Closed-Form Solution of the First Born Approximation for Direct Excitation in Simple Ion-Atom Collisions

Direct excitation, ionization, and electron transfer are the three basic processes in simple ion-atom collisions such as those between protons and hydrogen-like ions.  At high energies or high Z, the first two processes are described closely by the first Born approximation¹ and scale as 1/Z⁴ with target nuclear charge Z, while the third is described by a second Born approximation² scaling as 1/Z⁷.  The coupled-state, impact parameter approach developed by Bates¹ has been used by the author³ and others.  Bates stated¹ that “The impact parameter version of the first Born approximation [for excitation or ionization] does not readily lead to closed expressions for the various cross sections … but it is nevertheless of considerable value, the picture it gives of collisions being very instructive.”  Closed-form expressions for excitation have recently been found, which show the precise way the Born approximation manifests itself.  Remarkably, these analytic cross sections also satisfy simple first order differential equations.

¹D.R. Bates, in Atomic and Molecular Processes, edited by Bates (Academic, N.Y., 1962), pp. 549-621.

²J. Macek and S. Alston, Phys. Rev. A 26, 250 (1982).

³T. G. Winter, Phys. Rev. A 87, 032704 (2013) for Z = 2-6 and unpublished work for Z = 6-16.