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

In the paper's convention $\mathbf{J}$ is measured in units of $\hslash$, and $g$ is dimensionless. For $m=-J,-J+1,\dots ,J$, the magnetic moment and its Zeeman energy areIf $\mathbf{J}$ instead denotes dimensional angular momentum, its coefficient must contain $1/\hslash$. This resolves the units of the printed moment formula.

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]

For $g>0$, $B\to +\infty$ at fixed $T>0$ selects $m=-J$, so all moments align with the field:This is magnetic saturation. The Curie response of an arbitrary independent spin follows at weak field: $\coth z=z^{-1}+z/3+O(z^3)$ givesThis is the Curie law of independent moments with paramagnetism. Thermal populations are symmetric at zero field, so the model has no spontaneous magnetization. Both limits hold at fixed positive temperature; taking $T\to 0$ first is a different limiting procedure.