GENERALIZED EQUATION OF MOTION FOR A FERROMAGNET

We derive microscopically an equation of motion for a ferromagnetic substance at nonzero temperatures allowing for both transverse and longitudinal relaxation and generalizing the Landau-Lifhshitz equation. The consideration starts from the density matrix equation for a quantum spin interacting with the environment, which is within about 7% accuracy reduced to the closed equation for the first moment of the distribution function - the magnetization. The latter interpolates between the Landau-Lifshitz equation (S >> 1 and low temperatures) and the Bloch equation (S = 1/2 or high temperatures). For condensed magnetic media (i.e. a ferromagnet) one can replace in the spirit of the mean field theory the magnetic field by the molecular one containing the exchange field acting on a given magnetic ion from its neighbours, which results in a Landau-Lifshitz type equation of motion with a longitudinal relaxation term providing the Curie-Weiss static solution. Further we consider the mobility of a domain wall (DW) in a uniaxial ferromagnet at nonzero temperatures where the magnetization in the middle of the domain boundary is smaller than in the domains (the elliptic DW transforms to the linear one near T(c)). It is shown that longitudinal relaxation plays a crucial role in DW dynamics in a wide range of high temperatures.