Brillouin and Langevin functions

From Wikipedia, the free encyclopedia

1 Brillouin Function
2 Langevin Function
3 High Temperature Limit
4 High Field Limit
5 References

[edit] Brillouin Function

The Brillouin function is a special function that arises in the calculation of the magnetization of an ideal paramagnet. This function describes the dependency of the magnetization $M$ on the applied magnetic field $H$ and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:

$M = N g \mu_B J \cdot B_J(x)$

where $N$ is the Avogadro number, $g$ the Landé g-factor, and $μ B$ the Bohr magneton.

$B J$ is the Brillouin function^[1] which varies from -1 to 1:

$B_J(x) = \frac{2J + 1}{2J} \coth \left ( \frac{2J + 1}{2J} x \right ) - \frac{1}{2J} \coth \left ( \frac{1}{2J} x \right )$

where $x$ is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy $k B T$ :

$x = \frac{g \mu_B J B}{k_B T}$

where $k B$ is the Boltzmann constant and $T$ the temperature.

[edit] Langevin Function

Langevin function (red line), compared with

t a n h (x / 3)

(blue line).

In the classical limit, the moments can be continuously aligned in the field and $J$ can assume all values ( $J \to \infty$ ). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:

$L(x) = \coth(x) - \frac{1}{x}$

[edit] High Temperature Limit

When $x < < 1$ i.e. when $μ B B / k B T$ is small, the expression of the magnetization can be approximated by the Curie's law:

$M = C \cdot \frac{H}{T}$

where $C = \frac{N g^2 J(J+1) \mu_B^2}{3k_B}$ is a constant. One can note that $g\sqrt{J(J+1)}$ is the effective number of Bohr magnetons.

[edit] High Field Limit

When $x\to\infty$ , the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field: