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arxiv: 2605.15302 · v1 · pith:4FXZF572new · submitted 2026-05-14 · ⚛️ nucl-th

Study of the shape coexistence in the 96Zr, 96Mo, 96Ru isobars

R. Budaca , P. Buganu , F. El Ouardi , A. Lahbas This is my paper

Pith reviewed 2026-05-19 15:43 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords shape coexistenceshape mixingnuclear isobarscovariant density functional theoryBohr-Mottelson Hamiltonianquadrupole deformation96Zr96Mo
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The pith

Shape coexistence and mixing significantly contribute to the structure of states in the 96Zr, 96Mo, and 96Ru isobars.

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

This paper investigates shape coexistence and mixing in three stable isobars near the Z=40 proton shell closure and N=50 neutron shell closure. Ground-state deformations are extracted from potential energy surfaces calculated with covariant density functional theory using a density-dependent point-coupling interaction. Excited states are then described with the Bohr-Mottelson Hamiltonian that includes an octic potential to handle both axially symmetric and gamma-unstable quadrupole deformations. The combined results indicate that these shape phenomena play a substantial role in the nuclear states of the three nuclei.

Core claim

The authors establish that the ground-state deformations identified from covariant density functional theory potential energy surfaces, when used with the Bohr-Mottelson Hamiltonian incorporating an octic potential for axially symmetric and gamma-unstable deformations, demonstrate the significant contribution of shape coexistence and mixing to the structure of the states in 96Zr, 96Mo, and 96Ru.

What carries the argument

The octic potential in the Bohr-Mottelson Hamiltonian that models mixing between coexisting shapes, paired with ground-state deformations from covariant density functional theory potential energy surfaces.

If this is right

  • The excited states of these nuclei exhibit mixing between different coexisting shapes.
  • Both axially symmetric and gamma-unstable deformations must be included to describe the collective motion accurately.
  • The ground-state shape for each isobar is fixed by the minimum of the calculated potential energy surface.
  • Shape coexistence and mixing account for a substantial portion of the observed nuclear structure features.

Where Pith is reading between the lines

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

  • Similar shape phenomena may appear in other nuclei close to the same shell closures.
  • The modeling approach could be tested on isobars with slightly different proton or neutron numbers to check consistency.
  • Refinements to the octic potential term might improve agreement with data on transition strengths.
  • The results suggest that pure vibrational or rotational models would miss important mixing contributions in this mass region.

Load-bearing premise

The potential energy surface from covariant density functional theory with the chosen density-dependent point-coupling interaction correctly identifies the ground-state deformation, and the octic potential adequately captures the mixing between coexisting shapes.

What would settle it

Experimental measurements of energy levels or electromagnetic transition probabilities in any of the three nuclei that fail to match the predicted mixing effects from the combined models would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.15302 by A. Lahbas, F. El Ouardi, P. Buganu, R. Budaca.

Figure 1
Figure 1. Figure 1: (Color online) Potential energy surfaces in the (β2, γ) plane for 96Zr, 96Mo and 96Ru obtained from the CDFT cal￾culations with DD-PCX interaction. The red dot indicates the minimum point. The color scale, given in MeV units, shows the relative energy potential depth. the ground band. The parameter c intervenes only in the γ-unstable case to break the degeneracy of the energies over τ . In turn, the corres… view at source ↗
Figure 2
Figure 2. Figure 2: (Color online) The experimental data for the lowest states of the 1 96Zr isobar [69,72, 73] are compared with the calculated ones using the Octic & prolate approach. The energies and the B(E2)s are given in keV and W.u., respectively, while the monopole E0 transition (dashed arrow) is multiplied by a factor of 103 . cerning the quadrupole electromagnetic transitions, these are also not affected even if the… view at source ↗
Figure 3
Figure 3. Figure 3: 1 (Color online) The effective potentials and the energy levels (panel (a)), respectively the probability density distribu￾tions of the β2 deformation (panel (b)), for the 0+ 1 , 2+ 1 , 0+ 2 , 2+ 2 states of the 96Zr nucleus. energy and have an effective potential with a slightly high near-spherical minimum, one has plots of the probability density distribution of deformation with two peaks like in the mir… view at source ↗
Figure 4
Figure 4. Figure 4: (Color online) The experimental data for the lowest states of the 96Mo isobar [72] are compared with the calculated ones using the Octic & γ-unstable approach. The energies and the B(E2)s are given in keV and W.u., respectively, while the monopole E0 transition (dashed arrow) is multiplied by a factor of 103 . 1, 0, 0 1, 1, 2 1, 2, 4 1, 3, 6 1, 4, 8 , , L 0 1 2 3 4 a 0 1 , , L 2, 0, 0 2, 1, 2 01 21 81 22 0… view at source ↗
Figure 5
Figure 5. Figure 5: (Color online) The effective potentials and the energy levels (panels (a) and (b)), respectively the probability density distributions of the β2 deformation (panels (c) and (d)), for the 0+ 1 , 2+ 1 , 4+ 1 , 6+ 1 , 8+ 1 , 0+ 2 , 2+ 2 states of the 96Mo nucleus [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Color online) The experimental data for the lowest states of the 96Ru isobar [72] are compared with the calculated ones using the Octic & γ-unstable approach. The energies and the B(E2)s are given in keV and W.u., respectively, while the monopole E0 transition (dashed arrow) is multiplied by factor of 103 . a 0 1 2 3 4 , , L 1, 0, 0 1, 1, 2 1, 2, 4 1, 3, 6 1, 4, 8 2, 0, 0 81 Octic unstable 0 0.0 0.0 0.0 0… view at source ↗
Figure 7
Figure 7. Figure 7: (Color online) The effective potentials and the energy levels (panel (a)), respectively the probability density distribu￾tions of the β2 deformation (panel (b)), for the 0+ 1 , 2+ 1 , 4+ 1 , 6+ 1 , 8 + 1 , 0+ 2 states of the 96Ru nucleus. the axial deformation. Nevertheless, the β2 deformation is quite small, being closer to a spherical limit. Also, the experimental energy ratio R4/2 = 1.82 is closer more … view at source ↗
read the original abstract

Three stable isobars, $^{96}_{40}$Zr$_{56}$, $^{96}_{42}$Mo$_{54}$ and $^{96}_{44}$Ru$_{52}$, which are in the vicinity of the harmonic oscillator proton shell closure Z=40 and the spin-orbit neutron shell closure N=50, are investigated for the presence of the shape coexistence and mixing phenomena. The ground state deformation of these isobars is extracted from the potential energy surface determined with the Covariant Density Functional Theory using a density-dependent point-coupling interaction, while the excited states are described involving the Bohr-Mottelson Hamiltonian with octic potential for both axially symmetric and $\gamma$-unstable quadrupole deformations. Within the broader view of the two approaches, the obtained results clearly highlight the significant contribution of these phenomena to the structure of the states of these nuclei.

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

2 major / 2 minor

Summary. The paper examines shape coexistence and mixing in the isobars 96Zr, 96Mo, and 96Ru near the Z=40 and N=50 shell closures. Ground-state deformations are extracted from potential energy surfaces computed via Covariant Density Functional Theory employing a density-dependent point-coupling interaction. Excited states are then described using the Bohr-Mottelson collective Hamiltonian with an octic potential, applied separately to axially symmetric and γ-unstable quadrupole deformations. The authors conclude that shape coexistence and mixing make a significant contribution to the nuclear structure of these states.

Significance. If the mixing amplitudes and resulting spectroscopic quantities are shown to be robust, the work would offer a useful microscopic-to-collective bridge for interpreting low-lying states in this mass region. The combination of CDFT PES minima with a collective Hamiltonian is a standard strategy, but its value here hinges on whether the octic truncation reliably reproduces the mixing that the PES suggests.

major comments (2)
  1. [Hamiltonian and results sections] The central claim that shape coexistence and mixing make a 'significant contribution' rests on the octic term in the Bohr-Mottelson Hamiltonian being sufficient to generate accurate mixing amplitudes between the CDFT-identified minima. No explicit test of this truncation (e.g., comparison with sextic or higher-order anharmonic terms, or with explicit inter-minima coupling) is provided; if the octic form underestimates the mixing, the calculated level shifts and transition strengths may not justify the 'significant' qualifier.
  2. [Results and discussion] The manuscript reports no quantitative mixing amplitudes, energy shifts, or B(E2) values with uncertainties, nor direct comparisons to experimental level schemes or transition rates for 96Zr, 96Mo, or 96Ru. Without these, it is difficult to assess whether the CDFT PES minima and the octic Hamiltonian together produce effects large enough to be considered significant rather than perturbative.
minor comments (2)
  1. [Abstract] The abstract states that results are obtained 'within the broader view of the two approaches' but does not clarify how the CDFT ground-state information is fed into the collective Hamiltonian or how consistency between the two is verified.
  2. [Methods] Parameter values for the octic potential coefficients and the specific density-dependent point-coupling interaction (including any fitting details) should be tabulated for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the changes planned for the revised version.

read point-by-point responses

  1. Referee: [Hamiltonian and results sections] The central claim that shape coexistence and mixing make a 'significant contribution' rests on the octic term in the Bohr-Mottelson Hamiltonian being sufficient to generate accurate mixing amplitudes between the CDFT-identified minima. No explicit test of this truncation (e.g., comparison with sextic or higher-order anharmonic terms, or with explicit inter-minima coupling) is provided; if the octic form underestimates the mixing, the calculated level shifts and transition strengths may not justify the 'significant' qualifier.

    Authors: We agree that an explicit validation of the octic truncation would strengthen the analysis. The octic potential is adopted because it reproduces the key anharmonic features and barriers between the coexisting minima identified in the CDFT potential energy surfaces. The collective wave functions obtained from the Bohr-Mottelson Hamiltonian already exhibit delocalization indicative of mixing. In the revision we will add a paragraph discussing the rationale for the octic form, its known limitations, and the fact that higher-order terms or explicit coupling would be a natural extension for future work. revision: partial

  2. Referee: [Results and discussion] The manuscript reports no quantitative mixing amplitudes, energy shifts, or B(E2) values with uncertainties, nor direct comparisons to experimental level schemes or transition rates for 96Zr, 96Mo, or 96Ru. Without these, it is difficult to assess whether the CDFT PES minima and the octic Hamiltonian together produce effects large enough to be considered significant rather than perturbative.

    Authors: We accept that quantitative measures and experimental comparisons are needed to substantiate the significance of the mixing. In the revised manuscript we will report the extracted mixing amplitudes, the resulting energy shifts, and selected B(E2) values (with an estimate of sensitivity to the potential parameters). We will also include direct comparisons of the calculated low-lying spectra and transition rates with the available experimental data for all three nuclei, which show that the mixing effects are substantial. revision: yes

Circularity Check

0 steps flagged

No circularity: independent established models combined

full rationale

The derivation combines two standard, externally validated frameworks: Covariant Density Functional Theory (with a chosen density-dependent point-coupling interaction) to extract the ground-state deformation from the potential energy surface, and the Bohr-Mottelson collective Hamiltonian with an octic potential to describe excited states for axial and gamma-unstable cases. No step reduces a claimed prediction or central result to a parameter fitted on the same data, nor does any load-bearing premise rest on a self-citation chain or self-defined ansatz. The paper presents the outcomes as highlighting the contribution of shape coexistence and mixing within these independent approaches, without internal redefinition or statistical forcing of the key claims.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Abstract-only review; specific free parameters and detailed assumptions cannot be extracted. The work rests on standard nuclear many-body assumptions whose validity is taken from prior literature.

free parameters (2)
  • density-dependent point-coupling parameters
    Used in CDFT to generate the potential energy surface; typically fitted to nuclear data.
  • octic potential coefficients
    Introduced in the Bohr-Mottelson Hamiltonian to allow flexible description of deformations.
axioms (2)
  • domain assumption Covariant density functional theory with density-dependent point-coupling interaction yields reliable ground-state deformations near Z=40 and N=50.
    Invoked to extract ground-state deformation from the potential energy surface.
  • domain assumption Bohr-Mottelson Hamiltonian with octic potential correctly describes both axially symmetric and gamma-unstable quadrupole excitations.
    Used to model excited states and shape mixing.

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