Lattice Thermal Conductivity of Solid Helium

Abstract

Callaway's theory is applied to explain the thermal conductivity of solid ${\mathrm{He}}^4$ at different densities, i.e., $\rho =0.194, 0.262, \mathrm{and} 0.282$ g/${\mathrm{cm}}^3$, where the data are available on either side of the conductivity maximum. Good agreement between theory and experiment is obtained by taking $\alpha =2.4\mathrm{}$ for all the three cases for the exponential temperature dependence of umklapp processes given by $\propto e^{-\frac{\theta }{\alpha T}}$, which verifies the $\theta$ dependence of umklapp processes. The present calculations also reveal the importance of normal processes near the conductivity maximum where the correction term in Callaway's expression for thermal conductivity dominates. This also shows that the additivity of reciprocal relaxation times is not valid for single isotope pure solid helium. The calculations also suggest the presence of internal boundaries in the composition of the solid.