Ultrasonic equation of state of rhenium

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 °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, $K_T^{\prime }={\left(\frac{\partial K_T}{\partial P}\right)}_{P=0\mathrm{}}$, 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 $K_T^{\prime }$ 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 $K_T^{\prime } \left(\approx K_s^{\prime }\right)$, it is found that the x-ray value for $K_T^{\prime }$ 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 $C_H=\left(\frac{1}{6}\right)(C_{11}+C_{12}+2\mathrm{}{C}_{33}-4\mathrm{}{C}_{13})$. 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 Grüneisen-mode gamma ${\gamma }_H$ for Re, calculated from the $\frac{d{C}_{\mathrm{ij}}}{\mathrm{dP}}$, 1.83, which is significantly less than thermal gamma ${\gamma }_{\mathrm{th}} \mathrm{}\left(2.39\right)\mathrm{}$ at 300 °K.