Temperature Dependence of the Exchange Stiffness in Ferrimagnets

The temperature dependence of magnon frequencies is studied for ferrimagnets. The magnon frequency increases for the acoustical branch and decreases for the optical branch, both in low-temperature regions. Temperature variations obey the T5/2 law. In high-temperature regions, both acoustical- and optical-magnon frequencies decrease with increasing temperature. The sign change of the temperature variation of acoustical-magnon frequencies is due to optical-magnon population which increase with temperature. The sign change occurs at a lower temperature for a magnon with a shorter wavelength. This feature is in accordance with Riste's neutron-scattering experiments performed with magnetites. The second-order shift has proved to be small at low temperatures. In the course of its calculation, we give a simple expression for the matrix element of a magnon interaction. This bears a close relationship to Dyson's dynamical interaction in ferromagnets.