Transport processes in superfluidHe3−Bat low temperatures

Abstract

We derive and solve the Boltzmann equation for viscosity and diffusive thermal conductivity at low temperatures in the $B$ phase of superfluid $^{3}\mathrm{He}$. The viscosity $\ensuremath{\eta}$ is shown to tend towards a constant value as the temperature tends to zero, with the constant being inversely proportional to an angular average of the collision probability. A general expression for the collision probability valid at any temperature is given in terms of the singlet and triplet components of the normal-state scattering amplitude. If one takes for the normal-state amplitude the $s$- and $p$-wave approximation, the constant viscosity is found to equal about one third of its value at the transition temperature. The diffusive thermal conductivity ${\ensuremath{\kappa}}_{D}$ is found to vary as ${T}^{\ensuremath{-}1}$, as in the normal state, and with roughly the same coefficient of proportionality. We calculate as a function of pressure the viscosity and diffusive thermal conductivity in the normal state and in the superfluid at $T=0$, and the normal-state quasiparticle relaxation time at the Fermi energy. The results are compared with experimental data, and the adequacy of the $s$- and $p$-wave approximation for the normal-state scattering amplitude is discussed. Finite temperature corrections to $\ensuremath{\eta}$ and ${\ensuremath{\kappa}}_{D}T$ are obtained for a particularly simple normal-state scattering amplitude, showing that $\ensuremath{\eta}$ initially decreases with increasing temperature while ${\ensuremath{\kappa}}_{D}T$ increases.