Thermal Conductivity of Oriented Single Crystals of Hexagonal Close-Packed Helium 4

Abstract

Thermal conductivity measurements have been made on hcp single crystals of helium 4 between 0.36°K and 1.6°K. Data on crystals grown at 53.4, 85, and 125.8 atm exhibit a strong orientation dependence in the umklapp region, and Poiseuille flow at temperatures below the thermal conductivity maximum. In the umklapp region, the data are described by two thermal conductivities, $K_{\perp }=A_{\perp }\exp \left(\frac{{\Theta }_D}{2.58T}\right)$ and $K_{\parallel }=A_{\parallel }\exp \left(\frac{{\Theta }_D}{4.62T}\right)$, the components of the thermal conductivity tensor perpendicular and parallel to the $c$ axis. At 85 atm, $A_{\perp }=1.12\mathrm{}\times {10}^{-5\mathrm{}}$ W/cm ° K and $A_{\parallel }=2.31\mathrm{}\times {10}^{-4\mathrm{}}$ W/cm ° K. The umklapp data permit one to determine the orientation of the single crystals. The thermal conductivity in the Poiseuille region is proportional to $d^2{T}^6$, where $d$ is the sample diameter. The normal-process relaxation time determined from these data, ${\tau }_N=(2.1\mathrm{}\pm 0.4){\left(\frac{{\Theta }_D}{T}\right)}^3\times {10}^{-12\mathrm{}}$ sec, is in good agreement with the results of second-sound experiments.