Asymptotic Form for the Cross Section for the Coulomb Interacting Rearrangement Collisions

It is shown that in a rearrangement collision leading to the formation of highly excited hydrogenlike states the cross section at high energies behaves as $\frac{1}{n^2}$, with $n$ the principal quantum number, thus invalidating the Brinkman-Kramers approximation for large $n$. Similarly, in high-energy inelastic electron-hydrogenlike-atom collisions the exchange cross section for sufficiently large $n$ dominates the direct excitation cross section.