13 Jul 2026

Electron Spin Resonance and Determination of the g-Factor

practical pg-iv cmp esr lande-g-factor magnetic-resonance

Aim

To observe electron spin resonance in a paramagnetic sample and determine the electron $g$-factor.

Apparatus

ESR spectrometer, microwave source, Helmholtz coils, Hall probe, paramagnetic sample, and frequency meter.

Figure

Microwave excitation of the paramagnetic sample while the magnetic field is swept and measured.

Theory

An atom or ion with an unpaired electron has a spin magnetic moment. In an applied magnetic field, the two allowed spin orientations have different Zeeman energies. Their separation is

\[\Delta E=g\mu_BB.\]

When the sample is exposed to microwave radiation, absorption occurs when one photon supplies exactly this energy:

\[h\nu=g\mu_BB.\]

Measuring the resonant field at known frequency therefore gives

\[g=\frac{h\nu}{\mu_BB}.\]

A plot of resonance field against frequency should pass nearly through the origin; its slope is $h/(g\mu_B)$.

Observations

Microwave frequency (GHz)	Resonance field (mT)	$g$
9.10	324.5	2.00
9.20	328.0	2.01
9.30	331.5	2.00

Graph

Resonance field plotted against microwave frequency.

Calculation

For the 9.20 GHz reading, $\nu=9.20\times10^9$ Hz and $B=328.0$ mT $=0.3280$ T. Hence

\[g=\frac{h\nu}{\mu_BB}=\frac{(6.626\times10^{-34})(9.20\times10^9)}{(9.274\times10^{-24})(0.3280)}=2.00.\]

The three readings give values close to 2.00; their mean is used because field calibration and resonance-width uncertainty affect each reading slightly.

Result

The mean electron $g$-factor of the sample is

\[\boxed{g=2.00}.\]

Viva Questions

1. Why is a paramagnetic sample used? It contains unpaired spins that can absorb microwave energy.
2. What is resonance? Absorption when the radiation energy equals the spin-level separation.
3. Why is a Hall probe used? To calibrate the magnetic field at the sample position.

Maxima Code

Download the PG-IV ESR calculation.