JMLB - jmb22@nyu.edu

EQUATION OF MOTION OF THE MAGNETIZATION:

The dynamics of a magnetic dipole moment are described by the rate of change being proportional to the torque exerted on the moment $\vec{\mu}$ by the effective magnetic field $\vec{H}_{\mathrm{eff}}$ :

$\frac{-1}{\gamma} \ \frac{d \vec{\mu}}{dt} = \vec{\mu} \bullet \vec{H}_{\mathrm{eff}}$

where the constant of proportionality $\gamma^{-1}$ is the gyromagnetic ratio, a material dependent parameter which describes the origin of the magnetic moment with respect to electron spin and orbital motion. (For Cobalt: $\gamma=1.7 \times 10^6 \ \ \mathrm{cgs.}$ )
This relation generalizes to the net magnetic moment per unit area, or magnetization $\vec{M}$ , provided that the effective field is assumed to be uniform over the unit area:

$\frac{-1}{\gamma} \ \frac{d \vec{M}}{dt} = \vec{M} \bullet \vec{H}_{\mathrm{eff}}$

The equation of motion proposed by Landau and Lifshitz incorporates the damping of magnetic energy, by spin-spin and spin-orbit interactions, via the phenomenological damping constant $\lambda$ :

$\frac{-1}{\gamma} \ \frac{d \vec{M}}{dt} = \vec{M} \bullet \vec{H}_{\mathrm{eff}} \ - \ \frac{\lambda}{|\vec{M}|} \vec{M}\times(\vec{M}\times\vec{H}_{\mathrm{eff}})$

An alternative form was later given by Gilbert:

$\frac{-1}{\gamma} \ \frac{d \vec{M}}{dt} = \vec{M} \bullet \vec{H}_{\mathrm{eff}} \ - \ \frac{\alpha}{|\vec{M}|} \vec{M}\times(\frac{d\vec{M}}{dt})$

The two equations have been compared by Mallinson [J. C. Mallinson, IEEE Trans. Magn. 23 (4) (1987)] who used geometric arguments to show that the Gilbert equation is to be preferred due to physically implausible solutions of the Landau-Lifshitz equation for large damping, whereas the self-consistent damping mechanism of the Gilbert equation leads to plausible results for all values of $\alpha$ .