The nuclear monopole Hamiltonian

The monopole Hamiltonian $H_m$ is defined as the part of the interaction that reproduces the average energies of configurations. After separating the bulk contributions, we propose a minimal form for $H_m$ containing six parameters adjusted to reproduce the spectra of particle and hole states on doubly magic cores. The mechanism of shell formation is then explained. The reliability of the parametrization is checked by showing that the predicted particle-hole gaps are consistent with experimental data, and that the monopole matrix elements obtained provide the phenomenological cure made necessary by the bad saturation and shell properties of the realistic $\mathrm{NN}$ interaction. Predictions are made for the yet unobserved levels around ${}^{132}\mathrm{Sn},$ ${}^{22}\mathrm{O},$ ${}^{34,42}\mathrm{Si},$ ${}^{68,78}\mathrm{N}\mathrm{i},$ and ${}^{100}\mathrm{Sn}$ and for the particle-hole gaps in these nuclei.