Temperature-dependent crystal field effects in the spin-wave spectrum of terbium

A variational principle for the free energy is applied to the Hamiltonian for terbium in the ferromagnetic region to study the effect of interactions among spin waves having their origin in the strong crystalline electric fields. At high temperatures this `crystal field renormalization' has the effect of multiplying the crystal field parameters appearing in the spin-wave energies of the non-interacting theory by powers of the reduced magnetization. The powers are found to be close to those associated with the corresponding magnetic anisotropy coefficients. This leads to an appreciable temperature dependence of the spin-wave energies which, after allowing for exchange renormalization, appears to be in reasonable agreement with the inelastic neutron scattering results of Møller and Houmann. The ferromagnetic resonance measurements of Bagguley and Liesegang and of Stanford and Young have also been analysed and found to be consistent with this picture.