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 C₁₁, C^'=1/2(C₁₁-C₁₂), and C₄₄ 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 $P=0$ and $T=195$ ^∘{}K, the following values of the adiabatic elastic constants were obtained: C₁₁=81.3 kbar, C^'=6.65 kbar, C₄₄=51.0 kbar. At T=78^∘K, 195^∘K,273^∘K, and 299^∘K, the first two pressure derivatives of the effective elastic constants C₁₁, C^', and C₄₄ were computed by Cook's method from measurements of the pressure dependence of the transit times extending up to 9 kbar. The pressure derivatives dC₁₁dp=4.17±0.20 and dC^'dp=0.258±0.015 are independent of temperature for 78^∘K≤T≤299^∘K, while dC₄₄dp equals 1.74±{}0.08 for 195^∘K≤T≤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: ${-}5×{}{10}^{{-}4}{≤}{{C}^{{-}1}{d}²C}{d{p}²}{≤}{-}2×{}{10}^{{-}4}$ ${kbar}^{{-}2}$. To within experiment alerror, the fractional pressure derivatives of the adiabatic elastic constants decrease linearly with temperature in such a way that at 273^∘{}K, C₁₁^_^-1dC₁₁dp=0.054 kbar^-1, C^'-1dC^'dp=0.042 kbar^-1, and C₄₄^_^-1dC₄₄dp=0.039 kbar^-1, while at 78^∘{}K these quantities are equal to 0.049, 0.037, and 0.025 kbar^-1, respectively. Although the absolute values of the isothermal elastic constants BT=1/3(C₁₁+2C₁₂), C^', and C₄₄ as well as dBTdp and dC₄₄dp at $T=78$ ^∘{}K are in good agreement with simple theory, dC^'dp is much larger than expected.