Rearrangement collisions in highly excited states of atomic hydrogen

A new straightforward method for the evaluation of the rearrangement-scattering amplitude has been proposed to study the following reactions: ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}\to \mathrm{H}\mathrm{}\left(\mathrm{nlm}\right)+{\mathrm{H}}^{+}$; $e^{+}+\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}\to \left(e^{+}{e}^{-}\right)\mathrm{}\left(\mathrm{nlm}\right)+{\mathrm{H}}^{+}$, $2\le n\le \infty$ and $l=0,1\mathrm{}$, in the first Born approximation embodying the full interacting potential. A contour-integral representation has been used for the Laguerre polynomial that occurs in the final-state wave function. The amplitude containing only the electron-nucleus interaction has been evaluated analytically. But the amplitude for the projectile nucleus interaction involves a contour integration on a circular path, and this integral can easily be evaluated numerically. To provide an estimate of the asymptotic cross section, expressions for the transition amplitudes when $n\to \infty$ have also been given. It is shown explicitly that the charge-exchange cross sections asymptotically obey the inverse $n$-cube law irrespective of the incident energy. In proton-hydrogen collisions our computed results for the low-lying discrete states show excellent agreement with the earlier theoretical values of Mapleton and those of Band as well. For positronium (Ps) formation all the reported results, except for the $2s$ excitation, are quite new. Zeros of the angular distributions are seen to be almost independent of the principal quantum number $n$, but dependent only on the projectile energy. $n^3\sigma$ shows regular smooth behavior with increasing $n$, and the results for $n=18\mathrm{}$ and $n\to \infty$ agree within 0.5% throughout the energy range considered.