Quantum Mechanical (Phase Shift) Analysis of Differential Elastic Scattering of Molecular Beams

Abstract

For a spherically symmetrical intermolecular potential V(r)=ϵf(r/σ) the quantum calculation of the elastic scattering cross section dσ(Θ)/dΩ in the c.m. system is carried out as follows. For a given relative velocity (or deBroglie wavelength) and an assumed V(r), the radial wave equation is integrated for successive values of the angular momentum quantum number l, yielding the phase shifts ηι. Then dσ(Θ)dΩ is computed in terms of the series of ηι's in the standard way. A general computational program (following that of K. Smith) is outlined for the evaluation of the radial wave function and the phase shifts, utilizing an IBM 704 computer. Calculations are presented for the L‐J (12, 6) potential function. The results may be concisely represented using the framework provided by the semiclassical treatment of Ford and Wheeler, i.e., in terms of a set of reduced phase constants vs reduced angular momenta at various reduced relative kinetic energies K. Tables and graphs are presented from which the phases may be obtained, to a good approximation, for any given ϵ, σ and K. Computation of the differential and total cross sections from the phase shifts is then readily accomplished.The results are compared with the classical and semiclassical treatments. The problem of tunneling and orbiting is discussed.