Survival Probability in Dissociative Attachment

The survival probability in dissociative attachment is investigated with special attention to the ($e$, ${\mathrm{H}}_2$) system. It is shown that the simple expression for the dissociative-attachment cross section, as given by the product of a capture cross section and a survival probability, is equivalent to the $s$-wave approximation for the $g\to g$ dissociative attachment. This expression, however, does not constitute an approximation for the $g\to u$ dissociative attachment, since the parity of the initial rotational states of ${\mathrm{H}}_2$ is always opposite to that of the relative angular momentum states of H and ${\mathrm{H}}^{-}$ and the capture cross section appearing in the simple expression is identically zero. According to the Kronig selection rules and the symmetry requirements, only odd partial waves of the incident electron may contribute to the $g\to u$ dissociative attachment in the ($e$, ${\mathrm{H}}_2$) system. Consequently, the lowest contributing partial wave is not the $s$ wave but the $p$ wave of the incident electron. This, then, destroys the simple proportional dependence of the cross section on the survival probability. However, one may still express the cross section as a sum of products of a capture cross section and a survival probability for the various contributing angular momentum states of the constituent nuclei. The dependence of the survival probability on the angular momentum states of the constituent nuclei is also investigated for the ($e$, ${\mathrm{H}}_2$) system. It is observed that for the $g\to u$ dissociative attachment the survival probability depends strongly on the angular momentum states. This arises because the $g\to u$ dissociative attachment occurs at such a low energy that variations in the centrifugal barrier become comparable with the breakup energy of the constituent atoms. This then suggests a strong temperature dependence for the $g\to u$ dissociative attachment in the ($e$, ${\mathrm{H}}_2$) system. For the $g\to g$ dissociative attachment, such dependence is much weaker since here the process significant at a somewhat higher energy and the variation in centrifugal energy is overshadowed by the larger break-up energy of the constituent atoms. The validity of the commonly adopted approximation for survival probability (involving the auto-ionization width and relative velocity of the nuclei) is also examined.