that it describes an electron emitted into the direction k e . Without going into details this leads to (5.80) with an energy dependent anisotropy parameter (see Eq. (2) in C OOPER and Z ARE 1968a,b): β() =

1 2 + ( + 1)r, +1 ]

2 (2 + 1)[ r, −1

' 2 2 × ( − 1)r, −1 + ( + 1)( + 2)r, +1 ( − 6 ( + 1)r, +1 r, −1 cos δ +1 () − δ −1 () .

(5.90)

5

Fig. 5.10 Angular distribution of electrons at the photo-detachment from Cu− according to C OVINGTON et al. (2007)

Linewidths, Photoionization, and More e- yield / arb. un.

264

1

0 - 45o 0o

45o 90o 135o 180o

polarization angle γ

Obviously β is determined by the ratio r, −1 /r, +1 of the two radial matrix elements involved and by the phase differences δ +1 () − δ −1 () of the two partial waves. One easily verifies that for = 0 (and thus r, −1 = 0) β = 2 holds – as already found by the B ORN approximation. One also confirms without difficulty that in the general case −1 ≤ β ≤ 2. Figure 5.10 illustrates this by way of example for an experimentally determined angular distribution in the photo-detachment from an anion Cu− [Ar]3d 10 4s 2 1 S + ω → Cu[Ar]3d 10 4s 2 S + e− studied by C OVINGTON et al. (2007). Since the electron configuration of the copper anion may be considered He like (two s electrons on top of a closed 3d shell), and since one s electron is detached, one expects β = 2 and hence according to (5.80) a clean cos2 γ angular distribution. Figure 5.10 largely confirms this expectation. A small deviation cannot be excluded by the experiment (the minimum at 90◦ may not be exactly zero). One would attribute this to the fact that the description of the electron wave function by a pure product wave function of all orbitals (in this case a 4s electron has to be detached) is not completely correct for such a large atom. Rather one has to account for some configuration mixing as to be discussed in Sect. 10.2.3. Figure 5.11 shows the rather typical ion beam apparatus used in this experiment. The setup is in principle uncomplicated and self explaining. It requires, however, high experimental precision. The determination of the kinetic energy of the electrons is achieved by a 160◦ spherical sector analyzer (for a hemispherical analyzer see Appendix B.3 in Vol. 2) which is characterized by its high resolution and good focussing properties. The angular distribution shown in Fig. 5.10 has been recorded as usual in such experiments: one rotates the laser polarization which is much easier than turning the whole electron analyzer-detector setup. For θ = 0 the polarization direction is parallel to the direction of electron detection. Quantitative computations of cross sections for such multi-electron atoms require a major effort. First, the wave functions for bound and continuum states have to be determined accurately. Then the one electron dipole operator has to be replaced by (H.29) summing over all electron coordinates. For a realistic description of such an atom one also has to account for the spin and angular momentum coupling. The ensuing matrix elements have to be evaluated using the angular momentum algebra outlined in Appendix C.

5.5

Photoionization

265

4m insertable FARADAY cup

2.4 m insertable 90o selector insertable FARADAY FARADAY magnet cup cup

polarization einzel rotor lens einzel polarizer lens aperture am e l a b ser o ion bea m 45 e FARADAY laser power meter cup spherical electron energy analyzer (160o segment)

adjustable beam slits electrostatic steering elements

anion source

Fig. 5.11 Typical experimental setup for studying photo-detachment from anions, here according to C OVINGTON et al. (2007). The ion beam is focussed with several lenses and guided by deflectors. A 90◦ magnet selects one anion species which crosses in the interaction region a CW laser beam. The polarization of the latter may be rotated

V(r ) / Eh 0.5 - 15

Nħω

eWkin

15 r /a0

WI - 0.5 - 1.0

Fig. 5.12 Illustration of multi-photon ionization for the example of an H atom. In the continuum the electron has a kinetic energy Wkin ≡ corresponding to (5.91) depending on the number of absorbed photons N ω (the schematic is drawn to scale in a.u. for photons with a wavelength λ = 800 nm). The electrons kinetic energy Wkin is indicated by downward pointing red arrows, which indicate here that 0 to 5 additional photons beyond those necessary for ionization have been absorbed (ATI)

5.5.5

Multi-photon Ionization (MPI)

Processes of N th order (perturbation theory) correspond to the absorption of N photons as already discussed in Sect. 5.3 for excitation processes (and correspondingly for emission). Of course, ionization of a target atom or molecule (Tg) is also possible by such processes, even if the individual photon energy is below the ionization potential, ω < WI . This is sketched in the potential energy diagram Fig. 5.12.

266

5

Fig. 5.13 Photoelectron emission spectra for Xe according P ETITE et al. (1988), clearly showing ATI. Nd-YAG laser pulses of a wavelength 1064 nm (1.165 eV) at two different laser intensities I are compared. While (a) at I = 2.2 × 1012 W cm−2 only one or two extra photons above threshold are absorbed, (b) this number rises up to about nine if the intensity is increased by a factor of only five


I = 2.2×1012 Wcm-2

electron signal / arb. un.

(a)

1.165eV

I = 1.1×1013 Wcm-2

(b)

0

2

4

6 8 Wkin / eV

10 12 14

The process and its energy balance is now given by Tg + N ω → Tg+ + e− (Wkin ) with −WI + N ω = Wkin =

2 ke2 . 2me

(5.91)

Obviously the kinetic energy Wkin of the electron after ionization depends on the number N of photons absorbed. The probability for a multi-photon ionization process (MPI) depends on the photon flux Φ = I /(ω) – just as in the case of multiphoton excitation. In N th order perturbation theory one writes the transition rate from a bound initial state |a into the continuum state | as (N )

Rke ←a = σ (N ) Φ N ∝ I N ,

(5.92)

where σ (N ) is a generalized ionization cross section as we already know it from the excitation process described by (5.43). As illustrated in Fig. 5.12 it is possible that more photons are absorbed than necessary to reach the threshold for ionization – so called above-threshold ionization (ATI) process. A very clear experimental verification is shown by the photoelectron emission spectra in Fig. 5.13. In this, by now classic, experiment of P ETITE et al. (1988) Xe atoms were ionized with Nd-YAG laser pulses (λ = 1064 nm) of 130 ps duration. The photon energy is 1.165 eV and the spectra document that far more photons are absorbed than the 11 necessary for reaching the ionization threshold – clear evidence for the ATI process: the additional energy is found as kinetic energy Wkin of the electrons according to the energy balance in (5.91). Comparison of Fig. 5.13(a) and (b) shows that the number of additional photons absorbed rises quickly with increasing laser intensity. For MPI the angular distribution of the emitted electrons may become much more complicated than for one-photon ionization, which is described by (5.81) implying an angular momentum transfer of ±. If N photons are absorbed, angular momentum transfer of up to ±N is possible and leads to a correspondingly expanded

5.5

Photoionization

267

summation over all accessible continuum states from (5.84) which will be reflected in the differential cross section. Considering parity conservation, for a two-photon ionization process the partial waves = and ± 2 contribute (as long as ≥ 0), for a three photon process we have = ± 1 and ± 3 and so forth. When linearly polarized light is used the selection rule m = 0 remains valid, but for circularly polarized light also higher values of −N ≤ m ≤ N are accessible. It is obvious that the angular distribution of the emitted electrons is correspondingly more complex.16 As in multi-photon excitation, discussed in Sect. 5.3, the transition amplitudes are now to be treated at least in N th order perturbation theory. In the most simple case, two-photon ionization of an initial single electron n state to a final state of energy by linearly polarized light, the generalized differential ionization cross section according to C OOPER and Z ARE (1969) is (2) dσn (ω, θ ) m|r0 |γ m γ m|r0 |n m 2 ∝ , dΩ Wn − Wγ + ω

(5.93)

γ

using the transition amplitudes in the helicity basis introduced in (4.79) and (4.77). One has to average over all initial substates n m, to sum over all intermediate states γ m (in principle over all discrete bound states as well as over continuum states), and over all final continuum states m. One may guess that the evaluation of this expression may become a formidable enterprize. For circular polarization, or different polarization in each of the two steps, the expression becomes even more complicated. One sees that this gets even more intricate if one of the intermediate states may be excited resonantly or nearly resonantly (ω Wγ − Wn ). Then the typical damping terms must be included, and one has to account for possible saturation effects: in the limit one just populates (non-isotropically) an intermediate state from which ionization occurs in the second step. As a rule, (5.81) will not describe the angular distribution of the emitted electrons after MPI. Rather, the (generalized) differential cross section for N photon ionization will involve several β parameters: (N ) (N ) (ω, θ ) σn dσn 1 + β2 cos2 θ + β4 cos4 θ + · · · + β2N cos2N θ = . dΩ 4π 1 + β2 /3 + β4 /5 + · · · + β2N /(2N + 1)

(5.94)

Owing to the continuous progress with intense, tuneable and reliable laser systems the interest in studying such processes experimentally has increased tremendously over the paste decades. Multi-photon ionization, in particular via resonant intermediate states (so called resonantly enhanced multi-photon ionization, REMPI) is today a universally employed tool for studying the structure and dynamics of atoms, molecules, clusters and even biomolecules. 16 Mutatis

mutandis this also holds for single-photon ionization of non-isotropically populated initial states which may e.g. be prepared by additional photons in an optical pumping process with polarized light (see Appendix D in Vol. 2 or H ERTEL and S TOLL 1978).

5

electron signal

268

0

7p 2P3/2

8p 2P3/2

1+1 REMPI

1+1 REMPI

90

180


0 90 180 polarization angle θ / o

8d 2D5/2 2+1 REMPI

0

90

180

Fig. 5.14 Resonant multi-photon ionization of Cs atoms according to C OMPTON et al. (1984). Measured data (points) and computed (red lines) angular distributions of the photoelectron yield for the processes listed in (5.95)

We have to confine the discussion here on two examples. C OMPTON et al. (1984) have studied REMPI transitions in Cs atoms. The experimental setup is similar to that shown in Fig. 5.11, but much more simple since the target atoms are generated by a simple atomic beam oven. However, electron detection and polarization rotation are practically the same. The light source was a pulsed dye laser, tuneable to a number of wavelengths, with 10 ns pulse duration and an intensity of moderate 108 W cm−2 . Cs atoms have been excited with one or two photons being in resonance with the intermediate state – depending on selection rules. One further photon of the same wavelength served for ionization. Observed were the following processes: ⎧1ω 1ω 1 ⎪ −→ Cs(7p 2 P3/2 ) −→ Cs+ + e− ( s and p) ⎪ ⎨ 1ω 2 Cs 6s 2 S1/2 1ω −→ Cs(8p 2 P3/2 ) −→ Cs+ + e− ( s and p) ⎪ ⎪ ⎩2ω3 1ω −→ Cs(8d 2 D5/2 ) −→ Cs+ + e− ( s and p).

(5.95)

Figure 5.14 shows the measured and calculated angular distributions of photoelectrons. Comparison with Fig. 5.10 emphasizes the rich structure of the photoelectron signal as a function of polarization angle θ for the present MPI processes. This is what we expect according to (5.94). We shall return to this in the context of intense laser fields, e.g. in Sect. 8.5.3. In recent years so called electron imaging spectrometers (EIS) or velocity map imaging (VMI) devices have been developed. With these very elegant and efficient methods one collects in one ‘shot’ (e.g. for one laser pulse) a complete angular and energy distribution of the photoelectrons – such images have, of course, to be averaged over many events. Several variants of these techniques are used, all of them based on the fact that√the photoelectrons expand around their point of origin on spheres of radius r = t 2Wkin /me . At a well defined time t one projects this spherical shell onto an extended position sensitive detector and determines the kinetic energy Wkin – and by determining x/t and y/t also the emission angles θ and ϕ in respect of the polarization vector.

Photoionization

-160 V

pp

3 cm

0V

-1

00

(b)

0 -5 y

0

/ pix0 el

50

phosphorescent screen MCP +1600 V 6 cm

(a)

269 electron signal / arb. un.

5.5

0 00 10 -1

x

/p

ix

0 10 el

Fig. 5.15 Imaging photoelectron spectroscopy for H− according to R EICHLE et al. (2001). (a) Optical EIS system with a special field shape (red dotted the equipotential lines) and two trajectories for equal momenta p leading to the same detector position. (b) Photoelectron signal from H− in an infrared laser field with I = 1.7 × 1013 W cm2 . The polarization direction of the laser points into y-direction. The angular distribution of the photoelectrons is projected onto the xy plane, the distance x 2 + y 2 reflects the electron kinetic energy

Here we show in Fig. 5.15(a) a typical setup used for a nice experiment with atomic hydrogen anions by the group of Hanspeter H ELM, one of the pioneers of these methods. With a judiciously shaped electric field one optimizes the collection of electrons which are amplified by a multi-channel plate electron multiplier (MCP), detected by a phosphorescent screen, and finally optically registered and digitized by a CCD camera. Figure 5.15 shows a measured electron distribution for the photo-detachment from H− with a typical ATI structure: The distance from the centre represents Wkin , and each concentric angular distribution corresponds to one particular value of N according to (5.91) each with its own nontrivial angular distribution corresponding to (5.94). We shall come back to these powerful techniques at several occasions, e.g. in Chap. 5 and Appendix B.4, both in Vol. 2. Section summary

• Photoionization processes may be described as transitions between bound and continuum states, induced by the electromagnetic field. In dipole approximation the cross section (5.61) is simply proportional to the absolute square of the respective transition matrix element. It may be expressed in terms of the OOSD (5.62). • FBA may be used to obtain an estimate of the cross sections at photon energies ω > 5 to 10 × WI . At threshold (ω = WI ) the photoionization cross section is finite (in contrast to the prediction of FBA). • The angular distribution of the ejected electron is an important characteristic of the photoionization process, essentially imaging the continuum wave functions accessed. For ionization with linearly polarized light one observes in the general case a distribution ∝ [1 + βP2 (cos γ )], with β being the anisotropy parameter (−1 ≤ β ≤ 2) and γ the angle between polarization and ejected electron.

270

5


• Multi-photon ionization is a more complex process and must be treated at least in N th order perturbation theory (if N photons are absorbed), allowing for ATI where the excess energy is carried by the ejected electron as given by (5.91). The angular distributions of ATI electrons are correspondingly more intricate since angular momenta up to ±N may be exchanged between the electromagnetic field and the ejected electron.

Acronyms and Terminology ATI: ‘Above-threshold ionization’, in multi-photon ionization, if more photons are absorbed than necessary for ionization. a.u.: ‘atomic units’, see Sect. 2.6.2. CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). 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. EIS: ‘Electron imaging sprectrometer’, specifically for photoelectron spectroscopy. EPR: ‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2). FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sect. 6.6, Vol. 2 and Sect. 5.5.2, respectively). FWHM: ‘Full width at half maximum’. HITRAN: ‘High-resolution transmission molecular absorption database’, http:// www.cfa.harvard.edu/hitran (ROTHMAN et al. 2009). HWHM: ‘Half width at half maximum’. M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. M2: ‘Magnetic quadrupole’, transitions induced by the interaction of a magnetic quadrupole with the magnetic field of electromagnetic radiation. MCHF: ‘Multi configuration H ARTREE -F OCK’, method to determine wave functions for multi-electron systems (see Sect. 10.5.4). MCP: ‘Multi channel plate’, electron multiplier with many amplifying elements. MPI: ‘Multi-photon ionization’, ionization of atoms or molecules by simultaneous absorption of several photons. 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). ODE: ‘Ordinary differential equation’.

References

271

OOSD: ‘Optical oscillator strength density’, characterizes the strength of photoionization per energy interval (see Sect. 5.5.1). REMPI: ‘Resonantly enhanced multi-photon ionization’, ionization of atoms or molecules by several photons with one resonant intermediate state. 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, Vol. 2). VUV: ‘Vacuum ultraviolet’, spectral range of electromagentic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References B ETHE , H. A. and E. E. S ALPETER: 1957. Quantum Mechanis of One- and Two-Electron Atoms. Berlin, Göttingen, Heidelberg: Springer Verlag, 369 pages. B OUCHIAT , M. A.: 2007. ‘Linear stark shift in dressed atoms as a signal to measure a nuclear anapole moment with a cold-atom fountain or interferometer’. Phys. Rev. Lett., 98, 043003. B OUCHIAT , M. A. and C. B OUCHIAT: 1997. ‘Parity violation in atoms’. Rep. Prog. Phys., 60, 1351–1396. B URGESS , A. and M. J. S EATON: 1960. ‘A general formula for the calculation of atomic photoionization cross sections’. Mon. Not. R. Astron. Soc., 120, 121–151. C HANTLER , C. T., K. O LSEN, R. A. D RAGOSET, J. C HANG, A. R. K ISHORE, S. A. KO TOCHIGOVA and D. S. Z UCKER : 2005. ‘X-ray form factor, attenuation, and scattering tables (version 2.1)’, NIST. http://physics.nist.gov/ffast, accessed: 7 Jan 2014. C OMPTON , R. N., J. A. D. S TOCKDALE, C. D. C OOPER, X. TANG and P. L AMBROPOU LOS : 1984. ‘Photoelectron angular-distributions from multi-photon ionization of cesium atoms’. Phys. Rev. A, 30, 1766–1774. C OOPER , J. and R. N. Z ARE: 1968a. ‘Angular distribution of photoelectrons’. J. Chem. Phys., 48, 942–943. C OOPER , J. and R. N. Z ARE: 1968b. ‘Correction’. J. Chem. Phys., 49, 4252. C OOPER , J. and R. N. Z ARE: 1969. ‘Photoelectron angular distributions’. In: S. G ELTMAN et al., eds., ‘Lectures in Theoretical Physics’, vol. XI-C, 317–337. New York: Gordon and Breach. C OOPER , J. W.: 1962. ‘Photoionization from outer atomic subshells. A model study’. Phys. Rev., 128, 681–693. C OOPER , J. W.: 1988. ‘Near-threshold K-shell absorption cross-section of argon – relaxation and correlation-effects’. Phys. Rev. A, 38, 3417–3424. C OVINGTON , A. M. et al.: 2007. ‘Measurements of partial cross sections and photoelectron angular distributions for the photodetachment of Fe− and Cu− at visible photon wavelengths’. Phys. Rev. A, 75, 022711. FANO , U. and J. W. C OOPER: 1968. ‘Spectral distribution of atomic oscillator strengths’. Rev. Mod. Phys., 40, 441–507. G ÖPPERT-M AYER , M.: 1931. ‘Über Elementarakte mit zwei Quantensprüngen’. Ann. Phys. (Berlin), 9, 273–294. H ÄNSCH , T. W.: 2005. ‘N OBEL lecture: passion for precision’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/2005/hansch-lecture.html.

272

5


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. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. L AMBROPOULUS , P.: 1985. ‘Mechanisms for multiple ionization of atoms by strong pulsed lasers’. Phys. Rev. Lett., 55, 2141–2144. L IPELES , M., R. N OVICK and N. T OLK: 1965. ‘Direct detection of 2-photon emission from metastable state of singly ionized helium’. Phys. Rev. Lett., 15, 690–693. M ADDEN , R. P. and K. C ODLING: 1963. ‘New autoionizing atomic energy levels in He, Ne, and Ar’. Phys. Rev. Lett., 10, 516–518. M ANSON , S. T. and A. F. S TARACE: 1982. ‘Photo-electron angular-distributions – energydependence for s subshells’. Rev. Mod. Phys., 54, 389–405. N OVICK , R.: 1972. ‘2-Photon decay of metastable hydrogenic atoms’. Science, 177, 367. PALENIUS , H. P., J. L. KOHL and W. H. PARKINSON: 1976. ‘Absolute measurement of photoionization cross-section of atomic-hydrogen with a shock-tube for extreme ultraviolet’. Phys. Rev. A, 13, 1805–1816. P ETITE , G., P. AGOSTINI and H. G. M ULLER: 1988. ‘Intensity dependence of non-perturbative above-threshold ionization spectra – experimental-study’. J. Phys. B, At. Mol. Phys., 21, 4097– 4105. R EICHLE , R., H. H ELM and I. Y. K IYAN: 2001. ‘Photodetachment of H− in a strong infrared laser field’. Phys. Rev. Lett., 87, 243001. ROTHMAN , L. S. et al.: 2009. ‘The HITRAN 2008 molecular spectroscopic database’. J. Quant. Spectrosc. Radiat. Transf., 110, 533–572. S AHA , H. P.: 1989. ‘Threshold behavior of the M-shell photoionization of argon’. Phys. Rev. A, 39, 2456–2460. S CHÄFFER , H. W., R. W. D UNFORD, E. P. K ANTER, S. C HENG, L. J. C URTIS, A. E. L IVINGSTON and P. H. M OKLER : 1999. ‘Measurement of the two-photon spectral distribution from decay of the 1s2s S-1(0) level in heliumlike nickel. Phys. Rev. A, 59, 245–250. S CHMIDT , V.: 1992. ‘Photoionization of atoms using synchrotron radiation’. Rep. Prog. Phys., 55, 1483–1659. T ONG , D., S. M. FAROOQI, E. G. M. VAN K EMPEN, Z. PAVLOVIC, J. S TANOJEVIC, R. C OTE E. E. E YLER and P. L. G OULD: 2009. ‘Observation of electric quadrupole transitions to Rydberg nd states of ultracold rubidium atoms’. Phys. Rev. A, 79, 052509. W IKIPEDIA CONTRIBUTORS: 2013. ‘Atmospheric radiative transfer codes’, Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/wiki/Atmospheric_radiative_transfer_codes, accessed: 8 Jan 2014.

6

Fine Structure and L AMB Shift

We now refine our treatment of atoms with an essential step: we include the electron spin and its interactions. In this context we also meet the L AMB shift, very important in modern physics as it is one of the major testing grounds for QED. To study such fine details of atomic spectra one needs sophisticated spectroscopic methods into which we give a first introduction.

Overview

This chapter focuses on some key topics of atomic physics. It should be studied carefully. To start, Sect. 6.1 introduces some state-of-the-art experimental methods used today in high precision spectroscopy. Section 6.2 introduces into the phenomenology and terminology of spin-orbit interaction and angular momentum coupling, while Sect. 6.3 may be seen as the quantitative counterpart – perhaps somewhat strenuous, but important. The focus is on the H atom, described by the D IRAC theory and the only system for which fine structure (FS) splittings can be computed exactly. Alkali atoms serve as further examples. Section 6.4 is concerned with selection rules and intensities for FS transitions, a somewhat dry but necessary subject which the reader may perhaps skip on first reading and come back later. The L AMB shift is presented in Sect. 6.5 with the classical experiments as well as with some results from modern high precision spectroscopy for the H atom and highly charged ions. Understanding the physics should be supported by Sect. 6.5.6 with a very brief introduction to F EYNMAN diagrams and a glimpse on QED. Section 6.6 concludes the chapter with a short excursion into the physics of the electron magnetic moment anomaly.


273

274

6


6.1

Methods of High Resolution Spectroscopy

6.1.1

Grating Spectrometers

For precise measurements of atomic and molecular spectra correspondingly precise instruments and special detection schemes are needed. In absorption or emission a first approach will be the use of high resolution spectrometers. We take the opportunity to briefly recall some pieces from undergraduate optics, which are important but often forgotten. The most straight forward instruments used are monochromators (designed to generate quasi-monochromatic light from a broad band source) or spectrometers (to identify the wavelengths of spectral lines). Typically, an optical setup of focal length f images the entrance slit of a spectrometer onto its exit slit – after passing the light through suitable dispersive devices, based on refraction (prism spectrometers, see e.g. B ORN and W OLF 2006, Chap. 4.7) or diffraction (grating spectrometers). We review here only some facts about the latter which are most commonly used in modern spectroscopy (for more details see e.g. B ORN and W OLF 2006, Chap. 8.6.1). A variety of geometries may be applied. A particularly clear and efficient setup, illustrated in Fig. 6.1, uses a reflecting, spherical grating of radius R which acts both as focussing optics (focal length f = R/2) and dispersive element. While the ruling of spherical gratings is somewhat demanding, the simplicity of the setup is a great advantage and the reflecting geometry avoids losses in optical lenses and can be used also in the VUV and XUV. Grating, as well as entrance and exit slit are mounted on a so called ROWLAND circle which has the diameter R/2. This geometry is based on the “inscribed angle theorem” according to which the inscribed angles (here α and β) belonging to the same intercepted arcs on a circle (here AF and FB, respectively) do not depend on the position of their apex on the circle. Without diffraction we would have α = β and the entrance slit at A would be imaged by the grating onto B (0th order diffraction). Due to diffraction and interference (z = 1, 2, . . . ) the exit angle β depends on wavelength λ – as to be discussed in a moment – and the beam is focused onto the Fig. 6.1 ROWLAND circle

F A

focus of grating

entrance slit exit slit B

α ß spherical

α ß grating

6.1

Methods of High Resolution Spectroscopy GN

(a) ray 1

sin

(b)

275

(c)

GN

GN

FN

ray 2

FN sin

Fig. 6.2 (a) Standard geometry for a reflecting grating. Schematically indicated is the optical delay for two light “rays” on the way in (fat, red line) and out (fat black line). Angles of incidence (α) and diffraction (β) in respect of the grating normal (GN) are taken as positive to the left, negative to the right of GN. (b) Blazed grating with a blazing angle θB between the facette normal (FN) and GN. (c) Echelle grating in a near L ITTROW arrangement; shown is an R3 echelle with a blaze angle of 71.5◦

exit slit B. Typically, the grating is moved along the circle to vary the wavelengths which reaches B. The actual diffraction and interference geometry of a reflective grating is shown in Fig. 6.2(a). We see that the optical delay between “rays” 2 and 1 from neighbouring grooves of the grating at a distance d is given by s = d(sin α + sin β). Note that β < 0 in the arrangement of Fig. 6.2(a). We expect constructive interference in the outgoing wave front if the grating equation c λ p = z , or in frequency space ν = z d

s with p = (sin α + sin β), s = pd and z = 0, ±1, ±2, . . .

(6.1)

holds, with z being the order of the interference and pd the optical delay. An important characteristic of any spectrometer is the change of the deflection angle with wavelength. One derives the change of the interference maximum with wavelength (at a fixed angle of incidence α) from (6.1), and finds for a grating the angular dispersion

1 z dβ = . dλ cos β d

(6.2)

A detailed, rather straight forward calculation (B ORN and W OLF 2006, Chap. 8.6.1) shows that the (normalized) total intensity reflected from a grating with N coherently1 illuminated grooves (slits, facets) is

1 sin(N pkd/2) 2 T (p) = 2 T0 (p) sin(pkd/2) N 1 The

with k =

2π . λ

(6.3)

concept of lateral coherence will be discussed in some detail in Chap. 2, Vol. 2. Essentially, coherent illumination requires that s sin ϑd λ holds for the width s of the spectrometer entrance slit, where ϑd is the full opening angle of the optics illuminating the grating.

276

6 =0 1

=1

=2

/d

/d


(a)

0 2 1

(b)

0

Fig. 6.3 (a) Normalized interference function T (p)/T0 (p) of a diffraction grating with N coherently illuminated grooves in 0th, 1st and 2nd order for two wavelengths differing by λ; (b) both traces added, illustrating that the two wavelengths are (just) distinguishable when the principle maximum of one overlaps with the first side minimum of the other (here in 2nd order)

T0 (p) represents an overall normalized intensity function for a single groove of the grating.2 It depends on the shape and reflectivity of an individual groove and varies smoothly. Figure 6.3(a) shows the interference function T (p)/T0 (p) due to multi-ray interference for two, slightly different wavelengths λ and λ + λ, with maxima at p = zλ/d for z = 0, 1 and 2 according to (6.1). According to the so called R AYLEIGH criterium, illustrated in Fig. 6.3(b), the two wavelengths can just be distinguished when the principle maximum of the latter coincides with the first side minimum in the fringe pattern of the former (i.e. from N coherently interfering rays). With (6.1) and (6.3) the principle maxima are positioned at N pkd/2 = πN pd/λ = N zπ while the optical delay (p + p)d for the first side minima corresponds to πN (p + p)d/λ = (N z + 1)π

⇒

p =

λ . Nd

On the other hand, the principle maxima for a wavelength λ + λ correspond to p + p = z

λ + λ d

⇒

p = z

λ . d

By equating the two expressions for p we obtain the resolving power λ/ λ = N × |z|,

2 In

(6.4)

the most simple case, a transmission grating consisting of slits with a width s, this would be the single slit transmission function T0 (p) = I0 sinc2 (ksp/2),

with sinc(x) = sin(x)/x.

6.1


277

which holds for all interference based spectrometers. For grating spectrometers it determines the spectral resolution – usually given as difference λ of the wavelength between two neighbouring spectral lines which can just be distinguished. Modern gratings are fabricated by holographic methods or are ruled and may have several thousands of grooves per mm and a length of up to a few 100 mm, thus N may in principle be up to 100 000 or more, albeit difficult to illuminate coherently. Typically they operate in 1st or 2nd order. Special shape and reflectivity of the surface may help to channel all diffracted intensity into one interference order, thus avoiding ambiguities in the observed spectra. Specifically, blazed gratings have grooves with facets that are aligned such that the maximum diffraction angle in respect of the facet normal (FN) is equal to the entrance angle in respect of FN – as indicated in Fig. 6.2(b). One reads from the figure that this blaze angle is θB = (α + β)/2

(6.5)

where α + β is derived from (6.1) for a specific wavelength. The diffracted signal then appears as if directly reflected from the facets. Even though such geometric interpretation is somewhat questionable (typically the wavelengths are on the same order of magnitude as the groove distance) detailed calculations show that one may indeed maximize in this manner the light reflected for a specific order of interference and wavelength λB – one says the grating is blazed for that wavelength. At last we mention so called echelle gratings (from the French word “échelle”, stairs or ladder) which have experienced a true renaissance in recent years. In contrast to standard gratings they have only a moderate number of grooves N , but very large z 1 and large blaze angles. A typical setup is shown in Fig. 6.2(c). They may be viewed as an intermediate between grating spectrometer and interferometer. One fabricates them today by ruling or chemical etching. Characteristic for echelle spectrometers is the combination of high dispersion according to (6.2) and high resolving power according to (6.4). A disadvantage is that due to the high value of z the spectra may overlap in different orders. In frequency space we read from (6.1) that frequencies ν and ν + νFSR have a maximum of order z and z +1, respectively, at the same optical path difference s if

νFSR =

ν c = .

s z

(6.6)

Thus, frequencies differing by more than νFSR cannot be assigned unambiguously. Obviously, the higher the order of interference z for which the system is designed, the more serious is the problem. One may overcome it by crossing the echelle with a suitable second dispersive element (grating or prism) which leads to an unambiguous 2D pattern, ideally suited for state-of-the-art CCD devices for detection. Finally, one should note that in practice “the degree to which the theoretical resolving power is attained (by any grating spectrometer or monochromator) depends not only on the angles α and β (and on z and N ) but also on the optical quality of the grating surface, the uniformity of the groove spacing, the quality of the associated optics in the system, and (most importantly) on the width of the (entrance and

278

6


Fig. 6.4 M ICHELSON interferometer

reflecting mirror 2 s1 Δs=s2 - s1

collimating lens

light source

s2

beam splitter

reflecting mirror 1 focussing lens

detector

exit) slits (or detector elements). Any departure of the diffracted wavefront greater than λ/10 from a plane (for a plane grating) or from a sphere (for a spherical grating) will result in a loss of resolving power due to aberrations at the image plane. The grating groove spacing must be kept constant to within about 1 % of the wavelength at which theoretical performance is desired.” (PALMER and L OEWEN 2005, Chap. 2.4)

6.1.2

Interferometers

Expressions analogue to (6.4) also hold for all interference based spectrometric devices, such as M ICHELSON, M ACH -Z EHNDER, FABRY-P ÉROT (FPI) etc. interferometers. In an genuine interferometer N may vary from 2 beams in a M ICHELSON setup to several 100 in an FPI. On the other hand, the order of interference z is very large in all interferometers: For an optical path difference s of the interfering beams it is z = s/λ = ν s/c.

(6.7)

In the visible (VIS) a moderately large optical path difference of s = 6 cm corresponds already to z = 105 . One may thus in principle obtain very high resolution with interferometers. Typically one observes fringes of equal inclination (slightly different divergence angles lead to a modification of the optical path difference s). Since in various contexts M ICHELSON interferometers will play a role later on (e.g. in F OURIER transform spectrometers, see Sect. 5.3.2 in Vol. 2), we show such a setup in Fig. 6.4 for reference without entering here into details (N OBEL pize for M ICHELSON 1907). The key elements are a beam splitter and two reflecting mirrors between which the light travels an optical distance s1 and s2 , respectively. The optical path difference s = s2 − s1 may be varied by moving one of the mirrors with high precision. Unfortunately, spectra obtained with interferometers are not unambiguous (just as gratings used in higher order diffraction): assume that the central


279

Fig. 6.5 FABRY-P ÉROT interferometer

highly reflecting plane surfaces r',t' r,t

lens

3 C B θ

.

2θ L

2

θ P

A 1

possibly n >1

interference pattern (fringes of equal inclination)

6.1

maximum for the wavelengths λ1 (frequency ν1 ) corresponds to order z. A somewhat shorter wavelength λ2 will be observed at the same optical path difference s in z + 1 order if z = ν1 s/c

and z + 1 = ν2 s/c

=⇒

1 = (ν2 − ν1 ) s/c.

Completely equivalent to (6.6), we obtain for the frequency difference ν2 − ν1 of two neighbouring maxima the so called free spectral range νFSR = c/ s.

(6.8)

Hence, two lines can be distinguished in an interferogram only if their frequency difference is ν < νFSR . Since the resolution of a M ICHELSON interferometer (N = 2) is on the same order of magnitude as its free spectral range it cannot directly be used as a spectrometer. Quite generally, for spectroscopic applications of interferometers some type of additional selection has to be employed. FABRY-P ÉROT interferometers (FPI) are used for a number of purposes. Not least, each laser resonator is essentially an FPI as will be discussed in Sect. 1.1.2 in Vol. 2. But it is also an indispensable tool for high resolution spectroscopy. In Fig. 6.5 a typical setup is shown schematically with the characteristic elements. It consists of two plane parallel surfaces mounted in a distance of L with an amplitude reflectivity r and r , respectively. For the light intensity this corresponds to reflection coefficients R = |r|2 and R = |r |2 , respectively, and to transmission coefficients T = |t|2 and T = |t |2 . If the losses are negligible R + T = 1. The space between the mirrors may be empty or filled by a medium with an index of refraction n (this may e.g. be realized by one plane, coated, parallel glass plate). Light incident at an angle θ in respect of the surface normal is reflected many times at these highly reflecting surfaces and finally brought to interfere on the detector P . The optical path difference between neighbouring beams is given by the distance ABC in Fig. 6.5 (red dotted line) as s = 2Ln cos θ (note that it decreases with the tilt angle θ !). For perpendicular incidence the free spectral range (6.8) is thus

νFSR =

c , 2nL

(6.9)

280

6


or on the wavelengths scale

λFSR

λ2 . 2nL

(6.10)

The order of interference becomes z = 2nL/λ.

(6.11)

and the phase difference between neighbouring, interfering beams is: δ=

ν cos θ 2π s = 2kLn cos θ = 2π . λ

νFSR

(6.12)

The full field amplitude transmitted through the FPI is found from adding all partial amplitudes 2 Et = tt E0 + rr exp(iδ)tt E0 + rr exp(2iδ)tt E0 + · · · ,

(6.13)

which is a geometrical series that can be written in closed form. Neglecting losses, remembering that intensities are I ∝ |E|2 , and evaluating the geometrical series emerging from (6.13) one obtains the A IRY function

It 1 = I0 1 + F sin2 ( 2δ )

(6.14)

for the ratio of transmitted to incident intensity I0 as a function of frequency ν (using (6.12)). Here F is the so called finesse coefficient √ 4 RR . (6.15) F= √ (1 − RR )2 In the resonant case δ = (z + m)2π the full initial intensity is transmitted. Figure 6.6 shows some examples for the transmission through an FPI at different reflectivities, where R = R is assumed. The FWHM δh of the transmission lines √ of the FPI follows from (6.14) with It /I0 = 0.5 ⇒ F sin2 (δh /4) = 1 ⇒ δh 4/ F . Finally, with (6.12) the FWHM of√the frequency νh follows: at perpendicular incidence 2π νh / νFSR = δh = 4/ F holds and one defines for an FPI √ π R π√

νFSR (6.16) F= = the finesse F =

νh 2 1−R or on the wavelengths scale F =

λFSR

λh

With (6.10) the resolving power of an FPI is √ λ π R 2L = = F × z.

λh (1 − R) λ

(6.17)

6.1


Fig. 6.6 Transmission of a FABRY-P ÉROT interferometer as a function of the phase difference δ according to (6.12) for different finesse F (or reflection coefficients R = R ); R = 4.5 % is the reflectivity of uncoated glass surfaces). The phase change δ is given in respect of one transmission maximum; δ = 2π corresponds to

ν = νFSR

I/Imax

281 LORENTZ, F = 0.7

1.0 F = 0.7

(R = 4.5%) δh

0.5

F=6 (R = 60%)

F = 30 (R = 90%)

0

2

δ

3

4

Hence, the finesse F of an FPI corresponds to the (effective) number N of interfering beams in the general formula (6.4). In practice, F is limited by mechanical imperfections of the mirrors. As a rule of thumb the surfaces must be ground to at least λ/F precision over the whole surface. Typically, F may be up to a few 100. We also note, that the reflected intensity Ir is complementary to (6.14), so that Ir (δ) = I0 − It (δ). While in transmission (e.g. for slightly divergent light) one observes bright rings on a dark background, in reflection one finds dark rings on a bright background. Very important is the fact that in the interior of the resonator the radiation intensity is significantly enhanced: since only a fraction T = 1 − R of the internal intensity leaves the resonator, within the resonator the intensity is given by Iint = It /(1 − R). In this sense, a FABRY-P ÉROT resonator acts as a storage for light. This is exploited in the construction of lasers but also in spectroscopy: in the interior of a FABRY-P ÉROT resonator one may conduct highly sensitive spectroscopy using relatively low intensity light sources. This is understood simply by considering the light beam to pass a sample inside the resonator many times. During each passage the light beam is correspondingly attenuated. So called cavity ring down spectrometers (CRD) use this cleverly by feeding a laser pulse into an FPI resonator and following its decay (see Sect. 5.5.3 in Vol. 2). A F OURIER transform of this decay function allows a very sensitive determination of the samples absorption spectrum. In practice for lasers and spectroscopy one uses not only plane parallel mirrors. Very often concave, highly reflecting surfaces are employed which lead to stability (see Sect. 1.1.3 in Vol. 2) and an additional concentration of the available light inside the resonator. A particularly useful setup consists of two concave mirrors of radius R positioned in a distance R. A (nearly) plane wave (e.g. a laser beam) entering such a device is focused in the centre of the resonator and may be used efficiently for studying nonlinear processes such as multi-photon excitation of atoms and molecules: according to (5.43) such signals rise with some power of the intensity while the detected volume only decreases linearly with the cross section in the focus of the beam.

282

6

(a) skimmer

collimation aperture(s)


fluorescence detector

term and (b) excitation scheme for the Na D2 line

oven inclined incidence detector

3P 3/2

beam splitter CW dye laser beam tunable, perpendicular

FABRY-PEROT as marker

excitation

emission 3S1/2

Fig. 6.7 (a) Schematic of an atomic or molecular beam setup for high resolution spectroscopy. (b) Partial term scheme of Na, with the two Na D2 HFS lines recorded

6.1.3

D OPPLER Free Spectroscopy in Atomic Beams

The precise observation of finer effects in atomic spectra is not only limited by the resolving power of the spectrometers but very often also by the D OPPLER broadening (5.22). By studying spectra in atomic (or molecular) beams this handicap may be overcome. A typical experimental setup and detection scheme is sketched in Fig. 6.7: The atomic beam emerges from an “oven” (preferably at high pressures to allow for adiabatic cooling) and is collimated by a skimmer and some apertures to a divergence angle θ . The velocity distribution has its maximum at vm and a FWHM = v. Let the transition wavelength at resonance be λba and the angle of light incidence θ . With (1.32) the resulting D OPPLER shift νba and D OPPLER broadening δνD is then

νba = νba

vm vm cos θ = cos θ c λba

and δνD =

v cos θ, λba

(6.18)

respectively. When exciting (or detecting) perpendicular to the beam cos θ → cos(π/2 ± θ ) = ∓ sin θ ∓ θ, → 0, and only the perpendicular vehence the average shift vanishes, νba − νba locity components contribute to a small D OPPLER broadening:

δνD⊥ =

v 2 sin θ. λba

(6.19)

For illustration we show in Fig. 6.8 the hyperfine structure (HFS) of the Na–D2 line, recorded with the atomic beam setup Fig. 6.7(a). The term scheme is shown in Fig. 6.7(b); a detailed description of HFS will be given in Chap. 9. To demonstrate the dependence of the D OPPLER profiles on the angle of light incidence in respect of the atomic beam, two spectra for different angles of light incidence (θ = 90◦ and θ 40◦ ) are recorded simultaneously. The atomic beam is slightly supersonic with vm 1400 m / s and v 700 m / s. It is collimated to θ 2.5◦ . The exciting, narrow band light (λba 589 nm) comes from a fine tuneable dye-laser.

6.1

Methods of High Resolution Spectroscopy inclined irradiation

283

perpendicular irradiation 1800 MHz

1700 MHz

∆νD⊥ 900 MHz

×4

ν marker 100 MHz

Fig. 6.8 Hyperfine splitting of the Na D2 line (32 P3/2 ← 3 2 S1/2 transition) recorded with the atomic beam apparatus sketched in Fig. 6.7(a). One clearly sees the two hyperfine components with perpendicular (black) and with inclined (red) irradiation as function of the exciting frequency ν. The full red line gives 4× the signal extracted from the combined signal (red dashed). The marker shown at the bottom was simultaneously recorded with the FPI

One observes the whole fluorescence signal as a function of the incident frequency ν – and thus registers simultaneously the excitation spectrum for perpendicular and inclined incidence of the light beam. Both contributions can easily be separated as indicated in Fig. 6.8. While tuning the laser, a reference signal is simultaneously recorded from a small fraction of the laser beam which passes through an FPI. This signal appears and vanishes in intervals corresponding to the free spectral range of the FPI (here 100 MHz). It generates a convenient marker for calibrating the recorded spectrum. For perpendicular irradiation, the two HFS components of the Na D2 line are clearly separated by ca. 1700 MHz. The observed linewidth δνD⊥ 100 MHz corresponds to that predicted by (6.19). With some effort it can be further reduced. Inclined incidence (red line Fig. 6.8) leads to a D OPPLER shift νba − νba 1800 MHz and a broadening of δνD 900 MHz much larger than for perpendicular incidence – again in full agreement with the predictions of (6.18) for the given atomic beam conditions.

6.1.4

Collinear Laser Spectroscopy in Ion Beams

Laser spectroscopy on ions is often performed with energetic ion beams (the following also holds for high energy neutral particle beams). In such experiments the laser beam travels parallel or antiparallel to the fast ion or neutral beam. The method has two advantages: On the one hand, one may tune the absorption frequency of the ions by varying the energy of the beam and keeping the laser frequency fixed νL at some convenient laser line. On the other hand, D OPPLER broadening can be reduced considerably (so called D OPPLER narrowing). The D OPPLER shift of the absorption frequency νba as seen from the laser beam is according to (1.30) and (1.34) (6.20)

νba = νba wk + 2wk + wk2 ±νL 2wk (1 ± wk /2 + · · · ),

284

6


for ions with a rest mass m and a kinetic energy Wkin , with wk = Wkin /(mc2 ). The + and − signs refer to irradiation head-on or parallel to the ion beam, respectively. This shift may be considerable for ions and allows for calibrated scanning through the finer structures of spectra by tuning the acceleration voltage U , with Wkin = qeU for ions of charge qe. Typically the ions are prepared in an ion source with a finite energy width δWk at low kinetic energies. According to (6.20) this would lead to a D OPPLER broadening 2δWk δνD ±νba . (6.21) mc2 During acceleration of the ion beam δWk usually stays constant, and so does δwk = δWk /(mc2 ). Hence, for higher but not too high kinetic energies (δWk Wkin mc2 ) the D OPPLER width is derived from (6.20): δwk d νba νba δWk δwk νba √ (1 + 2wk + · · · ) . (6.22) dwk 2wk 2Wkin mc2 √ Comparison with (6.21) shows that the linewidth is reduced by a factor 4Wkin /δWk in respect of a measurement in the ion source due to D OPPLER narrowing. Typical for ion beam spectroscopy would be a 100 keV beam and an initial width δWk = 1 eV in the source (much above thermal energy but characteristic for ion sources), amounting to a reduction factor of 630. For illustration, in the case of He+ ions the thermal D OPPLER width at room temperature according to (5.21) is δνD (293 K) = νba × 6.1 × 10−6 . The linewidth in the 100 keV ion beam (δWk = 1 eV) would be δνD = νba × 3.66 × 10−8 – a factor of about 170 less than even the thermal D OPPLER width! This already allows a convenient resolution (even if not a high precision measurement) of the L AMB shift for the n = 2 states in He+ which is around 1.423 × 10−6 × νba . The D OPPLER shift according to (6.20) in this case is 0.007νba , which allows for sufficient tunability (in the per mill range) by changing the acceleration voltage. The method has been used (and is still used) extensively for the determination of HFS and FS structure as well as of the L AMB shift for heavy ions, specifically so for their unstable isotopes. δνD

6.1.5

Hole Burning

The basis of many modern methods of D OPPLER free spectroscopy is the so called hole burning by which a certain part of a population of the sample atoms or molecules is depleted by monochromatic excitation. This allows a nice illustration of the terms homogeneous and inhomogeneous linewidth. As we have seen in Sect. 5.1.4, in gas phase the D OPPLER width is typically much larger than the natural linewidth: δνD νnat . To record the absorption probability as a function of the frequency for an ensemble of atoms or molecules we may excite with a (quasi-monochromatic) narrow band tuneable laser in the neighbourhood of a resonance frequency νba . As discussed in Sect. 5.1.4 and Sect. 6.1.3 the absorption

6.1


(a)

285

(b) BOLTZMANN with hole

BOLTZMAN N

LORENTZ 0

(velocity ||

of light)

0

Fig. 6.9 Velocity distribution of atoms or molecules in the gas phase parallel to the wave vector k of the light. (a) M AXWELL -B OLTZMANN distribution prior to absorption and L ORENTZ profile for a particular velocity class vtest (dashed). (b) Velocity distribution with “hole”, which has been created by the radiation absorbed at vtest

frequency νba of an atom (in the thermally broadened distribution) moving with a velocity v is shifted by the D OPPLER effect to ν given by (5.19). When tuning the laser one observes an absorption signal which is proportional to the velocity distribution w(vx ) according to (5.20). This is shown in Fig. 6.9(a) by the full red line. We may turn the argument around and say that a given frequency ν is absorbed by atoms or molecules belonging to a velocity class

ν − 1 = λba (ν − νba ) (6.23) vx (ν) = v cos θ = c νba where vx is the projection of the atomic velocity onto the wave vector k of the light, with k x. More precisely, atoms with velocities v slightly smaller or larger than vx (ν) also absorb light of the frequency ν – with a probability given by the Lorentzian natural line profile on the velocity scale. This is illustrated in Fig. 6.9(a) by the red dashed line. Hence, when irradiating with a fixed test frequency νtest one only excites atoms of a particular velocity class around vtest = vx (νtest ) of the initial velocity distribution. If the light intensity is sufficiently high this results in a noticeable depletion of this velocity class in the initial state D OPPLER profile as shown in Fig. 6.9(b): the remaining velocity distribution of atoms in the initial state has then typically a “hole” at vtest . The width of the absorbing velocity class corresponds to the natural linewidth νnat = λba νnat : all atoms within the homogeneous linewidth participate in the depletion correspondingly. The same argument hold for collision broadened profiles of a width (5.17) if that is much larger than the natural linewidth.

6.1.6

D OPPLER Free Saturation Spectroscopy

One may exploit hole burning very elegantly for spectroscopy. This has first been realized by H ÄNSCH et al. (1971) who coined the term D OPPLER free saturation spectroscopy. A typical experimental setup is shown in Fig. 6.10 and the principle

286

6 las

er


beam splitter phase sensitive detection chopper

beam splitter

differential amplifier

mirror

mirror probe beam

pumpbeam

sample vapour (gas) in cell e.g. Na or H(2s,2p)

reference beam

photo detectors

Fig. 6.10 Experimental setup for D OPPLER free saturation spectroscopy according to H ÄNSCH et al. (1971) pump beam

probe beam

(a) (b) (c) Fig. 6.11 Principle of D OPPLER free saturation spectroscopy: circles with arrows indicate atoms and their velocity vectors, full colours indicate resonance, grey out of resonance. The incident frequency is (a) blue shifted in respect of νba , (b) exactly νba or (c) red shifted. Consequently (a) the probe beam excites atoms moving to the left, the pump beam only those moving to the right; (c) the probe beam excites atoms moving to the right, the pump beam only those moving to the left; (b) pump and probe beam excite only (the same) atoms at rest

of the method is illustrated in Fig. 6.11. One splits the laser beam into an intense pump beam and a weaker probe beam, and sends them counter-propagating through the cell with the sample. The pump beam “burns holes” into the D OPPLER profile of the target atoms in the ground state. Ideally the transition is then saturated for that frequency, i.e. nearly one half of the atoms in that particular velocity class are found in the excited state. All atoms not in resonance with the pump beam frequency are unaffected. Hence, in most situations the probe beam does not notice anything of this preparation since it comes from the opposite side and is in resonance with a different velocity class. Only if both, pump and probe beam match exactly a resonance transition for atoms at rest the probe beam ‘sees’ the same atoms as the pump beam. This leads to a reduction of the absorption exactly when the laser frequency corresponds to the atomic resonance frequency for atoms at rest. One calls this reduction “L AMB dip”. The signal may be recorded in comparison to a reference signal without pump-

6.1


287 crossover experiment: saturation signal probed in Na bulb

equivalent DOPPLER profile

ν

1700 MHz 100 MHz marker

Fig. 6.12 D OPPLER free saturation spectrum of the two main hyperfine lines in the 3 2 P3/2 ← 2 S1/2 transition of sodium (Na). The spectrum has been recorded by a most unsophisticated setup with Na in a bulb – to illustrate the robustness of the method. One recognizes the two characteristic L AMB dips showing the HFS splitting of 1700 MHz and one (physically insignificant) crossover signal in between Fig. 6.13 Fine structure of the Hα BALMER (n = 2 ↔ 3) transition. (a) BALMER series (n = 2 ↔ n ), overview. (b) Blow up of the (n = 2 ↔ 3) transition: sticks spectrum (theory) according to K RAMIDA (2010); D OPPLER broadened spectra simulated for 300 K (dashed) and 50 K (full line). (c) D OPPLER free saturation spectrum according to H ÄNSCH et al. (1974). More details are explained in Sect. 6.5.1

0

500 000

/ GHz

(a)

BALMER series

DOPPLER broadened

50 K 300 K shift

DOPPLER free saturation spectrum -5

(b)

crossover

0

5

(c) 10

/ GHz

ing and typically is registered by phase sensitive detection which synchronizes the detected signal with a chopper for the laser beam to minimize statistical fluctuations (using a so called lock-in amplifier). If several allowed transitions are accessible over the tuning range of the laser, correspondingly several L AMB dips appear. This is illustrated in Fig. 6.12 for the hyperfine structure of the Na atom which we have already encountered in Sect. 6.1.3, Fig. 6.8. In contrast to the latter experiments, the setup for recording the saturation spectrum Fig. 6.12 was particularly simple: just a bulb with Na vapour, no optical surfaces, and rather weak pump and probe laser beams. Consequently the two L AMP dips in the Na D2 line (3 2 P3/2 ← 3 2 S1/2 ) separated by ca. 1700 MHz HFS splitting are small, but clearly visible on the D OPPLER profile, and illustrate the robustness of the method. Unfortunately, the method is not completely unambigu-

288

6


ous: so called “crossover” signals may arise from two closely spaced lines exactly halfway between them. In any case, this D OPPLER free saturation spectroscopy is an excellent method to determine resonance lines with an accuracy on the order of the natural linewidth – irrespective of being taken in a gas cell with much larger D OPPLER width. A particular nice and important example (H ÄNSCH et al. 1972, 1974) is illustrated in Fig. 6.13. In this pioneering work the fine structure of the Hα line has been determined for the first time with high accuracy. To emphasize the resolution Fig. 6.13(a) gives an overview for the BALMER series of atomic hydrogen. For comparison (b) shows two D OPPLER broadened spectra in the frequency range of interest, simulated for 300 K and 50 K, respectively, which obviously do not allow to resolve the fine structure in detail. In contrast, saturation spectroscopy (c) gives a very clear picture of the fully resolved fine structure. We shall come back to it later.

6.1.7

R AMSEY Fringes

Even if one may overcome the D OPPLER broadening and other pitfalls of spectroscopic observations by clever methods, one often encounters the situation that the resolution is limited simply because not enough time is available to perform a measurement since, alas, the fundamental conditions imposed by the uncertainty relation (1.125) between time and energy provide apparently stringent boundaries:

W × t . If no other effect interferes, the accuracy W by which an atomic transition may be determined increases with the interaction time between radiation field and sample (in Sect. 4.3.5 we had simply assumed t → ∞). Norman R AMSEY (N OBEL prize 1989) has – in the context of magnetic resonance spectroscopy (1950) which will be detailed in Sect. 9.5.1 – invented and explored a particularly ingenious method which allows one in principle to prolong the interaction time effectively and thus to enhance the resolution. The idea is amazingly simple: in practice it will never be possible to expose the object of investigation to the radiation for an arbitrary long time. A finite interaction time, on the other hand, leads to a line profile with finite width as indicated in Fig. 4.11. However, if instead of one wave-train one uses two which are coherently phase coupled, this problem can be overcome. The result can easily be understood extending our earlier discussion using time dependent perturbation theory. If necessary, the matrix elements Dba of the perturbation operator have to be replaced depending on the type of interaction, but otherwise we start with (4.51). Integration has now to be carried out over the two wave-trains, i.e. from 0 to τ and in addition from T to T + τ . The system thus interacts with two coherent alternating fields of a duration τ each. One easily verifies that the excitation probability (4.59) (see also Fig. 4.11) which vanishes at ντ = 1 is now modulated by a rapidly oscillating amplitude of the frequency ντ/T . These so called R AMSEY fringes allow for a correspondingly sharper determination of the resonance frequency.

6.1


Fig. 6.14 R AMSEY fringes: Line profile as calculated by R AMSEY (1950) for the excitation with two spatially separated oscillating fields. The resolution is determined now by the time T between the two excitation processes (full red line, resonant, fast oscillations) while the overall linewidth corresponds to the much shorter interaction time τ with only one of these fields (black dashed line, not resonant)

289 1.22/ T

wab 0.6

∆ν = 0.64/ T

0.4

∆ν =1.4/ τ

0.2

0.0

(ν0 - ν)T (resonant) -1 1 3 -1.0 - 0.5 0 0.5 1.0 (ν0 - ν) τ (non resonant)

As a consequence one obtains an overall line profile as shown in Fig. 6.14. The practical realization for transitions in the radio frequency and microwave region caused no problem even in 1950 since there the phase coherence is a built in property of these waves. With modern laser techniques the required fixed phase relation between two different laser pulses is no problem either. We shall get to know a few practical, state-of-the-art realizations in Fig. 6.16 and Fig. 6.28 as well as in Sect. 9.5.1.

6.1.8

D OPPLER Free Two-Photon Spectroscopy

Another very elegant way to realize D OPPLER free spectroscopy is based on twophoton excitation (introduced in Sect. 5.3.1). The method has been applied experimentally for the first time in 1974 nearly simultaneously by several groups. For a detailed review the interested reader is referred to G RYNBERG and C AGNAC (1977). One uses a standing wave, e.g. a circularly polarized wave field, created by a FABRY-P ÉROT interferometer – advantageously in a confocal resonator with spherical mirrors so that focussing the laser beam provides sufficient intensity for efficient excitation of two-photon processes. The gas cell with the sample atoms or molecules is positioned into the focus of the mirrors – possibly the mirrors may even act as walls of the gas cell. The basic idea of the methods is the following. If v is the velocity of an atom or molecule in the gas cell and k the wave vector of the exciting light (in a FABRYP ÉROT resonator defined by the beam axis), then according to (1.33) the D OPPLER shift in 1st is k · v (angular frequency). For the inverse direction of light propagation (k → −k) the sign of the D OPPLER shift changes for this particular atom. Let us assume that a two-photon process is possible in the target atom between levels of energy Wa and Wb . An atom in a resonator with the two counter-propagating waves

290

6


thus interacts simultaneously with two waves of angular frequency ω − k · v and ω + k · v, respectively. Hence, the resonance condition for two-photon absorption is Wba = Wb − Wa = (ω − k · v) + (ω + k · v) = 2ω.

(6.24)

This is a remarkable finding: Independent of the individual velocity of an atom (or molecule) in the resonator all sample particles may be excited by absorbing two photons of the right energy ω = (Wb − Wa )/2. If both beams have the same polarization there is also a D OPPLER broadened background: each atom may also absorb two photons from one of the waves travelling in either direction (backwards and forwards) so that the terms ±k · v do not cancel. This signal is, however, much weaker and broadened so that it can be separated easily from the D OPPLER free signal. Often, by clever choice of the polarization, thus exploiting the M selection rules for two-photon transitions, the background may even be suppressed completely: for example, in a two-photon ns → n s transition M = q1 + q2 = 0 must hold with q1 and q2 being the components of the angular momentum of photon 1 and 2, respectively. If one uses RHC polarized light (q1 = −1) travelling in one direction, in the other direction, however, LHC polarized light (q2 = 1), then such a transition is possible. But an ns → n s transition may not be induced by two photons from one beam since in that case we should have M = ±2. These possibilities to record D OPPLER free spectra of atomic or molecular transitions in a simple gas cell continue to be of great practical value in today’s spectroscopy and are used increasingly for high precision measurements and more complex system. Initially, experiments have often been conducted for alkali atoms. Figure 6.15 illustrates as an example a high resolution measurements of the hyperfine structure for the transition 3s 2 S1/2 → 5s 2 S1/2 in the Na atom for which (a) shows the relevant parts of the term scheme. Figure 6.15(b) presents the results of a twophoton absorption experiment. For understanding the method we do not need to discuss details of HFS which will be subject to Chap. 9. We only note that F characterizes the total angular momentum of the system including electron and nuclear spin. For a two-photon process the total angular momentum may change by 0 or 2 so that in the present case only the transitions F = 2 → F = 2 and F = 1 → F = 1 may be accessed in a two-photon process. The absorption process is detected by registering the spontaneously emitted fluorescence 5s 2 S1/2 → 3p 2 P1/2 and 2 P3/2 at 615 nm and 616 nm which is emitted after the absorption process. A tuneable, pulsed dye laser has been used as light source. Similar to the experiment discussed in Sect. 6.1.3, for precise determination of the frequency differences a small part of the laser light is recorded simultaneously after passing through FABRY-P ÉROT reference interferometer. This produces calibration marks (c) at a distance of its free spectral range (here 300 MHz). One recognizes that with this relatively simple setup one can already resolve structures with an accuracy of a few MHz. Of fundamental importance is the precise determination of the energetic distance between ground state 1s 2 S1/2 and first excited s state 2s 2 S1/2 in atomic hydrogen. As already discussed in Sect. 5.3.2 the 2s 2 S1/2 state decays with a rate of 8.228Z 6 s−1 , which for the H atom amounts to a frequency linewidth of only 1.3 Hz.


3P1/2

3P3/2 0

m

fluo re s 330 n cence m

(b)

4P3/2 4P 61 1/2 6n m

m

1772 MHz

(a)

5n 61

laser wavelength 602.23 nm

4S1/2

291

589 n

5 S1/2

590 n m 330 n m

150 MHz

6.1

300 MHz

dye laser frequency

(c) 3S1/2

0

600 MHz

atomic transition frequency

Fig. 6.15 Two-photon transition between HFS states of Na, exemplified for the 3s 2 S1/2 → 5s 2 S1/2 transition, according to G RYNBERG and C AGNAC (1977). (a) Part of the term scheme with fluorescence lines (in nm) for detection; (b) measured two-photon spectrum; (c) calibration with a FABRY-P ÉROT interferometer: the marker lines are spaced by one free spectral range of the FPI of 300 MHz, corresponding to a change in absorption frequency of 600 MHz

This is a major challenge for precision spectroscopy. Two photons at a wavelength of 243 nm are needed. They may be are generated by frequency doubling (SHG) of a highly stabilized and calibrated CW dye laser at 486 nm. The detection of the excitation is typically achieved by “quenching” the excited, metastable 2s 2 S1/2 atoms (i.e. transferring them into the 2p 2 P states) and registering the emitted LYMAN-α radiation. We shall come to more details of this detection method in the context of Sect. 6.5.3. Figure 6.16 illustrates an interesting realization of this experiment by H ÄNSCH and coworkers (G ROSS et al. 1998) who employ a scheme with R AMSEY fringes (which we have introduced in Sect. 6.1.7). In the present case the R AMSEY fringes are generated by an AOM which chops the laser radiation in a sequence of pulses. Comparison of Fig. 6.16(b) and (c) shows that a substantial improvement of resolution may be achieved by a judicious choice of pulse durations and distances. Clearly, in addition to excellent resolution a precision measurement of also requires accurate calibration of the frequency. We shall discuss this aspect in Sect. 6.5.4. Finally, we want to point out that the two-photon process discussed here provides also the basis for a frequently used state-of-the-art method for H atom (radical) detection, e.g. in chemical gas phase reactions:in a so called 2 + 1 REMPI process one uses the resonant 2 photon excitation of the 2s 2 S1/2 state (243 nm) and then ionizes this state efficiently by a third photon, typically of the same wavelength. Section summary

• High resolution spectroscopy is the key to the investigation and understanding of finer details in atomic structure, such as FS and HFS. A brief summary of some essential tools has been given in this section.

292

6

Fine Structure and L AMB Shift vacuum chamber

(a)

gas discharge H atoms SHG

dye laser 486 nm

cold finger 2SFARADAY cage detector

AOM

243 nm

Fabry-Perot resonator

2S Signal / cps 7500 2S signal / cps

photon counter

2S Signal / cps

(b)

800 Hz @ 243 nm

100

(c)

5000 50

2500 0

0

- 80

- 40

0

40

detuning / kHz @ 121 nm

80

- 20

- 10

0

10

20

detuning / kHz @ 121 nm

Fig. 6.16 Two-photon excitation of the 1s 2 S1/2 → 2s 2 S1/2 transition in the H atom according to G ROSS et al. (1998). (a) Scheme of the experiment. An acousto-optic modulator (AOM) generates suitable pulse sequences at 243 nm. (b) Excitation probability as a function of detuning in respect of the 2hν resonance when irradiated by 50 μs pulses at 10 kHz repetition rate; (c) excitation rate 3.2 kHz, otherwise as (b)

• Interference based spectrometers such as grating spectrometers and interferometers have a resolving power λ/ λ = N × |z| where N is the number of interfering rays and z the order of interference. • A particular important device is the FABRY-P EROT interferometer. The effective √ number of interfering rays is given by its finesse F = νFSR / νh = π R/(1 − R), with νFSR = c/2L being the free spectral range between the two plates in a distance L, νh the FWHM of the transmitted line, and R the reflectivity of the plates. • The most straight forward way to avoid D OPPLER broadening of optical transitions in atoms and molecules is provided by molecular beam spectroscopy with perpendicular irradiation or emission. Collinear spectroscopy in high energy ion beams is another efficient method to dramatically reduce D OPPLER linewidths. • In addition, several ingenuous methods have been devised for state-of-the-art laser spectroscopy. Experimentally rather easy to realize are saturation spectroscopy (based on hole burning in intense radiation fields) and two-photon excitation, both allowing to achieve sub-D OPPLER resolution even in gas cells.

6.2

Spin-Orbit Interaction

293

• A particularly powerful method to prolong the interaction time in spectroscopy is the exploitation of R AMSEY fringes which are an important ingredient of modern atomic clocks.

6.2


6.2.1

Experimental Findings

In Chap. 3 we have described in detail how for alkali atoms the degeneracy of energy levels with the same principle quantum number n is removed for different orbital angular momentum quantum numbers . However, as we have already noted in the preceding section, on closer inspection one finds that even the n levels are again split into a finer substructure – and the same holds for the absorption and emission lines of the H atom. Quantitatively one finds that this so called fine structure splitting (FS) is on the order of magnitude Eh

WFS Z2 = (αZ)2 Wn me c2

(6.25)

with the atomic unit of energy Eh , the rest energy of the electron me c2 , and α = Eh /me c2 the fine structure constant (1.10) which we have encountered already on several occasions. This order of magnitude already suggests that FS must be of relativistic origin. For light atoms the splitting is rather small and non-trivial to measure in the gas phase. For the H atom in the first excited state FS splitting is of similar magnitude as D OPPLER broadening (5.22) at room temperature,

¯νFS ∼ 0.3 cm−1

and ¯νD = δωD /(2πc) ∼ (0.1−0.2) cm−1

in wavenumbers. Hence, some effort is necessary to see the splitting as already illustrated in Fig. 6.13. Much more convenient to observe is the FS splitting for alkali atoms. Form example, the D lines of Na (at ν¯ 17000 cm−1 ) are a doublet at wavelengths D1 :

λ1 = 589.0 nm

and D2 :

λ2 = 589.6 nm.

(6.26)

Figure 6.17 illustrates the term scheme. The splitting is about ¯ν 17 cm−1 while the D OPPLER width amounts to δ ν¯ D 0.06 cm−1 , so that spectroscopic separation is possible without problems. One needs an experimental resolution of about λ/ λ = ν¯ / ¯ν 1000. That can easily be achieved with a grating monochromator. All this observed fine structure is connected with the electron spin. As already discussed in Sect. 1.10 the spin (quantum number S) is accompanied by a magnetic moment which interacts with the orbital angular momentum (quantum number L) of the electron. This is called spin-orbit (or LS) interaction. Since two possible

294

6


Fig. 6.17 Fine structure transitions in the first excited state of Na

3p 2P3/2 3p 2P1/2

3s 2S1/2

orientations of the electron spin, sz = ±1/2, are possible in respect of a given axis here defined by the angular momentum, the levels are split into two components. For the example of a 3p electron shown in Fig. 6.17 the two states are characterized by a total angular momentum j = 3/2 and 1/2. For the 3s state the orbital angular momentum is zero, hence j = 1/2 and no splitting is observed. We shall discuss this in detail below.

6.2.2

Magnetic Moments in a Magnetic Field

We recall briefly the experimental findings communicated in Sect. 1.9–1.10. With the B OHR magneton μB 14 GHz T−1 according to (1.160), the magnetic moments of electron are for the orbit L = −gL μB L = − e M L, 2me

(6.27)

S = −ge μB S = − ge e S. M 2 me

(6.28)

and for the spin

The g factor of the orbit is gL ≡ 1, and for the spin it is exactly ge = 2 in D IRAC theory. On closer inspection there is, however, a small deviation from this value so that ge 2.0023 . . . , as will be elaborated in Sect. 6.6. Now we analyze the interaction of the combined magnetic moments with an external field B. For simplicity we choose B z. Each of them alone would have an interaction energy · B = −M z B B = −M V

(6.29)

L or M S onto B. With (6.28) and (6.27) the depending on the projections of M interaction potentials for orbital and spin magnetic moments thus are: L = gL μB Lz B with m |V L | m = gL μB Bm V S |sms = ge μB Bms . S = ge μB Sz B with sms |V V

(6.30) (6.31)

6.2


Fig. 6.18 Schematic illustration of the origin of spin-orbit interaction: the orbital magnetic field B L is viewed as arising from the nucleus circling around the electron

295 z

z

L + Ze x

-e

y

⇒

L

BL y

-e x

+Ze

If only the orbital magnetic moment or only the spin magnetic moment were acting, these expressions would represent the level splittings in an external B field, and one would observe • 2 + 1 possible orientations in space for the angular momentum (− ≤ m ≤ ) or alternatively • 2s + 1 = 2/2 + 1 = 2 for the spin (ms = ±1/2).

6.2.3

General Considerations About LS Interaction

The situation is further complicated by the interaction between the magnetic mo S of the electron spin and the magnetic field B L created by the orbiting ment M electron. To derive this interaction quantitatively one needs to know the effective magnetic field at the position of the electron spin. Thus, one transforms the motion into a coordinate system where the atomic nucleus rotates around an electron considered at rest – as indicated in Fig. 6.18. Without going into details of the underlying electrodynamics we note: The L ORENTZ force F = qv × B (1.105) onto a charge q moving with velocity v in a magnetic field B may be viewed as the result of an effective electric field E eff = v × B. Conversely, a charge moving in an electric field E generates a magnetic field B =−

v×E . c2

(6.32)

In the potential V (r) of the ionic core of an atom (atomic nucleus + core electrons) the electric field E experienced by an electron is derived from −eE = F =

r dV . r dr

Hence, the electron moving in this field generates according to (6.32) a magnetic field BL =

v r dV −1 1 dV −1 1 dV × = (r × me v) = L ec2 r dr eme c2 r dr eme c2 r dr

(6.33)

which obviously is proportional to the orbital angular momentum L. We now consider instead the positive nuclear charge circling around the electron, i.e. we sub-

296

6


stitute this expression with a positive sign into (6.29) and obtain with (6.28) and ge = 2: 2 LS = −MS · B L = 1 ge 1 dV L · S . V 2 2 4 me c r dr 2

(6.34)

We have introduced here – without proof – the so called T HOMAS factor 1/2 which LS into the nuclear follows directly from D IRAC equation when re-transforming V system at rest. Introducing Eh , a0 and the fine structure constant α = Eh /(me c2 ) the spinorbit (or LS) interaction is 2 LS = α a 3 ge ξ(r) L · S Eh V 0 4 2

with ξ(r) = a03

4πε0 dV . r dr

(6.35)

Note that for an attractive, pure C OULOMB potential ξ(r) = Zr −3 . To simplify the writing, in the remainder of this chapter all equations will be given in a.u. (unless otherwise indicated).

6.2.4

Magnitude of Spin-Orbit Interaction

The diagonal matrix element of (6.35) may be written (with ge ∼ = 2): LS = a L · S V with a = =

α2 ge n |ξ(r)|n 4

(6.36)

1 dV α2 α2 n | |n = 2 r dr 2

∞ 0

1 dV 2 2 r dr. Rn (r) r dr

Here we have introduced the spin-orbit coupling parameter a. Only for the H atom and H atom like ions it can be evaluated in closed form. The results are, however, characteristic and may readily be extended to quasi-one-electron systems such as the alkalis. For V (r) = −Z/r and ξ(r) = Zr −3 the coupling constant a can be derived from the expectation value (2.130) of r −3 for hydrogenic wave functions: Z4 α2 1 α2 . (6.37) a = ge Z 3 ∼ = 4 2 n3 ( + 1/2)( + 1) r This disappears for = 0. For > 0 we obtain a good estimate by approximating L · S s so that FS splitting is seen to be on the order of LS V

Z4 α2 2 n3 ( + 1/2)( + 1)

and

LS (αZ)2 V , |Wn | n( + 1/2)( + 1)

6.2


297

where |Wn | = Z 2 /2n2 has been introduced to obtain the relative perturbation. At small Z ≤ 11 it is very small, only about 10−6 . . . 10−3 for n ≤ 3. In general, the FS splitting decreases ∝ 1/n3 (for any given , spectroscopically relevant being ≤ 4). However, spin-orbit interaction becomes substantial for heavy atoms because of its proportionality to Z 4 . Perturbation theory will break down altogether for Zα 1. The physics emerging from such a situation is subject to current research with atoms of nuclear charges beyond the natural stability limit for 92 U. Such investigations are carried out today at heavy ion accelerators and storage facilities where this limit can easily be surpassed in highly energetic collisions.

6.2.5

Angular Momentum Coupling

Including spin-orbit interaction the Hamiltonian becomes 2 2 p 2 L α2 0 (r) + α ξ(r) FS (r) = H L · S = r + 2 + V (r) + ξ(r) L · S (6.38) H 2 2 2 2r 2 with p r2 /2 and L /(2r 2 ) being the radial and angular kinetic energy. To evaluate the L · S/2 we have to consider the coupling of spin and orbit fine structure term α 2 ξ(r) through their respective magnetic fields and derive the resulting eigenstates of the Hamiltonian (6.38). One formally introduces an operator of total angular momentum

z + Sz , J = L + S and its z component Jz = L

(6.39)

for which the standard commutation rules hold – in full analogy to those for orbital, (2.84), (2.75), and spin angular momenta, (2.91):

2 J , Jz = 0 Jx , Jy = iJz ,

Jy , Jz = iJx ,

Jz , Jx = iJy .

(6.40) (6.41)

This implies that one can measure simultaneously the magnitude and one of the components of the total angular momentum – but not two components. Specifically we shall use eigenstates |j mj for which 2 J |j mj = j (j + 1)2 |j mj and Jz |j mj = mj |j mj .

(6.42)

Correspondingly, J has 2j + 1 possible orientations, i.e. Jz is characterized by the quantum numbers 2

mj = −j, −j + 1, . . . , +j.

(6.43)

For the constituents L and S of this total angular momentum corresponding relations hold. However, their quantization axis is now the direction of J . One visualizes this

298

6


Fig. 6.19 Vector model for the coupling of spin s = 1/2 and orbital angular momentum = 2 forming a |j = 5/2 mj = 5/2 state (to scale)

Jz S

Fig. 6.20 Vector model for the possible projections Jz of a total angular momentum J = 3/2

Jz 3/2 ħ 1/2 ħ -1/2 ħ - 3/2 ħ

J S L

mj 3/2 1/2 -1/2 - 3/2

by a vector model as shown in Fig. 6.19. The following constraints must hold: | J | = j (j + 1), | L| = ( + 1), | S| = s(s + 1) (6.44) with = 0, 1, 2, . . . , n − 1, together with the triangular relation δ( sj ) = 1, or more explicitly: | − s|, | − s| + 1, . . . ≤ j ≤ | + s|

(6.45)

We have already encountered the triangular relation in Sect. 4.4.1 when discussing selection rules for dipole transitions and the photon spin sph = 1. For an electron spin s = 1/2 and > 0 it leads to j = ± 1/2, while for s = 1/2 and = 0 we have j = 1/2. The possible projections mj are illustrated in Fig. 6.20 for the case of j = 3/2.

Scalar Product of Angular Momentum Operators With the definition (6.39) the square of the total angular momentum is 2 2 2 L+ S)2 = L + 2 L · S + S J = (

(6.46)

and the scalar product of angular momentum and spin operators becomes 1 2 2 2 J −L −S . L · S= 2

(6.47)

The whole H AMILTON operator (6.38) can then be rewritten: 2 2 L p 2r α2 2 2 FS (r) = + 2 + V (r) + ξ(r) J − H L − S . 2 4 2r

(6.48)

Hence, we have to search for eigenstates |( s)j mj created by the coupling of L 2 2 2 and S which are at the same time eigenstates of J , L and S – and thus of L · S.

6.2


299

For these states (6.42) must be valid simultaneously with 1 2 2 2 J − L − S ( s)j mj L · S ( s)j mj = 2 1 = j (j + 1) − ( + 1) − s(s + 1) ( s)j mj . 2

(6.49)

During this evaluation we have exploited the fact that L and S commute, since they 2 operate on different spaces (position and spin space, respectively). Hence, also L 2 2 2 2 2 i S for any component i, and S commute. Since L L S Si L as well Si = i =L and since y z x L · S=L Sx + L Sy + L Sz .

(6.50)

2 2 2 S commute also with L · S and J . Hence, one may see with (6.46) that L and 2 2 2 FS according to (6.48) and j , and s are L , S and J all commute with H good quantum numbers (see Sect. 2.6.5). z and not with Sz as one sees from mulNote that L · S does not commute with L x nor L y commute with L z , and analogical z since neither L tiplying (6.50) with L for Sz . Thus, neither m nor ms are good quantum numbers. On the other hand, z + Sz does commute with L · S, since with (6.50) Jz = L

1 2 2 2 J , Jz − L , (Lz + Sz ) − S , (Lz + Sz ) = 0. [ L · S, Jz ] = 2 z + Thus, mj is a good quantum number, and we note that because of Jz = L Sz m + ms = mj

(6.51)

must always hold for eigenstates of the Hamiltonian (6.38), while neither m nor ms are good quantum numbers. FS which we are looking for are The eigenstates of the complete Hamiltonian H thus characterized by n, , s, j, mj : FS |n sj mj = Wn j |n sj mj . H We note, that (6.42) holds for these states. They are also eigenstates of L · S and 2 according to (6.47) we have (in units of ) n sj mj | L · S|n sj mj =

1 j (j + 1) − ( + 1) − s(s + 1) . 2

(6.52)

Eigenstates of the Total Angular Momentum The relations developed above allow to write the sought after states |n( s)j mj by linear combinations of product states |n sm ms = |n m |sms in such a way that L · S) m + ms = mj . This has to be done such that the Hamiltonian (and thus also

300

6


becomes diagonal. For spin and orbital angular momentum this can be achieved by using the definitions and commutation rules for angular momenta as sketched above. We refer to the relevant extensive literature on angular momentum algebra and to the brief extract of the most important formulas collected in Appendix B.2.3 The angular part of the eigenstates for (6.48) is written ( s)j mj = | m |sms m sms ( s)j mj or m ,ms

=

2j + 1

mj + −s

| m sms (−1)

m ,ms

m

s ms

j −mj

, (6.53)

using the so called C LEBSCH -G ORDAN coefficients sm ms | sj mj or 3j symbols (· · · ), respectively.

The Most Simple Example The most simple example for angular momentum coupling is offered by two electrons with spin quantum numbers s1 = s2 = 1/2 and zero angular momentum. Their total spin is S = S1 + S2 so that the triangular relation (4.72) with δ(1/2 1/2 S) = 1 allows for (2 × 1 + 1) = 3 states with total spin S = 1 (so called triplet state) and one state for the total spin S = 0 (singlet): (s1 s2 )SM =

1/2

s1 m1 s2 m1 |s1 s2 SM |s1 m1 s2 m2

m1 =−1/2 meter2 =−1/2

=

√ 2S + 1

1/2 m1 =−1/2 meter2 =−1/2

(−1)M

s1 m1

s2 m2

S −M

|s1 m1 s2 m2 .

With help of the readily accessible 3j symbols (e.g. with a Java applet from S VEN G ATO R EDSUN 2004) one verifies that the nonvanishing C LEBSCH -G ORDAN coefficients are 1 −1 1 −1 1 1 1 1 1 −1 = 1 2 2 2 2 11 = 1, 2 2 2 2 √ 1 1 1 −1 1 −1 1 1 10 = 2 2 2 2 2 2 2 2 10 = 1/ 2 √ 1 1 1 −1 00 = − 1 −1 1 1 00 = 1/ 2. 2 2 2 2 2 2 2 2 3 Several

slightly different definitions are used. We follow B RINK and S ATCHLER (1994).

6.2


301

Table 6.1 Triplet and singlet states with C LEBSCH -G ORDAN coefficients for the coupling of two electrons with spin 1/2. The symmetry of the spin functions is given as well as symbolic representations of the two spin 1/2 states involved Multiplicity

Symmetry

|χSM = |χ (1, 2)

S

M

|χ11 = |α(1)α(2)

1

1

⇒

equal

Triplet

symmetric

|α(1)β(2) + |β(1)α(2) |χ10 = √ 2

1

0

|χ1−1 = |β(1)β(2)

1

−1

⇒ phase

⇒

opposite

Singlet

antisymmetric

|α(1)β(2) − |β(1)α(2) |χ00 = √ 2

0

0

⇒ phase

The corresponding spin functions for triplet (symmetric) and singlet states (antisymmetric) are summarized and symbolized in Table 6.1, using the abbreviations |α = |1/2 1/2 and |β = |1/2 −1/2 as introduced in Sect. 2.5.4. The numbers in brackets behind the spin designations α and β refer to electron number (1) or (2). Note that the symmetry of a full wave function in spin and position is the product of the symmetries in spin and position space. As we shall elaborate on in Sect. 7.3.1 and Sect. 10.2.1 the PAULI principle demands the full wave function to be antisymmetric!

6.2.6

Terminology for Atomic Structure

To characterize atomic eigenstates in a compact and transparent manner an unambiguous terminology is used. It may readily be applied to atoms with a single active electron as well as to multi-electron systems. In the case of the so called RUSSEL S AUNDERS coupling (somewhat misleadingly also called LS coupling) one first composes as vectorial sums over all active electrons a total angular momentum L= Li (quantum number L) from all individual orbital angular momenta and a total spin S= S i (quantum number S) from all individual electron spins. As just discussed for a single electron, both angular momenta are then coupled to the overall total angular momentum J = S + L (quantum number J ). Note that all relations for angular momenta presented in Sect. 6.2.5 refer there to individual electrons with quantum numbers sm ms (uncoupled) and j mj (coupled). They do, however, also hold for the quantum numbers of the combined total angular momenta LSML MS and J MJ . In Sect. 7.3 we shall illustrate this in detail for the example of the helium atom, and also introduce there the alternative jj

302

6


coupling scheme which is relevant for heavier atoms where spin-orbit interaction is substantial. Here we just communicate the general terminology of LS coupling. The number of possible orientations of the total spin is called multiplicity = 2S + 1

(6.54)

of the state. For a one electron system such as alkali atoms we have S = s = 1/2 and the multiplicity is always 2 × 1/2 + 1 = 2, one calls this a doublet. As we have just seen in Sect. 6.2.5, atoms with two electrons have singlet and triplet states (S = 0 and S = 1, respectively). In systems with N electrons in principle multiplicities up to 2N × 1/2 + 1 could be constructed. Typically, quintuplets are the highest multiplicities observed (constructed from 4 electrons, i.e. half the number of electrons in a completely filled outer valence shell with two s and four p electrons). To distinguish the quantum number of a single electron from those of the full atom on characterizes them with lower case and capital letters, respectively. Hence, the total orbital angular momentum of an atomic state (quantum number L) is characterized by upright capital letters S, P, D, F, G for L = 0, 1, 2, 3, 4 (in analogy to s, p, d, f, g, . . . for the i quantum numbers of individual electrons). As superscript on the left of the angular momentum one writes the multiplicity, as subscript on the right the total angular momentum is given. In summary, an atomic state with N electrons in a configuration {n1 1 . . . nN N } (see Sect. 3.1.1), total orbital angular momentum L, total spin S and overall total angular momentum J is written as {n1 1 . . . nN N } 2S+1 LJ .

(6.55)

Often the electron configuration is dropped for brevity. As additional information for multi-electron spectra one often finds a superscript o for odd parity, especially if the latter cannot be identified uniquely from the angular momentum quantum number (more about parity is discussed in Appendix D). We give a few examples: • For the H atomas well as for alkali atoms s states with = 0 and j = J = | ± 1/2| = 1/2 are denoted as ns 2 S1/2 , • and p states with = 1 and j = J = | ± 1/2| = 1/2, 3/2 are correspondingly np 2 Po1/2 and np 2 Po3/2 , respectively, • while for a d state with = 2, j = | ± 1/2| = 3/2, 5/2 and hence nd 2 D3/2 or nd 2 D5/2 holds. • The ground state of the He atom (two 1s electrons, total spin = 0) is characterized by 1s 2 1 S0 , • while the first excited states are in the singlet system 1s2s 1 S0 and 1s2p 1 Po1 , • and the corresponding triplet states are 1s2p 3 Po2 , 1s2p 3 Po1 , 1s2p 3 Po0 and 1s2s 3 S1 . They will be discussed in detail later.

6.3

Quantitative Determination of Fine Structure

303

We mention, finally, that in the spectroscopic literature and relevant data collections the spectra of neutral atoms are marked with the roman number I, those of singly ionized atoms with II and generally the spectra of the N fold ionized species with roman N + 1. Thus, one speaks of the C I, O II, Fe III spectra and refers in this way to the spectra of the neutral C atom, and of the ions O+ and Fe++ , respectively. Section summary

• Spin-orbit (or LS) interaction (6.35) arises as magnetic dipole energy of the electron spin oriented in the magnetic field due to the electron orbital angular momentum. It is proportional to L · S and to the gradient of the potential in which the electron moves. • Quantitatively, the resulting fine structure (FS) splitting is on the order of α 2 Z 4 and decreases ∝ 1/n3 with the principal quantum number (its relative magnitude being ∝ (αZ)2 /n). • With spin-orbit interaction the orbital and spin angular momenta couple to a total angular momentum J = L+ S. With | m and |sms describing the orbital and spin angular momentum, respectively, the states of the coupled angular momenta |(ls)j mj are a sum of products of the former (involving so called C LEBSCH -G ORDAN coefficients or 3j symbols). • good quantum numbers are n j mj with mj = m + ms , while m and ms are not. The L · S interaction term is diagonal in this scheme, its eigenvalue being 2 ( /2)[j (j + 1) − ( + 1) − s(s + 1)]. • In multi-electron systems the angular momenta of several active electrons are coupled. For low atomic numbers Z the FS can be treated as a small perturbation, and RUSSEL -S AUNDERS coupling (also called LS coupling) gives a good description of the atomic structure: first all electron orbital momenta i couple to L (quantum number L) and all electron spin momenta s i couple to S (quantum number S). Then L and S couple to a total angular momentum J (quantum number J ). • A specific electron configuration it then characterized by 2S+1 LJ , with 2S + 1 giving the multiplicity of the spin states.

6.3


6.3.1

FS Terms from D IRAC Theory

As we have seen in Sect. 6.2.3 the spin-orbit splitting is on the order of α 2 Z 2 = Z 2 Eh /(me c2 ) – hence the name “fine structure constant” for α. The comparison with me c2 already indicates that this interaction is of relativistic nature. Hence, an exact treatment of the fine structure requires the solution of the D IRAC equation, which combines the principles of both, quantum mechanics and special relativity – in contrast to the S CHRÖDINGER equation. It describes fermions (such as the

304

6


electron) correctly by so called spinors (with four complex values) rather than by one complex wave function, and it provides automatically the correct electron spin s = 1/2 with its magnetic moment and ge = 2, as well as the concept of antimatter. Unfortunately, an exact solution and comparison with precision spectroscopy is only possible for the H atom and does even in this case – as we shall see in Sect. 6.5 – not lead to a complete agreement with experiment. Interactions with the vacuum field, to be treated with QED, enter into the term energies with small contributions only about an order of magnitude smaller than those introduced by the (relativistically correct) D IRAC theory. Thus, we refrain here from a detailed treatment of the D IRAC equation which is found in all major quantum mechanics textbooks. Clearly, computations with the D IRAC equation (even approximate ones) are substantially more complex than those with the S CHRÖDINGER equation. Hence, one tries to “make” the latter “relativistic”, i.e. to include electron spin and relativistic effects by two component spin functions (as outlined in Sect. 6.2.5), originally introduced by PAULI, and by adding appropriate interactions derived from D IRAC theory. As summarized below, there are three relativistic terms which have to be added to the S CHRÖDINGER equation. All of them are on the order (Zα)2 , i.e. of similar, rather small magnitude. If they cannot be treated exactly, perturbation theory usually leads to very satisfactory results for most problems in AMO physics and quantum chemistry. In combination with QED and higher order approximations, one can even explain state-of-the-art high precision experiments extremely well – for not too large atomic nuclei, i.e. as long as (αZ) 1 can be assumed in appropriate series expansions.

Relativistic Correction Term to the Kinetic Energy From the relativistic expression of the total energy (1.19) and with (1.22) one obtains the relativistic contribution to the kinetic energy in SI units p2 2 2 Wkin = m2e c4 + p 2 c2 − me c = me c 1 + 2 2 − me c2 (6.56) me c √ and approximates by expanding the root 1 + x 2 = 1 + x 2 /2 − x 4 /8 + · · · Wkin =

p2 1 p4 − . 2me 8 m3e c2

Thus, the relativistic, quantum mechanical correction term is to lowest order

2 2 2 p 2 1 p 4 1 p 1 Eh H1 = − 3 2 = − =− Eh , 2 2me me c2 2 me c2 2me Eh 8me c or in a.u. (with the fine structure constant α = Eh /me c2 ): 2 4 2 α2 1 = − α p =− H0 − V (r) H 2 4 2

0 (r) = with H

p 2 + V (r). 2

(6.57)

6.3


305

For H atoms this leads to

2 Z Z2 α2 α 2 2 H0 − V (r) = − H1 = − H0 + 2H0 + 2 . 2 2 r r 1 obviously depends only on r. In 1st order perturbation theory the energy shift is H 2 α2 02 + 2H 0 Z + Z |n n |H 2 2 r r

2 1 1 α Wn2 + 2ZWn + Z2 2 =− 2 r r

1 |n sj mj = − Vrel = n sj mj |H

with Wn = −Z 2 /(2n2 ). Inserting the matrix elements for the H atom according to (2.130) one obtains in a.u. Vrel = −Wn

(Zα)2 3 1 . − n 4n + 1/2

(6.58)

D ARWIN Term The so called DARWIN term emerges from D IRAC theory as in a.u. 2

3 = π Zα δ(r). H 2

(6.59)

This perturbation is non-zero only at the origin r = 0 and we only need to evaluate it for s states ( = 0) – the only states which have a finite probability at the origin. In 1st order perturbation theory we obtain Zα 2 ns|δ(r)|ns 2 2 2 (Zα)2 Zα 2 Zα 2 ψns (r) δ(r)d3 r = π ψns (0) = −Wn , VD = π 2 2 n 3 |n sj mj = π VD = n sj mj |H

(6.60)

where the last identity is derived for the H atom from inserting Rn0 (0) according to (2.122), using the explicit expressions for the L AGUERRE polynomials (2.123). We note that the relativistic correction as well as the DARWIN term represent ( independent) shifts of the energy levels and not splittings. Only if the term positions can be computed exactly (i.e. essentially only for the H atom and H like ions) one may hope to compare experiment and theory in respect of these shifts. Already for the alkali atoms, where degeneracy is significantly removed, such an absolute comparison of these very small effects is usually beyond the limits of the combined accuracy of theory and experiment.

306

6


Spin-Orbit Term LS (6.35) In contrast to the two terms discussed above the spin-orbit interaction V leads to a splitting which can be calculated and measured with good accuracy and is relevant for all atoms. In the L · S coupled scheme it is diagonal in respect of the sj mj quantum numbers: 2 LS = α n |ξ(r)|n sj mj | VLS = V L · S| sj mj . 2

(6.61)

With (6.52) and with the spin-orbit interaction parameter a defined in (6.36) one may write the energy change due to spin-orbit interaction as Wn j ≡ VLS =

a j (j + 1) − ( + 1) − s(s + 1) . 2

(6.62)

Obviously, the fine structure splitting depends on j and . States with = 0 and j = s do not split, all other states do. Systems with one active electron and s = 1/2 form doublet energy levels with j = ± 1/2. If the potential V (r) is known, the fine structure splitting is readily derived from the matrix element n |ξ(r)|n .

6.3.2

Fine Structure of the H Atom

For the H atom and H like ions and > 0 (6.62) becomes with (6.37): VLS =

" 1 Z4 α2 for j = + 1/2 × −( + 1) for j = − 1/2. 2n3 2 ( + 1/2)( + 1)

(6.63)

Interestingly, the total energy with all FS terms included depends only on j . The exact solution of the D IRAC equation (e.g. B JORKEN and D RELL 1964, Eq. 4.14) is usually expanded in powers of (Zα)2 and is Wnj

(Zα)2 1 3 (Zα)2 6 1+ − + O(Zα) me c2 , =− n j + 1/2 4n 2n2

(6.64)

in units of the electron rest energy, while in a.u. it reads

1 3 Z2 Z4 α2 − + · · · Eh . =− + 2n2 2n3 j + 1/2 4n

Wnj

(6.65)

In the following we stay with a.u. unless otherwise mentioned. The terms are split into FS levels with j = ± 1/2, if and only if ≥ 1, by

Wn =

1 Z4 α2 Eh . 2n3 ( + 1)

(6.66)

6.3


307

s1/2 0.730

p 3/2

0.091

0.122 2s, 2p

p1/2 p 3/2

0.213

0.243

s1/2

p 3/2

p1/2

p1/2 p3/2 1.186 Wnℓ BOHR SCHRÖDINGER

s1/2 p1/2

s1/2 Vrel relativistic

0.456

WNJ VD DARWIN

VLS spin orbit

DIRAC

Fig. 6.21 Fine structure of the H atom, n = 2 levels. All energy shifts in cm−1

Figure 6.21 summarizes the situation for the n = 2 levels of atomic hydrogen: The non-relativistic S CHRÖDINGER equation (as well as the B OHR model) predicts identical energies Wn for the 2s and 2p levels (left). The different contributions to Wnj from D IRAC theory are indicated as dotted lines. The final results Wnj = Wn + Vrel + VD + VLS , are shown on the right. Note that D IRAC theory predicts Wnj for s1/2 and p1/2 to be identical.

6.3.3

Fine Structure of Alkali and Other Atoms

The LS splitting decreases rapidly with n and as one would expect for an interaction ∝ 1/r 3 . For not too large (typical in spectroscopic studies) we memorize that fine structure splitting quite generally is ∝ Z 4 /n3 . Spin-orbit coupling is particularly important for larger atoms. As we shall see in the following chapters, RUSSEL -S AUNDERS (or LS) coupling works very well for light atoms where spinorbit coupling is small. It breaks down for large Z and other coupling schemes take over, such as the so called jj coupling. As already mentioned, only for H and hydrogen like ions He+ , Li++ , etc. (and a few special exceptions, see Sect. 6.5.5) absolute calculations of term energies are possible – with a precision to verify or falsify the energetic level positions shown in Fig. 6.21. However, the FS splittings due to L · S interaction for individual n levels may be computed with sufficient accuracy for comparison with experiment. Thus, LS according to (6.35) to the for practical applications it is usually sufficient to add V standard S CHRÖDINGER equation, which for effective one electron systems simply amounts to (6.62). For very heavy atoms it may become necessary to also account for the relativistic kinetic energy shift. For effective one electron systems such as alkali atoms and > 0 the energy levels split in doublets according to (6.62). The spin-orbit coupling parameter a (6.36) summarizes the specific FS properties of a given atom in a state characterized by n . Fine structure splitting for alkali atoms is significantly larger than for the

308

6


Table 6.2 Fine structure splitting of the first excited p states of H and the alkali atoms Atom

H

Li

Na

K

Rb

Cs

n

2p

2p

3p

4p

5p

6p

Wn j / cm−1

0.365

0.335

17.196

57.71

237.595

554.039

H atom. Table 6.2 gives a survey for the first resonance lines (ns 2 S1/2 ↔ np 2 P1/2 and np 2 P3/2 ) of different alkali atoms in comparison to the H atom. Note that for Cesium (Cs) FS splitting is already 5 % of the total transition energy – not really a small effect any more. We summarize a few general rules valid for H and the alkali atoms but also beyond: • The higher n and the smaller the FS splitting. For example, in Na already the 5p level is split by only 2.47 cm−1 and for the 3d level, where the electron hardly ever comes close to the nucleus it is only 0.05 cm−1 . • For term energies usually Wnj = +1/2 > Wnj = −1/2 holds – this is called normal ordering. For higher terms and heavier elements there are cases where the energies are inverted: the one electron picture is just an approximation which may break down under extreme conditions. • Expression (6.62) for the FS splitting is (approximately) valid in many cases also for more complex atoms. Then, of course, the overall total angular momentum J , the total orbital angular momentum L and the total spin S > 1/2 replace j , and s. The formula has proved to be valuable in particular for larger Z and n, at least as long as LS coupling is still valid. • Using (6.62) one derives the distance of two neighbouring fine structure levels in a multiplet, the so called L ANDÉ interval rule:

WFS = WJ − WJ −1 =

a J (J + 1) − (J − 1)J = aJ. 2

(6.67)

The energy differences between any two neighbouring FS levels with J and J − 1 in an FS multiplet are proportional to J . For doublets, (6.67) gives the splitting. Occasionally, equation (6.37) (valid for the H atom) is adapted by substituting Z with an effective nuclear charge Zeff . However, a comparison with experimental data for the alkali atoms shows that such a formula has only rather limited predictive power for FS splittings – even within one series of n at fixed . A much more consistent parametrization of a given n 2 L series for all n is obtained for the effective change of the quantum defect due to FS as derived from QDT and documented in Fig. 6.22. Using (3.25) we obtain

μj = n∗ 3 × Wn j

(6.68)

which may be applied since spin-orbit interaction scales with 1/n3 . In Fig. 6.22 we exemplify this for the FS splitting of the 2 P and 2 D series in Na and Rb, summarizing (as a function of binding energy) all data reported by NIST for the FS splitting Wn j in Na and Rb – effectively by two parameters for

6.3


309 0.4

0.02223

0.3688 np 2P

0.022 np 2P 0.3 0.020

0.00444 0.04

Na (Z=11)

Rb (Z=37) nd 2D

0.02

0.00042 nd 2D 0 -3.0

-2.0

-1.0

0.0

0.00 -3.0

0.00023 nf 2F

-2.0

-1.0

0.0

/ eV

Fig. 6.22 Effective change μj = n∗ 3 × Wn j of the quantum defect due to FS for Na and Rb. The experimental data points have been derived from the NIST data (K RAMIDA et al. 2013). Note that μj follows very clearly the linear least squares fits (red lines) as a function of the binding energy Wn – typical for quantum defects (compare Fig. 3.7)

each n 2 L series. As shown in Fig. 6.22, the linear least squares fits represent the experimentally observed data extremely well. We have to recall: μj is the change of the quantum defect due to FS splitting. It has to be compared to μ(n, ) as shown in Fig. 3.8 (for Wnl → 0, n 1). Obviously, for Na the FS splitting is a minor effect, in particular for the P1/2 − 2 P3/2 series where μ(∞, 1) 0.8551 and μj 0.022. In contrast, for the same series in Rb FS splitting with μj 0.369 is no longer negligible in comparison to μ(∞, 1) 2.6535. Section summary

• An exact treatment of fine structure requires the solution of the D IRAC equation. However, very good approximations can be achieved by incorporating electron spin and the three essential interaction terms into the S CHRÖDINGER equation: – Relativistic correction to the kinetic energy, according to (6.57), – the DARWIN term, non-zero only at the origin and thus relevant only for s electrons, and – spin-orbit interaction which can be diagonalized for one electron systems by for |( s)j mj states, and is given by (6.61) and (6.62). • The first two terms represent shifts and can be compared with experiment essentially only for H and H like ions. Spin-orbit interaction leads to splittings which are readily observed and compared to theory for all atoms. • For atomic H (and H like ions) D IRAC theory predicts a splitting between np3/2 and np1/2 states, but equal energies for the latter and ns1/2 .

310

6


• For larger alkali atoms FS splitting is no longer a small effect and amounts e.g. for the first resonance line in Cs already to 5 % of the transition energy.

6.4

Selection Rules and Intensities of Transitions

6.4.1

Introduction

For fine structure transitions all selection rules hold as derived and discussed for the general case in Sect. 4.4. In particular, for dipole allowed (E1) transitions the triangular rule δ(Ja Jb 1) = 1 holds for the total initial Ja and final Jb angular momenta. Their projection quantum numbers may change according to Ma − Mb = 0, ±1 depending on the polarization of the emitted or absorbed light. We use here capital letters, indicating that all selection rules also hold for systems with several electrons. In the following we shall apply the general expressions for transition probabilities derived in Sect. 4.6 to systems which ar well described by LS coupling. Specifically we are interested in the relative intensities of several lines within one multiplet. This requires some extension of the angular momentum algebra used so far. These tools – albeit they may appear somewhat uninspiring in the beginning – will later on prove very useful for describing various other, more complex situations. Details are presented in Appendixes C and B.3. To illustrate the present subject, Fig. 6.23 gives a schematic overview of the dipole allowed transitions between fine structure levels in the H atom up to n = 3 (term energies are not to scale). The thickness of the arrows indicates the strengths of the individual transitions within the different multiplets (only to be compared within one multiplet). All transitions between the doublet levels of the H atom (as well as of the alkalis) lead to double spectral lines or to triple lines one of which is typically very weak (see the 3d 2 D → 3p 2 P transition). We point out here that the relative intensities within a multiplet in emission differs from those in absorption. For example, the two components of the first resonance line in emission, 2p 2 P1/2 → 1s 2 S1/2 and 2p 2 P3/2 → 1s 2 S1/2 occur with of probability, while the ratio of the absorption cross sections for the inverse processes is 1 : 2.

6.4.2

Transitions Between Sublevels vs. Overall Transition Probabilities

We recall that LS coupling implies that orbital angular momentum L and spin angular momentum S combine to a total angular momentum J of the system. The following considerations are independent of whether L and S refer to a single active electron or represent several electrons which, in turn, are composed of orbital and spin angular momenta from several active electrons. Our first task is to calculate the line strengths S(Jb Ja ) in the (LS)J coupled scheme for transitions between individual fine structure levels with quantum numbers Ja and Jb . According to (4.112)


Fig. 6.23 Selection rules for fine structure transitions (emission) in the H atom

311

term energies

n not to scale

6.4

3

2

1

ℓ=0 3s1/2

ℓ=1

ℓ=2 3d5/2

3p3/2

3d3/2

3p1/2

2p3/2 2p1/2

2s1/2

1s1/2

it is proportional to the radial transition matrix element and the corresponding reduced matrix element of the renormalized spherical harmonics tensor C1 : 2 (6.69) S(Jb Ja ) = (2Jb + 1) γb |r|γa Lb SJb C1 La SJa 2 . The ratios of transition probabilities within individual multiplets (as indicated by the red arrow in Fig. 6.24) may be derived by further reduction of the reduced matrix element in (6.69). Since the dipole operator er responsible for E1 transitions does not act on the electron spin, S remains constant during the transition. In (6.69) this is formally born out by C1 , a spherical tensor of rank one which only acts onto the angular momentum part of the coupled scheme |La SJa . Hence, using (C.46) and the 6j symbols described in detail in Appendix B.3 we may pull out the spin: 2 S(Jb Ja ) = γb |r|γa (6.70) #2 " La Lb 1 Lb C1 La 2 . × (2Jb + 1)(2Ja + 1)(2Lb + 1) Jb Ja S From this we may derive the spontaneous lifetime(s) of the upper level(s). First, with (4.114) we obtain the individual spontaneous transition probability A(Ja Jb ) from an upper multiplet substate |Jb Mb into all substates |Ja Ma of one specific lower Ja level. This probability is independent of Mb and Ma but still depends on Jb and Ja . We then sum over all final Ja levels to obtain the total decay probability of the level Jb :

A(Ja Jb ) =

Ja

4αω3 S(Jb Ja ) ba 3c2 2Jb + 1

(6.71)

Ja

3 4αωba γb |r|γa 2 Lb C1 La 2 3c2 " La L b × (2Ja + 1)(2Lb + 1) Jb Ja Ja

1 S

#2 .

312

6


γb L b S

Jb A(Jb Ja) Ja

A(Lb La)

γa L a S

Fig. 6.24 Transitions between multiplet levels γb Lb S and γa La S, explaining the terminology for transition probabilities between individual J sublevels and overall L levels

The sign in the second line accounts for the radial transition matrix elements not being strictly independent of Ja and Jb and the angular frequencies ωba differing slightly for different multiplet components. Significant deviations occur only for heavy atoms.4 We now exploit the 6j orthogonality relation (B.67) and obtain:

A(Ja Jb ) =

Ja

A(La Lb ) =

3 4αωba γb |r|γa 2 Lb C1 La 2 = A(La Lb ) = τ −1 (6.72) 2 3c 3 4αωba S(Lb La ) . 2 3c 2Lb + 1

(6.73)

Note that (6.72) is not only independent of Mb but also of the initial Jb level: we thus have found the overall decay probability A(La Lb ) of the upper γb Lb S level into all substates of the lower γa La S level. It only depends on La and Lb and on the radial matrix element γb |r|γa . Thus A(La Lb ) is equal to the inverse spontaneous lifetime τ of the upper level (more precisely: of each substate |Jb Mb ) – as far as no other, competing decay channels exist. With (6.73) we have introduced also the overall line strength S(Lb La ) for the multiplet transition, analogously to (6.69). Obviously this relation is completely equivalent to (4.114) where an uncoupled scheme was assumed. Only the initial and final orbital angular quantum numbers Lb and La and the radial matrix element determine the excited state lifetime: τLa ←Lb = τ is the lifetime of the atomic transition. Take e.g. a sodium atom in the first excited 3p 2 P1/2 or 3p 2 P3/2 resonance state: both states decay into the 3s 2 S1/2 ground state and each of the 6 different excited substates has the same lifetime in respect of spontaneous decay. All above equations still contain the reduced matrix element of C1 . It may be further evaluated by angular momentum algebra. Specifically, for quasi-one-electron systems such as alkalis with S = s = 1/2 we put La = a and Lb = b , respectively, 4 The

lifetimes of the 3 2 P1/2 and 3 2 P3/2 states in Na are reported as 16.30 and 16.25 ns, respectively, while for Cs 14 % difference is observed with 34.7 ns for the 6 2 P1/2 level and 30.4 ns for the 6 2 P3/2 level (see e.g. S TECK 2010, and references there). This warrants an improved theoretical treatment with more accurate wave functions for each individual j state. These could e.g. be derived by QDT as outlined in Sect. 3.2.6 with the quantum defect differences among FS levels as presented in the last paragraphs of Sect. 6.3.3.

6.4


313

the angular momentum quantum numbers of the electron. The reduced matrix element Lb C1 La can be evaluated explicitly with (C.30): " 2 l S(nb b sjb − na a sja ) = nb b |r|na a (2jb + 1)(2ja + 1) a jb

2 b 1 a × (2 b + 1)(2 a + 1) . 0 0 0

lb ja

1 1/2

#2

(6.74)

The 6j symbol implies the triangular relation δ(ja jb 1) = 1, or equivalently the selection rule j = 0, ±1 (however jb = 0 ja = 0). The 3j symbol demands in addition that a + 1 + b is even, i.e. that b = a ± 1 ensuring parity conservation. Finally, with (B.53) the last line in (6.74) is simply written as ×( b + a + 1)/2.

(6.75)

In the more general case, where L is already a coupled orbital angular momentum as typical for more complex atoms, one has again to apply the reduction scheme described above onto that coupling scheme, do this step by step and exploit (C.30) where possible. To compute the radial matrix element nb b |r|na a one has to know the wave functions in detail. Useful programmes are now readily available. One may then obtain the spontaneous lifetime τ = τLb La from (6.72). Once τ is known (either from theory or from experiment) it is often useful to express all quantities in terms of it. Inserting (6.72) into (6.70) one obtains for the line strength S(Jb Ja ) =

" 3c2 2Lb + 1 La (2J + 1)(2J + 1) b a 3 Jb τ 4αωba

Lb Ja

1 S

#2 (6.76)

by which in turn all other relations maybe expressed: e.g. the state to state spontaneous transition probabilities A(Ja Ma ; Jb Mb ) according to (4.113) as well as the corresponding coefficients B(Ja Ma ; Jb Mb ) = B(Jb Mb ; Ja Ma ) for induced transition probabilities (4.124).

6.4.3

Some Useful Relations for Spectroscopic Practice

We communicate probabilities for transitions between the components of two multiplets, inserting (6.76) into the respective expressions from Sect. 4.6.5 The spontaneous emission probability from level Jb of a multiplet γb Lb S to a specific level Ja of a (lower lying) multiplet γa La S summed over all polarizations, emission angles 5 We

recall here that the direction of a transition is always read from right to left, e.g. B(Jb Ja ) implies a transition to Jb ← Ja from Ja .

314

6


and final projection quantum numbers Ma is obtained from (4.114) (independent of the initial Mb quantum number): " #2 3 4αωba S(Jb Ja ) La Ja S 2Lb + 1 A(Ja Jb ) = = (2Ja + 1) . (6.77) J b Lb 1 τ 3c2 2Jb + 1 For induced transitions, emission or absorption, the corresponding probability from level Jb of a multiplet γb Lb to level Ja of a multiplet γa La is obtained from (4.125) or (4.126), respectively. Averaged over an isotropic population of initial Mb substates and summed over all final Ma states (independent of the polarization employed) the B coefficient is #2 " 4π 2 αc S(Jb Ja ) π 2 c3 La Ja S 2Lb + 1 B(Ja Jb ) = = (6.78) (2Ja + 1) 3 J b Lb 1 3 2Jb + 1 τ ωba =

π 2 c3 3 ωba

(6.79)

A(Ja Jb ),

and for the inverse process (absorption or emission) " 4π 2 αc S(Jb Ja ) π 2 c3 La = B(Jb Ja ) = (2Jb + 1) 3 Jb 3 2Ja + 1 ωba =

Ja Lb

S 1

#2

2Jb + 1 B(Ja Jb ). 2Ja + 1

2Lb + 1 τ (6.80)

One sees that the relative intensities (branching ratios) to different final Ja or Jb levels depend on the degeneracies of the latter and on the squared re-coupling coefficients (6j symbols) − but not on M or on the polarization. We have already used the 6j orthogonality relation (B.67) to show that the spontaneous lifetime does not depend on the initial total angular momentum Jb . In the same way one may prove a corresponding relation for the induced processes: from (6.78) and (6.80) one finds that induced emission and absorption coefficients are independent of the initial Ja or Jb , respectively, if the final states are not resolved: B(La Lb ) =

B(Ja Jb ) =

π 2 c3 1 π 2 c3 = A(La Lb ) 3 τ 3 ωba ωba

(6.81)

B(Jb Ja ) =

π 2 c3 2Lb + 1 1 3 2L + 1 τ ωba a

(6.82)

Ja

B(Lb La ) =

Jb

=

2Lb + 1 B(La Lb ). 2La + 1

(6.83)

Obviously, the familiar E INSTEIN relations (4.127) are valid for the individual fine structure components (6.77)–(6.80) with j := J , as well as for the multiplet averaged quantities (6.81)–(6.83) where j := L.

6.4


315

Table 6.3 Branching ratios in 2 S ↔ 2 P and 2 P ↔ 2 D transitions calculated by (6.80) and (6.78). The relative intensities can only be compared within one multiplet. The sum over all transitions starting from one Ja or Jb are normalized to unity to

to

from 2S

1/2

2P

1/2

2P

2P

3/2

1/2

−

1

1

2P

2P

1/2

1/3

−

−

2P

2P

3/2

2/3

−

−

1

1

1

2S

from

1/2

3/2 2D 3/2 2D 5/2

1/2

2P

3/2

2D 3/2

2D 5/2

−

−

5/6

−

−

−

1/6

1

1

1/10

−

−

−

9/10

−

−

1

1

1

1

In the older literature, branching ratios within multiplets were often presented in extended sets of tables – or derived by equating different sums from (6.81) and (6.83). Today these relative line strengths and transition probabilities are evaluated readily using the 6j symbols (computed e.g. with S VEN G ATO R EDSUN 2004). Hence, we only communicate by way of example in Table 6.3 the branching ratios for 2 S ↔ 2 P and 2 P ↔ 2 D multiplets as relevant e.g. for H and the alkali atoms. It is interesting to note that for a 2 S ↔ 2 P transition the absorption probability for the 2S 2 2 1/2 → P3/2 is twice that for the S1/2 → 2P1/2 transition. In contrast, the inverse 2 2 2 processes P1/2 → S1/2 and P3/2 → 2 S1/2 have identical transition probabilities, as already mentioned in the context of Fig. 6.23. Section summary

• Selection rules for transitions between fine structure levels follow the general concepts derived in Sect. 4.4 from angular momentum conservation – now applied to the total angular momentum quantum number J and its projection M onto a given axis. • E1 transitions are allowed for J = 0, ±1 (but 0 0). For linearly and circularly polarized light M = 0 and ±1 must hold, with M referring to the directions of polarization and light propagation, respectively. In addition, parity of the whole system (photon plus atom) has to be conserved which for single electron systems requires = ±1. • Transition probabilities between individual J and J levels can be derived from line strengths in the usual manner, involving the radial matrix element between initial and final n and n states and the reduced matrix element Jb C1 Ja in the coupled system. Using angular momentum algebra the latter may be reduced to expressions which refer to the initial and final orbital angular momenta L and L . • Spontaneous emission probability from one specific initial substate |J M of level nL to all substates |J M of the final level n L is independent of J

316

6


(a)

(b)

(e) (a)

(f) (d)

(c)

(b)

(g)

crossover

LAMB shift 1057.846 MHz

(a)

(b) (c) 0

(d) (e) (f) (g) 10

/ GHz

Fig. 6.25 Fine structure and L AMB shift for the H atom. (a) Term scheme and transitions of the Hα BALMER line between the n = 3 and n = 2 levels (energies not to scale). (b) Measurement by D OPPLER free saturation spectroscopy and original assignment according to H ÄNSCH et al. (1974); positions and heights of the stick spectrum (vertical full red lines) give the most recent theoretical data according to K RAMIDA (2010)

and M. It only depends on L and L and on the corresponding radial matrix element. Thus, it is the inverse of the overall lifetime of the upper level. • Also, for induced emission and absorption, when averaging over all (equally populated) initial M levels, the transition probabilities are independent of the initial J and M and of the polarization.

6.5

L AMB Shift

6.5.1

Fine Structure and L AMB Shift for the Hα Line

In Sect. 6.1 we have already familiarized ourselves with optical methods for observing E1 transitions between the n = 2 and n = 3 levels in the H atom with high precision. Figure 6.25 shows (a) the relevant term scheme and reproduces in (b) again the well resolved spectrum from Fig. 6.13. The assignment of the spectral lines to transitions is indicated by lower case letters. For a quantitative comparison of line intensities one would use (6.74) with the radial dipole matrix elements from Appendix C.5. However, one important feature shown in Fig. 6.25 cannot be explained by the FS theory discussed so far: We remember that according to (6.65), the energy terms derived from D IRAC theory depend only on j but not on . This rule is obviously broken for the n = 2 levels states in H as revealed by the experiment, the 2p 2 S1/2 being slightly higher in energy than the 2p 2 P1/2 level: This important observation is called L AMB shift.

6.5

L AMB Shift

6.5.2

317

Microwave and RF Transitions – D OPPLER Free

The main enemy for optical precision spectroscopy of the H atom, with its small mass M and high average thermal velocity v , is the D OPPLER effect – which according to (5.22) also depends on the transition frequency νba : δνD ±νba × v /c = ±¯νba × v . If one tries to determine the FS splitting or other small effects from the difference of two optical transitions D OPPLER broadening enters proportionally to νba with dramatic influence on the result. However, in some cases it is possible to access the energetic difference between different |j mj states within one multiplet directly, typically by M1 transitions as described in Sect. 5.4. The necessary alternating magnetic fields can be generated without problems in microwave resonators. While the corresponding optical E1 transitions are observed in the VIS or UV region (at wavenumbers between (10 000 to 100 000) cm−1 ) the FS splittings amount to some cm−1 or even below. D OPPLER broadening can be neglected completely in this microwave regime (1 cm−1 = 30 GHz) and one may determine transition frequencies with orders of magnitude higher precision than by typical optical spectroscopy – and even finer effects may be measured with radio frequency transitions (RF) at frequencies from 1 MHz to 1 GHz, depending on the external magnetic field which one usually applies in addition.

6.5.3

Experiment of L AMB and R ETHERFORD

The most prominent, and in its consequences far reaching experiment was the first determination of the L AMB shift for which L AMB and K USCH (1955) obtained the N OBEL prize. In 1947, Willis E. L AMB and R. C. R ETHERFORD were looking for a transition between the 2s 2 S1/2 and 2p 2 P1/2 level – which according to D IRAC should be degenerate. Thereby they exploited the fact that the 2p 2 P1/2 state decays directly into the ground state (τ = 1.6 ns) while the 2s 2 S1/2 state is metastable. It may only decay by two-photon emission with a lifetime of τ ∼ 120 ms as we have learned in Sect. 5.3.2. The experimental scheme, sketched in Fig. 6.26, is interesting. A atomic beam (see footnote 24, Chap. 1) of hydrogen emerges from a hot oven where H2 dissociates. By electron impact a certain fraction of these H atoms is excited into the 2s 2 S1/2 and 2p 2 P levels. While the 2p 2 P1/2,3/2 atoms rapidly decay by spontaneous emission into the ground state, the metastable 2s 2 S1/2 atoms in the atomic beam reach a microwave resonator. There, the microwaves – if on resonance – induce the transition 2s 2 S1/2 → 2p 2 P1/2,3/2 . If one of these transitions occurs, the 2p 2 P1/2,3/2 states decay rapidly into the ground state. One detects the transition by the loss of metastable atoms reaching the detector. The L ANGMUIR -TAYLOR detector (see Sect. 1.9.3) is sensitive only to excited (2s 2 S1/2 ) atoms.

318

6


magnetic field B (variable) extrapolation to B = 0 electron beam

oven

ground state H atoms

excited H atoms

1s 2S1/2

2s 2S1/2

LANGMUIR TAYLOR detector (hot W metal sheet)

H+

Lyα is emitted, all atoms in tuneable microwave resonator 1...10 GHz

np states decay

16

QED DIRAC theory

2s 2 S1/2 (m j = 1/2) → 2 p 2 P3/2 (m j = 3/2)

12

~ 1060 MHz

8

2s 2 S1/2 (m j = 1/2) → 2 p P3/2 (m j = 1/2) 2s 2 S1/2 (m j = 1/2) → 2 p P1/2 (m j = -1/2)

4

2s 2 S1/2 (m j = 1/2) → 2 p P1/2 (m j = 1/2) 2s 2 S1/2 (m j = 1/2) → 2 p P3/2 (m j = -1/2)

}

transition frequency ν / GHz

Fig. 6.26 Scheme of the experimental setup of L AMB and R ETHERFORD for measuring microwave induced transitions from the 2s 2 S1/2 state of the H atom

0 0.1 = W (2s 2 S1/2 ) - W (2p 2 P1/2 )

0.2

0.3 B / T

Fig. 6.27 Results of the original L AMB shift experiment for atomic hydrogen. Transition frequencies between the 2s 2 S1/2 and the 2 p 2 P1/2 and 2p 2 P3/2 levels as a function of an external magnetic field B. Dashed lines: D IRAC theory; full lines: fits to the experiment and QED. The remaining energy difference between the 2s 2 S1/2 and 2p 2 P1/2 levels at B = 0 (ca. 1060 MHz) is called L AMB shift

The experiment was carried out at several fixed microwave frequencies and the splitting of the two states was registered by tuning this magnetic field so that resonant transitions occurred. In Fig. 6.27 the results are compared with D IRAC theory and quantum electrodynamics (QED). In the limit of vanishing magnetic field a finite value of 1060 MHz ( = 0.0353 cm−1 ) for the resonance frequency of the 2 2 2 S1/2 − 2 P1/2 transition remains – the L AMB shift, by which the 2 2 S1/2 level lies higher than the 2 2 P1/2 level. In contrast, D IRAC theory predicts exact agreement of both energies.

6.5

L AMB Shift

319

hydrogen atoms

cryostat

vacuum chamber

cooling laser

FARADAY cage chopper 243 nm 2×νF

2S detector

νF

time resolved photon counting

dye laser 486 nm probe laser

4/7 × νF

1/2 ×νF

Cesium clock

microwave resonator fluorescence detector

cooling lasers

frequency comb 70 fs Ti:Sapphire laser

9.2 GHz from Cs clock

cooling lasers

Fig. 6.28 Schematic of an advanced experiment for determining the 1S–2S transition frequency in the H atom by D OPPLER free two-photon spectroscopy according to N IERING et al. (2000). The setup in the vacuum chamber corresponds essentially to Fig. 6.16. The frequency comb is the heart of the experiment, providing a high precision frequency standard which is directly synchronized with an atomic Cesium fountain clock, sketched in the spirit of J EFFERTS and M EEKHOF (2011)

6.5.4

Precision Spectroscopy of the H Atom

In spite of these apparent advantages of microwave spectroscopy, optical methods have surpassed them by far – essentially on the basis of D OPPLER free two-photon spectroscopy as described in Sect. 6.1.8. Today, modern laser spectroscopy allows one – literally – to count the frequency of optical transitions with a never dreamed of precision. The pioneering work of Ted H ÄNSCH and his collaborators who devoted many years and a wealth of brilliant ideas on the spectroscopy of atomic hydrogen has led to an amazing accuracy. Jointly with John H ALL, Ted H ÄNSCH (2005) received the N OBEL prize in physics (shared with Roy G LAUBER) for “the development of laser-based precision spectroscopy, including the optical frequency comb technique”. We shall discuss this fascinating experimental technique in some detail in Sect. 1.4.3, Vol. 2. Briefly, one uses trains of extremely short, precisely clocked light pulses. Their repetition frequency is directly compared or synchronized with a time standard, e.g. with a Cs atomic clock (see e.g. N IERING et al. 2000). Instead of measuring the wavelengths of optical transitions by essentially analogue techniques as done by spectroscopists for the past two hundred years, one now compares the frequencies of the electromagnetic waves inducing the optical transitions in the VIS and UV spectral range directly to the comb frequencies. Thus, counting – an essentially digital technique – is employed ensuring orders of magnitude higher accuracy. Just to indicate the complexity and sophistication of such experiments, Fig. 6.28 reproduces roughly the scheme which was originally used by N IERING et al. (2000)

320

6


(for recent improvements see e.g. J ENTSCHURA et al. 2011; PARTHEY et al. 2011; M ATVEEV et al. 2013, the latter being a simultaneous effort of two institutes via a 920 km fiber link). Part of it, the two-photon excitation of an atomic H beam in a confocal resonator, has already been described in Sect. 6.1.8. A so called Cesium fountain atomic clock is used, which is currently employed world wide as primary time and frequency standard. It is sketched in Fig. 6.28 on the right. A compact description is found e.g. on the NIST Web-pages (J EFFERTS and M EEKHOF 2011). The special trick with this setup is the long effective interaction time T 1 s of the atoms with the microwave field: one exploits the method of R AMSEY fringes as described in Sect. 6.1.7. To optimize this concept the Cs atoms are first cooled with 6 diode lasers. By a brief detuning pulse applied to the bottom laser one “kicks” the atoms upwards, pushes them so to say through the microwave resonator. The grey trajectory in Fig. 6.28 shows schematically the path which the atoms take. Under earth gravity they move freely up to a classical turning point and then fall back into the resonator – indeed some kind of an “atom fountain”. One induces the ground state hyperfine transition in atomic Cs which is detected from a change of the fluorescence signal induced by the probe laser.6 As explained in Sect. 6.1.7 the total interaction time T in the frame work of the R AMSEY fringe method is given by the time of flight of the atoms from the resonator to the turning point and back to the resonator. The precision of the reference line is then ∝ 1/T , typically on the order of 1 Hz. Based on this type of optical measurements the spectroscopy of the H and D atom is today known with extreme precision. Specifically, the n = 2 L AMB shift W (2s 2 S1/2 ) − W (2p 2 P1/2 ) corresponds to for the H atom = 1057.847(9) MHz

and

(6.84)

for the D atom = 1059.28(6) MHz. Figure 6.29 summarizes the orders of magnitude of the different contributions to the n = 2 L AMB shift in the H atom. We add here: predicted by QED calculations but not seen in Fig. 6.29 is a tiny energy rise of the 22 P3/2 state by 0.00037 cm−1 while the 22 P1/2 state is lowered by 0.000479 cm−1 . The physical origin of the L AMB shift will be discussed in Sect. 6.5.6 where we shall also understand that these very small shifts are characteristic for P states. A particular challenge, for experiment and theory alike, is the L AMB shift of the 1s S1/2 ground state of hydrogen. In principle one may extract it from a very precise measurement of the 1S1/2 − 2S1/2 transition frequency and D IRAC theory. One problem is to avoid uncertainties in the fundamental constants needed to compute an exact value of the D IRAC prediction. One of the first ultra high precision measurements which circumvents this problem was reported by W EITZ et al. (1994). Without going into the sophisticated details of such an experiment, we just mention that the 1S1/2 − 2S1/2 frequency is compared with 4× that for the 2S − 4S/4D 6 We

recall that 1 s is defined as the duration of 9 192 631 770 periods of the transition frequency between the two hyperfine levels of the ground state in 133 Cs.

L AMB Shift

321

Fig. 6.29 Overview of the term positions of the n = 2 states for the H atom in S CHRÖDINGER, D IRAC and QED approximation; term distances are shown to scale and may be compared directly

wavenumber / cm-1

6.5

0

0.5

SCHRÖDINGER BOHR DIRAC j = 3/2

ℓ=1

QED ℓ=1

2P

3/2

0.365 cm-1 ℓ=0 j = 1/2

2S

1/2

2P

1/2

ℓ = 0, 1 ℓ=1 0.035 cm-1

transitions – thus deriving simultaneously an ultra precise value for the RYDBERG constant: 7 R∞ = 10 973 731.568539(55) m−1 R∞ c = 3.289841960364(17) × 1015 Hz.

(6.85) (6.86)

Using this value of R∞ , numerical values for the hydrogen 1S − 2S transition frequencies from B OHR /S CHRÖDINGER (1.149) or D IRAC theory (6.65) are obtained with appropriate accuracy: S CHRÖDINGER : D IRAC :

ν(1S − 2S) =

3R∞ c = 2.467381470272 × 1015 Hz, 4

ν(1S1/2 − 2S1/2 ) = 2.467411580803 × 1015 Hz .

(6.87)

The very recently confirmed (M ATVEEV et al. 2013) experimental value is ν(1S1/2,F =1 − 2S1/2,F =1 ) = 2.466061413187018(11) × 1015 Hz,

(6.88)

measured with a relative accuracy of 4.5 × 10−15 ! To compare this value measured for atomic H with the prediction from D IRAC theory (6.87) one has to correct it by the kinematic correction factor (1.146). With the present CODATA value for me /mp one obtains ν(1S1/2 − 2S1/2 ) × me /μ = 2.46740447220939(56) × 1015 Hz .

(6.89)

A distinct difference remains – on the one hand due to hyperfine interaction (in (6.88) indicated by the subscripts F = 1), and to the 1S1/2 L AMB shift on the other. A quantitative evaluation of HFS is relatively straight forward and experimentally well under control by measurements of the corresponding level splittings (we shall treat HFS in Chap. 9). In fact, the HFS splitting of the H 1sS1/2 ground state was for many years one of the best determined transition frequencies – until the ν(1S1/2 − 7 2010

CODATA values, M OHR et al. (2012), see also Appendix A.

322

6


2S1/2 ) transition acquired this merit. After all this is done one obtains the ground state L AMB shift W (1s 2 S1/2 ) − W (1s 2 S1/2 D IRAC) for the H atom = 8172.840(22) MHz

and

(6.90)

for the D atom = 8183.970(22) MHz, the presently agreed global average from several independent measurements. Note that this is again an upward shift of the D IRAC term level towards a slightly smaller binding energy – just as for the 2s 2 S1/2 level. We point out that the whole field of ultra high precision spectroscopy and the interpretation of its results by QED is still rapidly evolving. Several not fully resolved issues and discrepancies remain. The interested reader is referred to the literature for details. Unless otherwise specified, numerical values given here reflect the state-of-the-art according to some recent, extensive reviews by E IDES et al. (2001), K ARSHENBOIM (2005), K RAMIDA (2010). The critical reader may perhaps ask now, what this extreme precision might be good for. Let us quote from the master of precision spectroscopy, Ted H ÄNSCH et al. (2005): “Even beyond more precise measurements of fundamental constants and searches for their possible time variations, there are many good practical reasons for pushing the art of measuring time and frequency to the feasible limits. Advances in time and frequency metrology will enable the synchronization of clocks over large distances. Such synchronization is, e.g. needed in astronomy for very long baseline interferometry. Better clocks will improve the performance of satellite navigation systems. They are also crucial for the precise tracking of remote space probes. In telecommunications, better atomic clocks will be needed for network synchronization, as the need for bandwidth increases. Geologists can apply better clocks to study the variability of earth’s rotation or to follow the drifts of continents with millimeter precision. Astronomers can study irregularities in the periods of pulsars. For physicists interested in fundamental science, better clocks will permit new stringent tests of special and general relativity. Very likely, there will be new unexpected discoveries, as the art of precision spectroscopy and optical frequency metrology continues to advance.” And one may add in our present context, that progress in basic research has always been nourished by quantum jumps in precision. The L AMB shift itself is an excellent example: its discovery has paved the way for quantum electrodynamics (QED), today one of the cornerstones of modern physics.

6.5.5

LAMB Shift in Highly Charged Ions

Atomic physics with highly charged ions (HCI) is a very active and productive branch of modern physics. Advanced high energy ion storage rings and state-of-theart electron beam ion traps (EBIT), accessible as user facilities, offer challenging perspectives for a broad range of interesting fundamental science (see also Sect. 7.5,

6.5

L AMB Shift

323

Vol. 2) as well as for many technological, biological and medical applications. Critical tests of QED and of atomic-nuclear interactions with high Z ions are one exciting aspect of the spectroscopy of such systems (a compact review is given by B EIERS DORFER 2010). As we shall discuss in Sect. 6.5.6, the relevant coupling constant for QED is also αZ, and so called higher order loops (nth order perturbation theory) enter into the term energies with (αZ)n Thus QED may have a problem with highly charged ions where αZ is no longer really small, the fine structure constant being α 1/137. In the past, theory and experiment have mostly focussed on the 1S1/2 L AMB shift of hydrogen like ions. In singly charged He+ the experimental value for the 2 2 S1/2 –2 2 P1/2 transition is 14 041.13(17) MHz in excellent agreement with theory, giving 14 041.18(13) MHz. The main contribution from QED are the one-loop effects, self-energy and vacuum polarization. Effects from the finite size of the nuclei are included in comparisons of theory and experiment, and become equally important for larger nuclei. We shall discuss volume effects of this kind in Sect. 9.4 in the context of HFS. In total, one expects for hydrogen like, (Z − 1)-fold charged ions and ns levels a L AMB shift (see e.g. J OHNSON and S OFF 1985)

WLamb =

α (Zα)4 α Z4α2 2 F (Zα)m c = F (Zα)Eh , e π n3 π n3

(6.91)

given in terms of the electron rest mass energy and a.u., respectively. F (Zα) is a slowly varying function of Zα, but also depends slightly on n. For the neutral H atom F (1α) 10 as reflected in Fig. 6.29 for = 2. Inverting the experimental results, (6.84) and (6.90), we obtain F (1α) = 10.042037(27) and 10.39828(9) for the 1S1/2 and 2S1/2 L AMB shift, respectively. Note the proportionality WLamb ∝ Z 4 α 2 /n3 (in a.u.) which we have found to be characteristic also for fine structure (6.65). The additional factors α/π × F (Zα) make the L AMB shift about an order of magnitude smaller than FS. The overall trend of the L AMB shift is illustrated over the whole nuclear charge range from Z = 1 . . . 91 in Fig. 6.30(a). The dominating one-loop contributions and finite nuclear size effects are plotted together with the total theoretical value (red line) in units of (α/π)[(Zα)4 /n3 ]me c2 on a logarithmic scale. The vacuum polarization gives a negative contribution. Experimentally, spectroscopy in the hard X-ray regime in storage rings has provided impressive results for multiply charged, hydrogen like ions up to U91+ . Elegant methods have been employed in these studies, as sketched in Sect. 6.1.4. The 1S1/2 L AMB shift is typically determined by comparing the 1S1/2 − 2P1/2 and 1S1/2 − 2P3/2 transitions with D IRAC theory (making appropriate corrections for HFS and kinematics). Figure 6.30(b) according to S TÖHLKER (see G UMBERIDZE et al. 2005) compares the theory presented in (a) with these results from several sources on a linear scale. One may describe the agreement as quite “satisfactory”. However, so far truly high precision spectroscopy for HCI is not yet in sight, at least not one competing with the extreme accuracy of the 1s − 2s transition in the neutral H atom. Thus, the best test for two-loops QED contributions would presently

324

6 10

LAMB shift self-energy

1


10 8

- vacuum polarization

6 0.1

(a)

4

finite size of the nucleus

0.01

(b)

2

0.001 0

20 40 60 80 nuclear charge Z

100

0

20 40 60 80 nuclear charge Z

100

Fig. 6.30 1S1/2 L AMB shift for H like ions as a function of nuclear charge Z. (a) Main contributions to the L AMB shift according to J OHNSON and S OFF (1985) from so called one-loop QED effects (to be discussed in Sect. 6.5.6) and from the finite nuclear size. (b) Comparison of this theory with experiments according to G UMBERIDZE et al. (2005)

still be the L AMB shift of the H 1S1/2 ground state (known to be ca. 4 parts in 105 ) – were it is not hampered by the experimental uncertainties of the proton radius! On the other hand, for 91 fold ionized Uranium, U91+ , the 1S1/2 L AMB shift amounts to (460 ± 4.5) eV – by no means a tiny effect as in H atoms. And considering the difficulties of this experiment, the accuracy is impressive – but with about 1 % by far not yet good enough for a test of two-loop contributions. Recent theoretical and experimental advances appear to offer promising perspectives for a direct comparison of the 2s 2 S1/2 –2s 2 P1/2 L AMB shift in lithium like highly charged ions (B EIERSDORFER 2010). A number of interesting questions also remain to be clarified in respect of HFS in highly charged ions. Other fascinating possibilities and experimentally demanding challenges arise in this context from the advent of X-ray lasers, e.g. at XFEL P ROJECT (2011), for which theoretical predictions are made e.g. for laser induced fluorescence in the strong field regime (P OSTAVARU et al. 2011). In summary, the field remains open, highly active and exciting.

6.5.6

QED and F EYNMAN Diagrams

What is the physical origin of the L AMB shift? Very generally speaking: the interaction of the electron with the vacuum field. The fundamental theory for describing such processes, Quantum Electrodynamics QED and Quantum Field Theory, was developed by T OMONAGA, S CHWINGER and F EYNMAN. In 1965 they received the N OBEL prize for their ground breaking work, which is one of the cornerstones of modern theoretical physics including the standard model – i.e. far beyond its origin and the applications in atomic physics which we discuss here. However, even a brief introduction into QED would lead us far beyond the scope of the present textbook. Thus we confine ourselves here to a heuristic presenta-

6.5

L AMB Shift

Table 6.4 Most simple F EYNMAN graphs for e− , e+ and γ as well as for some elementary electromagnetic interactions

325 Graph

Description

1

Free electron e−

2

Free positron e+

3

Free photon γ

4

Electron emits photon

5

Electron absorbs photon

6

Positron emits photon

7

Positron absorbs photon

8

γ -annihilation by e− e+ pair generation

9

e− e+ -pair annihilation by γ generation

10

Self energy of the electron (one-loop)

11

Vacuum polarization (one-loop)

tion of F EYNMAN diagrams (or F EYNMAN graphs). They first appeared in Richard F EYNMAN’s famous 1949 paper on “Space-Time Approach to Quantum Electrodynamics” as a picturesque visualization representing the mathematical framework for describing the interactions of electrons, photons and other subatomic particles. They are a compact form of symbolizing perturbation integrals and will – that is our hope here – help us to understand what causes the L AMB shift and other related phenomena. QED expands interactions into a series of perturbation integrals accounting for the influence of the quantized electromagnetic field. These perturbation series are constructed for sequences of virtual or real events and warrant good book keeping to account for all possible processes which might occur – and this is done with the help of the F EYNMAN graphs. Each node (or vertex) in such a graph characterizes one individual interaction process, several subsequent nodes imply a higher order of perturbation theory. One speaks about one-loop, two-loop processes and so on. The movements of individual particles and their elementary interaction in space and time are represented by graphical symbols which are summarized in Table 6.4. The free electron (1) moves here from the left to the right, its antiparticle, the positron, (2) moves from right to left. The exchange particle, for electromagnetic interaction, the photon, (3) is characterized by a wiggly line. Graphs (4–9) are self explaining, graph (10) represents the typical interaction of an electron with the electromagnetic magnetic radiation in vacuum: a free electron moving in space constantly emits (virtual) photons which it quickly reabsorbs again. This is one of many

326

6


processes which contribute to the self energy of the electron. It plays a key role for the so called mass re-normalization by which QED overcomes problematic divergences. Another important process is vacuum polarization (11). In 1st order (oneloop) it generates one (virtual) e− e+ pair, which is then immediately annihilated again. Graph (11) may obviously be constructed from (8) and (9). We have somewhat loosely used the terms “quickly” and “immediately” which may be quantified by the energy-time uncertainty relation (1.125): the larger the energy of the emitted virtual photon, the shorter the time between emission and reabsorption, and similarly for pair formation and annihilation. From these basic diagrams one may construct more complex ones to describe various interaction processes between electrons and photons. The following rules have to be observed: • Energy and momentum conservation holds for each vertex. • Lines which leave the diagram are real particles. For all of them together the energy conservation law Wi = const must hold, with the particle energies in relativistic form: Wi2 = pi2 c2 + m2i c4 . • Lines which connect vertices represent “virtual particles”. For them, energy conservation does only need to hold within the uncertainty relation. However, these virtual particles cannot be observed! • Mathematically, vertices are represented by coupling constants, virtual particles by so called propagators. The fine structure constant α = Eh /(me c2 ), as defined in (6.103), is the coupling constant for electromagnetic interaction, specifying the interaction between electron and photon. The√interaction amplitude between an electron and a Z-fold charged particle is ∝ Z α. • In general, the “propagator” for bosons is f (q) ∝ 1/(|q|2 + m2boson ) with q being the momentum of the boson. Specifically for a photon f (q) ∝ 1/|q|2 . • The total amplitude of a graph is the product of all coupling constants, and propagators involved. Scattering cross sections are proportional to the square of the amplitude. In Table 6.5 F EYNMAN graphs are collected for several important and instructive examples of interactions. The graphs are more or less self explaining. We refrain here from a detailed description of the mathematical expressions represented by these graphs. Only for C OULOMB scattering we have indicated, how one may glean the underlying concepts: the well known RUTHERFORD scattering formula emerges in this case without any effort.

6.5.7

On the Theory of the L AMB Shift

According to QED the electromagnetic field has a finite amplitude, even in its vacuum ground state, determined by the zero point energy of the corresponding quantized harmonic oscillator. And the electron constantly emits and reabsorbs photons as indicated by the self energy graph in Fig. 6.31(a). Consequently, a force pulls

6.5

L AMB Shift

327

Table 6.5 Lowest order F EYNMAN graphs for several important interaction processes in atomic physics Example Process 1

Graph e-

C OULOMB scattering e− + Ze

√α

eprop

1/q 2 aga

Z√α

tor

+Ze

+Ze

√ √ The scattering amplitude at the two vertices is ∝ α and ∝ Z α, respectively. The √ virtual photon receives from √ the electron √ a 2recoil momentum q = 2me W sin θ/2. The total amplitude is thus ∝ α × Z α/q . Hence, the scattering cross section is: dσ Z 2 e4 Z2 α2 ∝ ∝ . 4 2 d" q W sin4 θ/2

(6.92)

This is the well known RUTHERFORDscattering formula. 2

e− e+ scattering

e-

e-

e+

e+

e-

e-

e+

e+

+

In addition to the C OULOMB term (as above), only two one-loop graphs are relevant: the annihilation and re-generation of an e− e+ pair by emission of a virtual photon. 3

C OMPTON scattering

γ

γ

+ e-

4

Pair production e+ e− → γ

ee-

γ

γ

γ

e-

e-

e+

+Ze The black cross symbolizes the atomic nucleus with which the electron interacts. The following diagrams also need the interaction with an atomic nucleus to allow for energy and momentum conservation. 5

6

Bremsstrahlung: e + Ze → e + Ze + γ

γ

e-

γ

+

e+Ze

ee-

+Ze

Electron bound in a C OULOMB potential. In the current literature the so called “D IRAC -C OULOMB propagator” is summarized by just two parallel lines, comprising several contributions e− + Ze

=

+

+

+ ...

the electron back and forth into a “Zitterbewegung” (trembling motion), so that effectively the electron is smeared out over a (still very small) finite volume instead of being point like. Due to this apparent “charge distribution” the electron ‘sees’

328

6

(a)


(b) self-energy

vacuum polarization

Fig. 6.31 F EYNMAN graphs of (a) one-loop and (b) two-loop radiative corrections for the L AMB shift. Graphs adapted from B EIERSDORFER (2010)

effectively a potential which – close to the origin – is slightly lifted up in respect of the ∝ −Z/R C OULOMB potential. And this in turn rises the energies of atomic states with wave functions that have a finite probability at the origin. This is mainly relevant for s electrons. And indeed the experimentally observed L AMB shift (6.84) (illustrated in Fig. 6.29) and (6.90) is positive and significant only for S states. QED can give reasonable estimates for the radius of this trembling motion and has profound tools to calculate WLamb in terms of a perturbation series – in principle with arbitrary accuracy, if all parameters are known with sufficient accuracy and computing power is not limited. As we cannot enter here into the depths of QED we turn the question around and try to glean an estimate for the “Zitter” radius rL from the measured L AMB shift. To keep it simple, we assume the electron to ‘see’ for r ≤ rL a constant potential −Z/rL , instead of −Z/r (in a.u.). In 1st order perturbation theory we thus obtain an estimate for the L AMB shift from the expectation value (3.41) of the perturbation U (r) = −1/rL − (−1/r) which is the difference of the smeared out potential and the original pure C OULOMB potential. Since rL is expected to be very small one can pull out the radial wave function at the origin and obtains for a state n m (all quantities in a.u.): Z Z − |n r rL 2 = ψn m (0) 4πZ

WLamb = n |

(6.93) rL 1 0

2 2 1 2 − r dr = ψn m (0) πZrL2 . r rL 3

According to (2.127) for the H atom |ψn00 (0)|2 = Z 3 /(πn3 ) so that

WLamb =

2 Z4 2 r . 3 n3 L

(6.94)

This clear result already explains the difference between ground state (n = 1) and first excited state (n = 2) L AMB shifts in atomic hydrogen by about a factor of 8 (see (6.84) and (6.90), respectively). Comparing (6.94) for n = 2 with hν from (6.84) gives an estimate for rL 1.39 × 10−3 a0 3.0 × 10−2 λC = 7.36 × 10−14 m,

6.5

L AMB Shift

329

with the C OMPTON wavelength λC of the electron (1.76). QED provides similar estimates (see e.g. E IDES et al. 2001, Eq. (28)) 2 2α 2α 3 ln (Zα)−2 λ¯ 2C = ln (Zα)−2 a02 . (6.95) r = π π This corresponds to r 2 3.4 × 10−2 λC for the characteristic effective spread of the electron – which obviously is below the C OMPTON wavelength but significantly above the proton radius (10−15 m). With (6.95) one would obtain a L AMB shift (in a.u.)

WLamb =

α Z4 α2 4 × ln (Zα)−2 . 3 π n 3

If we interpret 4/3 ln[(Zα)−2 ] as F (Zα), one may compare this directly to the general formula (6.91) – with numerical values 13 and 10, respectively which is not too bad for such a rough estimate. As a historical remark we point out that Hans B ETHE (1947), in his first calculation of the H 2S1/2 L AMB shift, obtained a very similar expression when calculating the self energy of the bound electron. Instead of 1/α 2 = me c2 /Eh as argument in the logarithm he obtained me c2 /(8.9Eh ) which led him to 1040 MHz – already pretty close to the experimental value (6.84). Today, of course, with several orders of magnitude better experimental accuracy and highly developed computational methods in QED it is obvious that in addition to the self-energy also the vacuum polarization has to be taken into account (see Fig. 6.31(a)) as well as higher order corrections. The curious reader may admire the diagrams for the relevant two-loop corrections in Fig. 6.31(b). Also a range of additional effects have to be considered. For the hydrogen atom in the n = 1 and 2 levels the present state-of-the-art is summarized in Table 6.6. We cannot enter into a deeper discussion which is ongoing among the experts. However, the agreement between theory and experiment is already now truly impressive. At the moment, the largest uncertainty which limits stringent tests of bound state QED still appears to be the proton radius. Its value rp = 0.8775(51) fm in 2010 CODATA (NIST 2010) already strongly depends on optical precision spectroscopy. Independent measurements from nuclear physics give different values and have much larger error bars. This is a startling situation, the proton being one of the most important building blocks of matter and its size so poorly known! The most recent and fascinating development in this context is a measurement of the L AMB shift in muonic hydrogen by P OHL et al. (2010). The muon (see Sect. 1.5.2) is the bigger brother of the electron, its mass being more than 200 times larger, with otherwise more or less identical properties. Hence, in muonic hydrogen, μp, all the physics of the H atom remains in principle unchanged – except for the magnitude of the measurable quantities. In particular, the B OHR radius is 200 times smaller in μp and thus the muons probability to be close to the proton is correspondingly higher. Consequently, the L AMB shift is influenced much more dramatically by the protons structure as in the case of atomic hydrogen. The

330

6


Table 6.6 Main QED contributions to the L AMB for the 1S, 2S and 2P states of atomic hydrogen, and comparison with the presently best experimental values Contribution

1s 1/2 MHz

Self energy (one-loop)

8383.339466(83)a

2s 1/2 − 2p 1/2 MHz 1072.958444f

2p 1/2 2p 3/2 MHz MHz −12.84692(2)e 12.54795(2)e

Vacuum pol. (one-loop) −214.816607(15)a −26.852075(2)g −0.00035e Two-loop corrections

0.7310(33)a

Proton sizea

1.253(50)a

Recoil

correctiona

Radiative recoil

corr.a

0.1566(62)g

2.401782(10)a 0.002(1)a

Recent theory (total)

8172.894(51)a

1057.844(2.5)d

Experiment

8172.840(22)b

1057.847(9)c

b E IDES

0

0

−0.0145e

−0.0177e

−0.0002e

0.0001e

−12.83599(8)e

12.51746(8)e

(2009)

et al. (2001)

c K RAMIDA

(2010)

d J ENTSCHURA

et al. (2005)

e J ENTSCHURA

and PACHUCKI (1996)

f J ENTSCHURA

et al. (2001) (non-perturbative)

g order

−0.01279(7)e

−0.0123(7)a

Other corrections

a B IRABEN

−0.00008e

0.02598(7)e

of magnitude: 1s1/2 scaled by 1/n3

μp 2S1/2 − 2P1/2 L AMB shift which the P OHL et al. collaboration succeeded to measure with high precision is 49881.88(76) GHz. On the basis of FS and HFS calculation and QED they deduced from their measurement a value for the proton radius rp = 0.84184(67) fm which claims an order of magnitude higher accuracy and differs by 5 standard deviations from the above CODATA value! Section summary

• The L AMB shift manifests itself most clearly by a splitting of the 2s1/2 and 2p1/2 terms in atomic H – for which D IRAC theory predicts identical energy. We memorize ν(2s1/2 ) − ν(2p1/2 ) 1060 MHz. • It was first discovered in 1947 by L AMB and R ETHERFORD using microwave spectroscopy and triggered the development of QED. Today two-photon spectroscopy of the H and D atom is the method of choice. It has achieved an amazing precision, not least due to the work of H ÄNSCH and collaborators, using frequency comb techniques and state-of-the-art atomic clocks. • Agreement with equally accurate QED computations is convincing. We have encountered a couple of F EYNMAN graphs which are useful to understand the origin of such radiative corrections for high precision atomic and molecular spectroscopy. In these terms, the L AMB shift arises dominantly from self energy, with some correction from vacuum polarization – both are single loop terms.

6.6

Electron Magnetic Moment Anomaly

331

• In a more picturesque language L AMB shift arises from the interaction of the atomic electron with the vacuum field of electromagnetic radiation. This causes a “Zitterbewegung” (trembling motion) of the electron which lets it ‘see’ an effectively smeared out C OULOMB potential. This, in turn, leads to a rise of the energy term for electrons which come close to the nucleus, i.e. essentially only for s electrons. • Heavy ion physics also provides a rich field for studies of the L AMB shift, in particularly so as QED expands radiative corrections in series of powers (αZ)n .

6.6


According to D IRAC theory the electron has a magnetic moment S = −ge μB s M

with s = 1/2, ge = 2 and μB =

e 2me

as introduced in (6.28). We have already mentioned on several occasions that the value ge = 2 differs slightly from reality. One defines the electron magnetic moment anomaly as ae = (ge − 2)/2.

(6.96)

After the preceding discussion we rightly suspect the origin of this deviation to be again the interaction of the electron with the vacuum field. But let us first have a look at the experimental determination of ae which in itself is fascinating. Again we encounter an experiment of unbelievable accuracy. All experiments to determine ge − 2 are essentially based on a very precise comparison of the (non-relativistic) cyclotron frequency of the electron in an external magnetic field B according to (1.111) e ωc = B (6.97) me with the L ARMOR precession frequency of its magnetic moment (1.162): ω s = ge ω L = ge

e B. 2me

(6.98)

Knowing both frequencies allows one to derive the anomaly by (ωs − ωc )/ωc = (ge − 2)/2. An older, but still instructive experiment is shown schematically in Fig. 6.32. A 100 ns, 100 keV electron beam is partially polarized by scattering from a gold foil. Electrons drift on helical trajectories into a magnetic field trap, their polarization being perpendicular to both, the field and their velocity. After a well defined time they are pulsed out of the trap and their polarization is measured by scattering from a second gold foil. For a hypothetical value ge = 2 the spin orientation of the electrons after each full circle would be the same as at the entrance. Due

332

6


unpolarized elektrons polarization foil

Fig. 6.32 Experimental setup for measuring g − 2 according to W ESLEY and R ICH (1971). For details see text

B

exit cylinder

trap region

injection cylinder

B

detector

analyzing foil

to the anomaly, spin precession time and circling time differ, thus allowing a direct determination of ae . Today’s most precise method goes back to D EHMELT and his collaborators (VAN DYCK et al. 1986). It too uses the difference between ωs and ωc , however, in a highly refined and compact manner. Thereby VAN DYCK et al. (1987) succeeded to store a single electron in a so called P ENNING trap. With this method they were, for the first time, able to measure ge − 2 with high precision. D EHMELT and PAUL (1989) received the N OBEL prize for the development and application of such ion traps – together with Norman R AMSEY, who was honoured for his method of separated field oscillators (which we have discussed in Sect. 6.1.7). Both inventions were instrumental for the development of precision spectroscopy and time measurement, today a standard in atomic clocks as well as for high resolution quadrupole mass spectrometers and many other applications. As indicated in Fig. 6.33(a), a magnetic field B of several Tesla (T), parallel to the axis of the trap, leads to cyclotron motion of the electron. In addition, an electric quadrupole field (equipotential surfaces x 2 + y 2 − 2z2 = const) keeps the electrons in the centre of the trap.8 The classical motion of an electron in such a trap is characterized by three eigenfrequencies: One axial frequency, ωz =

2eU0 me d02

with d02 = ρ02 + 2z02 ,

and two radial frequencies, ωc + ω¯ c = 2 and ω¯ m =

8 The

ωc − 2

ωc2 ωz2 − , 4 2

the modified cyclotron frequency,

ωc2 ωz2 − , 4 2

the magnetron frequency.

(6.99)

(6.100)

so called PAUL trap uses a similar design, however, instead of using a magnetic field the electrons are stabilized by superposing an oscillatory field onto the static electric quadrupole field.

6.6


333

positive potential on the ring negative potential on the lower cap

electron trajectories

(a)

energy

negative potential on the upper cap

(b)

Fig. 6.33 (a) P ENNING trap for the determination of g − 2 according to VAN DYCK et al. (1986). (b) Simplified scheme for the eigenfrequencies of Geonium

Quantum mechanically, the L ARMOR frequency (6.98) of the electron spin has to be added. For sufficiently low temperatures and high magnetic field it makes sense to look for quantum mechanical solutions to this system (using S CHRÖDINGER or D IRAC equation). The electron then assumes discrete energy levels Wnm which are a combination of electron spin states and harmonic oscillator, the so called L ANDAU levels:

2

1 ge 1 1 ω¯ c − δ n + + ms . (6.101) Wnm = ωc ms + n + 2 2 2 2 The first term describes the spin states with the projection quantum number ms , the second term contains the modified cyclotron resonances and the third term is the leading relativistic correction term with δ/ωc = ωc /me c2 . The name Geonium atom was coined for this system. Its lowest terms are shown in Fig. 6.33(b). By cooling the system to very low temperatures one may indeed “trap” one single electron and bring it into its energetically lowest states. Transitions are detected by induced currents. The experimental challenge is a most accurate determination of the anomaly frequency ωa = gωc /2 − ω¯ c between the states n = 0, ms = 1/2 and n = 1, ms = −1/2 as indicated in Fig. 6.33(b). From this the electron anomaly ae is obtained, if one knows ωc and the axial frequency ωz with sufficient accuracy. Such a high precision warrants a fundamental understanding of the physics in these P ENNING traps which has improved during the past two decades substantially. Many methodological improvements have been introduced. We cannot go into the details and have to refer the interested reader to the original literature, in particular to the work of G ABRIELSE and collaborators (P EIL and G ABRIELSE 1999; H ANNEKE et al. 2008) who were able to overcome several imperfections of the trap by a variety of clever experimental tricks. They improved the accuracy of the measurement by a factor of 10. In the course of these improvements they also succeeded to really document the population of the quantum states shown in Fig. 6.33(b) in the P ENNING trap. This is illustrated in Fig. 6.34 which shows the measured probability of quantum jumps between the individual n states. The number of quantum jumps observed depends directly on the thermal population of the excited states. It reduces with temperature

6


4.2 K

1.0

3.2 K

0.0 1.0

2.0 K

0

10

20 30 time / min

40

probability, Pn

cycltoron energy / ħ c

334

0.0 1.0

1.6 K

0.0 1.0

0.08 K

0.0 1.0

50

0.0

012 n

Fig. 6.34 Quantum jumps between lowest states of the one electron cyclotron oscillator. A dramatic reduction with cavity temperature is seen. Based on this, on the right a population analysis of the n states in the trap is shown. Adapted from P EIL and G ABRIELSE (1999)

as quantified by the population analysis on the right of Fig. 6.34. At the lowest temperatures, indeed, only the ground state is still populated and no quantum jumps are detected any more. We recall that we have discussed quantum jumps in atoms already in Sect. 4.7.3. However, in the present case we are discussing quantum jumps in a truly macroscopic quantum system. Thus, this experiment should be kept in mind in the context of a presently very fashionable discussion about the limits (or extensions) of quantum mechanics into the macroscopic world! In the most recent high precision experiment (O DOM et al. 2006; H ANNEKE et al. 2008) a one electron cyclotron oscillator (Geonium) was cooled down to below 100 mK and the magnetic field was 10.6 T. In this case νs ≈ ν¯ c ≈ 149 GHz while the axial frequency was tuned to νz ≈ 200 MHz. The magnetron frequency was νm ≈ 134 kHz. The resulting presently most accurate value is given by 2010 CODATA (NIST 2010) as9 ae = (ge − 2)/2 = 1.15965218076(27) × 10−3 .

(6.102)

In this context, the reader should bear in mind that the determination of fundamental physical constants is today based on an intricate system of weighted averages of a whole number of precision measurements for different observables (ae is one of them). They are thus not independent of each other and any change in the system note for completeness that the anomaly for the muon μ is also known with high accuracy (albeit to some orders of magnitude less well): aμ = 1.16592091(63) × 10−3 . 9 We

6.6

Electron Magnetic Moment Anomaly g=

1 2

2×1 +

DIRAC

SCHWINGER

335 +

2

×

...

+ ...

+ 5 more

... + ....

two-loop graphs

Fig. 6.35 F EYNMAN diagrams for ge − 2 in 1st order, 2nd order (one-loop) and some examples for 4th order (two-loop contributions). Note, the open photon propagator indicates interaction with the external magnetic field

of constants entering this CODATA system changes other constants as well. Specifically, the correlation coefficient between ae and α, the fine structure constant, is given as r = 0.9384 by the system. This means, the recommended numerical values for both fundamental constants depend very strongly on each other – and we have to add: on a precise theoretical interpretation. The theory of the electron magnetic moment anomaly has first been developed by J. S CHWINGER. According to this work the first nonvanishing term for ae is of 2nd order in QED and amounts to ae = α/2π = 1.161 × 10−3 . This is already very close to reality. Figure 6.35 shows the graphs up to 2nd order of the perturbation series completely, as well as two further correction terms of 4th order (the additional 5 possible graphs or 4th order are also two-loop contributions and are not difficult to guess). The result of such QED computations is often written as aeQED = A1 + A2 (me /mμ ) + A2 (me /mτ ) + A3 (me /mμ , me /mτ ), where even the virtual generation of μ and τ Leptons are included. Each of these individual terms is in turn expanded in a series: (2)

Ai = Ai

2 3 α (4) α (6) α + Ai + Ai + ··· . π π π

Clearly, the A1 series contributes the most (in the A1 series the term of 2nd order (2) is the S CHWINGER term A1 = 0.5). When evaluating the perturbation integrals behind these coefficients ab initio one major concern is good bookkeeping. To illustrate the efforts necessary: In 6th order there are already 72 graphs which contribute to A(6) 1 and in 8th order all 891 F EYNMAN graphs have been included. On that basis, according to G ABRIELSE ET al. (2006), the results for ae are inverted to determine today’s most precise value of the fine structure constant α. Based on AOYAMA et al. (2007) and H ANNEKE et al. (2008), who report 137.035999084(51),10 the 2010 10 More

recently, AOYAMA et al. (2012) have performed even 10th order QED calculations (involving 12672 diagrams) and report ae = 1.15965218178(77) × 10−3 from which they derive α −1 = 137.035999173(35).

336

6


CODATA data bank (M OHR et al. 2012) gives α=

e2 = 1/137.035999074(44). 4πε0 c

(6.103)

Section summary

• The deviation of the ge factor from 2 is given as the so called anomaly of the magnetic moment of the electron, ae = (ge − 2)/2 1.16 × 10−3 and can be determined today with an amazing accuracy of about 3 × 10−10 . • Experimentally it is measured by sophisticated techniques in electron traps (P ENNING trap, first used by D EHMELT in this context), where quasimacroscopic states of the electron are prepared – the so called Geonium. Essentially, one measures the difference between the L ARMOR and the cyclotron frequency. • The precise determination of the magnetic moment of the electron provides another stringent test for the potential of QED calculations. While the dominant contribution arises from the one second order perturbation term (single loop), presently the complete contributions up to 8th order have been computed.

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. 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). a.u.: ‘atomic units’, see Sect. 2.6.2. CCD: ‘Charge coupled device’, semiconductor device typically used for digital imaging (e.g. in electronic cameras). CRD: ‘Cavity ring down’ spectrometer (see Sect. 5.5.3, Vol. 2). 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. EBIT: ‘Electron beam ion trap’, source for highly charged ion beams see Sect. 7.5, Vol. 2. FPI: ‘FABRY-P ÉROT interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6). 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)’. HCI: ‘Highly charged ions’, see Sect. 7.5, Vol. 2.

References

337

HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9). LHC: ‘Left hand cicularly’, polarized light, also σ + light. M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. QDT: ‘Quantum defect theory’, interprets experimental spectra by phase shifts in the radial wave functions and makes predictions for scattering processes (see Sect. 3.2.6). QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. 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. RHC: ‘Right hand cicularly’, polarized light, also σ − light. SHG: ‘Second harmonic generation’, doubling of a fundamental frequency, for infrared or visible light typically by methods of nonlinear optics. 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. 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 electromagentic 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 AOYAMA , T., M. H AYAKAWA , T. K INOSHITA AND M. N IO: 2007. ‘Revised value of the eighthorder contribution to the electron g-2’. Phys. Rev. Lett., 99, 110 406. AOYAMA , T., M. H AYAKAWA , T. K INOSHITA AND M. N IO: 2012. ‘Tenth-order QED contribution to the electron g-2 and an improved value of the fine structure constant’. Phys. Rev. Lett., 109, 111 807. B EIERSDORFER , P.: 2010. ‘Testing QED and atomic-nuclear interactions with high-Z ions’. J. Phys. B: At. Mol. Phys., 43, 074 032.

338

6


B ETHE , H. A.: 1947. ‘The electromagnetic shift of energy levels’. Phys. Rev., 72, 339–341. B IRABEN , F.: 2009. ‘Spectroscopy of atomic hydrogen how is the Rydberg constant determined?’ Eur. Phys. J. ST, 172, 109–119. B JORKEN , J. D. AND S. D. D RELL: 1964. Relativistic Quantum Mechanics. New York: McGraw Hill. B ORN , M. AND E. W OLF: 2006. Principles of Optics. Cambridge University Press, 7th (expanded) edn. B RINK , D. M. AND G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3 edn., 182 pages. D EHMELT , H. G. AND W. PAUL: 1989. ‘The N OBEL prize in physics: for the development of the ion trap technique’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1989/. VAN DYCK , R. S., P. B. S CHWINBERG AND H. G. D EHMELT : 1986. ‘Electron magnetic-moment from geonium spectra – early experiments and background concepts’. Phys. Rev. D, 34, 722– 736. VAN DYCK , R. S., P. B. S CHWINBERG AND H. G. D EHMELT : 1987. ‘New high-precision comparison of electron and positron g-factors’. Phys. Rev. Lett., 59, 26–29. E IDES , M. I., H. G ROTCH AND V. A. S HELYUTO: 2001. ‘Theory of light hydrogenlike atoms’. Phys. Rep., 342, 63–261. F EYNMAN , R. P.: 1949. ‘Space-time approach to quantum electrodynamics’. Phys. Rev., 76, 769– 789. G ABRIELSE , G., D. H ANNEKE , T. K INOSHITA , M. N IO AND B. O DOM: 2006. ‘New determination of the fine structure constant from the electron g value and QED’. Phys. Rev. Lett., 97, 030 802. G ROSS , B., A. H UBER , M. N IERING , M. W EITZ AND T. W. H ÄNSCH: 1998. ‘Optical Ramsey spectroscopy of atomic hydrogen’. Europhys. Lett., 44, 186–191. G RYNBERG , G. AND B. C AGNAC: 1977. ‘Doppler-free multi-photonic spectroscopy’. Rep. Prog. Phys., 40, 791–841. G UMBERIDZE , A. et al.: 2005. ‘Quantum electrodynamics in strong electric fields: The groundstate lamb shift in hydrogenlike uranium’. Phys. Rev. Lett., 94, 223 001 – and personal communication from T. S TÖHLKER. H ANNEKE , D., S. F OGWELL AND G. G ABRIELSE: 2008. ‘New measurement of the electron magnetic moment and the fine structure constant’. Phys. Rev. Lett., 100, 120 801. H ÄNSCH , T. W.: 2005. ‘N OBEL lecture: Passion for precision’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/2005/hansch-lecture.html. H ÄNSCH , T. W., M. H. NAYFEH , S. A. L EE , S. M. C URRY AND I. S. S HAHIN: 1974. ‘Precisionmeasurement of Rydberg constant by laser saturation spectroscopy of Balmer alpha line in hydrogen and deuterium’. Phys. Rev. Lett., 32, 1336–1340. H ÄNSCH , T. W., I. S. S HAHIN AND A. L. S CHAWLOW: 1971. ‘High-resolution saturation spectroscopy of sodium D lines with a pulsed tunable dye laser’. Phys. Rev. Lett., 27, 707–710. H ÄNSCH , T. W., I. S. S HAHIN AND A. L. S CHAWLOW: 1972. ‘Optical resolution of Lamb shift in atomic-hydrogen by laser saturation spectroscopy’. Nat. Phys. Sci., 235, 63. H ÄNSCH , T. W. et al.: 2005. ‘Precision spectroscopy of hydrogen and femtosecond laser frequency combs’. Philos. Trans. R. Soc. Lond., Ser. A, 363, 2155–2163. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. J EFFERTS , S. AND D. M EEKHOF: 2011. ‘NIST-F1 cesium fountain atomic clock’, NIST. http:// www.nist.gov/physlab/div847/grp50/primary-frequency-standards.cfm, accessed: 8 Jan 2014. J ENTSCHURA , U. AND K. PACHUCKI: 1996. ‘Higher-order binding corrections to the Lamb shift of 2p states’. Phys. Rev. A, 54, 1853–1861. J ENTSCHURA , U. D., S. KOTOCHIGOVA , E. O. L. B IGOT , P. J. M OHR AND B. N. TAYLOR : 2005. ‘The energy levels of hydrogen and deuterium (version 2.1)’, NIST. http:// physics.nist.gov/PhysRefData/HDEL/, accessed: 8 Jan 2014. J ENTSCHURA , U. D., P. J. M OHR AND G. S OFF: 2001. ‘Electron self-energy for the K and L shells at low nuclear charge’. Phys. Rev. A, 63, 042 512.

References

339

J ENTSCHURA , U. D., A. M ATVEEV, C. G. PARTHEY, J. A LNIS, R. P OHL, T. U DEM, N. KO LACHEVSKY and T. W. H ÄNSCH : 2011. ‘Hydrogen-deuterium isotope shift: From the 1s–2stransition frequency to the proton-deuteron charge-radius difference’. Phys. Rev. A, 83, 042 505. J OHNSON , W. R. AND G. S OFF: 1985. ‘The Lamb shift in hydrogen-like atoms, 1 less-than-orequal-to Z less-than-or-equal-to 110’. At. Data Nucl. Data Tables, 33, 405–446. K ARSHENBOIM , S. G.: 2005. ‘Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants’. Phys. Rep., 422, 1–63. K RAMIDA , A. E.: 2010. ‘A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium’. At. Data Nucl. Data Tables, 96, 586–644. K RAMIDA , A. E., Y. R ALCHENKO , J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. L AMB , W. E. AND P. K USCH: 1955. ‘The N OBEL prize in physics: for his discoveries concerning the fine structure of the hydrogen spectrum, and for his precision determination of the magnetic moment of the electron, respectively’, Stockholm. http://nobelprize.org/nobel_prizes/physics/ laureates/1955/. M ATVEEV , A. et al.: 2013. ‘Precision measurement of the hydrogen 1s–2s frequency via a 920-km fiber link’. Phys. Rev. Lett., 110, 230 801. M ICHELSON , A. A.: 1907. ‘The N OBEL prize in physics: for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid’, Stockholm. http://www.nobelprize.org/nobel_prizes/physics/laureates/1907/. M OHR , P. J., B. N. TAYLOR AND D. B. N EWELL: 2012. ‘CODATA recommended values of the fundamental physical constants: 2010’. Rev. Mod. Phys., 2013, 1527–1605. http://physics.nist. gov/constants, accessed: 8 Jan 2014. N IERING , M. et al.: 2000. ‘Measurement of the hydrogen 1s–2s transition frequency by phase coherent comparison with a microwave cesium fountain clock’. Phys. Rev. Lett., 84, 5496– 5499. NIST: 2010. ‘Reference on constants, units, and uncertainties’, NIST. http://physics.nist.gov/ cuu/Constants/, accessed: 7 Jan 2014. O DOM , B., D. H ANNEKE , B. D’U RSO AND G. G ABRIELSE: 2006. ‘New measurement of the electron magnetic moment using a one-electron quantum cyclotron’. Phys. Rev. Lett., 97, 030 801. PALMER , C. AND E. L OEWEN: 2005. ‘Diffraction grating handbook’, New York: Newport corporation. http://www.gratinglab.com/Information/Handbook/Cover.aspx, accessed: 8 Jan 2014. PARTHEY , C. G. et al.: 2011. ‘Improved measurement of the hydrogen 1s–2s transition frequency’. Phys. Rev. Lett., 107, 203 001. P EIL , S. AND G. G ABRIELSE: 1999. ‘Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between fock states’. Phys. Rev. Lett., 83, 1287–1290. P OHL , R. et al.: 2010. ‘The size of the proton’. Nature, 466, 213–216. P OSTAVARU , O., Z. H ARMAN AND C. H. K EITEL: 2011. ‘High-precision metrology of highly charged ions via relativistic resonance fluorescence’. Phys. Rev. Lett., 106, 033 001. R AMSEY , N. F.: 1950. ‘A molecular beam resonance method with separated oscillating fields’. Phys. Rev., 78, 695–699. R AMSEY , N. F.: 1989. ‘The N OBEL prize in physics: for the invention of the separated oscillatory fields method and its use in the hydrogen maser and other atomic clocks and the separated oscillatory fields method’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1989/. S TECK , D.: 2010. ‘Alkali D line data’. http://steck.us/alkalidata/, accessed: 8 Jan 2014. S VEN G ATO R EDSUN: 2004. ‘3j6j9j-symbol java calculator’, Sven Gato Redsun. http://www. svengato.com/, accessed: 8 Jan 2014. T OMONAGA , S.-I., J. S CHWINGER AND R. P. F EYNMAN: 1965. ‘The N OBEL prize in physics: for fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles’, Stockholm. http://nobelprize.org/nobel_prizes/ physics/laureates/1965/.

340

6


W EITZ , M., A. H UBER , F. S CHMIDT-K ALER , D. L EIBFRIED AND T. W. H ÄNSCH: 1994. ‘Precision-measurement of the hydrogen and deuterium 1s ground state Lamb shift’. Phys. Rev. Lett., 72, 328–331. W ESLEY , J. C. AND A. R ICH: 1971. ‘High field electron g-2 measurement’. Phys. Rev. A, 4, 1341. XFEL P ROJECT: 2011. ‘European XFEL project, technical information’, Hamburg, Germany: DESY. http://xfel.desy.de/technical_information/photon_beam_parameter/, accessed: 8 Jan 2014.

7

Helium and Other Two Electron Systems

So far our discussions were restricted to atomic systems with effectively one active electron in an attractive C OULOMB or screened C OULOMB field. For the majority of atoms this simple model cannot be maintained, since the electrons repel each other pairwise and have to obey the PAULI principle. The helium atom (He) is the simplest and purest example for introducing the key problems which one encounters in true multi-electron systems, as well as the methods for describing and understanding them.

Overview

After a general introduction and a survey of the experimental observations (Sect. 7.1) the quantum mechanical basis for treating multi-electron systems is reviewed in Sect. 7.2. The reader should know about these tools or get used to them by reading this section. Section 7.3 expands this theme by introducing electron exchange and the characteristic excited state configurations. Fine structure interaction in He and He like ions are addressed in Sect. 7.4 – a consolidation of our knowledge acquired in Chap. 6 which will turn out to be essential also for later chapters. In Sect. 7.5 the most important selection rules for E1 transitions in multi-electron system are treated. This directly leads to “double excitation” in Sect. 7.6. It is of importance far beyond atomic physics: resonances of states imbedded into a continuum are found in all fields of physics. The ensuing interference structures, known as FANO resonances and their origin are described in a practical, easy to access approach. Finally, in Sect. 7.7 the alkaline earth atoms and the Hg atom are treated: they are related to He in an analogous manner as alkali atoms are related to the H atom.


341

342

7 Helium and Other Two Electron Systems

7.1

Introduction and Empirical Findings

7.1.1

Basics

Helium, with its atomic number (and nuclear charge) Z = 2, is the most simple of all multi-electron systems. Nevertheless, it eludes an exact calculation of its wave functions, energies and other properties – as indeed any multi-electron or manybody system in general, starting with the classical three body system. Helium is a very rare element. It owes its presence on earth mainly to the radioactive α decay of heavy elements: the He atom is an α particle which has captured two electrons. Typically it is isolated from natural gas, where at special sources, it may be found with concentrations up to 7 %. There are two natural isotopes: 42 He (relative probability of wrel = 99.999866(3) %, atomic mass m(42 He) = 4.00260325415(6) u, nuclear spin I = 0), and the very rare 32 He (wrel = 0.000134(3) %, m(32 He) = 3.0160293191(26) u, I = 1/2). In the ground state the two electrons of He are described by 1s 2 1 S0 . The first ionization potential for separating one of these electrons is 24.5873876 eV. For ionization of the second electron (in the hydrogen like ion He+ , spectroscopically called He II) an energy Z 2 Eh /2 = 54.4177630 eV is needed. The total binding energy of both electrons thus amounts to 79.0051506 eV. Correspondingly, the spectrum of He extends from the infrared spectral range (IR) through the visible (VIS), the ultraviolet (UV) far into the vacuum-ultraviolet spectral range (VUV). Figure 7.1 shows by way of example a typical spectrum in the UV/VIS spectral range which has been constructed with data from K RAMIDA et al. (2013). Marked in red are a few characteristic emission lines ending on the 1s2s 1,3 S and 1s2p 1,3 P states. This is, however, only a section of the full spectrum of He I and one can imagine clearly that, in the early phase of atomic physics, the interpretation of such spectra was by no means trivial. In particular since all optical emission and absorption lines from neutral He appear grouped into two seemingly independent sets of spectra – a singlet and a triplet system. Each of them demands an independent assignment of term energies. In the early time of spectroscopy one associated these even with two types of helium: para an ortho He. Today this puzzle is of course completely resolved: one observes two sets of spectra with different total spin quantum numbers: He with its two electrons is thus the prototype element for which all basic phenomena of multi-electron systems can be studied very clearly. 1s2s 1S - 1s 4p 1P° 1s2p 3P° - 1s5d 3D 1s2s 3S - 1s5p 3P°

200

300

1s2s 1S - 1s3p 1P° 1s2p 3P° - 1s 3d 3D

1s2p 1P° - 1s4d 1D

400

500

1s2p 3P° - 1s 3s 3S

600

700

800 nm

Fig. 7.1 Spectrum of He I in the VIS and UV spectral range, according to K RAMIDA et al. (2013). Only a few, particularly intense lines are designated here (red)

7.1

Introduction and Empirical Findings

343

Of particular importance (for all areas of modern physics) is the so called exchange interaction.

7.1.2

He I Term Scheme

In Fig. 7.2 the energy levels of the neutral He atom below the first ionization threshold are summarized. This diagram represents extensive material and detective work from generations of spectroscopists. We recognize in Fig. 7.2 the two term systems already mentioned. According to their fine structure splitting we designate them as singlet and triplet system. The ground state of He belongs to the singlet system. Optical dipole transitions (E1) are observed only within each of the two systems (indicated by red arrows in Fig. 7.2). However, one finds that collision processes may very well induce transitions between singlet and triplet system and vice versa – we shall discuss such processes in Sect. 7.2, Vol. 2. Obviously, degeneracy is removed in He: a consequence of a (partial) screening of the nuclear charge which each electron experiences by the other electron. As in the case alkali atoms one may characterize the term energies by an effective quantum number n∗ with a quantum defect μ = n − n∗ or by an effective charge Z ∗ :

+) / eV W W(He WW

Wn = −

Z 2 Eh Z ∗2 Eh =− . ∗2 2n 2n2

singlet

(7.1)

triplet

H

He+ + e0

1S

1P

1D

1F

3S

3P

3D

3F

-1

4 3F

n=5 n=4

2,3,4

n=3 -2

31S

0

31P1

31 D 2

3 3 S1

3 3P

0 1,2 0,1

3 3D

123 1,2,3

2 3 P0

-3 21P1

-4

-3.62318 eV

n=2

2 3P

0,1,2

21S0

23P1 -3.62331 eV

23S1 -24.59

2 3 P2

ground state (1s ) 2 1S0

Fig. 7.2 Energy levels of the neutral helium atom, He I (so called G ROTRIAN diagram). Note that the position of the ground state (1s)2 1 S0 is not drawn to scale. The spectra split into a singlet and a triplet system. The inset shows the (inverted) triplet splitting of the 1s2p 3 P0,1,2 states. For comparison the corresponding term energies of the H atom are shown: clearly, the quantum defects of the He terms are small and converge rapidly with n

344


Fig. 7.3 Coordinates for the two electron atom He

e-

z r12

r2

12

Z+

r1

ey

x

For the ground state of neutral He the measured binding energy 24.5873876 eV gives n∗ 0.7439 (or μ = 0.256) and Z ∗ 1.344, respectively. From the “view point” of one of the two electrons the C OULOMB potential of the nucleus is screened already in the ground state quite strongly. For all singly excited states of He the quantum defect μ lies significantly below 1 and converges for the 1sns 1 S0 levels rapidly to μ 0.139 only. Similarly Z ∗ rapidly converges to 1 and already for n > 3 and > 0 the energies of singlet and triplet terms are already very close to those in atomic hydrogen. If we assume the configuration of the He electrons to be 1sn this is easy to understand: the 1s electron finds itself in a nearly H like orbit (with Z = 2, however) and screens the charge of the nucleus very strongly. The second electron thus ‘sees’ essentially a nuclear charge Z = 1. Nevertheless, the excited electron in He behaves different from the valence electron e.g. in alkali atoms since the interaction with the second electron is still very direct. It is important to realize that both electrons can in principle not be distinguished! This will be a key issue in the quantum mechanical treatment. We shall find that this indistinguishability is the key to the puzzle with the two term systems of He. Section summary

• He is the prototype of two electron systems and indeed the basis for understanding any multi-electron system. Its first ionization potential is 24.6 eV, the second electron requires 54.4 eV for ionization. • The spectra of He (ranging from the IR to the far VUV region) show two characteristic types of term systems which in E1 transitions do not mix: a singlet and a triplet system. Understanding this particular feature will be a key to the understanding of electron-electron interaction and electron exchange: both electrons are in principle indistinguishable.

7.2

Some Quantum Mechanics of Two Electrons

7.2.1

HAMILTON Operator for the Two-Electron System

A two electron system is characterized by two position vectors r 1 and r 2 as sketched schematically in Fig. 7.3. Up to now we have not yet introduced any formalism for treating multi-electron systems. We follow the general rules already used for

7.2


345

deriving observables in the one electron case, specifically the Hamiltonian. We just have to add the components: the energies for the two individual electrons 1 and 2 1 = H

p 21 Ze2 − 2me 4πε0 r1

2 = and H

p 22 Ze2 − , 2me 4πε0 r2

(7.2)

respectively, and the repulsive C OULOMB energy between the two electrons ee = H

e2 . 4πε0 r12

(7.3)

The full H AMILTON operator for the two electron system is thus 2 + H ee = =H 1 + H H

p 21 p2 Ze2 Ze2 e2 − + 2 − + . 2me 4πε0 r1 2me 4πε0 r2 4πε0 r12

(7.4)

ee depends on Note that the interaction term H r12 = |r 1 − r 2 | = r12 + r22 − 2r1 r2 cos θ12 ,

(7.5)

and thus breaks the original spherical symmetry – an essential which we have exploited when solving the one electron case. In the remaining chapter we shall (unless mentioned otherwise) again write the H AMILTON operator in a.u.: 2 + H ee = − − Z − − Z + 1 . =H 1 + H H 2 r1 2 r2 r12

7.2.2

(7.6)

Two Particle Wave Functions

With (7.6) we have now a Hamiltonian which depends on two position vectors. We continue to use polar coordinates and the wave functions will now depend on the coordinates {r1 , θ1 , ϕ1 , r2 , θ2 , ϕ2 }. The most simple ansatz for a wave function would be Ψ (r 1 , r 2 ) = ψ1 (r1 , θ1 , ϕ1 )ψ2 (r2 , θ2 , ϕ2 ).

(7.7)

This product ansatz is strictly valid only for the model of independent particles, i.e. for two electrons that move completely uncorrelated and experience from each other only an average potential. This turns out to be a good 1st order approximation for the 1sn singly excited states of He. However, for the ground state and for doubly excited states where the two electrons interact closely this approximation is only of limited value. By consequently extending the probability definition of the wave function in the one particle case, as introduced with (2.5), the probability to find electron 1 at position r 1 and simultaneously electron 2 at position r 2 in a volume element d3 r 1 d3 r 2 is given by

346


2 dw12 = Ψ (r 1 , r 2 ) d3 r 1 d3 r 2 2 2 = ψ1 (r1 , θ1 , ϕ1 ) ψ2 (r2 , θ2 , ϕ2 ) d3 r 1 d3 r 2 .

(7.8)

From this, the probability to find electron 1 at position r 1 and electron 2 anywhere becomes 2 2 3 (7.9) dw1 = d r 1 Ψ (r 1 , r 2 ) d3 r 2 = ψ1 (r1 , θ1 , ϕ1 ) d3 r 1 . 2

As usual, we assume the one electron wave functions (orbitals) ψ1 (r1 , θ1 , ϕ1 ) and ψ2 (r2 , θ2 , ϕ2 ) to be orthogonal and normalized. The indices 1 and 2, more generally μ and μ , stand for the usual sets of quantum numbers: μ = {nμ μ mμ msμ }.

μ = {nμ μ mμ msμ },

(7.10)

For completeness and later use we also include here the projection quantum number ms = ±1/2 of the electron spin. They will become important in Sect. 7.3. In the compact bra-ket form, such two particle product states are written Ψμμ (1, 2) = ψμ (r 1 )ψμ (r 2 ) = |nμ μ mμ nμ μ mμ , (7.11) where the indices 1 and 2 (given in round brackets) are an abbreviation for the coordinates of the two electrons. Assuming that the single particle orbitals ψμ (r j ) are orthonormalized, this hold also for the two particle states: Ψμμ (1, 2)Ψλλ (1, 2) = ψμ (1)ψλ (1) ψμ (2)ψλ (2) (7.12) = nμ μ mμ |nλ λ mλ nμ μ mμ |nλ λ mλ = δμλ δμ λ . It is important to note that the Hamilton operator (7.6) for the two electron system in this form does not depend on electron spin! All influence of the relative orientation of the spins of the two electrons will enter through the anti-symmetrization rule to be discussed in Sect. 7.3. For the moment we ignore the spin and just note that the Hamiltonian does not change if we exchange the two electrons, i.e. their position coordinates r 1 and r 2 .

7.2.3

Zero Order Approximation: No e− e− Interaction

In the spirit of perturbation theory we first define the “unperturbed” problem and start with its solution. Thus, we neglect in (7.4) the repulsive screening term (7.3), to be treated later as perturbation, and write the Hamiltonian in 0th order in a.u.: 1 + H 2 0 = H H

i = with H

p 2i Z − 2 ri

(7.13)

7.2


347

i describes the eigenvalue problem of the one particle system, i.e. the hydrogen H like He+ ion. Its energies Wni are given by 2

i |ni i mi = Wni |ni i mi = − Z |ni i mi . H 2n2i

(7.14)

For the two particle system we thus have in 0th order 0 |Ψ = H 1 |ψ1 |ψ2 + H 2 |ψ2 |ψ1 H 1 |n1 1 m1 |n2 2 m2 + H 2 |n2 2 m2 |n1 1 m1 = H = (Wn1 + Wn2 )|n2 2 m2 |n1 1 m1 = (Wn1 + Wn2 )|Ψ , so that the energy of the He atoms in 0th order is approximation is

Z2 1 1 (0) . Wn1 n2 = Wn1 + Wn2 = − + 2 n21 n22

(7.15)

Note that these energies are calibrated in respect of the fully separated He++ + e− + e− system. Experimentally measured are spectral lines (i.e. energy differences). Tabulated one typically finds term energies Wn1 1 n2 2 − W1s 2 > 0 and ionization potentials WI > 0 in respect of the He(1s 2 ) ground state (e.g. K RAMIDA et al. 2013). (Alternatively, binding energies for the valence electron are reported in the literature in respect of the first ionization threshold Wn1 1 n2 2 + WI (I).) Figure 7.4 compares for He with Z = 2 the true, experimentally determined energies (left) with the 0th order approximation (right) – for the neutral atom (He I) as well as for the ionic system He+ + e− (He II). The ionization energy of the hydrogen like He+ ion (Z = 2) can be computed from S CHRÖDINGER equation as (0) WI (II) = Z 2 Eh /2 = 2Eh = 54.422 . . . eV. This is more ore less exact: according to K RAMIDA et al. (2013) at present the best measured value is1 WI (II) = WI He+ (1s) = 54.41776203 eV, (7.16) while the value given for the ionization potential of neutral He is WI (I) = WI He 1s 2 = 24.58738777 eV.

(7.17)

Thus, the (true) total energy of the neutral He ground state is W1s 2 = −WI (I) − WI (II) = −79.0051498 eV = −2.903387219Eh .

(7.18)

In contrast, 0th order approximation (7.15) gives (0)

W1s 2 = −

Z2 Eh × 2 = −4Eh = −108.8 eV 2

(7.19)

difference is due to the necessary kinematic corrections with m ¯ e /me for finite mass and further corrections such as the L AMB shift.

1 The

348


WI (II) + WI (I) =79.005 72.96 65.40 60

0

H0 without e- e- interaction

Wnℓn'ℓ'' / eV

He++ + 2e0 3snℓ 2snℓ

2s∞ℓ - 13.6 2s2ℓ - 27.2

autoionization

40 WI (I)= 24.59 20

^

experiment

W - W1s 2 / eV

3s∞ℓ 3s 2ℓ

- 40

He+ + e1s 2s 1S0 20.62eV 1s 2 1S 0.0eV

-13.60

1s2s 3S1 19.82eV

1s∞ℓ

- 54.42 - 60

1s2s,p - 80

He 0

- 20

- 54.42 - 100 - 108.84 1s1s

Fig. 7.4 Experimentally determined energies of He and He+ (left, black) and ansatz for perturbation theory in 0th order (right, red) (0)

(0)

and WI (I) = WI (II). And similarly, the 0th order binding energy of one electron in an n state is Z 2 (−(1 + 1/n2 ) + 1)/2Eh = −2Eh /n2 , specifically −Eh /2 = −13.6 eV in the 1s2s or 1s2p states (the excitation energy in 0th order would be −Eh /2 − (−2Eh ) = 1.5Eh ). As documented in Fig. 7.4 the real energies lie obviously much higher due to screening of the attractive C OULOMB potential by the repulsive 1/r12 term. Obviously, theory faces a major challenge if one aims at an accuracy matching that of the experiment! Even more complex is the situation when considering two excited electrons in excited n states. In 0th order a 2 2 configuration would lead to an excitation energy 3Eh – which is higher than the first ionization potential in 0th order. Hence, it is already embedded into the continuum of the singly ionized helium and has an excitation energy Eh in respect of the He+ 1s 2 S0 ground state – as indicated in Fig. 7.4. Such states really exist and we shall explore this kind of double excitation and the ensuing phenomenon of FANO resonances in some detail in Sect. 7.6.

7.2.4

The He Ground State – Perturbation Theory

Thus, the screening term (7.3) has substantial influence on the total energy of the He ground state. In a first step one may try to approach the problem by perturbation ee > 0 we expect indeed to come closer to the experimental value theory. Since H if we account for it. For the He(1s 2 ) ground state in 0th order both electrons are

7.2


Table 7.1 One electron binding energies / eV (or ionization potentials) of the ground state for He and He like ions: experimental data according to K RAMIDA et al. (2013) are compared with 0th and 1st order perturbation theory

349 Exp.

0th order

1st order

H−

0.76

13.6

003.4

He

24.58741

54.4

20.4

Li+

75.64018

122.4

71.4

Be++

153.8945

217.7

149.6

B3+

259.3752

340.1

255.1

C4+

392.0872

489.9

387.7

described by 1s hydrogen like wave functions (with Z = 2) and the wave function (7.11) is written as Ψ1s 2 (r 1 , r 2 ) = ψ1s (r 1 )ψ1s (r 2 ) or 1s 2 = |100 |100 . (7.20) In 1st order perturbation theory the energy correction term according to (3.41) is: (0) ee |Ψ1s 2 = Ψ1s 2 | 1 |Ψ1s 2

W = W1s 2 − W1s 2 = Ψ1s 2 |H (7.21) r12 2 2 1 1 = 100| 100| |100 |100 = d3 r 1 d3 r 2 ψ100 (r 1 ) ψ100 (r 2 ) . r12 r12

To evaluate this integral we insert the 1s wave function from Table 2.2: ψ100 (r) = Y00 (θ, ϕ)R1s (r) = (Z)3/2 π −1/2 e−Zr . With this the integral is written explicitly as −2Zr2 e (Z)6 d3 r 2 , e−2Zr1 d3 r 1

W = 2 r12 π and the inverse distance between the two electrons 1/r12 may be expanded into a power series according to (F.2) – involving L EGENDRE polynomials. Leaving the details as an exercise to the interested reader we simple convey that integration leaves only the first term from this series and the final result (with Z = 2) is simply 5

W = ZEh = 34.01 eV. 8 Thus, in 1st order perturbation theory the total energy of the He ground state is 5 (1) W1s 2 = −Eh Z 2 + ZEh = −74.79 eV. 8

(7.22)

Considering the crudeness of the ansatz this results is astonishingly close (about 5 %) to the experimental result (7.18). It is instructive, to compare the experimental binding energies (or ionization potentials) for several members of the He like isoelectronic sequence to WI (I) from (7.18) with 0th and 1st order perturbation theory according to (7.19) and (7.22). This is summarized in Table 7.1.

350


Obviously, the relative accuracy of 1st order perturbation theory improves with increasing nuclear charge Z. While for the H anion perturbation theory completely fails, it is quite satisfactory for the four fold ionized carbon where an agreement between experiment and perturbation theory to about 1 % is reached. We note here in passing, that the He like 1s 2 1 S0 state of H− is the only existing and experimentally documented, stable state of the hydrogen anion.

7.2.5

Variational Theory and Present State-of-the-Art

Variational methods are very important in quantum mechanics for the determination of energies, in particular for ground states. Helium is a particular clear example for its efficiency, and we shall use variational procedures in many other instances as well. The variational principle (also called the R ITZ or R AYLEIGH -R ITZ method after the inventors) makes use of an important theorem according to which the minimal value of energy W which is calculated for a given class of functions φ is always the best: ∗ Ψ d3 r |Ψ Ψ H Ψ |H W = min = min ∗ 3 (7.23) Ψ |Ψ Ψ Ψd r cμ ψμ (r 1 , r 2 ). with Ψ (r) = μ

Here r refers to the full configuration space of both electrons, and the coefficients cμ have to be varied until W reaches its minimum. In detail, one chooses a test wave function for the ground state which is in principle able to describes the properties of the system under study as well as possible. With this test wave function one calculates the energy as expectation value of the H AMILTON operator so that W is derived as a function of the parameters ck defining the wave function and one minimizes W according to the standard procedures of analysis. Specifically for He the so called H YLLERAAS wave functions Ψ (s, t, u) = exp(−ks)

N

c ,2m,n s t 2m un

(7.24)

mn

with s = r1 + r2 , t = r1 − r2 and u = r12 have proved to be very efficient. Thus, the energy is computed from Ψ (s, t, u) / Ψ (s, t, u)Ψ (s, t, u) W = Ψ (s, t, u)H

(7.25)

by variation of k and the coefficients c ,2m,n to obtain the minimum energy W . The resulting wave function obviously does not describe two independent electrons – for which the wave function would be given by the product of the individual orbitals. Rather, the electrons in a H YLLERAAS type wave function are highly correlated: they influence each other more directly as could be described by an averaged

7.3

PAULI Principle and Excited States in He

351

potential. With this ansatz and typically 5 parameters a very high accuracy can be achieved. H YLLERAAS already obtained W1s1s = −79.001 eV, to be compared with (7.18). Modern “state-of-the-art” calculations include all relevant corrections such as the finite He mass, L AMB shift, nuclear size and the recalculation to the present value of R∞ . The agreement between experiment and theory is on the order of 10−7 (D RAKE and M ARTIN 1998). The data presently reported in Section summary

• The Hamiltonian (7.6) of a two electron system is given by adding up all energies of the two single particle systems plus the mutual repulsion of the two electrons. • The most simple ansatz for two particle wave functions is a product of single particle wave functions for electron 1 and 2. Such a simple independent particle model is often a reasonable 0th and 1st order approximation. • Neglecting the 1/r12 screening potential it leads to energies for He which are substantially too low, but predicts interesting doubly excited states. • Including, however, the 1/r12 term in 1st order perturbation theory, a hydrogenic (1s)2 product wave functions already gives a total energy for the He(1s 2 ) ground state which agrees to within 5 % with experiment. • Excellent agreement is obtained by variational theory (7.23), also called R AYLEIGH -R ITZ method. For the He ground state, so called H YLLERAAS wave functions (7.24) have proved very useful. They imply strong correlation between the two electrons.

7.3


7.3.1

Exchange of Two Identical Particles

Up to now we have not yet explicitly accounted for the fact that electrons are fermions. Before clarifying this aspect specifically for He, we take the opportunity to discuss multiparticle wave functions in a more general context. It is straight forward to generalize the two particle wave function introduced in Sect. 7.2.2 to a system with N identical particles. To include the spin properties of the particle we introduce so called spin-orbitals which are characterized by the three quantum numbers n m referring to position space and the spin quantum number ms = ±1/2 (orientation of the spin in respect of the z-axis): ms s (ri , θi , ϕi )χ1/2 (si ). ψn mms (q i ) = ψn m

(7.26)

Here we have introduced a coordinate q i for each particle, representing the position vector r i and a symbolic spin variable si . The total wave function for the N particle quantum state is then written Ψ (q 1 , q 2 . . . q j . . . q N ). In Sect. 3.1.2 we have already mentioned the symmetry properties:

352


In respect of exchange of two particles, the state vectors (wave functions) of a quantum system of identical particles are symmetric for bosons, and antisymmetric for fermions. ij which exchanges particle i and j one may write Defining an exchange operator P this in mathematical form • for bosons ij Ψ (q 1 ..q i ..q j ..q N ) = Ψ (q 1 ..q j ..q i ..q N ) = Ψ (q 1 ..q i ..q j ..q N ) P (7.27) • and for fermions ij Ψ (q 1 ..q i ..q j ..q N ) = Ψ (q 1 ..q j ..q i ..q N ) = −Ψ (q 1 ..q i ..q j ..q N ). P (7.28) The latter relation is completely equivalent to the PAULI exclusion principle: assume two identical particles to be in the same quantum state, say particle i and particle j . In that case exchanging the two obviously leads to Ψ (q 1 ..q i ..q j ..q N ) = Ψ (q 1 ..q j ..q i ..q N ).

(7.29)

For fermions this contradicts (7.28) unless Ψ ≡ 0. Thus, two identical fermions cannot exist in the same spin-orbital – or in the classical formulation by PAULI J R . (1925): “There are never two or more equivalent electrons in an atom for which the values of all quantum numbers are identical.” In contrast, for bosons no conflict arises between (7.29) and (7.27) so that any number of identical bosons may be in the same spin-orbital, i.e. have the same set of quantum numbers.

7.3.2

Symmetries of Spatial and Spin Wave Functions

For a two electron system such as He and He like ions it is convenient to express the total wave function as a product2 Ψ (1, 2) ≡ Ψ (q 1 , q 2 ) = Ψ s (r 1 , r 2 )χ(s1 , s2 ) ≡ Ψ s (1, 2)χ(1, 2)

(7.30)

of a spatial part of the wave function, Ψ s (r 1 , r 2 ), which we have treated so far exclusively, and a spin function for the two electrons χ(1, 2). 2 If

no ambiguities can arise, we abbreviate here and in the following r j and sj as well as qj by j .

7.3


353

We may apply the exchange operator onto spatial and spin function separately. With 12 Ψ s (r 1 , r 2 ) = Ψ s (r 2 , r 1 ), P

(7.31)

and since the two electrons are indistinguishable, the states Ψ s (r 1 , r 2 ) and Ψ s (r 2 , r 1 ) belong to the same, non-degenerate eigenvalue. They may differ only by a scalar factor: 12 Ψ s (r 1 , r 2 ) = Ψ s (r 2 , r 1 ) = λΨ s (r 1 , r 2 ). P Applying this operation twice must reproduce the original state, hence λ2 = 1 and λ = ±1, i.e. Ψ s (r 2 , r 1 ) = ±Ψ s (r 1 , r 2 ).

(7.32)

The spatial part of the wave function, may thus be either symmetric in respect of exchange of the two electrons or antisymmetric. We shall call these wave functions Ψ+s (r 1 , r 2 ) and Ψ−s (r 1 , r 2 ), respectively. Since electrons are fermions (spin 1/2), for their total wave function in He Ψ (1, 2) = −Ψ (2, 1)

(7.33)

must hold, specializing (7.28). Thus, with (7.30) obviously follows that • the spatial part Ψ s (r 1 , r 2 ) of the wave function, must be symmetric if the spin function χ(1, 2) is antisymmetric • vice versa, the spatial part Ψ s (r 1 , r 2 ) of the wave function, must be antisymmetric if the spin function χ(1, 2) is symmetric. For the spin function we have already developed the necessary tools in Sect. 6.2.5: two electrons with spin s = 1/2 couple to a total spin S2 = S S1 +

(7.34)

with S = 1 or S = 0. The corresponding spin combinations and state vectors |χ(1, 2) = |χSMS are illustrated in Table 6.1. There, we have identified two different types of spin functions: the symmetric triplet functions with S = 1 and MS = −1, 0, +1 −1 1 χ χ = α(1)α(2) , = β(1)β(2) and 1 1 0 |α(1)β(2) + |β(1)α(2) χ = √ 1 2

(7.35)

and the antisymmetric singlet function with S = 0 and MS = 0 0 |α(1)β(2) − |β(1)α(2) χ = . √ 0 2

(7.36)

354


Again, the numbers (1) and (2) indicate which of two electrons is represented by the spin function α or β, with β(1)α(2) = − 1 1 and so on α(1)β(2) = 1 − 1 , 2 2 2 2 being compact writing for |ms1 ms2 = |s1 ms1 |s2 ms2 . Note the orthonormality of the total spin functions: M χS S χSMS = δS S δMS MS .

(7.37)

The total angular momentum of the system is also constructed from the individual angular momenta of the two electrons: L2 = L. L1 +

(7.38)

For the ground state we have so far only discussed the spatial part of the wave function (7.20), which is symmetric with respect to exchange of r 1 by r 2 . Thus, the corresponding spin function must be antisymmetric. Due to the PAULI principle (total anti-symmetry, at least one quantum number different) no triplet configuration is possible for the ground state and the complete ground state wave function is Ψ

1s 2 (1, 2)

s |α(1)β(2) − |β(1)α(2) . = Ψ1s √ 2 (r 1 , r 2 ) 2

(7.39)

In this case inclusion of the spin coordinates is just a formality which does not change the energy, the H AMILTON operator (7.6) as such being independent of the spins.3 In summary, the ground state of He with 1 = 2 = 0, a total spin S = 0 and a total orbital angular momentum L = 0 is a singlet 1s 2 1 S0 state. We use here the terminology introduced in Sect. 6.2.6, i.e. we characterize the total spin and the total orbital angular momentum with capital letters S and L and designate the states of He by n1 1 n2 2 2S+1 LJ . As already mentioned in Sect. 6.2.6 this coupling scheme is called RUSSEL -S AUNDERS coupling (or LS coupling). An alternative would be the jj coupling, where first the spin s and the orbital angular momentum of each individual electron couple to a single electron total angular momentum j ; and then the resulting j i would couple to a total angular momentum J of both electrons. In Sect. 7.3.4 we shall further analyze why He chooses to be LS and not jj coupled. In general, the total wave functions for He in LS coupling are singlet states

ΨS (1, 2) = Ψ+s (1, 2)χ00 (1, 2)

triplet states

ΨT (1, 2) = Ψ−s (1, 2)χ1 S (1, 2)

3 We

M

or with MS = −1, 0, 1.

(7.40) (7.41)

mention, however, a very remarkable property of this description of singlet states: the wave function (state) is not separable, i.e. we cannot write it as a simple direct product of states from the two separated electrons. One calls such states “entangled”, see also Appendix E.3.

7.3


355

We specialize this now for the excited states of He and consider electron configurations {n1 1 n2 2 } for which n1 = n2 . In the following we shall approximate the corresponding spatial part of the wave functions by products of orbitals of the type ψns1 1 (r 1 )ψns (r 2 ). Since both electrons are in principle indistinguishable the 2 2 spatial orbitals have to be combined to symmetric or antisymmetric linear superpositions: 1 Ψ+s (1, 2) = √ ψns1 1 m1 (r 1 )ψns2 2 m2 (r 2 ) + ψns1 1 m1 (r 2 )ψns2 2 m2 (r 1 ) 2 1 Ψ−s (1, 2) = √ ψns1 1 m1 (r 1 )ψns2 2 m2 (r 2 ) − ψns1 1 m1 (r 2 )ψns2 2 m2 (r 1 ) . 2

(7.42) (7.43)

We want to emphasize that this kind of ansatz is already an approximation, the so called independent particle model is used here. For the ground state of He it is not a very good approximation: as we have seen in Sect. 7.2.4 and Sect. 7.2.5 the two electrons are directly correlated. However, for singly excited states the independent particle model turns out to describe the system rather well.

7.3.3

Perturbation Theory for (Singly) Excited States

We now focus on such excited states in which one of the electrons remains in its 1s ground state configuration |100 , the other one, however, is raised into an excited state |n m . With the Hamiltonian (7.6) we obtain in 1st order perturbation theory for singlet (7.40) and triplet states (7.41) a total energy ee (r12 )|ΨS = W1 + W2 + Hee + WS = W (0) + ΨS |H (1)

and (7.44)

(1) ee (r12 )|ΨT = W1 + W2 + Hee WT = W (0) + ΨT |H −, (1)

respectively. We have to compute the diagonal matrix element Hee ± which determines the energy shift in respect of the undisturbed system (W (0) = W1 + W2 ). Since the H AMILTON operator does not depend explicitly on the spin and since the spin functions (7.37) are orthonormal, they can be pulled outs of the integral: (1) ee |ΨS,T = Ψ s (1, 2) 1 Ψ s (1, 2) . Hee ± = ΨS,T |H ± r12 ± Inserting (7.42) and (7.43) leads to (1) Hee ±=

1 s s s s ψ100 (r 1 )ψn m (r 2 ) ± ψ100 (r 2 )ψn m (r 1 ) 2 1 s s s s ψ (r 1 )ψn m (r 2 ) ± ψ100 (r 2 )ψn m (r 1 ) . r12 100

(7.45)

356


Knℓ

1snℓ

Jnℓ

1L

Knℓ 3L

Fig. 7.5 Singly excited He(1sn ) state with total angular momentum quantum number L: the term is raised by C OULOMB screening Jn and split into one singlet 1 L and three degenerate triplet states 3 L by exchange interaction Kn

One easily verifies that this perturbation term can be written as (1)

for singlet terms

Hee + = Jn + Kn

for triplet terms

Hee − = Jn − Kn ,

and

(1)

with the C OULOMB integral s ψ (r 1 )2 1 ψ s (r 2 )2 d3 r 1 d3 r 2 Jn = n m 100 r12 and the exchange integral 1 s s∗ s∗ s ψ100 (r 1 )ψn m (r 2 ) ψ100 (r 2 )ψn m (r 1 )d3 r 1 d3 r 2 . Kn = r12

(7.46) (7.47)

(7.48)

(7.49)

The influence of Jn and Kn on the total energy is illustrated in Fig. 7.5. The C OULOMB integral can be interpreted in a rather evident manner. For clarity we briefly switch to SI units and introduce the charge density of electron 1 at position r 1 the probability to find electron 2 at position r 2

2 ρ(r 1 ) = eψ s (r 1 ) 2 w(r 2 ) = ψ s (r 2 ) .

With these we rewrite (7.48) as # " ρ100 (r 1 ) . Jn = d3 r 2 wn m (r 2 ) e d3 r 1 4πε0 r12

and

(7.50)

One may thus read the C OULOMB integral as follows: {} is the repulsive interaction energy of electron 2 in position r 2 with the charge density of the other electron in state |100 – integrated over the whole charge distribution of electron 1. The external integration just averages this repulsion over the probability to find 2 anywhere in space. In total, Jn is the overall electrostatic repulsion energy of the two electrons. It leads to a raise of the terms as sketched in Fig. 7.5. In contrast, the exchange integral Kn is not open to an intuitive interpretation. Albeit also of electrostatic nature (1/r12 term) it is generated by a typical quantum mechanical effect, the exchange of the two electrons as a consequence of symmetrizing or anti-symmetrizing the

7.3


357

spatial part of the wave function. This becomes particularly evident if one formally rewrites the perturbation (7.46) and (7.47) as 1 (1) ee H S 1 S 2 )Kn , = Jn − (1 + 4 2

(7.51)

so that the total Hamiltonian becomes 2 + Jn − 1 (1 + 4 =H 1 + H S 2 )Kn . S 1 H 2

(7.52)

One easily verifies the identity of this expression with (7.6) using

1 2 1 2 2 2 1 2 3 3 S1S2 = S − S1 − S2 = S − − = 2 S −3 2 2 4 4 4 and the eigenvalue equation for the total spin S = S1 + S 2 (in a.u. = ) 2 S |SMS = S(S + 1)|SMS .

One finds that the expectation value of (7.51) is indeed identical with (7.46) and (7.47), respectively, depending on whether S = 0 or 1. One speaks about exchange interaction almost like its own kind of force which results from the anti-symmetrization rule for wave functions of fermions. It is this exchange interaction which enforces the splitting of singlet and triplet terms. The magnitude of this interaction energy is given by the exchange integral Kn – that is by a purely electrostatic interaction. The states which diagonalize the H AMILTON S 2 is diagonal, which implies diagoperator obviously have to be such that also S 1 2 onalization of S . Our initial ansatz to first couple the spins through S = S1 + S2 1 thus finds its justification by the operator 2 (1 + 4S 1 S 2 )Kn which enters the Hamiltonian. We shall expand this somewhat more precisely in the next section. We emphasize again that, according to the scheme sketched in Fig. 7.5, triplet states generally have lower energies than singlet states (for otherwise identical quantum numbers). A second look at Fig. 7.2 confirms this statement. The physical origin is also quite clear after the preceding discussion: in triplet states the spin function is symmetric, the spatial part of the wave function is antisymmetric. However, this implies that the probability for both electrons to come close to each other is smaller than for a symmetric spatial function. In a triplet state, the two electrons are never found at the same position simultaneously. Consequently, the average repulsive energy is smaller in triplets than in singlets, and hence triplet states lie lower than singlet states. Very generally, H UND’s rules hold: • For a given electron configuration the states with the highest multiplicity 2S + 1 have the lowest energy. • Among these, the states with highest L have the lowest energy. We point out that the special form (7.52) of writing the H AMILTON operator explicitly with exchange interaction is valid also for multi-electron systems where one

358


S j term over all pairs of electrons. This Hamiltonian plays a key has to sum the S i role in the H EISENBERG model of solid state ferromagnetism: it is the magnitude of the exchange interaction which determines whether it is energetically favourable for the spins to orient parallel to each other – and hence whether a material can be ferromagnetic or not. Finally, we mention a recent systematic and quantitative comparison between experiment and the above 1st order approximation for the 1sn series in singly excited He. H UTEM and B OONCHUI (2012) explicitly evaluated the C OULOMB and exchange integrals, using hydrogen orbitals and spherical harmonics series. They found e.g. for the first excited 1s2s 1 S and 3 S states a discrepancy of 5.3 % and 2.4 %, respectively, for the 1s2p 1 P and 3 P states 7.2 % and 4.1 % (as expected with the 1st order results higher than the experiment). For larger n the deviations get significantly smaller, but are still more than 1 % at the n = 4 level. Obviously, at the present level of precision in modern spectroscopy, a significantly better approximation is required. We shall come back to state-of-the-art computations in Chap. 10.

7.3.4

An Afterthought

What kind of force acts on the spins? It is definitively not magnetic, since, so far, we have not included any magnetic moments of the spin, spin orbit interaction and the like. Still, the spins prefer obviously to be oriented parallel or antiparallel; the following considerations should help to understand why. Turning this question around, one is tempted to search for a total wave function differing from (7.30) which, nevertheless, is consistent with the anti-symmetrization rule (7.33). One possible way to realize such states from a linear combination of orbitals are so called S LATER determinants. They will be discussed and used in some depth in Chap. 10. For the present case of a singly excited He atom in the configuration {1s, n } the following expressions are possible: s (1)α(1) 1 ψ s (1)α(1) ψn Ψ1 (1, 2) = √ 1s s s 2 ψ1s (2)α(2) ψn (2)α(2) s s (1)β(1) (1)α(1) ψn 1 ψ1s Ψ2 (1, 2) = √ s s 2 ψ1s (2)α(2) ψn (2)β(2) s s (1)α(1) (1)β(1) ψn 1 ψ1s Ψ3 (1, 2) = √ s (7.53) s 2 ψ1s (2)β(2) ψn (2)α(2) s s (1)β(1) (1)β(1) ψn 1 ψ1s . Ψ4 (1, 2) = √ s s 2 ψ1s (2)β(2) ψn (2)β(2) The numbers in brackets () refer again to the coordinates of electron 1 and 2. Obviously one may pull out α(1)α(2) and β(1)β(2) from Ψ1 (1, 2) and Ψ4 (1, 2), respectively. These linear combinations of atomic orbitals are thus identical to the triplet states ΨT (1, 2) = Ψ−s (1, 2)χ1±1 (1, 2) with an antisymmetric spatial wave functions

7.3


359

according to (7.41) – just as used in the previous description of the He excited states. However, the states Ψ2 (1, 2) and Ψ3 (1, 2) cannot be identified with any of the states used so far – albeit antisymmetric beyond any doubt: exchange of electron 1 and 2 implies interchanging two rows in the determinants, which by definition changes the sign of the determinant. To understand the significance of these state functions Ψ1 (1, 2) to Ψ4 (1, 2) let us use them to express the H AMILTON operator (7.6) in matrix representation. By simply multiplying this operator from the left with Ψj (1, 2)| and from the right with |Ψi (1, 2) and making use of the orthonormality relation (7.37) for the spin functions we obtain the 4 × 4 matrix ⎞ ⎛ 0 W1s + Wn + Jn − Kn 0 0 ⎟ ⎜ 0 0 0 0 ⎟ =⎜ H ⎠ ⎝ 0 0 0 0 0 0 0 W1s + Wn + Jn − Kn ⎛ ⎞ 0 0 0 0 ⎜ 0 W1s + Wn + Jn −Kn 0⎟ ⎟. +⎜ (7.54) ⎝0 W1s + Wn + Jn 0 ⎠ −Kn 0 0 0 0 We have written it in terms of a diagonal and a non-diagonal part. One readily verifies that Jn and Kn are indeed the C OULOMB and exchange integrals, (7.48) and (7.49), while W1s and Wn are the single particle energies in the 1s and n state, respectively. Obviously, the S LATER determinants (7.54) diagonalize the Hamiltonian only partially, they are thus not a complete set of eigenfunctions. We may, however, try to diagonalize the non-diagonal term, i.e. the 2 × 2 block in the middle of the second matrix in (7.54). Using the standard procedure we write the characteristic equation − W) = 0 det(H from which we obtain two solutions for W , the two missing eigenenergies W± = W1s + Wn + Jn ± Kn . And we happily note that this result is exactly the same 1st order result (7.44) with (7.46) and (7.47) that we have obtained with the “correct” state functions (7.40) and (7.41), respectively. The latter ones, i.e. Ψ+s (1, 2)χ00 (1, 2) and Ψ−s (1, 2)χ10 (1, 2), are of course also the result of the diagonalization procedure which we have sketched here: they simply are linear combinations of Ψ2 (1, 2) and Ψ3 (1, 2) to the eigenvalues W+ and W− while Ψ1 (1, 2) and Ψ4 (1, 2) are already part of the triplet system. Summarizing, it is the exchange interaction, recognized in (7.54) as the offdiagonal term −Knl , which enforces the parallel or antiparallel orientation of the electron spins! Combination of spin-orbitals which do not diagonalize the spin and orbital angular momentum simultaneously are no eigenvalues of the H AMILTON operator under discussion. Of course this statement is fully based on the fact that exchange interaction is by far the dominant perturbation – as assumed in (7.54). For high Z this will

360


change significantly with spin-orbit (or LS) interaction increasing. As we shall see in Sect. 10.4.1 LS interaction may indeed become of the same order of magnitude or even larger than exchange. Then it will be reasonable to diagonalize the Hamiltonian first in respect of the dominant LS interaction and later on add exchange interaction a minor perturbation. This explains without effort the transition from RUSSEL -S AUNDERS (or LS coupling) to jj coupling for atoms with high nuclear charge Z. Section summary

• The two electrons in the He atom are in principle indistinguishable, hence proper antisymmetrization of the He wave functions according to the PAULI principle is mandatory. This leads to singlet and triplet formation, with antiparallel and parallel electrons spins, respectively. • Consequently, the He 1s 2 ground state can only be a singlet state as the spatial quantum numbers of the two electrons are identical. • In contrast, for each singly excited 1sn configuration a singlet and a triplet state exists. To a good 1st order approximation their spatial part is described by the independent electron model as product wave functions of the respective single electron orbitals, symmetrized and antisymmetrized according to (7.40) for singlet and triplet states, respectively. • A reasonable estimate for the energy of these states is obtained in 1st order as W1sn = W1s + Wn + Jn ± Kn – with the C OULOMB integral, Jn , accounting for the repulsion of the two electrons, and the exchange integral, Kn , characterizing the electron exchange energy. • We recognize the exchange energy Kn to be responsible for the electrons to orient antiparallel or parallel as singlet or triplet system, respectively.

7.4

Fine Structure

Closer inspection shows of course that He levels with S, L > 0 may also have a fine structure (FS) due to spin-orbit (LS) interaction – quite analogous to that discussed in Chap. 6. One finds that only the He triplet levels (S = 1) are split while the singlet states (S = 0) remain un-split (as the name indicates). The order of magnitude of FS splitting in He is similar to that in atomic hydrogen. The largest splitting is found for the He 1s2p 3 P state (overall ca. 1.06 cm−1 or 0.00013 eV, see inset in Fig. 7.2). This has to be compared with the singlet-triplet splitting between the 21 P and 23 P state (2048 cm−1 or 0.25 eV). FS interaction is indeed a very small perturbation in comparison to exchange interaction and the description of the He levels in terms of singlet and triplet states is an excellent approximation. Thus, for a description of the FS states we have to start now with the total spin S 2 and the total orbital angular momentum L= L1 + L2 for the two elecS = S1 + trons which are already coupled by exchange interaction as just discussed. Under the influence of LS coupling a total angular momentum J = L + S of the system

7.4

Fine Structure

361

Fig. 7.6 Fine structure splitting of the 1s2p 3 P◦J state in He and He like ions. Note the inversion of the triplet states for small Z. All energies in cm−1 , different scale for each of the three systems

1s2 p 3 P oJ

J

0.0764 He I Z = 2

0 950

3.13

0.9879 1 2

2

J

0

2 2.27 1 Li II Z = 3

150

1 0

F VIII Z = 9

is formed for each configuration {n1 1 n2 2 LS}. The rules which have been used in Sect. 6.2.5 for the angular momenta of individual electrons apply with appropriate modifications. The corresponding eigenstates of the two electron system which di2 agonalize J and Jz are constructed in full analogy to (6.53) and may be written explicitly (LS)J MJ = ( 1 2 )LML (s1 s2 )SMS LML SMS (LS)J MJ . (7.55) m ,ms

This combined coupling scheme for the total L and S angular momenta is called RUSSEL -S AUNDERS coupling (or LS coupling), as already mentioned. Again, as in the single electron case, the multiplets split into 2S + 1 components (if S ≤ L). However, the magnitude of the splitting can no longer be derived simply from ξ(r) L S as in the single electron case (at least not for low Z). The Hamiltonian must include all magnetic interactions between individual electrons and will contain terms like LS = Si ξi (r i ) Li (7.56) H S1, S 2 (spin other orbit) as well as spin-spin L1 but also all terms of the type L2 S 2 but has to account for the correct dipoleinteraction. The latter is not simply ∝ S 1 dipole interaction. In Chap. 9 we shall discuss such kind of problems in some detail in the context of hyperfine interactions between electron spin and nuclear spin. The ensuing computations are rather complicated and cannot be described any longer by the simple interval rule (6.67). We do not want to go into the details here, but note that the deviations are particularly strong for small Z, specifically for He. Figure 7.6 (left) shows this for the case of the first excited triplet state 2 3 P with the highest FS in He. The term positions are completely inverted in this case, i.e. the highest J has the lowest energy. For comparison the corresponding triplet state splittings for the He like ions Li+ and F7+ are also shown. In the case of the fluorine ion at least the ordering of the states is again ‘normal’. Nevertheless, the interval rule (6.67), according to which the distances between the FS levels should be proportional to the higher J is not really fulfilled. For larger Z (and singly excited states) the spin-own-orbit interaction of the S 2 type dominates (2 refers to the excited electron) and the FS splitting L2 ξ2 (r 2 ) can be expressed again approximately by a (7.57) S 2 = J (J + 1) − L(L + 1) − S(S + 1) VLS = ξ2 (r 2 ) L2 2

362


Fig. 7.7 Fine structure splitting of the alkaline earth metals beryllium (Z = 4) and magnesium (Z = 12). The terms are normally ordered, i.e. for higher J the energy is higher. Note that for Mg the energy splittings are about

WFS ∝ J , i.e. the L ANDÉ interval rule holds

Be I 1s 2 2s2p 3 P oJ J=2

Mg I [Ne] 3s3p 3 P oJ J=2

2.345

40.614 1

1 0 0.645

20.059 0

in analogy to the one electron systems. We illustrate this in Fig. 7.7 for the example of alkaline earth metals. Their role in respect of He corresponds to that of the alkali atoms in respect of the hydrogen atom: they have two valence electrons on top of a closed shell core (see also Sect. 7.7.1). One sees that for Mg the interval rule holds again very well. We recall, however, that fine structure interaction grows according to (6.66) with 2 α Z 4 /n3 . For large Z and not too large n it may become substantial. We shall come back to this again in Chap. 10 and find that RUSSEL -S AUNDERS coupling breaks down for large Z. Section summary

• For He (as well as for other light atoms) spin-orbit interaction (7.57) is much smaller than exchange interaction (7.49). Hence, for light atom spin-orbit coupling can be considered a small perturbation which does not change the coupling of the spins. • A quantitative interpretation of the fine structure of two electron systems thus S 2 and total orbital angular momenta L= starts with total spin S = S1 + L1 + L2 which couple in turn to the overall angular momentum of the system J = L + S. • The overall magnitude for FS splitting in the case of He is on the same order of magnitude as for H. Unfortunately, the simple formulas used there for the splitting as well as the L ANDÉ interval rule cannot be applied in general (for light atoms): spin-spin interaction and spin-other-obit interactions have to be accounted for.

7.5

Electric Dipole Transitions

At the beginning of this chapter we have already introduced selection rules for E1 transitions in He. We want to explore some basic questions about photon absorption and emission in multi-electron systems, using the two electron He system as a (hopefully) still transparent example. We now have to expand our definition of the dipole operator (4.54). For one electron at position r i the dipole moment was just

7.5


363

D i = er i . For N active electrons the perturbation energies (4.55) for all potentially active electrons have to be added: /N 0 ieE0 U (r, t) = − (7.58) D i · E(t) = r i · ee−iωt + e∗ eiωt . 2 i=1

Here E(t) is again the electric field component of the wave and E0 represents the amplitude. The dipole transition operator (4.56) for a multi-electron system is corN (i) respondingly D = i=1 D with D(i) = −(D i · e)/e = r i · e. Time dependent perturbation theory has to be applied in full analogy to the one electron problem. Only the dipole transition matrix element is somewhat more complicated. The following considerations are based on the independent particle model. For He dipole transition matrix element has the form: Dba = Ψb (1, 2)r 1 + r 2 Ψa (1, 2) · e. (7.59) Explicitly with the wave functions according to (7.40) and (7.41) we obtain s s MS M Dba = Ψb± (1, 2)χSb b (1, 2)r 1 + r 2 Ψa± (1, 2)χSa Sa (1, 2) · e, abbreviating again (r 1 ) = (1) and (r 2 ) = (2). We first note that neither r 1 nor r 2 act directly on the spin component of the wave function. We may thus pull the latter out of the integral and apply the orthonormality relations for the spin functions: M s s MS Dba = Ψb± (1, 2)r 1 + r 2 Ψa± (1, 2) χSb b (1, 2)χSa Sa (1, 2) · e s s = Ψb± (1, 2)r 1 + r 2 Ψa± (1, 2) · eδSb Sa δMSb MSa .

(7.60)

This is a very important result: the spin state remains unchanged by E1 transitions in He. In particular, there are no E1 transitions between singlet and triplet system (these so called intercombination lines are forbidden). Correspondingly, the + and − symmetry of the spatial wave functions remain also unchanged in E1 transitions (this holds even for E transitions of all orders since the spin is not involved). This selection rule thus explains the experimental observation of the two quasi isolated systems of lines – historically “para and ortho helium”, i.e. transitions occur only within the singlet or the triplet system. For He this rule holds rather strictly. For He like ions with large Z one observes more and more also (weak) intercombination lines: as Z increases, spin-orbit interaction (7.57) also increases and has to be accounted for. This leads to a mixing of the pure singlet or triplet character of states in the independent particle model so that triplet and singlet components are added to the respective other multiplicity (so called configuration interaction, CI). For this very reason intercombination lines also become possible. Beyond the forbidden intercombination lines the model of independent particles supports another important selection rule: allowed are only pure one electron transitions even within the singlet or triplet system – i.e. only one of the quantum numbers is allowed to change in a one-photon E1 transition. The proof is quite instructive:

364


In the independent particle model the spatial part of the wave function is written as symmetric or antisymmetric products of one electron orbitals according to (7.42) or (7.43), respectively. Let the initial configuration be {a} = {1a, 2a} = {n1a 1a m1a , n2a 2a m2a }, the configuration after the transition {b} = {1b, 2b} = {n1b 1b m1b n2b 2b m2b }. The dipole transition matrix element between these is s s Dba = Ψb± (1, 2)r 1 + r 2 Ψa± (1, 2) · e (7.61) =

1 s s s s ψ1b (1)ψ2b (2) ± ψ1b (2)ψ2b (1)r 1 2 + r 2 ψ s (1)ψ s (2) ± ψ s (2)ψ s (1) · e. 1a

2a

1a

2a

We note that r 1 contributes to the matrix element only through that part of the wave function which depends on r 1 while r 2 is only relevant for the wave function depending on r 2 . Hence the components of (7.61) factorize. This results in expressions of the type Dba =

s s s 1 s ψ1b (1)ψ1a (1) ψ2b (2)r 2 · eψ2a (2) 2 s s s 1 s ± ψ1b (1)ψ2a (1) ψ2b (2)r 2 · eψ1a (2) 2 s s s 1 s ± ψ2b (1)ψ1a (1) ψ1b (2)r 2 · eψ2a (2) 2 s s s 1 s + ψ2b (1)ψ2a (1) ψ1b (2)r 2 · eψ1a (2) + · · · 2

and corresponding expressions with r 2 replaced by r 1 and vice versa, leading to identical results (one has to integrate over the whole space). Recalling that the one electron wave functions are orthonormal this implies s s Dba = δ1b1a ψ2b (r)r · eψ2a (r) (7.62) s s ± δ1b2a ψ2b (r)r · eψ1a (r) s s ± δ2b1a ψ1b (r)r · eψ2a (r) s s + δ2b2a ψ1b (r)r · e|ψ1a (r) , i.e. one of the quantum numbers has to remain the same before and after the transition. What remains of the double integrals over r 1 and r 2 are just single electron transition matrix elements: they are determined by the standard selection rules which we have developed in Sect. 4.4 for one electron systems. Thus, the model of independent particles allows only single electron transitions, i.e. only such transitions are allowed where one electron changes its quantum number while all others remains in its initial configuration. Specifically, if one electron is originally in the ground state 100, only transition {100n m} ←→ {100n m } or {100n m} ←→ {n m n m} may occur for which the usual selection rules = ±1 and m = ±1, 0 hold, together with Sb = Sa

7.6

Double Excitation and Autoionization

365

– the prohibition of intercombination transitions. Note however, that the selection rule (7.60) for the spin projection quantum numbers MS has to be replaced for fine structure transitions within the triplet system by the J and MJ rules which we have outlined in Sect. 6.4. Section summary

• The dipole transition operator for an N electron system is given by D= N (i) (i) i=1 D , with D = r i · e. • Exploiting the structure of the wave functions one finds that E1 transitions between singlet and triplet system (intercombination lines) are strictly forbidden. • The independent electron model allows only E1 transitions in which the quantum numbers of only one electron change. • The usual = ±1 and m = ±1, 0 selection rules hold. In addition, for fine structure transitions J = 0, ±1 with 0 0 must hold.

7.6


7.6.1

Doubly Excited States

In the previous discussion of excited He states at least one of the electrons was assumed to be in its 1s ground state, the other one in an excited n state. However, one may very well envisage configurations where two electrons are excited – irrespective of the question how such doubly excited states might be populated. The independent particle model predicts in 0th order perturbation theory energies given by (7.15). For the example of a series with the configuration 2 n (n ≥ 2) this would be (in a.u.)

Z2 1 Z2 1 (0) W2 n = − + 2 ≥− . 2 4 n 4 For He (Z = 2) these doubly excited states are thus expected at energies ≥ −1Eh = −27.2 eV, while the binding energy of the He+ ion is −2Eh = −54.4 eV. Hence, the doubly excited states {2 n } lie in the ionization continuum of He+ + e− and converge to the excited He+ (2s) state at −2Eh /4 = −13.6 eV. The {3 n } series has energies ≥ −13Eh /18 = −19.6 eV and converges to the He+ (3s) state at −2Eh /9 = −6.0 eV. C OULOMB and exchange interaction raise the lower limits of these series, but the general feature remains: these states are embedded in the ionization continuum. The expected limits for these series are schematically indicated in Fig. 7.4 right (red horizontal lines, marked 2s∞ and 3s∞ ). Series of absorption lines are indeed observed experimentally in an energy range indicated by the grey shaded areas in Fig. 7.4 on the left, marked autoionization (upper limits shown as black lines).

366


7.6.2

Autoionization, FANO Profile

Autoionization and related resonance phenomena are an important and fascinating topic – not only in atomic physics. It is thus important to recognize the characteristic features and to understand the physics behind the experimental observations. At first sight one wonders how such states might be populated at all, since double excitation (7.62) is forbidden for E1 transition in the independent particle model. However, in reality the concept of independent particles is obviously far from being perfect. We have seen this already for the ground state. In the case of doubly excited states the two electrons may get rather close to each other. Thus, it is no longer sufficient to account for the other electron by an averaged screening term. Modern theoretical calculations include the electron correlation by different techniques, e.g. by CI, i.e. by linear combination of several configurations, or – very elegantly – by a judiciously chosen, non-Cartesian coordinate system, where the distance between the two electrons is one of the free coordinates. Using such correlated wave functions one may indeed describe two electron E1 transitions of the type (7.63) hν + He 1s 2 → He n n . Beautiful examples for such spectra are obtained in absorption experiments with high resolution synchrotron radiation. A few samples, the 2 n and 3 n series, are shown in Fig. 7.8. Note that the characterization of the individual lines emphasizes already the complexity of the correlated electronic states which cannot be fully described by the configuration {2 n } and {3 n }. We cannot enter into the computational details here. Figure 7.8 immediately raises the question about the origin of the peculiar line shapes observed. The experiment directly records the He+ ion current as a function of the photon energy from a highly monochromatized synchrotron radiation beam which is passed through a He filled cell.4 We first note that the observed lines, i.e. the He(n n ) = He∗∗ states,5 are energetically indeed embedded in the ionization continuum of helium, He+ (1s) + e− . The two series 2 n and 3 n of doubly excited states shown here converge for n → ∞ towards the corresponding excited ionic states He+ (2 ) and He+ (3 ), respectively. These doubly excited states may in principle decay through optical transitions to a lower lying state by emission of a photon. Alternatively – and indeed much more efficiently – they can decay simply by ionization since they lie energetically within the ionization continuum: He∗∗ → He+ (1s) + e− .

(7.64)

Such process in which a state embedded into an ionization continuum decays by electron emission is called autoionization. The second electron returns here to the ground state, now of the He+ (1s) ion. The photon energy ω initially deposited 4 Note that the signals shown are not generated by differentiation as often done in spectroscopy for better assignment of line centres – the genuine line shapes are shown here. 5 Two

asterisks, short for double excitation.

7.6


367 WI (n = 2) 65.405

He (sp, 2 n ±) 1 P o

n = 2+

3+

3x 15

4+ 5+

photoionization yield

460

61

62

He (3,1n ) 4

5

n=3 q =1.45

6

63 WI (n =3) 72.963

5-

64

14

8

15

16

6+ 7+ 65

17

3,-13

19

Γ = 0.21 Wr = 69.88 70

71

72

72.88

72.90

72.92 72.94

excitation energy ħω / eV

Fig. 7.8 He autoionization spectra recorded with high resolution synchrotron radiation; top: (sp, 2n±) series (D OMKE et al. 1991), bottom left: (3, 1n ) series, inset bottom right: enlarged detail (S CHULZ et al. 1996). Red vertical lines indicate the limits He+ (n = 2) and He+ (n = 3) towards which the two series converge. For the state (3, 13 ) the red line shows by way of example a FANO profile fit (7.70) with the parameters q, Γ and Wr

for exciting the He∗∗ state is partially used for ionization (creating He+ (1s) from He(1s 2 )) while the excess energy, ω − WI [He(1s 2 )], is carried as kinetic energy Wkin by the ejected electron. The peculiar shape of the autoionizing lines documented very neatly in Fig. 7.8 is explained as a special kind of interference: the final state He+ + e− (Wkin ) may in principle be reached from the ground state through two different channels: • either one induces the ionization process directly ω + He 1s 2 → He+ (1s) + e− (Wkin ),

(7.65)

• or one excites one of these doubly excited states which subsequently decays into the continuum: ω + He 1s 2 → He∗∗ → He+ + e− (Wkin ). (7.66) In both cases the total energy balance is given by: ω = Wkin + WI He 1s 2 . The experimentally observed line shape results from the fact that both channels (direct ionization and ionization via the resonance He∗∗ ) can by principle not be distinguished in the experiment – a situation which is characteristic for any kind of interference experiment, we may e.g. think of YOUNG’s double slit experiment.

368


Let us call the probability amplitudes for the two processes cd and cr , respectively, written as cd = Aeiδ and cr = cr (W )eiφ(W ) . (7.67) According to the general principles of quantum mechanics these amplitudes for the two indistinguishable processes have to be added coherently. The probability for observing an absorption signal is then given by 2 S(W ) = |cd + cr |2 = Aeiδ + |cr |eiφ

(7.68)

= A2 + |cr |2 + 2A|cr | cos(δ − φ). Close to resonance the phase angle φ changes rapidly and one expects a characteristic interference pattern – as indeed observed in the autoionization spectra Fig. 7.8. A quantitative treatment of this phenomenon has been described for the first time by FANO (1961). Briefly, one considers amplitude A and phase δ for the direct photoionization process to be essentially constant and independent of the photon energy W = ω over an energy range relevant for a single resonance. For cr (W ) one assumes the typical behaviour of a (complex) resonance amplitude expected when W passes through the resonance, as e.g. for a harmonic oscillator. We have already discussed this in the context of ordinary optical resonance absorption in Sect. 5.1.2. As illustrated in Fig. 5.2(a), the magnitude |cr (W )| rises and decreases rapidly close to a resonance at energy Wr , its linewidth being Γ , while the phase changes from zero to π . Due to the superposition of cr (W ) with cd one obtains indeed very pronounced line shapes as observed in the experiment. The famous FANO line profile (more precisely the B EUTLER -FANO profile, B EUTLER being the spectroscopist who first observed such line shapes) is given by6 (q + )2 1 + 2

with =

W − Wr Γ /2

(7.70)

with the linewidth Γ , the reduced energy , and the FANO line shape parameter q. The latter includes the relative phase and coupling strength between the direct and the resonant process. We have fitted the He(3, 13 ) line shown in the experimental data Fig. 7.8 by way of example with this profile – and obtain a nearly perfect fit! In the Sect. 7.6.3 we investigate in a systematic manner different forms of such FANO resonance profiles and try to understand them in some more detail. 6 For

fitting a measured signal the original formula has to be slightly generalized, e.g. to S() =

q2

C (q + )2 C + A2 − 2 , 2 +1 1+ q +1

accounting for the background signal A2 and the strength C of the resonance.

(7.69)

7.6


369

Fig. 7.9 Resonance amplitude cr () in the complex plane, B = 1

Im = 0 (resonance) 1.0

= -1

0.5

=1

<< - 1

>>1 - 0.5

7.6.3

0

Re

0.5

Resonance Line Profiles

Resonances of the type discussed here are observed in many areas of physics. One finds them in optical spectroscopy as discussed here, in atomic and molecular collision physics with electrons, ions and atoms, but also in nuclear physics or high energy physics, as well as in solid state physics. They always occur when a final state may be reached on different pathways, one of which involves a resonant, quasi stable state. To understand the behaviour described by the FANO profile (7.70) we visualize the excitation amplitudes involved in the complex plane. With the normalized energy according to (7.70) one writes magnitude |cr ()| and phase φ() of the resonance amplitude cr – or alternatively real and imaginary part – according to (5.14a)–(5.15b), scaled with a resonance strength B: cr () = √ B 2 + 1 Re(cr ) = −

|cr ()|2 B

and and

1 tan φ() = − Im(cr ) =

or

|cr ()|2 . B

(7.71) (7.72)

We have already illustrated the resonance amplitude in Fig. 5.2. Figure 7.9 shows an alternative visualization in the complex plane: with (7.72) one easily verifies:

2 2 B 2 B Re(cr ) + Im(cr ) − = . (7.73) 2 2 Thus, when the reduced energy is tuned through resonance ( = 0), cr rotates in the complex plane on a circle with radius B/2 around a centre at (0, iB/2). We have now to add this resonance amplitude cr () to the (constant) direct amplitude cd as suggested by (7.68). With (7.71) the profile of the resonance line (i.e. the absorption or ionization cross section) becomes: iδ iφ() 2 iδ iB 2 −B + σ (W ) ∝ Ae + cr () e = Ae + 2 . (7.74) + 1 2 + 1 Autoionization resonances are observed close to resonance 0 (W = ω Wr ) on a background from direct ionization. Depending on the phase δ and the relative magnitude A/B of the direct amplitude vector addition of cd and cr () in the com-

370


(a)

(b)

= 90 o A = 0.25

Im

ii

ii iii

i

0.5

iii

ii i 1

-10

i

iii

0.5

i

0

-1 Re 1.25

i 10

ii 0.5

-0.5

iii

0.5

iii 0

= - 90 o A = 1.00

0.5

1.25

i

(d)

=206 o A = 0.56

0.5 0

-0.5 S( ) 1.5 ii

(c)

= - 26 o A = 0.56

-10

0

ii

ii

iii

i 10

-10

1.25

iii 0

i 10

-10

iii ii 0

10

Fig. 7.10 Top row: absorption amplitudes (full red arrows) in the complex plane as a sum of direct ionization amplitude cd = A exp(iδ) (dashed black arrow) and resonant amplitude cr () (dashed red circle) according to (7.71) with B = 1. Bottom row: Corresponding absorption profiles (ionization signals) S(). Depending on the parameters very different FANO line shapes emerge as a function of reduced energy = (W − Wr )/(Γ /2). In each case the complex amplitudes and the profiles are marked for different energies i < ii < iii

plex plane generates different line profiles as illustrated in Fig. 7.10.7 The shapes range from (a) a pure absorption profile for a phase angle δ = 90◦ (or whenever the direct amplitude vanishes), via dispersion like profiles (b) and (c) to completely destructive interference (d), i.e. to a transparent window in the absorption spectrum cd and cr () are of equal magnitude and opposite phase at resonance. Section summary

• Doubly excited states in He, embedded in the ionization continuum of He+ + e− may be excited by E1 due to a break down of the independent electron model. • Pronounced interference structures are observed due to coherent superposition of the amplitudes for direct ionization and autoionization. These can be described by the FANO profile (7.70). • It may be rationalized by addition of a direct and a resonance amplitude. While the former is constant, the magnitude of the latter changes rapidly over the resonance according to (7.71). 7 One

may derive from (7.74) with some algebra the modified FANO formula (7.69) with 2 4B C/(AB) + B/A + 2 sin δ B C = AB 4 + and q = − sin δ + for cos δ = 0 A A 2 cos δ

q = 0 for δ = π/2 and q → ∞ for δ = −π/2; C and thus q are in principle double valued.

7.7

Quasi-two-Electron Systems

371

7.7


7.7.1

Alkaline Earth Elements

WI = 13.282 13

5 4

Be+ ( 2S

1/2)

5 4 3

3 2sns 2snp 2snd 1S 1P o 1D 0 1 2

11 10

WI =

displaced terms 2pns 2pnp 2pnd 3P o 3P 3D 0 0

displaced terms 2pns 2pnp 2pnd 1Po 1 1 P1

Be+ ( 2p 2P3/2,1/2)

9.32270 8

n 5 4

5 4

4 3

3 7

3

3 2p 2 1S 0

2p 2 1D 2

2snp 2snd 3 Po 3D

2sns 3S 0 5 4

0 5 4

5 4 3

3

2p2 3P 0

3

6

excitation energy / eV

-20000

- 40000

4 Be I singlet

3 2

- 50000

2

- 60000

Be I triplet

1 0

- 10000

- 30000

2

5

binding energy / cm -1

The alkaline earth metal atoms are in respect of the He atom in the same role as the alkali atoms in respect of atomic hydrogen: their two potentially active valence electrons are screened from the nuclei by Z − 2 electrons in the completed filled ionic core levels. In full analogy to the alkali metal atoms (Sect. 3.2.3) with one electron, the Hamiltonian for two valence electrons of alkaline earth atoms is essentially that of He, except that in (7.6) the pure nuclear C OULOMB potential has to be replaced by a screened one which approaches 2/r1 (and 2/r2 ) for sufficiently large distance from the nucleus. Correspondingly, the term schemes and spectra are very similar to those of He, but in detail even richer, including also several series of doubly excited states. They are very well documented by K RAMIDA et al. (2013): energy levels, wavelength tables with transition probabilities, G ROTRIAN diagrams and simulated spectra can be obtained within seconds. Thus, here we show in Fig. 7.11 only the G ROTRIAN diagram of Be as one typical example.

2

1s 22s 2 1S0 Be ground state

- 75192.63

- 70000

Fig. 7.11 Term scheme of the beryllium atom derived from K RAMIDA et al. (2013) (G ROTRIAN diagram for the strongest transitions, Aab > 5 × 106 s−1 ). Singlet terms are marked by red, triplet terms by black lines

372


The ground state configuration of Be I is 1s 2 2s 2 , its first ionization potential is 9.32 eV, the ground state of the Be+ ion (Be II) being 1s 2 2s. To reach the first excited state of Be+ (2s 2 2p 2 P3/2,1/2 ) additional 3.96 eV, for full removal of the second valence electron 18.21 eV are needed. The doubly excited states of the alkaline earth atoms (also called “displaced terms”) partially start already below the first ionization potential as documented in Fig. 7.11 for the Be** 2pnp 1,3 P series. Doubly excited states above the ionization limit show the typical autoionization – with partially overlapping line structures. Singlet-triplet intercombination lines are forbidden as in the He case, although weak transitions are observed already in Mg and the prohibition is more and more released as Z increases: spin-orbit coupling induces the characteristic configuration interaction between singlet and triplet terms for larger Z so that the initial assumptions for the electron spin conservation rule (7.60) in E1 transitions is no longer valid. Hence the rule can be violated. The fine structure splitting in the first excited 3 PoJ state is for the Be atom still very weak, 0.645 cm−1 (J = 0 ↔ J = 1) and 2.345 cm−1 (J = 1 ↔ J = 2). For Mg, however, it is already 20.059 cm−1 and 40.714 cm−1 , respectively – in both cases thus normally ordered (contrary to He) and in the latter case in good agreement with L ANDÉ’s interval rule (6.67). For the heavier alkaline earth metals spinorbit interaction becomes rapidly more significant, with larger FS splitting and an enhancement of intercombination lines.

7.7.2

Mercury

As a last example for quasi-two-electron atoms we discuss mercury, Hg. Its ground state configuration is [Xe]4f 14 5d 10 6s 2 , i.e. the K to N shells are completely filled, and all subshells of the O shell are filled except for the 5f electrons. The buildingup of the P shell starts with the two 6s 2 electrons. Figure 7.12 shows part of the Hg emission spectrum, constructed from the NIST data – assuming here a somewhat unrealistically high electron temperature (Te = 15 000 K) which allows us to see also several transitions between higher lying states. In a standard low temperature gas discharges typically the 6s 2 1 S0 − 6s6p 3 Po2 is by far the dominant line due to a combination of a reasonable transition probability (Aab = 8 × 106 s−1 ) and low enough excitation energy Wexc = 4.89 eV. Hence, the population density of the excited 63 Po2 state ∝ exp(−Wexc /kB Te ) is high enough. Note that this line is an intercombination line which in strict RUSSEL -S AUNDERS coupling (LS coupling) is completely forbidden. Obviously, with its nuclear charge Z = 80 and a nucleon number from A = 196–204 Hg belongs to the heavy elements of the periodic system and LS coupling starts to break down as spin-orbit interaction increases ∝ Z 4 /n3 (see Sect. 6.3.2). Nevertheless, as shown in Fig. 7.13, one continues to classify the states as singlet or triplet system. The term scheme shows a remarkably large FS splitting of the 6 3 PJ states. With 1767 cm−1 (J = 0 ↔ J = 1) and 4631 cm−1 (J = 1 ↔ J = 2) the terms are normally ordered and in fair agreement with L ANDÉ’s interval rule. Its order of magnitude is getting close to the splitting between singlet and triplet system, which in the

300

6s 6p 1P o1 - 6s6d 3D2

6s6p 3P o2 - 6s7s 3S1

6s 6p 3P o1 - 6s7s 3S1

373 6s6p 3P o0 - 6s 7s 3S1

6s6p 3P o2 - 6s6d 3D3

6s 6p 3P o1 - 6s7d 3D2

6s 2 1S- 6s6p 3Po 200

6s6p 3Po - 6s 7d 3D


line intensity / arb. un

7.7

400 500 wavelength / nm

600

700

800

Fig. 7.12 Sticks spectrum of Hg in the UV and VIS after K RAMIDA et al. (2013) (S AHA-LTE spectrum, electron temperature 1.23 eV)

6snp 6snp 6snd 6snf WI = 1S 1P o 1D 1F o 0 1 2 3 10.4375 6 10 10 9 8 9 5 8 7 9 8 6 7 8

6snp 6snp 6snd 6snd 5d 9(2D5/2) 2 3S 3Po 3D 3Fo 0 2,1,0 3,2,1 4,3,2 6s 6p 10 9 8

7

7

7

9 8

8 7 6

6 5

excitation energy / eV

- 40000

3 Po

5

2 1 3 Po 0 3 Po

6

4 singlet

3.7

nm

- 50000

triplet

- 60000

25

2

- 70000

1 0

6

- 20000 - 30000

6

6

3

0

- 10000 complex terms

7

cm-1

[Xe] 4f 14 5d 10 6s 2 1S0 Hg I ground state

- 84184

- 80000

Fig. 7.13 Term scheme of the mercury atom derived from K RAMIDA et al. (2013) (G ROTRIAN diagram for the strongest transitions in Hg, Aab > 1 × 106 s−1 ). Singlet terms are marked by red, triplet terms by black lines. Intercombination lines are dashed black

case of 6 1 Po1 and 6 3 Po2 is about 10 026 cm−1 . Still, the line with the highest transition probability is 6s6p 3 Po2 − 6s6d 3 D3 (Aab = 1.3 × 108 s−1 ) – a clean LS allowed intra-triplet transition. Section summary

• The two valence electrons in alkaline earth metals and Hg essentially behave like those in He, with the nuclear charge effectively shielded by the filled inner shells.

374


• Even for Hg the characterization of valence states by RUSSEL -S AUNDERS (LS) coupling is still approximately correct. However, the FS splitting is large and a number of rather strong intercombination lines is observed.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2. 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). E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6). IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm 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. 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. 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 D OMKE , M. et al.: 1991. ‘Extensive double-excitation states in atomic helium’. Phys. Rev. Lett., 66, 1306–1309. D RAKE , G. W. F. and W. C. M ARTIN: 1998. ‘Ionization energies and quantum electrodynamic effects in the lower 1sns and 1snp levels of neutral helium (4 He I)’. Can. J. Phys., 76, 679–698. FANO , U.: 1961. ‘Effects of configuration interaction on intensities and phase shifts’. Phys. Rev., 124, 1866–1878. H UTEM , A. and S. B OONCHUI: 2012. ‘Evaluation of Coulomb and exchange integrals for higher excited states of helium atom by using spherical harmonics series’. J. Math. Chem., 50, 2086– 2102. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland.

References

375

K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. PAULI J R ., W.: 1925. ‘Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. Z. Phys., 31, 765–783. S CHULZ , K., G. K AINDL, M. D OMKE, J. D. B OZEK, P. A. H EIMANN, A. S. S CHLACHTER and J. M. ROST: 1996. ‘Observation of new Rydberg series and resonances in doubly excited helium at ultrahigh resolution’. Phys. Rev. Lett., 77, 3086–3089.

8

Atoms in External Fields

The interaction of atoms with external magnetic fields (Z EEMAN effect) has already been introduced in Chaps. 1 and 2, while radiation induced transitions where treated in Chaps. 4 and 5. Here we generalize and deepen what is already known, and develop the tools for a quantitative description of atoms and molecules in external magnetic and electric fields. Thus, this chapter provides the essential basis for understanding this type of interaction also in a more complex environment and gives first examples of how to approach macroscopic properties of matter, such as magnetism and dielectric polarization.

Overview

Sections 8.1–8.4 present the essentials on these topics. The reader should become well acquainted with these key tools in modern atomic physics – even though parts of it may seem somewhat strenuous at first sight. These concepts are essential also for molecular physics, as well as for a fundamental understanding of many properties of condensed matter and plasmas. This will be illustrated in Sects. 8.1.6, 8.1.7, 8.1.8, 8.2.10, 8.3 and 8.4.2 for a number of selected examples. In these presentations the reader will also find various references to modern developments in atomic physics which will brighten the study of these “classical” themes with currently hot topics – as e.g. “fast and slow” light in Sect. 8.4.4 or, in Sect. 8.5, with a first approach towards the rapidly developing modern research field of matter in intense and ultra-intense laser fields.

8.1

Atoms in a Static Magnetic Field

8.1.1

The General Case

Pioneering work on atoms in magnetic fields was already performed at the end of the 19th century (the second N OBEL prize was awarded to L ORENTZ and Z EEMAN 1902). Here we recall our previous discussion on magnetic moments in magnetic © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7_8

377

378

8


fields and on spin-orbit interaction. According to (6.35), spin-orbit interaction without external magnetic fields may be written in a.u. 2 LS = α ξ(r) L · S = a L · S V 2

(8.1)

and with (6.36) and (6.49) its expectation value (i.e. the FS splitting) is LS = VLS = V

a J (J + 1) − L(L + 1) − S(S + 1) 2

(8.2)

with the fine structure constant α 1/137 and the spin-orbit coupling parameter a = α 2 ξ(r) /2. We recall that ξ(r) = (dV /dr)/r. In the following, we shall use for clarity SI units and rewrite (8.1): LS = (αa0 )2 ξ(r) L · S/22 = a L · S/2 . V

(8.3)

The quantum numbers for orbital angular momentum, spin and total angular momentum are L, S and J , the corresponding projection quantum numbers ML , MS and MJ (also called magnetic quantum numbers). B depends on the magIn an external magnetic field B the interaction energy V netic moments ML and MS of orbital angular momentum L and electron spin S. They define the total magnetic moment MJ = ML + MS . Assuming B z we have to add (6.30) and (6.31) with the g factors gL = 1 (orbit) and ge = 2 (spin), and obtain (μB = e/2me is the B OHR magneton) J · B = −(M L + M S ) · B = μB (gL L B = −M z + ge V Sz )B z + 2 Sz Sz L Jz + J z · B. B = μB B = −M = μB

(8.4) (8.5)

J z = z + 2 Sz and defined the z-component M In the last steps we have used Jz = L J . μB (Jz + Sz )/ of M The quantum numbers for orbital angular momentum, electron spin and total angular momentum are again L, S and J , their projections onto the z-axis ML , MS and MJ , respectively (magnetic quantum numbers). We consider the problem for the completely unperturbed system as solved: 0 |nLML SMS = WnLS |nLML SMS H 0 comprises all electrostatic interactions (including exchange) but no spin-orbit H interaction: spin and orbit are still uncoupled in this 0th order solution, with n, L, S, ML and MS being good quantum numbers. We restrict our considerations to such cases where the perturbation by the magnetic field and/or spin-orbit interaction is small compared to the energy difference between neighbouring states n L S or n L S , so that only the substates of one nLS multiplet contribute to energetic shifts and level splittings.

8.1


379

Table 8.1 Limiting cases for the Z EEMAN effect B field

Perturbation

Optimal basis

Effect

low B a/μB

VB VLS

|LSJ MJ

anomalous Z EEMAN

high B a/μB

VLS VB

|LML SMS

PASCHEN -BACK

Using the binomial formula for L · S in (8.3) and with (8.5), the full Hamiltonian of an atom in a magnetic field becomes: LS B +V +V 2 2 2 L − S Sz Jz + J − +a + μB B 2 2

= H 0 H 0 =H unperturbed

spin-orbit interaction

(8.6)

magnetic interaction.

It is important to realize at this point that neither LML SMS nor LSJ MJ is a set of good quantum numbers for this combined Hamiltonian: as discussed in Sect. 6.2.5 the states |LML SMS do not diagonalize spin-orbit interaction. On the other hand, 2 Sz and J the |LSJ MJ states do not diagonalize the magnetic interaction in (8.6): do not commute since the latter contains also the components Sx and Sy . In contrast, 2 2 2 z + per definition Jz commutes with J as well as with L and S (since Jz = L Sz ) , Jz ] = 0. so that [H Thus, in the general case where magnetic and spin-orbit interactions are equally important, MJ = ML + MS is a good quantum number of the system, while J is not a good quantum number. If we want to treat the problem in perturbation theory we have to choose in 0th order the most appropriate coupling scheme, |LML SMS or |LSJ MJ – depending on which approximately describes the states best. This depends on the magnitude of the magnetic field as summarized in Table 8.1. The two limiting cases for very small and very large external magnetic field are called anomalous Z EEMAN effect and PASCHEN -BACK effect, respectively. Before entering into the details it is commendable to obtain an estimate for the order of magnitude of the magnetic interaction. According to (8.5) it will be in the range of μB B. Without much effort one may obtain in the laboratory magnetic fields up to 1 T to 2 T, using soft iron electromagnets or permanent magnets. With stateof-the-art super-conducting magnets one reaches 10 T to 30 T that are e.g. required for high resolution NMR and EPR spectroscopy. With μB = 5.788 × 10−5 eV T−1 , z + 2 B < 30 T and L Sz / 1 one thus expects splittings up to a maximum of VB < 2 × 10−3 eV = 14 cm−1 ,

(8.7)

corresponding to 10−3 –10−4 of typical electronic excitation energies for valence electrons. According to Table 6.2 this upper limit is much larger than the fine structure splitting of the 2p 2 P state in H or Li (0.37 and 0.36 cm−1 , respectively), and for He and Be we have seen an FS splitting of about 1 cm−1 . However, for the heavier alkalis, from Na upward, the splitting that can be achieved with laboratory magnetic

380

8


fields typically remains below FS splitting in the first excited states. To fully treat lower and higher Z elements and states with low as well as high principle quantum numbers n we have to consider both cases characterized in Table 8.1.

8.1.2

ZEEMAN Effect in Low Fields

In practice the case of low magnetic fields, B a/μB , is particularly impor0 + V LS whose eigenstates tant. In 0th order we start now with a Hamiltonian H |J MJ ≡ |LSJ MJ are spin-orbit coupled as described in Sect. 6.2.5. The 2S + 1 levels of a multiplet 2S+1 LJ are characterized by their total angular momenta J , each consisting of 2J + 1 degenerate magnetic substates |LSJ MJ . In an external B according to (8.5), which we treat in 1st order as magnetic field we have to add V a small perturbation. The additional magnetic energy thus becomes B = μB J MJ |Jz + VB = V Sz |J MJ B μB J MJ |Jz |J MJ + J MJ | = Sz |J MJ B.

(8.8) (8.9)

We now exploit that J MJ |Jz |J MJ = MJ and obtain

Sz |J MJ J MJ | μB MJ B. VB = 1 + MJ

In the next step we use some angular momentum algebra derived in Appendix C.1.2. By applying the projection theorem in the form (C.18) onto the matrix element of Sz the magnetic energy becomes

S · J |J MJ J MJ | μB MJ B. VB = 1 + J (J + 1)2

(8.10)

2 2 Using the binomial formula this can be further evaluated as 2 S· J = J + S − 2 2 2 ( J − S)2 = J + S − L . Inserting this into the above matrix element the eigen2 values J (J + 1) etc. emerge and one finds for the numerical factor in the square bracket

3J (J + 1) + S(S + 1) − L(L + 1) S · J |J MJ J MJ | . (8.11) = gJ = 1 + 2J (J + 1) J (J + 1)2

This is nothing but the L ANDÉ gJ factor phenomenologically introduced in (1.161). The final result for the magnetic energy of the total angular moment may now be written VB = gJ μB MJ B = −MJ z B,

(8.12)

8.1


381

with MJ z being the expectation value for the z-component of the total magnetic moment of the electron, defined in (8.5). Thus, in a (not to high) magnetic field each J level in a multiplet splits into 2J + 1 equally separated components. We point out here, that the above derivation also holds for multi-electron systems in RUSSEL -S AUNDERS (LS) coupling in low magnetic fields. The key assumption for deriving the gJ factor from the Hamiltonian (8.6) is that the magnetic moments of the electron orbits and spins are strictly proportional to the respective angular momenta, with gL = 1 and ge = 2. This implies that the total orbital angular momentum and the total spin are exactly L = i Li and S = i S i (where i refers to the electrons involved); only after that coupling, the total overall angular momentum J = L + S is composed. Note that assumptions also define the RUSSEL -S AUNDERS coupling scheme.

The Vector Model In Fig. 8.1 this somewhat abstract derivation is illustrated by the vector model as introduced in Sect. 6.2.5. We point out, however, that the logic of this essentially classic model is not completely compelling, i.e. one has to know somehow the correct quantum mechanical result in order to present it graphically. But it helps to make the relation between angular momentum addition and the resulting magnetic moments plausible. Again, the z-axis is taken parallel to B, and one starts with the definition of the total angular momentum: J = L + S. Since LS coupling is assumed to be much stronger than the magnetic interaction, L and S precess fast around J . Consequently, the magnetic moment of J is an averaged quantity MJ = ML + MS . The averaging corresponds to a projection of ML and MS onto the J direction. According to (8.4) the effective interaction with the magnetic field B is derived from this averaged magnetic moment MJ VB = − ML + MS · B = − MJ · B = −MJ z B.

(8.13)

In contrast, J precesses at a much slower frequency around B and the magnetic moments are projected onto the unit vector J /|J |. The magnitude of Fig. 8.1 Vector model for the “anomalous” Z EEMAN effect

z,B

slow

Jz

J

S fast L

ML MJz

MJ MS

382

8


the averaged moments are simply given by the respective scalar products M · J /|J |. In summary, with ML = −μB L/ and MS = −2μB S/ one writes (8.13)

ML · J J MS · J J + VB = − MJ · B = − ·B |J | |J | |J | |J |

(J + S) · J Jz μB L · J + 2S · J J ·B = μB B = gJ μB MJ B. = 2 2 J J Obviously, the ratio of scalar products to J 2 in the squared bracket is the gJ factor. Comparing this semiclassical relation to (8.11) and (8.12) one finds that they are equivalent: one just has to apply again the binomial formula for the scalar product S ·J , and interpret J 2 , S 2 and L2 by the respective quantum mechanical expectation values J (J + 1)2 , etc.

2

: J=2

1

: J=1

1P

MJ = +1 linearly polarized (no emission in z direction)

0 -

+1 0 -1

-1

"normal" ZEEMAN spectrum

(b) equal splitting

1D

MJ = +2 +1 0 -1 -2

gJ = 1

(a)

gJ = 1

Some Examples • Singlet system: We remember the so called ‘normal’ Z EEMAN effect which we have introduced already in Sect. 2.7. The standard experiment is illustrated in Fig. 4.18. We are now able to identify the special situation in which this classically expected splitting into three components is actually observed: the total spin has to be S = 0 so that no fine structure splitting is observed in this case. A typical example is He, where the spins of both electrons s1 = 1/2 and s2 = 1/2 may be composed to a triplet (S = 1) or a singlet (S = 0) system as discussed in detail in Sect. 7.3. In the latter case J = L which according to (8.11) is then characterized by a g factor gJ = 1. Hence, independent of J (or L) all J MJ substates in the singlet system have the same splitting μB B between neighbouring magnetic levels MJ and MJ + 1. In Fig. 8.2 this is illustrated for the example of

observed in xy direction

observed in z direction -

+

+

in z direction right and left circularly polarized

Fig. 8.2 So called ‘normal’ Z EEMAN effect exemplified by a 1 P ↔ 1 D transition (e.g. in the He singlet system). (a) Term scheme. (b) Spectrum (schematic); since gJ is the same for the upper and lower levels only the classical triplet of lines is observed (red: σ components, pink: π components)

8.1


383

Fig. 8.3 Triplet term scheme with allowed E1 transitions and the expected spectrum (schematic) below

MJ = +2 +1 0 -1 -2

3P

2

J=2 gJ = 3/2 3S

+1 0 -1

1

J=1 gJ = 2

σ

σ ν

a 1 P1 ← 1 D2 transition. Since gJ = 1 for all J one observes only three components for all transitions within a singlet system. • Triplet system: This is the counterpart to the singlet case just discussed. Figure 8.3 shows as an example a 3 S1 ↔ 3 P2 triplet transition. According to (8.11) the L ANDÉ g factors are different in the lower and upper state, g(3 S1 ) = 2 and g(3 P2 ) = 3/2, respectively, so that in the magnetic field all together nine lines of different frequency are observed. This is indicated below the term scheme in Fig. 8.3. • Doublet system: For S = 1/2 there are 2S + 1 = 2 fine structure levels with total angular momenta J = |L ± 1/2|. The g factor (8.11) is gJ =

J + 1/2 . L + 1/2

The most important 2 S1/2 ↔ 2 P3/2,1/2 transitions (e.g. the Na D line doublet) have g factors g(2 S1/2 ) = 2, g(2 P1/2 ) = 2/3 or g(2 P3/2 ) = 4/3. This leads to a splitting scheme in the magnetic field as shown in Fig. 8.4(a). Sketched in 2P

3/2

MJ = +3/2 +1/2 -1/2 - 3/2

gJ = 4/3

J = 3/2

(a) 2P

MJ = +1/2 -1/2

gJ = 2/3

1/2

J = 1/2

2S

1/2

+1/2

gJ = 2 J = 1/2

+

6 lines

-

+

-

-1/2

(b)

doublet spectra 2S

1/2 -

2P

3/2

2S

1/2 -

2P

1/2

2P

3/2 -

2D

5/2

2P

3/2 -

2D

3/2

2P

1/2 -

2D

3/2

4 lines

Fig. 8.4 Alkali doublets (a) splitting of the 2 S1/2 and 2 P1/2,3/2 states in a magnetic field with allowed E1 transitions (e.g. for the Na D1 and D2 lines). (b) corresponding doublet spectra (schematic) and more examples

384

8

σ


σ

Fig. 8.5 Stick spectrum (calculated) of transitions between magnetically split sublevels in a septet 7 S ← 7 P . This could e.g. be in Cr I between the configurations 3d 5 ( 6 S)4s − 3d 5 ( 6 S)4p) at a 3 4 wavelength 425.4 nm. Pink the π , red the σ ± components detected in the xy plane

Fig. 8.4(b) are the corresponding spectra. They may turn out to be rather complex. For reference we summarize the whole fine structure splitting with (8.8) and (8.2) explicitly for the example 2 P3/2,1/2 . In a low magnetic field with g3/2 = 4/3 and g1/2 = 2/3 we obtain by combining (8.8) and (6.62): * 4 μB BMJ + a2 for J = 32 , MJ = − 32 , − 12 , 12 , 32 (8.14) VB + VLS = 32 for J = 12 , MJ = ± 12 . 3 μB BMJ − a • Septet: For heavier atoms one also observes lines with high multiplicity. In Fig. 8.5 we just show as one arbitrary example the characteristic splitting of a line in the Cr I spectrum. (This calculation just demonstrates how complex such spectra can be, without going into detail for chromium which has a rather complicated electronic structure). We finally note, that the magnetic level splitting – as complicated as it may appear from the above examples – often serves as an important tool for revealing term configurations in the spectra of complex atoms and molecules.

Line Strengths For a quantitative evaluation of spectra one needs to calculate the transition probabilities between individual magnetic sublevels J MJ ↔ J MJ splitted in the magnetic field. These are easily obtained from (4.118) and (4.124) for spontaneous emission and absorption, respectively. One simply has to evaluate the corresponding 3j symbols for MJ = −q. The relative strengths of different lines J ↔ J within one multiplet nLS ↔ n L S require evaluation of (6.70), i.e. of the corresponding 6j symbols. With the tools communicated in Appendixes B.2 and B.3 this can be done without problems. Several Java applets for computing 3j , 6j and 9j symbols are available in the Internet (e.g. S VEN G ATO R EDSUN 2004).

8.1.3

PASCHEN-BACK Effect

The case of very high magnetic fields B a/μB can be of relevance for light atoms such as H, He or Li, as well as for all highly excited states since FS interaction decreases with the principle quantum number as 1/n3 (see (6.66)). Also in the extreme magnetic fields of some stellar atmospheres this case has to be considered.

8.1


385

Fig. 8.6 Vector model for Z EEMAN effect in high magnetic fields

z,B Sz Lz

MS

S L

ML

In 0th order we completely neglect now the spin-orbit coupling and start with the Hamiltonian: 0 + V B P B = H H z + 2 0 + μB (L =H Sz )B.

(8.15)

Then L and S are decoupled, and this situation is called the PASCHEN -BACK effect. z and In this approximation L Sz as well as Jz are well determined and commute with H . Thus LSML MS are good quantum numbers as well as MJ = ML + MS . In the vector model one may visualize this as illustrated in Fig. 8.6: both angular momenta precess around the external magnetic field B. We may compare this directly with Fig. 8.1, where the corresponding situation for low magnetic field is displayed (in both figures L = 1, S = 1, MJ = 3/2 is assumed and presented to scale). We thus choose the uncoupled states, short |ML MS , as basis and obtain with (8.15) z + 2 B |ML MS = μB B(L V Sz )|ML MS = μB B(ML + 2MS )|ML MS , and the splitting in the magnetic field is simply given by VB = μB B(ML + 2MS ).

(8.16)

This is an exact solution of the problem if spin-orbit interaction can be completely neglected in comparison to the magnetic splitting. Schematically this is summarized in Fig. 8.7 for 2 P3/2, 1/2 and 2 S1/2 states. In addition to the quantum numbers ML , MS and MJ = ML + MJ also the z-component υ = Mz /μB = ML + 2MS of the magnetic moment is displayed (in units of the B OHR magneton μB ) which determines the splitting. Note that the level characterized with υ = 0 is doubly degenerate since it may be composed in two different ways from ML and MS . Thus, the number of states remain the same as in the low field case, i.e. (2 × (3/2) + 1) + (2 × (1/2) + 1) = 6 for the 2 P3/2, 1/2 states. Conversely, MJ = ±1/2 refers to two levels with zero splitting as well as to splittings ±: albeit Jz is a good quantum number, it does not uniquely characterize the states.

386

8 ML MS

MJ 2P

3/2

2P

1/2

2S

1/2

+3/2 +1/2


υ

+1 +1/2 2 0 -1 +1 0

-1/2 +1/2 -1/2 - 3/2

+1/2 1 +1/2 0 -1/2 -1/2 -1

-1 -1/2 -2

0 +1/2 1

+1/2

without -1/2 σ+ π B field

σ-

0 -1/2 -1

Fig. 8.7 PASCHEN -BACK effect: splitting and E1 transitions in a high magnetic field, illustrated for the example 2 P3/2, 1/2 ↔ 2 S1/2 (cf. weak B field Fig. 8.4). One sees the classical line triplet as in the ‘normal’ Z EEMAN effect. Shown are the magnetic quantum numbers MJ = ML + MS as well as the projection of the magnetic moment onto the B or z-axis, υ = ML + 2MS , that determines the splitting. Different energies may belong to the same MJ , but one specific energy may also be realized for different MJ

In order to identify allowed optical transitions we have to account for the fact that L and S are now completely decoupled. And since the E1 dipole operator acts only on the orbital component L of the states and not onto the spin S, the selection rules for E1 transitions in a high magnetic field are

ML = 0, ±1,

MS = 0 and L = ±1

(8.17)

as indicated in Fig. 8.7. Since in the upper and lower level the splitting equally depends only on υ = ML + 2MS , six different transitions are possible but only three different frequencies are observed: in high magnetic fields one thus observes the classical ‘normal’ Z EEMAN triplet.

8.1.4

Do Angular Momenta Actually Precess?

Before generalizing the above considerations we want to address the conceptually important question whether the magnetic moments (and hence the angular momenta) really precess in a magnetic field, be it in an external field or in the internal magnetic field that causes spin-orbit coupling. In other words, we want to understand whether the suggestive images used in the vector model in Fig. 8.1 and Fig. 8.6 represent reality or, alternatively, only reflect the probability to find the angular momenta pointing into a specific direction. To answer the question let us consider just the spin of an electron in an isolated 2S 1/2 atom (L = ML = 0), and let us switch on an external field B z at time t = 0. Let the spin orientation in space at this moment be given by its polar and azimuthal

8.1


387

angles, θ0 ϕ0 . This may e.g. be achieved by a passing the atom through a S TERN G ERLACH magnet whose axes are aligned correspondingly in respect of B. This particular spin state prepared at t = 0, is described according to (E.16) by

ϕ0 1 1 θ0 ϕ0 1 1 θ0 + sin exp i − , (8.18) |χ = cos exp −i 2 2 2 2 2 2 2 2 in the basis of the spin states |SMS = | 12 ± 12 . In the magnetic field both spin components will change with time according to (2.16) as ∝ exp(−VB t/) corresponding to their energies VB in the magnetic field. With (8.16) VB = 2μB BMS and we obtain the spin state as a function of time:

χ(t) = cos θ0 exp −i ϕ0 exp −i μB B t 1 1 2 2 2 2

ϕ0 μB B 1 1 θ0 exp i t − . + sin exp i 2 2 2 2 This may also be written as

χ(t) = cos θ0 exp −i ϕ(t) 1 1 + sin θ0 exp i ϕ(t) 1 − 1 , 2 2 2 2 2 2 2 2 where ϕ(t) = ϕ0 +

2μB B t = ϕ0 + ωj t

(8.19)

with the (generalized) L ARMOR frequency ωj for the electron spin as defined in (1.162). This clearly shows that the azimuthal angle of the spin grows with ωj t while the polar angle remains unchanged: the spin thus precesses indeed with the L ARMOR frequency around the axis of the magnetic field – in fact just in the direction indicated in Fig. 8.1 and Fig. 8.6, the angular precession frequency ωj being independent of the polar angle θ . This result is in accordance with the classical model (Sect. 1.9.2, see in particular Fig. 1.36) of a spinning top with a magnetic moment onto which the external magnetic field exerts a torque – apart from the factor g ∼ = 2, which is needed to describe the intrinsic magnetic moment of the electron. We also recall at this point that it is this very precession with the L ARMOR frequency that is used for the ultra precise determination of the electron magnetic moment anomaly. As explained in Sect. 6.6 one essentially exploits the difference between L ARMOR frequency and cyclotron frequency to determine the very small deviation of the electron L ANDÉ factor from g = 2. To develop some feeling for the relevant time scales of this precession we consider a reasonably high field B = 0.5 T. In that case the L ARMOR precession time is h/2μB B = 71.4 ps. The inner atomic field which the electron spin ‘sees’ in the 2p state of atomic hydrogen due to the angular momentum is of the same order of magnitude (by dividing the spin-orbit splitting according to (6.66) by μB one estimates 0.26 T). However, as the splitting is proportional to Z 4 /n3 , for high Z atoms the magnetic field acting on the electron spin at the position of the nucleus may easily

388

8


become 30 T or even more – comparable to the highest fields which can be generated by supra conducting magnets. The precession times are reduced correspondingly. In the theory of magnetism one occasionally uses the term “exchange fields”, in particular in the context of ferromagnetism. To derive such (hypothetical) exchange fields one simply divides the exchange interaction – which is responsible for ferromagnetism – by μB , as we have done in the present context. The numerics of this procedure may lead to “fields” that are orders of magnitude larger than those discussed above. Of course they are just a mathematical construct! As we have learned in (7.49) in the context of the helium atom, exchange interaction is of purely electrostatic nature, enforced by anti-symmetrization of the electronic wave function, and definitively not generated by magnetic fields. The example of He illustrates very clearly that exchange interaction may be on the order of magnitude of an eV, in comparison to which spin-orbit splitting (Fig. 7.6) is minute.

8.1.5

In Between Low and High Magnetic Field

In the preceding sections we have approached the splitting of energy levels in an external magnetic field from two different limits: for spin-orbit coupling VLS being large compared to the interaction of the atom’s magnetic moment with the external magnetic field, VB , we started with LS coupled states |n(LS)J MJ as zero order approximation. In contrast, if VB is large compared to VLS it was appropriate to start from the uncoupled |nLML SMS basis and add VLS later on as a small perturbation. In between these two limits, i.e. if both interactions are of similar magnitude we cannot avoid a full diagonalization of Hamiltonian (8.6). We write =H 0 + H BLS H B + V LS BLS = V H

with a μB (Lz + 2 = 2 Sz )B. L · S+

(8.20)

Considering the unperturbed system as solved in the uncoupled basis |nLML SMS 0 |nLML SMS = WnLS |nLML SMS , H and assuming that only one principle quantum number n contributes, a linear combination of uncoupled states cML MS |LML SMS (8.21) |LSυMJ = ML MS

BLS are the is the obvious ansatz for the perturbed system. The eigenvalues of H sought-after splittings Vυ,MJ (B) of the energy levels in the magnetic field: BLS |LSυMJ = Vυ,MJ (B)|LSυMJ . H

(8.22)

z + Sz : Note that |LSυMJ are also eigenstates of Jz = L Jz |LSυMJ = MJ |LSυMJ

with MJ = ML + MS .

(8.23)

8.1


Fig. 8.8 Scheme of the transition from low to high B field. The level splittings are not drawn to scale. The dashed horizontal line indicates the energetic position of a hypothetical nL level without spin-orbit interaction and without external magnetic field

389

υ =ML+2MS ML MS

gJ = 4/3 2P

nP

MJ 3/2 1/2 - 1/2 - 3/2 1/2 - 1/2

3/2

a/2 a 2P

1/2

gJ = 2/3 unperturbed no field

anomalous ZEEMAN effect

2

+1 +1/2

1

0 +1/2

a/2 0 - 1 +1/2 0 +1 - 1/2 -1

0 - 1/2

- 2 - 1 - 1/2 PASCHEN-BACK effect

The magnetic quantum number MJ by itself does not yet fully specify an eigenstate of the total system. Thus, we use the additional parameter υ which is chosen such that in the limit of high fields υ = ML + 2MS = MJ + MS . Hence it corresponds z /μB of the magnetic moment onto the z-axis. This has to the projection υ = M already been exemplified for a 2 P3/2 state in Fig. 8.7 (see υ labels on the right). Note that there may be several solutions for each value MJ , as well as several MJ values for one specific υ. In detail Fig. 8.8 illustrates schematically the connection between quantum numbers and splittings in the transition from low to high B field: the states that have already been identified in Fig. 8.7 for the high field case are connected with the low field situation such that MJ = ML + MS . This is, however, not yet an unambiguous rule for the change from low to high field. In addition, we have applied the so called non-crossing rule according to which states with equal MJ must not cross! The latter rule emerges from the full treatment of the problem: we have to diagonalize the combined Hamiltonian (8.20) for magnetic and spin-orbit interaction by the expansion coefficients cML MS for the states |LSυMJ according to (8.21). BLS in matrix form: Using (8.16) we write H BLS |LML SMS (8.24) LML SMS H a = μB (ML + 2MS )B × δML ML δMS MS + 2 LML SMS L · S|LML SMS The matrix elements of L · S are derived in Appendix C.3.2. Specifically we use (C.59)–(C.62) and the important result that the off-diagonal matrix elements are non-zero only for MJ = ML + MS = ML + MS = MJ : BLS |LML SMS = · · · × δ(M +M )(M +M ) . LML SMS H (8.25) L S L S BLS matrix is thus very simple. Only states for The structure emerging for the H which MJ = MJ interact.

390

8


BLS for a 2 P state in the uncoupled |LML SMS basis Table 8.2 Matrix elements of H |1 12

|0 12

υ

2

1

0

0

−1

−2

MJ

3/2

1/2

1/2

−1/2

−1/2

−3/2

1 12 |

2μB B + a/2

0

0

0

0

0

0

0

0

1 − 12 |

0

μB B √ a/ 2

√ a/ 2

0

0 12 |

−a/2

0

0

−1 12 |

0

0

0

0 − 12 |

0

0

0

−a/2 √ a/ 2

√ a/ 2

0

−μB B

0

−1 − 12 |

0

0

0

0

0

−2μB B + a/2

ML MS

|1 − 12

|−1 12

|0 − 12

|−1 − 12

0

This H AMILTON matrix may easily be diagonalized. The ansatz (8.21) leads to the following system of linear equations ⎞ ⎛ c−L,−S ' ( ⎜ c−L+1,−S ⎟ ⎟ = 0, BLS − W LML SMS H (8.26) 1|LML SMS × ⎜ ⎠ ⎝ ... cL,S BLS − W with {. . . } representing the matrix corresponding to H 1. The eigenenergies W in the general case are then obtained from solving the secular equation ' ( BLS − W det ML , MS H 1|ML , MS = 0. (8.27) We want to discuss this in some more detail for the important case of a 2 P3/2,1/2 doublet, for which Table 8.2 represents (8.24). It has been obtained using Table C.1. Obviously, the secular equation (8.27) factorizes in this case into four rather convenient expressions: 2μB B + a/2 − W = 0 √ μB B − W a/ 2 √ =0 a/ 2 −a/2 − W √ −a/2 − W a/ 2 √ =0 a/ 2 −μB B − W −2μB B + a/2 − W = 0. These have six solutions Wi = Vυ,MJ (B) that are the sought-after energy splittings in the B field. With x = μB B/a they may be written: V2,3/2 /a = 2x +

1 2

(8.28)

2 1 3 2 1 4 x V1,1/2 /a = x − + 1+ x+ 2 4 4 9 3

(8.29)

8.1

Atoms in a Static Magnetic Field Vυ, M / a J

5

2P

3/2

391

-1/2

1/2

1/2

-5

2 1

1/2

0 2P

υ =ML+2MS

MJ = 3/2

-1/2 - 3/2

5

0 0

10 μB B / a

-1 -2

Fig. 8.9 Transition from a low to a high magnetic field B for the example of a 2 P3/2,1/2 system. Plotted as full red lines are the splittings VυMJ of the levels as a function of the field strength B. The parameters MJ = ML + MS and υ = ML + 2MS characterize the 6 states (see text). Dashed black (red) lines show the extrapolation in the case of a low (high) field. Two avoided crossings are marked by pink circles. The energy splitting VυMJ is given in units of the fine structure parameters a, the field strength B in units of a/μB

2 1 3 2 1 4 x V0,1/2 /a = x − − 1+ x+ 2 4 4 9 3 2 2 1 3 1 4 x V0,−1/2 /a = − x − + 1− x + 2 4 4 9 3 2 1 3 2 1 4 x V−1,−1/2 /a = − x − − 1− x + 2 4 4 9 3 1 V−2,−3/2 /a = −2x + . 2

(8.30)

(8.31)

(8.32) (8.33)

BLS is plotted to scale in Fig. 8.9. In a high The result of this diagonalization of H field the splitting is proportional to υB = (ML + 2MS )B, in a low field to gJ MJ B. An important consequence of the fact that states with equal MJ couple (as expressed by nonvanishing matrix elements in the Hamiltonian Table 8.2) is clearly seen in Fig. 8.9: There are no curve crossings for states with equal MJ = ML + MS . This implies a repulsion of the corresponding terms (here MJ = ±1/2). One sees that the linear splitting Vυ,MJ corresponding to g3/2 = 4/3 and g1/2 = 2/3 in the case of low fields is only valid over a rather small range of the magnetic field

392

8


Table 8.3 Limiting values for the energy splitting Vυ,MJ (B) of a 2 P3/2,1/2 doublet in very weak (left) and very high (right) magnetic field B, respectively. The levels are determined by the quantum numbers J and MJ in the weak field, and by υ = MJ + 2MS and MJ in the high field limit Weak field

Strong field

J

MJ

BμB a

BμB a

υ

MJ

3/2

3/2

2μB B + a/2

2μB B + a/2

2

3/2

3/2

1/2

(2/3)BμB + a/2

BμB

1

1/2

1/2

1/2

(1/3)BμB − a

−a/2

0

1/2

1/2

−1/2

−(1/3)BμB − a

−a/2

0

−1/2

3/2

−1/2

−(2/3)BμB + a/2

−BμB

−1

−1/2

3/2

−3/2

−2μB B + a/2

−2μB B + a/2

−2

−3/2

since it would lead to two crossings (marked with pink circles) which are avoided. While theses states with MJ = ±1/2 belonging to the 2 P1/2 and 2 P3/2 doublet levels strongly interact, the MJ = ±3/2 states of the 2 P3/2 term are not at all influenced by the 2 P1/2 level – the latter simply has no MJ = ±3/2 component. Thus the splitting of these terms is – at all field strengths – always linear in B: V±2,±3/2 = ±2μB B + a/2. With the eigenvalues (8.28)–(8.33) one may now solve the system of equations (8.26) for the coefficients cML MS which according to (8.21) determine the respective states |LSυMJ . In the case of small B fields these will approach the corresponding C LEBSCH -G ORDAN coefficients, for large fields, however, one of the |LML SMS states will dominate the linear superposition, in fact the state with ML = 2MJ − υ and MS = υ − MJ . We refrain from entering into the details of this somewhat tedious but trivial computation. However, it is instructive to verify that the general solutions (8.28)–(8.33) converge to the limiting cases for small and large fields as previously derived in Sect. 8.1.2 and 8.1.3, respectively. Table 8.3 summarizes the results of expanding the roots for BμB a and BμB a into a power series of x = BμB /a and 1/x = a/BμB ; the states are characterized in these limits by the quantum numbers J , MJ and υ, MJ , respectively. Comparing the three left columns in Table 8.3 with (8.14) we see that indeed the dependence of the term energies on the magnetic field is exactly reproduced. The three right most columns agree fully with (8.16) for the PASCHEN -BACK effect if one neglects the fine structure term for very high fields. The remaining small shifts of the terms energies are indicated schematically in Fig. 8.8 and Fig. 8.9 – most evident for the |LSυMJ = |1 1/2 0 1/2 and |1 1/2 0 −1/2 states.

8.1.6

Avoided Crossings

Problems of the type just discussed – how do degenerate or nearly degenerate energy levels of quantum systems change under the influence of external or internal fields (magnetic or electric) – is of quite general importance. It was first treated by B REIT and R ABI (1931) for the splitting of hyperfine levels in magnetic fields (so

8.1


393

called B REIT-R ABI formula, Eq. (9.53)). Prominent examples are found in magnetic resonance spectroscopy, i.e. in electron spin resonance (EPR) and nuclear spin resonance (NMR) which will be treated in some detail in Chap. 9. However, similar problems occur also in different context – e.g. in molecular bond formation or in atomic collision processes as we shall see in Vol. 2. Whenever two or more states, originally separated in energy, are exposed to an interaction that may become larger than their initial separation, one may in principle expect crossings of the energies as a function of the perturbation. Such a perturbation induced energy degeneracy warrants detailed inspection. We treat the problem for the general case, choosing, however, the most simple 0 the unperturbed situation of only two relevant states involved, |1 and |2 . Be H Hamiltonian and V (q) the perturbation as a function of a characteristic experimental parameter q (e.g. the strength of an electric field, the distance of two atoms and so on). Assume that in 0th order 0 1(0) = W (0) 1(0) and H 0 2(0) = W (0) 2(0) . H 1 2 We seek for eigenstates |a and eigenenergies Wa of the perturbed system: 0 + V (q) |a = Wa |a where a = 1 or 2 H 0 + V (q) − Wa |a = Wa(0) + V (q) − Wa |a = 0. or H By multiplication from the left with b| (where b = 1 or 2) one obtains a system of linear equations / 0 (0) W1 − W + V11 V12 c1 =0 (8.34) (0) c2 V∗ W − W + V22 12

2

for the expansion coefficients cab of the eigenstates: |1 = c11 1(0) + c12 2(0) and |2 = c21 1(0) + c22 2(0) . Solutions exist if and only if the secular equation holds: (0) V12 W1 − W + V11 =0 (0) ∗ V12 W2 − W + V22 (0) W1 − W + V11 W2(0) − W + V22 − |V12 |2 = 0.

(8.35)

The corresponding energies W1 and W2 thus are: (0)

W1,2 =

(0)

W2 + W1 V11 + V22 + 2 2 2 1 (0) (0) ± W1 − W2 + V11 − V22 + 4|V12 |2 . 2

(8.36)

394

8

W2


ΔW = 2V12

W1 qx perturbation parameter q

Fig. 8.10 Avoided crossing. As a function of a perturbation parameter q the energies W1 and W2 of two states are plotted: with (full lines) and without (dashed) interaction, respectively, i.e. with finite or negligible off-diagonal matrix element V12 (q). The perturbation parameter q may be an external magnetic or electric field, or e.g. the distance between two atoms ∗ = 0, the reIf the off-diagonal terms of the perturbation matrix vanish, V12 = V21 sults correspond to those obtained from perturbation theory in 1st order (for two non-degenerate states): (1)

(0)

W1 = W1 + V11

(1)

(0)

and W2 = W2 + V22 .

Thus in this situation (V12 = 0) and depending on the behaviour of V11 (q) and V22 (q) as a function of a perturbation parameter q, curve crossings are in principle possible, i.e. W (1) (q) = W1(0) − W2(0) + V11 − V22 = 0 may occur at a certain value of q = qx . If, however, the two states interact and V12 (qx ) = 0, the curve crossing is avoided as illustrated in Fig. 8.10. According to (8.36) the splitting at the approximate crossing point qx is

W = W1 − W2 = 2|V12 |.

(8.37)

We summarize: The potential energy curves of two states |a and |b do not cross as a function of a characteristic perturbation parameter q, if the interaction potential at the approximate crossing point is finite, i.e. if Vab = 0. Such a situation is called avoided crossing. Specifically, for the change of atomic energies in a magnetic field we have seen BLS do not vanish in Sect. 8.1.5 that the matrix elements (8.24) of the perturbation H if MJ = ML + MS = ML + MS = MJ : states with equal ML + MS = MJ couple. Hence, such states cannot cross as illustrated in Fig. 8.9 (non-crossing rule).

8.1.7

Paramagnetism

The magnetic properties of matter are determined by the magnetic dipole moments of its constituents (atoms and molecules). We give here a brief outline of how to

8.1


395

relate macroscopic quantities (volume magnetic susceptibility) to the microscopic magnetic properties which we have treated above in the present chapter. According to the basics of electrodynamics, the magnetic B field within a macroscopic material (e.g. an ensemble of atoms) is given by B = μ0 (H + M) = μ0 H + χM μ0 H = μr μ0 H .

(8.38)

H is the magnetic field strength in the material and M the so called magnetization, i.e. the magnetic moment per unit volume of this material (both measured in A m−1 ) with μ = μr μ0 being the magnetic permeability (μ0 = 4π × 10−7 N A−2 ), μr the relative magnetic permeability and χM = μr − 1 the magnetic susceptibility (both dimensionless). Occasionally one also finds χMmol given per mol (so called molar susceptibility). Thus, the present task is to express the magnetization M = (μr − 1)H = χM H = χM B/μ0

(8.39)

in microscopic terms. As long as we are interested in paramagnetism we may exploit the fact that χM is typically a very small quantity, so that to 0th order B ∼ = μ0 H . For of a single atom in an |J MJ state, MJ z = −gJ μB MJ is the effective magnetic moment, projected onto the direction of the external field – according to (8.12) valid for not too high magnetic fields. For the whole ensemble one has to average MJ z over the population of the magnetic substates that are populated in a thermal equilibrium (essentially) according to the B OLTZMANN distribution (1.54). With the magnetic energy VB (MJ ) of an atom in an |J MJ state we obtain for the average: MJ MJ z exp(−VB (MJ )/kB T ) . (8.40) MJ z = exp(−VB (MJ )/kB T ) The partition function in the denominator ensures correct normalization. At not too low temperatures T the energy (8.12) of an individual magnet VB (MJ ) is very small compared to the average thermal energy kB T – at room temperature we typically have VB (MJ ) 10−5 eV kB T 0.010 eV. Thus the partition function is 1 and the term in the nominator of (8.40) may be expanded as

VB (MJ ) MJ z exp −VB (MJ )/kB T −gJ μB MJ 1 − kB T = −gJ μB MJ + (μB gJ MJ )2

B . kB T

We insert this into (8.40) and see that positive and negative values of MJ compensate each other in 0th order while the 1st order terms in the lead to a finite positive contribution ∝ B/T . We finally obtain for the magnetization M = NMJ z =

C B, T

396

8


where N is the particle density of the material (dimension L−3 ). With (8.39) the well known C C URIE law χM = (8.41) T for paramagnetism follows, χM being the magnetic susceptibility. For not too dense matter the C URIE constant C may be easily evaluated for specific cases of L, S and J . If one extends the parameter range to very low values of the temperature T or to very high magnetic fields B the full sum (8.40) has to be used, the so called L ANGEVIN function, with the variable B/T . It leads to a saturation of the magnetization at large B/T . In the general case one has, however, also to account for the fact that a more rigorous treatment gives χM = ∂M/∂H . As a simple exercise the reader may evaluate in detail the case L = 0, S = J = 1/2. Paramagnetism requires a nonvanishing total angular momentum J of the material under investigation. In the electronic ground state in which we normally find matter the angular momentum is often zero and the paramagnetism of the atoms or molecules studied is determined by their electron spin. As it turns out, not many materials in nature happen to have unsaturated electron spins. Typical examples are the alkali metals and a few other metals (not the alkaline earth metals, however). Among the molecules are only a very few: one important example for molecular paramagnetism is oxygen, O2 in its 3 Σ1 ground state, as we shall see in Sect. 3.6.9, Vol. 2.

8.1.8

Diamagnetism

Diamagnetism is a general property of all matter. It is relevant mainly if the atoms or molecules under study do not have a permanent dipole moment. To understand diamagnetism we have to use the H AMILTON operator for the interaction with the electromagnetic field in its exact form expressed by the vector potential A of the field as explained in Appendix H.1. In summary, one replaces p → p + eA so that p2 LS = + V (r) + V H 2me in an external field becomes = H

p2 a L · S eA · p e2 2 + V (r) + + + A 2me me 2me 2 1 23 4 1 23 4 1 23 4 1 23 4 1 23 4 potential, kin. energy,

which correspond to

spin-orbit,

paramagn.,

diamagn.

etc. χ >0

etc. χ < 0.

(8.42)

As explicated in Sect. 4, Sect. 5.4 and Appendix H, the term eA · p /me leads J , respectively, to electric and magnetic dipole interaction, er · E and gJ μB B · which are responsible for electric polarizability (Sect. 8.2.10) and paramagnetism

8.1


397

(Sect. 8.1.7). It is also the cause for electric quadrupole interaction (Sect. 5.4). In contrast, the quadratic term in (8.42) leads to the so called ponderomotive potential in intense fields (to be discussed in Sect. 8.5.1), as well as to diamagnetism as we shall show now. In comparison to paramagnetism (as far as present in a material) diamagnetism is a very small effect, counteracting the external magnetic field according to L ENZ’s law. From a classical point of view it describes the deceleration of the electron orbits around the nucleus as a consequence of the external B field: the magnetic moment of one atom induced by the latter is MJ z = χ )M H = χ )M B/μ0

(8.43)

with χ )M being the magnetic susceptibility per particle (dimension L3 ). Since MJ z itself depends on B we have to read here the magnetic energy (6.29) as a differential equation, i.e. it builds up as the magnetic field increases: dVdmag = −MJ z dB = − Vdmag = −

)M 2 1χ B 2 μ0

or

χ )M B dB μ0

so that

χ )M = −μ0

1 dVdmag . B dB

(8.44)

Let us now derive χ )M from the quadratic term in (8.42). For a static B field according to (H.12) the vector potential is A = (B × r)/2. Thus, the diamagnetic component of the Hamiltonian is 2 2 dmag = e 1 (B × r) · (B × r) = e B 2 r 2 sin2 θ, V 2me 4 8me

with θ being the angle between B and r. For a spherically symmetric atom one averages over all solid angles using sin2 θ = 2/3 and obtains in 1st order perturbation theory the diamagnetic energy dmag = V

e2 2 2 2 e2 B r sin θ = B2 r2 8me 12me

(8.45)

with r 2 being the expectation value of the squared electron distance from the atomic centre. The diamagnetic energy (8.45) thus obtained has to be compared to (8.44). We finally obtain with μ0 = 1/(ε0 c2 ), the fine structure constant α and the atomic unit of length a0 : 2 2 r r e2 2 2π (8.46) χ )M = −μ0 r = − a03 α 2 2 = −1.12 × 10−4 a03 2 . 6me 3 a0 a0 The diamagnetic susceptibility is thus negative and extremely small – as expected. For atoms in their ground state the expectation value of r 2 is on the order of magnitude of a02 . (In the case of atomic hydrogen we may use the closed expressions according to Sect. 2.6.10.)

398

8


With the particle density N or alternatively the AVOGADRO constant NA the macroscopic quantities per volume or per mol are χM = χ )M N

and χMmol = χ )M NA ,

(8.47)

respectively. With this and (8.47) even for solid materials in the most dense closepacked arrangement of spheres the macroscopic diamagnetic susceptibility is only on the order of χM −0.83 × 10−4 . We mention in passing that while the value of χ )M increases with r 2 the particle density decreases with 1/ r 3 so that the macroscopic χM decreases with the size of the atoms. We finally note that in standard atomic spectroscopy the diamagnetic energy is minute and does not play a significant role. However, this is different when high lying RYDBERG states are studied. Since the radius of these atoms increases with n2 and r 2 a02 n4 the ratio of the diamagnetic term (8.45) to the normal magnetic field splitting (8.8) is dmag V B e Ba 2 n4 ∼ 10−6 . . . 10−7 n4 . VB 6gJ 0 T

(8.48)

For n = 34 at only 1 T it is already ∼1. Consequently, spectroscopy of RYDBERG atoms in magnetic fields cannot neglect the diamagnetic term. Section summary

• The interaction energy of an electron with an external magnetic field is deB = μB (L z + 2 scribed by V Sz )B/. LS = a B ) one • In low magnetic fields B (spin orbit interaction V L · S/2 V B as a small perturbation. The enuses the |(LS)J MJ basis and considers V ergy is then gJ μB MJ B with the L ANDÉ factor gJ in (8.11). • In the limit of high magnetic fields, i.e. for very low Z atoms, extremely large B in the uncoupled B fields, or high principle quantum numbers n, on treats V LS is a small perturbation. In this case, L, S, ML |LML SMS scheme, while V and MS are good quantum numbers as well as MJ = ML + MS . In addition, υ = ML + 2MS has to be introduced. • Vector models are useful to visualize the projection of magnetic moments onto each other and onto the external field. They are based on the precession of magnetic moments at L ARMOR frequency, which can be rationalized in quantum mechanical terms. • By diagonalizing the full Hamiltonian (8.6), its matrix form being given in Table 8.2, one may also treat the intermediate situation between low and high fields exactly. The energies are given by B REIT-R ABI type formulas, see (8.28)–(8.33). Characteristic are interactions between states with equal MJ which lead to the non-crossing rule for such states. • Avoided crossings are important also in a broader context: they occur if degenerate or quasi-degenerate states are coupled by a non-diagonal interaction which changes as a function of a characteristic parameter.

8.2

Atoms in an Electric Field

399

• From the microscopic energy of atomic dipole moments in a magnetic field one derives the macroscopic magnetic susceptibility χM of matter. Specifically C URIE’s law for the temperature dependence of χM = C/T in paramagnetic materials, and the diamagnetic susceptibility (χM < 0) has been obtained. The latter is very small, but rises ∝ n4 .

8.2


8.2.1

Introduction

The splitting of energy terms in static electric fields, the so called S TARK effect, has been discovered by Johannes S TARK in 1913 (N OBEL prize 1919). In traditional spectroscopy it plays only a minor role since static electric fields available in the laboratory (see Table 8.4) are typically very small compared to the inner atomic fields. The latter are on the order of magnitude Eatom =

Ze V = 5.14 × 1011 × Z . 2 m 4πε0 a0

(8.49)

This may be compared to the electric field strength for electric breakdown in air which is about 10 kV cm−1 = 1 × 106 V m−1 – which is thus far below the electric field strength needed to modify the atomic energies significantly.

8.2.2

Significance

Nevertheless, for quite a number of reasons it is important to discuss this theme thoroughly: 1. Electric fields play a key role in the structure and composition of matter: be it in molecular bonding, the forces in crystal lattices, soft matter or plasmas. The electric fields by which neighbouring atoms and ions interact are essential ingredients for our understanding of the properties of matter. Table 8.4 Typical electric field strengths Example

|E| / V m−1

In an electric power line

10−2

Close to a charged up plastic hair comb

103

Surface of the drum of a photocopier or laser printer

105

Electric breakdown in air

106 5 × 1011

In an H atom on the first B OHR orbit In focused short pulse laser at an intensity of I On the surface of a Uranium nucleus

= 1020 W cm−2

3 × 1013 3 × 1022

400

8


2. As an example, the polarizability of atoms approaching each other is a typical effect to be explained by electric fields. The electron clouds of two neighbouring atoms repel each other, thus forming two interacting dipoles. As we shall see in Sect. 8.3 this allows us to describe the long range potentials by which the atoms attract each other. Corresponding considerations are basic for understanding the formation of molecular bonds. 3. The polarizability of atoms and molecules is of fundamental importance in many areas of physics, specifically so in alternating electromagnetic fields. In the field of an electromagnetic wave polarizability leads to the refraction of light. The dependence of the index of refraction on the frequency (dispersion) can be explained on an atomistic basis as will be shown in Sect. 8.4.2. 4. In Chap. 6 we have documented by way of example that spectral lines may today be measured with a relative accuracy of up to 1013 or 1014 . In such precision measurements electric stray fields of only some V m−1 may influence the results and need to be considered seriously! 5. In modern atomic and molecular physics highly excited RYDBERG states play an important role. Since the radius of the excited states increases with n2 the inner atomic field decreases with n−4 . For n = 100 the inner atomic field is on the order of only some kV cm−1 . With corresponding external fields that may easily be generated in the laboratory the electronic energy levels of RYDBERG atoms and molecules can be influenced significantly. 6. An important field of current research is concerned with oscillating electromagnetic fields at very high intensity, as e.g. generated in the focus of intense ultrashort laser pulses. The amplitude of the electric field E0 is related to the laser intensity I by (4.3): I E0 = 2745 . (8.50) −1 Vm W cm−2 Present technology allows for intensities of up to at least 1020 W cm−2 . Thus, electric fields today available in the laboratory are far beyond inner atomic fields according to (8.49). And presently worldwide several laser systems are built up, to push the limits even further. The vision is to study matter in the laboratory under extreme conditions that otherwise only exist in the interior of stars. Buzz words are highly relativistic plasma dynamics (ion acceleration, nuclear fusion) or new approaches for particle physics (extreme energy densities can lead to particle generation).

8.2.3

Atoms in a Static, Electric Field

A static electric field – which is a polar vector field in contrast to the axial magnetic field – breaks the symmetry of the H AMILTON operator. The “perturbation” by the electric field may be written in 1st order approximation VE (r) = −D · E = er · E

(8.51)

8.2


401

(r) = H (−r). For convenient calculation of the matrix elements we and hence H rewrite (8.51) for Ez

as VE (r) = ezE = er cos θ = eErC10 (θ, ϕ)

(8.52)

making use of (4.75) and of the renormalized spherical harmonics Ckq according to 2 (B.29). Note that the squared orbital angular momentum operator L (which implies differentiation with respect to θ ) does not commute with this perturbation. Hence 2 and L is not any more a good quantum number L does no longer commute with H – in contrast to the situation for an atom in a magnetic field where according to (8.5) 2 z + 2 Sz , which commutes with L but the relevant interaction is proportional to L 2 no longer with J . z However, even with (8.52) included in the Hamiltonian it still commutes with L since C10 (θ, ϕ) = C10 (θ ) and VE (r) does not depend on ϕ, while Lz = i∂/∂ϕ acts only on ϕ, and Sz does not act on the spatial coordinates at all. Hence, ML and MS remain good quantum numbers in the electric field. As we shall see, the interaction matrix elements depend only on |ML |. Hence, it is reasonable to characterize the 2z and |ML |. In the extreme case we expect a superposition of many eigenstates by L orbitals with different L but a constant value of |ML |: a so called hybridization occurs – a phenomenon crucial for the understanding of chemical bonding. In the following considerations we shall concentrate for simplicity on systems with only one active electron (the valence electron). In the general case one has to replace in (8.51) the vector r by the sum r i over all electron coordinates. Typically, for coupled systems the reduction formulas for evaluating the matrix elements are accordingly more complicated.

8.2.4

Basic Considerations about Perturbation Theory

A comparison between the presently discussed perturbation (S TARK effect) with those previously treated is shown in Table 8.5, along with the most important effects related to these interactions. An estimate for the order of magnitude of the S TARK effect is obtained from (8.52) by VE (r) ea0 E, assuming that the angular dependent matrix elements are on the order of 1 and the radial matrix element with r a0 . For the breakdown field strength in air Emax = 106 V m−1 we thus find VE (r) < 5 × 10−5 eV = 0.4 cm−1 . The effect is thus expected to be indeed very small, by comparison to (8.7) even significantly smaller than the Z EEMAN effect under standard laboratory conditions. We distinguish two limiting cases depending on the non-C OULOMB term VnC in the perturbation hierarchy Table 8.5:

402

8


Table 8.5 Perturbation hierarchy for effective one electron atoms: Summary of interactions and consequences (gqn stands here for good quantum number) H like =H C (r) H

Alkali like +VnC (r)

FS splitting +VLS

Z EEMAN effect +VB

S TARK effect +VE (r)

pure C OULOMB Tkin + C/r

electrostatic not ∝ 1/r

spin-orbit ∝ L · S

ext. B field z + 2 μB (L Sz )B

ext. E field er · E

L degeneracy

L degeneracy removed (LS)J MJ coupling

M degeneracy removed , [H Sz ] = 0 J no longer gqn

|M| degeneracy removed 2 , [H L ] = 0 L no longer gqn

• VnC (r) VE (r) : In this case the electric field removes for the first time L degeneracy. This has to be treated by perturbation theory for degenerate states (see Sect. 3.3.4). One finds the so called linear S TARK effect. • VnC (r) VE (r) : L degeneracy is already removed when the E field is switched on. Due to the symmetry of the perturbation potential (z has odd parity) all diagonal matrix elements disappear. Hence, the energy does not change in 1st order perturbation theory and one has to resort to 2nd order for determining the energy shift. Consequently the so called quadratic S TARK effect ∝ E 2 is observed!

8.2.5

Matrix Elements

The perturbation VE by the electric field (8.51) does not act on the spin – in contrast to the B field (8.6). The spin S and its projection MS thus remain conserved. This facilitates the evaluation of the matrix elements γ J M VE (r)|γ J M (8.53) = eE γ J M rC10 (θ )γ J M = eE γ r|γ J M C10 (θ )|J M significantly. For later use we derive these now in detail. All quantum numbers related to the radial wave functions are summarized by γ while J M stands here for all angular momentum quantum numbers. Depending on the strength of the interaction as compared to FS interaction the states are again described most suitably in the uncoupled |LML SMS or in the coupled scheme |(LS)J MJ , respectively. Here L, S and J stand again for the total orbital angular momentum, the totals spin and the total angular momentum of the system, respectively, while the E field is assumed to act only on one active electron with the angular momentum . For the present derivations we shall assume for simplicity L = . However, the following may easily be extended to coupled orbital angular momenta. Obviously, the matrix elements (8.53) are essentially the same which we have already encountered in Chap. 4 in the context of describing E1 transitions induced by linearly polarized light. We thus obtain the same “selection rules” as in Sect. 4.4.

8.2


403

Strong Electric Field We first consider the case of a high electric field in the uncoupled scheme: γ S Ms L M VE |γ SMs LM = eErn n S Ms L M C10 |SMs LM = eErn n L M C10 |LM δS S δMs Ms ∞ Rn (r)Rn (r)r 3 dr with rn n = n r|n =

(8.54) (8.55)

0

and the radial wave function Rn (r). Using (C.29) and (C.30) we obtain

+M √ 1 L L 2L+L γ L M VE |γ LM = eErn n (−1) 2L + 1 −M 0 M 5 (8.56) × L 5C1 L γ L M VE |γ LM = eErn n δM M δL L±1 × (−1)M 2L + 1 (2L + 1)

1 L L L 1 L × , (8.57) −M 0 M 0 0 0 assuming for (8.57) a pure one electron system L = and L = and making use of the 3j symbols symmetries (B.37) and (B.49): Thus, the matrix element is only non-zero for L = L ± 1. Hence, as already mentioned, L is no longer a good quantum number, while the projection quantum number M = M is conserved. In contrast to the magnetic field where the angular momentum (an axial vector) explicitly is included in the perturbation, in the case of an electric field the quantity z (derived from the polar vector r) does not act on the projection of the angular momentum. Using (B.53) for the relevant 3j symbols we finally obtain: (L + 1)2 − M 2 γ (L + 1)M VE |γ LM = eErn n (8.58) (2L + 1)(2L + 3) L2 − M 2 γ (L − 1)M VE |γ LM = eErn n . (8.59) (2L − 1)(2L + 1) The matrix elements only depend on M 2 and thus on the absolute value |M| of the projection quantum number. This is a consequence of the fact that on inversion (thereby changing +M to −M) the sign of VE changes, while at the same time one (and only one) of the spherical harmonics of the wave functions changes since they have different parity due to L = L ± 1. According to (8.58) and (8.59) the influence of the electric field decreases with increasing |M| ≤ L. Thus, as we shall elaborate in Sect. 8.2.7, states with the highest projection quantum number show the weakest S TARK effect.

404

8


2 Again, the matrix elements show that L does not commute with z, hence neither . On the other hand, M is more specific as observable in an electric field. with H Thus, as detailed in Appendix D, it is convenient in such cases to use the real rather than the usual complex basis for angular momentum eigenstates.

Weak Electric Field We now compute the matrix elements for the S TARK effect in the coupled scheme |(LS)J M , i.e. for the case when the electric interaction is small compared to spinorbit interaction, VE VLS . Again the interaction operator VE does not act on the spin of the electron. The expression equivalent to (8.56) for the matrix element is obtained by making use of the tools developed in Sect. C.3.1. From (C.53) one obtains γ L SJ M VE |γ LSJ M = eErn n 2J + 1 (2J + 1) 2L + 1 (2L + 1)

" J 1 J L M−S × δM M δL L±1 × (−1) −M M 0 J

L J

1 S

#

1 L . 0 0 (8.60)

L 0

Specifically with (C.55) for a one electron system with S = 1/2 γ L SJ M VE |γ LSJ M

= eErn n δM M δL L±1 × (−1)M−3/2

J 1 J J J × 2J + 1 (2J + 1) −M M 0 −1/2 1/2

1 0

(8.61)

holds. Explicitly from (B.53) and (B.55) this can simply be written as γ L SJ M VE |γ LSJ M = eErn n δM M δL L±1 ⎧√ (J +1)2 −M 2 ⎪ ⎪ ⎪ ⎪ ⎨ (2J +2) 2J (−1)2M−1 (2J +1) × (−1) 2J (J +1)(2J +2) M ⎪ ⎪ √ ⎪ ⎪ J 2 −M 2 ⎩

2J

(8.62) for J = J + 1, for J = J, and for J = J − 1.

Here too the projection quantum number is conserved (in this case the projection of J ), and only the matrix elements with L = L ± 1 are non-zero. Even the rule that the states with highest projection quantum number |M| are least disturbed remains valid, since a quantitative evaluation of (8.62) shows that the opposite trend for J = J contributes only very little.

8.2


405

Fig. 8.11 Typical energy level positions in atoms; we consider the S TARK shift for the state marked with a black dot

W, n W n +1 Wn

W n −1

8.2.6

Perturbation Series

In the following we specify the S TARK effect in more detail for quasi-one-electron systems. Depending on whether the states under investigation are still essentially1 degenerate (for the H atom, H like atoms, high RYDBERG states) or not (e.g. for the alkali atoms) we have to apply perturbation theory for degenerate states or use the standard perturbation series, respectively. In the latter case energy and wave function of a state |a are changed as a|VE |b 2 and Wa = Wa(0) + a|VE |a + Wa − Wb b=a

a|VE |b (0) ψa = ψa(0) + ψ . Wa − Wb b b=a

According to (8.57) and (8.60) quite generally a|VE |b ∝ δ ±1 , i.e. the diagonal term vanishes. With (8.52) we have |zab |2 r 2 | a|C10 |b |2 ab = |eE|2 , (8.63) Wa − Wa(0) = |eE|2 Wa − Wb Wa − Wb b=a

b=a

and the change of energy depends on the square of the electric field: if the degeneracy is already removed, a quadratic S TARK effect is observed. For effective one electron systems, |a and |b represent |n m or |n sJ M , depending on the coupling case. Only states with equal m or M, respectively and with = ±1 interact (mix). Only the absolute values |m | or |M| of the projection quantum numbers matter.

8.2.7

Quadratic STARK Effect

Considering the typical term positions according to Fig. 8.11, equation (8.63) allows us to make already a few qualitative statements: (a) the S TARK effect will always lower the terms since the atomic level spacing typically decreases with increasing principle quantum number n as sketched in 1 Note that this depends on the precision of the measurement: for very weak electric fields and very high precision even the H atom levels of equal n are already split due to FS interaction.

406 Fig. 8.12 Estimating the relative magnitude of the matrix elements 2pz |C10 |2s (top) and 2px |C10 |2s (bottom). It is obvious that the components of the integral in the upper case overlap much more than in the lower case. The absolute value of the matrix element with the lower projection quantum number |m| = 0, i.e. 2pz |C10 |2s , thus has to be significantly larger

Fig. 8.13 Visualizing why an M = 0 state is more readily polarized than an |M| = 1 state

8

Atoms in External Fields z y x |M|= 0 |s〉

|M|= 0 |pz〉 C10 |M|= 1 |px〉

|M|= 0 |s〉

E

E

z y

+

x |M|= 0

|pz〉

z

+

y x

|M|= 1 |px〉

Fig. 8.11; thus the series in (8.63) contains always many more closely spaced levels for which Wa − Wb < 0 as compared to those levels where the reverse is true; this is particularly pronounced for the ground state; (b) higher lying levels are stronger influenced by the S TARK effect since Wa − Wb decreases with increasing principle quantum number n; (c) within one level states with larger |m | are lowered less – as already mentioned. In the uncoupled case the latter statement is read directly from the matrix elements (8.58) and (8.59). It is valid, however, also in respect of |M| in the coupled case as explicit evaluation of the matrix elements (8.62) shows. One may visualize this with the help of Fig. 8.12 where the components of the matrix elements 2pq |C10 |2s are illustrated. In physical terms positive and negative charges may be displaced by the E field more easily for the |2pz state along the z-axis than for the |2px or |2py state. In the latter case the positive charge would have to be more or less extracted from the negative charge cloud as illustrated in Fig. 8.13. Thus, for the quadratic S TARK effect one expects a characteristic dependence of the energies of different |M| states as a function of the applied electric field E as sketched in Fig. 8.14 for the example of a 2 P3/2,1/2 doublet. Corresponding to the splitting with |M| in emission or absorption spectra one finds in an electric field only two polarization components π and σ : these are linearly polarized (perpendicular to each other) since the electric field does not enforce an orientation onto the system – again in contrast to the situation in a magnetic field where σ + (LHC) and σ − (RHC) light is emitted at two different frequencies.

8.2


407

Fig. 8.14 Quadratic S TARK effect for the example of the Na 3 2 P1/2,3/2 states

W – W (0) 32P

3/2

|m|=3/2

3P |m|=1/2 32P1/2

|m|=1/2

without field

8.2.8

with field

E

Linear STARK Effect

The situation is completely different if the external E field is the dominant interaction which removes degeneracy. Thus, let us assume states of different parity are degenerate without the external E field. This is the case for the H atom and H like ions (as long as FS can be neglected) as well as for extremely high field strengths where the splitting is negligible (e.g. in molecular bond formation between several atoms) but also for highly excited RYDBERG states and high as we shall see in Sect. 8.2.9. For the 1s 2 S ground state of the H atom all matrix elements of VE vanish according to (8.57) or (8.62) because of δL L±1 . The H atom ground state has no linear S TARK effect. However, for the first excited state (with FS neglected) there are four degenerate states, written in the real basis, |2s0 ,

|2pz ,

|2px ,

|2py .

(8.64)

The states |2s0 , |2pz are characterized by M = 0, while |2px and |2py have a projection quantum number |M| = 1. According to (8.57) all diagonal matrix elements disappear, as well as those to different M and M . With (8.58) or (8.59) only two matrix elements are non-zero: 1 2pz |VE |2s0 = 2s0|VE |2pz = √ eEr2s2p . 3

(8.65)

The radial matrix element (8.55) between the states |2s and |2p may be derived√from the radial wave functions of the H atoms (Table 2.2) and is r2s2p = −(3 3/Z)a0 . The H AMILTON matrix is thus ⎛

2s0

(0) ⎜ W ⎜ ⎜ 0 + VE = ⎜ −3eEa0 H ⎜ ⎜ 0 ⎝ 0

2pz

2px

−3eEa0

0

W (0)

0

0

W (0)

0

0

2py

⎞ 0 ⎟ 2s0 ⎟ 2pz 0 ⎟ ⎟ , ⎟ 2px 0 ⎟ ⎠ (0) 2py W

(8.66)

408

8


0 |ψ = W (0) |ψ . The S CHRÖDINGER equation with H 0 + VE − W )|ψ = 0 (H may be solved algebraically. Since only two states couple, this amounts to solving a set of the linear equations for the coefficients cn ,m (0)

W − W −3eEa0 c2s0 = 0, (8.67) c2pz −3eEa0 W (0) − W for which the secular equation is (0) 2 W − W − (3eEa0 )2 = 0. It has two possible solution for the energy values W1 and W2 : W (0) − W1,2 = ±3eEa0

or

W1 = W (0) + 3eEa0

and

(8.68)

W2 = W (0) − 3eEa0 With (8.67) the coefficients c2s0 and c2pz are derived (properly normalized) √ √ (1) c2s0 = −c2pz = 1/ 2 and (2) c2s0 = c2pz = 1/ 2 for W1 and W2 , respectively. In summary, the eigenenergies and eigenstates of the H(n = 2) states, perturbed by an external electric field, are: √ (8.69a) (1) W (0) + 3eEa0 : |2− = |2s0 − |2pz / 2 √ (8.69b) (2) W (0) − 3eEa0 : |2+ = |2s0 + |2pz / 2 (3)

W (0) :

|2px

(8.69c)

(4)

W (0) :

|2py .

(8.69d)

These states correspond to the electronic charge distributions illustrated schematically in Fig. 8.15. The energy Wa in the electric field (including the quadratic S TARK effect which has not been evaluated here explicitly) depends on the field strength E as sketched in Fig. 8.16. We note that the asymmetric charge distribution of the states |2− and |2+ results in a finite dipole moment D at of these special, excited states in the electric field. We may thus explain the raising or lowering of the energies as due to the interaction of this dipole moment with the electric field. From (8.68) one reads the magnitude of the dipole moment directly from the energies: VE = Wa − Wa(0) = ±3eEa0 = D at · E.

(8.70)

Hence, the dipole moment of the H atom in the state |2− and |2+ states is 3ea0 and −3ea0 , respectively.

8.2


Fig. 8.15 Dipole states (so called S TARK states |2± of the excited H atom in an electric field as the sum and difference of |2pz and |2s , respectively

409

z

(|2pz〉 Fig. 8.16 S TARK effect for the H(2s, 2p) states as a function of the field strength. Linear S TARK effect of degenerate states (black dashed lines) and transition to the quadratic S TARK effect (full red lines). On the right the corresponding orbitals are indicated

8.2.9

−

=

+

=

±

|2s〉) / 2 =

|2±〉 E, z

W − W (0)

|2− 〉 x

2s, 2p

E, z

|2− 〉

|2px 〉 x

|2px,y〉 |2+ 〉

E, z

|2+ 〉

x

An example: RYDBERG States of Li

As an experimental example for the S TARK effect we shall discuss here highly excited atomic states (so called RYDBERG atoms). This is a wide field and still subject of present research. We present just one particularly impressive pioneering experiment of Z IMMERMAN et al. (1979). It is considered a benchmark experiment for pertinent theoretical investigations (see e.g. M ENENDEZ et al. 2005) and demonstrates a number of important aspects of the S TARK effect. The experimental setup is relatively simple, as illustrated schematically in Fig. 8.17. A Li atomic beam (see footnote 24 in Chap. 1) is excited by a resonant multi-photon process with three laser frequencies: the steps are 2s → 2p (671 nm), 2p → 3s (813 nm), and finally 3s → 15p (626 nm). At a fixed DC S TARK field, the latter laser is tuned through ca. 100 cm−1 . Excitation of the RYDBERG states is detected by field ionization in a pulsed electric field (HV pulser), applied shortly after the laser pulse. The spectra, observed for a number of electric field strengths, are reproduced in Fig. 8.18 (black, vertical traces with horizontal excitation lines). The red lines in Fig. 8.18 give the theoretically expected energy dependence of the relevant states as a function of the electric field strength. The observed absorption lines follow the theoretical understanding very impressively. There are two sets of states, for M = 0 and |M| = 1, which are excited by light polarized parallel and perpendicular to the E field (π and σ light), respectively. In the previously discussed case of the 2s − 2p

410

8


delayed trigger HV pulser DC atomic beam

laser beams collecting electronics

ion detector

Fig. 8.17 Experimental setup according to Z IMMERMAN et al. (1979) for RYDBERG spectroscopy in an electric field. Two laser systems tuned to the transitions 2s − 2p and 2p − 3s, respectively, excite the Li atoms into the 3s state. This is then further excited by a very narrow band, tuneable laser into the n = 15 region. Shortly after the excitation, in addition to the S TARK field a pulsed, high electric field (HV power supply) is applied which ionizes all RYDBERG atoms. Detected is the ion signal

binding energy / cm-1 440

Lithium I

(a)

(b)

M=0

M =1

16s 470 n =15 15p

15 15p

500

15s 530 0

2500

5000

0

2500

5000

electric field / Vm-1

Fig. 8.18 RYDBERG levels n 15 in an electric field as a function of the field strength, for atomic Li from Z IMMERMAN et al. (1979). (a) |M| = 0, (b) |M| = 1. Black spectra represent the experimental excitation probabilities, red lines give the calculated term positions

states this would correspond to excitation of the |2− , |2+ states (with π light) and the |2px and |2py states (with σ light). In the present case, for n = 15, practically all levels with 2 ≤ ≤ 14 = n − 1 are degenerate without electric field (a total of 13 levels). Thus, all these levels split with increasing electric field by linear S TARK effect. For the 15p state and very weak electric field (300 V m−1 ) one recognizes that the degeneracy is already removed and the energy decreases initially in a quadratic fashion with increasing field. As the electric field gets larger VE supersedes the initial splitting and the linear S TARK effect takes over also for the 15p state. The 15s state behaves analo-

8.2


411

gously, however, due to the larger initial splitting the linear region is reached only at about 2000 V m−1 . There, however, already numerous avoided crossings with the neighbouring levels n = 14 are encountered, as evident from Fig. 8.18 left (M = 0). A similar behaviour is seen at higher energies for the 16s levels. It is also interesting to compare the |M| = 0 and |M| = 1 levels. Even though the general pattern is very similar in both cases, one recognizes very clearly that the interaction – and hence the repulsion in the vicinity of avoided crossings for terms of equal symmetry – is much larger for |M| = 0 in comparison with |M| = 1: this clearly reflects the magnitude of the corresponding matrix elements according to (8.58) and (8.59), respectively.

8.2.10 Polarizability We have seen that the S TARK effect is quadratic when the degeneracy is already removed among interacting states of different parity. This is the case for almost all atoms in their ground and lower excited states at moderate electric fields. It does not only change the energy but also the wave functions, and thus the electronic charge distribution around the nucleus changes. This is called polarization. Quantitatively, the electric field induces an electric dipole D el = αE E

(8.71)

in each atom, with the (microscopic) polarizability αE being a characteristic property of the atom. Changing the electric field by dE changes the interaction energy by dW = −D el · dE = −αE EdE.

(8.72)

Thus, the total energy of the induced dipole is αE W − W (0) = − αE EdE = − E 2 . 2

(8.73)

Conversely, one may derive the polarizability αE from the energy change with changing field strength. With (8.72) −Del =

∂W ∂E

holds, and using (8.63) the polarizability (here of an atom in state |a ) becomes αE =

1 ∂Wa e2 ∂|E|2 |zab |2 Del =− =− E E ∂E E ∂E Wa − Wb b=a

= 2e2

b=a

|zab = 2e2 Wb − Wa |2

b=a

2 | a|C |b |2 rab 10

Wb − Wa

.

(8.74)

412

8


After averaging over all initial substates states with projection quantum number ma , inserting the transition frequencies ωba = Wba / with Wba = Wb − Wa , and using (opt) the oscillator strength fba as defined in (5.27), one may write the polarizability αE in compact form as:2 (opt) e2 fba αE = . 2 me ωba b=a

(8.75)

We note in passing that the above discussion refers to the so called induced polarization. The situation is different, if the medium consists of particles with a permanent dipole moment (e.g. water molecules). In that case the permanent electric dipole moments will tend to orient along the electric field. Similar to the discussion for magnetic moments in the context of paramagnetism, one has to consider the energetics and statistics involved in this orientation processes to obtain the orientation polarization. It is instructive to compare this quantum mechanically exact calculation with the classical model (J.J. T HOMSON) considering the atom as a harmonic oscillator of eigenfrequency ω0 . The driving mechanical force Fm = me ω02 z must be compensated by the electric force Fel = eE in the field E: Fm = me ω02 z = Fel = eE. The displacement z corresponds to a dipole moment D = ez =

e2 E, me ω02

from which the classical expression for the polarizability follows: αE =

e2 1 . me ω02

(8.76)

Comparison with (8.75) shows that the classical formula corresponds to an atom with only one single transition frequency ωba = ω0 . In contrast, the quantum mechanical result distributes the ability of the electrons to oscillate over all transition 2 Note

that the SI unit of the polarization is [αE ] = A2 s4 kg−1 = C m2 V−1 . In a.u. the polarization = αE /(4π0 a03 ). Often the esu system is still used in this context (see also Appendix A.3)

(au) αE

is with

(esu) = αE

αE 4π0

or

(esu) αE 106 = αE , 3 cm 4π0 m3

(esu) indicating that αE is usually given in cm3 . This scaling allows a direct comparison of the polarizability with the volume of the atoms that are polarized.

8.2


413

frequencies. The T HOMAS -R EICHE -K UHN sum rule (5.28), which for a one elec (opt) tron system reads fba = 1, may be seen as the mathematical manifestation of this picture. The classical formula (8.76) allows us to obtain a rough estimate for αE in the ground state of the H atom. Setting ω0 = Eh /2 leads to αE (4πε0 )4a03 .

(8.77)

Generalizing this with the aid of (8.75) one finds the (static) polarizability to be proportional to the third power of the extension a of the polarized object. This fully corresponds to the findings of classical electrodynamics.

8.2.11 Susceptibility In analogy to the magnetic properties Sect. 8.1.7 one derives the macroscopic dielectric susceptibility χ from the microscopic quantity polarizability αE . Electric displacement D and electric field E are related by D = ε0 E + χε0 E = εr ε0 E = ε0 E + P,

(8.78)

where ε = εr ε0 is the macroscopic dielectric permittivity, with the electric constant ε0 = 1/μ0 c2 and the relative dielectric permittivity εr (formerly dielectric constant). With (8.78) the (electric) polarization of a medium is3 P = (εr − 1)ε0 E = χε0 E,

(8.79)

measured in [P] = C m−2 . It relates to the polarizability by P = N αE E,

(8.80)

with N being again the particle density (number of atoms per volume). Thus, the dielectric susceptibility χ is χ = (εr − 1) =

N αE ε0

or per atom χa =

αE . ε0

(8.81)

To obtain some feeling for the order of magnitude of χ we make an estimate for our favourite Na atoms where nearly all oscillator strength is concentrated in the main transition 3s − 3p (ν¯ 3p−3s = 1.696 × 104 cm−1 ). Setting f3p−3s 1 we obtain with αE from (8.75) the static dielectric susceptibility per atom χa =

3 Note

e2 = 3.12 × 10−28 m3 . ε0 me (2πcν¯ 3p−3s )2

that this differs slightly from the scheme in (8.39) for the magnetization M.

(8.82)

414

8


This may be compared to the atomic volume. With rNa ∼ 3a0 it is Vat 1.7 × 10−29 m3 or roughly χa /4π . This is also what classical theory predicts. Section summary

• The interaction of atoms with a static electric field E is described by the dipole energy er · E = erC10 (θ )E, assuming the field to be parallel to z. This operator breaks the symmetry of the Hamiltonian, so that the orbital angular momentum L is no longer a good quantum number, in contrast to M. Interaction matrix elements are only finite between states with L and L ± 1. They are independent of the electron spin. • Consequently, the change of the energy of LM levels in an electric field (called S TARK effect) depends on M 2 . If L degeneracy for the energy levels is already removed, the S TARK effect is quadratic in E and negative. For L > 0 the levels split into L + 1 sublevels while no S TARK effect occurs for isolated s states. • If L degeneracy is not yet removed, or the splitting between different L levels is small compared to the dipole energy, the dipole operator mixes different L states, but M is conserved. The splitting of the new states is then linear in E (for not too high fields). • Typically, the S TARK splitting is a small effect in standard spectroscopy. However, since it depends quadratically on the radial matrix elements, it increases with principle quantum number ∝ n4 and becomes substantial for high RYD BERG states. • The polarizability of atoms is – following the quadratic S TARK effect – also derived from the same interaction. According to (8.75) it is obtained as sum over all relevant oscillator strengths divided by the respective squared transition frequencies. It is related to the relative dielectric permittivity εr and to the dielectric susceptibility χ by (8.81).

8.3

Long Range Interaction Potentials

We shall now address the question how two atoms or molecules (or their ions) interact as they approach each other. At this point we are interested in distances R large enough so that no chemical bonds can be formed yet, while on the other hand the respective charge distributions of the particles begin to influence each other. This range of distances is – so to say – the precursor for the formation of macroscopic matter and plays a key role in plasma physics, scattering physics, kinetic gas theory etc. A detailed derivation for the relevant interaction potentials has been given by B UCKINGHAM (1967). We summarize here the key results and make some plausibility considerations.

8.3


415

Monopole – Monopole: R −1 In the most elementary case one ion of charge q1 e interacts with another one of charge q2 e by the well known C OULOMB law: V (R) =

q1 q2 e 2 ∝ R −1 . 4πε0 R

(8.83)

Monopole – Permanent Dipole: R −2 Following the rules of electrostatics, the potential of a charge qe in the field E of a dipole (Fig. 8.19), e.g. a diatomic heteronuclear molecule, is given by: qeD qeD · eR =− cos θ V (R) = −D · E = − 2 4πε0 R 4πε0 R 2 ∝ −R −2

with eR = R/R.

Fig. 8.19 Geometry monopole – permanent dipole

(8.84)

R

q

D

+

θ

For later use we also note without proof the corresponding electric field that is obtained as gradient of (8.84): 1 E=− D − 3eR (D · eR ) . (8.85) 3 4πε0 R Permanent Dipole – Permanent Dipole: R −3 Two heteronuclear, diatomic molecules with permanent dipole moments present a prototypical example. As indicated in Fig. 8.20 both dipoles may be oriented differently in respect of R. Combining (8.84) and (8.85) gives: 1 D 1 · D 2 − 3(D 1 · eR )(D 2 · eR ) V (R) = −D 1 · E(D 2 ) = 3 4πε0 R =

D1 D2 [cos θ12 − 3 cos θ1 cos θ2 ] ∝ R −3 . 4πε0 R 3

Fig. 8.20 Geometry permanent dipole – permanent dipole

(8.86)

D1

D2

θ1

θ2 R

Monopole – Quadrupole: R −3 Without detailed reasoning we communicate that the same dependence on distance holds also for the interaction of a point charge with a quadrupolar charge distribution, the latter being e.g. a neutral atom in a p state or a homonuclear molecule: (8.87) V (R) ∝ R −3 .

416

8


Monopole – Induced Dipole: R −4 This is a particular important case, often encountered, e.g. when an electron or an ion of charge qe interacts with a neutral atom and polarizes its electron charge cloud as indicated in Fig. 8.21. With the polarizability αE of the electron shell according to (8.75) we have: 2 αE E(R) V (R) = − D ind · dE = − αE E · dE = − (8.88) 2

2 (au) αE q 2 qe αE αE (qe)2 1 =− = − or = − in a.u., 2 4πε0 R 2 2R 4 32π 2 ε02 R 4 Fig. 8.21 Geometry monopole – induced dipole

(au)

q

R

+

- - + -

(esu)

with αE = αE /(4πε0 a03 ) = αE /a03 (see footnote 2). Note that this so called polarization potential is always attractive. For an estimate of the order of magnitude we consider an H atom in the field of a singly charged ion and use (8.77) as an approximation for the polarizability. In this case V (R)/Eh = −2(R/a0 )−4 , so that in a distance of one B OHR radius the polarization potential would be just 2 a.u. Of course the formula holds strictly only for significantly larger distances. Quadrupole – Quadrupole: R −5 Important examples for this case are e.g. the interaction of two homonuclear, diatomic molecules as well as the interaction of one excited atom in a px state interacting with such a molecule. The geometry is shown in Fig. 8.22. Without entering into further details, we record the dependence on the distance: F (θ1 , θ2 ) V (R) = ∝ R −5 . (8.89) R5 Fig. 8.22 Geometry quadrupole – quadrupole

+ θ2

θ1

+ +

R

+

Permanent Dipole – Induced Dipole: R −6 An example for this case is the interaction of a heterogeneous molecule with a neutral atom as indicated in Fig. 8.23. This case too is easily derived since the induced dipole is always aligned parallel to the field of the permanent one. In analogy to the case monopole – induced dipole one obtains the potential as 2 αE V (R) = − D ind · dE = − αE E · dE = − E(R) 2

8.3


417

=−

2 1 αE − D − 3e (D · e ) R R 2 4πε0 R 3

=−

αE D 2 1 + 3 cos2 θ ∝ −αE R −6 , 2 6 2(4πε0 ) R

Fig. 8.23 Geometry permanent dipole – induced dipole

(8.90)

D θ

− − − R

+

where the second line follows with (8.85). In the limiting cases, D R and D ⊥ R, the angular dependence in the bracket gives a factor of 4 and 1, respectively. In any case one finds an attractive, non-isotropic potential proportional to R −6 . Induced Dipole – Induced Dipole: R −6 The famous VAN DER WAALS potential occurs in the interaction of all neutral atoms or molecules depending on their polarizability. It always ensures an attractive interaction at large distances – as far as not another of the above discussed cases dominates. Fig. 8.24 Geometry induced dipole – induced dipole

A -+- + -

R

- B + -

One may visualize this interaction to arise from spontaneous charge fluctuations that lead to dipole formation in one of the atoms. This initial dipole in turn induces by polarization a dipole in the other atom and so on, until a stable situation as depicted in Fig. 8.24 is reached. The resulting dipole – dipole interaction will lead to an expression similar to (8.90) – except for the fact that there is no preferred direction and one has to average over all alignment angles θ . In the quantum mechanical calculation one expands the electrostatic interaction of all charges involved (valence electrons and ion core) for large R into a 1/R N series. Summing over all electron coordinates r A (at atom A) di ∝ (er A )(er B )/R 3 that and r B (at atom B) one encounters typical dipole terms H correspond to (8.86). One has to treat these in 2nd order perturbation theory (in 1st order the dipole terms of a neutral charge distributions disappear). The interaction energy in 2nd order is the sought-after polarization potential: V (R) =

| a|H di |b |2 b=a

Wa − Wb

.

(8.91)

Here a specifies again the initial state (usually the ground state) and b are all intermediate states. Here too, as discussed in Sect. 8.2.7, for the vast majority of relevant terms Wa − Wb < 0 holds, and the sum is negative. Comparing this with (8.74) one

418

8


sees that this essentially amounts to compute the polarizability of both atoms. Since di ∝ R −3 and V (R) according to (8.91) depends quadratic on the matrix elements H di , the overall interaction potential is attractive and proportional to −R −6 : of H VAN DER

WAALS potential V (R) = −

C . R6

(8.92)

One may evaluate (8.91) approximatively (see B UCKINGHAM 1967), and finds V (R) = −

3 WA WB αA αB . 2(4πε0 )2 WA + WB R 6

(8.93)

The key parameters are the polarizabilities, αA and αB , together with average binding energies, WA and WB , of particle A and B, respectively – one typically uses the respective ionization energies. The force derived from VAN DER WAALS interaction is often called dispersion force as the polarizability αE (ω) depends on the frequency of the external electric field. Section summary

• The long range interaction potentials between ions, atoms and molecules play an important role in spectroscopy as well as in scattering physics. They are determined by the structure of the interacting particles and their plausibility. • For large R the interaction potential is ∝ 1/R for monopole-monopole interaction, ∝ −1/R 2 for monopole-permanent dipole, ∝ −1/R 3 for permanent dipole-permanent dipole, ∝ −1/R 4 for monopole-induced dipole, ∝ −1/R 5 for quadrupole-quadrupole, and ∝ −1/R 6 for dipole-induced dipole as well as induced dipole-induced dipole.

8.4

Atoms in an Oscillating Electromagnetic Field

8.4.1

Dynamic STARK Effect

How does an atom behave in an oscillating electromagnetic field? As discussed in detail in Chap. 4, optical transitions are induced only by resonant irradiation. However, even for non-resonant irradiation one may observe (much weaker) transitions of 2nd order. We shall discuss this so called R AMAN scattering in detail in Chap. 5, Vol. 2. In contrast, at present we want to ask whether also the eigenenergies of the atomic states do change in an oscillating electromagnetic field. The answer is of course yes, because the quadratic S TARK effect does not distinguish between positive and negative field. A more detailed consideration in the framework of QED shows that the treatment in the static case according to (8.63) has to be amended by introducing the photon energy in the resonance denominators. One speaks of dressed states. Even when considering only one atomic level, we have to allow for the possibility

8.4


419

that for a very short time t a “virtual” photon is emitted or absorbed, as we have discussed and visualized by F EYNMAN diagrams in Sect. 6.5.6. Such transient excitation is possible even for non-resonant irradiation since the energy W of states is defined only within the limits of the uncertainty relation, so that W > / t holds. On a very short time scale the levels thus get completely blurred and may be quasi “excited”. One often talks in this context – somewhat misguiding – about virtual intermediate levels. Thus, we expect a shift of the atomic levels by a modified quadratic S TARK effect. A clean derivation of the dynamic polarizability warrants some quite serious efforts. We reduce these here to a heuristic consideration by replacing the total energy of the system Wa prior to absorption or emission of a photon (angular frequency ω) by Wa → Wa + nω and the energies Wb of the intermediate states by Wb → Wb + (n ∓ 1)ω. The ∓ signs refer to absorption or emission, respectively. Introducing these energies into the expression (8.74) for the static polarizability, summing over emission and absorption terms, and averaging over all initial states, we obtain in the dynamic case:

|zba |2 1 |zba |2 . αE (ω) = e + ga Wb − Wa − ω Wb − Wa + ω 2

b=a,ma

The first term stands for absorption, the second for emission. As a last step we replace again Wb − Wa = ωba : αE (ω) = e2

1 |zba |2 / |zba |2 / + ga ωba − ω ωba + ω

(8.94)

b=a,ma

αE (ω) =

(opt) 2e2 1 ωba |zba |2 e2 fba = . 2 − ω2 ga me ωba ω2 − ω2 b=a,m b=a ba

(8.95)

a

(opt)

We have again introduced the optical oscillator strength fba according to (5.27). In the static limit ω → 0, expression (8.75) for the static polarizability is recovered. Conversely, in the limit of very high frequencies we make use of the T HOMAS (opt) R EICHE -K UHN sum rule (5.28) for fba (considering only one active electron) and obtain αE (ω) → −

e2 me ω2

for ω ωba ,

(8.96)

where the minus sign indicates that the induced dipoles have opposite direction as the polarizing electromagnetic field.

420

8


Finally, the shift of the atomic energy levels is derived from (8.73), now using the dynamic polarizability (8.94): Wa − Wa(0) = − Wa − Wa(0) =

αE (ω) 2 αE (ω) 2 αE (ω) I E =− E0 = − 2 4 2 ε0 c

(opt) e2 I fba . 2ε0 cme ω2 − ω2 b=a ba

(8.97)

We have averaged here over the square E(t) 2 = E0 cos(ωt) 2 of the electric field strength and introduced the intensity I of the electromagnetic radiation, using (4.2).

8.4.2

Index of Refraction

From elementary optics we know that the index of refraction n and the relative dielectric permittivity relate as √ n = εr (8.98) (assuming for the relative magnetic permeability μ = 1 in very good approximation). Hence, with the dynamic polarizability (8.95) and (8.81), the index of refraction n is obtained from n2 − 1 = εr − 1 = χ =

(opt) N αE (ω) N e2 fba = . 2 − ω2 ε0 ε0 me ω ba b=a

(8.99)

The resonance frequencies for typical optical materials are usually far away from the visible region and dispersion dn/dλ < 0 (so called normal dispersion). One may express (8.99) in terms of the wavelength, λ = 2πc/ω and write it as so called S ELLMEIER equation, which is often used in optics: n2 = 1 +

Bi λ2 . λ 2 − Ci

(8.100)

i

Usually one treats this as an empirical relation with typically three pairs of S ELL MEIER coefficients Bi , Ci obtained from experiment (a comprehensive set of data for optical materials is given by P OLYANSKIY 2012). For thin media (gases) n 1, so that n2 − 1 = (n − 1)(n + 1) 2(n − 1) and we obtain as a good approximation (opt) N e2 fba n1+ . 2ε0 me ω2 − ω2 b=a ba

(8.101)

We finally mention that in condensed matter the particle density is N 1/Vat , so that χ = χa N becomes on the order of 1 – quite different from magnetism where the magnetic susceptibility was a very small quantity. Thus one has to make

8.4


421

an important modification: in dense media the atoms experience an electric field, the so called L ORENTZ field, which is already modified as compared to the field in the surrounding vacuum. According to C LAUSIUS -M OSSOTI this leads to a modified formula: (opt) n2 − 1 N e2 fba = . n2 + 2 3ε0 me ω2 − ω2 b=a ba

(8.102)

In the limit of n 1, of course, (8.101) is recovered.

8.4.3

Resonances – Dispersion and Absorption

So far we have tacitly assumed that the frequency of the radiation is far away from resonances, ωba = ω. In order to include also frequency regions close to resonance we have to consider damping, i.e. we must account for the finite lifetimes τb = 1/γb of excited states. As in Sect. 5.1.1, we simply introduce a complex energy (or transition frequency) ωba → ωba − iγb /2 so that 1 ωba ± ω

⇒

1 . ωba ± ω − iγb /2

Inserting this into (8.94) and (8.99) leads to a complex index of refraction nc = n + iκ.

(8.103)

For thin media, (8.101) is now replaced by real and imaginary parts:

(opt) ωba − ω N e2 fba ωba + ω (8.104) + 2 4ε0 me ωba (ω − ω)2 + γb2 2 + γb (ω + ω) b=a ba ba 4 4

(opt) N e2 fba γb γb κ= . (8.105) + 2 4ε0 me 2ωba (ω − ω)2 + γb2 2 + γb (ω + ω) b=a ba ba 4 4 n=1+

Far from all resonances, i.e. for negligible γb , one recovers from (8.104) the original relation (8.101). In the general case the wave vector is now also complex and one has to replace k → (2π/λ0 )nc ek = (2π/λ0 )(n + iκ)ek , with ek being the unit vector in the direction of propagation. Thus, the electric field (4.1) in a medium becomes i E(r, t) = E0 eei(2πnz/λ0 −ωt) + e∗ e−i(2πnz/λ0 −ωt) e−2πκz/λ0 2

(8.106)

where we have assumed the light to propagate along the z-axis. The vacuum wavelength λ0 is shortened to λ = λ0 /n, while κ implies exponential damping of the wave (in physics texts κ is often called extinction coefficient). The intensity of the radiation I ∝ |E(r, t)|2 decreases with distance as I = I0 exp(−4πκz/λ0 ).

(8.107)

422

8 n

normal dispersion anomalous dispersion


(a)

1 absorption lines

(b)

0

Fig. 8.25 Schematic of the complex index of refraction nc as a function of the incident radiation frequency ω; (a) real part, n, and (b) imaginary part, κ ∝ μ. The absorption lines shown in the extinction coefficient κ clearly correspond to regions of anomalous dispersion in the index of refraction n

Comparing this to the L AMBERT-B EER law (4.21) we see that we have found a microscopic interpretation of the absorption coefficient μ introduced there. In the vicinity of an isolated resonance, ω ωba , only one term in the sums (8.104) and (8.105) dominates and the real index of refraction is (opt)

n=1+

N e2 fba (ω − ωba )/γb , 4ε0 me ωba γb [(ω − ωba )/γb ]2 + 1/4

(8.108)

showing close to resonance the characteristic anomalous dispersion region with dn/dλ > 0 (or dn/dω < 0). For the absorption coefficient we find from (4.21), (8.107) and (8.105) the familiar L ORENTZ profile: (opt)

fba N e2 4πκ . = μ= λba 4ε0 cme [(ω − ωba )/γb ]2 + 1/4

(8.109)

As illustrated in Fig. 8.25, the real part n (8.104) of the index of refraction may have several regions of normal and anomalous dispersion, while the imaginary part κ (8.105) represents absorption, with several L ORENTZ profiles of different strength. Clearly, these absorption lines correspond to regions of anomalous dispersion in the real part n(ω).

8.4.4

Fast and Slow Light

We do not want to end this discussion on the interaction of electromagnetic radiation with a medium without at least mentioning one fascinating aspect of modern optics that presently enjoys much attention (e.g. B OYD et al. 2010, and further publications in that special issue of J. Opt.), and may be of practical interest in the future. The essential ingredients that we briefly recall here, are already taught in undergraduate physics.

8.4


423

We consider the plane wave (8.106) propagating in +z-direction. For simplicity, we assume linear polarization, and use only the second exponential (with positive exponent) which we write4 E + (t, z) = E0 ei(ωt−k(ω)z)−μ(ω)z

(8.110)

where k(ω) = 2πn/λ0 = 2π/λ and μ(ω) = 2πκ/λ0 , with λ0 and λ = λ0 /n being again the wavelengths in vacuo and in the medium, respectively, while k(ω) is the real (propagation) and μ(ω) the imaginary part (absorption) of the wave vector in the medium, with n and κ according to (8.104) and (8.105). The positions of constant phase Φ = ωt − kz are given by z = (ω/k)t − Φ/k and propagate with the c dz ω phase velocity vp (ω) = = , (8.111) = dt Φ=const k(ω) n(ω) where we have used the familiar relations vp = νλ and c = νλ0 with ω = 2πν. To transport information, one has to imprint some recognizable temporal structure onto the wave. The amplitude E0 must depend on time and position, i.e. it becomes a field envelope. In the spirit of the so called SVE approximation (see footnote 1 in Appendix H) we consider a pulse of a quasi-monochromatic wave, with a carrier frequency ωc , and a pulse duration t 1/ωc . At a given position in space, say at z = 0, we thus write (8.110) as E + (t) = E0 (t)eiωc t .

(8.112)

As derived in Appendix I.2 we may write this as F OURIER transform (I.22), ∞ 1 + ) − ωc )eiωt dω, (8.113) E(ω E (t) = 2π −∞ where ) E(ω) =

∞

−∞

E0 (t)e−iωt dt,

is the F OURIER transform of the field envelope ∞ 1 iωt ) E0 (t) = dω. E(ω)e 2π −∞

(8.114)

(8.115)

When describing also the spatial dependence, we have to account for dispersion: as we have seen in the previous section, in a medium neither n nor vp are constant 4 We

use here the notation of Appendix I.2 where the relations to the full description are given – which in the present case would just be space consuming without leading to further insight.

424

8


with ω, and the wave vector k(ω) is not simply proportional to ω. However, as only a small interval of frequencies around ωc contributes, we may expand k around kc =

ωc dk ωc

ω : k(ω) = kc + ω = + , vp (ωc ) dω ωc vp (ωc ) c vg

(8.116)

with vg = dω/dk. For an arbitrary value of z we replace the time t by t − k(ω)z/ω, as in (8.110). We use the above expansion for k(ω) and abbreviate ω = ω − ωc to obtain from (8.113) E + (t) →E0+ (t − kz/ω) e−μz ω=∞ ) E( ω)ei(ωc + ω)t−i(kc + ω/vg )z d( ω) 2π ω=−∞

+∞ 1 i(t− vzg ) ω ) d( ω) . E( ω)e = ei(ωc t−kc z)−μz 2π −∞ In the last step we have exploited the fact that d( ω) = dω. Comparing the above expression in square brackets with (8.115) we realize that it is just the field amplitude shifted in time by z/vg . Hence, we may write the electric field of the wave at t − kz/ω:

z i(kc z−ωc t)−μz E0+ (t − kz/ω) = E0 t − . (8.117) e vg We clearly recognize that the carrier wave with carrier frequency ωc propagates with the wave vector kc just as (8.110), while the E0 (t − z/vg ) describes the form of the wave-packet. If E0 (t) has its maximum at t = 0, as determined by an appropriate ) choice of E(ω), at a different position z in space one finds this maximum E0 (0) obviously for z = vg t. Thus, characteristic structures of the wave-packet travel with the so called group velocity

vg =

c dω c = = . dn dk ng n + ω dω

(8.118)

Here we have used dω/dk = 1/(dk/dω) and differentiated k = ωn/c from (8.111) in respect of ω. In analogy to n = c/vp (which may be called phase index), one introduces a group index ng =

c dn =n+ω . vg dω

(8.119)

In the range of visible light for transparent materials as a general rule one finds n > 1 and dn/dω > 0 (normal dispersion), so that typically vg < vp < c: light pulses propagate at velocities smaller than in vacuum. So far the canonical discussion on phase and group velocity.

8.4


425

The discussion about slow and fast light starts with the fact that under certain conditions dispersion may be negative – as already illustrated in Fig. 8.25 for the standard dispersion behaviour of conventional materials close to a resonance.5 A look at the formula for the group index (8.119) shows, that one may then indeed find n to be smaller than 1 or even negative – which in principle implies the possibility of group velocities larger that the speed of light in vacuum vg > c. One speaks about superluminal propagation phenomena. And if n < 0, and hence vg < 0, this would even lead to light travelling backward before it has arrived. In contrast, in other regions of the spectrum vg may be extremely small so that light “crawls” so to say, or is being stopped. And indeed, such phenomena have been observed during the past years in fancy experimental setups (see e.g. B OYD and G AUTHIER 2002). To caution the reader quite in the beginning: E INSTEIN’s paradigm according to which information can never be communicated faster that the speed of light in vacuum is by no means violated as a consequence of group velocities vg > c. We want to exemplify these phenomena quantitatively on the basis of a simple case study. Let us consider Na vapour as medium through which light is supposed to propagate. For this consideration Na represents to a good approximation a two level system if we allow only light frequencies ω in the vicinity of the 3 2 S → 3 2 P transition (Na D-line). Let us assume a target density of N = 2 × 1013 cm−3 which may be achieved in a gas cell without problems. The other relevant parameters are the oscillator strength for the transition, fNaD = 0.98, and the decay probability for the excited state A = γ = 6.15 × 107 s−1 (corresponding to a lifetime τ = 16.2 ns), λ0 589 nm (in vacuo) and ω0 3.2 × 1015 s−1 . Hence, the prefactor in (8.108) is Ne2 fNaD /(4ε0 me ω0 γ ) = 0.08 and ω0 /γ = 5.2 × 107 . In Fig. 8.26 the situation close to the resonance line is depicted to scale as a function of ω − ω0 in units of the linewidth. Figure 8.26(a) shows the real index of refraction (phase index), and Fig. 8.26(b) the absorption coefficient. One recognizes that n undergoes the typical change between normal and anomalous dispersion, and finally back to normal – with values close to 1 for which (8.108) is valid. Absorption, on the other hand, is quite substantial contrasting the very moderate changes of the index of refraction. Figure 8.26(c) reports the group index ng according to (8.119). It shows dramatic changes in the resonance region. Correspondingly, the group velocity (8.108) also varies very fast across the resonance as shown in Fig. 8.26(d). When approaching a resonance vg becomes very small, increases again, passes through zero for ω − ω0 = ±γ /2 accompanied by two singularities in each case, and assumes a very small negative value exactly on resonance, ω − ω0 = 0. Generally speaking, experiments trying to observe the expected phenomena are extremely difficult, the main problem being that the most interesting effects, relying on |vg | c or |vg | > c, or even vg < 0, are expected where absorption is particularly high. Thus, typically instead of passive two level systems one investigates three and more levels schemes which also act as amplifiers when pumped with a suitable laser. Beyond atomic gases, today specially designed solid state materials are employed, 5 Today, in addition a variety of artificial, specially designed “smart”, “meta” and “nano” materials exist, as well as photonic fibres, with extended regions of quite unusual optical properties.

426

8 n

(a)

1.0

-2 0.96

-4

(ω − ω 0) / 2

2

-2

- 106

4

- 2×106

ng

4

- 3×106 - 4×106

(d)

0.92

(b)

106 (ω − ω 0) /

(c)

1.08 1.04

-4


2×10- 5

μ / 1000 cm-1

vg /c = 1/ng

1.0

15 1×10- 5

10 (ω − ω 0) /

5

0 - 4×10- 6

-4

-2

0

2

4

0.5 (ω − ω 0) / -4

-2 0

2000 4000

Fig. 8.26 (a) Absorption and (b) index of refraction (showing changing dispersion) in the vicinity of a resonance line. The data is shown to scale for the example of the Na D line at N = 2 × 1013 cm−3 . For comparison (c) the group index and (d) the group velocity are shown; note the different scales in the left and right half of (d), the overall profile being symmetric around ω − ω0 = 0

such as doped light transmitting fibres, photonic crystals, and assemblies of quantum dots. Propagation of light pulses faster than light as well as slowing down of such pulses has been observed. A detailed analysis allows to understand all observations reported in the literature from the basic principles of optics – even if they may appear as mysterious “superluminal” effects: information can be transported by such schemes, but not faster than the speed of light in vacuum. Genuine discontinuities within a wave always propagate with velocities less than c, while precursors of the wave may reach the observer much earlier. Without proof we mention that energy transport in an electromagnetic wave is always characterized by a propagation velocity 2n c, (8.120) n2 + 1 which is definitively less than c, the speed of light in vacuum. While such superluminal effects may just be a fascinating theme of fundamental research, the inverse case, light propagating at extremely low velocities, vg c, albeit perhaps less spectacular, could possibly be of much more practical relevance in data transmission technology: One may envisage special delay lines, data switches or optical storage devices that profit from these phenomena. One could e.g. slow down a data stream temporarily by a control laser if the traffic somewhere down the transmission lines is jammed – and later on release it again on demand. Figure 8.26(d) shows in principle, how low the group velocity can be for ω close to a resonance line. cf =


Fig. 8.27 Light extremely slowed down according to H AU et al. (1999). One sees the not delayed reference pulse (open circles) and the pulse delayed by ca. 7 μs in the B OSE -E INSTEIN condensate (full circles). The delay corresponds to a speed of light of ca. 32 m s−1

427

T = 450 nK Δτ = 7.05 ± 0.05 μs

25 light signal / arb. un.

8.4

L = 229 ± 3 μm

20

vg = 32.5 ± 0.5 ms-1

15 10 5 0 -2

0

2

4

τ / μs

6

8

10 12

An important breakthrough was achieved by H AU et al. (1999), who were able to demonstrate for the first time in a very cold atom gas (to be precise: in a B OSE E INSTEIN condensate) group velocities down to 17 m s−1 . Figure 8.27 shows an example from this work. One sees two light pulses in comparison: the undelayed reference pulse and a signal pulse delayed by ca. 7 μs! Since then, much progress has been made in this field and many ingenious concepts have been explored. These effects are now studied in a variety of media and technological applications appear feasible (we refer the interested reader to review and the recent special issue allready mentioned: B OYD and G AUTHIER 2002; B OYD et al. 2010, and to references there).

8.4.5

Elastic Scattering of Light

With the tools developed in the previous sections we are now also able to treat the important subject of elastic scattering of light. Light cannot only be absorbed, exciting atoms or molecules in the absorbing medium as presented in detail in Chap. 4. It can also be scattered elastically. We first note that the very intense light scattering one observes from dust and smoke particles or from small water droplets in vapour or fog (e.g. in the headlights of cars, in discos and laser shows) will not be subject to this section. This kind of elastic light scattering is called M IE scattering. It dominates if the scattering particles have dimensions on the order of the wavelength or larger. The angular distribution of M IE scattering depends on the size and form of the scattering particles and has to be calculated directly from M AXWELL’s equations with corresponding boundary conditions. Such computations may become quite complicated. For the identification and analysis of nano and micro particles M IE scattering is an often used important tool. Its theoretical treatment leads, however, beyond the scope of this textbook (a very detailed treatment is e.g. found in B ORN and W OLF 2006). Here we shall discuss light scattering from atoms (and molecules) which essentially arises from atoms being polarized by the electromagnetic radiation. More precisely, the oscillating electric field E(t) of the wave induces a time dependent dipole moment D(t) = αE E(t) in the atom, with αE being the polarizability of the

428

8


atoms discussed above. These dipoles in turn radiate at the frequency of the irradiating field. One may calculate the emission classically from (4.33). The power P emitted per solid angle is 2 |E|2 ω4 2 ¨ αE |D(t)| dP 2 sin Θ = sin2 Θ = dΩ (4π)2 ε0 c3 (4π)2 ε0 c3

(8.121)

and depends on the angle Θ between the polarization of the incident radiation and the direction of the detected radiation. The latter is often replaced by the angle ϑ between the polarizations of incident and scattered radiation, with sin2 Θ = cos2 ϑ . The differential cross section for this so called R AYLEIGH scattering of polarized light is obtained by dividing dP /dΩ by the incident light intensity I = cε0 |E|2 = cε0 E02 /2: 2 ω4 αE dσR 3 = σR cos2 ϑ. cos2 ϑ = 2 2 4 dΩ 8π 16π ε0 c

(8.122)

Integration over all azimuthal and polar angles gives a factor 8π/3, so that the integral R AYLEIGH cross section is given by σR =

2 ω4 αE

6πε02 c4

=

2 8π 3 αE

3ε02 λ4

.

(8.123)

Equation (8.122) represents again the typical doughnut like radiation characteristic which we have already encountered with resonant fluorescence in Sect. 4.5. One may observe R AYLEIGH scattering quite conveniently from a laser beam propagating in air. Depending on the laser intensity it can be seen in a darkened laboratory even in completely dry and dust free air, and one easily verifies that the emission parallel to the direction of linear polarization of the scattered light vanishes completely (i.e. perpendicular to the laser beam in one particular azimuthal direction). For unpolarized light one has to average (8.122) over the two directions of polarization and obtains 3 dσR = σR 1 + cos2 θ , dΩ 16π

(8.124)

where θ is now the angle between incident and scattered light direction. For low frequencies ω, in the IR and often even in the VIS spectral region, αE is essentially independent of incident frequency and corresponds to the static value (8.75). Thus, if we define a mean oscillator frequency ω0 , characteristic for each atom, by f 1 ba = , 2 2 ω0 b=a ωba (opt)

(8.125)

8.4


429

the total elastic scattering cross section for ω ω0 is given by σR = σe

ω4 . ω04

(8.126)

This is formally identical to the classical relation already found by Lord R AYLEIGH, who considered ω0 to be the eigenfrequency of the atomic electron. The proportionality σR ∝ λ−4 in (8.123) gives a very clear answer to the often posed question ‘why is the sky blue?’: blue light is scattered much more efficiently than red light – essentially by molecular oxygen and nitrogen being the main constituents of air. Their absorption bands are in the VUV region and do not influence the visible spectrum. The sky, which we register as elastically scattered light from the sun, thus appears blue. Conversely, this also explains the red colour of the sun rising and setting. Note, however, that for higher photon energies, in contrast to the classical low energy limit (8.126), the elastic scattering cross section (8.123) shows pronounced structures over a wide range of wavelengths as expected from the polarizability (8.94). In the vicinity of resonance frequencies particularly intensive scattering phenomena are observed – even where absorption is still negligible. For high enough photon energies, W = ω ω0 , the polarizability simply becomes αE = −e2 /(me ω2 ) according to (8.96), with ω0 being representative for the binding energy of any relevant electron in the atom. Then the integral cross section in (8.122)–(8.124) approaches the so called T HOMSON cross section (measured in units of “barn”, see Appendix A.2) σR −→ σe = ω ωab

e4 8π 4 2 8π 2 α a0 = r = 0.665 b, = 2 2 4 3 3 e 6πε0 me c

(8.127)

dσe = re2 sin2 Θ. dΩ

(8.128)

so that for polarized light

The electrons thus behave quasi like free electrons for which this cross section was first derived by J.J. T HOMSON on a fully classical basis. Here α is the fine structure constant, a0 the atomic units of length, and re the classical electron radius (see Appendix A). Note that (8.127) is a rather suggestive relation, but we recall that re has nothing to do with a radius of the electron – the latter being as point-like as we can measure today. So far, in all above discussion we have neither considered the particle properties of the photon nor used special relativity for treating its motion. Both will become important at high photon energies W , i.e. when γ = W/me c2 is no longer very small. For really free electrons energy and momentum conservation must be satisfied simultaneously, so that the electron will accept and the photon will loose some energy: C OMPTON scattering occurs, which we have introduced briefly in Sect. 1.4.2. We rewrite (1.75) as W 1 = W 1 + γ (1 − cos θ )

(8.129)

430

8


where W = ω and W = ω are the incident and scattered photon energy, respectively, while θ is the photon scattering angle in respect of the incoming light beam. Without derivation we communicate here the (relativistically correct) K LEIN N ISHINA formula for the differential C OMPTON scattering cross section:

W dσC re2 W 2 W 2 = − 2 + 4 cos θ . + dΩ 4 W W W

(8.130)

Note, that in the low energy, non-relativistic limit γ = W/me c2 1 the C OMPTON shift vanishes according (8.129), i.e. W /W 1, and the K LEIN -N ISHINA formula (8.130) reproduces the differential T HOMSON cross section (8.128) for elastic photon scattering – from a free electron. Still, the treatment of elastic light scattering from atoms (or molecules), i.e. from bound electrons is not yet complete. First, up to now we have treated only one single active electron. In the long wavelengths range and even in the region of atomic resonances this can (in principle) be cured by simply summing (8.125) or (8.95) over a sufficient number of absorption frequencies and thereby accounting for all Ne electrons of a system. Nevertheless, the scattering process will be dominated by only a few resonance transitions that are closest to the frequency of the incident radiation. At higher energies, W = ω ωba , specifically for the scattering of X- and gamma-rays where the wavelength becomes comparable or even smaller that atomic dimensions, and energy and momentum conservation has to be considered, the situation is much more complicated (a detailed discussion is found e.g. in K ANE et al. 1986). Most commonly, the form factor approximation is used: one sums explicitly over all electrons, thereby accounting for their probability density, and superposes the scattered radiation appropriately. Two cases are distinguished (see e.g. H ANSON 1986, who also describes explicitely how the polarization angle is converted into scattering angle): 1. The equivalent to R AYLEIGH scattering (elastic) at these energies is called coherent scattering. One argues that the recoil momentum q = k − k from the scatted photon can be accepted by the atom as a whole without significant energy exchange (with q = 2k sin(θ/2) = 4π sin(θ/2)/λ). Thus, the photon energy is conserved and the radiation from all electrons corresponding to their density is superposed coherently. From (8.127) one then derives the differential cross section for coherent scattering of unpolarized radiation r2 dσR = e F 2 (q, Z) 1 + cos2 θ , dΩ 2

(8.131)

where F(q, Z) is the atomic form factor as defined in (1.97), here emphasizing its dependence on atomic number Z. 2. C OMPTON scattering from atoms, called incoherent scattering, is described in a analogue manner, applying the K LEIN -N ISHINA formula (8.130) but now adding cross sections for photons of different final energy incoherently. As detailed e.g.

8.4


431

by H UBBELL et al. (1975) one generalizes the definition of atomic form factors to include excited states, with |a describing the ground and |b any excited state: F (b) (q, Z) = b|

Z

exp(iq · r i )|a

so that F (a) (q, Z) = F(q, Z). (8.132)

i=1

One defines an incoherent scattering function, essentially adding cross sections: F (b) (q, Z)2 . S(q, Z) = (8.133) b=a

The differential cross section for incoherent scattering of unpolarized radiation is then: 2

W r2 W dσaC W 2 = e S(q, Z) − sin + θ . (8.134) dΩ 2 W W W State-of-the-art numerical values of form factors for all atoms are found e.g. at C HANTLER et al. (2005) and B ERGER et al. (2010), including scattering angle integrated coherent and incoherent scattering cross sections, photoionization data etc. One finds that the integrated cross sections decrease with increasing photon energy for both, coherent and incoherent light scattering. We finally mention T HOMSON scattering from a relativistic electron beam as an interesting topic with potential applications for the generation of short pulsed X-ray radiation. We shall briefly discuss this in Sect. 10.6.2. Section summary

• In an oscillating electromagnetic field (angular frequency ω) the energy of atomic energy levels also changes (so called dynamic S TARK effect). Due to the quadratic nature of the interaction the effect is obtained by simply replac2 with ω2 − ω2 (ω denotes the relevant excitation frequencies). The ing ωba ba ba atomic polarizability αE (ω) in the field is obtained by the same substitution from its static limit αE . √ • The index of refraction n = εr , in a medium of density N , is obtained from the atomic polarizability by n2 = 1 + N αE (ω)/ε0 . • In spectral regions with dn/dω > 0 we speak about normal dispersion, while dn/dω < 0 corresponds to anomalous dispersion. The latter situation is associated with absorption. Introduction of the respective damping constants (linewidths of the transitions) leads to a complex index of refraction nc = n + iκ. • We distinguish phase and group velocity, vp = ω/k = c/n and vg = dω/dk = c/(n + ωdn/dω), respectively. Typically vg < vp < c. Near a resonance vg is very small. However, in the region of anomalous dispersion it rapidly changes and we can find vg > c or even vg < 0. Still, information transport can only occur with velocities ≤c.

432

8.5

8


Atoms in a High Laser Field

Modern developments in ultrafast laser science open new dimensions of light matter interaction and are the basis for an active, cutting edge field of research (a glimpse of the kind of processes studied today may be obtained e.g. from H ICKSTEIN et al. 2012). The radiation intensity I = W/(A t) in a focussed laser pulse scales with the total energy W of the pulse, its focal area A, and pulse duration t. Thus, the intensity of a 10 fs laser pulse is e.g. 106 times higher than that of a 10 ns pulse which contains the same energy W . As discussed in Sect. 8.2.2 and exemplified in Table 8.4, gigantic electric field strengths can be generated in this manner, presently surpassing the field that an electron experiences on the first B OHR orbit in the H atom by two orders of magnitude. And the experimental limits are being pushed further. Atoms and molecules exposed to such extreme conditions, react with a wealth of astonishing new phenomena – warranting also theoretical approaches quite different from that outlined in Sect. 8.2.

8.5.1

Ponderomotive Potential

As discussed previously, an electromagnetic field interacting with a free electron cannot just transfer photon energy directly onto the electron (C OMPTON scattering, as discussed in Sect. 8.4.5, is only relevant at very high photon energies ω). If, however, a third body (e.g. an atomic core) is present, energy and momentum conservation may in principle be achieved with the help of this third particle. Our discussion thus starts with a very conventional approach: we solve the classical, non-relativistic equation of motion of an electron oscillating in an electric field of amplitude E0 and frequency ω, me

dv = eE0 cos ωt, dt

and obtain velocity and kinetic energy of the electron in the stationary case: v(t) =

eE0 sin ωt me ω

⇒

e2 E02 1 me v 2 = sin2 ωt. 2 2me ω2

(8.135)

The deflection of the electron from its average position is given by x=−

eE0 cos(ωt) = −x0 cos(ωt). ω2 me

For the oscillation amplitude one calculates with (4.2) eλ2 2I 2I eE0 e = , = 2 x0 = 2 2 2 ω me ω me ε0 c 4π c me ε0 c

(8.136)

(8.137)

8.5


UP / eV

IH 1800 nm mec 2

106 104

433

(a)

102

x 0 / nm

10

1800 nm 800 nm 400 nm 200 nm 50nm

1

13nm

(b)

0.1

1 10-2

800 nm 400 nm 200 nm 50nm 13nm

0.01 1012

1014

1016 1018 I / Wcm-2

1020

1012

1014

1016

I / Wcm-2

1018

1020

Fig. 8.28 Ponderomotive potential (a) and maximum amplitude of an electron (b) in the field of a short laser pulse of intensity I and wavelength λ; the full red line corresponds to the wavelength λ = 800 nm of the Ti:Sapph laser

which in convenient units reads:

x0 / nm = 1.3607 × 10−7 [λ/ nm]2 I / 1012 W cm−2 .

(8.138)

The average energy Up inherent to this quiver motion is called ponderomotive potential: e2 E02 1 1 Up = me v 2 = = me ω2 x02 . 2 4me ω2 4 Inserting (4.2) one obtains e2 I e 2 I λ2 = ∝ I λ2 , or 2 2ε0 cme ω 8π 2 ε0 c3 me Up / eV = 9.3375 × 10−8 [λ/ nm]2 I / 1012 W cm−2 . Up =

(8.139) (8.140)

We point out that this expression is completely identical to (H.20), formally derived from the term quadratic in the vector potential A in the exact (semiclassical) Hamiltonian (H.1) for an atom in an electromagnetic field. The order of magnitude of Up and x0 is illustrated by Fig. 8.28 for a number of wavelengths λ as a function of laser intensity I . The full red lines refer to the fundamental of the Titanium-sapphire laser (short Ti:Sapph) at λ = 800 nm – the “work horse” of ultrafast laser science. As an example, an electron in the focus of a laser beam at an intensity of 1014 W cm−2 experiences a ponderomotive potential of Up = 5.976 eV, and according to (8.138) the corresponding excitation amplitude is x0 = 0.87 nm – a huge electron motion as compared to typical atomic radii of 0.1 nm to 0.25 nm (see Sect. 3.1.5). Clearly, electrons bound to an atom or molecule exposed to such field strengths will experience dramatic changes of their wave functions and term energies. We thus have to compare the ponderomotive potential (8.139) to the binding energies of the electrons in atoms. In the low intensity limit, we expect energy shifts as we have

434

8


derived them for the dynamic S TARK effect in (8.97). And indeed, a comparison (opt) with (8.139) shows that in the limit of high frequencies ω ωba , with fba = 1 both expressions become identical. However, for really intense laser fields, in particular at longer wavelengths as characterized in Fig. 8.28, a complete break down of the bound state description developed so far in atomic physics is expected. Two specific limits are indicated: on the one hand, the system becomes highly relativistic if Up > me c2 . The intensity necessary to reach this limit decreases according to (8.139) with the square of the wavelength. On the other hand, for intensities above IH the electric field in the laser focus is larger than the atomic field EH that an electron experiences in the H atom at a distance a0 from the nucleus. This intensity is independent of wavelength:

2 e ε0 c 2 ε0 c E = = 3.51 × 1016 W cm−2 . IH = 2 H 2 4πε0 a02

8.5.2

(8.141)

KELDISH Parameter

There are other aspects for considering a laser field to be high. One of these is derived from the ratio of ionization potential WI to ponderomotive potential Up . For reasons to be discussed in Sect. 8.5.4 one defines the so called WI ε0 cme ω2 WI K ELDYSH parameter γ = (8.142) = 2Up e2 I WI / eV 3 or in numerical terms γ = 2.31 × 10 I /1012 W cm−2 λ/ nm2 introduced in the pioneering work of K ELDYSH (1965). This parameter characterizes the transition from atom with laser field γ > 1 to a situation that one may describe as laser field with atom γ < 1. For the above example, an H atom with WI = 13.6 eV in a radiation field with I = 1014 W cm−2 at λ = 800 nm we find γ ∼ 1. At this intensity the atomic energy is thus comparable with the energy imposed onto the electron by the laser field. For an H atom, this may be called an intense laser field. We also emphasize at this point that the K ELDYSH parameter depends on the wavelength: the longer the wavelength, the more efficient is the laser field!

8.5.3

From Multi-photon Ionization to Saturation

Multi-photon ionization (MPI) has already been subject to our discussion in Sect. 5.5.5. There we have used perturbation theory: up to N th order for N photon absorption. As described there, the cross section for MPI depends on the laser

8.5


435

Fig. 8.29 Multi-photon ionization signal from Xe at 800 nm as a function of laser intensity according to L AROCHELLE et al. (1998). The slope in the log–log display gives with (5.92) an indication of the number of photons N involved in the process. Sketched in red are the slopes for processes with 9ω and 5ω, respectively. For the direct MPI of Xe at least 9 photons are needed – observed for the lowest intensities. At an intensity I = 1014 W cm−2 the process is saturated. The ion yield is compared with different theories

intensity as ∝ I N . However, if the laser field becomes comparable to inner-atomic fields this approach is bound to break down. Atoms and molecules often behave quite surprisingly in high laser fields: one even finds that the processes become more and more classical as intensities increase. For instance, at very high intensities atomic energy levels are shifted substantially and electrons can escape from the atoms by tunnelling or “above-barrier” processes, as we shall see in a moment. Their kinetic energies then increase as laser intensity increases – a phenomenon in direct contrast to the canonical observations with the photoelectric effect at low intensities. The transition between perturbative, tunnelling, and above-barrier region is, however, seamless. A “benchmark” type of experiment is shown in Fig. 8.29. It has been reported by L AROCHELLE et al. (1998), who very carefully measured the MPI yield from Xe with femtosecond laser pulses at 800 nm. The ionization potential of Xe is WI = 13.44 eV, and with WI /ω = 8.67 the number of photons required for ionization is 9. With an ion yield ∝ I N one would, in a double logarithmic plot, expect a slope of N = 9. As illustrated in Fig. 8.29 this is indeed the case for the lowest intensities, while at intermediate intensities the experimental data appear to follow a power law I 5 , as indicated in the graph. Xe is a quite complex atom with dense series of states above the first excited state, the latter requiring ca. 5 photons to be excited. Obviously at such high intensities the resonance condition is washed out due to the dynamic S TARK effect, and after this first step has been reached the subsequent ionization of this state occurs readily.

436

8


As also seen in Fig. 8.29 the rise of the ion yield above ca. 1014 W cm−2 continues to decrease dramatically. One may conclude that at these intensities almost all atoms are already ionized in the centre of the laser focus – one speaks of saturation. The continuing rise with intensity is essentially due to a geometrical effect: enhanced ionization now also occurs of the peripheral zone of the laser beam (having a Gaussian profile). Thus, the volume in which saturation intensity is reached increases, and with it the total ion yield. The upper scale in Fig. 8.29 gives the K ELDYSH parameter (8.142) for comparison. Saturation intensity obviously corresponds to γ 1, that is to say saturation sets in where the moderate field becomes a very high one.

8.5.4

Tunnelling Ionization

At very high intensities the internal atomic field will be modified substantially by the external (oscillating) electric field. Lets assume the atomic potential to be Coulombic with charge Ze. When adding a linearly polarized laser field, an atomic electron ‘sees’ a time dependent, overall potential V (r, t) = −

Ze2 − eE(t)z 4πε0 r

with z = r cos θ,

(8.143)

as illustrated in Fig. 8.30 for the time of maximum field (amplitude E0 ). A bound electron can thus “tunnel out” of the atom as indicated in Fig. 8.30(a) or may even leave the atom “above-barrier” (b), if the latter is lowered sufficiently. This happens at a critical intensity Icr when at the saddle point V (rs ) = −WI . From dV (r)/dr|rs = 0 one finds π 2 cε03 (WI )4 2Z 2 e6

4.0 × 109 WI 4 . eV Z2

Icr = Icr W cm−2

(8.144)

For an H atom (Z = 1, WI = Eh /2), the critical intensity is Ic = 1.37 × 1014 W cm−2 – which is easily reached in a focused femtosecond laser pulse. In this picture the K ELDYSH parameter may be visualized in an alternative interpretation: since the laser field oscillates, the crucial question is, whether the electron can escape the atom fast enough before the field reverses its sign. Considering Fig. 8.30(a), one estimates the distance tu through which the electron has to tunnel – for simplicity from a so called “zero range potential” (red dashed line). From Fig. 8.30(a) one reads: tu = WI /(eE0 )

(8.145)

8.5


437

V(r) / E h

(a) - eE0 z

(b)

V(r) / E h

- e E0 z

0.5

0.5 ℓtu 15

- 10

z / a0

WI - 0.5

- 1.0

10

- 10

z / a0

WI

e-

- 0.5

saddle point

e-

- 1.0

Fig. 8.30 Model to understand atomic ionization in a high electric field, in particular in an intense laser field: (a) tunnelling (b) electron emission “above-barrier”. Sketched are cuts through the potential parallel to the direction of the E field at the time of maximum field in z-direction

As √ the electron leaves the atom its kinetic energy is Wkin6 = WI , its velocity v = 2WI /me , and consequently the tunnelling time becomes √ √ me WI ε0 cme WI tu = √ . (8.146) ttu = = √ v 2eE0 2 e2 I In order to allow the electrons to leave that atom for good, the tunnelling time must be distinctively smaller than one half of the oscillation period, say ttu < 1/(2ω). One then defines the K ELDYSH parameter as ε0 cme ω2 WI γ = 2ωttu = , (8.147) e2 I in agreement with (8.142). Saturation of the ion signal, observed in Fig. 8.29 for high laser intensities (γ 1), is thus found to happen at intensities and frequencies for which the electron has sufficient time to escape when the oscillation field reaches its maximum value E0 . With this visualization of the ionization process it is obvious that ionization is more probable when the field oscillates less frequently, i.e. at larger wavelengths. The essentially classical ADK theory (A MMOSOV et al. 1986), often used with astonishingly good results for atoms, neglects the frequency dependence of the process completely. It considers saturation to be reached when the field is high enough for direct “above-barrier” ionization as sketched in Fig. 8.30(b). To give a typical value: in the H atom at λ = 800 nm, γ = 0.9 when this critical field intensity (8.144) is reached. A detailed understanding of the relevant processes is subject to current research. Some important concepts and consequences will be discussed in the following. 6 One should take this with a grain of salt: Tunnelling is a quantum mechanical process, while in a classical picture the electron can only leave the atom “above-barrier”.

438

8

105

103

10-1

seq

eq

seq

101

10-3

8.5.5

He 2+

He+

ns

ion signal / arb. un.

Fig. 8.31 Non-sequential double ionization of He by multi-photon processes with 160 fs pulses at 780 nm and comparison with different theories according to WATSON et al. (1997). The He+ (+) and He2+ (×) ion yield has been measured by WALKER et al. (1994). The dashed red lines (seq) represent a theory with one active electron only, the full black line (nseq) is a model calculation for non-sequential ionization


1014 1015 intensity / Wcm-2

1016

Recollision

If the timing between laser field and electron ejection is favourable, the electron may even return to the atom. This so called rescattering of electrons was first discussed in a pioneering paper by C ORKUM (1993). In a high, oscillating electric field the trajectory of an electron depends of course on the exact point in time when it starts. A simple classical calculation shows that the electron returns indeed to its starting point, if is has not yet travelled too far when the sign of the electric field is reversed. C ORKUM found, that (at the origin) the rescattered electron can acquire a kinetic energy of up to (el)

3.17 × Up ≥ Wkin .

(8.148)

This happens for an electron ejected at time t = 0, if the phase angle of the field E(t) ∝ cos(ωt + φ) is φ 17◦ . The physics of these backscattered electrons is very interesting and continues to be a hot topic in current research. One phenomenon associated with back scattered electrons is the ejection of a second electron, leading to the so called non-sequential double ionization. It can be recognized by a very special behaviour of the MPI cross section as a function of laser intensity, illustrated for He as an example in Fig. 8.31. Generally speaking, one expects processes of the following type for a multielectron system A: A + N1 ω → A+ + e− A+ + N2 ω → A2+ + e− ... Aq+ + Nq+1 ω → A(q+1)+ + e− .

8.5


Fig. 8.32 Visualization for the generation of high harmonics (HHG). If the electron is emitted “above-barrier” at the right time it may be back scattered by the inverting field with a kinetic energy up to Wkin ≤ 3.17Up . This energy + ionization potential is then available in principle for HHG

439 V(r ) / E h - e E0 r

r <0

eħωHHG ≤ 3.17Up+WI

0

≤ 3.17Up

r>0 r / a0 WI e-

If these processes occur consecutively one speaks of stepwise or sequential ionization. But in the case of strongly correlated systems one may also consider nonsequential ionization, i.e. the simultaneous emission of several electrons in one genuine multi-electron process. Alternatively, two electrons may be ejected by a recolliding electron as discussed – in summary (el) A + N1 ω → A+ + e− Wkin ≤ 3.17Up A2+ + 2e− ← A+ + e− ←&, which is also a non-sequential ionization process. Characteristic for non-sequential ionization is the kink in the double logarithmic plot of the ion yield as a function of intensity – very clearly seen for He++ in Fig. 8.31.

8.5.6

High Harmonic Generation (HHG)

Rescattered electrons cannot only eject a second electron. They may also be recaptured by the ion and emit electromagnetic radiation during this process: this leads to the generation of electromagnetic waves with frequencies that are multiples of the original laser frequency (fundamental). This process is called high harmonic generation (HHG) and has attracted worldwide considerable interest during the past years. The HHG mechanism is illustrated schematically in Fig. 8.32. The recolliding electron has a potentially high excess energy that may be emitted during the capture process as radiation. According to (8.148), the energy of the recolliding electron can be as high as 3.17Up . Thus, photon energies up to ωHHG ≤ 3.17Up + WI may be emitted upon capture of the electron. This HHG process is used in current research very successfully to generate short pulses in the soft X-ray region (XUV). An intense femtosecond laser pulse is focussed into a dense gas target (e.g. a gas jet, a gas filled cell or capillary). One

8

intensity

(a)

cutoff plateau

(3.17Up+WI)/ħ

2ω

HHG frequency

photons per band width / 107 nm-1

440


(b) 8 H31 4

H43 H37 H49 H55

0 26 24 22 20 18 16 14 wavelength / nm

Fig. 8.33 (a) Schematic HHG spectrum with plateau and cutoff at 3.17Up + WI . The frequency distances are 2ω. (b) Example of an experimentally observed HHG spectrum from BALCOU et al. (2002). 30 fs pulses at ca. 800 nm were focussed into a Ne gas jet. Different focussing conditions (full and dotted lines) lead to quite different efficiencies

obtains the XUV radiation in forward direction. Typically it contains a broad spectrum of harmonics ωHHG = (2N + 1)ω of the fundamental ω as shown schematically in Fig. 8.33(a). For symmetry reasons, usually only odd harmonics are emitted. The scheme indicates the particularly high efficiency for low harmonics, followed by a long “plateau” with frequencies at a distance of 2ω up to the so called cutoff at 3.17Up + WI , which is easily understood in view of Fig. 8.32. In Fig. 8.33(b) gives as a typical experimental example the spectrum of Ne. As shown in the figure one may modify the emitted output by judicious choice of focussing conditions. This is a consequence of the highly nonlinear process. Optimization of HHG generation for practical application is currently a hot topic in AMO research. Special temporal and spatial pulse shaping may be used to improve the conversion efficiency substantially. HHG is currently being used as a convenient, table top, time resolved short pulse radiation source in the near X-ray region. It has considerable application potential for X-ray spectroscopy. The shortest wavelengths achievable depend on the target, on the pump laser intensity, as well as on its frequency (since UP ∝ λ2 ). During the past decade the generation of attosecond laser pulses (1 as = 10−18 s) by superposition of several harmonics has developed very successfully (see e.g. the reviews by K RAUSZ and I VANOV 2009; S ANSONE et al. 2011). As it turns out, the harmonics generated are coherent; superposing them artfully (see e.g. T ZALLAS et al. 2003) and filtering the resulting radiation suitably corresponds to interference in a F OURIER series, leading to a sequence of pulses with an individual pulse duration below 1 fs. As always, a new method that improves earlier techniques by one or two orders of magnitude opens new perspectives with an unforeseeable potential in basic research and applications. Certainly we shall witness exciting developments of “attosecond science” in the years to come (as illustrated e.g. by S ANSONE et al. 2010; B OGUSLAVSKIY et al. 2012; V RAKKING and E LSAESSER 2012).


Fig. 8.34 ATI spectra of Ar according to PAULUS et al. (1994), obtained with 40 fs, 630 nm laser pulses at intensities of (a) 6 × 1013 W cm−2 , (b) 1.2 × 1014 W cm−2 , (c) 2.4 × 1014 W cm−2 and (d) 4.4 × 1014 W cm−2 (the traces are vertically slightly displaced for better visibility); the black arrows indicate the maximum classical back scattering energy of 3.17 × Up

8.5.7

441

107 3.17×Up count rate / arb. un.

8.5

105

(d)

γ =0.7 103 (b) (c) (a) γ = 1.33 γ =0.94 γ = 1.88

101 0

20 40 60 80 electron energy Wkin / eV

Above-Threshold Ionization in High Laser Fields

Before ending this chapter we return briefly to ATI processes which we have introduced already in Sect. 5.5.5. The question here is, how these processes change as the laser intensity is increasing – from MPI through the tunnelling regime and to above-barrier ionization? As a particularly suggestive example we show in Fig. 8.34 the spectra for Ar that were studied by PAULUS et al. (1994) with beautifully resolved ATI peaks. Argon has an ionization potential of 15.4 eV, according to (8.147) the laser intensities used here thus correspond to K ELDYSH parameters γ of (a) 1.88, (b) 1.33, (c) 0.94 and (d) 0.7 – a range covering the critical transition from moderate to above-barrier behaviour. This is reflected quite evidently in the electron spectra: while at the lowest intensity (a) an unspectacular ATI spectrum is observed, quite comparable to that shown in Fig. 5.13 for Xe, the higher intensities promote very pronounced structures as a function of electron energy, that remind us of the plateau seen in HHG which we have discussed in the last section. There, the cutoff was identified as corresponding to the maximum energy of rescattered electrons. Thus it appears self-evident to attribute the plateaus or beats in the ATI spectra also to recollision: obviously recolliding electrons too may absorb further photons. Without going into the finer details of these observations we indicate by arrows in Fig. 8.34 the maximum kinetic energy 3.17 × Up of the rescattered electrons. Obviously, there is even more structure and one should not over-stress the simple rescattering model for such a highly complex process. We just mention that serious quantum mechanical model calculations achieve very good agreement with the experimental data. It is interesting to note that ATI may also be observed when ionizing quite large molecules by (moderately) high intensity lasers. This is exemplified in Fig. 8.35 for C60 according to C AMPBELL et al. (2000). The ionization potential is in this case ca. 7.6 eV, much lower than for argon. The intensities are thus equivalent to those used in Fig. 8.34 as documented by the corresponding K ELDYSH parameters γ . Here too one may recognize, albeit weakly, something like a prolonged plateau for the higher

442

1.56eV (795nm) 105 count rate / arb. un.

Fig. 8.35 ATI spectra of C60 according to C AMPBELL et al. (2000); the laser intensities for the four traces is given in the legend. The vertical, grey lines spaced at a distance of the photon energy (for 795 nm) allow identification of the ATI peaks

8

103

101 0


I / Wcm-2 9.0×1013 7.5×1013 4.7×1013 3.7×1013

(γ = 0.85) (γ = 0.92) (γ = 1.17) (γ = 1.31)

3.17×Up

5 10 15 20 electron energy Wkin / eV

25

30

intensities. Clearly the decrease of the electron signal beyond the 3.17 × Up limit is significantly slower for higher intensities. It is also evident that in this case the laser intensity must not get too high as the clear ATI peaks in the electron spectrum smear out: this large, finite system has many active electrons (there are 240 valence electrons in C60 ). Interaction among them thermalizes the electron motion at the highest laser intensities. Similar trends may be recognized for Ar: in that sense, C60 with its high symmetry may be seen as a kind of “super atom”. Section summary

• Today’s short pulse lasers allow one to generate extremely high intensities I of electromagnetic radiation. The corresponding electric field strengths can easily surpass the inner atomic electric fields, even by orders of magnitude. Correspondingly, in the Hamiltonian the term quadratic to the vector potential of the field must be considered. It gives rise to the ponderomotive potential (8.140), Up ∝ I λ2 . • The K ELDYSH parameter γ = WI /2Up characterizes the field strengths: it is considered to be high for γ 1. • “Recollision” of the electron – ejected by the field and forced to return to the atom by the field – provides a useful concept for understanding non-sequential ionization, HHG and ATI in high intensity fields. A recolliding electron may acquire a kinetic energy up to 3.17 × Up . • HHG by intense femtosecond laser pulses offers excellent perspectives for time resolved spectroscopy with short X-ray pulses. It is also the basis attosecond pulses.

Acronyms and Terminology ADK: ‘A MMOSOV, D ELONE, and K RAINOV’, (1986) theory for strong field ionization (see e.g. Sect. 8.30).


443

AMO: ‘Atomic, molecular and optical’, physics. ATI: ‘Above-threshold ionization’, in multi-photon ionization, if more photons are absorbed than necessary for ionization. a.u.: ‘atomic units’, see Sect. 2.6.2. DC: ‘Direct current’, unidirectional electric voltage and current. 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). esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6). 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)’. HHG: ‘High harmonic generation’, in intense laser fields. HV: ‘High voltage’, electric voltages typically higher than 1000 V. IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). LHC: ‘Left hand circularly’, polarized light, also σ + light. MPI: ‘Multi-photon ionization’, ionization of atoms or molecules by simultaneous absorption of several photons. 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). QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. RHC: ‘Right hand circularly’, polarized light, also σ − light. 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. SVE: ‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38), Vol. 2). Ti:Sapph: ‘Titanium-sapphire laser’, the ‘workhorse’ of ultra fast laser science. 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).

444

8


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 A MMOSOV , M. V., N. B. D ELONE and V. P. K RAINOV: 1986. ‘Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field’. Sov. Phys. JETP, 64, 1191– 1194. BALCOU , P. et al.: 2002. ‘High-order-harmonic generation: towards laser-induced phase-matching control and relativistic effects’. Appl. Phys. B, 74, 509–515. B ERGER , M. J., J. H. H UBBELL, S. M. S ELTZER, J. C HANG, J. S. C OURSEY, R. S UKUMAR, D. S. Z UCKER and K. O LSEN: 2010. ‘XCOM: Photon cross sections database (version 1.5)’, NIST. http://physics.nist.gov/xcom, accessed: 8 Jan 2014. B OGUSLAVSKIY , A. E., A. E. B OGUSLAVSKIY, J. M IKOSCH, A. G IJSBERTSEN, M. S PANNER, S. PATCHKOVSKII, N. G ADOR, M. J. J. V RAKKING and A. S TOLOW: 2012. ‘The multielectron ionization dynamics underlying attosecond strong-field spectroscopies’. Science, 335, 1336–1340. B ORN , M. and E. W OLF: 2006. Principles of Optics. Cambridge University Press, 7th (expanded) edn. B OYD , R., O. H ESS, C. D ENZ and E. PASPALKALIS: 2010. ‘Slow light’. J. Opt., 12, 100301. B OYD , R. W. and D. J. G AUTHIER: 2002. ‘“Slow” and “fast” light’. In: ‘Progress in Optics’, vol. 43, 497–530. Amsterdam: Elsevier. B REIT , G. and I. I. R ABI: 1931. ‘Measurement of nuclear spin’. Phys. Rev., 38, 2082–2083. B UCKINGHAM , A. D.: 1967. ‘Permanent and induced molecular moments and long-range intermolecular forces’. Adv. Chem. Phys., 12, 107. C AMPBELL , E. E. B., K. H ANSEN, K. H OFFMANN, G. KORN, M. T CHAPLYGUINE M. W ITTMANN and I. V. H ERTEL: 2000. ‘From above threshold ionization to statistical electron emission: the laser pulse-duration dependence of C60 photoelectron spectra’. Phys. Rev. Lett., 84, 2128–2131. C HANTLER , C. T., K. O LSEN, R. A. D RAGOSET, J. C HANG, A. R. K ISHORE, S. A. KO TOCHIGOVA and D. S. Z UCKER : 2005. ‘X-ray form factor, attenuation, and scattering tables (version 2.1)’, NIST. http://physics.nist.gov/ffast, accessed: 7 Jan 2014. C ORKUM , P. B.: 1993. ‘Plasma perspective on strong-field multi-photon ionization’. Phys. Rev. Lett., 71, 1994–1997. H ANSON , A. L.: 1986. ‘The calculation of scattering cross-sections for polarized X-rays’. Nucl. Instrum. Methods A, 243, 583–598. H AU , L. V., S. E. H ARRIS, Z. D UTTON and C. H. B EHROOZI: 1999. ‘Light speed reduction to 17 meters per second in an ultracold atomic gas’. Nature, 397, 594–598. H ICKSTEIN , D. D. et al.: 2012. ‘Direct visualization of laser-driven electron multiple scattering and tunneling distance in strong-field ionization’. Phys. Rev. Lett., 109, 073004. H UBBELL , J. H., W. J. V EIGELE, E. A. B RIGGS, R. T. B ROWN, D. T. C ROMER and R. J. H OWERTON : 1975. ‘Atomic form factors, incoherent scattering functions, and photon scattering cross sections’. J. Phys. Chem. Ref. Data, 4, 471–538. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. K ANE , P. P., L. K ISSEL, R. H. P RATT and S. C. ROY: 1986. ‘Elastic-scattering of gamma-rays and X-rays by atoms’. Phys. Rep., 140, 75–159. K ELDYSH , L. V.: 1965. ‘Ionization in the field of a strong electromagnetic wave’. Sov. Phys. JETP, 20, 1307. K RAUSZ , F. and M. I VANOV: 2009. ‘Attosecond physics’. Rev. Mod. Phys., 81, 163–234.

References

445

L AROCHELLE , S., A. TALEBPOUR and S. L. C HIN: 1998. ‘Non-sequential multiple ionization of rare gas atoms in a ti:sapphire laser field’. J. Phys. B, At. Mol. Opt. Phys., 31, 1201–1214. L ORENTZ , H. A. and P. Z EEMAN: 1902. ‘The N OBEL prize in physics: in recognition of the extraordinary service they rendered by their researches into the influence of magnetism upon radiation phenomena’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1902/. M ENENDEZ , J. M., I. M ARTIN and A. M. V ELASCO: 2005. ‘The stark effect in atomic Rydberg states through a quantum defect approach’. Int. J. Quant. Chem., 102, 956–960. PAULUS , G. G., W. N ICKLICH, H. L. X U, P. L AMBROPOULOS and H. WALTHER: 1994. ‘Plateau in above-threshold ionization spectra’. Phys. Rev. Lett., 72, 2851–2854. P OLYANSKIY , M.: 2012. ‘RefractiveIndex.Info’, MediaWiki. http://refractiveindex.info, accessed: 10 Jan 2014. S ANSONE , G., L. P OLETTO and M. N ISOLI: 2011. ‘High-energy attosecond light sources’. Nat. Photonics, 5, 656–664. S ANSONE , G. et al.: 2010. ‘Electron localization following attosecond molecular photoionization’. Nature, 465, 763–767. S TARK , J.: 1919. ‘The N OBEL prize in physics: for his discovery of the Doppler effect in canal rays and the splitting of spectral lines in electric fields’, Stockholm. http://nobelprize.org/ nobel_prizes/physics/laureates/1919/. S VEN G ATO R EDSUN: 2004. ‘3j 6j 9j -symbol java calculator’, Sven Gato Redsun. http://www. svengato.com/, accessed: 8 Jan 2014. T ZALLAS , P., D. C HARALAMBIDIS, N. A. PAPADOGIANNIS, K. W ITTE and G. D. T SAKIRIS: 2003. ‘Direct observation of attosecond light bunching’. Nature, 426, 267–271. V RAKKING , M. J. J. and T. E LSAESSER: 2012. ‘X-ray photonics: X-rays inspire electron movies’. Nat. Photonics, 6, 645–647. WALKER , B., B. S HEEHY, L. F. D IMAURO, P. AGOSTINI, K. J. S CHAFER and K. C. K ULAN DER : 1994. ‘Precision-measurement of strong-field double-ionization of helium’. Phys. Rev. Lett., 73, 1227–1230. WATSON , J. B., A. S ANPERA, D. G. L APPAS, P. L. K NIGHT and K. B URNETT: 1997. ‘Nonsequential double ionization of helium’. Phys. Rev. Lett., 78, 1884–1887. Z IMMERMAN , M. L., M. G. L ITTMAN, M. M. K ASH and D. K LEPPNER: 1979. ‘Stark structure of the Rydberg states of alkali-metal atoms’. Phys. Rev. A, 20, 2251–2275.

9

Hyperfine Structure

We shall now discuss a further step of refinement in our understanding of atomic spectra. Hyperfine structure (HFS) arises from the interaction of the electrons in the atomic shell with the atomic nucleus. Aside from its general spectroscopic relevance in atomic, molecular and nuclear physics – in recent years also in ultra-cold boson and fermion gases – HFS is the basis for NMR and in detail also for EPR. Both are today among the most important methods for structural analysis in molecular physics, chemistry, biology, medicine and materials research.

Overview

This is not a particularly easy chapter to study. Nevertheless, the reader will – earlier or later – have to deal with this important and methodologically fundamental topic. After an introduction into the underlying interactions in Sects. 9.1 and 9.2 we shall discuss in Sect. 9.3 the Z EEMAN effect, again for low, high and arbitrary magnetic fields, following the discussion in Sects. 8.1.2–8.1.6 – there applied to the electron spin, here to the nuclear spin. Electric interactions (Sect. 9.4) and isotope shift (Sect. 9.4.2) are specialties in the interaction of atomic nuclei with atomic electron and require some mathematical effort – making use of formulas derived in the appendices. Finally, Sect. 9.5 introduces some state-of-the-art experimental techniques: molecular beam spectroscopy, EPR and finally NMR spectroscopy. All sections build on the previous ones, but the text may be read without having mastered each preceding detail.

9.1

Introduction

In Table 9.1 we review the perturbation hierarchy already discussed in previous chapters, now including hyperfine interaction between the atomic electron cloud and the atomic nucleus. The latter gives rise to a hyperfine structure (HFS) of atomic © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7_9

447

448

9

Hyperfine Structure

Table 9.1 Hierarchy and relevance of interactions in atomic physics (ordered according to relative strength) Interaction

Characteristic

Remarks

1.

C OULOMB

degeneracy

only for H atom and H like Ions

2.

deviation from Z/r potential

removal of degeneracy

for alkali atoms and all others

3.

spin-orbit J =L + S for low Z J = J i for high Z

fine structure splitting

(2S + 1) FS levels (S electron spin), each (2J + 1)-fold degenerate

4a.

external field

magnetic

(2J + 1)-fold degeneracy removed

4b.

external field

electric

(2J + 1)-fold degeneracy removed but states with M = ±|M| remain degenerate

Depending on field strength 3. and 4. may change sequence 5.

radiative corrections

L AMB shift

sensitive test for QED

6.

atomic nucleus – electron cloud (in analogy to 3. but much weaker)

hyperfine structure F =J +I

2I + 1 states (I nuclear spin), each (2F + 1)-fold degenerate

a

volume effects

form and mass of atomic nucleus

isotope shift of spectral lines

b

magnetic dipole – B j field of electron cloud

hyperfine splitting

c

electric quadrupole moment – E field of electron cloud

additional shift

7a.

external field

magnetic

(2F + 1)-fold degeneracy removed

7b.

external field

electric

analogue to 4b

6. with 7.: Important spectroscopic tool for nuclear physics, also for chemistry (atomic nuclei probe their chemical environment, NMR)

and molecular energy levels, subject of the present chapter. The magnetic moment of an atomic nucleus is are related to its spin I and the nuclear g factor as I = gI μN M I /

z = gI μN Iz / and has the projection M

(9.1)

with the nuclear magneton μN =

e me = μB 2mp mp

(9.2)

= 5.051 × 10−27 J T−1 = 3.152 × 10−8 eV T−1 = 7.623 MHz T−1 .

9.1

Introduction

449

Note the positive sign of the nuclear magnetic moment,1 in contrast to the magnetic moment of the electron: S = −gμB S/ M

(9.3)

with the B OHR magneton μB =

mp e = μN 2me me

(9.4)

= 927.4 × 10−26 J T−1 = 5.788 × 10−5 eV T−1 = 14.00 GHz T−1 . Precise values μN and μB are found in Appendix A. We emphasize that the nuclear magneton (9.1) is very small, μN μB /1836, and hence HFS is small too. As a quantitative basis for the following discussion, Table 9.2 presents a range of typical examples for nuclear moments. We remember that protons, neutrons and atomic nuclei are not elementary particles – in contrast to the electron. Hence, the nuclear g factors, gI = MI /I μN are not even approximately integer numbers, as documented in Table 9.2. The usual eigenvalue equations for angular momenta hold also for the nuclear spin. For the magnitude we have 2 I |I MI = 2 I (I + 1)|I MI ,

(9.5)

and 2I + 1 nuclear spin orientations are possible in respect of a given z-axis: Iz |I MI = MI |I MI with MI = −I, −I + 1, . . . I.

(9.6)

If no external magnetic field is present, hyperfine interaction leads to orientation of the nuclear spin I in respect of the total angular momentum J of the electron charge cloud. Thus, I and J couple and form the total angular momentum of the whole atom (electron cloud and nucleus) = F I + J.

(9.7)

In full analogy to FS coupling (Chap. 6) we just need to replace: L S J ↓↓↓ J IF

(9.8)

Correspondingly, in the coupled scheme (J I )F we have: authors use a different definition for gI :⇐⇒ −gI μN (see e.g. the review of A RIMONDO et al. 1977). 1 Some

450

9

Hyperfine Structure

Table 9.2 Properties of some hadrons and atomic nuclei (S TONE 2005). The notation A Z X refers to an atomic nucleus X with Z protons (atomic number) and a total of A nucleons (atomic mass number). The unit of area, 1 b (see Appendix A.2) corresponds to the area of an average size atomic nucleus Spin I

Landé factor gI = MI /(I μN )

Proton p

1/2

5.58569471(5)

Neutron n

1/2

−3.8260854(10)

Nucleon or atomic nucleus

Quadrupole momenta Q / eb

NMRb

2.79284736(2)

0

+

−1.9130427(5)

0

Magnetic moment MI /μN

Deuteron 21 D

1

0.8574382284

0.8574382284

0.0286(2)

3 He 2 4 He 2 6 Li 3 7 Li 3 12 C 6 13 C 6 14 N 7 15 N 7 16 O 8 19 F 9 23 Na 11 29 Si 14 31 P 15 39 K 19 67 Zn 30 85 Rb 37 129 Xe 54 133 Cs 55 199 Hg 80 201 Hg 80 235 92 U

1/2

−4.25499544(6)

−2.12749772(3)

0

0

−

0

0

1

0.8220473(6)

0.8220473(6)

−0.00083(8)

3/2

2.1709513(13)

3.256427(2)

−0.0406

0

−

0

0

1/2

+1.4048236(28)

+0.7024118(14)

0

1

0.40376100(6)

+0.40376100(6)

+0.02001(10)

1/2

−0.56637768(10)

−0.28318884(5)

0

0

−

0

0

1/2

+5.257736(16)

+2.628868(8)

0

3/2

1.478348(2)

+2.217522(2)

+0.109(3)

1/2

−1.11058(6)

−0.55529(3)

0

+ +

Electron

Muon μ−

+ +

1/2

+2.2632(6)

+1.13160(3)

0

3/2

0.26098(2)

+0.39147(3)

+0.049(4)

5/2

0.3501916(4)

+0.875479(9)

+0.150(15)

5/2

0.541192(4)

+1.35298(10)

+0.23(4)

1/2

−1.555952(16)

−0.777976(8)

0

7/2

0.7377214(9)

+2.582025(3)

−0.00371(14)

1/2

1.0117710(18)

+0.5058855(9)

0

3/2

−0.3734838(9)

−0.5602257(14)

+0.38(4)

7/2

−0.108(10)

−0.38(3)

4.936(6)

ge = |MS /(SμB )|

MS /μN

1/2

2.002319 . . .

−1838.2819709(8)

−

1/2

2.002331 . . .

−8.8905971(2)

−

Comparison e−

+

Note: for I = 0 or I = 1/2 the quadrupole moment is always Q ≡ 0 aA

precise definition for Q gives (9.71); see also Appendix A.2

b Isotopes

marked with “+” are particularly useful for NMR

|J I F MF = 2 F (F + 1)|J I F MF F

with

(9.9)

F = J − I, J − I + 1, . . . , J + I

for I < J

F = I − J, I − J + 1, . . . , J + I

for I > J.

and

9.1

Introduction

451 z

(a)

(b)

Fz

F

Fz

z

3ħ

I

1ħ - 1ħ J L

S

- 3ħ

MF F

3 2 1 0 -1 -2 -3

Fig. 9.1 (a) Vector model for coupling L and S to J , and of J and I to F , (b) F has 2F + 1 possibilities of orientation in space

The vector model in Fig. 9.1(a) visualizes these relations for the example of a 2 P3/2 level: first the orbital angular momentum L (here L = 1) couples with the spin S (here S = 1/2) forming a total angular momentum J (here J = 3/2) of the electron charge cloud. Orbital angular momentum and spin precess around J , which finally couples with the nuclear spin I (here I = 3/2) forming a total angular momentum F (here F = 3) of the atom. J and I precess around F , whose absolute value is √ |F | = F (F + 1) (here = 3.46). In turn, F has 2F + 1 possible orientations in space, MF = −F, −F + 1, . . . F , as illustrated in Fig. 9.1(b). Including spin-orbit (LS) and hyperfine interaction terms (J I ), as well as (if appropriate) magnetic interactions, we write the H AMILTON operator: LS + V MD + V B + V vol + V Q . =H 0 + V H

(9.10)

In the following we consider, one by one, the different contributions to HFS: the MD ), the HFS-Z EEMAN effect (V B ), volume shifts magnetic dipole interaction (V (Vvol ) and finally the electric quadrupole interaction (VQ ). Section summary

• Hyperfine structure is caused by the interaction between electron cloud and atomic nucleus. The nuclear magnetic properties are summarized by the nuclear magneton μN and the gI factor. While μN is about three orders of magnitude smaller than μB for the electron, the g factors are of similar magnitude. • Angular momentum coupling between the nuclear spin I and the total angular momentum J of the electron charge cloud to an overall angular momentum F of the atom occurs in complete analogy the coupling of electron spin S and orbital angular momentum L to J. • In addition to magnetic dipole interaction and interaction with external fields, HFS is also determined by volume shifts and electric quadrupole interaction.

452

9

9.2

Magnetic Dipole Interaction

9.2.1

General Considerations and Examples

Hyperfine Structure

The main contribution to HFS is due to the nuclear analogue of spin-orbit coupling in FS. It is treated in a similar spirit as the latter in Sect. 6.2.3. We write this magnetic dipole hyperfine interaction between the magnetic field of the atomic electron cloud B J and the magnetic dipole moment of the atomic nucleus MI as I · J I ·B J = − gI μ N MD = −M V I · BJ = A 2

with

the magnetic dipole HFS coupling constant A = gI μN βJ .

(9.11) (9.12)

J is the operator for the average magnetic field of the electron cloud, projected B onto J . It is caused by the electron orbital motion and their spin magnetic moments. Its relation with J is written as J = −βJ J /. B

(9.13)

MD may be diagonalized by coupling In analogy to the treatment of fine structure, V nuclear spin I and total angular momentum J of the electron cloud according to (9.7), so that in analogy to (6.47) 1 2 2 2 I · J= F −I −J . 2

(9.14)

The change of the eigenenergy is then derived from (9.11), and we obtain WMD =

A F (F + 1) − I (I + 1) − J (J + 1) , 2

(9.15)

corresponding to (6.62). This expression gives already a good phenomenological description of the empirically observed hyperfine structure splitting. The computation of βJ and thus A will be subject to the next sections. As for fine structure splitting, from (9.15) follows the L ANDÉ interval rule:

WMD = WMD (F ) − WMD (F − 1) = AF.

(9.16)

The energetic distance WMD of two neighbouring HFS levels, F and F − 1, in an HFS multiplet is proportional to F . We present some examples: For the ground and the first excited states of the H atom the hyperfine structure is depicted in Fig. 9.2. With I = 1/2 (proton) we obtain for the 1s 2 S1/2 ground state (J = 1/2) an HFS splitting into a doublet F = 0 and 1 (numerical values in the figure). The same holds for the excited 2s 2 S1/2 and 2p 2 P1/2 states. The 2p 2 P3/2 state (J = 3/2) also forms an HFS doublet, however, with F = 1 and 2.

9.2


n =2 2701 2p 3/2

2p 2P3/2

F =2 23.65

9911.201

13 670 2s1/2 , 2p1/2

453

F =1

2s 2S1/2

177.56

1057.847(9) 2p 2P1/2

F =1

59.17

F =0 243 nm

excited states

F =1 F =0

2 466 061 102. 474 851(34) n =1

2 466 061 413.187 074(34)

243 nm

43 770 1s 2S1/2 1s1/2 BOHR

DIRAC

8173 LAMB shift

F =1 1420.41 F =0

ground state

hyperfine structure

Fig. 9.2 Fine and hyperfine structure of the H atom (compare to Fig. 6.29). All energy splittings are given in MHz as taken from the compilation of K RAMIDA (2010). The nuclear spin of the proton (H+ ) is I = 1/2. The ground to excited state distance is not to scale. The scale for the splittings in more and more magnified towards the right.

The deuteron, d = pn, has a nuclear spin I = 1. As shown in Fig. 9.3, the HFS structure of the hydrogenic isotope deuterium (D) is thus quite different from atomic H. The 1s 2 S1/2 ground state and the excited states with J = 1/2 (e.g. 2s 2 S1/2 and 2p 2 P1/2 ) split also into doublets, however, with F = 1/2 and 3/2. In contrast, the 2p 2 P3/2 state forms a triplet with F = {J − I, J − I + 1, J + I } = {1/2, 3/2, 5/2}. Somewhat more complicated is the situation e.g. for sodium (Na), as shown in Fig. 9.4. With a nuclear spin I = 3/2 the maximum number of levels is (2I + 1) = 4. The 3s 2 S1/2 ground state is again an HFS doublet (since J = 1/2) with F = I ± J = {1, 2}. The same holds for the lower level of the excited FS doublet 3p 2 P1/2 . However, the 3p 2 P3/2 , J = 3/2 level forms a quartet with the components F = {J − I, J − I + 1, J − I + 2, J + I } = {0, 1, 2, 3}.

9.2.2

The Magnetic Field of the Electron Cloud

Atomic hyperfine structure has been studied in the past in great detail, and the relevant parameters are known for most atomic systems rather well. Today HFS is used as a very sensitive probe in molecular and condensed matter spectroscopy. For a quantitative comparison with theory, i.e. for the computation of A, one has to evaluate (9.13), i.e. to determine the magnetic field B J of the electron cloud at the position of the nucleus.

454 n =2

9 F = 5/2 F = 3/2

2p 2P3/2

2p 3/2

4.55 2.70

F = 1/2

9912.61 2s

Hyperfine Structure

2S

F = 3/2 40.92

1/2

1059.28

F = 1/2

2p 2P1/2

excited states

F = 3/2

2s1/2 , 2p1/2

13.63 2 466 732 407.521 71(15) F = 1/2

1s 2S1/2

n =1

F = 3/2 327.38 F = 1/2

8184

ground state

1s1/2 BOHR

DIRAC

LAMB shift

hyperfine structure

Fig. 9.3 Fine and hyperfine structure of the deuterium atom. The nuclear spin is I = 1. Otherwise as in Fig. 9.2

Fig. 9.4 Hyperfine structure of the sodium atom (Na) in the 32 S1/2 ground and 32 P1/2 ,3/2 excited states. The level splittings shown are not to scale, numerical values are given in MHz, with HFS splittings taken from A RIMONDO et al. (1977), while ground to excited state and FS splittings are derived from K RAMIDA et al. (2013)

3 2P3/2

F=3 62.5

F=2 F=1 F=0

3P 515 521.3

3 2P1/2

36.1 16.4

F=2

192

F=1

508 333 195.7 3S

3 2S1/2

F=2 1771.6

screened COULOMB potential

F=1 fine structure

hyperfine structure

In addition to the magnetic field from the electron orbits (similar to the situation for spin-orbit interaction treated in Sect. 6.2.3) we have to account for the magnetic field from the dipole moments of the electron spins. That makes it somewhat more complicated. We restrict the discussion to a single active electron. Fortunately, the magnetic fields of orbit and spin are additive and we treat these contribution, B L and B S , now one after the other.

9.2


455

For B L the situation is even somewhat less complicated than in the FS case: According to classical electrodynamics the magnetic field of a rotating shell with a and a radius r is inside the shell magnetic moment M B=

μ0 M 2 . 4π r 3

=M L = −μB L/, the field of the orbiting electron at the nuWith (1.150), M cleus is thus represented by the operator e 1 L = − μ0 2μB L = − B × 3 L, 3 2 4π r 4πε0 me c r

(9.17)

with the B OHR magneton μB = e/2me and μ0 = 1/ε0 c2 . The respective contribution to the hyperfine dipole energy is 2 a3 N = μ0 2μB gI μN 1 L · I = α gI me 0 2L · I Eh . LI = −B L · M V 4π 4 mp r 3 2 r 3 2

(9.18)

We may compare this to (6.35) for the spin-orbit interaction in the FS case.2 We point out that the above derivation of (9.18) has been obtained as energy of the N in the magnetic field B L of the electron orbit. It is interesting to nuclear dipole M realized that the same result is obtained if one computes the energy of the electronic L in the field B I of the localized nuclear dipole moment. dipole moment M s of is associated Let us now turn to the electron spin. Its magnetic moment M with a magnetic dipole field (e.g. JACKSON 1999, Eq. (5.56)) s · er − M s ), s = μ0 1 (3er M B 4π r 3

(9.19)

with er = r/r, being the unit vector in r direction. The resulting interaction does not only depend on the angle between S and I and on the distance r of the electron from the nucleus. It also depends on the angle between r and S. By inserting the s = −ge μB S/ of the electron spin S with ge = 2.0023, and magnetic moment M adding the orbital field B L (9.17), one obtains the total field B J of the electron cloud for r > 0. The singularities at r = 0 (vanishing distance between nucleus and electron) require special attention. We suppress here the details of the analysis and just communicate the result for the total magnetic field caused by the electron at arbitrary distance r:

L μ0 μB ge 4πge δ(r) . (9.20) + (3er S · er − S) + S BJ = − 2π r 3 2r 3 3 main difference is the ratio of electron to proton mass, me /mp 1/1836, determining the order of magnitude for the HFS splittings. The electron spin S has been replaced by I and correspondingly the ge factor by gI . We also note the factor 2, since no T HOMAS factor has to be applied presently (compensation of the coordinate transformation is not needed). And finally, a03 /r 3 has now replaced the more complicated ξ(r) – a consequence of the localized nuclear dipole.

2 The

456

9

Hyperfine Structure

The overall magnetic dipole hyperfine interaction (9.11) is then: ge MD = μ0 μB gI μN 2 S · er − I · er I · S) I · L + 3 (3 V 2 3 4π r r

8πge + δ(r)I · S . 3

(9.21)

The δ(r) term, only relevant at the nucleus, is called F ERMI contact term. Note that this expression is strictly valid only for effective one electron, one nucleus systems. In the multi-electron case one has to add the fields from all individual electron orz and Sz , respectively. In addition, for molecules bits and spins, projected onto L and solid state materials one has to sum over all relevant nuclei at their respective positions. We note in passing, that for L = 0 and apart from the F ERMI contact term, (9.21) is completely analogue to the electric dipole-dipole interaction, where (8.86) describes the interaction of two electric dipoles at a distance r. For vanishing angular momentum, L = 0 (a situation typical for many atoms and most, non-radical, organic molecules), one may write (9.21) MD = S · A V I, with the so called hyperfine coupling tensor

μ 0 ge μ B gI μ N 1 8π A= (3e e − 1) + δ(r) . r r 4π 3 2 r3

(9.22)

(9.23)

This tensor plays a key role in the theory of all magnetic resonance methods (EPR, NMR, etc.). It contains only operators derived from the components of the position vector r. In order to determine the hyperfine splitting in 1st order, one just has to determine the diagonal matrix elements of (9.22), i.e. one has to average over all angular momenta and all position coordinates. Averaging over the latter needs only to be done for the A tensor. As we shall see in a moment, for atoms which are described in spherical coordinates this implies averaging 1/r 3 , quite similar to the fine structure problem. In the case of complex molecules or solid state materials this averaging becomes, however, much more elaborate. Apart from the F ERMI contact term containing δ(r) to be treated below, one obtains an average hyperfine tensor ⎞ ⎛ 2 2 3xz 3x r−r

3xy

5 5 5 r r ⎟ μ 0 ge μ B gI μ N ⎜ 2 −r 2 ⎟ ⎜ 3yx 3y 3yz (9.24) A = ⎟. ⎜

5 5 5 r r ⎠ ⎝ r 4π 2 2 2 3zx

3zy

3z r−r

5 r5 r5 Averaging over the electron cloud is indicated by . . . . As evident from comparing with Table D.2 the elements of A are expressions of the quadrupole type that characterize the charge distribution of the electron density around the nucleus.

9.2


457

It is this local environment that determines the HFS. The nuclear spin is thus a sensitive probe for such charge distributions and hence an important tool for structural analysis of complex systems.

9.2.3

Nonvanishing Orbital Angular Momenta

Let us return to atoms and focus our attention on r > 0. This is reasonable if the orbital angular momentum is larger than zero. Electrons with > 0 almost never come close to the nucleus, and the terms containing δ(r) in (9.20) and (9.21) are obsolete when determining the magnetic field and the HFS dipole interaction, respectively. To compute the matrix elements in the coupled (effectively one electron) scheme |n(( S)J I )F MF some serious angular momentum algebra has to be applied, with 6j or even 9j symbols (as sketched in Appendix B.4, the latter allow the recoupling of schemes with four angular momenta, SJ I to F , in a similar manner as the 6j symbols for the coupling of three angular momenta discussed in the context of fine structure). Here we just outline the crucial steps of such an analysis. First we derive an estimate for the average magnetic field of the electron cloud B J by recalling the projection theorem (C.17) – used already successfully in the context of spin-orbit interaction. We apply it to the tensor operator B J , J · J |γ J M γ J M|B γ J M γ J M Jq |γ J M , BJ q |γ J M = J (J + 1)2

and write this symbolically for all components q in vector form J BJ · J . BJ = J (J + 1)

(9.25)

Comparing this expression with the original ansatz (9.13), we obviously have found an expression for BJ · J βJ = − . (9.26) J (J + 1) Inserting (9.20) for r > 0 and J = L + S, we have to evaluate: μB μ0 2 L + g (3 S · e e − S) · ( L + S) . βJ = e r r 4π r 3 J (J + 1)2 Since L = r × p and hence r · L = 0, this leads to 1 2 μB μ0 2 2L + ge 3( βJ = S · er )2 − S , 2 3 4π J (J + 1) r

(9.27)

(9.28)

where the averaging over the radial dependence and angular components can be carried out independently. For s = 1/2 we find with (2.105), that 2 +3( S · er )2 − S = 0,

(9.29)

458

9

Hyperfine Structure

2 and finally obtain with | L | = 2 ( + 1)

βJ =

2 ( + 1) μ0 μB n |r −3 |n , 4π J (J + 1)

(9.30)

and the magnetic dipole HFS coupling constant (9.12) for effective one electron systems with > 0 becomes A= =

2 ( + 1) μ0 μ B μ N gI n |r −3 |n , 4π J (J + 1)

or

(9.31)

1 α 2 me 2 ( + 1) n | 3 |n in a.u. gI 4 mp J (J + 1) r

In hindsight, this remarkable result justifies our somewhat relaxed dealings with averaging processes: since is a good quantum number in any coupling scheme, we have to consider only diagonal matrix elements of the HFS dipole interaction (as long as no electric field is active). For comparison with FS, and specifically with the spin-orbit coupling constant a according to (6.36), see footnote 2. The additional factor ( + 1)/J (J + 1) originates from the projection of the magnetic field B L onto J . Specifically, for H and H like ions we find with the expectation value 1/r 3 according to (2.130) A=

μ0 1 Z3 gI μ N μ B 3 , π J (J + 1)(2 + 1) a0 n3

= α2

Z3 me 1 gI mp J (J + 1)(2 + 1) n3

or

(9.32)

in a.u.

As we shall see in a moment, this special formula holds even for = 0.

9.2.4

The FERMI Contact Term

We return once more to the Hamiltonian (9.21), and discuss now the term with the delta function δ(r), the F ERMI contact term. It plays a role only for = 0, since according to (2.120) the radial wave function at small r is Rn (r) ∝ r . Only s states have a finite probability |ψn 0 (0)|2 at the origin. As just outlined, for the one electron case all other terms in (9.23) disappear for = 0. Hence, the contact term remains the only contribution to HFS, and (9.21) becomes MD = μ0 ge μB gI μN 2 δ(r) S · I , V 3 2

for = 0.

(9.33)

The diagonal matrix element is easily evaluated. In the |(LS)J I F MF coupling scheme we have as usual

9.2


459

1 S · I /2 = F (F + 1) − S(S + 1) − I (I + 1) and 2 2 2 δ(r) = ψn00 (r) δ(r)d3 r = ψn00 (0) in general, and

=

Z3 πa03 n3

specifically

(9.34) (9.35)

for H and H like ions according to (2.127). Thus, we finally obtain for effective one electron systems the HFS splitting for = 0 in 1st order perturbation theory: 2μ0 ge μB gI μN δ(r) S · I /2 (9.36) 3 2 2μ0 ge μB gI μN ψn00 (0) F (F + 1) − S(S + 1) − I (I + 1) . = 3

WMD =

With ge 2 and (9.15) the magnetic dipole HFS coupling constant becomes A=

2 4πα 2 me 2 16π μ0 μB gI μN ψn00 (0) = gI ψn00 (0) a03 Eh . 3 4π 3 mp

(9.37)

Specifically for hydrogen and hydrogen like ions, |ψn00 (0)|2 is given by (2.127). Interestingly, one obtains for the present case with = 0 (i.e. for J = S = 1/2) exactly the same expression (9.32) for A already found for > 0. For precision measurements one has, of course, to replace again a0 → a0 me /μm ¯ e.

9.2.5

Some Numbers

We compute a few numbers explicitly and first note: μ0 μN μB = 1.9082 × 108 Hz. π a03 h For the H atom in its ground state (F = 0 or 1 and J (J + 1) = 3/4) we obtain with gI 5.586 the hyperfine coupling constant A/ h 1421 MHz. With the interval rule (9.16) this value is identical to the splitting of the HFS doublet. The HFS splitting in the ground state of the H atom is thus nearly one order of magnitude smaller than the L AMB shift (6.90). Its experimental value is today one of the best known spectroscopic quantities.3 It can be determined directly as microwave M1 transition, and the first high precision measurement was carried out by R AMSEY and collaborators (C RAMPTON et al. 1963), using a hydrogen maser. The presently recommended best value (K ARSHENBOIM 2005) for the ground state HFS splitting of H is

νHFS (1s1/2 ) = 1420.405751768(1) MHz, 3 Presently

i.e. λ = c/ν 21.1 cm.

best known is the frequency of the 1s 2 S1/2 − 2s 2 S1/2 transition, see (6.88).

460

9

Hyperfine Structure

This famous 21 cm line of atomic hydrogen is very important in state-of-the-art radio astronomy. For instance, the abundance of atomic hydrogen in the universe has been mapped out by recording this transition. The HFS splitting in the excited states is much smaller. With (9.32) one calculates for H in the 2s 2 S1/2 state

νHFS (2s1/2 ) = νHFS (1s1/2 )/8 ∼ = 178 MHz and thus it is also about an order of magnitude smaller than the respective L AMB shift. For the 2p 2 P1/2 and 2p 2 P3/2 states the splitting is again smaller, by a factor 3 and 7.5, respectively. The HFS splitting of the deuterium atom in its ground state (F = 1 and 2) is significantly smaller than for hydrogen, due to the smaller gI = 0.8574382284 factor. Its value is also known with high precision:

νHFS (1s1/2 ) = 327.38435230(25)MHz.

(9.38)

In alkali atoms the HFS is comparatively large, in spite of the larger principle quantum number n. This is attributed to the much higher “effective” nuclear charge Zeff , which according to (9.32) enters as A ∝ (Zeff /n)3 . Also, the higher values of F lead to larger splitting. According to the interval rule (9.16) by a factor of 2 for 23 11 Na 1 in the ground state (I = 3/2 ⇒ F = 2) in comparison to 1 H (I = 1/2 ⇒ F = 1). On the other hand, the radius of the 3s electron in Na is larger by a factor of 7 as compared to the 1s radius in the H atom (see Fig. 3.4 in Sect. 3.2), reducing 1/r 3 and |ψn00 (0)|2 correspondingly. Finally, the g factor of the Na nucleus is only 1/4 of that for the proton. The experimental value

νHFS (3s1/2 ) 1772 MHz, already mentioned above, is plausible: from (9.32) we derive with this value an effective charge of Zeff ∼ 4.6 – to be compared with Z = 11 for Na.

9.2.6

Optical Transitions Between HFS Multiplets

Selection rules and probabilities for optical (E1) dipole transition between two HFS states |a and |b are derived in full analogy to those for FS transitions, as detailed in Sect. 6.4. One just has to replace the quantum numbers according to the scheme (9.8). In particular, the triangular relation δ(Fa Fb 1) = 1 holds as well as the selection rule for projection quantum numbers MF = 0, ±1 with the corresponding polarization of emitted or absorbed light. As an example, Fig. 9.5 shows the optically allowed HFS transitions for the 2 P1/2 , 2 P3/2 → 2 S1/2 doublets in Na(I = 3/2). Part of an experimental spectrum has already been presented in Sect. 6.1.3. Line intensities are expressed again by reduced matrix elements and 6j symbols from which intensity ratios are derived in the same manner as for FS multiplets. If one wants to determine the matrix elements quantitatively, one has to apply the

9.3

ZEEMAN Effect of Hyperfine Structure

Fig. 9.5 E1 transition between hyperfine components of a Na D doublet 2 P → 2 S with I = 3/2 (schematic, not to scale)

461

2P

2P

F=3 F=2 F=1 F=0

3/2

F=2 F=1

1/2

≈≈

≈

≈

≈ ≈≈ ≈ ≈ ≈ F=2

2S

1/2

F=1

reduction formalism twice in order to resolve the dependence on hyperfine and fine structure coupling. The final expressions can be evaluated with the aid of 6j and 3j symbols – except for the radial matrix elements that are characteristic for each individual atom and transition. They are responsible for the overall strength of a line. Of course, parity conservation also holds. In the case of one single active electron this implies b = a ± 1. Section summary

I · B MD = −M J is treated in the • Magnetic hyperfine dipole interaction V J = −βJ same spirit as spin-orbit interaction for FS, with B J / being the effective magnetic field of the electron cloud. Consequently, the level splitting is given by (9.15), in complete analogy to (6.62) for FS. • The main difference to FS interaction is the contribution of the electron spin J . This leads to somewhat more complex expressions for the interaction to B energy (9.21)–(9.23). • We distinguish the case of nonvanishing orbital angular momentum, in which expectation values 1/r 3 determine the splitting, while for zero orbital angular momentum the F ERMI contact term is relevant, i.e. the electron density at the origin. Interestingly, the final result (9.32) for HFS splitting in the case of H and H like ions is identical. • HFS splitting is nearly an order of magnitude less than the L AMB shift. • For optical transitions between HFS multiplets standard selection rules hold, i.e. δ(Fa Fb 1) = 1 and MF = 0, ±1.

9.3

ZEEMAN Effect of Hyperfine Structure

The problem is solved very similar as for the Z EEMAN effect of fine structure. However, some specialties and the fact that it forms the basis for NMR spectroscopy, make a detailed discussion advisable.

462

9

9.3.1

Hyperfine Structure

Hyperfine Hamiltonian with Magnetic Field

We have to add the interaction (6.29) between the external magnetic field B and J and nucleus M I to the H AMILTON the magnetic moments of electron cloud M operator (9.11) in the field free case: J ·B −M I ·B MD + V B = −MI · B J − M MDB = V H

I · J Jz Iz B. = A 2 + gJ μ B − gI μ N

(9.39)

In the second line, we have assumed B z as usual. The first term has just been treated in detail. And for simplicity – sufficient in many cases – one often neglects the third term, since the gyromagnetic ratios for electrons and nuclei differ by three orders of magnitude, gJ μB gI μN , and also A gI μN B for not too high B. Note, however, that this is not acceptable for high field NMR and EPR, using super-conducting magnets (B some tesla) – today standard methods in analytical organic chemistry. Also, the objects of NMR are often characterized by J = 0, so that the Z EEMAN splitting is determined exclusively by the interaction of the nuclear moment with the external field, which in turn is modified by the local field anisotropy due to a molecular environment. Thus, we shall – slightly different from the usual treatment in atomic physics text books – distinguish between weak, high, and very high external field, gJ μB B A,

gI μN B A gJ μB B,

and A gI μN B,

respectively. In the latter case we have to account fully for the term ∝ Iz B. For all cases, we note that Jz and Iz do not commute with I · J . Strictly speaking, neither F nor MF nor MS or MI are good quantum numbers.

9.3.2

Low Magnetic Fields

may be considered Nevertheless, for very low fields the coupling scheme J + I =F a good 0th order approximation with states |[(LS)J I ]F MF , and the term I · J is diagonalized as usual. Neglecting the term ∝ B Iz , the HFS Hamiltonian (9.39) becomes in analogy to (8.6): 2 Jz 2 2 MDB A I · J + gJ μB J · B = A F − J − I + gJ μ B B . H 2 2 2

(9.40)

The squared angular momenta have the well known eigenvalues 2 F · (F + 1), etc., and gJ μB B Jz / is treated as a perturbation.

9.3


463

In spite of the formal similarity with fine structure splitting in a magnetic field, there is a significant quantitative difference. To see this, we rewrite quite formally MD term in (9.39) in two ways: the V I I · J = g μ · B J or as I N 2 I · J J A 2 = gJ μ B · B I .

A

(9.41a) (9.41b)

In our above derivation we used view point (a): the nuclear spin I orients itself in the magnetic field B J of the electron cloud. The alternative view point (b) considers the splitting of the substates |J MJ of the electron cloud in the field B I of the nuclear moment. (a) From (9.41a) we get an estimate for the order of magnitude of the field BJ of the electron cloud at the nucleus. Specifically, for the H atom in its 1s 2 S1/2 ground state with A = 5.87 × 10−6 eV and gJ 2 we obtain: BJ A/(gI μN ) 33.3 T.

(9.42)

We recall: a soft iron magnet at saturation can typically generate a 2 T field, simple super-conducting magnets several T, and only with state-of-the-art high field magnets 30 T and more may be reached. Hence, the magnetic field of the charge cloud in an atom is by orders of magnitude larger than magnetic fields that can be generated conveniently in the laboratory. (b) Alternatively, from (9.41b) we estimate the average magnetic field of the nucleus at the position of the electron(s), and obtain a direct comparison between the first and second terms in (9.40): BI A/(gJ μB ) 0.05 T.

(9.43)

Thus, the transition from the low (B BI ) to high field case (B BI ) in HFS Zeeman splitting occurs at magnetic fields that can be achieved conveniently in the laboratory – in contrast to the PASCHEN -BACK effect for FS. For low magnetic fields in this sense, B BI according to (9.43), J , I , F and MF are approximately good quantum numbers, and, in analogy to FS, the term ∝ B in (9.40) can be treated as perturbation in the coupled scheme |J I F MF : A F (F + 1) − J (J + 1) − I (I + 1) WMD = 2 Jz + gJ μB J I F MF | |J I F MF B. (9.44) Evaluating the diagonal matrix element of Jz / in analogy to (8.10) gives gJ J I F MF |

|F MF F MF | J ·F Jz |J I F MF = gJ MF = g F MF F (F + 1)2

(9.45)

464

9

(a)

z

B

(b) F

Fz

I J

z

Hyperfine Structure

B

Jz J

BJz

S

L

Iz I

Fig. 9.6 (a) Vector model for the HFS Z EEMAN effect in a low magnetic B field, (b) Vector model for the HFS Z EEMAN effect in a high B field

with the HFS g factor gF = gJ

F (F + 1) + J (J + 1) − I (I + 1) . 2F (F + 1)

(9.46)

As shown in Fig. 9.6(a), one may visualize this again by a vector diagram. The J = gJ μ B magnetic moment M J / of the electron cloud is averaged due to the I may be neglected here . The contribution of M precession of J and I around F – in contrast to MS when treating the fine structure splitting: hence the small but important difference between the g factors for FS according to (8.11) and for HFS according to (9.46). The hyperfine splitting (9.44) in a low, external magnetic field B finally becomes: WMDB =

A F (F + 1) − J (J + 1) − I (I + 1) + gF μB BMF . 2

(9.47)

As a typical example Fig. 9.7 shows the HFS splitting of the Na D lines.

9.3.3

High and Very High Magnetic Fields

Let us now consider the opposite case: high external fields B BI , for which the is broken. However, the field is assumed to be not yet high coupling of I and J to F enough for breaking the spin-orbit coupling (LS)J . This implies B a/μB , a being the spin-orbit coupling parameter (6.36). We thus use the H AMILTON operator in its original form (9.39), and have to evaluate it somewhat differently: strongest is now the interaction of the electron clouds magnetic moment MJ with the external field B. As visualized by the vector model in Fig. 9.6(b), this leads to a fast precession of J around B, which is assumed parallel to the z-axis, as usual. This involves the formation of 2J + 1 (anomalous) Z EEMAN sublevels. At not too high external magnetic fields B < BJ the next smaller interaction is that between the magnetic


3

2P

F= 3 62.5 MHz

3/2

2

2

.0 n

0

-2

0.94 MHz / 10 - 4 T MF 2 1 0 -1 192 MHz -2 0.23 MHz / 10 - 4 T

F= 2 3 2P1/2

1 0 -1 0

1 0 -1

1

58

589

36.1 MHz 1 16.4 MHz

3 2 1 0 -1 -2 -3 1 0 -1

nm

MF =

WHFS

465

9.6

9.3

m

F= 2 3 2S1/2

MF =

2 1 0 -1 -2

0.7 MHz / 10-4 T

1772 MHz 1

1 0 -1

Fig. 9.7 Hyperfine structure of the Na D lines with splitting in a low magnetic field for the 3 2 S1/2 ground and the 3 2 P1/2 and 3 2 P3/2 excited states (schematic, not to scale)

moment MI of the nucleus and the field B J = −βJ J / of the electron cloud. We recall that BJ is on the order of 30 T. One may view B J as consisting of a static component B J z and one component which rotates in the xy plane. Since the precession of J and hence of the B J field of the electron cloud is much faster than that of the weakly interacting I , the components B J x and B J y average out. Thus, I precesses in the B J z field, i.e. also around the z-axis. Using the definition (9.11), (9.12) for the HFS coupling constant (A = gI μN βJ ) we thus rewrite the H AMILTON operator (9.39) with (B.8) in the spherical basis: MDB = A Jz Iz − J+ I− − J− I+ + gJ μB Jz − gI μN Iz B. H 2

(9.48)

The appropriate basis states are now of course |J MJ I MI for uncoupled J and I . In 1st order perturbation theory the HFS energies are given by the diagonal matrix MDB which in this basis are elements of H WMDB = AMJ MI + (gJ μB MJ − gI μN MI )B,

(9.49)

since the terms J+ I− and J− I+ have, according to (B.17), no diagonal components. As we have seen in the last section, they cannot be neglected for small fields. However, for high B fields their contribution averages out (the vanishing diagonal matrix elements are the mathematical equivalent for the fast rotation in the xy plane); hence, the term AMJ MI describes the remaining interaction of the electron

466

9

WHFS

MJ

Hyperfine Structure 3/2 1/2 MI - 1/2 - 3/2

+ 3/2

2μB B F 2P

3/2

I = 3/2

3 3 A 2 A A

2 1 0

MF 3 -3 2 -2 1 -1 0

+ 1/2 2μB B/3 2μB B/3 -1/2 2μB B

- 3/2 zero field

low field

high field

3/2 1/2 - 1/2 - 3/2 -3/2 -1/2 1/2 3/2 - 3/2 - 1/2 1/2 3/2

Fig. 9.8 Hyperfine splitting in low and high magnetic field, schematic for a 2 P3/2 state with I = 3/2 (e.g. Na)

cloud with the nuclear spin. We recognize (8.12) in the term gJ μB BMJ , which reflect spin-orbit with the external magnetic field B, while the nuclear spin interaction with B is given by gI μN MI B – relevant only for very high external fields. Figure 9.8 shows schematically the transition from low to (medium) high field gI μN B A gJ μB B for the already rather complex example of the Na 32 P3/2 state. According to (8.11) we have gJ = 4/3, from Table 9.2 I = 3/2 and gI = 1.478, and the experimental value of the HFS constant is A 20 MHz. For B 1.8 T this leads to gI μN B = A, while for B 10−3 T one finds gJ μB B = A. The right part in Fig. 9.8, where we have neglected gI μN MI B completely, is thus valid for B = 10−2 to ∼ 10−1 T. Each FS term is spit here into (2I + 1) = 4 sublevels. The distances between the terms are 3A/2 and A/2, for |MJ | = 3/2 and 1/2, respectively. For the transition from low to high field the “non-crossing rule” holds – just as for FS splitting in an external magnetic field: States with equal MJ + MI do not cross (corresponding to ML + MS in Sect. 8.1.5). This leads to the transitions from MF → MJ + MI as illustrated schematically in Fig. 9.8. For yet higher fields (above a few tesla), today accessible with superconduction magnets, one has to account also for the term gI μN MI B. However, even the total electronic angular momentum J is then no longer a good quantum number, since at B 10 T the spin-orbit coupling constant becomes a = gJ μB B. This makes the whole evaluation again more complicated. One obtains a picture similar to the PASCHEN -BACK effect in Fig. 8.8 on the right. However, each of the 6 sublevels shown there is now split again into 4 sub-sublevels. In each case the highest belongs to MI = −3/2.

9.3


9.3.4

467

Arbitrary Fields, BREIT-RABI Formula

As the splitting patterns in the case of high fields tends to become rather complex, we shall now focus on two significantly simpler model cases, and follow their term energies as a function of the magnetic field in detail. We have already noted that the magnetic fields needed to decouple I and J are not particularly high. The general case, where the field may neither be considered low nor high, is thus most often encountered when studying or applying HFS – in contrast to FS. One has to diagonalize the full Hamiltonian (9.39) – in analogy to the procedure for FS described in Sect. 8.1.5. And one may simultaneously also solve the case of very high fields. As a basis, either |(J I )F MF or |J MJ I MI states may be used. Using the |(J I )F MF basis, I · J is diagonal, and one has to evaluate the matrix elements of Jz and Iz . This is done in a manner quite similar to that in Sect. 8.1.5. We sketch this here for the most simple case, nuclear spin I = 1/2 and charge cloud angular momentum J = 1/2 with gJ 2. This is e.g. the case for the ground state of atomic H with = 0 and J = S = 1/2. As no complications from coupling electron spin and orbit arise, the splitting pattern becomes rather clear. The matrix elements Jz / obtained from (C.66) and (C.67) are summarized as follows: ⎛ J I F MF |Jz |J I F MF

F MF

⎜ ⎜F ⎜ ⎜ =⎜ 1 ⎜1 ⎜ ⎝1 0

MF 1 0 −1 0

11 − 12 0 0 0

1 0 1 −1 0 0 0

0 0

1 2

0

1 2

00

⎞

⎟ ⎟ ⎟ 0 ⎟ ⎟. 1 ⎟ 2 ⎟ 0 ⎠ 0

Since in this case J and I have the same value, the corresponding matrix for Iz is almost identical: only the off-diagonal matrix elements have inverse signs corresponding to (C.46) and (C.47). The field free term in (9.40) is diagonal in the (J I )F MF coupling scheme: MD |J I F MF J I F MF |V A = δMF MF F (F + 1) − I (I + 1) − J (J + 1) . 2 For J = I = 1/2 we have F = 1 and 0, and this becomes A/4 and −3A/4, respectively. With the abbreviation

gI gJ μ± = , μB ± μN 2 gJ the full Hamiltonian (9.40) becomes

(9.50)

468

9

⎛ MDB H

11

10

1 −1

A/4 − μ− B 0 0 0

0 A/4 0 μ+ B

0 0 A/4 + μ− B 0

F MF

⎜ ⎜1 ⎜ =⎜1 ⎜ ⎝1 0

1 0 −1 0

Hyperfine Structure

00

⎞

⎟ ⎟ ⎟ . μ+ B ⎟ ⎟ ⎠ 0 −3A/4 0

Obviously only states with MF = 0 mix. Thus, we have to diagonalize HFS − W )|ψ = 0, (H by solving the secular equation HFS − W ) = 0. det(H The solutions for MF = ±1 (with MJ = MI = 1/2 or MJ = MI = −1/2, respectively) are: A ± μ− B. 4 For the mixed terms MF = 0 (MJ = ±1/2 and MI = ∓1/2) one finds: A A W0± = − ± 1 + (2μ+ B/A)2 . 4 2 W±1 =

(9.51)

(9.52)

One verifies by insertion, that for low and intermediate fields the thus found four eigenenergies may be written in the form of the more general B REIT-R ABI formula (1931), valid for J = 1/2:4

A gJ μ B B 2 A 2MF gJ μB B ± + W =− 1+ . (9.53) 2(2I + 1) 2 I + 1/2 A A We recognize the similarity with the splitting of the fine structure in a magnetic field for a J = 3/2, 1/2 doublet according to (8.29)–(8.32). From (9.51) and expansion of the root in (9.52) for very low fields μB B A, one finds WHFS → WMD + gF μB BMF B→0

with WMD given by (9.15), while gI μN B is again neglected. We thus have retrieved the low field expression (9.47) – with gF = gJ /2 according to (9.46) for I = J = 1/2. In the opposite limit, B A/μB , expansion of the root in (9.53) recovers (9.49). Figure 9.9 illustrates (9.51) and (9.52) as a function of the external magnetic field B (with gI μN B being neglected). In analogy to fine structure splitting in a magnetic field, we see the linear behaviour for very small fields, and repulsion of the terms 4 Note that the ± in front of the root has to be applied judiciously: only (2J + 1)(2I + 1) different energies correspond to eigenstates.

9.3


469

WHFS / A

A× MJ MI +1/2 +1/2 MJ MI } A /4 } - A /4 1/2 -1/2

2 J = 1/2 1 I = 1/2 F=1 A /4 {

MF = 1

- 3A /4 F=0 -1

-1

2 μBB

0

{

1

B / AμB-1

2

-1/2 -1/2

0

} A /4 } - A /4

-2

-1/2 +1/2

Fig. 9.9 Z EEMAN effect for HFS: transition from low to high magnetic field for I = J = 1/2. Left: energies according to the B REIT-R ABI formula (9.53). Red and black, full lines refer to MI = +1/2 and −1/2, respectively. Dashed red lines show the energies one would expect without HFS. Right (not to scale): extrapolation to (moderately) high B fields according to (9.49) – however, without the small gI μN MI B term

WHFS / A

MJ +1/2

MI -1/2

} gI μN B - A / 2

NMR +1/2

+1/2

164 EPR

B / AμB-1 164 -1/2 NMR -1/2

-1/2

} gI μ B + A / 2 N

+1/2

Fig. 9.10 Z EEMAN effect for HFS at very high B fields. Left: extension of Fig. 9.9 to the (very) high field situation for the example of the H atom according to (9.51)–(9.52). For sufficiently high field strength B (9.49) holds, now including the gI μN MI B term (not to scale). Right: limiting case A gI μN B. Shown are also the potential M1 transitions in the microwave (EPR) and radio frequency range (NMR)

with equal total projection quantum number, here MF = 0, in the transition to high fields. While the term gI μN B is indeed negligible for low and moderately strong fields, it becomes important for extremely high fields – today exploited e.g. in EPR spectroscopy. This transition from moderately high to very high field is illustrated in Fig. 9.10. For the limit of very high fields the splitting is indicated on the right (not to scale): the sign of the nuclear spin orientation for the level with highest energy has changed as compared to lower magnetic fields! According to (9.49),

470 Fig. 9.11 Hyperfine splitting of 6 Li – a fermion with I = 1 and half integer J – in its electronic ground state 2s 2 S1/2 for low and intermediate magnetic fields. The energies are given in units of the hyperfine coupling constant (A h222 MHz)

9

MJ MI 1/2 1 1/2 0 1/2 - 1

WHFS / A 4 F

3/2 1/2

MF 3/2 1/2 - 1/2 - 3/2 1/2 - 1/2

Hyperfine Structure

2

0

20

40

60 B / mT

-2

-4

- 1/2 - 1 - 1/2 0 - 1/2 1

the crossover occurs when AMJ MI − gI μN MI B = 0, for atomic hydrogen at B 164Aμ−1 B 3 T. For later discussion, we have indicated in Fig. 9.10 transitions which are relevant for EPR and NMR spectroscopy, in the microwave and radio frequency region, respectively. All of them are M1 transitions. In all the above discussion we have chosen the coupled |J I F MF basis as a starting point. Of course one may also carry out the diagonalization in the uncoupled basis |J MJ I MI and obtain the same result. In this case the second and third term of the H AMILTON operator (9.39) are already diagonal, and one has to evaluate the matrix elements of I · J . In Fig. 9.11 we illustrate this for the HFS of 6 Li as a 6 second example. Li has recently obtained some interest in the physics of ultracold gases as a model fermion, since it offers the possibility to form a molecule through F ESHBACH resonances in a magnetic field. Its HFS is still relatively uncomplicated. The Hamiltonian, derived from (9.48) using (B.14) and (B.15), is displayed as matrix in Table 9.3. Again, one may diagonalize this Hamiltonian without problems, and finds a solution described by the B REIT-R ABI formula. Figure 9.11 shows the term energies as a function of the magnetic field applied. The ordering of the terms shown at the right in Fig. 9.11 is again only valid for intermediate fields. For very high B fields above some tesla the terms with positive energy shift, WHFS , invert their ordering, similar to Fig. 9.10. Section summary

• The hyperfine Hamiltonian (9.39) in an external magnetic field B consists of three terms: I · J interaction (∝ A), interaction of B with the electron spin (∝ Jz B) and interaction of B with the nuclear spin (∝ Iz B). • In weak magnetic fields, B BI , the latter may be neglected, the coupled (I J )F MF HFS-levels split linearly with B (with BI A/(gJ μB ) being the magnetic field the electron ‘feels’ from the nucleus).

9.4

Isotope Shift and Electrostatic Nuclear Interactions

471

HFS for 6 Li in the electronic ground state 2s 2 S1/2 , written in the uncoupled Table 9.3 H |J MJ I MI basis. The nuclear spin is I = 1, the HFS coupling constant A = h × 221.864(64) MHz (WALLS et al. 2003), and with (9.50) and Table 9.2 we have μ+ = 1.0016μB , and μ− = 1.0007μB 1 2 1 2

− 12 1

1 2

0

0 A 2

0

μB B √ A 2 2

− A2

0

0

0

0

0

0

0

1

1 2 1 A 2 + μ− B

0

0

MJ MI

− 12 1 1 2 −1 − 12 0 − 12 −1

1 2

0

√ 2 − μ+ B

−1

− 12 0

− 12 −1

0

0

0

0

0

0

0

0 A 2

√ 2

0

0

− A2 + μ+ B √ A 2 2

−μB B

0

0

0

0

A 2

0 − μ− B

• At very high fields B BJ the (I J )F coupling breaks and the uncoupled J MJ I MI levels split linearly with B (with BJ A/(gI μN ) being the magnetic field the nucleus ‘feels’ from the electron). • In between these limits typical B REIT-R ABI formulas (9.53) describe the splitting. Levels with equal MF = MJ + MI do not cross.

9.4


MD and external fields, atomic mass A Aside from magnetic dipole interactions V (for equal atomic number Z) and shape of the nuclei may also influence the energy levels of atoms. The so called isotope shifts (IS) and quadrupole interaction do not change the number of hyperfine levels, but modify their overall position and splittings. Most of the following considerations are presented again for effective one electron systems. They may, however, be generalized for multi-electron systems by summing over all electron coordinates.

9.4.1

Potential Expansion

The interaction energy between an atomic electron at position r and the charge distribution ρn (R) of the nucleus is read from Fig. 9.12: Ve−n (r) =

V (R, r)ρn (R)d3 R,

e 4πε0 |r − R| and the nuclear charge Ze = ρn (R)d3 R.

with V (R, r) = −

(9.54) (9.55) (9.56)

472

9

Fig. 9.12 Atomic nucleus and electron position with nuclear and electron coordinates R and r, respectively, describing the interaction potential (9.54)–(9.56)

Hyperfine Structure z

R r-

electron r

ρ n (R) R x

y

nucleus

Let us discuss the potential created by the electron close to the nucleus. Since the extension of the nuclear charge cloud is extremely small compared to the electron cloud, we can safely expand the potential around the origin, V (R, r) = −

3 3 e ∂V 1 ∂ 2V + X + α 4πε0 r ∂xα 0 2 ∂xα ∂xβ α=1

αβ

Xα Xβ + · · · ,

(9.57)

0

where x1 = x, x2 = y and x3 = z (also for Xα ). Thus, (9.54) becomes 3 ∂V e2 Z + ρn (R)Xα d3 R Ve−n (r) = − 4πε0 r ∂xα 0 α=1 1 ∂ 2 V + ρn (R)Xα Xβ d3 R + · · · . 2 ∂xα ∂xβ 0

(9.58)

(9.59)

αβ

With the first term in (9.58) we have recovered the C OULOMB energy of a point charge Ze, which is already included in the unperturbed atomic Hamiltonian (and – close to the nucleus – is correct also for multi-electron systems). The second term would be proportional to the electric dipole moment of the atomic nucleus – which is zero for symmetry reasons. The first nonvanishing terms, relevant in the present context, are the second order terms (9.59) – obviously tensor components. We can always find a coordinate system in which the axes are aligned such that this tensor is diagonal: 1 Vαβ ρn (R)Xα Xβ d3 R 2 αβ 1 = Vαα ρn (R)Xα2 d3 R (9.60) 2 α

2 1 2 3 2 3 Vαα (9.61) = ρn (R)R d R + ρn (R) 3Xα − R d R 6 α = Vvol (r) + VQ (r)

(9.62)

9.4

Isotope Shift and Electrostatic Nuclear Interactions A = 234 236 233 235

0

1

473

238

2 δW / cm-1

Fig. 9.13 Example for isotope shift (IS) of spectral lines in the VIS range, here for uranium where HFS IS. The measured isotope shift of a spectral line at 424.44 nm according to S MITH et al. (1951) is shown for different isotopes with nucleon number A – indicated by vertical black lines. For comparison a scaling ∝ A2/3 is shown by red vertical lines below (see explanation on p. 476)

The judiciously rewritten second line consists of two terms. The first is the so called volume term, the second is a sum over the components of a traceless tensor, the so called quadrupole tensor. The corresponding energy shifts are obtained in 1st order perturbation theory by multiplying Vvol (r) and VQ (r) with the electron probability distribution |ψel (r)|2 and integrating them over all r space. This will be explicated in the next two subsections.

9.4.2

Isotope Shift

Isotope effects arise for atoms with equal nuclear charge Z and different atomic mass numbers A. We distinguish two effects of spectroscopic relevance: the so called mass effect and the volume shift, which in turn are composed of different contributions. While the former is of simple kinematic nature, the volume shift identified by Vvol (r) in (9.61)–(9.62) is due to the finite size of the nucleus and reflects small deviations from a pure Z/r potential. It is responsible for the major part of IS for atoms with large Z. Valuable information about the atomic nuclei can be gleaned from the IS. A typical example for an isotope specific optical spectrum is shown in Fig. 9.13. Isotope shifts for atomic nuclei of different mass and shape have been determined since the 1950ies quite intensively, and have led to a wealth of spectroscopic data which contributed substantially to a systematic investigation of nuclear structure (see e.g. A NGELI 2004). Recent investigations focus on a comprehensive study of artificial isotopes over a large range of atomic numbers and mass numbers, allowing e.g. to determine the radii of such species. On the other hand, one performs high precision measurements for light atoms, in particular for the isotopes of He+ and Li+ . As already mentioned in Chap. 6, these quantities allow very sensitive tests of theory, including relativistic and radiative contributions (QED) to atomic energies. Below we present only a few details.

Mass Effect As explained already in Sect. 1.8.5 and Sect. 2.6.1, for a precise calculation of eigenenergies of states in one electron systems one has to replace the electron mass

474

9

Hyperfine Structure

me in the S CHRÖDINGER equation by the reduced mass μm ¯ e = me /(1 + me /M), with M being the nuclear mass. This allows one to separate the motion of centre of gravity, and the problem may be treated as a one particle problem. As the mass enters linearly into the H AMILTON operator p 2 /2me and me /M 1 one obtains, to a very good approximation, the resulting energy change (kinematic correction) by me δW = −δ . (9.63) W M Thus, to compare the optical transition frequencies of different isotopes one does not need to know the absolute values of the term energies. This fact is of particular importance for the precise interpretation of the spectra of small atomic numbers. The shift is largest of course for atomic hydrogen between 1 H and 1 D with

1 1 δW 1 = 2.72 × 10−4 . − W 1836 1 2 For the LYMAN α line this amounts to 22.4 cm−1 , in excellent agreement with experiment. A precise comparison requires of course a detailed knowledge of HFS and L AMB shift for both nuclei, as detailed in Sect. 9.2.5 and Sect. 6.5.4 (for a review on recent precision measurments see J ENTSCHURA et al. 2011). Quite generally, a good 1st approximation for the mass effect is obtained by rescaling a¯ 0 = a0 me /m ¯ e and E¯ h = Eh m ¯ e /me . It is largest for small atoms. However, strictly speaking, the above kinematic correction is only exact for one electron atoms. For multi-electron systems the separation of the nuclear motion is no longer trivial and one has to consider the motion of the nucleus in respect of all electrons (see e.g. D RAKE et al. 2005). A detailed analysis shows e.g. for the He atom that separation of the centre of mass motion leads to an additional term −(m ¯ e /M)∇ r 1 ∇ r 2 (in a.u.), the so called mass polarization. Its expectation value is not trivial to compute and depends on the electronic state of the system. Fortunately, it leads to significant contributions only for small A. Beyond the middle of the periodic system of elements one may apply (9.63) to very good approximation. (We note in passing that the mass effect for the uranium case, displayed schematically in Fig. 9.13, does amount to just 0.0012 cm−1 between 233 U and 238 U – as compared to the experimental value of 2.20 cm−1 , i.e. it is without any spectroscopic relevance). On the other hand, only for the small atoms one may hope at all to achieve a quantitative comparison of two isotopes with high precision. Recent work concentrates on the isotopes of He and Li where a precision below 1 MHz can be reached today. However, as it turns out, even for these seemingly simple atoms a theoretical calculation on such level of precision becomes a formidable task. In addition to a precise computation of mass polarization it requires inclusion of higher order radiative corrections (QED), a complete analysis of all HFS contributions as we have discussed them for the H atom, and finally of the volume terms to be treated next.

9.4


Fig. 9.14 Schematic illustration of modified C OULOMB potentials Vi (red) and Vk (pink) due to the finite size of the nuclei in two different isotopes i and k of an atom – with effective radii Ri and Rk . Black dashed line indicates the corresponding C OULOMB potential

475

V(r)

Ri Rk

r

Vk Vi

Volume Shift We now discuss the volume term in (9.61) which describes the interaction of the nuclear charge distribution with the field of the electron close to the origin. It is only relevant if the electron has a finite probability at the origin, i.e. for s states.5 We write the expectation value of R 2 for the atomic nucleus ρn (R)R 2 d3 R = Ze R 2 , and use the P OISSON equation for the electron charge cloud at r = 0, 2 ρel (r) e Vαα |0 ≡ V |0 = = ψel (0) . ε0 r=0 ε0 α Thus, the volume shift of the energy levels becomes 1 Vαα Wvol = 6

ρn (R)R d R = 2 3

2 Ze2 ψel (0) R 2 . 6ε0

(9.64)

The volume term complements the 0th order pure C OULOMB term of the interaction in (9.58), and contributes to the isotope shift: Due the small but finite extension of the atomic nuclei, s electrons experience close to the origin a potential which deviates from Z/r. For larger nuclei this becomes significant. The expectation value R 2 is directly related to the charge distribution, and thus to the number of nucleons in the atomic nucleus. It should be noted that (except for the simplest atoms) experimental measurements can only be compared with theory as difference δWvol between different isotopes with the same nuclear charge. However, in precision measurements even higher terms δWvol = C1 R 2 + C2 R 4 + · · · can be recorded (F RICKE et al. 1995). Figure 9.14 illustrates – very schematically – the potentials for two different isotopes A = i and A = k at equal Z, in and near to the nucleus resulting from this finite extension of the nuclei. One may estimate δWvol from the difference δ R 2 5A

relativistic treatment shows that, in addition to s1/2 electrons, a small contribution also arises for p1/2 electrons.

476

9

Hyperfine Structure

Table 9.4 Isotope shift for the example of 3 He and 4 He according to M ORTON et al. (2006). Measured and calculated (δWvol ) values and determination of the average nuclear radius Rc for 3 He, using the known value R (4 He) = 1.673(1) fm c Measurement / MHz

δWvol / MHz

Rc ( 3 He) / fm

3 1 3/2 ↔ 2 P0 1/2) − 4 He(2 3 S1 ↔ 2 3 P0 )

45394.413 (137)

42184.368 (166)

1.985 (41)

3 He(2 3 S

3 1 3/2 ↔ 2 P0 1/2) − 4 He(2 3 S1 ↔ 2 3 P1 )

1480.573 (30)

33668.062 (30)

1.963(6)

3 1 3/2 ↔ 2 P0 1/2) 4 3 − He(2 S1 ↔ 2 3 P2 ) 3 He(2 3 S 1/2 ↔ 3 3 P 1/2) 1 0 − 4 He(2 3 S1 ↔ 3 3 P2 )

810.599 (3)

33668.066 (3)

1.9643 (11)

45 394.413 (137)

42 184.368 (166)

1.985 (41)

Transition 3 He(2 3 S

3 He(2 3 S

Electron nucleon scattering

1.959 (30)

Nuclear theory

1.96 (1)

of the expectation values for the squared nuclear radii. Summing (9.64) over all s electrons in the electron charge cloud (j ) leads to δWvol =

2 Ze2 2 ψj (0) . δR 6ε0

The volume shift according to (9.64) should e.g. explain the isotope shift for uranium, shown in Fig. 9.13. The nuclear radius Rc may be estimated to a rough approximation by the liquid drop model for atomic nuclei with Rc = RLD A1/3 ,

(9.65)

which according to A NGELI (2004) reproduces the whole periodic system relatively well6 with RLD = 0.9542 fm (for comparison, the proton radius is Rp = 0.8775(51) fm). Hence, we expect the volume shift to be 2 Wvol ∝ R 2 RLD A2/3 . (9.66) As documented in Fig. 9.13, the isotope shifts for the 233 U to 238 U follow this scaling rather well. If one wants to derive reliable estimates for the squares of the nuclear radii R 2 from a measurement of the isotope shift, the electron density at the nucleus has to be known very precisely – and one has to know of course all other parameters which determine the measured value for a particular transition. Today’s experimental accuracy warrants correspondingly elaborate, non-perturbative computational efforts for some low Z atoms. Table 9.4 illustrates the state-of-the-art. Precision measurements for several 2 3 S1 ↔ 2, 3 3 P transitions in 3 He and 4 He are shown as derived including terms proportional to A−1/3 and to A−1 in (9.65) leads to better fits for the experimental data.

6 However,

9.4


477

by M ORTON et al. (2006). To extract from these data the nuclear radius Rc for the helium isotope 3 He, one has to account for the hyperfine structure of 3 He (transitions F = 3/2 ←→ F = 1/2 and F = 1/2 ←→ F = 1/2), and to use the known value Rc for 4 He.

9.4.3

Quadrupole Interaction Energy

While for electronic s states – and more generally for s1/2 and p1/2 states – the electrostatic HFS interaction between the electron and the nucleus is completely characterized by the volume shift (9.64), quadrupole interactions originate from a not completely spherical shape of the nucleus – which may be prolate or oblate. If the potential of the electron cloud has also a non-spherical component at the position of the nucleus (which requires j > 1/2), a quadrupole shift of the energies arises. (The quadrupole term disappears for electronic states with j = 0 or j = 1/2 which have no quadruple moment). The reformulation of (9.60) into (9.61) corresponds to writing it in irreducible form as outlined in Appendix F. It is thus advantageous to replace (9.57) by expanding the potential (9.55) in terms of reduced spherical harmonics according to (F.2). Outside the nucleus (i.e. for r > R), the quadrupole term VQ (r) in (9.61) may be written as a scalar product of two irreducible tensor operators of rank 2, as defined by (C.20). Explicitly, the energy shift due to quadrupole interaction is extracted from (F.5) and becomes in 1st order perturbation theory −Ze2 (−e) 1 (r) 2 (R) = C · ZeR C U2 · Q2 , (9.67) WQ = 2 2 3 4πε0 r 4πε0 where the brackets . . . indicate averaging, i.e. multiplication with the probability densities ρn (R)/(Ze) × |ψel (r)|2 and integration over electronic and nuclear coor(r) (R) dinates, r and R, respectively. The rank 2 tensor operators, C2 and C2 , are the renormalized spherical harmonics (B.29) in respect of the electronic and nuclear coordinates. We also have defined two rank 2 tensor operators, Q2 and U2 , which allow a concise description of the nuclear and electronic charge clouds and their interaction. Their components are7,8 for the nuclear quadrupole tensor Q2q = R 2 C2q (Θ, Φ)

(9.68)

1 C2m (θ, ϕ). r3

(9.69)

and for the electron cloud tensor U2m =

components Q2q with their expectation values Q2q , and the nuclear quadrupole moment Q whose definition (9.71) differs by a factor 2eZ from the expectation value Q20 . √ 8 While r k C (θ, ϕ) = r k 4π/(2k + 1)Y (θ, ϕ) = C (r) are the rescaled (regular) solid harkq kq kq √ monics, r −k−1 Ckq (θ, ϕ) = r −k−1 4π/(2k + 1)Ykq (θ, ϕ) = Ikq (r) are called (rescaled) irregular solid harmonics. 7 Note that we distinguish quadrupole tensor

478

9

Fig. 9.15 Atomic nuclei with quadrupole moment: (a) Q < 0 describes an oblate shape, (b) Q > 0 corresponds to a prolate shape

(a)

z

Iz

x

(b)

y

x

Hyperfine Structure z

Iz

y

The former definition is identical to (F.14) for = 2, while the latter, albeit similar to the quadrupole tensor of the electron charge cloud, differs from it by the weighting factor 1/r 3 instead of r 2 . Since electronic and nuclear coordinates, r and R, are independent of each other, the averaging . . . over the respective wave functions is done independently. Let us first consider the nucleus: Adopting the direction of the nuclear spin as axis of reference, and assuming rotational symmetry, the nuclear quadrupole tensor has only one nonvanishing component: Q20 = R 2 C20 (cos Θ) =

3X32 − R 2 2

(9.70)

Its expectation value in a well defined state |I M is Q20 = I M|Q20 |I M . In principle, this has to be averaged over the population probabilities w(M) – unless a pure |I M state is prepared. One defines the so called (9.71) nuclear quadrupole moment Q = ρn (R) 3X32 − R 2 d3 R = 2eZ Q20 , with the normalization (9.56). Numerical values are found e.g. in Table 9.2.9 Q may be computed if one knows the structure of the nucleus, i.e. the nuclear charge distribution ρn (R). However, in practice one rather uses the HFS to determine Q and to compare it with suitable nuclear structure models. We distinguish oblate nuclear shape (“pancake like”), shown in Fig. 9.15(a), which corresponds to Q < 0, and prolate (“cigar like”) shape with Q > 0, as illustrated in Fig. 9.15(b).10 The nuclear spin I may be considered responsible for such deviations of the shape from a pure sphere (see Table 9.2). With (F.25) the matrix e is not mentioned in published data. As the expectation values Q2q of the quadrupole tensor are correctly measured in units of b, to avoid misunderstandings we use for the quadrupole moment Q the (non-SI) unit eb (see Appendix A.2).

9 Usually,

10 It

is important to note that this definition of prolate and oblate only holds for the quadrupole moment (9.70), defined by the shape of the charge cloud. The sign changes when general tensor operators are used, constructed from angular momenta (Appendix F.3.2).

9.4


479

elements of Q20 can be reduced to 3M 2 − I (I + 1) I M|Q20 |I M = √ I Q2 I , (2I + 3)(I + 1)I (2I − 1)

(9.72)

with M being the projection quantum number of the nuclear spin I onto a given laboratory axis (defined e.g. by the z-axis of an oriented atom). The maximum value of I M|Q20 |I M is obtained for M = I . Multiplied with 2eZ it is called the spectroscopic (nuclear) quadrupole moment: I (2I − 1) I Q2 I . (9.73) Qs = 2eZ I I |Q20 |I I = 2Ze (2I + 3)(I + 1) Thus, the reduced matrix element11 of the nuclear quadrupole tensor is 2 Qs (2I + 3)(I + 1) . I Q2 I = R I C2 I = 2Ze I (2I − 1)

(9.74)

With this we may rewrite (9.72) for the nuclear quadrupole moment in a state |I M : Q(M) = 2Ze I M|Q20 |I M =

3M 2 − I (I + 1) Qs . I (2I − 1)

For further details we refer to Appendix F.3.1 and classical textbooks (e.g. B LATT and W EISSKOPF 1952). Summarizing, the spectroscopic nuclear quadrupole moment Qs is the expectation value of 2ZeQ20 in a nuclear state |I M = I – with respect to the laboratory frame. It has to be distinguished from the intrinsic nuclear quadrupole moment Qi with respect to the nuclear symmetry axis – which is not identical to the nuclear spin. Typically, the nuclear spin I does not point into the direction of the and the nuclear symmetry axis. If the nucleus is assumed to be rotationally symmetric (which is often the case but not necessarily so) spectroscopic and intrinsic nuclear quadrupole moments are related by (see e.g. N EYENS 2003) Qs =

2 − I (I + 1) 3MK Qi , (2I + 3)(I + 1)

(9.75)

where MK is the projection quantum number of I onto the nuclear symmetry axis. Even for the maximum value of MK = I we note that Qs < Qi . This is a consequence√of the nuclear axis being statistically oriented on a cone around the nuclear spin ( I (I + 1) > I ). Only in the limit of large I = MK both quantities become equal. 11 Note that in the nuclear physics literature several types of normalization of the reduced matrix elements may be found (see footnote 2 in Appendix C).

480

9

Hyperfine Structure

We emphasize again that the atomic nucleus (as well as the electron cloud) can have a quadrupole moment only if its spin is I > 1/2 (or its total angular momentum J > 1/2, respectively): the reduced matrix element obeys the triangular relation δ(I 2I ) = 1, i.e. I ≥ 1 must hold. (r) Let us now turn to the atomic electron charge cloud: The tensor U2 defined by (9.69), characterizes its electric field. Its zero component may be written U20 = −

3z2 − r 2 1 = 3 C20 (θ ), 5 2r r

(9.76)

and its matrix elements are J MJ |U20 |J MJ = √

2[3MJ2 − J (J + 1)] (2J + 3)(2J + 2)2J (2J − 1)

J U2 J .

(9.77)

The reduced matrix element 1 (r) J U2 J = − 3 J C2 J r

(9.78)

may be computed for one electron systems in a straight forward manner. Again, the expectation value of 1/r 3 has to be evaluated. In the s J coupled scheme (r) s J C2 s J is given by (C.51) and (C.54).

9.4.4

HFS Level Splitting

We can now evaluate the overall HFS level splitting due to magnetic dipole and electric quadrupole interaction. For the quadrupole interaction energy (9.67) we have to evaluate the matrix elements I J F MF |Q2 · U2 I J F MF of the of scalar product U2 · Q2 . Fortunately, according to (C.48), they are diagonal in F and MF and do not depend at all on MF . Thus, with (C.49) we can rewrite (9.67) as WQ =

Ze2 (2I + 1) (2J + 1) 4πε0 " # I I 2 × (−1)F +J +I J U2 J I Q2 I . J J F

(9.79)

This is evaluated by inserting (9.78), (9.74) and the explicit expressions for the 6j symbol (B.72). WQ is additive to the magnetic dipole interaction (9.15) as long as WQ WMD . Except for really heavy atomic nuclei this is usually a reasonable assumption. Summation of the two terms gives

9.4


Fig. 9.16 Example for the influence of quadrupole interaction on HFS as a function of the ratio of quadrupole coupling constant B and magnetic dipole hyperfine constant A; I = J = 3/2 is assumed

481 F =3 - 0.4

- 0.2

W HFS /A 2 0

2 1

-2

0

-4

0.2

0.4 B/A

1 WHFS = WMD + WQ = AK 2 +B

+ 1) − 2I (I + 1)J (J + 1) , 2I (2I − 1)2J (2J − 1)

3 2 K(K

(9.80)

with K = F (F + 1) − I (I + 1) − J (J + 1). A is the magnetic dipole hyperfine constant, given by (9.31) and (9.37) for > 0 and = 0, respectively, while for the so called quadrupole coupling constant one obtains e 2J − 1 1 2J − 1 a03 Qs B= Qs = Eh . (9.81) 4πε0 2J + 2 r 3 2J + 2 r 3 ea02 Sometimes (9.80) is cast into the more general form 2 2 I · J − 2 I 6( I · J )2 + 3 J , WHFS = AI · J + B 2I (2I − 1)2J (2J − 1)

(9.82)

where the averaging process is again interpreted in the spirit of the vector model, 2 2 and (9.80) is recovered by inserting I = I (I + 1), J = J (J + 1) and I · J = K/2. A typical example for the influence of quadrupole interaction on HFS is sketched in Fig. 9.16 as a function of B/A. The coupling corresponds to the first excited states in the main isotopes of the smaller alkali atoms: for the 3P3/2 state in 23 Na one finds B/A = 0.155, for the 4P3/2 state in 39 K, B/A = 0.472 (A RIMONDO et al. 1977). We emphasize again that (9.80) is only valid for I > 1/2, as the quadrupole moment disappears for I = 0 and 1/2. Correspondingly, (9.78) describing the electron cloud disappears if J = 0 or = 1/2, as well as for = 0. Section summary

• Mass, size and shape of the atomic nucleus contribute to shifts of hyperfine energy levels. • Only for small atoms, the simple kinematic mass effect due to the finite nuclear mass is of spectroscopic relevance. • To identify the influence of size and shape of the nucleus one expands the interaction potential between nucleus and electron in a multipole series and finds that quadratic terms give the first nonvanishing contributions.

482

9

Hyperfine Structure

• The finite size of the nucleus, decreases the overall binding energy of s electrons by the so called volume shift, Wvol ∝ R 2 which is on the order of δWvol /Wvol 10−4 for the largest atoms. For high Z it is the largest contribution to the isotope shift observed for atoms with equal Z and different A. • The quadrupole shift reflects the non-spherical shape of atomic nuclei with finite spin I > 1/2, represented by the nuclear quadrupole moment (9.71). It changes the energy of individual hyperfine levels F if the electron charge cloud has also a quadrupole moment (J > 1/2 and > 0). • The overall HFS splitting is given by (9.80), with the quadrupole and magnetic dipole HFS coupling constants, A (9.31) or (9.37) and B (9.81), respectively.

9.5

Magnetic Resonance Spectroscopy

In addition to high resolution, D OPPLER free optical spectroscopy, as discussed in Chap. 6, and related methods of laser induced fluorescence, a number of sophisticated radio frequency (RF) and microwave (MW) resonance methods have been devised to obtain information on atomic nuclei and their interaction of with the surrounding electron charge cloud (the pioneering experiments of R ABI 1944, were honoured with the N OBEL prize). They play an key role in modern molecular physics and chemical analytics. Since D OPPLER shift and broadening are proportional to the absorbed frequency, one may suppress them completely when determining FS and HFS by inducing the transitions between the respective levels directly. We have already discussed the key ideas in the context of the L AMB shift experiment in Sects. 6.5.2 and 6.5.3. Here we introduce molecular beam spectroscopy with RF and MW, EPR and finally NMR spectroscopy. A variety of other elaborate methods, including infrared and R AMAN spectroscopy, will be treated in Chap. 5, Vol. 2.

9.5.1

Molecular Beam Resonance Spectroscopy

R AMSEY (1989) and his collaborators have performed since 1946 ground breaking work in microwave and radio frequency resonance spectroscopy on magnetically or electrically prepared and guided atomic and molecular beams. Typical experimental equipment and methods which are used still today – even for quite complicated molecules – are still based essentially on the original design from the 1970ies (see e.g. G ALLAGHER et al. 1972). Of course, detection schemes, data acquisition and storage procedures have been improved by orders of magnitude in the mean time. Figure 9.17 illustrates the principle of an electrically focussing molecular resonance apparatus for studying magnetic dipole transitions (M1) in the RF and MW domain. First, one has to create a molecular beam of the atoms or molecules to be studied (see footnote 24 in Chap. 1). They are then state selected in electric quadrupole fields and focused onto the detector. Without going into details, we note that this

9.5

Magnetic Resonance Spectroscopy A quadrupole

z

483

C DC and HF field

beam source

B quadrupole detector

beam axis DC power supply ramp for tuning

mixer HF synthesizer

detection and data acquisiton

Fig. 9.17 Scheme of a R AMSEY molecular beam resonance apparatus for the observation of M1 transitions in molecules according to G ALLAGHER et al. (1972); for details see text

procedure is essentially an electrostatic equivalent to the S TERN -G ERLACH experiment (Sect. 1.9) where magnetic dipole moments are deflected in an inhomogeneous magnetic field. A quadrupole arrangement consists of four long metal rods, mounted symmetric and parallel to the atomic beam axis. Connected to static electric potentials (+ − +−), these rods generate an inhomogeneous electric field, close to the beam axis with a strength E ∝ ρ 2 , ρ being the radial distance from the axis. This field deflects the atoms/molecules corresponding to their electric dipole moment and orientation in space. It may be configured such that only one orientation of the dipole axis in respect of the z-axis is focused towards the beam axis. This mechanism is effective even if – as typical with atoms – the dipole moment has first to be generated by the S TARK effect (see Sect. 8.2.8). In a typical R AMSEY apparatus a specific dipole orientation is focused onto the detector by the combined action of the two quadrupole fields, A and B. In the region C in between A and B a parallel plate capacitor provides a homogeneous electric DC and an RF field, perpendicular to the z-axis (alternatively a microwave resonator may be placed at this position). Inducing an M1 transition in the atoms or molecules changes the component of the dipole moment in respect of the z-axis, so that they are no longer focused onto the detector. This leads to a corresponding reduction of the detected signal. Instead of tuning the RF, it is usually more convenient to change the DC field in region C, thus tuning the transition by means of the S TARK effect into resonance with the fixed RF. To illustrate the excellent resolution of this method, Fig. 9.18 shows the HFS spectrum of the highly polar, diatomic molecule lithium-iodide (LiI) – showing surprisingly rich structure for a molecule which at first sight appears to be a relatively simple system. We note the impressive agreement between the computed and measured spectra. Minimizing all other experimental inaccuracies, the limits of the resolution in such a measurement are determined by the uncertainty relation: the interaction region C has finite length , say for example = 1 m. At a typical beam velocity of 1000 m s−1 the interaction time is τ 1 ms. The frequency resolution ν will then at best be about 1 kHz. In terms of modern precision measurements this warrants improvement. It is, however, not trivial to increase the length of the interaction region C since various perturbations can easily obliterate the improvement. In particular,

9

Hyperfine Structure

signal

484

0 41705

41707 41709 transition frequency / kHz

Fig. 9.18 Experimentally observed and fitted radio frequency spectrum (red dots and lines) of nine S TARK split HFS transitions (F1 = 3/2 → 7/2) in LiI according to C EDERBERG et al. (2005). The LiI molecules are in the ground vibrational state (v = 0), slightly rotationally excited (J = 3). The black lines represent a computed “stick” spectrum

it is difficult to keep the RF or even an MW frequency or the DC field sufficiently constant over a longer distance. Thus, R AMSEY (1950) developed his ingenious fringe method, which effectively extends the interaction time without acquiring new uncertainties. We have discussed this scheme already in Sect. 6.1.7 and illustrated it for a modern atomic clock. In the setup sketched in Fig. 9.17 one simply splits region C into two short interaction regions which may be stabilized and controlled conveniently. In between the atoms or molecules travel freely over a time T . The resolution is thus improved as ν/ ν ∝ T .

9.5.2

EPR Spectroscopy

Electron paramagnetic resonance (EPR), also known as electron spin resonance (ESR), is the technical term for M1 transitions induced in a quantum system with one or more unpaired electrons in an external magnetic field. As a rule, the orbital angular momentum is = 0. The electron spin S orients itself with MS = ±1/2 parallel or antiparallel to the external B field. Without interaction of the electron and its molecular environment the energy of the system depends only on this orientation of the electron spin in respect of B. As described in Sect. 6.2.2, the interaction energy of the electron spin without orbital angular momentum in an external field B z is S B/ h = ge μB MS B/ h ±2 × 14 GHz 1 B/ T, W S / h = −M 2

(9.83)

where ± holds, depending on the spin orientation being parallel or antiparallel to the direction of the field. Transitions may be induced between both states by a microwave field. In this case, the electromagnetic field acts with its magnetic field component (see Appendix H.1.1) onto the magnetic moment of the spin and induces a magnetic dipole transition (M1) as described in Sect. 5.4. The selection rules do

9.5


485

EPR

NMR with 1H Wp / h MHz

MI = - 1/2

750 500

'state of the art' high field EPR

X band (3cm)

250 0 - 250 - 500

10

20 'state of the art' NMR up to 950 MHz

Ws / h GHz

MS = + 1/2

500

recent development 2007 ff. 30

300

B/T

MS = - 1/2 MI = +1/2

- 750

100 0 - 100 - 300 - 500

Fig. 9.19 Typical (M1) transition frequencies in NMR (left scale, black) and EPR (right scale, red) as a function of the external magnetic field B. In EPR the electron spin is flipped, in NMR the nuclear spin (specifically, this graph is for 1 H)

not require a change of , so that they typically occur within one n level (also possible for = 0). Of course, again the total angular momentum, including the photon angular momentum Jph = 1 is conserved: The triangular relation δ(Ja 1Jb ) = 1 must hold. The most simple case is Ja = Jb = S = 1/2, where for the spin projection we have

MS = ±1.

(9.84)

The microwave frequency needed to induce a typical EPR transition is thus

WS / h = 2WS / h = 2ge μB MS B/ h 28 GHz B/ T,

(9.85)

corresponding to the electrons L ARMOR frequency in the field B. Figure 9.19 displays schematically the energy splitting for the magnetic moment of a free electron in an external B field, and compares it to nuclear magnetic resonance (NMR) with protons (1 H). The latter will be discussed in Sect. 9.5.3. For both methods Fig. 9.19 indicates typical MW and RF frequency bands, used in present research together with the respective magnetic fields. For chemical analytics the X band is still used as a standard method. However, with increasing complexity of the molecules investigated, very high magnetic fields are used (high field EPR), which can only be reached with state-of-the-art super-conducting magnets. As the splitting is proportional to the external magnetic B field, one expects the resolution to increase accordingly. EPR spectroscopy lives of the fact that the magnetic moment of the electron is a very sensitive probe for magnetic fields in its molecular environment. In a com-

486

9

Hyperfine Structure

(b)

WS MW source

1

circulator

3

fixed frequency

detector

B 2

MW cavity with probe

(a)

reflected microwave power

reflected signal

irradiation from source

(c)

B

differentiated

signal ×(- 1)

Fig. 9.20 (a) Schematic of the microwave bridge in a CW-EPR spectrometer. The microwave enters at point 1 the so called circulator which guides it a point 2 completely onto the probe. In contrast, the reflected signal, entering at point 2, is routed towards point 3. (b) Spectra are recorded at fixed frequency, the magnetic field B is tuned so that the resonance energy WS changes (straight red lines). (c) The resulting absorption signal (full black line) as a function of B; the signal is differentiated so that a dispersion like line shape (red, dashed) emerges frequency counter and power meter

reference arm with attenuator detector diode

MW source

PSD

output signal

circulator MW cavity with probe

modulation unit modulation coils

main magnet

Fig. 9.21 Scheme of the whole CW-EPR spectrometer with the microwave bridge Fig. 9.20. The magnetic field is provided by the main magnet and the modulation coils. The probe is positioned in the resonator cavity and the signal is detected by a phase sensitive detector (PSD)

plex molecule the unpaired electrons interact e.g. with the nuclear spins of different atoms, leading to characteristic spectroscopic patterns, from which one may glean very detailed information about the molecular structure. In the standard, continuous EPR method (CW-EPR) a spectrum is obtained at fixed frequency of the microwave radiation by tuning the magnetic field. A characteristic setup is shown in Figs. 9.20–9.21. At each resonance, losses occur in the microwave resonator which can be detected with high sensitivity. In contrast to optical spectroscopy, one typically observes here differentiated signals (dispersion like) which often allow for better resolution of the structures. Figure 9.22 shows an example for an experimentally obtained EPR spectrum. We cannot go into the details here. As the resolution increases with B (requiring correspondingly higher resonance frequencies for the same transitions), modern EPR spectroscopy uses frequencies

9.5


487

0

EPR signal / arb. units

Fig. 9.23 Comparison of (a) an X band CW-EPR spectrum and (b) a pulsed high field EPR spectrum exemplified by a complex molecule according to P RISNER et al. (2007); note that the signal (a) is differentiated, however, (b) is not

CH3 — CH2 — O- H

absorption (differentiated)

Fig. 9.22 CW-EPR spectrum for the ethanol anion as an example obtained in the X band, after SDBS (2013). The different HFS coupling constants a for the various atoms lead to a pronounced structure of the spectra

20

40 60 ΔB / mT

(a)

Azz

80

(b)

gy

gz gx

332

336

gz

gx

gy

Azz 340 B / mT

6420

6440

up to 700 GHz and super-conducting magnets, today up to 30 T (high field EPR). This allows one to investigate structural details even in large molecules, specifically in radicals. This is illustrated in Fig. 9.23 for the example of a stable nitroxyl radicals (“TEMPO” in polystyrene). The spectrum recorded with conventional CW-EPR spectroscopy in the X band (a) and the pulsed high field EPR spectrum (b) show a similar magnitude of hyperfine coupling Azz – as expected from (9.39) (more precisely, Azz is the z-component of the HFS coupling tensor Eq. (9.22)). In contrast, the splitting of the lines, characterized by the g factor, scales with the magnetic field B. The gJ factor (which describes the superposition of the electron cloud and the electron spin) becomes itself a tensor in the field of a non-isotropic molecule. While the anisotropy of this g tensor is barely indicated in Fig. 9.23(a) it becomes evident and clearly measurable in the high field EPR spectrum (b).

9.5.3

NMR Spectroscopy

The frequencies for nuclear magnetic resonance (NMR) transitions with protons, 1 H, are also indicated in Fig. 9.19. For the energies of the nuclear spin states |I M I in an external magnetic field and the transitions between them, rules hold in complete analogy to those for the electron spin states. Specifically, for an uncoupled proton spin (e.g. in an atomic 1 S0 state), the energy splits into two levels in an external B field with 1 Wp / h = −gp μN MI B/ h = ∓42.56 MHz B/ T, 2

488

9 RF oscillator

Hyperfine Structure

RF receiver receiver coil

magnet pole

magnet pole probe

signal processing and control

tuning coils tuning

Fig. 9.24 Simplified schematic of an NMR spectrometer. A (magnetic) RF field at fixed frequency is applied perpendicularly to the main magnetic field B. The latter is tuned through the resonances which induce a current in the receiver coil due to the spin flip upon RF absorption. This signal is amplified, filtered and recorded in a PC

with gp = 5.58569471(5) being the g factor of the proton and MI = ±1/2 its spin projection quantum number (magnetic quantum number). Again, M1 transitions may be induced by interaction of the nuclear magnetic moment with magnetic field component of an electromagnetic wave – or with a local, oscillating magnetic field. The transition frequencies for these nuclear magnetic resonances lie in the RF range up to several 100 MHz:

Wp / h = gp μN B/ h = 42.56 MHz B/ T.

(9.86)

The experimental methods of NMR are in principle similar to those for EPR – except that microwave radiation and techniques are replaced by RF. Schematically, a typical NMR spectrometer is shown in Fig. 9.24. NMR spectroscopy has been developed to one of the most powerful tools for structural analysis of complex molecules, up to proteins, since the resonance frequency of each observable nucleus is highly sensitive to its local environment. Characteristic is a diamagnetic screening, the so called chemical shift δ, typically given in ppm in respect of the standard reference material tetramethylsilane (TMS), so that the modified NMR transition frequencies are given by:12

Wp / h = gp μN (1 − δ)B/ h.

(9.87)

Characteristic features in the spectra reflect the magnetic fields of neighbouring nuclear moments. Figure 9.25 shows a typical experimental 1 H-NMR spectrum for ethanol as example. It is characterized by the influence of neighbouring protons: the N = 3 protons of the methyl radical form a quartet (Itot = N × 1/2 = 3/2, degeneracy 2 × Itot + 1 = N + 1 = 4; so called N + 1 rule). They change the NMR signal 12 If the electron charge cloud involves unsaturated p orbitals, the chemical shift may also be paramagnetic.


Fig. 9.25 1 H-NMR spectrum of ethanol recorded at 89.56 MHz in CDCl3 , according to SDBS (2013)

489 CH3 — CH2 — OH

absorption

9.5

CH2 ‘sees’ CH3

CH3 ‘sees’ CH2

OH

4

3 2 NMR shift δ / ppm

1

0

of the protons in the methylene correspondingly. Conversely, the N = 2 protons of the methylene form a triplet and determine the NMR signal of the methyl radical. A certain problem of NMR spectroscopy is the small difference of population in the two states with MI = ±1/2. It is given by

Wp

Wp N+1/2 − N−1/2 = 1 − exp − . (9.88) N+1/2 kB T kB T A numerical value is ca. 6.9 × 10−6 for protons in a magnetic field of 1 T at room temperature. This means, the relative difference in population density is less than 10−5 . The expected absorption signal is then

S = hν(B12 N+1/2 − B21 N−1/2 )ρ(ν)

(9.89)

with the E INSTEIN Bba absorption coefficient and the spectral radiation density ρ(ν) (see Chap. 4). The number of absorption and emission processes is proportional to the occupation probability of the states. Hence, both processes occur nearly equally often. The net absorption rate which may be detected is thus very small. The fact that NMR is nevertheless very sensitive is due to the large number of absorbing molecules in the solid or liquid probe (order of magnitude 1022 –1023 ). In contrast, in atomic beam experiments one always deals with just a few particles, which may, however, be detected with a 100 % efficiency, due to the high population difference. With (9.88) and (9.89) one immediately sees that for a good NMR signal high magnetic fields and low temperatures and crucial. A significant enhancement of the detection probability for NMR (as well as for EPR) is achieved by pulsed excitation with an RF pulse of significant bandwidth. One detects the time dependent response of the system to this excitation and averages over many such measurements. The F OURIER transform of this signal generates the true NMR spectrum, providing simultaneously information on the relaxation properties of the system. Significant further improvements have been introduced by so called 2D or even multi dimensional procedures which employ several RF pulses of different frequencies in judiciously chosen time sequences. In this way, one may follow the systems response to an excitation of a specific nuclear spin, and identify correlations among different, active nuclei. Spatial anisotropy in molecular crystals may significantly blur the resolution, particularly so in solid state NMR

490

9

Hyperfine Structure

spectroscopy. This problem can be overcome today by the so called magic-anglespinning (MAS): similar to photoelectron spectroscopy (see Sect. 5.5.3) the influence of such anisotropy is described by the L EGENDRE polynomial P2 (cos θ ), with θ being here the orientation of the crystal axis. If one spins the probe sufficiently fast around an axis pointing into the direction of the magic angle θmag = 54.736◦ at which P2 disappears, such anisotropy averages out and one observes sharp lines. Apart from proton NMR also 13 C NMR plays an important role in molecular structure analysis, in particular so for large biomolecules. The carbon isotope 13 C occurs in nature with an abundance of 1.1 % and may also be enriched if necessary. It has a nuclear spin I = 1/2, however, with gI = 1.4048236(28) its gyromagnetic ratio is only 1/4 of that for the proton, gp = 5.58. Thus, corresponding to (9.86), the splitting of the levels is only Wp / h = gI μN B = 10.71 MHz B/ T. On the one hand, according to (9.88) this leads to a lower sensitivity. On the other hand, this also implies a weaker interaction with the environment and brings the advantage that – in contrast to 1 H NMR – sharp lines are observed even in large, complex molecular structures, while the small splitting may be overcome by superconductive high field magnets. Thus, a rich and flexible NMR ‘toolbox’ is available for structural investigations. The combination of specific, site selective probes (isotope labelling), multidimensional methods and MAS allows today an extremely precise determination of the spatial structure of even quite large biomolecules in a solid state environment. EPR and NMR spectroscopy are today highly developed, powerful methods for structural analysis, available in many different varieties. In addition to the methods briefly indicated above, we have to mention sophisticated double resonance methods with simultaneous or sequential optical transitions, as well as the combination of electron spin and nuclear spin flips, and various time resolved and/or imaging procedures. In summary, we refer to a wide range of research with a host of applications in physics, chemistry, biology, medicine and materials sciences. Excellent monographs and reviews provide profound introduction into and in-depth information about the field, and a series of N OBEL prices e.g. to E RNST (1991), W ÜTHERICH (2002), and L AUTERBUR and M ANSFIELD (2003) illustrates important steps of progress and the far reaching significance of magnetic resonance spectroscopies. Section summary

• Magnetic resonance spectroscopy exploits M1 transitions between different magnetic sublevels of one n level in the RF and MW region (thus avoiding D OPPLER broadening). Today, magnetic fields up to 30 T are used. The transition frequencies for electron spin and proton spin flip are summarized in Fig. 9.19. • ‘Classical’ molecular beam spectroscopy, still used today, employs electric or magnetic deflection and focussing to detect atoms and molecules sensitively in well defined magnetic substates. These are depleted in a magnetic field by the transitions studied.


491

• In EPR spectroscopy one studies electron spin flip transitions in the MW spectral region. It is applicable to molecules with unpaired electrons in the liquid or solid state. The g tensor of the electron acts as a highly selective probe for structural analysis of its chemical environment, e.g. in large molecular radicals. • NMR spectroscopy is today one of the most powerful methods for chemical analysis. Mostly 1 H spin flips are used, but other nuclei such as 13 C are also serve as probes in large molecules. The observed transition frequencies are very sensitive to the local environment of the nuclei.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2. CW: ‘Continuous wave’ (as opposed to pulsed) light beam, laser radiation etc. DC: ‘Direct current’, unidirectional electric voltage and current. 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). FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6). 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)’. HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9). IS: ‘Isotope shift’, of spectral lines due to different atomic number (part of the HFS). M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. MAS: ‘Magic angle spinning’, fast rotation of a solid state probe in NMR to overcome spatial inhomogenities. 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 and 300 GHz; ISO 21348 (2007) defines it as the wavelength range between 1 mm to 15 mm. 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). QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction.

492

9

Hyperfine Structure

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. VIS: ‘Visible’, spectral range of electromagnetic radiation. Wavelengths between 380 nm and 760 nm according to ISO 21348 (2007).

References A NGELI , I.: 2004. ‘A consistent set of nuclear RMS charge radii: properties of the radius surface r(N, Z)’. At. Data Nucl. Data Tables, 87, 185–206. A RIMONDO , E., M. I NGUSCIO and P. V IOLINO: 1977. ‘Experimental determination of hyperfinestructure in alkali atoms’. Rev. Mod. Phys., 49, 31–75. B LATT , J. M. and V. F. W EISSKOPF: 1952. Theoretical Nuclear Physics. New York: John Wiley & Sons, Inc., 1979, Springer-Verlag New York edn. B REIT , G. and I. I. R ABI: 1931. ‘Measurement of nuclear spin’. Phys. Rev., 38, 2082–2083. C EDERBERG , J. et al.: 2005. ‘An anomaly in the isotopomer shift of the hyperfine spectrum of LiI’. J. Chem. Phys., 123, 134321. C RAMPTON , S. B., N. F. R AMSEY and D. K LEPPNER: 1963. ‘Hyperfine separation of groundstate atomic hydrogen’. Phys. Rev. Lett., 11, 338. D RAKE , O. W. F., W. N ÖRTERSHÄUSER and Z.-C. YAM: 2005. ‘Isotope shifts and nuclear radius measurements for helium and lithium’. Can. J. Phys., 83, 311–325. E RNST , R. R.: 1991. ‘The N OBEL prize in chemistry: for his contributions to the development of the methodology of high resolution nuclear magnetic resonance (NMR) spectroscopy’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/laureates/1991/. F RICKE , G., C. B ERNHARDT, K. H EILIG, L. A. S CHALLER, L. S CHELLENBERG, E. B. S HERA and C. W. D EJAGER: 1995. ‘Nuclear ground state charge radii from electromagnetic interactions’. At. Data Nucl. Data Tables, 60, 177–285. G ALLAGHER , T. F., R. C. H ILBORN and N. F. R AMSEY: 1972. ‘Hyperfine spectra of 7 Li35 Cl and 7 Li37 Cl’. J. Chem. Phys., 56, 5972–5979. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. JACKSON , J. D.: 1999. Classical Electrodynamics. New York: John Wiley & sons, 3rd. edn., 808 pages. J ENTSCHURA , U. D., A. M ATVEEV, C. G. PARTHEY, J. A LNIS, R. P OHL, T. U DEM, N. KO LACHEVSKY and T. W. H ÄNSCH : 2011. ‘Hydrogen-deuterium isotope shift: from the 1s–2stransition frequency to the proton-deuteron charge-radius difference’. Phys. Rev. A, 83, 042505. K ARSHENBOIM , S. G.: 2005. ‘Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants’. Phys. Rep., 422, 1–63. K RAMIDA , A. E.: 2010. ‘A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium’. At. Data Nucl. Data Tables, 96, 586–644. K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. L AUTERBUR , P. C. and S. P. M ANSFIELD: 2003. ‘The N OBEL prize in physiology or medicine: for their discoveries concerning magnetic resonance imaging’, Stockholm. http://nobelprize. org/nobel_prizes/medicine/laureates/2003/. M ORTON , D. C., Q. W U and G. W. F. D RAKE: 2006. ‘Nuclear charge radius for 3 He’. Phys. Rev. A, 73, 034502. N EYENS , G.: 2003. ‘Nuclear magnetic and quadrupole moments for nuclear structure research on exotic nuclei’. Rep. Prog. Phys., 66, 633–689.

References

493

P RISNER , T. F., M. B ENNATI and M. M. H ERTEL: 2007. ‘Experimental example of an EPR spectrum in the X band and high field EPR’, private communication. 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/. R AMSEY , N. F.: 1950. ‘A molecular beam resonance method with separated oscillating fields’. Phys. Rev., 78, 695–699. R AMSEY , N. F.: 1989. ‘The N OBEL prize in physics: for the invention of the separated oscillatory fields method and its use in the hydrogen maser and other atomic clocks and the separated oscillatory fields method’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1989/. SDBS: 2013. ‘Spectral database for organic compounds’, National Institute of Advanced Industrial Science and Technology (AIST), Japan. http://sdbs.db.aist.go.jp, accessed: 8 Jan 2014. S MITH , D. D., G. L. S TUKENBROEKER and J. R. M C NALLY J R .: 1951. ‘New data on isotope shifts in uranium spectra: U236 and U234 ’. Phys. Rev., 84, 383–384. S TONE , N. J.: 2005. ‘Table of nuclear magnetic dipole and electric quadrupole moments’. At. Data Nucl. Data Tables, 90, 75–176. WALLS , J., R. A SHBY, J. J. C LARKE, B. L U and W. A. VAN W IJNGAARDEN: 2003. ‘Measurement of isotope shifts, fine and hyperfine structure splittings of the lithium D lines’. Eur. Phys. J. D, 22, 159–162. W ÜTHERICH , K.: 2002. ‘The N OBEL prize in chemistry: for his development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/ laureates/2002/.

10

Multi-electron Atoms

The last chapter in this volume is dedicated to eigenstates and energies of atoms with many electrons. For these heavy, complex atoms, in principle all electrons have to be treated equally. The repulsion of the electrons among each other is of the same order of magnitude as the C OULOMB attraction of the nucleus. So, this problem can no longer be treated by perturbation theory.

Overview

In this final chapter of Vol. 1 much comes together what has been introduced in the preceding text. In Sects. 10.1–10.2 we give an overview on the classical methods for computing multi-electron wave functions, applying what we have learned in Chap. 3 and Chap. 7. In Sect. 10.3 a short excursion into density functional theory is added. With increasing atomic number Z not only the number of electrons grows – and hence the complexity of the problem. Also the significance of different types of interaction changes: for light atoms exchange interaction was dominant, and LS coupling gave a good description as described in Chap. 6. With increasing spin-orbit interaction, LS coupling is no longer adequate, as we shall illustrate in Sect. 10.4. The energy scale as a whole changes – roughly ∝ Z 2 . Transition energies for quantum jumps within the outermost electron shells are still in the VIS or UV spectral range; the reader may familiarize him- or herself by way of a few characteristic examples with the ‘zoo’ of energy levels and coupling schemes in complex atoms. Changes in the inner shells are associated with emission or absorption of X-ray radiation, which treated in Sect. 10.5 – complementing Chap. 4. In addition, our understanding of photoionization (Chap. 5) is deepened by some examples of multi-electron systems. Finally, Sect. 10.6 introduces the reader to modern sources for the generation of X-rays.


495

496

10.1

10


Central Field Approximation

The most generally used approach for solving the multi-electron problem remains the independent particle model – as in essentially all the preceding chapters: Each of the N electrons moves in the C OULOMB potential of the nucleus and in the averaged potential of all other (N − 1) electrons (also called mean field). Any direct correlation between the electrons due to their momentary, individual coordinates is neglected – this is the key assumption. The motion of each electron thus depends on those of the other electrons only in as far as they contribute to the average repulsion potential. The character of radial wave functions (electron orbitals) in such a screened C OULOMB potential will be very similar to those in the hydrogen atom. This will be manifested e.g. in the asymptotic behaviour at small and large r, and in the number of zero-crossings. Nonetheless, pure hydrogen like orbitals are typically no longer a good 0th order approximation for larger N and Z. One thus has to solve the S CHRÖDINGER equation numerically, and then to refine the independent particle model step by step. The most prominent observations may be described already in this manner without further effort.

10.1.1 Hamiltonian for a Multi-electron System We consider an atom with nuclear charge Z and N electrons and start again by just including electrostatic forces. For the moment, we neglect are all magnetic interactions as well as relativistic and QED effects. Following the approach taken in Chap. 7 for the two electron system He, the total energy is obtained as the sum over all energies of each electron: kinetic energy, C OULOMB attraction by the nucleus, and C OULOMB repulsion from all other electrons. Extending (7.4), the H AMILTON operator for the multi-electron system becomes: = H

N 2 pˆ j j =1

Ze2 − 2me 4πε0 rj

N

+

N

1 e2 2 4πε0 rj k

(10.1)

j k=j

with rj k = |r j − r k |.

(10.2)

The factor 1/2 in front of the double sum compensates the double appearance of the repulsive terms 1/rkj and 1/rj k , thus allowing to write the equation symmetric in j and k. In atomic units (10.1) reads = H

N j =1

N

N

1 1 hej + 2 |r j − r k |

(10.3)

j k=j

1 Z with hej = − j − , 2 rj

(10.4)

10.1


497

the latter representing the unscreened single electron H AMILTON operator in the field of the nucleus. The full S CHRÖDINGER equation to be solved is Ψ (q 1 , q 2 , . . . , q N ) = W Ψ (q 1 , q 2 , . . . , q N ), H

(10.5)

q j referring to position (rj , θj , ϕj ) and spin (sj ) coordinates of electron j . In contrast to the simple models for He in singly excited states, for multi-electron systems in general, the repulsion terms in (10.3) can no longer be regarded as a (small) perturbation – especially when summed over all electrons. Strictly speaking, the terms ∝ |r j − r k |−1 make the potential non-isotropic and different for each electron. In addition, we have to recall that the N electrons are indistinguishable and occupy N different sets of quantum numbers, each of which is characterized by a specific set of quantum numbers, different from all others (PAULI exclusion principle). The total wave function must be antisymmetrized with respect to exchange of any two electrons. This all leads to substantial complications, and any attempt to solve the S CHRÖDINGER equation (10.5) in a direct way – even in numerical form – will be prohibitively elaborate. One thus has to resort to intelligent approximation strategies, the two most important of which will be sketched in Sect. 10.2 and Sect. 10.3.

10.1.2 Centrally Symmetric Potential However, we first introduce some of the underlying essential concepts – ignoring for the moment the complications arising from the requirement of antisymmetrization. The key idea of the central field approximation is, to represent the repulsion among electrons by an averaged, centrally symmetric potential. This centrally symmetric potential allows one – at least in essence – to separate the problem, and to solve the S CHRÖDINGER equation (10.5) by a product of single particle wave functions (orbitals) for each electron: Ψ (r 1 , r 2 , . . . , r j , . . . , r N ) = ψ1 (r 1 )ψ2 (r 2 ) · · · ψμ (r j ) · · · ψN (r N ).

(10.6)

Basically, this is the mathematical formulation of the independent particle model. The indices 1, 2, . . . , μ, . . . , N of the one-particle wave functions represent the usual sets (7.10) of quantum numbers μ = {nμ μ mμ msμ }

(10.7)

for the N occupied orbitals of the N atomic electrons. Together, they characterize the total wave function1 Ψ (1, 2, . . . , μ, . . . , N ) and the total energy W = W {1, 2, . . . , μ, . . . , N }. The individual orbitals are assumed to be orthonormal ψμ (r)ψλ (r) = δμλ . and in the following the index μ stands for a set of quantum numbers {nμ μ m μ } (later also for {nμ μ m μ msμ }. If no ambiguities can arise, we abbreviate q j , as well as r j and sj by j – as we also have done in Chap. 7.

1 Here

498

10


For electron j the centrally symmetric potential may be written as Vj (rj ) = Ψ |

N k=j

1 |Ψ , |r j − r k |

(10.8)

where Ψ | |Ψ stands for integration over all other electrons k = j . For very small as well as for very large rj this potential will behave as discussed for the active electron in alkali atoms (see (3.10) and Fig. 3.10). With this potential the full Hamiltonian (10.3) may be rewritten as composed of a centrally symmetric part c = H

N 1 e hj + Vj (rj ) 2

(10.9)

j =1

and a (hopefully) small perturbation term which is the difference between (10.3) and (10.9): , +N N 1 1 nc = H −H c = H − Vj (rj ) (10.10) 2 rj k j

k=j

nc contains all non-spherical parts of the interaction. Its diagonal term disappears H in 1st order perturbation theory and is neglected in the central field approximation. nc as a perturbation in higher order. Typically it However, one may considered H is small if one has already used ‘reasonable’ orbitals for determining Vj (rj ). The so called M ØLLER -P LESSET perturbation theory accounts for just this remaining difference, by computing 2nd (MP2), 3rd (MP3) and 4th order (MP4) corrections.

10.1.3 HARTREE Equations and SCF Method Obviously one has to find the one-particle wave functions (orbitals) ψ1 (r), ψ2 (r) to ψN (r) for all N electrons – ignoring for the moment the spin dependence. In order to derive the corresponding one-particle S CHRÖDINGER equations, one follows a variational approach, similar to that introduced in Sect. 7.2.5. Starting from the product ansatz (10.6) and the Hamiltonian (10.9) one minimizes the total energy |ψ of the system. We do not go into details, but the general idea outlined here ψ|H leads to plausible results. The orbitals ψμ (r) are found as eigenfunction of N S CHRÖDINGER equations e h + Vμ (r) ψμ (r) = Wn ψμ (r), (10.11) differing by the repulsive potential Vμ (r) which according to (10.8) is averaged over all electrons, except electron μ. Equation (10.11) is solved by the usual type of spatial wave functions ψn m (r) = Rn (r)Y m (θ, ϕ).

(10.12)

10.1


499

Of course, the radial functions Rn (r) are here not simply hydrogenic wave functions. Rather, one has to compute Rn (r) = un (r)/r numerically as already practiced for quasi one electron systems in Sect. 3.2.4 from the radial S CHRÖDINGER equation

d2 un Z ( + 1) un (r) = 0. + 2 Wn + − Vμ (r) − (10.13) r dr 2 r2 So far, this all looks straight forward. However, the determination of the averaged centrally symmetric repulsive potentials according to (10.8) is by no means trivial: in principle one already needs to know the orbitals of all electrons for the averaging procedure! Assume for a moment we know the orbitals. Then, averaging over the repulsion from all other electrons k = j implies to insert (10.6) into (10.8) from left and right and to integrate over all other coordinates. As the orbitals are orthonormal, the relatively simple expressions Vμ (r) =

N

ψ r μ

μ =μ

N 2 1 3 |ψμ (r )| ψ r = r d μ |r − r | |r − r |

(10.14)

μ =μ

emerge. While μ refers to the quantum numbers of the orbital for electron j , the summation has to be carried out over all other N − 1 occupied orbitals, from μ = 1 to N . Now, one may formally insert this expression back into (10.11) and obtains for every occupied orbital μ = 1 to N : he ψμ (r) +

+ N

3 |ψμ (r

d r

μ =μ

)|2

|r − r |

, ψμ (r) = E¯ h ψμ (r).

(10.15)

These coupled integro-differential equations for ψμ (r) are called H ARTREE equations. Each of them contains the solutions ψμ (r ) of all others! It’s a major task to solve them. In practice this is done iteratively, as we shall discuss in the next section. We point out again that the potentials Vμ (r) may be different for each electron orbital. After all this is done, in 1st order (H ARTREE approximation) the total wave function Ψ defined by (10.6) is obtained from the N orbitals ψμ . The corresponding expectation value of the total energy WHartree {1, 2, . . . , N } of the system is derived from (10.9), using (10.14) for the mean repulsive potentials: c |Ψ = WHartree = Ψ |H

N μ=1

μ +

N N 1 Jμμ 2 μ=1 μ =μ

(10.16)

500

10


Fig. 10.1 Scheme for finding the self-consistent field potential for a multi-electron system (SCF method) – illustrated for the H ARTREE approximation

e with μ = ψμ (r) h ψμ (r) =

d3 rψμ∗ (r) he ψμ (r)

−1 and Jμμ = ψμ (r)ψμ r r − r ψμ (r)ψμ r |ψμ (r)|2 |ψμ (r )|2 . = d3 rd3 r |r − r |

(10.17) (10.18)

Here μ is the energy a single electron would have in the unscreened C OULOMB potential of the nucleus, and Jμμ the averaged repulsions between two electrons which we have already introduced as C OULOMB integral in Sect. 7.3.3. Note that WHartree is not equal to the sum over all orbital energies E¯ h from (10.15) – due to the factor 1/2. Without this factor the repulsive energies would be counted twice.

10.1.4 HARTREE Method The search for the best set of one-particle wave functions ψμ for all N occupied orbitals rests upon an iteration procedure first introduced and explored by H ARTREE. It was later on justified by S LATER, based on the variational method. It is sketched schematically in Fig. 10.1. One first ‘guesses’ a plausible 0th order approximation for the central field potentials Vμ (r) for all electrons of the configuration under investigation.

10.1


501

A possible ansatz for this potential is the T HOMAS -F ERMI model, to be discussed in the next section. With this potential one computes all electron orbitals in the configuration, and uses these in turn to derive improved potentials Vμ according to (10.14). With these potentials one repeats the procedure. At the end of each integration one checks how well the wave functions obtained in the calculation agree with those at the input. As long as the agreement is poor, one repeats the computation until it converges, i.e. until the potential reproduces itself within a preset limit. The whole procedure is called self-consistent field method (SCF).

10.1.5 THOMAS-FERMI Potential The classical textbook approach for the interaction potential in 0th order is the so called T HOMAS -F ERMI model. One assumes that the (many) electrons inside the atom follow essentially the rules for a F ERMI electron gas, as introduced in Sect. 2.4.3. However, in the present case the electrons are not contained in a box. Rather they are distributed in a potential V (r) < 0 which is determined by the C OULOMB potential of the nucleus and the screening from all electrons, just as outlined in Sect. 3.2.3. Thus, the maximum kinetic energy, F , electrons may have inside the atom is assumed to be position dependent, and given by F (r) = −V (r). The electron density (2.59) becomes also position dependent:

3/2 1 2me 3/2 . V (r) Ne (r) = 3π 2 2

(10.19)

On the other hand, one assumes the electrostatic potential −V (r)/e to be related to the charge density −eNe (r) by the P OISSON equation: e2

−V (r) = − Ne (r). ε0 We may eliminate Ne (r) and obtain the differential equation 3/2 e2 8 2m3e −V (r) − V (r) = or 3 4πε0 3π √ 3/2 8 2 −V (r) − V (r) = in atomic units, 3π

(10.20)

(10.21) (10.22)

from which V (r) can be derived. Since the potential is considered spherically symmetric, it depends only on r and = (1/r)d2 /dr 2 . The boundary conditions at very small and very large r are those for the screened C OULOMB potential given by (3.10). Thus, one scales this potential as Z V (r) = − Φ(r/b), r

(10.23)

502

10

0 -20 V(r) / Eh

Fig. 10.2 T HOMAS -F ERMI potential for the atomic core of neon (Z = 10). The full red line gives −(9Φ(r/b) + 1)/r: that is the averaged potential which a valence electron ‘sees’. For comparison also shown is the C OULOMB potential of the neon nucleus −10/r (limit for r → 0) and the completely screened potential −1/r (limit for r → ∞)

0.2

0

0.4

Multi-electron Atoms 0.6

0.8

1.0 r /a 0

-1/r THOMAS-FERMI model

-40 -60 -80

-Z/r

-100

with b being a Z dependent parameter:

3π 2/3 1 = 0.8853Z −1/3 . b= √ 1/3 Z 8 2

(10.24)

With this, and x = r/b one finally obtains from (10.22) the so called T HOMAS -F ERMI equation

d2 Φ Φ 3/2 = . dx 2 x 1/2

(10.25)

The limiting values of Φ(x) are 1 and 0 for x → 0 and x → ∞, respectively. The T HOMAS -F ERMI equation is treated in the literature on differential equations (see e.g. Z WILLINGER 1997) and describes with (10.23) a universal potential which scales with Z. Its relation to the electron density (in a.u.) is 3/2 Φ Z for x ≥ 0. (10.26) Ne (x) = 4πb3 x In the literature, one finds numerical data for the T HOMAS -F ERMI function Φ(x) as well as approximate analytical formulas, e.g. in L ATTER (1955): Φ(x) 1 + 0.02747x 1/2 + 1.243x − 0.1486x 3/2 (10.27) −1 + 0.2302x 2 + 0.007298x 5/2 + 0.006944x 3 . Note that this Φ(x) with (10.23) gives the full atomic model potential with the nuclear charge Z, screened by Z electrons – as experienced by an infinitely small probe charge. For one of the atomic electrons one might e.g. use the potential [−(Z − 1)Φ(r/b) − 1]/r as a initial input into the SCF calculation of atomic orbitals. Figure 10.2 shows this model potential for the example of Ne (Z = 10), in comparison to the pure C OULOMB limits for very small and very large distances. Section summary

• The Hamiltonian (10.1) for a multi-electron system with nuclear charge Z and N electrons contains – in addition to a sum over all single electron Hamiltoni-

10.2

•

• •

•

10.2

HARTREE-FOCK Method

503

ans – the repulsive potentials among the N electrons which make it the overall interaction potential non-isotropic. A surprisingly good key approximation is the independent particle model, according to which the total wave function may be written as product (10.6) of all single electron orbitals. It is based on the central field approximation which represents the electron repulsion by a centrally symmetric screening potential. Mathematically this may be cast into a set of N coupled integro-differential equations. Without antisymmetrization (i.e. neglecting electron exchange) these are the H ARTREE equations (10.15). Typically, they are solved by an iterative procedure in which at each iteration step the N orbitals are obtained from numerical integration of the radial equation; the screening potentials are obtained from averaging over the orbitals of the previous iteration step. This is repeated until the orbitals are reproduced in each step, i.e. until they are self consistent (SCF approximation). An initial estimate for the repulsive screening potential may be obtained from the T HOMAS -F ERMI model, which is based on a statistical electron distribution in the C OULOMB potential Z fold charged nucleus. A closed expression may be obtained from (10.23) with (10.27).

HARTREE-FOCK Method

The above discussion of the H ARTREE SCF method was meant as a first introduction on how to calculate wave functions and energies for multi-electron systems. So far, we have completely neglected the PAULI principle. It demands for some major additions to the procedure, first introduced by F OCK and independently by S LATER – the so called H ARTREE -F OCK (HF) method (also HF approximation or HF theory). It provides the basis for essentially all advanced methods of atomic and molecular structure computation and modern quantum chemistry. In detail it is a rather sophisticated theory and we cannot give here a tutorial for those who actually want to use it (we refer to the numerous textbooks on quantum chemistry, e.g. S ZABO and O STLUND 1996). In the following we just sketch some of its essential ingredients.

10.2.1 PAULI Principle and SLATER Determinant To account for the fact that the N electrons in an atom are indistinguishable fermions, we have to anti-symmetrize the total wave function in the S CHRÖDINGER equation (10.5). The H AMILTON operator is still represented by (10.3) – it is not explicitly spin dependent – and we start with product wave functions similar to (10.6). However, HF theory introduces so called spin-orbitals ψμ (q μ ) = ψμs (r j )χμ (j ) (as already applied for He in Sect. 7.3.1) and uses products of these:

504

10


Ψ (q 1 , . . . , q N ) = ψ1 (q 1 )ψ2 (q 2 ) . . . ψμ (q 2 ) . . . ψN (q N ) s (r N )χN (N ). = ψ1s (r 1 )χ1 (1)ψ2s (r 2 )χ2 (2) . . . ψμs (r j )χμ (j ) . . . ψN

The indices μ = {nμ μ mμ msμ } represent again the standard sets of four quantum numbers for each electron. Antisymmetrization is now realized most conveniently by the so called S LATER determinant (of an N × N matrix) which generates an antisymmetric sum of product wave functions: Ψ (q 1 , q 2 , . . . , q N ) ψ1 (q 1 ) ψ1 (q 2 ) 1 . . . =√ N ! ψ1 (q j ) ... ψ1 (q ) N

ψ2 (q 1 ) ψ2 (q 2 ) ... ψ2 (q j ) ... ψ2 (q N )

... ... ... ... ... ...

ψμ (q 1 ) ψμ (q 2 ) ... ψμ (q j ) ... ψμ (q N )

. . . ψN (q 1 ) . . . ψN (q 2 ) ... . . . . . . . ψN (q j ) ... . . . . . . ψN (q N )

(10.28)

Exchange of two particles corresponds to exchanging two rows of the determinant. This changes the sign of the total wave function. Thus, Ψ (q 1 , q 2 , . . . , q N ) is indeed antisymmetric in respect of exchanging two electrons. One verifies again the PAULI principle (see Sect. 7.3.1): if any two sets of quantum numbers, e.g. 1 and 2, are equal, two whole columns are equal, and the determinant becomes ≡ 0. The explicit form of the S LATER determinant (10.28) is very clear, but somewhat lengthy to write and read. Thus, one usually abbreviates it. We write it as a state vector: 1 s |Ψ = √ det ψ1s (1)χ1 (1) . . . ψμs (j )χj (j ) . . . ψN (N )χN (N ) . N!

(10.29)

Examples – Restricted vs. Unrestricted HF Let us discuss again He as the most simple example, using this terminology. We know already the results from Chap. 7 to compare with. The general (approximate) H ARTREE -F OCK wave function of He reads: Ψ (1, 2) = √1 det ψ s (1)χ1 (1)ψ s (2)χ2 (2) . 1 2 2!

(10.30)

In general, the spatial orbitals for different electrons will be different. If one fully allows for this possibility, one speaks of unrestricted H ARTREE -F OCK (UHF). For closed shells, one may assume that each spatial orbital is occupied twice, once by an electron with α spin and once by an electron with β spin. This approximation is called (closed shell) restricted H ARTREE -F OCK (RHF). It is obviously a

10.2

HARTREE-FOCK Method

505

reasonable approximation for the He ground state, since the two electrons are absolutely identical, both being in a 1s orbital: 1 s s Ψ1s 2 (1, 2) = √ det ψ1s (1)α(1)ψ1s (2)β(2) 2! α(1)β(2) − β(1)α(2) s s . (r 1 )ψ1s (r 2 ) = ψ1s √ 2 This is exactly the result (7.39) for the He ground state: a singlet state with total spin S = 0, MS = 0. Things are more complicated in the excited state – apart from the trivial cases α(1) = α(2) and β(1) = β(2). The S LATER determinant in UHF reads now 1 s s Ψ1sn (1, 2) = √ det ψ1s (1)α(1)ψn (2)β(2) 2! =

s (r )ψ s (r )α(1)β(2) − ψ s (r )ψ s (r )β(1)α(2) ψ1s 1 n 2 1 1s 2 . √ n 2

Note that this is neither a singlet nor a triplet state. Thus, as we know from Sect. 7.3, these are not eigenstates of the Hamiltonian (10.9) for the excited He system. We have already discussed in Sect. 7.3.4 how to amend this fact by constructing singlet and triplet states from linear combinations of S LATER determinants. Now let us have a brief look at Li. Its ground state configuration being 1s 2 2s, the appropriate S LATER determinant would read 1 s s s Ψ1s 2 2s (1, 2, 3) = √ det ψ1s (1)α(1)ψ1s (2)β(2)ψ2s (3)α(3) 3! 1 s s s or alternatively = √ det ψ1s (1)α(1)ψ1s (2)β(2)ψ2s (3)β(3) . 3! In the first case electron 1 and 3 have identical spin, in the second case electron 1 and 3 have opposite, 2 and 3 identical spin. Obviously the two electrons in the 1s orbital are no longer fully equivalent (as indicated by 1s ). Thus it makes sense to apply UHF. We shall see in a moment how this works out when actually determining the spin-orbitals in a H ARTREE -F OCK calculation. Clearly, the UHF procedure will be more expensive than RHF, and hence one often restricts calculations for large system with many electrons to RHF. We finally mention, that again orthonormality is required for all sets of orbitals. However, orthogonality between orbitals with opposite spin is automatically ensured since αβ = βα = 0. This eventually leads to one set of orthogonal spatial orbitals for α spin and one orthogonal set for β spin – which are not mutually orthogonal.

506

10


Simultaneous Measurement of Observables A few words on the simultaneous measurement of observables are now required: 2 Lj commute according to for electron j the angular momentum operators Lj and 2 (B.2). Since Lj is part of the j operator and commutes with any function of r j , also Lj commutes with the one particle H AMILTON operator. Thus, also L= Lj 2 z commute with the N particle Hamiltonian. And since the and hence L and L latter is completely independent of the spins, it also commutes with S= S j as 2 well as with S and with Sz . • One thus may generate fully antisymmetric states |γ LSML MS , which are eigenstates of the H AMILTON operator and simultaneously eigenstates of the total angular momentum and of the total spin. We have already constructed such states for atomic He. For the ground state of He the above S LATER determinant provides such a state. For the excited states, c linear combinations of several however, one needs for the diagonalization of H S LATER determinants in order to obtain RUSSEL -S AUNDERS (LS) coupled eigenstates |γ LSML MS . In Sect. 7.3.4 we have discussed this already in detail for the example of He atoms in their first excited states.

10.2.2 HARTREE-FOCK Equations In the H ARTREE -F OCK approximation (HF) one determines the S LATER determinant (10.29) in such a way that it minimizes the expectation value of the H AMILTON operator (10.3) – which (to 1st order) is then the total energy of the system, replacing (10.16):2 |Ψ = WHF = Ψ |H

N μ=1

μ +

N N 1 [Jμμ − Kμμ ]. 2

(10.31)

μ=1 μ =1

As in the H ARTREE approximation, μ is the energy (10.17) of a single electron in the unscreened C OULOMB field of the nucleus, while Jμμ is the C OULOMB integral (10.18). The difference between (10.31) and (10.16) is due to the characteristic exchange integrals −1 (10.32) Kμμ = ψμ (q)ψμ q r − r ψμ (q)ψμ q s∗ s s∗ s ψμ (r)ψμ (r )ψμ (r)ψμ (r ) = χμ |χμ 2 . d3 rd3 r |r − r | may wonder why the sum does not exclude μ = μ. Note, however, that in this case C OULOMB and exchange integral just cancel each other.

2 One

10.2

HARTREE-FOCK Method

507

We have met these already in Sect. 7.3.3 for the He case. They are – so to say – representatives of exchange interaction. The scalar product in front of the integral is =1 if the orbitals have the same spin, it is =0 if they have opposite spin. We cannot go into details of the variational procedure. The main outcome is again a set of N coupled integro-differential equations, this time for the spin-orbitals. These H ARTREE -F OCK equations may be written as he ψμ (r) +

+ N μ =1

, Vμd μ (r)

ψμ (r) −

+ N

, Vμex μ (r)ψμ (r)

= E¯ h ψμ (r). (10.33)

μ =1

The summations are over all occupied spin-orbitals (see footnote 2). The functions Vμd μ (q j ) and Vμex μ (q j ) are called direct potentials (or C OULOMB terms) and exchange potentials, respectively. They are, however, not potentials in the standard sense. Rather, they are defined through the spin-orbitals ψμ (q j ), with q = r, s, which in turn are to be determined from the HF equations. The direct potential corresponds essentially to the screening term in the central potential (10.14) of the H ARTREE method: −1 Vμd μ r = ψμ q r − r ψμ q =

d3 r

ψμ∗ (r )ψμ (r ) |r − r |

.

(10.34)

In principle, this matrix element of 1/|r − r | implies integration over the spatial coordinates r and the scalar product of spin functions. However, since bra and ket vector in Vμd are the same, the spin product always gives 1. In contrast, the exchange potential is defined as a so called non-local potential −1 Vμex μ (r) = ψμ q r − r ψμ q (10.35) s∗ ∗ s ψμ (q )ψμ (q ) ψ (r )ψμ (r ) 2 3 μ |χμ = dq = χ , r d μ |r − r | |r − r | with the characteristic exchange of the sets of quantum numbers μ vs. μ. It involves the orbital μ to be determined by (10.33) (and not just the average over all other orbitals μ as the direct potential). Thus, (10.33) is not even an eigenvalue equation for the spin-orbitals. We mention in passing that, very formally, one defines a so called F OCK operator which allows one to rewrite (10.33) so that it looks like an eigenvalue equation. Since from our perspective no deeper insights into atomic and molecular structure are gained by this concept, we suppress further details. The solution of the H ARTREE -F OCK equations for the spin-orbitals may in principle be done by iteration – following closely the scheme of the H ARTREE method described in Sect. 10.1.4. Alternatively, one introduces predefined basis sets (e.g. special Gaussian type functions) from which the orbitals are constructed as linear

508

10


combinations. The variational procedure is then applied to the coefficients, and thus an algebraization of the problem is achieved. In any case, one finally obtains, in addition to the binding energies Wμ of all electrons, the symmetries of the states, their orbitals, and the effective self consistent potentials. We recall again, that the total energy is then obtained from (10.31) and not (!) by simple summation over all orbital energies E¯ h (which would account for the repulsive terms twice). The S LATER determinants (HF states) are, however, not yet necessarily a correct description of the atom. Rather, they provide, a good starting point. As mentioned above, in the next step one has to diagonalize the total Hamiltonian. As long as spinorbit interaction does not play a major role (i.e. for light atoms where the L · S terms can be neglected or may be treated as small perturbations), one may diagonalize 2 2 L and S by linear combinations of several HF states within one configuration, invoking some angular momentum algebra with C LEBSCH -G ORDAN coefficients. We have explained this in detail in Sect. 7.3.4 for the He atom. The states thus obtained describe the system as good as possible in the framework of the independent particle model and Russel-Saunders coupling (LS coupling).

10.2.3 Configuration Interaction (CI) Irrespective of this ‘trivial’ LS diagonalization, the model of independent electrons which simply move in a screened central field, is an approximation which often does not meet demands for state-of-the-art accuracy: correlations between individual electrons are not included in simple product wave functions from one single configuration. Various ‘post HF’ approaches are available, such as the M ØLLER P LESSET perturbation theory discussed in Sect. 10.1.2. In respect of exchange interaction, the non-locality of the potential is a key property, which makes a simple perturbation approach difficult or impossible. One often tries nevertheless with clever local approximations, to estimate a quasi-potential for exchange (see e.g. Sect. 10.3). Some serious consequences from neglecting correlations have already been discussed in the context of autoionizing doubly excited states of He (see Sect. 7.6.1). One may try to include such correlations by a linear combination of orbitals from several configurations. This approach is called configuration interaction (CI). One optimizes the additions from other configurations e.g. by variational methods. The difference between the HF result (if necessary including the spin diagonalization) and the exact energy is defined as correlation energy: Wcor = Wexact − WHF .

(10.36)

For He in its ground state it amounts to Wcor = −0.114 eV (1.4 %), for neon already −10.3 eV (already 3 % of the total energy Wexact −3507 eV). Clearly, for atoms of the size of Ne and larger, efforts beyond the HF method are required. For even larger atoms the additional problem of spin-orbit coupling and relativistic effects

10.2

HARTREE-FOCK Method

509

have to be accounted for. They have to be included in a CI calculation. Yet another, serious problem arises from the fact that in spectroscopy one always determines transition energies, which are the difference between typically large total binding energies for all electrons. This may lead to rather big errors in the calculated energy difference, which is to be compared with the directly measured value – even if the relative precision of the total calculated energy appears astonishingly high.

10.2.4 KOOPMAN’s Theorem Helpful in this context is the so called KOOPMAN’s theorem: It states that the energies E¯ h calculated for each orbital with the HF equations (10.33) for each electron (more precisely for each set of quantum numbers) is approximately equal to the ionization energy of that electron. Consequently the excitation energy of an atom is approximately equal to the difference of the respective orbital energies. This is, however, just a 1st order approximation, since changes in the atomic structure due the excitation or ionization process are not accounted for. Today, the H ARTREE -F OCK method is almost exclusively used as an initial basis for often very elaborate CI procedures. These are by no means limited to atoms. Similar and much extended methods are used in molecular physics and quantum chemistry, as well as in solid state physics. A variety of powerful programmes are even commercially available (see also Sect. 3.6.8, Vol. 2). A detailed description would, however, reach far beyond the scope of the present book. Section summary

• The H ARTREE -F OCK (HF) method employs electron wave functions correctly antisymmetrized as S LATER determinants (10.28) constructed from spin orbitals ψμ (q μ ) = ψμs (r j )χμ (j ). • The latter are derived by solving the H ARTREE -F OCK equations (10.33), a set of coupled integro-differential equations which include a non-local exchange potential. • The total HF energy is obtained from (10.31) as a sum of single (unscreened) electron energies, C OULOMB screening energies and exchange integrals. However, in the general case one has to diagonalize the Hamiltonian (in LS 2 2 coupling together with L and S ) by a linear superposition of S LATER determinants for a given electron configuration. • These HF energies are still limited by the model of independent electrons moving in a central field. To account for electron correlations one has to go further, in the simplest case to M ØLLER -P LESSET perturbation theory. A rigorous treatment requires CI, i.e. a linear superposition of several electron configurations.

510

10


• KOOPMAN’s theorem gives a very useful 1st order approximation to ionization potentials and excited state energies for a single active electron.

10.3

Density Functional Theory

Walter KOHN (1998) received the N OBEL prize for the development of density functional theory (DFT). This method is today the method of choice for a range of applications in atomic, molecular and solid state physics (in their ground states) for energy and structural determination of complex systems. Different from the procedures discussed so far, DFT does not focus on the wave functions of a system – for physical interpretations these are anyhow often not of primary relevance. Rather, DFT as its name already implies, is a method to determine electron densities. The key goal is the characterization of the ground state properties for (interacting) many-particle systems, based on their single electron densities. The starting point of DFT is the fact (as proven by H OHENBERG and KOHN), that knowing the electron density ρ(r) of the ground state for an electronic system (with or without interaction) characterizes the system completely. In the model of independent particles, which interact only via an averaged potential with each other and with an external field or the field of many atoms (in a molecule or solid state material), the electron density of a system with N electrons is given by ρ(r) =

N ψj (r)2 .

(10.37)

j =1

The so called KOHN -S HAM orbitals ψj are obtained from the eigenvalue equations (in a.u.)

1 − + vKS (r) ψj (r) = j ψj (r). (10.38) 2 The density dependent (effective) KOHN -S HAM potential vKS (r) = v0 (r) +

ρ(r )d3 r + vxc ρ(r) |r − r |

(10.39)

is composed by an external potential v0 (for a single atom of nuclear charge Z this is simply −Z/r), the repulsion of the electrons among each other (the second term) and a non-classical exchange potential vxc (ρ(r)). The latter is the ‘functional derivative’ of the exchange energy function Wxc (ρ(r)) in respect of ρ(r): vxc (r) =

δWxc (ρ(r)) . δρ(r)

The nature of exchange interaction has been discussed already in Chap. 7.

(10.40)

10.3

Density Functional Theory

511

Equations (10.37)–(10.39) are called the KOHN -S HAM equations. Without exchange they are equivalent to the H ARTREE method. The key problem, and the main efforts of DFT thus is, to find an adequate treatment of the exchange potential. In the most simple variety, the so called local density approximation (LDA), one chooses a local potential derived from the local density functional which – similar as in the T HOMAS -F ERMI model (Sect. 10.1.5) – is directly related to the electron density: Wxc ρ(r) = xc ρ(r) ρ(r)d3 r. (10.41) Here xc (ρ) is the exchange correlation energy per electron at constant electron density ρ. In any case, the total state of the system is described by the corresponding S LATER determinant Ψρ(r) constructed with KOHN -S HAM orbitals according to (10.28). The procedure then follows a variational approach: the electron density ρ(r), which ultimately defines the KOHN -S HAM orbitals, is optimized until the expectation value of the H AMILTON operator (10.3) |Ψρ(r) W = Ψρ(r) |H

(10.42)

reaches a minimum. In mathematical form one writes |Ψρ(r) = 0. δW ≡ δ Ψρ(r) |H

(10.43)

Then (10.42) gives the best ground state energy within the framework of DFT, and the corresponding ρ(r) describes the best electron distribution for the ground state. Unfortunately we cannot go into more detail of this method which becomes increasingly important in modern structure theory of atoms, molecules and solid states. Figure 10.3 shows some typical results for atomic electron densities obtained with a simple PC programme (S CHUMACHER 2011) found on the Internet. According to the authors, the programme uses the “Xalpha” functional and “The model is not spin polarized and non-relativistic. Overall, it has an accuracy between LDA and LSDA”. Plotted in Fig. 10.3 is the radial electron density 4πr 2 Ne (r) = 4π nj j r 2 Rn2j j (r), summed over all occupied orbitals nj j so that ∞ 4π 0 r 2 Ne (r)dr = Z. The maxima of this distribution show the different, occupied shells. For comparison the completely smooth T HOMAS -F ERMI electron density (10.26) is also plotted – clearly it may only be used as a 0th order approximation. A standard textbook on DFT is PARR and YANG (1989). In the mean time, this powerful method has developed in a variety of brands. Particularly important progress has been made in recent years with time dependent density functional theory (TDDFT) which has become a key tool for physical and quantum chemical structure calculations and simulations of dynamic processes in complex systems. With certain tricks it is now possible to study excited states and, very important, optical transitions. The review of M ARQUES and G ROSS (2004) provides an excellent introduction, while more recent advances are summarized by C ASIDA (2009).

512

10 1s 2

4πr 2 Ne(r) / a0-1

6

6C

4 2s 2 2p 2 2 0

0

2


2 1s 2 2s 2 2p 6 12 1s 11Na 20 10 2s 2 2p 6 × 10 15 8 1s 2 6 2s 2 2p 6 10 4 3s 2 3p 6 × 10 5 2 2 3s only 1s 0 0 0 2 4 0 2 4 r/ a0

18Ar 10 Ne 2He

4

Fig. 10.3 Computed radial electron density for several atoms (full lines) in atomic units. For comparison the densities obtained from the T HOMAS -F ERMI potential (dashed lines). For Na, with its isolated valence electron in a 3s orbital, its radial density is multiplied by a factor of 10 (black, dash dotted line)

Section summary

• The density functional theory (DFT) provides a very efficient, self consistent, widely used method for electronic structure calculations (mainly for electronic ground states) in atoms, molecules and condensed matter. • It concentrates on electron densities, rather than on wave functions, and approximates exchange by different types of judiciously chosen functionals, in the simplest case by a local potential (LDA).

10.4

Complex Spectra

10.4.1 Spin-Orbit Interaction and Coupling Schemes Up to now we have restricted our description of multi-electron systems completely onto electrostatic interactions. As already discussed for the He atom, exchange interaction (which is of such electrostatic origin) leads to coupling of spin and orbital angular momenta of the N individual electrons, such that they combine to a total spin and a total angular momentum of the atom: S=

N i=1

Si

and L=

N

Li .

i=1

We recall that this kind of coupling by exchange interaction dominates for small nuclear charge Z and is called RUSSEL -S AUNDERS coupling or (somewhat ambiguously) LS coupling. In contrast, spin-orbit interaction is of magnetic origin. As long as it is small (e.g. in the case of He) it may be added as a perturbation, typically being proportional to L · S and to Z 4 (see Sect. 6.2.4). Under the influence of this perturbation, total

Complex Spectra

513

Fig. 10.4 Doublet splitting in the 3rd main group of the periodic system as a function of the nuclear charge Z. Red for the ground state ns 2 (1 S0 )np 2 P01/2,3/2 and grey for the ns 2 (1 S0 )(n + 1)p 2 P0 1/2, 3/2 excited state

fine structure splitting / cm-1

10.4

8000

Tl

... ns 2(1S0) np 2Po1/2, 3/2

6000 4000 In 2000

... ns 2 (1S0) (n+1)p 2Po1/2,3/2

Ga B Al

0

0

20

40

60

80

Z

spin and total orbital angular momentum finally couple to an overall total angular momentum of the active electrons: J = L + S. Except for very small Z one typically considers only the interaction of spins with their own orbit and neglects spin-spin as well as spin-other-orbit interactions. This leads to fine structure splittings which may be described very similar to those of quasi-one-electron systems. This also holds for the Z EEMAN splitting observed in a magnetic field. However, for large Z spin-orbit interaction terms become comparable or even larger than exchange interaction so that RUSSEL -S AUNDERS (LS) coupling does no longer describe the physical reality appropriately. Strictly speaking, L and S are then no longer good quantum numbers – even if they are often still used to characterize the atomic states. We have already reported deviations from the standard (LS) selection rules for optical transitions – and we shall see more of it below, e.g. in the case of the Hg atom, where strong intercombination lines are observed even though in the term scheme one still distinguishes a singlet and a triplet system (Fig. 7.13). In the extreme case for very high Z one typically finds spin-own-orbit interaction to dominate and the coupling scheme changes completely. Most appropriately it is described by jj coupling of the N atomic electrons: First all individual orbital Sj angular momenta Li (quantum number j ) couple with their respective spin (sj = 1/2) to form an individual total angular momentum J i (quantum number ji ). Only then the individual J i couple together (under the influence of exchange interaction which may now be considered to be a perturbation) and form the total angular momentum J of the atom: Li + Si Ji =

⇒

J=

N

J i.

(10.44)

i=1

There is a variety of cases in between LS and jj coupling – which we shall get to know by way of example in the following. First, we illustrate in Fig. 10.4 the increasing fine structure splitting for the elements in the 3rd main group of the

0

Sn

0

1P

1

Ge

spin-orbit interaction

4000

2000


configuration ... np(2P°) (n+1)s exchange interaction

Fig. 10.5 Exchange interaction and fine structure splitting as a function of Z for the example of the terms np(n + 1)s in the 4th main group of the periodic system. Plotted is the difference of the term energies in respect of the 3 P state 0

10 W (2S+1PJ) - W ( 3P0 ) / cm-1

514

3P

2

C

Si 1P

1

3P

3P

2

10

20

30

40

1

50

3P

0

Z

periodic system (Table 3.3). While FS is essentially negligible for the element boron (B), in the case of thallium (Tl) it reaches an order of magnitude of eV. A direct comparison of spin-orbit and exchange interaction becomes possible in the 4th main group: in Fig. 10.5 the excited ns 2 np(n + 1)s configuration for singlet and triplet states with their respective fine structure splitting are shown. For the carbon atom (small Z) exchange interaction clearly dominates, separating the 1 P1 and 3 PJ states by nearly 0.2 eV, while the fine structure splitting of the 3 PJ states is nearly negligible. In contrast, for large Z fine structure splitting (i.e. spin-orbit interaction) becomes comparable to the singlet-triplet splitting (i.e. to exchange interaction) or even larger – in particular so for tin (Sn). Thus, RUSSEL -S AUNDERS coupling hardly describes reality any longer correctly. The next element in this group, lead (Pb), is not shown here. Event though spectroscopic data are still denominated as singlets and triplets (K RAMIDA et al. 2013), missing terms in this scheme clearly document the transition from LS to jj coupling.

10.4.2 Examples of Complex Spectra The following Figs. 10.6–10.10 represent some characteristic examples for complex atomic spectra. We shall discuss these briefly. The presentation used here to show the term positions together with observed transitions are called G ROTRIAN diagrams. However, to avoid confusion we communicate only a few of the most important transitions and by no means all known energy terms. The diagrams contain in any case a lot of information as a result of many years of spectroscopic work of many research teams. The full data sets are collected by K RAMIDA et al. (2013), who have analyzed and evaluated the literature sources critically and presented them in a user friendly manner. We start with a typical and particularly important example, carbon (C), an element of the 4th group, for which a G ROTRIAN diagram is shown in Fig. 10.6. According to H UND’s rules, the ground state is a triplet with the configuration 1s 2 2s 2 2p 2 3 P0 , where the two spins of the 2p electrons are parallel (symmetric), the orbital wave function is, however, antisymmetric (PAULI principle). Since carbon is a light element (small Z = 6), spin-orbit interaction is weak, and hence the

Complex Spectra

515

2S

2D

2 2

o

2 22 o

P 3P

1

S 1 P 1 D 3 S 3P 3D

5 4

3So

o

5

17

o

o

o

120 110 2P 100 90 80 70 60 50 4P 40 30 2 2 2 5P 20 3 10 0

3Po

.5 94 3 13 156

8

24

16 6

3 19

2

o

P 1D 1Fo 3P 3D 3F

14

3

1

4 3

1189

940

1Po

2 22

11 6

1

C+ ionic ground state

1/2,3/2

2 22

5 4 70 60 50 40 30 20 10 0

2P

6 14 131 6

I

22

2

120 110 100

3

2

3Do

displaced terms 5So

-1 I / 1000 cm

(2 3P) / 1000 cm-1

C+ ionic excited state: 2 2

143

10.4

2 2

[He] 2

2 2 2 3P

C ground state o

Fig. 10.6 G ROTRIAN diagram for carbon, 6 C. WI (→ 2 P1/2 ) = 90820.42 cm−1 . Energies are given in 1000 cm−1 : left excitation energies of the neutral atom (C I), right the excitation energies of the ion (C II); wavelengths marked on the transitions are given in nm (rounded to integers)

splitting within the 3 PJ multiplet is relatively small: W (J = 2) − W (J = 0) = 43.40 cm−1 . On the left side of the diagram the terms are shown for which one of the two 2p electrons is excited. The configurations are 2s 2 2p ns (left most), 2s 2 2p np (middle) and 2s 2 2p nd (right). The spectra resemble somewhat those of the simple spectra of the alkali and alkaline earth atoms. The terms converge to the lowest ionization limit at which the C+ ion in its ground state 2s 2 2p 2 P is formed. The energetically next higher terms belong to the configuration 2s 2p 3 and 2s 2p 2 ns. Here a 2s electron is excited – one speaks about displaced terms. They converge to an ionization limit which belongs to excited states of the C+ ion as indicated on the right side of Fig. 10.6. Typically, only the lowest displaced terms are well known from experiments. The term diagrams for the heavier elements of the 4th group of the periodic system (Si, Ge, Sn and Pb) are similar – except for the increasing spin-orbit interaction as discussed already summarized in Fig. 10.5 for the ns 2 np (n + 1)s configurations as examples. Atomic nitrogen (N), for which the G ROTRIAN diagram is shown in Fig. 10.7, is an example for elements of the 5th group of the periodic system. The ground state is a quartet term (4 S0 ) with a configuration [He]2s 2 2p 3 . The excited states in which

516

10

Multi-electron Atoms 2s 2 p 3

[He]

2s 2 2p 2 3P

N+

0,1,2

5 So

ionic ground state

150 ns

0 11

17

40

2

0

20

2

10 0

4 11

5

12

30

displaced terms 528 4P 1487

3

50

410

13

60

8 11

W - W(2 4S) / 1000 cm-1

859 9

70

3 3 14

80

1558

4

90

5 4 3

6

100

5 4

5 4

40

30 2s 22p 2(1D) 2s2p 4 20 ns np 2D 2Fo 2 10 D 2Po 2Do 0

12

110

nd

13

120

np

4 o 4 o 2 o 2So D 4Po S 2Do P 2P 4F 2F 4P 4D 2D

123 132 95 10 7

WI

2 P 4P

2s 22p 2(3P)

50

W - WI / 1000 cm-1

160

2

[He] 2s 2 2 p 3 4S0 N ground state

Fig. 10.7 G ROTRIAN diagram for nitrogen, 7 N. WI (→ 3 P0 ) = 117 225.7 cm−1 , WI (→ 1 D) = 132 541.9 cm−1 . Otherwise as Fig. 10.6

one of the three 2p electrons is excited may be divided into three groups, related to the N+ ionic state towards which they converge: 3 P, 1 D or 1 S. On the left of the diagram the configurations 2s 2 2p 2 ns, 2s 2 2p 2 np and 2s 2 2p 2 nd with their respective multiplets are given, converging to the 2s 2 2p 2 3 P configuration of the N+ ion. On the right of these the less well known configurations 2s 2 2p 2 ns and 2s 2 2p 2 np are displayed. They converge towards the N+ (2s 2 2p 2 1 D) state. Terms converging to the 2s 2 2p 2 1 S state of the N+ ion are not known for nitrogen. However, such terms are observed for the heavier elements of the 5th group (e.g. for As). The 4 P and 2 D states on the right side of the diagram belong to the configuration 2s 2p 4 , where one of the stronger bound 2s electrons is excited. As for carbon these are called displaced states. The G ROTRIAN diagram of oxygen (O), typical for the 6th group in the periodic system, is shown in Fig. 10.8. The ground state configuration 1s 2 2s 2 2p 4 leads to the same three states 3 P, 1 D and 1 S as in the case of the 4th group. The two extra electrons have opposite spin and fill two orthogonal p orbitals. The ordering of the 3 P multiplet is now inverted: for oxygen the 3 P2 term is energetically lowest, while for carbon the terms are normally ordered. Since the O+ ion is isoelectronic with neutral N, three terms (4 S, 2 D and 2 P) are possible in the O+ ionic ground state configuration 2s 2 2p 3 . Thus, there are three ionization limits towards which the excited state configurations of the neutral atom 2s 2 2p 3 nl may converge.

Complex Spectra

I (→

2Po)

I (→

2Do)

517

O+ ion ground state o [He] 2 2 2 3 4S

2 22

3(2Po)

60

1Po o 1S 1,3D 1,3 o o 3P 1 P 1,3D

P

= 150.3 1Do 3Do

2 22

= 136.6

3(4S)

1,3Po 1,3Fo

1,3D 1P

5 4

99

95 103

2 2

20

3Po

88

80

79 displaced terms

2s2 2p4 1S

92 20 0

5

1 30

40

98

60

1,3G

395

11 5

8 77

3

D

40

o

0

3

94

80

S

o

4 3

77 80

I

100

1,3 o 1,3

1,3F

5 4 3P 5 P 120 o o o 3D 5Do 3 S 5S

( 2 3P) / 1000 cm-1

2 22

3(2Do)

-1 I / 1000 cm

10.4

2s2 2p4 1D [He] 2s2 2p4 3P

O ground state o

Fig. 10.8 G ROTRIAN diagram for oxygen, 8 O. WI (→ 4 S3/2 ) = 109 837.02 cm−1 . Otherwise as Fig. 10.6

Next we discuss the term scheme of neon (Ne) as a typical representative of the rare gases. All rare gases are characterized by a closed rare gas outer shell (ns 2 np 6 ). The neutral ground state is thus always a 1 S0 state. The G ROTRIAN diagram of neon is shown in Fig. 10.9. Its neutral ground state configuration is 1s 2 2s 2 2p 6 . Excitation of one electron from the 2p shell leads to terms which converge to the Ne+ ion ground state 2s 2 2p 5 2 Pj with j = 3/2 and 1/2. The coupling scheme cannot be expressed in the form of the typical RUSSEL -S AUNDERS coupling, rather one uses a j − scheme: In this case the ionic ground state configuration 2s 2 2p 5 2 Pj couples with the excited n electron to form an angular momentum K. Each [K] state splits under the influence of spin-orbit coupling (K couples with the electron spin of the excited electron) into a doublet with J = K ± 12 . One uses the terminology (2S+1 Lj )nl 2S+1 [K]J , e.g. the first excited state of neon is (2 P3/2 )3s 2 [ 32 ]2 . In all rare gases some of the first excited states, those with J = 0, 2 or 3, are metastable: they cannot decay to the ground 1 S0 state due to the selection rule J = 0 ± 1, but 0 0. The famous red line of the helium-neon laser is a transition between two higher lying excited states in neon: (2 P1/2 )3p 2 [ 32 ]2 ← (2 P1/2 )5s 2 [ 12 ]1 , marked red in Fig. 10.9. The population of the upper level is achieved by collisions with metastable He in the 1s2s 1 S0 excited state. Depopulation of the lower laser level is supported by collisions with the glass wall of the laser cell.

518

10


Ne+ ionic excited state: 2s2p 6 2S

390

np 5 1 Po 4

380

220 210 200

370

3 190

360 230 [He] 2s 2 2p 5 2Po1/2, 3/2 Ne+ ionic ground states 2s 2 2p 5 (2Po

210

ns

np

200

W - W(2 1 S0) / 1000 cm-1

3/2)

nf

ns

np

nd

40

1/2)

nf

30

2[7/2]o 2 [9/2]o

27

20

2[5/2]o 2[5/2]o 2[7/2]o

190 WI

nd

50

2s 2 2p 5 (2Po

2[3/2]o 2[3/2]o 2[5/2]o

2[3/2]o 2[5/2]o 2[7/2]o

180

2[3/2]o 2 [1/2]o 2[1/2]o 2[3/2]o

2[1/2]o 2[1/2]o 2[3/2]o 2[5/2]o

170

6 5

160

4

150

6 5 4

10 0

4 3

4

1838

W - WI / 1000 cm-1

220

627

3

140 130

3 62

63

displaced terms

120 110

74 [He] 2s 2 2p 6 1S0 Ne ground state

0 o

o

Fig. 10.9 G ROTRIAN diagram of neon, 10 Ne. WI (→ 2 P3/2 ) = 173 929.75 cm−1 , WI (→ 2 P1/2 ) = 174 710.17 cm−1 . The so called j - coupling scheme holds. Otherwise as Fig. 10.6

In neon one also observes some weak absorption lines in the XUV range. These transitions lead from the ground state to highly excited states with the configuration 2s 2p 6 np. These are again displaced terms where one of the 2s electrons is excited to an np level. Such terms are also found for heavier rare gases. As a last example we consider aluminum (Al). Its G ROTRIAN diagram is shown in Fig. 10.10. The ground state configuration [Ne]3s 2 3p contains only a single electron in the 3p shell. The excited states where this 3p electron is excited thus form a simple, alkali like term scheme as indicated on the left in the diagram Fig. 10.10. All these excited terms converge towards the Al+ ionic ground state 3s 2 1 S0 . In addition, there are series of complex terms, all except one embedded in the ionization continuum, as shown on the right hand side of the diagram. A 3s electron is excited in this case. There are three terms with the electron configuration 3s 3p 2 , the 4 P multiplet having the lowest energy. Terms with higher excited configurations J 3s 3p n finally converge all towards the 3s 3p 3 P excited state of the Al+ ion.

X-Ray Spectroscopy and Photoionization o

Al+ ionic ground state: 3s 3p(3P ) 3p nd ns np

100 3s 2 1S

[Ne] 0 Al+ ionic ground state

o

WI (→ 3P ) = 85.7

60

2 S 2P 4 P

3s 2(1S0) ns np nd nf 2S 2 2 3 1/2 P3/2,1/2 D5/2,1/2 F7/2,5/2

WI

4 Po

5

2S 2 2 P D

5 4

4

2D

o4

2F

50 o4

P D

40

5 4

30

3

20 10 0

40

6 5

4

5 4 3 5 131

30 20

0

5 4 08 3 5 112

34

3

displaced terms

4

4

10

4

39

W - W(3 2 P1/2) / 1000 cm-1

70

519

3

[Ne] 3s 2 3p 2P1/2 Al ground state

W - WI / 1000 cm-1

10.5

Fig. 10.10 G ROTRIAN diagram for aluminium, 13 Al. WI (→ 1 S0 ) = 48 278.37 cm−1 . Otherwise as Fig. 10.6

Section summary

• With increasing atomic number spin-orbit interaction increases ∝ Z 4 and RUSSEL -S AUNDERS (LS) coupling gives a less and less appropriate description. • In the limit if very high Z the orbital angular momentum and spin of each electron couple with each other to j under the influence of spin-orbit interaction, and only at the end the different j couple by the weaker exchange interaction in a jj coupling scheme. • We have discussed several examples of term schemes (as G ROTRIAN diagrams) for more complex atoms with rich structure and increasing complexity. Excited states of noble gases are described by a j − coupling scheme. • We memorize that C, N and O each have 3 lowest multiplet levels with the same electron configuration 2s 2 2p 2 , 2s 2 2p 3 , and 2s 2 2p 4 , respectively. These levels are spaced by about 1000 to 2000 cm−1 and follow H UND’s rules. In the case of O the lowest level is 3 P followed by 1 D and 1 S.

10.5

X-Ray Spectroscopy and Photoionization

Today, X-ray spectroscopy is not only an important tool in modern research for detailed structural analysis – be it in the context of biomolecules, nano-materials, soft or crystalline matter – it is also indispensable for analytical purposes in physics, chemistry, medicine and technology. Analytical chemistry and physics makes use of X-ray spectroscopy in many areas of application, e.g. in archaeological science to determine the origin of historical findings, or to reveal whether some precious objects of art are original or fake – to mention just two not so mundane examples. One

520

10


important asset for today’s manifold applications of X-ray spectroscopy is the abundant accessibility to state-of-the-art electron storage rings which have been built exclusively for the generation and applications of synchrotron radiation, in particular in the VUV, XUV and X-ray spectral range. Since the methods for generating and detecting this kind of electromagnetic radiation may typically be considered part of optical physics (in its broadest sense) and since atoms with high Z play an important role in this context, this subject has to be discussed here. We can, however, only touch some aspects of this important area of research and collect some of the most important basics. In the present section we focus on terminology, methods, and key observations of spectroscopy. In Sect. 10.6 we shall then elaborate on some of the pertinent sources for X-ray radiation as prerequisite for spectroscopy. For more details the reader is referred to specialized literature (e.g. the rather comprehensive book of ATTWOOD 2007).

10.5.1 Absorption and Emission from Inner Shells Up to now we have been concerned almost exclusively with the spectra of valence electrons, i.e. from electrons in the outermost atomic shell. We now turn to the inner shells, in particular to those of larger atoms. Due to the characteristic dependence of atomic energies on the square of the nuclear charge Z, we expect the corresponding spectra at very short wavelengths. In uranium, e.g., an electron in the K shell has a binding energy of more than 110 keV. Interestingly, in certain respects these spectra are much easier to understand than those of the complex outer shells of intermediate size atoms which we have discussed in the last section with the help of G ROTRIAN diagrams: since all inner shells are filled, transitions between these levels can only occur when a hole is created in one of the shells – typically by collisions or by photoionization. We discuss these processes on the basis of a very schematic term scheme for an element “Tg” (target) with high atomic number Z, as sketched in Fig. 10.11. Orbitals and total angular momentum for each electron are again characterized by the quantum numbers n j , the designation of the shells as K, L, M, N, . . . and their respective subshells, following the usual schematic which we have introduced in Sect. 3.1.3. Let us assume that all these inner shells are filled. Three types of processes may be observed: • Absorption of a photon from one of the inner shell levels can (essentially) occur only into the continuum, since all inner shells are occupied. This leads to photoionization Tg(n ) + ω → Tg+ (n )−1 + e− (10.45) with the energy balance ω − WI = , where WI = WK , WL , WM etc. is the ionization potential from the respective shell. The kinetic energy of the electron ejected is . The initial electron configuration is described by (n ), and the symbol (n )−1 indicates that a corresponding

10.5


(a)

521

(b)

absorption (continua)

emission (lines)

n ℓ j

shells OI - O...

OI ...

NI ... WM

MI MII ... edges

WL

WK

LI LII LIII edges

K edge

3 3 75/2 3 3 5/2 3 2 3/2 3 2 3/2 3 1 1/2 3 1 1/2 3 0 /2

NVII

2 2 5/2 2 2 3/2 2 1 3/2 2 1 1/2 2 0 1/2

MV

NV NIII NI

MIII MI

2 1 3/2 2 1 1/2 2 0 1/2 Kα1 1 0 1/2

β2 β1 α2

LI

LII

LIII

NVI NIV NII

MIV MII

LIII LII LI

L series

K series

Fig. 10.11 Absorption and emission of X-ray radiation: (a) absorption is only possible into the ionization continuum, since the intermediate levels are fully occupied; (b) emission can only occur after generation of “holes” in the K, L, M, . . . shells. Lines with red arrows correspond to photon energies, black lines characterize the energetics of the system

“hole” has been created in the atomic shell. Since the energy balance has to be maintained by the ejected electron, one observes as a function of the photon energy so called absorption edges: for energies below WI there is no absorption. As soon as ω > WI , the corresponding shell can be ionized and the cross section for photoionization jumps to a finite value. For still larger ω it decays again, as we have already discussed in Sect. 5.5. We shall come back to this in a moment. But first, we discuss the other two processes. • If such a whole exists in an inner shell it may be refilled by spontaneous decay with an electron from a higher shell n . This process generates so called characteristic emission lines ' ( ' −1 ( + ω Tg+ (n )−1 . . . n → Tg+ n n (10.46) with the energy balance Wn − Wn = ω. In contrast to the absorption edges just described, one now observes a spectrum of discrete emission lines. Due to the well known ν 3 factor (4.38) this process is particular efficient in the X-ray region – and many orders of magnitude more

522

10

Fig. 10.12 AUGER electron emission. Three processes are relevant: (a) generation of a whole in a lower shell by photo-absorption (dashed red upward arrow), (b) electron transition from a higher level refilling this hole (red downward), (c) exchange of a virtual photon (wiggly grey line) and emission of an electron (red upward arrow), called an AUGER electron


o

o

Wn'ℓ' Wn''ℓ''

Wnℓ

(a)

(b)

(c)

probable than the absorption of a second photon. The corresponding line spectra may be read in principle from Fig. 10.11. They are essentially one electron spectra which may be understood more or less as directly as the spectra of hydrogen like ions – or better, alkali like ions if we want to account for the screening of the nuclear Z/r potential. One may also discuss these spectra in terms of holes being excited – instead of electron transitions. We come back to this concept quantitatively in a moment. • Finally a third process may occur. The hole is refilled by an electron from a higher shell, but – instead as in reaction (10.46) where a photon is emitted – the excess energy is used to eject a second electron. This so called AUGER electron emission process may be visualized as ' ( ' −1 ( Tg+ (n )−1 . . . n → Tg+ n . . . n + (ω) ' −1 −1 ( (10.47) n + e− , → Tg++ n . . . n where (ω) indicates a “virtual” photon which is exchanged between the two electrons, giving the latter sufficient energy to leave that atom. As one reads from Fig. 10.12 the energy balance is given by = Wn + Wn − Wn .

(10.48)

10.5.2 Characteristic X-Ray Spectra – MOSLEY’s Law In comparison to the rich material obtained from spectroscopy of outer shells, far less detailed data exist for the absorption and emission of X-ray radiation from inner shells – for obvious reasons: only during the past decades dedicated synchrotron radiation sources have provided intense, tuneable X-ray sources of excellent quality. Fortunately, the available material is well documented and excellently backed up by theoretical computations. We use here the NIST X-ray data-bank (C HANTLER et al. 2005), which offers characteristic emission lines in tabulated form, as well as

10.5


523 K

10 data

5 fit parameters (n*, qs)

)

.56

(0.924 , 7

LI LII

K

4

LIII

(Wedge

1.4 9)

1

.95

(1

.06

2,

3

MI

(1

keV) 1/2

)

.21

5 ,1

2

) ) .82 .98 MV , 21 99 , 24 0 6 (4.1 (3.5 NI ) 29.26 , 4 NIV 5 (6.4 NVII .89) 5 6 ) , 3 9 40.0 (5.01 (7.305 ,

5)

1

, 98

4.9

5

(2.

(5.1

13 ,

11.7

8)

0 0

10

20

30

40

50

60

70

80

90

Z

Fig. 10.13 M OSLEY diagram for the lowest X-ray absorption edges for all elements (compiled from the tables in C HANTLER et al. 2005). Grey and black lines give the results of fits with (10.50). The numbers in brackets are the parameters (n∗ , qs ). One sees that the inner shells cannot be fitted optimally over the whole periodic system with one set of parameters

absorption edges and photoionization cross sections over a wide range of energies, for all elements. The data are based mostly on state-of-the-art ab initio computations which have been tested extensively with the available experimental material. Figure 10.13 shows the X-ray absorption edges of the K, L, and some higher shells of all natural elements of the periodic system. This so called M OSLEY diagram is based on the assumption that the transition energies are obtained in analogy to the RYDBERG -R ITZ formula (1.149) for an effective one electron system with an effective nuclear charge Z ∗ = Z − qs : W =−

(Z − qs )2 Eh . 2 n∗2

(10.49)

Note that this also implies KOOPMAN’s theorem (see Sect. 10.2.4) – assuming that holes behave just complementary to electrons. The idea is thus to interpret the empirically determined ionization energies for the different shells (i.e. the position of the absorption edges) grosso modo by a combination of a screening parameter qs and the quantum defect n − n∗ . One plots the square root of the energies as a function of Z, as suggested by (10.49): E h Z ∗ Z − qs E h . (10.50) |Wn | = = 2 n∗ n∗ 2

524

10


We have already used this so called M OSLEY formula (3.34) for the alkali metal atoms. Ideally one would expect straight lines, as roughly confirmed by Fig. 10.13. However, the K and L edges cannot be fitted optimally with one set of parameters over the whole periodic system. For example, at low Z the effective quantum number of the K shell is nicely n∗ = 1.062 and the screening parameter qs = 1.49 is moderate. In contrast, for larger Z the best fit is obtained with n∗ = 0.924 and substantial screening qs = 7.56. For Z > 13 one finds a somewhat crude, but rather useful rule of thumb for obtaining the Kα energy: WKα (Z)/ eV 14 × (Z − 3)2 .

(10.51)

For aluminum (Al) with Z = 13 this gives 1400 eV (true value 1560 eV, soft Xray) and for tungsten (W) with Z = 74 the formula leads to 70 574 eV (true value 71 676 eV, hard X-ray).

10.5.3 Cross Sections for X-Ray Ionization We come now to a somewhat more quantitative discussion of photoionization for multi-electron atoms. It is by far the most important cause for the absorption of X-ray radiation by matter for photon energies up to some 100 keV. In this energy range photoionization cross sections, σa , are typically given in barn, [σa ] = 1 b (see Appendix A.2). Often one also finds the absorption coefficient μ (dimension L−1 ) tabulated. We have introduced it in Sect. 1.3.2 with the L AMBERT-B EER law I = I0 exp(−μd) for the reduction of intensity I by passage through matter of thickness d, with μ = Na σa , where Na is the particle density of the absorbers. The so called mass absorption coefficient μ/ρ (its dimension being L2 M−1 ) refers to the absorbing mass per unit area instead to d. With the density ρ = Na ma (ma = atomic mass) this gives μ/ρ = σa /ma . In Sect. 5.5 we have already treated photoionization in some detail. For high energies, the first B ORN approximation should give a first, rough guess, even though we do not expect exact predictions. According to (5.77) the cross section for photoionization in FBA is σns ∝

Z5 n3 (2ω)7/2

(10.52)

for one electron in an H atom like s orbital, at photon energies ω |Wn |. Clearly, the cross section increases strongly with atomic number Z, and decreases with the principle quantum number n (and correspondingly with the respective electron shell). Modified formulas will have to be applied for higher orbital angular momenta (p, d etc.). Also, for the overall cross section one has to multiply (10.52) with the number Nn of active electrons in each shell, so that σa = Nn σn for a given shell n . The B ORN approximation is naturally only a first, very rough approximation indicating a trend. One finds various approximation formulas in the literature. However, since excellent, theory and experiment based data banks exist in the literature

10.5


525

BORN L 106

Al

L BORN K K photo ionization

104

100

elastic (coherent)

σa / b

sum 1

COMPTON

photo ionization

pair

production 0.01

100 1 photon energy / keV

104

Fig. 10.14 Photo-absorption cross section (in barn) for aluminum as a function of photon energy. The main contribution is due to photoionization. At higher energies pair production dominates (according to C HANTLER et al. 2005). Black dashed lines give the photoionization cross section in B ORN approximation

one should not loose time with such approximation when attacking practical problems. We use he NIST X-ray data bank of C HANTLER et al. (2005) and B ERGER et al. (2010) and show in Fig. 10.14 the total photo-absorption cross section σa for aluminum (Al) over a broad range of photon energies. Al is an example for a light atom (Z = 13). In contrast, Fig. 10.15 presents σa for lead (Pb), a heavy element (Z = 82) well known as a good X-ray absorber and frequently used for protection from X- and γ -ray radiation. For the L edge of lead at 16 keV one reads e.g. in Fig. 10.15 σa 52 000 b (roughly 200× that for Al at the same photon energy). With ma = 207 u this corresponds to a mass absorption coefficient of ca. μ/ρ = 15.0 m2 / kg. This implies e.g. that a protection west with just 67 g lead per m2 the irradiation with a 16 keV X-ray source is already reduced to 1/e 37 %. As indicated Fig. 10.14 and Fig. 10.15 by dashed lines, the trend and the order of magnitude of the B ORN approximation. The real cross section differs quantitatively somewhat and is of course much more structured. With increasing energy the cross section drops typically ∝ (ω)−5/2 ∝ λ2.5 (and not as predicted by B ORN approximation ∝ (ω)−7/2 ). It jumps, however, at each absorption edge significantly: i.e. the cross section assumes a finite value whenever the energy of the photon is high enough to photoionize the respective shell – as evident from Fig. 10.11. We remember: considering the energetics of individual electrons independently from the reaction of the whole atom is the very content of KOOPMAN’s theorem according to Sect. 10.2.4. The edge energy is then simply the HF orbital energy Wn according

526

10


O N

M

106

BORN L

Pb

photo ionization L BORN K 104

elastic (coherent)

K

100

σa / b

sum

1

photo ionization

COMPTON

0.01

pair production

100 1 photon energy / keV

104

Fig. 10.15 Photo-absorption cross section (in barn) for lead as a function of photon energy. Otherwise as Fig. 10.14

to (10.33). One recognizes the K and L edges very clearly for Al, for Pb in addition also the N, M and O edges. Figure 10.14 and Fig. 10.15 illustrate that energetic photons are not only attenuated by photoionization. Without going into details we summarize the most important mechanisms for absorption and scattering of X-ray photons: 1. Photoionization (photoelectric effect) Tg + ω → Tg+ + e−

Eq. (10.45)

2. C OMPTON scattering e + hν −→ e + hν

see Sect. 8.4.5

3. Pair production ω −→ e− + e+

for ω ≥ 2me c2

4. T HOMSON scattering (elastic) ω −→ ω; k → k

see Sect. 8.4.5.

Obviously, photoionization dominates the absorption cross section at photon energies below some 100 keV. Above 2me c2 = 1.022 MeV an electron positron pair can be generated (pair production). Since both, energy and momentum conservation

10.5


527

must be fulfilled simultaneously, pair production can only occur in the presence of a third particle, preferentially an atomic nucleus with high Z. Correspondingly the absorption cross section increases again for hard γ radiation, as seen in particular for Pb. For energies around 1 MeV incoherent photon scattering (C OMPTON effect) plays a central role, as described by the K LEIN -N ISHINA formula (8.130). In addition to these three processes elastic (also coherent or T HOMSON) scattering of photons plays a (minor) role. As discussed in Sect. 8.4.5, at energies significantly below 2me c2 = 1.022 MeV, but far above typical atomic resonances one expects cross sections σel Z 2 σe . With the T HOMSON cross section σe = 0.665 b one verifies this for Fig. 10.14 (Z = 13, Al) and Fig. 10.15 (Z = 82, Pb) at the lowest energies. As energy increases into the relativistic regime (L ORENTZ factor γ 1) the elastic cross section drops dramatically (as expected: oscillation amplitudes of relativistic electrons are much smaller than at classical energies). One roughly finds σel ∝ (ω)−2 ∝ 1/γ 2 .

10.5.4 Photoionization at Intermediate Energies In Sect. 5.5 we have discussed in some detail how (in principle) to compute photoionization cross section – specifically for the particularly simple example of the H atom using the B ORN approximation. In the previous section we have presented the photo-absorption cross sections for two selected metal atoms and focussed on an overview from VUV to hard X-ray photons. We now present in some more detail, by way of example, a non-trivial atom of intermediate nuclear charge. We take the rare gas atom argon (Ar) with Z = 18 in the interesting range of photon energies from threshold up to some keV. Figure 10.16 shows the dependence of the photoionization cross section for Ar in various ranges of energy. Experimental results and theoretical calculations from a number of different groups are compared. They supplement each other obviously quite well, even though a perfect agreement cannot be expected for such a rather complex element. We do not enter into a discussion of the many details, both in experimental data as well as in theory. In recent years, high quality data are obtained almost exclusively at synchrotron radiation electron storage rings (see Sect. 10.6.2), and in Fig. 10.16 these data are recognized by their small error bars and the consistent trends reported. Notwithstanding this progress, many data have still been obtained with selected lines from spectral lamps or X-ray tubes. Figure 10.16(a) gives a nice example. Shown there is the photoionization process Ar [Ne]3s 2 3p 6 1 S + ω −→ Ar+ [Ne]3s(3s)−1 3p 6 + e− (p) (10.53) from a single subshell, the MI shell (3s 2 ). One distinguishes this process from others by measuring the kinetic energy of the outgoing, free electron. The final state is here a 1 P0 . The term scheme of Ar is quite similar to that of Ne (see Fig. 10.9). In addition to the full K and L shells, [Ne] = 1s 2 2s 2 2p 6 , in Ar the M shell is filled with 3s 2 and 3p 6 electrons. The first ionization potential, related to the configuration [Ne]3s 2 3p 5 , is WI = 15.76 eV. To ionize a 3s electron, additional 13.48 eV are

528

10

Ar 3s

30 photionizaton cross section σ / Mb

1.5 dipole-length appr. dipole-velocity Houlgate et al. 1.0 Samson and Gardner 0.5

(a)

1.0 0.8 0.6 0.4 0.2 0.0


50

70

6

90

50

6

2p

3

4

(c')

1.0

100

150

200

(d)

2p

0 2

(b)

245

250

0.5

2s

(c)

0

0.0

300

500

600

800

(e') 0.20

0.10

1s → np

0.08

0.15 0.10 0.05

3205

(e)

3210

(f)

0.06 0.04

K edge

K+M double excitation

0.00

1000 J.H. McCrary G.V.Marr A.J.Bearden CXRO ZCZ06 R.H.Millar FFAST XCOM

0.02 0.00

3210

3230

100

3250

2400

3600

4800

6000

(g)

1.00 0.01 1E-4

from FFAST (NIST)

1E-6 10

100

1000

104

105

photon energy ħω / eV

Fig. 10.16 Photoionization cross section of argon (Ar) in different energy regimes – comparison of experiment and theory adapted from several authors. S AHA (1989): (a) partial cross section for the 3s ionization above-threshold, showing a typical C OOPER minimum with MCHF theory (full and dashed lines). S UZUKI and S AITO (2005): (b) total cross section up to 200 eV (experiment only), (c) experiments from L edge up to 500 eV, (c ) blow up close to L edge, (d) 500 eV to 1150 eV. Z HENG et al. (2006): (e) near K edge with high and (e ) very high resolution, (f) K edge over a broader energy range. Derived from the FFAST data bank (C HANTLER et al. 2005): (g) overview for whole energy range

necessary. The data shown in Fig. 10.16(a) show the energy dependence of the process (10.53) directly above its energetic threshold. Note that this particular process contributes only very little to the total photoionization cross section.

10.5


529

However, this particular partial cross section shows in an impressive manner how the photoionization cross section starts with the usual jump to a finite value a threshold, drops down rapidly to nearly zero just above 40 eV photon energy (ca. 12 eV electron energy), and finally rises again. This so called C OOPER minimum is not unusual in the photoionization of complex atoms above ionization threshold. It can be understood easily from the theoretical background: The photoionization cross section (5.88) contains two matrix elements (5.89) that essentially describe the overlap between bound state and continuum wave functions (weighted by the distance r). Depending on the positions of the nodes in the wave functions they may be positive ore negative – what can change with the energy of the continuum electron. Thus, it may happen, that the two matrix elements in (5.88) just compensate for one particular energy. Alternatively one dominant matrix element may change from a positive to a negative value, passing necessarily zero at some energy which then leads to zero contribution from this matrix element. In the present case things are even a little bit more complex since these wave functions are, strictly speaking, multielectron wave functions comprising more than the ideal configuration according to (10.53). Here the dominant matrix element is . . . 3s . . . 1 S|r|p 1 P0 where the contribution from the 3s 2 3p 6 configuration is partially compensated by strong CI with the 3s 2 3p 5 3d configuration. Correspondingly expensive is the multi configuration H ARTREE -F OCK (MCHF) calculation shown in Fig. 10.16(a), which interprets the experimental results rather well – not differing much in its dipole lengths and velocity form which is a characteristic for any high quality calculation (see footnote 10 in Chap. 4). Figure 10.16(b) shows the total photoionization cross section for the energy range between 40 and 100 eV, which is dominated by ionization of the six 3p electrons. Here too one recognizes a C OOPER minimum, this time at 50 eV, followed by a maximum at ca. 80 eV – so to say its counter part. In Fig. 10.16(c) and (c ) the details in the vicinity of the L edge are illustrated: the ionization thresholds (terminology see Fig. 10.11) for LIII , LII and LI at 248.4, 250.6 and 326.2 eV, respectively. The contribution of the 2s electrons (LI ) is very small – similar as in the M shell. Below the LIII and LII thresholds one sees in (c ) indications of autoionizing resonances, which we got to know in Sect. 7.6.2. Figure 10.16(d) shows the rather boring decrease above the L edge, but still significantly below the K edge. The latter is finally reached at 3205.9 eV as shown in detail in Fig. 10.16(e), (e ) and (f). In (e) one recognizes the double excitation of K and M shell 1s3p → 4p 2 (again an autoionizing state). The most pronounced structure in this energy range is doubtless the excitation of RYDBERG states 1s → np for (n ≥ 4), shortly below the K edge, shown on an expanded scale in (e ). Figure 10.16(g), finally, gives a full survey over the whole energy range from threshold up to 100 keV in a log–log plot, extracted from the B ERGER et al. (2010) data bank. These data do, of course, only reproduce the rough tendencies. We cannot end this section without mentioning that in addition many measurements of the anisotropy parameter β have been reported (see Sect. 5.5.3), containing additional information – we refrain, however, from reproducing such data here.

530

10


Section summary

• Absorption and emission of X-ray radiation from (and to) inner atomic shells is governed by the PAULI principle which forbids transitions into completely filled electron shells. • Absorption (at not too high photon energy, ω) is mostly due to photoionization. The absorption cross section as a function of ω shows a typical edge structure displaying the ionization potentials of different shells. Beyond these, the cross section decreases rapidly (Figs. 10.14 and 10.15). • At intermediate photon energies, photoionization cross sections of complex atoms show a rich structure. E.g. so called C OOPER minima may occur, shortly above an ionization threshold. They are attributed to cancellation of two radial matrix elements for bound-free transitions, or to zero passage of one dominant matrix element. • The absorption √ edges WI are reasonably well described by M OSLEY’s law, predicting WI ∝ Z − qs . • Based on this, a very rough estimate for the energy of Kα radiation is obtained from (10.51). • At ω > 100 keV, the cross section starts to be dominated by C OMPTON scattering, while for ω > 1 MeV pair production sets in. • Transitions of electrons from higher to lower levels can only occur into electron holes. This leads to characteristic X-ray emission spectra, similar to those from quasi-one-electron spectra (alkali atoms). • Alternatively, the filling of inner shell holes may be accompanied by emission of AUGER electrons which carry the excess energy.

10.6

Sources for X-Rays

10.6.1 X-Ray Tubes The classical device for generating X-rays is the X-ray tube (originally RÖNTGENRöhre, named after its inventor RÖNTGEN 1901, who received the N OBEL prize for the discovery of X-rays). Even today this very simple method is still widely used. A typical example of a state-of-the-art, small X-ray tube for analytical purposes is illustrated by a photo of the device as well as by a schematic in Fig. 10.17. An electron beam generated by a hot cathode wire an a negatively biased W EHNELT cylinder hits a metal anode. Inside the metal it is strongly decelerated in the electric field of atomic nuclei (preferably with high Z). This leads to X-ray “bremsstrahlung” (the German verb for decelerating is “bremsen”). At the same time, a fraction of the atoms in the anode is ionized by electron impact, which leads to hole formation in the inner shells of the atoms. As a consequence emission of characteristic X-ray radiation becomes possible. Figure 10.18 shows characteristic spectra emitted from a Rhenium anode (Rh) at different electron energies. The measured X-ray spectrum presented here has been

10.6

Sources for X-Rays

Fig. 10.17 Example of a modern X-ray tube (photo and schematic have been kindly provided by H ASCHKE and L ANGHOFF 2007). Top: Photo of a mini X-ray tube with side exit of the radiation. Bottom: construction of this tube schematic

531

oil insulation glass body of the tube cathode focal area anode

e-

filament

e-

WEHNELT cylinder Be window

HV

screening X-ray radiation

recorded at 90◦ detection angle to reduce the signal for the spectral analysis with a crystal monochromator. One detects elastically as well as inelastically (C OMPTONEffekt) scattered X-ray radiation. Indicated in Fig. 10.18 are the literature values for the Kα , Kβ and Lα,β emission lines for Rh as well as the energies expected according to (1.75) for inelastically scattered photons. The weak signal at 8.02 keV (Cu−Kα line) originates from a small impurity in the anode. The spectrum clearly documents that the X-ray bremsspectrum cannot have a higher energy than the generating electron beam at the anode, with Wkin = eU corresponding to the voltage U (HV in Fig. 10.18) applied to the anode. This implies that the shortest wavelength emitted in the X-ray bremsspectrum follows the so called rule of D UANE -H UNT: λmin =

hc . eU

(10.54)

10.6.2 Synchrotron Radiation, Introduction Today, for the spectroscopy with VUV, XUV and X-ray radiation electron storage rings are available and intensively used at many places around the world, dedicated to generate synchrotron radiation. In so called 3rd generation storage rings one uses extremely well focussed, highly relativistic electrons. Typically, the electrons are stored in a large, overall circular or elliptic structure in which they are guided by deflection magnets (magnetic dipole fields). They are arranged with long straight sections in between, where the electrons are focussed with different magnetic structures, mainly with quadrupole magnets, but other structures such as sextupole or octupole magnets are also needed to control the chromaticity, i.e. the variation of

532

10 Kα (inel.)

Rh X-ray emiss. / arb. un.


Cu - Kα(inel.)

Kα (elast.)

40 keV Kβ (inelast.) 20 keV L α Lβ 0

5

Kβ (elast.) 10 keV 10 15 20 ћω / keV

25

30

Fig. 10.18 X-ray bremsstrahlung and characteristic radiation (top) from the tube shown in Fig. 10.17 with a rhenium (Rh) anode at three different electron energies (the spectra have been kindly provided by H ASCHKE and L ANGHOFF 2007). The bremsstrahlung’s continuum as well as characteristic radiation (Kα , Kβ and Lα,β ) can be clearly recognized

focussing properties with the spread in electron momentum.3 Electromagnetic radiation in a wide spectral range, so called synchrotron radiation (SR) is generated where the electrons are accelerated, e.g. as they are deflected by the dipole magnets. The overall energy of the electron beam is kept constant, compensating the radiative losses by iterative re-acceleration in special microwave cavities somewhere in the beam path. Typical data presented in the following refer (as far as not otherwise mentioned) by way of example to BESSY II, the Berlin electron storage ring for synchrotron radiation in Berlin-Adlershof. In Fig. 10.19 the topography of BESSY II is sketched schematically. The ring has a circumference of 240 m. In it electrons of a nominal energy of 1.7 GeV are stored with a current of 100 mA to 400 mA. 32 dipole magnets keep the beam on its ring trajectory. In between 16 about 4 m long straight sections are arranged into which undulators or wigglers may be inserted. The orbital period of the electrons amounts to ca. 0.8 μs, and the electrons are kept in bunches at a temporal distance of ca. 2 ns, each having a pulse duration typically below 20 ps. Under normal operating conditions about 350 such bunches circle in the ring, each followed by a ca. 100 ns gap. They are initially prepared and accelerated in a smaller synchrotron which is indicated in Fig. 10.19 inside the ring. After injection into the storage ring electrons are stored for several hours. The most recent mode of operation features continuous refilling of the ring to compensate for lost electrons. Typically, some 50 user stations, so called beamlines are installed for general use. Key advantages of such synchrotron sources are their high brilliance and the broad spectrum generated, typically from the terahertz region at the long wavelength end down to soft or even hard X-ray radiation. By using suitable monochromators 3 The

quality of the focussing is characterized by the so called emittance. That is the spatial extension of the electron beam multiplied by its divergence angle. As an example, BESSY II is characterized by an emittance of (3 to 6) nm rad in horizontal and <0.1 nm rad in vertical direction.

10.6

Sources for X-Rays

target hall

533 high frequency frequ resona nator ators rs

i jecction inje inj i

electron gun

target hall

deflection magnets

synchrotron circumference ce 96m

undulatorr

line

s

beamlines

target hall

a be

mli

ne

s

nes

am

mli

be

storage g ring g circumference 240 m 32 deflection magnets 16 straight sections

bea

deflection magnets

target hall

Fig. 10.19 Schematic of the topography in BESSY II. The electrons are pre-accelerated in a microtron, further accelerated in a small synchrotron ring to the nominal energy, and then injected into the storage ring. The radiation energy emitted is compensated by high frequency generators. In between the deflection magnets, marked in red, are straight sections are arranged for “insertion devices” and multipole focussing magnets. The “beam lines” – exit tubes of radiation for the users – are shown here for clarity only in the lower half of the figure. Only one undulator is indicated as an example for an insertion device (We thank Prof. JAESCHKE et al. 2007, for providing details and sketches)

one may thus obtain intense, conveniently tunable, quasi-monochromatic radiation of low divergence – ideally suited for spectroscopic experiments of various kinds as well as for X-ray diffraction experiments (see e.g. Fig. 1.19). Other, increasingly important applications of synchrotron radiation are microscopy, holography and lithography with X-ray radiation. Brilliance is here not just a buzzword but a measurable characteristic of SR. Eventually one is interested in the number of photons N emitted per unit time t and area A (which is called photon flux Φ, given in SI units [Φ] = s−1 m−2 ) and per solid angle Ω ([Ω] = sr, see Sect. A.4). This leads to the standard definition for the radiance (see Table 1.6) R=

ωdΦ ωd3 N = , dtdAdΩ dΩ

(10.55)

with ω = W = hc/λ = hcν¯ being the photon energy. The spectral radiance 2 )W = ωd Φ , R dΩdW

(10.56)

534

10


i.e. it refers to radiance per unit energy interval (or per frequency, wavenumber or wavelength). The synchrotron radiation community uses a special quantity called spectral brilliance B=

3 Φ ,

Ω W/W

(10.57)

which refers to the relative bandwidth dW/W = dν/ν = dλ/λ. Usually, brilliance is measured in units 1photon/ s mm2 × mrad2 ×0.1

% bandwidth

= 1 Sch.

(10.58)

We abbreviate this somewhat complicated unit4 by “Sch” in honour of Julian S CHWINGER (N OBEL prize in physics 1965 together with T OMONAGA and F EYN MAN ). As we shall see, S CHWINGER has provided the theoretical background for this kind of radiation. Figure 10.20(a) gives a survey of spectral brilliance for a variety of synchrotron radiation sources of the 3rd and 4th generation, including insertion devices and undulators as well as for the new “Free Electron Laser” sources (FEL) – some already operating, some under construction (dashed lines). For comparison the brilliance achievable at dipole deflection magnets is shown at the bottom in Fig. 10.20(a). Even this radiation is – with ca. 3 × 1014 Sch still by orders of magnitude higher than e.g. bright sun light on earth (maximum in the visible spectral range ca. 3 × 1011 Sch), standard X-ray tubes with 107 to 1010 Sch and other conventional laboratory sources – with the exception of some laser systems. Brilliance does not only describe how many photons are available per solid angle, area and wavelength interval. It also defines how well they can be focussed: the less extended a light source is, and the smaller its emittance angle, the better it may be focussed onto a target. As detailed in Appendix B.2, Vol. 2, according to the H ELMHOLTZ -L AGRANGE relation (for paraxial beams equivalent to the A BBE sine law) the product of n2 S Ω is a constant when imaging light, electron or ion beams. With the index of refraction n 1 being a constant in 4 This ‘unit’ may be somewhat confusing. What it implies is simply that B gives the number of photons emitted per second into a solid angle Ω = 1 mrad2 from a source area S = 1 mm2 into an energy interval W = 10−3 W . Thus, into an arbitrary solid angle and energy interval the source emits photons

Ω

W

S × × −3 . =B× s mm2 10 W mrad2

Expressed in terms of standard spectral radiance (10.56) one would write )W photons R = × S × Ω × W, s ω so that with (10.56), (10.57) and W = ω B = 10−3 W

d2 Φ )W . mrad2 mm2 = 10−3 × R dΩdW

Sources for X-Rays

100

1

wavelength / nm 1000 100 10 1 0.1

0.01

Y XFEL FLASH DESY (seeded)

1033

LCSL

1029

DESY FLASH PETRA III

1025

20 m ID X-ray Und.

6 plasma lasers 10

HHG

SPring-8

XUV Und.

1023

BESSY U-49

1021 U-125

1015 1013

109

DESY FLASH

1027

1017

1012

LCSL

1031

1019

DESY XFEL

fs-Ti:Sapph

ALS

APS

dye laser

103

peak power / W

peak brillance / Sch (see text)

535

Rubin exzimer

10.6

U49 U125

BESSY 100

U5.0

(a)

(b)

BESSY dipole deflecton magnet

101 102 103 104 photon enenergy / eV

105

106

1E-3 0.01

0.1

1

10

Fig. 10.20 (a) Brilliance for various synchrotron radiation sources of the third generation (including insertion devices ID and specifically undulators U) and for Free Electron Lasers (FEL) as a function of photon energy. (b) comparison of the peak power in these sources with a variety of laser systems

the present case (and assuming that no photons are lost), B is thus a constant too. Synchrotron radiation from third generation sources typically emerges from an area of only some 10−2 mm2 and is extremely well collimated. Compared to conventional light sources, SR sources are superior by many orders of magnitude. Note, however, that the angular divergence of laser sources is usually even better than SR. Also, from an experimental point of view the peak power of a source is often more relevant than its brilliance – for obtaining an experimental signal. Figure 10.20(b) thus compares typical laser systems with representative SR sources from this point of view. Peak power determines the total number of photons available for an experiment – per unit time. Also the total number of photons, i.e. the available measuring time may be crucial, so that even X-ray tubes are still in use and allow long time experiments, in particular for such experiments where SR peak power cannot be exploited – e.g. due to damaging side effects on delicate targets.

536

10

(a)

p

(b)

z'

θ ρ

k

. p

p

. p

LORENTZ transformation

x'


x

z

δθ = kx / kz = 1 / 2 γ kz = 2 γ kz'

kz'

(c)

δθ

(d)

kx' = kz'

A

(e)

2 δθ

ρ

ρ 2 δθ

δθ

k

p

}

k'

kx = kx' = kz'

time window for observation 2 δ t ħ ω = ρ 2sin( δ θ )/c 2 δ te = ρ 2 δθ /v

Fig. 10.21 For understanding the generation of synchrotron radiation (SR): polar plots for the angular distribution of (a) the classical radiation characteristic from an electron on a circular orbit, observed in its rest frame and (b) its L ORENTZ transform; (c) components of the wave vector in a frame moving with the electron and (d) the same in the laboratory system; (e) ‘observation window’ for the radiation from this electron

Nevertheless, in addition to brilliance, excellent tunability, broad wavelength range, as well as relatively good accessibility and user friendliness of the publicly funded facilities make SR to an unsurpassed radiation source for a wide range of applications. This is specifically true for the VUV, XUV and soft as well as hard X-ray region, where SR is clearly superior even to laser sources.

10.6.3 Synchrotron Radiation, Quantitative Relations Starting point for the strictly relativistic derivation of the intensity and angular distribution of SR is (4.32) which gives the radiation characteristic in the moving rest frame of the electron. This has to be L ORENTZ transformed into the laboratory system. Consequently a highly collimated radiation characteristic tangential to the orbit of the electrons is emitted. The not completely trivial theory has been developed for the first time in a famous paper by S CHWINGER (1949). The formula derived there for the emission characteristic is so precise that synchrotron radiation is used today as a calibration standard in radiation metrology. Without going into details we just present the key ingredients by some hand waving arguments, summarized in Fig. 10.21.

10.6

Sources for X-Rays

537

We apply the terminology introduced in Sect. 1.2. Dashed quantities refer to the electron rest frame, un-dashed quantities to the laboratory system. Let the total energy of the electron be We with γ = We /(me c2 ) = (1 − β 2 )−1/2 , the energy of the photons emitted is ω and ω (corresponding to wavenumbers k and k), respectively. In the field B of a deflection magnet the highly relativistic electrons (β ∼ = 1) are accelerated onto a circular orbit with an instantaneous radius of curvature ρ γ me c/(eB) according to (1.109). In its moving rest frame x y z (one chooses z p) the electron emits, as indicated in Fig. 10.21(a), the characteristic radiation of a H ERTZ dipole, with an angular characteristics ∝ cos2 θ according to (4.32). If one transforms this radiation pattern into the laboratory system, its frequency in z-direction and its wave vector component kz are dramatically magnified, while the x-component kx = kx remains constant. Thus, the relativistic D OPPLER shift (1.29) leads to a collimation of the radiation as indicated in Fig. 10.21(b) into the direction of the electron trajectory – a key characteristic of synchrotron radiation. For β 1 one obtains kz = 2γ kz according to (1.31), which corresponds to a very small (full) divergence angle 2δθ 2

1 kx = kz γ

(10.59)

for the radiation emitted, as one reads from Fig. 10.21(c) and (d). To illustrate this by numbers: at BESSY II with its electron energy of 1.7 GeV one has γ 3300 and the full divergence angle of the radiation at a deflection magnet is 2δθ 1 (in words: one arc minute!).5 From (10.59) one may even obtain a simple estimate of the bandwidth. One has to realize that an observer at large distance ‘sees’ the radiation from a single electron only for a very short time. As sketched in Fig. 10.21(e) this time represents the time the electron needs to pass an arc segment 2δθρ. The electron needs a time δte = ρ2δθ/v, the light the time 2δtω = sin(2δθ )/c 2δθ/c. An observer, looking at the electron tangentially from a far distance, ‘sees’ the light emitted from that electron smeared out over a time

1 1 1−β 1 1 − = − = 3 , δt 2δθ v c γ v c γc β 2γ c where we have used (1.15) for β 1.6 Applying the uncertainty relation (1.125) for time and energy one estimates from this the characteristic bandwidth of SR to 5 Of course, the emission angle effective for the experiment also depends crucially on the experimental setup, which includes a certain length of the electron orbit. The angle δθ given here thus only holds for the vertical divergence perpendicular to the plane of the ring. In horizontal direction the experimental apertures limit the SR beam. 6 Clearly,

this time refers to the light emitted from a single electron and has nothing to do with the actually measurable duration of the SR light pulses from bunches of electrons. Typical pulse duration of modern synchrotrons are several ps.

538

10


be W /δt = 2γ 3 c/. The exact calculation confirms the order of magnitude from this crude estimate: A so called critical energy Wc = ωc =

3c γ 3 3γ 2 eB , = 2 2me

(10.60)

enters as key parameter into the S CHWINGER formula for the intensity distribution. The corresponding critical wavelength is λc =

4π 4πme c . = 3γ 3 3γ 2 eB

(10.61)

To characterize the angular distribution of the radiation, instead of referring to polar (θ ) and azimuthal angle (ϕ) one uses for practical purpose two orthogonal angles χ (in the ring plane) and ψ (normal to it). In the literature the original formula (Eq. (II.32) in the paper of S CHWINGER 1949) for the emitted photon flux is rewritten in a variety of different ways. A rather concise form is the following:7 2 d3 Φ 3α I /e ω 2 2 γ 1 + x2 = 2 dχdψdω/ω 4π S ωc

x2 2 K [ξ ] . × K2/32 [ξ ] + 1 + x 2 1/3

(10.62)

Here I is the electron current in the ring and S the emitting area. The unit Sch (S CHWINGER), introduced above, is obviously compatible with this formula written essentially dimensionless (apart from cm−2 s−1 ). Kn is the generalized secondorder B ESSEL function, implemented in standard mathematical programs, and the variables x and ξ are defined by x =γψ

and ξ =

3/2 ω 1 + x2 . 2Wc

(10.63)

_ _2 _ _2 As it turns out, the two components in (10.62), ∝ K2/3 (ξ/2) and ∝ x 2 K1/3 (ξ/2) 2 (1 + x ), respectively, represent distributions of radiation intensity for two different polarizations: the first one describes light linearly polarized parallel to the plane of the ring, while the second term corresponds to a polarization normal to the ring – which is nonvanishing only outside the ring plane – and has a phase shift of π/4 in respect of that in the plane. Superposition of both components thus makes SR

7 For

readers who want to follow these reformulations: S CHWINGER uses the esu system, and his formula (II.32) gives the energy which an electron emits along the orbit over χ = 2π . For the determination of the brilliance it has to be divided by 2π and by the photon energy. Now, 2πρ/c = te is the period time for an electron on a circular orbit and, on the other hand, Ne = 2πρI /(ec) gives the number of electrons in the ring at a current I . With an emitting area S and the fine structure constant α = e2 /(4πε0 c) one obtains (10.62).

10.6

Sources for X-Rays

539

light emitted above or below the ring plane elliptically polarized: it is a unique potential of SR to provide short wavelength, tuneable radiation with elliptic or even circular polarization. Typically such radiation is offered at dedicated beam lines. For compact description of the radiation characteristic one usually presents the radiation with polarization parallel to the ring plane as a function of the photon energy ω, i.e. one sets ψ = 0 and obtains as a convenient numerical relation: d3 Φ I/A (We / GeV)2 Shor (ξ ) Sch. = 1.326 × 1013 dχdψdω/ω

S/ mm2

(10.64)

Alternatively one integrates over the angle: d2 Φ I/A = 2.4577 × 1013 (We / GeV)Sint (ξ ) mrad Sch. dχdω/ω

S/ mm2

(10.65)

While from (10.62) one obtains Shor (ξ ) directly for x = γ ψ = 0 and ξ = W/2Wc , one finds Sint (ξ ) after some reformulation by integration: 2 (ξ ) Shor (ξ ) = 4ξ 2 K2/3 ∞ Sint (ξ ) = 2ξ K5/3 (z)dz.

(10.66)

ξ

Both functions are illustrated in Fig. 10.22(a). One sees that the maximum brilliance of the SR is reached just below the critical energy, at ξ = W/2Wc 1. For BESSY II e.g., Wc = 2.5 keV, and the useful spectral range at a dipole deflection magnet extends up to about 10 keV. Finally, by integrating (10.62) over all photon energies ω, one obtains the overall angular distribution in respect of ψ (angle normal to the ring plane) which is plotted in Fig. 10.22(b):

5 x2 dΦ 2 −5/2 1+ . ∝ 1+x dψ 7 1 + x2

(10.67)

The FWHM angle is 1.3/γ , in fair agreement with the rough estimate 2δψ = 1/γ according to (10.59). The total radiation emitted may – after appropriate rewriting – be obtained from (4.32) or by integration of (10.65) over all frequencies: e2 γ 2 dp 2 e2 c γ 4 e4 We2 B 2 P= = = . 6πε0 ρ 2 6πε0 m4e c3 6πε0 m2e c3 dt This expression is obviously the relativistic analogue to (4.32), where in the last steps (1.110) and γ = We /(me c2 ) has been inserted. Note the strong, inverse proportionality on the 4th power of the radiating particle’s mass, here the electron. Thus, the electron is the particle of choice, if one wants to generate synchrotron radiation.

540

10

S int (ξ ) S hor (ξ ) 1

(a)

0.1

2.146 ξ

1/3

2.910 ξ

2/3


(b)

dΦ dψ

1.0 0.8 0.6

1.339 ξ 1/2 e- ξ

F W H M ~_ 1.3/ γ

0.4

0.01 0.001 0.0001

0.2 0.01

1.00

ξ = ħ ω /2 Wc

-2

0

2

γψ

Fig. 10.22 Characteristic energy and angular distribution of SR generated at a deflection magnet. (a) Brilliance as a function of photon energy in the plane of the ring, Shor corresponding to (10.64) and integrated over angle, Sint corresponding to (10.65). (b) Energy integrated intensity (10.67) as a function of the angle above and below the plane of the ring, showing a characteristic angular FWHM 1.3/γ

10.6.4 Undulators and Wigglers An even more dramatic enhancement of brilliance and extension of the wavelength range can be achieved by using so called insertion devices, specifically undulators and wigglers. These are a sequence of periodically positioned dipole deflection magnets, which are poled alternately in north-south and south-north direction. They are mounted into the straight sections of the storage ring and force electrons onto rapidly oscillating orbits – which again leads to emission of radiation. Schematically this is shown in Fig. 10.23(a) and illustrated in Fig. 10.23(b) by a photograph of an undulator, type U49, implemented at BESSY II. One computes the radiation emitted from the electron passing the undulator magnets – quite analogously to deflection magnets – by L ORENTZ transformation from the electron rest frame into the laboratory. This again leads to a strongly forward peaked radiation characteristic. A rough estimate for the wavelength is obtained as follows: the electron ‘sees’ the periodic distance λu of the dipole magnets, due to L ORENTZ contraction (1.27) shortened to λ = λu /γ . In the emission process the relativistic D OPPLER shift (1.29) again reduces this wavelength to λu λu λ = γ (1 − β cos θ ) λu 1 − β + βθ 2 /2 2 1 + γ 2 θ 2 , γ 2γ where cos θ has been expanded for small emission angles θ , and (1.15) was inserted for β ∼ = 1. The exact calculation leads to the undulator equationfor the so called resonance wavelength:

K2 λu λr = 2 1 + + γ 2θ 2 . (10.68) 2 2γ

10.6

Sources for X-Rays

(a) e-

541

u

N S N S N S

θc =

1 √ u

intensity

N S N S N S

Δ

≈ 1/ u

(c)

ħω

(b)

Fig. 10.23 (a) Scheme of an undulator with Nu (here =3) periods of aligned, alternately poled magnetic fields. They force the electron√onto an oscillatory trajectory. The opening angle of the central cone is narrowed by a factor 1/ Nu as compared to a deflection magnet. (b) Photograph of a U49 undulator for X-ray spectroscopy at BESSY II. (c) Typical spectrum from an undulator in the oscillation plane with odd harmonics. (We thank Eberhard JAESCHKE et al. 2007, for the photograph)

Here K is the dimensionless magnetic deflection or undulator parameter K=

eBλu B λu = 0.93373 2πme c T cm

(10.69)

with B being the magnetic field in the undulator magnets. The resonance wavelength λr thus depends on the undulator wavelength λu , the beam energy γ , the undulator parameter K, and, remarkably, it also depends on the emission angle θ . A detailed analysis shows that one has to distinguish two quite different situations: undulators with K ≤ 1 and wigglers for which K 1. Essential for an undulator (K ≤ 1) is, that the oscillation amplitudes of the electrons remain small, so that the amplitudes of the radiation emitted at each of the Nu magnet pairs can interfere constructively. This induces partially coherent radiation with a substantially reduced bandwidth ω/ω = λ/λ 1/Nu around the resonance frequency ωr = c/(2πλr ) – similarly to multi-beam interference at a grating. This is accompanied by a corresponding enhancement of the intensity and a reduction of the angular spread θ → θc 1/γ Nu . At the same time, higher harmonics ωn = nωr of the resonance frequency may be emitted, as indicated in Fig. 10.23(c). The computation shows that in forward direction only odd harmonics n = 2j + 1 are emitted, with the fundamental j = 0 dominating. At synchrotron radiation sources of the third or forth generation, undu-

542

10


lators are very popular for intense short wavelength applications due to their superior performance (see Fig. 10.20). In contrast, for K 1 the emission cones from the individual undulations become wider and do no longer superpose coherently. Such a situation is then called a wiggler. A key feature is the generation of very high harmonics with broader bandwidth. With increasing K they dominate the overall wavelengths spectrum which then becomes a broad continuum. In principle, this spectrum does not differ much from that at a bending magnet – except for the fact that it is much more intense and is shifted to much higher photon energies than achievable at bending magnets with a given electron energy. Wigglers thus provide convenient sources for soft and hard X-rays.

10.6.5 Free Electron Laser (FEL) In a long undulator with many periods of magnets and suitable design the coherently superposed radiation from all the magnets may become so intense that it acts back onto the electron beam which has generated it: a bunch structure emerges, and the electrons ‘ride’ so to say on the electromagnetic wave. This implies the feedback and amplification of the radiation which is needed to achieve complete coherence and which is characteristic for a laser. Such devices are called free electron laser (FEL). The setup is very similar to that for an undulator shown in Fig. 10.23, however, with many magnets and typically a highly relativistic electron beam designed and dedicated to this purpose. Such a facility is, in principle, a single user facility, even if as rule precautions are taken that several users may work quasi in parallel at such a system (i.e. intermittently). The first systems which have been working very successfully for many years by now had been designed for the infrared spectral range. Presently, worldwide a number of FELs are under construction or already in operation. They generate intense soft or even hard X-ray short pulses (some fs). A variety of techniques are applied, one of these being the SASE principle (Self-Amplified Spontaneous Emission), which was used e.g. at DESY for the VUV laser FLASH (Free Electron LASer in Hamburg) and will also be implemented at the XFEL (Xray Free Electron Laser) there. More efficient, but also more delicate, are concepts where the FEL action is ‘seeded’ initially by an already coherent, albeit weak, laser pulse which may be generated from high harmonics (HHG) of a Titanium-sapphire laser, as discussed in Sect. 8.5.6. Such concepts are planned, under construction or even already in operation at all FEL facilities worldwide. A look at Fig. 10.20 reveals that with these new sources at methodological ‘quantum jump’ for the interaction of light with matter of various kinds is under way. Not only is the brilliance at these facilities many orders of magnitude higher than anything one may have dreamed of some years ago. Perhaps even more important is the temporal structure of such sources, which are designed for coherent femtosecond pulses in the soft and hard X-ray radiation. They will allow for a new generation of experiments which we may look forward to with great expectations. To mention just one out of a host of dream experiments: it should become possible to generate a

10.6

Sources for X-Rays

543

complete X-ray diffraction image from a single, large biomolecule during one single laser pulse – and thus one would circumvent many efforts and imponderabilities which arise from the present necessity to crystallize such objects.

10.6.6 Relativistic THOMSON Scattering Without entering into great details we just mention this interesting and timely subject, following the introduction to coherent, elastic light scattering in Sect. 8.4.5. The basic idea is quite simple: if one collides a laser beam (photon energy ωL ) with a highly relativistic electron beam (γ = We /me c2 1) the radiation is exposed to the relativistic D OPPLER shift twice – on the way in as well as upon reemission. Thus, for a head on interaction of laser and electron beam and observation in the direction of the electron beam, the emitted radiation has, according to (1.31), an energy ωX 4ωL γ 2 .

(10.70)

One easily verifies that even at moderate electron energies multi keV radiation may be obtained in such a setup, e.g. from a 50 MeV electron beam, Titanium-sapphire laser radiation (800 nm) is scattered as 60 keV X-ray photons. The problem with this concept as a short pulsed X-ray source is of course the low magnitude of the T HOMSON cross section. However, if intense laser fields are used, nonlinear effects come into play (nonlinear, relativistic T HOMSON scattering), essentially related to the magnetic field component of the radiation. The topic is subject to state-of-the-art research (see e.g. C HUNG et al. 2011, and references there).

10.6.7 Laser Based X-Ray Sources As already indicated these big, new facilities will provide only limited amount of measuring time, and many experiments stand already in line waiting for their turn. Thus, it is worthwhile to search for alternatives, which do not necessarily offer the ultimate specifications of the new XFELs, but are more readily accessible. One class of such sources are laser based X-ray sources. We specifically mention such setups where a strongly focussed, very short and intense Titanium-sapphire laser pulse generates a hot plasma of elements with high Z. The electrons thus generated may induce bremsstrahlung or characteristic X-ray radiation just as in an X-ray tube – with the essential difference that in this cases an X-ray pulse is generated with a duration on the order of some 100 fs. This allows one to study dynamical processes. One example of such a source is shown in Fig. 10.24(a). Its essential ingredients are a Cu target which is locally ionized by an intense laser pulse (5 mJ, 50 fs, 800 nm), focussed onto a few μm2 . Since the target is vaporized during the laser pulse, one has to use a moving band-target. A second, this time transparent, rapidly moving tape protects the laser focussing optics from all debris particles which emerge from the Cu target. As illustrated by the measured X-ray spectrum, shown in Fig. 10.24(b),

544


(b)

titanium sapphire laser 1 kHz, 5 W, 45 fs @ 800 nm Kα

count rate / arb. units

(a)

10

tranparent protection against debris

Cu band target

Kβ

2

4

6

8

10

ħω / keV 12

Fig. 10.24 Laser based X-ray source for short pulses according to Z HAVORONKOV et al. (2005). (a) Scheme of an experimental setup with a moving Cu tape source, and moving transparent tape to protect the optics for focussing the Titanium-sapphire laser pump beam. (b) Measured characteristic X-ray radiation spectrum

and demonstrated by time resolved X-ray diffraction experiments, one obtains a sufficiently intense X-ray pulse to produce a measurable signal for studying the ultrafast dynamics of non-equilibrium states of matter. The further development will certainly provide interesting new applications to which one may look forward – especially in the context of the competition from the emerging new XFEL facilities.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2. BESSY: ‘Berlin Electron Strorage ring for Synchrotron Radiation’, Germany’s third generation synchrotron radiation source in Berlin-Adlerhof. CI: ‘Configuration interaction’, mixing of states with different electronic configurations in atomic and molecular structure calculations, using linear superposition of S LATER determinants (see Sect. 10.2.3). DFT: ‘Density functional theory’, today one of the standard methods for computing atomic and molecular electron densities and energies (see Sect. 10.3). esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3). FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sect. 6.6, Vol. 2 and Sect. 5.5.2, respectively). FEL: ‘free electron laser’, laser like radiation source using amplification in a spatially oscillating, relativistic electron beam, (see Sect. 10.6.5). FLASH: ‘Free Electron LASer in Hamburg’. FS: ‘Fine structure’, splitting of atomic and molecular energy levels due to spin orbit interaction and other relativistic effects (Chap. 6).


545

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)’. HF: ‘H ARTREE -F OCK’, method (approximation) for solving a multi-electron S CHRÖDINGER equation, including exchange interaction. HHG: ‘High harmonic generation’, in intense laser fields. LDA: ‘Local density approximation’, simplest version of density functional theory. LS: ‘L’, for orbital and S for spin angular momenta, unfortunately this is used in two opposite contexts: (i) LS interaction characterizes the spin-orbit interaction energy, while (ii) LS coupling denotes an angular momentum coupling scheme for multi-electron systems where all orbital angular momenta and all spin angular momenta of all electrons are first separately coupled together to L and S, respectively, and finally L and S are coupled to J . LSDA: ‘Local spin density approximation’, similar to LDA but for electrons with one spin orientation only. MCHF: ‘Multi configuration H ARTREE -F OCK’, method to determine wave functions for multi-electron systems (see Sect. 10.5.4). MP2: ‘M ØLLER -P LESSET correction of 2nd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials. MP3: ‘M ØLLER -P LESSET correction of 3rd order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials. MP4: ‘M ØLLER -P LESSET correction of 4th order’, perturbative approach to correct HF energies for contributions from non-spherical repulsive potentials. NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. QED: ‘Quantum electrodynamics’, combines quantum theory with classical electrodynamics and special relativity. It gives a complete description of light-matter interaction. RHF: ‘Restricted H ARTREE -F OCK’, assuming all spatial wave functions in a given closed shell to be equal when computing atomic wave functions. SASE: ‘Self-Amplified Spontaneous Emission’, scheme for FEL operation. 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. SR: ‘Synchrotron radiation’, electronmagnetic radiation in a broad range of wavelengths, generated by relativistic electrons on circular orbits. TDDFT: ‘Time dependent density functional theory’, a modern variety of DFT, allowing also for excited state calculations. UHF: ‘Unrestricted H ARTREE -F OCK’, allowing different spatial wave functions for each orbital when computing atomic orbitals (specifically in closed shells). UV: ‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007).

546

10


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). XFEL: ‘X-ray free electron laser’, as FEL but specifically designed to generate coherent, short pulse X-ray laser like radiation, for details see Sect. 10.6.5. 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 ATTWOOD , D.: 2007. Soft X-rays and Extreme Ultraviolet Radiation, Principles and Applications. Cambridge, UK: Cambridge University Press. B ERGER , M. J., J. H. H UBBELL, S. M. S ELTZER, J.C HANG, J. S. C OURSEY, R. S UKUMAR, D. S. Z UCKER and K. O LSEN: 2010. ‘XCOM: photon cross sections database (version 1.5)’, NIST. http://physics.nist.gov/xcom, accessed: 8 Jan 2014. C ASIDA , M. E.: 2009. ‘Time-dependent density-functional theory for molecules and molecular solids’. J. Mol. Struct., Theochem, 914, 3–18. C HANTLER , C. T., K. O LSEN, R. A. D RAGOSET, J. C HANG, A. R. K ISHORE, S. A. KO TOCHIGOVA and D. S. Z UCKER : 2005. ‘X-ray form factor, attenuation, and scattering tables (version 2.1)’, NIST. http://physics.nist.gov/ffast, accessed: 7 Jan 2014. C HUNG , S. Y., H. J. L EE, K. L EE and D. E. K IM: 2011. ‘Generation of a few femtosecond keV X-ray pulse via interaction of a tightly focused laser copropagating with a relativistic electron bunch’. Phys. Rev. Spec. Top., Accel. Beams, 14, 060705. H ASCHKE , M. and N. L ANGHOFF: 2007. ‘Mini power X-ray tube and spectrum – private communication’, see also: IfG – Institute for Scientific Instruments GmbH. http://www.ifgadlershof.de/, accessed: 8 Jan 2014. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. JAESCHKE , E., W. E BERHARD and M. S AUERBORN: 2007. ‘Technical data, sketches, and photographs about BESSY II’. We acknowledge grateful the material provided. KOHN , W.: 1998. ‘The N OBEL prize in chemistry: for his development of the density-functional theory’, Stockholm. http://nobelprize.org/nobel_prizes/chemistry/laureates/1998/. K RAMIDA , A. E., Y. R ALCHENKO, J. R EADER and NIST ASD T EAM: 2013. ‘NIST Atomic Spectra Database (version 5.1)’, NIST. http://physics.nist.gov/asd, accessed: 7 Jan 2014. L ATTER , R.: 1955. ‘Atomic energy levels for the Thomas-Fermi and Thomas-Fermi-Dirac potential’. Phys. Rev., 99, 510–519. M ARQUES , M. A. L. and E. K. U. G ROSS: 2004. ‘Time-dependent density functional theory’. Annu. Rev. Phys. Chem., 55, 427–455. PARR , R. G. and W. YANG: 1989. Density Functional Theory of Atoms and Molecules. International Series of Monographs on Chemistry. New York, Oxford: Oxford University Press, 333 pages. RÖNTGEN , W. C.: 1901. ‘The N OBEL prize in physics: in recognition of the extraordinary services He has rendered by the discovery of the remarkable rays subsequently named after him’, Stockholm. http://nobelprize.org/nobel_prizes/physics/laureates/1901/. S AHA , H. P.: 1989. ‘Threshold behavior of the M-shell photoionization of argon’. Phys. Rev. A, 39, 2456–2460.

References

547

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. S CHWINGER , J.: 1949. ‘On the classical radiation of accelerated electrons’. Phys. Rev., 75, 1912– 1925. S UZUKI , I. H. and N. S AITO: 2005. ‘Total photoabsorption cross-section of Ar in the sub-keV energy region’. Radiat. Phys. Chem., 73, 1–6. S ZABO , A. and N. S. O STLUND: 1996. Modern Quantum Chemistry. Dover, first revised edn. Z HAVORONKOV , N., Y. G RITSAI, M. BARGHEER, M. W ÖRNER, T. E LSÄSSER, F. Z AMPONI, I. U SCHMANN and E. F ÖRSTER: 2005. ‘Microfocus Cu Kα source for femtosecond X-ray science’. Opt. Lett., 30, 1737–1739. Z HENG , L., M. Q. C UI, Y. D. Z HAO, J. Z HAO and K. C HEN: 2006. ‘Total photoionization crosssections of Ar and Xe in the energy range of 2.1–6.0 keV’. J. Electron Spectrosc., 152, 143–147. Z WILLINGER , D.: 1997. Handbook of Differential Equations. Boston, MA: Academic Press, 3rd. edn.

Appendices

Overview

These appendices offer a collection of concepts and relations which belong to the standard mathematical ‘tools’ in atomic, molecular and optical physics. On many occasions we refer to these in the main text. We do not attempt here to derive them in a closed, axiomatic manner – rather a compact summary of useful and important formulas and equations is presented. Usually they are accompanied by some hints about their origins and connections to make them plausible for the reader without requesting the substantial efforts needed to go into the depth of derivations or proofs. Literature references are given for further information. Appendix A offers a tabulated overview of current values for the most important physical constants used in atomic, molecular and optical physics – and in these textbooks. Some remarks about systems of units and dimensional analysis are also included. Appendix B defines the angular momentum operators and introduces some basics about angular momentum algebra with 3j symbols (respectively C LEBSCH G ORDAN coefficients), 6j symbols (respectively R ACAH functions), and 9j symbols. Appendix C is concerned with the rules of evaluating and transforming matrix elements of irreducible tensor operators. This includes the very important W IGNER E CKART theorem, matrix elements for electromagnetically induced transitions and the reduction of coupling schemes as well as some useful geometrical relations. Finally we give some directions for the evaluation of radial matrix elements. Appendix D formulates the concept of parity and introduces – as an alternative to the usual complex spherical harmonics – real basis states for angular momenta, typically used in quantum chemistry. The concept is also extended to half integer angular momenta. Appendix E offers a recipe type of access to the important theme of coordinate rotation for angular momenta and irreducible tensor operators.

550

Appendices

Appendix F introduces the concept of multipole expansions and gives the most important relations derived from it. Two alternatives to define multipole tensor operators are discussed: based on spatial coordinates and on combinations of angular momentum operators. Appendix G describes the concept of convolution and illustrates it for the most important distribution functions. Appendix I presents some essentials about F OURIER transformations with special emphasis on electromagnetic fields. Appendix H provides the formal basis for treating electromagnetically induced transitions – are presented in the main text somewhat heuristically. It also derives the important T HOMAS -R EICHE -K UHN sum rule. In Appendix J, finally, we discuss the behaviour and the proper normalization of continuum wave functions. Both are important for a correct treatment of ionization and scattering processes.

A

Constants, Units and Conversions

A.1

Fundamental Physical Constants and Units

Modern physics relies on the knowledge of a relatively large number of so called “fundamental constants” which are measured with high precision in an ongoing, cooperative international effort. Table A.1 gives a survey on the most important ones, used throughout these books. Table A.1 2010 CODATA recommended values of physical constants extracted from NIST (2010), published by M OHR et al. (2012) Constant

Value

Equation

Universal constants speed of light in vacuum

c

2.99792458 × 108 m s−1

defined

magnetic constant

μ0

4π × 10−7 N A−2

defined

electric constant

ε0

8.854187817 . . . × 10−12 F m−1

1 μ0 c2

characteristic vacuum impedance

Z0

376.730313461 . . . "

cμ0 = √ = μ0 /ε0

Newtonian constant of gravitation

G

6.67384(80) × 10−11 m3 kg−1 s−2

G/c

6.70837(80) × 10−39 (GeV /c2 )−2

P LANCK constant

h

6.62606957(29) × 10−34 J s

=

107 4π c2 1 ε0 c

4.135667516(91) × 10−15 eV s

1.054571726(47) × 10−34 J s

c

197.3269631(49) MeV fm

P LANCK mass

mP

2.17651(13) × 10−8 kg

√ c/G

P LANCK length

P

1.616199(97) × 10−35 m

/(mP c)

P LANCK temperature

TP

1.416833(85) × 1032 K

mP c2 /kB

P LANCK time

tP

5.39106(32) × 10−44 s

P /c

e

1.602176565(35) × 10−19 C

angular momentum quant

h/2π

Electromagnetic constants elementary charge

© Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7

551

552

A Constants, Units and Conversions

Table A.1 (Continued) Constant B OHR magneton

μB

Value

Equation

927.400968(20) × 10−26 J T−1

e/(2me )

5.7883818066(38) × 10−5 eV T−1 μB / h

13.99624555(31) × GHz T−1

μN

5.05078353(11) × 10−27 J T−1

μN / h

3.1524512605(22) × 10−8 eV T−1 ˙ 7.62259357(17) MHz T−1

KJ

483597.870(11) × 109 Hz V−1

2e/ h

RK

25812.8074434(84) Ω

h/e2

me

9.10938291(40) × 10−31 kg

me

5.4857990946(22) × 10−4 u

ge

2.00231930436153(53)

μe μB

g factor anomaly

ae

1.15965218073(28) × 10−3

g/2 − 1

proton mass

mp

1.672621777(74) × 10−27 kg

proton-electron mass ratio

me /mp

1836.15267245(75)

neutron mass

mn

1.674927351(74) × 10−27 kg

mass of He nucleus

mα

6.64465675(29) × 10−27 kg

unified atomic mass unit

1u

1.660538921(73) × 10−27 kg

≡ m(12 C)/12

atomic unit of energy

Eh

27.21138505(60) eV

e4 me /(4πε0 )2

atomic unit of length (B OHR radius)

a0

0.52917721092(17) × 10−10 m

4πε0 2 /me e2

RYDBERG constant

R∞

10 973 731.568539(55) m−1

Eh /(2hc)

atomic unit of time

t0

2.418884326502(12) × 10−17 s

/Eh

fine structure constant new datab

α

7.2973525698(24) × 10−3

e2 /(4πε 0 c) = Eh /me c2

classical electron radiusc

re

2.8179403267(27) × 10−15 m

α 2 a0

C OMPTON wavelength

λC

2.4263102389(16) × 10−12 m

h/me c = 2παa0

T HOMSON cross section

σe

˙ 0.6652458734(13) × 10−28 m2

(8π/3)α 4 a02 = (8π/3)re2

Proton radius

Rp

0.8775(51) × 10−15 m

B OLTZMANN constant

kB

1.3806488(13) × 10−23 J K−1

S TEFAN-B OLTZMANN constant

σB

5.670373(21) × 10−8 W m−2 K−4

AVOGADRO constant

NA

6.02214129(27) × 1023 mol−1

volumed

nuclear magneton

J OSEPHSON constant VON

K LITZING constant

e/(2mp )

Atomic and nuclear constants electron mass electron g factora

Eh a02 = 2 /me

1/137.035999074(44) 1/137.035999037(91)

Vm

22.413953(21) × 10−3 m3 mol−1

L OSCHMIDT constantd

NL

2.6516462(24) × 1025 m−3

a.u. of dipole moment

ea0

8.47835326(19) × 10−30 C m

molar

5 4 2 π kB 15 h3 c2

NA /Vm

A.2

SI and Atomic Units

553

Table A.1 (Continued) Constant

Value

energy equivalents

1 eV

1.602176565(35) × 10−19 J

e × 1V

1J

6.24150934(14) × 1018 eV

1 J /1 eV

1u

0.931494061(21) GeV

1 u c2 /1 eV

me

0.510998928(11) MeV

me c2 /1 eV

1 cm−1

1.239841930(27) × 10−4 eV

hc/1 eV

eV in a.u.

1 eV

1 eV /2R∞ hc

eV in chem. units

1 eV

3.674932379(81) × 10−2 Eh ˙ 96.4853364(48) kJ mol−1 23.061 kcal mol

Equation

1 eV ×NA

−1

eV in wavenumbers

1 eV

8 065.54429(19) cm−1

1 eV / hc

eV in degree K

1 eV

1.1604519(11) × 104 K

1 eV /kB

a As

usual in the literature, g is defined positive, see (1.164), in contrast to NIST

b By

B OUCHENDIRA et al. (2011)

c This

is just a constructed quantity: at this distance the C OULOMB energy of two electrons is equal to the rest energy me c2 of an electron

d In

ideal gas at 273.15 K and 100 kPa

A.2

SI and Atomic Units

In general, throughout these text books we consequently use the SI units, meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd) – the international standard of units. In addition to these 7 “base units” (NIST, 2000a, Table 1), we employ • standard prefixes (NIST, 2000b), e.g. cm, mg, fs, • SI “derived units” (NIST, 2000a, Tables 2 and 3) when useful or common custom, such as joule (J), volt (V) or degree Celsius (◦ C), • several of the “accepted units outside the SI” (NIST, 2000c), such as electronVolt, 1 eV = e × V = 1.602 × 10−19 J, or barn, 1 b = 100 fm2 = 10−28 m2 . . . , • and make one exception for the occasionally used (non-SI) unit electron-barn, 1 eb = e × 1 b = 1.6022×10−47 A s m−2 . We point out here that all values of constants and physical quantities given in the table above and elsewhere in these text books refer to the currently valid definitions of the SI. These definitions may possibly change in 2014 and be replaced by a new set of definitions, completely based on fundamental constants (see Table A.1) as has been decided by the CGPM (2011) (see also W IKIPEDIA CONTRIBUTORS, 2013). Alternatively, we often use atomic units (a.u.) which are commonly used in atomic and molecular physics. We consider these, however, to be just simplifying

554


abbreviations: one reformulates equations such, that for lengths, times and energies only the combinations r/a0 ,

t/t0 ,

and W/Eh ,

respectively,

(A.1)

occur in the final expressions – which then become otherwise free of units and include only some dimensionless parameters, such as the fine structure constant α. It is often helpful to leave the equations in this form which simultaneously includes a dimensional analysis. However, for simplicity of writing we occasionally omit the respective units a0 , t0 , and Eh (e.g. in a S CHRÖDINGER equation), if no ambiguities can arise. The respective quantities are then measured in a0 , t0 , and Eh . For more details on a.u. see Sect. 1.8.3. We emphasize, however, that it is rather misleading, to talk about “setting = e = me = 1” – as occasionally formulated in (mostly theoretical) literature. The same holds for relativistic expressions where the confusing notion that “one sets c = 1” is often used – even in general texts, specifically on particle physics.

A.3

SI and GAUSS Units

The attentive reader will find a number of useful references to standard (mostly older) and sometimes important literature, where G AUSS, electrostatic (esu) or electromagnetic (emu) units are still used (G AUSS units are equivalent to esu and emu for electrostatic and electromagnetic units, respectively). Even in current literature, specifically in theoretical papers, these systems are still used. Obviously, the seeming advantage of writing a few equations in a simpler way than SI demands, appears so attractive to some authors that they renounce general compatibility. Thus, Table A.2 communicates some conversion rules between the SI and the G AUSS systems, following essentially the Appendix on units and dimensions in JACKSON (1999).

A.4

Radian and Steradian

Radian (rad) and steradian (sr) are two somewhat ‘conspicuous’ units which are frequently used in AMO physics. They are, however, genuine “SI derived units (NIST, 2000a)”. A brief clarification may be useful. Radian and steradian may be visualized as shown in Fig. A.1. The radian (rad) is defined in two dimensions for the plane angle ϕ by the length of an arc on a circle of radius r, divided by this radius r: ϕ= . r

(A.2)

With the full circumference of the circle being 2πr, the full angle thus becomes ϕ = 2π rad = 360 ◦ , so that 1 rad = 360 ◦ /(2π) = 57.296 ◦ .

A.4

Radian and Steradian

555

Table A.2 Relations and conversions between the SI and the Gaussian system Quantity/relation

SI

√

G AUSS

velocity of light

c=

dielectric permittivity

εr ε0

εr

magnetic permeability

μr μ0

μr

1/μ0 ε0

c

electric field E, displacement D, D = ε0 E +√P = εr ε0 E polarization P unit E esu = E SI 4πε0

D = E + 4πP √ = εr E E SI = E esu / 4πε0

magnetic H and B field, magnetization M unit

B = μ0 (H + M) = μr μ0 H √ B emu = B SI 4π/μ0

B = H + 4πM √ = μr H B SI = B emu μ0 /4π

M AXWELL equations

∇D = ρ

∇D = 4πρ

∇ × E = − ∂B ∂t

∇ × E = − 1c ∂B ∂t

∇B = 0

∇B = 0

∇×H =J +

∇ × H = 1c (J +

∂D ∂t

∂D ∂t )

charge, current, etc. unit

F = q(E + v × B) √ q esu = q SI / 4πε0 1C = 1As = 2.9979 × 109 esu

C OULOMB potential

V (r) =

unit electric dipole moment D EBYE

Cm 1D = 3.33564 × 10−30 C m

esu cm 1D = 10−18 esu cm

Polarizability

esu = α SI /(4πε ) αE 0 E

SI = (4πε )α esu αE 0 E

B OHR magneton μB =

e/(2me )

e/(2me c)

L ORENTZ force

(a)

F = q(E + × B) √ q SI = √ 4πε0 q esu 1esu =1 √dyn cm √ √ = 4πε0 10−9 N m −10 C = 3.3356 × 10 v c

q1 q2 4π ε0 r

V (r) =

q1 q2 r

(b)

rad =

— m

Fig. A.1 On the definition of (a) radian and (b) steradian

In three dimensions the steradian (sr) is defined as the solid angle Ω which characterizes any surface area of the size S on a sphere of radius r. In other words, the solid angle Ω is given as the surface S on a sphere of radius r divided by the square

556


of this radius: S . r2 For a cone with a full opening angle 2θ as shown in Fig. A.1 we have θ Ω= dΩ = 2π sin θ dθ = 2π(1 − cos θ ). Ω=

(A.3)

(A.4)

0

With the surface of a sphere being 4πr 2 the full solid angle is 4π sr, and 1 sr corresponds to an angle θ = arccos(1 − 1/(2π)) = 0.57195 rad = 32.77◦ – or equivalently to a cone with a full angle ϑ = 2θ = 65.54◦ . Note that expressed in SI base units, both rad = m m−1 and sr = m2 m−2 have the unit 1, i.e. are dimensionless. The symbols rad and sr may thus be used where advantageous to distinguish the respective observable quantities from others . . . or be omitted wherever the expressions are unambiguous. For small angles (A.4) may be expanded as lim Ω = πθ 2 + O θ 4 . θ→0

The divergence of well collimated beams (light beams, electron or ion beams etc.) is often given in mrad2 , i.e. in terms of the full divergence angle ϑ = 2θ measured in mrad. In the case of axially symmetry, i.e. for paraxial rays in a conical beam, the solid angle of divergence is thus

2 ϑ π sr. Ω = 10−6 4 mrad Thus, a (full) divergence angle of 1 mrad corresponds to a solid angle of Ω = (π/4)10−6 sr. Note, however, that for a rectangular beam with a small divergence ϑ 2 , measured in mrad2 , the solid angle is just Ω = 10−6 (ϑ/ mrad)2 sr.

A.5

Dimensional Analysis

In physics and technology dimensional analysis is often a useful tool to • make a first check on the validity of a physical expression, • guess, what the relation between several physical quantities might be, • obtain orders of magnitude estimates from a given relation. Throughout these textbooks we frequently make use of these possibilities. Thus, we summarize here the terminology used in this context: 1. Physical quantities, such as the electron mass typically have • a symbolic name, such as “me ”, • a numerical value, here 9.1094 × 10−31 , and


557

• a unit, here “kg”. • Together we write for this quantity: me = 9.1094 × 10−31 kg. 2. In dimensional analysis we try to avoid specific units for mass, length, and so on, since physical laws must be independent of the specific choice of units. Rather we use a basic set of dimensions: mass (M), length (L), time (T), charge (Q), and temperature (') – all written in block letters without serifs. For most of our purposes all other dimensions of physical quantities can be expressed as a product of powers of these basic dimensions, e.g. the dimension of force is MLT−2 , of energy ML2 T−2 , of electric current QT−1 , or with the thermal energy being Wtherm = kB T , the B OLTZMANN constant kB has the dimension ML2 T−2 '−1 and so on. 3. However, for convenience or for clarity we shall sometimes use more letters or abbreviations for the dimensions of physical quantities, e.g. for volume V = L3 , for energy Enrg = ML2 T−2 or R = L−3/2 for the dimension of a radial wave function. 4. The units of a physical quantity A or of a dimension A are represented by square brackets, [A] and [A], respectively. For instance the unit of a potential V (r) would be [V (r)] = eV – and more generally we have [Enrg] = eV or [Enrg] = J, both are acceptable.

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. a.u.: ‘atomic units’, see Sect. 2.6.2. emu: ‘electromagnetic units’, old system of unities, equivalent to the G AUSS system for magnetic quantities (see Appendix A.3). esu: ‘electrostatic units’, old system of unities, equivalent to the G AUSS system for electric quantities (see Appendix A.3). NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html. 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.

References B OUCHENDIRA, R., P. C LADE, S. G UELLATI -K HELIFA, F. N EZ and F. B IRABEN: 2011. ‘New determination of the fine structure constant and test of the quantum electrodynamics’. Phys. Rev. Lett., 106, 080801. CGPM: 2011. ‘Resolution 1 of the General Conference on Weights and Measures (2011) CGPM: On the possible future revision of the International System of Units, the SI’, Bureau International des Poid et Messure (BIPM). http://www.bipm.org/en/si/new_si/, accessed: 8 Jan 2014.

558


JACKSON, J. D.: 1999. Classical Electrodynamics. New York: John Wiley & sons, 3rd edn., 808 pages. M OHR, P. J., B. N. TAYLOR and D. B. N EWELL: 2012. ‘CODATA recommended values of the fundamental physical constants: 2010’. Rev. Mod. Phys., 2013, 1527–1605. http://physics.nist.gov/ constants, accessed: 8 Jan 2014. NIST: 2000a. ‘Reference on constants, units, and uncertainties: SI base and derived units’, NIST. http://physics.nist.gov/cuu/Units/units.html, accessed: 8 Jan 2014. NIST: 2000b. ‘Reference on constants, units, and uncertainties: SI prefixes’, NIST. http://physics. nist.gov/cuu/Units/prefixes.html, accessed: 8 Jan 2014. NIST: 2000c. ‘Reference on constants, units, and uncertainties: Units outside the SI’, NIST. http://physics.nist.gov/cuu/Units/outside.html, accessed: 8 Jan 2014. NIST: 2010. ‘Reference on constants, units, and uncertainties’, NIST. http://physics.nist.gov/ cuu/Constants/, accessed: 7 Jan 2014. W IKIPEDIA CONTRIBUTORS: 2013. ‘Proposed redefinition of SI base units’, Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units, accessed: 8 Jan 2014.

B

Angular Momenta, 3j and 6j Symbols

We note here the most important relations for angular momenta in compact form for reference and use throughout these books. Closely related are the 3j , 6j and 9j symbols relevant for the coupling of angular momenta. They play an important role in atomic and molecular physics. We follow here – with one important exception concerning the normalization of the spherical components of angular momenta – strictly the definitions of B RINK and S ATCHLER (1994) and E DMONDS (1996). Explicit expressions for the 3j , 6j and 9j symbols are somewhat complicated but well documented in both references as well as in many textbooks on quantum mechanics. For practical evaluation purposes one may use today readily accessible, compact 3j , 6j and 9j calculators (see e.g. S VEN G ATO R EDSUN, 2004). Also, they are often integrated in standard mathematical programmes. Thus, we refrain here from extensive tabulations and restrict ourselves to definitions, symmetry properties, orthogonality relations, a few matrix elements and some important special cases (as collected from several sources: E DMONDS, 1996; B RINK and S ATCHLER, 1994; W EISSTEIN, 2004a,b).

B.1

Angular Momenta

B.1.1

General Definitions

The generalized angular momentum operator J is defined as vector operator with three real, Hermitian components Jx , Jy and Jz with Ji† = Ji for i = x, y, z,

(B.1)

which satisfy the commutation relations [Jx , Jy ] = iJz ,

[Jz , Jx ] = iJy ,

[Jy , Jz ] = iJx

and

2 J , Ji = 0,

or in compact form: © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7

559

560

B

J × J = i J and


2 J = 0. J ,

(B.2)

Alternatively one may construct J from three spherical components J± and J0 (so called helicity basis) which are an irreducible representation of the rotation group of rank 1. They are derived from the real components:1 1 J+ = − √ (Jx + iJy ), 2

1 J− = + √ (Jx − iJy ) 2

and J0 ≡ Jz .

(B.3)

The inverse relations are: 1 Jx = − √ (J+ − J− ), 2

i Jy = √ (J+ + J− ). 2

(B.4)

We note that these operators are constructed in complete analogy to the spherical components (4.75) of the position operator r in the helicity basis which we have introduced in the context of dipole transitions in Sect. 4.4. The spherical components J± are not Hermitian, with J+† = −J−

and J−† = −J+ ,

(B.5)

and their commutation relations are derived from (B.2): [J0 , J± ] = ±J± and [J+ , J− ] = −J0 , 2 2 J , J± = 0 and J , J0 = 0.

(B.6) (B.7)

The scalar product of two angular momenta J 1 and J 2 is given by J1 · J 2 = J1x J2x + J1y J2y + J1z J2z = −J1+ J2− + J10 J20 − J1− J2+

(B.8)

in accordance with the general definition of scalar products of tensor operators 2 (C.20). Hence, J is expressed alternatively in the real or in the helicity basis: 2 J = Jx2 + Jy2 + Jz2 = −J+ J− + J02 − J− J+ =

+1

(−1)q Jq J−q .

(B.9)

q=−1

The operators J+ and J− play a key role in the construction of angular momentum states. Using their commutation relations (B.6)–(B.7) one may construct angular momentum states without using spherical harmonics (2.79)–(2.83) – which 1 Very

often one finds in textbooks (even in B RINK and S ATCHLER, 1994) these ladder operators defined as J± = Jx ± iJy . These are, however, not spherical components of the J vector operator and lead to asymmetric commutation relations. With the present definition – which is also used e.g. by B LUM (2012), W EISSBLUTH (1978) – J± fulfill both functions: as ladder (or stepping) operators and spherical components of J.

B.1

Angular Momenta

561

exclusively represent orbital angular momenta with integer quantum numbers . All relevant algebra for angular momentum states may be derived in that way. Specifically, if |J M are eigenstates of Jz = J0 with eigenvalues M, it follows from (B.6) that J+ |J M and J− |J M are also eigenstates of J0 with eigenvalues (M + 1) and (M − 1), respectively: J0 J± |J M = J± J0 |J M ± J± |J M = (M ± 1)J± |J M .

(B.10)

2 J . The operators With (B.7) the new states J± |J M are also eigenfunctions of J+ and J− are thus raising and lowering operators (or ladder operators) for the projection quantum number M. According to (B.10) M increases or decreases in steps of one. One may show (here without proof) that M has a maximum and a minimum value

−J ≤ M ≤ J,

(B.11)

and clearly, this maximum and minimum value M = ±J can only be connected by M = −J, −J + 1, . . . , J − 1, J if both M and J are either integer or half integer numbers – in complete accordance with the experimental observation: integer values arise from orbital angular momenta, and half integer values reflect the involvement of spin 1/2 particles. The states thus generated obey the standard eigenvalue equations and are of orthonormalized: 2 J |J M = 2 J (J + 1)|J M J M J M = δ J J δM M .

and J0 |J M = M|J M

(B.12) (B.13)

With some algebra one may show that the only nonvanishing matrix elements of the spherical angular momentum components J0 , J+ and J− are J M|J0 |J M = M

and

J M ± 1|J± |J M = ∓ √ a(±M) 2 with a(M) = J (J + 1) − M(M + 1).

(B.14) (B.15) (B.16)

Equivalently one may write: J± |J M = ∓ √ a(±M)|J M ± 1 2

and J0 |J M = M|J M .

(B.17)

We finally note, that all relations communicated in this section for the general angular momentum J are completely equivalent to those for the orbital angular momentum L which we have derived in Sect. 2.5 (one may of course also introduce ± ). In contrast to L, the general angular momentum is not spherical components L explicitly referring to position space and angles, and is not restricted to integer values of J and M.

562

B

B.1.2


Orbital Angular Momenta – Spherical Harmonics

One special kind of angular momenta are the orbital angular momenta introduced in Sect. 2.5. As described there, their respective operators may be represented in terms of spatial coordinates – more precisely, the operator of the squared orbital angular 2 momentum L and the components of L can be expressed as differential operators with respect to the azimuthal and polar angles in polar coordinates. Of particular importance are

1 ∂ ∂ 1 ∂2 2 2 2 2 2 L = Lx + Ly + Lz = − (B.18) sin θ + 2 sin θ ∂θ ∂θ sin θ ∂ϕ 2 z = −i ∂ and L ∂ϕ

(B.19)

with the spherical harmonics Y m (θ, ϕ) as joint eigenfunctions. These operators have the eigenvalues 2 ( + 1) and m, respectively, corresponding to (B.12). Here = 0, 1, 2, . . . and m is any integer value between − ≤ m ≤ . Without proof we communicate here the spherical harmonics, m 2 + 1 ( − m)! m Y m (θ, ϕ) = (−1) P (cos θ ) exp(imϕ) (B.20) 4π ( + m)! 2π π ∗ with dϕ Y m (θ, ϕ)Y m (θ, ϕ) sin θ dθ = δ δm m . (B.21) 0

0

P m (x) are the associated L EGENDRE polynomials P m (x) =

m dm P (x) d +m 2 1 2 m 1 − x2 1 − x x −1 , = m +m dx 2 ! dx

(B.22)

which in turn are derived from the ordinary L EGENDRE polynomials P (x) =

1 d 2 x −1 . 2 ! dx

(B.23)

As a function of cos θ the first L EGENDRE polynomials are: P0 (cos θ ) = 1 P1 (cos θ ) = cos θ 3 1 cos2 θ − 2 2 5 3 P3 (cos θ ) = cos3 θ − cos θ 2 2 1 P4 (cos θ ) = 35 cos4 θ − 30 cos2 θ + 3 . 8

P2 (cos θ ) =

(B.24)

B.1

Angular Momenta

563

We also note the orthogonality relation P (cos θ )P (cos θ )dΩ = 4π

4π δ . 2 + 1

With P0 (cos θ ) = 1 follows for all L EGENDRE polynomials with > 0 P (cos θ )dΩ = 0.

(B.25)

4π

The definition (B.20) of the spherical harmonics Y m (θ, ϕ) implies the so called standard phase convention according to C ONDON and S HORTLEY (1951) with the factor (−1)m . One finds that the c.c. is ∗ (θ, ϕ) = (−1)m Y −m (θ, ϕ), Y m

(B.26)

and inversion at the origin (r → −r) leads to Y m (π − θ, π + ϕ) = (−1) Y −m (θ, ϕ).

(B.27)

The spherical harmonics are said to have positive or negative parity depending on whether is even or odd, respectively (see detailed discussion in Appendix D). In compact form one obtains – well suited for computational use – d +m (sin θ )2 (−1) +m (2 + 1)( − m)! (sin θ )m exp(imϕ) Y m (θ, ϕ) = 2 ! 4π( + m)! d(cos θ ) +m d −m (sin θ )2 (−1) (2 + 1)( + m)! = exp(−imϕ). (sin θ )−m 4π( − m)! 2 ! d(cos θ ) −m (B.28) For simplicity of writing one often also uses renormalized spherical harmonics C m (θ, ϕ) =

4π Y m (θ, ϕ). 2 + 1

(B.29)

We note the special values Y m (0, ϕ) =

(2 + 1)/(4π)δm,0

and C m (0, ϕ) = δm,0 .

(B.30)

Explicit expressions for the renormalized spherical harmonics up to = 3 are given in Table B.1. A graphical illustration of their angular dependence is shown in Fig. 2.6.

564

B


Table B.1 Spherical harmonics according to (B.28) and (B.29)

m

Symbol

2 L

Lz

Renormalized √spherical harmonics C k (θ, ϕ) = 4π/(2 + 1)Y m (θ, ϕ)

0

0

s

0

0

C00 = 1

0

p0

22

0

±1

p±1

22

±

C10 = cos θ C1±1 = ∓ 12 sin θ · e±iϕ

0

d0

62

0

±1

d±1

62

±

±2

d±2

62

±2

0

f0

122

0

±1

f±1

122

±

±2

f±2

122

±2

C30 = 12 (5 cos3 θ − 3 cos θ) √ C3±1 = ∓ 14 3e±iϕ sin θ(5 cos θ 2 − 1) 2 ±2iϕ C3±2 = 15 8 cos θ sin θe

±3

f±3

122

±3

C3±3 = ∓

1

2

3

C20 = 12 (3 cos2 θ − 1) C2±1 = ∓ 32 sin θ cos θ · e±iϕ C2±2 = 38 sin2 θ · e±i2ϕ

√

5 4

sin3 θ · e±3iϕ

B.2

Coupling of Two Angular Momenta: CLEBSCH-GORDAN Coefficients and 3j Symbols

B.2.1

Definitions

When coupling two angular momenta J 1 and J 2 to generate a third one J1 + J2 = J the eigenstates |J M of the latter are constructed from the uncoupled states |J1 M1 and |J2 M2 as |J1 M1 J2 M2 J1 M1 J2 M2 |J1 J2 J M . (B.31) |J1 J2 J M = M1 M2

Formally, one may read the sum |J1 M1 J2 M2 J1 M1 J2 M2 | ≡ 1 simply as quantum mechanical unit operator (see Sect. 2.3.1) written in front of |J1 J2 J M . J1 , J2 and J are the corresponding (integer or half integer) angular momentum quantum numbers, M1 , M2 and M the corresponding projection quantum numbers in respect of the z-axis. The inverse transformation is |J1 J2 J M J1 J2 J M|J1 M1 J2 M2 , (B.32) |J1 M1 J2 M2 = JM

and the coefficients J1 J2 J M|J1 M1 J2 M2 are the conjugate complex ones of J1 M1 J2 M2 |J1 J2 J M . According to the general convention one chooses the phase such that the coefficients are real: J1 J2 J J1 J2 J M|J1 M1 J2 M2 = J1 M1 J2 M2 |J1 J2 J M = CM . 1 M2 M

(B.33)

B.2

Coupling of Two Angular Momenta

565

They are called C LEBSCH -G ORDAN coefficients. They are non-zero if – and only – two conditions are fulfilled: 1. for the Ji quantum numbers δ(J1 J2 J ) = 1 must be fulfilled, with the triangular relation defined by: " δ(J1 J2 J ) =

1 0

if |J2 − J1 |, |J2 − J1 | + 1, . . . ≤ J ≤ |J2 + J1 | else

(B.34)

2. for the projection quantum numbers Mi = M1 , M2 and M −Ji , −Ji + 1, . . . ≤ Mi ≤ Ji

and

(B.35)

M = M1 + M 2 must hold. The C LEBSCH -G ORDAN coefficients disappear if the triangular relation is not satisfied, i.e. if δ(J1 J2 J ) = 0, or if M = M1 + M2 . Alternative to the C LEBSCH -G ORDAN coefficients the W IGNER 3j symbols

J1 M1

J2 M2

J M

are used. They are defined completely symmetric in respect of the three angular momenta, and are related to the C LEBSCH -G ORDAN coefficients by J1 −J2 +M

J1 M1 J2 M2 |J M = (−1)

(2J + 1)

1/2

J1 M1

J2 M2

J −M

. (B.36)

Note the minus sign in front of M in the last column. The 3j symbols disappear unless δ(J1 J2 J ) = 1 and M1 + M2 = M.

B.2.2

(B.37)

Orthogonality and Symmetries

The following orthonormality relations hold:

J1 M1 J2 M2 J M J1 M1 J2 M2 |J M = δM1 M δM2 M δ(J1 J2 J ) 1

2

(B.38)

JM

J1 M1 J2 M2 J M J1 M1 J2 M2 |J M = δMM δJ J δ(J1 J2 J ).

M1 M2

(B.39)

566

B


Correspondingly, the 3j symbols satisfy the orthogonality relations:

J1 J2 J J1 J2 J = δM1 M δM2 M δ(J1 J2 J ) (2J + 1) 1 2 M1 M2 M M1 M2 M JM

J1 (2J + 1) M1 M1 M2

J2 M2

and thus also

J

J1 M1

J2 M2

M

J1 M1

J2 M2

J M

J M

M1 M2

(B.40)

= δJ J δMM δ(J1 J2 J ) (B.41)

2 =

δ(J1 J2 J ) . (2J + 1)

(B.42)

The 3j symbols are symmetric in respect of cyclic exchange of their columns. Their sign changes according to (−1)J1 +J2 +J when two columns are exchanged or the sign of all projection quantum numbers is changed:

J2 J J1 J1 J2 J J J1 J2 = = (B.43) M2 M M1 M M1 M2 M1 M2 M

J2 J1 J etc. (B.44) = (−1)J1 +J2 +J M2 M1 M

J2 J J1 J1 +J2 +J . (B.45) = (−1) −M1 −M2 −M

B.2.3

General Formulae

The 3j symbols can be computed either by suitable recursion formulae or from the R ACAH formula:

J1 J2 J (B.46) M1 M2 M = (−1)J1 −J2 −M δ(M1 +M2 )−M (J1 J2 J ) × (J1 + M1 )!(J1 − M1 )!(J2 + M2 )!(J2 − M2 )!(J + M)!(J − M)! ' × (−1)t t! (J − J2 + t + M1 )!(J − J1 + t − M2 )! t

(−1 × (J1 + J2 − J − t)!(J1 − t − M1 )!(J2 − t + M2 )! with the so called triangular function

(abc) =

(a + b − c)!(a − b + c)!(−a + b + c)! . (a + b + c + 1)!

(B.47)

B.2

Coupling of Two Angular Momenta

567

One has to sum over all values of t which lead to non-negative terms for the faculty expressions in (B.47). The 3j symbols can be obtained very fast from simple computing programmes, as already mentioned (e.g. S VEN G ATO R EDSUN, 2004, Java script, also for 6j and 9j symbols).

B.2.4

Special Cases

Closed expressions for some frequently used cases are often useful. The 3j symbol becomes specially simple if one of the angular momenta is zero:

J1 M1

J2 M2

0 0

=

(−1)J1 −M1 δJ1 J2 δ−M1 M2 . √ 2J1 + 1

(B.48)

From (B.45) one finds the rule:

J1 J 2 J = 0 if S = J1 + J2 + J is odd. 0 0 0

(B.49)

A rather compact expression is also obtained if S = J1 + J2 + J is even and all projection quantum numbers are zero:

J1 0

J2 0

J 0

= (−1)S/2 ×

(S − 2J1 )!(S − 2J2 )!(S − 2J )! (S + 1)!

1/2 (B.50)

(S/2)! . (S/2 − J1 )!(S/2 − J2 )!(S/2 − J )!

Frequently used relations are: /

J+ M

1 2

J +1 M

J −M −

J −M − 1

J M

1 1

0

J −M−1/2

= (−1)

= (−1)J −M−1

J − M + 1/2 (2J + 1)(2J + 2)

(J − M)(J − M + 1) (2J + 1)(2J + 2)(2J + 3)

2(J − M + 1)(J + M + 1) (2J + 1)(2J + 2)(2J + 3)

(J − M)(J + M + 1) J 1 = (−1)J −M −M − 1 1 J (2J + 1)(2J + 2)

M J 1 = (−1)J −M √ −M 0 J (J + 1)(2J + 1)

+1 J M −M J M

1 2

1 2 1 2

1 0

= (−1)J −M−1

(B.51)

(B.52)

(B.53)

(B.54) (B.55)

568

B


J 2 (B.56) −M − 2 2 6(J − M − 1)(J − M)(J + M + 1)(J + M + 2) = (−1)J −M (2J + 3)(2J + 2)(2J + 1)2J (2J − 1)

J J 2 (B.57) M −M − 1 1 6(J + M + 1)(J − M) = (−1)J −M (1 + 2M) (2J + 3)(2J + 2)(2J + 1)2J (2J − 1)

2[3M 2 − J (J + 1)] J J 2 = (−1)J −M √ . (B.58) M −M 0 (2J + 3)(2J + 2)(2J + 1)2J (2J − 1) J M

Note: Formula (B.54) is slightly erroneous in B RINK and S ATCHLER (1994).

B.3

RACAH Function and 6j Symbols

B.3.1

Definition

Often one has to couple three angular momenta. A typical case is the hyperfine structure (HFS) of atoms in one electron systems, where orbital angular momentum, electron spin and nuclear spin have to be considered. In the case of two electrons, one may have no orbital angular momentum (ns orbital), the other electron may be in a n configuration, e.g. Hg(6sn ). Here the spins of the two electrons couple to a total spin and the latter couples with the orbital angular momentum of the n electron. In molecular physics too, a host of problems involves similar coupling requirements, e.g. the molecular rotation may couple with the electron’s orbital angular momentum and its spin. And even in optical transitions between spin-orbit coupled atomic states the photon adds an extra angular momentum from which the optical selection rules are derived. Generally, in order to determine the states of three angular momenta one starts with the product states |j1 m1 |j2 m2 |j3 m3 = |j1 m1 j2 m2 j3 m3 in an uncoupled representation. Formally, the total angular momentum vector operator is given by J2 + J 3, J = J1 + 2 and one has to find eigenstates of J and Jz with the eigenvalues J (J + 1)2 and M. This may be done in two alternative schemes, based on the coupling of two angular momenta described in Appendix B.2. One may first couple J 1 and J2 to J 12 and then combine J 12 with J 3 to J . According to (B.31) from the product states |j1 m1 |j2 m2 one obtains

B.3

RACAH Function and 6j Symbols

(j1 j2 )J12 M12 =

569

|j1 m1 |j2 m2 j1 m1 j2 m2 |J12 M12 ,

m1 m2

from which with |j3 m3 (j1 j2 )J12 M12 |j3 m3 J12 M12 j3 m3 |J M (j1 j2 )J12 j3 ; J M =

(B.59)

M12 m3 2 follows. This state is simultaneously eigenstate of J , Jz and J 12 with J12 being an additional quantum number of the system. Alternatively one couples first J 2 with J 3 to give J 23 and combines then J 23 with J 1 to J . The eigenstate for the total 2 angular momentum J and its z-component Jz is then j1 (j2 j3 )J23 ; J M = |j1 m1 (j2 j3 )J23 M23 j1 m1 J23 M23 |J M . (B.60) m1 M23 2 This state is also eigenstate of J 23 , and in this case J23 is an additional quantum number. These two representations are of course not linearly independent from each other – they describe the same subspace of angular momentum states. Thus, one may express each state in one of the two coupling schemes by a linear combination of the states in the other scheme: (j1 j2 )J12 j3 ; J M (B.61) j1 (j2 j3 )J23 ; J M j1 (j2 j3 )J23 ; J M (j1 j2 )J12 j3 ; J M . = J23

The expansion coefficients are scalar quantities and independent of M. To write them in a symmetric manner one uses the so called W -function after R ACAH (1942) or the 6j symbol: j1 (j2 j3 )J23 ; J M (j1 j2 )J12 j3 ; J M

= (2J12 + 1)1/2 (2J23 + 1)1/2 W (j1 j2 Jj3 ; J12 J23 ) " −j1 −j2 −j3 −J j1 with W (j1 j2 Jj3 ; J12 J23 ) = (−1) j3

B.3.2

(B.62)

j2 J

# J12 . (B.63) J23

Orthogonality and Symmetries

For the 6j symbols not to disappear, the 6 angular momenta must be coupled according to the above definitions. This requires triangular relations between the angular momenta. Schematically the following triangular relations must be satisfied: " # " # " # " # * * * * * * , , , . (B.64) * * * * * *

570

B


The 6j symbols are invariant against exchanging any two columns or any two rows: " # " # " # # " j1 j2 J12 j1 J12 j2 J J23 j1 j2 J12 j1 = = , etc. = j3 J J23 j3 J23 J J J23 j3 j2 J12 j3 (B.65) Finally, a number of sum rules hold for the 6j symbols. We note # " j1 j2 k 2k = 1, (B.66) (−1) (2k + 1) j2 j 1 J k

and the orthogonality relation " j1 (2J + 1) 2J + 1 j3 J

B.3.3

j2 j4

J J

#"

j1 j3

j2 j4

# J = δJ J . J

(B.67)

General Formulae

One may develop explicit expressions for the 6j symbols from (B.59)–(B.61). One finds according to R ACAH: " # j1 j2 J12 (B.68) j3 J J23 =

(j1 j2 J12 ) (j1 J J23 ) (j3 j2 J23 ) (j3 J J12 )

(−1)t (t + 1)! t

f (t)

with f (t) = (t − j1 − j2 − J12 )!(t − j1 − J − J23 )!(t − j3 − j2 − J23 )! × (t − j3 − J − J12 )!(j1 + j2 + j3 + J − t)! × (j2 + J12 + J + J23 − t)!(J12 + j1 + J23 + j3 − t)! and the triangular function (abc) according to (B.47). The summation is again over all nonvanishing terms. Again it is commendable to use one of the available Java-applets for practical evaluations. The 6j symbols may also be expanded as sums over products of 3j symbols. An important relation is the contraction formula:

j3 1 j2 j1 +j2 +j3 +μ1 +μ2 +μ3 (−1) m1 μ2 −μ3 μ1 μ2 μ3

2 j3 j2 3 j1 j1 × −μ1 m2 μ3 μ1 −μ2 m3

" # 1 2 3 1 2 3 . = m1 m2 m3 j1 j2 j3

(B.69)

B.4

Four Angular Momenta and 9j Symbols

571

For the special case that 1 + 2 + 3 is even and j3 = 1/2, a particularly simple relation emerges:

j2 3 j1 (B.70) = −(−1)− 1 − 2 −j2 −j1 (2 1 + 1)(2 2 + 1) −1/2 1/2 0 #

" 1 2 3 1 2 3 . × j2 j1 1/2 0 0 0

B.3.4

Special Cases

Again, some closed expressions for special cases are occasionally useful. With s = a + b + c and X = b(b + 1) + c(c + 1) − a(a + 1): " # (−1)s δ(abc)δbe δcd a b c (B.71) =√ 0 d e (2b + 1)(2c + 1) " # 2(−1)s+1 X a b c =√ (B.72) 1 c b 2b(2b + 1)(2b + 2)2c(2c + 1)(2c + 2) " # 2(−1)s a b c (B.73) =√ 2 c b (2b − 1)2b(2b + 1)(2b + 2)(2b + 3)

*

*

B.4

a+ b+

1 2 1 2

a+ b

1 2

[3X(X − 1) − 4b(b + 1)c(c + 1)] ×√ (2c − 1)2c(2c + 1)(2c + 2)(2c + 3) 1 a (s − 2c + 1)(2 + s) (−1)s+1 2 = 2 (2a + 1)(a + 1)(2b + 1)(b + 1) b c + 12 1 s+1 a (s − 2c + 1)(2 + s) (−1) 2 = . 1 2 (2a + 1)(a + 1)(2b + 1)(b + 1) b+ 2 c

(B.74)

(B.75)

Four Angular Momenta and 9j Symbols

Very briefly, we mention the coupling of four angular momenta, j1 , j2 , j3 and j4 , which again may be accomplished in different sequences. E.g. one may couple |(j1 j2 )j12 , (j3 j4 )j34 ; j m or alternatively |(j1 j3 )j13 , (j2 j4 )j24 ; j m . The two schemes relate to each other by a unitary transformation: (j1 j3 )j13 , (j2 j4 )j24 ; j m (B.76) (j1 j2 )j12 , (j3 j4 )j34 ; j m (j1 j2 )j12 , (j3 j4 )j34 ; j (j1 j3 )j13 , (j2 j4 )j24 ; j . = j12 ,j34

572

B


As a short notation for the recoupling coefficients (j1 j2 )j12 , (j3 j4 )j34 ; j (j1 j3 )j13 , (j2 j4 )j24 ; j

⎧ 1/2 ⎨ j1 j3 = (2j12 + 1)(2j34 + 1)(2j13 + 1)(2j24 + 1) ⎩ j13

j2 j4 j24

(B.77) ⎫ j12 ⎬ j34 ⎭ j

the so called W IGNER 9j symbols are used {given in curly brackets}. Each column and each row has to satisfy a triangular relation, and an odd number of permutations of rows or columns only changes the sign according to (−1) all j ’s . The orthonormality relation reads (2j12 + 1)(2j34 + 1)(2j13 + 1)(2j24 + 1) j13 ,j24

⎧ ⎨ j1 × j3 ⎩ j13

j2 j4 j24

⎫⎧ j12 ⎬ ⎨ j1 j34 j3 ⎭⎩ j j13

j2 j4 j24

⎫ j12 ⎬ j34 = δj13 j δj24 ,j . 13 24 ⎭ j

For later use, we also communicate here one useful formula. In the case that one of the j ’s is zero, the 9j symbol reduces to a 6j symbol: ⎧ ⎫ " # ⎨a b c ⎬ δcf δgh (−1)b+d+c+g a b c d e f = . (B.78) ⎩ ⎭ [(2c + 1)(2g + 1)]1/2 e d g g h 0

Acronyms and Terminology c.c.: ‘complex conjugate’. HFS: ‘Hyperfine structure’, splitting of atomic and molecular energy levels due to interactions of the active electron with the atomic nucleus (Chap. 9).

References B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics, vol. 64. Berlin, Heidelberg: Springer Verlag, 3rd edn., 343 pages. B RINK, D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3rd edn., 182 pages. C ONDON, E. U. and G. S HORTLEY: 1951. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 441 pages. E DMONDS, A. R.: 1996. Angular Momentum in Quantum Mechanics. Princeton, NJ, USA: Princeton University Press, 154 pages. R ACAH, G.: 1942. ‘Theory of complex spectra II’. Phys. Rev., 62, 438–462. S VEN G ATO R EDSUN: 2004. ‘3j 6j 9j -symbol java calculator’, Sven Gato Redsun. http://www. svengato.com/, accessed: 8 Jan 2014.

References

573

W EISSBLUTH, M.: 1978. Atoms and Molecules. Student Edition. New York, London, Toronto, Sydney, San Francisco: Academic Press, 713 pages. W EISSTEIN, E. W.: 2004a. ‘Wigner 3j -symbol’, Wolfram Research, Inc., Champaign, IL, USA. http://mathworld.wolfram.com/Wigner3j-Symbol.html, accessed: 8 Jan 2014. W EISSTEIN, E. W.: 2004b. ‘Wigner 6j -symbol’, Wolfram Research, Inc., Champaign, IL, USA. http://mathworld.wolfram.com/Wigner6j-Symbol.html, accessed: 8 Jan 2014.

C

Matrix Elements

In this appendix some tools are collected which are useful for evaluating matrix elements of irreducible tensor operators. These matrix elements are needed to compute transition probabilities for optical excitation of atomic and molecular states as well as for the analysis of collision induced processes.

C.1

Tensor Operators

C.1.1

Definition

According to R ACAH (1942) one defines as irreducible tensor operator Tk of the rank k each operator whose 2k + 1 components Tkq (q = −k, −k + 1, . . . , k) satisfy the same commutation rule in respect of J as the spherical harmonic operators Ykq (θ, ϕ). Following R ACAH this implies in our notation (B.3)1 for the spherical components of the angular momentum operator J: [Jz , Tkq ] = q Tkq and 1 [J± , Tkq ] = ∓ (k ∓ q)(k ± q + 1)Tkq±1 . 2

(C.1) (C.2)

B RINK and S ATCHLER (1994) define – equivalently – a spherical tensor (operator) Tk of rank k as a quantity defined by 2k + 1 components, which transforms upon co-

ordinate rotation in the same way as the irreducible representation Dk of the rotation group (see Appendix E): = (C.3) Tkq Tks Dksq (αβγ ). s

refer to J± = ∓ √1 (Jx ± iJy ) while R ACAH gives the relation for ∓(Jx ± iJy ). See also 2 footnote 1 in Appendix B.

1 We


575

576

C Matrix Elements

Here Tkq is the component of the tensor in the new coordinate system which arises by rotating the old, un-dashed system through the E ULER angles αβγ . It is represented in term of the components Tks of the tensor in the old system. Important examples of tensor operators are the spherical harmonics Y m (θ ϕ) (rank ) themselves, or somewhat more general the eigenstates |j m of the angular momentum operator (rank j ). † of a tensor operator Tkq is The adjoint (or Hermitian conjugate) operator Tkq defined by (see Sect. 2.3.1) † † ∗ JM , J M J M = J M|Tkq (C.4) J M Tkq |J M = Tkq and with the C ONDON and S HORTLEY (1951) phase convention † = (−1)q Tk−q . Tkq

(C.5)

† ∗ , as e.g. in the = Tkq If Tkq is simply a complex function this is equivalent to Tkq case of the spherical harmonics (B.26). † = Tkq . Obviously, with (C.4) A tensor is called Hermitian or self adjoint if Tkq the diagonal matrix elements (and hence the eigenvalues) are then real: Hermitian operators describe measurable, real quantities. Tensors of rank 1 are called vector operators, e.g. the position vector r, the unit vector of polarization e, the dipole moment D, as well as all angular momentum . The components of a vector operator V in the operators, such as S, L, J, I or F spherical basis are

x ± iV y ]; ±1 = ∓ √1 [V V 2

C.1.2

0 = V z . V

(C.6)

WIGNER-ECKART Theorem

When evaluating matrix elements for a tensor operator, the W IGNER -E CKART theorem is very useful. It says, that for any irreducible tensor operators of a given rank k, the matrix elements between states |J M and |J M depend in identical form on the projection quantum numbers M, M , and q: γ J M Tkq |γ J M 5 = J Mkq J M γ J 5Tk γ J (C.7)

√ J k 5 J γ J 5Tk γ J (C.8) = (−1)J −k+M 2J + 1 −M M q

√ k J 5 J J −M γ J 5Tk γ J . (C.9) 2J + 1 = (−1) −M q M

C.1

Tensor Operators

577

The thus defined reduced matrix element2 γ J Tk γ J is independent of M , M and q. Due to the properties of the 3j symbol the matrix elements are non-zero only if M = M − q and if the triangular relation δ(J kJ ) = 1 holds! In practice, to determine the reduced matrix element one has to evaluate just one nonvanishing matrix element (preferably a particularly simple one) and to divide it by the corresponding C LEBSCH -G ORDAN coefficient, e.g. J Tk J = J 0|Tk0 |J 0 / J 0|J k00 if J + J + k is even. An important example is the reduced matrix element of an angular momentum. With J M Jz |J M = δJ J δM M M (C.10)

√ 1 J 5 J J 5 = (−1)J −M 2J + 1 J J −M 0 M and (B.55) for the 3j symbol, one obtains: 5 J J = δJ J J (J + 1). J 5

(C.11)

The W IGNER -E CKART theorem allows one to relate two matrix elements of different tensor operators, Sk and Tk of the same rank k with each other in a simple manner. The ratio R of the reduced matrix elements of two tensor operators of the same rank k may be computed from any pair of (nonvanishing) matrix elements of their components: γ J Tk γ J γ J M |Tkq |γ J M R J , J, T, S = = . γ J Sk γ J γ J M | Skq |γ J M with (C.7) we write Specifically for vector operators V V q |γ J M = R γ J M J q |γ J M , γ J M

(C.12)

(C.13)

q . which is independent of M and M and holds for all components V The diagonal matrix elements of the scalar product V · J may be written γ J M · γ J M · γ J M|V J |γ J M , (C.14) γ J M|V J |γ J M = M

where we have simply inserted the quantum mechanical unit operator 1 = M |γ J M γ J M |. With (C.13), this expression becomes 2

Our definition of reduced matrix elements follows B RINK and S ATCHLER (1994), asthat these reduced suming, however, integer rank only, so that (−1)2k = 1. √ Note and J , rather + 1) γ J T γ J = (2J matrix elements are not symmetric in J k √ (1942), E DMONDS (1996) and others use a symmetric (2J + 1) γ J Tk γ J . R ACAH √ definition: γ J Tk γ J Racah = 2J + 1 γ J Tk γ J .

578

C Matrix Elements

· γ J M|V J |γ J M = R γ J M| J γ J M γ J M J |γ J M

(C.15)

M 2 = R γ J M| J |γ J M = RJ (J + 1)2 . 2 In the second line the eigenvalue (B.12) of J has been used, so that finally

R=

· γ J M|V J |γ J M . J (J + 1)2

(C.16)

Inserting this back into (C.13) gives the very useful projection theorem · γ J M|V J |γ J M γ J M γ J M Jq |γ J M . Vq |γ J M = 2 J (J + 1)

(C.17)

2 If one √ realizes that (V · J )J /J (J + 1) is the projection of V onto the unit vector J / J (J + 1), its physical relevance becomes evident. The projection theorem is used e.g. in a coupled scheme of angular momenta, to express the matrix elements of individual components in terms of the total angular momentum J. Specifically for q = 0 = z one obtains with γ J M |Jz |γ J M = MδM M

z |γ J M = γ J M|V

C.2

· γ J M|V J |γ J M M. J (J + 1)

(C.18)

Products of Tensor Operators

Direct products of matrices are an important concept in matrix algebra (see e.g. B LUM, 2012, Appendix A). If A is N dimensional and B is n dimensional, then the direct product A ⊗ B is N n dimensional. Similarly, from tensor operators Rk1 and Uk2 of rank k1 and k2 , respectively, one may construct (2k1 + 1)(2k2 + 1) components of new tensor operators. These can be expressed in irreducible form again as direct product Rk2 ⊗ Uk2 by tensors TK of rank K with |k1 − k2 | ≤ K ≤ |k1 + k2 |: k1 q1 U k2 q2 k1 q1 k2 q2 |KQ R TKQ = [Rk2 ⊗ Uk2 ]KQ = q1 q2

=

k1 q U k2 Q−q k1 qk2 Q − q|KQ . R

(C.19)

q

Here k1 k2 q1 q2 |KQ is a C LEBSCH -G ORDAN coefficient as introduced by (B.31) in the context of angular momentum coupling. Specifically, with (B.36) and (B.48) the scalar product (rank K = 0) of two tensor operators of the same rank k becomes simply kq U k−q . Rk · Uk = (−1)q R (C.20)

C.2


579

This is the generalization of the standard scalar product of two vectors R and S (see e.g. (B.8)). The result is a single scalar quantity.

C.2.1

Products of Spherical Harmonics

Application of√ the relations given above to the renormalized spherical harmonics (B.29), Ckq = 4π/(2k + 1)Ykq , leads to a number of useful expansion formulae. (1) (1) With (C.20) the scalar product of the two tensor operators C = C (θ1 , ϕ1 ) and (2) (2) C = C (θ2 , ϕ2 ) at different angles becomes: (1)

(2)

C · C =

(−1)m C m (θ1 , ϕ1 )C −m (θ2 , ϕ2 ).

(C.21)

m=−

If the coordinate system is such that θ2 ϕ2 defines the z-axis, we have C −m (θ2 , ϕ2 ) = δm0 , and the sum reduces to C 0 (γ 0) = P (cos γ ), with γ being the angle between the two direction in space characterized by (θ1 , ϕ1 ) and (θ2 , ϕ2 ). Rewriting ∗ (−1)m C −m = C m

we obtain the addition theorem for the renormalized spherical harmonics: P (cos γ ) =

∗ C m (θ1 , ϕ1 )C m (θ2 , ϕ2 )

(C.22)

m=−

=

C mp (θ1 , ϕ1 )C mp (θ2 , ϕ2 ).

(C.23)

m=0,p=±1

The second line is easily verified with the definition of real tensor operators (D.5) applied to the (renormalized) real spherical harmonics C mp (θ, ϕ) (D.7). One also derives from (C.19) a useful expansion formula for the product of two spherical harmonics at the same angle

kq m m C m (θ, ϕ)C m (θ, ϕ) = k0 00 Ckq (θ, ϕ),

(C.24)

m m

where the normalization constant k0| 00 has been derived from the definition of spherical harmonics at θ = ϕ = 0. Using the orthogonality relations (B.38) for the C LEBSCH -G ORDAN coefficients one also obtains the inversion of this relation: C ∗ m (θ, ϕ)C m (θ, ϕ) = Ckq (θ, ϕ) kq m m k0 00 . (C.25) kq

580

C Matrix Elements

C.2.2

Matrix Elements of the Spherical Harmonics

The often used matrix elements of the spherical harmonics in an | m basis, normalized as √ (2 + 1)(2 + 1) m Ckq | m = C ∗ m (θ, ϕ)Ckq (θ, ϕ)C m (θ, ϕ)dΩ, 4π (C.26) may be evaluated by replacing the latter two functions with the expansion (C.25): √ (2 + 1)(2 + 1) m Ckq | m = 4π

× C ∗ m (θ, ϕ)Ck q (θ, ϕ)dΩ k q k qm k 0k 00

=

k q

2 + 1 δk δq m k q k qm k 0k 00 . 2 + 1 kq

Thus, we finally obtain:

2 + 1 k qm m k 00 0 2 + 1 m Ckq | m = (−1)m 2 + 1 (2 + 1) × δm m+q δ k

k k × 0 0 0 −m q m

√ k 5 −m 5Ck . 2 + 1 m Ckq | m = (−1) −m q m

m Ckq | m =

(C.27) (C.28)

(C.29)

The last equation is simply the W IGNER -E CKART theorem for the matrix element. Comparing the latter two equalities gives the reduced matrix element of Ck : 5 5 Ck = (−1) (2 + 1) 0

k 0

. 0

(C.30)

We recall that this expression is zero if + k + is odd. We also note that √ 5 5 − (2 + 1) 5 5 Ck . Ck = (−1) √ (2 + 1) For k = 0 this implies

5 5C0 = δ ,

(C.31)

(C.32)

C.2


and more generally for = one obtains with (B.50) (−1)k/2 k!( + k/2)! (2 + 1)(2 − k)! . Ck = (2 + k + 1)! (k/2)!2 ( − k/2)!

581

(C.33)

Note that this expression only holds for even k. The reduced matrix element vanishes for odd k. For the special case C2 one obtains: √ ( + 1) . (C.34) C2 = − √ (2 − 1)(2 + 3) Note the minus sign! Of special importance in respect of the selection rules for electric dipole transitions is according to (4.79) the case k = 1. Because of (B.34) and (B.49) parity conservation holds: = ± 1

(C.35)

(see also Appendix D). With (B.50) the only nonvanishing matrix elements are ⎧ +1 ⎨ for = + 1 5 2 +3 5C1 = (C.36) ⎩− (2 −1) for = − 1, and because of (B.43)–(B.2.3) m C1q | m = (−1)q −m|C1q −m = −m C1−q | −m = (−1)q m|C1−q m .

(C.37)

Hence, in dipole transitions the selection rule for the projection quantum number becomes m = q. The matrix elements vanish unless m = m ± 1 or m = m. A detailed evaluation of (C.29), using (B.52)–(B.55) leads to: ( − m + 1)( + m + 1) (C.38) ( + 1)mC10 | m = (2 + 1)(2 + 3) ( + m + 1)( + m + 2) ( + 1)(m + 1)C11 | m = (C.39) 2(2 + 1)(2 + 3) ( − m + 1)( − m + 2) ( + 1)(m − 1)C1−1 | m = (C.40) 2(2 + 1)(2 + 3) ( − m)( + m) ( − 1)mC10 | m = (C.41) (2 − 1)(2 + 1) ( − 1 − m)( − m) ( − 1)(m + 1)C11 | m = − (C.42) 2(2 − 1)(2 + 1)

582

C Matrix Elements

( − 1 + m)( + m) . ( − 1)(m − 1)C1−1 | m = − 2(2 − 1)(2 + 1)

(C.43)

For the particularly simple case of s ↔ p transitions one finds that all allowed transitions have the same amplitudes: 10|C10 |00 = 00|C10 |10 = 11|C11 |00 = 1 −1|C1−1 |00 = 00|C11 |1 −1 = 00|C1−1 |11 =

C.3

(C.44) 1/3.

Reduction of Matrix Elements

Often matrix elements of tensor operators TKQ have to be evaluated in a scheme of coupled angular momenta, such as |(LS)J M . Typically, these operators may be k1 q1 (1) which acts only composed according to (C.19) from two tensor operators, R on the first component of the eigenstate, and Uk2 q2 (2) which acts only on the second component. With the W IGNER -E CKART theorem (C.9), the matrix element of TKQ between state |γ LSJ M and |γ L SJ M is: L S J M TKQ |LSJ M

√ J K 5 J J −K+M L S J 5TK LSJ . 2J + 1 (C.45) = (−1) −M M Q For evaluating the reduced matrix elements of product tensors angular momentum algebra provides a useful toolbox. In simple cases, with some algebra and exploiting 6j symbols (B.61), the reduced matrix elements in the coupled scheme |(LS)J M may be recast into the uncoupled scheme |LML |SMS . We communicate here only (1) some results. If Rk acts only on the first angular momentum of the coupled scheme (2) and Uk ≡ 1, we have: 5 (1) L S J 5Rk LSJ = δS S (−1)k+L +S+J (2J + 1) 2L + 1 # " L L k 5 L 5Rk L . × J J S

(C.46)

(1) Conversely, if Rk ≡ 1 and U acts only onto the second angular momentum in the coupled scheme, we obtain: 5 (2) L S J 5Uk LSJ = δL L (−1)k+L+S+J (2J + 1) 2S + 1 # " S S k 5 S 5Uk S . (C.47) × J J L

C.3


583

Note: due to the well defined phases of the C LEBSCH -G ORDAN coefficients the sequence of the angular momenta is not arbitrary. For the scalar product (rank K = 0) of two tensor operators of rank k the reduction becomes particularly clear. We first notice that according to (C.7) 5 j1 j2 J M (Rk · Sk )00 |j1 j2 J M = δM M δJ J j1 j2 J 5Rk · Sk j1 j2 J , (C.48) where we have used the explicit expression (B.48) for the 3j symbol. Explicitly (C.48) may be evaluated by inserting on the left side into the scalar product Rk · Sk the quantum mechanical unit operator. Then each part may be reduced according to (C.46). By applying the orthogonality relations according to R ACAH (1942) (if one has only to average over the angular momentum part) one finds for the reduced matrix element of the scalar product: 5 j1 j2 J 5Rk · Sk j1 j2 J = 2j1 + 1) 2j2 + 1 (C.49) " # 5 k 5 j j j1 5Rk j1 · j2 5Sk j2 . × (−1)J +j1 +j2 1 1 j2 j 2 J A somewhat different situation is encountered when a tensor TK is the product of two tensor operators Rk1 and Sk2 which are constructed from the same coordinates, momenta etc., and thus act on the same system. From simple angular momentum recoupling, E DMONDS (1996) obtains in this case (Eq. (7.1.1)): √ 5 j 5(Rk1 Sk2 )K j = 2K + 1(−1)K+j +j " k1 k2 K # 5 j 5Rk j¯ j¯Sk j . × j j j¯

(C.50)

j¯

C.3.1

Matrix Elements of the Spherical Harmonics in LS Coupling

Most often required√ are the matrix elements of the spherical harmonics Yk (or of the renormalized Ck = 4π/(2k + 1)Yk ). We have described these in Appendix C.2.2 already in a basis of pure n states. For LS coupled orbital angular momenta with orbital quantum numbers and the reduced matrix elements Ck are given by (C.28).3 In the coupled scheme with a total electron spin S the reduced matrix element (C.46) is: 3 The

following expressions are also valid in multi-electron systems as long as there is only one active electron (while further electrons have zero orbital angular momentum as it is the case for alkali atoms, or for He below the first ionization limit). Here and is identical to the total orbital angular momenta L and L, respectively, while the total spin S ≥ 1/2 of the system can in principle be composed from several particles.

584

C Matrix Elements

5 S J 5Ck SJ = δS S (2J + 1) 2 + 1 (2 + 1) " # k × (−1)−S− − −J J J S 0

(C.51) k 0

. 0

In this case, inverting the reduced matrix element leads to √ 5 5 J −J (2J + 1) 5 SJ 5Ck SJ . SJ Ck SJ = (−1) √ (2J + 1)

(C.52)

This expression is only non-zero if + + k is even. Thus, (C.45) reads now S J M Ckq | SJ M (C.53) = δS S 2J + 1 (2J + 1) 2 + 1 (2 + 1)

" #

k k J k J × (−1)M −S . −M M q J J S 0 0 0

Special Case: Spin 1/2 For the case S = s = 1/2 (e.g. for an effective one particle system with spin 1/2 – i.e. electron in an atom or a single active proton in a nucleus), using (B.70) reduces (C.51) to: 5 5

5 5 j k j 1 5 5 1 j −1/2+k j 5Ck 5 j = (−1) (2j + 1) × . (C.54) 1/2 −1/2 0 2 2 This expression is even independent of and (of course + + k must be even) and one obtains for (C.53): 1 1 j m Ckq j m = (−1)j −m +j −1/2+k 2j + 1 (2j + 1) 2 2

k j k j j j × . (C.55) m −m q 1/2 −1/2 0 For the evaluation of line strengths in E1 transitions, we report here some values for the reduced matrix element of C1 in the case of a one electron systems: j

5 5 5 5 1 j 5C1 5 j 5 2 : 2 1 5

j 1/2 √ 1/2 − 1/3 √ 3/2 − 1/3 5/2 0

3/2 √ 1/3 √ − 1/15 √ − 2/5

5/2 0 √ 2/5 √ − 1/35

(C.56)

C.3


C.3.2

585

Scalar Products of Angular Momentum Operators

Scalar products of coupled angular momenta are most conveniently evaluated by using binomial formulas. We exemplify this for the product of angular momentum L and spin S which is the crucial operator in spin-orbit interaction. As discussed in 2 Chap. 6, the operator L · S commutes with the square J and the projection Jz of the total angular momentum J = L+ S. In the scheme of coupled angular momenta, i.e. when the system is described by eigenstates |LSJ MJ , the good quantum numbers are L, S, J and MJ . From the first binomial formula one obtains: LSJ MJ L · S|LSJ MJ 2 1 2 2 LSJ MJ L − S |LSJ MJ J − 2 2 J (J + 1) − L(L + 1) − S(S + 1) δJ J δMJ MJ . = 2 =

(C.57)

However, one often needs to express the matrix elements in the uncoupled basis of eigenstates |LML SMS , e.g. in the case of strong magnetic fields as described in Sect. 8.1.3. Thus, for uncoupled L and S one returns to the definition of the scalar product according to (B.8) L · S|LML SMS LML SMS = − LML SMS (C.58) S− |LML SMS L+ − LML SMS S+ |LML SMS + LML SMS Sz |LML SMS , L− Lz and uses the matrix elements (B.14)–(B.17) for the angular momenta. Since the q act only onto the LML component of the state vector and Sq only onto operators L the SMS component one sees that the matrix elements are only nonvanishing for MJ = ML + MS = ML + MS = MJ .

(C.59)

Hence, MJ = ML + MS is a good quantum number, albeit J is not. Three types of nonzero matrix elements exist in the uncoupled system: L · S|LML SMS L(ML + 1)S(MS − 1) = L(ML + 1)S(MS − 1) L+ S− |LML SMS (C.60) = 2 L(L + 1) − ML (ML + 1) S(S + 1) − MS (MS − 1) /4 L · S|LML SMS L(ML − 1)S(MS + 1) − = − LS ML − 1 MS + 1|L S+ |LML SMS = 2 L(L + 1) − ML (ML − 1) S(S + 1) − MS (MS + 1) /4

(C.61)

586

C Matrix Elements

Table C.1 Matrix elements of L · S/2 for a 2 P state in the uncoupled basis |LML SMS , with υ = ML + 2MS . |1 12

|0 12

|1 − 12

|−1 12

|0 − 12

|−1 − 12

υ

2

1

0

0

−1

−2

MJ

ML MS

3/2

1/2

1/2

−1/2

−1/2

−3/2

1 12 |

1/2

0

0

0

0

0 12 |

0

0

0

0

0

1 − 12 | −1 12 | 0 − 12 | −1 − 12 |

0

√ 1/ 2

√ 1/ 2

0

−1/2

0

0

0

0

0

0

0

−1/2 √ 1/ 2

√ 1/ 2

0

0

0

0

0

0

0

0

0

1/2

0

LML SMS | L · S|LML SMS z = − LML SMS |L Sz |LML SMS = 2 ML MS .

(C.62)

Table C.1 provides as an example all LS matrix elements for a 2 P state (L = 1, S = 1/2).

C.3.3

Components of Angular Momenta

As a further example we compute the z-component of an angular momentum which acts only on one part of the coupled system. Typical applications are the components Sz or Lz in a spin-orbit coupled scheme |(SL)J MJ , or Iz and Jz in the hyperfine coupled scheme |(I J )F MF . We use here the latter quantum numbers, which for other application just have to be replaced appropriately. We first derive the reduced matrix element of J according to (C.46): 5 I J F 5 J I J F (C.63) # " J J 1 5 J 5J J = (−1)1+I +J +F δI I (2F + 1) 2J + 1 F F I # " J J 1 , = (−1)1+I +J +F δI I δJ J (2F + 1)J (J + 1) 2J + 1 F F I where in the last step (C.11) was used. The nonvanishing matrix elements of the components of J may now be written with the W IGNER -E CKART theorem (C.7), e.g. (C.64) J I F MF Jz |J I F MF

√ 5 F 1 F J I F 5 = (−1)F −1+MF δMF MF 2F + 1 J J I F . −MF MF 0 Inserting (C.63) yields:

C.4

Electromagnetically Induced Transitions

587

J I F MF Jz |J I F MF

= (−1)2F +MF +I +J δMF MF F × 2F + 1 (2F + 1)(2J + 1)(J + 1)J −MF

F MF

1 0

"

J F

(C.65) # J 1 . F I

Thus, these matrix elements are diagonal in MF , I and J . We may finally insert the explicit expressions of the 3j symbols from (B.55) and (B.54). We have to distinguish two cases: (a) F = F J I F MF |Jz |J I F MF √ " J 3F +I +J MF (2F + 1)(2J + 1)(J + 1)J = (−1) √ F F (F + 1) =

F (F + 1) + J (J + 1) − I (I + 1) (−1)2I +2J +1 MF 2 F (F + 1)

J F

1 I

#

(C.66)

except for F = 0 where J I F MF |Jz |J I F MF / = 0. (b) F = F + 1 J I (F + 1)MF |Jz |J I F MF " # J J 1 = (−1)3F +I +J +1 F F +1 I 2(F + MF + 1)(F − MF + 1)(2J + 1)(J + 1)J × (2F + 2)

(−1)4F +2I +2J +2 MF 2 = 1− (C.67) 2 F +1 (J + I − F )(F + J + I + 2)((F + 1)2 + (J − I )2 ) × . (2F + 1)(2F + 3) In the last step we also have inserted the explicit expression for the 6j symbols (B.72) to obtain these useful, compact relations.

C.4


Probably the most important application for the tools described here are transition matrix elements for transitions induced by electromagnetic radiation. We shall discuss these for electric dipole (E1) and quadrupole (E2), as well as for magnetic dipole (M1) transitions.

588

C.4.1

C Matrix Elements


Let us begin with emission or absorption of a photon in an E1 transition. As discussed in detail in Chap. 4, the probability of such a transition between state |a = |γ J M and state |b = |γ J M is proportional to the absolute square of matrix elements of the type 1 2 2 2 γ J M r · e|γ J M = γ r|γ J M C1q |J M · e∗q . (C.68) q=−1

Since C1 may be seen as a tensor of rank 1, this expression contains in principle the (2 × 1 + 1)(2 × 1 + 1) components of the product tensor. One may express these in irreducible form by tensors of rank 0 (basically the total intensity), rank 1 (so called orientation) and rank 2 (so called alignment). We shall come back to this in Appendix C, Vol. 2. For low Z atoms, LS coupling is characteristic according to the scheme |γ LSJ M and |γ L S J M . In this case the selection rules are determined by the matrix elements S L J M |C1q |SLJ M . Equation (C.53) factors this matrix element into its important components. In an electromagnetically induced transition, the total electron spin does not change, i.e. S = 0. For E1 transitions we have k = 1 and with q = 0, ±1 one reads in (C.53) from the first 3j symbol the selection rules for J and M: M = q and δ(J 1J ) = 1. Thus M = M ± 1 or M = M and J = J ± 1 or J = J – however, 0 0 is strictly forbidden. Finally, the last 3j symbol recovers the parity selection rule L = L ± 1.

C.4.2

Electric Quadrupole Transitions

According to (5.56) the crucial matrix element for E2 transitions is 1 a|T0 T|b (E2) ∝ b|Q2q+ |a = √ b|r 2 |a (−1)q b|C2q |a + b|C2−q |a 2 (C.69) and

1 ∝ b|Q2q− |a = √ b|r 2 |a (−1)q b|C2q |a − b|C2−q |a , 2i

where the definitions (D.5) have been inserted. Since k = 2, we note that q = 0, ±1, or ±2. In LS coupling we use again (C.53) for evaluation. Again S = 0, and considering the triangular rule δ(J J 2) = 1, we read from the 3j symbols that the total angular momentum J may now change by 0, by ±1 or ±2 units of , now with the restriction 0 0 and 1/2 1/2. With M = q follows in this case, that transitions are possible with M = M, as well as with M ± 1 and even with M ±2. This leads to correspondingly structured quadrupole absorption and emission patterns. In addition, for quadrupole transitions, parity conservation now implies L = L ± 2.

C.4


C.4.3

589

Magnetic Dipole Transitions

Selection rules for M1 transitions are derived from the matrix element (5.53) a|eE0 D|b (M1) ∝ b| L + 2 S|a B = b| J + S|a B ,

(C.70)

where the index B indicates that the coordinate frame for orbital angular momentum L and spin S is chosen here with its z-axis parallel to the magnetic RF field inducing the transition. This particular polarization dependence can be expressed again by the W IGNER -E CKART theorem (C.9). Since we discuss a dipole transition with k = 1 we obtain: √ (C.71) b| L + 2 S|a B = γ γ (−1)J −1+M 2J + 1

5 J 1 5 J SJ . J 5 × LJ + 2 J 5 −M M q With γ |γ we describe the overlap integral of radial part of the wave function between initial and final state. The 3j symbol determines the selection rule for the magnetic quantum numbers M, with q referring in this case to the direction of the magnetic field vector (rather than the electric field in the E1 case). In spectroscopic experiments, typically, radio frequency (RF) or microwave radiation (MW) is used in external, static magnetic field B st , defining the z-axis. If the radiation field vector B points into this z-direction (i.e. if the polarization vector e and the wave vector k lie in the xy plane), then q = 0 and transitions with M = M occur. To induce M = ±1 transition (q = ±1), the B vector of the inducing field has to be perpendicular to static field B st . The two reduced matrix elements in (C.71) determine the selection rules for the quantum numbers J , L and S. They have to be evaluated (for not too high static magnetic fields) according to system. With (C.11) √(C.46) and (C.47) in the coupled√ we have L LL = δL L L(L + 1) and S SS = δS S S(S + 1). Thus we obtain 5 LLSJ = δS S δL L (−1)k+L +S +J (2J + 1) 2L + 1 L S J 5 # " L L 1 (C.72) L(L + 1) and × J J S 5 SLSJ = δS S δL L (−1)k+L+S+J (2J + 1) 2S + 1 L S J 5 # " S S 1 S(S + 1). (C.73) × J J L Hence, the total spin and the total orbital angular momentum do not change during a M1 transition in LS coupling! In contrast, transitions with J = J ± 1 or J = J are possible, provided 0 0. We recall that the radial wave functions for different principal quantum numbers and otherwise identical configuration are orthogonal.

590

C Matrix Elements

Hence, only within one electronic state γ |γ = 0, so that M1 transitions do not occur between different electronic states. Of course, this strictly holds only for pure LS coupling. In the case of configuration mixing or jj coupling one has examine the situation in detail. We finally note, that all above considerations may be applied, mutatis mutandis, to a hyperfine coupled scheme as well – typically prior to evaluating the LS coupling.

C.5

Radial Matrix Elements

To obtain quantitative values for transition probabilities it is necessary to also compute the radial parts of the matrix elements, as indicated by γ . . . |Tkq |γ . . . above. In general, the radial part may completely be pulled out of the matrix element, and can be computed independently. For single electron wave functions the radial matrix elements are ∞ r 3 drRn (r)Rn (r). (C.74) γ r|γ = 0

They are needed when evaluating probabilities for E1 transitions as described in Chap. 4 but also for a quantitative description of the S TARK effect discussed in Chap. 8. The radial eigenfunctions have to be known and these depend on – even though the orbital angular momentum L has dropped out of the angular part of the matrix element (C.54). In closed form, these matrix elements may only be derived for the hydrogen atom and H like ions. Using the known radial functions (2.122) one finds 4 (n − − 1)! (n − − 1)! (C.75) n r|n = 2 2 n n [(n + )!]3 [(n + )!]3 ∞ +1 2 +1 −ρ/2 −ρ /2 2 × L2 e ρ ρ r dr, n + ρ Ln+ (ρ)re

0

with ρ = 2Zr/(na0 ) and ρ = 2Zr/(n a0 ) and the associated L AGUERRE polynomials 2 +1 (ρ) = Ln+

n− −1

(−1)k+1

k=0

ρk [(n + )!]2 . (n − − 1 − k)!(2 + 1 + k)! k!

For not too large values of n and one may evaluate (C.75) in closed form, using standard analytical programs like Maple, Mupad4 or Mathematica on a PC. We assemble in Table C.2 some characteristic results for n |r|n in (atomic, Z scaled) units a0 /Z. 4 Available

as part of the Symbolic Math Toolbox add-on for MATLAB.

C.5

Radial Matrix Elements

Table C.2 Radial transition matrix elements for the hydrogen atom an H-like atoms

591 n1

1

n2

2

1

0

2

1

1

0

3

1

1

0

4

1

1

0

5

1

2

0

3

1

2

0

4

1

2

0

5

1

2

1

3

0

2

1

3

2

n |r|n a0 /Z √ 128 6/243 1.290 √ 27 6/128 0.516 √ 6144 15/78 125 0.305 √ 250 30/6561 0.209 √ 13 824 12/15 625 3.065 √ 512 30/2187 1.282 √ 576 000 60/5764 801 0.774 √ 3456 18/15 625 0.938 √ 27 648 180/78 125 4.748

If very high values for n and are required (RYDBERG atoms), there are suitable alternatives. Typically, it is sufficient to compute n , − 1|r|n, for n = n. Based on earlier work a closed form has been given e.g. by T OWLE et al. (1996) (again in units of a0 /Z):

(−)n − (n + − 1)!(n + )! 1/2 n , − 1r|n, = 4Z (n − )!(n − − 1)! (4n n) +1 (4n n)ν (n − n )N −ν−1 N ! × (n + n)n +n (2 − 1 + ν)!(N − ν)!

(C.76)

× 2 F1 (−ν, −2 + 1; N − ν + 1; x)

−

(−4n n)ν (n − n )N −ν −1 N ! (n + n )2 (2 − 1 + ν )!(N − ν )!

× 2 F1 −ν , −2 + 1 − ν ; N − ν + 1; x . Here 2 F1 (α, β; γ ; x) is the hypergeometric series 2 F1 (α, β; γ ; x) =

(α)t (β)t t

(γ )t t!

xt

(C.77)

with (α)t = α(α + 1) · · · (α + t − 1) 2 x = − n − n /4n n ν = min nr , nr N = max nr , nr , ν = min nr , nr + 2 N = max nr , nr + 2 , and nr = n − and nr = n − − 1 are radial quantum numbers. The hypergeometric series always terminates since α and β are negative numbers. Extended tables on line strengths S(n sj − n sj ) or intensities (2j + 1)A(jj ) (with the E INSTEIN

592

C Matrix Elements

coefficient A(jj ) corresponding to (4.114)) are found in T OWLE et al. (1996) for H like RYDBERG atoms. Since for sufficiently high n and all atoms become hydrogen like, these results are of general relevance.

Acronyms and Terminology 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. 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)’. M1: ‘Magnetic dipole’, transitions induced by the interaction of a magnetic dipole with the magnetic field component of electromagnetic radiation. 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. 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.

References B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics, vol. 64. Berlin, Heidelberg: Springer Verlag, 3rd edn., 343 pages. B RINK, D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3rd edn., 182 pages. C ONDON, E. U. and G. S HORTLEY: 1951. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 441 pages. E DMONDS, A. R.: 1996. Angular Momentum in Quantum Mechanics. Princeton, NJ, USA: Princeton University Press, 154 pages. ISO 21348: 2007. ‘Space environment (natural and artificial) – Process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland. R ACAH, G.: 1942. ‘Theory of complex spectra II’. Phys. Rev., 62, 438–462. T OWLE, J. P., P. A. F ELDMAN and J. K. G. WATSON: 1996. ‘A catalog of recombination lines from 100 GHz to 10 microns’. Astrophys. J. Suppl. Ser., 107, 747–760.

D

Parity and Reflection Symmetry

The concept of parity conservation plays a key role in modern physics and is used throughout this book. We present here a short introduction to parity and reflection symmetry from an atomic and molecular physics perspective.

D.1

Parity

refers to spatial symmetry of states and is defined by: The parity operator P Pψ(r) = ψ(−r).

(D.1)

We have described the central force problem in polar coordinates. With r → −r they change as (r, θ, ϕ) → (r, π − θ, ϕ + π). As long as the interaction is spatially isotropic (e.g. the C OULOMB potential) inversion has no influence on the Hamilto and H commute: nian, hence P H ] = 0. [P, Thus, the eigenfunctions of the Hamiltonian are also eigenfunctions of the parity operator. We distinguish states with positive and negative parity (alternatively also has eigenvalues Pat = ±1: called odd and even), i.e. P Pψ(r) = ψ(−r) = ±ψ(r). The product of two functions with unequal parity has negative parity, while the product of two functions with the same parity has positive parity. We note that the integral over all space disappears for a function F (odd) (r) with odd parity: 2π π F (odd) (r)dΩ ≡ 0. (D.2) 0

0

For atomic orbitals we write explicitly n (r)Y m (θ, ϕ) = Rn (r)Y m (π − θ, ϕ + π). n m (r) = PR Pψ © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7

593

594

D Parity and Reflection Symmetry

With the definition of the spherical harmonics (B.28) one finds Y m (π − θ, ϕ + π) = (−1) Y m (θ, ϕ), so that the parity of the states ψn m (r) is Pat = (−1) , hence, spherical harmonics which differ by = ±1 have opposite parity. Specifically, the relation = −r Pr

(D.3)

holds, and we may say that the dipole operator D = −er has odd parity. As trivial as that statement may seem, it has significant consequences in the context of selection rules for optical transitions. As discussed in detail in Chap. 4, for an optical 2π π transition to occur, the integral 0 0 Y b mb (θ, ϕ)rY a ma (θ, ϕ)dΩ must not disappear. Thus, with (D.2) and (D.3) the product Y b mb (θ, ϕ)Y a ma (θ, ϕ) must have odd parity and transitions are thus only allowed if the parity of initial | a ma and final state | b mb is different. A dipole transition thus connects only states with opposite parity. The selection rule for single electron transitions is = ±1. Somewhat more general – expressing the particle point of view – the photon is said to have odd parity, Pph = −1. The selection rule (4.86) for dipole transitions may thus be inferred directly and elegantly from a general parity conservation rule for the whole system – including atom and photon. Since the photon with its negative parity disappears during absorption and appears in emission, absorption or emission of a photon must be accompanied by a change of parity in the atomic system, hence the selection rule = ±1.

D.2

Multi-electron Systems

The above discussion tacitly assumed a single electron, characterized by | m , ignoring its spin. The parity concept may, however, be extended naturally to multielectron systems, assuming that the spatial part of the total wave function |LM can be written as a product of single electron functions (or as a linear combination of products such as a Slater determinant). The total parity of an N electron system is then the product of the parities of all its electron orbits. Thus, a configuration {n1 1 n2 2 . . . nk k . . . nN N } has even or odd parity depending on whether N

(−1)

1

i

= (−1)λ = ±1

is positive or negative, respectively (or alternatively whether λ = 0 or 1). In principle one has to sum over all electrons of an atom; however, in practice it usually suffices to sum just over all electrons in open shells.

D.3

Reflection Symmetry of Orbitals – Real and Complex Basis States

D.3

595


At this point some general remarks about the choice of basis states or wave functions and their symmetry properties are in order. For simplicity we first discuss again a single electron and ignore its spin. To study different atomic problems some choices of basis states may be more suitable than others. The one most commonly used in atomic physics for the angular part of the wave functions (states) is the representation by (complex) spherical harmonics. There are, however, a variety of cases where another symmetries may be more adaptable to a given problem, such as the interaction with an external electric field (see Sect. 8.2). In that case the eigenstates are linear combinations of states with ±M and |M| remains a good quantum number, while the sign of M is indefinite. Similarly, in optical transitions the polarization of the exciting light may lead to particular linear superpositions of different basis states as discussed in Chap. 4. All these linear combinations are per definition also pure states, and in principle, one can find an infinite number of such states which, properly orthonormalized, could also serve as a basis set. In standard quantum mechanical textbooks the | m (or |LM ) states are somewhat overemphasized and one is led to believe that these are the only suitable basis states for atomic systems. That is only true if a system is 2 z . characterized by the magnitude of angular momentum L and its z-component L In addition, the measurement to be described must select states which diagonalize both operators simultaneously, so that 2 L |LM = L(L + 1)2 |LM

z |LM = M|LM . L

(D.4)

And indeed, in the presence of a magnetic field in z-direction a typical atomic Hamiltonian remains diagonal in this basis as discussed in Sects. 2.7 and 8.1. Such properties of the complex angular momentum basis, together with the fact that the corresponding matrices for coupling and rotating angular momenta are extensively tabulated, makes them very attractive. The price to be paid is that linear combinations have to be used already for rather simple applications, such as atoms in external electric fields, dipole excitation by linearly polarized light, as well as for most of molecular physics and chemistry. Thus, one may construct real tensor operators Tkqp (observables, spherical harmonics etc.) as linear combinations from the complex ones ∗ with Tk−q = (−1)q Tkq for 0 < q ≤ k and p = ±1:

i(p−1)/2 i(p−1)/2 ∗ (−1)q Tkq + p Tk−q = √ (−1)q Tkq + p Tkq (D.5) Tkqp = √ 2 2 while for q = 0: Tk0+ = Tk0 and Tk0− = 0.

596


Table D.1 Renormalized real spherical harmonics Ckq for k = 1 to 3 k

p = +1

0

C00+ = 1

C00− = 0

1

C10+ = cos θ

C10− = 0

1

C11+ = sin θ cos ϕ

C11− = sin θ sin ϕ

2

C20+ = (3 cos2 θ − 1)/2 √ C21+ = 3 sin θ cos θ cos ϕ √ C22+ = ( 3/2) sin2 θ cos 2ϕ

C21− =

C30+ = (5 cos3 θ − 3 cos θ)/2 √ C31+ = ( 6/4) sin θ(5 cos2 θ − 1) cos ϕ √ C32+ = ( 15/2) cos θ sin2 θ cos 2ϕ √ C33+ = ( 10/4) sin3 θ cos 3ϕ

C30− = 0 √ C31− = ( 6/4)(5 cos2 θ − 1) sin ϕ √ C32− = ( 15/2) cos θ sin2 θ sin 2ϕ √ C33− = ( 10/4) sin3 θ sin 3ϕ

2 2 3 3 3 3

p = −1

C20− = 0 √ 3 sin θ cos θ sin ϕ √ C22− = ( 3/2) sin2 θ sin 2ϕ

In complete analogy the real angular momentum states are defined by |J Mp =

i(p−1)/2 (−1)q |J M + p|J − M √ 2

|J 0+ = |J 0 and |J 0− = 0

for 0 < M ≤ J

(D.6)

for M = 0.

Note that the relations between real and complex unit polarization vectors (4.7) are also constructed according to that scheme. And by inserting (B.20) into (D.5) we obtain the real spherical harmonics 2k + 1 Ykqp (θ, ϕ) = Ckqp (θ, ϕ) where 4π 2(k − q)! q Ckq+ (θ, ϕ) = P (cos θ ) cos(qϕ) for 0 < q ≤ k (D.7) (k + q)! k 2(k − q)! q P (cos θ ) sin(qϕ) for 0 < q ≤ k Ckq− (θ, ϕ) = (k + q)! k Ck0+ (θ, ϕ) = Pk0 (cos θ ) and Ck0− (θ, ϕ) ≡ 0 for q = 0, q

with the associated L EGENDRE polynomials Pk which can be derived from (B.22). Some explicit expressions for the renormalized real spherical harmonics Ckqp (θ, ϕ) are given in Table D.1. Their orthogonality relations are

2π

dϕ 0

0

π

∗ Ckqp (θ, ϕ)Ck q p (θ, ϕ) sin θ dθ =

4π δkk δqq δpp . 2k + 1

(D.8)

Rotation of the real tensor operators, real states and real spherical harmonics is achieved by (E.25) using the real rotation matrices (E.26)–(E.28).

D.3


597

2 2z , While the real spherical harmonics are eigenfunctions of L and L 2 L Ykqp = k(k + 1)2 Ykqp

2z Ykqp = q 2 2 Ykqp L

with q = 0, 1, . . . , k,

(D.9)

z nor of L x or L y . However, the real spherical they are neither eigenfunctions of L harmonics and the corresponding real states (electron orbitals) are characterized by a well defined reflection symmetry – not only in respect of the xy plane (which is also a plane of reflection symmetry for the complex basis) but also in respect of the xz and yz plane. Reflection of a wave function at the xz plane e.g. is equivalent to replacing ϕ by −ϕ. In case of the complex basis this leads to Ykq (θ, −ϕ) = ∗ (θ, ϕ) = (−1)q Y Ykq k−q (θ, ϕ). However, by applying this relation to the definition of the real spherical harmonics (D.5) one finds for a reflection at the xz plane σv (xz)Ykqp = pYkqp

with p = ±1,

(D.10)

where we have introduced the reflection operator σv (xz). Often one wants to express (D.7) in Cartesian coordinates. We apply a very useful relation communicated by C AOLA (1978). In our notation the so called solid harmonics Ykq (x, y, z) are (renormalized): Ckq (x, y, z) 4π Ykq (x, y, z) = r k Ckq (θ, ϕ) = 2k + 1 = (−1)q (k − q)!(k + q)!

(k−q)/2 j =max(−q,0)

(D.11) (−1)j (x + iy)j +q (x − iy)j zk−q−2j . 22j +q (j + q)!j !(k − q − 2j )!

Inserting this into (D.5) leads to explicit expressions for the renormalized real solid harmonics Ckqp . Table D.2 summarizes the most important cases. Figure D.1 illustrates graphically the angular dependence of the three real basis orbitals for k = = 1: |px , |py , and |pz (short: px , py and pz orbital, respectively) and compares them to the usual complex basis orbitals |1 ±1 and |10 . Obviously, the real spherical harmonics are represented by dumbbell like states parallel to the x-, y-, and z-axes of the atom, respectively. In contrast the complex basis consists of two doughnut like orbitals (|1 ±1 with opposite sense of orientation) while the |10 orbital is also dumbbell like and identical to |pz . One may think of preparing these basis states by optical dipole excitation. However, since electrons have also a spin, and m are typically not good quantum numbers. Thus, a complete optical preparation of the orbitals shown in Fig. D.1 is only possible in very special cases. If e.g. one can make use of a

598 Table D.2 Renormalized real solid harmonics Ckq for k = 1 to 3

D Parity and Reflection Symmetry k

p = +1

p = −1

0

C00+ = 1

C00− = 0

1

C10+ = z

C10− = 0

1

C11+ = x

C11− = y

− r 2 )/2

C20− = 0

2

C20+

2

C21+ =

C21− =

2

√ 3zx √ 2 C22+ = 3(x − y 2 )/2

3

C30+ = z(5z2 − 3r 2 )/2

C30− = 0

3

C31+ =

C31− =

3 3

Fig. D.1 Angular part of atomic p orbitals (spherical harmonics) in the complex (top row) and real (bottom row) basis. Plotted are the squared moduli |Y1q |2 and |Y1q± |2 , respectively. The colour shading indicates regions of different signs of the functions. The ⊕ and $ signs indicate positive and negative reflection symmetry in respect of the xy plane, while reflection symmetry in respect of the xz plane according to (D.10) is characterized by red p = +1 and p = −1

= (3z2

√ 6x(5z2 − r 2 )/4 √ C32+ = 15z(x 2 − y 2 )/2 √ C33+ = 10x(x 2 −3y 2 )/4

⊕z y

⊕z

⊕z

z |0〉

x

x

√ 6y(5z2 − r 2 )/4 √ C32− = 15zxy √ C33− = 10y(3x 2 −y 2 )/4

p =+1

|+1〉

| px 〉

√ 3zy √ C22− = 3xy

|-1〉

x

y

p =+1

x

y

⊕z p = – 1

p =+1 | pz 〉

y

x

z

| py 〉

y

x

y

→ 1 P1 transition, it is indeed possible to prepare all five different basis states individually by optical dipole excitation. In such a case, for example in a two electron system like He, the electron spin is compensated by another electron in the system, but only one electron is excited:

1S

0

(a) The complex basis set requires circularly polarized light for its preparation: left and right hand circularly polarized, propagating along the z-axis to excite the |1 +1 (equivalent to Y1+1 ) and the |1 −1 (equivalent to Y1−1 ) states, respectively. Linearly polarized light propagating perpendicularly to the z-axis and with its E vector aligned parallel to the z-axis excites the |10 state (equivalent to Y10 ). (b) The real basis set requires linearly polarized light for its preparation: the |pz state (identical to |10 ) is excited as just described, while the preparation of |px

D.4

Reflection Symmetry in the General Case

Fig. D.2 Schematic illustration of a reflection σv (xz) in the xz plane being constructed from an inversion at the origin (P ) and a rotation through π about the y-axis

599

z

z

z

^

y

P

y

x

x

y x

^

σv (xz)

and |py requires linearly polarized light with its E vector parallel to x and y, respectively. Generally speaking, the real basis functions are the basis of choice when the sign of the angular momentum is irrelevant, i.e. whenever there is no specific preferred orientation. Thus, they are relevant if external electric fields are present. They play a particularly important role in molecular physics and quantum chemistry as discussed in detail in Chaps. 3–4 of Vol. 2, but also in collision problems. There, the electrostatic field between different, neighbouring atoms defines a suitable z-axis and is typically very strong so that spin quantum numbers can be treated as a mere afterthought. Thus, the orbital angular momentum quantum numbers and the absolute value of their projection onto this axis become good quantum numbers. Real atomic orbitals (AO’s), as depicted in the bottom row of Fig. D.1, and linear combinations of them – attached to different atoms – are the basis for a quantitative treatment of electronic wave functions in molecules. Reflection symmetry is also an important property in the context of collision problems. As we shall discuss in Chaps. 6–8, Vol. 2, reflection symmetry in respect of the scattering plane is conserved during a binary collision for the combined wave function of the interacting particles. Thus, either reflection symmetry of both particles remains constant during the collision, or both particles change their reflection symmetry in respect of the collision plane.

D.4


Since reflection symmetry is such an important property of atomic and molecular states, we now generalize the concept. So far, the discussion was restricted to the orbital part of the state vectors, i.e. to the integer values of the total angular momentum J . In these cases reflection symmetry is an intuitively obvious property of the wave functions. However, it is possible to extend this concept to half integer angular momenta as well. We start by noting that a reflection σv in respect of a given plane may be viewed as an inversion at the origin with subsequent rotation through an angle π around the axis perpendicular to this plane. This is illustrated in Fig. D.2. Inversion at the multiplies the state vectors |J M with origin implies, that the parity operator P (−1)λ , where λ is an odd or even integer depending on whether the parity of the whole system is odd or even (typically determined by the orbital angular momentum of the wave function). The rotation through π is achieved with the help of rotation

600


matrices described in Appendix E. Thus, the state |J M after reflection in the xy plane becomes σv (xy)|J M = (−1)λ

J M

DJM M (π, 0, 0)J M = e−iMπ (−1)λ |J M

= (−1)λ+M |J M .

(D.12)

We note here in passing that with this relation the |+1 and |−1 p states in Fig. D.1 have both positive reflection symmetry in respect of the xy plane, while the |0 state has negative reflection symmetry – as intuitively obvious. Reflection at the xz plane leads to σv (xz)|J M = (−1)λ DJM M (0, π, 0)J M M J = (−1)λ d−MM (π)|J −M = (−1)λ+M+J |J −M .

(D.13)

Here we have used the symmetry relation (E.9) for the reduced rotational matrix. From (D.12) and (D.13) we see that the standard complex |J M representation are eigenstates of the xy plane reflection operator but not of the operator for reflection in the xz plane. In analogy to (D.5) one may, however, construct also for the general case (J integer or half integer) states of specified reflection symmetry in respect of the xz plane: |J Mp = v(p, M) (−1)M |J M + (−1)λ+J p|J −M with (D.14) ⎧ for M = 0 ⎨ 1/2 √ 2 for 0 < M ≤ J and p = 1 or +i v(p, M) = 1/ √ (D.15) ⎩ −i/ 2 for 0 < M ≤ J and p = −1 or −i. σv (xz) in respect of Here p = ±(−1)J is the eigenvalue of the reflection operator the xz plane. Applying (D.13) to (D.14) one easily verifies that |J Mp are orthonormalized eigenstates of σv (xz): σv (xz)|J Mp = p|J Mp .

(D.16)

For integer values of J = k, M = q and λ = J the previous relations (D.5) are recovered, and p = ±1. For half integer values of J the eigenvalues of the σv (xz) reflection are obviously p = ±i. The same eigenvalues are found for reflection in the xy plane. However, half integer states with ±M belong to different reflection symmetry, while for integer values of J (spherical harmonics) reflection symmetry depends on |M|. In general, the |J Mp states are not eigenstates of any of the components of the angular momentum operator. The matrix elements of Jx , Jy , and Jz may be derived from the definition (D.14) with (B.4) and (B.14)–(B.17). In general one has to distinguish different cases such as J, M = 0, 1/2, and 1 as well as p = ±1 or ±i. Thus, we note here only a few special aspects.

D.4


601

1. The states defined by (D.14) are eigenstates of Jz2 . One easily derives " +1 for p = +1 or +i Jz |J Mp = iM J M(−p) × −1 for p = −1 or −i so that Jz2 |J Mp = M 2 2 |J Mp .

(D.17)

2. Jz changes the reflection symmetry from +(−1)J to −(−1)J and vice versa, and so does Jx : (D.18) Jx |J Mp = a(M)J (M + 1)(−p) + a(−M)J (M − 1)(−p) as one verifies for J ≥ 1 and M > 0 easily with some algebra. In contrast Jy does not change reflection symmetry: (D.19) Jy |J Mp = a(M)J (M + 1)p − a(−M)J (M − 1)p where a(M) is typically1 given by a(M) = √ J (J + 1) − M(M + 1). 2 3. As specific examples, we have already discussed the p orbitals |px = |11+ , |py = |11− and |pz = |10+ depicted in the lower part of Fig. D.1. Without proof we note here in passing the relations: Jy |px = −i|pz Jy |pz = +1|px

(D.20)

Jy |py = 0. 4. The simplest, and somewhat exceptional case with half integer J is a pure spin state with λ = 0, J = 1/2, describing e.g. an electron in a 1 S1/2 state. The basis states in the |J M representation are |1/2 1/2 and |1/2 −1/2 , with spin pointing in the +z- and −z-direction, respectively. Reflection on the xy plane gives according to (D.12) 1 1 1 1 1 1 1 1 σv (xy) = i and σv (xy) − = −i − . (D.21) 22 22 2 2 2 2 We see that the up and down spin states have opposite reflection symmetry in respect of the xy plane. The corresponding states with defined xz reflection symmetry are obtained from (D.14) (λ = even) with p = ±i: 1 These formulas hold for all J ≥ 1 and M ≥ 1. The other cases are easily derived individually from the definition (D.14) with (B.4) and (B.14)–(B.17).

602


1 1 1 1 1 − − |+i = √ i 2 2 2 22

1 1 1 11 |−i = √ i + − . 2 2 i 2 22

(D.22) (D.23)

These states are identical to (E.17) and (E.18), respectively, obtained by coordinate rotation for a spin state pointing into ±y direction. Reflection at the xz plane gives with (D.13): σv (xz)|−i = −i|−i . σv (xz)|+i = i|+i and

(D.24)

In this particular case (and only in this one), these xy reflection symmetric states are also eigenvectors of Jy . With (B.4) and (B.14)–(B.17) one easily verifies that Jy |±i = ± |±i . 2

(D.25)

We finally apply the concept of reflection symmetry to optical transitions treated in detail in Chap. 4. In Sect. 4.1.2 we have introduced linear (ex , ey ) and circular (e+1 , e−1 ) polarization vectors. We may write |ex , |ey and |e+1 , |e−1 as the real and complex representation of the corresponding photon states, respectively. Then the definitions (4.4)–(4.7) are in complete agreement with (D.14). Since for the photon J = λ = 1, M = ±1, we have p = +1 for |ex and p = −1 for |ey , i.e. |ex and |ey have positive and negative reflection symmetry in respect of the xz plane. Since the (fully quantized) Hamiltonian for the system photon + atom is invariant under reflection at any plane through the centre of the atom, the reflection symmetry of the state vectors describing the combined system must conserve reflection symmetry. Depending on whether we use light which is polarized in the x- or y-direction, the atom changes or conserves its reflection symmetry in an optical transition as indicated by the following scheme: |J Mp; ex ←→ |J Mp; . . . |J Mp; ey ←→ J M(−p); . . . . For example, the prototype 1 S0 ↔ 1 P1 transition induced by light linearly polarized in the x-direction induces no change in the reflection symmetry: The 1 S0 state has + reflection symmetry in respect of the xz plane as well as the 1 P1 dumbbell pointing into x-direction (see Fig. D.1). The y-polarized light excites the dumbbell pointing into the y-direction, having negative reflection symmetry. A somewhat less trivial example is depicted in Fig. D.3. In a 2 S1/2 → 2 P1/2 transition the initial state |Ja = |+i corresponds to (D.22), with its spin pointing into the +y-direction it has p = +i reflection symmetry in respect of the xz plane. Since after the excitation to a P state, the parity of the atom changes (λ = 0 → 1), the final state |Jb with angular momentum pointing into +y-direction has now p = −i symmetry. When exciting with ey light, the combined system |Ja |ey prior

Acronyms and Terminology Fig. D.3 Illustration of reflection symmetry changing and non-changing processes in a 2 S1/2 ↔ 2 P1/2 transition induced by light linearly polarized in x- and y-directions, respectively

603 z

z ⎢Ja

ey

x k

2S

z ⎢Ja ⎢–Jb e y

⎢Jb y

y

1/2

z

x → 2P1/2

x

y

x

x k 2S

1/2

→ 2P1/2

to the transition has +i × (−1) = −i reflection symmetry, thus the final state must also have −i symmetry, which corresponds to |Jb pointing into +y-direction. In contrast, for excitation with ex polarized light, the combined initial system |Ja |ex has +i × (+1) = i symmetry, which must be conserved and corresponds to reversing the orientation of the spin into |−Jb . Circularly polarized photons |e+1 or |e−1 propagating into z-direction have both positive reflection symmetry in respect of the xy plane (in analogy to the |+1 and |−1 p states in Fig. D.1) and cannot change this reflection symmetry of the atomic states. Formally one sees this from (D.12), since in a dipole transition both λ and M change by ±1 in respect of the “photon frame” where z k. Thus, (−1)λ+M remains constant during the excitation process and reflection symmetry in respect of the xy plane does not change.

Acronyms and Terminology 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)’.

References C AOLA, M. J.: 1978. ‘Solid harmonics and their addition theorems’. J. Phys. A, Math. Gen., 11, L23–L25.

E

Coordinate Rotation

We compile here the most important formulas for rotation of coordinate systems and their consequences on angular momentum states and tensor operators – without going into details of the derivation.

E.1

EULER Angles

One specifies a coordinate rotation by three E ULER angles (α, β, γ ). As shown in Fig. E.1 the rotation is performed in three steps: (xyz) → x y z → x y z → x # y # z# . First, one rotates the system around its original z-axis (z = z) through an angle α. The following rotation through an angle β around the new y axis (y = y ) brings z into the direction z and x into direction x . The final rotation through an angle γ is around the thus tilted z = z# axis (x → x # and y → y # ).

Fig. E.1 Definition of the Euler angles (α, β, γ ) for rotation of a coordinate system xyz

z, z' z'',

z#

β

y#

γ

y' y''

α y

α β

x x'

γ

x#

x''


605

606

E

E.2

Coordinate Rotation

Rotation Matrices

The operator achieving this rotation is derived from the angular momentum operator J: D(αβγ ) = exp(−iγ Jz ) exp(−iβ Jy ) exp(−iα Jz ). The corresponding rotation matrix elements, more precisely the matrix elements J of the rotation group, may be derived with of the irreducible representation D some angular momentum algebra (following the notation of B RINK and S ATCHLER, 1994),1 and are written as J DJMN (αβγ ) = J M|D(αβγ )|J N = exp(−iαM)dMN (β) exp(−iγ N ),

(E.1)

with the reduced rotation matrix: J dMN (β) =

[(J + M)!(J − M)!(J + N )!(J − N )!]1/2 (−1)t (J + M − t)!(J − N − t)t (t + N − M)! t

(E.2)

× (cos β/2)2J +M−N −2t (sin β/2)2t+N −M . Special cases:

4π Y ∗ (β, α) 2k + 1 J M

(E.3)

(J − M)! M P (cos β) if M > 0 (J + M)! J

(E.4)

DJM0 (αβγ ) = CJ∗ M (β, α) = J dM0 (β) = (−1)M

J and d00 (β) = PJ (cos β).

(E.5)

We note some important symmetry properties DJMN (αβγ )∗ = (−1)M−N DJ−M−N (αβγ ) = DJN M (−γ − β − α), J J J dMN (β) = (−1)M−N dNJ M (β) = d−N −M (β) = dN M (−β)

J J = (−1)J −M dM−N (π − β) = (−1)J +M dM−N (π + β). J dMN (π) = (−1)J +M δ−MN

and

J J dMN (0) = δMN = (−1)2J dMN (2π).

(E.6) (E.7) (E.8) (E.9) (E.10)

1 Here the coordinate system is rotated. An alternative convention, also found in the literature, is to rotate the states.

E.2

Rotation Matrices

607

Explicitly we communicate formulas for the frequently used cases J = 1/2, J = 1, and J = 2 according to B RINK and S ATCHLER (1994): 1 1 1 1 β β d 2 1 1 = −d 12 1 = sin (E.11) d 12 1 = d 2 1 1 = cos − − − − 2 2 2 2 2 2 2 2 2 2 β 2 β 1 1 1 1 2 d11 = d−1 −1 = cos d1−1 = d−11 = sin 2 2 (E.12) sin β 1 1 1 1 1 d00 = cos β. d01 = d−10 = −d0−1 = −d10 = √ 2 2 2 d22 = d−2−2 = cos4 (β/2)

(E.13)

2 = d2 = d2 2 d20 02 −20 = d0−2 √ = ( 3/8) sin2 β

2 = −d 2 2 2 d21 −1−2 0 = −d−2−1 = d12

= − sin β(1 + cos β)/2 2 2 0 = −d 2 2 d2−1 = d1−2 −21 = −d−12

2 2 d2−2 = d−22 = sin4 (β/2) 2 d11

=

2 d−1−1

2 d1−1

= (2 cos β − 1)(1 + cos β)/2

= sin β(1 + cos β)/2 2 = d−11

= (2 cos β + 1)(1 − cos β)/2 2 = d2 2 2 d10 0−1 = −d01 = −d−10 √ = ( 3/2) sin β cos β.

2 = (3 cos2 β − 1)/2 d00

An angular momentum state |J N # in the new system may be expressed by eigenstates |J M in the original system as: J |J N = |J N # = D

|J M J M|D(αβγ )|J N =

M

= exp(−iγ N )

|J M DJMN (αβγ )

M J |J M exp(−iαM)dMN (β).

(E.14)

M

The components Tkq (θ # , ϕ # ) of tensor operators of rank k (e.g. Ykq or a vector with k = 1) can be expressed by tensor components in the old system Tkq θ # , ϕ # = Tkq (θ, ϕ)Dkqq (αβγ ) q

k = exp −iγ q Tkq (θ, ϕ) exp(−iαq)dqq (β)

(E.15)

q

with θ , ϕ and θ # , ϕ # being the angles in the old and new system, respectively. We illustrate the procedure by a particularly simple and important example, a spin s = 1/2 state |1/2 1/2 # pointing into the ϕ = α, θ = β direction in the orig1/2 inal coordinate frame xyz. Inserting for this case dMN (β) from (E.11) into (E.14) we obtain:

608

E

Coordinate Rotation

(a)

(b)

Fig. E.2 Rotation to describe a spin s pointing into (a) +y-direction, (b) +x-direction

#

1 iα β 1 iα β 1 iγ − exp − cos + exp sin . = exp − 2 2 2 2 2 2 2 2 (E.16) (For simplicity of writing we have dropped here J = 1/2.) The third rotation angle γ around the final z# axis is somewhat arbitrary and just contributes a free phase factor.2 Figure E.2(a) indicates the rotation angles for one specific case where the spin s points into the +y-direction of the (xyz) coordinate system. Thus, the obligatory rotation angles are α = ϕ = π/2 and β = θ = π/2. We choose γ = −3π/2, make use of (E.11)–(E.13), and thus obtain from (E.16) for this particular spin state

1 #(y) 1 1 1 + . = √ i+ − − 2 2 2 2

(E.17)

If the spin s points into the −y-direction, we make exactly the same rotation and find from (E.14)

1 #(y) 1 1 1 − − + + (E.18) i = √ 2 2 . i 2 2 Somewhat more compact, we designate the spin states parallel and antiparallel to the y-axis with by |α (y) and |β (y) , respectively, and those which refer to the z-axis with |α and |β . Equations (E.17) and (E.18) may then be written: 1 |α (y) = √ i|α − |β 2 2 Note, however, that for spin

1 and |β (y) = √ i|α + |β . i 2

(E.19)

1/2 this rotation angle is defined modulo 4π – rather than modulo 2π (one more turn around the z axis changes the sign). For example, the angle γ = −3π/2 in Fig. E.2 has not the same effect as γ = π/2.

E.3

Entangled States

609

For completeness we also describe a state where the spin s points into the ±xdirection. As illustrated in Fig. E.2(b) the E ULER angles in this case are (0, π/2, π) and we obtain 1 #(x) ∓1 ± (E.20) = √ |α ± |β . 2 2 All these states are of course eigenstates of the squared spin angular momentum S 2 , with spin quantum number s = 1/2. The eigenvalue of S 2 are 2 s(s + 1) = 2 3 /4. And specifically |α and |β are also eigenstates of Sz , the spin projection onto the z-axis, with eigenvalues /2 and −/2, respectively. What about the states |±1/2 (x,y) ? Here is a little exercise for the active reader, which can be solved with a few lines of algebra (for complex quantities): Using (B.4), (B.16) and (B.17), show that the spin states |α (y) and |β (y) are indeed eigenstates of the spin component in y direction with eigenvalues ±/2: Sy |α (y) = |α (y) 2

and Sy |β (y) = − |β (y) . 2

(E.21)

And the same holds of course for the corresponding spin states pointing in ±xdirection.

E.3

Entangled States

For many-particle systems we have to construct spin functions which account for the fact that identical particles are indistinguishable. We have discussed this in the main text on several occasions, see e.g. Table 6.1 in the context of fine structure and Sect. 7.3 where specifically the He atom is treated. Let us take the latter example, i.e. the two electrons in the He ground state (1s)2 1 S0 , or any singlet state of two electrons in the continuum after ionization (with a symmetric spatial wave function). The two spins of the electrons are anti-parallel and since we cannot distinguish which of these two fermions has which spin we have to construct an antisymmetric description of their joint state, according to (7.36) 0 χ = √1 α(1)β(2) − β(1)α(2) , 0 2

(E.22)

where the numbers in brackets refer to electron (1) and electron (2), respectively. Obviously, this state of the two electrons is inseparable, i.e. it cannot be written as product of two wave functions of the individual electrons. One calls such states entangled. Equation (E.22) refers to a particular coordinate frame and |α and |β reflect the spin components with respect to its z-axis. The total (spin) angular momentum

610

E

Coordinate Rotation

operator is S = S1 + S 2 , and the singlet state (E.22) is constructed such that the 2 respective eigenvalues of S and Sz are zero:3 2 S1z + S χ00 = 0. S2z )χ00 = 0 and Sz χ00 = (

(E.23)

One may now be curious what happens if we define this entangled, antisymmetric spin state with respect to another axis, say the y-axis. This may be done in complete analogy to (E.22) from spin states with angular momentum parallel and antiparallel to the y-axis: 0 χ = √1 α(1)β(2) (y) − β(1)α(2) (y) . y 2

(E.24)

In full analogy to (E.23) we have now Sy |χy0 = 0, i.e. the projection of the total spin onto the y-axis is also zero. The relation (E.19) between the |α (y) and |β (y) states and the |α and |β states parallel to the z-axis may now be inserted into (E.24): with just a few lines of algebra (in which many terms cancel) one recovers again (E.22), i.e. the singlet state with respect to the z-axis. Thus, the definition of this entangled state does not depend on the coordinate axis and its angular projection is zero in all coordinate systems. So much about coordinate rotation for entangled states. Entanglement is a very active field of current research with far reaching consequences, not only for fundamental conceptual and philosophical reasons (dating back to the famous E INSTEIN , ROSEN , P ODOLSKY 1935 paper, quoted by now over 5000 times), but also in view of a number of potential future applications such as quantum cryptography or quantum computation. We cannot enter here in a detailed discussion of this wide field but refer the reader to some relatively recent reviews and references quoted there (T ICHY et al., 2011; H ORODECKI et al., 2009; R AIMOND et al., 2001), see also the respective chapters in B LUM (2012).

E.4

Real Rotation Matrices

For reference we also communicate the rotation procedure (FANO, 1960) for the real harmonics, real states and real tensor operators as defined in Appendix D. In this case, the transformation is achieved by # # Tkq = Tkqp (θ, ϕ)Dkqp,q p (αβγ ) p θ , ϕ

(E.25)

qp

first relation is obvious, since each of the operators S1z and S2z acts only onto one of the states which have the eigenvalues ±/2, i.e. they compensate each other. The latter relation may easily be verified as an exercise with the help of (B.8) and (B.17).

3 The

References

611

with p, p = ±1. The real rotation matrices are obtained by applying the complex rotation (E.15) onto the defining equations (D.5) and finally back-transformation to the real basis. One obtains:4 For k ≥ q, q > 0:

(E.26) Dkqp,q p (αβγ ) ⎧ k k ⎪ (−1)q dqq (β) cos(αq + γ q ) + p dq−q (β) cos(αq − γ q ) ⎪ ⎪ ⎨ if p = p = (−1)−q k (β) sin(αq + γ q ) + d k ⎪ (−1)q p dqq ⎪ q−q (β) sin(αq − γ q ) ⎪ ⎩ if p = p

For q or q = 0: Dkqp,q p (αβγ ) = (−1)−q+q

(E.27)

"

√ k cos(qα + q γ ) if p = p 2dqq (β) sin(qα − q γ ) if p = p

specifically Dkqp,0+ (αβγ ) = Ckqp (β, α) k For q = q = 0: Dk0+,0+ (αβγ ) = d00 (β).

(see Eq. (D.7)) (E.28)

References B LUM, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics, vol. 64. Berlin, Heidelberg: Springer Verlag, 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 INSTEIN, A., B. P ODOLSKY and N. ROSEN: 1935. ‘Can quantum-mechanical description of physical reality be considered complete?’ Phys. Rev., 47, 777–780. FANO, U.: 1960. ‘Real representations of coordinate rotations’. J. Math. Phys., 1, 417–423. 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. H ORODECKI, R., P. H ORODECKI, M. H ORODECKI and K. H ORODECKI: 2009. ‘Quantum entanglement’. Rev. Mod. Phys., 81, 865–942. R AIMOND, J. M., M. B RUNE and S. H AROCHE: 2001. ‘Colloquium: Manipulating quantum entanglement with atoms and photons in a cavity’. Rev. Mod. Phys., 73, 565–582. 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.

4 See also

H ERTEL and S TOLL (1978), Eq. (15) (some small typos there have been corrected here).

Multipole Expansions and Multipole Moments

One often needs suitable approximations for potentials which at large distances are nearly symmetric around a centre, but not completely. For scalar as well as for vector potentials, multipole expansions may be derived. We treat here the important case of an electrostatic potential arising from a charge distribution localized in a small region around the origin – as e.g. encountered when describing the interaction of atomic nuclei with the electron charge cloud of an atom (see Fig. 9.12 in Chap. 9). In the final subsection we shall also introduce a more general definition of multipole tensor operators and multipole moments.1

F.1

Laplace Expansion

We first note that the inverse distance between two points r and r in space may be expanded for r r into a power series of x = r /r: −1/2 1 1 = 1 − 2x cos θ + x 2 (F.1) |r − r | r

3 5 1 1 3 cos2 θ − + x3 cos3 θ − cos θ · · · = 1 + x(cos θ ) + x 2 r 2 2 2 2 where θ is the angle between r and r . With (B.24) we identify the angle dependent terms in brackets as L EGENDRE polynomials. With this result in mind, we communicate the L APLACE expansion which holds for r < r: ∞

r k 1 = Pk (cos θ ) |r − r| r k+1 k=0

1 The term “multipole moments” is used in the literature for a variety of slightly different definitions. As far as necessary for distinction, we shall add some specification, indicated in (brackets).


613

F

614

F

=

∞ ∞ k r k ∗ r k (r) (r ) θ = C (θ, ϕ)C , ϕ C · Ck kq kq r k+1 r k+1 k q=−k

k=0

=


∞ r k r k+1 k=0

k

(F.2)

k=0

Ckqp (θ, ϕ)Ckqp θ , ϕ .

q=0,p=±1

In the second line of (F.2) we have made use of the addition theorem (C.22) for the renormalized spherical harmonics, with θ , ϕ and θ , ϕ being the angles of the position vectors r and r , respectively. In the following equality we have made use (r) (r ) of (C.21) and the scalar product notation (C.20) for the two tensors Ck and Ck . The third line of (F.2) introduces the real (renormalized) spherical harmonics as defined in Appendix D.3.

F.2

Electrostatic Potential

With these relations we write the electrostatic interaction energy dV12 (r, r ) of a charge dq, located at a position r , with a point charge −e located at r dV12 r, r =

∞ 1 −e (r) k 1 −edq (r ) = Ck · r dq Ck . k+1 4πε0 |r − r| 4πε0 r

(F.3)

k=0

For an extended electric charge with density ρ(r ), localized around the origin, the charge element is dq = ρ(r )d3 r , and the total interaction energy with the point charge −e is obtained from −e ρ(r ) 3 d r. V (r) = dV12 r, r = (F.4) 4πε0 |r − r| With (F.3) we may write this as a multipole expansion V (r) =

∞ k −e 1 Qkq Ckq (θ, ϕ) 4πε0 r k+1 k=0

in real form

=

∞ −e 1 4πε0 r k+1 k=0

or

(F.5)

q=−k

k

Qkqp Ckqp (θ, ϕ),

(F.6)

q=0,p=±1

with (charge) multipole moments Qkq = Qkqp =

Ckq θ , ϕ r k ρ r d3 r , Ckqp θ , ϕ r k ρ r d3 r =

or

(F.7) Ckqp r ρ r d3 r ,

(F.8)

F.2

Electrostatic Potential

615

where we have used in the second row for abbreviation the renormalized real solid harmonics Ckqp as introduced in Appendix D.3. These multipole expansions, derived from (F.3), are most appropriate if r r. Thus, it is suitable e.g. for describing the field of an atomic nucleus as seen by an electron far away from the nucleus (i.e. for ≥ 1), or for the far field of an atom in a well defined polarization state (with ≥ 1). The total charge is given by q = Q00 =

ρ r d3 r .

(F.9)

For an electronic charge cloud of an atom with N electrons it is q = −N e, and for an atomic nucleus with atomic number Z (protons in the nucleus) we have q = Ze. If the charge distribution can be described by an electronic or nuclear wave function as ρ(r ) = q|ψn m (r )|2 = q|Rn (r )C m (θ , ϕ )|2 the angular part of the matrix elements in (F.7) may be evaluated as described in Appendix C.2.2 – if necessary applying the recoupling rules from Appendix C.3. In this case, we find with (C.30) that only even multipole moments are nonvanishing: k = 0 (monopole), k = 2 (quadrupole), k = 4 (octupole), etc. More generally, for an arbitrary charge distributions, convenient expressions for the real multipole moments (F.8) are obtained with the renormalized real solid harmonics (D.11). The components of the dipole moment are: Dz = Q10+ = Dx = Q11+ =

z ρ r d3 r x ρ r d3 r

Q10− = 0 Dy = Q11− =

y ρ r d3 r .

(F.10)

If only positive or only negative charges are involved, the dipole moments are made to disappear by suitable choice of the origin, thus defining a charge centre. They also disappear if both charges have the same centre (characteristic for atoms, unless they are placed in an electric field, see Sect. 8.2.8). In contrast, if positive and negative charges with different centres are to be described (typical for polyatomic and heteronuclear diatomic molecules) the dipole moment is finite. With (F.8) and Table D.2 the real quadrupole moments are: 1 2 3z − r 2 ρ r d3 r 2 √ 3 Q21+ = 3 z x ρ r d r √ 3 2 Q22+ = x − y 2 ρ r d3 r 2 Q20+ =

Q20− = 0 √ 3 3 zyρ r d r √ 3 Q22− = x y ρ r d3 r . 2 Q21− =

(F.11)

616

F


Note that a variety of slightly different terminologies is used in the literature.2 In nuclear physics one defines (assuming symmetry around the nuclear z-axis) the nuclear quadruple moment Q=

3z2 − r 2 ρ r d3 r = 2Q20+ .

(F.12)

Explicitly, in Cartesian coordinates the potential expansion (F.6) reads V (r) =

C2qp (x, y, z) −e q D · r 1 + 3 + Q2qp + · · · . 4πε0 r 2 qp r r5

(F.13)

The components (F.10) of the electric dipole moment D are zero for charges centred at the origin, while the quadrupole moments Q2qp according to (F.11) are finite for an anisotropic charge distribution. The renormalized real solid harmonics C2qp (x, y, z) are found in Table D.2.

F.3

Multipole Tensor Operators

Closely related to the (charge) multipole moments just discussed, but without any reference to the charge of the particles, one defines irreducible spatial multipole tensor operators of rank k. They are constructed from the position vector and have the components Qkq (r) = r k Ckq (θ, ϕ) = Ckq (x, y, z) for −k ≤ q ≤ k.

(F.14) (F.15)

using again the (renormalized) solid harmonics (D.11). They may also be written as real quantities according to (D.5): Qkqp (r) = r k Ckqp (x, y, z) = Ckqp (x, y, z) for 0 ≤ q ≤ k and p = ±1.

(F.16) (F.17)

For explicit expressions we refer to Table D.2. General rules for tensor operators have been laid out in Appendix C. 2 E.g.

following JACKSON (1999), in classical electrodynamics one usually expands in terms of the standards spherical harmonics, multiplying the terms in the sum (F.7) by 4π/(2 + 1), so that the (complex) multipole moments are given as q m =

∗ Y m (Θ, Φ)R ρ(R)d3 R.

F.3


617

The expectation values Qkq , i.e. the matrix elements γ J M|Q2q |γ J M averaged over the population probabilities w(M) of |γ J M states, are called (spatial) multipole moments: Qkq = r 2 w(M) J M|Ckq (θ )|J M . (F.18) M

They characterize the particle distribution (e.g. for the protons in an atomic nucleus, for the electrons in an atom, or for electrons and nuclei in a molecule). For a given angular momentum J of a quantum state the above definitions (F.14) and (F.16) allow to construct multipole tensor operators and multipole moments of rank 0 ≤ k ≤ 2J . They all have a well defined parity kq (r) = (−1)k Qkq (r) PQ

(F.19)

but also a reflection symmetry in respect of the xy plane (see also Appendix D.4): σv (xy)Qkq = (−1)k+q Qkq .

(F.20)

Here σv (xy) is the reflection operator in respect of the xy plane. The real tensor operators (F.16) and the corresponding multipole moments often allow a more flexible and self evident interpretation of physical situations. They are constructed such that, in addition to parity (F.19) and reflection symmetry in respect of the xy plane (F.20), they also have a well defined reflection symmetry p = ±1 in respect of the xz plane: σv (xz)Qkqp = pQkqp .

(F.21)

They are appropriate tools to express symmetries e.g. in molecules and interaction processes such as excitation by photons or collisions. For rotations of the real multipole moments in space (E.25)–(E.28) may be used.

F.3.1

The Quadrupole Tensor

Most commonly used is the quadrupole tensor Q2 with its components Q2q = C2q (r, θ, ϕ) according to Table D.2. Higher moments become increasingly complicated. If the quantum state |J M of an object is known exactly, the components of the quadrupole moment are obtained from the respective matrix elements of the quadrupole tensor Q2 . These matrix elements can be derived from the W IGNER E CKART theorem (C.9):

√ J J 2 J −M J Qk J (F.22) γ J M Q2q |γ J M = (−1) 2J + 1 M −M q with J Q2 J = J C2 J γ J |r 2 |γ J .

(F.23)

Clearly, systems with angular momenta j = 0 or 1/2 do not have a quadrupole moment, since the 3j symbols must satisfy the triangular relation δ(j j 2) = 1.

618

F


Specifically, the Q20 component of the quadrupole tensor is Q20 = C20 (θ ) =

3 cos2 θ − 1 3z2 − r 2 = r2 . 2 2

(F.24)

With (B.58) its diagonal matrix element is 3M 2 − J (J + 1) γ J M|Q20 |γ J M = √ J C2 J r 2 , (2J + 3)(J + 1)J (2J − 1)

(F.25)

and if the radial wave functions Rγj (r) are known, the radial matrix element r 2 is obtained from 2 2 r = γ J |r |γ J = r 4 |Rγj |2 dr. The extrema of (F.25) for M = J and M = 0 are: J (2J − 1) J C2 J r 2 and γ J J |Q20 |γ J J = (2J + 3)(J + 1) J (J + 1) J C2 J r 2 . γ J 0|Q20 |γ J 0 = − (2J + 3)(2J − 1)

(F.26)

(F.27)

The reduced matrix element J C2 J still depends on the coupling scheme of the states under discussion. For LS coupling detailed expressions for the renormalized spherical harmonics have been derived in Sect. C.3.1. Specifically, for a quasi-one-fermion system with spin s = 1/2 (one active atomic electron or nuclear proton) we use (C.55) to obtain the diagonal matrix elements: 1 1 j mQ20 j m 2 2 1 1 = j mC20 j m γj |r 2 |γj 2 2

j j 2 j j 2 2 2j −m−1/2 (2j + 1) r = (−1) m −m 0 1/2 −1/2 0 =−

1 3m2 − j (j + 1) 2 r . 4 j (j + 1)

(F.28)

For the last equality we have used (B.58) for the 3j symbols and the fact that j is half integer, so that (−1)2j = −1. The highest (positive) value is found for m = 0, the lowest (negative) value for m = j : 1 1 j 0Q20 j 0 = r 2 /4 (F.29) 2 2 1 1 1 2j − 1 2 jj Q20 jj = − r . (F.30) 2 2 4 j +1

F.3


619

The expectation values of the quadrupole tensor, i.e. the (spatial) quadrupole moments according to (F.18), may be related to the (charge) quadrupole moments defined in Sect. F.2 by Q2q = q Q2q = r 2 J C2 J w(M) J M20|J M . (F.31) For a prolate (cigar shaped) charge distribution, dominated by low M states, (F.31) leads to Q > 0, while an oblate (pancake like) distribution is characterized by large |M| and Q < 0 (see e.g. Fig. 9.15). In nuclear physics, the quadrupole moment of a nucleus (assuming cylindrical symmetry around the z-axis) is traditionally defined by Q = 2eZ Q2q . Note the factor 2eZ (overall charge eZ) as already introduced in (F.12): averaging γ J M|Q20 |γ J M over state populations w(M) is equivalent to integration over the charge distribution ρ(r) = eZ |ψ(r)|2 . The probability density |ψ(r)|2 refers to the total wave function for the charged particles of interest, while the population of the states w(M) depends on the specific experiment.3 For a single active spin 1/2 particle system (i.e. a proton in the nucleus) in a pure |jj state, instead of (F.30) the nuclear quadrupole moment thus reads (see also B OHR and M OTTELSON, 1998, Eq. (3-27)) 1 2j − 1 1 γj |r 2 |γj . Qsp = 2 jj Q20 jj = −eZ 2 2 2j + 2

F.3.2

General Multipole Tensor Operators

Often it is desirable to use a more general form of irreducible tensor operators in multipole expansions. Rather than constructing them from position operators one may generate such operators from angular momenta. One straight forward procedure is to “polarize” the solid harmonics as introduced by FALKOFF and U H LENBECK (1950) for applications in nuclear physics. Using the normalization of M ACEK and H ERTEL (1974) for the solid harmonics (FANO, 1960), these tensor operators are obtained from Tkq± =

3

(Ji1 Ji2 . . . Jik )∇i1 ∇i2 . . . ∇ik Ckq± (r),

(F.32)

i1 i2 ...ik =1

where ij = 1 to 3 stands for x, y and z. The angular momenta are given in a.u., i.e. in units of . Applying this operation to the explicit expression for the (renormalized) solid harmonics Table D.2 we obtain Table F.1. 3 For

a general formalism see Chap. 9 in Vol. 2.

620

F


Table F.1 Multipole tensor operators constructed from angular momenta for k = 1 to 3 (only the zero component is given for the octupole moment k = 3); for reference we also communicate the relations with the FANO and M ACEK (1973) orientation (O1− ) and alignment parameters (A1+ and A2+ ) k 0 1 1 2 2 2 3

p = +1 T00+ = 1

p = −1 T00− = 0

T10+ = Jz T11+ = Jx

T10− = 0 T11− = Jy = J (J + 1)O1− T20− = 0 √ T21− = 3(Jz Jy + Jy Jz ) √ T22− = 3(Jx Jy + Jy Jx )

2 T20+ = (3Jz2 − J ) √ √ T21+ = 3(Jz Jx + Jx Jz ) = J (J + 1) 3A1+ √ √ T22+ = 3(Jx2 − Jy2 ) = J (J + 1) 3A2+ 2 T30+ = (15Jz3 − 9 J Jz + 3Jz )

etc.

The Tk0 (= Tk0+ ) components are obviously diagonal in J and M. Explicitly, the matrix elements for k = 1 are M J T1 J J M|T10 |J M = M = √ J (J + 1)

(F.33)

and describe the orientation of the state (nonvanishing angular momentum). The latter equality follows from the W IGNER -E CKART theorem (C.9) and (B.55). The matrix elements of T20 characterize the alignment (quadrupole moment or the anisotropy) of the charge density distribution in a given state: J M|T20 |J M = 3M 2 − J (J + 1) =√

(F.34)

2[3M 2 − J (J + 1)] J T2 J . (2J + 3)(2J + 2)2J (2J − 1)

The latter follows again from the W IGNER -E CKART theorem, this time using (B.58). The reduced matrix elements for rank k = 1 and k = 2 follow directly from (F.33) and (F.33), respectively: J T1 J = J (J + 1) and (F.35) (F.36) J T2 J = (2J + 3)(J + 1)J (2J − 1). For an arbitrary rank k one finds (see e.g. M ACEK and H ERTEL, 1974, and references given there; note, however, that the factor (2J + 1)−1/2 is specific for the B RINK and S ATCHLER notation of the reduced matrix elements): (2J + k + 1)! k! J Tk J = k . (F.37) 2 (2J + 1)(2J − k)! We point out that these (general) multipole tensor operators and their expectation values, the (general) multipole moments, may be finite also for odd rank – in contrast


621

to the usual (charge or spatial) multipole moments as discussed above. This is a direct consequence of their construction by angular moments, rather than by position vectors. The W IGNER -E CKART theorem provides a one to one relation between the matrix elements of any irreducible tensor operator of rank k for a set of basis states |j m – say for the spatial multipole tensor operators (F.14) Qkq constructed from position coordinates – to any other irreducible tensor of the same rank in that basis – say the Tkq multipole tensor operator. With (C.12) this relation is simply given by J Tk J J M Qkq |J M , J M Tkq |J M = J Qk J

(F.38)

with J Qk J defined by (F.23). We emphasize that (F.38) also holds for the respective expectation values (the multipole moments). Often one is only interested in the expectation values (multipole moments) of these tensors in an orbital angular momentum basis with sharp = (see e.g. Appendix D.2 in Vol. 2). We may then use the explicit expression (C.33) for the reduced matrix elements of the renormalized spherical harmonics. These are nonvanishing only for even k and we note that their sign is always negative – in contrast to (F.37)! Thus, the multipole moments Tkq (constructed from angular momenta) have the opposite sign from Qkq (which are built from spatial coordinates)! Expectation values of these multipole tensor operators are widely used to characterize anisotropy and orientation of atomic or molecular systems – in particular if these cannot be described by pure states. Strictly speaking, they are measured in units of k but one usually drops the unit. We shall refer to them as multipole moments as long as no confusion can arise with the (spatial) multipole moments constructed from position vectors.4 More details will be discussed in the context of the density matrix in Chap. 9 and Appendix C, Vol. 2. In addition, we shall introduce there yet another kind of irreducible tensor operators to characterize anisotropic state populations, the so called statistical tensor operators. Their expectation values are the state multipoles which again can be related by simple numerical factors with any other irreducible representation of tensor operators, in analogy to (F.38).

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2.

is important to keep in mind that the sign of Tk0 is opposite to that of Qk0 . When speaking about shapes of charge distributions we have to note that oblate implies Q20 < 0 and T20 > 0, while a prolate shape is encountered if Q20 > 0 and T20 < 0.

4 It

622

F


References B OHR, A. and B. R. M OTTELSON: 1998. Nuclear Structure, vol. 1: Single-Particle Motion. Singapore: World Scientific, reprint from 1969 edn., 471 pages. B RINK, D. M. and G. R. S ATCHLER: 1994. Angular Momentum. Oxford: Oxford University Press, 3rd edn., 182 pages. FALKOFF, D. L. and G. E. U HLENBECK: 1950. ‘On the directional correlation of successive nuclear radiations’. Phys. Rev., 79, 323–333. FANO, U.: 1960. ‘Real representations of coordinate rotations’. J. Math. Phys., 1, 417–423. FANO, U. and J. H. M ACEK: 1973. ‘Impact excitation and polarization of emitted light’. Rev. Mod. Phys., 45, 553–573. JACKSON, J. D.: 1999. Classical Electrodynamics. New York: John Wiley & Sons, 3rd edn., 808 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.

G

Convolutions and Correlation Functions

In this appendix all distribution functions f (x) are normalized such that +∞ f (x)dx = 1. −∞

(G.1)

In other chapters of these textbooks a different type of normalization may be more convenient and are used whenever appropriate, e.g. normalization to the maximum max[f (x)] = 1.

G.1

Definition and Motivation

The mathematical operation called convolution quantifies the overlap between two mathematical functions f1 (x) and f2 (x) as one is reversed and shifted in respect of the other. A convolution is defined as ∞ f1 (ξ )f2 (x − ξ )dξ, (G.2) (f1 ∗ f2 )(x) := −∞

which describes an important mathematical procedure, often required and used in experimental physics and measuring techniques. The cross-correlation of two (possibly complex) functions is defined by ∞ G(x) = (f1 * f2 )(x) := f1∗ (ξ )f2 (ξ + x)dξ ≡ f1∗ (−x) ∗ f2 (x). (G.3) −∞

Thus, both procedures are identical if one of the functions is even, f (−x) = f (x), which is the case in most of the physical applications.1 If f1 ≡ f2 one calls G(x) = (f * f )(x) an autocorrelation function. alternative method to solve (G.3) is given by F (f1 * f2 ) = 2π F (f1 )F (f2 ) where F (f ) describes the F OURIER transform of f (see Appendix I).

1 An


623

624

G


S1 S2 x, ξ

Fig. G.1 Convolution scheme of two rectangular profiles as manifested by two slits S1 and S2 of unequal size. Their respective transmission profiles are shown as full black and red dashed lines. As slit S2 is shifted over slit S1 in x, ξ direction, the overlap (grey area) changes. The resulting total transmission is shown as heavy red line (convolution)

Convolution and cross-correlation are well documented in mathematical tables as well as in the web (often with illustrative animations, e.g. W EISSTEIN, 2011). Thus we keep this brief recollection focused on the applications most often encountered in AMO physics. Convolutions occur e.g. if a physical observable is analyzed and detected that may assume a range of values described by a probability distribution f1 (ξ ) as the variable ξ is changed. The measurement device too, will never be sensitive at one and only one value of ξ . Rather the signal will be detected with a varying detection probability f2 (ξ ). Hence, the detector is tuned over the region of interest by shifting its detection efficiency through x. The resulting signal is described by (G.3). Two rectangular profiles are the simplest case – e.g. manifested as transmission of light through two slits. This is shown for illustration of the principle in Fig. G.1, starting with the situation in the leftmost cartoon with both slits separated from each other where transmission is zero. The signal will increase (linearly) when S2 is shifted into +x-direction as soon as the areas of the slits begin to overlap. The maximum is reached when the overlap is complete and stays constant as long as the slits fully overlap. Further shift will reduce the signal again linearly. The overall result is a capped triangular shape of the transmission profile (heavy red line in Fig. G.1). It will be triangular if the two slits have equal width. Typical applications are found in any kind of spectroscopy on atoms, molecules, solids or in nuclear and elementary particle physics: the system absorbs, emits, reflects – photons, electrons, atoms or elementary particles – as a function of energy, frequency or wavelength ξ , characteristic for the physics studied. A suitable detector registers this signal with a specific detection probability which depends on ξ . This may e.g. be controlled by the slit position and width of a spectrograph, the transmission curve of a frequency filter or particle energy analyzer. Recording a spectrum implies changing ξ → ξ − x and thus shifting the transmission maximum of the spectrograph – knowing that on the left and right of that maximum still some signal is transmitted. The detector finally sums over all of this transmitted signal. Convolution is also at work in nature when different processes may occur simultaneously in one system and overlay each other. A standard example is the broadening of spectral lines emitted or absorbed by atoms and molecules. They may e.g. move with a velocity described by the B OLTZMANN distribution and at the same time undergo collisions. While the first effect leads to a significant, inhomogeneous spectral broadening due to the D OPPLER effect, collisions may also disturb

G.2

Correlation Functions and Degree of Coherence

625

the emission or absorption process and add to the broadening. The combination of both effects is described by a convolution of the respective line profiles. The physics involved in all these processes and examples will be discussed in some detail in Sect. 5.1.1, and Chaps. 1, 2 in Vol. 2, and in other parts of these volumes. In the present appendix we summarize a few useful mathematical expressions which are needed to describe such measurements quantitatively. At the end one aims of course at an intelligent de-convolution of the signal detected as a function of x, i.e. one tries to extract the original profile f1 (ξ ) form the convoluted signal. This is no trivial task unless the shapes of the two profiles involved are both well known. Often one is content with just measuring the convolution and gleaning the width of the profile f1 (ξ ) from known convolution formulas.

G.2

Correlation Functions and Degree of Coherence

In physics, the cross-correlation (G.3) – also called (first-order) correlation function – is often used to ‘correlate’ an observable at different points in space r and/or time t. Typical observables are the electric field, or the intensity of an electromagnetic wave. They may represent pulses (wave-packets), random sequences of pulses, stationary signals (e.g. continuous light beams) or even more or less statistical noise. Correlation functions describe the coherence of these observables in space and/or time – in more colloquial terms one might say: the correlation function measures traces of similarities over a distance in space and/or time. The propagation of electromagnetic waves is described by its wave vector k and angular frequency ω in the combination kr − ωt. Thus, with EA (t) = E(r, t) and EB (t ) = E(r , t ) representing the field at two positions in space2 and time, one may keep r constant, focus on the temporal coherence, and write the correlation function in dependence of the time delay δ = t − t G (δ) = (1)

EA∗ (t)EB (t

1 + δ) = Tav

Tav /2 −Tav /2

EA∗ (t)EB (t + δ)dt.

(G.4)

The angle brackets . . . indicate averaging, the second equality giving one possible recipe how to accomplish this averaging for the stationary case (e.g. for a CW light beam or for continuous noise). Of course, integration has to be over a sufficiently long time Tav so that all statistical short term fluctuations are smoothed out. If EA and EB are square integrable functions, instead to average one has to integrate over all times as in (G.3) and the 1/Tav factor becomes redundant. In normalized form, the correlation function is called first-order degree of coherence: g (1) (δ) =

2 The

EA∗ (t)EB (t + δ) . [ |EA (t)|2 |EB (t)|2 ]1/2

(G.5)

further treatment is independent of whether EA and EB relate actually to two different fields or represent the same field at different positions in space.

626

G


Correlation functions are e.g. used in state-of-the-art ultrafast science for the determination of shape and duration of short laser pulses as to be detailed in Chap. 1, Vol. 2. In such a measurement, the electric field amplitude EA (t) (or the intensity ∝ |EA (t)|2 ) of the pulse to be characterized is convoluted with a second light pulse of well known profile EB (t) (or intensity ∝ |EB (t)|2 ). This is achieved by delaying one pulse for a variable time δ in respect of the other. Both signals are multiplied and integrated over a long time Tav . The signal is then recorded as a function of the delay time δ and the result may finally be de-convoluted. If EA (t) and EB (t) are identical (or just displaced in time) the first-order degree of coherence (G.5) simply becomes the normalized autocorrelation function g (1) (δ) =

E ∗ (t)E(t + δ) E ∗ (t − δ)E(t) = = g (1) (−δ). |E(t)|2 |E(t)|2

(G.6)

The important symmetry with respect zero follows directly from the definition (G.4) for A = B. If the shape of a pulse is already well known, it allows one to determine directly the pulse duration (see below for special examples). Note that here the 1/Tav factor cancels out in (G.5) and (G.6): these definitions hold for CW fields as well as for a single pulses or a pulse sequence. In practice, the averaging may not be that trivial, especially if E(t) is not given in closed analytical form. For CW fields the average . . . can be performed over times at a given position in space – or alternatively over the whole ensemble in space at a given fixed time. One calls a system ergodic if the temporal average is equal to the average over a representative ensemble – which is generally true for reasonable physical systems. Usually the temporal average is more convenient to perform – while determining the ensemble average (averaging over the whole space) would be a rather difficult task. According to the ergodicity theorem,both averages are identical.

G.3

Gaussian Profile

In reality, experimental profiles are not just rectangular. To be specific, we start by discussing the Gaussian profile – characteristic for a number of physical phenomena and often also a good approximation for analyzer profiles. We write it in the form most often used in atomic and laser physics: 2 1 fG (x; w, x0 ) = exp −2(x − x0 )2 /w 2 (G.7) πw √ with a FWHM of x = 2 ln 2w = 1.177w. (G.8) We note that at x − x0 = w gives the distance at which fG has decreased to 1/e2 . One easily verifies that fG (x; w, x0 ) is normalized according to (G.1). The mean value x and the variance σ 2 = (x − x )2 are given by 2 1 +∞ −2(x−x0 )2 /w2 x = xe dx = x0 , and (G.9) π w −∞

G.4

Hyperbolic Secant

2 σ 2 = x − x =

627

2 1 πw

+∞

−∞

(x − x0 )2 e−2(x−x0 )

2 /w 2

dx =

w2 . 4

(G.10)

√ The standard deviation is thus σ = w/2 and the FWHM is 8 ln 2σ 1.2w. An important case is the B OLTZMANN distribution for velocities in a gas. The 1D distribution in respect of the component vx , say, is given by

mvx2 m exp − dvx w(vx )dvx = (G.11) 2πkB T 2kB T with the particle mass m, the absolute temperature T , and the B OLTZMANN constant kB . The average velocity is vx = 0, and the variance vx2 = kB T /m. D OPPLER broadening (Sect. 5.1.4) is a direct consequence of this Gaussian dis2 = ω2 k T /(mc2 ) with the angular tribution. We recall from (5.21) its variance ωD ba B transition frequency ωba and the speed of light c. Interestingly, the convolution (here identical to the cross-correlation) of two Gaussian profiles with 1/e2 at x = w1 and w2 , respectively, is again a Gaussian profile, with w 2 = w12 + w22 . One verifies easily (using e.g. SWP 5.5, 2005) (fG * fG )(x; w2 , x2 ) ∞ 2 1 exp −2(ξ − x1 )2 /w12 × exp −2(ξ − x2 − x)2 /w22 dξ = π w1 w2 −∞

(x − (x1 − x2 ))2 2 = , (G.12) exp −2 π(w12 + w22 ) w22 + w12 the convoluted FWHM thus being the geometric mean of both widths

x = 1.177w = 1.177 w12 + w22 = ( x1 )2 + ( x2 )2 .

(G.13)

Thus, the autocorrelation function or the first-order degree of coherence (G.6) of a Gaussian temporal profile with a FWHM t has a √ FWHM: t auto = 2 t. (G.14)

G.4

Hyperbolic Secant

The hyperbolic secant squared is another, often used line profile, in particular for laser pulse intensities. Normalized according to (G.1) it is 1 sech2 [(x − x0 )/w] = 2 2w 2w cosh [(x − x0 )/w]

2 2 1 , = 2w e(x−x0 )/w + e−(x−x0 )/w √ with a FWHM of x = −2 ln( 2 − 1) w = 1.763w.

fH (x; w, x0 ) =

(G.15)

(G.16)

628

G


The convolution is a not completely trivial integral. For the autocorrelation function one finds (properly normalized) (fH * fH )(x; w) =

1 (x/w) cosh(x/w) − sinh(x/w) w [sinh(x/w)]3

(G.17)

3 sech4 [x/(2.24445w)] sech2 [x/(1.5429w)] ∼ = 4 × 2.24445w 2 × 1.5429w all with a FWHM of x = 2.720w.

(G.18)

The first equality is exact, the sech4 gives an excellent approximation, and the sech2 is still a good approximation for small |x| ≤ 1.7; in the far wings, however, it is somewhat too high.

G.5

LORENTZ Profile

Another profile often encountered is the L ORENTZ profile, describing e.g. the natural line profile or collision broadening (see Sects. 5.1.1 and 5.1.3, respectively). We write the L ORENTZ profile fL (x; γ , x0 ) =

γ 2 /4 2 γ π (x − x0 )2 + γ 2 /4

with a FWHM x = γ ,

(G.19) (G.20)

again properly normalized according to (G.1). The L ORENTZ profile is a particularly wide profile (see Fig. 5.5) and its variance x 2 obviously diverges since fL (x; γ , x0 ) → 1/x 2 for large x, while its mean value is x = x0 . The convolution of one L ORENTZ profile with another leads again to a L ORENTZ profile (for a proof see e.g. Appendix I.4.5): (fL * fL )(x; γ2 , x2 ) =

γ 2 /4 2 γ π (x − (x1 − x2 ))2 + γ 2 /4

(G.21)

with the FWHM of the convolution x = γ = γ1 + γ2 .

G.6

VOIGT Profile

Convolution of a L ORENTZ with a G AUSS profile is required e.g. for optical line shapes of atoms or molecules in the gas phase if D OPPLER broadening and collision broadening are on the same order of magnitude (or at low temperatures D OPPLER and natural linewidth). This so called VOIGT profile +∞ 2 2 2 1 γ e−2ξ /w dξ, (G.22) fV (x; σ, γ ) = (fG * fL )(x; γ , 0) = π w 2π −∞ (ξ − x)2 + γ 2 /4


629

is an integral which cannot be evaluated by standard integration formulas. With some advanced functions this may, however, be achieved. Extended literature exists on the subject. We essentially follow NIST-DLMF (2013) and W IKIPEDIA CON TRIBUTORS (2014). Properly normalized according to (G.1) and with the definitions for w and γ used above, the VOIGT function can be expressed as 2 Re(exp(−z2 ) erfc(−iz)) fV (x; w, γ ) = (G.23) π w √ 2 (x + iγ /2) (G.24) with z = w where exp(−z2 ) erfc(−iz) is the complex error function (also FADEEVA function). The complementary error function erfc(x) = 1 − erf(x) is usually implemented in modern symbolic mathematics programmes (we use SWP 5.5, 2005). Even an approximate linewidth (FWHM) for the VOIGT profile is found in the literature as (G.25)

ωV = 0.5346 ωL + 0.2166( ωL )2 + ( ωD )2 , supposed to be accurate to within 0.02 %, where ωL and ωD are the FWHM of L ORENTZ and D OPPLER profile, respectively. One example of this convoluted D OPPLER and collision profile for ωL = ωD is shown in Fig. 5.5. In this case the formula gives ωV = 1.638 ωL .

Acronyms and Terminology AMO: ‘Atomic, molecular and optical’, physics. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. FWHM: ‘Full width at half maximum’. NIST: ‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html.

References NIST-DLMF: 2013. ‘Digital library of mathematical functions: §7.19 Voigt functions’, NIST. http://dlmf.nist.gov/7.19, accessed: 9 Jan 2014. SWP 5.5: 2005. ‘Scientific work place’, Poulsbo, WA 98370-7370, USA: MacKichan Software, Inc. http://www.mackichan.com/, accessed: 9 Jan 2014. W EISSTEIN, E. W.: 2011. ‘Convolution’, Wolfram Research, Inc., Champaign, IL, USA. http:// mathworld.wolfram.com/Convolution.html, accessed: 9 Jan 2014. W IKIPEDIA CONTRIBUTORS: 2014. ‘Voigt profile’, Wikipedia, The Free Encyclopedia. http://en. wikipedia.org/wiki/Voigt_profile, accessed: 9 Jan 2014.

H

Vector Potential, Dipole Approximation, Oscillator Strength

H.1

Interaction of the Field of an Electromagnetic Wave with an Electron

In view of its clearness and compactness we use in this book – apart from a few exceptions – in general the dipole length approximation for describing the interaction of electromagnetic waves with atoms and molecules. It is thus appropriate to justify this approach in some detail. In addition, we shall provide the basis for some useful concepts related to the dipole approximation.

H.1.1 Vector Potential In the quantum mechanically correct formulation for the interaction of an electromagnetic field with charged particles one replaces the momentum operator p= −i∇ of the particle (charge q) by p field = p − qA, where A = A(r, t) is the vector potential of the field. The Hamiltonian of an electron in the field thus becomes = H =

p 2field 1 + V (r) = ( p + eA)2 + V (r) 2me 2me

(H.1)

e p·A e2 2 p2 + V (r) + + A 2me me 2me

(H.2)

where the so called C OULOMB gauge of the field has been used: ∇ · A = 0.

(H.3)

For simplicity we have restricted the discussion to a single active electron. One easily generalizes (H.2) to multi-electron systems by replacing momentum p and position vector r with p i and r i , respectively, for each electron i, and summing over all electrons. © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7

631

632

H Vector Potential, Dipole Approximation, Oscillator Strength

The vector potential A relates to the electric and magnetic field vectors E(r, t) = −

∂ A(r, t) + ∇Vext ∂t

and B(r, t) = ∇ × A(r, t)

(H.4)

with Vext (r, t) describing an additional, external potential. If such a field is present, one has to add Vext to V (r) in (H.1) and (H.2). Not included in this description are of course all interactions related to the electron spin which follow from the D IRAC equation, including spin-orbit interaction, as briefly discussed in Chap. 6. In vacuum, we write the vector potential, electric field, and magnetic field for a plane electromagnetic wave as: A0 i(kr−ωt) ee + e∗ e−i(kr−ωt) = A0 e cos(kr − ωt) 2 iE0 i(kr−ωt) ee E(r, t) = − e∗ e−i(kr−ωt) = E0 e sin(kr − ωt) 2 k iB0 k i(kr−ωt) × ee B(r, t) = − e∗ e−i(kr−ωt) = B0 × e sin(kr − ωt). 2 k k A(r, t) =

(H.5) (H.6) (H.7)

The relations among the amplitudes are derived from (H.4) E0 = ωA0 ,

B0 = kA0 =

ω A0 , c

and B0 = E0 /c.

(H.8)

H.1.2 Intensity The P OINTING vector S =E×H =

E×B μ0

gives the energy flux. With E ⊥ B and ε0 μ0 = 1/c2 we obtain its absolute value, called intensity: 2 ε0 c[E0 (r, t)]2 −iωt ee − e∗ eiωt 4 2 ε0 c[E0 (r, t)] 1 + sin(2β) cos(2ωt) . = 2

If (r, t) = |S| = ε0 c|E|2 =

(H.9) (H.10)

Here we have inserted for e the general unit polarization vector (4.15) with the ellipticity angle β. Obviously, depending on β, the intensity If (r, t) may oscillate fast (with 2ω) in position space and time, as emphasized by the index ‘f’. However, independent of β one obtains the cycle averaged intensity 2 2 I (r, t) = If (r, t) = ε0 c E(r, t) = ε0 cω2 A(r, t) (H.11) 2 ε0 cω2 2 2 c ε0 c E0 (r, t) = A0 (r, t) = B0 (r, t) . = 2 2 2μ0

H.1


633

In principle, the amplitude E0 (r, t) (and A0 and B0 ) as well as the averaged intensity I (r, t) may still depend slowly on time t and position r. This is called SVE.1 More about cycle averaged intensities is discussed in Sect. 1.3.1, Vol. 2. The electric field is measured in units [E0 ] = V m−1 , the intensity in [I ] = W m−2 . Note that all field quantities used here represent quasi-monochromatic waves: their bandwidth is assumed to be much narrower than any atomic or molecular absorption line studied. The transition to a continuous spectrum is made by replacing I → I˜(ω)dω, where the spectral intensity distribution I˜(ω) (or the spectral energy density u(ω) ˜ = I˜(ω)/c) refer to the unit of angular frequency interval. Finally, one has to integrate over all frequencies: . . . dω.

H.1.3 Static Magnetic Field For a static, homogeneous external magnetic field B one derives the vector potential by inverting (H.4): 1 1 A = − r × B = B × r. 2 2

(H.12)

This may be verified by inserting this expression into (H.4).2 For such a constant B field the Hamilton operator (H.2) becomes e p2 e 2 A2 = + V (r) + , H L·B + 2me 2me 2me

(H.13)

where for the third term we have used the identity (B × r) · p = B · (r × p ) for the scalar triple product and the definition of angular momentum L=r × p . This term corresponds exactly to (6.29) with (6.27), i.e. we have derived here correctly the interaction potential of an external magnetic field with the magnetic moment of the orbital angular momentum. A heuristic derivation is given in Chaps. 1 and 6. The final term in (H.13) represents a (usually) small correction e 2 A2 e2 2 2 = r B sin θ, 2me 8me

(H.14)

1 The SVE approximation demands that |∂E

0 /∂t| ωc E0 as well as |∂E0 /∂z| E0 /λc etc. Then, second order derivatives may be neglected in the general wave equation.

2 The

triple vector product is expanded as

1 1 ∇ × A = ∇ × − r × B = − [B · ∇r − r · ∇B + r∇ · B − B∇ · r]. 2 2

The first two terms are directional gradients. B · ∇r = B and for a homogeneous B field r · ∇B = 0. The third term also disappears since ∇ · B = div B ≡ 0. With div r = 3 the forth term = −3B. Finally, the whole right hand side gives −(1/2)(B − 3B) = B, q.e.d.

634


where θ is the angle between the external B field and the position vector r of the electron in the atom. We estimate for, say, 30 T and r = a0 a maximum value of ca. 5 × 10−8 eV. Thus, in spectroscopy this term only plays a role if extreme accuracy is asked for or very high magnetic fields and very large orbital radii are involved (i.e. high lying RYDBERG states). On the other hand, it is this very term which is responsible for all diamagnetism of matter.

H.1.4 Relation Between Matrix Elements of p and r For the following considerations we derive an important relation between the matrix elements of momentum and position. For electrons i and j the standard commutation rules for canonically conjugated observables (momentum and position) hold: [xi , p yj ] = 0 and [xi , p xj ] = iδij ,

etc.

(H.15)

With these and the identity a, b2 = [ a, b] b + b[ a, b]

(H.16)

we can write ] = i p ] = i xi and [r, H p, (H.17) [xi , H me me p= p i . Thus, for the matrix elements of p between eigenwhere r = r i and states |a and |b of the Hamiltonian we derive b| p |a =

me ]|a = me (Wa − Wb ) b|r|a . b|[r, H i i

(H.18)

With (Wa − Wb )/ = −Wba / = −ωba the sought-after relation between the matrix elements of p and r is: b| p |a = ime ωba b|r|a .

(H.19)

H.1.5 Ponderomotive Potential Before actually evaluating the matrix elements of the interaction in the Hamiltonian (H.2) in detail, we have a second look at the term proportional to A2 , now for an atom or molecule in an electromagnetic wave. Clearly, this term is a time dependent additional energy in the Hamiltonian. By averaging A2 over one period we obtain with (H.11) an estimate for its dependence on intensity I : Up =

e2 A2 e2 I = . 2me 2ε0 cme ω2

(H.20)

H.1


635

This is identical to the expression (8.139) discussed in Sect. 8.5.1 for the ponderomotive potential Up . There, a completely classical picture of an electron oscillating in the electric field of the wave is used to ‘derive’ it. In respect of standard laser spectroscopy, the ponderomotive potential is usually negligible. As described in Sect. 8.5.1, however, it leads to very interesting phenomena if the objects studied are exposed to very intense electromagnetic radiation which can be generated without difficulties by today’s short pulse lasers.

H.1.6 Series Expansion of the Perturbation and the Dipole Approximation now discuss the term proportional to A · p in (H.2) which is responsible for electromagnetically induced transitions. With (H.5) it is e A0 ( U p , r, t) = p · eei(kr−ωt) + e∗ e−i(kr−ωt) me 2 eE0 −iωt = De − D∗ e+iωt . 2

(H.21)

D We abbreviate the transition operator

D=

eik·r −ieik·r p·e= ∇·e ωme ωme

(H.22)

(with the dimension L) and the amplitude eE0 = eωA0 = ecB0 = e 2I /(cε0 )

(H.23)

(with the dimension of a force MLT−2 ). For electromagnetic waves in the IR, VIS, UV, and VUV spectral range the wavelength is usually very large compared to the objects studied. We thus can assume k · r 1 and expand the exponential function in (H.22): D=

1 (1 + ik · r + · · · ) p · e. ωme

(H.24)

In the electric dipole approximation (short: dipole approximation) only the first term is taken into account and one speaks of E1 transitions. Thus, the dipole transition matrix elements of D between two eigenstates |a and |b is: Dab =

1 a|∇|b · e = i a|r|b · e. a| p |b · e = −i ωba me ωba me

(H.25)

For the last equality we have made use of (H.19). Strictly mathematical, both forms of Dab are completely equivalent – if the wave functions used are exact solutions of

636


the unperturbed Hamiltonian. However, since we usually know the wave functions only approximatively (except for some special cases such as the H atom) both formulas (H.26) and (H.27) lead to slightly different results. Both variants are used in the literature: they are called dipole velocity approximation (since v = p /me ), and the dipole length approximation, respectively. The matrix elements of the interaction potential (H.21) are thus e 2I (ω) i ab (t) = a|U (r, t)|b = − U a|∇|b · ee−iωt + e∗ e+iωt (H.26) 2 ωme cε0 or alternatively in dipole length approximation (r, t)|b = i E0 a|er|b · ee−iωt − e∗ e+iωt . ab (t) = a|U U 2

(H.27)

The latter form is identical to that obtained from the heuristic considerations presented in Sect. 4.3.4. All expressions above refer to a single active electron which is excited or deexcited in an external electromagnetic field. For larger atoms and molecules more than one electron (in principle all) can participate in the interaction – even simultaneously. Such events typically lead to interesting phenomena (see e.g. autoionization, Sect. 7.6). Thus, in general one has to sum the respective interaction potentials for all electrons. The transition operator (H.22) will then read N N 1 ik·r i −i ik·r i e pi · e = e ∇ i · e. ωme ωme

D=

i=1

(H.28)

i=1

Specifically, in dipole approximation for a multi-electron system the transition matrix element (H.25) has to be replaced by Dab = −i

N

N

i=1

i=1

a| ∇ i |b · e = i a| r i |b · e. ωba me

(H.29)

The first and second equality refer to the dipole velocity and dipole length form of the transition operator, respectively. They are identical if (and only if) the wave functions are exact. The sum has to be carried out in principle over all electrons i, with N being the total number of electrons in the system.

H.2

Line Strength and Oscillator Strength

H.2.1 Definitions Several, slightly differing, quantities are used in the literature to characterize the overall strength of a dipole transition between levels a and b with angular momentum quantum numbers ja ma and jb mb , and degeneracies ga = 2ja + 1 and

H.2


637

gb = 2jb + 1, respectively. In principle, j may refer to the orbital (L) or to the total angular momentum (J ), whichever is appropriate. Note that we abbreviate the quantum numbers γa ja ma := a, but we also use a and b for designating the respective energy levels – as long as no confusion can arise. We essentially3 follow C ONDON and S HORTLEY (1951) and define the line strength symmetric in respect of initial and final energy levels and as a sum over all polarizations: γb jb mb |r|γa ja ma 2 |r ba |2 ≡ S(jb ja ) ≡ S(ja jb ) := mb ma

mb ma

b|x|a 2 + b|y|a 2 + b|z|a 2 . =

(H.30)

mb ma

The dimension of the line strength is L2 . In spherical coordinates (see 4.75) this may be written even more explicitly: γb jb mb |rq |γa ja ma 2 ≡ S(ja jb ) S(jb ja ) = mb ma q

2 jb mb |C1q |ja ma 2 . = γb |r|γa

(H.31)

q mb ma

Using the W IGNER -E CKART theorem in the form (4.82) and the orthogonality relations of the 3j symbols (B.42) we obtain the compact relations:4 2 S(jb ja ) = (2jb + 1) γb |r|γa jb C1 ja 2 (H.32) 2 = (2ja + 1) γa |r|γb ja C1 jb 2 ≡ S(ja jb ). If LS coupling is appropriate, one may recouple the reduced matrix elements Ja La SC1 Jb Lb S 2 using (C.46). Exploiting the orthogonality relation (B.67) for the 6j symbols one obtains the relation S(Jb Ja ) Ja

2Jb + 1

S(Lb La ) , 2Lb + 1

(H.33)

where the equality sign holds in so far as the radial matrix element does not depend on the individual fine structure levels J but only on the orbital quantum numbers L and n – which is a good approximation for light atoms. line strength S CS (jb ja ) used by C ONDON and S HORTLEY (1951) is related by S CS (jb ja ) = eS(jb ja ) to the quantity used here – which leads to more compact formulas for the A and B (opt) coefficients and for the oscillator strength fba . 3 The

4 Note

that a factor 3 arising here from the sum over all polarizations q is compensated by a factor 1/(2 × 1 + 1) = 1/3 from the orthogonality relations of the C LEBSCH -G ORDAN coefficients when summing over ma and mb .

638


Complementary to the line strength, which is symmetric and has a dimension L2 , the so called oscillator strength f(opt) (e) is defined asymmetrically and dimensionless for a given polarization with unit vector e. We first define it for a specific transition from one initial substate |a = |ja ma to one final substate |b = |jb mb : (opt)

fjb mb ←ja ma (e) =

2Wba 2Wba r ba 2 2 | D | = · e ba . E h a0 Eh a02

(H.34)

Here Dba is the dipole transition matrix element (H.29). In the second equality we use explicitly the dipole length form for a single electron system, with r ba as defined by (4.79). The oscillator strength is thus proportional to the transition probability (4.65). And since it is also proportional to the absorbed (or emitted) photon energy Wb − Wa = Wba = ωba , its value is positive for absorption and negative for emission. For the particular case of linear polarization with elin z this gives a simple expression for the oscillator strength: (opt) fjb mb ←ja ma

Wba zba 2 me ωba |zba |2 . =2 =2 E h a0

(H.35)

It may be further evaluated with (4.75) and the W IGNER -E CKART theorem in the form (4.82): 2 |zba |2 = γb jb mb |r0 |γa ja ma 2 = (2jb + 1) γb |r|γa jb C1 ja 2

ja −ma

1 jb 0 mb

2

(H.36) .

Note that this becomes zero for mb = ma . In an experiment one typically averages over all initial substates |ja ma and sums isotropically one obover all final substates |jb mb . If the initial level is populated tains for an arbitrary unit polarization vector e = q aq eq for the transition between from level jb to level ja with (H.34) and (4.93) (opt) fjb ja

1 (opt) 2 Wba r ba 2 = f (e) = · e ga m m jb mb ←ja ma ga E h m m a 0 a

a

b

(H.37)

b

1 2 1 2 Wba γb jb mb |rq |γa ja ma e∗q · = aq e q ga E h m m a

=

b

q=−1

q =−1

1 2 2 2 Wba |aq |2 γb |r|γa jb mb |C1q |ja ma . ga E h m m a

b

q=−1

(H.38)

H.2


639

Using (4.82), the orthogonality relation (B.42) for the 3j symbols, and (unit vector e) the sums can be carried out: (opt) fjb ja

q

2 1 1 Wba gb r jb C1 ja 2 γ =2 | |γ

|aq |2 b a E h ga a0 3

|aq |2 = 1

(H.39)

q=−1

(opt)

fjb ja =

2 2 Wba gb r 2 γ | |γ

b a jb C1 ja . 3 E h ga a0

(H.40)

Finally by comparing this with (H.32) we obtain the expression 2 Wba S(jb ja ) 2me ωba S(jb ja ) = 3ga Eh 3 ga a02

(opt)

fjb ja =

(H.41)

as used in Sect. 5.2.2. We note here one very important result: The averaged oscillator strength for an isotropically populated initial level – and hence the probability for absorption or induced emission – is independent of the polarization. Thus, we also may simply average (H.35) (opt) fjb ja

2 Wba zba 2 2me ωba 1 = = |zba |2 ga E h m a 0 ga m a

(H.42)

a

where we have exploited that here q = 0, and thus only contributions from ma = mb arise. For an initial singly degenerate ns state (H.42) is reduced to (H.35). Equivalent expressions are derived from (H.41) with (H.30) and the dipole moment D = −er: (opt)

fjb ja =

2 2me ωba γb jb mb |r|γa ja ma 2 3ga m m a

=

(H.43)

b

2 r 8π 2 me Wba 2 Wba γ j m | |γ j m

|D ba |2 . b b b a a a = 3ga Eh m m a0 3ga h2 e2 m m a

b

a

b

(H.44) Note the factor 3 in the denominator in comparison to (H.42), compensating for the equal contributions from the components of |r|2 = |x|2 + |y|2 + |z|2 .

H.2.2 THOMAS-REICHE-KUHN Sum Rule We sum the oscillator strength according to (H.35) over all existing final states |b = |γb jb mb and rewrite the expression suitably:

640


(opt)

fjb mb ←ja ma =

2me ωba

b

b

=

|zba |2 =

2me ωba

b

a|z|b b|z|a

(H.45)

1 ime ωba a|z|b b|z|a − ime ωab a|z|b b|z|a . i b

Now we apply (H.19) and obtain b

(opt)

fjb mb ←ja ma =

1 a|z|b b|pz |a − a|pz |b b|z|a i b

(H.46) a|zpz − pz z|a 1 = a|zpz |a − a|pz z|a = = 1, i i where we have exploited completeness 1 = b |b b|. The last step realizes the commutation rules (H.15) and normalization a|a = 1. We note that this derivation does not depend on the specific initial substate |a = |ja ma since summation is always over all final states |b . Hence, (H.46) is also valid (opt) for the average oscillator strength fjb ja . Thus, we have derived the very important T HOMAS -R EICHE -K UHN sum rule: (opt) fjb ja = 1. (H.47) b

The oscillator strength is often used in atomic and molecular physics to characterize dipole transitions. It allows one to compare the strength of different transitions in (opt) different atoms or molecules. For systems with one active electron fjb ja ≤ 1 strictly holds, the classical reference being an oscillating electron which has an oscillator strength = 1. For systems with Ne active electrons one has to replace in (H.45) the single e (i) and obtains the sum rule: electron coordinate z by N i=1 z (opt) fjb ja = Ne . (H.48) b

Finally, we point out that the summation just discussed must include the ionization continuum if the basis is to be complete. In this context, continuum states are usually normalized in the energy scale (see Appendix J). Correspondingly one defines the optical oscillator strength density (OOSD), df(opt) /d, with the dimension Enrg−1 , where is the energy of the emitted electron in the continuum. Thus, (H.48) includes summation over discrete states and integration for energies beyond the ionization potential WI : b

(opt)

fjb ja →

discrete b

(opt)

fjb ja +

∞ WI

(opt)

dfja d

d.

Obviously for high energies, limW →∞ (df(opt) /d) = 0 must hold.

(H.49)


641

Acronyms and Terminology E1: ‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation. IR: ‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between 760 nm and 1 mm according to ISO 21348 (2007). OOSD: ‘Optical oscillator strength density’, characterizes the strength of photoionization per energy interval (see Sect. 5.5.1). SVE: ‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38), Vol. 2). 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 electromagentic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

References C ONDON, E. U. and G. S HORTLEY: 1951. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 441 pages. ISO 21348: 2007. ‘Space environment (natural and artificial) – process for determining solar irradiances’. International Organization for Standardization, Geneva, Switzerland.

FOURIER Transforms and Spectral Distributions of Light

I.1

Short Summary on FOURIER Transforms

The F OURIER transform (FT) is a mathematical tool widely used in physics and engineering. We collect here the most important definitions and relationships and give a few practical examples, focussing on the description of short light pulses. F OURIER transforms are based on the complex version of the F OURIER integral (here without proof): ∞ ∞ 1 X(t) = eiωt dω X(τ )e−iωτ dτ (I.1) 2π −∞ −∞ ∞ ∞ 2πiνt or = e dν X(τ )e−2πiντ dτ. −∞

−∞

Various, slightly different notations can be found in the literature. We use the standard notations of modern physics with the F OURIER transform1 ) X(ω) = F X(t) =

∞

−∞

X(t)e−iωt dt

(I.2)

of the time dependent function X(t). Conversely, X(t) is recovered by the inverse F OURIER transform 1 ) X(t) = F −1 X(ω) = 2π

∞

−∞

iωt ) dω. X(ω)e

The complex conjugate of the F OURIER transform of X(t) is ∞ ∗ ) X (ω) = X ∗ (t)eiωt dt, −∞

1 In

other notations the factor 1/2π is applied symmetrically as using frequency ν instead of angular frequency ω = 2πν.

√

(I.3)

(I.4)

1/2π , or completely avoided by


643

I

644

I


and correspondingly X ∗ (t) =

1 2π

∞

−∞

)∗ (ω)e−iωt dt. X

(I.5)

By rewriting (I.1) as

∞ 1 iω(t−τ ) X(τ ) e dω dτ X(t) = 2π −∞ −∞

∞

one identifies the expression in square brackets as the D IRAC delta function: 1 δ(t − τ ) = F −1 e−iωτ = 2π and equivalently δ(ωc − ω) =

1 1 iωc t = F e 2π 2π

∞

−∞ ∞ −∞

eiω(t−τ ) dω

(I.6)

e−i(ω−ωc )t dt.

(I.7)

With this, one verifies the important relationship

X(t)2 dt = 1 2π −∞ ∞

∞

−∞

2 ) X(ω) dω,

(I.8)

called P LANCHEREL’s theorem (sometimes PARCEVAL’s theorem – which actually refers to F OURIER series – or R AYLEIGH’s theory). To proof it, one just has to insert (I.3) and (I.2), rearrange the integrations and recover the D IRAC delta function. We leave it to the reader to work out the details. As we shall see below, for electromagnetic radiation P LANCHEREL’s theorem essentially states energy conservation. In the context of describing short pulses of light, we note a very useful relation for the F OURIER transform of an oscillation around a carrier frequency ωc with an ) envelope function X(t): With X(ω) = F[X(t)] we can write X(t)e

iωc t

=e =

1 2π ∞

iωc t

1 2π

−∞

∞

1 iωc t iωt ) e dω ≡ X(ω)e 2π −∞

∞ −∞

) − ωc )ei(ω−ωc )t dω X(ω

) − ωc ) . ) − ωc )eiωt dω = F −1 X(ω X(ω

(I.9)

Thus, the carrier frequency just shifts the F OURIER transform F[X(t) exp(iωc t)] towards positive frequencies in respect of F[X(t)]. This formally also resolves the problem one might envisage with negative frequencies in the spectral distribution ) X(ω): since for any physically relevant distribution with a FWHM= γ one expects ωc γ , there will be no contributions from negative frequencies in (I.2). An analogue relation applies to a shift in time. Another set of useful relations concerns F OURIER transforms of convolutions (see Appendix G). The proof is similar to that for P LANCHEREL’s theorem.

I.1

Short Summary on FOURIER Transforms

645

1. The convolution theorem states that the F OURIER transform of a convolution of ) two functions X(t) and Y (t) is the product of their F OURIER transforms X(ω) ) and Y (ω): ) Y )(ω). F (X ∗ Y )(t) = F X(t) F Y (t) = X(ω) (I.10) This is proven as follows: F (X ∗ Y )(t) =

∞ ∞

∞ ∞

=

−∞

−∞

−∞

−∞

=

∞

−∞

−iωt X t Y t − t dt e dt

−iωt X t e dt Y t − t e−iω(t−t ) dt

X t e−iωt dt

∞ −∞

Y t e−iωt dt .

2. The F OURIER transform of a product of two functions is the convolution of their F OURIER transforms: 1 1 ) ) F X(t)Y (t) = F X(t) ∗ F Y (t) = (X ∗ Y )(ω). 2π 2π

(I.11)

And the proof for this relation is only slightly more complex, involving the D IRAC delta function: F X(t)Y (t)

∞ ∞ ∞ iω t iω t −iωt 1 1 ) ) = dt X ω e dω Y ω e dω e 2π −∞ −∞ 2π −∞

∞ ∞ iω t 1 ) = X ω e dω −∞ 2π −∞

∞ 1 ) ω − ω ei(ω −ω )t dω e−iωt dt × Y 2π −∞ ∞ ∞ ∞ 1 1 ) ω − ω dω dω ei(ω −ω)t dt ) ω Y = X 2π 2π −∞ −∞ −∞ ∞ ∞ 1 ) ω − ω dω dω )ω Y = δ ω − ω X 2π −∞ −∞ ∞ 1 ) ω − ω dω . ) ω Y = X 2π −∞ The inverse relations of course also hold: )∗Y ))(ω) = 2πX(t)Y (t) F −1 (X ) Y )(ω) = (X ∗ Y )(t). F −1 X(ω)

and

(I.12) (I.13)

646

I


3. In analogy to the convolution theorems, the cross-correlation theorems ∗ )∗ (ω)Y )(ω) and F (X * Y )(t) = F X(t) F Y (t) = X 1 ) ) 1 F X(t) * F Y (t) = (X * Y )(ω) F X ∗ (t)Y (t) = 2π 2π

(I.14) (I.15)

hold, as well as their inverse relations. From these expressions follows directly a very important relationship between F OURIER transform and autocorrelation function, the W IENER -K HINCHIN theorem: ∞ 2 −1 ) F X(ω) = X ∗ (τ )X(τ + t)dτ or (I.16) −∞

2 ) X(ω) = F

I.2

∞ −∞

X ∗ (τ )X(τ + t)dτ .

(I.17)

How Electromagnetic Fields are Written

With these tools at our command we want to describe wave-packets, i.e. short, coherent light pulses. At this point, we emphasize again that real electromagnetic fields are real observables, i.e. measurable quantities! One often finds in the literature that electromagnetic wave fields are ‘for simplicity’ written in complex form. However, albeit the greater mathematical simplicity of the complex form is indeed attractive, one may easily overlook some important aspects by a too naive use of the complex form – as we have documented in Chap. 4, when describing absorption and induced emission of light in quantum mechanical terms. Thus, following (4.1) we write the electric field vector E(r, t) of a wave-packet as a real quantity, explicitly using its two conjugate complex terms. For simplicity of writing we ignore here the r dependence2 in E0 (r, t) and keep the position in space constant kr = 0: 1 E(t) = E0 (t) ee−i(ωc t−φc ) + e∗ ei(ωc t−φc ) (I.18) 2 ∗ 1 ≡ eE − (t) + e∗ E + (t) with E + (t) = E0 (t)ei(ωc t−φc ) = E − (t) . 2 In the spirit of the so called SVE approximation (see footnote 1 in Appendix H), we have introduced here E0 (t), the time dependent envelope of the field – now a real function depending on time. The phase φc may also depend on time and position (see e.g. (1.46) in Vol. 2). It allows to specify the relative phase of the carrier oscillation ±ωc t → ∓k c r ± ωc t . Similarly, we keep the r dependence of the envelope in the back of our mind and set E ± (t) → E ± (r, t) when required.

2 Where necessary, we shall simply reintroduce the wave vector by replacing

I.2

How Electromagnetic Fields are Written

647

in respect of the envelope E0 (t) as illustrated in Fig. 1.24 in Vol. 2. In nonlinear optics and ultrafast science it plays a significant role for a variety of processes. To make things not too complicated here, we assume in the following φc to be constant with time, and we shall focus on the E + (r, t) term, so that we can still exploit the advantages of complex arithmetic. Within the SVE approximation we are interested in the temporal dependence of the intensity of the electromagnetic field. According to (H.11), ignoring the position dependence, the average intensity is ε0 c 2 ε0 c − I (t) = If (t) = E0 (t) = E (t)E + (t). (I.19) 2 2 Here If (t) indicates cycle averaging over fast oscillations. For reference we also communicate the cycle average energy density of the field If (t) ε0 − = E (t)E + (t). uf (t) = c 2

(I.20)

The electric field of a light pulse may be expressed as inverse F OURIER transform, i.e. as a linear superposition of oscillators with frequencies ω around the carrier frequency ωc in a bandwidth δω ωc : ∞ 1 iωt ) dω. (I.21) E(ω)e E0 (t) = 2π −∞ To explicitly show the carrier frequency ωc and phase φc , we apply (I.9): ∗ E + (t) = E0 (t)ei(ωc t−φc ) = E − (t) ∞ e−iφc ∞ ) 1 )+ (ω)eiωt dω. = E(ω − ωc )eiωt dω = E 2π −∞ 2π −∞ Conversely, the F OURIER transform of the field envelope is ∞ ) E0 (t)e−iωt dt, E(ω) = −∞

(I.22)

(I.23)

or alternatively one may write3 − ) − ωc )e−iφc = E ) (−ω) ∗ )+ (ω) = E(ω E ∞ ∞ = E + (t)e−iωt dt = e−iφc E0 (t)ei(ωc −ω)t dt. −∞

3 Even

−∞

)+ (ω))∗ = E )− (ω), since though (E + (t))∗ = E − (t), note that (E ∞

∗ ∞ + ∗ + ∗ iωt ) (ω) = E E (t) e dt E + (t)e−iωt dt = =

−∞

∞

−∞

−∞

)− (−ω). E − (t)eiωt dt = E

(I.24)

648

I


) Note that in this terminology E0 (t) is a real quantity. E(ω) will also be real if and )+ (ω) is usually complex. only if E0 (t) is symmetric around its maximum, while E )+ (ω) typically has a pronounced resonance at However, with these definitions E ω > 0 so that contributions from negative frequencies to the inverse FT (I.22) can safely be neglected. In other words: artificial truncations of the limits of integrals are not needed, since anyhow only positive frequencies contribute when working )+ (ω). These relationships will be illustrated below for specific examples. with E

I.3

The Intensity Spectrum

If one inserts E ± (t) according to (I.22) into expression (I.19) for the cycle averaged intensity and integrates over all times, one recovers P LANCHEREL’s theorem (I.8) ∞ ∞ ∞ + 2 ε0 c 1 E (ω) dω = I (t)dt = I˜(ω)dω = F, (I.25) 2 2π −∞ −∞ −∞ leading to the fluence F (i.e. the total pulse energy Wtot per unit area). With the second equality we have defined the spectral intensity distribution: 2 ε0 c )+ 2 ε0 c ) I˜(ω) = E (ω) = E(ω − ωc ) . 4π 4π

(I.26)

The assumption that this indeed describes the spectrum of the radiation is by no means as trivial as commonly assumed. Note, that this expression is not the F OURIER transform of the intensity I (t)! One may, however, rationalize this definition by remembering how optical (or other) spectra of electromagnetic radiation are measured: Assume e.g. an interferometric measurement (FABRY-P ÉROT interferometer, diffraction grating etc.), where the signal S is generated by superposition of two (or many) amplitudes. A typical interference pattern emerges from, say two wave fields E + (t) and E + (t + δ), which have traversed different optical path length cδ (corresponding to a delay time δ). The signal is then S ∝ E − (t) + E − (t + δ) E + (t) + E + (t + δ) = 2I0 + 2 Re E − (t + δ)E + (t) ,

(I.27)

and only the interference term [in square brackets] is relevant for the measurement of the spectrum. If such a measurement is made with short pulses, one obviously integrates over all times (or sufficiently long times), thus an autocorrelation function of the field is recorded. To obtain the spectrum one has to F OURIER transform this signal which according to (I.17) leads to 1 )+ 2 ˜ F E − (t) * E + (t) = E (ω) ∝ I (ω). 2π

I.4

Examples

649

If a prism spectrograph is used a similar argument can be based on the wave optical interpretation of the index of refraction. Hence, the definition of the spectrum according (I.26) is justified. Proper normalization is provided by the P LANCHEREL theorem in the form (I.25), which simply states energy conservation: the fluence F is independent of whether we integrate intensity over all times or spectral intensity distribution over all angular frequencies. The units are [F ] = J m−2 , [I (t)] = W m−2 , and [I˜(ω)] = J s m−2 , respectively. It is important to note that the above use of P LANCHEREL’s theorem is only possible for square integrable functions, i.e. for pulses of electromagnetic radiation with finite duration. If we want to describe CW light sources the integration over all times has to be replaced by an average over a sufficiently long time Tav . P LANCHEREL’s theorem (I.25) must then be replaced by ∞ Tav /2 1 ε0 c Tav /2 + 2 E (t) dt = I (t)dt = I˘(ω)dω = I, (I.28) Tav −Tav /2 2Tav −Tav /2 −∞ where I˘(ω) is now the spectral intensity of the ensemble, measured in units [I˘(ω)] = W s m−2 , while I is the average intensity with [I ] = W m−2 . More about stationary, quasi-monochromatic or chaotic light sources, their spectra and their coherence properties will be found in Sect. 2.1, Vol. 2.

I.4

Examples

In the following we shall present the most commonly used temporal and spectral profiles of light pulses. The F OURIER transforms collected here can be derived by analytic integration, where necessary with the help of a suitable computer programme (e.g. SWP 5.5, 2005). They are also found in standard textbooks or at the web (e.g. W EISSTEIN, 2012; W IKIPEDIA CONTRIBUTORS, 2014). We write the field envelope as (I.29) E0 (t) = E0 h t/|t0 | with h(0) = 1 and I0 =

ε0 c 2 E 2 0

(I.30)

where E0 is the maximum field amplitude, |t0 | a characteristic width, and I0 the cycle averaged maximum intensity. We point out that the spectral intensity distribution as defined in (I.26) refers to a single pulse. It has the dimension [I˜(ω)] = J s m−2 . Integrated over all frequencies it gives the fluence F per pulse, i.e. the total pulse energy per unit area: ∞ ∞ I (t)dt = I0 h2 (t/t0 )dt. (I.31) F= −∞

−∞

If one wants to extend the notion of the spectral intensity distribution to a CW beam, one has to modify the normalization. As detailed in Sect. 2.1, Vol. 2, one may assume that the beam consists of a statistical ensemble of single pulses which

650

I


are characterized by individual time constants τi and their individual spectra I˜i (ω). One then has to average these spectra and to renormalize in respect of the average duration τc of the pulses. One obtains an overall spectrum: 1 ∞ w(τi )I˜i (ω)dτi . (I.32) I˘(ω) = I˜i (ω) = τc 0 This spectrum has now the dimension W s m−2 and the integration over all frequencies gives the average, stationary intensity I of the beam.

I.4.1

Gaussian Distribution

Gaussian pulse shapes are probably the most popular ones, mainly due to their convenient mathematics. Starting from (I.18) we write the field envelope 2

t (I.33) E0 (t) = E0 h(t) = E0 exp − 2 τG √ with a FWHM= 2 ln 2τG = 1.665τG . The corresponding cycle averaged intensity (I.19) is now

t2 I (t) = I0 exp −2 2 τG with a FWHM t1/2 =

√ 2 ln 2τG = 1.177τG .

We also note the overall fluence

∞ π τG π t2 2 ε0 cE0 = τG I0 . F = I0 exp −2 2 dt = 2 2 2 τG −∞ The F OURIER transform of the field envelope is again a Gaussian:

√ 1 2 2 ) and E(ω) = τG π exp − τG ω 4

√ )+ (ω) = E(ω ) − ωc )e−iφ = τG π exp − 1 τG2 (ω − ωc )2 − iφ . E 4

(I.34)

(I.35)

(I.36) (I.37)

Thus, with (I.26) the (experimentally measurable) intensity spectrum becomes

2 τ τG2 I0 ω − ωc 2 , (I.38) exp − G (ω − ωc )2 = 2 exp − 2 2 ωG ωG √ with ωG = 2/τG and a FWHM √ √

ω1/2 = 2 ln 2ωG = 2 2 ln 2/τG = 2.355/τG . (I.39) I˜(ω) = I0

I.4

Examples

651

∞ One easily verifies that −∞ I˜(ω)dω = F as in (I.35). In frequency space ν1/2 = √ 2 ln 2/(πτG ) and the so called time-bandwidth product is

t1/2 ν1/2 =

I.4.2

2 ln 2 = 0.441. π

(I.40)

Hyperbolic Secant

Another, quite popular distribution function for describing the time dependence of short light pulses is the hyperbolic secant E0 (t) = E0 sech with a FWHM of −2 ln(2 − tion is

2E0 t E0 = = τs cosh(t/τs ) et/τs + e−t/τs

(I.41)

√ 3)τs = 2.634τs . The corresponding intensity distribu

2 2 I (t) = I0 sech (t/τs ) = I0 t/τ , e s + e−t/τs √ with a FWHM t1/2 = −2 ln(−1 + 2)τs = 1.763τs . 2

This normalization leads to a fluence ∞ F = I0 sech2 (t/τs )dt = 2τs I0 .

(I.42) (I.43)

(I.44)

−∞

The F OURIER transform of the field envelope is again a hyperbolic secant 2E0 πτs πτs ω ) = πτ ω/2 E(ω) = E0 πτs sech 2 e s + e−πτs ω/2 )+ (ω) = E0 πτs e−iφc sech πτs (ω − ωc ) . E 2

or

The spectral intensity profile for a pulse with the carrier frequency ωc is

ε0 c 2 2 2 πτs ˜ πE0 τs sech (ω − ωc ) . I (ω) = 4 2 In angular frequency space it has a FWHM √

ω1/2 = 4 arcsech(1/ 2)/(πτs ) = 1.122/τs

(I.45) (I.46)

(I.47)

(I.48)

and thus ν1/2 = 0.179/τs , so that the time-bandwidth product is

t1/2 ν1/2 = 0.315.

(I.49)

652

I.4.3

I


Rectangular Wave-Train

For a number of situations a rectangular wave-train is a useful model of a light pulse. Assume an (angular) carrier frequency ωc = 2π/Tc , let the pulse begin at time t = ti and end at ti + τi ; during this time interval the amplitude is constant and zero elsewhere, its relative phase φi may be chosen freely. In our standard format (I.18) the electric field of the wave is explicitly " 1 E0 ei[ωc (t−ti )−φi ] for ti < t < ti + τi + c.c. (I.50) E i (t) = e∗ 0 else 2 The intensity distribution has the same temporal shape with I0 = ε0 cE02 /2. It is convenient to directly evaluate the F OURIER transform (I.24) of Ei+ (t): t0 +τr )+ (ω) = E0 exp i(ωc t − ωc ti − φi − ωt) dt E i t0

exp[i(ωc − ω)τi ] − 1 = E0 τr exp −i(ωti + φi ) i(ωc − ω)τi )i (ω) = E0 τi exp[−iωti ] 1 − exp[−iωτi ] . E iωτi

(I.51) (I.52)

The spectral intensity distribution (I.26) is thus

τi (ω − ωc ) ε0 c )+ 2 I0 τi2 sinc2 , I˜i (ω) = Ei (ω) = 4π 2π 2

(I.53)

with the sinc x = (sin x)/x function. The overall fluence is ∞ ∞ F= I (t)dt = I˜(ω)dω = I0 τi . −∞

−∞

The FWHM of I˜i (ω) is given by ω1/2 = 5.566/τi or in frequency space ν1/2 = 0.886/τi so that the time-bandwidth product is

t1/2 ν1/2 = 0.886,

(I.54)

which is much broader than for the Gaussian or sech2 distributions. Finally, we note in passing that all above expressions apply to a simple rectangular pulse (without oscillations) when one just sets ωc = 0 and φi = 0.

I.4.4

Rectangular Spectrum

Another important case is a rectangular spectrum. Such a situation may arise, e.g. if one spectrally filters a broad band light CW source with a sharp band-pass filter (e.g. in stellar interferometry). The spectrum would then be

I.4

Examples

653

* I˘(ω) =

I / ω 0

for ωc − 12 ω < ω < ωc + 12 ω else,

(I.55)

where I is the average stationary intensity of the beam. The integral (I.3) can easily be carried out in this case and one obtains the inverse F OURIER transform:

sin x I iωc t

ωt with sinc(x) = F −1 I˘(ω) = e sinc . (I.56) 2π 2 x This formula turns out to be useful e.g. for understanding stellar interference spectroscopy (see Sect. 2.1.4, Vol. 2).

I.4.5

Exponential and LORENTZ Distributions

The One-Sided Exponential Distribution * h(x) =

0

if x < 0

exp(−x)

if x ≥ 0

(I.57)

is used to describes an exponentially decaying field (oscillating at a carrier frequency ωc ) and its cycle averaged intensity, respectively:

t and E(t) = E0 h 2τe t with FWHM t1/2 = τe ln 2 = 0.693τe . I (t) = I0 h (I.58) τe The overall fluence is simply F = τe × I0 . The F OURIER transform of the field envelope is complex in this case (asymmetric distribution): ) E(ω) =

2τe E0 , 1 + 2iτe ω

)+ (ω) = e−iφ E

and

2τe E0 . 1 + 2iτe (ω − ωc )

The intensity spectrum becomes a L ORENTZ distribution I0 τe2 γ 2 /4 I0 1 = , I˜(ω) = π 1 + [2τe (ω − ωc )]2 πγ 2 γ 2 /4 + (ω − ωc )2 with a FWHM ω1/2 = γ = 1/τe .

(I.59) (I.60)

654

I


In frequency space this gives ν1/2 = 1/(2τe π) = 0.159/τe , so that the timebandwidth product becomes

t1/2 ν1/2 = 0.110.

(I.61)

This is an extraordinarily small time-bandwidth product in comparison with the Gaussian or sech2 case. Note, however, that the temporal shape of this “pulse” is characterized by an extremely wide ∞ wing at positive times. Normalization I˜(ω)dω = F is verified easily with −∞ dx/(1 + x 2 ) = π .

The Two Sided Exponential Distribution E0 (t) = E0 exp −|t|/(2τee ) is a somewhat pathological pulse envelope, with E + (t) = E0 exp i(ωc t − φ) − |t|/(2τee ) .

(I.62)

(I.63)

It has an intensity profile I (t) = I0 exp −|t|/τee

(I.64)

with a fluence F = 2I0 τee ,

(I.65)

and a FWHM t1/2 = 2 ln 2τee = 1.386τee . Die F OURIER transform of the field envelope is now a (real) L ORENTZ distribution ) E(ω) =

4τee E0 , 1 + (2τee ω)2

)+ (ω) = E0 e−iφ E

or alternatively

γ , γ 2 /4 + (ω − ωc )2

(I.66) (I.67)

with an FWHM γ = 1/τee . The intensity spectrum is given by

2 I0 γ ˜ I (ω) = 2π γ 2 /4 + (ω − ωc )2 √ with a FWHM ω1/2 = 2 − 1γ = 0.644γ = 0.643/τee ,

(I.69)

so that the time-bandwidth product becomes in this case √ 2 − 1(ln 2)/π = 0.142.

tL ν1/2 =

(I.70)

(I.68)

We point out that (I.68) is not a L ORENTZ profile, and may, e.g. not easily be convoluted. We emphasize that in any case this is not a typical profile for characterizing a real light pulse: it has a cusp at time zero and extremely wide tails, both in the time and in the frequency domain.

I.5

Fourier Transform in Three Dimensions

655

For reference we also write (I.67) in normalized form γ 1 ˜ L(ω, γ)= 2 2π γ /4 + (ω − ωc )2 with

∞

(I.71)

˜

= 1. The inverse FT is given by (I.63) as ∞ 1 1 −γ |t|/2+iωc t ˜ ˜ F −1 L(ω, e γ) = L(ω, γ )eiωt dω = 2π −∞ 2π

−∞ L(ω, γ )dω

(I.72)

with γ = 1/τee . We have set here φ = 0. We finally can now supplement the proof for the additivity of linewidths when convoluting two L ORENTZ profiles, as introduced by (G.21). According to (I.12) the inverse FT of a convolution of two L ORENTZ profiles is proportional to the products of their inverse FT: F −1 L˜ 1 (ω, γ1 ) ∗ L˜ 1 (ω, γ2 ) (I.73) = 2π ×

1 −γ1 |t|/2−iω1 t 1 −γ1 |t|/2+iω2 t 1 −(γ1 +γ2 )|t|/2+i(ω2 −ω1 )t e e e × = . 2π 2π 2π

Now we just transform this back, compare with (I.72), and obtain: L1 (x, γ1 ) ∗ L1 (x, γ2 ) = =

1 −(γ1 +γ2 )|t|/2+i(ω2 −ω1 )t F e 2π

(I.74)

(γ1 + γ2 )2 /4 2 . (γ1 + γ2 )π (x − (ω2 − ω1 ))2 + (γ1 + γ2 )2 /4

Thus, the convolution of one L ORENTZ profile with another one leads again to a L ORENTZ profile. The FWHM of the two profiles are just added: γ = γ1 + γ2 .

I.5

(I.75)

Fourier Transform in Three Dimensions

F OURIER transforms can be extended into n dimensions. Three dimensional FTs will be employed when 3D wave-packets are introduced (Eq. (1.106) in Sect. 1.4, Vol. 2). There, in principle, one has to integrate over plane waves exp(−ikr) in full momentum space k. The same holds for FBA (see Sects. 6.6, 8.2, and 8.4.4, Vol. 2). Practical evaluation of these 3D FTs and their inverse is significantly more complex than in one dimension. Complete operational tool-sets for the 2D and 3D cases have recently been developed by BADDOUR (2009) and (2010), respectively. We briefly summarize some key results for the 3D case. The definition of the 3D F OURIER transform – in analogy to (I.2) – reads ) ) X(k) = X(kx , ky , kz ) = X(r)e−ikr d3 r (I.76) R3

656

I


and its inverse – in analogy to (I.3) – is 1 X(r) = X(x, y, z) = (2π)3

R3

ikr 3 ) d k. X(k)e

We note in passing that the D IRAC delta function in three dimensions is 1 δ k−k = ei(k−k )r d3 r. (2π)3 R3

(I.77)

(I.78)

Its characteristic property is expressed by f x δ x − x d3 x f (x) = R3

2π

=

π

dϕ

∞

sin θ dθ

0

0

f x δ x − x x 2 dx ,

(I.79)

δ x − x d3 x = 1.

(I.80)

0

specifically this ensures

R3

In view of (I.79) one may also express the 3D delta function as δ(x − x ) δ θ − θ δ ϕ − ϕ . δ x − x = 2 x sin θ

(I.81)

For problems with some kind of spherical symmetry (e.g. in atomic physics) it is useful to rewrite (I.76) in spherical polar coordinates, with standard position coordinates (r, θr , ϕr ) and coordinates in wave vector space (k, θk , ϕk ): ) θk , ϕk ) = X(k,

2π

0

π

0

∞

X(r, θr , ϕr )e−ikr r 2 sin θr drdθr dϕr .

(I.82)

0

One expands X(r, θr , ϕr ) into a series of spherical harmonics X(r) = X(r, θr , ϕr ) =

∞

where

(I.83)

∗ X(r, θr , ϕr )Y m (θr , ϕr ) sin θr dθr dϕr .

(I.84)

X m (r)Y m (θr , ϕr ),

=0 m=−

X m (r) = 0

2π

π 0

By expanding also the plane wave (I.82) according to (J.13) one obtains the 3D F OURIER transform in k space ) = X(k, ) θk , ϕk ) = 4π X(k)

∞

+

(−i) X m (k)Y m (θk , ϕk ),

=0 m=−

(I.85)


657

with the th order spherical H ANKEL transform of X m (r) ∞ + ' ( X m (k) = S X m (r) = X m (r)j (kr)r 2 dr.

(I.86)

0

The inverse 3D F OURIER transform – in analogy to (I.3) – is defined as ∞ 1 ikr 3 ) X(r) = d k, X(k)e (2π)3 −∞

(I.87)

which – complementary to (I.82) – may again be expressed in spherical coordinates. ) Explicitly, one expands X(k) into a series of spherical harmonics ) = X(k, θk , ϕk ) = X(k) ) m (k) = X

2π 0

∞

) m (k)Y m (θk , ϕk ), X

where

(I.88)

=0 m=−

π 0

∗ ) θk , ϕk )Y m (θk , ϕk ) sin θk dθk dϕk . X(k,

) m (k) Using now the inverse spherical H ANKEL transform of X + 2 ∞) ) X m (r) = X(k)j (kr)k 2 dk, π 0

(I.89)

(I.90)

one finally obtains the inverse 3D F OURIER transform in spherical coordinates: X(r) = X(r, θr , ϕr ) =

∞ 1 + ) m (k)Y m (θk , ϕk ). (i) X 4π

(I.91)

=0 m=−

Comparing (I.85) and (I.88) one also finds that + ' ( ) m (k) = 4π(−i) X m (k) = 4π(−i) S X m (r) X

from which the F OURIER expansion coefficients may be obtained.

Acronyms and Terminology c.c.: ‘complex conjugate’. CW: ‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc. FBA: ‘First order B ORN approximation’, approximation describing continuum wave functions by plane waves; used in collision theory and photoionization (see Sect. 6.6 in Vol. 2 and Sect. 5.5.2, respectively). FT: ‘F OURIER transform’, see Appendix I. FWHM: ‘Full width at half maximum’. SVE: ‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38), Vol. 2).

658

I


References BADDOUR, N.: 2009. ‘Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates’. J. Opt. Soc. Am. A, 26, 1767–1777. BADDOUR, N.: 2010. ‘Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates’. J. Opt. Soc. Am. A, 27, 2144–2155. SWP 5.5: 2005. ‘Scientific work place’, Poulsbo, WA 98370-7370, USA: MacKichan Software, Inc. http://www.mackichan.com/, accessed: 9 Jan 2014. W EISSTEIN, E. W.: 2012. ‘Fourier transform’, Wolfram Research, Inc., Champaign, IL, USA. http://mathworld.wolfram.com/FourierTransform.html, accessed: 9 Jan 2014. W IKIPEDIA CONTRIBUTORS: 2014. ‘Fourier transform’, Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/wiki/Fourier_transform, accessed: 9 Jan 2014.

J

Continuum

J.1

Normalization of Continuum Wave Functions

Many physical problems require the inclusion of continuum states in addition to discrete bound states – e.g. scattering processes, photoionization, electron impact ionization an the like. In all these cases the normalization of the continuum states poses a specific problem. We briefly indicate here how this problem may be solved – as one way among several possible approaches. For the discrete spectrum the radial wave functions Rn (r) are normalized in the usual fashion: Rn (r)Rn (r)r 2 dr = un (r)un (r)dr = δnn . (J.1) In the following we use the radial functions in the form u(r) = rR(r). In the continuous spectrum, following B ETHE and S ALPETER (1957) Eq. (4.11ff), they are normalized in T scale by demanding

∞

druT (r) 0

T + T T − T

uT (r)dT = 1.

(J.2)

Here T (k) is any function of the electron wavenumber k in the continuum, possibly k itself. In the theoretical treatment of photoionization normalization in scale is frequently used, with being the energy of the electron in the continuum. The integration over dT needs only to be over a small interval 2 T around T , since all other contributions cancel.1 With such normalized continuum functions one may represent any wave function including the continuum as partial wave expansion (for simplicity written here only for one electron systems): 1 Summation of (J.1) over all n states in the discrete spectrum would be the equivalent to this second

integration in (J.2). Since only one state contributes, the result is also =1. © Springer-Verlag Berlin Heidelberg 2015 I.V. Hertel, C.-P. Schulz, Atoms, Molecules and Optical Physics 1, Graduate Texts in Physics, DOI 10.1007/978-3-642-54322-7

659

660

J

ϕ(r, θ, ϕ) =

+

1 Y m (θ, ϕ) r m

∞

an m un (r) +

Continuum

,

∞

(J.3)

dT (k)a m (T )uT (r) . k=0

n= +1

The relation between normalization in T scale and in k scale is given by −1/2 1/2 dT dk = uk (r) × . uT (r) = uk (r) × dk dT

(J.4)

Specifically, for normalization in energy scale, with 2 k 2 me a02 k 2 = = Eh 2 2M¯ M¯

and k =

M¯ 1 me a0

2 Eh

(J.5)

M¯ 1 . me a0 (2Eh )1/2

(J.6)

¯ we obtain (for a system with reduced mass M) d me = Eh a02 k dk M¯

and

M¯ 1 dk = = d me Eh a02 k

And for electrons with M¯ me and = k 2 /2 we obtain most simply in a.u. d/dk = k.

(J.7)

To illustrate this by way of example, we discuss a general radial wave function in the continuum. In a.u. it is asymptotically (see e.g. B URKE, 2006)

Z (J.8) u (r) = b sin kr − π/2 + ln(2kr) + σ + δ . k Here Ze is the charge of a C OULOMB field (if present) which enters into σ = arg Γ ( + 1 − iZ/k) and the slowly varying logarithmic C OULOMB phase. The standard phase shift δ reflects the influence of further, non-Coulombic interactions. For a free particle (Z/k) ln(2kr) as well as σ and δ vanish. To obtain the normalization factor b in k scale normalization, we neglect the logarithmic phase, write δ˜ = − π/2 + σ + δ , and evaluate k+ k k+ k b dk u (r) = i dk exp i k r + δ˜ − c.c. 2 k− k k− k = 2b sin(kr + δ˜ )

sin kr . r

Inserting this into (J.2), and substituting the rapidly oscillating term sin2 (kr + δ˜ ) by its average value 1/2, one may integrate over all r ∞ π sin kr 2 = b2 , 2b dr sin(kr + δ˜ ) sin(kr + δ˜ ) r 2 0

J.2

Plane Waves in 3D

661

∞ √ where the identity 0 sin(|a|r) dr = π2 has been applied, so that b = 2/π . Hence, r normalized in k scale the radial wave function becomes

Z 2 u (r) = (J.9) sin kr − π/2 + ln(2kr) + σ + δ . π k To determine the dimension of uk (r), we write the units in the expression under the integral in (J.2) [udrdk] = 1, or as dimensional equation u2 L1 L−1 = 1 so that u = 1, i.e. uk (r) in k scale normalization is dimensionless. For clarity it might be useful to explicitly write out the a.u., i.e. to replace r → r/a0 and Z/k → Z/(ka0 ) in (J.9), so that the expression becomes independent of the units used. For normalization in scale we use (J.4)–(J.7) and obtain in a.u.:

2 dk Z u (r) = sin kr − π/2 + ln(2kr) + σ + δ π d k

Z 2 or explicitly (J.10) sin kr − π/2 + ln(2kr) + σ + δ = πk k

me Eh /2 1/4 2 Z = sin kr − π/2 + ln(2kr) + σ + δ . πEh a0 M¯ ka0 (J.11) The latter expression is independent of the units used. The prefactor (Eh a0 )−1/2 is again obtained from a dimensional analysis: normalized in scale from (J.2) we have [u2 drd] = 1, so that u2 = Enrg−1 L−1 and u = Enrg−1/2 L−1/2 . Note that for electron scattering the mass factor me /M¯ 1 disappears. Different authors use slightly different notations and energy units (often without mentioning which).2

J.2

Plane Waves in 3D

J.2.1

Expansion in Spherical Harmonics

Without proof we note that a plane wave in 3D can be written as ψ (k) (r) = eik·r = 4π

∞ =0

2 For

i j (kr)

∗ Y m (θk , ϕk )Y m (θr , ϕr ),

(J.13)

m=−

example, expression (J.11) was introduced by C OOPER (1962) to photoionization as uε (r) → π −1/2 ε −1/4 sin ε 1/2 r − π/2 + Zε 1/2 ln 2ε 1/2 r + δ .

Here, ε = 2/Eh , i.e. energies are measured in RYDBERG units = Eh /2, lengths in a0 .

(J.12)

662

J

Continuum

with the polar coordinates (r, θr , ϕr ) representing r, while (k, θk , ϕk ) gives the magnitude and direction of the wave vector k, and j (kr) = u (kr)/(kr) are spherical B ESSEL functions. They are solutions of the radial S CHRÖDINGER equation (2.110) for vanishing potential:

2 ( + 1) 1 d2 u k u (r) = 0. + − (J.14) 2 dr 2 2 2r 2 As one easily verifies, the most simple ones are: j0 (x) = sinc x =

sin x x

and j1 (x) =

sin x cos x . − x x2

(J.15)

All others may in principle be derived from the recursion formula: j +1 (x) =

2 + 1 j (x) − j −1 (x). x

Asymptotically the following relations hold: " x /[(2 + 1)(2 − 1)(2 − 3) · · · ] j (x) = sin(x − π/2)/x

# for x . for x

(J.16)

(J.17)

With the addition theorem (C.22) and the angle γ between k and r one may also write (J.13) as: ψ (k) (r) = eik·r =

∞ (2 + 1)i j (kr)P (cos γ ).

(J.18)

=0

J.2.2

Normalization in Momentum and Energy Scale

The probability density |ψ (k) |2 of the plane waves (J.13) or (J.18) is given per wavenumber to the third power (dimension L3 ) and per volume (L−3 ), so they are overall dimensionless. They may be normalized in k scale (see e.g. OVCHINNIKOV et al., 2004): )(k) (r) = ψ

1 eik·r = r|k . (2π)3/2

(J.19)

To verify this normalization we first integrate over position space using (I.78): )(k)∗ (r)ψ )(k) (r)d3 r a03 δ ka0 − k a0 . k 2 |k 1 k = ψ 2 1 This is the continuum equivalent to the usual orthogonality a|b = δab of bound state wave functions. Since ka0 is dimensionless, the expression is given per wavenumber interval cubed (dimension L3 ) and thus reflects the built in density


663

of states. Finally, with (I.80), integration over k spaces confirms the correct normalization of (J.19): 3 k k k d k = a03 δ ka0 − k a0 d3 k ≡ 1. To obtain 3D normalization in the scale, we note that d3 k = k 2 dkdΩ = k 2

dk ddΩ. d

Hence (J.4) becomes in 3D )(k) (r) = k ψ

dk d

1/2

)(k) (r). ψ

(J.20)

Specifically for conversion into energy scale with (J.6) d3 k =

1 kddΩ, Eh a02

holds, and the 3D plane wave (J.19) normalized in energy scale becomes ka0 ik·r (k) −3/2 ) (r) = (2π) ψ e . a03 Eh

(J.21)

(J.22)

)(k) (r)|2 is Its dimension is now Enrg−1/2 L−3/2 so that the probability density |ψ correctly given per energy and volume.

Acronyms and Terminology a.u.: ‘atomic units’, see Sect. 2.6.2. c.c.: ‘complex conjugate’.

References B ETHE, H. A. and E. E. S ALPETER: 1957. Quantum Mechanis of One- and Two-Electron Atoms. Berlin, Göttingen, Heidelberg: Springer Verlag, 369 pages. 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 OOPER, J. W.: 1962. ‘Photoionization from outer atomic subshells. A model study’. Phys. Rev., 128, 681–693. 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.

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


665

666 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

667 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

668 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

669 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

670 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

671 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

672 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

673 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

674 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

675 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

676 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

677 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

678 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

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


679

680 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

681 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

682 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

683 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

684 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

685 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

686 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

687 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

688 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

689 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

Atomic Molecular Optical Physics by Hertel C Schulz , Volume 1

Recommend Documents