Heat capacity of a classical magnetic dipole ($C_B=k_B[1-x^2/{\sinh }^{2}x]$)

A freely orientable classical magnetic dipole moment of fixed magnitude $m$ in a uniform magnetic field $B$ has angular canonical partition function $Z=4\pi \sinh x/x$, where $x=mB/(k_{B}T)$. Its internal energy is $U=-mB\coth x+k_{B}T$, and its fixed-field heat capacity is $C_B=k_B[1-x^2/{\sinh }^{2}x]$. At low temperature, two locally Cartesian tangent coordinates on the orientation sphere each contribute $k_{B}T/2$ by the equipartition theorem, giving $C_B\to k_B$. A discrete quantum spin in a nonzero magnetic field instead has a spectral gap and its heat capacity vanishes exponentially at sufficiently low temperature.