Diagrammatic theory of spin excitations in an anisotropic ferromagnet with spin-phonon interactions

Using a generalization of the VLP technique to spin-phonon systems the effect of spin-phonon interactions on the magnetic excitations of an S=1 Heisenberg ferromagnet having both exchange and single-ion anisotropy is studied. The calculations are carried out using a semi-invariant expansion of the spin Green function. The diagrams are classified according to the high density expansion (1z), where z is the number of spins with which a given spin interacts. The renormalized energy and damping of the magnetic excitations are calculated from the poles of the transverse spin Green function evaluated to order (1z)2 and to second order in the spin-phonon interaction. Results applicable over a wide range of temperatures up to the transition temperature are obtained. The role of the exchange anisotropy is emphasized, particularly in its contribution to the damping of the long wavelength magnetic excitations.