High-Energy Scattering at Backward Angles

Abstract

The previous attempt by Schiff to describe large-angle potential scattering is shown to be inaccurate for strong potentials. The reason for this is that a small ripple in partial-wave amplitude, ${t}_{l}$, of the order of ${(\mathrm{ka})}^{\ensuremath{-}1}{(\ensuremath{-}1)}^{l}{t}_{l}$ is neglected in the eikonal approximation. This ripple may produce a strong coherent effect in the backward direction. To overcome this difficulty the partial-wave sum for the second Born approximation is carried out exactly, and it is shown that a stationary phase, hitherto neglected, occurs in the oscillatory integrals. An application to the square-well potential shows that the error in the eikonal approximation is of order ${(\mathrm{ka})}^{\ensuremath{-}\frac{1}{2}}$ rather than ${(\mathrm{ka})}^{\ensuremath{-}1}$ as previously thought. Even more important is the fact that this occurs in the second Born term and that the cross section is now increased by a factor of ${(\frac{V}{E})}^{2}(\mathrm{ka})$. Investigation of higher-order terms shows that one is, of course, describing the square-well glory, but it is quite clear that these effects persist for potentials other than those with discontinuities. Finally, in a numerical check, a marked improvement in the fit to the correct differential cross section is observed when second-and third-order contributions are added to Schiff's essentially first-order result.