Thermal Conductivity in Isotopic Mixtures of Solid Helium

Abstract

This paper reports thermal-conductivity measurements in isotopic mixtures of solid helium at concentrations of 1%, 4%, 10%, and 50% of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$ and vice versa. The data were taken at $V=19.5$ ${\mathrm{cm}}^{3}$/mole and $V=20.2$ ${\mathrm{cm}}^{3}$/mole; the data at 19.5 ${\mathrm{cm}}^{3}$/mole are analyzed using the phenomenological equation of Callaway. Resistance to heat flow is assumed to be due to boundary scattering, normal-process scattering, umklapp-process scattering, and a Rayleigh-like scattering mechanism which may be a combination of mass-fluctuation scattering and lattice-distortion scattering. The strength of these scattering mechanisms is determined by the analysis. The major contribution to the thermal resistance at low temperature is found to be the lattice distortion around the site of an isotopic impurity. The intrinsic strength of lattice-distortion scattering is evaluated and found to depend on concentration more strongly than the $x(1\ensuremath{-}x)$ expected by analogy to mass-fluctuation scattering. Two arguments are given in explanation of this increased concentration dependence.