Elastic constants and Debye temperature of bccHe3

Measurements of the longitudinal, the fast-transverse, and the slow-transverse sound velocities have been made in single crystals of bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ at 24.45 ${\mathrm{cm}}^3$/mole and at 0.4 K. The quality and orientation of the crystals were determined using x rays. The reduced elastic constants, in units of ${10}^9$ ${\mathrm{cm}}^2$/${\sec }^2$, $\frac{C_{11}}{\rho }=1.634\mathrm{}\pm 0.020$, $\frac{C_{12}}{\rho }=1.356\mathrm{}\pm 0.033$, and $\frac{C_{44}}{\rho }=0.753\mathrm{}\pm 0.010$ resulted from a least-squares fitting of the data. These constants yield a compressibility which is in agreement with the results of thermodynamic measurements and a Debye temperature which is consistent with ${\Theta }_D^{max}$, the maximum value of the temperature-dependent calorimetric Debye temperature. Existing sound-velocity data at 21.66 ${\mathrm{cm}}^3$/mole were reanalyzed with the results, in units of ${10}^9$ ${\mathrm{cm}}^2$/${\sec }^2$: $\frac{C_{11}}{\rho }=2.727\mathrm{}\pm 0.033$, $\frac{C_{12}}{\rho }=2.477\mathrm{}\pm 0.058$, and $\frac{C_{44}}{\rho }=1.422\mathrm{}\pm 0.021$. These constants also yield a compressibility in agreement with thermodynamic measurements. Only $C_{12}$ is significantly different (4%) from the constants previously reported. As an important consequence, however, $\frac{(C_{11}-C_{12})}{2\rho }$, the modulus corresponding to the slow-transverse velocity in the $〈110〉$ direction, is considerably larger. The Debye temperature calculated using these constants is not inconsistent with ${\Theta }_D^{max}$. This result is contrary to that previously reported. At neither density is there evidence to suggest that any of the anomalous specific heat must be due to an anomalous phonon dispersion.