Modified Born Approximation and Elastic Scattering by Weak Central Potentials

When only a few partial waves are substantially phase shifted and yet many partial waves are slightly phase shifted, it is possible to use the direct Born approximation, provided that one projects out the inaccurate lower partial waves and replaces them by accurate theoretical or phenomenological phase shifts. We test this technique for central potentials with two different well strengths, i.e., one which will fail to bind the 1S wave, and one which can bind the 1S state. We compare numerically generated angular distributions and total cross sections with those obtained from a modified form of the direct Born approximation. The technique would be useful for weak forces, e.g., the nucleon-nucleon and electron-atom interactions, but inapplicable for strong forces, e.g., atom-atom interaction.