共鳴 Magnetic Resonance · Laboratory experiment

Electron spin resonance: Landé g-factor of DPPH

Aim: to observe electron spin resonance of the free radical DPPH in a radio-frequency coil placed between Helmholtz coils, measure the resonance field at several frequencies, and determine the Landé g-factor. An X-band (9.5 GHz) field-sweep spectrometer with lock-in detection is included for comparison, together with a nitroxide radical that shows hyperfine structure.

Apparatus

Oscilloscope

Observation table

Graph: resonance frequency against field

Results

Theory

An unpaired electron with spin S = ½ in a field B has two Zeeman levels, \(E = m_s g\mu_B B\) with \(m_s = \pm\tfrac12\). Radiation of frequency ν is absorbed when

$$ h\nu = g\,\mu_B\,B \qquad\Rightarrow\qquad g = \frac{h\nu}{\mu_B B},\qquad \frac{\nu}{B} = \frac{g\mu_B}{h} \approx 28.04\ \text{GHz/T}\ (g=2.0036).$$

For a free electron, g = 2.002319 (QED). In DPPH the spin is almost entirely spin-like, with a quenched orbital moment, so g = 2.0036. Its line is exchange-narrowed and shows no resolved hyperfine structure.

Helmholtz coils

$$ B = \left(\tfrac45\right)^{3/2}\frac{\mu_0 N I}{R} = 4.23\ \text{mT/A}\quad (N = 320,\ R = 6.8\ \text{cm}).$$

Detection

A 50 Hz current added to the DC current sweeps B through resonance twice per cycle. On the oscilloscope (Y–t), the absorption peaks are equally spaced exactly when \(B_{dc} = B_{res}\). In X–Y mode, the two peaks (field rising and falling) coincide when the phase shifter compensates the coil's inductive lag.

Earth's field (about 20–40 µT along a typical coil axis) adds to B. Fitting ν = (gμB/h)(B + B⊕) with an intercept separates it from g.

Procedure

  1. Insert the DPPH sample in the RF coil, set ν ≈ 30 MHz, and switch on the 50 Hz modulation.
  2. Increase the DC current until absorption peaks appear. Adjust until they are equally spaced (Y–t) or coincide (X–Y).
  3. Record ν and I. Repeat for 8–10 frequencies between 15 and 130 MHz.
  4. Plot ν against B and fit a straight line: g = h × slope / μB.
  5. X-band: set ν ≈ 9.5 GHz, sweep around 339 mT, click the derivative line to find its centre, and compute g directly. Switch to TEMPO to see the 1 : 1 : 1 triplet from the ¹⁴N nucleus (I = 1), with aN = 1.60 mT.

Linewidth and relaxation

For a Lorentzian line, the peak-to-peak width of the derivative ΔBpp gives \(T_2 = \dfrac{2}{\sqrt3\,\gamma\,\Delta B_{pp}}\), with γ = gμB/ħ.

Viva questions

Why use a free radical?

Its unpaired electron gives a strong, narrow line with g close to the free-electron value, which makes it the standard g-marker in ESR.

Why is lock-in detection used at X-band?

Field modulation at about 100 kHz with phase-sensitive detection rejects noise outside a narrow band. The output is the first derivative of the absorption.

What determines the number of hyperfine lines?

n equivalent nuclei of spin I give 2nI + 1 lines. One ¹⁴N (I = 1) gives three lines of equal intensity.

Compare ESR and NMR frequencies at 0.34 T.

ESR: 9.5 GHz. Proton NMR: 14.5 MHz. The ratio equals the ratio of the electron and proton magnetic moments, about 658.