Grand-canonical spin susceptibility of a half-filled single level ($\mu =ϵ,\chi ={\mu }_B^2/(2k_{B}T)$)

One noninteracting spatial orbital with two spin energies $ϵ\pm {\mu }_{B}B$, in the grand canonical ensemble with mean occupation one, has chemical potential $\mu =ϵ$ for every positive temperature. Its spin population difference is $-\tanh [{\mu }_{B}B/(2k_{B}T)]$, giving total-moment response $\chi ={\mu }_B^2/(2k_{B}T)$ 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 $\chi ={\mu }_B^2/(k_{B}T)$. The factor of two is an ensemble effect in this small system.