Sound Propagation in X-Ray-Oriented Single Crystals of hcp Helium-4 and bcc Helium-3

Measurements of the longitudinal and transverse sound velocities in x-ray-oriented single crystals of hcp ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ (19.0-20.5 ${\mathrm{cm}}^3$/mole) and bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ (21.6 ${\mathrm{cm}}^3$/mole) are reported. These data are used to compute the respective second-order adiabatic elastic moduli. The moduli for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ imply an extremely soft transverse mode in the directions and thus demonstrate that the low-temperature bcc-${}_{}{}^3{}_{}{}^{}\mathrm{He}$ specific-heat anomaly is due to an anomalous phonon spectrum. The Debye temperatures and volume compressibilities determined by the elastic moduli for both bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and hcp ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are in agreement with values determined by other methods.