**1. Introduction**

Among the multitude of experimental techniques used in nuclear-structure physics, low-energy Coulomb excitation is one of the oldest and, still to this day, one of the most widely employed. The reason for its success is twofold. On the one hand, this technique requires ion beams with relatively low energy (a few MeV per nucleon) and the large cross sections of the Coulomb-excitation process can compensate for low beam intensity. For these reasons, it was widely used for experimental nuclear-structure studies in their early days and, at present, leads the way at new-generation radioactive ion beam (RIB) facilities. On the other hand, low-energy Coulomb excitation is particularly sensitive to nuclear collective properties, such as the shape. Specifically, this method can be used to determine reduced transition probabilities between low-lying states and their spectroscopic quadrupole moments. As it relies on the well-known electromagnetic interaction, all these observables can be extracted in a model-independent way. Furthermore, the unique and model-independent information on relative signs of *E*2 matrix elements, achievable solely with this technique, makes it possible to link transitional and diagonal *E*2 matrix elements to Hill–Wheeler parameters (*β*2, *γ*) describing a quadrupole shape, via non-energy weighted quadrupole sum rules [1]. Hence, low-energy Coulomb excitation constitutes a powerful tool to study phenomena such as shape coexistence, shape transitions, superdeformation, and octupole collectivity (see [2–4] for recent examples).

This paper aims to outline how the results of low-energy Coulomb-excitation measurements can be used to benchmark the Shell Model and inspire further theoretical developments. In the next Section the method is briefly introduced, and first-order and higher-order effects, giving rise to sensitivity to transitional and diagonal electromagnetic matrix elements, are discussed. The following Section presents examples of low-energy Coulomb-excitation experiments that provided outcomes particularly relevant for the Shell Model. The aim of this paper is not to provide a comprehensive review of low-energy Coulomb-excitation studies, as these can be found elsewhere (see, for instance, [5,6]).

**Citation:** Rocchini, M.; Zieli ´nska, M. Low-Energy Coulomb Excitation for the Shell Model. *Physics* **2021**, *3*, 1237–1253. https://doi.org/10.3390/ physics3040078

Received: 7 August 2021 Accepted: 28 September 2021 Published: 15 December 2021

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