Complex potential and electron spectrum in atomic collisions involving fast electronic transitions: Penning and associative ionization

Abstract

Beginning with a Feshbach projection-operator analysis, formulas for the local complex potential in the entrance-channel radial wave equation are reviewed and developed in detail for inelastic atom-atom collisional processes that involve an electronic transition between Born-Oppenheimer states. Penning and associative ionization are used as illustrative examples in the presentation. A complete derivation of $T$-matrix elements involving radial wave functions is then presented in the context of transition rates and inelastic differential cross sections (both for energy and angle). Two approximate formulas for the energy spectrum of the particle ejected in the electronic transition are developed in which quantities obtained from exact, complex radial wave functions are replaced by those obtained from approximate, totally real radial wave functions. Numerical computations for the electron energy spectrum of the Penning process $\mathrm{He}(1s2s, 2{}_{}{}^3{}_{}{}^{}S)+\mathrm{H}(1s, 1{}_{}{}^2{}_{}{}^{}S)\to \mathrm{He}(1\mathrm{}{s}^2, 1{}_{}{}^1{}_{}{}^{}S)+{\mathrm{H}}^{+}+e^{-}$ are then presented to probe the effects of the imaginary width of the complex potential. It is found that the real component of the entrance-channel radial wave function is not much affected by the imaginary width when the width is small compared to other energy terms. In such cases, the imaginary component of the wave functions is then $\frac{\pi }{2}$ out of phase with respect to the real component, even at fairly close separations, owing to their asymptotic boundary conditions. This result also leads to a method of estimating the contribution of the imaginary radial wave functions to matrix elements using information obtained only from approximate, totally real wave functions. Finally, the eigenenergies of all 149 rotational-vibrational states of $\mathrm{He}{\mathrm{H}}^{+}\left({}_{}{}^1{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}\right)$ are reported, along with the computed cross sections of associative ionization to each state.