Electron Capture from Atomic Nitrogen and Oxygen by Protons. I

Abstract

The BK approximation (Born approximation using only the proton-active electron interaction, $\mathrm{ne}$) is used to derive the cross sections for $p$-orbital capture from atomic nitrogen and atomic oxygen by protons. These processes are represented symbolically as follows: ${\mathrm{H}}^{+}+A[{p}^{n}(\mathrm{LS})]\ensuremath{\rightarrow}\mathrm{H}(^{2}S)+{A}^{+}[{p}^{n\ensuremath{-}1}({L}^{\ensuremath{'}}{S}^{\ensuremath{'}})].$ Russell-Saunders ($\mathrm{LS}$) coupling is assumed for the description of the atoms and ions, and only terms of the ground configurations are considered. Cross sections are derived for the set of processes for which the multiplicity is conserved between the initial state, ${\mathrm{H}}^{+}+A[{p}^{n}(\mathrm{LS})]$, and the final state, $\mathrm{H}(^{2}S)+{A}^{+}[{p}^{n\ensuremath{-}1}({L}^{\ensuremath{'}}{S}^{\ensuremath{'}})]$. Approximate cross sections for the inverse processes are expressed in terms of the cross sections for the corresponding direct processes. Furthermore, it is shown that all these cross sections are approximately related to one another for each atom. From this same analysis it is found that the cross sections for the direct processes are proportional to each of the following quantities: ${(2L+1)}^{\ensuremath{-}1}$; $n$, the number of $p$ orbitals in the target atom; the square of a coefficient of fractional parentage; a sum of squares of vector coupling coefficients. Detailed numerical calculations are presented for the following processes: ${\mathrm{H}}^{+}+\mathrm{N}(^{4}S)\ensuremath{\rightarrow}\mathrm{H}(1s)+{\mathrm{N}}^{+}(^{3}P); {\mathrm{H}}^{+}+\mathrm{O}(^{3}P)\ensuremath{\rightarrow}\mathrm{H}(1s)+{\mathrm{O}}^{+}(^{4}S; ^{2}D; ^{2}P); {\mathrm{H}}^{+}+\mathrm{O}(^{3}P)\ensuremath{\rightarrow}\mathrm{H}(2s)+{\mathrm{O}}^{+}(^{4}S).$ The cross sections for the last process are approximately $\frac{1}{8}$ of the cross sections for the second process, and thus they obey the ${n}^{\ensuremath{-}3}$ law of Oppenheimer. This fact supports the use of this rule for estimating the cross sections for capture into all $s$ states of hydrogen for each residual ion. Estimates of ($2s$) and ($1s$) orbital capture are also obtained; the cross sections for ($2s$) orbital capture from $\mathrm{N}(^{4}S)$ and $\mathrm{O}(^{3}P)$ are compared, and it is found that the process becomes significant relative to $p$-orbital capture for an impact energy somewhat below 1 MeV for nitrogen, and somewhat above 1 MeV for oxygen; however, ($2s$) orbital capture dominates ($2p$) orbital capture from both atoms for impact energies above 8 MeV. Estimates of the Born cross sections (Born approximation using the proton-nucleus interaction together with all proton-electron interactions, $nn+ne$) are obtained from the relation, ${Q}_{\mathrm{B}e}(A)=R(\mathrm{H}; \mathrm{He}){Q}_{\mathrm{BK}}(A)$. $R$ is the ratio, $\frac{{Q}_{\mathrm{B}}}{{Q}_{\mathrm{BK}}}$, previously calculated for atomic hydrogen and helium, and ${Q}_{\mathrm{BK}}(A)$ are the calculated BK cross sections of this paper. These Born estimates, ${Q}_{\mathrm{B}e}$, do not differ a great deal from the experimental cross sections per gas atom for capture from the corresponding diatomic molecule.