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 $(g{\mu }_B/2)\tanh (y/2)$.

The Brillouin function gives the normalized magnetization of independent angular-momentum-$J$ moments. It is odd, increasing, has slope $(J+1)/(3J)$ at zero and tends to $\pm 1$ at large positive or negative argument. Explicitly,It follows by differentiating the logarithm of ${\sum }_{m=-J}^J\exp (mx/J)$, the one-ion partition function. Its limiting value gives magnetic saturation, and its initial slope gives the Curie law.