Dissipation

The coupling of spins results in a transfer of energy from one spin to other spins and the lattice (and results in a resonance absorption line of finite width). Ferromagnetic resonance experiments take advantage of this interaction and allow the measurement of the gyromagnetic constant or the damping constant.

Consequently a phenomenological term has to be introduced and several suitable damping terms have been proposed:

The damping terms employed by Bloch in paramagnetic resonance studies,

$$\left(-\frac{M_x}{\tau_2}, -\frac{M_y}{\tau_2}, -\frac{M_z - M_s}{\tau_1}\right)$$

are one possibility. $\tau_1$ and $\tau_2$ represent the relaxation times for the longitudinal and transverse directions.

Another expression was introduced much earlier by Landau and Lifshitz:

$$-\frac{\alpha\vert\gamma_0\vert}{\vert\mbox{\bf M}\vert}\mbox{\bf M}\times \left(\mbox{\bf M}\times\mbox{\bf H}\right)$$

Typical values of $\alpha$ are in the range of 10^-2 to 10^-1 for many magnetic materials. And the above damping term is only applicable if $\alpha \ll 1$ and therefore the inertia term is larger than the damping term.

But strictly speaking the damping motion should act not only on the precession motion but also on the motion induced by the damping term. In other words the damping should act on the resultant motion of the magnetization. Thus Gilbert proposed:

$$-\frac{\alpha}{\vert\mbox{\bf M}\vert} \left(\mbox{\bf M}\times\frac{d\mbox{\bf M}}{dt}\right)$$

The LL-damping term can be derived from Gilbert's expression by neglecting the higher order terms in $\alpha^2$.

Landau-Lifshitz equation:

$$\frac{d\mbox{\bf M}}{dt}= -\vert\gamma_0\vert\mbox{\bf M} \... ...bf M}\vert} \mbox{\bf M}\times(\mbox{\bf M}\times\mbox{\bf H})$$