= Heat capacity of a classical magnetic dipole
{title2=$C_B=k_B[1-x^2/\sinh^2x]$}
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 $x=mB/(k_BT)$. Its <internal energy> is $U=-mB\coth x+k_BT$, and its fixed-field <heat capacity> is $C_B=k_B[1-x^2/\sinh^2x]$. At low <temperature>, two locally Cartesian tangent coordinates on the orientation <sphere> each contribute $k_BT/2$ 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>.
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