LiquidHe3. V. Calculations of Landau Parameters by Brueckner Theory

Abstract

Using the reaction matrix or $G$ matrix obtained by Brueckner theory for liquid $^{3}\mathrm{He}$, rearrangement terms are evaluated, and the Landau $f$ function is estimated from microscopic theory. Taking the second variational derivative with respect to the particle occupation number of the total energy, written in terms of Brueckner theory, an approximative expression for the $f$ function is obtained. It includes the $G$ matrix, and the first and second derivatives of the $G$-matrix elements. The density dependence of the $G$ matrix, i.e., the derivatives with respect to the Fermi momentum, is evaluated numerically, and the spin-independent and spin-dependent parts of the Landau $f$ function are calculated. The effective interaction changes completely, from an average attractive $G$ matrix to an average repulsive $f$ function. Also, the coefficients of the expansion of the Landau $f$ function in terms of Legendre polynomials are estimated, and the calculated values are in fair agreement with experimental results. In lowest order, the calculations give 5.1 to 8.7 for ${F}_{0}$, 2.8 to 3.7 for ${F}_{1}$, - 0.8 to - 0.4 for ${Z}_{0}$, and - 1.2 to - 0.4 for ${Z}_{1}$. The experimental values are, respectively, 10.77, 6.25, - 0.665, and - 0.72. The value - 0.72 for ${Z}_{1}$ is, however, obtained from the exclusion-principle sum rule for the scattering amplitude in a way which is, at best, very uncertain. According to our calculated coefficients for $L>1$, the experimental value for ${Z}_{1}$ is underestimated. The sign is correct, but the absolute value should be larger.