EPR spectroscopy

Resonance condition

An unpaired electron (S = \tfrac12) in field B has two levels, m_S = \pm\tfrac12, split by g\mu_B B. Microwaves of fixed frequency are applied while the field is swept, and absorption occurs at

Typical X-band spectrometers use about 9.5 GHz and fields around 0.34 T. Spectra are recorded as the first derivative of absorption because field modulation with phase-sensitive detection improves sensitivity.

The g-value

The free electron has g_e = 2.0023. Spin–orbit coupling mixes in orbital angular momentum, shifting g. Organic radicals have g close to 2.003; transition-metal ions deviate much more. In frozen solutions or solids, g is anisotropic: an axial Cu(II) complex with a d_{x^2-y^2} ground state shows g_\parallel > g_\perp > 2.0023, which identifies the singly occupied orbital.

Hyperfine coupling

The electron spin interacts with nuclear spins. One nucleus of spin I splits each line into 2I+1 components of equal intensity. For n equivalent nuclei, there are

lines. For I = \tfrac12 intensities follow Pascal's triangle.

• Methyl radical, ·CH_3: three equivalent protons give 4 lines, 1:3:3:1, with a \approx 2.3 mT.
• Benzene radical anion: six equivalent H give 7 lines, 1:6:15:20:15:6:1.
• Nitroxides (TEMPO): one ^{14}N (I = 1) gives 3 equal lines.
• Cu(II): ^{63}Cu/^{65}Cu (I = \tfrac32) give 4 lines.

Inequivalent sets multiply: two sets of protons (with n_1 and n_2 nuclei) give (n_1+1)(n_2+1) lines if all couplings are resolved.

What hyperfine reveals

Isotropic coupling arises from Fermi contact spin density at the nucleus, so it maps the singly occupied orbital. For aromatic π radicals the McConnell relation a_H = Q\rho_C (with |Q| a little over 2 mT) converts ring-proton couplings into spin densities on the attached carbons; the benzene anion's coupling of about 0.375 mT corresponds to \rho = \tfrac16. Anisotropic (dipolar) hyperfine, seen in solids, gives distances and orbital orientation.

Practical uses

• Spin trapping: short-lived radicals (·OH, superoxide) are captured by nitrones such as DMPO to form persistent nitroxides with diagnostic hyperfine patterns.
• Quantification: double integration of the derivative spectrum against a standard gives spin concentration.
• Systems with S \geq 1 show zero-field splitting; Kramers ions (odd electron count) always retain an EPR-observable doublet, while non-Kramers ions may be EPR silent at normal frequencies.

Modern EPR is dominated by pulsed and multi-frequency methods that extract couplings and distances hidden in continuous-wave spectra.

Distance measurements. Pulsed dipolar spectroscopy, especially DEER (also called PELDOR), measures the dipolar coupling between two spin labels and hence distance distributions of about 1.5–8 nm, extended further with deuteration. Site-directed spin labelling with nitroxides, trityls or Gd(III) tags allows conformational ensembles of proteins and nucleic acids to be mapped, including in cells. Converting time traces to distance distributions is an ill-posed inverse problem; Tikhonov regularisation and newer model-based and neural-network approaches are compared in community benchmarks.

Hyperfine spectroscopy. ENDOR (electron–nuclear double resonance), ESEEM and HYSCORE resolve small couplings to ^1H, ^2H, ^{14}N, ^{13}C and ^{17}O. They identify ligands and substrates at metalloenzyme active sites, for instance in nitrogenase and photosystem II intermediates, and are routinely interpreted with DFT-computed hyperfine tensors.

High field. W-band (94 GHz) and higher-frequency instruments resolve small g-anisotropies, separate overlapping radicals and access large zero-field splittings in high-spin systems and single-molecule magnets.

Time-resolved EPR. Photoexcitation creates spin-polarised triplets and radical pairs with emissive and absorptive lines; this is used in photosynthetic reaction centres, organic photovoltaics and studies of the radical-pair mechanism proposed for avian magnetoreception.

Materials and quantum technologies. EPR characterises defects in semiconductors, radicals in batteries, and molecular qubits, where coherence times (T_m) are the figure of merit and design rules involve removing nuclear spins and rigidifying the environment.

Persistent challenges include sensitivity at room temperature for fast-relaxing metal centres, accurate prediction of g-tensors and zero-field splittings for heavy elements, and quantitative interpretation of broad distance distributions.