The method of polarized orbitals for the elastic scattering of slow electrons by ionized helium and atomic hydrogen

Abstract

The method of polarized orbitals is analyzed for scattering by hydrogen or ionized helium, and is shown to require modification for p -wave scattering; with this change the polarizedorbital phase shifts for all angular momenta are associated with a variational principle. As the additional terms in the p -wave equation involve the first derivative, a novel numerical method (similar to Numerov’s method) was devised for the convenient and rapid solution of linear second-order differential equations containing the first derivative, and extended to the integro-differential equations arising in the non-iterative solution of the present problem . Polarized-orbital phase shifts are presented for the singlet and triplet scattering by ionized helium of 8- , p- and d -wave electrons, at energies up to the ionization threshold. At low energies these agree closely with the predictions of the quantum-defect method, provided that the modified equation is used for p -wave scattering. Electron-hydrogen p -wave phase shifts were also obtained with this equation; the new phase shifts increase slightly (≤ 5%) the total cross-sections calculated by Temkin & Lamkin (1961). The simpler exchange polarization approximation yields phase shifts in quite good accord with those of the (modified) method of polarized orbitals.