First-principles thermoelasticity of transition metals at high pressure: Tantalum prototype in the quasiharmonic limit

Abstract

The thermoelastic properties of tantalum have been investigated over its theoretical high-pressure bcc solid phase (up to $26\phantom{\rule{0.2em}{0ex}}000\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ at $10\phantom{\rule{0.3em}{0ex}}\mathrm{Mbar}$) using an advanced first-principles approach that accurately accounts for cold, electron-thermal, and ion-thermal contributions in materials where anharmonic effects are small. Specifically, we have combined ab initio full-potential linear-muffin-tin-orbital electronic-structure calculations for the cold and electron-thermal contributions to the elastic moduli with phonon contributions for the ion-thermal part calculated using model generalized pseudopotential theory. For the latter, a summation of terms over the Brillouin zone is performed within the quasiharmonic approximation, where each term is composed of a strain derivative of the phonon frequency at a particular $\mathbf{k}$ point. At ambient pressure, the resulting temperature dependence of the Ta elastic moduli is in excellent agreement with ultrasonic measurements. The experimentally observed anomalous behavior of ${C}_{44}$ at low temperatures is shown to originate from the electron-thermal contribution. At higher temperatures, the main contribution to the temperature dependence of the elastic moduli comes from thermal expansion, but inclusion of the electron- and ion-thermal contributions is essential to obtain quantitative agreement with experiment. In addition, the pressure dependence of the moduli at ambient temperature compares well with recent diamond-anvil-cell measurements to $1.05\phantom{\rule{0.3em}{0ex}}\mathrm{Mbar}$. Moreover, the calculated longitudinal and bulk sound velocities in polycrystalline Ta at higher pressure and temperature in the vicinity of shock melting $(\ensuremath{\sim}3\phantom{\rule{0.3em}{0ex}}\mathrm{Mbar})$ agree well with data obtained from shock experiments. However, at high temperatures along the melt curve above $1\phantom{\rule{0.3em}{0ex}}\mathrm{Mbar}$, the ${B}^{\ensuremath{'}}$ shear modulus becomes negative, indicating the onset of unexpectedly strong anharmonic effects. Finally, the assumed temperature dependence of the Steinberg-Guinan strength model obtained from scaling with the bulk shear modulus is examined at ambient pressure.