Impurities in an imperfect bose gas II. Quasi-particle scattering

The problem of quasi-particle scattering by a hard sphere in an imperfect Bose gas is examined in order to gain insight into the behavior of ions in He II. Wave functions and phase shifts are calculated exactly using a simple approximation for the condensate wave function. For wavelengths characteristic of temperatures down to about 0.1°K the consequent cross sections are much like those of classical sound waves. (The depression of the condensate near the sphere plays a crucial role in this result.) At longer wavelengths the total cross section approaches a constant. Such an effect in He II would lead, for temperatures less than about 0.1°K, to ion mobilities substantially smaller than one would predict using acoustic cross sections. The more complicated problem of the scattering of density fluctuations (phonons) is considered briefly, and, with the help of further approximations, it is verified that the phonon scattering exhibits the acoustic characteristics calculated for the single-particle scattering.