Brillouin and Langevin functions

Brillouin and Langevin functions

=Brillouin Function=

The Brillouin functionC. Kittel, "Introduction to Solid State Physics" (8th ed.), pages 303-4] [Citation
last = Darby
first = M.I.
author-link =
last2 =
first2 =
author2-link =
title = Tables of the Brillouin function and of the related function for the spontaneous magnetization
journal = Brit. J. Appl. Phys.
volume = 18
issue =
pages = 1415-1417
date =
year = 1967
url =
doi =
id =
] is a special function defined by the following equation:

:B_J(x) = frac{2J + 1}{2J} coth left ( frac{2J + 1}{2J} x ight ) - frac{1}{2J} coth left ( frac{1}{2J} x ight )
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 "x" approaches +∞ and -1 as "x" approaches -∞.

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 M on the applied magnetic field B 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 number of atoms per unit volume,
*g the g-factor,
*mu_B the Bohr magneton,
*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}
*k_B is the Boltzmann constant and T the temperature.

:

Langevin Function


thumb|right|Langevin function (red line), compared with">tanh(x/3) (blue line).

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

:L(x) = coth(x) - frac{1}{x}

High Temperature Limit

When x << 1 i.e. when mu_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 gsqrt{J(J+1)} is the effective number of Bohr magnetons.

High Field Limit

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

:M = N g mu_B J

References


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