Asymptotic form of the charge-exchange cross section in three-body rearrangement collisions

A three-body general-type rearrangement collision is considered where the initial and final bound states are described by hydrogenlike wave functions. The solution obtained is for the first Born approximation where the full interaction potential is taken into account. When the initial state is the ground state, it is shown that for $\frac{n}{\mu Z}\gg 1$, where $n$, $\mu$, and $Z$ are, respectively, the principal quantum number, the reduced mass, and the nuclear charge of the formed atom, the capture cross section at all incident energies and for capture into the $s$, $p$, and $d$ angular momentum states behaves as $\frac{C}{n^3}+O\left(\frac{1}{n^5}\right)$, where $C$ depends on masses and charges of the particles, the final angular momentum, and the incident energy. An analytic expression for $C$ is given. It is shown that for the low-lying levels the $\frac{1}{n^3}$ scaling law at all incident energies is only approximately satisfied. The only exception is for capture into the $s$ states according to the Oppenheimer-Brinkman-Kramers approximation. The case for the symmetric collisions is considered and it is shown that for high $n$ and high incident energy $E$, the cross section behaves as $\frac{1}{E^3}$. Zeros and minima in the differential cross sections are given in the limit of high $n$ for electron capture by protons from atomic hydrogen, and for positronium formation by proton-atomic hydrogen collisions.