Variation of the Elastic Constants of Sodium with Temperature and Pressure

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 ${C}_{11}$, ${C}^{\ensuremath{'}}=\frac{1}{2}({C}_{11}\ensuremath{-}{C}_{12})$, and ${C}_{44}$ appear to decrease linearly with temperature between 150 and 299\ifmmode^\circ\else\textdegree\fi{}K, while their second temperature derivatives are all negative between 78 and 150\ifmmode^\circ\else\textdegree\fi{}K. At $P=0$ and $T=195$ \ifmmode^\circ\else\textdegree\fi{}K, the following values of the adiabatic elastic constants were obtained: ${C}_{11}=81.3$ kbar, ${C}^{\ensuremath{'}}=6.65$ kbar, ${C}_{44}=51.0$ kbar. At $T=78\ifmmode^\circ\else\textdegree\fi{}K, 195\ifmmode^\circ\else\textdegree\fi{}K,273\ifmmode^\circ\else\textdegree\fi{}K, \mathrm{and} 299\ifmmode^\circ\else\textdegree\fi{}K$, the first two pressure derivatives of the effective elastic constants ${C}_{11}$, ${C}^{\ensuremath{'}}$, 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 $\frac{d{C}_{11}}{\mathrm{dp}}=4.17\ifmmode\pm\else\textpm\fi{}0.20$ and $\frac{d{C}^{\ensuremath{'}}}{\mathrm{dp}}=0.258\ifmmode\pm\else\textpm\fi{}0.015$ are independent of temperature for $78\ifmmode^\circ\else\textdegree\fi{}K\ensuremath{\le}T\ensuremath{\le}299\ifmmode^\circ\else\textdegree\fi{}K$, while $\frac{d{C}_{44}}{\mathrm{dp}}$ equals 1.74\ifmmode\pm\else\textpm\fi{}0.08 for $195\ifmmode^\circ\else\textdegree\fi{}K\ensuremath{\le}T\ensuremath{\le}299\ifmmode^\circ\else\textdegree\fi{}K$, decreasing to 1.17\ifmmode\pm\else\textpm\fi{}0.07 at 78\ifmmode^\circ\else\textdegree\fi{}K. The following bounds on the second pressure derivatives of all the elastic constants were established: $\ensuremath{-}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\ensuremath{\le}\frac{{C}^{\ensuremath{-}1}{d}^{2}C}{d{p}^{2}}\ensuremath{\le}\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ ${\mathrm{kbar}}^{\ensuremath{-}2}$. To within experiment alerror, the fractional pressure derivatives of the adiabatic elastic constants decrease linearly with temperature in such a way that at 273\ifmmode^\circ\else\textdegree\fi{}K, $\frac{C_{11}^{}{}_{}{}^{\ensuremath{-}1}d{C}_{11}}{\mathrm{dp}}=0.054$ ${\mathrm{kbar}}^{\ensuremath{-}1}$, $\frac{{C}^{\ensuremath{'}\ensuremath{-}1}d{C}^{\ensuremath{'}}}{\mathrm{dp}}=0.042$ ${\mathrm{kbar}}^{\ensuremath{-}1}$, and $\frac{C_{44}^{}{}_{}{}^{\ensuremath{-}1}d{C}_{44}}{\mathrm{dp}}=0.039$ ${\mathrm{kbar}}^{\ensuremath{-}1}$, while at 78\ifmmode^\circ\else\textdegree\fi{}K these quantities are equal to 0.049, 0.037, and 0.025 ${\mathrm{kbar}}^{\ensuremath{-}1}$, respectively. Although the absolute values of the isothermal elastic constants ${B}_{T}=\frac{1}{3}({C}_{11}+2{C}_{12})$, ${C}^{\ensuremath{'}}$, and ${C}_{44}$ as well as $\frac{d{B}_{T}}{\mathrm{dp}}$ and $\frac{d{C}_{44}}{\mathrm{dp}}$ at $T=78$ \ifmmode^\circ\else\textdegree\fi{}K are in good agreement with simple theory, $\frac{d{C}^{\ensuremath{'}}}{\mathrm{dp}}$ is much larger than expected.