Ultrasonic equation of state of rhenium

Abstract

The elastic constants ${C}_{\mathrm{ij}}$ of single-crystal Re and their pressure derivatives $\frac{d{C}_{\mathrm{ij}}}{\mathrm{dP}}$, measured to 4.2 kbar at 25 \ifmmode^\circ\else\textdegree\fi{}C, are, respectively; ${C}_{11}: 6177.3 \mathrm{and} 8.65$; ${C}_{33}: 6828.2 \mathrm{and} 8.45$; ${C}_{13}: 2055.7 \mathrm{and} 2.98$; ${C}_{44}: 1605.4 \mathrm{and} 1.49$; ${C}_{66}: 1714.1 \mathrm{and} 1.56$. The initial pressure derivative of the isothermal bulk modulus, ${K}_{T}^{\ensuremath{'}}={(\frac{\ensuremath{\partial}{K}_{T}}{\ensuremath{\partial}P})}_{P=0}$, calculated from the Voigt-Reuss-Hill approximation, is 5.43, which is significantly larger than the previously reported value of 2.93 based on x-ray-diffraction data. The contributions to ${K}_{T}$ and ${K}_{T}^{\ensuremath{'}}$ from long-range Fermi energy and short-range ion-core repulsive interactions (${K}_{F}$ and ${K}_{\mathrm{SR}}$, respectively) are evaluated by assuming the Born-Mayer potentials for the latter contribution. Using an empirical relationship between atomic volume and ${K}_{T}^{\ensuremath{'}} (\ensuremath{\approx}{K}_{s}^{\ensuremath{'}})$, it is found that the x-ray value for ${K}_{T}^{\ensuremath{'}}$ is probably too low. Evaluation of the Born-Mayer parameters leads to the conclusion that the short-range contribution, rather than the Fermi contribution, dominates ${K}_{T}$, and shear moduli ${C}_{44}$ and ${C}_{66}$, and their pressure derivatives, whereas the Fermi energy is a major contributory factor in the case of shear modulus ${C}_{H}=(\frac{1}{6})({C}_{11}+{C}_{12}+2{C}_{33}\ensuremath{-}4{C}_{13})$. The latter conclusion is strengthened by the nonlinearity of ${C}_{H}$ with respect to pressure, and by the anomalous variation of the superconducting temperature ${T}_{c}$ with pressure, which effect is related to the change in Fermi-surface topology. The value of the Gr\"uneisen-mode gamma ${\ensuremath{\gamma}}_{H}$ for Re, calculated from the $\frac{d{C}_{\mathrm{ij}}}{\mathrm{dP}}$, 1.83, which is significantly less than thermal gamma ${\ensuremath{\gamma}}_{\mathrm{th}} (2.39)$ at 300 \ifmmode^\circ\else\textdegree\fi{}K.