4He Impurities in 3He: Energetics and Dynamics

Abstract

We have developed the optimized Fermi Hypernetted Chain Theory of a single impurity atom in a Fermi liquid, and have applied the theory to a 4He atom in bulk 3He. Previous applications of the theory for one component have produced excellent agreement with the experimental equation of state of 4He; the resulting equation of state of bulk 3He is about 0.4 K above the experimental one. Within the same theory, we obtain the pressure dependence of the chemical potential of 4He in bulk 4He, 3He in 3He, as well as the chemical potential of a 4He impurity in 3He. The pressure dependence of the impurity chemical potential agrees well with the experimental data, but we have a constant energy offset of about 1.2 K that disallows conclusive statements. This offset is partly explained by the relative inaccuracy of the 3He equation of state. We then calculate the self-energy of a single 4He impurity in 3He. Our results for the effective mass m*4 fall within the experimental error of the best available data; they increase from about m*4/m4≈1.4 at zero pressure, to m*4/m4≈1.6 at p=20 atm. We show that this effective mass enhancement is, to about equal parts, due to hydrodynamic backflow and to the coupling to particle-hole excitations. When the latter are turned off in the “collective” approximation of the impurity-background correlations one obtains a significantly lower effective mass, m*4/m4≈1.2−1.35.