Roton backflow and quasiparticle scattering at ${}^4\mathrm{He}$ surfaces

We study the effect of roton backflow on the scattering of quasiparticles and atoms at the free surface of superfluid ${}^4\mathrm{He}$ at $T=0\mathrm{}\hspace{0.5em}\mathrm{K}.$ As a starting point, we use Beliaev’s formalism and include backflow semiphenomenologically in the form of a backflow potential. We derive equations of motion for the bulk quasiparticles and the free atoms. Assuming that all the quasiparticles travel ballistically, we solve the equations of motion numerically for oblique incidence and calculate probabilities for all the one-to-one surface scattering processes allowed by the conservation laws. We compare the results with those obtained when backflow is neglected. Use of some of the calculated rates in the simulations of experiments shows that the calculated scattering rates with backflow included are in improved agreement with experiments.