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 g_e . 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 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 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: :(g_\mathrm{iso})^2 = (g_{xx})^2 + (g_{yy})^2 + (g_{zz})^2 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 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. 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 \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. In many cases, the isotropic hyperfine splitting pattern for a radical freely tumbling in a solution (isotropic system) can be predicted.
Multiplicity • For a radical having
M equivalent nuclei, each with a spin of
I, the number of EPR lines expected is 2
MI + 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 2
MI + 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 (2
M1
I1 + 1) (2
M2
I2 + 1). As an example, the methoxymethyl radical, 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 (2
M1
I1 + 1) (2
M2
I2 + 1) = [2(2)(1/2) + 1] [2(3)(1/2) + 1] = 3×4 = 12, again as observed. • 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 2
MI + 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. As a second example, the methoxymethyl radical, H3COCH2
. the OC
H2 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 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}. == Applications ==