Atom-surface scattering at nonzero temperature: An example of chaotic scattering with three degrees of freedom

We study the chaotic scattering of a helium atom off a surface of copper at nonzero temperature using a three degrees of freedom classical Hamiltonian model. The model is a natural extension of a two degrees of freedom model and includes the effect of the temperature as a harmonic vibration of the copper surface. To study the three degrees of freedom model it is convenient to analyze first the uncoupled two degrees of freedom model for different values of the initial 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 central fixed point in the Poincaré map for different values of the initial energy. Knowing the stable and unstable manifolds of the two degrees of freedom model it is possible to construct the stable and unstable manifolds for the model that includes the vibration of the surface with three degrees of freedom. It is a deformation of a stack of the two degrees of freedom system. Also for the three degrees of freedom system the resulting invariant manifolds have the correct dimension to divide the phase space. By this construction we can understand the scattering phenomena in analogy to the ones in two degrees of freedom Hamiltonian systems. In particular, we explain the connection between the set of singularities of the scattering function and the cross section and their behaviour when the coupling with the vibration is switched on.