He4 effective interaction for dilute solutions of He4 in liquid He3 at low temperatures

The authors present for the first time an effective interaction between two $^{4}\mathrm{He}$ atoms in a bath of liquid $^{3}\mathrm{He}$ at temperatures below 0.1 K. This interaction turns out to be, in its long-range limit, an oscillatory function of the $^{4}\mathrm{He}$ interatomic separation, the oscillations arising essentially from the sharp discontinuity in the momentum distribution at the Fermi surface. In addition, an expression is formulated for the oscillations in the density of the $^{3}\mathrm{He}$ bath induced by one $^{4}\mathrm{He}$ impurity, as well as a modified Friedel sum rule, which relates the relative amount of matter displaced by this impurity to the phase shifts pertaining to the impurity-$^{3}\mathrm{He}$ quasiparticle scattering at the Fermi surface. Finally, the dependence of the results on temperature and on the $^{3}\mathrm{He}$ superfluidity gap is discussed.