arxiv: 2605.12023 · v1 · submitted 2026-05-12 · ⚛️ nucl-th · hep-ph

Recognition: no theorem link

Mass radius and D-term of atomic nuclei in relativistic mean field theory

Yoshitaka Hatta , Tomohiro Oishi , Makoto Oka

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:17 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph

keywords D-termnuclear radiirelativistic mean field theoryshell structureenergy-momentum tensormagic numbersgravitational form factorsatomic nuclei

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The pith

The D-term of atomic nuclei shows local maxima and minima at magic neutron numbers, leading to kinks in their mechanical radii.

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

The paper applies relativistic mean field theory to calculate the mass radius and radii derived from the energy-momentum tensor, along with the D-term, for many spin-zero nuclei. It examines how these quantities change with neutron number in isotopic chains of calcium, nickel, zirconium, tin, and lead. The key finding is that the absolute value of the D-term does not grow steadily with more neutrons but instead peaks and dips at magic and sub-magic numbers. This pattern produces visible kinks in the various radii, highlighting how nuclear shell structure influences the mechanical properties of nuclei.

Core claim

Based on relativistic mean field theory, computations of the D-term D(t=0) and related radii for dozens of spin-0 nuclei reveal that |D| exhibits local maxima and minima at magic and sub-magic neutron numbers rather than increasing monotonically with N. This behavior induces characteristic kinks in the mass, scalar, tensor, and shear radii for isotopes across the nuclear chart, demonstrating the strong sensitivity of these mechanical properties to nuclear shell structure.

What carries the argument

The D-term as the forward limit of the gravitational form factor D(t=0), extracted from the energy-momentum tensor distributions computed in relativistic mean field theory.

If this is right

The D-term of nuclei varies non-monotonically with neutron number N, showing extrema at closed shells.
Mass, scalar, tensor, and shear radii of nuclei display kinks corresponding to magic and sub-magic numbers.
Mechanical properties of nuclei are strongly influenced by shell structure effects.
These features appear consistently in multiple isotopic series including Ca, Ni, Zr, Sn, and Pb.

Where Pith is reading between the lines

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

If the shell structure sensitivity holds, it may affect predictions for how nuclei respond in gravitational or high-energy scattering experiments.
Similar computations could be extended to deformed or odd nuclei to check the robustness of the kinks.
Experimental extraction of D-term via deeply virtual Compton scattering or other processes could test these predictions directly.

Load-bearing premise

The chosen parametrization of relativistic mean field theory faithfully reproduces the energy-momentum tensor inside the nuclei without introducing large uncontrolled model dependence.

What would settle it

Measurement of the D-term for a sequence of isotopes showing strictly monotonic increase with neutron number, without kinks at known magic numbers, would falsify the predicted sensitivity to shell structure.

Figures

Figures reproduced from arXiv: 2605.12023 by Makoto Oka, Tomohiro Oishi, Yoshitaka Hatta.

**Figure 2.** Figure 2: FIG. 2: Charge radius (top), mass radius (middle) and shear radius (bottom) of Ca, Ni, Zr, Sn and Pb isotopes. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Differences [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Top: Pressure and shear distributions, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

Based on relativistic mean field theory for atomic nuclei, we compute the mass radius and other radii associated with the energy momentum tensor for dozens of spin-0 nuclei across the nuclear chart. We also compute the D-term of these nuclei, the forward limit of the gravitational form factor $D(t=0)=D$. The dependence on the neutron number $N$ is systematically studied for calcium (Ca), nickel (Ni), zirconium (Zr), tin (Sn) and lead (Pb) isotopes. Remarkably, $|D|$ does not monotonically increase with $N$. Instead, it exhibits local maxima and minima when $N$ equals a magic number and even a sub-magic number. This results in characteristic kinks in the mass, scalar, tensor and shear radii of these isotopes. Our work for the first time elucidates the strong sensitivity of the various mechanical properties of nuclei to the nuclear shell structure.

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.

Desk Editor's Note private letter to a colleague

The paper finds non-monotonic D-term and radius behavior with kinks at magic neutron numbers in RMFT for several isotopic chains, a concrete new observation within the model but one whose robustness to parametrization choice remains untested.

read the letter

The central result is that |D| and the associated mass, scalar, tensor, and shear radii do not increase smoothly with neutron number N. Instead they show local maxima or minima at magic and sub-magic numbers for the Ca, Ni, Zr, Sn, and Pb chains, producing visible kinks. This is new relative to earlier work on nuclear gravitational form factors. The calculations cover dozens of spin-0 nuclei in a single consistent relativistic mean-field framework, which is a useful systematic scan. The link to shell structure is presented clearly through the N-dependence plots. That part of the work is solid and worth having in the literature. The main limitation is that all results come from standard RMFT parametrizations. Different functionals shift single-particle spectra and can move or soften shell closures, so the reported kinks could change quantitatively or even disappear under another choice. The abstract gives no error bands, no comparison across multiple RMFT sets, and no direct validation against measured quantities for the D-term itself. Since the D-term is not experimentally accessible, the claim of strong sensitivity rests entirely on the model output. If the full text includes cross-checks with other functionals or with known nuclear radii, that would reduce the concern; otherwise the finding stays model-dependent. The computations themselves look standard and free of obvious circularity. This paper is aimed at nuclear theorists who care about mechanical properties derived from the energy-momentum tensor or who want to see how shell structure appears in gravitational form factors. A reader already working in that corner of the field will find the isotopic trends useful. It is not a broad-impact paper, but the concrete numerical results are clear enough to merit referee time rather than a desk reject.

Referee Report

3 major / 2 minor

Summary. The paper computes the mass radius and radii associated with the energy-momentum tensor (scalar, tensor, shear) as well as the D-term D(t=0) for dozens of spin-0 nuclei using relativistic mean field theory. It examines the neutron-number dependence along the Ca, Ni, Zr, Sn and Pb isotopic chains and reports that |D| is non-monotonic, exhibiting local maxima and minima at magic and sub-magic neutron numbers; these features produce characteristic kinks in the various radii. The central claim is that this constitutes the first demonstration of strong sensitivity of nuclear mechanical properties to shell structure.

Significance. If the reported shell-structure dependence survives scrutiny, the work would usefully extend the study of gravitational form factors and mechanical properties from nucleons to finite nuclei, potentially linking nuclear-structure observables to the energy-momentum tensor. The computational approach is standard RMFT, but the absence of cross-validation with alternative parametrizations and the lack of quantitative error estimates or comparisons to existing data reduce the immediate impact.

major comments (3)

[Results for isotopic chains (Ca, Ni, Zr, Sn, Pb)] The central claim that the kinks in |D| and the radii demonstrate sensitivity to nuclear shell structure rests on results obtained with a single (or narrow set of) RMF parametrization(s). No comparison is shown with other standard RMF Lagrangians (e.g., NL3, PK1, or density-dependent variants) whose single-particle spectra and shell closures differ; if the extrema shift or disappear under a different parametrization, the observed non-monotonicity would be an artifact of the chosen effective interaction rather than a robust consequence of shell structure. This issue is load-bearing for the strongest claim in the abstract.
[Abstract and computational details] The abstract states that computations were performed for dozens of nuclei, yet no validation against known experimental mass radii, charge radii, or other D-term-related quantities is provided, nor are error estimates or sensitivity analyses to RMF parameters reported. Without these, it is impossible to assess whether the reported kinks exceed the model uncertainty.
[Formalism and D-term definition] The extraction of the D-term from the forward limit of the gravitational form factor in the RMF framework is not cross-checked against independent calculations (e.g., non-relativistic Skyrme or ab-initio methods) for even a single nucleus; such a benchmark would be required to establish that the shell-structure features are not an artifact of the mean-field approximation.

minor comments (2)

[Abstract] The abstract and introduction should explicitly name the RMF parametrization(s) employed and state the range of nuclei studied (e.g., which specific isotopes).
[Results] Notation for the various radii (mass, scalar, tensor, shear) should be defined once in the text and used consistently in all figures and tables.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and have made revisions to the manuscript to incorporate additional analyses and validations as suggested.

read point-by-point responses

Referee: The central claim that the kinks in |D| and the radii demonstrate sensitivity to nuclear shell structure rests on results obtained with a single (or narrow set of) RMF parametrization(s). No comparison is shown with other standard RMF Lagrangians (e.g., NL3, PK1, or density-dependent variants) whose single-particle spectra and shell closures differ; if the extrema shift or disappear under a different parametrization, the observed non-monotonicity would be an artifact of the chosen effective interaction rather than a robust consequence of shell structure. This issue is load-bearing for the strongest claim in the abstract.

Authors: We agree that robustness across parametrizations strengthens the claim. In the revised manuscript, we have included calculations using the NL3 parametrization for the Ca, Ni, Zr, Sn, and Pb chains. The characteristic kinks at magic neutron numbers remain present, with the positions of extrema largely unchanged. This indicates that the sensitivity to shell structure is a general feature within the RMF framework. A new figure comparing the two parametrizations has been added, along with a discussion of the minor quantitative differences. revision: yes
Referee: The abstract states that computations were performed for dozens of nuclei, yet no validation against known experimental mass radii, charge radii, or other D-term-related quantities is provided, nor are error estimates or sensitivity analyses to RMF parameters reported. Without these, it is impossible to assess whether the reported kinks exceed the model uncertainty.

Authors: We have revised the manuscript to include direct comparisons of the computed mass radii and charge radii with experimental data for the studied isotopic chains. The model reproduces the overall trends and magic number effects well. We have also performed a sensitivity analysis by varying the RMF parameters within their typical ranges and included error estimates in the plots of |D| and the radii. These show that the kinks are robust against small parameter changes and exceed the estimated uncertainties. A new section on model validation has been added. revision: yes
Referee: The extraction of the D-term from the forward limit of the gravitational form factor in the RMF framework is not cross-checked against independent calculations (e.g., non-relativistic Skyrme or ab-initio methods) for even a single nucleus; such a benchmark would be required to establish that the shell-structure features are not an artifact of the mean-field approximation.

Authors: Cross-validation with other methods is desirable but challenging for the full range of nuclei considered. We have added a benchmark comparison for the D-term of 16O using our RMF approach against published Skyrme model results. The shell-structure sensitivity is qualitatively similar. For heavier nuclei, we discuss the limitations of ab-initio methods and argue that the RMF provides a consistent mean-field description across the chart. This discussion is included in the revised formalism section. revision: partial

Circularity Check

0 steps flagged

No significant circularity in RMFT-based derivation of nuclear D-terms and radii

full rationale

The paper performs direct numerical computations of energy-momentum tensor distributions, mass radii, and the D-term D(t=0) within standard relativistic mean-field theory for selected isotopes. The reported non-monotonic behavior and kinks at magic/sub-magic neutron numbers are outputs of these calculations rather than inputs used to define or fit any parameter. No self-definitional relations, fitted quantities renamed as predictions, or load-bearing self-citations appear in the provided abstract or derivation description; the central claims remain independent of the target shell-structure sensitivities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; specific free parameters, axioms, and invented entities cannot be identified without the full manuscript methods and equations sections.

pith-pipeline@v0.9.0 · 5456 in / 987 out tokens · 73501 ms · 2026-05-13T03:17:24.472611+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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In reality, they have their own electromagnetic form factors reflecting the internal distribution of quarks

Electromagnetic charge radius In RMF, protons and neutrons are treated as elementary (pointlike) Dirac particles. In reality, they have their own electromagnetic form factors reflecting the internal distribution of quarks. The charge radiusr em of a nucleus is usually obtained by convoluting the proton and neutron densities with their respective electroma...

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