Spin-1 paramagnet

For $N$ independent spins with $s_z\in \{-1,0,1\}$ and one-spin Hamiltonian $-\mu B{s}_z$,Writing $x=\beta \mu B$, its heat capacity is

For $N$ noninteracting units with energies $-ϵ,0,ϵ$, put $x=\beta ϵ$. Thenand

For one three-level unit with energies $-\mu B,0,\mu B$, the high-temperature heat capacity is $C\simeq 2{\mu }^2{B}^2/(3k_B{T}^2)$. A measured coefficient $C=a/T^2$ therefore gives the displayed magnetic moment. If the heat capacity instead represents $N$ independent units, replace $a$ by $a/N$; the normalization must be established before extracting a single-unit moment.