Nonadiabatic Theory of Electron-Hydrogen Scattering

Abstract

A rigorous theory of the $s$-wave elastic scattering of electrons from hydrogen is presented. The Schr\odinger equation is reduced to an infinite set of coupled, two-dimensional partial differential equations. A zeroth order scattering problem is defined by neglecting the coupling terms of the first equation. An exact relation is derived between the phase shift of this zeroth order problem and the true phase shift. The difference between these is contained in a rapidly convergent series whose terms correspond adiabatically to multipole distortions of the hydrogen by the incoming electron. The physical significance of the zeroth order problem is discussed, and its recognition is considered basic to the understanding of the scattering problem. The exchange approximation for $s$-wave scattering is shown to be a variational approximation of the zeroth order problem. A perturbation theory is introduced to calculate the higher order corrections. The dipole correction has an increasingly important quantitative effect in the limit of zero energy. The effect of the long-range part of this correction on the scattering length can be expressed by a formula in terms of inverse powers of a long-range parameter $R$. Phase shifts for both singlet and triplet scattering are calculated, including up to quadrupole terms. The convergence is such that this number of terms should yield better than four-place accuracy. Uncertainties in our calculated values decrease the accuracy to approximately three significant figures.