Theory of Dilute Mixtures ofHe3in LiquidHe4at Low Temperatures

Abstract

Dilute mixtures of ${\mathrm{He}}^3$ in liquid ${\mathrm{He}}^4$ are regarded as a set of ${\mathrm{He}}^3$ atoms intereacting with elementary excitations of the ${\mathrm{He}}^4$. The coupling between the ${\mathrm{He}}^3$ and the ${\mathrm{He}}^4$ is chosen phenomenologically and it is not weak but, for low temperatures, its main effects may be eliminated by means of two canonical transformations. The equations which determine the transformations describe the motion of a ${\mathrm{He}}^4$ excitation in the presence of one or two ${\mathrm{He}}^3$ atoms and, in the latter case, they resemble Faddeev's equations in the three-body problem. At low temperatures, there are few ${\mathrm{He}}^4$ excitations and many of the properties of the mixture are determined by the effective ${\mathrm{He}}^3$ Hamiltonian produced by the transformation. In this respect, the system behaves like a Fermi liquid. The ${\mathrm{He}}^3$-${\mathrm{He}}^3$ potential is essentially momentum-independent and consists of the Van der Waals interaction together with an effectively repulsive potential induced by the ${\mathrm{He}}^4$. It is suggested that a Fermi-liquid analysis is the best way of obtaining information about this potential, and the possible existence of a fermion superfluid phase transition is discussed from that point of view.