# 朗道-利夫希兹方程

## 朗道-利夫希茲方程

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \mathbf {M} \times \mathbf {H_{\mathrm {eff} }} -\lambda \mathbf {M} \times \left(\mathbf {M} \times \mathbf {H_{\mathrm {eff} }} \right)}$

(1)

${\displaystyle \lambda =\alpha {\frac {\gamma }{M_{\mathrm {s} }}},}$

## 朗道-利夫希茲-吉爾伯特方程

1955年吉爾伯特由一個依賴於磁場的時間導數取代了朗道-利夫希茲的阻尼項：

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \left(\mathbf {M} \times \mathbf {H} _{\mathrm {eff} }-\eta \mathbf {M} \times {\frac {d\mathbf {M} }{dt}}\right)}$

(2b)

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma '\mathbf {M} \times \mathbf {H} _{\mathrm {eff} }-\lambda \mathbf {M} \times (\mathbf {M} \times \mathbf {H} _{\mathrm {eff} })}$

(2a)

${\displaystyle \gamma '={\frac {\gamma }{1+\gamma ^{2}\eta ^{2}M_{s}^{2}}}\qquad {\text{and}}\qquad \lambda ={\frac {\gamma ^{2}\eta }{1+\gamma ^{2}\eta ^{2}M_{s}^{2}}}.}$

## 方程形式

### 普通形式

${\displaystyle {\dot {\vec {m}}}(x,t)={\vec {m}}\times \nabla ^{2}{\vec {m}}}$

### 协变形式

${\displaystyle D_{t}{\vec {m}}(x,t)={\vec {m}}\times \nabla ^{2}{\vec {m}}}$

## 参考文献

• Landau-Lifshitz equation, B Guo and S Ding, World Scientific, ISBN 109812778756
1. ^ Yang, Bo. Numerical Studies of Dynamical Micromagnetics. [8 August 2011].
2. ^ http://wpage.unina.it/mdaquino/PhD_thesis/main/node47.html
3. ^ Aharoni 1996
4. ^ Brown 1978
5. ^ Chikazumi 1997
6. ^ T. Iwata, J. Magn. Magn. Mater. 31–34, 1013 (1983); T. Iwata, J. Magn. Magn. Mater. 59, 215 (1986); V.G. Baryakhtar, Zh. Eksp. Teor. Fiz. 87, 1501 (1984); S. Barta (unpublished, 1999); W. M. Saslow, J. Appl. Phys. 105, 07D315 (2009).