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Brillouin Function

The Brillouin function[1][2] is a special function defined by the following equation:

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

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 \to +\infty and -1 as x \to -\infty.

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:[1]

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

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.

Note that in the SI system of units B given in Tesla stands for magnetic induction, B=\mu_0 H, where H is the applied magnetic field given in A/m and \mu_0 is the permeability of vacuum.

Langevin Function
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 $$( J \to \infty$$). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:

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

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

$$L(x) = \tfrac{1}{3} x - \tfrac{1}{45} x^3 + \tfrac{2}{945} x^5 - \tfrac{1}{4725} x^7 + \dots$$

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

$$L(x) = \frac{x}{3+\tfrac{x^2}{5+\tfrac{x^2}{7+\tfrac{x^2}{9+\ldots}}}}$$

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

The inverse Langevin function can be approximated to within 5% accuracy by the formula[3]

$$L^{-1}(x) \approx x \frac{3-x^2}{1-x^2},$$

valid on the whole interval (-1, 1). For small values of x, better approximations are the Padé approximant

$$L^{-1}(x) = 3x \frac{35-12x^2}{35-33x^2} + O(x^7)$$

and the Taylor series[4]

$$L^{-1}(x) = 3 x + \tfrac{9}{5} x^3 + \tfrac{297}{175} x^5 + \tfrac{1539}{875} x^7 + \dots$$

High Temperature Limit

When $$x \ll 1$$ i.e. when $$\mu_B B / k_B T$$ is small, the expression of the magnetization can be approximated by the Curie's law:

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

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.
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:

$$M = N g \mu_B J$$

References

^ a b c C. Kittel, Introduction to Solid State Physics (8th ed.), pages 303-4 ISBN 978-0471415268
^ 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
^ Cohen, A. (1991). "A Padé approximant to the inverse Langevin function". Rheologica Acta 30 (3): 270–273. doi:10.1007/BF00366640.
^ 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.

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