Brillouin and Langevin functions

The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics.

The Brillouin function is a special function defined by the following equation:

The function is usually applied (see below) in the context where x is a real variable and J is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as ${\displaystyle x\to +\infty }$ and -1 as ${\displaystyle x\to -\infty }$.

The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization ${\displaystyle M}$ on the applied magnetic field ${\displaystyle B}$ and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:

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