Spin Waves in a Model Ferromagnetic Metal

Calculations of the coefficient of $q^2$ in the expression for the energy $\epsilon$ of a long-wavelength spin wave have been performed for a model of a ferromagnetic metal with short-range particle interactions in the low-density limit. A single $s$-like energy band of tight-binding form is considered in a face-centered cubic crystal. First- and second-neighbor interaction integrals are included in $\epsilon \left(\mathrm{k}\right)$ as arbitrary parameters. In contrast to previous work, the effective electron interaction which enters the spin-wave calculation is not the bare potential, but is closely related to the two-particle $t$ matrix. It is found to be possible to satisfy the spin-wave stability criterion for the ferromagnetic state as well as the $t$-matrix criterion for the occurrence of ferromagnetism at reasonably low particle densities for a wide range of values of the band parameters, and for interaction strengths of the order of the bandwidth.