Path-integral theory of the scattering ofHe4atoms at the surface of liquidHe4

The path-integral theory of the scattering of a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom near the free surface of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, which was originally formulated by Echenique and Pendry, has been recalculated with use of a physically realistic static potential and atom-ripplon interaction outside the liquid. The static potential and atom-ripplon interaction are based on the variational calculation of Edwards and Fatouros. An important assumption in the path-integral theory is the ‘‘impulse approximation’’: that the motion of the scattered atom is very fast compared with the motion of the surface due to ripplons. This is found to be true only for ripplons with wave vectors smaller than $q_m$∼0.2 A${̊}^{\mathrm{-}1}$. If ripplons above $q_m$ made an important contribution to the scattering of the atom there would be a substantial dependence of the elastic reflection coefficient on the angle of incidence of the atom. Since this is not observed experimentally, it is argued that ripplons above $q_m$ give a negligible effect and should be excluded from the calculation. With this modification the theory gives a good fit to the experimental reflection coefficient as a function of the momentum and angle of incidence of the atom. The new version of the theory indicates that there is a substantial probability that an atom may reach the surface of the liquid without exciting any ripplons. The theory is not valid when the atom enters the liquid but analysis of the experiments shows that, once inside the liquid, the atom has a negligible chance of being scattered out again.