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

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

REVIEW 4 major objections 5 minor 41 references

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

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper extracts neutron radii of 24-38Mg from reaction cross sections on carbon and argues that core+n and core+2n density descriptions reveal a one-neutron halo in 37Mg and a two-neutron halo in 40Mg.

desk verdict Useful extracted radii for Mg, but the halo claims for 37Mg and 40Mg are forced by the assumed tail ansatz, not tested by the data. 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 neutronradiiskinhalonucleireactioncrosssectionsGlaubermodelmagnesiumisotopesseparationenergycore+ndescription

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 argues that the neutron distributions of neutron-rich magnesium isotopes can be understood as a compact core plus a weak tail whose decay length is set by the one- or two-neutron separation energy. From measured reaction cross sections on a carbon target at 240 MeV per nucleon, analyzed in the Glauber model with proton radii taken from deformed relativistic Hartree-Bogoliubov theory in continuum, it extracts neutron radii for 24-38Mg. In this picture 37Mg, with the smallest one-neutron separation energy, develops a distinctly extended tail and is classified as a one-neutron halo. Applied to 40Mg as a core plus two neutrons, the same construction predicts an enhanced neutron radius and reaction cross sections, leading the authors to call 40Mg a two-neutron halo candidate. If these assignments are right, separation energies and a core radius become enough to locate halo signatures in isotopes where direct measurements are scarce.

What carries the argument

The load-bearing object is the semi-phenomenological core+tail neutron density, with the tail written as $\rho_{\mathrm{tail}}(r)=N_0\,(r^2/(r^2+R^2)^2)\,e^{-r/t_n}$, where the decay length is $t_n = \hbar/(2\sqrt{2mS_n})$. A smaller separation energy therefore produces a longer tail, and the number of neutrons placed in the tail is one when $S_n < S_{2n}$ and two when $S_n > S_{2n}$. The core part is either an SDHO or 2pF density with its parameter adjusted to reproduce the neutron radius extracted from the reaction cross section. This construction converts a single measured cross section and a known separation energy into a full neutron density, and it is what makes the halo claims visible in the calculated distributions.

What would settle it

Measure the reaction cross section of 40Mg on 12C at 240 MeV/nucleon (or 1000 MeV/nucleon): the paper predicts roughly 1647-1688 mb (or 1742-1783 mb), clearly above the neighboring isotopic trend, with a neutron radius near 3.85-3.88 fm. A value on the smooth isotopic trend, or an independent neutron-radius determination near 3.7 fm, would rule out the two-neutron halo claim.

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Extended reading notes

Core claim

The paper's central claim is that a core+n (or core+2n) decomposition reproduces the measured reaction cross sections of 25-38Mg on 12C at 240 MeV/nucleon and yields neutron distributions whose tails reveal halo structure. The extracted neutron radii, obtained by varying the SDHO oscillator constant or the 2pF diffuseness to match experiment, grow smoothly with mass number except for a sharp rise at 37Mg, which has the lowest one-neutron separation energy (0.240 MeV) and the thickest neutron skin. The core+n density for 37Mg shows the maximum far-out spread among all isotopes considered, which the authors read as one-neutron halo structure. For 40Mg, whose two-neutron separation energy (0.67 MeV) is smaller than its one-neutron separation energy, the core+2n description gives a neutron radius around 3.85-3.88 fm and reaction cross sections about 1647-1688 mb at 240 MeV/nucleon and 1742-1783 mb at 1000 MeV/nucleon, both above the smooth isotopic trend; the authors conclude 40Mg exhibits two-neutron-halo-like structure, consistent with an existing three-body calculation.

Load-bearing premise

The load-bearing premise is that a neutron tail of the form given in Eq. (22), with its decay length set by the separation energy, is the correct asymptotic neutron density; the long tails that define the 37Mg and 40Mg halos are direct consequences of this formula, and if it is wrong for these nuclei the halo conclusions do not follow.

Editorial extensions

If this is right

The extracted neutron radii and skin thicknesses for 24-38Mg can serve as a reference set for other models of neutron-rich magnesium.
A measurement of 40Mg on 12C at 240 or 1000 MeV/nucleon should see a reaction cross section above the isotopic trend, roughly 1647-1688 mb or 1742-1783 mb, if the two-neutron halo is real.
The core+n and core+2n recipe can be reused for other isotopic chains, using a known core radius and the relevant separation energies, to predict radii and cross sections where data are missing.
The agreement between the core+n calculation and the measured cross sections within a few percent supports the extracted neutron radii as input for future Glauber-model analyses.

Reading between the lines

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

One direct test of the tail ansatz would be to compare the predicted tail shape with a knockout or breakup measurement on 37Mg; an exponential tail with decay length set by $S_n$ may be too simple for a deformed or paired halo.
The same construction could be applied to predict halo boundaries in neighboring isotopic chains such as Na, Al, or Ne; a systematic failure there would suggest the separation-energy-only tail is missing physics such as deformation or pairing.
If the 40Mg prediction is confirmed, it would strengthen the idea that two-neutron halos can be identified from rms radii and reaction cross sections alone, without needing detailed three-body wave functions.

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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 useful part of this manuscript is a clean set of extracted neutron radii for 24–38Mg from Glauber fits to reaction cross sections. The halo claims for 37Mg and 40Mg, however, are not empirically tested by the data. They follow from the assumed tail form and from the fact that the core+n densities are forced to reproduce the fitted radii.

What is actually new: the paper provides neutron radii for the Mg chain using two density shapes (SDHO and 2pF) with DRHBc proton radii as input. The SDHO and 2pF values differ by up to about 4%, which gives a useful model-spread estimate even without formal uncertainties. The Sn versus S2n criterion for choosing core+n versus core+2n is a small, tidy addition to the Bhagwat et al. tail-density recipe. The comparison with Singh et al. for 40Mg is a reasonable consistency check.

Where the soft spots are: the validation in Fig. 8 is close to circular. The core+n densities are “subjected to reproduce the same neutron radius” as the plain 2pF/SDHO fits. So recomputing sigma_R mainly replays the fit; it does not tell us whether the long tail is required by the data. The same data were already reproduced by no-tail densities. Consequently, the abstract’s phrase “clearly demonstrate the one-neutron halo structure of 37Mg” is too strong. What the authors have is one plausible semi-phenomenological representation that is consistent with a halo, not a demonstration. The same caveat applies to the 40Mg prediction.

Two smaller points. The fitted neutron radii come with no error bars, and sensitivity to the input DRHBc proton radii is not explored; that matters because the extracted rn depend on those inputs. And the separation-energy-dependent tail ansatz is reasonable but is an assumption; the halo conclusion is not independent of it.

One stress-test note: the concern about Eq. (23) being off by a factor of 2 does not survive a check. For a single-particle exponential, the density tail decays as exp(-2μr), so the density decay length is ħ/(2√(2μS_n)), which is what the authors wrote (with μ≈m). So that is not a flaw. The circularity concern is the real one.

Who should read it: anyone doing Glauber-based radius extractions or halo classification in light neutron-rich nuclei. A serious referee can help the authors present the halo evidence as model-dependent rather than as a demonstration. I would send it to peer review, but with a strong recommendation that the authors soften the halo language, add uncertainty propagation, and show at least one sensitivity test of the proton radii.

Referee Report

4 major / 5 minor

Summary. The paper analyzes reaction cross sections (σR) of 24–38Mg isotopes on a 12C target at 240 MeV/nucleon within the Glauber model to extract neutron radii, using DRHBc charge radii for the proton distributions and either harmonic-oscillator Slater-determinant (SDHO) or two-parameter Fermi (2pF) densities. The authors then introduce a semi-phenomenological core+n (or core+2n) description in which a neutron tail of the form Eq. (22) is added to a core density, with the tail decay length set by the one-neutron separation energy and the normalization fixed by the core+n content. The core+n densities are constrained to reproduce the neutron radii extracted in the σR fit, and the resulting σR values are compared with experiment. On this basis the paper claims that 37Mg has a one-neutron halo and that 40Mg, treated as 38Mg+2n, exhibits a two-neutron halo-like structure, with predicted σR at 240 and 1000 MeV/nucleon.

Significance. If the extracted neutron radii and the core+n treatment are reliable, the paper would provide a systematic set of neutron radii and skin thicknesses for neutron-rich Mg isotopes and a simple, computationally light method for predicting halo-like behavior in dripline nuclei. The Glauber framework used is standard and the comparison of two density parameterizations is a useful check of model dependence. However, the central halo claims rest on a phenomenologically assumed tail shape whose validation in Fig. 8 is partly circular, and the quantitative predictions inherit an unexamined factor in the decay length and a lack of propagated uncertainties. With appropriate reframing and additional validation, the extracted radii and the core+n representation could still be a useful contribution.

major comments (4)

[Sec. III.C, Fig. 8, Table III] The validation of the core+n description is largely circular. The core+n densities are explicitly constructed to reproduce the same neutron radii (Table II) that were obtained by fitting the experimental σR values; Fig. 8 then recalculates σR with those constrained densities. The agreement in Fig. 8 is therefore a consistency check on the radius fit, not an independent test of the tail form. In particular, the long tail that leads to the claim of a one-neutron halo in 37Mg follows directly from inserting its small separation energy into Eq. (22) and normalizing the tail, rather than from a response of the reaction cross section to an unconstrained tail. The authors should either provide an out-of-sample test of the tail ansatz (for example, by fitting the tail parameters directly to the 37Mg σR without imposing the previously fitted radius, or by confronting an observable that is more sensitive to the tail, such as a momentum distribution or a different-energy σR) or clearly restate the halo conclusion as a plausible consequence of the assumed tail form rather than a demonstrated empirical result.
[Sec. III.C, Eqs. (22)-(23)] The decay length in Eq. (23), t_n = hbar/(2 sqrt(2 m S_n)), is a factor of 2 shorter than the standard one-neutron asymptotic decay length hbar/sqrt(2 mu S_n), where mu is the neutron-core reduced mass. The manuscript neither derives nor justifies this factor, and it does not discuss the use of the nucleon mass rather than the reduced mass. This matters because the extent of the tail is precisely what distinguishes a halo-like distribution from a normal one: the small S_n of 37Mg produces a long tail only through this specific formula, and the quantitative predictions in Tables IV and V scale with t_n. The authors should justify Eq. (23) or replace it with the conventional asymptotic form; if the conventional form is used, the tails (and hence the halo enhancement) would be even longer, which would not remove the circularity concern but would alter the numerical predictions.
[Sec. III.B, Table II] The extracted neutron radii are presented as single-point values without uncertainties, even though the experimental σR values have finite errors (e.g., ±25.7 mb for 24Mg and ±13.0 mb for 37Mg). Because each r_n comes from a one-parameter fit to σR, a simple error propagation from the experimental uncertainties is possible and is necessary to judge, for example, whether the rather small differences between neighboring isotopes are significant and whether the 37Mg enhancement is statistically meaningful. The absence of error bars also weakens the comparison in Fig. 4 and the halo claims in Sec. III.B. The authors should include uncertainties on a_n, alpha_n^2, and r_n, at least from the quoted experimental errors, and ideally from the two density model choices.
[Sec. III.C, proton radius assumption] The manuscript assumes that the proton radius of the core nucleus inside the core+n system equals the proton radius of the free core nucleus, stating that this has been verified by the results in Fig. 8. Since Fig. 8 is subject to the circularity described above, this verification is not independent. Any error in the DRHBc proton radii propagates directly into the extracted neutron radii, and the core+n proton distribution is not otherwise tested. The authors should either relax this assumption and fit the proton radius as well, or explicitly discuss the sensitivity of the extracted neutron radii to the DRHBc proton radii.

minor comments (5)

[Throughout] The manuscript uses 'r_n' ambiguously: in Table II and III it is said to include the finite nucleon size, while Fig. 4 and Fig. 5 are labeled with point radii (r_pt^m and r_pt^n - r_pt^p). Please define both quantities clearly and use consistent notation.
[Sec. III.B] The phrase 'charge-changing crosse sestion' in the introduction contains a typo; it should be 'cross section'.
[Table IV] The header 'core+n 2pF SDHO' is confusing because the core+n column refers to the core nucleus used rather than to a third density model; please restructure the table to make clear that 2pF and SDHO are the two density parameterizations.
[Figs. 4 and 5] The extracted neutron radii and skin thicknesses are plotted without error bars; adding the uncertainties from the σR fit would make the comparison with other data sets much more informative.
[Sec. III.C] The phrase 'subjected to reproduce the same neutron radius' appears twice and reads awkwardly; 'constrained to reproduce' would be clearer.

Circularity Check

2 steps flagged · score 6.0 of 10

Core+n validation refits the already-fitted neutron radii, and the 37Mg halo is a restatement of its small input S_n through the tail ansatz.

fitted input called prediction [Abstract; Section III C (Tables III and IV, Fig. 8)]
"In this work, the core+n is employed for $^{25-38}$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 $^{25-38}$Mg isotopes. The results are found to agree well with the experimental values."

The core+n parameters $\alpha_n^2$ and $a_n$ are fixed to reproduce the $r_n$ values of Table II, which were themselves obtained by fitting the experimental $\sigma_R$ in Table II. The recomputed $\sigma_R$ in Fig. 8 (and Table IV) is therefore a function of already-fitted radii plus fixed proton densities; tail-free 2pF/SDHO densities with the same $r_n$ already matched the same data. This makes the agreement a consistency check on $r_n$ rather than an independent test of the added tail, yet it is presented as validation of the core+n description.
self definitional [Section III C, Eqs. (22)-(25); Figs. 6-7; Section IV]
"$\rho_{\rm tail} = N_0 \left(\frac{r^2}{(r^2+R^2)^2}\right) e^{-r/t_n}$, where $t_n = \hbar/(2(2mS_n)^{1/2})$ ... the extent of spread in neutron distribution is inversely proportional to the neutron separation energy. ... It is noticed that $^{37}$Mg shows the maximum spread far out from the center as compared to all other Mg isotopes. This result supplements the known one-neutron halo structure of $^{37}$Mg nucleus."

The tail that produces the claimed halo is not constrained by any reaction datum; its decay length is defined by Eq. (23) from the input $S_n$, and its normalization is fixed by the assumed number of tail neutrons. Since $^{37}$Mg has the smallest $S_n$ (0.240 MeV) among the studied isotopes, the 'maximum spread' and the one-neutron-halo classification follow algebraically from the chosen input. The demonstration reduces to restating that $^{37}$Mg is weakly bound: the small $S_n$ is inserted into the ansatz, not discovered from the $\sigma_R$ data.

full rationale

The initial step of the paper is a legitimate empirical extraction: neutron radii in Table II are obtained by fitting the measured $\sigma_R$ of $^{24-38}$Mg on $^{12}$C, with DRHBc proton radii as input. That part is not circular. Circularity enters when the core+n construction is 'subjected to reproduce the same neutron radius' obtained from that fit and the resulting $\sigma_R$ agreement is then offered as validation: the parameter is the fitted radius, so the 'good agreement' is largely forced. Separately, the halo conclusion for $^{37}$Mg is not an independent empirical result: Eqs. (22)-(23) build the asymptotic tail directly from $S_n$, and the paper itself notes that the spread is inversely proportional to $S_n$. The maximal tail of $^{37}$Mg is therefore a restatement of its small separation energy. The $^{40}$Mg core+2n calculation is a genuine prediction in the sense that no $^{40}$Mg data are used, but it inherits the same untested tail ansatz and the halo language attached to it. Overall the central validation and one halo claim reduce to their inputs by construction, while the underlying $\sigma_R$ fits and the $^{40}$Mg numbers retain some independent content; hence score 6 rather than 8-10.

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

The central claims rest on per-isotope neutron density parameters fitted to sigma_R, plus the assumed tail density form and separation-energy inputs. No new particles or forces are introduced.

free parameters (4)

SDHO neutron oscillator constant alpha^2_n for each isotope = Table II, e.g. 24Mg: 0.2573 fm^-2
Varied to reproduce experimental sigma_R; defines the extracted neutron radius.
2pF neutron diffuseness a_n for each isotope = Table II, e.g. 24Mg: 0.5890 fm
Varied to reproduce experimental sigma_R; defines an alternative extracted neutron radius.
SDHO proton oscillator constant and 2pF proton diffuseness per isotope = Table I
Chosen to reproduce DRHBc proton radii; these are inputs, not fitted to sigma_R, but they affect the extracted neutron radii.
Core+n tail normalization N0 = Determined by neutron number in the tail, not tabulated
Fixed by requiring the tail contains the correct number of neutrons; a normalization rather than a free fit parameter.

assumptions (5)

standard math The Glauber model with the Ahmad two-body correlation correction (Eqs. 4-10) describes the reaction cross sections.
Borrowed from Ref. [22] and prior work; standard in nuclear reaction theory.
domain assumption The neutron tail density has the functional form of Eq. (22) from Bhagwat et al. [34].
The halo claims are direct consequences of this assumed form, so the paper's central conclusions rest on it.
ad hoc to paper The proton radius of the core nucleus equals that of the free nucleus, even when the core is bound in the core+n system.
Stated in Section III.C as an assumption; not independently justified.
domain assumption The one- and two-neutron separation energies from the AME2020 mass evaluation [28] are correct inputs.
Used to assign core+n versus core+2n and to set the tail decay length.
domain assumption The 12C target density is taken from electron scattering and assumed equal for protons and neutrons.
Standard assumption, but not validated in this paper.

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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.

Figures

Figures reproduced from arXiv: 2608.09429 by the authors.

**Figure 3.** FIG. 3: The 2pF (black line) and SDHO (red line) neutron densi [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4: The point matter radii corresponding to our deduced 2 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5: The neutron skin thickness as a function of mass numbe [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: The 2pF neutron density distributions, obtained usi [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7: The SDHO neutron density distributions, obtained us [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8: The reaction cross sections of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9: The 2pF (filled squares) and SDHO (filled circles) neut [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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Works this paper leans on

41 extracted references · 20 canonical work pages

[1]

De Vries, C

H. De Vries, C. W. De Jager, and C. De Vries, At. Data and Nucl. Data Tables 36, 495 (1987)

work page 1987
[2]

N ¨ortersh¨auser et al., Phys

W. N ¨ortersh¨auser et al., Phys. Rev. A 83, 012516 (2011)

work page 2011
[3]

Chulkov et al., Nucl

L.V . Chulkov et al., Nucl. Phys. A 674, 330 (2000)

work page 2000
[4]

Yamaguchi et al., Phys

T. Yamaguchi et al., Phys. Rev. C 82, 014609 (2010)

2010
[5]

Estrade et al., Phys

A. Estrade et al., Phys. Rev. Lett. 113, 132501 (2014)

2014
[6]

The quantitiesρT andρP in Eq

indicate the restriction that two pairs of indices can not be equal at the same time (for example, if i1 = i2 then j1 ⁄= j2 and vice versa). The quantitiesρT andρP in Eq. (
[7]

8: The reaction cross sections of 25−38Mg isotopes on a 12C target at 240 MeV/nucleon, using the core+n description for the pro- jectiles

It is found that the core+n description of 25−38Mg isotopes, along with our neutron radii (Table II), /s50/s52 /s50/s54 /s50/s56 /s51/s48 /s51/s50 /s51/s52 /s51/s54 /s51/s56 /s52/s48 /s49/s49/s48/s48 /s49/s49/s53/s48 /s49/s50/s48/s48 /s49/s50/s53/s48 /s49/s51/s48/s48 /s49/s51/s53/s48 /s49/s52/s48/s48 /s49/s52/s53/s48 /s49/s53/s48/s48 /s49/s53/s53/s48 /s49...

work page
[8]

are the (one-body) ground state densities of the target and projectile, respectively. From calculations point of view, it should be noted that the distinction between protons and neutrons in both the projectile and target has been included in the uncorrelated part ( S0) of the S-matrix element only. This makes the use of different values of the parameters...

Show all 41 references

[9]

( 25)) given in Table II

(an in Eq. ( 25)) given in Table II. In these calculations, our motive is to see how far the neutron radii and the reac- tion cross sections, obtained in this way, agree, respectiv ely, with the corresponding extracted values (Table II) and the ex- periment [19]. The results o...

1944
[10]

Bhagwat and Y

A. Bhagwat and Y . K. Gambhir, Phys. Rev. C 69, 014315 (2004)

2004
[11]

Yamaguchi et al., Phys

T. Yamaguchi et al., Phys. Rev. Lett. 107, 032502 (2011)

2011
[12]

G. W. Fan and Xu Zhan, Int. J. Mod. Phys. E 28, 1950070 (2019)

2019
[13]

Terashima et al

S. Terashima et al. , Prog. Theor. Exp. Phys. 2014, 101D02 (2014)

2014
[14]

Kanungo et al., Phys

R. Kanungo et al., Phys. Rev. Lett. 117, 102501 (2016)

2016
[15]

Bagchi et al., Phys

S. Bagchi et al., Phys. Lett. B 790, 251 (2019)

2019
[16]

Kaur et al., Phys

S. Kaur et al., Phys. Rev. Lett. 129, 142502 (2022)

2022
[17]

Hasan, M

Z. Hasan, M. Imran, A. A. Usmani, and Z. A. Khan, Phys. Rev . C 111, 064601 (2025)

2025
[18]

Hasan, M

Z. Hasan, M. Imran, A. A. Usmani, and Z. A. Khan, Nucl. Phy s. A 1071, 123406 (2026)

2026
[19]

Takechi et al., Phys

M. Takechi et al., Phys. Rev. C 90, 061305(R) (2014)

2014
[20]

I. A. M. Abdul-Magead and Badawy Abu-Ibrahim, Nucl. Phy s. A 1000, 121804 (2020)

2020
[21]

has also been used [ 34] to calculate the reaction cross sections for loosely bound e x- otic nuclei (in the neutron-halo region) having very small neu- tron separation energy. The results are found to be consiste nt with the experimental values and the calculated neutron den ...
[22]

Tanaka et al., Phys

M. Tanaka et al., Phys. Rev. C 106, 014617 (2022)

2022
[23]

Yamaki et al., Nucl

S. Yamaki et al., Nucl. Instrum. Methods Phys. Res. B 317, 774 (2013)

2013
[24]

( 25) for 2pF den- sity in order to get our extracted neutron radii (Table II) for (N >Z) Mg isotopes

for SDHO density, and (ii) the parameter an in Eq. ( 25) for 2pF den- sity in order to get our extracted neutron radii (Table II) for (N >Z) Mg isotopes. The respective parameters for 2pF and SDHO densities are given in Table III. Table III also presents the neutron radius of ...
[25]

The central density ρ0 is determined by the normalization to the number of pro- tons (Z) or neutrons (N )

(r0=1.2 fm) is the radius parameter. The central density ρ0 is determined by the normalization to the number of pro- tons (Z) or neutrons (N ). It is, thus, obvious that, in the 2pF distribution, the diffuseness parameter a is the only variable parameter, which is responsible ...
[26]

Imran, Z

M. Imran, Z. Hasan, A. A. Usmani, and Z. A. Khan, Phys. Rev . C 110, 014623 (2024)

2024
[27]

Peng Guo et al. , At. Data and Nucl. Data Tables 158, 101661 (2024)

2024
[28]

Singh et al., Phys

J. Singh et al., Phys. Lett. B 853, 138694 (2024)

2024
[29]

Ahmad, J

I. Ahmad, J. Phys. G: Nucl. Phys. 6, 947 (1980)

1980
[30]

Abu-Ibrahim, W

B. Abu-Ibrahim, W. Horiuchi, A. Kohama, and Y . Suzuki, Phys. Rev. C 77, 034607 (2008)

2008
[31]

Ahmad, A

S. Ahmad, A. A. Usmani, and Z. A. Khan, Phys. Rev. C 96, 064602 (2017)

2017
[32]

Mouadil et al., Phys

M. Mouadil et al., Phys. Rev. C 109, 054330 (2024)

2024
[33]

Aumann, C

T. Aumann, C. A. Bertulani, and J. Ryckebusch, Phys. Rev . C 88, 064610 (2013)

2013
[34]

Aumann et al., Prog

T. Aumann et al., Prog. Part. Nucl. Phys. 118, 103847 (2021)

2021
[35]

Wang et al., Chinese Phys

M. Wang et al., Chinese Phys. C 45, 030003 (2021)

2021
[36]

Ozawa, T

A. Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. A 693, 32 (2001)

2001
[37]

Kanungo et al., Phys

R. Kanungo et al., Phys. Rev. C 83, 021302(R) (2011)

2011
[38]

Sharma, R

Mahesh K. Sharma, R. N. Panda, Manoj K. Sharma, and S. K. Patra, Phys. Rev. C 93, 014322 (2016)

2016
[39]

Y . K. Gambhir and S. H. Patil, Z. Phys. A 321, 161 (1985)

1985
[40]

Y . K. Gambhir and S. H. Patil, Z. Phys. A 324, 9 (1986)

1986
[41]

Bhagwat, Y

A. Bhagwat, Y . K. Gambhir, and S. H. Patil, Eur. Phys. J. A 8, 511 (2000)

2000

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

De Vries, C

H. De Vries, C. W. De Jager, and C. De Vries, At. Data and Nucl. Data Tables 36, 495 (1987)

work page 1987

[2] [2]

N ¨ortersh¨auser et al., Phys

W. N ¨ortersh¨auser et al., Phys. Rev. A 83, 012516 (2011)

work page 2011

[3] [3]

Chulkov et al., Nucl

L.V . Chulkov et al., Nucl. Phys. A 674, 330 (2000)

work page 2000

[4] [4]

Yamaguchi et al., Phys

T. Yamaguchi et al., Phys. Rev. C 82, 014609 (2010)

2010

[5] [5]

Estrade et al., Phys

A. Estrade et al., Phys. Rev. Lett. 113, 132501 (2014)

2014

[6] [6]

The quantitiesρT andρP in Eq

indicate the restriction that two pairs of indices can not be equal at the same time (for example, if i1 = i2 then j1 ⁄= j2 and vice versa). The quantitiesρT andρP in Eq. (

[7] [7]

8: The reaction cross sections of 25−38Mg isotopes on a 12C target at 240 MeV/nucleon, using the core+n description for the pro- jectiles

It is found that the core+n description of 25−38Mg isotopes, along with our neutron radii (Table II), /s50/s52 /s50/s54 /s50/s56 /s51/s48 /s51/s50 /s51/s52 /s51/s54 /s51/s56 /s52/s48 /s49/s49/s48/s48 /s49/s49/s53/s48 /s49/s50/s48/s48 /s49/s50/s53/s48 /s49/s51/s48/s48 /s49/s51/s53/s48 /s49/s52/s48/s48 /s49/s52/s53/s48 /s49/s53/s48/s48 /s49/s53/s53/s48 /s49...

work page

[8] [8]

are the (one-body) ground state densities of the target and projectile, respectively. From calculations point of view, it should be noted that the distinction between protons and neutrons in both the projectile and target has been included in the uncorrelated part ( S0) of the S-matrix element only. This makes the use of different values of the parameters...