Exact second-order amplitudes from products of first-order amplitudes

Abstract

One of the techniques which has been used to simplify the evaluation of second-order plane-wave Born amplitudes for elastic scattering is to retain only the delta function part of the Green function in the interaction operator. In this approximation, the second-order amplitude can be expressed in terms of products of first-order amplitudes. The authors investigate this possibility for second-order amplitudes for both elastic and inelastic scattering. They show that the second-order amplitude for either atomic elastic scattering or excitation can be split into a part containing phase and phaseshift information times a complex amplitude whose real part may be obtained without approximation from a product of amplitudes of the first-order type. The relative importance of the part of the second-order amplitude which can be obtained from first order amplitudes is examined for electron excitation of the 2s and 2p states of hydrogen and it is found that the magnitude of the product rule approximation amplitudes for 2s excitation are good to within about 10% while 2p amplitudes would generally be unreliable in the intermediate energy range.