Temperature dependence of the phonon entropy of vanadium

Abstract

The phonon density-of-states ͑DOS͒ of elemental vanadium was measured at elevated temperatures by inelastic neutron scattering. The phonon softening predicted by thermal expansion against the bulk modulus is much larger than the measured shifts in phonon energies. We conclude that the phonon anharmonicities associated with thermal expansion are largely canceled by effects from phonon-phonon scattering. Prior measurements of the heat capacity and calculations of the electronic entropy of vanadium are assessed, and consistency requires an explicit temperature dependence of the phonon DOS. Using data from the literature, similar results are found for chromium, niobium, titanium, and zirconium.

Electron-phonon interactions and high-temperature thermodynamics of vanadium and its alloys

“…The measurements on pure V with the Pharos and LRMECS time-offlight neutron spectrometers are in very good agreement at all temperatures, and also with the three-axis data reported previously. 16 The anomalous lack of softening of the phonons in pure V is clearly seen in all measurements. The phonon DOS of V is essentially constant from 10 K to 1273 K, subject only to broadening.…”

“…The phonon DOS of V is essentially constant from 10 K to 1273 K, subject only to broadening. In the three-axis measure-ments, it was found that this behavior changes at higher temperatures, and the phonon DOS of V softens between 1273 K and 1673 K. 16 The anomalous behavior from 10 K to 1273 K is particularly strong for the high-energy longitudinal modes between 26 meV and 31 meV, whose energy clearly increases with temperature. On the other hand, the low-energy transverse phonons soften between 10 K and 1273 K, but this softening is smaller than expected from thermal expansion over such a wide temperature range ͑see discussion below on expected quasiharmonic behavior͒.…”

Neutron scattering measurements of phonons in nickel at elevated temperatures

“…We expect the broadening of the phonons to take the form of a damped harmonic oscillator function [24][25][26] 22 …”

“…Measurements of the phonon dispersions of bodycentered-cubic ͑bcc͒ transition metals [21][22][23][24][25][26] have shown that the quasiharmonic model cannot always be reconciled with the experimentally determined entropy of a solid. For chromium, considering only thermal expansion against the bulk modulus severely underestimates the entropy given by the softening of the phonon DOS.…”

“…21 On the other hand, the phonon spectrum of vanadium is largely unchanged over a wide range of temperatures, despite normal thermal expansion. 22 Wallace 3 and Eriksson et al 5 evaluated the harmonic, quasiharmonic, and anharmonic entropies of transition metals. In the case of nickel, they divide the electronic entropy into a contribution from the noninteracting electrons S E and a contribution from magnetism S M such that…”