  We present a microscopic calculation of magnetization damping for a magnetic
"toy model." The magnetic system consists of itinerant carriers coupled
antiferromagnetically to a dispersionless band of localized spins, and the
magnetization damping is due to coupling of the itinerant carriers to a phonon
bath in the presence of spin-orbit coupling. Using a mean-field approximation
for the kinetic exchange model and assuming the spin-orbit coupling to be of
the Rashba form, we derive Boltzmann scattering integrals for the distributions
and spin coherences in the case of an antiferromagnetic exchange splitting,
including a careful analysis of the connection between lifetime broadening and
the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract
dephasing and magnetization times T_1 and T_2 from initial conditions
corresponding to a tilt of the magnetization vector, and draw a comparison to
phenomenological equations such as the Landau-Lifshitz or the Gilbert damping.
We also analyze magnetization precession and damping for this system including
an anisotropy field and find a carrier mediated dephasing of the localized spin
via the mean-field coupling.
