# Landau–Lifshitz–Gilbert equation

Landau–Lifshitz–Gilbert equation

In physics, the Landau-Lifshitz-Gilbert equation , named for Lev Landau and Evgeny Lifshitz and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization M in a solid. It is a modification by Gilbert of the original equation of Landau and Lifshitz.

Landau-Lifshitz ordinary differential equation

The LLE introduced by harvtxt|Landau|Lifshitz|1935 to model the precessional motion of magnetization M in a solid with an effective magnetic field H and with damping. It is the ordinary differential equation:$frac\left\{dmathbf\left\{M\left\{dt\right\}= gamma left \left[mathbf\left\{M\right\},mathbf\left\{H\right\} ight\right] -alphafrac\left\{gamma\right\}\left\{M\right\}left \left[mathbf\left\{M\right\},left \left[mathbf\left\{M\right\},mathbf\left\{H\right\} ight\right] ight\right] .qquad \left(1\right)$The constants α is the Landau-Lifshitz phenomenological damping parameter and depends on the solid, and γ is the electron gyromagnetic ratio.

Landau-Lifshitz-Gilbert equation

The LLE above works fairly well when the damping is small, but fails for large damping. To fix this problem harvtxt|Gilbert|1955 suggested a different damping term.The resulting equation is called the Landau-Lifshitz-Gilbert equation, and is given by:$frac\left\{dmathbf\left\{M\left\{dt\right\}= gamma \left[mathbf\left\{M\right\},mathbf\left\{H\right\}\right] -frac\left\{alpha\right\}\left\{M\right\}left \left[mathbf\left\{M\right\},frac\left\{dmathbf\left\{M\left\{dt\right\} ight\right] .qquad$where α is called the Gilbert damping constant. Any solution also satisfies the Landau-Lifshitz equation for &gamma; replaced by &gamma;/(1+α2).

ee also

* Micromagnetism
* Magnet
* Ferromagnetism

* [http://www.bama.ua.edu/~tmewes/Java/dynamics/MagnetizationDynamics.shtml Magnetization dynamics applet]

References

*citation|first=T.L.|last= Gilbert|title=A Lagrangian formulation of the gyromagnetic equation of the magnetic field|journal= Physical Review|volume= 100 |year=1955|page= 1243 This is only an abstract; the full report is "Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956", but was never published. A description of the work is given in citation|first=T. L.|last= Gilbert|journal = IEEE Trans. Mag.
volume=40|issue=6
pages= 3443-3449
year=2004
doi=10.1109/TMAG.2004.836740|title=A phenomenological theory of damping in ferromagnetic materials

*citation|authorlink1=Lev Landau|authorlink2=Evgeny Lifshitz|first=L.D. |last=Landau|first2= E.M.|last2= Lifshitz|title= Theory of the dispersion of magnetic permeability in ferromagnetic bodies|journal= Phys. Z. Sowietunion|volume= 8, 153 |year=1935
*citation|title=The Landau-Lifshitz equation revisited
first=G V|last= Skrotskiĭ |year=1984 |journal=Sov. Phys. Usp.|volume= 27 |pages=977-979 |doi=10.1070/PU1984v027n12ABEH004101

*citation|ISBN= 978-9812778758
title=Landau-Lifshitz Equations |series=Frontiers of Research With the Chinese Academy of Sciences
first=Boling |last=Guo |first2= Shijin |last2=Ding|year=2008
publisher=World Scientific Publishing Company

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