EPR Spectroscopy: g-Factor and Hyperfine Lines

By Aniket Bhardwaj · 29 October 2026 · Inorganic

Electron Paramagnetic Resonance (EPR) spectroscopy, or Electron Spin Resonance (ESR), probes paramagnetic species with unpaired electrons. In examinations like CSIR-NET, GATE, IIT-JAM, and CUET-PG, numerical problems test calculating resonance fields, determining Landé g-factors, and evaluating hyperfine lines. Mastering these calculations is key to scoring in spectroscopy.

Exam details

Source: practice problem written for this article (not a past-paper question)
Relevant exams: CSIR-NET Chemical Sciences, GATE Chemistry, IIT-JAM Chemistry, CUET-PG
Topic: Electron Paramagnetic Resonance (EPR) Spectroscopy: Resonance Condition, g-Factor, and Hyperfine Splitting
Difficulty: Advanced

Resonance Condition and the Landé g-Factor

In a magnetic field B0, unpaired electron spin (S = 1/2) splits into two Zeeman levels: ms = −1/2 and ms = +1/2. Perpendicular microwave radiation of frequency ν induces transitions when photon energy hν equals ΔE under selection rule Δms = ±1.

The Landé g-factor probes local electronic structure. For a free electron, ge = 2.0023. In complexes, spin-orbit coupling alters g. When a d subshell is less than half-filled (d1 Ti3+), spin-orbit coupling lowers g below ge (g < ge). When more than half-filled (d9 Cu2+), g rises above ge (g > ge).

Nuclear Hyperfine Splitting and Multiplicity Rules

Hyperfine splitting arises when electron spin couples to magnetic nuclei of spin I. Under selection rules Δms = ±1 and ΔmI = 0, each nuclear spin state splits the signal into peaks separated by coupling constant a.

Single nucleus of spin I: Yields 2I + 1 lines of equal intensity.
n equivalent nuclei of spin I: Yields 2nI + 1 lines. When I = 1/2 (like 1H), peak intensities follow Pascal's triangle.
Non-equivalent groups: Total lines follow the product N = ∏ (2ni Ii + 1).

Key EPR Formulas:
Zeeman Condition: ΔE = hν = g μB B0
Resonant Field: B0 = (hν) / (g μB)
Landé g-factor: g = (hν) / (μB B0)
Hyperfine Lines: N = 2nI + 1 (single group) or N = ∏ (2ni Ii + 1) (multiple groups)
Constants: h = 6.62607 × 10−34 J s, μB = 9.27401 × 10−24 J T−1, ge = 2.0023.
Units: ν in Hz; B0 in T or mT (1 T = 1,000 mT = 10,000 G).

Step-by-Step Worked Numerical Examples

Question: An EPR spectrometer operates at an X-band microwave frequency of ν = 9.500 GHz. Calculate the resonant magnetic field B0 (in millitesla) for a free radical with g = 2.0023.

Step 1: Identify given values: ν = 9.500 × 10^9 Hz, g = 2.0023, h = 6.62607 × 10−34 J s, μB = 9.27401 × 10−24 J T−1.

Step 2: Use resonance formula B0 = (hν) / (g μB).

Step 3: Compute numerator and denominator:
hν = (6.62607 × 10−34 J s) × (9.500 × 10^9 s−1) = 6.294767 × 10−24 J
g μB = 2.0023 × (9.27401 × 10−24 J T−1) = 1.856935 × 10−23 J T−1

Step 4: Calculate B0:
B0 = (6.294767 × 10−24 J) / (1.856935 × 10−23 J T−1) = 0.338987 T
Converting to millitesla: B0 = 0.338987 × 1,000 mT = 338.99 mT.

Step 5: Second-route check using gyromagnetic ratio γ = (g μB) / h = 2.80247 × 10^10 Hz T−1:
B0 = ν / γ = (9.500 × 10^9) / (2.80247 × 10^10) = 0.33899 T = 338.99 mT.

Answer: 338.99 mT

Question: A copper(II) complex exhibits an EPR absorption line at 0.3050 T when irradiated with a microwave frequency of 9.350 GHz. Calculate the Landé g-factor of this complex.

Step 1: Given data: ν = 9.350 × 10^9 Hz, B0 = 0.3050 T, h = 6.62607 × 10−34 J s, μB = 9.27401 × 10−24 J T−1.

Step 2: Rearrange Zeeman condition: g = (hν) / (μB B0).

Step 3: Calculate product values:
hν = (6.62607 × 10−34) × (9.350 × 10^9) = 6.195375 × 10−24 J
μB B0 = (9.27401 × 10−24) × 0.3050 = 2.828573 × 10−24 J

Step 4: Compute g:
g = (6.195375 × 10−24) / (2.828573 × 10−24) = 2.1903.

Step 5: Second-route check using ratio k = h / μB = 7.14477 × 10−11 T s:
g = k × (ν / B0) = 7.14477 × 10−11 × (9.350 × 10^9 / 0.3050) = 2.1903. Value rounds to 2.190.

Answer: 2.190

Question: Determine the number of hyperfine lines and relative intensity ratios for: (a) [VO(acac)2] containing one 51V nucleus (I = 7/2), and (b) 1,4-benzosemiquinone radical anion containing four equivalent 1H nuclei (I = 1/2).

Step 1: For [VO(acac)2], vanadium is V(IV) (d1, S = 1/2). The central nucleus is 51V with I = 7/2.
Number of lines: N = 2I + 1 = 2(7/2) + 1 = 8 lines.
Because only one nucleus couples, each line has equal transition probability. Intensity ratio is 1:1:1:1:1:1:1:1.

Step 2: For 1,4-benzosemiquinone radical anion, the electron couples to n = 4 equivalent protons (1H, I = 1/2).
Number of lines: N = 2nI + 1 = 2(4)(1/2) + 1 = 5 lines.
For I = 1/2, intensities follow binomial coefficients for n = 4: 1:4:6:4:1.

Step 3: Second-route check by successive proton coupling:
Proton 1: 1:1; Proton 2: 1:2:1; Proton 3: 1:3:3:1; Proton 4: 1:4:6:4:1 (total lines = 5). Results match exactly.

Answer: (a) 8 lines with 1:1:1:1:1:1:1:1 ratio; (b) 5 lines with 1:4:6:4:1 ratio

Common Mistakes Students Make

Confusing electron spin S with nuclear spin I: In N = 2nI + 1, always use nuclear spin I of the coupling nucleus, never electron spin S.
Applying Pascal's triangle to nuclei with I > 1/2: Pascal's triangle applies only to I = 1/2 nuclei. For 14N (I = 1), two equivalent nuclei give 5 lines with intensities 1:2:3:2:1, not 1:4:6:4:1.
Mixing magnetic field units: 1 Tesla = 10,000 Gauss = 1,000 millitesla. Entering Gauss directly without converting to Tesla causes huge errors.
Assuming g is fixed at 2.0023: In transition metal complexes, spin-orbit coupling shifts g significantly from 2.0023.

Exam Relevance and Topic Overview

Examination	Question Focus	Recommended Strategy
CSIR-NET Chemical Sciences	Multi-nuclear hyperfine coupling, Kramers ions, and anisotropic g-values (g∥, g⊥).	Use the multiplicative rule ∏(2ni Ii + 1) and remember that d1 has g < ge while d9 has g > ge.
GATE Chemistry	Quantitative calculations of resonance frequency, B0, and Landé g-factors.	Memorise h = 6.626 × 10−34 J s and μB = 9.274 × 10−24 J T−1 for fast numerical execution.
IIT-JAM Chemistry	Resonance condition hν = g μB B0, counting hyperfine lines, and Pascal intensity ratios.	Practice line counting for organic radicals with equivalent protons.
CUET-PG Chemistry	Basic principles of electron Zeeman splitting and identifying EPR active species.	Remember that only species with unpaired electrons (S ≠ 0) show EPR spectra.

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