Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Interacting boson model studies of collectivity in even-even and odd-even Te isotopes

E. A. Cederlöf1,2,*, W. Teng3, Nazeef1,4, T. Wan1,5, C. Qi1,†, and Y. Zhang3

  • *Contact author: ebbace@kth.se
  • †Contact author: chongq@kth.se

Phys. Rev. C 113, 054312 – Published 20 May, 2026

DOI: https://doi.org/10.1103/vmvz-9m19

Abstract

Background: With the recent progress in lifetime measurements, there has been strong experimental and theoretical interest in studying the evolution of the collectivity in the long Te isotopic chain.

Purpose: The purpose of the present work is to investigate the structure of low-lying excited states in even-even Te nuclei as well as the νh11/2 band in the odd-mass systems within the interacting boson model framework.

Methods: A new open-source Python package is developed to evaluate the eigenvalues and eigenfunctions of a Hamiltonian which contains both the standard n̂d and Q̂·Q̂ terms as well as higher-order terms which could be essential for triaxial deformation and mixed symmetry effects.

Results: Systematic calculations of low-spin yrast and nonyrast states in even-even Te108−124 and odd-even Te109−125 nuclei have been carried out using the newly developed Python package and compared with the neighboring Cd isotopes. From the resulting eigenfunctions, the reduced transition probabilities B(E2) are calculated and compared to adopted values. Due to the vibrational-like structure of the spectra of the Te nuclei, the Hamiltonian is dominated by the vibrational term, which is reflected in the B(E2) values. This alone does not explain the low B4/2 ratios observed in some Te nuclei. However, it is demonstrated via the contour plot that the B4/2 ratios can be highly sensitive to the higher-order terms of the Hamiltonian within specific parameter domains.

Conclusions: We demonstrate that inclusion of higher-order interactions can provide a possible explanation of the anomaly observed in certain Te isotopes. The results highlight both the importance of extended IBM Hamiltonians and the utility of pyIBM as a modern computational tool for collective nuclear structure studies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (84)

  1. F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, UK, 1987).
  2. A. Bohr and B. R. Mottelson, Nuclear Structure (Benjamin, New York, NY, 1975), Vol. 2.
  3. F. Iachello and P. Van Isacker, The Interacting Boson-Fermion Model (Cambridge University Press, Cambridge, UK, 1991).
  4. M. Homma and K. Nomura, Sensitivity analysis of β-decay half-life predictions for Ge, As, Zr, and Mo nuclei within the mapped interacting boson model, Phys. Rev. C 111, 044302 (2025).
  5. S. Pascu et al., Increasing octupole collectivity across the Z=64 isotopic chain: B(E3) values in Gd150, Phys. Rev. Lett. 134, 092501 (2025).
  6. S. Pascu et al., Collective excitations in Gd150, Phys. Rev. C 111, 034302 (2025).
  7. B. Maheshwari, K. Nomura, and P. Van Isacker, Nuclear size, electric monopole transitions, and the location of 02+ states, Phys. Rev. C 111, 024317 (2025).
  8. L. Lotina, K. Nomura, R. Rodríguez-Guzmán, and L. M. Robledo, Quadrupole-hexadecapole correlations in neutron-rich samarium and gadolinium isotopes, Phys. Rev. C 111, 024301 (2025).
  9. K. Nomura, Role of octupole shape degree of freedom in neutron-rich odd-mass xenon isotopes, Phys. Rev. C 110, 064306 (2024).
  10. L. Knafla, K. Nomura, A. Esmaylzadeh, A. Harter, J. Jolie, V. Karayonchev, Y. H. Kim, U. Köster, M. Ley, C. Michelagnoli, A. Pfeil, J.-M. Régis, and F. von Spee, Investigating the prolate-to-oblate shape phase transition: Lifetime measurements and γ spectroscopy of the low-lying negative parity structure in Os193, Phys. Rev. C 109, 014313 (2024).
  11. A. Esmaylzadeh, A. Blazhev, K. Nomura, J. Jolie, M. Beckers, C. Fransen, R.-B. Gerst, A. Harter, V. Karayonchev, L. Knafla, M. Ley, and F. von Spee, Investigation of γ softness: Lifetime measurements in Ru104,106, Phys. Rev. C 106, 064323 (2022).
  12. A. Pfeil, K. Nomura, N. Gavrielov, J.-M. Régis, U. Köster, Y. H. Kim, A. Esmaylzadeh, A. Harter, J. Jolie, L. Knafla, M. Ley, and V. Karayonchev, Lifetime measurements in Nb99 and Zr99: Investigation of shape coexistence, Phys. Rev. C 108, 034310 (2023).
  13. K. Nomura, D. Vretenar, Z. P. Li, and J. Xiang, Interplay between pairing and triaxial shape degrees of freedom in Os and Pt nuclei, Phys. Rev. C 104, 024323 (2021).
  14. K. Nomura, R. Rodríguez-Guzmán, and L. M. Robledo, Quadrupole-octupole coupling and the evolution of collectivity in neutron-deficient Xe, Ba, Ce, and Nd isotopes, Phys. Rev. C 104, 054320 (2021).
  15. A. Esmaylzadeh, V. Karayonchev, K. Nomura, J. Jolie, M. Beckers, A. Blazhev, A. Dewald, C. Fransen, R.-B. Gerst, G. Häfner, A. Harter, L. Knafla, M. Ley, L. M. Robledo, R. Rodríguez-Guzmán, and M. Rudigier, Lifetime measurements to investigate γ softness and shape coexistence in Mo102, Phys. Rev. C 104, 064314 (2021).
  16. K. Nomura and C. M. Petrache, Questioning the wobbling interpretation of low-spin bands in γ-soft nuclei within the interacting boson-fermion model, Phys. Rev. C 105, 024320 (2022).
  17. K. Nomura, D. Vretenar, Z. P. Li, and J. Xiang, Pairing vibrations in the interacting boson model based on density functional theory, Phys. Rev. C 102, 054313 (2020).
  18. K. Nomura, D. Vretenar, Z. P. Li, and J. Xiang, Coupling of pairing and triaxial shape vibrations in collective states of γ-soft nuclei, Phys. Rev. C 103, 054322 (2021).
  19. C. E. Jones et al., Lifetime measurements in Mo102 interpreted in the interacting boson model and the X(5) symmetry, Phys. Rev. C 111, 034325 (2025).
  20. N. Gavrielov, J. E. García-Ramos, P. Van Isacker, and A. Leviatan, Persistent vibrational structure in 110–116Cd, Phys. Rev. C 108, L031305 (2023).
  21. K. Nomura, Mapped interacting boson model for nuclear structure studies, Eur. Phys. J. A 61, 139 (2025).
  22. E. Maya-Barbecho and J.-E. García-Ramos, An intrinsic-state formalism for the interacting boson-fermion model with configuration mixing, Phys. Lett. B 868, 139724 (2025).
  23. P. Vasileiou, D. Bonatsos, and T. J. Mertzimekis, Mean-field-derived IBM-1 Hamiltonian with intrinsic triaxial deformation, Phys. Rev. C 110, 014313 (2024).
  24. Y. Zhang, Y. X. Liu, F. Pan, and J. P. Draayer, Algebraic description of the triaxially to axially rotational shape phase transition, Phys. Rev. C 111, 054303 (2025).
  25. T. Wang, Typical new spherical-like γ-soft spectra in Pd104,106,108, Phys. Rev. C 112, 034301 (2025).
  26. Y. Zhang, Y. W. He, D. Karlsson, C. Qi, F. Pan, and J. P. Draayer, A theoretical interpretation of the anomalous reduced E2 transition probabilities along the yrast line of neutron-deficient nuclei, Phys. Lett. B 834, 137443 (2022).
  27. Y. Zhang, S.-N. Wang, F. Pan, C. Qi, and J. P. Draayer, Triaxial rotor modes in finite-N boson systems, Phys. Rev. C 110, 024303 (2024).
  28. W. Teng, Y. Zhang, and C. Qi, A novel approach for the anomalous collectivity in neutron-deficient Os isotopes, Chin. Phys. C 49, 014102 (2025).
  29. F. Pan, Y. Zhang, Y. Wu, L. Dai, and J. P. Draayer, b(e2) anomaly along the yrast line in neutron-deficient a≈170 even-even nuclei induced by a triaxial rotor term, Phys. Rev. C 110, 054324 (2024).
  30. T. Wang, A collective description of the unusually low ratio B4/2=B(E2;41+→21+)/B(E2;21+→01+), Europhys. Lett. 129, 52001 (2020).
  31. T. Wang, B(E2) anomaly cannot be explained with O(6) higher-order interactions, Phys. Rev. C 107, 064303 (2023).
  32. W. Teng, S.-N. Wang, Y. Zhang, and L. Fortunato, Triaxial rotor in the O(6) limit of the interacting boson model, Phys. Scr. 99, 015305 (2024).
  33. W. Teng, S.-N. Wang, Y. Zhang, X.-Z. Zhao, X. Deng, and X.-T. Li, B(E2) anomaly and triaxial deformation within a two-fluid SU(3) symmetry, Chin. Phys. C 49, 084106 (2025).
  34. W. Teng, S. N. Wang, X. Z. Zhao, and Y. Zhang, The IBM description of the B(E2) anomaly: Dynamical triaxiality and configuration mixing, Nucl. Phys. A 1063, 123214 (2025).
  35. B. Cederwall and C. Qi, The B(E2) anomaly: Evidence for a low-lying mixed-symmetry collective excitation mode (2025), arXiv:2512.11555.
  36. G. J. Fu and C. Qi, Novel triaxiality-driven collective feature in atomic nuclei investigated via the two-body random ensemble, Phys. Rev. C 112, L061301 (2025).
  37. T. Grahn et al., Excited states and reduced transition probabilities in Os168, Phys. Rev. C 94, 044327 (2016).
  38. B. Cederwall et al., Lifetime measurements of excited states in Pt172 and the variation of quadrupole transition strength with angular momentum, Phys. Rev. Lett. 121, 022502 (2018).
  39. B. Sayğı et al., Reduced transition probabilities along the yrast line in W166, Phys. Rev. C 96, 021301 (2017).
  40. A. Goasduff, J. Ljungvall, T. R. Rodríguez, F. L. Bello Garrote, A. Etile, G. Georgiev, F. Giacoppo, L. Grente, M. Klintefjord, A. Kuşoğlu, I. Matea, S. Roccia, M.-D. Salsac, and C. Sotty, B(E2) anomalies in the yrast band of Os170, Phys. Rev. C 100, 034302 (2019).
  41. O. Möller, N. Warr, J. Jolie, A. Dewald, A. Fitzler, A. Linnemann, K. O. Zell, P. E. Garrett, and S. W. Yates, E2 transition probabilities in Te114: A conundrum, Phys. Rev. C 71, 064324 (2005).
  42. C. Qi, The anomalous quadrupole collectivity in Te isotopes, Phys. Rev. C 94, 034310 (2016).
  43. M. J. Gómez, Nuclear structure studies on quadrupole and octupole correlations in the vicinity of heavy N = Z nuclei with AGATA and NEDA, Ph.D. thesis, Universitat de Valencia (2022).
  44. I. Zanon et al., Anomalous B4/2 ratio in the yrast band of Os167, Phys. Rev. C 111, 034323 (2025).
  45. K. Arnswald et al., Enhanced collectivity along the N = Z line: Lifetime measurements in Ti44, Cr48, and Fe52, Phys. Lett. B 772, 599 (2017).
  46. J. Kern, P. E. Garrett, J. Jolie, and H. Lehmann, Search for nuclei exhibiting the U(5) dynamical symmetry, Nucl. Phys. A 593, 21 (1995).
  47. P. E. Garrett, J. L. Wood, and S. W. Yates, Critical insights into nuclear collectivity from complementary nuclear spectroscopic methods, Phys. Scr. 93, 063001 (2018).
  48. K. Heyde and J. L. Wood, Shape coexistence in atomic nuclei, Rev. Mod. Phys. 83, 1467 (2011).
  49. P. E. Garrett, M. Zielińska, and E. Clément, An experimental view on shape coexistence in nuclei, Prog. Part. Nucl. Phys. 124, 103931 (2022).
  50. S. Leoni, B. Fornal, A. Bracco, Y. Tsunoda, and T. Otsuka, Multifaceted character of shape coexistence phenomena in atomic nuclei, Prog. Part. Nucl. Phys. 139, 104119 (2024).
  51. P. E. Garrett and J. L. Wood, On the robustness of surface vibrational modes: Case studies in the Cd region, J. Phys. G: Nucl. Part. Phys. 37, 064028 (2010).
  52. E. A. Cederlöf et al., Lifetime measurements of low-lying states in odd-mass isotopes Te117,119 and the nature of vibration in Te isotopes, Phys. Rev. C 112, 014321 (2025).
  53. E. A. Cederlöf, From ideal vibrations to possible anomalies: Insights from tellurium lifetime measurements and modelling, Ph.D. dissertation, KTH Royal Institute of Technology/Uppsala University, 2025.
  54. S. Heinze, A method for the solution of arbitrary bosonic and fermionic many-particle systems, Ph.D. thesis, University of Cologne, 2008.
  55. R. J. Casperson, IBAR: Interacting boson model calculations for large system sizes, Comput. Phys. Commun. 183, 1029 (2012).
  56. R. F. Casten, Nuclear Structure from a Simple Perspective, 2nd ed. (Oxford University Press, Oxford, UK, 2001).
  57. C. Xiu, Y. Zhang, M. J. Li, J. Yang, and Y.-X. Chen, Effects of an odd particle on shape phase transitions in odd-even systems, Chin. Phys. C 45, 084106 (2021).
  58. D. D. Warner and R. F. Casten, Predictions of the interacting boson approximation in a consistent Q framework, Phys. Rev. C 28, 1798 (1983).
  59. F. Iachello and O. Scholten, Description of odd-mass nuclei in the interacting boson-fermion model, Phys. Rev. Lett. 43, 679 (1979).
  60. C. Bahri, D. Rowe, and J. Draayer, Programs for generating Clebsch–Gordan coefficients of SU(3) in SU(2) and SO(3) bases, Comput. Phys. Commun. 159, 121 (2004).
  61. E. Ahlgren Cederlöf, W. Teng, Nazeef, C. Qi, and Y. Zhang, pyIBM: A user-friendly Python package for IBM/IBFM-1 calculations, https://github.com/ebbace/pyIBM.
  62. E. Ahlgren Cederlöf, W. Teng, Nazeef, C. Qi, and Y. Zhang, pyIBM: A user-friendly Python package for IBM/IBFM-1 calculations, https://zenodo.org/records/17084202.
  63. S. Pascu, N. V. Zamfir, G. Căta-Danil, and N. Mărginean, Structural evolution of the Z=52–62 neutron-deficient nuclei in the interacting boson approximation framework, Phys. Rev. C 81, 054321 (2010).
  64. J. B. Gupta, Spectral features of vibrational Te isotopes, Phys. Rev. C 107, 034315 (2023).
  65. K. Suzuki and K. Nomura, Shape transition and coexistence in Te isotopes studied with the quadrupole collective Hamiltonian based on a relativistic energy density functional, Phys. Rev. C 110, 064304 (2024).
  66. K. Kaneko, Y. Sun, N. Shimizu, and T. Mizusaki, Quasi-SU(3) coupling induced oblate-prolate shape phase transition in the Casten triangle, Phys. Rev. Lett. 130, 052501 (2023).
  67. E. A. Coello Pérez and T. Papenbrock, Effective field theory for nuclear vibrations with quantified uncertainties, Phys. Rev. C 92, 064309 (2015).
  68. E. A. Cederlöf et al., Lifetime measurement of the yrast 2+ state in Te118, Eur. Phys. J. A 59, 300 (2023).
  69. S. Pascu et al., Evolution of quadrupole collectivity in Te isotopes, Phys. Rev. C 112, 024312 (2025).
  70. F. von Spee, M. Beckers, A. Blazhev, A. Dewald, F. Dunkel, A. Esmaylzadeh, C. Fransen, G. Hackenberg, J. Jolie, L. Knafla, C. D. Lakenbrink, M. Schiffer, N. Warr, and M. Weinert, Structure of low-lying states in Te116, Phys. Rev. C 109, 024325 (2024).
  71. M. Doncel et al., Spin-dependent evolution of collectivity in Te112, Phys. Rev. C 96, 051304(R) (2017).
  72. C. B. Li et al., Lifetime measurements of the first 2+ states in Te116,118, Phys. Rev. C 109, 034310 (2024).
  73. M. Doncel, T. Back, D. M. Cullen, D. Hodge, C. Qi, B. Cederwall, M. J. Taylor, M. Procter, K. Auranen, T. Grahn, et al., Lifetime measurement of the first excited 2+ state in Te112, Phys. Rev. C 91, 061304(R) (2015).
  74. M. Saxena, R. Kumar, A. Jhingan, S. Mandal, A. Stolarz, A. Banerjee, R. K. Bhowmik, S. Dutt, J. Kaur, V. Kumar, M. Modou Mbaye, V. R. Sharma, and H.-J. Wollersheim, Rotational behavior of Te120,122,124, Phys. Rev. C 90, 024316 (2014).
  75. M. Reece, B. J. Coombes, A. J. Mitchell, A. E. Stuchbery, G. J. Lane, A. Gargano, V. U. Bashu, L. J. Bignell, C. Gautam, L. J. McKie, N. J. Spinks, and J. A. Woodside, Coulomb excitation of Te124: Emerging collectivity and persisting seniority structure in the 61+ level, Phys. Rev. C 112, 034311 (2025).
  76. National Nuclear Data Center, NNDC nuclear data services, https://www.nndc.bnl.gov/.
  77. J. L. Wood, K. Heyde, W. Nazarewicz, M. Huyse, and P. van Duppen, Coexistence in even-mass nuclei, Phys. Rep. 215, 101 (1992).
  78. R. M. Clark, M. Cromaz, M. A. Deleplanque, M. Descovich, R. M. Diamond, P. Fallon, I. Y. Lee, A. O. Macchiavelli, H. Mahmud, E. Rodriguez-Vieitez, F. S. Stephens, and D. Ward, Searching for E(5) behavior in nuclei, Phys. Rev. C 69, 064322 (2004).
  79. F. Iachello, Dynamic symmetries at the critical point, Phys. Rev. Lett. 85, 3580 (2000).
  80. S. F. Hicks, G. K. Alexander, C. A. Aubin, M. C. Burns, C. J. Collard, M. M. Walbran, J. R. Vanhoy, E. Jensen, P. E. Garrett, M. Kadi, A. Martin, N. Warr, and S. W. Yates, Intruder structures observed in Te122 through inelastic neutron scattering, Phys. Rev. C 71, 034307 (2005).
  81. S. F. Hicks, J. R. Vanhoy, P. G. Burkett, B. R. Champine, S. J. Etzkorn, P. E. Garrett, S. W. Yates, and M. Yeh, Lifetimes in Te124: Examining critical-point symmetry in the Te nuclei, Phys. Rev. C 95, 034322 (2017).
  82. A. A. Pasternak, J. Srebrny, A. D. Efimov, V. M. Mikhajlov, E. O. Podsvirova, C. Droste, T. Morek, S. Juutinen, G. B. Hagemann, M. Piiparinen, S. Törmänen, and A. Virtanen, Lifetimes in the ground-state band and the structure of Te118, Eur. Phys. J. A 13, 435 (2002).
  83. P. Van Isacker and J.-Q. Chen, Classical limit of the interacting boson Hamiltonian, Phys. Rev. C 24, 684 (1981).
  84. B. Sorgunlu and P. Van Isacker, Triaxiality in the interacting boson model, Nucl. Phys. A 808, 27 (2008).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation