Pulsed electron paramagnetic resonance (EPR) is an electron paramagnetic resonance technique that involves the alignment of the net magnetization vector of the electron spins in a constant magnetic field. This alignment is 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. Fourier transformation of the microwave signal yields 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.
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) is a spectroscopic technique widely used in biology, chemistry, medicine and physics to study systems with one or more unpaired electrons. Because of the specific relation between the magnetic parameters, electronic wavefunction and the configuration of the surrounding non-zero spin nuclei, EPR and ENDOR provide information on the structure, dynamics and the spatial distribution of the paramagnetic species. However, these techniques are limited in spectral and time resolution when used with traditional continuous wave methods. This resolution can be improved in pulsed EPR by investigating interactions separately from each other via pulse sequences.
Historical overview
R. J. Blume reported the first electron spin echo in 1958, which came from a solution of sodium in ammonia at its boiling point, -33.8˚C.[1] A magnetic field of 0.62 mT was used requiring a frequency of 17.4MHz. The first microwave electron spin echoes were reported in the same year by Gordon and Bowers using 23GHz excitation of dopants in silicon.[2]
Much of the pioneering early pulsed EPR was conducted in the group of W. B. Mims at Bell Labs during the 1960s. In the first decade only a small number of groups worked the field, because of the expensive instrumentation, the lack of suitable microwave components and slow digital electronics. The first observation of electron spin echo envelope modulation (ESEEM) was made in 1961 by Mims, Nassau and McGee.[3] Pulsed electron nuclear double resonance (ENDOR) was invented in 1965 by Mims.[4] In this experiment, pulsed NMR transitions are detected with pulsed EPR. ESEEM and pulsed ENDOR continue to be important for studying nuclear spins coupled to electron spins.
In the 1980s, the upcoming of the first commercial pulsed EPR and ENDOR spectrometers in the X band frequency range, lead to a fast growth of the field. In the 1990s, parallel to the upcoming high-field EPR, pulsed EPR and ENDOR became a new fast advancing magnetic resonance spectroscopy tool and the first commercial pulsed EPR and ENDOR spectrometer at W band frequencies appeared on the market.
The basic principle of pulsed EPR and NMR is similar. Differences can be found in the relative size of the magnetic interactions and in the relaxation rates which are orders of magnitudes larger (faster) in EPR than NMR. A full description of the theory is given within the quantum mechanical formalism, but since the magnetization is being measured as a bulk property, a more intuitive picture can be obtained with a classical description. For a better understanding of the concept of pulsed EPR consider the effects on the magnetization vector in the laboratory frame as well as in the rotating frame. As the animation below shows, in the laboratory frame the static magnetic field B0 is assumed to be parallel to the z-axis and the microwave field B1 parallel to the x-axis. When an electron spin is placed in 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 ωL.[5]
where γ is the gyromagnetic ratio and B0 the magnetic field. The electron spins are characterized by two quantum mechanical states, one parallel and one antiparallel to B0. Because of the lower energy of the parallel state more electron spins can be found in this state according to the Boltzmann distribution. This imbalanced population results in a net magnetization, which is the vector sum of all magnetic moments in the sample, parallel to the z-axis and the magnetic field. To better comprehend the effects of the microwave field B1 it is easier to move to the rotating frame.
Animation showing the rotating frame. The red arrow is a spin in the Bloch sphere which precesses in the laboratory frame due to a static magnetic field. In the rotating frame the spin remains still until a resonantly oscillating magnetic field drives magnetic resonance.
EPR experiments usually use a microwave resonator designed to create a linearly polarized microwave field B1, perpendicular to the much stronger applied magnetic field B0. The rotating frame is fixed to the rotating B1 components. First we assume to be on resonance with the precessing magnetization vector M0.
Therefore, the component of B1 will appear stationary. In this frame also the precessing magnetization components appear to be stationary that leads to the disappearance of B0, and we need only to consider B1 and M0. The M0 vector is under the influence of the stationary field B1, leading to another precession of M0, this time around B1 at the frequency ω1.
This angular frequency ω1 is also called the Rabi frequency. Assuming 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 and is given by:
Here tp is the duration for which B1 is applied, also called the pulse length. The pulses are labeled by the rotation of M0 which they cause and the direction from which they are coming from, since the microwaves can be phase-shifted 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 degrees 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 laboratory 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 laboratory frame it becomes a rotating magnetization in the x-y plane at the Larmor frequency. This rotation generates a 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).[6]
Another assumption we have made was the exact resonance condition, in which the Larmor 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 leaves magnetization in the xy-plane, but since the microwave field (and therefore the rotating frame) do not have 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.
If Δω is 0 then the microwave field rotates as fast as the magnetization vector and both appear to be stationary to each other. If Δω>0 then the magnetization rotates faster than the microwave field component in a counter-clockwise motion and if Δω<0 then the magnetization is slower and rotates clockwise. 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 laboratory frame. Here 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 Beff, which originates from the vector sum of B1 and B0. The magnetization is then tipped around Beff at a faster effective rate ωeff.
This leads directly to the third consequence that the magnetization can not be efficiently tipped into the xy-plane because Beff does not lie in the xy-plane, as B1 does. The motion of the magnetization now defines a cone. That means as Δω becomes larger, the magnetization is tipped less effectively into the xy-plane, and the FID signal decreases. In broad EPR spectra where Δω > ω1 it is not possible to tip all the magnetization into the xy-plane to generate a strong FID signal. This is why it is important to maximize ω1 or minimize the π/2 pulse length for broad EPR signals.
So far the magnetization was tipped into the xy-plane and it remained there with the same magnitude. However, in reality the electron spins interact with their surroundings and the magnetization in the xy-plane will decay and eventually return to alignment with the z-axis. This relaxation process is described by the spin-lattice relaxation time T1, which is a characteristic time needed by the magnetization to return to the z-axis, and by 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 the system to return to thermal equilibrium after it has been perturbed by the B1 pulse. Return of the magnetization parallel to B0 is achieved through interactions with the surroundings, that is spin-lattice relaxation. The corresponding relaxation time needs to be considered when extracting a signal from noise, where the experiment needs to be repeated several times, as quickly as possible. 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 xy-plane to create a significant signal.
The spin-spin relaxation time, also called the transverse relaxation time, is related to homogeneous 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 that results in the decay of the EPR signal. The other packets contribute to the transverse magnetization decay due to the homogeneous broadening. In this process all the spin in one spin packet experience the same magnetic field and interact with each other that 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 the frequency spectrum is encoded in the motion of the transverse magnetization. The frequency spectrum is reconstructed using the time behavior of the transverse magnetization made up of y- and x-axis components. It is convenient that these two can be treated as the real and imaginary components of a complex quantity and use the Fourier theory to transform the measured time domain signal into the frequency domain representation. This is possible because both the absorption (real) and the dispersion (imaginary) signals are detected.
The FID signal decays away and for very broad EPR spectra this decay is rather fast due to the inhomogeneous broadening. To obtain more information one can recover the disappeared signal with another microwave pulse to produce a Hahn echo.[7] After applying a π/2 pulse (90°), the magnetization vector is tipped into the xy-plane producing an FID signal. Different frequencies in the EPR spectrum (inhomogeneous broadening) cause this signal to "fan out", meaning that the slower spin-packets trail behind the faster ones. After a certain time t, a π pulse (180°) is applied to the system inverting the magnetization, and the fast spin-packets are then behind catching up with the slow spin-packets. A complete refocusing of the signal occurs then at time 2t. An accurate echo caused by a second microwave pulse can remove all inhomogeneous broadening effects. After all of the spin-packets bunch up, they will dephase again just like an FID. In other words, a spin echo is a reversed FID followed by a normal FID, which can be Fourier transformed to obtain the EPR spectrum. The longer the time between the pulses becomes, the smaller the echo will be due to spin relaxation. When this relaxation leads to an exponential decay in the echo height, the decay constant is the phase memory time TM, which can have many contributions such as transverse relaxation, spectral, spin and instantaneous diffusion. Changing the times between the pulses leads to a direct measurement of TM as shown in the spin echo decay animation below.
Applications
ESEEM [3][5] and pulsed ENDOR[4][5] are widely used echo experiments, in which the interaction of electron spins with the nuclei in their environment can be studied and controlled.
A popular pulsed EPR experiments currently is double electron-electron resonance (DEER), which is also known as pulsed electron-electron double resonance (PELDOR).[5] In this experiment, two frequencies control two spins to probe their coupling. The distance between the spins can then be inferred from their coupling strength. This information is used to elucidate structures of large bio-molecules. PELDOR spectroscopy is a versatile tool for structural investigations of proteins, even in a cellular environment.[8]
Dynamic nuclear polarization (DNP) results from transferring spin polarization from electrons to nuclei, thereby aligning the nuclear spins to the extent that electron spins are aligned. Note that the alignment of electron spins at a given magnetic field and temperature is described by the Boltzmann distribution under the thermal equilibrium. It is also possible that those electrons are aligned to a higher degree of order by other preparations of electron spin order such as: chemical reactions, optical pumping and spin injection. DNP is considered one of several techniques for hyperpolarization. DNP can also be induced using unpaired electrons produced by radiation damage in solids.
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spins excited are those of the electrons instead of the atomic nuclei. EPR spectroscopy is particularly useful for studying metal complexes and organic radicals. EPR was first observed in Kazan State University by Soviet physicist Yevgeny Zavoisky in 1944, and was developed independently at the same time by Brebis Bleaney at the University of Oxford.
In Fourier transform nuclear magnetic resonance spectroscopy, free induction decay (FID) is the observable NMR signal generated by non-equilibrium nuclear spin magnetization precessing about the magnetic field. This non-equilibrium magnetization can be created generally by applying a pulse of radio-frequency close to the Larmor frequency of the nuclear spins.
Ferromagnetic resonance, or FMR, is coupling between an electromagnetic wave and the magnetization of a medium through which it passes. This coupling induces a significant loss of power of the wave. The power is absorbed by the precessing magnetization of the material and lost as heat. For this coupling to occur, the frequency of the incident wave must be equal to the precession frequency of the magnetization and the polarization of the wave must match the orientation of the magnetization.
In MRI and NMR spectroscopy, an observable nuclear spin polarization (magnetization) is created by a homogeneous magnetic field. This field makes the magnetic dipole moments of the sample precess at the resonance (Larmor) frequency of the nuclei. At thermal equilibrium, nuclear spins precess randomly about the direction of the applied field. They become abruptly phase coherent when they are hit by radiofrequency (RF) pulses at the resonant frequency, created orthogonal to the field. The RF pulses cause the population of spin-states to be perturbed from their thermal equilibrium value. The generated transverse magnetization can then induce a signal in an RF coil that can be detected and amplified by an RF receiver. The return of the longitudinal component of the magnetization to its equilibrium value is termed spin-latticerelaxation while the loss of phase-coherence of the spins is termed spin-spin relaxation, which is manifest as an observed free induction decay (FID).
Insensitive nuclei enhancement by polarization transfer (INEPT) is a signal enhancement method used in NMR spectroscopy. It involves the transfer of nuclear spin polarization from spins with large Boltzmann population differences to nuclear spins of interest with lower Boltzmann population differences. INEPT uses J-coupling for the polarization transfer in contrast to Nuclear Overhauser effect (NOE), which arises from dipolar cross-relaxation. This method of signal enhancement was introduced by Ray Freeman in 1979. Due to its usefulness in signal enhancement, pulse sequences used in heteronuclear NMR experiments often contain blocks of INEPT or INEPT-like sequences.
In magnetic resonance, a spin echo or Hahn echo is the refocusing of spin magnetisation by a pulse of resonant electromagnetic radiation. Modern nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) make use of this effect.
In physics, the spin–spin relaxation is the mechanism by which Mxy, the transverse component of the magnetization vector, exponentially decays towards its equilibrium value in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). It is characterized by the spin–spin relaxation time, known as T2, a time constant characterizing the signal decay. It is named in contrast to T1, the spin–lattice relaxation time. It is the time it takes for the magnetic resonance signal to irreversibly decay to 37% (1/e) of its initial value after its generation by tipping the longitudinal magnetization towards the magnetic transverse plane. Hence the relation
During nuclear magnetic resonance observations, spin–lattice relaxation is the mechanism by which the longitudinal component of the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium state to thermodynamic equilibrium with its surroundings (the "lattice"). It is characterized by the spin–lattice relaxation time, a time constant known as T1.
In magnetic resonance imaging (MRI), the k-space or reciprocal space is obtained as the 2D or 3D Fourier transform of the image measured. It was introduced in 1979 by Likes and in 1983 by Ljunggren and Twieg.
Magnetization transfer (MT), in NMR and MRI, refers to the transfer of nuclear spin polarization and/or spin coherence from one population of nuclei to another population of nuclei, and to techniques that make use of these phenomena. There is some ambiguity regarding the precise definition of magnetization transfer, however the general definition given above encompasses all more specific notions. NMR active nuclei, those with non-zero spin, can be energetically coupled to one another under certain conditions. The mechanisms of nuclear-spin energy-coupling have been extensively characterized and are described in the following articles: Angular momentum coupling, Magnetic dipole–dipole interaction, J-coupling, Residual dipolar coupling, Nuclear Overhauser effect, Spin–spin relaxation, and Spin saturation transfer. Alternatively, some nuclei in a chemical system are labile and exchange between non-equivalent environments. A more specific example of this case is presented in the section Chemical Exchange Magnetization transfer.
In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx, My, Mz) as a function of time when relaxation times T1 and T2 are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. Sometimes they are called the equations of motion of nuclear magnetization. They are analogous to the Maxwell–Bloch equations.
The Bloch–Siegert shift is a phenomenon in quantum physics that becomes important for driven two-level systems when the driving gets strong.
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field and respond by producing an electromagnetic signal with a frequency characteristic of the magnetic field at the nucleus. This process occurs near resonance, when the oscillation frequency matches the intrinsic frequency of the nuclei, which depends on the strength of the static magnetic field, the chemical environment, and the magnetic properties of the isotope involved; in practical applications with static magnetic fields up to ca. 20 tesla, the frequency is similar to VHF and UHF television broadcasts (60–1000 MHz). NMR results from specific magnetic properties of certain atomic nuclei. Nuclear magnetic resonance spectroscopy is widely used to determine the structure of organic molecules in solution and study molecular physics and crystals as well as non-crystalline materials. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI). The original application of NMR to condensed matter physics is nowadays mostly devoted to strongly correlated electron systems. It reveals large many-body couplings by fast broadband detection and it should not to be confused with solid state NMR, which aims at removing the effect of the same couplings by Magic Angle Spinning techniques.
Electron nuclear double resonance (ENDOR) is a magnetic resonance technique for elucidating the molecular and electronic structure of paramagnetic species. The technique was first introduced to resolve interactions in electron paramagnetic resonance (EPR) spectra. It is currently practiced in a variety of modalities, mainly in the areas of biophysics and heterogeneous catalysis.
In magnetic resonance imaging, the flip angle, also called the nutation angle, pulse angle, tip angle, or excitation angle, refers to the angle that the magnetization vector is tipped or rotated away from the z-axis by a radiofrequency pulse, relative to the main magnetic field. The flip angle is generally measured in degrees or radians; for example, a 90-degree flip angle will deflect all longitudinal magnetization into the transverse, or xy plane. Changing the amplitude or duration of the pulse will change the flip angle.
Magnetic resonance imaging (MRI) is a medical imaging technique mostly used in radiology and nuclear medicine in order to investigate the anatomy and physiology of the body, and to detect pathologies including tumors, inflammation, neurological conditions such as stroke, disorders of muscles and joints, and abnormalities in the heart and blood vessels among others. Contrast agents may be injected intravenously or into a joint to enhance the image and facilitate diagnosis. Unlike CT and X-ray, MRI uses no ionizing radiation and is, therefore, a safe procedure suitable for diagnosis in children and repeated runs. Patients with specific non-ferromagnetic metal implants, cochlear implants, and cardiac pacemakers nowadays may also have an MRI in spite of effects of the strong magnetic fields. This does not apply on older devices, and details for medical professionals are provided by the device's manufacturer.
Acoustic paramagnetic resonance (APR) is a phenomenon of resonant absorption of sound by a system of magnetic particles placed in an external magnetic field. It occurs when the energy of the sound wave quantum becomes equal to the splitting of the energy levels of the particles, the splitting being induced by the magnetic field. APR is a variation of electron paramagnetic resonance (EPR) where the acoustic rather than electromagnetic waves are absorbed by the studied sample. APR was theoretically predicted in 1952, independently by Semen Altshuler and Alfred Kastler, and was experimentally observed by W. G. Proctor and W. H. Tanttila in 1955.
In quantum mechanics, magnetic resonance is a resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum (azimuthal) quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.
Adiabatic radio frequency (RF) pulses are used in magnetic resonance imaging (MRI) to achieve excitation that is insensitive to spatial inhomogeneities in the excitation field or off-resonances in the sampled object.
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