Introduction to the Technique of Electron Spin Resonance (ESR) Spectroscopy

Introduction to the Technique of  Electron Spin Resonance (ESR) Spectroscopy Physics Laboratory Course Dr. Dr. B. Simov Simoviˇ iˇc

April 20, 2004

Abstract Electron spin resonance (esr) spectroscopy deals with the interaction of electromagnetic radiation with the intrinsic magnetic moment of electrons electrons arising from their spin. It has found applications in a wide range of different fields spanning from chemistry and biology to the novel and stimulating areas of quantum computa computation and ”single ”single spin” detection. This laboratory tory course provides provides a basic introduction to esr. It is intended to motiva motivate students students eager eager to learn learn about about experimen experiment tal aspects of spectroscopy.

Chapter 1 Introduction 1.1

What is ESR?

The phenomenon of electron spin resonance spectroscopy can be explained explained by considering the behavior of a free electron. According to quantum theory the electron has a spin which can be understood as an angular momentum momentum leading to a magnetic moment. Consequently, the negative charge that the electron carries carries is also also spinni spinning ng and constitute constitutess a circul circulatin atingg electri electricc curren current. t. The circulating current induces a magnetic moment µS which, if the electron is subjected to a steady magnetic field H0  z, causes the electron to experience a torque tending to align the magnetic moment with the field. The relation between the magnetic moment and the spin vector is µs

=−

gµ B 

S

(1.1)

where µB is the Bohr magneton1 and g is the Land´ e factor.2 The energy of the system depends upon the projection of the spin vector along H0 . Quantum Quantum theory stipulates stipulates that only two values values are permitted for an electron S z = ±/2, which means that the electron magnetic moment can only assume two projections onto the applied field as shown on Fig.1.1 Fig.1.1 Consequently, 1 µz = ± gµ B 2 1 2

 µB = 2em = 9, 2741010−24 Joule.Tesla−1 g = 2.0023 for a free electron

2

(1.2)

H0

µ

E + = ½ gµB/  

¡ 

ω0

E - = - ½ gµB/  

e(a)

(b)

Figure 1.1: (a) Schematic representation of a single electron spin in a steady magnetic field H0 (b) Correspondi Corresponding ng energy-l energy-level evel scheme.

and the ensemble of energy levels therefore reduce to the two values 1 E ± = ± gµ B H 0 2

(1.3)

If electromagnetic radiation is applied at a frequency that corresponds to the separation between the permitted energies equal to ∆E  ∆E  = E + − E − = gµ B H 0 = ω , energy is absorbed from the electromagnetic field. This is the phenomenon of ESR.3 For electrons bound into an atom or a molecule, the phenomenon of ESR may not be observed at all, because electron spins pair off in atomic or molecular orbitals so that virtually no net spin magnetism is exhibited and the material is said to be diamagnetic. When an atom or a molecule has an odd number of electrons, however, complete pairing is clearly not possible and the material is said to be paramagnetic. In that case ESR can be observed. So far we have considered a single electron interacting with an external magneti magneticc field field.. In the presen presentt exper experim imen ent, t, how however, ever, we deal with a macroscopic sample which means a statistical ensemble of magnetic moments. Therefore, we need to consider the relative populations of the energy levels N + and N − , which are given by the Boltzmann distribution: 3

The relation between the frequency of resonance ν  and the ampli amplitude tude H 0 of the magnetic field is : ν  = 1.3996106 (g H 0 )(Hz, Gauss)

3

Mn

2+

ion I=5/2 Iz -5/2 -3/2

Sz =1/2

-1/2 1/2 3/2 5/2

5/2 3/2

Sz = -1/2

1/2 -1/2 -3/2 -5/2

Figure 1.2: Splitting of the ESR line in Mn 2+ owi owing ng to hyperfin hyperfinee inte interaction. raction.

N + ∆E = exp(− exp(− ) (1.4) N − kB T where ∆E  ∆E  = E + − E −, kB Boltzmann’s constant, and T the absolute temperature. perature. Since Since it is the absorptio absorption, n, due to the slightly slightly greater greater populatio population n of the lower level, that is observed, this difference between the two populations tions should should therefore therefore be made made as large as possible possible.. At room temperature temperature N +  N − for a Zeeman splitting corresponding to a frequency of 10 GHz.

1.2 1.2

The The hyperfi yperfine ne inte intera ract ctio ion n

Hyperfine interaction is the interaction between the magnetic moment of an electron with the magnetic moment of the nucleus in its vicinity. Nuclei individually associated with the electron spin system often have a magnetic moment I  which also has different allowed orientations (2I  (2I  + 1) in H0 . The magnetic field associated with the nuclear moment then can add to or subtract from the applied field experienced by the electron spin system associated with with it. In the bulk sample sample some electron electronss will therefore therefore be subject to an 4

NO2

N

N

NO2

NO2

Figure 1.3: DDPH-Molecule(C6 H5 )2 N − NC6H2 (NO2 )3

increase increased d field and some some to a reduce reduce field. field. Con Conseq sequen uently tly,, the original original elecelectron resonance line is split into (2I  (2I + + 1) components. components. For example, example, when the electronic spin of a transition metal or a free radical4interacts with its own nuclear spin the hyperfine interaction is described by the Hamiltonian term H hf hf s = AI.S

(1.5)

with A the coupling constant. constant. The hyperfine coupling constant constant varies with the nuclear species, and it is a measure of the strength of the interaction between between the nu nucle clear ar and electron electronic ic spins. spins. Fig. Fig.1.2 illustrates well the phenomenon: nomenon: the hyperfin hyperfinee intera interacti ction on between between the electro electronic nic spins spins5 and the nuclear spin I=5/2 in the Mn2+ ion splits the resonance line of the 3d electrons into six sub-levels. In molecules, the unpaired electron circulates between several atoms and the resulting hyperfine structure is the result of a Hamiltonian term of the form H hf (1.6) hf s = ΣAi mi where the projection mi of the ith nuclear spin on the magnetic field direction may take on the following 2I  2I i + 1 value values: s: I i , I i − 1, I i − 2, .... ....,, 1 − I i , −I i . For example, the hyperfine interaction with the two equally coupled nitrogen nuclei (I=1) in DPPH molecule (see Fig.1.3 Fig.1.3)) leads to a splitting of the resonance into five components of respective intensity 1:2:3:2:1. 4

A fre freee rad radica icall is an ato atom, m, mol molecu ecule le or ion contain containing ing one unp unpair aired ed electron electron.. By contrast, a transition ion can have several unpaired electrons. 5 The electronic configuration of the free ion Mn 2+ is 3d5 .

5

1.3 1.3

The The dipo dipole le-d -dipo ipole le inte intera ract ctio ion n

For a large concentration of electronic spins, the electronic magnetic moments also interact appreciably with each other, and this can alter considerably the ESR spectra. The interaction interaction is mediated by the dipolar field associated with the magnetic moment µS 

H(r) =

µ0 1 3(µS .r)r ( − µS  + ) 4π r3 r2

(1.7)

Combining equation1.2 equation1.2 and 1.3 1.3,, we see that the energy of dipole-dipole interaction between two adjacent electrons distant of r of  r lays between E dd dd and −E dd dd with µ0 2 2 1 E dd g µB 3 (1.8) dd = 8π r The dipolar interaction induces therefore a broadening of the resonance line, which increases with the concentration of dipole moments.

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Chapter 2 ESR spectrometer We need four essential components to build an ESR spectrometer: • A monochromatic microwave source • A waveguide for guiding the microwave power to the sample • A cavity designed to ensure a proper coupling between the sample and the incoming wave. • A detector for microwave power to detect the response of the sample to microwave irradiation. A schematic drawing of the ESR spectrometer is shown in Fig.2.1 Fig.2.1

2.1 2.1

Micr Micro owave pa part rtss

The different parts used in this experiment are listed below: 1. A gun oscillator is a monochr monochromat omatic ic source source of micro microwa wave ve.. The fundamental frequency is here ν   10GHz. Tuning of the frequency is achieved by slowly turning the screw on the top of the metallic case of the oscillator oscillator.. The frequency frequency can be read out with the frequency frequency counter located next to the source. 2. A calibrated attenuator is use to control the level of microwave power from the source. The scale is logarithmic: 7

Magnet

Gun

WG

Short

WG

A t     t      e n  u  a t      o r  

WG

T

Cavity

Detector

Figure 2.1: ESR spectrometer

P 0 P  with P 0 the output of the microwave source. xdB = 10Log

3. ”A T-hybrid” is the 4-ports device sketched in Fig.2.2 Fig.2.2.. A wave entering from the source at input 3 splits equally into two waves travelling to 1 and 2. The port 4 being orthogonal, no transmission from port 3 to port 4 is allowe allowed. d. Also, Also, no reflectio reflection n occurs at port p ort 3 and 4 owing owing to the presence of the source and the detector.1 Waves are therefore reflect reflected ed only only from ports ports 1 and 2. Let ∆φ ∆φ be the difference between these these two two reflected reflected wave waves. s. If ∆φ ∆φ = 0 then the two recombine in port 3. If ∆φ ∆φ = π , the two two recombin recombinee in port 4. In this experimen experiment, t, the relative phase and amplitude of inputs 1 and 2 can be controlled with an attenuator and a moveable short located on the right arm of the hybrid. 1

Reflections occur because of impedan Reflections impedance ce mismatch mismatch along the path of the wave. wave. How How-ever. ev er. both the det detect ector or and the source source are des design igned ed to ke keep ep the impe impedan dance ce equal to its free-space value at their entrance plane.

8

To Detector £  £ 

£ 

Reflection      

¢  ¢  ¢ 

To moveable short

To cavity

¡  ¡  ¡ 

From source

Figure 2.2: Magic T.

4. The detector is a crystal rectifier (diode) which consists of a semiconducting conducting material. The incident microwav microwavee power causes the current current to flow. flow. The current current I increase increasess with with the microwa microwave ve power power P and the sensitivity of the detection strongly depends on the slope dI/dP which is specific to each diode. 5. The waveguide is a rectangular opened-ended metallic tube delimiting a dielectric media in which electromagnetic waves propagate according to Maxwell equations. equations. Boundary conditions conditions have have to be fulfilled fulfilled by the electrical and magnetic components of the wave on the metallic walls. Consequently the propagation is restricted to a set of modes occurring at well-defined frequencies which are the characteristic values of  the wave equation. There is a cut-off wavelength above which no propagation is allowed and which corresponds for a rectangular waveguide of width a, to λc = a/2. a/2. 6. The cavity is a closed metallic box with an iris to allow the microwave to couple in and out (see Fig.2.1 Fig.2.1)) Any cavity possesses resonant frequencies at which the energy stored reaches large values. These frequencies are related related to the dimensio dimensions ns of the cavity cavity.. The quality quality factor factor Q of a cavity measures the frequency width of the resonance or equivalently 9

f0

Quality factor

Total reflection

Absorption

Frequency

Figure Figu re 2. 2.3: 3: Th Thee mod modee of the ca cavi vity ty used used in our ES ESR R spe spectr ctrom omete eterr can be monitored with a network-analyzer which measures the reflected power versus frequenc frequency y. The vertica verticall scale is logarithm logarithmic. ic. The sharp dip shows shows the absorption of part of the incident power by the cavity. The amplitude of the dip quantifies quantifies the amount of absorbed power by the cavity. cavity. Note that both the amplitude of the dip and the frequency at which the mode occurs change when a sample is introduced in the cavity.

its selectivity. It is defined like Q = ω0

Energy stored Energy loss

(2.1)

Q-values in general are of the order of magnitude of the volume-tosurface ratio of the resonator, divided by the skin depth in the conductor ductor at the frequenc frequency y of resonanc resonance. e. Figure Figure 2.3 shows the detailed characteristics of the rectangular cavity used in the present experiment.

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2.2 2.2

Elec Electr tron onic ic equi equipm pmen ents ts

1. An oscilloscope with its X and Y channels. 2. A plotter also with its X and Y channels. 3. The pre-amplifier amplifi amplifies es in two two successi successive ve steps: steps: a first first dc amplification and an ac amplification through a 115 Hz-bandpass filter. Beware of not saturating the dc stage of amplification. 4. The GUN power supply delivers 15V. 5. A Hall-probe measures the static magnetic field. As indicated by its name, the principle of operation is based on the Hall-effect. The probe should therefore be positioned vertically relative to the magnetic field in order to get the maximum sensitivit sensitivity y. Be careful when manipulating manipulating the probe. It is indeed fragile and expensive. 6. The gaussmeter converts the voltage measured from the Hall-probe into a value of magnetic field (the maximum deviation corresponds to 1V). 7. The two sets of coils I and II generate the static magnetic field H0 and the small modulating field Hm respectively. 8. The magnet power supply supplies the current to the pair of coils I which produces the static field. It is controlled by the ramp-generator. The current is extremely stable in order to avoid spurious noise that could interfere with the measurement. 9. The low frequenc frequency y oscilla oscillator tor creates a sinusoidal current at the frequency of 115 Hz in the second pair of coil (II). This produces the field Hm and provides the external reference signal for the lock-in amplifier 10. The ramp-generator produces a ramp of magnetic field by varying continuously the current in the pair of coils I. The voltage output of the ramp can be connected to the X-channel of the plotter if the variation 11

of static fields are too small to be accurately read from the gaussmeter. For larger amplitude of change, the analogue output of the gaussmeter can be converted to a digital signal and sent to the X-channel of the plotter. 11. The lock-in amplifier amplifies signals at frequencies close to the frequenc frequency y of a referenc referencee signal signal.. More details details concerni concerning ng the princi principle ple of operation operation of a lock-in lock-in are given given in the follow following ing section. section. Here Here we emphasize emphasize that the reference signal should come from the low frequency oscilla oscillator. tor. Be careful careful to connect connect the 115Hz 115Hz signal signal to the proper input of the lock-in amplifier.

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Chapter 3 Detection scheme 3.1

ESR signal

In a resonance experiment, the phenomenon of absorption of electromagnetic energy by the sample is generally translated into a variation of the complex impedance Z  = R+ jX  of an oscillating oscillating circuit. When the sample experiences a time-varying field, the absorptive component of the susceptibility to this field changes the resistivity R while the dispersive components changes the reactance X .1 ♦ For a complex susceptibility  χ sketch the variation through resonance of  χ and  χ and also δR and  δX  as a function of frequency. 



The sensitivity of a given detection scheme to absorption or dispersion can thus be defined as the change of the measured quantity - a voltage, a current or a power - arriving at the detector for given values of  δR and δX . δX . In the ESR spectrometer designed in Fig.2.1 Fig.2.1,, we detect variations in the power pow er reflected reflected by the cavit cavity y. In other words, words, we work with a reflecti reflectiononcavity spectrometer. Hence we need to consider the relationship between the impedance defined as the entrance plane of the cavity Z c and the reflection coefficient Γ: Z c − Z 0 Γ= (3.1) Z c + Z 0 1

We recall here that the absorption (dispersion )is related to the imaginary (real) part of the complex susceptibility and changes in phase quadrature (in phase) with the timevarying field.

13

ω

Γ  Γ 

b δΓ 

0

a Γ  Γ0 

a’ Absorption aa’ Dispersion ba’

Figure 3.1: This construction shows how the absorption and dispersion components of the signal are connected to the variations of the reflection coefficient Γ. with Z 0 the characte characteris ristic tic impedance impedance of the wave. wave. At resonance resonance,, Γ = 0 for a cavity on tune (X=0) and matched with the waveguide (Z  (Z c = Z 0 ). Far off  from resonance the cavity is equivalent to a short, nothing goes through the iris, and consequently Γ ≈ 1. Cha Changes nges in the reflection reflection coefficient coefficient are given by the vector ∂ Γ ∂ Γ δΓ = δR + δX (3.2) ∂R ∂X  and the ESR signal (S) is therefore defined as the scalar product S  = Γ0 .δ Γ

(3.3)

This means that the extent to which the absorption and dispersion components affect the modulus mo dulus |Γ| depends on the phase of changes in δ Γ expressed relative to Γ. This can be readily seen in Fig.3.1 Fig.3.1.. ♦ Based on Fig.3.1 Fig.3.1 and what has been said above explain how to detect  the dispersive component (ba’) of the signal.

3.2 3.2

Fiel Field d modu modula lati tion on

When the magnetic field is scanned through the region of resonance, the spin system in the resonant cavity absorbs a small amount of energy from 14

the microwave magnetic field H 1 , and produces a slight change in the resonant frequency of the microwave cavity. The DC detection of ESR is severely limited, however, by the drift of the amplifier and the 1/f noise. For these reasons, most ESR spectrometers incorporate magnetic field modulation which transfers the relevant signal from DC to AC. The principle is the following. When the magnetic field is modulated at the angular frequency ωm , an alternating alternating field 1 H m sin ωm t (3.4) 2 superimposed on the constant magnetic field H 0 + H δ . H δ is the local field induced induced by the surroun surroundin dingg of the conside considered red electron. electron. It is therefor thereforee H δ which determines the broadening of the ESR line. The ”constant” magnetic field is normally swept over the range ∆H  ∆H 0 from (H 0 − 12 ∆H 0 ) to (H  (H 0 + 12 ∆H 0 ) in a time t0 where H 0 is the magnetic field strength at the center of the scan. At any time t during the scan, the instantaneous magnetic field strength H  is given by H  = H 0 + H δ + H mod ∆ H 0 ( mod = H 0 + ∆H 

t 1 1 − ) + H m sin ωm t t0 2 2

(3.5)

where

t 1 − ) (3.6) t0 2 Consequently, the signal at the input of the detector is sinusoidal with the frequency ωm and with an amplitude proportional to the derivative ∂s/∂H . Note that in order to consider the field (H  (H 0 + H δ ) as constant, constant, the scan should be slow enough so that there are many cycles of the modulation frequency during the passage between the peak-to-peak (or half amplitude) points of  the resonance line. H δ = ∆H 0 (

♦ Sketch the effect of field modulation on the ESR line.

3.3 3.3

Prin Princi cipe pe of a phase phase-s -sen ensi siti tiv ve detec detecti tion on

A lock-in detector compares the ESR signal from the detector with a reference signal and only passes the components of the former that have the proper frequency and phase. If the reference voltage comes from the same oscillator 15

that produces the field modulation voltage, the ESR signal passes through while noise is suppressed. suppressed. Thus a lock-in detector only accepts signals signals that ”lock ” to the reference reference signal. signal. Hence the name of ”phase-sensiti phase-sensitive ve detector”. detector”. The operation of a lock-in detector is simple. A reference oscillator produces a reference signal vr vr = V r cos(ωt cos(ωt + φ) (3.7) where φ is a phase angle. At resonance the sample absorbs microwa microwave ve energy and produces the ESR signal voltage es es = E s cos(ωt cos(ωt + φs )

(3.8)

where φs is another phase angle that may be close to φ. Th Thee mult multip ipli lier er produces an output that is the product es vr of the ESR signal and modulated voltages 1 es vr = E s V r cos(ωt cos(ωt + φs )cos(ωt )cos(ωt + φ) = E s V r [cos(2ωt [cos(2ωt + φ + φs )+cos(φ )+cos(φ − φs )] 2 (3.9) The low pass filter removes the first term to produce the dc output voltage V out out 1 V out = E s V r cos(φ cos(φ − φs ) (3.10) out 2 and we see immediately that the sensitivity is maximal if  φ is set equal to φs , which gives 1 (3.11) V out out = 2 E s V r When the electron spin resonance signal is measured by the lock-in amplifier plifier it still still contains contains a consider considerabl ablee amount amount of noise. noise. A large part of this noise noise may may be remove removed d by passing passing the signal signal through through a low-p low-pass ass filter. filter. The filter has associated with it a time constant or response time τ 0 , which is a measure measure of the cutoff frequency frequency of the filter. Another Another way way to say say this is that the filter fails to pass frequencies that are much greater than the inverse of the time constant 1/τ  1/τ 0 ; it attenuates, distorts, and retards those incoming waveforms that have frequencies in the vicinity of  τ 0, and it transmits undisturbed those frequencies considerably below 1/τ  1/τ 0 . The wave waveform form that that is impressed on the ESR response filter may be considered as the derivative of the absorption (or dispersion) line, and for comparison with the above 16

criteria, its effective frequency may be taken as the inverse of the time that it takes takes to scan through through the resonance resonance from one peak to the next. next. In other words, if the time that it takes to scan through the magnetic field range is very short compared to the time constant τ 0 , then no signal will appear on the recorder; if this time equals the time constant, then a distorted signal will result; while if one waits many time constants to complete the scan, then the recorder will faithfully reproduce the true lineshape.

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Chapter 4 Experiment The following questions are of course suggestions and should not substitute the initi initiati ative ve of the experi experime ment ntal alis ist. t. Th Thee cabli cabling ng of the the spectr spectrome ometer ter is described in Fig.4.1 Fig.4.1.. 1. Read the manual and by doing so try to answer each of the exercises marked with the symbol ♦ 2. Measure the magnetic field as a function of the current in the magnet for large and small, slow and fast field changes. Observe the hysteresis. 3. Measure the amplitude of the modulation of the field as a function of  the modulating current. 4. Measure the ESR signal on a powder sample of DPPH in absorption and dispersion. Estimate the linewidth and the g-factor. 5. Measure the ESR signal of DPPH in solution for different concentration. Estimate Estimate the dipole-dipole dipole-dipole coupling constant. constant. From that estimate estimate the range of concentration where the dipolar interaction becomes stronger than the hyperfine interaction and compare with experiments. 6. Repeat 6 and 7 for Mn2+ in solution. 7. Observe and discuss the effect of varying the amplitude of the modulating field on different ESR lines.

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POWER SUPPLY

Magnet

Gun

WG

WG

short

A t     t      e n  u  a t      o r  

WG

T

Cavity

AC

DC or AC

AMPLIFICATOR AC

Detector

DC

y

GAUSSMETER

POWER SUPPLY

OSCILLATOR

ω m

Scope x RAMP GENERATOR Plotter

Y LOCK-IN

x

Figure 4.1: Schem Schematic atic representation representation of the cabling of the spectrometer. Note that both DC and AC dete detectio ction n sc schem hemes es are represente represented. d. The dashed lines show the path of the AC signal.

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4.1 4.1

Bibl Biblio iogr grap aph hy

All the books listed below are in principle available at the library of the physics department at ETH. 1. H. A. Atwater, Introduction to microwave theory , McGraw-Hill (1962) 2. C. P. Poole.Jr, Electron Spin Resonance, Resonance, Interscience Publishers (1967) 3. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Ions, Clarendon Press . Oxford (1970) 4. D. J. E. Ingram, Radio and Microwave Spectroscopy , Butterworth & Compagny (1976) 5. C. P. Poole.Jr. and H. A. Farach, Theory of Magnetic Resonance, Resonance, 2nd edition, edition, Interscience Interscience Publishers (1987) 6. C. P. Slichter, Principle of Magnetic Resonance, Resonance, 3rd edition, Springer Verlag (1996)

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Contents 1 Intr Introdu oduct ctio ion n 1.11 Wh 1. What at is ES ESR? R? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The hy hyperfin perfinee in intera teracti ction on . . . . . . . . . . . . . . . . . . . . . 1.3 The dipo dipolele-dipol dipolee int interac eraction tion . . . . . . . . . . . . . . . . . . .

2 2 4 6

2 ESR ESR spec spectr trom omet eter er 2.1 Micr Microw owav avee part partss . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Elec Electron tronic ic equ equipm ipmen ents ts . . . . . . . . . . . . . . . . . . . . . .

7 7 11

3 Detect Detection ion sc schem heme e 13 3.11 ES 3. ESR R si sign gnal al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.22 Fi 3. Fiel eld d mod modul ulati ation on . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Prin Principe cipe of a phase-s phase-sensi ensitiv tivee detecti detection on . . . . . . . . . . . . . 15 4 Exper Experim imen entt 18 4.11 Bi 4. Bibl blio iogra graph phy y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

21

List of Figures 1.11 1.

(a) Schem Schemati aticc rep repres resen entat tatio ion n of a si singl nglee el elect ectron ron spin spin in a steady magnetic field H0 (b) Corresponding energy-level scheme. 3 1.2 Spl Splitti itting ng of of the the ESR ESR line line in Mn2+ owing to hyperfine interaction. 4 1.3 DDP DDPH-M H-Molec olecule ule(C (C6 H5 )2 N − NC6H2 (NO2 )3 . . . . . . . . . . 5 2.1 2.1 2.22 2. 2.33 2.

ESR spe spectr ctrom omete eterr . . . . . . . . . . . . . . . . . . . . . . . . . 8 Magi Ma gicc T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The mode mode of the cavit cavity y used in our ESR spectro spectromet meter er can be monitored with a network-analyzer which measures the reflected power versus frequency. frequency. The vertical scale is logarit logarithhmic. The sharp dip shows the absorption absorption of part of the incident power by the cavity cavity. The amplitude of the dip quantifies quantifies the amountt of absorbed power by the cavity amoun cavity. Note that both the amplitude of the dip and the frequency at which the mode occurs change when a sample is introduced in the cavity. . . . 10

3.1 This This constructi construction on shows shows how the absorptio absorption n and dis dispersi persion on components of the signal are connected to the variations of  the reflection coefficient Γ. . . . . . . . . . . . . . . . . . . .

14

4.1 Schema Schematic tic represen representati tation on of the cabl cabling ing of the spectromete spectrometer. r. Note that both b oth DC and AC detection detection schemes are represented. represented. The dashed lines show the path of the AC signal. . . . . . . .

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22