Spin waves in the heavy-rare-earth metals Gd, Tb, Dy, and Er

The transverse-spin-wave dispersion relation and neutron-scattering cross section for strongly anisotropic magnets is formulated in terms of effective renormalized exchange and anisotropy parameters. The spin-wave dispersion for the cone structure is considered in detail, including a diagonalization of off-diagonal wave-vector-dependent terms and renormalization effects caused by the crystal field. The derived expressions require no a priori knowledge about the crystal-field and magnetoelastic parameters. This is required for the application of the standard-basis-operator theory. A comprehensive analysis of the spin-wave data on Gd, Tb, Dy, and Er yields the interatomic exchange parameters and the effective anisotropy constants. The exchange parameters obey the de Gennes scaling and decrease as the cubed inverse distance, as expected for the isotropic Rudermann-Kittel interaction. Therefore, there is not much room for an unresolved two-ion anisotropy. The cone spin-wave data for Er can be accounted for on the basis of an isotropic exchange interaction and crystal-field parameters with magnitudes in quantitative agreement with those derived from static measurements on dilute ${\mathrm{Er}}_c-{\mathrm{Y}}_{1-c}$ alloys. No convincing evidence for a giant non-symmetry-breaking two-ion anisotropy has been found in Tb, Dy, and Er. A qualitative detection of a small symmetry-breaking two-ion anisotropy has been made in Tb and possibly also Dy and Er. The calculated and measured neutron-scattering intensities for Er are in good agreement, and the renormalization parameters calculated from known crystal-field parameters compare closely with those obtained from the fit to spin waves.