Longitudinal zero-sound dispersion, thermal conduction, and the LandauF ₂ in³He

Abstract

The Boltzmann equation for the distribution of Landau quasiparticles on the surface of the Fermi sphere is solved by an expansion in Legendre polynomials having symmetry appropriate to longitudinal sound. It is assumed that the collision operator can be replaced by three finite relaxation times and that the quasiparticle interaction includes a nonzero Landau parameterF ² _s . From the solution, the heat flux and temperature gradient in a zero-sound wave can be computed, and comparison with a corresponding phenomenological generalization of Fourier's law yields an expression relating thermal conductivity, λ,F ² _s , and the parameter ξ-c ₀ /V _F , wherec ₀ is the longitudinal zero-sound velocity andv _F the Fermi velocity. This expression should hold simultaneously with a second equation expressing the condition that zero sound should propagate undamped atO K, and thus we can solve for ξ andF ² _s . We obtain the valueF ² _s =−2.99 (v _F =56.62 m/sec), which depends hardly at all on the experimental value of λ, specific heat, and sound absorption used. These estimates agree with earlier ones from transverse zero sound as to sign ofF ² _s , but it appears that complete quantitative consistency may necessitate invoking Landau parameters of third and higher order.