Resonances in the Scattering of Electrons from Atoms

A formal procedure is developed for describing the resonances in electron scattering from atoms caused by metastable, auto-ionizing states. It is shown that the position of a particular resonance is given exactly by the usual Rayleigh-Schrödinger perturbation series pertaining to an unperturbed discrete state embedded in an unperturbed continuum. An expression is given for the width which shows explicitly what initial state $i$ and final state $f$ should be used in the "golden rule" as applied to auto-ionization: ${\Gamma }_{i\to f}=2\mathrm{}\pi {\left|〈i\right|V\left|f〉\right|}^2$. The formal results are applied to the lowest ${}_{}{}^1{}_{}{}^{}P_u^{}{}_{}{}^{}$ resonance in ${\mathrm{He}}^{+}+e^{-}$ scattering; i.e., the $2s2p {}_{}{}^1{}_{}{}^{}P_u^{}{}_{}{}^{}$ metastable state of He. The resonance energy obtained is 60.28 eV above the ground state of He, and the calculated width is 0.038 eV. These results are seen to be in excellent agreement with the experimental values.