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

Thermal conductivity measurements have been made on hcp single crystals of helium 4 between 0.36\ifmmode^\circ\else\textdegree\fi{}K and 1.6\ifmmode^\circ\else\textdegree\fi{}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, ${K}_{\ensuremath{\perp}}={A}_{\ensuremath{\perp}}\mathrm{exp}(\frac{{\ensuremath{\Theta}}_{D}}{2.58T})$ and ${K}_{\ensuremath{\parallel}}={A}_{\ensuremath{\parallel}}\mathrm{exp}(\frac{{\ensuremath{\Theta}}_{D}}{4.62T})$, the components of the thermal conductivity tensor perpendicular and parallel to the $c$ axis. At 85 atm, ${A}_{\ensuremath{\perp}}=1.12\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ W/cm \ifmmode^\circ\else\textdegree\fi{} K and ${A}_{\ensuremath{\parallel}}=2.31\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ W/cm \ifmmode^\circ\else\textdegree\fi{} 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, ${\ensuremath{\tau}}_{N}=(2.1\ifmmode\pm\else\textpm\fi{}0.4){(\frac{{\ensuremath{\Theta}}_{D}}{T})}^{3}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ sec, is in good agreement with the results of second-sound experiments.

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