Electron Capture from Atomic Nitrogen and Oxygen by Protons. I

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: ${\mathrm{H}}^{+}+A\left[p^n\mathrm{}\right(\mathrm{LS}\left)\right]\to \mathrm{H}\left({}_{}{}^2{}_{}{}^{}S\right)+A^{+}\left[p^{n-1\mathrm{}}\right(L^{\prime }{S}^{\prime }\left)\right].\mathrm{}$ 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, ${\mathrm{H}}^{+}+A\left[p^n\mathrm{}\right(\mathrm{LS}\left)\right]$, and the final state, $\mathrm{H}\left({}_{}{}^2{}_{}{}^{}S\right)+A^{+}\left[p^{n-1\mathrm{}}\right(L^{\prime }{S}^{\prime }\left)\right]$. 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: ${\mathrm{}(2L+1)\mathrm{}}^{-1\mathrm{}}$; $n$, 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: