Basics of Pulsed Magnetic Resonance
CW vs. FT Magnetic Resonance. In order to understand pulsed EPR, first the
difference between CW (continuous wave) and FT (Fourier Transform) techniques must
be realized. We can compare these techniques by using an analogy which relates them to
musical instruments, where our sample is a guitar which is continuously playing a unique
chord, and our magnet is a piano. For CW techniques, we would play each key on the
piano in succession, detecting resonances between the frequencies in the guitar chord and
frequencies of the piano notes along the way. We basically perform the same experiment
in CW EPR, only the magnetic field is swept instead of its frequency and we detect any
resonances in our sample. An alternative approach is to strike each key on the piano at
the same time and Fourier transform the resulting sound to obtain the frequency spectrum
of the guitar; this fact is called the multiplex advantage and is fundamental to pulsed
magnetic resonance experiments. In pulsed EPR, we apply a short and intense
microwave pulse consisting of a finite bandwidth of frequencies, digitize the signals
coming from our sample, and perform a Fourier transformation to obtain the EPR
spectrum in the frequency domain.
Spin Behavior in Static Magnetic Fields. One must understand the equilibrium
nature of electron spins in an external magnetic field before the behavior of the spins
during and after a microwave pulse can be comprehended. All electrons have the
intrinsic property of spin, which gives rise to a magnetic moment. When an electron spin
is placed in a magnetic field, a torque is exerted on the spin, causing its magnetic moment
to precess about the magnetic field like a spinning top about a gravitational field. The
angular frequency of this precession is called the Larmor frequency and is related to the
magnetic field by:
ωL = -γB0
where ωL is the Larmor frequency, γ is the gyromagnetic ratio, and B0 is the external
magnetic field. If we consider a large number of electron spins precessing about a
magnetic field aligned along the z axis, each electron spin will assume one of two
quantum states and align either parallel (spin quantum number ms = -½) or antiparallel
(ms = +½) to the magnetic field. The parallel state has lower energy, and thus at thermal
equilibrium there is a surplus of electron spins in the parallel state according to the
Boltzmann distribution. This yields a net magnetization M0 along the +z axis (Figure 1).
Because the magnetization is a vector sum of all the individual magnetic moments in the
sample, for a large enough number of spins the x-y component will cancel out and the
resultant net magnetization will be along the +z axis and exactly parallel to the external
field.
Pulses and the FID. As B0 is our static magnetic field used to establish the
equilibrium condition, we need an additional magnetic field B1 to perform our
experiment. Additionally, our detection coils must lie in the x-y plane so that magnetic
effects from our sample after perturbation from equilibrium are not masked by the
external field. Thus, only transverse magnetization, or magnetization having an x-y
component, will give a signal. Typically B1<<B0 and the heart of pulse EPR experiments
lies in the manipulation of the magnetization by B1 pulses that have specific tip angles,
and then the subsequent detection of the magnetic behavior during its return to
equilibrium. Pulses are often named by their tip angles, and the most commonly
employed tip angles are π/2 (90 degrees) and π (180 degrees). A π/2 pulse will tip the
magnetization into the x-y plane, and is also called a saturating pulse because the
magnetization along the z axis goes to zero, i.e. the population difference between
parallel and antiparallel states goes to zero. A π pulse is also called an inversion pulse,
because it tips the magnetization 180 degrees, exchanging the populations of the quantum
states (Figure 2). The utility of a π pulse will be made evident later.
Upon tipping the magnetization into the x-y plane with a π/2 pulse, the
magnetization is stationary* and the spins establish phase coherence. However, because
the electron spins interact with their surroundings, the transverse magnetization will
decay away and return to equilibrium through relaxation processes after the pulse ends.
During this event, the stationary magnetization rotates in the x-y plane at the Larmor
frequency as it relaxes back to equilibrium, which generates currents and voltages in the
resonator that are detected as our signal. This is called a Free Induction Decay (FID),
which encodes the frequency-domain spectrum.
Relaxation. After an intense microwave pulse, the magnetization will interact
with its surroundings and return to equilibrium. This process is called relaxation, and is
characterized by two time constants, T1 and T2. The spin-lattice relaxation time, T1,
describes how quickly the magnetization recovers its longitudinal component along the z
axis, in other words, it is the amount of time in which the energy absorbed from the pulse
is dissipated to the lattice as the system returns to equilibrium. By rule, T2 is faster than
T1, and thus T1 determines the repetition rate of our experiment.
* When viewed from the rotating frame. For simplicity, the rotating frame representation
will be used throughout this work when discussing the behavior of non-equilibrium
magnetization.
The transverse relaxation time, T2, describes how quickly the net magnetization
dissipates in the x-y plane, i.e. how quickly the spins lose coherence and fan out 360
degrees into randomized precession. Because our detection coils lie in the x-y plane,
transverse relaxation usually determines the length of time that we can acquire a signal
(Figure 3).
Two main processes contribute to relaxation, and they are related to different
broadening mechanisms: homogeneous and inhomogeneous broadening. In a
homogeneously broadened spectrum, the lineshape is the sum of many lines each having
the same Larmor frequency and linewidth (Figure 4); that is, all the spins experience the
same magnetic field. The spins interact with each other, resulting in random and mutual
spin flip-flops (called spin-spin relaxation). T1 and T2 both result in homogenous
broadening. Molecular motion can also contribute to this effect, and the overall result is
an exponential decay of the magnetization.
An inhomogeneously broadened spectrum is broadened because the spins in
various parts of the sample experience slightly different effective magnetic fields; e.g. B1
inhomogeneities and unresolved hyperfine structure. As seen in fig. 4, the
inhomogeneously broadened lineshape is determined by unresolved couplings amongst
individual spin packets that are themselves homogeneously broadened and shifted in
frequency with respect to each other, so that at any point in time only a small fraction of
the spins are in resonance. This distribution of frequencies results in several magnetic
components that cancel each other out and decrease the transverse magnetization. Thus,
the overall contribution to transverse relaxation can be described more accurately by the
time-constant T2*:
(T2*)-1 = (T2)-1 + (Tinhom)-1
Unlike homogeneous broadening, the shape of this decay is not exponential but rather
reflects the shape of the EPR spectrum.
Lorentzian
Gaussian
Echoes. As discussed earlier, a single π/2 pulse produces an FID, and the Fourier
transform of the FID will give the frequency domain EPR spectrum. Immediately after
the pulse is turned off, relaxation effects begin and our signal starts to disappear. Ideally,
we could begin to collect our signal immediately after the pulse is turned off; however,
the spectrometer cannot acquire the signal directly after the microwave pulse because the
receiver would be destroyed by the high power. Additionally, our signal is in the nW
range while the applied pulses are about 1 kW, so our signal would be masked. Thus,
there is a lag between the end of the pulse and when the signal can be measured; this is
called dead time and is typically ~80 ns, depending on the quality factor, or Q, of the
resonator.
While homogeneous broadening results in random and irreversible events,
inhomogeneous broadening is reversible and static (no net energy is lost). Because our
EPR spectrum is usually inhomogeneously broadened, we can resurrect the disappeared
signal with an additional pulse to produce an echo. If we apply the second pulse at a time
t after dephasing begins, then the echo maximum will occur at time t after the second
pulse. This second pulse produces a signal after the dead time, and the echo shape
resembles two back-to-back FIDs. For simplicity, we will consider what happens during
a Hahn echo (Figure 5).
The decay of the FID in an inhomogeneously broadened spectrum is due to the dephasing
of spin packets of different frequencies, causing the magnetization to fan out in the x-y
plane. If we look down the z axis at the x-y plane (fig. 5 top), there will be faster moving
spin packets and slower moving spin packets, depending on their relative frequencies.
When we apply a π pulse, the magnetization is inverted 180 degrees yet still rotates in the
same direction with the same speed. Now, instead of moving away from the slower spin
packets, the faster moving spin packets move towards the slower spin packets. The
magnetization eventually refocuses to form what is called an echo before it resumes
relaxation.
In comparison to the FID decay, there are more contributions to the echo decay
than simply transverse relaxation that affect our signal intensity and duration. The time
constant for echo decay is called the phase memory time, or TM. Many processes
contribute to the phase memory time, such as spin-spin relaxation, spectral diffusion, spin
diffusion, and instantaneous diffusion. Other than spin-spin relaxation, spectral diffusion
can often be a large contributor to a short TM. Processes such as nuclear spin flip-flops,
molecular motion, and molecular rotation can cause spin packets to change their
frequency. This frequency change affects the ability of the spin packet to refocus
properly, and thus the echo intensity decreases.
The DEER Experiment
Introduction to ELDOR and DEER. ELDOR (ELectron DOuble Resonance) is a
two-frequency technique in which we observe at one frequency the effects on the
inhomogeneous EPR spectrum upon perturbation of the EPR spectrum at a second
“pumping” frequency. In order for the signal to change at the observe frequency, there
must exist a mechanism that links the observed and pumped regions of the EPR
spectrum. Examples of such mechanisms include the dipolar interaction between spins,
molecular motion, or forbidden hyperfine transitions.
When two spins are far enough apart, the exchange interaction drops off and we
are left with only the dipolar interaction, which consists of an r-3 distance dependence.
For distances longer than ~1.5 nm, the dipolar interaction is hidden within the
inhomogeneous linewidth, however DEER (Double Electron Electron Resonance) in an
ELDOR technique which is able to resolve this interaction in order to measure distances
between unpaired electrons.
The dipolar coupling is dependent on both the distance and the angle between the
spin-to-spin vector and external field; in most cases the radicals are randomly oriented
and the interaction can be described by a Pake pattern (Figure 6).
The Pake pattern is a Fourier transform of the DEER signal, and the splitting of the two
prominent features represents twice the dipolar coupling frequency. Once the dipolar
coupling frequency is known, the interspin distance can be calculated. As will be
explained later, the interspin distance is typically not found via the Pake pattern anymore,
because small distortions in the time-domain data may have large effects on the
computation of the distance distribution. For this reason, mathematical algorithms and
regularizations are applied to stabilize the solution against noise-induced artifacts.
Four-pulse DEER. The DEER signal is a modulation of echo intensity of one
spin population as the pump pulse timing of a second spin population is changed (Figure
7). The timing of the three observe frequency pulses remains constant, however the
pump frequency pulse starts at a time slightly before the first Hahn echo, and the timing
of its application is varied over several repetitions of the DEER pulse sequence. The
resulting oscillations in the integrated echo intensity occur at the dipolar interaction
frequency. Because the dipolar frequency is inversely proportional to the cube of the
interspin distance, longer distances yield lower frequency oscillations, which can
sometimes be difficult to characterize. For this reason, the range of obtainable distances
with the DEER experiment is 2-8 nm (CW EPR can be used for shorter distance
measurements). The DEER signal is composed of two parts: an exponential decay due to
intermolecular interactions of spins, and the periodic oscillations due to the
intramolecular interactions of paired spins. As we want only the intramolecular
contribution (the “form factor”), the decay from intermolecular interactions (the
“background factor”) can be fitted to an exponential curve and subtracted out after the
experiment.
During the four-pulse DEER experiment (Figure 7), an initial π/2 pulse tips the
magnetization into the x-y plane. The A spins precess at frequency ωA (but appear
stationary in the rotating frame). After the pulse ends, the presence of field
inhomogeneities and different resonance fields dephase the spins. One of these
contributions stems from the dipolar interaction, which adds or subtracts to the angular
frequency a value of ± ½ωAB, depending on the quantum state of the coupled B spin. The
subsequent observe frequency π pulse reverses the dephasing and refocuses the A spins to
ωA, resulting in an undetected Hahn echo. Once the spins again resume relaxation, the
dipolar frequency is extracted by the application of an inversion “pump” pulse to the B
spins, which changes the dipolar contribution experienced by the A spins from +½ωAB to
-½ωAB, and vice versa. The change of the dipolar contribution causes these spins to
acquire a phase lag, which affects their ability to be refocused by the final π pulse applied
at the observe frequency—and therefore the resulting echo intensity is also affected. The
amount of phase lag imposed on the A spins is a function of both the time at which the
pump pulse is applied to the B spins, and the dipolar frequency. Since the dipolar
frequency is a static contribution, these two variables can be decoupled by changing the
time at which the pump pulse is applied. The resulting DEER echo intensity varies with
the cosine of the phase lag, and its oscillations are proportional to the dipolar coupling
frequency.
Practical Considerations. The optimization of experimental conditions depends
on many factors, so only a few important ones will be discussed here: temperature,
concentration, delay times, and data analysis.
The signal-to-noise ratio (S/N) varies with temperature due to the temperature
dependence of spin transitions and relaxation times. The polarization of spin transitions
is a Boltzmann distribution of level populations, which is inversely proportional to
temperature. In addition, relaxation becomes slower at lower temperatures, so one may
think that performing the DEER experiment under the lowest attainable temperature
would be ideal. However, after a certain point the gain in S/N due to optimal polarization
and phase memory times is overcompensated for by the loss in S/N due to a long spinlattice relaxation time. This is because T1 determines the repetition rate of our
experiment, and for a given measurement time the S/N increases with the square root of
the number of repetitions. Usually the optimal temperature for DEER is between 50-65
K, however these temperatures require the use of expensive helium as a cryogen instead
of nitrogen. Consequently, nitrogen is usually used at a temperature of 80 K unless the
relaxation times are too quick at that temperature.
The signal amplitude increases with concentration up to a certain point where
instantaneous diffusion kills the phase memory time. When refocusing with π pulses,
instantaneous diffusion owes an additional exponential decaying factor to the echo.
Sufficient S/N can be realized in DEER experiments of doubly-labeled proteins with a
concentration of 100 µM at X-band frequencies.
The dipolar frequency is inversely proportional to the cube of the interspin
distance, and so long distances yield DEER echoes with low frequency oscillations. For
this reason, a longer delay time d2 is required to adequately characterize the oscillation in
echo intensity with the pump pulse (Figure 7). However, a short phase memory time may
limit the delay time that can be allotted between the second and third observer π pulses;
once the transverse magnetization has completely fanned out such that there is no Mxy,
the spins cannot be refocused. For this reason, in order to resolve longer distances, the
TM must be maximized so that d2 can be extended sufficiently (Figure 8).
d2 (µs) – 0.1 nm
d2 (µs) – 0.2 nm
d2 (µs) – 0.5 nm
resolution
resolution
resolution
2.0
1.0
0.5
0.2
2.5
2.5
1.3
0.5
3.0
5.2
2.6
1.0
3.5
9.6
4.8
1.9
4.0
16.4
8.2
3.2
4.5
26.3
13.7
5.2
5.0
40
20
8.0
rAB (nm)
Figure 8. Delay times required for a given resolution in the distance distribution at a
given distance.
Analysis of the DEER data will reveal one or more probability distributions
corresponding to the interspin distances. If the spin-labeled protein backbone is rather
immobile, then the distance distribution will be narrow compared to a more flexible
backbone. Narrower distance distributions yield more oscillations in the echo intensity,
because there is less dephasing. For broad distributions, the multiple dipolar frequencies
over that distribution will dephase each other, leading to destructive interference which
dampens the oscillation in echo intensity quicker. The dynamical information is as
important as the distances in DEER experiments, therefore it is desirable to extend the TM
and d2 as long as possible in order to obtain the best agreement between the experimental
and true distance distributions (fig. 8). Further uncertainties in the distance distribution
of up to 1.6 nm may be introduced as a result of conformational distributions of spin label
rotamers. In order to circumvent this problem, we may use derivatized spin labels.
Computing the distance distribution from the Pake pattern or time-domain signal
is an ill-posed problem. Small distortions in the DEER signal or Pake spectrum such as
noise, or deviations from the ideal Pake spectrum due to orientation selection can have
large effects on the distance distribution. For this reason, mathematical algorithms must
be applied that stabilize the solution against noise-induced artefacts by requiring a certain
smoothness of the distance distribution. Tikhonov regularization is the most widely used
approach in science and engineering to solve this ill-posed problem. In this approach, the
smoothing is applied in such a way that allows for a compromise between the
minimization of the mean square deviation of the theoretical and experimental form
factors, and the minimization of the roughness. This compromise is defined by the
minimization of the target function:
Gα(P) = ρ + αη
Where ρ is the mean square deviation between the theoretical and experimental form
factors, η is the roughness, and α is the regularization parameter. The optimal α which
minimizes the target function is found by the L curve criterion, which is a parametric plot
of logη(α) vs. logρ(α); the optimum α corresponds to the corner of the L-shaped curve.
Tikhonov regularization involves a compromise, and if distance distributions involve
both broad and narrow components, then it may not be the best approach. In these cases,
more accurate distance distributions can be obtained by the superposition of several
Gaussian peaks.
Although there are many improvements that must be made in terms of the reliable
interpretation of data, the DEER experiment is a powerful method for measuring
intramolecular distances in terms of sensitivity and specificity. The DEER experiment
can complement techniques such as FRET, NMR, and X-ray crystallography.
Glossary
B1 inhomogeneity – Ideally, the applied microwave field would be uniform across the
resonator; however, this is not the case and different parts of our sample can experience
slightly different magnetic fields.
Coherence – A condition in which two signals maintain a fixed phase relationship
relative to each other and therefore have identical frequencies (or sets of frequencies).
Coherence transfer – The crucial mixing process in DEER in which the pump pulse
transfers coherence between two dipolar coupled spins, so that electron coherence which
was formerly evolving with ωA=ΔωA + ½ωAB before the pump pulse now evolves with
ωA=ΔωA - ½ωAB, and vice versa (ωA is the resonance frequency of the observed spins,
ωAB is the dipolar coupling frequency, and ΔωA is the resonance offset of the observed
spins).
Fourier transformation – A process in which one function is transformed into a different
function. Fourier analysis has many applications in math and science, but in magnetic
resonance the Fourier transform is a complex integral which is used to switch data
between its frequency-domain and time-domain representations.
Instantaneous diffusion – In circumstances wherein the distance between spins is small
(such as at high spin concentrations or in aggregates), a quickly inverted spin changes the
local field at a nearby spin and causes its frequency to shift. Because a frequency shift
also results in a phase shift, this interferes with that spin’s ability to be refocused. This
makes the phase memory time, TM, shorter.
Pulse – An abrupt application of magnetic radiation; a pulse is characterized by its
strength, duration, and frequency range.
Quality Factor (Q) – The Q of a resonator is the ratio of the energy stored to the power
dissipated by the resonator, and is related to the impedance match between the resonator
and the microwave source. In CW EPR we want a high Q, or a critically coupled
resonator, because high Q’s are good at converting magnetization into a detectable signal.
In pulsed EPR, we want a low Q, or an overcoupled resonator, in order to dissipate the
power quicker and minimize the dead time. Low Qs also increase the frequency
bandwidth of the resonator, which is important in ELDOR experiments.
Resonator – Optical absorptions are much stronger than magnetic resonances due to
greater population differences between quantum states and the relative strength of the
electric dipole transition moment versus the magnetic dipole transition moment. For this
reason, we use resonators in magnetic resonance experiments, which enhance the
sensitivity. Resonators do this by converting the microwaves into B1, focusing this
applied field onto our sample, and converting the resulting magnetic behavior into a
detectable signal.
Rotating Frame – The entire three-dimensional coordinate system is rotated at an angular
velocity equal to the applied microwave frequency; as a result, when we are on resonance
both the magnetic components precessing at the Larmor frequency and B1 appear to be
stationary. This makes the visualization and mathematical treatment of the magnetic
behavior easier, especially when one is new to these concepts. If we are off resonance,
i.e. the Larmor frequency is not exactly equal to the applied microwave frequency, then
the magnetization vector will appear to rotate in the x-y plane at the resonance offset Δω
= ω – ωmw. This is usually the case because of different resonance fields and B1
inhomogeneities.
Spectral diffusion – A process by which the frequency of a spin packet changes with
time, resulting in a broadening of the spectrum. Spectral diffusion can be caused by
molecular motion, exchange interactions, nuclear spin flip-flops, or electron-nuclear
cross-relaxation.
Spin diffusion – A process by which spins undergo the continuous exchange of energy via
spin flip-flops, causing energy to be dissipated throughout the sample rather than being
lost to the lattice. This effect reduces polarization differences between quantum states to
a timescale which is much shorter than the intrinsic relaxation rate.
Spin packet – One of the many individual homogeneously broadened lines that contribute
to an inhomogeneously broadened spectrum.
Tip angle – The angle by which the magnetization is rotated by a pulse. The tip angle
depends on both the strength of the magnetic field and the duration of its application.
References
1. Most of the figures and information are from Ralph T. Weber, Ph.D., at Bruker
Biospin Corporation: www.bruker-biospin.com
2. Hemminga, M., Berliner, L.J. ESR Spectroscopy in Membrane Biophysics.
Springer, vol. 27. 2007.
3. Jeschke, G. and Polyhach, Y. Distance measurements on spin-labelled
biomacromolecules by pulsed electron paramagnetic resonance. Phys. Chem.
Chem. Phys. 9: 1895-1910. 2007.
4. Pannier, M., Veit, S., Godt, A., Jeschke, G., Spiess, H.W. Dead-time free
measurement of dipole-dipole interactions between electron spins. J. Mag. Res.
142: 331-340. 2000.
5. Cafiso lab group website: http://ernst.chem.virginia.edu/dsclab