Lattice dynamics of hcp Hf

Inelastic neutron scattering techniques have been used to study the lattice dynamics of hcp Hf. The phonon dispersion curves along the [001], [100], and [110] symmetry directions were determined at 295 and 1300 K, and a selected number of phonon frequencies were measured also at 800 K. As the temperatures decreases we observe a rather large increase in the frequencies of all but the [001] LO branch. The zone-center mode of the [001] LO branch, on the other hand, softens appreciably, and at room temperature this branch exhibits a dip at the zone center. These features of the phonon dispersion curves of Hf are similar to those of hcp Zr and Ti. The data were used to evaluate the lattice specific heat at constant pressure as a function of temperature. The calculated total specific heat, obtained by taking into account the electronic contribution, was found to agree, to within experimental uncertainties, with the results of specific-heat measurements. We find that the phonon anomalies (and their temperature dependence) in the dispersion curves of the superconducting elements of the IV column of the Periodic Table can be understood qualitatively as originating from the splitting about the Fermi level of doubly degenerate bands by the lattice distortion corresponding to the [001] LO mode. We argue that this mechanism may also be responsible for the phonon anomalies observed in other superconducting elements (Tc) and compounds (La${\mathrm{Sn}}_3$).