MOMENT THEORY INVESTIGATIONS OF BORN APPROXIMATION SCATTERING PHENOMENA

This dissertation is primarily concerned with the application of moment theory techniques to the study of inelastic scattering in the first Born approximation. The task of constructing the complete set of discrete and continuum target eigenstates customarily required to calculate the doubly differential Born inelastic scattering cross section can be avoided by replacing the complete set of eigenstates with a suitably chosen complete, discrete, and quadratically integrable set of pseudostates. These pseudostates, when properly chosen, generate a discrete representation of the cross section whose energy moments converge to those of the actual cross section. Given these moments, the properties of the actual cross section, as well as the cross section itself, can be extracted using moment theory.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;is evaluated, the hydrogen atom's multipole oscillator strength distribution is derived, and FORTRAN computer codes are listed and described for the various calculations performed.;Chapter 1 introduces those aspects of moment theory used in this work by deriving generalized Gaussian integration, generalized Radau integration, and the Tchebycheff density approximation, showing how the moments of a nonnegative distribution can be used both to evaluate integrals over the distribution and to approximate the distribution itself. In chapter 2 Born approximation scattering is introduced, showing the central role played in inelastic Born scattering by the target's generalized oscillator strength distribution, or GOSD, a nonnegative distribution for a target in its ground state which describes the response of the target to the scattering event. The Tchebycheff density technique, coupled with the technique of recurrence coefficient extension, is used to generate convergent images of the GOSD from its moments. Generalized Gaussian and Radau integration and the recurrence coefficient extension technique are used to accurately Fourier transform the GOSD from its moments, generating the inelastic component of the target's generalized position time autocorrelation function. In chapter 3 a suitably chosen set of pseudostates is used to calculate the moments of the GOSD; these moments are then used to image the GOSD using the Tchebycheff density and recurrence coefficient extension procedures. The ability of the pseudostates to reproduce the local behavior of the eigenstates is examined, showing that the multipole transition density to a pseudostate at any given energy converges to the multipole transition density to the corresponding eigenstate as the number of pseudostate basis functions increases. In chapter 4 the static and binary encounter approximations to the Born inelastic scattering cross sections singly differential in scattering angle and in momentum transfer are investigated. The reasons for the breakdown of the approximations are studied and the ranges of validity of the approximations are examined. Throughout the dissertation detailed computational applications are made to non-relativistic electron scattering from the hydrogen atom neglecting exchange, allowing the techniques and approximations investigated to be readily compared to accurate analytic or numerical results. In the appendices the generation of classical Gaussian integration points and weights is described, the energy moments of the hydrogen atom's GOSD are accurately calculated, the integral.