**Preface**

The shell model (SM) entered nuclear physics almost 70 years ago after Maria Goeppert Mayer and Hans Jensen—who shared the Nobel prize in 1963 for their work—were able to explain the mystery behind the magic numbers associated with the large stability in the ground state of some nuclides.

The SM is widely considered the basic scheme for the microscopic description of the nucleus, and, starting from its introduction; it has been successfully applied for investigating a variety of nuclear structure phenomena, which have important implications in our understanding of both astrophysics and physics beyond the standard model.

In the last two decades, thanks to high-performance computing facilities and the implementation of very efficient codes and methods, large-scale SM calculations have become a well-established approach to investigating medium- and heavy-mass nuclei whose description involves many valence nucleons in large model space. Today, it is possible to deal with huge matrices of dimension 10<sup>11</sup> - 1012, which was unthinkable until a few years ago.

Over about the same period, the extraordinary improvements in sensitivity and efficiency of the experimental tools and the development of a large variety of radioactive beams have allowed the exploration of new regions of the nuclide chart towards the drip lines. The richness of data emerging from these experimental studies has probed the reliability and robustness of the SM in describing the new phenomena observed far from stability, such as the onset of collectivity at the historical magic numbers, the development of islands of inversion and the appearance of new magic numbers, the evidence of shape coexistence. These studies have revealed modifications in the shell structure as a function of proton and neutron numbers, leading to a paradigm shift away from the universality of the magic numbers. Large-scale SM calculations are required to investigate these phenomena and to understand the role of the different components of the nuclear force in determining the evolution of the shell structure towards the driplines.

While the SM has been used predominantly with empirical effective Hamiltonians, substantial progress has been achieved in recent years in deriving the SM Hamiltonians from realistic bare interactions, including two and three-body forces based on chiral effective field theory; this has given a further impulse towards a fully microscopic description of atomic nuclei starting from the quantum chromodynamics degrees of freedom. Within this context, valence-space Hamiltonians can be derived using many-body perturbation theory, and only very recently, nonperturbative approaches have been introduced.

This Special Issue collects 14 contributions of leading experts in the field, intending to provide, starting from the historical setting, a clear overview of the status and future developments of the nuclear shell model, including its applications in describing various nuclear structure phenomena. The book is primarily addressed to the nuclear physics community, but it may also interest other branches of physics.

In closing, we would like to take this opportunity to express our deep gratitude to all authors for their valuable contributions. We also warmly acknowledge the MDPI Book staff and the Atomic Physics Section Editorial team. Thanks to Ms. Ling Yang, the Managing Editor, for her assistance during the Volume preparation.

> **Angela Gargano, Giovanni De Gregorio, and Silvia Monica Lenzi** *Editors*
