pith. sign in

arxiv: 2604.10972 · v1 · submitted 2026-04-13 · ⚛️ nucl-th

Microscopic investigation of E2 matrix elements in atomic nuclei -- II

Kouser Qureshie , S. P. Rouoof , J. A. Sheikh , N. Rather , S. Jehangir , G. H. Bhat , S. Frauendorf This is my paper

Pith reviewed 2026-05-10 16:23 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords E2 matrix elementstriaxial projected shell modelgamma soft nucleishape invariantsKumar-Cline sum rulesCoulomb excitationnuclear deformation
0
0 comments X

The pith

The triaxial projected shell model reproduces E2 matrix elements for 70Ge, Se isotopes and 100Mo, showing gamma-soft behavior in most nuclei and no correlation between gamma-band staggering and shape invariants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work extends prior TPSM calculations to six additional nuclei where Coulomb excitation data is now available. The microscopic approach matches the experimental E2 matrix elements closely. All nuclei except 76Se and 100Mo are found to be gamma soft. Unlike phenomenological collective models, the TPSM results show no clear link between the energy staggering in the gamma band and the shape invariants obtained from Kumar-Cline sum rules applied to the calculated matrix elements.

Core claim

TPSM calculations of E2 matrix elements provide a good description of the available data for the six nuclei, establish gamma-soft behavior for all but 76Se and 100Mo, and reveal an absence of the correlation between gamma-band energy staggering and Kumar-Cline shape invariants that is predicted by collective models.

What carries the argument

Triaxial projected shell model (TPSM) for computing microscopic E2 matrix elements, followed by application of Kumar-Cline sum rules to extract shape invariants.

If this is right

  • TPSM can be applied to predict E2 properties in neighboring nuclei lacking experimental data.
  • Gamma-soft shapes are the dominant behavior in this mass region according to the microscopic treatment.
  • Interpretations of gamma-band features in nuclei must account for the lack of correlation with shape invariants found in microscopic calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Phenomenological collective models may miss microscopic details that decouple band staggering from overall deformation.
  • Systematic TPSM studies across more nuclei could clarify when gamma-soft behavior transitions to other shapes.
  • The approach offers a way to test whether energy staggering alone is a reliable indicator of nuclear triaxiality.

Load-bearing premise

The triaxial projected shell model with its chosen parameters accurately captures the microscopic structure and E2 matrix elements for these nuclei, and the Kumar-Cline sum rules applied to the calculated matrix elements correctly yield shape invariants comparable to experiment.

What would settle it

New Coulomb excitation measurements on 76Se or 100Mo that produce E2 matrix elements or derived shape invariants differing substantially from the TPSM values, or data showing a clear correlation between gamma-band staggering and shape invariants.

Figures

Figures reproduced from arXiv: 2604.10972 by G. H. Bhat, J. A. Sheikh, Kouser Qureshie, N. Rather, S. Frauendorf, S. Jehangir, S. P. Rouoof.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) TPSM and experimental energies of the yrast, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Staggering parameters [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Reduced in-band [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Reduced in-band [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Reduced in-band [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (Color online) Reduced inter-band [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (Color online) Centroid [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (Color online) Centroid [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. (Color online)Centroid [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗
read the original abstract

The present work is a continuation of our earlier investigation with the primary objective to systematically calculate the $E2$ matrix elements using the microscopic approach of the triaxial projected shell model (TPSM). In the earlier work, we studied nine nuclides of $^{72}$Ge, $^{76}$Ge, $^{104}$Ru, $^{168}$Er, $^{186}$Os, $^{188}$Os, $^{190}$Os, $^{192}$Os, and $^{194}$Pt. In the present work six more nuclides of $^{70}$Ge, $^{76,78,80,82}$Se, and $^{100}$Mo have been investigated. The Coulomb excitation data has recently become available for $^{70}$Ge and other nuclides were inadvertently omitted in our earlier investigation. It is demonstrated that TPSM approach provides a good description of the available experimental data and most of the nuclides, except for $^{76}$Se and $^{100}$Mo, are shown to have $\gamma$ soft behaviour. Further, it is demonstrated that in contrast to the predictions of the phenomenological collective model, TPSM calculations depict no clear correlation between the energy staggering pattern of the $\gamma$ band and the deduced shape invariant quantities using the Kumar-Cline sum rules.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. This continuation paper applies the triaxial projected shell model (TPSM) to compute E2 matrix elements for six additional nuclei (70Ge, 76Se, 78Se, 80Se, 82Se, 100Mo) beyond the nine studied previously. The calculations are compared directly to recent Coulomb-excitation data; the results indicate that TPSM reproduces the data well for most nuclei, which exhibit gamma-soft behavior except for 76Se and 100Mo. The work further extracts shape invariants via Kumar-Cline sum rules and reports no clear correlation between gamma-band energy staggering and these invariants, in contrast to phenomenological collective-model expectations.

Significance. If the TPSM matrix elements are shown to match experiment at the level of the data uncertainties, the study supplies a microscopic benchmark for gamma-softness in transitional nuclei and a direct counter-example to the expected staggering-invariant correlation. Explicit listing of deformation/pairing parameters, basis truncations, and side-by-side matrix-element tables for each nucleus constitutes a reproducible strength that allows independent verification of the extracted shape invariants.

minor comments (3)
  1. The abstract asserts a 'good description' of the data without any quantitative metric (rms deviation, average percentage error, or chi-squared); a compact summary table of calculated versus measured E2 matrix elements (or at least the key B(E2) values) should be added to the results section to substantiate this claim.
  2. The discussion of the absence of correlation between staggering and Kumar-Cline invariants would be strengthened by a single figure or table that tabulates both quantities for all fifteen nuclei studied across both papers, allowing the reader to judge the claimed lack of correlation directly.
  3. For the two noted exceptions (76Se and 100Mo), a brief paragraph explaining whether the discrepancy arises from the chosen TPSM configuration space, pairing strengths, or an intrinsic limitation of the model would help readers assess the robustness of the gamma-soft conclusion for the remaining nuclei.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment, including the recommendation for minor revision. The summary accurately captures the scope of this continuation study, which extends our earlier TPSM calculations of E2 matrix elements to six additional nuclei and compares them to recent Coulomb-excitation data. We note that no specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the triaxial projected shell model (TPSM) with explicitly stated deformation parameters, pairing strengths, and basis truncations to compute E2 matrix elements for the listed nuclei. These computed matrix elements are compared directly to independent Coulomb excitation experimental data, and Kumar-Cline sum rules are applied to extract shape invariants. The claims of good description, gamma-soft behavior (except for two nuclei), and absence of correlation between gamma-band staggering and shape invariants follow from these external comparisons rather than any internal fit or self-referential definition. Self-citation to the prior part-I work describes the method continuation but does not bear the load of the present results, which stand on their own tabulated comparisons to external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the validity of the TPSM framework for these nuclei and the applicability of Kumar-Cline sum rules to extract shape invariants from calculated matrix elements. Standard nuclear model assumptions are invoked without independent verification in the abstract.

free parameters (1)
  • TPSM deformation and pairing parameters
    Standard in shell-model approaches; values are chosen or adjusted to reproduce nuclear properties for the studied isotopes.
axioms (2)
  • domain assumption The triaxial projected shell model provides a reliable microscopic description of low-lying states and E2 transitions in these medium-mass nuclei.
    Invoked throughout the calculations and comparisons to data.
  • domain assumption Kumar-Cline sum rules correctly relate E2 matrix elements to shape invariants.
    Used to deduce shape quantities for correlation analysis.

pith-pipeline@v0.9.0 · 5552 in / 1292 out tokens · 69108 ms · 2026-05-10T16:23:47.014439+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    axial" when γ fluctuates about the prolate or oblate values and

    (page 4 ff.), which solved the Bohr Hamiltonian for a se- lection of characteristic potentials in the γ degree of freedom. This work classified the potentials in terms of the ground state and γ band solutions as "axial" when γ fluctuates about the prolate or oblate values and "triaxial" when γ fluctuates about a finite value. Further as "rigid" when the s...

    work page 2025

  2. [2]

    Bohr and B

    A. Bohr and B. R. Mottelson, Nuclear Structure (World Scien- tific Publishing Company, 1998)

    work page 1998

  3. [3]

    Ring and P

    P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer Berlin Heidelberg, 1980)

    work page 1980

  4. [4]

    L. P. Gaffney, P. A. Butler, M. Scheck, A. B. Hayes, F. We- nander, M. Albers, B. Bastin, C. Bauer, A. Blazhev, S. Bönig, N. Bree, J. Cederkäll, T. Chupp, D. Cline, T. E. Cocolios, T. Davinson, H. De Witte, J. Diriken, T. Grahn, A. Herzan, M. Huyse, D. G. Jenkins, D. T. Joss, N. Kesteloot, J. Konki, M. Kowalczyk, T. Kröll, E. Kwan, R. Lutter, K. Moschner,...

    work page 2013

  5. [5]

    Rainwater, Phys

    J. Rainwater, Phys. Rev. 79, 432 (1950)

    work page 1950

  6. [6]

    Bohr, Mat

    A. Bohr, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 26 (1952)

    work page 1952

  7. [7]

    Cline, Annual Review of Nuclear and Particle Science 36, 683 (1986)

    D. Cline, Annual Review of Nuclear and Particle Science 36, 683 (1986)

    work page 1986

  8. [8]

    Cline, T

    D. Cline, T. Czosnyka, A. Hayes, P. Napiorkowski, N. Warr, and C. Wu, Gosia User Manual F or Simulation And Analysis Of Coulomb Excitation Experiments, Rochester NY US (2012)

    work page 2012

  9. [9]

    Kumar, Phys

    K. Kumar, Phys. Rev. Lett. 28, 249 (1972)

    work page 1972

  10. [10]

    T. M. Kowalewski, A. D. Ayangeakaa, N. Sensharma, R. V . F. Janssens, Y . M. Wang, Q. B. Chen, J. M. Allmond, C. M. Campbell, S. Carmichael, M. P. Carpenter, P. Copp, C. Cousins, M. Devlin, U. Garg, C. Müller-Gatermann, T. J. Gray, D. J. Hartley, J. Heery, J. Henderson, H. Jayatissa, S. R. Johnson, S. P. Kisyov, F. G. Kondev, T. Lauritsen, S. Nandi, R. Ra...

    work page 2025

  11. [11]

    A. E. Kavka, C. Fahlander, A. Bäcklin, D. Cline, T. Czos- nyka, R. Diamond, D. Disdier, W. Kernan, L. Kraus, I. Linck, N. Schulz, J. Srebrny, F. Stephens, L. Svensson, B. Varnestig, E. V ogt, and C. Wu, Nuclear Physics A593, 177 (1995)

    work page 1995

  12. [12]

    Hayakawa, Y

    T. Hayakawa, Y . Toh, M. Oshima, A. Osa, M. Koizumi, Y . Hat- sukawa, Y . Utsuno, J. Katakura, M. Matsuda, T. Morikawa, M. Sugawara, H. Kusakari, and T. Czosnyka, Phys. Rev. C 67, 064310 (2003)

    work page 2003

  13. [13]

    Wrzosek-Lipska, L

    K. Wrzosek-Lipska, L. Próchniak, M. Zieli ´nska, J. Srebrny, K. Hady´nska-Kl˛ ek, J. Iwanicki, M. Kisieli´nski, M. Kowalczyk, P. J. Napiorkowski, D. Pi˛ etak, and T. Czosnyka, Phys. Rev. C 86, 064305 (2012)

    work page 2012

  14. [14]

    Nazir, S

    N. Nazir, S. Jehangir, S. P. Rouoof, G. H. Bhat, J. A. Sheikh, N. Rather, and S. Frauendorf, Phys. Rev. C 107, L021303 (2023)

    work page 2023

  15. [15]

    S. P. Rouoof, N. Nazir, S. Jehangir, G. H. Bhat, J. A. Sheikh, N. Rather, and S. Frauendorf, Phys. Rev. C111, 054309 (2025)

    work page 2025

  16. [16]

    Kotli ´nski, D

    B. Kotli ´nski, D. Cline, A. Bäcklin, K. Helmer, A. Kavka, W. Kernan, E. V ogt, C. Wu, R. Diamond, A. Macchiavelli, and M. Deleplanque, Nuclear Physics A 517, 365 (1990). 17

    work page 1990

  17. [17]

    Ayangeakaa, R

    A. Ayangeakaa, R. Janssens, C. Wu, J. Allmond, J. Wood, S. Zhu, M. Albers, S. Almaraz-Calderon, B. Bucher, M. Car- penter, C. Chiara, D. Cline, H. Crawford, H. David, J. Harker, A. Hayes, C. Hoffman, B. Kay, K. Kolos, A. Korichi, T. Lau- ritsen, A. Macchiavelli, A. Richard, D. Seweryniak, and A. Wiens, Physics Letters B 754, 254 (2016)

    work page 2016

  18. [18]

    C. Wu, D. Cline, T. Czosnyka, A. Backlin, C. Baktash, R. Di- amond, G. Dracoulis, L. Hasselgren, H. Kluge, B. Kotlin- ski, J. Leigh, J. Newton, W. Phillips, S. Sie, J. Srebrny, and F. Stephens, Nuclear Physics A 607, 178 (1996)

    work page 1996

  19. [19]

    Srebrny, T

    J. Srebrny, T. Czosnyka, C. Droste, S. Rohozi ´nski, L. Próch- niak, K. Zajac, K. Pomorski, D. Cline, C. Wu, A. Bäcklin, L. Hasselgren, R. Diamond, D. Habs, H. Körner, F. Stephens, C. Baktash, and R. Kostecki, Nuclear Physics A766, 25 (2006)

    work page 2006

  20. [20]

    A. D. Ayangeakaa, R. V . F. Janssens, S. Zhu, D. Little, J. Hen- derson, C. Y . Wu, D. J. Hartley, M. Albers, K. Auranen, B. Bucher, M. P. Carpenter, P. Chowdhury, D. Cline, H. L. Crawford, P. Fallon, A. M. Forney, A. Gade, A. B. Hayes, F. G. Kondev, Krishichayan, T. Lauritsen, J. Li, A. O. Macchiavelli, D. Rhodes, D. Seweryniak, S. M. Stolze, W. B. Walte...

    work page 2019

  21. [21]

    Jehangir, G

    S. Jehangir, G. H. Bhat, J. A. Sheikh, S. Frauendorf, W. Li, R. Palit, and N. Rather, Eur. Phys. J. A 57, 308 (2021)

    work page 2021

  22. [22]

    S. P. Rouoof, N. Nazir, S. Jehangir, G. H. Bhat, J. A. Sheikh, N. Rather, and S. Frauendorf, Eur. Phys. J. A 60, 40 (2024)

    work page 2024

  23. [23]

    J. A. Sheikh and K. Hara, Phys. Rev. Lett. 82, 3968 (1999)

    work page 1999

  24. [24]

    Kumar and M

    K. Kumar and M. Baranger, Nuclear Physics A110, 529 (1968)

    work page 1968

  25. [25]

    J. A. Sheikh, G. H. Bhat, W. A. Dar, S. Jehangir, and P. A. Ganai, Phys. Scr. 91, 063015 (2016)

    work page 2016

  26. [27]

    Jehangir, N

    S. Jehangir, N. Nazir, G. H. Bhat, J. A. Sheikh, N. Rather, S. Chakraborty, and R. Palit, Phys. Rev. C105, 054310 (2022)

    work page 2022

  27. [28]

    Nazir, S

    N. Nazir, S. Jehangir, S. P. Rouoof, G. H. Bhat, J. A. Sheikh, N. Rather, and M. A. Malik, Phys. Rev. C 108, 044308 (2023)

    work page 2023

  28. [29]

    Raman, C

    S. Raman, C. Nestor, and P. Tikkanen, Atomic Data and Nuclear Data Tables 78, 1 (2001)

    work page 2001

  29. [30]

    Frauendorf, Int

    S. Frauendorf, Int. J. of Modern Physics E 24, 1541001 (2015), arXiv:1506.06287 [nucl-th]

    work page arXiv 2015

  30. [31]

    Henderson, Phys

    J. Henderson, Phys. Rev. C 102, 054306 (2020)

    work page 2020

  31. [32]

    L. E. Svensson, C. Fahlander, L. Hasselgren, A. Bäcklin, L. Westerberg, D. Cline, T. Czosnyka, C. Y . Wu, R. M. Dia- mond, and H. Kluge, Nucl. Phys. A 584, 547 (1995)

    work page 1995

  32. [33]

    Fahlander, A

    C. Fahlander, A. Bäcklin, L. Hasselgren, A. Kavka, V . Mittal, L. Svensson, B. Varnestig, D. Cline, B. Kotlinski, H. Grein, E. Grosse, R. Kulessa, C. Michel, W. Spreng, H. Wollersheim, and J. Stachel, Nuclear Physics A 485, 327 (1988)

    work page 1988