Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Extension of mixed of symmetries method in the calculation of electric quadrupole transition rate of Cadmium isotopic chain

Document Type : Research Paper

Authors

H. Sabri 1
A.R. Jabbari 1
M. Sayedi 2

1 Department of Physics, University of Tabriz, P.O.Box: 51664, Tabriz - Iran

2 Faculty of Basic Sciences, Ilam University, P.O.Box: 69315-516, Ilam - Iran

https://doi.org/10.24200/nst.2024.1521.1991
Abstract
In this paper, in the framework of the interacting boson model (IBM), the method of mixed U(5) and SO(6) symmetry limits has been used to calculate the electric quadrupole transition rates between different levels in the isotopic chain of Cadmium (110-104Cd) and in the E < 3 MeV region. In this method, wave functions of different intruder levels are considered as a combination of wave functions in the two N and N+2 bosons spaces and the effect of electric quadrupole transition's operator is determined by them. The effective charges of the transition operator are extracted in comparison with the experimental values of the quadrupole transition rates and then, theoretical predictions of two methods, using only U(5) dynamical limit and mixed of symmetries are compared with each other. The results show that the accuracy of the predictions are increased via mixed of symmetries method in all of considered transitions. On the other hand, the U(5) dynamical limit is only successful in examining the quadrupole transitions between the levels of the ground band and levels with a longer half-life and confirms the existence of a closed neutron sub-shell in the 106Cd nucleus. Also, for such inter-bands transition and also transitions are originated from intruder states, the exactness of the mixed symmetries prediction are obvious.

Highlights

  1. Sambataro M. A study of Cd and Te isotopes in the interacting boson approximation. Nucl. Phys. A. 1993;380:365.

 

  1. Duval P.D, Barrett B.R. Configuration mixing in the interacting boson model. Physics Letter B. 1981;100:223.

 

  1. Meyer R.A, Peker L. Evidence for the coexistence of shape in even-mass Cd nuclei. Z. Phys. A. 1977;283:379.

 

  1. Heyde K, Van Isacker P, Waroquier M, Wenes G. Description of the low-lying levels in 112,114Cd. Phys. Rev. C. 1982;25:3160.

 

  1. Heyde K, Wood J.L. Shape coexistence in atomic nuclei. Rev. Mod. Phys. 2011;83:1467.

 

  1. Bohr A.N, Mottelson B.R. Collective and individual aspects of nuclear structures. Mat. Fys. Medd. 1953;27:1.

 

  1. Iachello F, Armira A. The Interaction Boson Model. Cambridge University Press, Cambridge, England. 1987.

 

  1. Elliott J.P. The interacting boson model of nuclear structure. Rep. Prog. Phys. G. 1985;48:171.

 

  1. Elliott J.P. Collective Motion in the Nuclear Shell Model. l. Classification Scheme for States of Mixed Configuration. Proc. Roy. Soc. Lond. Ser. A. 1958;245:128.

 

  1. Lehmann H, Garrett P.E, Jolie J, McGrath C.A, Yeh M, Yates S.W. On the nature of three-phonon excitations in 112Cd. Physics Letters. B. 1996;387:259.

 

  1. Deleze M, Drissi S, Jolie J, Kern J, Vorlet J.P. Systematic study of mixed ground-state and intruder bands in 110, 112, 114Cd. Nucl. Phys. A. 1993;554:1.

 

  1. Rowe D.J. Phase transitions and quasi-dynamical symmetry in nuclear collective models: I. The U(5) to O(6) phase transition in the IBM. Nucl. Phys. A. 2004;745:47.

 

  1. Kern J, Garrett P.E, Jolie J, Lehmann H. Search for nuclei exhibiting the U(5) dynamical symmetry. Nucl. Phys. A. 1995;593:21.

 

  1. Bengtsson T, Dudek J, Leander G, Nazarewicz W, Zhang J.Y. Shape coexistence and shape transitions in even-even Pt and Hg isotopes. Physics Letters B. 1987;183:1.

 

  1. Lobach Y.N, Efimov A.D, Pasternak A.A. Lifetime and configuration mixing in 110Cd. Eur. Phys. J. A. 1999;6:131.

 

  1. Heyde K, Coster C.D, Wood J.L, Jolie J. Proton 2p-2h intruder excitations and the modified vibrational intensity and selection rules. Phys. Rev. C. 1992;46:2113.

 

  1. Julin R, Helariutta K, Muikku M. Intruder states in very neutron-deficient Hg, Pb and Po nuclei. Journal of Physics G: Nuclear and Particle Physics. 2001;27:R109.

 

  1. Heyde K, Jolie J, Lehmann H, De Coster C, Wood J.L. Coexistence in even-even Cd nuclei: global structure and local perturbations. Nucl. Phys. A. 1995;586:1.

 

  1. Wood J.L, Heyde K. A focus on shape coexistence in nuclei. Journal of Physics G: Nuclear and Particle Physics. 2016;43:020402.

 

  1. DeCoster C, Heyde K, Decroix B, Van Isacker P, Jolie J, Lehmann H, Wood J.L. Particle-hole excitations in the interacting boson model (I) General structure and symmetries. Nucl. Phys. A. 1996;600:251.

 

  1. Heyde K, Jolie J, Fossion R, Baerdemacker S.De, Hellemans V. Phase transitions versus shape coexistence. Phys. Rev. C. 2004;69:054304.

 

  1. Seidi M, Sabri H, Gholami R. Solution of Bohr Hamiltonian in the Z(5) critical point by using Morse potential for 190Hg nucleus. Journal of Nuclear Science and Technology. 2022;43(2):61-70 [In Persian].

 

  1. Ghapanvari M, Sayedi M. Experimental observables of shape phase transition from vibrational to Gamma-unstable in the odd and even nuclei in the odd and even nuclei. Journal of Nuclear Science and Technology. 2022;43(4):17-27 [In Persian].

 

  1. Garett P.E, Green K.L. Breaking of vibrational motion in the isotopes 110-116Cd. Phys. Rev. C. 2008;78:044307.

 

  1. Garrett P.E, Rodríguez T.R, Varela A.D, Green K.L, Bangay J, Finlay A, Yates S.W. Multiple Shape Coexistence in 110-112Cd. Physical review letters. 2019;123(14):142502.

 

  1. Garrett P.E, Rodríguez T.R, Diaz Varela A, Green K.L, Bangay J, Finlay A, Yates S.W. Shape coexistence and multiparticle-multihole structure in 110-112Cd. Physical Review. C. 2020;101(4):044302.

 

  1. Garett P.E. Shape coexistence at low spin in the Z=50 region and its spectroscopic signatures. Journal of Physics G, Nuclear and Particle Physics. 2016;43:084002.

 

  1. Garrett P.E, Wood J.L, Yates S.W. Critical insights into nuclear collectivity from complementary nuclear spectroscopic methods. Physica Scripta. 2018;93:063001.

 

  1. Garrett P.E, Green K.L, Lehmann H, Jolie J, McGrath C.A, Yeh M, Yates S.W. Properties of 112Cd from the (n,n'γ) reaction: Lifetimes and transition rates. Phys. Rev. C. 2007;75:054310.

 

  1. Nomura K, Rodríguez-Guzmán R, Robledo L.M. Shape evolution and the role of intruder configurations in Hg isotopes within the interacting boson model based on a Gogny energy density functional. Phys. Rev. C. 2013;87:064313.

 

  1. Garrett P.E, Bangay J, Diaz Varela A, Ball G.C, Cross D.S, Demand G.A, Finlay P, Garnsworthy A.B, Green K.L. Detailed spectroscopy of 110Cd: Evidence for weak mixing and the emergence of γ-soft behavior. Phys. Rev. C. 2012;86:044304.

 

  1. Lehmann H, Jolie J. The U(5)-O(6) model: an analytical approach to shape coexistence. Nucl. Phys. A. 1995;588:623.

 

  1. Jolie J, Lehmann H. On the influence of the O(5) symmetry on shape coexistence in atomic nuclei. Physics Letters B. 1995;342:1.

 

  1. Fortune H.T. Coexistence and configuration mixing in 112Cd. Nucl. Phys. A. 2021;1014:122233.

 

  1. Blachot J. Nuclear Data Sheets for A = 104. Nuclear Data Sheets. 2007;108:2035.

 

  1. Frenne D.De, Negret A. Nuclear Data Sheets for A = 106. Nuclear Data Sheets. 2007;109:943.

 

  1. Blachot J. Nuclear Data Sheets for A = 108. Nuclear Data Sheets. 2000;91:135.

 

  1. G~urdal G, Kondev F.G. Nuclear Data Sheets for A = 110. Nuclear Data Sheets. 2011;113:1315.

 

  1. Lehmann H, Jolie J, Coster C.De, Decroix B, Heyde K. Particle-hole excitations in the interacting boson model (II): The U(5)-O(6) coupling. Nuclear Physics A. 1997;621:767.

 

  1. Coster C.De, Decroix B, Heyde K, Wood J.L, Jolie J, Lehmann H. Particle-hole excitations in the interacting boson model (III): The O(6)-SU(3) coupling. Nucl. Phys. A. 1997;621:802.

 

  1. Coster C.De, Decroix B, Heyde K, Jolie J, Lehmann H, Wood J.L. Particle-hole excitations in the interacting boson model (IV). The U(5)-SU(3) coupling. Nucl. Phys. A. 1999;651:31.

 

  1. Pan F, Li D, Cheng G, Qiao Z, Bai J, Draayer J.P. Exactly solvable configuration mixing scheme in the vibrational limit of the interacting boson model. Phys. Rev. C. 2018;97:034316.

 

  1. Pan F, Yuan S, Qiao Z, Bai J, Zhang Y, Draayer J.P. γ-soft rotor with configuration mixing in the O(6) limit of the interacting boson model. Phys. Rev. C. 2018;97:034326.

 

  1. Dai L, Pan F, Feng Z, Zhang Y, Cui S, Draayer J.P. Exact solution of U(5)–O(6) transitional description in interacting boson model with two-particle and two-hole configuration mixing. Chinese Physics C. 2020;44:064102R.

 

  1. Hosseinnezhad A, Majarshin A.J, Luo Y.A, Ahmadian D, Sabri H. Deformation in 92-128Pd isotopic chain. Nucl. Phys. A. 2022;1028:122523.

 

  1. Corminboeuf F, Brown T.B, Genilloud L, Hannant C.D, Jolie J, Kern J, Warr N, Yates S.W. Characterization of Three-Phonon States in 110Cd. Phys. Rev. Lett. 2000;84:4060.

 

  1. Garrett P.E, Lehmann H, McGrath C.A, Yeh M, Yates S.W. First observation of mixed-symmetry states in a good U(5) nucleus. Phys. Rev. C. 1996;54:2259.

 

  1. Sabri H, Jahangiri Z, Mohammadi M.A. Investigation of shape coexistence in 118–128Te isotopes. Nucl. Phys. A. 2016;946:28.

 

  1. Leviatan A, Gavrielov N, García-Ramos J.E, Van Isacker P. Quadrupole phonon in cadmium isotopes. Phys. Rev. C. 2018;98:031302.

 

  1. Leviatan A, Gavrielov N, Garcia-Ramos J.E, Van Isacker P. Partial dynamical symmetry and the vibrational structure of Cd isotopes. EPJ Web of Conferences. 2018;178:05003.

 

  1. Leviatan A, Shapira D. Algebraic benchmark for prolate-oblate coexistence in nuclei. Phys. Rev. C. 2016;93:051302.

 

  1. Rastgar M, Sabri H, Ezzati A.O. Combination of SU(1,1)-transitional Hamiltonian and O(6) Casimir operator for description of intruder states in 112Cd nucleus. Int. J. Mod. Phys. E. 2023;32:2350034.

 

  1. Cejnar P, Jolie J, Casten R.F. Quantum phase transitions in the shapes of atomic nuclei. Reviews of Modern Physics. 2010;82(3):2155-212.

Keywords

Mixed of symmetries
Electric quadrupole transition rate
Intruder state
Interacting boson model (IBM)
  1. Sambataro M. A study of Cd and Te isotopes in the interacting boson approximation. Nucl. Phys. A. 1993;380:365.

 

  1. Duval P.D, Barrett B.R. Configuration mixing in the interacting boson model. Physics Letter B. 1981;100:223.

 

  1. Meyer R.A, Peker L. Evidence for the coexistence of shape in even-mass Cd nuclei. Z. Phys. A. 1977;283:379.

 

  1. Heyde K, Van Isacker P, Waroquier M, Wenes G. Description of the low-lying levels in 112,114Cd. Phys. Rev. C. 1982;25:3160.

 

  1. Heyde K, Wood J.L. Shape coexistence in atomic nuclei. Rev. Mod. Phys. 2011;83:1467.

 

  1. Bohr A.N, Mottelson B.R. Collective and individual aspects of nuclear structures. Mat. Fys. Medd. 1953;27:1.

 

  1. Iachello F, Armira A. The Interaction Boson Model. Cambridge University Press, Cambridge, England. 1987.

 

  1. Elliott J.P. The interacting boson model of nuclear structure. Rep. Prog. Phys. G. 1985;48:171.

 

  1. Elliott J.P. Collective Motion in the Nuclear Shell Model. l. Classification Scheme for States of Mixed Configuration. Proc. Roy. Soc. Lond. Ser. A. 1958;245:128.

 

  1. Lehmann H, Garrett P.E, Jolie J, McGrath C.A, Yeh M, Yates S.W. On the nature of three-phonon excitations in 112Cd. Physics Letters. B. 1996;387:259.

 

  1. Deleze M, Drissi S, Jolie J, Kern J, Vorlet J.P. Systematic study of mixed ground-state and intruder bands in 110, 112, 114Cd. Nucl. Phys. A. 1993;554:1.

 

  1. Rowe D.J. Phase transitions and quasi-dynamical symmetry in nuclear collective models: I. The U(5) to O(6) phase transition in the IBM. Nucl. Phys. A. 2004;745:47.

 

  1. Kern J, Garrett P.E, Jolie J, Lehmann H. Search for nuclei exhibiting the U(5) dynamical symmetry. Nucl. Phys. A. 1995;593:21.

 

  1. Bengtsson T, Dudek J, Leander G, Nazarewicz W, Zhang J.Y. Shape coexistence and shape transitions in even-even Pt and Hg isotopes. Physics Letters B. 1987;183:1.

 

  1. Lobach Y.N, Efimov A.D, Pasternak A.A. Lifetime and configuration mixing in 110Cd. Eur. Phys. J. A. 1999;6:131.

 

  1. Heyde K, Coster C.D, Wood J.L, Jolie J. Proton 2p-2h intruder excitations and the modified vibrational intensity and selection rules. Phys. Rev. C. 1992;46:2113.

 

  1. Julin R, Helariutta K, Muikku M. Intruder states in very neutron-deficient Hg, Pb and Po nuclei. Journal of Physics G: Nuclear and Particle Physics. 2001;27:R109.

 

  1. Heyde K, Jolie J, Lehmann H, De Coster C, Wood J.L. Coexistence in even-even Cd nuclei: global structure and local perturbations. Nucl. Phys. A. 1995;586:1.

 

  1. Wood J.L, Heyde K. A focus on shape coexistence in nuclei. Journal of Physics G: Nuclear and Particle Physics. 2016;43:020402.

 

  1. DeCoster C, Heyde K, Decroix B, Van Isacker P, Jolie J, Lehmann H, Wood J.L. Particle-hole excitations in the interacting boson model (I) General structure and symmetries. Nucl. Phys. A. 1996;600:251.

 

  1. Heyde K, Jolie J, Fossion R, Baerdemacker S.De, Hellemans V. Phase transitions versus shape coexistence. Phys. Rev. C. 2004;69:054304.

 

  1. Seidi M, Sabri H, Gholami R. Solution of Bohr Hamiltonian in the Z(5) critical point by using Morse potential for 190Hg nucleus. Journal of Nuclear Science and Technology. 2022;43(2):61-70 [In Persian].

 

  1. Ghapanvari M, Sayedi M. Experimental observables of shape phase transition from vibrational to Gamma-unstable in the odd and even nuclei in the odd and even nuclei. Journal of Nuclear Science and Technology. 2022;43(4):17-27 [In Persian].

 

  1. Garett P.E, Green K.L. Breaking of vibrational motion in the isotopes 110-116Cd. Phys. Rev. C. 2008;78:044307.

 

  1. Garrett P.E, Rodríguez T.R, Varela A.D, Green K.L, Bangay J, Finlay A, Yates S.W. Multiple Shape Coexistence in 110-112Cd. Physical review letters. 2019;123(14):142502.

 

  1. Garrett P.E, Rodríguez T.R, Diaz Varela A, Green K.L, Bangay J, Finlay A, Yates S.W. Shape coexistence and multiparticle-multihole structure in 110-112Cd. Physical Review. C. 2020;101(4):044302.

 

  1. Garett P.E. Shape coexistence at low spin in the Z=50 region and its spectroscopic signatures. Journal of Physics G, Nuclear and Particle Physics. 2016;43:084002.

 

  1. Garrett P.E, Wood J.L, Yates S.W. Critical insights into nuclear collectivity from complementary nuclear spectroscopic methods. Physica Scripta. 2018;93:063001.

 

  1. Garrett P.E, Green K.L, Lehmann H, Jolie J, McGrath C.A, Yeh M, Yates S.W. Properties of 112Cd from the (n,n'γ) reaction: Lifetimes and transition rates. Phys. Rev. C. 2007;75:054310.

 

  1. Nomura K, Rodríguez-Guzmán R, Robledo L.M. Shape evolution and the role of intruder configurations in Hg isotopes within the interacting boson model based on a Gogny energy density functional. Phys. Rev. C. 2013;87:064313.

 

  1. Garrett P.E, Bangay J, Diaz Varela A, Ball G.C, Cross D.S, Demand G.A, Finlay P, Garnsworthy A.B, Green K.L. Detailed spectroscopy of 110Cd: Evidence for weak mixing and the emergence of γ-soft behavior. Phys. Rev. C. 2012;86:044304.

 

  1. Lehmann H, Jolie J. The U(5)-O(6) model: an analytical approach to shape coexistence. Nucl. Phys. A. 1995;588:623.

 

  1. Jolie J, Lehmann H. On the influence of the O(5) symmetry on shape coexistence in atomic nuclei. Physics Letters B. 1995;342:1.

 

  1. Fortune H.T. Coexistence and configuration mixing in 112Cd. Nucl. Phys. A. 2021;1014:122233.

 

  1. Blachot J. Nuclear Data Sheets for A = 104. Nuclear Data Sheets. 2007;108:2035.

 

  1. Frenne D.De, Negret A. Nuclear Data Sheets for A = 106. Nuclear Data Sheets. 2007;109:943.

 

  1. Blachot J. Nuclear Data Sheets for A = 108. Nuclear Data Sheets. 2000;91:135.

 

  1. G~urdal G, Kondev F.G. Nuclear Data Sheets for A = 110. Nuclear Data Sheets. 2011;113:1315.

 

  1. Lehmann H, Jolie J, Coster C.De, Decroix B, Heyde K. Particle-hole excitations in the interacting boson model (II): The U(5)-O(6) coupling. Nuclear Physics A. 1997;621:767.

 

  1. Coster C.De, Decroix B, Heyde K, Wood J.L, Jolie J, Lehmann H. Particle-hole excitations in the interacting boson model (III): The O(6)-SU(3) coupling. Nucl. Phys. A. 1997;621:802.

 

  1. Coster C.De, Decroix B, Heyde K, Jolie J, Lehmann H, Wood J.L. Particle-hole excitations in the interacting boson model (IV). The U(5)-SU(3) coupling. Nucl. Phys. A. 1999;651:31.

 

  1. Pan F, Li D, Cheng G, Qiao Z, Bai J, Draayer J.P. Exactly solvable configuration mixing scheme in the vibrational limit of the interacting boson model. Phys. Rev. C. 2018;97:034316.

 

  1. Pan F, Yuan S, Qiao Z, Bai J, Zhang Y, Draayer J.P. γ-soft rotor with configuration mixing in the O(6) limit of the interacting boson model. Phys. Rev. C. 2018;97:034326.

 

  1. Dai L, Pan F, Feng Z, Zhang Y, Cui S, Draayer J.P. Exact solution of U(5)–O(6) transitional description in interacting boson model with two-particle and two-hole configuration mixing. Chinese Physics C. 2020;44:064102R.

 

  1. Hosseinnezhad A, Majarshin A.J, Luo Y.A, Ahmadian D, Sabri H. Deformation in 92-128Pd isotopic chain. Nucl. Phys. A. 2022;1028:122523.

 

  1. Corminboeuf F, Brown T.B, Genilloud L, Hannant C.D, Jolie J, Kern J, Warr N, Yates S.W. Characterization of Three-Phonon States in 110Cd. Phys. Rev. Lett. 2000;84:4060.

 

  1. Garrett P.E, Lehmann H, McGrath C.A, Yeh M, Yates S.W. First observation of mixed-symmetry states in a good U(5) nucleus. Phys. Rev. C. 1996;54:2259.

 

  1. Sabri H, Jahangiri Z, Mohammadi M.A. Investigation of shape coexistence in 118–128Te isotopes. Nucl. Phys. A. 2016;946:28.

 

  1. Leviatan A, Gavrielov N, García-Ramos J.E, Van Isacker P. Quadrupole phonon in cadmium isotopes. Phys. Rev. C. 2018;98:031302.

 

  1. Leviatan A, Gavrielov N, Garcia-Ramos J.E, Van Isacker P. Partial dynamical symmetry and the vibrational structure of Cd isotopes. EPJ Web of Conferences. 2018;178:05003.

 

  1. Leviatan A, Shapira D. Algebraic benchmark for prolate-oblate coexistence in nuclei. Phys. Rev. C. 2016;93:051302.

 

  1. Rastgar M, Sabri H, Ezzati A.O. Combination of SU(1,1)-transitional Hamiltonian and O(6) Casimir operator for description of intruder states in 112Cd nucleus. Int. J. Mod. Phys. E. 2023;32:2350034.

 

  1. Cejnar P, Jolie J, Casten R.F. Quantum phase transitions in the shapes of atomic nuclei. Reviews of Modern Physics. 2010;82(3):2155-212.

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