Fermi-Gas Model of the Kapitza Resistance between a Solid and LiquidHe3

We present a simple Fermi-gas model for the low-temperature Kapitza resistance (KR) of a nonmagnetic liquid-${\mathrm{He}}^3$—solid interface in which the vibrational quanta of the solid surface atoms are carried away by single fermion states. A one-dimensional calculation of the energy-exchange process is made assuming that a fermion interacts with a single solid atom through a Morse potential. For temperatures $T$ small compared to the Fermi temperature $T_F$, the KR $R$, is found to be of the form $R{T}^3=\frac{a_0}{\left\{1+\left[46{\left(\frac{k_f}{a}\right)}^2-1.8\mathrm{}\right]{\left(\frac{T}{T_F}\right)}^2\right\}}$ where $k_F$ denotes the Fermi wave vector and $a$ the inverse range parameter of the Morse potential. The constant of proportionality $a_0$ is generally independent of the nature of the fermion—solid-atom interaction potential and depends on the properties of the Fermi gas only through its dependence on the fourth power of $k_F$. As a function of increasing $T$, $R{T}^3$ decreases rapidly on account of the largeness of the ${\left(\frac{T}{T_F}\right)}^2$ correction. The theoretical results can therefore account for the pressure and temperature dependences of the KR of the liquid-${\mathrm{He}}^3$—cooper interface observed by Anderson et al. The success of the model tentatively suggests that in the phenomenon of the liquid-${\mathrm{He}}^3$ KR, the vibrational quanta of the surface atoms are carried away by individual quasiparticle states rather than by the zero-sound modes considered in the acoustic mismatch theory of Bekarevich and Khalatnikov.