Phonon-Quasiparticle Interactions in Dilute Solutions of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>in Superfluid<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>: I. Phonon Thermal Conductivity and Ultrasonic Attenuation

Expressing the interaction between long-wavelength phonons and ${\mathrm{He}}^{3}$ quasiparticles in dilute solutions of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$ entirely in terms of macroscopic observables, we calculate the attenuation of first sound due to the ${\mathrm{He}}^{3}$ viscosity, and the rate of scattering of phonons from the ${\mathrm{He}}^{3}$. The results for the attenuation are in excellent agreement with recent experiments. In addition, we calculate the thermal conduction due to phonons for $T\ensuremath{\lesssim}0.6$ \ifmmode^\circ\else\textdegree\fi{}K; the conductivity is limited at very low temperatures by ultrasonic attenuation, while at higher temperatures it is predominantly limited by elastic phonon-quasiparticle scattering.

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