Abstract
The exact solutions for the Schr\odinger equation with a central potential of the form $V(r)=\frac{2Z}{r}\ensuremath{-}\ensuremath{\Delta}$, $r\ensuremath{\le}{r}_{1}$; $V(r)=\frac{2}{r}$, $r\ensuremath{\ge}{r}_{1}$; ${r}_{1}=\frac{2(Z\ensuremath{-}1)}{\ensuremath{\Delta}}$ are developed. $Z$ and $\ensuremath{\Delta}$ are chosen to fit observed term values for various atoms, and the parametrized continuum orbitals are used to calculate the photo ionization cross section of these atoms. For certain transitions the photo-ionization cross-section matrix element can be written as an exact expression. This is done for valence photo-ionization in He and the alkalis, for $4d$-shell photo-ionization in the elements indium to xenon, and for the $3d$ shell in krypton. The relationship between this approach and the Coulomb method of Burgess and Seaton is briefly discussed.
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