Spin-one-half paramagnetic magnetization ($M=n\mu \tanh [\mu B/(k_{B}T)]$)

For independent localized two-state moments of projections $\pm \mu$ along a field, the one-ion partition function is $Z_1=2\cosh [\mu B/(k_{B}T)]$. Its thermal mean gives $M=n\mu \tanh [\mu B/(k_{B}T)]$. A $J=1/2$ ion has $\mu =g_J{\mu }_B/2$, so the special value $\mu ={\mu }_B$ additionally assumes $g_J=2$. At weak field the SI volume susceptibility is ${\chi }_H={\mu }_{0}n{\mu }^2/(k_{B}T)$, neglecting demagnetizing corrections.