Pulsed electron paramagnetic resonance

Pulse Electron paramagnetic resonance (EPR) is very powerful spectroscopic technique related to common nuclear magnetic resonance (NMR). Its most basic form involves the alignment of the net magnetization vector of the electron spins in a constant magnetic field. This alignment gets perturbed by applying a short oscillating field, usually a microwave pulse. One can then measure the emitted microwave signal which is created by the sample magnetization. Using Fourier transformation the microwave signal gives an EPR spectrum in the frequency domain. With a vast variety of pulse sequences it is possible to gain extensive knowledge on structural and dynamical properties of paramagnetic compounds. Pulsed EPR techniques such as Electron spin echo envelope modulation (ESEEM) or pulsed Electron nuclear double resonance (ENDOR) can reveal the interactions of the electron spin with its surrounding nuclear spins.

Scope

Electron paramagnetic resonance (EPR) or Electron spin resonance (ESR) is an spectroscopic technique widely used in biology, chemistry, medicine and physics to study systems with one or more unpaired electrons. Due to the relation between the magnetic parameters, electronic wavefunction and the configuration of the surrounding non-zero spin nuclei, EPR and ENDOR provide exclusive information on the electronic structure. This leads to detailed information on the structure, dynamics and the spatial distribution of these paramagnetic species. Although EPR spectroscopy shows a huge potential, it has limitations in spectral and time resolution when used with traditional continuous wave methods. Pulse EPR can distinctively address a given problem and offers high resolution by investigating interactions separately from each other via pulse sequences. Nowadays a multitude of such experiments is available, leading to numerous possibilities in increasing spectral resolution or designing new methods for solving application problems.

Historical Overview

The first works in pulsed EPR were conducted in the group of W.B. Mims at Bell laboratories during the 1960`s. In the first decade only a small number of groups worked the field, because of the too expensive instrumentation, the lack of suitable microwave components and slow digital electronics. After the first pulsed ENDOR experiments in 1965 another cornerstone of pulsed EPR emerged, the ESEEM technique. In the 1980`s, the upcoming of the first commercial pulse EPR and ENDOR spectrometers in the X-Band frequency range, lead to a fast growth of the field. In the 1990`s, parallel to the upcoming high-field EPR, pulse EPR and ENDOR became a new fast advancing magnetic resonance spectroscopy tool and the first commercial pulse EPR and ENDOR spectrometer at W-band frequencies came on to the market. Today pulse EPR is a very active field with emphasis on high-frequency pulse ENDOR spectroscopy. ^[1]

Principle

The basic principle of pulsed EPR is closely comparable with NMR spectroscopy. Differences can be found in the relative size of the magnetic interactions and in the relaxation times which are around three orders of magnitudes shorter in EPR than NMR. A full description of the theory is given in the quantum mechanical picture but since we are measuring the magnetization as a bulk property we can give a more intuitive picture in the classical description. For a better understanding of the concept of Pulse EPR we need to explain the effects on the magnetization vector in the laboratory frame as well as in the rotating frame. In the lab frame we assume the static magnetic field B₀ to be parallel to the z-axis and the microwave field B₁ parallel to the x-axis. When an electron spin is placed in an magnetic field it experiences a torque which causes its magnetic moment to precess around the magnetic field. The precession frequency is known as the Larmor frequency ω_L.

$\omega _{L}=-\gamma B_{0}$

where γ is the gyromagnetic ratio and B₀ the magnet field. The electron spins are characterized by two quantum mechanical states, one parallel and one antiparallel to B₀. Due to the lower energy of the parallel state more electron spins can be found in this state according to the Boltzmann distribution. This results in a net magnetization, which is the vector sum off all magnetic moments in the sample, to be parallel to the z-axis in alignment with the magnetic field. To better comprehend the effects of the microwave field B₁ it is easier to move to the rotating frame. EPR experiments usually use a microwave resonator designed to create a linear polarized microwave field B1, perpendicular to the much stronger applied Magnetic field B₀. The linear polarized microwaves which can be seen as a magnetic field oscillating at the microwave frequency. In the rotating frame we rotate with one of the rotating B₁ components. First we assume to be on resonance with our precessing magnetization vector M₀.

$\omega _{L}=\omega _{0}$

Therefore the component of B₁ will appear stationary. In this frame also the precessing magnetization components appear to be stationary which leads to the disappearance of B₀ and we need only to concern ourselves with B₁ and M₀. Now the M₀ vector is under the influence of the stationary appearing, field B₁, leading to another precession of M₀, this time around B₁ at the frequency ω₁.

$\omega _{1}=-\gamma B_{1}$

This angular frequency ω₁ is also called the Rabi frequency. If we assume B₁ to be parallel to the x-axis , the magnetization vector will rotate around the +x-axis in the zy-plane as long as the microwaves are applied. The angle by which M₀ is rotated, is called the tip angle α, given by:

$\alpha =-\gamma |B_{1}|t_{p}$

Here t_p is the duration at which B₁ is applied, also called the pulse length. The pulses are labelled by the rotation of M₀ which they cause and the direction from which they are coming from, since we can phaseshift the microwaves from the x-axis on to the y-axis. For example a +y π/2 pulse means that a B₁ field, which has been 90 degree phase shifted out of the +x into the +y direction, has rotated M₀ by a tip angle of π/2, hence the magnetization would end up along the –x-axis. That means the end position of the magnetization vector M₀ depends on the length, the magnitude and direction of the microwave pulse B₁. In order to understand how the sample emits microwaves after the intense microwave pulse we need to go back to the lab frame. In the rotating frame and on resonance the magnetization appeared to be stationary along the x or y-axis after the pulse. In the lab frame it becomes a rotating magnetization in the x-y plane at the Lamour frequency. This rotation generates our signal which is maximized if the magnetization vector is exactly in the xy-plane. This microwave signal generated by the rotating magnetization vector is called Free induction decay (FID). Another assumption we have made was the exact resonance condition, in which the Lamour frequency is equal to the microwave frequency. In reality EPR spectra have many different frequencies and not all of them can be exactly on resonance, therefore we need to take off-resonance effects into account. The off-resonance effects lead to three main consequences. The first consequence can be better understood in the rotating frame. A π/2 pulse kicks our magnetization in the xy-plane, since the microwave field and therefore our rotating frame have not the same frequency as the precessing magnetization vector, the magnetization vector rotates in the xy-plane, either faster or slower than the microwave magnetic field B₁. The rotation rate is governed by the frequency difference Δω.

$\Delta \omega =\omega -\omega _{0}$

If Δω is 0 the microwave field rotates as fast as the magnetization vector and both appear to be stationary to each other. If Δω>0 the magnetization rotates faster than the microwave field component in a counter-clockwise motion and if Δω<0 the magnetization is slower and rotates in a clockwise motion. This means that the individual frequency components of the EPR spectrum, will appear as magnetization components rotating in the xy-plane with the rotation frequency Δω. The second consequence appears in the lab frame. Here one understands that B₁ tips the magnetization differently out of the z-axis, since B₀ does not disappear when not on resonance due to the precession of the magnetization vector at Δω. That means that the magnetization is now tipped by an effective magnetic field B_eff, which results out of the vector sum of B₁ and B₀. The magnetization is then tipped around B_eff at the faster effective rate ω_eff.

$\omega _{eff}^{2}=(\omega _{1}^{2}+\Delta \omega ^{2})^{1/2}$

This leads directly to the third consequence that we cannot tip the magnetization efficiently into the xy-plane because B_eff does not lie in the xy-plane, as B₁ does. The motion of the magnetization defines now a cone. That means as Δω becomes larger, the less effective the magnetization can be tipped in the xy-plane which decreases the FID signal. In broad ERP spectra where Δω > ω₁, it is not possible to tip all the magnetization into the xy-plane to create a good FID signal. This is why it is important to maximize ω₁ or minimize the π/2 pulse length for broad EPR signals. So as B₁ gets larger and the pulses shorter we can detect more of our EPR spectrum.

So far we have tipped our magnetization into the xy-plane and it remained there with the same magnitude. But in reality the electron spins interact with their surroundings and the magnetization in the xy-plane will decay away and eventually return to alignment with the z-axis. This relaxation process is describes by the spin-lattice-relaxation time T₁, which describes the time needed by the magnetization to return to the z-axis and the spin-spin-relaxation time T₂, which describes the vanishing time of the magnetization in the xy-plane. The spin-lattice relaxation results from the urge of our system to return to the thermal equilibrium state after it has been hit by the B₁ pulse. To return the magnetization parallel to B₀, the system uses interactions with the surroundings leading to the spin lattice relaxation. This relaxation time needs to be considered when extracting the FID signal from the noise, where the experiment needs to be repeated as quickly as possible for several times. Therefore, in order to repeat the experiment, one needs to wait until the magnetization along the z-axis has recovered, because if there is no magnetization in z direction, then there is nothing to tip into the x-plane to create a good FID signal. The spin-spin relaxation time also called transverse relaxation time is related to homogenous and inhomogeneous broadening. An inhomogeneous broadening results from the fact that the different spins experience local magnetic field inhomogeneities (different surroundings) creating a large number of spin packets characterized by a distribution of Δω. As the net magnetization vector precesses, some spin packets slow down due to lower fields and others speed up due to higher fields leading to a fanning out of the magnetization vector which make the signal in the EPR spectrum decay. The other contribute to the transverse magnetization decay is due to the homogenously broadening in which all the spin in one spin packet experience the same magnetic field and interact with each other, which can lead to mutual and random spin flip-flops. These fluctuations contribute to a faster fanning out of the magnetization vector.

All the information about our frequency spectrum are somehow encoded in the motion of the transverse magnetization. To reconstruct the frequency spectrum we can study the time behaviour of the transverse magnetization made up of with a y and x-axis component. It is convenient that these two can be treated as the real and imaginary components of a complex quantity. The reason for this representation is that one can now use Fourier theory to transform the measured time domain signal into the frequency domain representation. This is possible since we are detecting both the absorption (real) and the dispersion (imaginary) signals.

As we have seen before, the FID that we measures decays away. For very broad EPR spectra this decay is very fast, due to the inhomogeneous broadening. To get more out of the measurement one can recover the disappeared signal with another microwave pulse to produce a Hahn echo. After applying a π/2 pulse (90°), the magnetization vector is tipped into the xy-plane producing a FID signal. Due to different frequencies in the EPR spectrum (inhomogeneous broadening) this signal "fans out". That means that the slower spinpackets trail behind the faster ones. If we now, after a certain time t apply a π pulse (180°), we basically turn the whole xy-plane for the fanned out FID signal and the fast spinpacket are then behind catching up with the slow spinpackets. A complete refocusing of our signal occurs than at 2t. An accurate echo caused by a second microwave pulse can remove all inhomogeneous broadening effects. After all spinpackets bunch up, they will dephase again just like a FID. In other words a spin echo is a reversed FID followed by a normal FID, which we can Fourier transform and obtain our EPR spectrum. As longer the time between the pulses becomes, the lower the echo will be due to the relaxation of the magnetization vector back into alignment with the z-axis. This relaxation leads to an exponential decay in the echo height. The decay constant is the phase memory time T_M, which has many contributors such as transverse relaxation, spectral, spin and instantaneous diffusion. Changing the times between the pulses leads to a direct measurement of T_M and one can gain valuable information on the studied system.

Applications

The most popular echo experiments is ESEEM in which the interaction of the electron spins with the surrounding periodic dephasing nuclei causes a periodic oscillation of the echo height. With this one can identify nearby nuclei and their distances from the electron spin leading to extensive knowledge on the local environment. Another echo experiment is given by pulsed ENDOR. In contrast to continuous wave ENDOR, in which microwave and radio frequency fields are continuously applied, both fields are only applied in short pulses in pulsd ENDOR.

References

^ Schweiger, A. ; Jeschke, G. (2001). Principles of pulse electron paramagnetic resonance. Oxford University Press, New York.{{cite book}}: CS1 maint: multiple names: authors list (link)

External links

Pulse EPR Bruker BioSpin

[multiple-1] Schweiger, A. ; Jeschke, G. (2001). Principles of pulse electron paramagnetic resonance. Oxford University Press, New York.{{cite book}}: CS1 maint: multiple names: authors list (link)

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