Temperature Dependence of the Elastic Constants

The following two equations are proposed for the temperature dependence of the elastic stiffness constants: $c_{\mathrm{ij}}=c_{\mathrm{ij}}^0-\frac{s}{(e^{\frac{t}{T}}-1)}$ and $c_{\mathrm{ij}}=a-\frac{b{T}^2}{\mathrm{}(T+c)\mathrm{}}$, where $c_{\mathrm{ij}}^0$, $s$, $t$, $a$, $b$, and $c$ are constants. The applicability of these two equations and that of Wachtman's equation is examined for 57 elastic constants of 22 substances. The first equation has a theoretical justification and gives the best over-all results. Neither of the three equations give the theoretically expected $T^4$ dependence at low temperatures, and therefore they are not expected to give very accurate results at very low temperatures ($\lesssim \frac{{\Theta }_D}{50}$). A new melting criterion is also examined.