Modified Born model for excitation of atoms by electrons

A modification of the Born approximation is employed to compute the differential cross section for excitation of the $2s$ and $2p$ states of hydrogen by electrons at energies of 100, 200, 500, 1000, and 1500 eV. The modification consists of expanding the exact wave function of the total Hamiltonian in terms of Coulomb functions as opposed to plane waves in the normal Born approximation. The differential cross sections thus obtained are orders of magnitude larger at back angles and thus in much better agreement with experiment.