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REVIEW 4 major objections 5 minor 60 references

Neutron skin thickness for $^{nat}$Pb, $^{159}$Tb and effects of deformation on matter radii for Mg, Na, Ar isotopes

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Proton reaction cross sections, reanalysed with a density-scaling folding model, place natural lead's neutron skin at 0.271±0.008 fm and separate deformed from spherical matter radii.

desk verdict The deformation analysis is worth publishing, but the natPb skin claim's 0.008 fm precision rests on an unverified A=208 assumption that the paper itself hedges. read the letter →

arxiv 2506.18936 v1 pith:YPUDDNSQ submitted 2025-06-22 nucl-ex nucl-th

Shingo Tagami , Takaykui Myo , Masanobu Yahiro This is my paper · ORCID
classification nucl-exnucl-th
keywords neutronskinthicknessreactioncrosssectioninteractionfoldingmodelnucleardeformationmatterradiusdensityscaling159Tb
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

The paper asks whether proton reaction cross sections alone can pin down the neutron skin of heavy nuclei, and what exactly the two standard cross-section observables measure. Using the chiral (Kyushu) $g$-matrix folding model with a density-scaling procedure, it extracts $r_{\rm skin}=0.271\pm0.008$ fm for natural lead (treating natPb as $A=208$), consistent with the PREX-II value for ${}^{208}$Pb, and $r_{\rm skin}=0.273\pm0.073$ fm for ${}^{159}$Tb. For Mg, Na, and Ar isotopes, it shows that matter radii extracted from interaction cross sections $\sigma_{\rm I}$ are spherical-limit values, while those from reaction cross sections $\sigma_{\rm R}$ include deformation effects. The result matters because a scattering-based measurement can match parity-violating electron-scattering precision, and because radius tables that mix the two cross-section types conflate deformed and spherical ground-state information.

What carries the argument

The load-bearing mechanism is the density-scaling procedure combined with a folding potential. Given a target density from D1S-GHFB+AMP or SLy7, the model scales the neutron (and sometimes proton) density radially via $\rho_{\rm scaling}(r)=\rho(r/\alpha)/\alpha^3$ so that the folded potential reproduces the measured $\sigma_{\rm R}$ while the proton radius is fixed to the electron-scattering value $r_p$; the scaled neutron radius then defines $r_{\rm skin}$. Two interactions supply the folded potential: the chiral (Kyushu) $g$-matrix for projectile energies up to about 450 MeV, and the Love-Franey $t$-matrix above that, with an ESP-f fine-tuning factor to remove systematic normalization. This combination is what converts a one-dimensional cross-section measurement into a radius difference at a precision of about 0.01 fm.

What would settle it

Measure reaction cross sections for p+${}^{206}$Pb and p+${}^{207}$Pb separately and extract each skin with the same folding model; if the two values differ by more than about 0.02 fm, the constant-skin prediction fails and the reported 0.271 fm for natural lead cannot be taken at face value. A parity-violating electron-scattering run on natural lead analysed with the full isotopic composition would settle the same question directly.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the density-scaling folding model can turn existing proton reaction cross sections into neutron-skin values with a 0.008 fm uncertainty for natural lead once natural lead is approximated as $A=208$, and the extracted $0.271\pm0.008$ fm agrees with the PREX-II value $0.283\pm0.071$ fm for ${}^{208}$Pb. The same framework yields $r_{\rm skin}=0.273\pm0.073$ fm for the deformed nucleus ${}^{159}$Tb, and shows that for Mg, Na, and Ar isotopes the matter radius determined from interaction cross sections lies close to the spherical-limit radius, while the reaction-cross-section radius sits at the deformed value. The ${}^{24}$Mg case is analysed quantitatively: the 0.087 fractional gap between $\sigma_{\rm R}$ and $\sigma_{\rm I}$ is only 0.037 from deformation alone, so part of the published $r_{\rm m}(\sigma_{\rm I})$ appears to be a fluctuation.

Load-bearing premise

The natural-lead extraction treats natural lead as pure ${}^{208}$Pb, relying on the prediction that lead neutron skins are nearly constant across mass number; if that constancy fails, the quoted $r_{\rm skin}$ is not a well-defined natural-lead value.

Editorial extensions

If this is right

  • Natural lead has a neutron skin of $0.271\pm0.008$ fm, consistent with PREX-II's $0.283\pm0.071$ fm for ${}^{208}$Pb.
  • The skin of ${}^{159}$Tb is $0.273\pm0.073$ fm; subtracting the deformation effect ($\beta_2=0.309$) leaves a spherical-limit skin near 0.11 fm.
  • For Mg isotopes, 21,23Na, and 40Ar, matter radii from reaction cross sections include deformation, while those from interaction cross sections give the spherical-limit radius.
  • For ${}^{24}$Mg, the difference between $r_{\rm m}(\sigma_{\rm R})=3.03\pm0.08$ fm and $r_{\rm m}(\sigma_{\rm I})=2.79\pm0.15$ fm is only about 40% accounted for by deformation, implying part of the published interaction-cross-section value is a statistical fluctuation.
  • The folding model with ESP-f scaling can extract neutron skins from proton reaction cross sections at a precision comparable to parity-violating electron scattering.

Reading between the lines

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

  • A corollary the paper leaves implicit: if lead skins are really $A$-independent, the same analysis applied to p+${}^{206}$Pb and p+${}^{207}$Pb data should return skins within about 0.02 fm of ${}^{208}$Pb; this is checkable with existing data and no new theory.
  • The difference between $\sigma_{\rm R}$- and $\sigma_{\rm I}$-based radii could serve as a deformation diagnostic for any nucleus with both measurements, converting the fractional cross-section excess into an effective $\beta_2$ and cross-checking the empirical formula beyond Mg, Na, and Ar.
  • Radius compilations that merge interaction and reaction cross-section values into one matter radius mix spherical and deformed states; future tables should report the observable or apply a deformation correction.
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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 / 5 minor

Summary. The manuscript uses the chiral (Kyushu) g-matrix folding model and the Love-Franey (LF) t-matrix folding model, together with density scaling, to extract neutron skin thicknesses for natPb and 159Tb, and matter radii for Mg, Na, and Ar isotopes. For natPb the target is treated as pure 208Pb and the extracted r_skin=0.271±0.008 fm is presented as agreeing with the PREX-II value for 208Pb. For 159Tb, r_skin=0.273±0.073 fm is obtained. For 24Mg, the paper compares r_m(σ_R)=3.03±0.08 fm with r_m(σ_I)=2.79±0.15 fm and interprets the difference as evidence that σ_I-based radii give spherical-limit matter radii while σ_R-based radii include deformation; the same interpretation is applied to Na and Ar isotopic chains. A fine-tuning factor F=0.955, calibrated to the 24Mg σ_I datum, is used for the high-energy LF extractions for Na and Ar.

Significance. If the central claims held at their quoted precision, the paper would show that proton reaction cross sections can determine neutron skins of heavy nuclei to 0.008 fm, and that the comparison of σ_R and σ_I data can separate deformation effects from spherical-limit matter radii. The model machinery is not new: the Kyushu g-matrix folding model has been benchmarked in prior work on 12C, Ca, Sn, and Pb, and the D1S-GHFB+AMP and SLy7 densities are standard. The manuscript is also transparent about several of its own limitations, including the statement in Sec. IV that 'Prediction A may be true' and the observation in Sec. III.E that the 37,38Ar depression may be a data fluctuation. However, the headline natPb precision is conditional on an untested assumption about the constancy of Pb skins across A, the Na/Ar extractions are calibrated to a single 24Mg anchor point, and the 24Mg 'too small' conclusion is a sub-sigma statement. These issues substantially limit the significance of the results as currently presented.

major comments (4)
  1. [Sec. III.A / Table III / Sec. II.D] The quoted r_skin=0.271±0.008 fm for natPb treats natural lead as a pure 208Pb target, and the 0.008 fm is the scaling-fit error only. Natural lead is 54.4% 208Pb, 22.1% 207Pb, 22.1% 206Pb, and 1.4% 204Pb. The only support for the A=208 assumption is the prediction that Pb skins are almost constant as a function of A, but no calculation or measured constraint for 204,206,207Pb is shown, and the Sn evidence in Fig. 1 has 30% experimental uncertainties and a 50% model spread from A=118 to 124. If the true Pb skins vary by even ~0.02 fm across A, the abundance-weighted σ_R would differ from the pure-208 calculation and the fitted r_skin would shift by more than the quoted 0.008 fm. The authors should either demonstrate the constancy of Pb skins, model natPb as a composition average, or add a systematic uncertainty that covers the composition ambiguity. As written, the abstract's precision claim is not supported.
  2. [Sec. III.D / Sec. III.E] The fine-tuning factor F=0.955 is fixed by requiring the LF t-matrix folding model to reproduce the central value of r_m(σ_I)=2.79±0.15 fm for 24Mg, and this same F is then applied to all Na isotopes (Sec. III.D) and all 32-40Ar isotopes (Sec. III.E). Consequently the r_m(σ_I) values in Tables VI and VIII are calibrated to the 24Mg input rather than being independent extractions. The uncertainty of the 24Mg anchor (0.15 fm) is not propagated into the quoted errors of the Na and Ar radii, and no validation of F on a system where both σ_I and σ_R are available is given. The deformation conclusions for Na and Ar therefore rest on a single calibration point; please propagate the F uncertainty or provide an independent check.
  3. [Sec. III.C / Abstract] The abstract states that for 24Mg, r_m(σ_I)=2.79±0.15 fm does not include effects of deformation, and Sec. III.C concludes that the central value of r_m(σ_I) is 'too small' compared with r_m,0(P)=2.929 fm. The difference is 0.14 fm against a 1σ error of 0.15 fm, i.e., less than a one-sigma deviation. The wording overstates the strength of the evidence; a quantitative significance statement or a softened conclusion is needed.
  4. [Sec. III.A / Sec. I] The natPb fit includes σ_R data at E_lab=341 MeV, but the Kyushu g-matrix folding model is stated in Sec. I to be benchmarked for proton scattering in the range 20≤E_lab≤180 MeV (Ref. [29]). The extension to 341 MeV is justified only by a general statement about cutoff effects and by 12C+12C tests, not by a proton-scattering benchmark in this energy range. A model-uncertainty contribution from this energy extrapolation should be included in the quoted 0.008 fm error or the fit should be repeated with the high-energy points excluded to show robustness.
minor comments (5)
  1. [Sec. IV, Eq. (34)] Equation (34) as printed, r_m^2 = r_m,0^2[1+5β2^2/(4π)] = r_m,0^2, is internally inconsistent for β2≠0; please correct the intended relation and recompute the derived r_m,0=5.065 fm if needed.
  2. [Table VI, A=26 row] The entry '0075' should read '0.075'.
  3. [Table VIII / Figs. 11-12] The table heading uses r_skin while Figs. 11 and 12 use r_skin.p for the same quantity; please make the notation consistent.
  4. [Sec. III.C, Eq. (33)] The expression for σ_inel in Eq. (33) does not obviously follow from Eqs. (29)-(32) with the stated definitions of r_m(P), r_m,0(P), x, and c; please check the algebra and correct or clarify.
  5. [Table II] Table II lists r_m(σ_R)=2.352±0.013 fm for 12C, while Sec. II.C and Sec. III.C use r_m(12C)=2.338 fm from Ref. [55]; please clarify which value is used in the double-folding calculations and why the values differ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the extracted radii are outputs of an inverse folding-model fit, not inputs renamed as predictions.

full rationale

The paper's derivation chain is not circular. For natPb, 159Tb, Na, Mg, and Ar, the reported radii are obtained by the scaling procedure of Eq. (27): given input densities (D1S-GHFB+AMP, SLy7, or a density scaled to r_n(PREX-II)), the folding model cross sections are computed and the densities are scaled until sigma_R or sigma_I matches the measured data. The radii are therefore outputs of the fit, not quantities already present in the inputs. The fine-tuning factor F is admittedly fitted to the central value of r_m(sigma_I)=2.79+/-0.15 fm for 24Mg, and f is fitted to the average sigma_R(exp)/sigma_R(th) for p+40Ar; but these are openly introduced normalization constants, and the per-nucleus radii are subsequently determined from each nucleus's own cross-section data by the same scaling procedure. The reported consistency with Suzuki and Ozawa radii is a comparison of independently extracted values, not an identity forced by construction. For natPb, the analysis assumes A=208 and starts from the PREX-II neutron density, but the final r_skin=0.271+/-0.008 fm is a small shift away from the PREX-II input, and the data could have required a larger shift; thus the agreement with PREX-II is not an input-output tautology. The genuinely load-bearing but unverified element is 'Prediction A' that Pb skins are almost constant across A, which the paper itself hedges in the Summary as 'Prediction A may be true'; that is a composition-modeling assumption and a systematic-risk caveat, not a circular reduction of the result to its own inputs. No fitted parameter is renamed as a prediction, and no self-citation or uniqueness theorem forces the central claims.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central extraction depends on the folding model's validity, the density shapes from D1S-GHFB+AMP or SLy7, the one-parameter density scaling, and the calibration factors F and f. No new entities are introduced. The A=208 treatment of natPb is an ad hoc assumption.

free parameters (3)
  • Fine-tuning factor F for LF t-matrix model = 0.955
    Determined so the LF folding model reproduces r_m(sigma_I)=2.79±0.15 fm for 24Mg+12C from Suzuki et al. (Sec. III.D). Used for all Na and Ar sigma_I analyses.
  • Fine-tuning factor f (ESP-f) for p+40Ar = 1.0327
    Chosen by averaging sigma_R(exp)/sigma_R(th) over E_lab to scale p+40Ar cross sections (Sec. III.F). Calibrates model to the data it then extracts radii from.
  • Per-nucleus density scaling factor alpha = e.g., alpha_n=0.986 for natPb
    Eq. (27) scales proton/neutron densities to reproduce sigma_R(exp); the scaled radius is then reported as the extracted value.
assumptions (5)
  • domain assumption The Kyushu chiral g-matrix folding model gives a valid optical potential for proton-nucleus scattering between 20 and 450 MeV.
    Assumed throughout Sec. II.B and III.A/B; supported by the authors' prior tests, not by derivations in this paper.
  • standard math The Brieva-Rook localization of the exchange potential is accurate.
    Invoked in Sec. II.B (Eq. 8 and the sentence 'It can be localized with the Brieva-Rook approximation'); justified by Ref. [53].
  • ad hoc to paper Scaling a density by a single factor alpha preserves the true shape of the ground-state density.
    This is the core extraction assumption behind Eq. (27); no justification is given that halo or deformation-tail shapes are correctly captured by isotropic scaling.
  • domain assumption For high-energy scattering, sigma_I(exp) can be computed as sigma_R in the folding model.
    Assumed in Sec. II.C ('Glauber-model calculations regard sigma_R as sigma_I... We take the same assumption').
  • ad hoc to paper A=208 for natPb; skin values of 204,206,207,208Pb are nearly identical.
    Stated in Sec. II.D and Sec. III.A; supports the natPb-PREX-II comparison and is the weakest assumption.

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Cite this review

Pith. "Pith review of Neutron skin thickness for $^{nat}$Pb, $^{159}$Tb and effects of deformation on matter radii for Mg, Na, Ar isotopes." pith.science (2026). https://pith.science/paper/YPUDDNSQ

@misc{pith2026250618936,
  author       = {Pith},
  title        = {Pith review of: Neutron skin thickness for $^nat$Pb, $^159$Tb and effects of deformation on matter radii for Mg, Na, Ar isotopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPUDDNSQ}},
  note         = {Machine review of arXiv:2506.18936}
}
abstract

Data on reaction cross section $\sigma_{\rm R}$ of proton scattering are available for $^{nat}$Pb (natural lead) and $^{159}$Tb. In addition, data on $\sigma_{\rm R}$ and/or interaction ones $\sigma_{\rm I}$ are available for Mg, Na, Ar isotopes. Our aim is to determine neutron skin thickness $r_{\rm skin}$ for $^{nat}$Pb, $^{159}$Tb and to investigate effects of deformation for Mg, Na, Ar isotopes. We use the chiral (Kyushu) $g$-matrix folding model for lower energies and the folding model based on the Love-Franey (LF) $t$-matrix for higher energies. For $^{159}$Tb, our skin value is $r_{\rm skin}=0.273 \pm 0.073$~fm. As for $^{24}$Mg, our matter radius $r_{\rm m}(\sigma_{\rm R})=3.03 \pm 0.08$~fm determined from $\sigma_{\rm R}$ includes effects of deformation, whereas the corresponding $r_{\rm m}(\sigma_{\rm I})=2.79 \pm 0.15$~fm does not. Effects of deformation are seen for Mg isotopes, $^{21, 23}$Na, $^{40}$Ar. Our result $r_{\rm skin}=0.271 \pm 0.008$~fm for $^{nat}$Pb agrees with $r_{\rm skin}^{208}({\rm PREX\text{-}II}) = 0.283\pm 0.071$~fm for $^{208}$Pb, where we assume $A=208$ for $^{nat}$Pb.

Figures

Figures reproduced from arXiv: 2506.18936 by the authors.

Figure 1
Figure 1. FIG. 1. Skin values of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. 3NFs in NNLO. Diagram (a) corresponds to the Fujita [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. This is investigated with D1S-GHFB+AMP and [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figures from the paper (9 more)
Figure 3
Figure 3. Figure 3: shows Elab dependence of σR for p+natPb scat￾tering in Elab = 21.1 ∼ 341 MeV. We assume A = 208 for natPb. The σR(PREX-II) are calculated with the Kyush g-matrix folding model with the D1S-GHFB+AMP neu￾tron density scaled to r 208 n (PREX-II), where r 208 p (D1S) = r 2…
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: show our rm(σI), the rm(σSuzuki) of Refs. [18, 22], the rm(SLy7) for Na isotopes as a function of A. Our rm(σI) are consistent with the rm(σSuzuki). The rm(SLy7) are parallel to our rm(σI) and have natural A dependence, i.e., rm(SLy7) = 1.1263A1/3 − 0.3243 with good ac…
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: shows σI as a function of A. The LF t-matrix fold￾ing model with the SLy7 densities overestimates the data [23]. The F σI(LF) monotonically increase as a function of A, whereas the data [23] have a depression in 37,38Ar. The de￾pression in 37,38Ar may be a fluctuation …
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: shows our results on proton skin thickness rskin.p for 32–40Ar. There is a bump at A = 37, 38. This is related to the fact that the central values of σI(exp) are slightly smaller than those of the neighborhood nuclei. If we take the upper￾bound values of the data σI(e…
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]
Figure 15
Figure 15. Figure 15: We find that effects of deformation on rn are not negligible. 2.4 2.6 2.8 3.0 3.2 3.4 18 20 22 24 26 28 30 32 34 Na rn (σI ) rn (σR ) rp (exp) rn (σI), rn (σ R), rp(exp) [fm] mass number A FIG. 15. A dependence of the rn(σI), the rn(σR), the rp(exp) for Na isotopes. S…

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