Solution of a System of Differential Equations Associated with the Theory of Atomic Collisions in Solids

Dr. James K. Baird

Department of ChemistryUniversity of Alabama in Huntsville

November 22, 1996

Abstract

The Space Shuttle in its orbit 300 km above the surface of the earth encounters the earth's residual atmosphere, which consists mostly of atomic oxygen. The Shuttle and the O-atoms collide with a relative velocity of 8 km/sec. The energy transferred in these collisions is enough to break chemical bonds and cause a slow, atomic scale surface erosion of organic materials such as plastics and adhesives. To make sense of this phenomenon at the atomic level, we take an incident O-atom to be a structureless mass and the atoms in the target surface to be other structureless masses coupled by chemical bonds represented as springs. Applications of Newton's laws of motion to the collisions leads to an initial value problem for an infinite set of coupled, second order ordinary differential equations. We will show how this system can be solved by a generating function method, which it is claimed [R.W. Zwanzig, J. Chem. Phys. 32, 1173 (1960)] was introduced into mathematical physics by E. Schrodinger (1887-1961), who was one of the founders of quantum mechanics.