High-Energy Scattering at Backward Angles
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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, tl, of the order of (ka)-1(-1)ltl 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 (ka)-12 rather than (ka)-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 (VE)2(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. 1972 The American Physical Society.
