The continuum solid and compliance functions in gas-surface low-energy collisions

Abstract

The paper presents a model and calculations for the scattering of atomic and molecular beams from solid surfaces at low energies. The formulation permits the study of momentum and energy exchange phenomena including multiple collisions and the capture, i.e. adsorption of the projectile. The model uses a hybrid continuum-discrete representation for the system. An elastic continuum is used for the representation of the generation of phonons in the solid through collisions and the resulting momentum and energy exchange processes. on the other hand, the particulate nature of the lattice, as manifested in the corrugation is retained. This hybrid continuum-discrete representation of the solid is a suitable description for relaxation dynamics of adsorbates on surfaces and low energy collisions of gas particles with surfaces. For high energy collisions, the collision time is too short for the phonons created at the target to reach other lattice points. Therefore, the representation of the target solid by a relatively small cluster is an adequate model and there is no need to include the collaborative response of the lattice as a whole. For the low energy collisions on the other hand, the collision time is long enough such that the phonons have sufficient time to propagate over a substantial region in the lattice. Consequently, the collaborative response of the solid as a collection of a large number of lattice particles is essential. The present model uses a continuum model for the response of the solid as an accurate and convenient representation. The continuum hypothesis is validated by the predominance of long or equivalently, low frequency waves among the phonons generated during the collision. The formulation is presented for an atomic particle as projectile. Possible extensions to cover projectiles like molecules or clusters are briefly indicated. The phonons generated by the projectile are described in terms of compliance functions. These are basically Green's functions for the response of a semi-infinite solid to forces acting on its boundary. In physical terms, the compliance functions can be viewed as a frequency dependent effective spring with damping coefficients for the motion of the target point embedded in the solid. This makes the handling of the solid extremely practical by reducing the representation of its collaborative response to that of one point. The full set of compliance coefficients for all possible surface forces and moments are available. Within this picture, the discreteness of the lattice enters through the corrugated gas particle-surface interaction potential. As applications, both exact numerical integration of the equations as well as perturbation calculations are presented within a classical framework, although the formulation and calculations presented here are all within a classical context. Specific results are given for the projectile trajectories as well as momentum and energy exchange. The numerical, i.e. exact, trajectory calculations show the capability of the model in allowing realistic simulations for phenomena that involve rather complicated dynamics, including multiple collisions, skidding along the surface and eventual capture of the projectile. The perturbation solutions on the other hand, while limited to single collisions, provide attractive analytical expressions that capture basic features of the physics in the energy exchange through collision Particularly, the availability of thermal averages through simple quadratures is a valuable asset considering the gigantic computational tasks involved in exact simulations. The quadratures involved can be evaluated in closed form for both the exponential repulsive and Morse potentials. The specific results presented use the parameters of the He + LiF system as well as models with deeper wells and softer solids.