Strong Coupling Corrections to the Ginzburg-Landau Theory of Superfluid ³He

In the Ginzburg-Landau theory of superfluid $^{3}$He, the free energy is expressed as an expansion of invariants of a complex order parameter. Strong coupling effects, which increase with increasing pressure, are embodied in the set of coefficients of these order parameter invariants\cite{Leg75,Thu87}. Experiments can be used to determine four independent combinations of the coefficients of the five fourth order invariants. This leaves the phenomenological description of the thermodynamics near $T_{c}$ incomplete. Theoretical understanding of these coefficients is also quite limited. We analyze our measurements of the magnetic susceptibility and the NMR frequency shift in the $B$-phase which refine the four experimental inputs to the phenomenological theory. We propose a model based on existing experiments, combined with calculations by Sauls and Serene\cite{Sau81} of the pressure dependence of these coefficients, in order to determine all five fourth order terms. This model leads us to a better understanding of the thermodynamics of superfluid $^{3}$He in its various states. We discuss the surface tension of bulk superfluid $^{3}$He and predictions for novel states of the superfluid such as those that are stabilized by elastic scattering of quasiparticles from a highly porous silica aerogel.