Collisions of Slow Electrons and Positrons with Atomic Hydrogen

We investigate the scattering of electrons and positrons by atomic hydrogen for projectile energies in the range from 11.0 to 54.4 eV. We calculate (a) the differential and total cross sections for elastic and inelastic scattering, (b) quantities related to polarization and correlation of electron spins, and (c) the polarization of radiation emitted in various electromagnetic transitions. A close-coupling approximation is used in which the total wave function is expanded in hydrogen eigenstates and only terms corresponding to the $1s$, $2s$, and $2p$ states are retained; the wave function is symmetrized or antisymmetrized explicitly in the case of electron collisions. In positron interactions, positronium formation is neglected. The coupled integro-differential equations that result from the approximate wave function are integrated numerically on an IBM-709 or 7090 computer, subject to standard boundary conditions, to yield the reactance matrix elements in each total spin and total angular-momentum state. In the case of electron scattering, the integral terms are treated by means of an iteration procedure. We find for elastic $1s\ensuremath{-}1s$ electron-hydrogen scattering that the inclusion of the $2p$ state in the close-coupling wave function modifies some partial-wave contributions at lower energies; however, the effect on the total cross section is small. The $1s\ensuremath{-}1s$ cross section has a maximum computed value of about $6\ensuremath{\pi}a_{0}^{}{}_{}{}^{2}$ at the second quantum excitation energy, and the differential cross section is strongly peaked in the forward direction. For elastic $2s\ensuremath{-}2s$ scattering of electrons, calculated total cross sections are exceptionally large, attaining values of the order of $400\ensuremath{\pi}a_{0}^{}{}_{}{}^{2}$ at 11.00 eV; here, too, the differential cross section is strongly peaked in the forward direction. Our calculated inelastic electron-hydrogen $1s\ensuremath{-}2p$ cross sections are in disagreement with experimental results, sometimes by as much as a factor of two. The calculated cross section reaches a maximum of $1.3\ensuremath{\pi}a_{0}^{}{}_{}{}^{2}$ at about 20 eV. The predictions for polarization of photons emitted by hydrogen atoms excited by electron bombardment yield a result that, near the $n=2$ threshold, is a rapidly varying nonmonatomic function of energy; again, over-all agreement with experimental results is poor. We support our belief that these discrepancies probably can not be reconciled by any close-coupling calculation. We also present results for the $1s\ensuremath{-}3p$ excitation cross section calculated with a $1s\ensuremath{-}3p$ close-coupled wave function; there are no experimental data for comparison here but we point out the consequences these results have for $1s\ensuremath{-}2s$ excitations. Our calculated total $1s\ensuremath{-}2s$ excitation cross sections show little difference as a result of including the $2p$ state in the close-coupling wave function. Agreement with experiment is again poor although measurements are subject to possible errors in normalization and we suggest further investigation of normalization procedures. As in the elastic case, the differential $1s\ensuremath{-}2s$ cross section is strongly peaked in the forward direction. Measurements of the spin-flip cross section and our calculation of it are in fair agreement at the $n=2$ threshold. The effect of the $2p$ state on elastic positron-hydrogen scattering is quite pronounced, especially for energies immediately above the $n=2$ threshold. For $1s\ensuremath{-}2s$ excitations by positrons, the same effect is seen, but it manifests itself over a wider energy range. Calculated values of reactance matrix elements are provided in tabular form for electron-hydrogen scattering at six energies above threshold.