Heat exchange in liquid helium by phonon tunneling through very thin plates

Abstract

Thin plates with a thicknessd smaller than the phonon mean-free path can be used to increase the heat exchange in³He-⁴He refrigerators at very low temperatures. The resistanceR for the heat current density of these plates is theoretically investigated. For plates withd still larger than the dominant thermal-phonon wavelength λ _T in the plate,R is given by twice the Kapitza resistance. Ford ≈ λ _T the resistanceR decreases due to phonon tunneling. Ford ≪ λ _T there exists a region whereR varies asT ⁻¹, in contrast to theT ⁻³ dependence for thicker plates. Numerical calculations are carried out for copper plates immersed in pure⁴He. The results show that one obtains a large increase of the heat exchange due to phonon tunneling for plates with a thickness of a few microns at temperatures in the millidegree range.