ABSTRACT
The most complicated nuclear systems occur in nuclei with both valence protons and valence neutrons. In the interesting cases, those valence nucleons occupy several j-orbits. In actual cases, the number of two-nucleon matrix elements required to specify the effective interaction, becomes very large. The order of the interaction matrices to be diagonalized becomes so large that exact diagonalization by present day computers is practically impossible. At the end of Section 24 an example of such a situation was presented. In 62 154 Sm 92 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq4236.tif"/> the 12 valence protons may occupy the 5 orbits in the 50–82 major shell and the 10 valence neutrons may occupy the 6 orbits in the next major shell, 82–126. To determine the effective interaction in this range of nuclei, 1307 two nucleon matrix elements are needed, 290 of which are diagonal. There is no consistent way to determine them from experimental data. The matrix to be diagonalized for obtaining the positions of J = 0 states with positive parity in 154Sm is of order 41, 654, 193, 517, 797. The one for the J = 2 states with positive parity is even bigger, its order is 346, 132, 052, 934, 889. It is clear that the straightforward application of the shell model cannot be made in such cases. A very drastic truncation scheme must be used within the 848framework of the shell model to reduce considerably the dimensions of the problem.
