From static to dynamic

In order to study dynamical phenomena we can combine the equations above with the work of Landau, Lifshitz and Gilbert. Taking Brown's equations for energy and the effective field $\ensuremath{\mathbf{H}}_{\mathrm{eff}}$:

$$\mathcal{E} = \mathcal{E}_{\mathrm{ex}} + \mathcal{E}_{\mathrm{an}} + \mathcal{E}_{\mathrm{Ze}} + \mathcal{E}_{\mathrm{di}}$$ (2.31)

$$= -\int \mu_0 \ensuremath{\mathbf{H}}_{\mathrm{eff}}(\ensuremath{\mathbf{r}})\cdot\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}}) \mathrm{d}^3r$$ (2.32)

where

$$\ensuremath{\mathbf{H}}_{\mathrm{eff}} = - {1 \over \mu_0} \ensuremath{\mathbf{\nabla}}_{\ensuremath{\mathbf{M}}} \mathcal{E}$$ (2.33)

then the time development of the magnetisation can be written as (Landau and Lifshitz, 1935):

$${\mathrm{d}\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}}) \over \mathrm{d}t} = \gamma\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})\times\ensu... ...thbf{r}})\times\ensuremath{\mathbf{H}}_{\mathrm{eff}}(\ensuremath{\mathbf{r}}))$$ (2.34)

where $\gamma\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})\times\ensuremath{\mathbf{H}}_{\mathrm{eff}}(\ensuremath{\mathbf{r}})$ is representative of the precession of $\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})$ in a local field $\ensuremath{\mathbf{H}}_{\mathrm{eff}}(\ensuremath{\mathbf{r}})$ and ${\bar\alpha \over M_s} \ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}})\times(... ...thbf{r}})\times\ensuremath{\mathbf{H}}_{\mathrm{eff}}(\ensuremath{\mathbf{r}}))$ is an empirical damping term.

The damping constant $\bar\alpha$ is not well understood but at zero temperature it is due to spin waves quantised as magnons (Blundell, 2001, p122), and at finite temperature due to atomic lattice oscillations quantised as phonons.