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 useful for analyzing metal ions and organic radicals (compounds with unpaired electrons). The technique reveals some structural information but often simply provides a characteristic "finger print". The measurement requires a large magnet into which the sample is placed. Signals are detected using microwaves. In contrast to NMR and infrared (IR) spectroscopy, EPR spectroscopy is less common. For a given sample, some of the parameters of interest are g-values (analogous to chemical shift), anisotropy (asymmetry), hyperfine coupling constants (analogous to coupling constant J), and relaxation times.
History 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.
Theory Every electron has a magnetic moment and spin quantum number , with magnetic components or . In the presence of an external magnetic field with strength , the electron's magnetic moment aligns itself either antiparallel () or parallel () to the field, each alignment having a specific energy due to the Zeeman effect:
where is the electron's so-called g-factor (see also the Landé g-factor), for the free electron, is the Bohr magneton.
Therefore, the separation between the lower and the upper state is for unpaired free electrons. This equation implies (since both and are constant) 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 change its electron spin by either absorbing or emitting a photon of energy such that the resonance condition, , is obeyed. This leads to the fundamental equation of EPR spectroscopy: .
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 and 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 9388.4 MHz, the predicted resonance occurs at a magnetic field of about = 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 where and depend on the nucleus under study.)
Field modulation
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. 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.
The same idea underlies the Pound–Drever–Hall technique for frequency locking of lasers to a high-finesse optical cavity.
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 Boltzmann distribution:
where is the number of paramagnetic centers occupying the upper energy state, is the Boltzmann constant, and is the thermodynamic temperature. At 298 K, X-band microwave frequencies ( ≈ 9.75 GHz) give ≈ 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 ) depends on the photon frequency according to
where is a constant, is the sample's volume, is the unloaded quality factor of the microwave cavity (sample chamber), is the cavity filling coefficient, and is the microwave power in the spectrometer cavity. With and being constants, ~ , i.e., ~ , where ≈ 1.5. In practice, 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 and a large number of spins. Therefore, the required parameters are: A high spectrometer frequency to minimize 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.
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: An unpaired electron can gain or lose angular momentum, which can change the value of its g-factor, causing it to differ from . This is especially significant for chemical systems with transition-metal ions. Systems with multiple unpaired electrons experience electron–electron interactions that give rise to "fine" structure. This is realized as zero field splitting and exchange coupling, and can be large in magnitude. 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. Additional smaller splittings from nearby nuclei is sometimes termed "superhyperfine" coupling. 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. These effects (g-factor, hyperfine coupling, zero field splitting, exchange 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 but also to any local magnetic fields of atoms or molecules. The effective field experienced by an electron is thus written
where includes the effects of local fields ( can be positive or negative). Therefore, the resonance condition (above) is rewritten as follows:
The quantity is denoted and called simply the g-factor, so that the final resonance equation becomes
This last equation is used to determine in an EPR experiment by measuring the field and the frequency at which resonance occurs. If does not equal , 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 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: 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 does the frequency of the resonance. For a large ensemble of randomly oriented spins (as in a fluid solution), the EPR spectrum consists of three peaks of characteristic shape at frequencies gxxB0, gyyB0 and gzzB0.
In first-derivative spectrum, the low-frequency peak is positive, 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").
At higher temperatures, the three peaks coalesce to a singlet, corresponding to giso, for isotropic. The relationship between giso and the components is:
One elementary step in analyzing an EPR spectrum is to compare giso with the g-factor for the free electron, ge. Metal-based radicals giso is typically well above ge whereas organic radicals, giso ~ ge.
The determination of the absolute value of the g factor is challenging due to the lack of a precise estimate of the local magnetic field at the sample location. Therefore, typically so-called g factor standards are measured together with the sample of interest. In the common spectrum, the spectral line of the g factor standard is then used as a reference point to determine the g factor of the sample. For the initial calibration of g factor standards, Herb et al. introduced a precise procedure by using double resonance techniques based on the Overhauser shift.
Hyperfine coupling