Third-Order Elastic Constants of Copper at Low Temperature

The six third-order elastic constants of copper were determined at 295, 77, and 4.2°K from measurements of the change in the velocity of sound accompanying the application of a uniaxial stress. Irradiation with neutrons or $\gamma$ rays as well as dilute alloy additions was used in an attempt to pin the dislocations so that their effects could be eliminated. It was found that dislocation still contributed to the measurements even though the experimental data were linear and reproducible and showed no changes in ultrasonic attenuation with stress. The ultimate criterion for determining when the dislocation effects had been eliminated was the comparison of the pressure derivatives of the second-order elastic constants calculated from the uniaxial stress measurements with the pressure derivatives obtained by direct measurement. Only the neutron-irradiated sample passed this test, and all the results presented were obtained on this sample. At room temperature the third-order elastic constants agree with those obtained by Hiki and Granato. At 4.2°K, the results are $C_{111}=-20\mathrm{}\pm 2$, $C_{112}=-12\mathrm{}\pm 1.5$, $C_{123}=-5\mathrm{}\pm 1.5$, $C_{144}=-1.3\mathrm{}\pm 0.2$, $C_{166}=-7.1\mathrm{}\pm 0.25$, and $C_{456}=+0.25\mathrm{}\pm 0.08{10}^{12}$ dyn/${\mathrm{cm}}^2$. It should be noted that these results do not satisfy the Cauchy relations $C_{112}=C_{166}$ and $C_{123}=C_{144}=C_{456}$. A value of the Grüneisen $\gamma$ in the limit of low temperatures was calculated and found to be within the experimental error of the value obtained from thermal-expansion measurements.