Threshold Limit of Electron and Positron Impact Ionization of Hydrogen Atom in the Effective Charge Model

Abstract

We have presented here a method of evaluation of the matrix element for electron and positron impact ionization in hydrogen atom near the threshold within the framework of the effective charge model. The final state is represented by the product of two-Coulomb wavefunctions; for electron impact, both are due to the attractive electron–proton interactions, while for positron impact one of these corresponds to the positron–proton repulsive interaction and the other one is due to the attractive electron–proton interaction. The final expression for the matrix element for electron impact is left as a one dimensional integral to be done numerically. Interestingly in positron impact ionization the exponentially vanishing factor associated with the normalization constant of the wave function corresponding to repulsive interaction is exactly compensated for. For the Geltman model of no screening we have calculated the total cross section near the threshold. We obtain a linear threshold law for electron impact ionization as expected whereas for positron impact our result shows a square threshold law in contrast to the exponentially vanishing result of Geltman (1983 J. Phys. B: At. Mol. Phys. 16 1525).