Elastic Scattering of Low-Energy Electrons by Atomic Hydrogen

The differential and total cross sections for elastic scattering of electrons by atomic hydrogen are calculated below the threshold for excitation of the second target quantum level (10. 2 ev). 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. The coupled set of integro-differential equations resulting from the approximate wave function is integrated numerically on an IBM 709 computer, subject to standard boundary conditions, to yield the phase shift in each total spin and total angular-momentum state. The solution involves an iteration procedure to treat the integral terms, and a specialized integration scheme, including an asymptotic expansion of the solution, to overcome certain numerical difficulties associated with low-incident electron energy. The results of this calculation agree reasonably well, in regions where comparison is possible, both with previous analyses and with experiment---although in states in which short-range correlation effects are important, the close-coupling expansion is seen to converge very slowly. It is suggested that this situation may be rectified either by including continuum hydrogen eigenstates in the wave function, or by replacing the close-coupling approximation by some different method, such as the alternative expansion suggested in the present work. The most striking feature of our results is the appearance in many of the spin and angular-momentum states of pronounced, extremely narrow Breit-Wigner resonances at energies slightly below the second quantum excitation threshold. The resonance lying lowest in energy has been analyzed in most detail; it occurs in the singlet S state. It is found to have a full width at half maximum of 0. 109 ev, and to be centered at 9. 61 ev. The long-range polarization effect is found to be dominant only at and very near zero energy for S and P states; for D states it is important up to 6 or 7 ev, and for F states it is of central importance for almost the entire range of energy below threshold. The results of the calculation differ most from previous calculations at small scattering angles in the differential cross section, and in the existence of resonances near threshold. It is suggested that the former discrepancies can be resolved by differential cross-section measurements at angles of 30 deg and less, whereas the latter phenomenon requires electron-energy resolutions less than, or of the order of, 0. 1 ev for experimental verification.