The determination of the critical concentration for a dilute heisenberg ferromagnet from the low-energy spin waves

The critical concentration for a dilute Heisenberg ferromagnet is determined by finding the concentration for which the ferromagnetic ground state is unstable with respect to the formation of long-wavelength spin waves. An upper bound to the spin-wave energies is determined using a variational procedure and this is used to obtain a lower bound on the critical concentration. This result is considerably higher than the corresponding Ising result.