Rearrangement Collisions at Very High Energies

Abstract

The energy dependence of the charge-exchange cross section in proton—hydrogen-atom collisions at very high energies, within the framework of the Schrödinger equation, has long been a subject of controversy. We attempt to determine the energy dependence of the forward-scattering cross section at very high energies in an arbitrary rearrangement collision involving two heavy particles of masses $M_a$ and $M_b$ and a light particle of mass $m$, with $m$ initially bound to $M_a$ and finally bound to $M_b$. In the limits $\frac{m}{M_a}$, $\frac{m}{M_b}\to 0$ the scattering is entirely in the forward direction and the cross section $\sigma$ for forward capture is given exactly by the impact-parameter treatment. A Born-type expansion is developed in the impact-parameter treatment for the forward-capture amplitude $A$. Thus, $A$ is written as a sum of a finite number of Born terms plus a remainder $R$. The Schwarz inequality can be used to bound $R$ since there are no non-normalizable plane-wave functions—the motion of $M_a$ and $M_b$ is treated classically. We can thereby show that for a certain class of nonsingular interactions the second Born term provides the dominant contribution to $\sigma$ at high energies, whether or not the Born series converges. (This may be the first example for which it has been shown that the second Born term dominates.) This result makes plausible the dominance of the second Born term in $p-\mathrm{H}$ forward charge exchange.