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REVIEW 4 major objections 6 minor 1 cited by

Shell effects in nuclear charge radii based on Skyrme density functionals

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single Cooper-pair-condensation correction to the Skyrme HFB charge-radius formula reproduces the measured shell-closure kinks at N=28, 82, and 126 and the shell quenching at N=50 in even-even Ca, Ni, Sn, and Pb isotopes.

desk verdict First Skyrme-HFB test of the pair-condensation charge-radius correction shows real kinks, but the unfitted constant a0 makes the result vulnerable; worth refereeing with a demand for sensitivity analysis. read the letter →

arxiv 2506.20414 v1 pith:ALR7CJXO submitted 2025-06-25 nucl-th

Rong An , Shuai Sun , Xiang Jiang , Na Tang , Li-Gang Cao , Feng-Shou Zhang This is my paper · ORCID
classification nucl-th MSC 81V35 PACS 21.10.Ft21.60.Jz21.30.Fe
keywords nuclearchargeradiiSkyrmedensityfunctionalsHartree-Fock-BogoliubovCooperpaircondensationshellclosurequenchingdifferentialisotopeshifts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Skyrme density functionals have long failed to reproduce the abrupt kinks in nuclear charge radii at shell closures. This paper argues that the failure is cured by adding one term, $a_0\sqrt{A}|D_n-D_p|$, to the charge-radius formula, where $D_{n,p}=\sum_{k>0}u_kv_k$ measures the Cooper-pair condensation of neutrons or protons near the Fermi surface. With the coefficient $a_0=0.561$ taken unchanged from earlier relativistic calculations, Skyrme HFB with the SLy5 and SkM$^*$ forces reproduces the trend of differential charge radii in even-even Ca, Ni, Sn, and Pb isotopes, including kinks at $N=28$, 82, and 126 and shell quenching at $N=50$ in Ni and Sn. If correct, this identifies the neutron-proton pairing asymmetry, rather than an artifact of one relativistic model, as the common origin of these shell effects in nuclear size.

What carries the argument

The load-bearing object is the correction term $a_0\sqrt{A}\,|D_n-D_p|$ appended to the mean-square charge radius formula. Here $D_{n,p}=\sum_{k>0}u_kv_k$ is a sum over quasiparticle amplitudes in the canonical HFB basis, so it measures the occupancy spread, or Cooper-pair condensation, of neutrons and protons near the Fermi surface, and $|D_n-D_p|$ is the neutron-proton asymmetry of that pairing condensate. The $D$ values come directly from the HFB self-consistent solution, while the overall strength $a_0=0.561$ is a fixed phenomenological coefficient fitted earlier to potassium and calcium charge radii and transferred without adjustment to Skyrme HFB. The term adds extra charge radius where neutron and proton pairing differ most, which is exactly at and beyond shell closures, turning the smooth Skyrme HFB curves into curves with kinks and quenching.

What would settle it

Refit $a_0$ independently for each of the four isotopic chains and for each Skyrme force; if the best-fit values scatter well outside the uncertainty of 0.561, the universal transfer premise is false. A complementary experiment is a high-precision charge radius of $^{100}$Sn, where HFB$^*$ predicts a kink that plain HFB does not.

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Extended reading notes

Core claim

The central claim is that the modified charge-radius formula $R_{\rm ch}^2=\langle r_p^2\rangle+0.7056+a_0\sqrt{A}\,|D_n-D_p|$ with $a_0=0.561$ makes non-relativistic Skyrme HFB calculations reproduce the measured differential charge radii of even-even Ca, Ni, Sn, and Pb isotopes, whereas the same calculations without the extra term give smooth, monotonic curves that miss the shell closures. In the modified calculations, the kinks at $N=28$, 82, and 126 appear, the $N=50$ shell quenching appears in Ni and Sn, and the inverted parabolic-like shapes between filled shells are captured and weaken from Ca toward Pb. The authors take this as evidence that the difference between neutron and proton Cooper-pair condensation around the Fermi surface, computed self-consistently from the HFB wave functions, is the mechanism behind the discontinuous behaviour of nuclear charge radii.

Load-bearing premise

The argument rests on one fitted number: $a_0=0.561$, obtained from potassium and calcium charge radii in a relativistic model, must stay valid in Skyrme HFB and across Ca, Ni, Sn, and Pb.

Editorial extensions

If this is right

  • The same formula and the same $a_0$ work in both relativistic and non-relativistic mean-field frameworks, so the correction offers a unified way to add shell effects to mean-field charge radii.
  • Skyrme HFB acquires the ability to describe fine structure in charge radii, including the kinks at $N=28$, 82, and 126 and the $N=50$ quenching, without changing the energy functional.
  • The inverted parabolic-like radii between filled shells are reproduced and are predicted to weaken progressively from Ca to Pb isotopic chains.
  • A shell-closure kink at $^{100}$Sn is predicted by both HFB$^*$ and RHB$^*$, giving a falsifiable target for future precision measurements.
  • The residual deviations in Ca and Pb show that the correction is not complete by itself; the authors attribute part of the Pb mismatch to the neglect of shape deformation toward neutron-deficient regions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the natural next test is to apply the same $|D_n-D_p|$ term to odd-A and odd-odd nuclei, where the parent formula produces odd-even staggering in Ca and K; matching the Ni, Sn, and Pb staggering data would test whether the same mechanism covers both shell kinks and odd-even effects.
  • Editorial inference: because $a_0$ is fixed and universal in the paper, the formula makes concrete predictions for other magic-adjacent chains, for example Kr, Sr, or Zr near $N=50$ and 82, before those radii are precisely measured.
  • Editorial inference: the Pb failure toward neutron-deficient isotopes suggests a direct extension in which the same pair-condensation term is included on top of deformed Skyrme HFB; if the $N=126$ kink survives deformation, the mechanism is robust, and if not, the missing physics lies in the interplay of pairing and shape.
  • Editorial inference: the self-consistent $D_{n,p}$ values could be used as a cheap diagnostic of pairing saturation in functional fits, separate from their role in charge radii.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript extends a phenomenological correction to the nuclear charge radius formula, R_ch^2 = <r_p^2> + 0.7056 + a0 sqrt(A) |D_n - D_p| with a0 = 0.561, to Skyrme Hartree-Fock-Bogoliubov calculations with the SLy5 and SkM* functionals. The correction term, taken from Ref. [69], is intended to account for neutron and proton Cooper-pair condensation around the Fermi surface. The authors compute differential mean-square charge radii and the three-point indicator Delta_r^(2N) for even-even Ca, Ni, Sn, and Pb isotopic chains, compare with experimental data and with relativistic RHB(NL3) results, and conclude that the modified model reproduces the shell-closure kinks at N = 28, 82, and 126 and predicts shell quenching at N = 50 in Ni and Sn.

Significance. If the claimed result is robust, the paper provides a useful extension of a phenomenological pairing-condensation correction to non-relativistic energy density functionals, a class of models that has difficulty describing kinks and odd-even effects in charge radii. The manuscript is honest about several limitations, uses two Skyrme forces and a relativistic comparison, and explicitly identifies that a0 was fitted in Ref. [69]. The main value is as a systematic test; however, the central claim rests on an untested transfer of a fitted constant across frameworks and chains, and the evidence is presented only through visual comparison. No quantitative validation metric, no benchmark of the Delta D scales, and no code or reproducibility details are provided, so the improvement, while plausible, is not yet demonstrated. The paper would be significantly strengthened by adding transferability checks and error measures.

major comments (4)
  1. [Section II, Eq. (3) and Table I] The load-bearing assumption is that a0 = 0.561, fitted in Ref. [69] to potassium and calcium charge radii within a relativistic mean-field framework, transfers without renormalization to Skyrme HFB calculations that use chain-dependent pairing strengths V0. The quantity D_{n,p} = sum_{k>0} u_k v_k depends on the pairing interaction, the quasiparticle cutoff, and the mean-field model, so the scale of |D_n - D_p| in Skyrme HFB need not match the RMF scale that fixed a0. The manuscript does not compare D_n and D_p between HFB and RHB for common nuclei, does not refit a0 within Skyrme HFB, and does not test the sensitivity of the predicted kinks to the value of a0. Please add a transferability test, for example by varying a0 by +/-20% or by fitting a0 to one chain and checking the others, and report representative D_n and D_p values for the four chains.
  2. [Section III and Figs. 1-4] The central claims that HFB* 'reproduces' the trends, that deviations are 'slight', and that SkM* behaves better than SLy5 for tin are based entirely on visual inspection. No RMS deviation, mean absolute error, chi-square, or other quantitative metric is given for the differential charge radii or for Delta_r^(2N). Without such metrics the reader cannot judge whether the improvement over HFB is statistically meaningful, whether the overestimates in 46-50Ca and the Pb slopes are acceptable, or which force is more successful. Please add per-chain error measures for HFB, HFB*, RHB, and RHB* relative to the experimental data, and report the numerical slopes around N = 28, 50, 82, and 126.
  3. [Section III, calcium discussion] The calcium chain is presented as a validation of the model, but this is partly circular because a0 = 0.561 was adjusted in Ref. [69] to reproduce the inverted parabolic shape and the odd-even staggering in potassium and calcium charge radii. The authors state this in Section II and then in Section III interpret the Ca agreement as support for the model. The Ca results should be framed as a consistency check of the cross-framework transfer rather than as an independent test. The independent validation should rest on the Ni, Sn, and Pb chains, or the authors should refit a0 using Skyrme HFB on calcium and then judge the Ca comparison on that basis.
  4. [Section II, Eq. (4) and Section III, Table I] The numerical implementation lacks the details needed to reproduce D_{n,p}. The value of D_{n,p} depends on the quasiparticle energy cutoff, the spatial box or basis size, and the pairing strengths, but only the V0 values are listed in Table I. Since the correction term and its framework dependence are the core of the paper, please specify the HFB numerical parameters (cutoff, box size, number of basis states, and the neutron/proton pairing gaps used to fit V0). Reporting D_n, D_p, and Delta D for representative nuclei such as 40Ca, 48Ca, 68Ca, 100Sn, 132Sn, and 208Pb would also help the reader assess the transferability concern raised above.
minor comments (6)
  1. [Figs. 1 and 2 captions] In Figs. 1 and 2, HFB* and RHB* are both denoted by 'open diamond', so the two model families cannot be distinguished in black-and-white print; please use different markers or line styles and define them clearly in each caption.
  2. [Fig. 1 caption and Section III] The reference nucleus for the nickel chain is inconsistent: Fig. 1 says 'relative to the references 40Ca and 56Ni', while the text says 'with respect to reference nuclei 40Ca and 58Ni'. Please unify the definition.
  3. [Eq. (3)] The constant 0.7056 is the square of the proton charge radius, presumably r_p = 0.84 fm. Please state this explicitly so that the value is not a magic number, and give the relevant references at that point.
  4. [Throughout] There are numerous grammatical and typographical errors, such as 'gives raise to', 'the influence of new term', 'the theoretical results obtained by Eq. (3) are labeled by HFB*', and 'combining the existing literatures'. A careful language edit is needed.
  5. [Section IV] The summary states that shell closure effects are 'slightly distorted' in the Ca and Pb isotopes, but the body text describes overestimated 46-50Ca radii and deviating Pb slopes. Please quantify these deviations with the metrics requested in the major comments so that 'slightly' is meaningful.
  6. [Figs. 2 and 4] The text and captions refer to the lead reference nucleus as '182Pb'; please verify whether this is the intended experimental reference and state it consistently with the data compilation used.

Circularity Check

1 steps flagged · score 4.0 of 10

Calcium validation inherits the K/Ca-fitted a0; the Skyrme transfer and Ni/Sn/Pb kinks remain independent.

  1. fitted input called prediction [Eq. (3) and Section II (a0); Section IV (Ca claim)]
    "The parameter set a0 = 0.561 is adjusted by reproducing the inverted parabolic-like shape and odd-even oscillation behaviors in charge radii of potassium and calcium isotopes [69]. In our calculations, the same parameter set a0 shown in Eq. (3) is further applied to depict the local variations of nuclear charge radii within Skyrme EDFs. ... The modified model can reproduce the trend of changes of differential charge radii in the calcium, nickel, tin, and lead isotopes."

    The coefficient a0 in Eq. (3) is not derived here; it was fitted, in the same-author Ref. [69], to reproduce the inverted parabolic shape and odd-even staggering of K and Ca charge radii. The present paper then counts the Ca chain among the checks of the model and claims the modified model reproduces the Ca trend. For the Ca leg, the observable used as confirmation is the same observable that fixed a0, so the agreement is partly inherited from the fit rather than being an independent Skyrme prediction. This is only a partial reduction, not a full one, because the transfer from RMF to Skyrme HFB changes Delta D and a0 is not refit here; the Ni, Sn, Pb, and N=50/82/126 results remain independent checks.

full rationale

The central Skyrme-transfer claim has genuine independent content: a0 is fixed across all four chains, pairing strengths are anchored to empirical gaps (Table I), and the Ni, Sn, and Pb comparisons are out-of-sample relative to the K/Ca calibration. The N=50, 82, and 126 kinks are therefore not statistically forced by a fit performed in this paper. The one genuinely circular element is the use of Ca as a validation chain after a0 was adjusted to reproduce K and Ca charge radii in Ref. [69]; for that chain the success inherits the calibration. The paper itself acknowledges that the added term is phenomenological, so there is no hidden first-principles claim. Score 4 reflects this partial circularity while recognizing that the rest of the claimed phenomenology is independently tested.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central calculation depends on a small set of fitted or selected parameters: a0 (fitted to K/Ca), per-chain pairing strengths V0 (fitted to pairing gaps), and the mixed pairing choice. The correction term itself is an ad hoc ansatz imported from earlier work by the same group, and its transfer to Skyrme HFB is assumed. No new entity is introduced.

free parameters (4)
  • a0 (global charge-radius correction amplitude) = 0.561
    Fitted in Ref. [69] to reproduce the inverted parabolic shape and odd-even staggering in K and Ca charge radii; reused unchanged here. It is the main free parameter of the correction term in Eq. (3).
  • V0, SLy5 pairing strength (per chain) = Ca: 245.2, Ni: 362.8, Sn: 410.0, Pb: 468.0 MeV fm^3
    Table I: adjusted to reproduce empirical pairing gaps of 44Ca, 64Ni, 122Sn, and 198Pb. Pairing strengths affect HFB occupations and thus the Delta D correction.
  • V0, SkM* pairing strength (per chain) = Ca: 262.8, Ni: 313.5, Sn: 366.2, Pb: 396.0 MeV fm^3
    Table I: same calibration as the SLy5 set; chain-dependent pairing strength is an input to the central calculation.
  • Pairing mixing parameter eta = 0.5 (mixed-type pairing)
    Chosen by hand in Section II; no sensitivity study on eta is reported, though it changes the pair condensation Delta D.
assumptions (5)
  • domain assumption Spherical symmetry and time-reversal symmetry are assumed; only even-even nuclei are studied.
    The HFB framework is restricted to spherical even-even nuclei (Section II), so deformation effects, known to matter in neutron-deficient Pb, are excluded. The authors acknowledge this in Section III.
  • ad hoc to paper The charge radius correction term a0 sqrt(A) |Dn - Dp| is a valid phenomenological representation of pair condensation effects.
    Eq. (3) is taken from Ref. [69]; the paper gives no derivation from the Skyrme functional and calls the term phenomenological in Section III.
  • domain assumption Dn,p = sum over canonical states u_k v_k measures Cooper-pair condensation relevant to charge radii.
    Eq. (4) uses HFB occupation amplitudes; the link between this pairing measure and charge radii is asserted from prior work, not demonstrated here.
  • ad hoc to paper The globally fitted constant a0 = 0.561 transfers from relativistic K/Ca fits to Skyrme HFB for all four chains.
    Section II states the same parameter set is applied; no re-fitting or sensitivity analysis is given.
  • domain assumption Mixed-type pairing (eta = 0.5) with a density-dependent zero-range force is appropriate for all chains.
    Section II chooses mixed pairing and calibrates V0 to pairing gaps; no comparison with volume or surface pairing is shown.

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Pith. "Pith review of Shell effects in nuclear charge radii based on Skyrme density functionals." pith.science (2026). https://pith.science/paper/ALR7CJXO

@misc{pith2026250620414,
  author       = {Pith},
  title        = {Pith review of: Shell effects in nuclear charge radii based on Skyrme density functionals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALR7CJXO}},
  note         = {Machine review of arXiv:2506.20414}
}
abstract

A unified description of the charge radii throughout the entire nuclide chart plays an essential role for our understanding of nuclear structure and fundamental nuclear interactions. In this work, the influence of new term, which catches the spirit of neutron and proton pairs condensation around Fermi surface, on the charge radii has been investigated based on the Skyrme density functionals with the effective forces SLy5 and SkM$^{*}$. The differential charge radii of even-even Ca, Ni, Sn, and Pb isotopes are employed to evaluate the validity of this theoretical model. Meanwhile, the results obtained by the relativistic density functional with the effective Lagrangian NL3 are also shown for the quantitative comparison. The calculated results suggest that the modified model can improve the trend of changes of the differential charge radii along Ca, Ni, Sn, and Pb isotopic chains, especially the shell closure effect at the neutron numbers $N=28$, 82 and 126. The shell quenching phenomena of charge radii can also be predicted at the neutron number $N=50$ along the corresponding Ni and Sn isotopes, respectively. The inverted parabolic-like shapes between the two fully filled shells can also be observed, but the amplitude is gradually weakened from Ca to Pb isotopic chains. Combining the existing literatures, it suggests that the discontinuous behavior in nuclear charge radii can be described well by considering the influence of neutron Cooper pairs condensation around Fermi surface.

Figures

Figures reproduced from arXiv: 2506.20414 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) Mean-square charge radii differences o [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Mean-square charge radii differences o [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) The values of ∆ [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) The values of ∆ [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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