Towards a direct numerical solution of Schrodinger's equation for (e, 2e) reactions

The finite-difference method for electron-hydrogen scattering is presented in a simple, easily understood form for a model collision problem in which all angular momentum is neglected. The model Schrodinger equation is integrated outwards from the atomic centre on a grid of fixed spacing h. The number of difference equations is reduced each step outwards using an algorithm due to Poet, resulting in a propagating solution of the partial-differential equation. By imposing correct asymptotic boundary conditions on this general, propagating solution, the particular solution that physically corresponds to scattering is obtained along with the scattering amplitudes. Previous works using finite differences (and finite elements) have extracted scattering amplitudes only for low-level transitions (elastic scattering and n = 2 excitation). If we are to eventually extract ionisation amplitudes, however, the numerical method must remain stable for higher-level transitions. Here we report converged cross sections for transitions up to n = 8, as a first step towards obtaining ionisation (e; 2e) results.