*1.1. Isospin Symmetry in Nuclear Structure*

Atomic nuclei are unique quantum many-body systems composed of two sorts of fermions—protons and neutrons, which are known to have similar masses and possess similar properties with respect to the strong interactions. It was Heisenberg [1] (see English translation in Ref. [2]) who soon after the discovery of the neutron, introduced an *isospin formalism* similar to the ordinary spin formalism as an elegant mathematical tool for dealing with protons and neutrons. Nucleons are considered to be isospin *t* = 1/2 particles and represented by two-component spinors spanning an abstract vector space where the isospin operator, ˆ**t**, acts. The neutron and the proton are two eigenstates of ˆ*t*<sup>3</sup> (the third component of the isospin operator):

$$
\psi\_{\mathbb{H}}(\vec{r}) = \psi(\vec{r}) \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \\
\psi\_{\mathbb{P}}(\vec{r}) = \psi(\vec{r}) \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
$$

with eigenvalues *mt*= ±1/2, respectively, and*r* the radius vector. The three components of the isospin operator, analogues of the Cartesian components, generate an isospin SU(2) algebra:

$$[\hat{\mathbf{f}}\_{\rangle}, \hat{\mathbf{f}}\_{k}] = \mathbf{i}\epsilon\_{jkl}\hat{\mathbf{f}}\_{l} \,\tag{1}$$

**Citation:** Smirnova, N.A. Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments. *Physics* **2023**, *5*, 352–380. https://doi.org/ 10.3390/physics5020026

Received: 2 February 2023 Revised: 5 March 2023 Accepted: 7 March 2023 Published: 31 March 2023

**Copyright:** © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where *j*, *k*, *l* = 1, 2, 3,  *jkl* is the Levi-Civita symbol, and the square of the isospin operator,

$$
\mathfrak{f}^2 = \hat{\mathfrak{f}}\_1^2 + \hat{\mathfrak{f}}\_2^2 + \hat{\mathfrak{f}}\_3^2 \,. \tag{2}
$$

commutes with each of the components: [ˆ**t** 2, ˆ*tj*] = 0.

Operators corresponding to various physical observables can be conveniently expressed using isospin formalism. For example, the third component of the isospin operator ˆ*t*<sup>3</sup> allows one to express the nucleon charge operator,

$$\boldsymbol{\hat{q}} = \left(\frac{1}{2} - \hat{\boldsymbol{t}}\_3\right) \mathbf{e}\_{\prime\prime}$$

and the ladder operators <sup>ˆ</sup>*t*±,

$$
\hat{\mathfrak{f}}\_{\pm} = \hat{\mathfrak{f}}\_1 \pm i \hat{\mathfrak{f}}\_2,\tag{3}
$$

transforming a proton into a neutron and vice versa, can be useful to formulate nuclear *β* decay. Here, "e" denotes the elementary charge.

Nowadays, isospin symmetry is an important concept in particle physics describing a symmetry between *u* and *d* quarks with respect to the strong interaction and their similarly light masses as compared to the other known quarks. The isospin character of nucleons, and of other hadrons composed from *u* and/or *d* quarks, is a consequence of isospin coupling.

Based on the conservation of charge and the approximate charge-independence of the nuclear forces, Wigner [3] introduced the total isospin operator for an *A*-nucleon system arising from the coupling of the individual isospin operators:

$$\mathbf{\dot{T}} = \sum\_{k=1}^{A} \mathbf{\dot{t}}(k)\_{\,\,\,t}$$

or for the components:

$$
\hat{T}\_{\pm} = \sum\_{k=1}^{A} \hat{t}\_{\pm}(k) \,, \quad \hat{T}\_{\mp} = \sum\_{k=1}^{A} \hat{t}\_{\mp}(k) \,, \tag{4}
$$

with *<sup>T</sup>*(*<sup>T</sup>* <sup>+</sup> <sup>1</sup>) and *MT* = (*<sup>N</sup>* <sup>−</sup> *<sup>Z</sup>*)/2 being eigenvalues of **<sup>T</sup>**<sup>ˆ</sup> <sup>2</sup> and *<sup>T</sup>*<sup>ˆ</sup> 3, respectively, *N* the neutron number, and *Z* the atomic number. A charge-independent nuclear Hamiltonian would commute with **T**ˆ ,

$$[[\hat{H}\_{\text{nucl}}, \hat{\mathbf{T}}] = 0, \prime$$

or

$$[\hat{H}\_{\text{nucl}}, \hat{T}\_{\pm}] = [\hat{H}\_{\text{nucl}}, \hat{T}\_3] = 0 \dots$$

An additional isospin quantum number *T* appears to label *A*-nucleon states besides the total angular momentum, *J*, and parity, *π*. The spectrum of *H*nucl thus consists of degenerate isobaric multiplets, which can be labeled by (*Jπ*, *T*) in nuclei with the same mass number *A* and *MT* = −*T*,..., *T*, called isobaric analogue states (IAS).

It was realized long ago that electromagnetic interactions destroy this degeneracy. However, as it was shown by Wigner [4], this leads mainly to dynamical breaking of the isospin SU(2) symmetry. Indeed, the Coulomb interaction between protons, which is the main source of the isospin-symmetry breaking on the nuclear level, can be represented

as a linear combination of an isoscalar (*V*ˆ (0)), an isovector (*V*ˆ (1)) and an isotensor (*V*ˆ (2)) operator:

$$\mathcal{V}\_{\text{Coul}} = \sum\_{i
$$\mathcal{V}\_{\text{D}} = \sum\_{i$$
$$

By estimating the effect of this charge-dependent operator on the isobaric multiplets within the lowest order perturbation theory (due to its expectation value within the states of a given isospin, *T*) and applying the Wigner–Eckart theorem in the isospace, one gets an expression quadratic in *MT*:

$$
\begin{split}
\langle\etaTM\_{\mathrm{T}}|\mathcal{V}\_{\mathrm{Coul}}|\etaTM\_{\mathrm{T}}\rangle &=& \frac{\langle TM\_{\mathrm{T}}00|TM\_{\mathrm{T}}\rangle}{\sqrt{2T+1}} \langle\etaT||\mathcal{V}^{(0)}||\etaT\rangle \\ &+& \frac{\langle TM\_{\mathrm{T}}10|TM\_{\mathrm{T}}\rangle}{\sqrt{2T+1}} \langle\etaT||\mathcal{V}^{(1)}||\etaT\rangle \\ &+& \frac{\langle TM\_{\mathrm{T}}20|TM\_{\mathrm{T}}\rangle}{\sqrt{2T+1}} \langle\etaT||\mathcal{V}^{(2)}||\etaT\rangle,
\end{split} \tag{6}
$$

where double bar denotes reduction in the isospin space; (*TMTλμ*|*TMT*) are the Clebsch– Gordan coefficients; and *η* refers to other quantum numbers characterizing an isobaric multiplet: *η* = (*A*, *Jπ*,...). By inserting Clebsch–Gordan coefficients, one gets:

$$\langle \eta T M\_T | V\_{\text{Coul}} | \eta T M\_T \rangle = E^{(0)}(\eta, T) + E^{(1)}(\eta, T) M\_T + E^{(2)}(\eta, T) \left[ 3 M\_T^2 - T(T + 1) \right], \tag{7}$$

where *E*(*λ*)(*η*, *T*) are related to the reduced in isospace matrix elements of isotensors, as seen from Equation (6). This expression remains valid if leading-order terms of chargedependent forces of nuclear origin are included, as discussed in Section 1.2. Such a dependence, re-written for nuclear masses, is known as the isobaric-multiplet mass equation (IMME) [4],

$$\mathcal{M}(\eta, T, M\_T) = a(\eta, T) + b(\eta, T)M\_T + c(\eta, T)M\_{T'}^2 \tag{8}$$

with M being an atomic mass excess. Experimental *a*, *b* and *c* coefficients can be deduced from available data on nuclear masses and spectra of up to about *A* = 71 [5,6].

Interestingly, Equation (8) holds exceptionally well, even for isobaric multiplets with more than three members (*T* > 1). This makes the IMME a powerful tool for predicting the nuclear masses of nuclei along the *N* = *Z* line, as illustrated in Section 3. Deviations from the quadratic form are rare and small. They are specifically searched for in experiments, as they can bring important information on the presence of charge-dependent many-body forces or witness strong isospin mixing.

From a group-theoretical point of view [7], Equation (7), or equivalently, Equation (8), expresses a reduction of the isospin SU(2) group to its SO(2) subgroup. The eigenstates of the full Hamiltonian, *H*ˆ nucl + *V*ˆ Coul, can still be characterized by the isospin quantum number *T*, but the (2*T* + 1)-fold degeneracy inherent to the isotopic multiplets is now removed. This effect is analogous to a Zeeman splitting of atomic levels in the presence of a magnetic field.

As every symmetry, isospin symmetry proposes a number of selection rules for various transition operators, on the basis of their tensorial character with respect to the SU(2) group in isospace. For example, allowed *β*-decay, governed by the vector or axial-vector weak currents, is described by Fermi (F) or Gamow–Teller (GT) operators, respectively. In the impulse approximation, these operators read

$$\mathcal{O}\_{\rm F}(\beta^{\pm}) = \sum\_{k=1}^{A} \hat{\mathfrak{f}}\_{\pm}(k) \,, \quad \mathcal{O}\_{\rm GT}(\beta^{\pm}) = \sum\_{k=1}^{A} \mathfrak{d}(k)\hat{\mathfrak{f}}\_{\pm}(k) \,. \tag{9}$$

Both operators are seen to be isovector components. The Fermi operator is a scalar, and the Gamow–Teller operator is a vector in the ordinary spin space (*σ*ˆ is the Pauli spin operator). The Wigner–Eckart theorem establishes angular momentum parity, and isospin selection rules can be established for transitions between an initial state (*J πi <sup>i</sup>* , *Ti*) and a final state (*J πf <sup>f</sup>* , *Tf*). For Fermi transitions, one has:

$$
\Delta f = 0,\\
\Delta T = 0,\\
\Delta \pi = 0,\\
$$

and for Gamow–Teller transitions, one has:

$$\begin{array}{l} \Delta f = 0, 1, \,\Delta T = 0, 1, \,\Delta \pi = 0 \\ \text{(no } f\_i = 0 \to f\_f = 0) \,. \end{array}$$

From this one can conclude that *Ji* = 0 → *Jf* = 0 decay can be only by the Fermi type.

A similar analysis can be performed for electromagnetic operators. Assuming a one-body structure of nucleonic convection and spin currents and point-like nucleons, electromagnetic operators can be shown to be a linear combination of an isoscalar and an isovector operator [8], e.g., for an operator of multipolarity *<sup>L</sup>*, one has *<sup>O</sup>*<sup>ˆ</sup> *LM* <sup>=</sup> *<sup>O</sup>*<sup>ˆ</sup> (0) *LM* <sup>+</sup> *<sup>O</sup>*<sup>ˆ</sup> (1) *LM*, where *M* = −*L*, ... , *L*. Therefore, their matrix elements between states of given isospin can be expressed as

$$\begin{split} \langle \langle f\_f M\_f; T\_f M\_T | \mathcal{O}\_{LM} | j\_i M\_{i\cdot}; T\_i M\_T \rangle = \delta\_{T\_i T\_f} \langle f\_f M\_f | \mathcal{O}\_{LM}^{(0)} | j\_i M\_i \rangle \\ + \frac{(T\_f M\_T 10 | T\_f M\_T)}{\sqrt{2 T\_f + 1}} \langle f\_f M\_f; T\_f || \mathcal{O}\_{LM}^{(1)} || j\_i M\_{i\cdot}; T\_i \rangle \end{split} \tag{10}$$

where *δTiTf* is the Kronecker delta.

From Equation (10) one immediately gets the isospin selection rules for electromagnetic transitions [8].


$$
\begin{split}
\langle\rangle\_{f}\langle M\_{f};TM\_{f}|\hat{\mathcal{O}}\_{LM}||j\_{i}M\_{i};TM\_{T}\rangle&=\langle J\_{f}M\_{f}|\hat{\mathcal{O}}\_{LM}^{(0)}||j\_{i}M\_{i}\rangle\\ &+\frac{M\_{T}}{\sqrt{T(T+1)(2T+1)}}\langle J\_{f}M\_{f};T||\hat{\mathcal{O}}\_{LM}^{(1)}||j\_{i}M\_{i};T\rangle
\end{split}
\tag{11}
$$

• Another specific rule can be established for electric dipole operator. In the lowest order of the long-wavelength approximation, the electric-dipole (*E*1) operator is an isovector operator:

$$\hat{O}(E1) = \sum\_{k=1}^{A} \mathbf{e}(k)\vec{r}(k) = \sum\_{k=1}^{A} \left(\frac{1}{2} - \hat{\mathbf{r}}\_{3}(k)\right) \mathbf{e}\vec{r}(k) \,. \tag{12}$$

Hence, *E*1 transitions between the states of the same isospin (*Ti*=*Tf* =*T*) in *N* = *Z* nuclei are forbidden by the isospin symmetry because of the vanishing Clebsch– Gordan coefficient, (*T* 010| *T* 0 ) = 0 (see Equation (11)).

Finally, isospin selection rules govern also nuclear reactions (see also, e.g., Refs. [9–11], for specific topics). Restricting ourselves to nuclear decays, only nucleon, two-nucleon and *α*-particle emission are mentioned here: for example, for isospin-allowed proton emission, the difference in isospin between the initial and final states is Δ*T* = 1/2; for two-proton emission, it is Δ*T* = 1; *α* emission should be consistent with Δ*T* = 0.

Observation of isospin-forbidden decay modes indicates explicit isospin-symmetry breaking and the presence of isospin mixing in nuclear states.

## *1.2. Isospin-Symmetry Breaking*

Although isospin symmetry proved to be quite a useful concept in nuclear and particle physics, which helps to simplify theoretical modeling of the nucleon–nucleon interaction and provides an efficient framework for the nuclear many-body problem, experimental evidence has been accumulated on the breaking of isospin symmetry.

First, it is known that isobaric multiplets are not degenerate. The differences in energy between states forming an isobaric multiplet are called *Coulomb displacement energies*, since the Coulomb interaction is the main contributor to the effect. Such splittings can be explained within *dynamical breaking* of isospin symmetry, as was pointed out in Section 1.1. However, observation of isospin-forbidden decays, i.e., decays which break isospin selection rules, indicates that isospin is not a good quantum number, and there is a certain amount of isospin mixing in nuclear states. To describe such phenomena, one must introduce an explicit breaking of isospin symmetry within a nuclear structure model. Development of microscopic approaches for an accurate description of isospin-symmetry breaking is important not only for understanding the structure and decay of proton-rich nuclei, but also for the evaluation of nuclear-structure corrections to weak processes in nuclei. Taking isospin-symmetry breaking into account may also help to improve our knowledge of certain reactions involving proton-rich nuclei, which are crucial for nuclear astrophysics.

At the nuclear level, isospin symmetry is broken mainly due to the Coulomb interaction among protons (a long-range component of the electromagnetic interaction between protons), and to a minor extent by the proton and neutron mass difference and the presence of the charge-dependent forces of nuclear origin (short-range). At the quark level, these causes can be rooted to the *u* and *d* quark mass difference and electromagnetic interactions between the quarks. The need for charge-dependent forces of nuclear origin was established long ago from the analysis of the nucleon–nucleon (*NN*) scattering data. For example, it is known that there are differences in the neutron–neutron (*ann*), proton–proton (*app*, with electromagnetic effects being subtracted) and neutron–proton (*anp*) <sup>1</sup>*S*<sup>0</sup> (a *T* = 1 channel) scattering lengths [12,13]. Namely, the difference of *ann* and *app*,

$$a\_{nn} - a\_{pp} = 1.6 \pm 0.6 \text{ fm} \,\text{/} \tag{13}$$

is a signature of *charge-symmetry breaking* of the strong *NN* force; and the even larger difference between *anp* and the average of *ann* and *app*,

$$\frac{1}{2}(a\_{nn} + a\_{pp}) - a\_{np} = 5.64 \pm 0.40 \text{ fm} \,\text{/s} \tag{14}$$

is known as the *charge-independence breaking* property.

Moreover, still long ago, Nolen and Schiffer [14] noticed that the Coulomb force alone cannot satisfactorily explain the binding energy differences in mirror nuclei if one requires the model to reproduce nuclear charge radii and vise versa (the so-called Nolen– Schiffer anomaly). The insufficiency of the two-body Coulomb interaction in reproduction of splittings of isobaric multiplets was also demonstrated in more refined shell-model

calculations (e.g., Refs. [15–17]). Many-body approaches must therefore, take into account short-range charge-dependent components of the nucleon–nucleon interaction.

Henley and Miller [18] proposed to divide two-nucleon forces into four classes according to their isospin characters, namely,


If, as an example the two-body Coulomb interaction, acting between protons, is considered, one may notice that it comprises terms of classes I, II and III, as seen in Equation (5). It is important to note that although class II and class III forces commute with the two-nucleon isospin operator, such forces do violate the isospin symmetry in an *A*-nucleon system with *A* > 2.

Isospin-symmetry breaking two-nucleon forces have been constructed and explored in earlier meson-exchange models [12,19] and within the modern chiral effective field theory *χ*EFT) [13,20–22]. The details of various contributions from hadronic mass splittings and electromagnetic processes can be found in the above-given references. From *χ*EFT, the following hierarchy was deduced [20]: *VI* > *VI I* > *VIII* > *VIV*. In addition, chargedependent three-nucleon (3*N*) forces have been constructed within *χ*EFT see, e.g., the review [13] and references therein). Those may contribute to possible deviations of the IMME from its quadratic form, as discussed in Section 3.1 below.

Although charge-dependent realistic inter-nucleon interactions are frequently used in many-body calculations, in particular, in ab initio approaches, there have been few studies specifically focused on the degree of isospin-symmetry breaking. Nevertheless, ab initio Green's function Monte Carlo calculations with charge-dependent forces from the realistic Argonne *v*<sup>18</sup> *NN* + Illinois-7 3*N* potential supplemented by more refined chargedependent terms have been performed [23]. Quite good reproduction of the binding-energy differences in a few pairs of light mirror nuclei and the expected amount of isospin-mixing in 8Be were reported. A significant feature of those calculations is that they introduced and demonstrated the role of class IV forces. Charge-dependent *NN*+3*N* forces from *χ*EFT are used in state-of-the-art no-core shell model calculations for light nuclei [24,25], and the validity of isospin symmetry in electric quadrupole moments of mirror nuclei has been probed within the same theoretical approach in Ref. [26].

This review is devoted rather to the description of isospin-nonconserving phenomena in spectra and decays of heavier nuclei, for which a solution of the nuclear many-body problem needs an approach requiring effective charge-dependent interactions. Various theoretical frameworks aimed at a reliable description of isospin-symmetry breaking have been developed to deal with the problem. Among them are state-of-the-art shell-model calculations [15–17,27–35], including its no-core realization [36] and continuum-coupling extension [37], mean-field approaches and beyond (e.g., [38–47]) and others. Earlier comprehensive reviews on isospin symmetry and its breaking can be found in Refs. [9,48–50].

The present paper focuses rather on a particular theoretical approach to the problem, namely, on the nuclear shell model (e.g., see books [51–54]). Indeed, the shell model conserves all fundamental symmetries of atomic nuclei (such as angular momentum and particle number) and describes quite accurately individual states and transitions at low energies. This makes it an adequate approach for searching for tiny isospin-symmetry breaking effects. In the following sections, we highlight recent progress achieved by the isospin nonconserving shell model. A short summary of selected results has already been published in the proceedings of EuNPC2018 [55].

#### **2. Formalism**

The starting point of the shell model is a non-relativistic Hamiltonian for point-like nucleons containing nucleon kinetic energies and effective *NN* interactions (only two-body interactions are considered here):

$$
\hat{H} = \sum\_{k=1}^{A} \hat{T}\_{\text{kin}}(k) + \sum\_{k$$

By adding and subtracting a one-body spherically symmetric potential (e.g., a harmonicoscillator potential), one can rewrite the Hamiltonian as a sum of an independent-particle Hamiltonian (*H*ˆ 0) and a residual interaction (*V*ˆ ):

$$\hat{H} = \sum\_{k=1}^{A} \left[ \hat{\mathcal{I}}\_{\text{kin}}(k) + \hat{\mathcal{U}}(k) \right] + \left[ \sum\_{k$$

The eigenstates of *H*ˆ (*H*ˆ Ψ*<sup>m</sup>* = *Em*Ψ*m*) are searched for in terms of a complete orthonormal set of eigenfunctions of *H*ˆ <sup>0</sup> (*H*ˆ <sup>0</sup>Φ*<sup>m</sup>* = *E*0*m*Φ*m*):

$$\Psi\_m = \sum\_{m'} C\_{mm'} \Phi\_{m'} \dots$$

Using this expansion, the eigenproblem is reduced for *H*ˆ to the diagonalization of the Hamiltonian matrix, Φ*m*|*H*<sup>ˆ</sup> <sup>|</sup>Φ*m*, computed from single-particle energies of valencespace orbitals, *ε <sup>p</sup>*,*n*(*a*), and two-body matrix elements (TBMEs) of the residual interaction, *ab*; *JMTMT*|*V*<sup>ˆ</sup> <sup>|</sup>*cd*; *JMT MT* (*a*, *b*, *c*, *d* run over valence-space orbitals in a spherically symmetric mean field, i.e., *a* = (*nala ja*) and so on). As a result, one gets eigenvalues *Em* and the corresponding sets of expansion coefficients {*Cmm*}. If the nuclear Hamiltonian, which is rotational invariant, is also taken to be charge-independent (the proton and neutron single-particle energies are identical and TBMEs are independent from *MT* with *T* = *T* ), its eigenstates are characterized by the angular momentum and isospin quantum numbers (*JMTMT*), thereby forming degenerate spin (isospin) multiplets.

Since the model's space dimensions grow quickly as the number of particles increases, only for light nuclei can the shell model problem be solved for all nucleons considered in a model space comprised of many harmonic-oscillator shells. When using realistic internucleon interaction, the approach is referred to as an ab initio no-core shell model [24]. For heavier nuclei, the shell-model problem is formulated for valence nucleons only, occupying a model space consisting of one or two oscillator shells beyond a closed shell core. This restriction of the model space has been proved to be sufficient for low-energy nuclear structures. However, because of a severely truncated model space, one needs to derive a so-called *effective interaction*.

In this context, the isospin formalism helps to reduce the number of parameters. Nevertheless, construction of robust valence-space effective Hamiltonians remains a challenging and a long-standing problem of nuclear theory. Microscopic effective interactions have been constructed, for example, within the many-body perturbation theory, starting from the pioneering work in 60s [56,57] and continuing on into recent times (for reviews, see Refs. [58–60]). In spite of important advances, microscopic interactions are known to be less successful than more phenomenological parametrizations, based on the adjustment of TBMEs to selected data on nuclear spectra from a given model space. In particular, with two-nucleon forces only, the resulting effective interaction suffers from serious deficiencies in their monopole component [61]. This feature was ascribed to missing 3*N* forces. In addition, a number of theoretical issues in application of many-body perturbation theory to nuclear effective interaction problem have been raised regarding convergence of the expansion [62], which have not convincingly been answered yet.

In the last decade, new non-perturbative approaches to the construction of effective valence-space Hamiltonians have been put forward, based on unitary transformation techniques—the in-medium similarity-renormalization group approach (IMSRG) [60,63] and the Okubo–Lee–Suzuki transformations of no-core shell-model solutions [64,65]. In addition, similar ideas have been implemented within the coupled-cluster method [66–68]. Moreover, some of these approaches, including modern many-body perturbation theory [69, 70], have successfully incorporated three-nucleon forces in their frameworks, producing state-of-the-art microscopic effective valence-space interactions from first principles.

In spite of all these developments, phenomenological effective interactions still remain a benchmark. Therefore, let us start the discussion of isospin-nonconserving (INC) Hamiltonians from a phenomenological perspective.

## *2.1. Phenomenological Approaches*

Phenomenological effective Hamiltonians are typically isospin-conserving; therefore, the Coulomb contribution is usually evaluated and subtracted from the data before it is used in a fit. The resulting interactions are called realistic, and they can provide high accuracy in the description of nuclear excited states and transitions at low energies for a large set of nuclei (ideally, all nuclei) from a given model space. The most famous examples are the Cohen–Kurath Hamiltonians [71] in the *p* shell; the universal *sd* shell (USD) family of Hamiltonians [72,73], and Kuo-Brown modified KB3G [74] and GXPF1A [75] Hamiltonians in the *p f* shell.

An attractive option to construct an accurate INC Hamiltonian is thus to adopt a well-established charge-independent Hamiltonian as a lowest-order approximation and to add an INC term. The latter must contain the two-body Coulomb interaction and effective charge-dependent *NN* forces (*V*ˆ CD), at least of classes II and III (no class IV forces are discussed here, but eventually, the framework can be extended to include them as well). Such an operator is a sum of an isoscalar, an isovector and an isotensor term:

$$
\hat{\mathcal{V}}\_{\text{INC}} = \hat{\mathcal{V}}\_{\text{Coul}} + \hat{\mathcal{V}}\_{\text{CD}} = \sum\_{\lambda = 0, 1, 2} \hat{\mathcal{V}}\_{\text{INC}}^{(\lambda)}, \quad \text{where} \quad \begin{cases}
\hat{\mathcal{V}}\_{\text{INC}}^{(0)} = (\upsilon\_{pp} + \upsilon\_{nn} + \upsilon\_{np}^{T=1})/3, \\
\hat{\mathcal{V}}\_{\text{INC}}^{(1)} = \upsilon\_{pp} - \upsilon\_{nn}, \\
\hat{\mathcal{V}}\_{\text{INC}}^{(2)} = (\upsilon\_{pp} + \upsilon\_{nn})/2 - \upsilon\_{np}^{T=1}.
\end{cases}
$$

To describe the Coulomb effects of the core, an isovector one-body term is added which gives rise to the so-called *isovector single-particle energies*, *ε*˜(*a*)=*ε <sup>p</sup>*(*a*)−*εn*(*a*), where *a* runs over model-space orbitals. In lowest-order perturbation theory, the splitting of the isobaric multiplets is due to the expectation value of this operator; therefore, it is expressed by a quadratic polynomial in *MT*, similarly to Equation (7):

$$
\langle \Psi\_{TM\_T} | \hat{\mathcal{V}}\_{\text{INC}} | \Psi\_{TM\_T} \rangle = E^{(0)}(\eta, T) + E^{(1)}(\eta, T)M\_T + E^{(2)}(\eta, T) \left[ 3M\_T^2 - T(T + 1) \right].
$$

In order to find the best set of parameters of *V*ˆ INC and isovector single-particle energies *ε*˜*a*, one can perform a fit requiring that theoretical isovector and isotensor components allow one to reproduce experimentally deduced *b* and *c* IMME coefficients for a wide selection of lowest and excited isobaric multiplets with *T* = 1/2, 1, 3/2, .... This procedure was first proposed in Ref. [15] and was used in the later work related to the *sd*-shell [16,27] and *p f*-shell and heavier nuclei [28]. Among various possible forms of *V*ˆ CD, modelization of that term either by a *ρ*-exchange Yukawa-type potential (with a scaled meson mass) or by the *T* = 1 term of the isospin-conserving Hamiltonian in the isovector and isotensor channels resulted in similar quality fits [15,27]. At the same time, the use of the *π*-exchange potential was found to require much stronger renormalization of the two-body Coulomb force, and therefore, it was not retained.

Figure 1 shows the *b* coefficients for the lowest doublets and *c* coefficients for the lowest triplets obtained from such phenomenological interactions for *sd*-shell and *p f*-shell nuclei, in comparison with the experimental values. It is evident that the agreement between theory and experiment is remarkable. The root-mean-square (rms) deviations between theory and experiment represented in Figure 1 are 30 keV (95 keV) for *b* coefficients in the

*sd* (*p f*) shell and around 9 keV (25 keV) for the *c*-coefficients in the *sd* (*p f*) shell. One can observe that the description of the *p f*-shell *b* coefficients worsens towards the middle of the shell. By excluding data for *A* = 59, 61, 63, the rms deviation reduces to 55 keV. This problem seems to be linked to the difficulty in the description of nuclei from the upper part of the *p f* shell because of large dimensions involved, and may not be related to the form of the INC terms. Note also that more realistic forms of *V*ˆ CD did not help to improve the fit [16].

**Figure 1.** Experimental ("Exp") [5,76] and theoretical ("Theory") IMME *b* coefficients for the lowest doublets (**left**) and *c* coefficients for the lowest triplets (**right**) in the *sd* and *p f* shells. The *sd*shell results were quoted from Ref. [27], and *p f*-shell calculations were performed with GX1Acd interaction [77]. See text for details.

As seen in Figure 1, the shell model well reproduces both the general trends and the fine structure of *b* and *c* coefficients. The latter considers the staggering *c* coefficients as a function of *A*, as well visible in Figure 1 (right): the *c* coefficients in *A* = 4*n* + 2 multiplets are systematically larger than those in *A* = 4*n* (*n* being a positive integer). Similarly, the *b* coefficients in doublets and quartets form two families for *A* = 4*n* + 1 and *A* = 4*n* + 3, with opposite phases, however (for doublets, *b* coefficients are largest in *A* = 4*n* + 1 nuclei, and for quartets, they are largest for *A* = 4*n* + 3 nuclei). To amplify the effect, in Figure 2, the differences in *b* coefficients between *A* and *A* − 2 nuclei are plotted.

**Figure 2.** Experimental [5,76] (**left**) and theoretical (**right**) differences in IMME *b* coefficients (Δ*b*) for the ground-state, first-excited and second-excited natural-parity *T* = 1/2 multiplets in the *sd* and *p f* shells. The *sd*-shell results were obtained with the interaction from Ref. [27], and *p f*-shell calculations were performed with GX1Acd interaction [77].

The staggering was noticed long ago and explained by the interplay between the Coulomb force and the pairing TBMEs [78]. It should be visibly present in *b* coefficients of multiplets with half-integer *T* and *c* coefficients of multiplets with integer values of *T*. The same conclusions have been reached [79] within a simpler macroscopic approach supplemented by different proton and neutron average pairing gaps, which made it possible to grasp the main features of staggering.

Modern microscopic approaches [27,28,34,44,45] using realistic interactions well reproduce the effect. The main advantage of the shell-model type approaches is that they can describe *b* and *c* coefficients of excited states as well. Figures 2 and 3 show the differences among *b* coefficients, Δ*<sup>b</sup>* and *c* coefficients for the three lowest positive-parity multiplets in doublets and triplets, respectively. Interestingly, that the amplitudes of oscillations diminish with excitation energy. This hints that the pairing effect gradually weakens as systems become more and more excited.

**Figure 3.** Experimental [5,76] (**left**) and theoretical (**right**) IMME *c* coefficients for the lowest, firstexcited and second-excited *T* = 1 multiplets in the *sd* and *p f* shells. The *sd*-shell results were obtained with the interaction from Ref. [27], and *p f*-shell calculations were performed with GX1Acd interaction [77]. For *A* = 42, the data are given for *J<sup>π</sup>* = 0+, 2+, 4<sup>+</sup> states. See text for details.

The approach described above can rather well reproduce an extended set of *b* and *c* coefficients and provides an attractive tool with which to predict binding energies and states in mirror systems using a method of Coulomb energy differences, described in Section 3.1 below. At the same time, a few drawbacks exist—namely, that it (i) does not allow one to predict nuclear masses on purely theoretical grounds, (ii) does not account for the so-called Thomas–Ehrman shift and (iii) it does not provide enough accuracy in the description of the differences in excitation energies of analogue states, usually referred to as Coulomb energy differences.

Another strategy was put forward by Zuker, Lenzi and collaborators in a series of papers starting from [17] (see also Refs. [80–82] for a recent review). The idea consists in modeling charge-dependent forces of nuclear origin with a few TBMEs, adjusted to reproduce the differences in excitation energies of isobaric multiplets relative to the lowest in energy multiplet. Those quantities are known as mirror energy differences (MEDs) and triplet energy differences (TEDs) in *T* = 1 multiplets, and they are related to the differences in *b* or *c* coefficients between the lowest multiplet and an excited one. For example, for triplets,

$$\begin{array}{l} \text{MED}(f) = -2(b(f) - b\_0) \, , \\ \text{TED}(f) = 2(c(f) - c\_0) \, , \end{array}$$

where *b*<sup>0</sup> (*c*0) is a *b* (*c*) coefficient of the lowest triplet. Considered as a function of *J* along an excitation band (a pattern of excited states linked by pronounced electromagnetic transitions), MEDs and TEDs can bring pertinent information on nuclear structure effects. A vary accurate description has been achieved [17,80] of the *p f* shell by a phenomenological parameterization of various physical effects, such as changes in nuclear radius (or deformation) and electromagnetic corrections to the single-particle energies, with *V*ˆ CD being modeled by a few *J* = 0 TBMEs in isovector and isotensor channels.

In Ref. [28], it was shown that modelization of *V*CD by two *J* = 0, *T* = 1 TBMEs in the *f*7/2 orbital and theoretically calculated single-particle energies was sufficient to reproduce the staggering behavior of *b* and *c* coefficients. This may not be surprising, since we understand the staggering effect is due to the Coulomb contribution to the pairing-type matrix elements.

Later on, the approach was generalized to other model spaces using a more extended form of a charge-dependent term of nuclear origin as a number of TBMEs in Refs. [31,32]. MED and TED appear to be sensitive tools to unveil the structure of excited states, and in particular, TEDs and MEDs can shed light onto pair alignment process or on the shape evolution. Detailed study of the heavy *N* = *Z* region allowed researchers to understand co-existing shapes and other effects in *A* = 66, 70, 74, 78 (e.g., [29,83]).

Moreover, MEDs have been shown [84] to depend linearly on the difference between neutron and proton radii, known as "neutron skin", and that they strongly correlate with the *s*1/2-orbital occupation. In general, low-*l* orbitals, especially *s*1/2 orbitals, are characterized by an extended radius and play thus a special role in nuclear structure. In particular, it was noted that MEDs of states having higher occupation of *s*1/2 are unusually large. It turns out that states in proton-rich nuclei having high occupation of such an orbital experience a stronger shift with respect to their mirror states in neutron-rich partners. This is the essence of the so-called Thomas–Erhman effect [85,86]. Parameterizations of the charge-dependent forces mentioned above do not necessarily include this effect, which thus requires special care. In order to account for the Thomas–Ehrman shift, several approaches have been developed. For example, one can vary the energy of the proton *ε*(*s*1/2) single-particle orbital (e.g., Ref. [87]) or quench TBMEs which involve *s*1/2 orbitals [88]. Recently, a direct construction of TBMEs based on a simultaneous fit of isoscalar, isovector and isotensor terms has been undertaken, which lead to a few new types of USD interactions [34], aiming at consistent description of proton-rich and neutron-rich nuclei on similar grounds.
