Thermal boundary conductance to a two-dimensional helium film

The thermal boundary conductance between vitreous quartz and a thin, superfluid helium film has been measured for the first time using third-sound resonance. This conductance, $K_B$, can be described as $0.25\times {10}^{-4\mathrm{}}{T}^{6.3\pm 1.0}$ ${\mathrm{W}/\mathrm{c}\mathrm{m}}^2$ K, between 0.7 and 0.1 K. Both the temperature dependence and magnitude preclude any bulk, acoustic-mismatch explanation. Three-phonon processes which conserve both energy and momentum produce a $T^7$ temperature dependence. It is concluded that the heat-transport mechanism must involve the solid helium layer as an intermediate. At low temperatures the third-sound resonance $Q$ saturates, becoming temperature independent below 0.4 K. To explain this behavior an additional parameter, $\Gamma$, which characterizes intrinsic dissipation in a flowing film, must be added to the two-dimensional hydrodynamics. The low-temperature value of $\Gamma$ decreases from 1.6 ${\sec }^{-1\mathrm{}}$ for a 1.35-layer film to 0.4 ${\sec }^{-1\mathrm{}}$ for a 4.40-layer film.