Brillouin and Langevin functions

The Brillouin function is a special function that arises in the calculation of the magnetization of an ideal paramagnet. The magnetization $M$ is given by:

$M=Ng\mu _{B}SB_{S}(x)$

where $N$ is the Avogadro number, $g$ the Landé g-factor, $\mu _{B}$ the Bohr magneton, and $S$ the magnitude of magnetic spin.

$B_{S}$ is the Brillouin function:

$B_{S}(x)={\frac {2S+1}{2S}}\coth \left({\frac {2S+1}{2S}}x\right)-{1 \over 2S}\coth \left({x \over 2S}\right)$

$x$ is given by:

$x={\frac {g\mu _{B}SH}{k_{B}T}}$

where $H$ is the applied magnetic field, $k_{b}$ is the Boltzmann constant, and $T$ is the temperature.