# 居里定律

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

${\displaystyle \mathbf {M} }$是磁化强度
${\displaystyle \mathbf {B} }$磁感应强度
${\displaystyle T}$是温度，以开尔文为单位
${\displaystyle C}$是材料的居里常数

## 用量子力学推导

${\displaystyle E=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}$

### 双态 (自旋-½)粒子

${\displaystyle E_{0}=-\mu B}$

${\displaystyle E_{1}=\mu B.}$

${\displaystyle \left\langle \mu \right\rangle =\mu P\left(\mu \right)+(-\mu )P\left(-\mu \right)={1 \over Z}\left(\mu e^{\mu B\beta }-\mu e^{-\mu B\beta }\right)={2\mu \over Z}\sinh(\mu B\beta ),}$

${\displaystyle Z=\sum _{n=0,1}e^{-E_{n}\beta }=e^{\mu B\beta }+e^{-\mu B\beta }=2\cosh \left(\mu B\beta \right).}$

${\displaystyle \left\langle \mu \right\rangle =\mu \tanh \left(\mu B\beta \right).}$

${\displaystyle M=n\left\langle \mu \right\rangle =n\mu \tanh \left({\mu B \over kT}\right)}$

${\displaystyle \left({\mu B \over kT}\right)\ll 1}$

${\displaystyle \mathbf {M} (T\rightarrow \infty )={n\mu ^{2} \over k}{\mathbf {B} \over T},}$

${\displaystyle M\approx {\frac {\mu _{0}\mu ^{2}n}{k}}{\frac {H}{T}},}$

${\displaystyle \chi ={\frac {\partial M}{\partial H}}\approx {\frac {M}{H}}}$

${\displaystyle \chi (T\to \infty )={\frac {C}{T}},}$

### 一般情况

${\displaystyle C={\frac {\mu _{B}^{2}}{3k_{B}}}ng^{2}J(J+1)}$ [2]

## 用经典统计力学推导

${\displaystyle E=-\mu B\cos \theta ,}$

${\displaystyle Z=\int _{0}^{2\pi }d\phi \int _{0}^{\pi }d\theta \sin \theta \exp(\mu B\beta \cos \theta ).}$

${\displaystyle Z=2\pi \int _{-1}^{1}dy\exp(\mu B\beta y)=2\pi {\exp(\mu B\beta )-\exp(-\mu B\beta ) \over \mu B\beta }={4\pi \sinh(\mu B\beta ) \over \mu B\beta .}}$

${\displaystyle \left\langle \mu _{z}\right\rangle ={1 \over Z}\int _{0}^{2\pi }d\phi \int _{0}^{\pi }d\theta \sin \theta \exp(\mu B\beta \cos \theta )\left[\mu \cos \theta \right].}$

${\displaystyle \left\langle \mu _{z}\right\rangle ={1 \over Z\beta }{\frac {\partial Z}{\partial B}}={1 \over \beta }{\frac {\partial \ln Z}{\partial B}}}$

（这种方法也可以用于上面的模型，但计算非常简单，所以没有那么有用。）

${\displaystyle \left\langle \mu _{z}\right\rangle =\mu L(\mu B\beta ),}$

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

## 参考资料

1. ^ Coey, J. M. D.; Coey, J. M. D. Magnetism and Magnetic Materials. Cambridge University Press. 2010-03-25 [2022-02-23]. ISBN 978-0-521-81614-4. （原始内容存档于2022-02-23） （英语）.
2. ^ Kittel, Charles. Introduction to Solid State Physics, 8th Edition. Wiley. : 304. ISBN 0-471-41526-X.