Periodic orbits and the homoclinic tangle in atom-surface chaotic scattering

In this paper the phase-space structure of a realistic chaotic scattering system, namely, the collisions of He atoms off Cu surfaces with different degrees of corrugation, is investigated. We demonstrate that the homoclinic tangle generated by a principal unstable periodic orbit, which corresponds to the unperturbed motion of the He atom traveling parallel to the surface in the asymptotic region, determines the entire scattering dynamics of the system. The fractal properties and some physical invariant features of the system can be understood using suitable Poincar\'e surfaces of section. Moreover, in this paper we also analyze in detail the periodic orbit structure in the interaction region, and show how the homoclinic chaotic trajectories can be organized in a similar fashion to the well-known Farey tree organization for resonances. The consequences of this analogy for the different scaling laws observed in chaotic scattering problems are discussed.