Properties of Dilute Solutions ofHe3in LiquidHe4at Low Temperatures

Using an effective interaction or reaction matrix obtained from a modified Brueckner theory, various low-temperature properties are calculated or estimated for dilute solutions of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The system is regarded as a low-density Fermi liquid with ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles created by ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atoms in superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The single-particle energy spectrum is given by an effective mass, and an effective interaction between the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ quasiparticles is derived. The calculations are done for two different two-body potentials: an Yntema-Schneider potential given by Brueckner and Gammel, and a Frost-Musulin potential given by Bruch and McGee. The Landau $f$ function is estimated from the reaction matrix, and the coefficients of the expansion of the Landau $f$ function in terms of Legendre polynomials are calculated. The estimated values are in reasonably good agreement with experimental results. The exclusion-principle sum rule is also roughly satisfied by the lowest-order coefficients. Low-temperature properties, such as the compressibility, the quasiparticle effective mass or specific-heat ratio, and the magnetic susceptibility, are estimated; results are in fair agreement with experimental values. The various properties are also given as functions of the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration in the solution, and the maximum solubility of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ is estimated in good agreement with the experimental value. Also, transport coefficients - i. e., viscosity, thermal conductivity and spin diffusion - are estimated after identification of the reaction matrix with the scattering amplitude in the formulas developed by Abrikosov and Khalatnikov and by Hone. The agreement with experimental results is surprisingly good, considering that the results depend very much on the value chosen for the effective mass.