The EPR g Value

Introduction

An EPR spectrometer usually holds the microwave frequency fixed and sweeps the magnetic field until absorption occurs. The field at which a signal appears depends on the instrument frequency, so raw field positions cannot be compared between laboratories. The g value removes that dependence. It plays the same role in EPR that chemical shift plays in NMR: a normalised number that reports the electronic environment of the unpaired electron and allows spectra recorded on different spectrometers to be compared directly.

Core explanation

The resonance condition. An unpaired electron in a magnetic field B has two spin states, mₛ = +½ and −½, separated in energy by gμBB, where μB is the Bohr magneton (9.274 × 10⁻²⁴ J T⁻¹). Absorption occurs when the microwave photon energy matches this gap:

hν = gμBB

Rearranging gives g = hν/(μBB). Because h and μB are constants, g can be computed from any measured pair of frequency and field. At X-band (about 9.5 GHz) a signal with g ≈ 2 appears near 0.34 T (340 mT); at Q-band (about 34 GHz) the same signal moves to about 1.2 T, yet g is unchanged.

The free-electron value. For a completely isolated electron, relativistic quantum electrodynamics gives gₑ ≈ 2.0023. In a molecule the electron is not isolated. It occupies an orbital, and orbital angular momentum can couple to spin angular momentum. In most ground states orbital angular momentum is largely quenched by the surrounding ligands or bonds, but spin–orbit coupling mixes in small contributions from excited states. These contributions shift g away from 2.0023.

Size and sign of the shift. A simple perturbation picture gives Δg roughly proportional to λ/ΔE, where λ is the spin–orbit coupling constant and ΔE is the energy gap to the excited state that is mixed in. Two practical consequences follow:

- Light-atom organic radicals have small λ and large ΔE, so g lies very close to gₑ, typically 2.002–2.006. Radicals with spin density on oxygen, sulfur or heavier atoms show larger shifts. - Transition-metal ions have larger λ and smaller ligand-field gaps, so g can differ markedly from 2. For a d-shell less than half full (for example d¹ vanadyl, VO²⁺), g is usually below 2.0023; for a shell more than half full (for example d⁹ copper(II)), g is usually above it.

The g tensor. Because spin–orbit mixing depends on which orbitals are involved, g generally depends on the orientation of the molecule relative to the field. It is therefore described by a tensor with three principal values, gₓ, gᵧ and g z. In fluid solution rapid tumbling averages these to an isotropic value, g iso = (gₓ + gᵧ + g z)/3. Symmetry sets the pattern: cubic sites give one value, axial sites give g∥ and g⊥, and lower symmetry gives three distinct values.

Measuring g accurately. In a first-derivative spectrum, g is taken at the point where the derivative crosses the baseline for a symmetric line. Accurate work needs a calibrated frequency counter and field probe, or a standard of known g recorded under the same conditions. Higher microwave frequencies spread g differences over a larger field range, improving the resolution of nearby g values.

Formulae

hν = gμBB, so g = hν/(μBB). A convenient working form is g = 71.448 × ν(GHz)/B(mT). The isotropic average is g iso = (gₓ + gᵧ + g z)/3.

Step-by-step reasoning

To extract and interpret a g value:

1. Record the exact microwave frequency and the field at the line centre. 2. Convert units consistently (GHz and mT in the working formula). 3. Calculate g and compare it with 2.0023. 4. Judge the size and sign of Δg: a small shift suggests a light-atom radical; a large shift suggests a metal centre or heavy-atom involvement. 5. If the sample is frozen or solid, check whether several g components are present.

Visual explanation

Picture two diverging straight lines on an energy-versus-field plot, one for each spin state, with slopes of ±½gμB. A fixed microwave photon energy is a vertical gap. The field at which the lines are separated by exactly that gap is the resonance field. A larger g means steeper lines, so resonance is reached at lower field.

Real-world analogy

A shop receipt printed in different currencies shows different numbers for the same basket. Converting every total to one reference currency lets you compare them. The g value is that reference currency: it converts field positions measured at different microwave frequencies into one comparable number.

Real-world example

Biochemists study copper proteins such as plastocyanin and azurin by EPR. Their g∥ values, around 2.2–2.3, together with unusual hyperfine patterns, showed that the copper sites differ electronically from simple copper(II) salts in water. The g values helped reveal the strong covalency of the copper–sulfur bond in these electron-transfer proteins.

Why?

Why does copper(II) show g above 2 while vanadyl shows g below 2? The sign of the spin–orbit coupling constant λ changes between shells less than half filled and shells more than half filled. For a d⁹ ion, the effective contribution behaves like a "hole" and adds to g; for d¹ it subtracts from g.

Common misconception

"Every EPR signal near g = 2 must come from an organic radical." Some metal centres, such as certain low-spin or strongly covalent complexes and some defect centres in solids, also give signals near 2. The g value must be combined with hyperfine structure, linewidth and temperature behaviour before assigning a species.

Worked example

Question: A solution spectrum recorded at 9.500 GHz shows a single line centred at 339.0 mT. Calculate g and suggest what type of species is present.

Reasoning: g = 71.448 × ν(GHz)/B(mT) = 71.448 × 9.500/339.0 = 678.76/339.0 = 2.0022. This is essentially the free-electron value, so spin–orbit contributions are tiny.

Answer: g ≈ 2.002, consistent with a carbon-centred organic radical with little heavy-atom character.

Quick check

1. If a spectrometer's frequency is raised from X-band to Q-band, what happens to the resonance field and to the g value of a given radical? Answer: The resonance field increases roughly in proportion to frequency, but the g value stays the same because it is a property of the species.

Exam focus

Be ready to rearrange hν = gμBB with correct units, to state gₑ = 2.0023, and to explain Δg qualitatively through spin–orbit coupling. Examiners often ask why g for transition-metal ions deviates much more than for organic radicals, and how g iso relates to the principal values.

Advanced insight

Precise g-tensor calculations now use relativistic density functional theory, and high-field EPR at 94 GHz or above can resolve g anisotropy in organic radicals smaller than 0.001. Such resolution distinguishes tyrosyl from tryptophanyl radicals in enzymes and reveals hydrogen bonding to oxygen-centred radicals, which shifts the g component along the C–O bond.

Summary

The g value, defined by hν = gμBB, converts a resonance field into a frequency-independent number. Its deviation from gₑ = 2.0023 arises from spin–orbit coupling, scaling roughly as λ/ΔE. Organic radicals lie close to 2.002–2.006, while metal ions deviate strongly, with sign depending on shell filling. In solids, g is a tensor with up to three principal values.

Practice questions

1. Calculate the resonance field for a species with g = 2.0023 at 34.00 GHz. Answer: B = 71.448 × 34.00/2.0023 = 1213 mT, or about 1.21 T. 2. A vanadyl complex gives g iso = 1.96. Explain why g is below 2.0023. Answer: Vanadyl is d¹, a shell less than half full, so spin–orbit coupling makes a negative contribution to g. 3. A frozen sample shows gₓ = 2.02, gᵧ = 2.05 and g z = 2.20. What g would be observed in fast-tumbling solution? Answer: g iso = (2.02 + 2.05 + 2.20)/3 = 2.09. 4. Why do organic radicals containing sulfur often show larger g shifts than purely carbon-centred radicals? Answer: Sulfur has a much larger spin–orbit coupling constant than carbon, so spin density on sulfur mixes in more orbital angular momentum and shifts g further from gₑ.