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REVIEW 2 major objections 4 minor 1 cited by

Proton shell closures make neutron pairing gaps asymmetric across N=82.

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2026-07-11 04:22 UTC pith:44637WYB

load-bearing objection Solid new Sb masses plus a clean even-Z reading of OES asymmetry at Z=50 and Z=64; HFB works through mid-shell but deformation is prescribed and odd-Z remains open. the 2 major comments →

arxiv 2607.05647 v1 pith:44637WYB submitted 2026-07-06 nucl-ex nucl-th

Interplay between Nuclear Shell Structure and Pairing around Doubly Magic ¹³²Sn

Rane Simpson , Georgios Palkanoglou , Ellen Brisley , Jaime Cardona , Annabelle Czihaly , Sakshi Kakkar , Makar Simonov , Ethan Taylor
show 15 more authors
Coulter Walls Pavithra Weligampola Chris Chambers Fernando Maldonado Millan Ali Mollaebrahimi Dwaipayan Ray Andrew Weaver Jiajun Yu Iris Dillman Alexandros Gezerlis Gerald Gwinner Augusto Machiavelli Stephan Malbrunot-Ettenauer Moritz Pascal Reiter Anna A. Kwiatkowski
This is my paper
classification nucl-ex nucl-th
keywords nuclear shell structurepairing correlationsodd-even staggering132SnHartree-Fock-Bogolyubovmass measurementsneutron-rich nuclei
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Near the doubly magic nucleus 132Sn, the energy cost of adding or removing an unpaired neutron does not change smoothly across the N=82 shell. Existing mass data show that this odd-even staggering is strongly asymmetric at Z=50 and again at the Z=64 proton subshell, and nearly symmetric in proton mid-shell nuclei. Original Hartree-Fock-Bogolyubov calculations reproduce the even-Z pattern by letting neutron pairing grow with proton number; the stronger pairing washes out the N=82 discontinuity away from proton shell closures. New TITAN mass measurements of 137,138Sb extend the map into odd-Z nuclei, where the same competition appears but is harder to capture with the same mean-field tools. The work therefore isolates a concrete, measurable signature of how proton shell structure modulates neutron pairing far from stability.

Core claim

Proton shell structure enhances the asymmetry of neutron odd-even staggering across N=82: the difference between the three-point indicators Δ(3)(N=81) and Δ(3)(N=83) is largest at the proton (sub)shell closures Z=50 and Z=64, and HFB calculations attribute this to neutron pairing correlations that grow with Z and smear the N=82 closure away from those closures.

What carries the argument

The three-point odd-even staggering indicator Δ(3)(N) extracted from binding energies (or from the HFB pairing correlation energy Ecorr), which isolates the pairing contribution to the gap across the N=82 shell and thereby maps how proton shell structure modulates neutron pairing.

Load-bearing premise

The deformed HFB calculations rely on fixed, Z-independent quadrupole deformations taken from an external mass model rather than being determined self-consistently for each nucleus.

What would settle it

A self-consistent deformed HFB calculation (or an independent shell-model calculation with the same single-particle spectrum) that fails to recover the experimental rise of Δ(3)(N=83) from Z=50 to the mid-shell, or new mass measurements of more neutron-rich Sn, Sb or I isotopes that erase the reported recovery of staggering for odd-Z nuclei at N=85,86.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper interprets existing mass data near 132Sn to show that proton (sub)shell structure at Z=50 and Z=64 enhances the asymmetry between neutron odd-even staggering Δ^(3)(N=81) and Δ^(3)(N=83). Original spherical and deformed HFB calculations for even-Z nuclei attribute this to neutron pairing correlations that grow with Z and smear the N=82 closure away from proton shell closures. New TITAN MR-TOF mass measurements of 137Sb (improved precision) and 138Sb (first determination) extend the analysis to odd-Z nuclei, where the shell-pairing interplay is harder to capture phenomenologically or with HFB, motivating further work.

Significance. The result, if robust, clarifies how proton shell structure modulates neutron pairing far from stability, a competition of direct relevance to nuclear structure and r-process modeling. Strengths include independent experimental masses (with calibrant checks and AME consistency for 137Sb), HFB that reproduces Sn OES and even-Z isotonic trends through Z≈60 without ad-hoc fitting to the asymmetry itself, and an explicit link between single-particle gaps (including the Z=64 subshell) and the observed OES pattern. The new 138Sb mass and the even-Z HFB success constitute concrete, falsifiable advances.

major comments (2)
  1. [Supplemental Material, deformed HFB; main text Fig. 1(e)] Supplemental Material (deformed HFB paragraph and Fig. 1(e) discussion): The quantitative agreement for Δ^(3)(N=83) up to Z≈60 relies on prescribed, Z-independent quadrupole deformations β2 taken as FRDM averages rather than variational minima. While the paper notes that this treatment renders E_corr-based Δ^(3) unreliable for Z>60, the main-text claim of successful deformed-HFB description should more explicitly flag this external input as a caveat, because the mid-shell success is not fully self-contained.
  2. [OES including odd-Z nuclei; Summary] OES including odd-Z nuclei section and Summary: The statement that both spherical and deformed HFB are inconsistent with experimental odd-Z Δ^(3)(N) is central to the motivation for future work, yet no quantitative comparison (curves, residuals, or table) is provided. Adding the corresponding HFB results (even if only in the Supplemental Material) would make the acknowledged limitation concrete and allow readers to judge its severity.
minor comments (4)
  1. [Fig. 1 and Phenomenological interpretation] Fig. 1(c)–(e) and related text: The vertical dashed lines marking Z=50 and Z=64 are helpful, but the near-degeneracy argument for the absence of a Z=58 subshell (≲0.7 MeV) would be clearer if the single-particle spectrum of Supplemental Fig. 2 were referenced more explicitly in the main text.
  2. [Experiment; Table I] Table I and Experiment section: The mass-resolving power R≈390 000 and the hyper-EMG fitting procedure are stated, but a brief note on how the 14 counts for 138Sb still yield a reliable centroid (and the assigned 68 keV uncertainty) would aid non-specialist readers.
  3. [Phenomenological interpretation] Eq. (1) and surrounding text: The three-point formula is standard, yet a short remark that higher-order formulas were checked and give consistent trends would preempt questions about finite-difference artifacts near shell closures.
  4. [References / Supplemental Material] References: Recent mass work on nearby chains (e.g., the 2025 Sn and Ce results already cited) is well covered; adding a pointer to the latest AME updates or FRDM deformation tables used for β2 would improve reproducibility.

Circularity Check

0 steps flagged

No significant circularity: experimental OES trends are independent data; HFB is a parameter-free application of the authors' prior formulation validated on Sn, not fitted to the Z-asymmetry being explained.

full rationale

The paper's central phenomenological claim (proton (sub)shell closures at Z=50 and Z=64 maximize the neutron Δ^(3)(N=81)–Δ^(3)(N=83) asymmetry) is read directly from existing AME and recent mass data; no derivation is involved. The HFB calculations that interpret the even-Z trends compute E_corr from a Woods-Saxon + pairing Hamiltonian taken from the authors' earlier formulation, then form the three-point OES of E_corr; Sn isotopic OES is used only as a post-hoc validation check that shows agreement without any ad-hoc parameters adjusted to the multi-Z asymmetry. Deformation parameters are imported as Z-independent FRDM averages rather than variationally optimized or tuned to the OES data, and the paper itself flags the resulting unreliability for Z>60. Odd-Z HFB is acknowledged to fail, so the new Sb masses serve only as experimental extension. No step reduces a claimed prediction to a fitted input or to a self-citation of the target result itself. Minor self-citation of the HFB method exists but is not load-bearing for the observed pattern.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central even-Z claim rests on standard nuclear mean-field machinery (Woods-Saxon single-particle spectrum, HFB pairing, three-point OES formula) plus externally prescribed deformations and the identification of a Z=64 proton subshell gap from the model spectrum. No new particles or forces are invented. Free parameters are those of the trapping potential and the imported β2 values; the OES asymmetry itself is not fitted.

free parameters (3)
  • Woods-Saxon depth V0(N,Z) and surface diffuseness a
    Single-particle spectrum is generated from a Woods-Saxon potential with N–Z-dependent depth and fixed a=0.67 fm; these set the shell gaps that drive the claimed Z=50/64 imprint.
  • Prescribed quadrupole deformations β2(N)
    Deformed HFB uses Z-independent β2 averages from FRDM (e.g. β2(83)=−0.032) rather than variational minimization; absolute Δ^(3) for N=83 depends on this choice.
  • Pairing interaction strength in HFB Δij
    Two-body pairing matrix elements set the scale of E_corr; the paper states no ad-hoc OES fit but inherits the interaction from the prior Gezerlis/Palkanoglou formulation.
axioms (5)
  • domain assumption Three-point finite-difference formula Δ^(3)(n) measures the pairing gap contribution to binding-energy odd-even staggering.
    Eq. (1) and surrounding text; standard in the field but known to mix mean-field and pairing contributions.
  • domain assumption Pairing correlation energy E_corr = E_HFB − E_NS is a valid proxy for the pairing part of experimental OES after slowly varying mean-field pieces cancel in Δ^(3).
    Pairing theory section; used to compare theory to experiment in Fig. 1(b–e).
  • domain assumption Spherical or axially deformed Woods-Saxon plus spin-orbit potential adequately generates the single-particle spectrum near 132Sn, including a ~2–3 MeV proton gap at Z=64.
    Supplemental single-particle spectrum and main-text claim that Z=64 is a pronounced subshell.
  • domain assumption Odd-even (odd-odd) nuclei can be treated as one- (two-) quasiparticle HFB states with imposed number parity.
    Pairing theory section; standard blocking approximation.
  • domain assumption Proton–neutron pairing is negligible for most neutron-rich cases here because Fermi surfaces are separated by several MeV.
    Stated in pairing theory; authors note up to ~20% pn triplet contribution only on the proton-rich side.

pith-pipeline@v1.1.0-grok45 · 21434 in / 3384 out tokens · 29818 ms · 2026-07-11T04:22:57.386083+00:00 · methodology

0 comments
read the original abstract

Shell structure in finite quantum systems gives rise to sudden changes in observable properties, while pairing correlations often compete against such discontinuities. The region near the doubly magic nucleus $^{132}$Sn provides a fertile ground for testing the combined effect of shell structure and pairing. Here, we provide a novel phenomenological interpretation of existing mass data in the vicinity of the $Z=50$ and $N=82$ shell closures, which we further investigate by performing original Hartree-Fock-Bogolyubov (HFB) mean-field calculations for even-$Z$ nuclei: we find that the proton shell structure enhances an asymmetry of the neutron odd-even staggering in binding energies. We also report mass measurements of $^{137,138}$Sb, including the first experimental mass determination of $^{138}$Sb, performed using TRIUMF's Ion Trap for Atomic and Nuclear Science (TITAN). Together with existing experimental data, our results reveal an interplay between shell structure and pairing in odd-$Z$ nuclei which is more challenging to interpret phenomenologically or using HFB, thereby motivating future experimental and theoretical pairing studies in heavy neutron-rich nuclides.

Figures

Figures reproduced from arXiv: 2607.05647 by Alexandros Gezerlis, Ali Mollaebrahimi, Andrew Weaver, Anna A. Kwiatkowski, Annabelle Czihaly, Augusto Machiavelli, Chris Chambers, Coulter Walls, Dwaipayan Ray, Ellen Brisley, Ethan Taylor, Fernando Maldonado Millan, Georgios Palkanoglou, Gerald Gwinner, Iris Dillman, Jaime Cardona, Jiajun Yu, Makar Simonov, Moritz Pascal Reiter, Pavithra Weligampola, Rane Simpson, Sakshi Kakkar, Stephan Malbrunot-Ettenauer.

Figure 1
Figure 1. Figure 1: FIG. 1. Odd-even staggering of binding energies around [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Time-of-Flight spectra (TOF) recorded by TITAN [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Experimental odd-even staggering of binding en [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The pairing correlation energy of even- [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Odd-even staggering of binding energies: The [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Cited by 1 Pith paper

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  1. Exploring the shell structure of trapped superfluid gases

    nucl-th 2026-07 conditional novelty 5.0

    Shell structure in trapped two-component Fermi gases vanishes as interactions approach unitarity, with the interaction's effective range — not the scattering length — controlling how much shell structure survives.

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