# 布里渊函数和郎之万函数

## 布里渊函数

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

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

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

## 郎之万函数

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

$ = bN \left ( \coth(fb/k_BT) - \frac{1}{fb/k_BT} \right )$

x为小量时，郎之万函数可由其截断的泰勒级数近似：

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

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

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

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

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

## 高温极限

$x \ll 1$ 时，即$\mu_B B / k_B T$ 很小，磁矩可以由居里定律近似：

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

## 强场极限

$x\to\infty$，布里渊函数的值趋于 1，材料的磁化强度饱和，磁矩的取向完全沿外场方向，于是有

$M = N g \mu_B J$

## 参考文献

1. ^ 1.0 1.1 C. Kittel, Introduction to Solid State Physics (8th ed.), pages 303-4 ISBN 978-0-471-41526-8
2. ^ Darby, M.I. Tables of the Brillouin function and of the related function for the spontaneous magnetization. Brit. J. Appl. Phys. 1967, 18 (10): 1415–1417. Bibcode:1967BJAP...18.1415D. doi:10.1088/0508-3443/18/10/307.
3. ^ Michael Rubinstein and Ralph H. Colby. Polymer Physics. Oxford University Press. 2003: 76. ISBN 978-0-19-852059-7.
4. ^ Cohen, A. A Padé approximant to the inverse Langevin function. Rheologica Acta. 1991, 30 (3): 270–273. doi:10.1007/BF00366640.
5. ^ Johal, A. S.; Dunstan, D. J. Energy functions for rubber from microscopic potentials. Journal of Applied Physics. 2007, 101 (8): 084917. Bibcode:2007JAP...101h4917J. doi:10.1063/1.2723870.