Landau–Lifshitz–Gilbert equation

The Landau - Lifshitz -Gilbert equation (sometimes with English transcription in German also known as Landau - Lifshitz -Gilbert equation) describes in electrodynamics the behavior of the magnetic moments of ferromagnetic material in an effective magnetic field. It is named after Lev Davidovich Landau, Evgeny Mikhailovich Lifshitz and TL Gilbert. This is an ordinary differential equation, from which, however, this effective field a complicated integro-differential equation arises by considering the non-local nature with regard to the interaction of the Magnetisierungsdipole.

Landau - Lifshitz equation

The original Landau - Lifshitz equation was set up in 1935. Describes both the precession of the magnetization and the dissipation occurring. is the constant value of the vector, the so-called " saturation magnetization ".

With the gyromagnetic ratio and the phenomenological damping parameter. However, this formula fails for the case of large damping ().

Landau - Lifshitz -Gilbert equation

1955 replaced the Gilbert damping term and led a kind of a viscous force. There was the so-called Landau - Lifshitz -Gilbert equation:

Which can also be written in simpler equivalent form ( exakt! ):

With, the Gilbert damping parameter and the identification ( unit vector ). It can be shown that the resulting Landau - Lifshitz last -Gilbert equation is identical to that cited in the previous sub-chapter original Landau - Lifshitz equation, if one identifies λ; the crucial difference is, however, except for the larger formal simplicity that is in " fits " not now and but and used. Formal is replaced by; the last term contains all damping terms.

In contrast to the Landau - Lifshitz equation, the magnetic moment is now directed asymptotically toward the field from where now the damping effect as in mechanics, the " damped oscillator " on the precession frequency. For the case of small damping, the Landau - Lifshitz -Gilbert equation becomes the Landau - Lifshitz equation.

The " effective field "

Landau and Lifshitz 1935 have not specified how the vector of all four interactions involved (the " magnetic exchange energy ", the " dipole -dipole energy ," the " anisotropy " and the " Zeeman energy " ) depends. In details can not be discussed here.

Like spin waves

Using the Landau - Lifshitz -Gilbert equations can, inter alia, dynamic states (eg spin waves, as in the picture ) will be treated realistically, with all relevant geometries ( for example, thin-film geometries ) and interactions (including the very long-range magnetic dipole -dipole interaction as well) can be fully taken into account, when high memory requirements and corresponding computing times takes into account in the computer simulations.

The dispersion relations in these systems - these are the relationships between frequency and wavelength of the excitation states - are very complex due to the high number of characteristic lengths of the system and the angle involved.

References and footnotes

• Magnetism
• Ordinary Differential Equations