THEORY OF HE-3-HE-4 MIXTURES - ENERGETICS, STRUCTURE, AND STABILITY

The properties and the stability of mixtures of the two quantum liquids He-3 and He-4 are characterized by a delicate balance between the interparticle interaction, quantum-mechanical zero-point motion, and the Pauli exclusion principle. In order to calculate the microscopic structure and properties of such mixtures from first principles, a highly precise theory is needed. The necessary tools for such a precise theory are provided by the extended Jastrow-Feenberg variational method. We start with a variational ansatz for the ground state wave function which incorporates both pair and triplet correlation functions. The (Fermi)-hypernetted chain method for the summation of infinite classes of diagrams is reviewed in detail for a binary mixture of fermions and bosons where the fermion component is dilute. Optimized correlation functions are obtained by solving Euler equations for the pair and triplet correlations. Our theoretical results for the ground state energetics of the mixture agree with experimental data within better than 0.03 K in the whole physically accessible range of densities and concentrations. Such a high accuracy is needed to calculate derived quantities like the chemical potentials of both species, or the critical density where the mixture becomes (locally) unstable against phase separation. The problem of the stability of the mixture against infinitesimal density and concentration fluctuations (''local stability'') is closely related to the collective excitations of the system and the existence of solutions of the Euler equations for the correlation functions. In pure Bose systems, we prove that the existence of solutions of the Euler equation is a necessary (but not sufficient) condition for the stability of the collective excitations. The same is not true for the Fermion-Boson mixture. To cure this formal inconsistency we improve the variational theory by non-orthogonal perturbation theory in a correlated basis (CBF perturbation theory). We calculate all CBF ring diagrams and demonstrate how CBF corrections can be incorporated at all orders in the Fermion channels of the hypernetted chain summations.