Microscopic versus empirical effective interactions in a single j shell

Effective-interaction matrix elements within a single j shell can be determined directly from experimental data. Such matrix elements have been found to be much different from those calculated microscopically within a complete major shell. The recent work of Moszkowski et al. showed that this difference is at least partially due to the severe truncation to a single j shell. We have constructed the microscopic effective interaction for a single j shell using two different methods: (1) simple perturbation theory and (2) explicit projection from the full major shell to a single j shell. We conclude that the empirical matrix elements and the full major-shell matrix elements should not agree, since the model spaces are different. On top of this the empirical values and the microscopic results for a single j shell would be expected to be similar only when the configurations vertical-bar ( j2) JT> are very pure