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# Electron paramagnetic resonance

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### Electron paramagnetic resonance

Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) 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.

## Contents

• Theory 1
• Origin of an EPR signal 1.1
• Maxwell–Boltzmann distribution 1.2
• Spectral parameters 2
• The g factor 2.1
• Hyperfine coupling 2.2
• Resonance linewidth definition 2.3
• Pulsed EPR 3
• Applications 4
• Miniature electron spin resonance spectroscopy with Micro-ESR 5
• High-field high-frequency measurements 6
• References 8
• External links 9

## Theory

### Origin of an EPR signal

Every electron has a magnetic moment and spin quantum number s = \tfrac{1}{2} , with magnetic components m_\mathrm{s} = + \tfrac{1}{2} and m_\mathrm{s} = - \tfrac{1}{2} . In the presence of an external magnetic field with strength B_\mathrm{0} , the electron's magnetic moment aligns itself either parallel ( m_\mathrm{s} = - \tfrac{1}{2} ) or antiparallel ( m_\mathrm{s} = + \tfrac{1}{2} ) to the field, each alignment having a specific energy due to the Zeeman effect:

E = m_s g_e \mu_\text{B} B_0,

where

Therefore, the separation between the lower and the upper state is \Delta E = g_e \mu_\text{B} B_0 for unpaired free electrons. This equation implies that the splitting of the energy levels is directly proportional to the magnetic field's strength, as shown in the diagram below.

An unpaired electron can move between the two energy levels by either absorbing or emitting a photon of energy h \nu such that the resonance condition, h \nu = \Delta E , is obeyed. This leads to the fundamental equation of EPR spectroscopy: h \nu = g_e \mu_\text{B} B_0 .

Experimentally, this equation permits a large combination of frequency and magnetic field values, but the great majority of EPR measurements are made with microwaves in the 9000–10000 MHz (9–10 GHz) region, with fields corresponding to about 3500 G (0.35 T). Furthermore, EPR spectra can be generated by either varying the photon frequency incident on a sample while holding the magnetic field constant or doing the reverse. In practice, it is usually the frequency that is kept fixed. A collection of paramagnetic centers, such as free radicals, is exposed to microwaves at a fixed frequency. By increasing an external magnetic field, the gap between the m_\mathrm{s} = + \tfrac{1}{2} and m_\mathrm{s} = - \tfrac{1}{2} energy states is widened until it matches the energy of the microwaves, as represented by the double arrow in the diagram above. At this point the unpaired electrons can move between their two spin states. Since there typically are more electrons in the lower state, due to the Maxwell–Boltzmann distribution (see below), there is a net absorption of energy, and it is this absorption that is monitored and converted into a spectrum. The upper spectrum below is the simulated absorption for a system of free electrons in a varying magnetic field. The lower spectrum is the first derivative of the absorption spectrum. The latter is the most common way to record and publish EPR spectra.

For the microwave frequency of 9388.2 MHz, the predicted resonance occurs at a magnetic field of about B_0 = h \nu / g_e \mu_\text{B} = 0.3350 teslas = 3350 gausses.

Because of electron-nuclear mass differences, the magnetic moment of an electron is substantially larger than the corresponding quantity for any nucleus, so that a much higher electromagnetic frequency is needed to bring about a spin resonance with an electron than with a nucleus, at identical magnetic field strengths. For example, for the field of 3350 G shown at the right, spin resonance occurs near 9388.2 MHz for an electron compared to only about 14.3 MHz for 1H nuclei. (For NMR spectroscopy, the corresponding resonance equation is h\nu = g_\mathrm{N} \mu_\mathrm{N} B_0 where g_\mathrm{N} and \mu_\mathrm{N} depend on the nucleus under study.)

### Maxwell–Boltzmann distribution

In practice, EPR samples consist of collections of many paramagnetic species, and not single isolated paramagnetic centers. If the population of radicals is in thermodynamic equilibrium, its statistical distribution is described by the Maxwell–Boltzmann equation:

\frac{ n_\text{upper} }{ n_\text{lower} } = \exp{ \left( -\frac{ E_\text{upper}-E_\text{lower} }{ kT } \right) } = \exp{ \left( -\frac{ \Delta E }{ kT } \right) } = \exp{ \left( -\frac{ \epsilon }{ kT } \right) } = \exp{ \left( -\frac{ h\nu }{ kT }\right) }, \qquad \text{(Eq. 1)}

where n_\text{upper} is the number of paramagnetic centers occupying the upper energy state, k is the Boltzmann constant, and T is the thermodynamic temperature. At 298 K, X-band microwave frequencies (\nu ≈ 9.75 GHz) give n_\text{upper} / n_\text{lower} ≈ 0.998, meaning that the upper energy level has a slightly smaller population than the lower one. Therefore, transitions from the lower to the higher level are more probable than the reverse, which is why there is a net absorption of energy.

The sensitivity of the EPR method (i.e., the minimal number of detectable spins N_\text{min}) depends on the photon frequency \nu according to

N_\text{min} = \frac{k_1V}{Q_0k_f \nu^2 P^{1/2}}, \qquad \text{(Eq. 2)}

where k_1 is a constant, V is the sample's volume, Q_0 is the unloaded quality factor of the microwave cavity (sample chamber), k_f is the cavity filling coefficient, and P is the microwave power in the spectrometer cavity. With k_f and P being constants, N_\text{min} ~ (Q_0\nu^2)^{-1}, i.e., N_\text{min} ~ \nu^{-\alpha}, where \alpha ≈ 1.5. In practice, \alpha can change varying from 0.5 to 4.5 depending on spectrometer characteristics, resonance conditions, and sample size.

A great sensitivity is therefore obtained with a low detection limit N_\text{min} and a large number of spins. Therefore, the required parameters are:

• A high spectrometer frequency to maximize the Eq. 2. Common frequencies are discussed below
• A low temperature to decrease the number of spin at the high level of energy as shown in Eq. 1. This condition explain why spectra are often recorded on sample at the boiling point of liquid nitrogen or liquid helium.

## Spectral parameters

In real systems, electrons are normally not solitary, but are associated with one or more atoms. There are several important consequences of this:

1. An unpaired electron can gain or lose angular momentum, which can change the value of its g-factor, causing it to differ from g_e . This is especially significant for chemical systems with transition-metal ions.
2. The magnetic moment of a nucleus with a non-zero nuclear spin will affect any unpaired electrons associated with that atom. This leads to the phenomenon of hyperfine coupling, analogous to J-coupling in NMR, splitting the EPR resonance signal into doublets, triplets and so forth.
3. Interactions of an unpaired electron with its environment influence the shape of an EPR spectral line. Line shapes can yield information about, for example, rates of chemical reactions.
4. The g-factor and hyperfine coupling in an atom or molecule may not be the same for all orientations of an unpaired electron in an external magnetic field. This anisotropy depends upon the electronic structure of the atom or molecule (e.g., free radical) in question, and so can provide information about the atomic or molecular orbital containing the unpaired electron.

### The g factor

Knowledge of the g-factor can give information about a paramagnetic center's electronic structure. An unpaired electron responds not only to a spectrometer's applied magnetic field B_0 but also to any local magnetic fields of atoms or molecules. The effective field B_\text{eff} experienced by an electron is thus written

B_\text{eff} = B_0(1 - \sigma),

where \sigma includes the effects of local fields (\sigma can be positive or negative). Therefore, the h \nu = g_e \mu_\text{B} B_\text{eff} resonance condition (above) is rewritten as follows:

h\nu = g_e \mu_B B_\text{eff} = g_e \mu_\text{B} B_0 (1 - \sigma).

The quantity g_e(1 - \sigma) is denoted g and called simply the g-factor, so that the final resonance equation becomes

h \nu = g \mu_\text{B} B_0.

This last equation is used to determine g in an EPR experiment by measuring the field and the frequency at which resonance occurs. If g does not equal g_e , the implication is that the ratio of the unpaired electron's spin magnetic moment to its angular momentum differs from the free-electron value. Since an electron's spin magnetic moment is constant (approximately the Bohr magneton), then the electron must have gained or lost angular momentum through spin–orbit coupling. Because the mechanisms of spin–orbit coupling are well understood, the magnitude of the change gives information about the nature of the atomic or molecular orbital containing the unpaired electron.

In general, the g factor is not a number but a second-rank tensor represented by 9 numbers arranged in a 3×3 matrix. The principal axes of this tensor are determined by the local fields, for example, by the local atomic arrangement around the unpaired spin in a solid or in a molecule. Choosing an appropriate coordinate system (say, x,y,z) allows to "diagonalize" this tensor, thereby reducing the maximal number of its components from 9 to 3: gxx, gyy and gzz. For a single spin experiencing only Zeeman interaction with an external magnetic field, the position of the EPR resonance is given by the expression gxxBx + gyyBy + gzzBz. Here Bx, By and Bz are the components of the magnetic field vector in the coordinate system (x,y,z); their magnitudes change as the field is rotated, so as the frequency of the resonance. For a large ensemble of randomly oriented spins, the EPR spectrum consists of three peaks of characteristic shape at frequencies gxxB0, gyyB0 and gzzB0: the low-frequency peak is positive in first-derivative spectra, the high-frequency peak is negative, and the central peak is bipolar. Such situation is commonly observed in powders, and the spectra are therefore called "powder-pattern spectra". In crystals, the number of EPR lines is determined by the number of crystallographically equivalent orientations of the EPR spin (called "EPR center").

### Hyperfine coupling

Since the source of an EPR spectrum is a change in an electron's spin state, it might be thought that all EPR spectra for a single electron spin would consist of one line. However, the interaction of an unpaired electron, by way of its magnetic moment, with nearby nuclear spins, results in additional allowed energy states and, in turn, multi-lined spectra. In such cases, the spacing between the EPR spectral lines indicates the degree of interaction between the unpaired electron and the perturbing nuclei. The hyperfine coupling constant of a nucleus is directly related to the spectral line spacing and, in the simplest cases, is essentially the spacing itself.

Two common mechanisms by which electrons and nuclei interact are the a" or "A" are used for isotropic hyperfine coupling constants, while "B" is usually employed for anisotropic hyperfine coupling constants.[2]

In many cases, the isotropic hyperfine splitting pattern for a radical freely tumbling in a solution (isotropic system) can be predicted.

• For a radical having M equivalent nuclei, each with a spin of I, the number of EPR lines expected is 2MI + 1. As an example, the methyl radical, CH3, has three 1H nuclei, each with I = 1/2, and so the number of lines expected is 2MI + 1 = 2(3)(1/2) + 1 = 4, which is as observed.
• For a radical having M1 equivalent nuclei, each with a spin of I1, and a group of M2 equivalent nuclei, each with a spin of I2, the number of lines expected is (2M1I1 + 1) (2M2I2 + 1). As an example, the methoxymethyl radical, H2C(OCH3), has two equivalent 1H nuclei, each with I = 1/2 and three equivalent 1H nuclei each with I = 1/2, and so the number of lines expected is (2M1I1 + 1) (2M2I2 + 1) = [2(2)(1/2) + 1][2(3)(1/2) + 1] = 3×4 = 12, again as observed.
Simulated EPR spectrum of the CH3 radical
• The above can be extended to predict the number of lines for any number of nuclei.

While it is easy to predict the number of lines a radical's EPR spectrum should show, the reverse problem, unraveling a complex multi-line EPR spectrum and assigning the various spacings to specific nuclei, is more difficult.

In the oft-encountered case of I = 1/2 nuclei (e.g., 1H, 19F, 31P), the line intensities produced by a population of radicals, each possessing M equivalent nuclei, will follow Pascal's triangle. For example, the spectrum at the right shows that the three 1H nuclei of the CH3 radical give rise to 2MI + 1 = 2(3)(1/2) + 1 = 4 lines with a 1:3:3:1 ratio. The line spacing gives a hyperfine coupling constant of aH = 23 G for each of the three 1H nuclei. Note again that the lines in this spectrum are first derivatives of absorptions.

Simulated EPR spectrum of the H2C(OCH3) radical

As a second example, consider the methoxymethyl radical, H2C(OCH3). The two equivalent methyl hydrogens will give an overall 1:2:1 EPR pattern, each component of which is further split by the three methoxy hydrogens into a 1:3:3:1 pattern to give a total of 3×4 = 12 lines, a triplet of quartets. A simulation of the observed EPR spectrum is shown at the right and agrees with the 12-line prediction and the expected line intensities. Note that the smaller coupling constant (smaller line spacing) is due to the three methoxy hydrogens, while the larger coupling constant (line spacing) is from the two hydrogens bonded directly to the carbon atom bearing the unpaired electron. It is often the case that coupling constants decrease in size with distance from a radical's unpaired electron, but there are some notable exceptions, such as the ethyl radical (CH2CH3).

### Resonance linewidth definition

Resonance linewidths are defined in terms of the magnetic induction B and its corresponding units, and are measured along the x axis of an EPR spectrum, from a line's center to a chosen reference point of the line. These defined widths are called halfwidths and possess some advantages: for asymmetric lines, values of left and right halfwidth can be given. The halfwidth \Delta B_h is the distance measured from the line's center to the point in which absorption value has half of maximal absorption value in the center of resonance line. First inclination width \Delta B_{1/2} is a distance from center of the line to the point of maximal absorption curve inclination. In practice, a full definition of linewidth is used. For symmetric lines, halfwidth \Delta B_{1/2} = 2\Delta B_h, and full inclination width \Delta B_\text{max} = 2\Delta B_{1s}.

## Pulsed EPR

The dynamics of electron spins are best studied with pulsed measurements.[3] Microwave pulses typically 10–100 ns long are used to control the spins in the Bloch sphere. The spin–lattice relaxation time can be measured with an inversion recovery experiment.

As with pulsed NMR, the Hahn echo is central to many pulsed EPR experiments. A Hahn echo decay experiment can be used to measure the dephasing time, as shown in the animation below. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence, which is not refocused by the \pi pulse. In simple cases, an exponential decay is measured, which is described by the T_2 time.

## Applications

EPR/ESR spectroscopy is used in various branches of science, such as UV light. In many cases, the reactions to make the radicals and the subsequent reactions of the radicals are of interest, while in other cases EPR is used to provide information on a radical's geometry and the orbital of the unpaired electron.

## Miniature electron spin resonance spectroscopy with Micro-ESR

Miniaturisation of military radar technologies allowed the development of miniature microwave electronics as a spin-off by the California Institute of Technology. Since 2007 these sensors have been employed in miniaturized electron spin resonance spectrometers called Micro-ESR.

The high cost, large size, and difficult maintenance of electron spin resonance spectrometers has limited their use to specialized research centers with highly trained personnel. Micro-ESR makes ESR feasible for nonspecialists to determine oxidation by directly measuring of free radicals.

Applications include real-time monitoring of free radical containing asphaltenes in (crude) oils, biomedical R&D to measure oxidative stress, evaluation of the shelf life of food products.

• Electron Magnetic Resonance Program National High Magnetic Field Laboratory
• Electron Paramagnetic Resonance (Specialist Periodical Reports) Published by the Royal Society of Chemistry
• Using ESR to measure free radicals in used engine oil

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