⁴He impurities in ³He: Energetics and dynamics

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.