# Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments

## Abstract

**:**

## 1. Introduction

#### 1.1. Isospin Symmetry in Nuclear Structure

- For $\Delta T=1$ transitions (${T}_{f}={T}_{i}\pm 1$), the (reduced) matrix elements of analogue transitions in mirror nuclei or between respective analogue states should be identical, since they are governed only by the isovector term.
- In transitions between the states of the same isospin (${T}_{i}={T}_{f}=T$), both isoscalar and isovector terms contribute, and the matrix element for analogue transitions within an isobaric multiplet exhibits a linear trend as a function of ${M}_{T}$:$$\begin{array}{c}{\displaystyle \langle {J}_{f}{M}_{f};T{M}_{T}|{\widehat{O}}_{LM}|{J}_{i}{M}_{i};T{M}_{T}\rangle =\langle {J}_{f}{M}_{f}|{\widehat{O}}_{LM}^{\left(0\right)}|{J}_{i}{M}_{i}\rangle}\\ {\displaystyle +\frac{{M}_{T}}{\sqrt{T(T+1)(2T+1)}}\langle {J}_{f}{M}_{f};T||{\widehat{O}}_{LM}^{\left(1\right)}||{J}_{i}{M}_{i};T\rangle \phantom{\rule{0.166667em}{0ex}}.}\end{array}$$
- Another specific rule can be established for electric dipole operator. In the lowest order of the long-wavelength approximation, the electric-dipole ($E1$) operator is an isovector operator:$$\widehat{O}\left(E1\right)=\sum _{k=1}^{A}\mathrm{e}\left(k\right)\overrightarrow{r}\left(k\right)=\sum _{k=1}^{A}\left(\frac{1}{2}-{\widehat{t}}_{3}\left(k\right)\right)\mathrm{e}\overrightarrow{r}\left(k\right)\phantom{\rule{0.166667em}{0ex}}.$$Hence, $E1$ transitions between the states of the same isospin (${T}_{i}={T}_{f}=T$) in $N=Z$ nuclei are forbidden by the isospin symmetry because of the vanishing Clebsch–Gordan coefficient, $(T\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0|\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}})=0$ (see Equation (11)).

#### 1.2. Isospin-Symmetry Breaking

- class I (${V}_{I}$) are charge-independent forces $\{1,\widehat{\mathbf{t}}\left(1\right)\xb7\widehat{\mathbf{t}}\left(2\right)\}$;
- class II (${V}_{II}$) are forces which break the charge independence, but preserve the charge symmetry of the two-nucleon system, $\left\{{\widehat{t}}_{3}\left(1\right){\widehat{t}}_{3}\left(2\right)\right\}$;
- class III (${V}_{III}$) are charge-symmetry breaking forces, which vanish in the neutron-proton system, $\{{\widehat{t}}_{3}\left(1\right)+{\widehat{t}}_{3}\left(2\right)\}$;
- class IV (${V}_{IV}$) are forces which do not conserve the isospin of the two-nucleon system: $\{\widehat{\mathbf{t}}\left(1\right)\times \widehat{\mathbf{t}}\left(2\right),{\widehat{t}}_{3}\left(1\right)-{\widehat{t}}_{3}\left(2\right)\}$.

## 2. Formalism

#### 2.1. Phenomenological Approaches

#### 2.2. Semi-Phenomenological Approaches

#### 2.3. Microscopic Approaches

## 3. Structure and Decay of Neutron-Deficient Nuclei

#### 3.1. IMME Coefficients for Masses and Excitation Spectra of Proton-Rich Nuclei

#### 3.2. Isospin-Forbidden Decays

#### 3.2.1. Isospin-Forbidden $\beta $-Decay

#### 3.2.2. Signatures of Isospin-Symmetry Breaking from Electromagnetic Transitions

#### 3.2.3. $\beta $-Delayed Proton, Diproton or $\alpha $ Emission

## 4. Theoretical Isospin-Symmetry Breaking Corrections to Weak Processes in Nuclei

#### 4.1. Superallowed Fermi $\beta $-Decay

#### 4.2. $\beta $-Decay between Mirror $T=1/2$ States

#### 4.3. Gamow–Teller Transitions in Mirror Nuclei

## 5. Astrophysical Applications

## 6. Conclusions and Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CD | charge-dependent |

CKM | Cabbibo-Kobayashi-Moskawa |

CNO | carbon-nitrogen-oxygen |

CVC | conserved vector current |

$E1$, $E2$ | electric-dipole, electric-quadrupole |

EFT | effective field theory |

F | Fermi |

GT | Gamow-Teller |

h.c. | hermitian congugate |

HF | Hartree–Fock |

IAS | isobaric analogue state |

IMME | isobaric-multiplet mass equation |

IMSRG | in-medium similarity-renormalization group |

INC | Isospin-nonconserving |

$M1$ | magnetic-dipole |

MED | mirror energy difference |

N${}^{3}$LO | next-to-next-to-next-to-leading |

$NN$ | nucleon–nucleon |

rms | root mean square |

TED | triplet energy difference |

TMBE | two-body matrix element |

V–A | vector–axial vector |

WS | Wood-Saxon |

USD | universal $sd$ shell |

$\chi $EFT | chiral effective field theory |

$3N$ | three-nucleon |

## References

- Heisenberg, W. Über den Bau der Atomkerne. I. Z. Phys.
**1932**, 77, 1–11. [Google Scholar] [CrossRef] - Hesenberg, W. On the structure of atomic nuclei. I. In Nuclear Forces; Brink, D.M., Ed.; Pergamon Press: Oxford, UK, 1965; p. 214. [Google Scholar]
- Wigner, E. On the consequences of the symmetry of the nuclear Hamiltonian on the spectroscopy of nuclei. Phys. Rev.
**1937**, 51, 106–119. [Google Scholar] [CrossRef] - Wigner, E. Isotopic spin—A quantum number for nuclei. In Proceedings of the Robert A. Welch Foundation Conference on Chemical Research; Milligan, W.O., Ed.; Welch Foundation: Houston, TX, USA, 1957; Volume 1, pp. 67–91. [Google Scholar]
- Lam, Y.H.; Blank, B.; Smirnova, N.A.; Antony, M.S.; Bueb, J. The isobaric multiplet mass equation for A≤71 revisited. At. Data Nucl. Data Tables
**2013**, 99, 680–703. [Google Scholar] [CrossRef] - MacCormick, M.; Audi, G. Evaluated experimental isobaric analogue states from T=1/2 to T=3 and associated IMME coefficients. Nucl. Phys. A
**2014**, 925, 61–95. [Google Scholar] [CrossRef] [Green Version] - Frank, A.; Jolie, A.; Van Isacker, P. Symmetries in Atomic Nuclei; Springer Science+Business Media, LLC: New York, NY, USA, 2009. [Google Scholar] [CrossRef] [Green Version]
- Warburton, E.K.; Weneser, J. The role of isospin in electromagnetic transitions. In Isospin in Nuclear Physics; Wilkinson, D.H., Ed.; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1969; pp. 152–228. [Google Scholar]
- Wilkinson, D.H. (Ed.) Isospin in Nuclear Physics; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1969. [Google Scholar]
- Harney, H.L.; Richter, A.; Weidenmüller, H.A. Breaking of isospin symmetry in compound-nucleus reactions. Rev. Mod. Phys.
**1986**, 58, 607–645. [Google Scholar] [CrossRef] - Fujita, Y.; Rubio, B.; Gelletly, W. Spin–isospin excitations probed by strong, weak and electro-magnetic interactions. Prog. Part. Nucl. Phys.
**2011**, 66, 549–606. [Google Scholar] [CrossRef] - Machleidt, R. The meson theory of nuclear forces and nuclear structure. In Advances in Nuclear Physics. Volume 19; Negele, J.W., Vogt, E., Eds.; Plenum Press: New York, NY, USA, 1989; Chapter 2. [Google Scholar] [CrossRef]
- Epelbaum, E.; Hammer, H.-W.; Meißner, U.-G. Modern theory of nuclear forces. Rev. Mod. Phys.
**2009**, 81, 1773–1825. [Google Scholar] [CrossRef] - Nolen, J.A.; Schiffer, J.P. Coulomb energies. Ann. Rev. Nucl. Sci.
**1969**, 19, 471–526. [Google Scholar] [CrossRef] - Ormand, W.E.; Brown, B.A. Empirical isospin-nonconserving Hamiltonians for shell-model calculations. Nucl. Phys. A
**1989**, 491, 1–23. [Google Scholar] [CrossRef] - Nakamura, S.; Muto, K.; Oda, T. Isospin-forbidden beta decays in ls0d-shell nuclei. Nucl. Phys. A
**1994**, 575, 1–45. [Google Scholar] [CrossRef] - Zuker, A.P.; Lenzi, S.M.; Martinez-Pinedo, G.; Poves, A. Isobaric multiplet yrast energies and isospin nonconserving forces. Phys. Rev. Lett.
**2002**, 89, 142502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Henley, E.M.; Miller, G.A. Meson theory of charge-dependent nuclear forces. In Mesons in Nuclei. Volume 1; Rho, M., Wilkinson, D.H., Eds.; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1979; pp. 405–434. [Google Scholar]
- Miller, G.A.; Nefkens, B.M.K.; Slaus, I. Charge symmetry, quarks and mesons. Phys. Rep.
**1990**, 194, 1–116. [Google Scholar] [CrossRef] - van Kolck, U.L. Soft Physics: Applications of Effective Chiral Lagrangians to Nuclear Physics and Quark Models. Ph.D. Thesis, The University of Texas at Austn, Austin, TX, USA, 1993. Available online: https://www.proquest.com/openview/b885fad2126b5b81a16dca7d226f854a/ (accessed on 7 March 2023).
- Epelbaum, E. Few-nucleon forces and systems in chiral effective field theory. Prog. Part. Nucl. Phys.
**2006**, 57, 654–741. [Google Scholar] [CrossRef] [Green Version] - Machleidt, R.; Entem, D.R. Chiral effective field theory and nuclear forces. Phys. Rep.
**2011**, 503, 024001. [Google Scholar] [CrossRef] [Green Version] - Wiringa, R.B.; Pastore, S.; Pieper, S.C.; Miller, G.A. Charge-symmetry breaking forces and isospin mixing in
^{8}Be. Phys. Rev. C**2013**, 88, 044333. [Google Scholar] [CrossRef] [Green Version] - Barrett, B.R.; Navrátil, P.; Vary, J.P. Ab initio no core shell model. Prog. Part. Nucl. Phys.
**2013**, 57, 654–741. [Google Scholar] [CrossRef] [Green Version] - Maris, P.; Epelbaum, E.; Furnstahl, R.J.; Golak, J.; Hebeler, K.; Hüther, T.; Kamada, H.; Krebs, H.; Meißner, U.-G.; Melendez, J.A.; et al. Light nuclei with semilocal momentum-space regularized chiral interactions up to third order. Phys. Rev. C
**2021**, 103, 054001. [Google Scholar] [CrossRef] - Caprio, M.A.; Fasano, P.J.; Maris, P.; McCoy, A.E. Quadrupole moments and proton-neutron structure in p-shell mirror nuclei. Phys. Rev. C
**2021**, 104, 034319. [Google Scholar] [CrossRef] - Lam, Y.H.; Smirnova, N.A.; Caurier, E. Isospin nonconservation in sd-shell nuclei. Phys. Rev. C
**2013**, 87, 054304. [Google Scholar] [CrossRef] [Green Version] - Kaneko, K.; Sun, Y.; Mizusaki, T.; Tazaki, S. Variation in displacement energies due to isospin-nonconserving forces. Phys. Rev. Lett.
**2013**, 110, 172505. [Google Scholar] [CrossRef] [Green Version] - Kaneko, K.; Sun, Y.; Mizusaki, T.; Tazaki, S. Isospin-nonconserving interaction in the T=1 analogue states of the mass-70 region. Phys. Rev. C
**2014**, 89, 031302. [Google Scholar] [CrossRef] [Green Version] - Holt, J.D.; Menendez, J.; Schwenk, A. Three-body forces and proton-rich nuclei. Phys. Rev. Lett.
**2013**, 110, 022502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bentley, M.; Lenzi, S.M.; Simpson, S.A.; Diget, C.A. Isospin-breaking interactions studied through mirror energy differences. Phys. Rev. C
**2015**, 92, 024310. [Google Scholar] [CrossRef] [Green Version] - Lenzi, S.M.; Bentley, M.; Lau, R.; Diget, C.A. Isospin-symmetry breaking corrections for the description of triplet energy differences. Phys. Rev. C
**2018**, 98, 054322. [Google Scholar] [CrossRef] [Green Version] - Ormand, W.E.; Brown, B.A.; Hjorth-Jensen, M. Realistic calculations for c coefficients of the isobaric mass multiplet equation in 1p0f shell nuclei. Phys. Rev. C
**2017**, 96, 024323. [Google Scholar] [CrossRef] [Green Version] - Magilligan, A.; Brown, B.A. New isospin-breaking “USD” Hamiltonians for the sd shell. Phys. Rev. C
**2020**, 101, 064312. [Google Scholar] [CrossRef] - Martin, M.S.; Stroberg, S.R.; Holt, J.D.; Leach, K.G. Testing isospin symmetry breaking in ab initio nuclear theory. Phys. Rev. C
**2021**, 104, 014324. [Google Scholar] [CrossRef] - Caurier, E.; Navrátil, P.; Ormand, W.E.; Vary, J.P. Ab initio shell model for A=10 nuclei. Phys. Rev. C
**2002**, 66, 024314. [Google Scholar] [CrossRef] - Michel, N.; Nazarewicz, W.; Płoszajczak, M. Isospin mixing and the continuum coupling in weakly bound nuclei. Phys. Rev. C
**2010**, 82, 044315. [Google Scholar] [CrossRef] - Sagawa, H.; Van Giai, N.; Suzuki, T. Effect of isospin mixing on superallowed Fermi β decay. Phys. Rev. C
**1996**, 53, 2163–2170. [Google Scholar] [CrossRef] - Liang, H.; Van Giai, N.; Meng, J. Isospin corrections for superallowed Fermi β decay in self-consistent relativistic random-phase approximation approaches. Phys. Rev. C
**2009**, 79, 064316. [Google Scholar] [CrossRef] - Petrovici, A. Isospin-symmetry breaking and shape coexistence in A≈70 analogs. Phys. Rev. C
**2015**, 91, 014302. [Google Scholar] [CrossRef] - Satuła, W.; Dobaczewski, J.; Nazarewicz, W.; Rafalski, M. Microscopic calculations of isospin-breaking corrections to superallowed beta decay. Phys. Rev. Lett.
**2011**, 106, 132502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Satula, W.; Dobaczewski, J.; Nazarewicz, W.; Rafalski, M. Isospin-breaking corrections to superallowed Fermi β decay in isospin- and angular-momentum-projected nuclear density functional theory. Phys. Rev. C
**2012**, 86, 054316. [Google Scholar] [CrossRef] [Green Version] - Satuła, W.; Baçzyk, P.; Dobaczewski, J.; Konieczka, M. No-core configuration-interaction model for the isospin- and angular-momentum-projected states. Phys. Rev. C
**2016**, 94, 024306. [Google Scholar] [CrossRef] [Green Version] - Baçzyk, P.; Dobaczewski, J.; Konieczka, M.; Nakatsukasa, T.; Sato, K.; Satula, W. Isospin-symmetry breaking in masses of N≈Z nuclei. Phys. Lett. B
**2018**, 778, 178–183. [Google Scholar] [CrossRef] - Baczyk, P.; Satula, W.; Dobaczewski, J.; Konieczka, M. Isobaric multiplet mass equation within nuclear density functional theory. J. Phys. G
**2019**, 46, 03LT01. [Google Scholar] [CrossRef] [Green Version] - Roca-Maza, X.; Colò, G.; Sagawa, H. Nuclear symmetry energy and the breaking of the isospin symmetry: How do they reconcile with each other? Phys. Rev. Lett.
**2018**, 120, 202501. [Google Scholar] [CrossRef] [Green Version] - Naito, T.; Colò, G.; Liang, H.; Roca-Maza, X.; Sagawa, H. Toward ab initio charge symmetry breaking in nuclear energy density functionals. Phys. Rev. C
**2022**, 105, L021304. [Google Scholar] [CrossRef] - Bertsch, G.F.; Mekjian, A. Isospin impurities in nuclei. Ann. Rev. Nucl. Sci.
**1972**, 22, 25–64. [Google Scholar] [CrossRef] - Raman, S.; Walkiewicz, T.A.; Behrens, H. Superallowed 0
^{+}→0^{+}and isospin-forbidden J^{π}→J^{π}Fermi transitions. At. Data Nucl. Data Tables**1975**, 16, 451–494. [Google Scholar] [CrossRef] - Auerbach, N. Coulomb effects in nuclear structure. Phys. Rep.
**1983**, 98, 273–341. [Google Scholar] [CrossRef] - Brussaard, P.J.; Glaudemans, P.W.M. Shell-Model Applications in Nuclear Spectroscopy; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1977. [Google Scholar]
- Heyde, K.L.G. The Nuclear Shell Model; CRC Press/Taylor & Francis Group: Boca Raton, FL, USA, 2004. [Google Scholar] [CrossRef]
- Suhonen, J. From Nucleons to Nucleus; Springer: Heidelberg/Berlin, Germany, 2007. [Google Scholar] [CrossRef] [Green Version]
- Caurier, E.; Martínez-Pinedo, G.; Nowacki, F.; Poves, A.; Zuker, A.P. The shell model as a unified view of nuclear structure. Rev. Mod. Phys.
**2005**, 77, 427–488. [Google Scholar] [CrossRef] [Green Version] - Smirnova, N.A. Isospin-symmetry breaking in nuclear structure. Nuovo Cim. C
**2019**, 42, 54. [Google Scholar] [CrossRef] - Bertsch, G.F. Role of core polarization in two-body interaction. Nucl. Phys.
**1965**, 74, 234–240. [Google Scholar] [CrossRef] - Kuo, T.T.S.; Brown, G.E. Structure of finite nuclei and the free nucleon-nucleon interaction. An application to
^{18}O and^{18}F. Nucl. Phys.**1966**, 85, 40–86. [Google Scholar] [CrossRef] - Hjorth-Jensen, M.; Kuo, T.T.S.; Osnes, E. Realitic effective interactions for nuclear systems. Phys. Rep.
**1995**, 261, 125–270. [Google Scholar] [CrossRef] - Coraggio, A.; Covello, A.; Gargano, A.; Itaco, N.; Kuo, T.T.S. Shell-model calculations and realistic effective interactions. Prog. Part. Nucl. Phys.
**2009**, 62, 135–182. [Google Scholar] [CrossRef] [Green Version] - Stroberg, S.R.; Hergert, H.; Bogner, S.; Holt, J.D. Nonempirical interactions for the nuclear shell model: An update. Ann. Rev. Nucl. Part. Sci.
**2019**, 69, 307–362. [Google Scholar] [CrossRef] [Green Version] - Poves, A.; Zuker, A.P. Theoretical spectroscopy and the fp shell. Phys. Rep.
**1981**, 70, 235–314. [Google Scholar] [CrossRef] - Barrett, B.R. Theoretical approaches to many-body perturbation theory and challenges. J. Phys. G: Nucl. Part. Phys.
**2005**, 31, S1349–S1355. [Google Scholar] [CrossRef] - Stroberg, S.R.; Calci, A.; Hergert, H.; Holt, J.D.; Bogner, S.; Roth, R.; Schwenk, A. Nucleus-dependent valence-space approach to nuclear structure. Phys. Rev. Lett.
**2017**, 118, 032502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dikmen, E.; Lisetskiy, A.F.; Barrett, B.R.; Maris, P.; Shirokov, A.M.; Vary, J.P. Ab initio effective interactions for sd-shell valence nucleons. Phys. Rev. C
**2015**, 91, 064301. [Google Scholar] [CrossRef] [Green Version] - Smirnova, N.A.; Barrett, B.R.; Kim, Y.; Shin, I.J.; Shirokov, A.M.; Dikmen, E.; Maris, P.; Vary, J.P. Effective interactions in the sd shell. Phys. Rev. C
**2019**, 100, 054329. [Google Scholar] [CrossRef] [Green Version] - Jansen, G.R.; Engel, J.; Hagen, G.; Navrátil, P.; Signoracci, A. Ab initio coupled-cluster effective interactions for the shell model: Application to neutron-rich oxygen and carbon isotopes. Phys. Rev. Lett.
**2014**, 113, 142502. [Google Scholar] [CrossRef] [Green Version] - Jansen, G.R.; Schuster, M.D.; Signoracci, A.; Hagen, G.; Navrátil, P. Open sd-shell nuclei from first principles. Phys. Rev. C
**2016**, 94, 011301. [Google Scholar] [CrossRef] [Green Version] - Sun, Z.H.; Morris, T.D.; Hagen, G.; Jansen, G.R.; Papenbrock, T. Shell-model coupled-cluster method for open-shell nuclei. Phys. Rev. C
**2018**, 98, 054320. [Google Scholar] [CrossRef] [Green Version] - Fukui, T.; De Angelis, L.; Ma, Y.Z.; Coraggio, A.; Gargano, A.; Itaco, N.; Xu, F. Realistic shell-model calculations for p-shell nuclei including contributions of a chiral three-body force. Phys. Rev. C
**2018**, 98, 044305. [Google Scholar] [CrossRef] [Green Version] - Ma, Y.Z.; Coraggio, A.; De Angelis, L.; Fukui, T.; Gargano, A.; Itaco, N.; Xu, F. Contribution of chiral three-body forces to the monopole component of the effective shell-model Hamiltonian. Phys. Rev. C
**2019**, 100, 034324. [Google Scholar] [CrossRef] [Green Version] - Cohen, S.; Kurath, D. Effective interactions for the 1p shell. Nucl. Phys.
**1965**, 73, 1–24. [Google Scholar] [CrossRef] - Wildenthal, B.H. Empirical strengths of spin operators in nuclei. Prog. Part. Nucl. Phys.
**1984**, 11, 5–51. [Google Scholar] [CrossRef] - Richter, W.A.; Brown, B.A. New “USD” Hamiltonians for the sd shell. Phys. Rev. C
**2006**, 85, 045806. [Google Scholar] [CrossRef] [Green Version] - Poves, A.; Sanchez-Solano, J.; Caurier, E.; Nowacki, F. Shell model study of the isobaric chains A=50, A=51 and A=52. Nucl. Phys. A
**2001**, 694, 157–198. [Google Scholar] [CrossRef] [Green Version] - Honma, M.; Otsuka, T.; Brown, B.A.; Mizusaki, T. New effective interaction for pf-shell nuclei and its implications for the stability of the N=Z=28 closed core. Phys. Rev. C
**2004**, 69, 034335. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.H.; Zhang, P.; Zhou, X.H.; Wang, M.; Litvinov, Yu.A.; Xu, H.S.; Xu, X.; Shuai, P.; Lam, Y.H.; Chen, R.J.; et al. Isochronous mass measurements of T_z=-1fp-shell nuclei from projectile fragmentation of
^{58}Ni. Phys. Rev. C**2018**, 98, 014319. [Google Scholar] [CrossRef] [Green Version] - Brown, B.A.; Rae, W.D.M. The shell-model code NuShellX. Nucl. Data Sheets
**2014**, 120, 115–118. [Google Scholar] [CrossRef] - Jänecke, J. Vector and tensor Coulomb energies. Phys. Rev. C
**1966**, 147, 735–742. [Google Scholar] [CrossRef] - Klochko, O.; Smirnova, N. A. Isobaric-multiplet mass equation in a macroscopic-microscopic approach. Phys. Rev. C
**2021**, 103, 024316. [Google Scholar] [CrossRef] - Bentley, M.A.; Lenzi, S.M. Coulomb energy differences between high-spin states in isobaric multiplets. Prog. Part. Nucl. Phys.
**2007**, 59, 497–561. [Google Scholar] [CrossRef] - Warner, D.D.; Van Isacker, P.; Bentley, M.A. The role of isospin symmetry in collective nuclear structure. Nat. Phys.
**2006**, 2, 311–318. [Google Scholar] [CrossRef] - Bentley, M.A. Excited states in isobaric multiplets—Experimental advances and the shell-model approach. Physics
**2022**, 4, 995–1011. [Google Scholar] [CrossRef] - Lenzi, S.M.; Poves, A.; Macchiavelli, A.O. Isospin symmetry breaking in the mirror pair
^{73}Sr -^{73}Br. Phys. Rev. C**2020**, 102, 031302. [Google Scholar] [CrossRef] - Boso, A.; Lenzi, S.M.; Recchia, F.; Bonnard, J.; Zuker, A.P.; Aydin, S.; Bentley, M.A.; Cederwall, B.; Clement, E.; de France, G.; et al. Neutron skin effects in mirror energy differences: The case of
^{23}Mg–^{23}Na. Phys. Rev. Lett.**2018**, 121, 032502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Thomas, R.G. An analysis of the energy levels of the mirror nuclei, C
^{13}and N^{13}. Phys. Rev.**1952**, 88, 1109–1125. [Google Scholar] [CrossRef] - Ehrman, J.B. On the displacement of corresponding energy levels of C
^{13}and N^{13}. Phys. Rev.**1951**, 81, 412–416. [Google Scholar] [CrossRef] - Longfellow, B.; Gade, A.; Brown, B.A.; Richter, W.A.; Bazin, D.; Bender, P.C.; Bowry, M.; Elman, B.; Lunderberg, B.E.; Weisshaar, D.; et al. Measurement of key resonances for the
^{24}Al(p,γ)^{25}Si reaction rate using in-beam γ-ray spectroscopy. Phys. Rev. C**2018**, 97, 054307. [Google Scholar] [CrossRef] [Green Version] - Cenxi, Y.; Qi, C.; Xu, F.; Suzuki, T.; Otsuka, T. Mirror energy difference and the structure of loosely bound proton-rich nuclei around A=20. Phys. Rev. C
**2014**, 89, 044327. [Google Scholar] [CrossRef] [Green Version] - Pape, A.; Antony, M.S. Masses of proton-rich T
_{z}<0 nuclei with isobaric mass equation. At. Data Nucl. Data Tables**1988**, 39, 201–203. [Google Scholar] [CrossRef] - Brown, B.A. Diproton decay of nuclei on the proton drip line. Phys. Rev. C
**1991**, 43, 1513. [Google Scholar] [CrossRef] [Green Version] - Ormand, W.E. Mapping the proton drip line up to A=70. Phys. Rev. C
**1997**, 55, 2407–2417. [Google Scholar] [CrossRef] [Green Version] - Brown, B.A.; Clement, R.R.C.; Schatz, H.; Volya, A.; Richter, W.A. Proton drip-line calculations and the rp process. Phys. Rev. C
**2002**, 65, 045802. [Google Scholar] [CrossRef] [Green Version] - Richter, W.A.; Brown, B.A.; Signoracci, A.; Wiescher, M. Properties of
^{26}Mg and^{26}Si in the sd shell model and the determination of the^{26}Al(p,γ)^{26}Si reaction rate. Phys. Rev. C**2011**, 83, 065803. [Google Scholar] [CrossRef] [Green Version] - Benenson, W.; Kashy, E. Isobaric quartests in nuclei. Rev. Mod. Phys.
**1979**, 51, 527–540. [Google Scholar] [CrossRef] - Zhang, Y.H.; Xu, H.S.; Litvinov, Yu.A.; Tu, X.L.; Yan, X.L.; Typel, S.; Blaum, K.; Wang, M.; Zhou, X.H.; Sun, Y.; et al. Mass measurements of the neutron-deficient
^{41}Ti,^{45}Cr,^{49}Fe, and^{53}Ni nuclides: First test of the isobaric multiplet mass equation in fp-shell nuclei. Phys. Rev. Lett**2012**, 109, 102501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brodeur, M.; Kwiatkowski, A.A.; Drozdowski, O.M.; Andreoiu, C.; Burdette, D.; Chaudhuri, A.; Chowdhury, U.; Gallant, A.T.; Grossheim, A.; Gwinner, G.; et al. Precision mass measurements of magnesium isotopes and implications for the validity of the isobaric mass multiplet equation. Phys. Rev. C
**2017**, 96, 034316. [Google Scholar] [CrossRef] [Green Version] - Bertsch, G.F.; Kahana, S. Tz3 term in the isobaric multiplet equation. Phys. Lett. B
**1970**, 33, 193–194. [Google Scholar] [CrossRef] - Signoracci, A.; Brown, B.A. Effects of isospin mixing in the A=32 quintet. Phys. Rev. C
**2011**, 84, 031301. [Google Scholar] [CrossRef] [Green Version] - Kamil, M.; Triambak, S.; Magilligan, A.; García, A.; Brown, B.A.; Adsley, P.; Bildstein, V.; Burbadge, C.; Diaz Varela, A.; Faestermann, T.; et al. Isospin mixing and the cubic isobaric multiplet mass equation in the lowest T=2, A=32 quintet. Phys. Rev. C
**2022**, 104, L061303. [Google Scholar] [CrossRef] - Barker, F.C. Intermediate coupling shell-model calculations for light nuclei. Nucl. Phys.
**1966**, 83, 418–448. [Google Scholar] [CrossRef] - Smirnova, N.A.; Blank, B.; Brown, B.A.; Richter, W.A.; Benouaret, N.; Lam, Y.H. Theoretical analysis of isospin mixing with the β decay of
^{56}Zn. Phys. Rev. C**2016**, 93, 044305. [Google Scholar] [CrossRef] [Green Version] - Hoyle, C.D.; Adelberger, E.G.; Blair, J.S.; Snover, K.A.; Swanson, H.E.; Von Lintig, R.D. Isospin mixing in
^{24}Mg. Phys. Rev. C**1983**, 27, 1244–1259. [Google Scholar] [CrossRef] - Orrigo, S.; Rubio, B.; Fujita, Y.; Blank, B.; Gelletly, W.; Agramunt, J.; Algora, A.; Ascher, P.; Bilgier, B.; Cáceres, L.; et al. Observation of the β-delayed γ-proton decay of
^{56}Zn and its impact on the Gamow-Teller strength evaluation. Phys. Rev. Lett.**2014**, 112, 222501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hagberg, E.; Koslowsky, V.T.; Hardy, J.C.; Towner, I.S.; Hykawy, J.G.; Savard, G.; Shinozuka, T. Tests of isospin mixing corrections in superallowed 0
^{+}→0^{+}β decays. Phys. Rev. Lett**1994**, 73, 396–399. [Google Scholar] [CrossRef] [PubMed] - MacLean, A.D.; Laffoley, A.T.; Svensson, C.E.; Ball, G.C.; Leslie, J.T.; Andreoiu, C.; Babu, A.; Bhattacharjee, S.S.; Bidaman, H.; Bildstein, V.; et al. High-precision branching ratio measurement and spin assignment implications for
^{62}Ga superallowed β decay. Phys. Rev. C**2020**, 102, 054325. [Google Scholar] [CrossRef] - Schuurmans, P.; Camps, J.; Phalet, T.; Severijns, N.; Vereecke, B.; Versyck, S. Isospin mixing in the ground state of
^{52}Mn. Nucl. Phys. A**2000**, 672, 89–98. [Google Scholar] [CrossRef] - Severijns, N.; Vénos, D.; Schuurmans, P.; Phalet, T.; Honusek, M.; Srnka, D.; Vereecke, B.; Versyck, S.; Zákoucký, D.; Köster, U.; et al. Isospin mixing in the T=5/2 ground state of
^{71}As. Phys. Rev. C**2005**, 71, 064310. [Google Scholar] [CrossRef] [Green Version] - Farnea, E.; de Angelis, G.; Gadea, A.; Bizzeti, P.G.; Dewald, A.; Eberth, J.; Algora, A.; Axiotis, M.; Bazzacco, D.; Bizzeti-Sona, A.M.; et al. Isospin mixing in the N=Z nucleus
^{64}Ge. Phys. Lett. B**2004**, 551, 56–62. [Google Scholar] [CrossRef] - Bizzeti, P.G.; de Angelis, G.; Lenzi, S.M.; Orlandi, R. Isospin symmetry violation in mirror E1 transitions: Coherent contributions from the giant isovector monopole resonance in the
^{67}As–^{67}Se doublet. Phys. Rev. C**2012**, 86, 044311. [Google Scholar] [CrossRef] [Green Version] - Lisetskiy, A.F.; Schmidt, A.; Schneider, I.; Friessner, C.; Pietralla, N.; von Brentano, P. Isospin mixing between low-lying states of the odd-odd N=Z nucleus
^{54}Co. Phys. Rev. Lett.**2002**, 89, 012502. [Google Scholar] [CrossRef] - Prados-Estevez, F.M.; Bruce, A.M.; Taylor, M.J.; Amro, H.; Beausang, C.W.; Casten, R.F.; Ressler, J.J.; Barton, C.J.; Chandler, C.; Hammond, G. Isospin purity of T=1 states in the A=38 nuclei studied via lifetime measurements in
^{38}K. Phys. Rev. C**2007**, 75, 014309. [Google Scholar] [CrossRef] [Green Version] - Giles, M.M.; Nara Singh, B.S.; Barber, L.; Cullen, D.M.; Mallaburn, M.J.; Beckers, M.; Blazhev, A.; Braunroth, T.; Dewald, A.; Fransen, C.; et al. Probing isospin symmetry in the (
^{50}Fe,^{50}Mn,^{50}Cr) isobaric triplet via electromagnetic transition rates. Phys. Rev. C**2019**, 99, 044317. [Google Scholar] [CrossRef] [Green Version] - Bizzeti, P.G.; Bizetti-Sona, A.M.; Cambi, A.; Mandò, M.; Maurenzig, P.R.; Signorini, C. Strength of analogue E2 transitions in
^{30}Si and^{30}P. Lett. Nouvo Cim.**1969**, 16, 775. [Google Scholar] [CrossRef] - Ekman, J.; Rudolph, D.; Fahlander, C.; Zuker, A.P.; Bentley, M.A.; Lenzi, S.M.; Andreoiu, C.; Axiotis, M.; de Angelis, G.; Farnea, E.; et al. Unusual isospin-breaking and isospin-mixing effects in the A=35 mirror nuclei. Phys. Rev. Lett.
**2004**, 92, 132502. [Google Scholar] [CrossRef] [PubMed] - Pattabiraman, N.S.; Jenkins, D.G.; Bentley, M.A.; Wadsworth, R.; Lister, C.J.; Carpenter, M.P.; Janssens, R.V.F.; Khoo, T.L.; Lauritsen, T.; Seweryniak, D.; et al. Analog E1 transitions and isospin mixing. Phys. Rev. C
**2008**, 78, 024301. [Google Scholar] [CrossRef] [Green Version] - von Neumann-Cosel, P.; Gräf, H.-D.; Krämer, U.; Richter, A.; Spamer, E. Electroexcitation of isoscalar and isovector magnetic dipole transitions in
^{12}C and isospin mixing. Nucl. Phys. A**2000**, 669, 3–13. [Google Scholar] [CrossRef] - Corsi, A.; Wieland, O.; Barlini, S.; Bracco, A.; Camera, F.; Kravchuk, V.L.; Baiocco, G.; Bardelli, L.; Benzoni, G.; Bini, M.; et al. Measurement of isospin mixing at a finite temperature in
^{80}Zr via giant dipole resonance decay. Phys. Rev. C**2011**, 84, 041304. [Google Scholar] [CrossRef] [Green Version] - Ceruti, S.; Camera, F.; Bracco, A.; Avigo, R.; Benzoni, G.; Blasi, N.; Bocchi, G.; Bottoni, S.; Brambilla, S.; Crespi, F.C.L.; et al. Isospin mixing in
^{80}Zr: From finite to zero temperature. Phys. Rev. Lett.**2016**, 115, 222502. [Google Scholar] [CrossRef] [Green Version] - Gosta, G.; Mentana, A.; Camera, F.; Bracco, A.; Ceruti, S.; Benzoni, G.; Blasi, N.; Brambilla, S.; Capra, S.; Crespi, F.C.L.; et al. Probing isospin mixing with the giant dipole resonance in the
^{60}Zn compound nucleus. Phys. Rev. C**2021**, 103, L041302. [Google Scholar] [CrossRef] - Brown, B.A. Isospin-forbidden β-delayed proton emission. Phys. Rev. Lett.
**1990**, 65, 2753–2756. [Google Scholar] [CrossRef] - Dossat, C.; Adimi, F.; Aksouh, F.; Becker, F.; Bey, A.; Blank, B.; Borcea, C.; Borcea, R.; Boston, A.; Caamano, M.; et al. The decay of proton-rich nuclei in the mass A=36-56 region. Nucl. Phys. A
**2005**, 792, 18–86. [Google Scholar] [CrossRef] - Blank, B.; Borge, M.J.G. Nuclear structure at the proton drip line: Advances with nuclear decay studies. Prog. Part. Nucl. Phys.
**2008**, 60, 403–483. [Google Scholar] [CrossRef] - Ormand, W.E.; Brown, B.A. Isospin-forbidden proton and neutron emission in 1s-0d shell nuclei. Phys. Lett. B
**1986**, 174, 128–132. [Google Scholar] [CrossRef] - Smirnova, N.A.; Blank, B.; Richter, W.A.; Brown, B.A.; Benouaret, N.; Lam, Y.H. Isospin mixing from β-delayed proton emission. Phys. Rev. C
**2017**, 95, 054301. [Google Scholar] [CrossRef] [Green Version] - Saxena, M.; Ong, W.-J.; Meisel, A.; Hoff, D.E.M.; Smirnova, N.; Bender, P.C.; Burcher, S.P.; Carpenter, M.P.; Carroll, J.J.; Chester, A.; et al.
^{57}Zn β-delayed proton emission establishes the^{56}Ni rp-process waiting point bypass. Phys. Lett. B**2022**, 829, 137059. [Google Scholar] [CrossRef] - Towner, I.S.; Hardy, J.C. Currents and their couplings in the weak sector of the Standard Model. In Symmetries and Fundamental Interactions in Nuclei; Henley, E.M., Haxton, W.C., Eds.; World Scientific: Singapore, 1995; pp. 183–249. [Google Scholar] [CrossRef] [Green Version]
- Severijns, N.; Beck, M.; Naviliat-Cuncic, O. Tests of the standard electroweak model in nuclear beta decay. Rev. Mod. Phys.
**2006**, 78, 991–1040. [Google Scholar] [CrossRef] [Green Version] - González-Alonso, M.; Naviliat-Cuncic, O.; Severijns, N. New physics searches in nuclear and neutron β-decay. Prog. Part. Nucl. Phys.
**2019**, 104, 165–223. [Google Scholar] [CrossRef] [Green Version] - Towner, I.S.; Hardy, J.C. The evaluation of V
_{ud}and its impact on the unitarity of the Cabibbo–Kobayashi– Maskawa quark-mixing matrix. Rep. Prog. Phys.**2010**, 73, 046301. [Google Scholar] [CrossRef] [Green Version] - Hardy, J.C.; Towner, I.S. Superallowed 0
^{+}→0^{+}nuclear β decays: 2020 critical survey, with implications for V_{ud}and CKM unitarity. Phys. Rev.**2020**, 102, 045501. [Google Scholar] [CrossRef] - Seng, C.-Y.; Gorchtein, M.; Patel, H.H.; Ramsey-Musolf, M.J. Reduced Hadronic Uncertainty in the Determination of V
_{ud}. Phys. Rev. Lett.**2018**, 121, 241804. [Google Scholar] [CrossRef] [Green Version] - Ormand, W.E.; Brown, B.A. Isospin-mixing corrections for fp-shell Fermi transitions. Phys. Rev. C
**1995**, 52, 2455–2460. [Google Scholar] [CrossRef] [Green Version] - Damgaard, J. Corrections to the ft-values of 0
^{+}→0^{+}superallowed β-decays. Nucl. Phys. A**1969**, 130, 233–240. [Google Scholar] [CrossRef] - Auerbach, N. Coulomb corrections to superallowed β decay in nuclei. Phys. Rev. C
**2009**, 79, 035502. [Google Scholar] [CrossRef] [Green Version] - Xayavong, L.; Smirnova, N.A. Radial overlap correction to superallowed 0
^{+}→0^{+}β decay reexamined. Phys. Rev. C**2018**, 97, 024324. [Google Scholar] [CrossRef] [Green Version] - Miller, G.A.; Schwenk, A. Isospin-symmetry-breaking corrections to superallowed Fermi β decay. Formalism and schematic models. Phys. Rev. C
**2008**, 78, 035501. [Google Scholar] [CrossRef] [Green Version] - Miller, G.A.; Schwenk, A. Isospin-symmetry-breaking corrections to superallowed Fermi β decay: Radial excitations. Phys. Rev. C
**2009**, 80, 064319. [Google Scholar] [CrossRef] [Green Version] - Towner, I.S.; Hardy, J.C. Improved calculations of isospin-symmetry breaking corrections to superallowed Fermi β decay. Phys. Rev. C
**2008**, 77, 025501. [Google Scholar] [CrossRef] [Green Version] - Hardy, J.C.; Towner, I.S. Superallowed 0
^{+}→0^{+}nuclear β decays: 2014 critical survey, with precise results for V_{ud}and CKM unitarity. Phys. Rev.**2015**, 91, 025501. [Google Scholar] [CrossRef] [Green Version] - Ormand, W.E.; Brown, B.A. Corrections to the Fermi matrix element for superallowed β decay. Phys. Rev. Lett.
**1989**, 62, 866–869. [Google Scholar] [CrossRef] [Green Version] - Towner, I.S.; Hardy, J.C. Comparative tests of isospin-symmetry breaking corrections to superallowed 0
^{+}→0^{+}nuclear β decay. Phys. Rev. C**2010**, 82, 065501. [Google Scholar] [CrossRef] [Green Version] - Ormand, W.E.; Brown, B.A. Calculated isospin-mixing corrections to Fermi β-decays in 1s0d-shell nuclei with emphasis on A=34. Nucl. Phys. A
**1985**, 440, 274–300. [Google Scholar] [CrossRef] - Xayavong, L.; Smirnova, N.; Bender, M.; Bennaceur, K. Shell-model calculation of isospin-symmetry breaking correction to super-allowed Fermi beta decay. Acta Phys. Pol. B. Proc. Supp.
**2017**, 10, 285–290. [Google Scholar] [CrossRef] [Green Version] - Xayavong, L.; Smirnova, N.A. Radial overlap correction to superallowed 0
^{+}→0^{+}nuclear β decays using the shell model with Hartree-Fock radial wave functions. Phys. Rev. C**2022**, 105, 044308. [Google Scholar] [CrossRef] - Naviliat-Cuncic, O.; Severijns, N. Test of the conserved vector current hypothesis in T=1/2 mirror transitions and new determination of V
_{ud}. Phys. Rev. Lett.**2009**, 102, 142302. [Google Scholar] [CrossRef] [Green Version] - Towner, I.S. Mirror asymmetry in allowed Gamow-Teller β-decay. Nucl. Phys. A
**1973**, 216, 589–602. [Google Scholar] [CrossRef] - Smirnova, N.A.; Volpe, M.C. On the asymmetry of Gamow-Teller β-decay rates in mirror nuclei in relation with second-class currents. Nucl. Phys. A
**2003**, 714, 441–462. [Google Scholar] [CrossRef] [Green Version] - Grenacs, L. Induced weak currents in nuclei. Ann. Rev. Nucl. Part. Sci.
**1985**, 35, 455–499. [Google Scholar] [CrossRef] - Minamisono, K.; Nagatomo, T.; Matsuta, K.; Levy, C.D.P.; Tagishi, Y.; Ogura, M.; Yamaguchi, M.; Ota, H.; Behr, J.A.; Jackson, K.P.; et al. Low-energy test of second-class current in β decays of spin-aligned
^{20}F and^{20}Na. Phys. Rev. C**2011**, 84, 055501. [Google Scholar] [CrossRef] [Green Version] - Langanke, K.; Martinez-Pinedo, G. Nuclear weak-interaction processes in stars. Rev. Mod. Phys.
**2003**, 75, 812–862. [Google Scholar] [CrossRef] [Green Version] - Jose, J.; Hernanz, M.; Iliadis, C. Nucleosynthesis in classical novae. Nucl. Phys. A
**2006**, 777, 550–578. [Google Scholar] [CrossRef] [Green Version] - Wallace, R.K.; Woosley, S.E. Explosive hydrogen burning. Astrophys. J. Supp. Ser.
**1981**, 45, 389–420. [Google Scholar] [CrossRef] - Schatz, H.; Aprahamian, A.; Görres, J.; Wiescher, M.; Rauscher, T.; Rembges, J.F.; Thielemann, F.K.; Pfeiffer, B.; Möller, P.; Kratz, K.-L.; et al. rp-process nucleosynthesis at extreme temperature and density conditions. Phys. Rep.
**1998**, 294, 167–263. [Google Scholar] [CrossRef] - Fowler, W.A.; Hoyle, F. Neutrino processes and pair formation in massive stars and supernovae. Astrophys. J. Supp.
**1964**, 9, 201–319. [Google Scholar] [CrossRef] - Herndl, H.; Görres, J.; Wiescher, M.; Brown, B.A.; Van Wormer, L. Proton capture reaction rates in the rp process. Phys. Rev. C
**1995**, 52, 1078–1094. [Google Scholar] [CrossRef] [Green Version] - Fisker, J.L.; Barnard, V.; Görres, J.; Langanke, K.; Martinez-Pinedo, G.; Wiescher, M. Shell-model based reaction rates for rp-process nuclei in the mass range A=44-63. At. Data Nucl. Data Tables
**2001**, 79, 241–292. [Google Scholar] [CrossRef] - Richter, W.A.; Brown, B.A. Shell-model studies of the rp reaction
^{35}Ar(p,γ)^{36}K. Phys. Rev. C**2012**, 85, 045806. [Google Scholar] [CrossRef] [Green Version] - Lam, Y.H.; Herger, A.; Lu, N.; Jacobs, A.M.; Smirnova, N.A.; Kurtukian-Nieto, T.; Johnston, T.; Kubono, S. The regulated NiCu cycles with the new
^{57}Cu(p,γ)^{58}Zn reaction rate and its influence on type I X-ray bursts: The GS 1826-24 clocked burster. Astrophys. J.**2022**, 929, 73–88. [Google Scholar] [CrossRef] - Brown, B.A.; Richter, W.A.; Wrede, C. Shell-model studies of the astrophysical rapid-proton-capture reaction
^{30}P(p,γ )^{31}S. Phys. Rev. C**2014**, 89, 062801. [Google Scholar] [CrossRef] - Richter, W.A.; Brown, B.A.; Longland, R.; Wrede, C.; Denissenkov, P.; Fry, C.; Herwig, F.; Kurtulgil, D.; Pignatari, M.; Reifarth, R. Shell-model studies of the astrophysical rp-process reactions
^{34}S(p,γ)^{35}Cl and^{34g,m}Cl(p,γ)^{35}Ar. Phys. Rev. C**2020**, 102, 025801. [Google Scholar] [CrossRef]

**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 $pf$ shells. The $sd$-shell results were quoted from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77]. See text for details.

**Figure 2.**Experimental [5,76] (

**left**) and theoretical (

**right**) differences in IMME b coefficients (${\Delta}_{b}$) for the ground-state, first-excited and second-excited natural-parity $T=1/2$ multiplets in the $sd$ and $pf$ shells. The $sd$-shell results were obtained with the interaction from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77].

**Figure 3.**Experimental [5,76] (

**left**) and theoretical (

**right**) IMME c coefficients for the lowest, first-excited and second-excited $T=1$ multiplets in the $sd$ and $pf$ shells. The $sd$-shell results were obtained with the interaction from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77]. For $A=42$, the data are given for ${J}^{\pi}={0}^{+},{2}^{+},{4}^{+}$ states. See text for details.

**Figure 4.**Schematic picture of $\beta $-delayed p, $\gamma $, $2p$ and $\alpha $ emission from an IAS. See text for details.

**Figure 6.**Schematic picture of the Fermi strength distribution in the daughter nucleus due to the isospin-symmetry breaking effects, as can be viewed from the shell-model’s perspective.

**Left**: depletion of the Fermi strength from an IAS because of non-analogue transitions.

**Right**: insertion of the intermediate states to better constrain the radial part of the single-particle wave functions.

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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

**AMA Style**

Smirnova NA.
Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments. *Physics*. 2023; 5(2):352-380.
https://doi.org/10.3390/physics5020026

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2023. "Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments" *Physics* 5, no. 2: 352-380.
https://doi.org/10.3390/physics5020026