The Approach to Asymptotic Behavior in Photoionization

Abstract

AbstractThe high energy behavior of many atomic processes can be understood in terms of the theory of asymptotic Fourier transforms (AFT). In high energy atomic processes initial or final state wave functions will include continuum states of asymptotic plane wave character, and these will often result in a matrix element which can be viewed as a (multi‐dimensional) Fourier transform. The result will be a behavior of matrix elements and cross sections as inverse powers of the asymptotic momenta, associated with singular behaviors of atomic wave functions, which in turn are associated with the singular (Coulomb) behavior of the interaction of the atomic constituents. The situation is complicated by the Coulomb functions which modify these plane wave behaviors; these result in slowly convergent factors (we call them Stobbe factors) which may be explicitly factored out, yielding a matrix element as a product of a Stobbe factor and a residual matrix element which converges rapidly to its asymptotic form. We will illustrate these ideas here in a discussion of single and double non‐relativistic photoionization, though the approach may also be applied to electron scattering (and with excitation or ionization) and in the relativistic case. The main ideas of the approach can already be illustrated in the discussion of ionization of a particle in a central potential. Within this approach we explain both (for single ionization) the persistent high energy deviations from independent particle approximation predictions and (for double ionization) the shake–off and quasi–free contributions. We also discuss the importance of final state interactions and retardation corrections.