Generalized Langevin equation approach for atom/solid-surface scattering: General formulation for classical scattering off harmonic solids

TLDR

Classical scattering off harmonic lattices is the current domain of study, and the framework is grounded in a lattice dynamics formulation linked to Kubo–Mori generalized Brownian motion, making it broadly applicable beyond this specific case. The authors present a general theoretical framework to incorporate many‑body or lattice effects into gas/solid scattering. They formulate a nonperturbative generalized Langevin equation that explicitly includes the gas atom and a few directly struck surface atoms, while treating the remaining lattice as a harmonic heat bath via a friction kernel and Gaussian random force, and solve it stochastically using (n+1)-particle trajectories. The stochastic solution yields thermally averaged probability distribution functions from which temperature‑dependent gas–surface cross sections are obtained, and in the zero‑friction limit it offers an efficient alternative to Monte‑Carlo sampling for atom–oscillator cross sections that can be extended to other collision problems.

Abstract

A general theoretical framework for introducing many-body or lattice effects into gas/solid scattering is presented. The theory is presently restricted to classical scattering off harmonic lattices but is otherwise completely general. It is nonperturbative and valid for arbitrary lattice temperature. The theory is based on a formulation of lattice dynamics suggested by and related to the Kubo–Mori theory of generalized Brownian motion. This formulation leads to a generalized Langevin equation (GLE) in which only the coordinates of the gas atom and the n∼1–6 surface atoms directly struck by the gas atom appear explicitly. The remainder of the lattice, which functions as a harmonic heat bath, affects the collision through a friction kernel and a Gaussian random force appearing in the GLE. The GLE can be solved in terms of a tractable number of (n+1) -particle gas–surface trajectories using approximate stochastic techniques. Stochastic solution yields thermally averaged temporal gas particle probability distribution functions (pdf). From the long time limit of these pdf’s all temperature dependent gas–surface cross sections can be found. In the limit of zero friction, the theory gives a convenient method for calculating atom–oscillator thermally averaged cross sections which circumvents laborious Monte Carlo classical trajectory sampling and which can be generalized to treat other gas phase collision problems.