= Spin-one-half paramagnetic magnetization
{title2=$M=n\mu\tanh[\mu B/(k_BT)]$}
For independent localized two-state moments of projections $\pm\mu$ along a field, the one-ion partition function is $Z_1=2\cosh[\mu B/(k_BT)]$. Its thermal mean gives $M=n\mu\tanh[\mu B/(k_BT)]$. 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 $\chi_H=\mu_0n\mu^2/(k_BT)$, neglecting demagnetizing corrections.
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