Neutron radii and semi-phenomenological treatment of neutron distributions for Mg isotopes

Govind Kumar; M. Imran; Z. A. Khan; Z. Hasan

Neutron radii and semi-phenomenological treatment of neutron distributions for Mg isotopes

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Neutron radii of Mg isotopes point to a two-neutron halo in 40Mg

desk verdict A useful but flawed extraction of Mg neutron radii; the 40Mg halo claim depends on an inconsistent tail parameter and an in-sample validation. read the letter →

arxiv 2608.09429 v2 pith:T4WFYLP5 submitted 2026-08-10 nucl-th

Govind Kumar , M. Imran , Z. Hasan , Z. A. Khan This is my paper · ORCID

classification nucl-th PACS 21.10.Gv24.10.Ht25.60.Bx25.70.-z

keywords neutronradiiMgisotopesGlaubermodelreactioncrosssectionscore+ndescriptiontwo-neutronhaloskinDRHBc

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 aims to extract the neutron radii of $^{24-38}$Mg by fitting measured reaction cross sections on a $^{12}$C target at 240 MeV/nucleon within the Glauber model, using proton radii from the deformed relativistic Hartree-Bogoliubov theory in continuum as the fixed proton input. The authors test two density shapes, a harmonic-oscillator Slater-determinant form and a two-parameter Fermi form, and find that both reproduce the data but give neutron radii that differ by up to about 0.13 fm for the same isotope. They then construct a semiphenomenological core+n (or core+2n) description of the neutron tail, controlled by one- and two-neutron separation energies, and show it reproduces the measured cross sections for $^{25-38}$Mg while exposing the one-neutron halo of $^{37}$Mg. Applying the same construction to $^{40}$Mg as $^{38}$Mg+2n, they predict an enhanced neutron radius and reaction cross section that point to a two-neutron halo. If the extraction is right, these radii give a model-dependent but useful map of neutron skin and halo development across the magnesium isotopic chain.

What carries the argument

The load-bearing object is the semiphenomenological neutron density built from a core plus a separation-energy tail: $\rho_n(r) = \rho_{\rm core}(r) + N_0 (r^2/(r^2+R^2)^2) e^{-r/t_n}$ with $t_n = \hbar/(2(2mS_n)^{1/2})$. Whether the tail carries one neutron or two is decided by comparing the one-neutron and two-neutron separation energies, $S_n$ and $S_{2n}$. The core density is either the harmonic-oscillator Slater-determinant form (SDHO) or the two-parameter Fermi form (2pF), and its single parameter is tuned so that the full density reproduces the neutron radius extracted from the Glauber fits. The Glauber model itself, with its $S_0 + S_2$ correlation treatment and the parametrized nucleon-nucleon profile, supplies the map from density parameters to $\sigma_R$.

What would settle it

Measure the reaction cross section of $^{40}$Mg on $^{12}$C at 240 MeV/nucleon: the paper predicts about 1650 mb (2pF) and 1690 mb (SDHO), well above the trend set by the lighter Mg isotopes. A measured value consistent with the lighter-isotope trend, or a measured neutron radius near 3.6 fm instead of the predicted 3.85-3.88 fm, would rule out the claimed two-neutron halo.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a separation-energy-guided description of neutron distributions in neutron-rich Mg isotopes that connects measured reaction cross sections to halo structure. Taking the DRHBc proton radii as given, the authors vary one parameter in each density family—the oscillator constant in the SDHO density or the diffuseness in the 2pF density—until the Glauber-model reaction cross section for each isotope matches the experimental $\sigma_R$ on $^{12}$C at 240 MeV/nucleon. The resulting neutron radii in Table II rise from about 3.02 fm for $^{24}$Mg to about 3.87 fm for $^{38}$Mg in the SDHO case, with the 2pF values smaller by up to 0.13 fm. The core+n construction, which adds a tail $\rho_{\rm tail}(r) = N_0 r^2/(r^2+R^2)^2 e^{-r/t_n}$ with $t_n = \hbar/(2(2mS_n)^{1/2})$, is then forced to reproduce the extracted radius; it reproduces the measured cross sections for $^{25-38}$Mg and makes $^{37}$Mg the most extended neutron distribution, consistent with a one-neutron halo. For $^{40}$Mg, where $S_n > S_{2n}$, the core+2n description with a $^{38}$Mg core predicts $r_n \approx 3.85$ to 3.88 fm and $\sigma_R \approx 1650$ to 1690 mb at 240 MeV/nucleon, and the paper reads the enhanced radius and cross-section trend as evidence for a two-neutron halo, in line with a previous three-body $^{38}$Mg+n+n calculation.

Load-bearing premise

The load-bearing premise is that the true neutron density of each Mg isotope is close enough to one of the two adopted shapes (the harmonic-oscillator Slater form or the two-parameter Fermi form) that varying a single parameter to match the measured reaction cross section determines the neutron radius; since the two shapes give radii differing by up to 0.13 fm for the same isotope, a density that is neither shape would bias the extracted radii and the $^{40}$Mg halo prediction.

Editorial extensions

If this is right

The extracted radii imply a smooth increase of neutron skin from $^{24}$Mg to $^{38}$Mg, with $^{37}$Mg standing out as the most extended neutron distribution.
The core+n description, with the tail spread set by the one-neutron separation energy, reproduces the experimental $\sigma_R$ for $^{25-38}$Mg within a few percent, so it can be used to test other proposed neutron radii.
For $^{40}$Mg, the core+2n construction predicts $\sigma_R$ on $^{12}$C of about 1650 mb (2pF) and 1690 mb (SDHO) at 240 MeV/nucleon, and about 1740 and 1780 mb at 1000 MeV/nucleon; these are the signatures to look for in a measurement.
Because $S_n > S_{2n}$ for $^{40}$Mg, the tail in this recipe carries two neutrons, making the predicted structure a two-neutron halo rather than a sequential one-neutron halo.

Reading between the lines

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

The 0.13 fm spread between the SDHO and 2pF extractions sets a model uncertainty that could be narrowed by adding a third density family or by folding in charge-changing cross sections for Mg if they become available.
The same core+n/core+2n recipe could be applied to other neutron-rich isotopic chains with measured interaction cross sections, such as Ne, Na, or Si, to flag candidate halo nuclei before direct radius measurements exist.
A future $\sigma_R$ measurement for $^{40}$Mg would constrain the tail parameter $t_n$ directly, testing whether the separation-energy scaling assumed for the tail is correct; if confirmed, it would also motivate knockout or momentum-distribution studies of the two-neutron halo.
The extraction also inherits the model dependence of the DRHBc proton radii, since charge-changing cross-section data for Mg are not yet available; a future measurement would remove that anchor uncertainty.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper extracts neutron radii for 24–38Mg by fitting the Takechi et al. reaction cross sections at 240 MeV/nucleon in the Glauber model, then uses a semiphenomenological core+n/core+2n density to describe the neutron tail and concludes that 40Mg has a two-neutron halo. The extraction is workmanlike and the tables are useful, but the halo prediction rests on an inconsistent tail parameter and the validation is largely in-sample. I would not trust the 40Mg numbers as they stand.

What is genuinely useful: the paper gives a systematic set of neutron radii from two density forms (SDHO and 2pF), with fitted parameters, and a recipe for extrapolating to unmeasured isotopes. The core+n construction is a small variant of Bhagwat-Gambhir-Patil [34], differing by anchoring the core parameter to the fitted neutron radius rather than the charge radius. The authors are transparent about the shape dependence—the two forms differ by up to 0.13 fm for the same data—and they compare their matter radii with earlier work.

The soft spots are real. First, Eq. (23) fixes the tail decay length from the one-neutron separation energy Sn. For 40Mg, the authors explicitly note Sn > S2n and use a core+2n description, but they still use Sn = 1.30 MeV in the tail formula instead of the two-neutron separation energy S2n = 0.67 MeV. For a two-neutron halo, the asymptotic tail should be governed by the pair separation energy; using Sn shortens the tail and makes Table V quantitatively unreliable. Second, the validation in Sec. III.C is partly circular: Table III tunes the core+n parameter to reproduce the radii extracted from sigma_R, so the recalculated cross sections in Fig. 8 are not an independent test. Table IV is a better consistency check, but the parameters ultimately come from the same data.

The extracted radii for 24–38Mg are plausible and the paper could be a useful survey, but the 40Mg halo claim and the predicted cross sections need stronger support. The flaws are fixable: use S2n for the two-neutron tail, add a systematic error budget, and present an out-of-sample test (for instance, a known 2n halo such as 11Li or 19C). I would send it to a referee rather than desk-reject, because the core analysis is standard and the issues can be addressed in revision.

Referee Report

3 major / 5 minor

Summary. The paper analyzes reaction cross sections of 24-38Mg on 12C at 240 MeV/nucleon within the Glauber model to extract neutron radii, using DRHBc proton radii as input and two density parametrizations (SDHO and 2pF). It then introduces a semiphenomenological core+n description for 25-38Mg (Sn<S2n), constrains the tail parameters to reproduce the extracted neutron radii, and recalculates reaction cross sections as a claimed validation. For 40Mg, the paper assumes a core+2n description and predicts its neutron radius and reaction cross sections at 240 and 1000 MeV/nucleon, concluding that 40Mg exhibits a two-neutron halo-like structure.

Significance. If the extracted neutron radii and the 40Mg halo prediction were quantitatively robust, the work would provide useful systematics for neutron skins in the Mg chain and a simple semiphenomenological tool for neutron tails. The paper has strengths: it uses a standard Glauber framework, compares results with earlier analyses (Takechi, Ozawa, Kanungo, Sharma), and explicitly documents the density-shape dependence in Table II. However, the central quantitative outputs are not secure because the extracted radii are shape-dependent without a systematic error estimate, the core+n validation is in-sample and largely circular, and the core+2n prediction relies on an untested extrapolation with an internally inconsistent tail-decay parameter. These issues affect the main claims rather than only presentation.

major comments (3)

[Sec. III.B, Table II] The extracted neutron radii are not robust to the choice of density shape. For the same experimental sigma_R, the 2pF and SDHO densities give r_n values that differ by up to about 0.13 fm (e.g., 24Mg: 3.0529 fm vs 3.1815 fm, roughly 4%), and no systematic uncertainty is assigned to this model dependence. Since the two forms are arbitrary members of a parametric family, the quoted r_n values in Table II are shape-dependent, and this ambiguity propagates directly into the core+n and core+2n constructions and the final 40Mg claim.
[Sec. III.C, Eqs. (24)-(25), Fig. 8] The claimed validation of the core+n description is circular. In Table III, the parameter an (or alpha^2_n) in Eqs. (24)-(25) is adjusted so that the core+n density reproduces the very neutron radius that was extracted from the measured sigma_R of the same isotope in Table II. Figure 8 then shows that this same sigma_R data set is reproduced. This is an in-sample consistency check, not an independent test of the tail model; the agreement in Fig. 8 and Table IV is largely by construction and cannot be used as evidence that the core+n tail form is physically correct.
[Sec. III.D, Eqs. (22)-(23), Table V] For 40Mg, the tail decay length t_n is computed from the one-neutron separation energy Sn = 1.30 MeV using Eq. (23), but the paper explicitly defines the core+2n regime by Sn > S2n and states that in this regime the tail contains two neutrons. Since 40Mg has S2n = 0.67 MeV, the decay length governing a two-neutron tail should be set by the two-neutron separation energy, not by Sn. Using Sn in Eq. (23) gives a shorter tail than S2n would, so Table V is quantitatively insecure even if the qualitative conclusion might survive. In addition, the core+2n prescription is validated on no nucleus with Sn > S2n; all validation cases are core+n isotopes with Sn < S2n. The 40Mg halo prediction therefore rests on an untested extrapolation compounded by an internal inconsistency in the governing parameter.

minor comments (5)

[Title and abstract] The title contains a typo ('trea tment'), and the abstract and main text contain similar spelling errors (e.g., 'cross sestion' in Sec. III.B).
[Sec. II, Eq. (23)] The formula for t_n is typeset ambiguously as 'tn = ℏ 2(2mSn)1/2'; the intended division and square-root structure should be clarified.
[Fig. 4] The extracted matter radii in Fig. 4 are shown without error bars, which is particularly problematic given the 4% spread between SDHO and 2pF values in Table II.
[Table V] The column layout of Table V is confusing: the 2pF and SDHO entries for r_n, r_m, and sigma_R are interleaved without clear subheadings, making it difficult to read the predictions for each density form.
[Sec. III.D] The comparison with Singh et al. [21] is qualitative; a quantitative comparison of the predicted density tails or skin thicknesses would strengthen the claim of agreement.

Circularity Check

3 steps flagged · score 6.0 of 10

Core+n 'validation' is in-sample: the tail parameters are forced to reproduce the very sigma_R-fitted neutron radii, so the recalculated sigma_R agreement is by construction; the 40Mg halo claim is an extrapolation built on that circular validation.

fitted input called prediction [Abstract and Sec. III.B]
"In this work, the core+n is employed for 25-38Mg isotopes, and is subjected to reproduce the same neutron radius of the given isotope, as we obtained from sigma_R calculations. To validate the core+n description, we have revisited the reaction cross sections of 25-38Mg isotopes. The results are found to agree well with the experimental values."

The neutron radii in Table II were obtained by fitting the 240 MeV/nucleon sigma_R data (Sec. III.B: 'varying ... that fit the experimental values'). The core+n construction is then deliberately constrained to reproduce those same fitted radii. Consequently, the 'validation' consisting of recomputing sigma_R from the core+n density is not an independent test: the model input was derived from the very data being reproduced. The good agreement is a restatement of the fit, not a prediction.
fitted input called prediction [Sec. III.C, Eqs. (24)-(25) and Fig. 8]
"let us now vary (i) the parameter alpha2_n in Eq. (24) for SDHO density, and (ii) the parameter an in Eq. (25) for 2pF density in order to get our extracted neutron radii (Table II) for (N>Z) Mg isotopes. ... To verify the utility of our extracted neutron radii for Mg isotopes (Table II), and to establish credibility for the use of core+n description for 25-38Mg isotopes, we have revisited the reaction cross sections of 25-38Mg from a 12C target at 240 MeV/nucleon."

The core+n parameters are adjusted to hit the r_n values that were themselves obtained by fitting the same sigma_R data. Recalculating sigma_R with those parameters therefore returns approximately the sigma_R values used in the fit. The paper calls this 'verification' and 'credibility', but the agreement is inherited from the fit rather than being an out-of-sample test of the tail model.

1 more flagged steps

fitted input called prediction [Sec. III.C, Table IV]
"the core+n description has also been used to find the neutron radii of 25-38Mg isotopes and predict their reaction cross sections on a 12C target at 240 MeV/nucleon, involving the (free state) parameter alpha2_n in Eq. (24) (an in Eq. (25)) given in Table II. ... It is found that the results on neutron radii and reaction cross sections, respectively, agree well with the corresponding extracted (Table II) and experimental [19] values."

Table II parameters are the ones fitted to sigma_exp_R. Feeding those exact fitted parameters into the core+n density and then 'predicting' sigma_R is a closed loop: the output is a function of the input that was tuned to match that output. The resulting small (Delta sigma_R)% values are therefore a consistency check of the interpolation, not a validation of the model's predictive power.

full rationale

The quantitative neutron radii in Table II are fit parameters, not predictions: Sec. III.B states they are obtained by varying the density parameters 'that fit the experimental values' of sigma_R. The core+n construction then deliberately reproduces those fitted radii (abstract and Sec. III.C), and the recalculated sigma_R (Fig. 8, Table IV) is presented as validation. Because the same sigma_R data were used to set r_n, this validation is in-sample: the model is being tested on the data that produced its input. This is the central circular loop. The 40Mg core+2n prediction is a genuine extrapolation (no 40Mg sigma_R data are fit), but the paper invokes the 'successful use' of core+n for 25-38Mg as motivation; that success is not independently established. I find no load-bearing self-citation chain: references [16-18,24] provide standard formalism and densities, but the circularity is the fit-feedback loop, not the citations. The inconsistency that t_n in Eq. (23) uses S_n while the core+2n regime is defined by S_n > S_2n is a correctness/consistency concern, not a circularity, so it does not change the score. Overall score 6: partial circularity, because a central 'prediction' (recalculated sigma_R, Table IV) reduces to its fitted input, while the 40Mg claim is weakened rather than definitionally forced.

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

The central derivation is a fit anchored to external inputs (DRHBc proton radii, NN amplitudes, experimental sigma_R). The neutron density parameters are free and fitted per isotope; the core+n construction adds no new physics but reuses the same fitted radii, so the ledger is dominated by per-isotope fitting parameters and the unvalidated tail form.

free parameters (3)

2pF neutron diffuseness a_n per isotope = 0.5085-0.6798 fm (Table II)
Adjusted for each of 24-38Mg so that the Glauber sigma_R matches experiment.
SDHO neutron oscillator constant alpha^2_n per isotope = 0.2191-0.2992 fm^-2 (Table II)
Adjusted for each isotope to fit the same sigma_R data.
core+n/2n core density parameter (a_n or alpha^2_n) = Table III values
Varied so that the core+tail total neutron radius equals the radius already extracted in Table II, not independently determined.

assumptions (5)

domain assumption The Glauber S-matrix can be truncated after the two-body correlation term (Eqs. 4-10)
The central formula for sigma_R is taken from the authors' own Ref. [16] without independent derivation in this paper; if this expansion is inaccurate, all extracted radii shift.
domain assumption The SDHO and 2pF functional forms span the true neutron density shape
The extraction fits only one parameter per shape; the systematic spread between shapes in Table II shows the results hinge on this assumption.
domain assumption DRHBc proton radii (Ref. [20]) are the correct charge radii for all Mg isotopes
The paper states model-dependent estimates are necessary because CCCS data are absent; errors in these inputs propagate directly into the neutron radii.
domain assumption The Bhagwat-Gambhir-Patil tail form (Eq. 22) with the separation-energy length t_n describes the asymptotic neutron density
Adopted from Ref. [34]; the paper provides no derivation or test against microscopic densities other than the circular sigma_R check.
ad hoc to paper The proton radius of the core nucleus is unchanged by adding the valence neutron(s)
Stated explicitly in Sec. III.C ('we assumed similar value of the proton radius for the core nucleus'), justified only by the resulting sigma_R agreement.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Neutron radii and semi-phenomenological treatment of neutron distributions for Mg isotopes." pith.science (2026). https://pith.science/paper/T4WFYLP5

@misc{pith2026260809429,
  author       = {Pith},
  title        = {Pith review of: Neutron radii and semi-phenomenological treatment of neutron distributions for Mg isotopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4WFYLP5}},
  note         = {Machine review of arXiv:2608.09429}
}

abstract

Involving the charge radii of \rm Mg isotopes, as calculated using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), we have extracted the neutron radii of $^{24-38}$\rm Mg isotopes by studying their reaction cross sections ($\sigma_{R}$) from $^{12}$\rm C at 240 MeV/nucleon within the framework of Glauber model. The calculations use (i) descriptions of nuclei in terms of the Slater determinant involving harmonic oscillator single-particle wave functions (SDHO), and (ii) two-parameter Fermi (2pF) shape of density distribution, with the aim to assess the density dependence of neutron skin in $^{24-38}$\rm Mg isotopes. To understand the asymptotic behavior (spread) of neutron distribution, we propose to introduce the use of core+n ($S_{n}<S_{2n}$) or core+2n ($S_{n}>S_{2n}$) description for stable as well as unstable isotopes; $S_{n}$ ($S_{2n}$) is the one-neutron (two-neutron) separation energy of the considered isotope. The core+n (core+2n) is treated semi-phenomenologically. In this work, the core+n is employed for $^{25-38}$\rm Mg isotopes, and is subjected to reproduce the same neutron radius of the given isotope, as we obtained from $\sigma_{R}$ calculations. To validate the core+n description, we have revisited the reaction cross sections of $^{25-38}$\rm Mg isotopes. The results are found to agree well with the experimental values. Moreover, the core+n neutron distributions clearly demonstrate the one-neutron halo structure of $^{37}$\rm Mg. These findings motivated us to use the core+2n description for the neutron distribution of $^{40}$\rm Mg in predicting its neutron radius, and $\sigma_{R}$ from $^{12}$\rm C at 240 and 1000 MeV/nucleon. The trend of the neutron radius and $\sigma_{R}$ suggests that $^{40}$\rm Mg exhibits two-neutron halo like structure.