The last decade has witnessed both quantitative and qualitative progress in shell-model studies, which have resulted in remarkable gains in our understanding of the structure of the nucleus. Indeed, it is now possible to diagonalize matrices in determinantal spaces of dimensionality up to ${10}^9$ using the Lanczos tridiagonal construction, whose formal and numerical aspects are analyzed in this review. In addition, many new approximation methods have been developed in order to overcome the dimensionality limitations. New effective nucleon-nucleon interactions have been constructed that contain both two- and three-body contributions. The former are derived from realistic potentials (i.e., potentials consistent with two-nucleon data). The latter incorporate the pure monopole terms necessary to correct the bad saturation and shell-formation properties of the realistic two-body forces. This combination appears to solve a number of hitherto puzzling problems. The present review concentrates on those results which illustrate the global features of the approach: the universality of the effective interaction and the capacity of the shell model to describe simultaneously all the manifestations of the nuclear dynamics, either single-particle or collective in nature. The review also treats in some detail the problems associated with rotational motion, the origin of quenching of the Gamow-Teller transitions, double-$\beta$ decays, the effect of isospin nonconserving nuclear forces, and the specificities of neutron-rich nuclei. Many other calculations—which appear to have “merely” spectroscopic interest—are touched upon briefly, although the authors are fully aware that much of the credibility of the shell model rests on them.

DOI:https://doi.org/10.1103/RevModPhys.77.427