Variation of the Elastic Constants of Sodium with Temperature and Pressure

Abstract

The variation with temperature and pressure of the velocities of 10-MHz elastic waves propagating in monocrystalline bcc sodium was measured by the phase-comparison technique. At atmospheric pressure, the elastic constants $C_{11}$, $C^{\prime }=\frac{1}{2}(C_{11}-C_{12})$, and $C_{44}$ appear to decrease linearly with temperature between 150 and 299°K, while their second temperature derivatives are all negative between 78 and 150°K. At $P=0\mathrm{}$ and $T=195\mathrm{}$ °K, the following values of the adiabatic elastic constants were obtained: $C_{11}=81.3\mathrm{}$ kbar, $C^{\prime }=6.65\mathrm{}$ kbar, $C_{44}=51.0\mathrm{}$ kbar. At $T=78\mathrm{}°K, 195°K,273\mathrm{}°K, \mathrm{and} 299°K$, the first two pressure derivatives of the effective elastic constants $C_{11}$, $C^{\prime }$, and $C_{44}$ were computed by Cook's method from measurements of the pressure dependence of the transit times extending up to 9 kbar. The pressure derivatives $\frac{d{C}_{11}}{\mathrm{dp}}=4.17\mathrm{}\pm 0.20$ and $\frac{d{C}^{\prime }}{\mathrm{dp}}=0.258\mathrm{}\pm 0.015$ are independent of temperature for $78°K\le T\le 299°K$, while $\frac{d{C}_{44}}{\mathrm{dp}}$ equals 1.74±0.08 for $195°K\le T\le 299°K$, decreasing to 1.17±0.07 at 78°K. The following bounds on the second pressure derivatives of all the elastic constants were established: $-5\mathrm{}\times {10}^{-4\mathrm{}}\le \frac{C^{-1\mathrm{}}{d}^{2}C}{d{p}^2}\le -2\mathrm{}\times {10}^{-4\mathrm{}}$ ${\mathrm{kbar}}^{-2\mathrm{}}$. To within experiment alerror, the fractional pressure derivatives of the adiabatic elastic constants decrease linearly with temperature in such a way that at 273°K, $\frac{C_{11}^{}{}_{}{}^{-1\mathrm{}}d{C}_{11}}{\mathrm{dp}}=0.054\mathrm{}$ ${\mathrm{kbar}}^{-1\mathrm{}}$, $\frac{C^{\prime -1\mathrm{}}d{C}^{\prime }}{\mathrm{dp}}=0.042\mathrm{}$ ${\mathrm{kbar}}^{-1\mathrm{}}$, and $\frac{C_{44}^{}{}_{}{}^{-1\mathrm{}}d{C}_{44}}{\mathrm{dp}}=0.039\mathrm{}$ ${\mathrm{kbar}}^{-1\mathrm{}}$, while at 78°K these quantities are equal to 0.049, 0.037, and 0.025 ${\mathrm{kbar}}^{-1\mathrm{}}$, respectively. Although the absolute values of the isothermal elastic constants $B_T=\frac{1}{3}(C_{11}+2\mathrm{}{C}_{12})$, $C^{\prime }$, and $C_{44}$ as well as $\frac{d{B}_T}{\mathrm{dp}}$ and $\frac{d{C}_{44}}{\mathrm{dp}}$ at $T=78\mathrm{}$ °K are in good agreement with simple theory, $\frac{d{C}^{\prime }}{\mathrm{dp}}$ is much larger than expected.