- Open Access
Quadrupole strength in isobaric triplets
Phys. Rev. C 112, 064311 – Published 8 December, 2025
DOI: https://doi.org/10.1103/5wkc-7gmf
Abstract
The dependence of the matrix elements on isospin projection is linked to the conservation of the isospin symmetry. To study this conjecture, we calculated the rates for the even-even mirror nuclei with 42 98 within nuclear density functional theory, employing the generalized Bohr Hamiltonian, and carrying out angular momentum projection. We demonstrated that collective effects are crucial for describing experimental data near the line without invoking explicit beyond-Coulomb isospin symmetry-breaking corrections. We also determined the values for odd-odd nuclei and in doubly blocked configurations. We discussed the requirements for accurately describing isobaric analog states and emphasized how current theoretical results should be interpreted within the study of isospin symmetry across isospin triplets.
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References (68)
- R. D. O. Llewellyn, M. A. Bentley, R. Wadsworth, H. Iwasaki, J. Dobaczewski, G. de Angelis, J. Ash, D. Bazin, P. C. Bender, B. Cederwall, B. P. Crider, M. Doncel, R. Elder, B. Elman, A. Gade, M. Grinder, T. Haylett, D. G. Jenkins, I. Y. Lee, B. Longfellow et al., Establishing the maximum collectivity in highly deformed nuclei, Phys. Rev. Lett. 124, 152501 (2020).
- S. Lenzi and M. Bentley, Test of isospin symmetryisospin symmetry along the line, in The Euroschool Lectures on Physics with Exotic Beams, Vol. III, edited by J. Al-Khalili and E. Roeckl (Springer, Berlin/Heidelberg, 2009), pp. 57–98.
- W. Satuła and W. Nazarewicz, Isospin effects in nuclei in extended density functional theory, Phys. Scr. 91, 023013 (2016).
- M. A. Bentley, Excited states in isobaric multiplets—Experimental advances and the Shell-model approach, Physics 4, 995 (2022).
- N. A. Smirnova, Isospin-symmetry breaking within the nuclear Shell model: Present status and developments, Physics 5, 352 (2023).
- K. Wimmer, W. Korten, P. Doornenbal, T. Arici, P. Aguilera, A. Algora, T. Ando, H. Baba, B. Blank, A. Boso, S. Chen, A. Corsi, P. Davies, G. de Angelis, G. de France, J.-P. Delaroche, D. T. Doherty, J. Gerl, R. Gernhäuser, M. Girod et al., Shape changes in the mirror nuclei and , Phys. Rev. Lett. 126, 072501 (2021).
- M. M. Giles, B. S. Nara Singh, L. Barber, D. M. Cullen, M. J. Mallaburn, M. Beckers, A. Blazhev, T. Braunroth, A. Dewald, C. Fransen, A. Goldkuhle, J. Jolie, F. Mammes, C. Müller-Gatermann, D. Wölk, K. O. Zell, S. M. Lenzi, and A. Poves, Probing isospin symmetry in the isobaric triplet via electromagnetic transition rates, Phys. Rev. C 99, 044317 (2019).
- A. Boso, S. Milne, M. Bentley, F. Recchia, S. Lenzi, D. Rudolph, M. Labiche, X. Pereira-Lopez, S. Afara, F. Ameil, T. Arici, S. Aydin, M. Axiotis, D. Barrientos, G. Benzoni, B. Birkenbach, A. Boston, H. Boston, P. Boutachkov, A. Bracco et al., Isospin dependence of electromagnetic transition strengths among an isobaric triplet, Phys. Lett. B 797, 134835 (2019).
- G. L. Zimba, P. Ruotsalainen, G. De Gregorio, G. de Angelis, J. Sarén, J. Uusitalo, K. Auranen, A. D. Briscoe, Z. Ge, T. Grahn, P. T. Greenlees, A. Illana, D. G. Jenkins, H. Joukainen, R. Julin, H. Jutila, A. Kankainen, J. Louko, M. Luoma, J. Ojala et al., Isospin symmetry breaking in the triplet, Phys. Rev. C 110, 024314 (2024).
- G. L. Zimba, P. Ruotsalainen, D. G. Jenkins, W. Satuła, J. Uusitalo, R. Wadsworth, X. P. Lopez, K. Auranen, A. D. Briscoe, B. Cederwall, S. Chen, G. de Angelis, M. Doncel, A. Ertoprak, T. Grahn, P. T. Greenlees, A. Illana, H. Joukainen, R. Julin, H. Jutila et al., First identification of excited states in and implications for isospin nonconserving forces in nuclei, Phys. Rev. Lett. 134, 022502 (2025).
- E. K. Warburton and J. Weneser, in Isospin in Nuclear Physics edited by D. H. Wilkinson (North-Holland, Amsterdam, 1969), Chap. 5.
- P.-G. Reinhard and W. Nazarewicz, Nuclear charge densities in spherical and deformed nuclei: Toward precise calculations of charge radii, Phys. Rev. C 103, 054310 (2021).
- A. M. Bernstein, V. R. Brown, and V. A. Madsen, Isospin decomposition of nuclear multipole matrix elements from decay rates of mirror transitions: Test of values obtained with hadronic probes, Phys. Rev. Lett. 42, 425 (1979).
- W. Satuła, J. Dobaczewski, W. Nazarewicz, and M. Rafalski, Microscopic calculations of isospin-breaking corrections to superallowed beta decay, Phys. Rev. Lett. 106, 132502 (2011).
- W. Satuła, J. Dobaczewski, W. Nazarewicz, and T. R. Werner, Isospin-breaking corrections to superallowed fermi decay in isospin- and angular-momentum-projected nuclear density functional theory, Phys. Rev. C 86, 054316 (2012).
- M. Anguiano, J. Egido, and L. Robledo, Coulomb exchange and pairing contributions in nuclear Hartree-–Fock-–Bogoliubov calculations with the Gogny force, Nucl. Phys. A 683, 227 (2001).
- G. F. Bertsch, C. A. Bertulani, W. Nazarewicz, N. Schunck, and M. V. Stoitsov, Odd-even mass differences from self-consistent mean field theory, Phys. Rev. C 79, 034306 (2009).
- S. T. Belyaev, Effect of pairing correlations on nuclear properties, Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd. 31, 11 (1958).
- A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II (W. A. Benjamin, Reading, MA, 1975).
- M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky, and C. Y. Wong, Funny hills: The shell-correction approach to nuclear shell effects and its applications to the fission process, Rev. Mod. Phys. 44, 320 (1972).
- P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, Berlin, 1980).
- S. Ćwiok, W. Nazarewicz, and W. Zych, The dependence of Coulomb displacement energy on deformation of a nucleus, Acta. Phys. Pol. B 11, 445 (1980).
- W. Satuła and R. Wyss, Microscopic structure of fundamental excitations in nuclei, Phys. Rev. Lett. 87, 052504 (2001).
- S. Głowacz, W. Satuła, and R. A. Wyss, Cranking in isospace, Eur. Phys. J. A 19, 33 (2004).
- D. J. Rowe and J. L. Wood, Fundamentals of Nuclear Models: Foundational Models (World Scientific Publishing Company, Singapore, 2010).
- P. Klüpfel, J. Erler, P.-G. Reinhard, and J. A. Maruhn, Systematics of collective correlation energies from self-consistent mean-field calculations, Eur. Phys. J. A 37, 343 (2008).
- L. Próchniak and S. Rohoziński, Quadrupole collective states within the Bohr collective Hamiltonian, J. Phys. G 36, 123101 (2009).
- M. Girod and B. Grammaticos, The zero-point energy correction and its effect on nuclear dynamics, Nucl. Phys. A 330, 40 (1979).
- A. Baran, J. Sheikh, J. Dobaczewski, W. Nazarewicz, and A. Staszczak, Quadrupole collective inertia in nuclear fission: Cranking approximation, Phys. Rev. C 84, 054321 (2011).
- L. Prochniak, Microscopic description of collective properties of even-even Xe isotopes, Phys. Scr. 90, 114005 (2015).
- D. Muir, L. Próchniak, A. Pastore, and J. Dobaczewski, Structure of Krypton isotopes using the generalised Bohr Hamiltonian method, J. Phys.: Conf. Ser. 1643, 012147 (2020).
- J. Dobaczewski et al., Solution of the skyrme-Hartree-Fock-Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.:(VI) hfodd (v2.40h): A new version of the program, Comput. Phys. Commun. 180, 2361 (2009).
- J. Dobaczewski, P. Bączyk, P. Becker, M. Bender, K. Bennaceur, J. Bonnard, Y. Gao, A. Idini, M. Konieczka, M. Kortelainen, L. Próchniak, A. M. Romero, W. Satuła, Y. Shi, L. F. Yu, and T. R. Werner, Solution of universal nonrelativistic nuclear DFT equations in the Cartesian deformed harmonic-oscillator basis. (IX) hfodd (v3.06h): A new version of the program, J. Phys. G: Nucl. Part. Phys. 48, 102001 (2021).
- J. Dobaczewski et al., code hfodd (unpublished).
- D. Vretenar, A. Afanasjev, G. Lalazissis, and P. Ring, Relativistic Hartree-Bogoliubov theory: Static and dynamic aspects of exotic nuclear structure, Phys. Rep. 409, 101 (2005).
- S. A. Giuliani and L. M. Robledo, Non-perturbative collective inertias for fission: A comparative study, Phys. Lett. B 787, 134 (2018).
- W. Ryssens, G. Scamps, S. Goriely, and M. Bender, Skyrme-Hartree-Fock-Bogoliubov mass models on a 3D mesh: II. Time-reversal symmetry breaking, Eur. Phys. J. A 58, 246 (2022).
- X. Sun, J. Dobaczewski, M. Kortelainen, J. Sadhukhan, A. Sánchez-Fernández, and H. Wibowo, Moments of inertia of rare-earth nuclei and the nuclear time-odd mean fields within exact solutions of the adiabatic theory, Phys. Lett. B 868, 139685 (2025).
- P.-G. Reinhard, B. Schuetrumpf, and J. Maruhn, The axial Hartree–Fock + BCS code SkyAx, Comput. Phys. Commun. 258, 107603 (2021).
- J. Erler, P. Klüpfel, and P.-G. Reinhard, Self-consistent nuclear mean-field models: Example Skyrme-Hartree-Fock, J. Phys. G 38, 033101 (2011).
- P.-G. Reinhard, The zero-point energy for rotation, Z. Phys. A 285, 93 (1978).
- A. Gozdz, K. Pomorski, M. Brack, and W. Werner, The mass parameters for the average mean-field potential, Nucl. Phys. A 442, 26 (1985).
- K. Hagino, P.-G. Reinhard, and G. Bertsch, Projection and ground state correlations made simple, Phys. Rev. C 65, 064320 (2002).
- P.-G. Reinhard and K. Goeke, The generator coordinate method and quantised collective motion in nuclear systems, Rep. Prog. Phys. 50, 1 (1987).
- S. J. Krieger, P. Bonche, H. Flocard, P. Quentin, and M. S. Weiss, An improved pairing interaction for mean–field calculations using Skyrme potentials, Nucl. Phys. A 517, 275 (1990).
- J. A. Sheikh, J. Dobaczewski, P. Ring, L. M. Robledo, and C. Yannouleas, Symmetry restoration in mean-field approaches, J. Phys. G: Nucl. Part. Phys. 48, 123001 (2021).
- W. Nazarewicz, J. Dudek, R. Bengtsson, T. Bengtsson, and I. Ragnarsson, Microscopic study of the high-spin behavior in selected nuclei, Nucl. Phys. A 435, 397 (1985).
- Nuclear Structure of the Zirconium Region, edited by J. Eberth, R. A. Meyer, and K. Sistemich (Springer, Berlin/Heidelberg, 1988).
- B. Pritychenko, M. Birch, B. Singh, and M. Horoi, Tables of E2 transition probabilities from the first states in even–even nuclei, At. Data Nucl. Data Tables 107, 1 (2016).
- K. Hadyńska-Klek, P. J. Napiorkowski, M. Zielińska, J. Srebrny, A. Maj, F. Azaiez, J. J. Valiente Dobón, M. Kicińska Habior, F. Nowacki, H. Naïdja, B. Bounthong, T. R. Rodríguez, G. de Angelis, T. Abraham, G. Anil Kumar, D. Bazzacco, M. Bellato, D. Bortolato, P. Bednarczyk, G. Benzoni et al., Superdeformed and triaxial states in , Phys. Rev. Lett. 117, 062501 (2016).
- M. M. Giles, B. S. Nara Singh, L. Barber, D. M. Cullen, M. J. Mallaburn, M. Beckers, A. Blazhev, T. Braunroth, A. Dewald, C. Fransen, A. Goldkuhle, J. Jolie, F. Mammes, C. Müller-Gatermann, D. Wölk, K. O. Zell, S. M. Lenzi, and A. Poves, Erratum: Probing isospin symmetry in the isobaric triplet via electromagnetic transition rates, Phys. Rev. C 104, 029901(E) (2021).
- C. Morse, H. Iwasaki, A. Lemasson, A. Dewald, T. Braunroth, V. Bader, T. Baugher, D. Bazin, J. Berryman, C. Campbell, A. Gade, C. Langer, I. Lee, C. Loelius, E. Lunderberg, F. Recchia, D. Smalley, S. Stroberg, R. Wadsworth, C. Walz et al., Lifetime measurement of the state in and isospin properties of quadrupole transition strengths at , Phys. Lett. B 787, 198 (2018).
- Evaluated Nuclear Structure Data File (ENSDF), http://www.nndc.bnl.gov/ensdf/.
- K. Wimmer, P. Ruotsalainen, S. Lenzi, A. Poves, T. Hüyük, F. Browne, P. Doornenbal, T. Koiwai, T. Arici, K. Auranen, M. Bentley, M. Cortés, C. Delafosse, T. Eronen, Z. Ge, T. Grahn, P. Greenlees, A. Illana, N. Imai, H. Joukainen et al., Isospin symmetry in the triplet, Phys. Lett. B 847, 138249 (2023).
- P.-G. Reinhard, D. J. Dean, W. Nazarewicz, J. Dobaczewski, J. A. Maruhn, and M. R. Strayer, Shape coexistence and the effective nucleon-nucleon interaction, Phys. Rev. C 60, 014316 (1999).
- J. Sadoudi, M. Bender, K. Bennaceur, D. Davesne, R. Jodon, and T. Duguet, Skyrme pseudo-potential-based EDF parametrization for spuriousity-free MR EDF calculations, Phys. Scr. T154, 014013 (2013).
- K. Bennaceur, J. Dobaczewski, and F. Raimondi, New density-independent interactions for nuclear structure calculations, EPJ Web Conf. 66, 02031 (2014).
- F. Raimondi, K. Bennaceur, and J. Dobaczewski, Nonlocal energy density functionals for low-energy nuclear structure, J. Phys. G: Nucl. Part. Phys. 41, 055112 (2014).
- S. Frauendorf and A. O. Macchiavelli, Overview of neutron–proton pairing, Prog. Part. Nucl. Phys. 78, 24 (2014).
- A. M. Romero, J. Dobaczewski, and A. Pastore, Symmetry restoration in the mean-field description of proton-neutron pairing, Phys. Lett. B 795, 177 (2019).
- J. Engel, S. Pittel, M. Stoitsov, P. Vogel, and J. Dukelsky, Neutron-proton correlations in an exactly solvable model, Phys. Rev. C 55, 1781 (1997).
- G. Martínez-Pinedo, K. Langanke, and P. Vogel, Competition of isoscalar and isovector proton-neutron pairing in nuclei, Nucl. Phys. A 651, 379 (1999).
- W. Satuła and R. Wyss, Competition between and pairing in proton-rich nuclei, Phys. Lett. B 393, 1 (1997).
- E. Perlińska, S. G. Rohoziński, J. Dobaczewski, and W. Nazarewicz, Local density approximation for proton-neutron pairing correlations: Formalism, Phys. Rev. C 69, 014316 (2004).
- K. Sato, J. Dobaczewski, T. Nakatsukasa, and W. Satuła, Energy-density-functional calculations including proton-neutron mixing, Phys. Rev. C 88, 061301(R) (2013).
- J. A. Sheikh, N. Hinohara, J. Dobaczewski, T. Nakatsukasa, W. Nazarewicz, and K. Sato, Isospin-invariant skyrme energy-density-functional approach with axial symmetry, Phys. Rev. C 89, 054317 (2014).
- B. C. Backes, J. Dobaczewski, V. Guillon, and K. Bennaceur, The nuclear DFT proton-neutron paired states in homogeneous and isotropic infinite nuclear matter (unpublished).
- Database for the findings of this article, https://webfiles.york.ac.uk/HFODD/Projects/Isobaric_Triplets/.