Brillouin and Langevin functions

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

Brillouin function [edit]

The Brillouin function[1][2] 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 and -1 as .

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 on the applied magnetic field and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:[1]

where

  • is the number of atoms per unit volume,
  • the g-factor,
  • the Bohr magneton,
  • is the ratio of the Zeeman energy of the magnetic moment in the external field to the thermal energy :
  • is the Boltzmann constant and the temperature.

Note that in the SI system of units given in Tesla stands for the magnetic field, , where is the auxiliary magnetic field given in A/m and is the permeability of vacuum.

Langevin function [edit]

Langevin function (blue line), compared with (magenta line).

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

For small values of x, the Langevin function can be approximated by a truncation of its Taylor series:

An alternative better behaved approximation can be derived from the Lambert's continued fraction expansion of tanh(x):

For small enough x, both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from Loss of significance.

The inverse Langevin function L−1(x) is defined on the open interval (−1, 1). For small values of x, it can be approximated by a truncation of its Taylor series[3]

and by the Padé approximant

Graphs of relative error for x ∈ [0, 1) for Cohen and Jedynak approximations

Since this function has no closed form, it is useful to have approximations valid for arbitrary values of x. One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen:[4]

This has a maximum relative error of 4.9% at the vicinity of x = ±0.8. Greater accuracy can be achieved by using the formula given by R. Jedynak:[5]

valid for x ≥ 0. The maximal relative error for this approximation is 1.5% at the vicinity of x = 0.85.

Interesting and comprehensive studies of the well-known approximation formulas of the inverse Langevin function can be found in the paper written by Jedynak.[5]

High-temperature limit [edit]

When i.e. when is small, the expression of the magnetization can be approximated by the Curie's law:

where is a constant. One can note that is the effective number of Bohr magnetons.

High-field limit [edit]

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

References[edit]

  1. ^ a b c C. Kittel, Introduction to Solid State Physics (8th ed.), pages 303-4 ISBN 978-0-471-41526-8
  2. ^ Darby, M.I. (1967). "Tables of the Brillouin function and of the related function for the spontaneous magnetization". Brit. J. Appl. Phys. 18 (10): 1415–1417. Bibcode:1967BJAP...18.1415D. doi:10.1088/0508-3443/18/10/307. 
  3. ^ Johal, A. S.; Dunstan, D. J. (2007). "Energy functions for rubber from microscopic potentials". Journal of Applied Physics. 101 (8): 084917. Bibcode:2007JAP...101h4917J. doi:10.1063/1.2723870. 
  4. ^ Cohen, A. (1991). "A Padé approximant to the inverse Langevin function". Rheologica Acta. 30 (3): 270–273. doi:10.1007/BF00366640. 
  5. ^ a b Jedynak, R. (2015). "Approximation of the inverse Langevin function revisited". Rheologica Acta. 54 (1): 29–39. doi:10.1007/s00397-014-0802-2.