Past exam of the condensed matter physics course of the University of Oxford / 2016 / 3 / a / Solution

Write the two allowed moment projections as $\pm \mu$, where $\mu =g{\mu }_B/2$ for spin one-half. Their Zeeman energies in the prescribed flux density are $\mp \mu B$. The single-particle partition function and the mean moment along the field areFor noninteracting moments, magnetization is moment per volume:For electron-like spin-only moments $g\simeq 2$, these become $M_s=N{\mu }_B/V$ and $\eta ={\mu }_B/k_B$.

Differentiate with respect to the specified $B$ to obtainThis $B$ derivative is not a dimensionless SI volume susceptibility. In the weak-feedback approximation $B\simeq {\mu }_{0}H$, the usual dimensionless differential susceptibility is ${\chi }_{\mathrm{diff}}\simeq {\mu }_0{\chi }_B$. The small-field limit gives Curie's law ${\chi }_0={\mu }_{0}N{\mu }^2/(V{k}_{B}T)$. At finite bias a secant response $M/H\simeq {\mu }_{0}M/B={\chi }_0\tanh (x)/x$ differs from the differential slope ${\chi }_0{\mathrm{sech}}^{2}x$. For an internal $B={\mu }_0(H+M)$ retained self-consistently, the exact local conversion is ${\chi }_H={\mu }_0{\chi }_B/(1-{\mu }_0{\chi }_B)$; demagnetizing geometry can further affect an applied-field measurement. Stating the field convention avoids a hidden factor of ${\mu }_0$ or confusion between a slope and a ratio.