Simple model for the quasiparticle interactions inHe3. I. The superfluid transition temperatures

Taking as a basis both the $s-p$ and paramagnon approximations, we explore the consequences of a phenomenological model for the low-frequency effective interaction (vertex function) in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ of the form (unsymmetrized) ${\Gamma }^k\sim -J\left(q\right)\mathrm{}\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{S}}+V\left(q\right)\mathrm{}$. Here $q$ represents the magnitudes of the momentum and energy transfer and $S=\frac{1}{2}$. Using the known Landau scattering amplitudes, we demonstrate that within this ansatz it is possible to (i) understand the magnitude and pressure dependence of the superfluid transition temperature $T_c$, (ii) give a partial justification for the validity of the $s-p$ approximation as applied to transport properties, and (iii) understand qualitatively the stabilization of the superfluid $A$ phase within a more general framework than the paramagnon model. The present ansatz yields the same expression for the scattering amplitudes $A(\theta ,\phi )$, as does the $s-p$ approximation at $\theta =\pi$, which limit is most significant for transport properties. We find that the values of the Landau scattering amplitudes, (i) and (iii) are consistent with the picture that $J\left(q\right)\mathrm{}$ is peaked around $q=0\mathrm{}$ and that $J\left(0\right)\mathrm{}$ is larger than the potential term, $V$. This description is similar to that used in spin-fluctuation theories. Choosing the parameters in ${\Gamma }^k$ to yield reasonable fits to the measured $\left\{A_l^{s,a}\right\}$, we find by numerical solution of the Eliashberg equations that $T_c\sim 0.1T_F{e}^{-\frac{9}{A_1^s}}$. This expression which is in good agreement with experiment at all pressures provides confirmation of the importance of spin fluctuations in $T_c$. Numerical examples of the pressure dependent $\left\{A_l^{s,a}\right\}$ and $T_c$ are given for a number of different models for the functions $J$ and $V$.