**1. Introduction**

The approximate charge symmetry and charge independence of the nucleon-nucleon (NN) interaction [1] results in elegant symmetries in the behaviour of the otherwise exceptionally complex nuclear system. Examining and exploiting these isospin-related symmetries, and determining the extent to which they are broken, has become a rich field of nuclear structure physics over the last 30 years. When the symmetries are slightly broken, this provides an opportunity to observe nuclear behaviour through the lens of the wellunderstood electromagnetic interaction, providing a probe of nuclear structure phenomena such as pairing, particle alignments, shape changes and radii. It may even be possible to learn about the charge-dependent components of the nuclear interaction itself. Moreover, it is possible to exploit the often near-perfect isospin symmetry between pairs of analogue states to extract information other phenomena; in this review such an example is provided in the study of neutron skins.

Wigner's isospin concept [2] provided the conceptual and mathematical foundation for describing these symmetries. All states are assigned an isospin,**T**, quantum number, *T*, with a projection defined by *Tz* = ∑*<sup>i</sup> tz*(*i*)=(*N* − *Z*)/2, where *N* denotes the number of neutrons and *Z* the number of protons in a nucleus. In this formalism, the nucleon is treated as two states of the same particle with quantum number *<sup>t</sup>* and projection *tz* <sup>=</sup> <sup>∓</sup><sup>1</sup> 2 for the proton/neutron respectively. With the concept of isospin established, we now have a powerful isospin classification scheme, which enables us to map out, in isospin space, the resulting symmetries—visualised in Figure 1. Crucially, the mathematical formalism of isospin enables the treatment of the two types of fermion in the same system, allowing predictions based on the assumption of pure isospin symmetry, and the tools to model the observed deviations from that symmetry.

This short review focuses on the energy differences between excited isobaric analogue states—i.e., analogue states of the same isospin *T* in different members of an isobaric multiplet (different *Tz*). With perfect isospin symmetry, and in the absence of electromagnetic effects, the excitation energies would be identical. In reality the electromagnetic effects, and any other isospin-non conserving interactions, such as charge-dependent nuclear forces, lift the degeneracy. The study, and modelling, of these differences is discussed here. Two types of energy difference are usually measured: mirror energy differences (MED)

**Citation:** Bentley, M.A. Excited States in Isobaric Multiplets—Experimental Advances and the Shell-Model Approach. *Physics* **2022**, *4*, 995–1011. https://doi.org/10.3390/ physics4030066

Received: 12 July 2022 Accepted: 4 August 2022 Published: 5 September 2022

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for mirror nuclei (*Tz* = ±*T*) or triplet energy differences (TED) for isobaric triplets (*T* = 1, *Tz* = 0, ±1). MED and TED, the differences in excitation energy, *E*∗, are defined by:

$$\text{MED}\_{l,T} = E\_{l,T,T\_z = -T}^\* - E\_{l,T,T\_z = T'}^\* \text{ and} \tag{1}$$

$$\text{TED}\_{f,T=1} = E\_{f,T\_z=-1}^\* + E\_{f,T\_z=1}^\* - 2E\_{f,T\_z=0}^\* \tag{2}$$

respectively, with *J* the total angular momentum quantum number.

Developments of experimental technique, especially in the *γ*-ray spectroscopy of excited states in proton-rich nuclei, have led to a wealth of new data in recent years, allowing for experimental measurements of MED, e.g., [3–11], and TED, e.g., [12–15]. It is, however, the interpretation of these observations through shell-model analysis that has energised this field of study (e.g., [16–21]). This has allowed detailed nuclear structure phenomena, and especially their evolution with angular momentum and excitation along the yrast line, to be investigated in detail. This review outlines some experimental advances in Section 2 including specific case studies. The shell-model approach is outlined in Section 3 and some recent advances, made through shell-model interpretation, are discussed in Section 4.

**Figure 1.** A schematic visualisation of the classification of nuclear states according to the total isospin quantum numbers *T*, *Tz*. Each circle represents a set of states, of given isospin, which are allowed by the Pauli principle. Note that the diagram assumes that the lowest-energy set of states in any nucleus have the lowest allowed value of isospin. This is usually, but not always, true, e.g., odd-odd *N* = *Z* nuclei (equal and odd numbers of neutrons, *N*, and protons, *Z*).

#### **2. Advances in Experimental Techniques and Selected Case Studies**

The key challenge, in experimental measurements of energy differences between excited states of isobaric multiplets, is the typically low cross sections for, or low production rates of, the required proton-rich (i.e., *Z* ≥ *N*, *Tz* ≤ 0) nuclei. Two reaction mechanisms are generally employed: fusion-evaporation reactions with stable beams at near Coulomb-barrier energies and knockout reactions from relativistic radioactive beams. For fusion-evaporation reactions, the major difficultly is the low production cross section

of the neutron-evaporation channels that lead to the required proton-rich systems, leading to cross sections often less than 1 <sup>μ</sup>b—i.e., representing a fraction of <<sup>1</sup> × <sup>10</sup>−<sup>6</sup> of the total reaction cross section. The experimental challenge is therefore the clean selection of the reaction channel to remove the huge background from proton-emission channels. In the second method, knockout from fast radioactive beams, the knockout cross sections are reasonable (∼few mb) and the identification of the desired proton-rich fragment is straightforwardly achieved with post-target magnetic spectrometers. However, here the experimental challenge comes from the potentially low secondary beam rates and from performing high-resolution *γ*-ray spectroscopy at high beam velocities, *v* (*v*/*c* ∼0.35–0.55, where *c* is the speed of light), with the associated Doppler-broadening issues. Since the last reviews of this topic, e.g., [3,4], progress has been made in addressing these two sets of challenges, which have in turn led to advances in our understanding of MED and TED.

In the following Sections 2.1–2.3, three example cases studies are presented which highlight the recent experimental advances. The impact of these case studies on our shellmodel based interpretation of isospin-symmetry breaking, in mirror nuclei and *T* = 1 isobaric triplets, is discussed in Section 4.

#### *2.1. Prompt Tagging of Fusion-Evaporation Channels and a Case Study: The A = 23, Tz* <sup>=</sup> <sup>±</sup><sup>1</sup> 2 *Mirror Nuclei*

In fusion-evaporation reactions, the required proton-rich nuclei are populated with low cross sections and, following the evaporation of at least one prompt neutron, often at the same time as evaporated charged particles. One method of selection of the desired reaction channel is to surround the target with high-efficiency neutron- and chargedparticle detectors, in addition to the high-resolution *γ*-ray array. As a case study, we use the example of the mass number *<sup>A</sup>* <sup>=</sup> 23, *Tz* <sup>=</sup> <sup>±</sup><sup>1</sup> <sup>2</sup> mirror nuclei 23Mg/23Na [11]. This mirror pair was studied at GANIL (Grand Accélérateur National d'Ions Lourds), Caen, France, using an 16O beam on 12C target with the nuclei of interest populated through the *α*, *n* and *α*, *p* reaction channels, respectively. The prompt *γ* rays were detected with the EXOGAM array [22]. The prompt evaporated neutrons were detected with the Neutron Wall [23], an array of 50 liquid scintillator detectors. The proton and alpha particles were detected with DIAMANT [24], an array of 80 CsI scintillators. These highly-efficient detectors enabled a clean channel selection through the full identification of all emitted particles, allowing for the event-by-event tagging of the *γ* rays from the nuclei of interest. In this case study, the cleanliness of the channel selection allowed for the confident assignment of states in proton-rich 23Mg up to angular momentum/parity of *J<sup>π</sup>* = <sup>15</sup> 2 <sup>+</sup>, through a *γ*–*γ* coincidence analysis and using comparisons with the mirror nucleus, on which an identical analysis was performed. The identification of these states enabled MED to be determined up to high spin, and this proved crucial in the subsequent shell-model analysis. The impact of this measurement, and of the resulting shell-model analysis, connected to radii and neutron skins, is discussed in Section 4.2.

For the study of heavier proton-rich or *N* = *Z* nuclei, and especially where *N* = *Z* beam/target combinations are not possible, prompt particle tagging of the nuclei of interest becomes more challenging due to very low production cross sections and the need to identify more than one evaporated neutron. The development of more efficient, highly modular, neutron detector arrays such as NEDA [25], coupled to the improvements in high-resolution and high-efficiency *γ*-ray measurement afforded by the AGATA *γ*-ray array [26], provide exciting possibilities (e.g., [27]). The recent in-beam spectroscopy of 88Ru [28] through a 2*n* evaporation channel, using the AGATA, DIAMANT, Neutron Wall and NEDA arrays, provides a characteristic example.
