Spin waves in terbium. I. Two-ion magnetic anisotropy

The energies of spin waves propagating in the $c$ direction of Tb have been studied by inelastic neutron scattering, as a function of a magnetic field applied along the easy and hard directions in the basal plane, and as a function of temperature. From a general spin Hamiltonian, consistent with the symmetry, we deduce the dispersion relation for the spin waves in a basal-plane ferromagnet. This phenomenological spin-wave theory accounts for the observed behavior of the magnon energies in Tb. The two $\overrightarrow{\mathrm{q}}$-dependent Bogoliubov components of the magnon energies are derived from the experimental results, which are corrected for the effect of the direct coupling between the magnons and the phonons, and for the field dependence of the relative magnetization at finite temperatures. A large $\overrightarrow{\mathrm{q}}$-dependent difference between the two energy components is observed, showing that the anisotropy of the two-ion coupling between the magnetic moments in Tb is substantial. The $\overrightarrow{\mathrm{q}}$-dependent anisotropy deduced at 4.2 K is of the same order of magnitude as the isotropic part, and depends strongly on the orientation of the moments in the basal plane. The rapid decrease of both the axial- and the basal-plane anisotropy with increasing temperatures implies that the two-ion coupling is effectively isotropic above ∼ 150 K. We present arguments for concluding that, among the mechanisms which may introduce anisotropic two-ion couplings in the rare-earth metals, the modification of the indirect exchange interaction by the spin-orbit coupling of the conduction electrons is of greatest importance.