Semiclassical Theory of Inelastic Scattering

The Schiff approximation is applied to the problem of collisions of two asymmetric molecules. The treatment includes effects of energy interchange between translational and internal degrees of freedom, and it includes transitions to states not directly coupled by the intermolecular potential. The differential and partial scattering cross sections are obtained in terms of a phase-shift matrix which is a generalization of the WKB phase shift. This matrix may be obtained simply as an integral of the matrix of the intermolecular potential over the classical trajectory. Semiclassical approximations similar to those used in the case of elastic scattering are made, and the angular distribution reduces to the previously obtained classical expression. The approximations used in the derivation are that l and k (the orbital angular momentum quantum number and the wavenumber) must be large and Δk/k must be small. The calculations are reasonably simple for scattering below about 30° where the scattering may be considered as a perturbation on a straight-line trajectory.