 Brillouin magnetization formula

For independent distinguishable moments with angular momentum $J$, define $y=g{\mu }_{B}B/(k_{B}T)$. The one-site partition function is $Z=\sinh [(J+1/2)y]/\sinh (y/2)$. Its logarithmic derivative gives $⟨{\mu }_z⟩=g{\mu }_B[(J+1/2)\coth ((J+1/2)y)-\frac{1}{2}\coth (y/2)]$. At $J=1/2$ this reduces to $(g{\mu }_B/2)\tanh (y/2)$.