High-Energy Approximations in Potential Scattering Theory

Abstract

The relationship between the eilronal approximation of Glauber and the impact-parameter approximation of Blankenbecler and Goldberger is clarified. This is clone by comparing the approximate forms for the scattering amplitude for nonrelativistic scattering by a potential with the Born expansion, which at high energy assumes the natural form of a series of terms in inverse powers of the center-of-mass ~nomentum p. I t is found that both approximations reproduce the first Born term and, to order l i p , the on-energy-shell part of the second Born term, for all momentum transfers. Furthermore, the Glauber approximation correctly gives the leading contribution (in powers of l i p ) to the on-energy-shell part of erevy term in the Born series, whereas the Blankenbecler-Goldberger approximation fails to do this for Born terms beyond the second. However, since the comparison to the Born series shows that both approximations neglect contributions of order 1/p2, neither approximation, in the simple forms commonly used, is correct beyond terms of order 119.

Applications of the Glauber approximation to atomic collisions

“…Therefore, Moore's comparison sheds light on the accuracy of Glauber's formula in the high energy limit. Moore (1970) finds that for all specified finite momentum transfers the Glauber approximation reproduces the first Born term (as we already know) and, to order 1/K, the on-energy-shell part? of the second Born term; moreover, still for all finite momentum transfers, the Glauber approximation reproduces the leading contribution (to order Yn/kn-l) to the on-energy-shell part?…”

“…as a-tm (Moore 1970). In this case, as Moore shows, those contributions to the nth Born series term which are not included in Glauber's formula-namely, for n 2 2, the off-shell parts and the portions of order A-" in the on-shell parts-vanish as a+ CO.…”

On the classical-trajectory eikonal approximation for electron capture into multicharged ions

“…On the other hand, the classical-trajectory eikonal approximation from the outset describes close-distant-distant ... collisions (in the prior form, the first collision refers to the projectile, the others to the target nucleus) which have a leading velocity dependence as v -12 but should be o dominant in the intermediate energy regime owing to --b-b the larger phase space available, b It should be emphasized that the situation in capture o reactions is entirely different from the situation encountered in elastic scattering and collisional excic tation. For these processes the Glauber cross section o approaches the Born cross section (and the experith mental cross section) at high energies and specifically th for potential scattering it has been shown [7] that, to leading order in v, the on-shell part of the Born ~, 2 amplitude is identical in each order to the corresponding Glauber amplitude. There is a variety of other approximations concerned with high-energy electron capture 9 We refer to Refs.…”

The eikonal approximation revisited

“…If one approximates G ~ Go in (3.22) and sums the resulting Born series, one then obtains the eikonal amplitude (3.16) [11,12]. Glauber [8] has discussed the angular range of the EA.…”