Systematics of shape resonances in reactive collisions

Abstract

Wentzel-Kramers-Brillouin (WKB) methodology is used to undertake a systematic analysis of shape resonances in exoergic reactive collisions at low energies $E$, and examples investigated are $\mathrm{Li}+{\mathrm{H}}^{+}, \mathrm{He}(2{\phantom{\rule{0.16em}{0ex}}}^{3}S)+\mathrm{H}$, and $\mathrm{He}(2{\phantom{\rule{0.16em}{0ex}}}^{3}S)+\mathrm{Mu}$ (muonium). In so doing, the resonance positions should be measured by the tunneling parameter $\ensuremath{\alpha}$ and not by $E$. The resonance peak height ${P}_{\mathrm{res}}$ of the reaction probability and the resonance width times the vibrational period of the quasibound resonance state can be given by simple closed-form expressions with two variables, $\ensuremath{\alpha}$ and ${P}_{0}$, the latter of which is a reaction probability given at collision energies much above a centrifugal barrier top and is determined by only short-range interactions. The resonance promotes the reaction maximally (i.e., ${P}_{\mathrm{res}}=1$) when the resonance satisfies $\ensuremath{\alpha}={\ensuremath{\alpha}}_{0}={(2\ensuremath{\pi})}^{\ensuremath{-}1}ln[(1\ensuremath{-}{P}_{0})/{P}_{0}]$, in other words, when the transmission coefficient of tunneling through the centrifugal barrier happens to be equal to ${P}_{0}$ at the resonance energy. If ${\ensuremath{\alpha}}_{0}\ensuremath{\gtrsim}1$, the reaction system is rich in tunneling resonances. If ${\ensuremath{\alpha}}_{0}\ensuremath{\lesssim}0$, the prominent resonances are mostly an over-barrier type. The resonances occurring at $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\alpha}}_{0}$ are of no significance in the reaction.