Polar Spin Waves in Ferromagnetic Metals

A simple one-band model of the electronic structure of a ferromagnetic metal is examined to determine the existence and properties of polar spin-wave states, i.e., "optical" spin waves due to the finite range of the electron-electron interaction. The model Hamiltonian is an extension of the Hubbard Hamiltonian, including Coulomb interaction between electrons occupying neighboring-site Wannier functions. The equation-of-motion method is used in the random-phase approximation to determine the various spin-wave collective excitations. For a finite-range Coulomb interaction, the equations can be transformed into the equations of the Koster-Slater impurity problem with nearest-neighbor-potential matrix elements. Polar spin waves are found to exist for a wide range of band occupancies, tending to merge with the continuum with decreasing occupancy of the spin-up band. It is found that near wave vector $q=0\mathrm{}$, inelastic neutron scattering by the acoustic spin-wave mode dominates scattering by all other spin-wave and Stoner modes; however, a finite polar-mode scattering cross section does exist away from $q=0\mathrm{}$. Since polar spin waves carry an electric moment, Raman scattering of light from polar spin waves is also examined as a possible way of observing these states. A procedure is described for identifying the various Raman lines corresponding to various polar spin-wave states.