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. These functions are named after French physicists Paul Langevin and Léon Brillouin, who contributed to the microscopic understanding of magnetic properties of matter.

The Langevin function is derived using statistical mechanics and describes how magnetic dipoles are aligned by an applied field. The Brillouin function was developed later to give an explanation that considers quantum physics. The Langevin function could then be seen as a special case of the more general Brillouin function if the quantum number

J

{\displaystyle J}

was infinite (

J

→

∞

{\displaystyle J\to \infty }

).

Brillouin function for paramagnetism

The Brillouin function arises when studying magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization

M

{\displaystyle M}

on the applied magnetic field

B

{\displaystyle B}

, defined by the following equation:

The function

B

J

{\displaystyle B_{J}}

is usually applied in the context where

x

{\displaystyle x}

is a real variable and a function of the applied field

B

{\displaystyle B}

. In this case, the function varies from -1 to 1, approaching +1 as

x

→

+

∞

{\displaystyle x\to +\infty }

and -1 as

x

→

−

∞

{\displaystyle x\to -\infty }

.

The total angular momentum quantum number

J

{\displaystyle J}

is a positive integer or half-integer. Considering the microscopic magnetic moments of the material.