Spin waves, phonons, and crystal-field excitations in thulium

The magnetic excitations propagating along the c axis in thulium metal have been studied experimentally by inelastic neutron scattering. At low temperatures, in the seven-layered periodic magnetic structure, the spin-wave energies are centered around 9 meV, with a bandwidth of about 2 meV. A strong coupling to the transverse phonons is revealed through the magnetic scattering cross section. Near ${\mathit{T}}_{\mathit{N}}$ clear evidence of crystal-field excitations is found, both in the basal plane and in the c-axis component of the scattering function. A theory based on the random-phase approximation (RPA), which describes the spin waves and the crystal-field excitations in the unusual magnetic structure of Tm, and their coupling to the phonons, is developed. The neutron-scattering results, combined with known magnetization data, have allowed us to derive values for all the relevant coupling parameters. The most important is the two-ion exchange coupling, and the Fourier transform of this interaction along the c direction is presented for all the magnetic heavy rare-earth metals. Except for the linewidth phenomena, which are neglected in the RPA, the calculations account accurately for most experimental details in the antiferromagnetic phase and for the crystal-field excitations close to ${\mathit{T}}_{\mathit{N}}$.