Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B1 / b / Solution

Write $y=g{\mu }_{B}B/(k_{B}T)$. Summing the Boltzmann factors gives the one-atom partition functionSince $⟨m⟩=-{\partial }_y\log Z_1$, the Brillouin magnetization formula isHere $M$ is the total magnetic moment used by the question; the magnetization per unit volume is $M/V$. For independent distinguishable atoms, $Z_N=Z_1^N$, with no indistinguishability factor.

Putting $J=1/2$ yieldsThe apparent singularities of the hyperbolic cotangents at $B=0$ cancel; the finite sum makes the zero-field limit explicit.^[ref]