Excluded bound state in theS12α−Ninteraction and the three-body binding energies ofHe6andLi6

The effect on the ground-state $A=6\mathrm{}$ three-body binding energies of representing the $S_{\frac{1}{2}}$ $\alpha -N$ interaction by a repulsive potential compared to an attractive, excluded-bound-state potential is examined. The theory underlying construction of an attractive, excluded-bound-state potential is reviewed and then applied in the construction of an $S_{\frac{1}{2}}$ $\alpha -N$ interaction. The role of Levinson's theorem from potential theory as opposed to its modified form for composite-particle scattering is stressed in comparing the repulsive and attractive-excluded-bound-state interactions. The $A=6\mathrm{}$ three-body equations are generalized to accommodate an attractive $S_{\frac{1}{2}}$ $\alpha -N$ interaction with a forbidden bound state and it is shown that the limit to exclude the forbidden state leads to a set of well-defined three-body equations containing no spurious, deeply-bound solutions. Results for the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ binding energies suggest that the attractive, excluded-bound-state interaction gives a better representation of Pauli-exclusion effects in the $S_{\frac{1}{2}}$ $\alpha -N$ interaction than the repulsive form, based on the marked improvement in the predicted ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ binding energy compared to experiment with essentially no degradation in the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ value. This conclusion can be further tested by using the new wave functions to calculate the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}\to \alpha +d$ momentum distribution which is sensitive to the components in the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ wave function that are present solely due to the $S_{\frac{1}{2}}$ $\alpha -N$ interactions.