Electron paramagnetic resonance

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

${\displaystyle E=m_{s}g_{e}\mu _{\text{B}}B_{0},}$

where

• ${\displaystyle g_{e}}$ is the electron's so-called g-factor (see also the Landé g-factor), ${\displaystyle g_{\mathrm {e} }=2.0023}$ for the free electron,[3]
• ${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton.

Therefore, the separation between the lower and the upper state is ${\displaystyle \Delta E=g_{e}\mu _{\text{B}}B_{0}}$ for unpaired free electrons. This equation implies (since both ${\displaystyle g_{e}}$and ${\displaystyle \mu _{\text{B}}}$ are constant) that the splitting of the energy levels is directly proportional to the magnetic field's strength, as shown in the diagram below.

Splitting of electron spin states

An unpaired electron can change its electron spin by either absorbing or emitting a photon of energy ${\displaystyle h\nu }$ such that the resonance condition, ${\displaystyle h\nu =\Delta E}$, is obeyed. This leads to the fundamental equation of EPR spectroscopy: ${\displaystyle h\nu =g_{e}\mu _{\text{B}}B_{0}}$.

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 ${\displaystyle m_{\mathrm {s} }=+{\tfrac {1}{2}}}$ and ${\displaystyle m_{\mathrm {s} }=-{\tfrac {1}{2}}}$ 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 continuous wave EPR spectra.

For the microwave frequency of 9388.2 MHz, the predicted resonance occurs at a magnetic field of about ${\displaystyle B_{0}=h\nu /g_{e}\mu _{\text{B}}}$ = 0.3350 T = 3350 G

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 above, 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 ${\displaystyle h\nu =g_{\mathrm {N} }\mu _{\mathrm {N} }B_{0}}$ where ${\displaystyle g_{\mathrm {N} }}$ and ${\displaystyle \mu _{\mathrm {N} }}$ depend on the nucleus under study.)

The field oscillates between B₁ and B₂ due to the superimposed modulation field at 100 kHz. This causes the absorption intensity to oscillate between I₁ and I₂. The larger the difference the larger the intensity detected by the detector tuned to 100 kHz (note this can be negative or even 0). As the difference between the two intensities is detected the first derivative of the absorption is detected.

As previously mentioned an EPR spectrum is usually directly measured as the first derivative of the absorption. This is accomplished by using field modulation. A small additional oscillating magnetic field is applied to the external magnetic field at a typical frequency of 100 kHz.[4] By detecting the peak to peak amplitude the first derivative of the absorption is measured. By using phase sensitive detection only signals with the same modulation (100 kHz) are detected. This results in higher signal to noise ratios. Note field modulation is unique to continuous wave EPR measurements and spectra resulting from pulsed experiments are presented as absorption profiles.

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:

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

where ${\displaystyle n_{\text{upper}}}$ is the number of paramagnetic centers occupying the upper energy state, ${\displaystyle k}$ is the Boltzmann constant, and ${\displaystyle T}$ is the thermodynamic temperature. At 298 K, X-band microwave frequencies (${\displaystyle \nu }$ ≈ 9.75 GHz) give ${\displaystyle n_{\text{upper}}/n_{\text{lower}}}$ ≈ 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 ${\displaystyle N_{\text{min}}}$) depends on the photon frequency ${\displaystyle \nu }$ according to

${\displaystyle N_{\text{min}}={\frac {k_{1}V}{Q_{0}k_{f}\nu ^{2}P^{1/2}}},\qquad {\text{(Eq. 2)}}}$

where ${\displaystyle k_{1}}$ is a constant, ${\displaystyle V}$ is the sample's volume, ${\displaystyle Q_{0}}$ is the unloaded quality factor of the microwave cavity (sample chamber), ${\displaystyle k_{f}}$ is the cavity filling coefficient, and ${\displaystyle P}$ is the microwave power in the spectrometer cavity. With ${\displaystyle k_{f}}$ and ${\displaystyle P}$ being constants, ${\displaystyle N_{\text{min}}}$ ~ ${\displaystyle (Q_{0}\nu ^{2})^{-1}}$, i.e., ${\displaystyle N_{\text{min}}}$ ~ ${\displaystyle \nu ^{-\alpha }}$, where ${\displaystyle \alpha }$ ≈ 1.5. In practice, ${\displaystyle \alpha }$ 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 ${\displaystyle N_{\text{min}}}$ 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 explains why spectra are often recorded on sample at the boiling point of liquid nitrogen or liquid helium.

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 ${\displaystyle B_{0}}$ but also to any local magnetic fields of atoms or molecules. The effective field ${\displaystyle B_{\text{eff}}}$ experienced by an electron is thus written

${\displaystyle B_{\text{eff}}=B_{0}(1-\sigma ),}$

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

${\displaystyle h\nu =g_{e}\mu _{B}B_{\text{eff}}=g_{e}\mu _{\text{B}}B_{0}(1-\sigma ).}$

The quantity ${\displaystyle g_{e}(1-\sigma )}$ is denoted ${\displaystyle g}$ and called simply the g-factor, so that the final resonance equation becomes

${\displaystyle h\nu =g\mu _{\text{B}}B_{0}.}$

This last equation is used to determine ${\displaystyle g}$ in an EPR experiment by measuring the field and the frequency at which resonance occurs. If ${\displaystyle g}$ does not equal ${\displaystyle g_{e}}$, 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 one to "diagonalize" this tensor, thereby reducing the maximal number of its components from 9 to 3: g_xx, g_yy and g_zz. For a single spin experiencing only Zeeman interaction with an external magnetic field, the position of the EPR resonance is given by the expression g_xxB_x + g_yyB_y + g_zzB_z. Here B_x, B_y and B_z are the components of the magnetic field vector in the coordinate system (x,y,z); their magnitudes change as the field is rotated, so does 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 g_xxB₀, g_yyB₀ and g_zzB₀: the low-frequency peak is positive in first-derivative spectra, the high-frequency peak is negative, and the central peak is bipolar. Such situations are 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").

Since the source of an EPR spectrum is a change in an electron's spin state, the EPR spectrum for a radical (S = 1/2 system) would consist of one line. Greater complexity arises because the spin couples with nearby nuclear spins. The magnitude of the coupling is proportional to the magnetic moment of the coupled nuclei and depends on the mechanism of the coupling. Coupling is mediated by two processes, dipolar (through space) and isotropic (through bond).

This coupling introduces additional 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.[6]

Two common mechanisms by which electrons and nuclei interact are the Fermi contact interaction and by dipolar interaction. The former applies largely to the case of isotropic interactions (independent of sample orientation in a magnetic field) and the latter to the case of anisotropic interactions (spectra dependent on sample orientation in a magnetic field). Spin polarization is a third mechanism for interactions between an unpaired electron and a nuclear spin, being especially important for ${\displaystyle \pi }$-electron organic radicals, such as the benzene radical anion. The symbols "a" or "A" are used for isotropic hyperfine coupling constants, while "B" is usually employed for anisotropic hyperfine coupling constants.[7]

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, H
  ₂C(OCH
  ₃) 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, the reverse problem, unraveling a complex multi-line EPR spectrum and assigning the various spacings to specific nuclei, is more difficult.

In the often 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 a_H = 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 H
₂C(OCH
₃) radical

As a second example, the methoxymethyl radical, H₃COCH₂^. the OCH₂ center 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 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 ${\displaystyle \Delta B_{h}}$ 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 ${\displaystyle \Delta B_{1/2}}$ 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 ${\displaystyle \Delta B_{1/2}=2\Delta B_{h}}$, and full inclination width ${\displaystyle \Delta B_{\text{max}}=2\Delta B_{1s}}$.

The microwave bridge contains both the microwave source and the detector.[21] Older spectrometers used a vacuum tube called a klystron to generate microwaves, but modern spectrometers use a Gunn diode. Immediately after the microwave source there is an isolator which serves to attenuate any reflections back to the source which would result in fluctuations in the microwave frequency.[22] The microwave power from the source is then passed through a directional coupler which splits the microwave power into two paths, one directed towards the cavity and the other the reference arm. Along both paths there is a variable attenuator that facilitates the precise control of the flow of microwave power. This in turn allows for accurate control over the intensity of the microwaves subjected to the sample. On the reference arm, after the variable attenuator there is a phase shifter that sets a defined phase relationship between the reference and reflected signal which permits phase sensitive detection.

Most EPR spectrometers are reflection spectrometers, meaning that the detector should only be exposed to microwave radiation coming back from the cavity. This is achieved by the use of a device known as the circulator which directs the microwave radiation (from the branch that is heading towards the cavity) into the cavity. Reflected microwave radiation (after absorption by the sample) is then passed through the circulator towards the detector, ensuring it does not go back to the microwave source. The reference signal and reflected signal are combined and passed to the detector diode which converts the microwave power into an electrical current.

At low energies (less than 1 μW) the diode current is proportional to the microwave power and the detector is referred to as a square-law detector. At higher power levels (greater than 1 mW) the diode current is proportional to the square root of the microwave power and the detector is called a linear detector. In order to obtain optimal sensitivity as well as quantitative information the diode should be operating within the linear region. To ensure the detector is operating at that level the reference arm serves to provide a "bias".

In an EPR spectrometer the magnetic assembly includes the magnet with a dedicated power supply as well as a field sensor or regulator such as a Hall probe. EPR spectrometers use one of two types of magnet which is determined by the operating microwave frequency (which determine the range of magnetic field strengths required). The first is an electromagnet which are generally capable of generating field strengths of up to 1.5 T making them suitable for measurements using the Q-band frequency. In order to generate field strengths appropriate for W-band and higher frequency operation superconducting magnets are employed. The magnetic field is homogeneous across the sample volume and has a high stability at static field.

The microwave resonator is designed to enhance the microwave magnetic field at the sample in order to induce EPR transitions. It is a metal box with a rectangular or cylindrical shape that resonates with microwaves (like an organ pipe with sound waves). At the resonance frequency of the cavity microwaves remain inside the cavity and are not reflected back. Resonance means the cavity stores microwave energy and its ability to do this is given by the quality factor Q, defined by the following equation:

${\displaystyle Q={\frac {2\pi ({\text{energy stored}})}{({\text{energy dissipated}})}}}$

The higher the value of Q the higher the sensitivity of the spectrometer. The energy dissipated is the energy lost in one microwave period. Energy may be lost to the side walls of the cavity as microwaves may generate currents which in turn generate heat. A consequence of resonance is the creation of a standing wave inside the cavity. Electromagnetic standing waves have their electric and magnetic field components exactly out of phase. This provides an advantage as the electric field provides non-resonant absorption of the microwaves, which in turn increases the dissipated energy and reduces Q. To achieve the largest signals and hence sensitivity the sample is positioned such that it lies within the magnetic field maximum and the electric field minimum. When the magnetic field strength is such that an absorption event occurs, the value of Q will be reduced due to the extra energy loss. This results in a change of impedance which serves to stop the cavity from being critically coupled. This means microwaves will now be reflected back to the detector (in the microwave bridge) where an EPR signal is detected.[23]