Atom scattering off a vibrating surface: An example of chaotic scattering with three degrees of freedom

In this article, we study the classical chaotic scattering of a He atom off a harmonically vibrating Cu surface. The three degrees of freedom (3- dof) model is studied by first considering the non-vibrating 2-dof model for different values of the energy. We calculate the set of singularities of the scattering functions and study its connection with the tangle between the stable and unstable manifolds of the fixed point at an infinite distance to the Cu surface in the Poincar\'e map for different values of the initial energy. With these manifolds, it is possible to construct the stable and unstable manifolds for the 3-dof coupled model considering the extra closed degree of freedom and the deformation of a stack of maps of the 2-dof system calculated at different values of the energy. Also, for the 3-dof system, the resulting invariant manifolds have the correct dimension to divide the constant total energy manifold. By this construction, it is possible to understand the chaotic scattering phenomena for the 3-dof system from a geometric point of view. We explain the connection between the set of singularities of the scattering function, the Jacobian determinant of the scattering function, the relevant invariant manifolds in the scattering problem, and the cross-section, as well as their behavior when the coupling due to the surface vibration is switched on. In particular, we present in detail the connection between the changes in the structure of the caustics in the cross-section and the changes in the zero level set of the Jacobian determinant of the scattering function.