Brillouin and Langevin functions

The Langevin function ( according to the physician Paul Langevin ) is a mathematical function used for the calculation of orientation of polarization, polarization, magnetization and resistance.

Definition

The Langevin function is defined by

With the cotangent hyperbolic respectively.

An application

The best-known application is the semi-classical description of a paramagnet in an external magnetic field. For this, the Langevin parameter is introduced:

The individual symbols stand for the following sizes:

• M - Magnetic moment of a particle
• B - amount of the applied external magnetic field
• - Boltzmann constant
• T - absolute temperature

For the magnetization M of a paramagnet is then as follows:

N stands for the amount of substance, and m is the magnetic moment of the individual spins of the paramagnet. A further description of the quantum mechanical paramagnetism is given by the Brillouin function.

Approximations

An approximation of the Langevin function for is

For the approximation is valid