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

Abstract We study the classical chaotic scattering of a He atom off a harmonically vibrating Cu surface. The three degree of freedom (3-dof) model is studied by first considering the non-vibrating 2-dof model for different values of the energy. The set of singularities of the scattering functions shows the structure of the tangle between the stable and unstable manifolds of the fixed point at an infinite distance to the Cu surface in the Poincare map. These invariant manifolds of the 2-dof system and their tangle can be used as a starting point for the construction of the stable and unstable manifolds and their tangle for the 3-dof coupled model. When the surface vibrates, the system has an extra closed degree of freedom and it is possible to represent the 3-dof tangle as deformation of a stack of 2-dof tangles, where the stack parameter is the energy of the 2-dof system. 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 relation between the changes as a function of the energy 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.