Effectiveα+αandα+NPotentials from Resonating-Group Calculations

The method of the resonating-group structure is used to study the $\alpha +\alpha$ and $\alpha +N$ systems. With a nucleon-nucleon potential of a near-Serber exchange mixture, it is found that good agreement with the empirical phase-shift data can be obtained. Using the resultant radial scattering functions, effective potentials between the clusters are derived, which yield the same phase shifts as does the resonating-group calculation. These potentials are quite similar to the phenomenological potentials obtained by Ali and Bodmer, and Darriulat et al. in the $\alpha +\alpha$ case and by Swan and Pearce in the $\alpha +N$ case. They consist of a hard core with a radius which is weakly energy-dependent but strongly $l$-dependent and an attractive long-range part which has only a weak dependence on the relative orbital angular momentum and the energy in both even-and odd-$l$ states. In addition, it is found that the long-range part has an odd-even feature which has previously been noted by Gammel and Thaler in their phenomenological study of the $\alpha +p$ scattering data, but has not been commonly considered in the usual local-potential approach to a scattering problem. Based on the experience here, an effective potential is also proposed for those cases where a straightforward application of the resonating-group method is impractical.