Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 35C / b / Solution

For $N$ independent spins with $s_z\in\{-1,0,1\}$ and one-spin Hamiltonian $-\mu Bs_z$,
$$
Z=\left(1+2\cosh(\beta\mu B)\right)^N.
$$
Writing $x=\beta\mu B$, its heat capacity is
$$
C=Nk_Bx^2\frac{2(\cosh x+2)}
{(1+2\cosh x)^2}.
$$


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