Pulse EPR Theory

In order to describe a physical phenomenon, we need to establish an axis system or reference frame. The reference frame which most people are familiar with is the lab frame which consists of three stationary mutually perpendicular axes. The lab frame in EPR is usually defined as in Figure 4.1. The magnetic field, 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 x and z axes. All discussions of the electronic magnetization in this section will be described in this axis system.

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

ω_L = -γB₀

equation 4.1

where ω_L is the Larmor frequency, γ is the constant of proportionality called the gyromagnetic ratio, and B₀ is the magnetic field. The sense of rotation and frequency depend on the value of γ and B₀. A free electron has a γ/2π value of approximately -2.8 MHz/Gauss, resulting in a Larmor frequency of about 9.75 GHz at a field of 3480 Gauss. The Larmor frequency corresponds to the EPR frequency at that magnetic field.

EPR experiments are usually performed with a resonator using linearly polarized microwaves. The microwave resonator is designed to produce a microwave field, B₁, perpendicular to the applied magnetic field, B₀. In most cases, |B₁| << |B₀|.

Linearly polarized microwaves can be thought of as a magnetic field oscillating at the microwave frequency. (See the upper series of Figure 4.3.) An alternative way of looking at linearly polarized microwaves which is more useful when using the rotating frame is shown in the lower series of Figure 4.3. The sum of two magnetic fields rotating in opposite directions at the microwave frequency will produce a field equivalent to the linearly polarized microwaves. As we shall see, only one of the rotating components is important in describing the FT-EPR experiment.

Alas, the effect that B₁ has on the magnetization is very difficult to envision when everything is moving simultaneously as in the first picture in Figure 4.4. To avoid vertigo, we can observe what is happening from a rotating coordinate system in which we rotate synchronously with one of the rotating B₁ components. We shall assume that we are at resonance, i.e.


ω_L = ω₀

equation 4.2

where ω₀ is the microwave frequency. By rotating the coordinate system at an angular velocity of ω₀, we can make one of the components of B₁ to appear stationary. (See second picture of Figure 4.4) The other component will appear to be rotating at an angular velocity of 2ω₀ and can be neglected. (The reasons for neglecting the fast component is based on effective fields and will be covered later in this chapter.) The rotating frame also makes the magnetization components precessing at the Larmor frequency to appear stationary. Using Equation 4.1 and assuming the magnetization is not precessing in the rotating frame (ω = 0), the field B₀ disappears in the rotating frame. In the rotating frame, we need only to concern ourselves with a stationary B₁ and M₀.

We have already looked at the interaction of a static magnetic field with the magnetization; the magnetization will precess about B₁ at a frequency,

ω₁ = - γ B₁

equation 4.3

where ω₁ is also called the Rabi frequency. Let us assume that B₁ is parallel to the x axis. The magnetic field will rotate the magnetization about the +x axis as long as the microwaves are applied. (See Figure 4.5.)

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

α = -γ|B₁|t_p

equation 4.4

where t_p is the length of the pulse. Pulses are often labeled by their tip angle, i.e. a π/2 pulse corresponds to a rotation of M₀ by π/2. The most commonly used tip angles are π/2 and π (90 and 180 degrees). The tip angle is dependent on both the magnitude of B₁ and the length of the pulse. For example, a B₁ of 10 Gauss can often be obtained, resulting in a π/2 pulse length of approximately 9 ns. The effect of a π/2 pulse is shown in Figure 4.6; it results in a stationary magnetization along the -y axis. If we were to make the pulse twice as long, we would have a π pulse and the magnetization would be rotated to the -z axis.

Because B₁ is parallel to +x it is known as a +x pulse. If we were to shift the phase of the microwaves by 90 degrees, B₁ would then lie along the +y axis and the magnetization would end up along the +x axis. Microwave pulses are therefore labeled not only by their tip angle but also by the axis to which B₁ is parallel.

In the introduction, it was mentioned that the sample emitted microwaves after the intense microwave pulse. How this happens is not completely clear if viewed from the rotating frame. If viewed from the lab frame, the picture is much clearer. The stationary magnetization along -y then becomes a magnetization rotating in the x-y plane at the Larmor frequency. This generates currents and voltages in the resonator just like a generator. (See Figure 4.8 and Figure 4.9.) The signal will be maximized for the magnetization exactly in the x-y plane. This microwave signal generated in the resonator is called an FID (Free Induction Decay).

So far we have been dealing with exact resonance conditions, i.e. the Larmor frequency is exactly equal to the microwave frequency. EPR spectra contain many different frequencies so not all parts of the EPR spectrum can be exactly on-resonance simultaneously. Therefore, we need to consider what happens to the magnetization when we are off-resonance.

First, we shall look at the rotating frame behavior of transverse magnetization having a frequency ω following a π/2 pulse. Initially the magnetization will be along the -y axis, however, because ω is not equal to ω₀, the magnetization will appear to rotate in the x-y plane. This means that the magnetization either is rotating faster or slower than the microwave magnetic field, B₁. The rotation rate will be equal to the frequency difference:

Δω = ω - ω₀

equation 4.5

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

This frequency behavior gives us a clue as to how the EPR spectrum is encoded in the FID. The individual frequency components of the EPR spectrum will appear as magnetization components rotating in the x-y plane at the corresponding frequency, Δω. If we could measure the transverse magnetization in the rotating frame, we could extract all the frequency components and hence reconstruct the EPR spectrum.

A second consequence of not being exactly on-resonance is that the microwave magnetic field B₁ actually tips the magnetization into the x-y plane differently because B₀ does not disappear when we are not on-resonance. We determined that B₀ disappears in the rotating frame when we were on-resonance because our magnetization is no longer precessing. When we are off-resonance, the magnetization is precessing at Δω and therefore:

B₀ = Δω / -γ

equation 4.6

Now the magnetization is not tipped by B₁ but by the vector sum of B₁ and B₀, which is called B_eff or the effective magnetic field. The magnetization is then tipped about B_eff at the faster effective rate ω_eff:

ω_eff = (ω₁² + Δω²)^1/2

equation 4.7

Another consequence is that we cannot tip the magnetization into the x-y plane as efficiently because B_eff does not lie in the x-y plane as B₁ does. The magnetization does not move in an arc as it does on-resonance, but instead its motion defines a cone. In fact, it can be shown that the magnetization that can be tipped in the x-y plane exhibits an oscillatory and decreasing behavior as |Δω| gets larger:

M_-y = M₀ * 1/(1 + (Δω/ω₁)²)^1/2 * sin ( (1 + (Δω/ω₁))²)^1/2 * π/2 )

equation 4.8

One thing is evident from Figure 4.12, if we have a very broad EPR spectrum (Δω > ω₁), we will not be able to tip all the magnetization into the x-y plane to create an FID. This is why it is important to maximize ω₁ (or equivalently to minimize the π/2 pulse length) for broad EPR signals. As B₁ gets larger (and the pulse lengths get shorter), we can successfully detect more of our EPR spectrum. (See Figure 4.13.)

We have already seen that electron spins in a magnetic field are characterized by two quantum mechanical states, one with the magnetic moment parallel and the other state with the magnetic moment anti-parallel to the magnetic field. The moments will be randomly distributed between parallel and anti-parallel with slightly more in the lower energy parallel state because the electronic system obeys Boltzmann statistics when it is in thermal equilibrium. Then, the ratio of populations of the two states is equal to:

n_{anti-parallel} / n_parallel = e^-ΔE/kT 

equation 4.9

where n represents the populations of the two states, ΔE is the energy difference between the two states, k is Boltzmann’s constant and T is the temperature.

The magnetization that we have been discussing so far is actually 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, n_parallel - n_{anti-parallel} and will be aligned along the z axis. To get an idea of the size of the population differences, if we are working at X-band (~ 9.8 GHz) at room temperature (300 K) with a sample with 10,000 spins, on average 5,004 spins would be parallel and 4996 spins would be anti-parallel resulting in a population difference of only 8. At room temperature and X-band, we are dealing with a small population difference between the two states. When we apply a π/2 pulse to our sample, we no longer have thermal equilibrium. How does this happen? When B₁ rotates the magnetization into the x-y plane, the magnetization along the z axis goes to zero, i.e. the population difference goes to zero. (See Figure 4.14.) If we were to use Equation 4.9 to estimate the temperature of our spins, we would obtain T = infinity. Our spin system is obviously not in thermal equilibrium and through its interactions with the surroundings, it will eventually return to thermal equilibrium. This process is called spin-lattice relaxation.

We could go even one step further and apply a π pulse. This will actually rotate the magnetization anti-parallel to the z-axis, corresponding to more magnetic moments aligned along the -z axis. (This is why a π pulse is often referred to an inversion pulse.) If we use Equation 4.9, we actually calculate a negative temperature.

The rate constant at which M_z recovers to thermal equilibrium is T₁, the spin-lattice relaxation time. The magnetization will exhibit the following behavior after a π/2 pulse:

M_z(t) = M₀ * ( 1 - e^-(1/T₁))

equation 4.10

or after a π pulse:

M_z(t) = M₀ * ( 1 - 2 * e^-(1/T₁))

equation 4.11

In order to extract our signals from the noise, we must signal average the FID by repeating the experiment as quickly as possible and adding up the individual signals. What does “as quickly as possible” mean? We must wait until the magnetization along the 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 FID will be maximum and the following FIDs will eventually approach a limit value that is smaller than the initial value. (See Figure 4.16.)

The transverse relaxation time corresponds to the time required for the magnetization to decay in the x-y plane. There are two main contributions to this process and they are related to different broadening mechanisms: homogeneous and inhomogeneous broadening.

In an inhomogeneously broadened spectrum, the spectrum is broadened because the spins experience different magnetic fields. These different fields may arise from unresolved hyperfine structure in which there are so many overlapping lines that the spectrum appears as one broad signal. (See Figure 4.17.) Typically 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 of spin-packets characterized by a distribution of Δω in the rotating frame. As shown in Figure 4.10, the magnetization of an individual spin-packet will rotate if Δω not equal to 0 and the larger Δω is, the faster it rotates. If we sum up all the components of the individual spin-packets, we see that many components cancel each other out and decrease the transverse magnetization. (See Figure 4.18.) The shape of this transverse magnetization decay (actually a FID) is in general not an exponential decay but instead reflects the shape of the EPR spectrum. The characteristic time constant for the decay is called T₂ * . (T two star.)

So far, all our discussions have been very geometric. It was mentioned that the information about the frequency spectrum was somehow encoded in the transverse magnetization in the rotating frame. One means of reconstructing the frequency spectrum is to study the time behavior of the transverse magnetization. (See Figure 4.19.) The component of the transverse magnetization along the -y axis will vary as:

M_-y = M * cos Δωt

equation 4.14

where Δω is the frequency offset ω-ω0 and t is the time after the microwave pulse. The component along +x will vary as:

M_x = M * sin Δωt

equation 4.15

A common mathematical convenience is to treat these two components as the real and imaginary components of a complex quantity:

M_t = Me^iΔωt 

equation 4.16

where

e^iφ = cos φ + i sin φ 

equation 4.17

and

i= (-1)^1/2 

equation 4.18

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

We do not necessarily have to understand these equations in great detail. Any functions related by Equation 4.19 and Equation 4.20 form what is called a Fourier transform pair. The pairs that we shall encounter frequently are shown in Figure 4.21. The important points to learn are:

• Though a function may be purely real, it will in general have a complex Fourier transform.
• Even functions (f(-t) = f(t) also called symmetric) have a purely real Fourier transform. (See Figure 4.21 a.)
• Odd functions (f(-t) = -f(t) also called anti-symmetric) have a purely imaginary Fourier transform. (See Figure 4.21 b.)
• An exponential decay in the time domain is a lorentzian in the frequency domain. (See Figure 4.21 c.)
• A gaussian decay in the time domain is a gaussian in the frequency domain. (See Figure 4.21 d.)
• Quickly decaying signals in the time domain are broad in the frequency domain.
• Slowly decaying signals in the time domain are narrow in the frequency domain.
• These pairs are reciprocal, i.e. a lorentzian in the time domain results in a decaying exponential in the frequency domain.

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

f(t) + g(t) ⇔ F(ω) + G(ω) 

equation 4.21

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

f(t -Δt)  ⇔ F(ω) * e^-ωΔt 

equation 4.22

f(t) * e^iΔωt ⇔ F(ω-Δω)

equation 4.23

When time shifting, we obtain the original frequency domain signal with a frequency dependent phase shift. As we can see from Figure 4.23, the phase shift transfers some of the real signal to the imaginary and vice versa. This effect leads to the well known linear phase distortion (and correction) in Fourier transform spectroscopy. We start off in Figure 4.23 with a purely real signal (remember that a symmetric signal has a purely real Fourier transform) and after the time delay we obtain an oscillating mixture of real and imaginary components. Because of the reciprocal nature of Fourier transform pairs, similar behavior in the time domain signal is observed when the frequency is shifted in the frequency domain signal.

The convolution integral appears frequently in a number of scientific disciplines. The convolution of two functions is defined as:

f(t) * g(t) = ∫f(τ)g(t-τ)dτ

equation 4.24

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 two functions. In the limit of a Dirac delta function (i.e. a spike), the convolution can be graphically represented as in Figure 4.24. We are placing a copy of our function at each of the spikes.

Now its time to start applying what we have learned in the previous sections to a concrete problem, predicting what a time domain signal (e.g. a FID) looks like if we are given a frequency domain signal (e.g. an EPR spectrum). As an example, we consider a three line EPR spectrum such as a nitroxide. (See Figure 4.25.) We assume that the magnetic field is set so that the center line is on-resonance, the lines are lorentzian, and the splitting is equal to A. Remember that in an FT experiment we are detecting both the absorption (real) and dispersion (imaginary) signals.

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

We know from the convolution theorem that the time domain signal is simply the product of the two transformed functions. (See Equation 4.26.) We already know the Fourier transform for a lorentzian:

e^-t/T2 

equation 4.27

Next we have to calculate the Fourier transform of the three line stick spectrum. One thing that helps is that this signal is symmetric, yielding a purely real time domain signal. Using the additive properties of Fourier transforms, we express the three line stick spectrum as the sum of two signals with known Fourier transforms. Adding the two time domain signals gives us the Fourier transform of the stick spectrum.

Multiplying the two time domain functions gives us the result in Figure 4.28. This is the FID of the three line EPR spectrum.

The next three figures are examples of what happens to the FID when the EPR signal changes.

A little bit of care is required when comparing conventional field swept spectra and frequency spectra obtained by FT-EPR. The field and frequency axes run in opposite directions. Here are two spectra of the same sample. The upper spectrum is a frequency spectrum acquired by Fourier transforming the echo (To be discussed in the next section.). The lower spectrum was acquired in a conventional field swept experiment.

In Figure 4.33 we see the Larmor frequencies when the field is set so the center line is on-resonance. The higher field line actually has a lower (negative) Larmor frequency than the center line. We need to apply more magnetic field to increase its Larmor frequency so that it would be on-resonance with the microwaves. The lower field line has a higher Larmor frequency.

As we have seen in the previous sections, one microwave pulse produces a signal that decays away (FID). If our EPR spectrum is inhomogeneously broadened, we can recover this disappeared signal with another microwave pulse to produce a Hahn echo.

How does the echo bring back our signal? The decay of the FID is due to the different frequencies in the EPR spectrum causing the magnetization to fan out in the x-y plane of the rotating frame. When we apply the π pulse, we flip the magnetization about the x axis. The magnetization still rotates in the same direction and speed. This almost has the effect of running the FID backwards in time. The higher frequency spin packets will have traveled further than the lower frequency spin packets after the first pulse. However, because the higher frequency spin packets are rotating more quickly, they will eventually catch up with the lower frequency spin packets along the +y axis after the second pulse. (See Figure 4.35.)

After all the spin packets bunch up, they will dephase again just like a FID. So one way to think about a spin echo is a time reversed FID followed by a normal FID. Therefore, if we Fourier transform the second half of the FID, we obtain the EPR spectrum.

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

echo height(τ) α e^-2τ/T_m

equation 4.28

where T_M, the phase memory time, is the decay constant. Many processes contribute to T_M such as T₂ (spin-spin relaxation), as well as spectral, spin, and instantaneous diffusion.

Notice the factor of two in Equation 4.28 which is not in the expression for the FID. This is because dephasing starts after the first pulse and the echo occurs at 2 after the first pulse. So by studying the echo decay as we increase τ, we can measure T_M.

Spectral diffusion often is a large contributor to T_M. Nuclear spin flip-flops, molecular motion, and molecular rotation can cause spin packets to suddenly change their frequency. A faster spin packet far from the +y axis will suddenly become a slower spin packet without the needed speed to catch up with the other spin packets in their race to refocus. Therefore, we are not refocusing all the magnetization. In Figure 4.37 we see that after the runner marked with an asterisk has a shifted frequency, we only get four of the five runners lining up to refocus.

A very important class of echo experiments is ESEEM (Electron Spin Echo Envelope Modulation). The electron spins interact with the nuclei in their vicinity and this interaction causes a periodic oscillation in the echo height superimposed on the normal echo decay. The modulation or oscillation is caused by periodic dephasing by the nuclei. If we subtract the decay of the spin echo and Fourier transform the oscillations, we obtain the splittings due to the nuclei. Armed with this information, you can identify nearby nuclei and their distances from the electron spin and shed light on the local environment of the radical or metal ion.

Hahn or two pulse echoes are not the only echoes to occur. If we apply three π/2 pulses we obtain five echoes. Three of the echoes are simply two pulse echoes produced by the three pulses. The stimulated and refocused echoes only occur when you have applied more than two pulses.

In Pulse EPR spectroscopy, we often can excite only a small portion of our EPR spectrum. This fact simplifies things when performing echo experiments. First, if we have a very broad EPR spectrum, within the range of our excitation of the spectrum it looks almost flat and therefore approximately symmetric. As a consequence, our echo will be purely real with no imaginary component. Second, the echo width is approximately equal to the pulse width.

Quite often it is more convenient to use two equal length pulses instead of the traditional π/2 - π pulse sequence. The reason for doing this is the π pulse is twice as long as the π/2 pulse and therefore will limit the amount of the EPR spectrum we can excite. (See Figure 4.21e.) With a bit of calculus, it can be shown that the maximum echo height for two equal length pulses is achieved with two 2π/3 (120°) pulses. (See Figure 4.41.) The narrower 2π/3 pulses excite a broader portion of our spectrum than the π pulse can.

Sometimes both hard (short) and soft (long) pulses are combined together in one experiment. For example, to perform a Davies pulse ENDOR experiment, you use a soft π pulse to burn a narrow hole in the EPR spectrum (See Figure 4.42.) and two narrow pulses to detect it.

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