= Grand-canonical spin susceptibility of a half-filled single level
{title2=$\mu=\epsilon,\quad\chi=\mu_B^2/(2k_BT)$}
One noninteracting spatial orbital with two <spin> energies $\epsilon\pm\mu_BB$, in the <grand canonical ensemble> with mean occupation one, has <chemical potential> $\mu=\epsilon$ for every positive <temperature>. Its <spin> population difference is $-\tanh[\mu_BB/(2k_BT)]$, giving total-moment response $\chi=\mu_B^2/(2k_BT)$ at zero field. Empty and doubly occupied orbital states are permitted. Fixing exactly one <Electron> instead leaves only the two singly occupied states and gives $\chi=\mu_B^2/(k_BT)$. The factor of two is an ensemble effect in this small system.
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