# 布里渊函数和郎之万函数

（重定向自布里渊函数

## 布里渊函数

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

${\displaystyle M=Ng\mu _{B}J\cdot B_{J}(x)}$

${\displaystyle x={\frac {g\mu _{B}JB}{k_{B}T}}}$

## 郎之万函数

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

${\displaystyle =bN\left(\coth(fb/k_{B}T)-{\frac {1}{fb/k_{B}T}}\right)}$

x为小量时，郎之万函数可由其截断的泰勒级数近似：

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

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

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

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

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

## 高温极限

${\displaystyle x\ll 1}$  时，即${\displaystyle \mu _{B}B/k_{B}T}$  很小，磁矩可以由居里定律近似：

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

## 强场极限

${\displaystyle x\to \infty }$ ，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

${\displaystyle M=Ng\mu _{B}J}$

## 参考文献

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.