Theory of the Temperature Dependence of the Spin-Wave Excitation Energies in the Rare-Earth Metals

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Abstract
The temperature dependence of the spin-wave excitation energies in the ferromagnetic heavy rare-earth metals, due to spin-wave interactions originating in the strong crystalline electric fields, is studied from both macroscopic and microscopic viewpoints. Consistency between an interacting spin-wave theory, linearized in the Hartree-Fock approximation, and a suitable extension of Smit's macroscopic theory is obtained. Both approaches are consistent with Kanamori and Tachiki's semiphenomenological spin-wave theory. The resonance frequency may be expressed as a geometric mean of the axial and planar anisotropy fields, provided that the low-temperature behavior of the anisotropy constants is modified to allow for elliptical spin precession. At higher temperatures, the results are consistent with Cooper's phenomenological theory. For Tb, agreement with experiment is excellent, but there is a discrepancy in the magnitude of the resonance frequency in Dy.