PULSEDEPRPRIMER

Dr. Ralph T. WeberBruker Instruments, Inc.Billerica, MA USA

Pulsed EPR Primer

Copyright © 2000 Bruker Instruments, Inc.

The text, figures, and programs have been worked out with the utmost care. However, we cannot accept either legal responsibility or any liability for any incorrect statements which may remain, and their consequences. The following publication is protected by copyright. All rights reserved. No part of this publication may be reproduced in any form by photocopy, microfilm or other proce-dures or transmitted in a usable language for machines, in particular data processing systems with-out our written authorization. The rights of reproduction through lectures, radio and television are also reserved. The software and hardware descriptions referred in this manual are in many cases registered trademarks and as such are subject to legal requirements.

PrefaceI gratefully acknowledge the many useful comments and discus-sions with Professor Gareth Eaton of the University of Denverand Dr. Peter Hoefer of Bruker Analytik.

This document is best viewed with Adobe AcrobatTM 4.0. If youare using a PC running Microsoft WindowsTM, you can alsoview the several animations that are interspersed throughout thetext. Simply click the movie poster with the blue frame and themovie will play in a floating window. To close the floating win-dow, place your cursor in the frame and press the <Esc> key.

A Movie Poster

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Table of Contents

1 Pulse EPR Theory..................................................................31.1 The Rotating Frame..................................................................................... 3

Magnetization in the Lab Frame ................................................................... 4Magnetization in the Rotating Frame ........................................................... 6Viewing the Magnetization from Both Frames: The FID ........................... 10Off-Resonance Effects ................................................................................ 11

1.2 Relaxation Times....................................................................................... 15Spin Lattice Relaxation Time ..................................................................... 15Transverse Relaxation Time ....................................................................... 19

1.3 A Few Fourier Facts .................................................................................. 21The Fourier Transform ................................................................................ 24Fourier Transform Pairs .............................................................................. 24Fourier Transform Properties ...................................................................... 26The Convolution Theorem .......................................................................... 28A Practical Example ................................................................................... 29

1.4 Field Sweeps vs. Frequency Spectra ......................................................... 331.5 Multiple Pulses = Echoes .......................................................................... 34

How Echoes Occur ..................................................................................... 34ESEEM ....................................................................................................... 37Stimulated Echoes ....................................................................................... 38Pulse Lengths and Bandwidths ................................................................... 39

2 Pulse EPR Practice..............................................................412.1 The Pulse EPR Bridge ............................................................................... 42

Excitation .................................................................................................... 43Detection .................................................................................................... 44

2.2 The Pulse Programmer...............................................................................482.3 Data Acquisition ........................................................................................50

Point Digitizers ............................................................................................51Integrators ....................................................................................................52Transient Recorders .....................................................................................55Aliasing .......................................................................................................56Dynamic Range ...........................................................................................58Signal Averaging .........................................................................................59

2.4 Resonators..................................................................................................612.5 Phase Cycling.............................................................................................63

4 Step Phase Cycle ......................................................................................63Unwanted Echoes & FIDs ...........................................................................66

3 Bibliography......................................................................... 673.1 NMR ..........................................................................................................673.2 EPR ............................................................................................................683.3 Pulsed ENDOR ..........................................................................................71

vi

Pulsed EPR PrimerThis primer is an introduction to the basic theory and practice ofPulse EPR spectroscopy. It gives you sufficient background tounderstand the basic concepts. In addition, we strongly encour-age the new user to explore some of the texts and articles at theend of this primer. You can then fully benefit from your particu-lar pulse EPR application or think of new ones.

A common analogy for describing CW (Continuous Wave) andFT (Fourier Transform) techniques is in terms of tuning a bell.We are assigned the task of measuring the frequency spectrum ofthe bell. In one scheme for tuning the bell, we use a frequencygenerator and amplifier to drive the bell at one specific fre-quency. In order to obtain a frequency spectrum of the bell, weslowly sweep the frequency in order to detect any acoustic reso-nances in the bell. We essentially perform a similar experimentin CW EPR: the field is slowly swept and we detect any reso-nances in the sample. This does not seem like the best means fortuning because we know from everyday experience that if westrike a bell with a hammer, it will ring (i.e. resonate acousticallyat multiple frequencies). So an alternative approach is to strikethe bell, digitize the resultant sound, and Fourier transform thedigitized signal to obtain a frequency spectrum. Only one shortexperiment is required to obtain the frequency spectrum of thebell. This fact is often called the multiplex advantage. InFT-EPR, we apply a short but very intense microwave pulse(analogous to a hammer strike) and digitize the signals comingfrom the sample. After Fourier transformation, we obtain ourEPR spectrum in the frequency domain.

EPR has traditionally been a CW (Continuous Wave) spectros-copy. The NMR spectroscopist enjoyed substantial gains in sen-sitivity with a correspondingly drastic reduction in measurementtime by moving to a pulse FT technique because they have alarge number of very narrow lines spread over a wide (comparedto the linewidth) frequency range. In most cases, the EPR spec-

troscopist is unable to enjoy these sensitivity improvementsbecause EPR spectra are usually broad and not as numerous.Why would EPR spectroscopists wish to switch to a pulse meth-odology without the promise of increased sensitivity? NMRspectroscopists soon discovered by measuring in the timedomain and using multi-dimensional techniques, they were ableto extract much more information than they ever could possiblyimagine. We can enjoy these same advantages in EPR as well.

Perhaps one of the most common pulse EPR applications isESEEM (Electron Spin Echo Envelope Modulation) in whichyou obtain information regarding interactions of the electronspin with the surrounding nuclei. Interpretation of the data yieldsimportant structural information, particularly for large metallo-proteins for which no single crystals are available for X-ray dif-fraction and the molecules are too large to perform highresolution NMR experiments.

Pulse experiments measure relaxation times more directly thanCW techniques such as saturation. The relaxation time measure-ments offer you dynamical as well as distance information forthe samples you are studying.

As interest in measuring longer distances between paramagneticcenters increases, the techniques of 2 plus 1, DEER (DoubleElectron Electron Resonance), and ELDOR (ELectron DOubleResonance) are invaluable in measuring particularly long dis-tances in very large molecules.

Quite often there are events that take place on time-scales thatdo not influence the relaxation times and hence the lineshapes.EXSY (EXchange SpectroscopY) measures rates for slow inter-and intra-molecular chemical exchange, homogeneous electrontransfer, and molecular motions.

2

The Rotating Frame

Pulse EPR Theory 1Though Pulse EPR may seem a bit daunting in the beginning,there are a few simple principles that help you understand pulseEPR experiments. The first important principle to master is therotating frame. Since Pulse EPR involves going between thetime and frequency domains, we shall also discuss some of theimportant relations in Fourier theory. You will find that we willoften use these simple principles throughout the coming chap-ters. The treatment is not mathematical, but intended to give youan intuitive understanding of the phenomena.

The Rotating Frame 1.1The magnetization of your sample can often undergo very com-plicated motions. A useful technique, widely used in both CWand FT EPR and NMR, is to go to a rotating coordinate system,referred to as the rotating frame. From this alternative point ofview, much of the mathematics is simplified and an intuitiveunderstanding of the complicated motions can be gained.

A simple analogy for the rotating frame involves a carousel andtwo people trying to have a conversation. One person is ridingon the carousel and the other person is standing still on theground. Because the carousel is moving, the two people will beable to speak to each other only once per revolution and nomeaningful conversation is possible. If, however, the person onthe ground walks at the same speed as the carousel is rotating,the two people are next to each other continuously and they cancarry on a meaningful conversation because they are stationaryin the rotating frame.

The presentation is based on classical mechanics; the classicalpicture is often clearer and more productive than the quantummechanical picture. Even though the phenomenon on a micro-scopic level is best described by quantum mechanics, we are

Pulse EPR Theory 3

Magnetization in the Lab Frame

measuring a bulk property of the sample, namely the magnetiza-tion, which is nicely described from a classical point of view.

Magnetization inthe Lab Frame

In order to describe a physical phenomenon, we need to estab-lish an axis system or reference frame. The reference framewhich most people are familiar with is the lab frame which con-sists of three stationary mutually perpendicular axes. The labframe in EPR is usually defined as in Figure 1. The magneticfield, B0 is parallel to the z axis, the microwave magnetic field,B1, is parallel to the x axis, and the y axis is orthogonal to the xand z axes. All discussions of the electronic magnetization inthis section will be described in this axis system.

When an electron spin is placed in a magnetic field, a torque isexerted on the electron spin, causing its magnetic moment toprecess about the magnetic field just as a gyroscope precesses ina gravitational field. The angular frequency of the precession iscommonly called the Larmor frequency and it is related to themagnetic field by

ωL = -γ B0 , [1]

where ωL is the Larmor frequency, γ is the constant of propor-tionality called the gyromagnetic ratio, and B0 is the magnetic

Figure 1 Definition of the lab axis system.

4

The Rotating Frame

field. The sense of rotation and frequency depend on the value ofγ and B0. A free electron has a γ/2π value of approximately-2.8 MHz/Gauss, resulting in a Larmor frequency of about 9.75GHz at a field of 3480 Gauss. The Larmor frequency corre-sponds to the EPR frequency at that magnetic field.

Let us consider a large number of electron spins in a magneticfield, B0, aligned along the z axis. (See Figure 2.) The electronspins are characterized by two quantum mechanical states, onewith its magnetic moment parallel to B0 and one antiparallel.The parallel state has lower energy and at thermal equilibrium,there is a surplus of electron spins in the parallel state accordingto the Boltzmann distribution. Therefore, there should be a netmagnetization parallel to the z axis. (The magnetization is thevector sum of all the magnetic moments in the sample.) Theelectron spins are still precessing about the z axis, however theirorientations are random in the x-y plane as there is no reason toprefer one direction over another. For a very large number ofelectron spins, the various transverse (i.e. in the x-y plane) com-ponents of the magnetic moments cancel each other out. Theresult is a stationary magnetization, M0, aligned along B0.

Figure 2 The Larmor precession and the resultant station-ary magnetization.

Pulse EPR Theory 5

Magnetization in the Rotating Frame

Magnetization inthe Rotating

Frame

EPR experiments are usually performed with a resonator usinglinearly polarized microwaves. The microwave resonator isdesigned to produce a microwave field, B1, perpendicular to theapplied magnetic field, B0. In most cases, |B1| << |B0|.

Linearly polarized microwaves can be thought of as a magneticfield oscillating at the microwave frequency. (See the upperseries of Figure 3.) An alternative way of looking at linearlypolarized microwaves which is more useful when using therotating frame is shown in the lower series of Figure 3. The sumof two magnetic fields rotating in opposite directions at themicrowave frequency will produce a field equivalent to the lin-early polarized microwaves. As we shall see, only one of therotating components is important in describing the FT-EPRexperiment.

Alas, the effect that B1 has on the magnetization is very difficultto envision when everything is moving simultaneously as in thefirst picture in Figure 4. To avoid vertigo, we can observe whatis happening from a rotating coordinate system in which we

Figure 3 Linearly polarized microwaves represented as two circularly polarizedcomponents.

6

The Rotating Frame

rotate synchronously with one of the rotating B1 components.We shall assume that we are at resonance, i.e.

ωL = ω0, [2]

where ω0 is the microwave frequency. By rotating the coordinatesystem at an angular velocity of ω0, we can make one of thecomponents of B1 to appear stationary. (See second picture ofFigure 4.) The other component will appear to be rotating at anangular velocity of 2ω0 and can be neglected. (The reasons forneglecting the fast component is based on effective fields andwill be covered later in this chapter.) The rotating frame alsomakes the magnetization components precessing at the Larmorfrequency to appear stationary. Using Equation [1] and assumingthe magnetization is not precessing in the rotating frame (ω = 0),the field B0 disappears in the rotating frame. In the rotatingframe, we need only to concern ourselves with a stationaryB1and M0.

Figure 4 The microwave magnetic field in both reference frames.

Pulse EPR Theory 7

Magnetization in the Rotating Frame

We have already looked at the interaction of a static magneticfield with the magnetization; the magnetization will precessabout B1 at a frequency,

ω1 = - γ B1 , [3]

where ω1 is also called the Rabi frequency. Let us assume thatB1 is parallel to the x axis. The magnetic field will rotate themagnetization about the +x axis as long as the microwaves areapplied. (See Figure 5.)

The angle by which M0 is rotated, commonly called the tipangle, is equal to,

α = - γ |B1| tp, [4]

where tp is the length of the pulse. Pulses are often labeled bytheir tip angle, i.e. a π/2 pulse corresponds to a rotation of M0 byπ/2. The most commonly used tip angles are π/2 and π (90 and180 degrees). The tip angle is dependent on both the magnitudeof B1 and the length of the pulse. For example, a B1 of 10 Gausscan often be obtained, resulting in a π/2 pulse length of approxi-mately 9 ns. The effect of a π/2 pulse is shown in Figure 6; itresults in a stationary magnetization along the -y axis. If we

Figure 5 Rotating the magnetization.

For a given tip angle,as B1 gets larger, thepulse leng th getsshorter.

8

The Rotating Frame

were to make the pulse twice as long, we would have a π pulseand the magnetization would be rotated to the -z axis.

Because B1 is parallel to +x it is known as a +x pulse. If we wereto shift the phase of the microwaves by 90 degrees, B1 wouldthen lie along the +y axis and the magnetization would end upalong the +x axis. Microwave pulses are therefore labeled notonly by their tip angle but also by the axis to which B1 is paral-lel.

Figure 6 The effect of a π/2 pulse.

α α = π/2

Figure 7 Four different pulse phases.

Pulse EPR Theory 9

Viewing the Magnetization from Both Frames: The FID

Viewing theMagnetization

from Both Frames:The FID

In the introduction, it was mentioned that the sample emittedmicrowaves after the intense microwave pulse. How this hap-pens is not completely clear if viewed from the rotating frame. Ifviewed from the lab frame, the picture is much clearer. The sta-tionary magnetization along -y then becomes a magnetizationrotating in the x-y plane at the Larmor frequency. This generatescurrents and voltages in the resonator just like a generator. (SeeFigure 8 and Figure 9.) The signal will be maximized for themagnetization exactly in the x-y plane. This microwave signalgenerated in the resonator is called a FID (Free InductionDecay).

Figure 8 Generation of a FID.

Figure 9 Rotation of the magnetization acting like a gen-erator.

A π/2 pulse maxi-mizes the magnetiza-tion in the x-y planeand therefore maxi-mizes the signal.

ω

10

The Rotating Frame

Off-ResonanceEffects

So far we have been dealing with exact resonance conditions,i.e. the Larmor frequency is exactly equal to the microwave fre-quency. EPR spectra contain many different frequencies so notall parts of the EPR spectrum can be exactly on-resonancesimultaneously. Therefore, we need to consider what happens tothe magnetization when we are off-resonance.

First, we shall look at the rotating frame behavior of transversemagnetization having a frequency ω following a π/2 pulse. Ini-tially the magnetization will be along the -y axis, however,because ω ≠ ω0, the magnetization will appear to rotate in thex-y plane. This means that the magnetization either is rotatingfaster or slower than the microwave magnetic field, B1. Therotation rate will be equal to the frequency difference:

∆ω = ω − ω0 . [5]

In the case of ∆ω = 0, the rotation rate is zero (i.e. stationary),which is precisely what we would expect for a system exactlyon-resonance. If ∆ω > 0 the magnetization is gaining and willrotate in a counter-clockwise fashion. Conversely, if ∆ω < 0 themagnetization is lagging and will rotate in a clockwise fashion.

Figure 10 The magnetization in the rotating frame exactlyon-resonance and ∆ω off-resonance.

On-resonance Off-resonance

Pulse EPR Theory 11

Off-Resonance Effects

This frequency behavior gives us a clue as to how the EPR spec-trum is encoded in the FID. The individual frequency compo-nents of the EPR spectrum will appear as magnetizationcomponents rotating in the x-y plane at the corresponding fre-quency, ∆ω. If we could measure the transverse magnetization inthe rotating frame, we could extract all the frequency compo-nents and hence reconstruct the EPR spectrum.

A second consequence of not being exactly on-resonance is thatthe microwave magnetic field B1 actually tips the magnetizationinto the x-y plane differently because B0 does not disappearwhen we are not on-resonance. We determined that B0 disap-pears in the rotating frame when we were on-resonance becauseour magnetization is no longer precessing. When we are off-res-onance, the magnetization is precessing at ∆ω and therefore:

[6]

Figure 11 The effective microwave magnetic field in therotating frame.

Quadrature detec-tion (to be discussedin the Detection sec-t i o n o n pa g epage 44) is a meansfor measuring bothtransverse magneti-zation componentsin the rotating frame.This g ives us therequired amplitudeand phase informa-tion to transform thesignals into a fre-quency representa-tion.

B0∆ω

γ–--------=

12

The Rotating Frame

in the rotating frame. Now the magnetization is not tipped by B1but by the vector sum of B1 and B0, which is called Beff or theeffective magnetic field. The magnetization is then tipped aboutBeff at the faster effective rate ωeff:

[7]

Another consequence is that we cannot tip the magnetizationinto the x-y plane as efficiently because Beff does not lie in thex-y plane as B1 does. The magnetization does not move in an arcas it does on-resonance, but instead its motion defines a cone. Infact, it can be shown that the magnetization that can be tipped inthe x-y plane exhibits an oscillatory and decreasing behavior as|∆ω| gets larger:

[8]

Figure 12 The transverse magnetization as a function ofthe offset after a π/2 pulse.

ωeff ω12 ω∆ 2+=

The tip angle is thena function of the off-set ∆ω. The π/2 tipangle is only strictlyvalid exactly on res-onance.

M y– M01

1 ω∆ω1-------

2+

----------------------------- 1 ω∆ω1-------

2+ π

2---⋅

sin⋅ ⋅=

Pulse EPR Theory 13

Off-Resonance Effects

One thing is evident from Figure 12, if we have a very broadEPR spectrum (∆ω > ω1), we will not be able to tip all the mag-netization into the x-y plane to create an FID. This is why it isimportant to maximize ω1 (or equivalently to minimize the π/2pulse length) for broad EPR signals. As B1 gets larger (and thepulse lengths get shorter), we can successfully detect more ofour EPR spectrum. (See Figure 13.)

Figure 13 The effect of pulse length on an FT-EPR spec-trum of the perinaphthenyl radical.

A h a nd y ru l e o fthumb is that the sig-na l i n t e ns i t y a t∆ω = ω1 will be afactor of two smallerthan when ∆ω = 0.

14

Relaxation Times

Relaxation Times 1.2So far our description is a bit unrealistic because when we tippedthe magnetization into the x-y plane, it remained there with thesame magnitude. Because the electron spins interact with theirsurroundings, the magnetization in the x-y plane will decayaway and eventually the magnetization will once more return toalignment with the z axis. This process is called relaxation and ischaracterized by two constants, T1 and T2. The spin lattice relax-ation time, T1, describes how quickly the magnetization returnsto alignment with the z axis. The transverse relaxation time, T2describes how quickly the magnetization in the x-y plane (i.e.transverse magnetization) disappears.

Spin LatticeRelaxation Time

We have already seen that electron spins in a magnetic field arecharacterized by two quantum mechanical states, one with themagnetic moment parallel and the other state with the magneticmoment anti-parallel to the magnetic field. The moments will berandomly distributed between parallel and anti-parallel withslightly more in the lower energy parallel state because the elec-tronic system obeys Boltzmann statistics when it is in thermalequilibrium. Then, the ratio of populations of the two states isequal to:

, [9]

where n represents the populations of the two states, ∆E is theenergy difference between the two states, k is Boltzmann’s con-stant and T is the temperature.

The magnetization that we have been discussing so far is actu-ally the vector sum of all the magnetic moments in the sample.Since the moments can only be either parallel or anti-parallel,the magnetization is simply proportional to the difference,

nanti parallel–nparallel

------------------------------- eE∆

kT-------–

=

Pulse EPR Theory 15

Spin Lattice Relaxation Time

nparallel - nanti-parallel and will be aligned along the z axis. To getan idea of the size of the population differences, if we are work-ing at X-band (~ 9.8 GHz) at room temperature (300 K) with asample with 10,000 spins, on average 5,004 spins would be par-allel and 4996 spins would be anti-parallel resulting in a popula-tion difference of only 8. At room temperature and X-band, weare dealing with a small population difference between the twostates.

When we apply a π/2 pulse to our sample, we no longer havethermal equilibrium. How does this happen? When B1 rotatesthe magnetization into the x-y plane, the magnetization along thez axis goes to zero, i.e. the population difference goes to zero.(See Figure 14.) If we were to use Equation [9] to estimate thetemperature of our spins, we would obtain T = ∞. Our spin sys-tem is obviously not in thermal equilibrium and through itsinteractions with the surroundings, it will eventually return tothermal equilibrium. This process is called spin-lattice relax-ation.

Technically speak-ing, temperature isno t d e f in ed i n anon -e qu i l i b r i umcondition, so nega-t i ve and i n f i n i t e“temperatures” donot violate any ther-modynamic laws.

Figure 14 Populations before and after π/2 and π pulses.

16

Relaxation Times

We could go even one step further and apply a π pulse. This willactually rotate the magnetization anti-parallel to the z-axis, cor-responding to more magnetic moments aligned along the -z axis.(This is why a π pulse is often referred to an inversion pulse.) Ifwe use Equation [9], we actually calculate a negative tempera-ture.

The rate constant at which Mz recovers to thermal equilibrium isT1, the spin-lattice relaxation time. The magnetization willexhibit the following behavior after a π/2 pulse:

[10]

or after a π pulse:

. [11]

Mz t( ) M0 1 et

T1-----–

–⋅=

Mz t( ) M0 1 2 e⋅t

T1-----–

–⋅=

Figure 15 Recovery of the magnetization after a microwave pulse.

Pulse EPR Theory 17

Spin Lattice Relaxation Time

In order to extract our signals from the noise, we must signalaverage the FID by repeating the experiment as quickly as possi-ble and adding up the individual signals. What does “as quicklyas possible” mean? We must wait until the magnetization alongthe z axis has recovered, because if there is no z magnetization,you cannot tip it into the x-y plane to create a FID. The first FIDwill be maximum and the following FIDs will eventuallyapproach a limit value that is smaller than the initial value. (SeeFigure 16.)

The limit value as a function of T1 and SRT (the Shot RepetitionTime, which is the time between individual experiments) isequal to:

. [12]

One important fact is that if SRT = 5 x T1, 99% of the magneti-zation will have recovered before the next experiment.

Figure 16 Repeating a FID experiment too quickly.

For best results, youshould use a ShotRepetition Time of5 x T1.

Mz SRT( ) M0 1 eSRTT1

-----------––⋅=

18

Relaxation Times

TransverseRelaxation Time

The transverse relaxation time corresponds to the time requiredfor the magnetization to decay in the x-y plane. There are twomain contributions to this process and they are related to differ-ent broadening mechanisms: homogeneous and inhomogeneousbroadening.

Figure 17 a) Homogeneous broadening. The lineshape isdetermined by the relaxation times and thereforelorentzian lineshapes are a common result. (SeeEquation [13] and Figure 21.) The EPR spec-trum is the sum of a large number of lines eachhaving the same Larmor frequency and line-width.

b) Inhomogeneous broadening. The lineshape isdetermined by unresolved couplings because theEPR spectrum is the sum of a large number ofnarrower individual homogeneously broadenedlines that are each shifted in frequency withrespect to each other. Gaussian lineshapes are acommon result.

Pulse EPR Theory 19

Transverse Relaxation Time

In an inhomogeneously broadened spectrum, the spectrum isbroadened because the spins experience different magneticfields. These different fields may arise from unresolved hyper-fine structure in which there are so many overlapping lines thatthe spectrum appears as one broad signal. (See Figure 17.) Typ-ically this type of broadening results in a Gaussian lineshape,which we shall discuss in the next section.

This distribution of local fields gives us a large number ofspin-packets characterized by a distribution of ∆ω in the rotatingframe. As shown in Figure 10, the magnetization of an individ-ual spin-packet will rotate if ∆ω ≠ 0 and the larger ∆ω is, thefaster it rotates. If we sum up all the components of the individ-ual spin-packets, we see that many components cancel eachother out and decrease the transverse magnetization. (SeeFigure 18.) The shape of this transverse magnetization decay(actually a FID) is in general not an exponential decay butinstead reflects the shape of the EPR spectrum. The characteris-tic time constant for the decay is called T2

*. (T two star.)

A spin-packet is oneof the many individ-ual homogeneouslybroadened EPR linesthat contributes to aninhomogeneouslybr o ad e ne d E P Rsp e c t r um . (Se eFigure 17.)

Figure 18 Fanning out of the transverse magnetization and the decrease of the trans-verse magnetization.

20

A Few Fourier Facts

In Figure 17, each of the individual spectra (or spin-packets)which comprise the inhomogeneously broadened line are homo-geneously broadened. In a homogeneously broadened spectrum,all the spins experience the same magnetic field. The spins inter-act with each other, resulting in mutual and random spinflip-flops. Molecular motion can also contribute to this relax-ation. These random fluctuations contribute to a faster fanningout of the magnetization. This broadening mechanism results inLorentzian lineshapes which we shall discuss in the next section.The decay of the transverse magnetization (FID) from thismechanism is in general exponential:

[13]

where T2 is often called the spin-spin relaxation time.

A Few Fourier Facts 1.3So far, all our discussions have been very geometric. It was men-tioned that the information about the frequency spectrum wassomehow encoded in the transverse magnetization in the rotatingframe. One means of reconstructing the frequency spectrum is tostudy the time behavior of the transverse magnetization. (SeeFigure 19.) The component of the transverse magnetizationalong the -y axis will vary as:

, [14]

where ∆ω is the frequency offset ω−ω0 and t is the time after themicrowave pulse. The component along +x will vary as:

. [15]

Un l ike the s ta t i ceffects of inhomoge-neous broadening,homogeneous broad-ening results fromrandom and irrevers-ible events. This factwill become impor-tant when we discussspin echoes.

M-y et

T2-----–

=

M-y M ωt∆cos⋅=

Mx M ωt∆sin⋅=

Pulse EPR Theory 21

Transverse Relaxation Time

A common mathematical convenience is to treat these two com-ponents as the real and imaginary components of a complexquantity:

, [16]

where

[17]

and

. [18]

Figure 19 Time behavior of the transverse magnetization.

Mt Mei ω∆ t=

eiφ φcos i φsin+=

i 1–=

22

A Few Fourier Facts

The transverse magnetization can then be represented by a vec-tor in the x-y plane. It has both a magnitude M and a directionrepresented by the phase angle φ.

The reason why we go to this representation is because we cannow use Fourier theory. Fourier theory relates a time domainsignal with its frequency domain representation via the Fouriertransform. This transform is the means by which we extract ourEPR spectrum from the FID.

It is not the purpose of this primer to make you an expert in thearcane secrets of Fourier theory, A few theorems and identitiescan offer you an intuitive and visual understanding of manythings you will encounter in pulse EPR.

Figure 20 Representation of the transverse magnetizationas a complex quantity.

Pulse EPR Theory 23

The Fourier Transform

The FourierTransform

We can represent a function either in the time domain or the fre-quency domain. It is the Fourier transform which convertsbetween the two representations. The Fourier transform isdefined by the expression:

[19]

There is also an inverse Fourier transform:

[20]

Fourier TransformPairs

We do not necessarily have to understand these equations ingreat detail. Any functions related by Equation [19] and Equa-tion [20] form what is called a Fourier transform pair. The pairsthat we shall encounter frequently are shown in Figure 21. Theimportant points to learn are:

• Though a function may be purely real, it will in general havea complex Fourier transform.

• Even functions (f(-t) = f(t) also called symmetric) have apurely real Fourier transform. (See Figure 21 a.)

• Odd functions (f(-t) = -f(t) also called anti-symmetric) have apurely imaginary Fourier transform. (See Figure 21 b.)

• An exponential decay in the time domain is a lorentzian inthe frequency domain. (See Figure 21 c.)

• A gaussian decay in the time domain is a gaussian in the fre-quency domain. (See Figure 21 d.)

We shall use lowercase letters to denotethe time domain rep-resentation, f(t), andupper case letters tode n o te t h e f r e -quency domain rep-resentation, F(ω).

F ω( ) f t( )e iωt– dt∞–

+∞

∫=

f t( ) 12π------ F ω( )eiωt dω

∞–

+∞

∫=

The real part of thefrequency domainsignal corresponds tothe absorption andthe imaginary partcorresponds to thedispersion signal.

24

A Few Fourier Facts

• Quickly decaying signals in the time domain are broad in thefrequency domain.

• Slowly decaying signals in the time domain are narrow in thefrequency domain.

• These pairs are reciprocal, i.e. a lorentzian in the time domainresults in a decaying exponential in the frequency domain.

Figure 21 Useful Fourier transform pairs. For simplicity,F(ω) normalization constants are omitted.

Notice the similarityof the function inFigure 21 e withthat in Figure 12.

Pulse EPR Theory 25

Fourier Transform Properties

Fourier TransformProperties

One important property that we shall need is that the Fouriertransform of the sum of two functions is equal to the sum of theFourier transforms:

. [21]

Another important property is how the frequency domain signalchanges as we time shift (delay or advance the signal in time)the time domain signal or how the time domain signal changes ifwe frequency shift the frequency domain signal. After a bit ofmath, we obtain the following Fourier transform pairs:

[22]

. [23]

Figure 22 The addition property of the Fourier transform.

f(t) + g(t) F(ω) + G(ω)⇔

f(t-∆t) F(ω) e iω∆t–⋅⇔

f(t) ei∆ωt⋅ F(ω ∆ω )–⇔

26

A Few Fourier Facts

When time shifting, we obtain the original frequency domainsignal with a frequency dependent phase shift. As we can seefrom Figure 23, the phase shift transfers some of the real signalto the imaginary and vice versa. This effect leads to the wellknown linear phase distortion (and correction) in Fourier trans-form spectroscopy. We start off in Figure 23 with a purely realsignal (remember that a symmetric signal has a purely real Fou-rier transform) and after the time delay we obtain an oscillatingmixture of real and imaginary components. Because of the recip-rocal nature of Fourier transform pairs, similar behavior in thetime domain signal is observed when the frequency is shifted inthe frequency domain signal.

Figure 23 The time shift properties of the Fourier trans-form.

Pulse EPR Theory 27

The Convolution Theorem

The ConvolutionTheorem

The convolution integral appears frequently in a number of sci-entific disciplines. The convolution of two functions is definedas:

. [24]

It can also be shown that f(t) * g(t) = g(t) * f(t).

It is difficult to envision exactly what the convolution is doing,but it can be interpreted loosely as a running average of the twofunctions. In the limit of a Dirac delta function (i.e. a spike), theconvolution can be graphically represented as in Figure 24. Weare placing a copy of our function at each of the spikes.

The convolution theorem states that the Fourier transform of theconvolution of two functions is equal to the product of the Fou-rier transforms of the individual functions. We now have twonew Fourier transform pairs:

[25]

. [26]

So the convolution theorem gives us an easy way to calculate aconvolution integral if we know the individual Fourier trans-forms. More importantly, it offers us a powerful means of envi-sioning time signals in the frequency domain and vice versa.

f(t) * g(t) f(τ ) g(t-τ) dτ∞–

+∞

∫=

Figure 24 The convolution of two functions.

f(t) * g(t) F(ω) G(ω)⇔

F(ω) * G(ω) f(t) g(t)⇔

28

A Few Fourier Facts

A PracticalExample

Now its time to start applying what we have learned in the previ-ous sections to a concrete problem, predicting what a timedomain signal (e.g. a FID) looks like if we are given a frequencydomain signal (e.g. an EPR spectrum). As an example, we con-sider a three line EPR spectrum such as a nitroxide. (SeeFigure 25.) We assume that the magnetic field is set so that thecenter line is on-resonance, the lines are lorentzian, and the split-ting is equal to A. Remember that in an FT experiment we aredetecting both the absorption (real) and dispersion (imaginary)signals.

The first thing to notice is that we can deconvolute the spectruminto a stick spectrum and a Lorentzian function.

Figure 25 A three line EPR spectrum with both absorptiveand dispersive components.

Figure 26 Deconvoluting a three line EPR spectrum into astick spectrum and a lorentzian function.

Pulse EPR Theory 29

A Practical Example

We know from the convolution theorem that the time domainsignal is simply the product of the two transformed functions.(See Equation [26].) We already know the Fourier transform fora lorentzian:

. [27]

Next we have to calculate the Fourier transform of the three linestick spectrum. One thing that helps is that this signal is symmet-ric, yielding a purely real time domain signal. Using the additiveproperties of Fourier transforms, we express the three line stickspectrum as the sum of two signals with known Fourier trans-forms. Adding the two time domain signals gives us the Fouriertransform of the stick spectrum.

Multiplying the two time domain functions gives us the result inFigure 28. This is the FID of the three line EPR spectrum.

Figure 27 The Fourier transform of a three line stick spec-trum obtained as the sum of two functions.

Figure 28 FID of a three line EPR spectrum.

et– T2⁄

30

A Few Fourier Facts

On this and the next page are examples of what happens to theFID when the EPR signal changes.

Figure 29 The effect of linewidth.

Figure 30 The effect of line splittings.

As the linewidth oft h e E P R s i g n a lincreases, the FIDdecays more quickly.

As the splitting oft h e E P R s i g n a ldecreases, the oscil-lations in the FIDbecome slower.

Pulse EPR Theory 31

A Practical Example

These practical examples demonstrate that if we make use of theFourier transform pairs, properties, and convolution theorem, wecan easily envision how signals appear in both time and fre-quency domains. We do not have to perform any complicatedmathematical operations to Fourier transform our signals. Wecan visually estimate the appearance of signals in both the timeand frequency domains. Even though this intuitive ability is notmandatory, it comes in very handy later on when adjustingparameters and processing data.

Figure 31 The effect of a frequency shift.

If we are not exactlyon resonance withthe center of a sym-metr ic s ignal , wewill get an oscilla-tion between the realand imaginary com-ponents.

32

Field Sweeps vs. Frequency Spectra

Field Sweeps vs. Frequency Spectra 1.4A little bit of care is required when comparing conventional fieldswept spectra and frequency spectra obtained by FT-EPR. Thefield and frequency axes run in opposite directions. Here are twospectra of the same sample. The upper spectrum is a frequencyspectrum acquired by Fourier transforming the echo (To be dis-cussed in the next section.). The lower spectrum was acquired ina conventional field swept experiment.

In Figure 33 we see the Larmor frequencies when the field is setso the center line is on-resonance. The higher field line actuallyhas a lower (negative) Larmor frequency than the center line. Weneed to apply more magnetic field to increase its Larmor fre-quency so that it would be on-resonance with the microwaves.The lower field line has a higher Larmor frequency.

Figure 32 Field sweep and frequency spectrum are mirrorimages of each other.

Figure 33 Larmor frequencies when B0 is set for reso-nance on the center line.

Pulse EPR Theory 33

How Echoes Occur

Multiple Pulses = Echoes 1.5As we have seen in the previous sections, one microwave pulseproduces a signal that decays away (FID). If our EPR spectrumis inhomogeneously broadened, we can recover this disappearedsignal with another microwave pulse to produce a Hahn echo.

Echoes are important in EPR because FIDs of very broad spectradecay away very quickly. We shall see in the second part of thischapter that we cannot detect signals during an approximately80 ns period after the microwave pulse. This period of time iscalled the deadtime. If the FID is very short, it will disappearbefore the deadtime ends. If we make τ long enough, we canensure that the echo appears after the deadtime.

How EchoesOccur

How does the echo bring back our signal? The decay of the FIDis due to the different frequencies in the EPR spectrum causingthe magnetization to fan out in the x-y plane of the rotatingframe. When we apply the π pulse, we flip the magnetizationabout the x axis. The magnetization still rotates in the samedirection and speed. This almost has the effect of running theFID backwards in time. The higher frequency spin packets willhave travelled further than the lower frequency spin packetsafter the first pulse. However, because the higher frequency spinpackets are rotating more quickly, they will eventually catch up

Figure 34 A Hahn echo.

Echo

FID

34

Multiple Pulses = Echoes

with the lower frequency spin packets along the +y axis after thesecond pulse. (See Figure 35.)

After all the spin packets bunch up, they will dephase again justlike a FID. So one way to think about a spin echo is a timereversed FID followed by a normal FID. Therefore, if we Fou-rier transform the second half of the FID, we obtain the EPRspectrum.

Figure 35 Refocusing of the magnetization during an echo.

Figure 36 Magnetization behavior during an echo experiment.

Pulse EPR Theory 35

How Echoes Occur

In all we have said so far, we should be able to make τ, the pulseseparation, very long and still obtain an echo. Transverse relax-ation leads to an exponential decay in echo height:

, [28]

where TM, the phase memory time, is the decay constant. Manyprocesses contribute to TM such as T2 (spin-spin relaxation), aswell as spectral, spin, and instantaneous diffusion.

Notice the factor of two in Equation [28] which is not in theexpression for the FID. This is because dephasing starts after thefirst pulse and the echo occurs at 2τ after the first pulse. So bystudying the echo decay as we increase τ, we can measure TM.

Spectral diffusion often is a large contributor to TM. Nuclearspin flip-flops, molecular motion, and molecular rotation cancause spin packets to suddenly change their frequency. A fasterspin packet far from the +y axis will suddenly become a slowerspin packet without the needed speed to catch up with the otherspin packets in their race to refocus. Therefore, we are not refo-cusing all the magnetization. In Figure 37 we see that after therunner marked with an asterisk has a shifted frequency, we onlyget four of the five runners lining up to refocus.

Quite often, the echodecay is not a sim-p l e ex p o n en t i a lowing to the manyprocesses that cancont r ibu t e t o th eecho decay.

Echo Height(τ ) e2τ– Tm⁄

∝

Figure 37 Dephasing due to a sudden frequency shift. The asterisk marks the runnerwhose frequency has suddenly become less.

36

Multiple Pulses = Echoes

ESEEM A very important class of echo experiments is ESEEM (ElectronSpin Echo Envelope Modulation). The electron spins interactwith the nuclei in their vicinity and this interaction causes a peri-odic oscillation in the echo height superimposed on the normalecho decay. The modulation or oscillation is caused by periodicdephasing by the nuclei. If we subtract the decay of the spinecho and Fourier transform the oscillations, we obtain the split-tings due to the nuclei. Armed with this information, you canidentify nearby nuclei and their distances from the electron spinand shed light on the local environment of the radical or metalion.

Figure 38 Modulation of the echo height with τ due to ESEEM.

Figure 39 The Fourier transform of the ESEEM showing proton couplings.

Pulse EPR Theory 37

Stimulated Echoes

Stimulated Echoes Hahn or two pulse echoes are not the only echoes to occur. If weapply three π/2 pulses we obtain five echoes. Three of the ech-oes are simply two pulse echoes produced by the three pulses.The stimulated and refocused echoes only occur when you haveapplied more than two pulses.

The stimulated echo is particularly important because it alsoexhibits ESEEM effects when t1 is varied. A Hahn echo decayswith a time constant of TM/2 whereas the stimulated echo decayswith a time constant of approximately T1. (Spin and spectral dif-fusion contributions causes the stimulated echo to decay some-what faster than T1.) TM is often much shorter than T1, so theESEEM decays more slowly in a stimulated echo than in a Hahnecho experiment. Therefore, a three pulse ESEEM experimentusually gives superior resolution than a two pulse ESEEMexperiment.

Figure 40 Echoes and timing in a three pulse experiment.

Remembering ourFourier theory, broadin the time domainmeans narrow in thefrequency domain.

38

Multiple Pulses = Echoes

Pulse Lengths andBandwidths

In Pulse EPR spectroscopy, we often can excite only a small por-tion of our EPR spectrum. This fact simplifies things when per-forming echo experiments. First, if we have a very broad EPRspectrum, within the range of our excitation of the spectrum itlooks almost flat and therefore approximately symmetric. As aconsequence, our echo will be purely real with no imaginarycomponent. Second, the echo width is approximately equal tothe pulse width.

Quite often it is more convenient to use two equal length pulsesinstead of the traditional π/2 - π pulse sequence. The reason fordoing this is the π pulse is twice as long as the π/2 pulse andtherefore will limit the amount of the EPR spectrum we canexcite. (See Figure 21e.) With a bit of calculus, it can be shownthat the maximum echo height for two equal length pulses isachieved with two 2π/3 (120°) pulses. (See Figure 41.) The nar-rower 2π/3 pulses excite a broader portion of our spectrum thanthe π pulse can.

Figure 41 Simulated echo shapes for different tip angles.

Pulse EPR Theory 39

Pulse Lengths and Bandwidths

Sometimes both hard (short) and soft (long) pulses are combinedtogether in one experiment. For example, to perform a Daviespulse ENDOR experiment, you use a soft π pulse to burn a nar-row hole in the EPR spectrum (See Figure 42.) and two narrowpulses to detect it.

The resulting echo can be a bit puzzling at first glance. It is actu-ally the sum of two echoes: one is a narrow positive going echofrom the broad EPR spectrum and the other is a broad negativegoing echo from the narrow hole. In order to adjust the π pulse,the microwave power is varied until the area of the broad nega-tive going echo is as negative as possible.

Holeburning meansto excite a narrowfrequency range ofan EPR spectrum.The r e s u l t an treduced Mz leads toless detected EPRintensity in that nar-row range, therebycreating a “hole” inthe spectrum.

Figure 42 Echo shapes in a hole burning experiment.

40

Multiple Pulses = Echoes

Pulse EPR Practice 2Modern pulse EPR spectrometers perform an amazing feat.They detect tiny (< 1 nW) signals tens of nanoseconds after apowerful (> 1 kW) microwave pulse and can repeat this featevery 1 µs. This section describes how the Bruker E580 spec-trometer accomplishes this feat.

Figure 43 shows a photograph of an E580 spectrometer. Thecomponents are identified in the block diagram.

Figure 43 A photograph and block diagram of a Bruker E580 spectrometer.

Pulse EPR Practice 41

Pulse Lengths and Bandwidths

Many of the components such as the magnet, resonator, etc.should be familiar from your experience with a CW EPR spec-trometer. The TWT (Travelling Wave Tube) is a high powermicrowave amplifier that produces the 1 kW microwave pulses.There are more components to be controlled in a pulse bridge, soa second Bridge controller is required in addition to the standardMBC (Microwave Bridge Control) board. The pulse program-mer produces pulses that orchestrate all the events to producehigh power microwave pulses, protect receivers, and triggeracquisition devices. The digitizer captures and averages the FIDand echo signals.

The Pulse EPR Bridge 2.1The microwave bridge creates the microwave pulses and detectsthe FIDs and echoes. Because of this two-fold duty for the pulsebridge it is a good idea to separate the two functions in our dis-cussions. A few of the parts are actually required for both excita-tion and detection.

Figure 44 A block diagram of the bridge separated into its two functions.

42

The Pulse EPR Bridge

Excitation In order to excite or produce an FID or echo, we need to create ashort high power microwave pulse. Typical pulse lengths are12-16 ns for a π/2 pulse with up to 1 kW of microwave power.This is achieved by supplying low power microwave pulses tothe TWT where they are amplified to very high power. (SeeFigure 45.)

The MPFU (Microwave Pulse Forming Unit) produces the lowpower microwave pulses. Each unit consists of two “arms” withindividual attenuators and phase shifters to adjust the relativeamplitudes and phases in the two arms. To create a +x pulse, the+x PIN (P-type Intrinsic N-type) diode switch passes micro-waves through for the specified pulse length. For a -x pulse, the-x PIN diode switch is used instead. If additional phases oramplitudes are needed, more MPFU are installed in parallel withthe first MPFU.

Two PIN diode switches are required to turn the microwavessufficiently off, so there is a second switch (Pulse Gate) in serieswith the MPFU. The transmitter level attenuator controls theoverall power for input to the TWT. After the TWT amplifies the

Figure 45 The excitation portion of the pulse bridge.

Pulse EPR Practice 43

Detection

microwave pulses, the HPP (High Power Pulse) attenuatorallows you to change the amplitude of the high power micro-wave pulses.

In normal operation, most of the attenuators and phase shiftersare kept fixed except for the HPP attenuator. This attenuatoradjusts the B1 that we apply to our sample. Because B1 is pro-portional to the square root of the microwave power, we need todecrease the HPP attenuator by 6 dB in order to double B1.

Detection

The FIDs and echoes are very low level signals so we need apreamplifier to lift them up out of the noise. This is a bit trickyhowever, because we are using high power microwave pulsesand the reflected pulses as well as the resonator ringdown (oneof the causes of the so-called deadtime) can easily burn out ourpreamp. To avoid destroying it, we use a PIN diode switch(known as the defense diode) to block the high power micro-wave pulses from reaching the preamp. We cannot measure the

Figure 46 The detection portion of the pulse bridge.

44

The Pulse EPR Bridge

signals until the high power microwaves are dissipated and wecan turn the defense diode on again. (See Figure 47.)

The amplified signal then proceeds to the quadrature detector.Quadrature detection is simply an electronic means for measur-ing both transverse magnetization components in the rotatingframe. This gives us the required amplitude and phase informa-tion to transform the signals into a frequency representation.(See Figure 48 and Figure 19.)

Figure 47 The defense pulse and the deadtime.

Figure 48 Quadrature detection.

Pulse EPR Practice 45

Detection

The outputs from the quadrature detector correspond to the realand imaginary components of the magnetization and are com-monly labeled Channel a and Channel b. There is a phase shifterto adjust the reference phase for the quadrature detection. Thisphase rotates the detection axes and therefore changes theappearance of the signal. In Figure 49, we start with an on-reso-nance FID and the reference phase adjusted so that we only havea signal in Channel a. If we were to change the reference phase,some of the signal in Channel a appears in Channel b and viceversa.

Figure 49 The effect of the reference phase on the signal.

46

The Pulse EPR Bridge

The quadrature detection is followed by one more stage ofamplification and filtering by the VAMP (Video Amplifier).Both the gain and bandwidth of the VAMP are adjustable. SixdB steps are required to change the signal amplitude by a factorof two.

The bandwidth is normally kept at the maximum value, 200MHz. Narrower bandwidth reduces the noise, but also distortshigher frequency signals. There are a few cases (See page 56.)where the bandwidth must be reduced. Figure 50 shows theeffect of bandwidth reduction on the FT-EPR spectrum. Notethat there is both a time shift and an attenuation of higher fre-quency components of the spectrum at narrower bandwidth.

Figure 50 The effect of bandwidth reduction on anFT-EPR spectrum. Note: this does not effectfieldswept spectra.

Pulse EPR Practice 47

Detection

The Pulse Programmer 2.2In order to excite and detect FIDs and echoes, many events mustbe orchestrated. First, because the TWT is a pulse amplifier, itmust be turned on a little before the microwave pulse. Themicrowave pulse must be supplied to the TWT at a precise timeafter the TWT is turned on. This pulse is produced by turning the+x and pulse gate PIN diodes on and off at precisely the sametime. While the high power microwaves are on, the defensediode must protect the preamp. Lastly we must trigger the digi-tizer to acquire the signal.

The PatternJetTM pulse programmer supplies all the signals thatorchestrate all the individual components so that each event

Figure 51 The timing for a pulse experiment.

48

The Pulse Programmer

occurs precisely at the right moment. It would be very difficultindeed if we had to determine all the delays and pulse lengths toperform each experiment. This is why the XeprTM software, bydefault, automatically calculates everything for us after calibra-tion at the initial spectrometer installation. All we have to supplyare the length of the microwave pulses, the time between them,and the starting time for the data acquisition. The software doesall the rest of the work for us.

Pulse EPR Practice 49

Detection

Data Acquisition 2.3Once we obtain a signal from the detection portion of the bridge,we need to digitize it somehow to process the signal with a com-puter. There are three different classes of digitizer required forpulse EPR spectroscopy; point digitizer, integrator, and transientrecorder. (See Figure 52.) The SpecJetTM digitizer performsthese three classes of experiments as well as signal averaging toimprove the signal to noise ratio of the signal.

Figure 52 The three classes of acquisition devices used in pulse EPR.

50

Data Acquisition

Point Digitizers In the point digitizer mode of the SpecJetTM, the digitizer onlysamples one point (<2 ns) in the FID or echo at a time, therebyrequiring multiple acquisitions for measuring signals. (SeeFigure 53.) The most common measurements requiring thismode are ESEEM and relaxation measurements experimentswhere only the height of the echo needs to be measured.

For example, in a two pulse experiment, we generate the signalby measuring the echo height for the initial τ value; then step outτ, digitize the second point of our signal; and so on until we haveacquired the entire echo decay.

Figure 53 Acquisition of an echo decay with a point digi-tizer exponential.

Pulse EPR Practice 51

Integrators

Integrators The point digitizer method is often called non-selective detec-tion, whereas the integration method is called selective detec-tion. We shall see why this is so.

Because of the limited excitation bandwidth in pulse EPR, wecannot always Fourier transform an FID or echo to obtain abroad EPR spectrum. (See Figure 13.) We could, however, mea-sure the echo height as we sweep the magnetic field to generatea broad EPR spectrum. There is a slight problem which is calledpower broadening. (This effect is different from power broaden-ing in CW EPR.) We can easily achieve a B1 of 10 Gauss in therotating frame. If we have features narrower than 10 G, in ananalogous fashion to field overmodulation, the power broaden-ing will decrease our resolution. In CW EPR, we turn down thefield modulation. In pulse EPR, we can use softer pulses toachieve the need for better resolution. (See Figure 54.)

Figure 54 Linewidths for different pulse lengths withnon-selective detection for echo detectedfield-swept spectra.

Soft pulses, oftenca l l e d s e l e c t i vepulses, are lower B1and power pulses,and there fore arelonger pulses.Hard pulses, oftencalled non-selectivepulses, are higher B1and power pulse, andtherefore are shorterpulses.

52

Data Acquisition

What we have essentially done is limit the bandwidth of excita-tion. By using an integrator, we can also limit the bandwidth ofdetection. It is the off-resonance high frequencies that contributeto the power broadening. If we are able to filter the high fre-quency components out, we can regain our resolution even withhard pulses. By integrating the area under the echo, we canachieve this filtering. How this filtering is accomplished can beseen in Figure 55. On-resonance, the area under the echo islarge and positive. If we go off-resonance, we obtain the highfrequency components with negative going contributions. Thesenegative signals cancel out the positive signals when we inte-grate the echo, effectively achieving the desired filtering effect.The longer period of integration time, the more effective andselective the bandwidth limitation becomes. (See Figure 56 andnotice the similarity with Figure 54.)

Figure 55 Suppression of off-resonance effects by signalintegration.

Pulse EPR Practice 53

Integrators

Figure 56 Linewidths for different integration times withselective detection for echo detected field-sweptspectra.

54

Data Acquisition

TransientRecorders

The transient recorder is extremely efficient at recording andsignal averaging FIDs and echoes because it captures a completesignal in one acquisition.In this mode, the SpecJet is functioninglike a digital oscilloscope.

Figure 57 Capturing of a signal in one acquisition with atransient recorder.

Pulse EPR Practice 55

Aliasing

Aliasing To use a digitizer effectively, we need to be careful about the rateat which we sample the signals. We must make sure that we ful-fill the Nyquist criterion:

, [29]

where νmax is the highest frequency in our signal and theNyquist frequency is:

, [30]

where ∆t is the time between the points in the digitized signals.If we do not comply with this condition, we get fold over oraliasing when we Fourier transform the signal. (See Figure 58.)A lower frequency component equally fits the digitized pointsand the signal will appear as a lower frequency.

This foldover effect or aliasing is one of the reasons for limitingthe detection bandwidth in the video amplifier. By using a nar-rower bandwidth, the high frequency signals that could causeproblems are filtered out before they can be digitized.

νmax νN<

νN 1 2∆t( )⁄=

56

Data Acquisition

Figure 58 Fold over effects from not digitizing with sufficient resolution. Quadraturesignals are shown in the left-hand column.

Pulse EPR Practice 57

Dynamic Range

Dynamic Range In the digitization process, the signal is converted into a streamof integers. How well this data represents our signal depends onthe amplitude resolution of the conversion. The SpecJet has adynamic range of ± 0.5 Volts and separates this range into 256(8 bits) equally spaced steps. The digitizer determines which ofthese 256 steps best matches the voltage of the signal. If we wishto distinguish between two signals that are very close in voltage,the voltage difference must be larger than the separation of adja-cent steps of our digitizer. If we do not supply a large enoughsignal, we obtain noisy data exhibiting jagged step-like or digiti-zation noise. (See Figure 59.) It is important to use a videoamplifier gain that is sufficient to supply approximately a ±0.5Volt signal to use the digitizer fully.

Figure 59 The effect of video amplifier gain on the digi-tized signal.

58

Data Acquisition

Signal Averaging A commonly used technique to increase the signal to noise ratioof a signal is to repeat the experiment and average the results ofthe repeated experiments. The signal will grow linearly whereasthe noise will grow with the square root of the number of aver-ages. Over all, the sensitivity increases with the square root ofthe number of averages.

Signal averaging not only increases the signal to noise ratio, butits also increases the effective dynamic range. If we need toresolve two signals that have almost the same voltage, the noiseactually helps when we signal average. The noise randomly per-turbs the signal up and down, so as we average the signals, wefill the space between the 256 equally spaced steps described in

Figure 60 Signal to noise improvement as a function of thenumber of averages.

Pulse EPR Practice 59

Signal Averaging

the previous section. If the signal is closer to one step than theother, statistically the upper step will be measured more oftenthan the lower step.

As we average more, we obtain better amplitude resolution.

Figure 61 Improvement in amplitude resolution with sig-nal averaging ten times.

Figure 62 Dependence of amplitude resolution on thenumber of averages.

60

Resonators

Resonators 2.4Resonators are perhaps the most critical element of a pulse EPRspectrometer. They convert the microwave power into B1 andalso convert the transverse magnetization into a FID or echo. InCW EPR, we typically use high Q cavities because they are effi-cient at converting spin magnetization into a detectable signal.This is not an option for pulse EPR because high Qs contributeto long deadtimes. The Q is the ratio of the energy stored and thepower dissipated in the resonator. We need to dissipate the highpower microwave pulses very quickly (the so called ring-downtime) so that it does not interfere with the detection of the veryweak FID and echo signals. Another requirement of the resona-tor is bandwidth so that we do not distort broad EPR signals. Wetherefore have two very good reasons to keep the Q as low aspossible.

We still need to convert the microwave power into B1 and thetransverse magnetization into signals efficiently. The efficiencyis proportional to √Q. We cannot increase Q, so we mustincrease the proportionality constant. It is optimized (for a givensample diameter) in small resonators such as dielectric andsplit-ring resonators.

Figure 63 Two types of resonators Bruker uses for pulseEPR. The high range of the Q values are for amatched resonator. The low range is for an over-coupled resonator.

Pulse EPR Practice 61

Signal Averaging

In CW EPR, we normally critically couple the resonator. Thetwo pulse resonators still have too high a Q when matched, sowe need to further decrease the Q by overcoupling the resonator.This does mean some of the microwave power is reflected back,thereby decreasing the power to the sample, but we need to com-promise and minimize the deadtime.

Figure 64 Tuning mode patterns and reflected power forcritically coupled and overcoupled resonators.Notice that no microwave power is reflectedwhen on-resonance and critically coupled.

62

Phase Cycling

Phase Cycling 2.5Phase cycling serves two purposes: to suppress artefacts due toimbalances in the quadrature detection and to eliminateunwanted FIDs and echoes. The phases of the microwave pulsesare changed in a prescribed fashion while the two quadraturedetection channels are added, subtracted, and exchanged toachieve the desired net effect.

4 Step PhaseCycle

Imbalances in the quadrature detector can distort the Fouriertransformed signal. We assume that both detectors in Figure 48have exactly the same gains, the reference phases are π/2phase-shifted from each other, and there are no DC offsets. Thisis very difficult to realize in practice. The imbalance in phaseand amplitude causes aliasing in which positive frequency sig-nals start appearing at negative frequencies and vice versa. TheDC offsets appear as large features at zero frequency.

The four step phase cycle (See Figure 65.) suppresses all ofthese quadrature artefacts. In the first step of the phase cycle, weapply a +x pulse and store the channel a signal as the real dataand the channel b signal as the imaginary data. Next we apply a-x pulse, causing our signals to changes sign. Therefore, we sub-tract the second set of signals in order that our FID does not can-cel but instead becomes twice as large. This step of the phasecycle eliminates the zero frequency artefact because the DC off-sets are unaffected by the phase of the microwaves, thereforesubtraction cancels it out.

Pulse EPR Practice 63

4 Step Phase Cycle

The next two steps require application of a +y or -y pulse. Thisthen exchanges the signal that originally was in channel a withthe channel b signal. We now add and subtract the channel b sig-nals with our previous real results and the channel a signals withthe imaginary results. These two steps suppress the aliasing arte-

Figure 65 Changes in the FID during a four step phasecycle.

64

Phase Cycling

facts because we have sent identical signals through both chan-nels a and b now, thus averaging the gain and reference phaseimbalances to approximately zero. After Fourier transformingthe FID, we now obtain a nice spectrum with no artefacts. (SeeFigure 66.)

Figure 66 The effect of the four step phase cycle upon thefrequency spectrum.

Pulse EPR Practice 65

Unwanted Echoes & FIDs

Unwanted Echoes& FIDs

We saw in Figure 40 that three microwave pulses create fiveechoes. In a three pulse ESEEM experiment, we are only inter-ested in the stimulated echo. The other echoes only give us arte-facts as they run through our stimulated echo. There is a phasecycle that leaves the stimulated echo intact but subtracts theother echoes away. (See Figure 67.) Almost all pulse EPRexperiments are performed with some type of phase cycling inorder to focus on the one echo or FID in which we are interested.

Figure 67 Cancellation of unwanted echoes by phasecycling.

66

NMR

Bibliography 3This chapter is a brief overview of the basic theory and practiceof pulse EPR spectroscopy. If you would like to learn more,there are many good books and articles that have been written onthese subjects. We recommend the following:

NMR 3.1The Principles of Nuclear Magnetic ResonanceA. AbragamOxford at the Clarendon Press 1978

Principles of Nuclear Magnetic Resonance in One and Two DimensionsR. R. Ernst, G. Bodenhausen and A. Wokaun Oxford Science Publications 1987

A Handbook of Nuclear Magnetic ResonanceR. FreemanLongman Scientific & Technical 1987

Two Dimensional Nuclear Magnetic Resonance in LiquidsA. BaxDelft University Press 1982

Principles of High Resolution NMR in SolidsM. MehringSpringer Verlag1983

Experimental Pulse NMR: A Nuts and Bolts ApproachE. Fukushima and S.B.W. RoederAddison-Wesley 1981

Bibliography 67

Unwanted Echoes & FIDs

Pulsed Magnetic Resonance: NMR, ESR and OpticsD.M.S. Bagguley (Ed.)Oxford Science Publications 1992

EPR 3.2Electron Paramagnetic Resonance of Transition IonsA. Abragam and B. BleaneyDover Publications, New York 1970

Transition Ion Electron Paramagnetic ResonanceJ.R. PilbrowOxford Science Publications 1990

Electronic Magnetic Resonance of the Solid StateJ.A. Weil (Ed.)The Canadian Society for Chemistry, Ottawa 1987

Structural Analysis of Point Defects in SolidsJ.M. Spaeth, J.R. Niklas and R.H. BartramSpringer Verlag 1992

Electron Spin EchoesW.B. Mimsin "Electron Paramagnetic Resonance", Ed. S. GeschwindPlenum Press, New York 1972

Time Domain Electron Spin ResonanceL. Kevan and R.N. SchwartzWiley & Sons 1979

Pulsed EPR: A New Field of ApplicationsC.P. Keijers, E.L. Reijerse and J. Schmidt (Eds.)North Holland 1989

68

EPR

Advanced EPR: Application in Biology and BiochemistryA.J. Hoff (Ed.)Elsevier 1989

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Notes

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