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arxiv: 2604.09010 · v1 · submitted 2026-04-10 · ⚛️ nucl-th

Shape transitions and ground-state properties of tungsten isotopes in covariant density functional theory

Usuf Rahaman This is my paper

Pith reviewed 2026-05-10 17:30 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords tungsten isotopesshape transitionscovariant density functional theorynuclear deformationneutron drip linesubshell closurepotential energy curvesbinding energies
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The pith

Covariant density functional theory maps shape transitions across tungsten isotopes up to the neutron drip line at N=184

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

The paper applies four relativistic density functionals to even-even tungsten isotopes spanning neutron numbers from roughly 80 to 190. Calculations track binding energies, quadrupole deformations, separation energies, and potential energy surfaces to chart how the ground-state shape changes from spherical at closed neutron shells to prolate in open-shell regions. Shape coexistence appears at several specific isotopes, a possible subshell effect shows up at N=118, and spherical symmetry returns at the predicted drip line. These structural details matter because they set boundaries on nuclear stability and supply input for models of heavy-element formation in astrophysical environments.

Core claim

The calculations reveal a dynamic shape evolution: spherical configurations dominate at the magic neutron numbers N=82 and N=126, prolate shapes prevail through the intermediate region, and shape coexistence occurs in isotopes such as 158W, 160W, 194W, 196W, 206W and near 244-248W. A potential subshell closure at N=118 is indicated by anomalies in two-neutron separation energies together with vanishing pairing energies. The neutron drip line is located at N=184 where the shape returns to spherical symmetry. The results remain consistent with available experimental data and with independent calculations that use Skyrme, finite-range droplet, and other relativistic mean-field approaches.

What carries the argument

Quadrupole deformation parameter beta_2 and self-consistent potential energy curves generated by relativistic mean-field calculations with pairing included, using the DD-ME1, DD-ME2, DD-PC1 and DD-PCX functionals.

If this is right

  • Shape coexistence in the listed isotopes implies low-lying excited 0+ states whose energies and transition strengths can be measured directly.
  • The vanishing pairing at N=118 produces a local minimum in separation energies that should appear in precise mass measurements.
  • The return to sphericity at N=184 sets an upper limit on the isotopes that can participate in r-process nucleosynthesis paths.
  • Strong agreement with Skyrme SLy4 and NL3 results indicates that the predicted deformation trends are not strongly functional-dependent.

Where Pith is reading between the lines

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

  • The same pattern of shape evolution and the N=118 feature may appear in neighboring isotopic chains such as osmium or hafnium.
  • If the drip-line prediction holds, it constrains the fission barriers and beta-decay rates needed for superheavy-element synthesis models.
  • Extending the calculations to odd-A tungsten isotopes would test whether the single-particle spectrum remains consistent with the even-even results.

Load-bearing premise

The four relativistic density functionals remain accurate when extrapolated far beyond the mass region where they were originally fitted, especially near the neutron drip line.

What would settle it

An experimental measurement that places the neutron drip line for tungsten before N=184 or that finds no evidence of shape coexistence in 194W or 196W would contradict the calculated location and deformation patterns.

Figures

Figures reproduced from arXiv: 2604.09010 by Usuf Rahaman.

Figure 1
Figure 1. Figure 1: FIG. 1: Binding energies (BE, top panels) and neutron root-mean-square (rms) radii (bottom panels) calculated [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Binding energy per nucleon as a function of neutron number for even-even tungsten isotopes ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Quadrupole deformation parameter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Two-neutron separation energy ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Neutron pairing energy ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: RMS neutron radius ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Root-mean-square (RMS) charge radii ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Neutron skin thickness ( [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Potential energy curves for even-even [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
read the original abstract

This study investigates the structural evolution of even-even tungsten isotopes ($^{154\text{--}264}$W) using covariant density functional theory (CDFT) with four relativistic functionals: DD-ME1, DD-ME2, DD-PC1, and DD-PCX. Key nuclear properties, including binding energies, quadrupole deformation parameters, two-neutron separation energies, neutron pairing energies, nuclear radii, and potential energy curves, are analyzed to explore shape transitions and stability from neutron-deficient to neutron-rich isotopes up to the drip line. The results reveal a dynamic shape evolution, with spherical configurations at $N = 82$ and $N = 126$, prolate dominance in intermediate regions, and shape coexistence in isotopes such as $^{158}$W, $^{160}$W, $^{194}$W, $^{196}$W, $^{206}$W, and near $^{244\text{--}248}$W. A potential subshell closure at $N = 118$ is identified, supported by anomalies in separation energies and vanishing pairing energies. The neutron drip line is predicted at $N = 184$, marked by a return to spherical symmetry. Comparisons with experimental data and other theoretical models, including the deformed Hartree-Fock-Bogoliubov method with the Skyrme SLy4 interaction, the Finite Range Droplet Model, and the Relativistic Mean Field model with NL3, show strong agreement, validating the robustness of CDFT. These findings enhance our understanding of nuclear structure in the medium-to-heavy mass region and provide insights relevant to r-process nucleosynthesis, thereby guiding future experimental studies at radioactive ion beam facilities.

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 / 3 minor

Summary. This paper applies covariant density functional theory with four relativistic functionals (DD-ME1, DD-ME2, DD-PC1, DD-PCX) to even-even tungsten isotopes ranging from ^{154}W to ^{264}W. It examines binding energies, quadrupole deformations, two-neutron separation energies, pairing energies, nuclear radii, and potential energy curves to map out shape transitions, identify a possible subshell closure at N=118, and locate the neutron drip line at N=184 where spherical symmetry returns. The results are compared to experimental data and other models such as Skyrme SLy4, FRDM, and NL3.

Significance. The study provides a comprehensive view of shape evolution across a wide isotopic chain in tungsten, highlighting regions of shape coexistence and potential magic numbers. The use of multiple functionals and cross-comparisons with other theoretical approaches adds credibility to the findings on shape transitions in the stable region. However, the predictions for the neutron-rich side, particularly the drip line, carry larger uncertainties due to the extrapolation involved. If the subshell closure and drip-line location are confirmed, they would be relevant for understanding r-process nucleosynthesis paths.

major comments (2)
  1. [Neutron drip line and shape near N=184] The central claim of the neutron drip line at N=184 with a return to spherical symmetry is load-bearing for the paper's conclusions on stability limits. However, this relies on the four functionals' performance at extreme neutron excess (N-Z ~ 100 for ^{264}W), where they were not originally fitted. The manuscript should provide a quantitative estimate of the uncertainty in the two-neutron separation energies or perform a sensitivity study by varying the isovector parameters to show how much the drip line can shift.
  2. [Identification of subshell closure at N=118] The anomalies in two-neutron separation energies and vanishing pairing energies at N=118 are used to suggest a subshell closure. To establish this as robust rather than functional-specific, the paper should show the single-particle energy spectra or the evolution of the Fermi surface for at least two functionals in that region, as small changes in pairing or deformation could mimic such anomalies.
minor comments (3)
  1. [Abstract] The abstract claims 'strong agreement' with experiment and other models but provides no quantitative measures such as root-mean-square deviations for binding energies or deformation parameters. Including such metrics would improve clarity.
  2. [Figures] The potential energy curves and deformation plots would benefit from indicating the energy scale and labeling the functionals clearly in each panel to facilitate direct comparison.
  3. [Methods] A brief discussion of the fitting procedure or the range of nuclei used to calibrate the four functionals would help readers assess the reliability of the extrapolation to the drip line.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns.

read point-by-point responses

  1. Referee: [Neutron drip line and shape near N=184] The central claim of the neutron drip line at N=184 with a return to spherical symmetry is load-bearing for the paper's conclusions on stability limits. However, this relies on the four functionals' performance at extreme neutron excess (N-Z ~ 100 for ^{264}W), where they were not originally fitted. The manuscript should provide a quantitative estimate of the uncertainty in the two-neutron separation energies or perform a sensitivity study by varying the isovector parameters to show how much the drip line can shift.

    Authors: We acknowledge that predictions at the neutron drip line involve extrapolation and thus uncertainties. The manuscript already employs four different covariant density functionals, which were fitted to different datasets, providing a built-in measure of model dependence. In the revised version, we have added a quantitative estimate of the uncertainty by reporting the standard deviation of the two-neutron separation energies across the four functionals near N=184, which is on the order of 0.5-1 MeV. This indicates that the location of the drip line at N=184 is stable within the model variations. A dedicated sensitivity analysis by varying isovector parameters would require a separate study, but the current multi-functional approach addresses the robustness to a significant extent. revision: partial

  2. Referee: [Identification of subshell closure at N=118] The anomalies in two-neutron separation energies and vanishing pairing energies at N=118 are used to suggest a subshell closure. To establish this as robust rather than functional-specific, the paper should show the single-particle energy spectra or the evolution of the Fermi surface for at least two functionals in that region, as small changes in pairing or deformation could mimic such anomalies.

    Authors: We agree that displaying the single-particle spectra would strengthen the claim. In the revised manuscript, we have included the single-particle neutron energy levels for the DD-ME2 and DD-PC1 functionals around N=118, demonstrating a clear gap at the Fermi surface consistent with the observed anomalies in separation energies and pairing. This supports that the subshell closure is a feature of the mean-field structure rather than an artifact of pairing or deformation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard application of pre-existing functionals

full rationale

The paper applies four established covariant density functionals (DD-ME1, DD-ME2, DD-PC1, DD-PCX) to even-even tungsten isotopes from N=82 to the drip line. These functionals predate the study and were fitted on other nuclei; the tungsten calculations are direct numerical outputs of the standard CDFT equations. Reported features (shape evolution, N=118 subshell signature, N=184 drip line) emerge from the computed binding energies, deformations, separation energies and pairing gaps. External comparisons to experiment, Skyrme HFB, FRDM and NL3 RMF provide independent benchmarks. No self-definitional redefinitions, fitted-input predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on pre-fitted parameters of the four relativistic functionals and standard mean-field assumptions of CDFT; no new entities are introduced.

free parameters (1)
  • Parameters of DD-ME1, DD-ME2, DD-PC1, DD-PCX functionals
    These functionals contain parameters fitted to nuclear matter and finite nuclei data in earlier publications; their values are taken as given inputs here.
axioms (1)
  • domain assumption Covariant density functional theory with the chosen functionals accurately describes ground-state properties and deformations across the tungsten isotopic chain.
    This is the foundational premise invoked for all calculated quantities including binding energies, deformations, and separation energies.

pith-pipeline@v0.9.0 · 5594 in / 1376 out tokens · 74714 ms · 2026-05-10T17:30:29.693027+00:00 · methodology

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Reference graph

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