Lattice dynamics of hcp Hf

Abstract

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 temperature 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 (LaSn3).

On the surface relaxation of transition metals

“…We have also examined the splitting of the bands in terms of atom localization, and found that both bands are equally localized on the two atoms in the hcp unit cell in all cases. 4 [11,12]. The most striking feature of the calculated phonon frequencies is the anomalously low values for Tc and Re, especially when compared to those obtained for Ru and Os.…”

“…the [001] longitudinal optical phonon excitations. It has been known for a long time that in Tc and Re, and to a somewhat less extent in Zr and Hf as well, the [001] longitudinal optical phonon branches have anomalously low frequency and show anomalous temperature dependence [11,12]. The origin of these anomalies was ascribed to the electronic bands near the Fermi level [10].…”

“…Since there is no known anomaly in the temperature dependence of the [100] LO phonon branches, one would expect no anomaly in the layer relaxation based on the results above. We have found that for this surface the relaxation decays fast in both Re and Os: it starts at roughly −17 % relaxation for d 12 , and drops to around 1-2 % in the subsequent layer distances. This nding indicates that indeed in Re the anomalous relaxation along the hexagonal c axis has a bulk rather than a surface origin, and the surface relaxation is not an isotropic property of the transition metals.…”

Anisotropic distortion and Lifshitz transition inα-Hf under pressure

“…In fact, the E 2g optical phonon mode and the coupled elastic shear modulus C 44 were experimentally found to show an anomalous pressure-dependent behavior [19], but up to now this issue has not been clarified yet. It was also reported that the lattice distortion will shift the degeneracy of the bands in the A-H-L plane, and for the bands close to the Fermi level this splitting will reduce the electronic energy and a phonon anomaly will occur [20]. Despite the anomalous behavior mentioned in the studies above, no clear theoretical clues have been provided, i.e., the relation between mechanism and phenomena is still not established.…”

“…The phonon dispersion curves for α-Hf under pressure from 0 GPa (black curves), 10 GPa (blue curves), 15 GPa (green curves), to 40 GPa (red curves) respectively. The pink stars present the experimental data measured by neutron spectroscopy at 295 K [20].The red arrow points out the E 2g mode in the phonon spectra.…”