Diffraction and angular momentum effects in semiclassical atomic scattering theory

The semiclassical scattering theory of Mott and Massey and Ford and Wheeler is here extended to multichannel scattering as occurs at a crossing or pseudocrossing of the transient molecule formed by the colliding atoms. The generalized theory incorporates both interference and diffraction phenomena, but the emphasis in this work is on diffraction. For small-angle scattering, diffraction effects become broader, not narrower, as the collision energy increases: $\Delta b\Delta \tau \ge \hslash {\left[\frac{E_{\mathrm{inc}}}{\mathrm{}\left(2m\right)\mathrm{}}\right]}^{\frac{1}{2}}$ relates the uncertainties in impact parameter $b$ and reduced scattering angle $\tau =E_{\mathrm{inc}}\theta$, and determines the range in $b$ required to resolve a structure in the deflection function of height $\Delta \tau$. In the kilovolt range of collision energies, the effects of local maxima and minima in the deflection function are washed out, and the Airy-function approximation of Ford and Wheeler is inappropriate to describe the differential cross section. More generally, it is shown that at keV collision energies the stationary-phase approximation, heretofore essential in the reduction to the semiclassical limit, breaks down in the vicinity of a level crossing. An approximate theorem is proposed which remains valid in this region and elsewhere reduces to the standard stationary-phase approximation. Several illustrative examples are considered. A separate development treats the effect on the differential scattering cross section of a change in electronic angular momentum when electronic excitation occurs.