Spin Hall response at finite wave vector in ferromagnets

Spin Hall effect at finite wave vector in a ferromagnetic conductor is theoretically studied by calculating the spin density as the linear response to an applied electric field. The cases of a spin-orbit interaction due to random impurities and a localized Rashba interaction are considered. It is shown that the spin Hall effect has a broad response for the wave vector $q\lesssim 2\kf$ where $\kf $ is the Fermi wave vector. This fact confirms the local nature of the spin-charge conversion effects.