Improved calculations of Kapitza resistance: Combined effects of phonon attenuation and impedance matching on Kapitza resistance

We calculate the Kapitza resistance $R_K\mathrm{}\left(T\right)\mathrm{}$ for a Cu-${\mathrm{He}}^4$ interface using the acoustic mismatch theory of Khalatnikov and of Mazo and Onsager. Included in the calculation are the effects of: phonon attenuation in the Cu as well as impedance matching due to the increased He density near the interface. We calculate $R_K\mathrm{}\left(T\right)\mathrm{}$ for several values of attenuation in the Cu; detailed calculations are displayed for the case of equal attenuations for longitudinal and transverse waves in the solid. Also considered are different attenuation profiles in the Cu. We use a density profile calculated from compressibility data and the Van der Waals attractive force between He and the Cu substrate. Our model includes the effects of a solid layer of He at the Cu surface, i.e., in such a He layer both longitudinal and transverse waves are allowed. Included in our calculations are the effects of different density profiles for the He. The calculations indicate that for suitable choices of the physical parameters, the theoretical results for $R_K\mathrm{}\left(T\right)\mathrm{}$ agree in magnitude as well as $T$ dependence with the experimental data. Unfortunately, the important physical properties have not yet been experimentally determined and definitive test of this theory must await such data.