Spin-wave theory of ferromagnetic crystals containing impurities

A spin-wave theory of ferromagnetic crystals containing a low concentration of impurity atoms with different spin and exchange integrals is developed by a Green function method. The Green function of one-magnon excitations is calculated by expanding in a series in powers of the perturbation introduced by the impurity and then averaging the terms of the series over all possible configurations of the impurities. Those diagrams which correspond to the scattering of a spin wave by one impurity atom in all orders of the perturbation are chosen, corresponding to the approximation for the self-energy linear in the concentration. By an investigation of the poles of the Green function it is shown that the dispersion curves of the spin waves have a non-monotonic character near the points corresponding to energies of the resonance modes of s, p or d types, and the damping, as a function of spin-wave energy, shows a resonance near these points. It is shown that the density of states in the impurity bands corresponding to the energy levels of the localized states in the single-impurity problem is a non-analytic function of the concentration in agreement with a general result of Lifshitz.