Ballistic phonons and the transition to second sound in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$

We have studied the propagation of ballistic phonons in solid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ using the fast-heat-pulse technique. At sufficiently low temperatures (≤0.1 K), the intensities and velocities of the ballistic phonons are in good agreement with phonon focusing and ultrasonic data. The temperature dependence of the velocity and intensity of the slow transverse mode in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ (molar volume range 23.7-24.5 ${\mathrm{cm}}^3$/mole) shows anomalous behavior in the vicinity of 0.2 K as this mode borrows weight from the higher lying branches. The implications of these measurements for specific-heat data are discussed. In addition, the numerical value of the measured second-sound velocity (which is independent of orientation) for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is shown to be in close agreement with that calculated using first-sound velocities determined by ultrasonics, but significantly different from $\frac{v_{\mathrm{Debye}}}{\surd 3}$; the result in the isotropic case.