Low-energy spin waves for impure Heisenberg ferromagnets

A ferromagnetic Heisenberg system with a random arrangement of two different types of atom with spins s1 and s2 and exchange couplings J11(r), J12(r) and J22(r) is studied in the limit of very low energies. The eigenstate with k = 0 is known from the symmetry of the Hamiltonian, and so a set of basic states are used which will reduce to the exact result in this limit. These states are used in a variational calculation to obtain an expansion for the spin-wave energies in the low-energy limit. In the limit of a very low concentration of one type of atom the energy can be evaluated exactly. Approximate results are obtained which can over the entire concentration range except when the spins or exchange constants differ from each other by very large amounts.