Past exam of the condensed matter physics course of the University of Oxford / 2014 / 4 / a / Solution

The SI volume magnetic susceptibility is the weak-field response ${\chi }_M=\partial M/\partial H$, or $M/H$ for a linear isotropic sample with no spontaneous moment. It is dimensionless; a molar susceptibility would be a different quantity. For weak magnetization, $B\simeq {\mu }_{0}H$. Neglect local-field and demagnetizing corrections in this independent-spin model.

A spin-$1/2$ ion has moment projections $\pm g{\mu }_B/2$. The requested coefficient assumes a spin-only $g\simeq 2$, giving projections $\pm {\mu }_B$ and Zeeman energies $\mp {\mu }_{B}B$. With $x={\mu }_{B}B/(k_{B}T)$, the one-ion partition function and magnetization areFor $|x|\ll 1$, $\tanh x\simeq x$, soHere $N$ is a number density, as specified, rather than total ion number. The small-field condition ismeaning the Zeeman splitting $2{\mu }_{B}B$ is small compared with the thermal scale up to an irrelevant order-one convention. It is not a temperature-independent absolute upper bound on $B$. For a different $g$ value the coefficient contains $(g{\mu }_B/2{)}^2$ instead of ${\mu }_B^2$.