Deformed magic numbers at $N=$178 and $Z=$120, 124 in the 112 $\leq N \leq $ 190 superheavy region from Skyrme mean-field calculations

Kok-Siong Khoo; Mastura Syamimi Abdullah; Meng-Hock Koh; W. Asous

REVIEW 4 major objections 5 minor 77 references

Deformed magic numbers at $N=$178 and $Z=$120, 124 in the 112 $\leq N \leq $ 190 superheavy region from Skyrme mean-field calculations

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Neutron number N=178 and proton numbers Z=120, Z=124 are identified as deformed magic numbers in superheavy nuclei.

desk verdict Solid systematic Skyrme survey of superheavy magic candidates, but a sign inconsistency in the separation-energy differential definition undercuts the central ridge identification until resolved. read the letter →

arxiv 2506.02684 v1 pith:AQIWFFI7 submitted 2025-06-03 nucl-th

W. Asous , Mastura Syamimi Abdullah , Meng-Hock Koh , Kok-Siong Khoo This is my paper

classification nucl-th

keywords superheavynucleideformedmagicnumbersSkyrmemean-fieldHartree-Fock-BCSalpha-decayhalf-livestwo-nucleonseparationenergyoblatedeformationN=178

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 where extra stability, or magic numbers, sits in the superheavy region beyond the known spherical closures. Using Skyrme mean-field calculations with three parametrizations, it claims that neutron number $N=178$ acts as a deformed magic number for proton numbers around $114\le Z\le 118$, and that proton numbers $Z=120$ and $Z=124$ act as deformed magic numbers near $N=172$–$178$, all at oblate ground-state shapes. If correct, these nuclei would be more stable against $\alpha$ decay than their neighbours and could serve as stepping stones toward the hypothesized doubly magic nucleus $^{310}_{126}$. The claim matters because deformed shells in this region are not reliably predicted by other models, so identifying where they occur guides which superheavy nuclei to search for experimentally.

What carries the argument

The central machinery is the Skyrme energy density functional solved in a Hartree-Fock-plus-BCS scheme, with ground states located by constrained axial quadrupole calculations. Two-nucleon separation energy differentials $\delta S_{2n}$ and $\delta S_{2p}$, defined as second differences of binding energies, act as shell-gap detectors, while single-particle spectra at the oblate, spherical, and prolate minima trace the underlying energy gaps. Alpha-decay energies $Q_\alpha$ and half-lives from eight semi-empirical formulas corroborate the shell structure, and the dimensionless deformation $\beta_{20}$ attaches each magic-number candidate to a specific shape.

What would settle it

Compute the ground states of $^{292}_{114}$, $^{294}_{116}$, $^{296}_{118}$, and $^{298}_{120}$ with a fully unrestricted triaxial and octupole-allowed energy minimization and check whether the oblate minima and the $N=178$, $Z=120$ shell gaps survive; alternatively, measure the $\alpha$-decay energies and half-lives across the $N=178$ isotones near $Z=114$–$120$ and look for the predicted dip in $Q_\alpha$ and peak in $T_{1/2}$ relative to $N=176$ and $N=180$.

Watch

Extended reading notes

Core claim

Within the Skyrme Hartree-Fock-plus-BCS framework, using the SkM*, SLy5, and SLy4 parametrizations, the paper finds a ridge in the two-neutron separation energy differential at $N=178$ for $112\le Z\le 122$, backed by a large single-particle energy gap at oblate deformation; this identifies $N=178$ as a deformed neutron magic number for $Z=114$–$118$. On the proton side, ridges in the two-proton separation differential at $Z=120$ and $Z=124$, backed by proton energy gaps and by peaks in $\alpha$-decay half-life differences, identify $Z=120$ and $Z=124$ as deformed magic numbers near $N=172$–$178$. Simultaneous peaks at $Z=126$ and $N=184$ reinforce the spherical doubly magic nucleus $^{310}_{126}$. The paper therefore claims a set of deformed shell closures located at oblate ground-state shapes, while weaker candidates such as $N=174$ and $N=172$ appear only for some parametrizations.

Load-bearing premise

The load-bearing premise is that every ground state is axially symmetric and reflection-symmetric; if triaxial or octupole distortions lower the energy for any of these nuclei, the claimed oblate deformed magic numbers could shift or disappear.

Editorial extensions

If this is right

If $N=178$ is a deformed magic number, isotones near $Z=114$–$118$ with $N=178$ should show longer alpha-decay half-lives and lower $Q_\alpha$ than neighbouring isotones, giving a concrete experimental signature.
If $Z=120$ and $Z=124$ are deformed proton magic numbers, the most stable proton-rich superheavy candidates cluster around $N=172$–$178$, narrowing the search space for synthesis.
The simultaneous peaks in $\delta S_{2n}$ and $\delta S_{2p}$ at $N=184$ and $Z=126$ reinforce the case for $^{310}_{126}$ as a doubly magic spherical nucleus.
Because the $N=174$ candidate appears only with SLy5 and SLy4 while SkM* reproduces the experimental trend, the paper's central candidate $N=178$ is the one that survives cross-force comparison.
The parametrization-dependent appearance of $N=172$ and $N=174$ sub-magic numbers means that only experimental data on these isotones can settle whether those weaker shell effects are real.

Reading between the lines

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

Beyond the paper: because the ground-state search was restricted to axial, parity-even shapes, triaxial or octupole minima could in principle lower some of these nuclei below the oblate minima, and checking that with an unrestricted deformation search would either harden or overturn the $N=178$ and $Z=120$, $Z=124$ identifications.
Beyond the paper: the disagreement among the three Skyrme forces about $N=174$ versus $N=178$ suggests that deformed shell strength in this region is fragile; a measured alpha-decay chain crossing $N=178$ would discriminate between the parametrizations.
Beyond the paper: the same shell-gap machinery could be applied to odd-neutron and odd-odd nuclei in the region, where pairing and blocking might shift the deformed gaps and change which isotope chains are the most practical synthesis targets.

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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

Colleague,

This is a systematic Skyrme HF+BCS survey of the N=170-190, Z=112-130 region, identifying N=178 and Z=120,124 as deformed magic numbers at oblate deformation. The calculations are standard and careful: basis-convergence tests, comparison with HFB results for BE/A and radii, and eight alpha-decay half-life formulas. The tables are useful, and the Z=124 candidate is genuinely new. So there is real content here.

But there is a load-bearing sign problem. Eq. (13) defines S2n = EB(N-2)-EB(N), the negative of the usual separation energy. Eq. (14) then defines δS2n = S2n(N)-S2n(N+2), which algebraically is the negative of the standard three-point gap. Eq. (15) gives that standard gap, 2EB(N)-EB(N-2)-EB(N+2). These two are not the same quantity. If the code used Eq. (15), then for SkM* at Z=114 the table gives δS2n(178)=0.31 MeV, with 0.59 at N=176 and 0.32 at N=180: a local minimum, not a ridge. The claimed N=178 ridge in Fig. 5 is not visible for that chain. If the code used Eq. (14), the plotted ridges are in the negative gap, and Eq. (15) is a typo. Either way, the central observable needs to be fixed and the figures regenerated with a clearly stated convention. This is not cosmetic; the magic-number identification rests on these ridges.

Other soft spots are less severe. The peak identification is heuristic—no quantitative threshold for a 'ridge'. The ground state is restricted to axial/parity-even shapes, so a triaxial instability would shift these conclusions. And the argument that N=174 is not magic leans on SkM* matching the experimental Qα trend, a reasonable but model-dependent selection. None of those are disqualifying on their own, but the sign issue is.

Who this is for: practitioners working on superheavy structure and synthesis planning will find the tables and the Z=124 suggestion worth engaging with. But before citing or using the magic-number assignment, I would want the sign issue resolved. If the authors clarify the convention and confirm which quantity generated Fig. 5, the paper is a solid candidate for publication. As it stands, it deserves a serious referee, but the referee should demand that correction.

Recommendation: send to peer review, with the sign inconsistency as the first major comment.

Referee Report

4 major / 5 minor

Summary. This manuscript reports Skyrme-HF+BCS calculations for even-even superheavy nuclei with 112 ≤ Z ≤ 130 and 172 ≤ N ≤ 190, using the SkM*, SLy5, and SLy4 parametrizations. It evaluates ground-state quadrupole deformations, binding energies per nucleon, two-nucleon separation-energy differentials δS2q, α-decay energies, and α-decay half-lives from eight semi-empirical formulas. On this basis the authors propose N = 178 as a neutron deformed magic number near Z = 114–118 and Z = 120 and Z = 124 as proton deformed magic numbers near N = 172–178, all associated with oblate ground states, and they reconfirm N = 184 and Z = 126 as spherical magic numbers. The calculations include a basis-convergence test (Table 3) and comparison with HFB binding energies (Table 4).

Significance. If the predictions survive closer scrutiny, the paper is a useful systematic map of possible shell closures in a region relevant to superheavy-element synthesis. Its strengths are the use of three standard Skyrme forces, explicit basis-convergence checks, comparison against HFB results, and the combination of several observables (δS2q, Qα, T1/2, single-particle gaps) that are, in part, independently falsifiable by future α-decay measurements. The central new candidates (N = 178, Z = 120, Z = 124) are clearly stated. However, the load-bearing δS2q analysis is compromised by an internal sign inconsistency (Eq. (15) vs Eq. (14)), and the restriction to axial/parity-symmetric shapes is an acknowledged assumption that is directly relevant to the deformation labels attached to the magic numbers.

major comments (4)

[Sec. 2.3, Eqs. (14)–(15), Fig. 5, Sec. 3.4] Eq. (14) defines δS2n = S2n(N,Z) − S2n(N+2,Z), which with Eq. (13) gives δS2n = EB(N−2)+EB(N+2)−2EB(N). Eq. (15) defines δS2n = 2EB(N) − EB(N−2) − EB(N+2), i.e., the exact negative. For a closed shell the standard two-nucleon differential has a local maximum; the quantity in Eq. (15) has a local minimum. A direct check using Table 4 for SkM* at Z = 114 gives, via Eq. (15), δS2n = 0.59 MeV at N = 176, 0.31 MeV at N = 178, and 0.32 MeV at N = 180, i.e., a local minimum at N = 178, not the ridge claimed in Sec. 3.4 and Fig. 5. The authors must either correct Eq. (15) or, if the figure uses the positive gap, state explicitly which convention is plotted. As it stands, the central evidence for N = 178 as a magic number from δS2n is inverted by the written definition.
[Sec. 2.1] The paper states that 'The ground-state is assumed to be axially and parity symmetric shape.' Since the abstract and conclusions identify the magic numbers with oblate (or prolate) ground-state deformation, a triaxial or octupole instability at any of the candidate nuclei could change both the deformation label and the size of the shell gaps. Please test representative candidates (e.g., 296118 and 298120, which are highlighted in Sec. 3.6) with triaxial and reflection-asymmetric degrees of freedom, or cite work showing these are not important in this region, and adjust the claims if the ground states change.
[Sec. 2.1, Table 2, Sec. 3.3] The pairing strengths are averaged over the 86 nuclei considered, and Sec. 3.3 (Fig. 4) shows that the root-mean-square and charge radii are strongly sensitive to the choice of pairing strength. Because pairing affects the two-nucleon separation-energy differentials and the single-particle gaps used to identify magic numbers, the paper should demonstrate that the δS2q ridges and the assigned magic numbers are stable with respect to a reasonable variation of the averaged pairing strengths (e.g., using the two sets already discussed in Sec. 3.3 and Table 2).
[Sec. 3.4 and Sec. 4] The evidence for Z = 124 as a deformed proton magic number is not presented consistently. In Sec. 3.4 the δS2p maxima for Z = 124 are identified at N = 176 (SkM*) and N = 180 (SLy5), while Fig. 6 shows a proton gap for Z = 124 only for SLy5 at N = 180; the abstract and conclusion list a broader range N = 172–178. Please specify, for each force, the neutron range in which Z = 124 is supported by each observable.

minor comments (5)

[Sec. 2.1] There is a typo 'and and ωz' in the definition of the oscillator frequencies, and the definition of ω0 is incomplete ('ω3_0 = ω2_⊥ωz').
[Fig. 5 caption] The caption contains the typo 'Skryme' and should read 'Skyrme'.
[Table 10] In row Z = 124, N = 180, the column T_mR shows '27.43', which is clearly erroneous since neighboring values are around −6; this value would corrupt the averaged half-lives and error bars if used literally.
[Sec. 3.6] The sentence 'the SLy4 parametrization shows peaks at N = 174 for Z = 124 and 126 indicating that N = 172 to be magic' is confusing; please clarify the parent/daughter logic and the intended magic number.
[References and Appendix] Several references are incomplete or misspelled (e.g., 'O. Y. Ts' in Refs. 2–4 should be 'Yu. Ts. Oganessian'), and the Appendix uses 'paramterization' for 'parametrization'; a careful proofread is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: magic-number candidates emerge from Skyrme mean-field observables, not from fitted inputs or self-citation; a sign inconsistency in δS2n is a correctness issue, not a circular one.

full rationale

The paper's derivation chain is self-contained against the model: Skyrme HF+BCS calculations with fixed literature parametrizations (SkM*, SLy5, SLy4) and averaged pairing strengths calibrated to the Madland gap formula over the 86 studied nuclei. These averaged strengths are not tuned to the claimed magic numbers N=178, Z=120, 124, so the subsequent δS2q, Qα, and T1/2 patterns are model outputs rather than imposed results. The alpha-decay half-life formulae are taken from external fits. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The self-citations (Refs. 69 and 72, by co-author Koh) are not load-bearing: Ref. 69 is cited alongside multiple external works for the separation-energy differential method, and Ref. 72 is mentioned only as a possible future extension. The central conclusions are instead cross-checked against single-particle gaps, Qα trends, half-life systematics, and experimental data. Two caveats are noted, but they are correctness risks rather than circularity. First, the restriction to axially symmetric, parity-even shapes (Sec. 2.1) could alter deformation assignments if triaxial or octupole minima exist. Second, Eq. (15) is the algebraic negative of Eq. (14): Eq. (14) gives δS2n = EB(N−2)+EB(N+2)−2EB(N), while Eq. (15) gives 2EB(N)−EB(N−2)−EB(N+2). If the code implements Eq. (15) literally, the claimed 'ridges' at N=178 and Z=120, 124 would be local minima of the standard two-nucleon gap, inverting the magic-number identification. This internal sign inconsistency undermines the evidence but does not make the conclusion equivalent to an input or fitted parameter; it requires correction and reanalysis of the affected figures. Overall, no circular step is exhibited.

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

The central claim rests on the validity of the Skyrme mean-field approach for this region, the axial/parity shape restriction, and the heuristic identification of magic numbers through peaks in observables. The fitted parameters are the averaged pairing strengths and the basis truncation; none are tuned to the claimed magic numbers.

free parameters (5)

Pairing strength G_n for SkM* = 14.79 MeV
Fitted per nucleus to Madland pairing gap formula (Eq. 2) for 86 nuclei, then averaged; value from Table 2.
Pairing strength G_p for SkM* = 13.72 MeV
Fitted per nucleus to Madland pairing gap formula (Eq. 3) for 86 nuclei, then averaged; value from Table 2.
Pairing strength G_n for SLy5 and SLy4 = 16.72 MeV
Same procedure as above; value from Table 2.
Pairing strength G_p for SLy5 and SLy4 = 14.25 MeV
Same procedure as above; value from Table 2.
Harmonic oscillator basis truncation N0 = 18
Selected from convergence of Q_alpha for Og-294 (Table 3); N0=18 chosen as Q_alpha nearly constant beyond.

assumptions (4)

domain assumption Skyrme energy density functional with the chosen parametrizations (SkM*, SLy5, SLy4) accurately describes the superheavy region.
The entire calculation rests on the validity of these mean-field functionals for nuclei far from stability; SLy5 is known to have spin-instability, and SLy4 is said to perform well in the superheavy region (Sec. 2.1).
domain assumption Ground states are axially symmetric and parity even.
Stated in Sec. 2.1: 'The ground-state is assumed to be axially and parity symmetric shape.' Triaxial or octupole shapes are not explored, which could alter deformation and magic-number assignments.
domain assumption Peaks in delta S2q, Q_alpha, and T1/2 indicate magic numbers.
The analysis identifies magic numbers from peaks in these observables (Secs. 3.4-3.7); this is a standard but heuristic criterion.
domain assumption Semi-empirical alpha-decay formulas (VS, PS, Royer, Brown) are valid for extrapolation to superheavy nuclei.
Eight literature formulas are used and averaged; their spread is treated as uncertainty (Sec. 2.2, Appendix A).

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Cite this review

Pith. "Pith review of Deformed magic numbers at $N=$178 and $Z=$120, 124 in the 112 $\leq N \leq $ 190 superheavy region from Skyrme mean-field calculations." pith.science (2026). https://pith.science/paper/AQIWFFI7

@misc{pith2026250602684,
  author       = {Pith},
  title        = {Pith review of: Deformed magic numbers at $N=$178 and $Z=$120, 124 in the 112 $\leq N \leq $ 190 superheavy region from Skyrme mean-field calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQIWFFI7}},
  note         = {Machine review of arXiv:2506.02684}
}

abstract

Background: Various motivations for exploration of superheavy region revolve around the question on whether 126 is a spherical proton magic number, as is the case for neutrons. In exploring this region, identification of nuclei with relatively longer half-life as compared to its neighbours is crucial for experimental studies. Such information is provided from theoretical predictions, which are however, heavily dependent on the theoretical model used and observable quantities under investigation. Purpose: Limiting ourselves to the Skyrme Hartree-Fock-plus-Bardeen-Cooper-Schrieffer approach, we aimed to analyse the appearance of a nuclear region with relatively high stability associated with emergence of spherical and deformed magic numbers in the region of $170 \le N \le 190$ ($112 \le Z \le 130$) based on various observables. Methods: Three Skyrme parametrizations namely the SkM* frequently employed for fission calculations, and the SLy5 and SLy4 commonly used for superheavy region, are considered to provide comparisons within the Skyrme mean-field approach. We evaluated the variation of electric quadrupole deformation ($\beta_{20}$), binding energy per nucleon ($BE/A$), two-nucleon separation energy differential ($\delta S2_{q}$), alpha-decay energy ($Q_{\alpha}$) and alpha-decay half-lives ($T_{1/2}$). Conclusion: Our analyses suggest that neutron number $N = 178$ is candidate for deformed magic number around proton number $114 \le Z \le 118$. For protons, $Z = 120$ and $124$ appears to be a candidate for deformed magic number at around $N = 172 \sim 178$. Both sets of deformed magic numbers appear at oblate ground-state deformation.

Figures

Figures reproduced from arXiv: 2506.02684 by the authors.

**Figure 2.** Shows plots of binding energy per nucleon ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Top plots shows root-mean-square radii ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: The figure shows plots for (a) root-mean-square radii ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: shows the two-neutron separation energy differential δS2n and two-proton separation energy differential δS2p. Simultaneous evaluation of both quantities is necessary to pinpoint the emergence of shell closures, as suggested by5 . It is obvious from [PITH_FULL_IMAGE:fi…

**Figure 6.** Figure 6: Single-particle levels at the oblate (Q20 < 0), spherical (Q20 = 0) and prolate (Q20 > 0) minimum obtained with the SkM* and SLy5 parametrizations for the Z = 114, N = 174 nucleus. The single bottom panel shows proton single-particle levels obtained with the SLy5 param…

**Figure 7.** Figure 7: Alpha-decay energies, Qα-values (in MeV) for isotopic series 114 ≤ Z ≤ 126 and 172 ≤ N ≤ 190 for the SkM*, SLy5, and SLy4 Skyrme parametrizations as a function of N. Available experimental values taken from4 and75 are plotted in star shaped according to their designate…

**Figure 8.** Figure 8: Shows overlapped between the ground-state [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Shows the log10 T1/2 of all Skyrme parametrization are plotted from averaged of 8 different formulae against (N) and the error bar are the maximum and minimum T1/2 range for isotopic series 114 ≤ Z ≤ 130 (172 ≤ N ≤ 190). The experimental log T α 1/2 are plotted using m…

**Figure 10.** Figure 10: Shows difference in alpha-decay half-life log [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: Shows even-even nuclei in the superheavy region of nuclear chart. The boxes in grey rep [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: Analysis of 8 different formulae for isotopes [PITH_FULL_IMAGE:figures/full_fig_p044_12.png]

**Figure 13.** Figure 13: Analysis of 8 different formulae for isotopes [PITH_FULL_IMAGE:figures/full_fig_p045_13.png]

**Figure 14.** Figure 14: Analysis of 8 different formulae for isotopes [PITH_FULL_IMAGE:figures/full_fig_p046_14.png]

**Figure 15.** Figure 15: Analysis of Brown formulae type (B, MB1, and MB2) on [PITH_FULL_IMAGE:figures/full_fig_p047_15.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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