Ballistic phonons and the transition to second sound in dilute mixtures ofHe3in liquidHe4

Using the fast-heat-pulse technique we have studied the transition from ballistic phonon propagation to second sound in dilute mixtures of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. At low temperatures and pressures the phonons are scattered primarily by the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles. Phonon-quasiparticle scattering times were determined by analyzing the pulse shapes in the transition region. These scattering times were found to be in reasonably good agreement with those theoretically calculated for Rayleigh-like scattering by Baym and Ebner, although the temperature dependence of our data was somewhat weaker than the $T^{-4\mathrm{}}$ behavior predicted by the theory. Our data are also in good agreement with the effective scattering times inferred from recent thermal conductivity measurements in mixtures by Rosenbaum et al. The phonon-quasiparticle scattering times were found to increase significantly with increasing pressure in qualitative agreement with both the theory and the thermal conductivity results. The velocity of second sound in the mixtures was measured and compared with calculated velocity curves assuming various models for the excitation spectrum. It was shown that using the Landau-Pomeranchuck spectrum for the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles and the normal ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ excitation spectrum leads to a velocity curve which, while in qualitative agreement with the experimental data, falls somewhat below the data at $T\gtrsim 0.7$ K. The so-called "${}_{}{}^3{}_{}{}^{}\mathrm{He}$ roton" model yields a velocity curve which is in even greater disagreement with the experimental results. These measurements suggest the necessity for more detailed theoretical work on the elementary excitation interactions in mixtures, particularly in the region near the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ roton minimum.