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.

For moment projections $\pm \mu$, $p_{+}=(1+\tanh x)/2$ but $M/M_s=\tanh x$, with $x=\mu B/(k_{B}T)$. At zero field half the moments occupy each projection while the net magnetization is zero. An effective Curie moment is also distinct from the saturated projection used in this two-state model.

For independent moments with magnetic energies $\mp \mu B$, $Z=(2\cosh x{)}^N$ and $S=N{k}_B[\ln (2\cosh x)-x\tanh x]$, where $x=\mu B/(k_{B}T)$. This counts the magnetic orientations only, excluding translational and lattice entropy.