Macroscopic Ferromagnetic Dynamics

In metals with finite magnetization $\vec{M}$, experiment shows that transverse polarized dc spin currents $\vec{J}_{i}$ both decay and precess on crossing a finite sample thickness. The present work uses Onsager's irreversible thermodynamics, with $\vec{M}$ and $\vec{J}_{i}$ as fundamental variables, to develop a theory with aspects of the Landau-Lifshitz theory solely for the $\vec{M}$ of (charged) electronic ferromagnets, and of the Leggett theory for the $\vec{M}$ and $\vec{J}_{i}$ of (uncharged) nuclear paramagnets. As for the ferromagnet of Landau-Lifshitz, $\partial_{t}\vec{M}$ includes a characteristic decay time $\tau_{M}$. As for the nuclear paramagnet, $\partial_{t}\vec{J}_{i}$ includes a characteristic decay time $\tau_{J}$, is driven by the gradient of a (vector) spin pressure, and precesses about a mean-field proportional to $\vec{M}$. The spin pressure has a coefficient $G$ proportional to a velocity squared, and $D_{0}\equiv \frac{1}{2}G\tau_{J}$ serves as an effective diffusion coefficient. These equations apply when spin currents are generated. Using the derived dynamical equations for the magnetization and for the spin current, we obtain the steady state (dc limit) solution whose transverse wavevector squared is complex, with real part from diffusion and imaginary part from precession. The ac case is also considered.