Past exam of the condensed matter physics course of the University of Oxford / 2013 / 4 / c / Solution

Let $x=g{\mu }_{B}B/(k_{B}T)$. The single-ion partition function and mean projected magnetic moment areFor $J>0$, the geometric sum also givesIn the high-temperature, weak-polarization regime $|x|J\ll 1$, expand the Boltzmann factors. The numerator's zero-order term vanishes because $\sum m_J=0$, and the first-order term uses $\sum m_J^2=J(J+1)(2J+1)/3$. The denominator is $2J+1+O(x^2)$. ThereforeThis fixed-field per-ion Curie coefficient describes mean moment per ion at fixed field, so it includes $B$. It is not the field-independent volume-susceptibility Curie constant. For number density $n$ and $B\simeq {\mu }_{0}H$, the latter is $C_{\chi }=n{\mu }_0(g{\mu }_B{)}^{2}J(J+1)/(3k_B)$, giving ${\chi }_H=C_{\chi }/T$. At $J=0$, the specified moment is zero and both expressions vanish.