Landau–Lifshitz–Gilbert equation

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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.

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[edit] Landau-Lifshitz ordinary differential equation

The LLE introduced by 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{d\mathbf{M}}{dt}= \gamma \left[\mathbf{M},\mathbf{H} \right]-\alpha\frac{\gamma}{M}\left[\mathbf{M},\left[\mathbf{M},\mathbf{H}\right]\right].\qquad (1)

The constants α is the Landau-Lifshitz phenomenological damping parameter and depends on the solid, and γ is the electron gyromagnetic ratio.

[edit] Landau-Lifshitz-Gilbert equation

The LLE above works fairly well when the damping is small, but fails for large damping. To fix this problem Gilbert (1955) suggested a different damping term. The resulting equation is called the Landau-Lifshitz-Gilbert equation, and is given by

\frac{d\mathbf{M}}{dt}= \gamma[\mathbf{M},\mathbf{H}]-\frac{\alpha}{M}\left[\mathbf{M},\frac{d\mathbf{M}}{dt}\right].\qquad

where α is called the Gilbert damping constant. Any solution also satisfies the Landau-Lifshitz equation for γ replaced by γ/(1+α2).



[edit] See also

[edit] References

  • Gilbert, T.L. (1955), “A Lagrangian formulation of the gyromagnetic equation of the magnetic field”, Physical Review 100: 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 Gilbert, T. L. (2004), “A phenomenological theory of damping in ferromagnetic materials”, IEEE Trans. Mag. 40 (6): 3443-3449, DOI 10.1109/TMAG.2004.836740 
  • Landau, L.D. & Lifshitz, E.M. (1935), “Theory of the dispersion of magnetic permeability in ferromagnetic bodies”, Phys. Z. Sowietunion 8, 153 
  • Skrotskiĭ, G V (1984), “The Landau-Lifshitz equation revisited”, Sov. Phys. Usp. 27: 977-979, DOI 10.1070/PU1984v027n12ABEH004101 
  • Guo, Boling & Ding, Shijin (2008), Landau-Lifshitz Equations, Frontiers of Research With the Chinese Academy of Sciences, World Scientific Publishing Company, ISBN 978-9812778758