Temperature dependence of the vibrational modes of molybdenum

Inelastic neutron scattering techniques have been used to study the temperature dependence of the phonon dispersion curves of bcc Mo. We find that, with increasing temperature, the largest relative decrease in phonon frequencies (∼12%) occurs for the $\mathrm{L}\left[\xi \xi \xi \right]$ branch in the vicinity of $\xi =\frac{2}{3}$. The phonon frequencies in the vicinity of the $H$ point in the Brillouin zone, on the other hand, increase slightly with increasing temperature. As a result, at high temperatures, the anomalous drop in the phonon frequencies at $H$ decreases in magnitude and the $\mathrm{L}\left[\xi \xi \xi \right]$ branch develops a dip at $\xi =\frac{2}{3}$. The experimental results also indicate that the phonon anomaly in the vicinity of point $N$ is less pronounced at high temperatures. For the purpose of comparison with the experimental results, we have performed first-principle "frozen-phonon" calculations for the longitudinal-phonon frequencies of Mo at the ($\frac{2}{3}$,$\frac{2}{3}$,$\frac{2}{3}$) and $H$ points of the Brillouin zone. The results show that thermal expansion and anharmonicity can account for the temperature dependence of the ($\frac{2}{3}$,$\frac{2}{3}$,$\frac{2}{3}$) longitudinal-phonon frequency. For the $H$ phonon, on the other hand, the thermal expansion and anharmonicity terms cannot account for the observed frequency shift and we suggest that the contributions arising from the electron-phonon renormalization and the thermal disorder of the lattice must be included.