Boundary-limited thermal conductivity of hcp<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>

We have been able to grow constant-volume hcp $^{4}\mathrm{He}$ polycrystalline samples that exhibited Poiseuille flow of phonons as determined by thermal-conductivity measurements. These measurements were also extended well into the boundary scattering region to an order of magnitude lower temperature than previously reported. We find the conductivity is $k\ensuremath{\propto}{T}^{3}$ throughout this region. We were also interested in searching for any significant departures from this ${T}^{3}$ behavior that may have indicated the onset of specular reflection of phonons, or a supersolid transition. Such phenomena have been predicted for sufficiently low temperatures, and in view of our results, 30 mK would now appear to be an upper bound for any occurrence of these phenomena.

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