= Fixed-number low-temperature Pauli susceptibility
{title2=$\chi/(2\mu_B^2)=g_F+\frac{\pi^2}{6}(g_F''-g_F'^2/g_F)(k_BT)^2+O(T^4)$}
For a smooth single-spin <density of states> $g$ near a positive <Fermi energy> and fixed total <particle number>, the <Sommerfeld expansion> gives $\mu=\epsilon_F-(\pi^2/6)(g_F'/g_F)(k_BT)^2+O(T^4)$. The <spin> contribution to the total-moment <magnetic susceptibility>, defined by differentiating with respect to $B$, is $2\mu_B^2\int g(E)[-n'(E)]dE$. Expanding about $\mu$ and including the shift of <chemical potential> gives the displayed formula. The $g_F''$ term is thermal broadening; the $-g_F'^2/g_F$ term enforces fixed <particle number>. A band edge or singular <density of states> within $k_BT$ of $\epsilon_F$ requires a separate analysis. Orbital magnetic response is excluded.
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