Three body clusters in finite nuclei (I). 4He

A method is developed for the treatment of the Bethe-Faddeev three body cluster equations in finite nuclei. A matrix method is employed to sum the three hole line graphs in ⁴He. For each value of a constant shift C in the intermediate state oscillator spectrum we have calculated: (i) the two body bound state binding energy self-consistently in the Brueckner approximation, (ii) the energy contribution from three hole line graphs, and (iii) the effect of single particle potential insertion in particle lines. The most important dependence on C comes from those graphs containing single particle insertions and their effect is to make the sum of (i) + (ii) + (iii) much less dependent upon C than (i) alone. The three hole line contribution for ⁴He comes mainly from third order graphs. Effects of truncation of the matrix are severe and calculations with a larger matrix could alter the quantitative but probably not the qualitative results.