# Electron paramagnetic resonance

Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials with unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but it is electron spins that are excited instead of the spins of atomic nuclei. EPR spectroscopy is particularly useful for studying metal complexes or organic radicals. EPR was first observed in Kazan State University by Soviet physicist Yevgeny Zavoisky in 1944, [1] [2] and was developed independently at the same time by Brebis Bleaney at the University of Oxford.

## Theory

### Origin of an EPR signal

Every electron has a magnetic moment and spin quantum number ${\displaystyle s={\tfrac {1}{2}}}$, with magnetic components ${\displaystyle m_{\mathrm {s} }=+{\tfrac {1}{2}}}$ and ${\displaystyle m_{\mathrm {s} }=-{\tfrac {1}{2}}}$. In the presence of an external magnetic field with strength ${\displaystyle B_{\mathrm {0} }}$, the electron's magnetic moment aligns itself either parallel (${\displaystyle m_{\mathrm {s} }=-{\tfrac {1}{2}}}$) or antiparallel (${\displaystyle m_{\mathrm {s} }=+{\tfrac {1}{2}}}$) 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 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 ${\displaystyle h\nu }$ 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 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 ${\displaystyle h\nu =g_{\mathrm {N} }\mu _{\mathrm {N} }B_{0}}$ where ${\displaystyle g_{\mathrm {N} }}$ and ${\displaystyle \mu _{\mathrm {N} }}$ depend on the nucleus under study.)

### Field modulation

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 spectra 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.

### 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:

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

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.
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