Dispersion Calculation of Resonant Photo-Ionization Line Shapes: Application to Helium

Abstract

For resonant photo-ionization of spherical atoms, unitarity and simple assumptions about the analytic properties of inelastic scattering amplitudes allow calculation of the photoionization line shape from the elastic final-state phase shift. Explicitly, the Fano shape factor, $q$, is given by ${cot\ensuremath{\delta}}^{\mathrm{nr}}$, where ${\ensuremath{\delta}}^{\mathrm{nr}}$ is the nonresonant elastic phase shift for the final state. This result predicts that if ${\ensuremath{\delta}}^{\mathrm{nr}}$ is constant for a series of resonances below an inelastic threshold, the values of $q$ will be constant for the series and independent of the type of resonance. For photo-ionization of helium, values of $q$ calculated from the elastic phase shifts are seen to be in fair agreement with those calculated from wave functions.