**1. Introduction**

The shell model has served as the most fundamental view one possesses when looking at the structure of nuclei. With its inception, at the hands of Maria Goeppert-Mayer [1] and Hans Jensen and colleagues [2] in 1949, at "three-score years and ten", it is not going to die. It is based on the premise of independent-particle motion in a spherical mean field with strong spin–orbit coupling. The quantum mechanical solution, at the level of independent-particle motion in a harmonic-oscillator potential, can be obtained using methods that all senior-year undergraduate students should be able to handle. It provides a far-reaching language for talking about nuclear structure. With the "gift" of the harmonic oscillator potential to the mathematics of quantum physics, the symmetries that emerge are without equal in the quantum domain. Thus, why question "shell model" in its verbal (i.e., operative) form?

The problem is correlations. Correlations are the antithesis of independent-particle motion in quantum many-body systems. The problem in nuclei is: Just how deeply do correlations influence what we are studying? A shell modeler must start by assuming a correlation-free basis: a complete set of states, which are many copies of single-particle states each labelled by a principal quantum number (*N*), an angular momentum quantum number (*l*), a directional component of angular momentum (*ml*), and spin plus direction-ofspin quantum numbers (*s*, *ms*). (Spin–orbit coupling favors a *j*-coupled basis, |*N*, *j*, *l*, *mj*, where *j* and *mj* are the total angular momentum and its projection.) However, pairing correlations immediately dominate singly closed-shell nuclei; and most nuclei are deformed

**Citation:** Stuchbery, A.E.; Wood, J.L. To Shell Model, or Not to Shell Model, That Is the Question. *Physics* **2022**, *4*, 697–773. https://doi.org/ 10.3390/physics4030048

Received: 30 November 2021 Accepted: 8 March 2022 Published: 29 June 2022

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in their ground states while many that have spherical ground states exhibit low-energy deformed states. A simple extension of the shell model philosophy to a deformed mean field, the Nilsson model, augmented with adiabatic rotational degrees of freedom, provides an enormously powerful organizing principle for handling large amounts of data, in the guise of the unified or Bohr–Mottelson model. However, a very large number of nuclei do not separate into this simple adiabatic factorization. Such nuclei are often called "transitional nuclei". Herein lies the biggest challenge that remains in order to achieve a unified view of nuclear structure. Transitional nuclei are "sandwiched" between the shell model [3] and the unified model [4], and correlations are dominant. How do we develop theories applicable to such nuclei? To shell model or not to shell model?

The use of the term "to shell model" here is in reference to the time-honoured theoretical approach to nuclear structure which uses a basis of spherical independent-particle states, truncated at a small number of shell model energy shells, and a residual two-body interaction. The shell model is therefore a configuration interaction problem. The question then is which correlations are important, and how can one ensure that the relevant correlations emerge in the calculations.

The shell model approach is straightforward for handling all nuclei: start by introducing two-body interactions. Indeed, at the level of pairing interactions, this leads to the quasispin and seniority concepts. Quasispin is a formulation that manifestly illustrates what is meant by correlations in a quantum mechanical many-body system. With a simple approximation (by use of quasispin coherent states) this leads to the Bardeen–Cooper– Schrieffer (BCS) theory of superconductivity (see Section 4.5.3 in [5]). In finite many-body systems, as applied to nuclei, the language only needs some simple constraints to accommodate shell structure. Seniority, and its implied quasispin structure, dominates excitation patterns in singly closed shell nuclei. However, seniority breaks down immediately, at low spin, when both protons and neutrons are active. This is again due to correlations, but these correlations are not yet well understood: this is the point where nuclear deformation emerges. This nexus is the focus of the present review.

The shell model provides the most fundamental language one possesses for discussion of nuclear structure. This conceptual basis is often called the "shell model". Here, as defined, the term "shell model" is adopted in its more restricted usage as a computational model, where a Hamiltonian defined by residual interactions is diagonalized in a spherical independent-particle basis. Our view is that, with sufficient computing power, a suitable basis, and appropriate interactions, all structural details of nuclei would likely emerge. The issue, apart from the magnitude and complexity of the problem, is whether the structures in the output would be evident and intelligible. Here, the task of discussing the emergent structures in nuclei and the use of algebraic models to understand them is adopted in the context of the nuclear shell model. Therefore, the experimental data are broadly reviewed and the cases where simple models based on phenomenology and algebraic models give insights that would not be evident in a complex large-scale shell model approach are highlighted.
