{"id":"9862a308-2ea6-4f88-a9cd-4a3b5f37aeca","arxiv_id":"2506.20414","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A pair-condensation correction added to Skyrme HFB reproduces shell-closure kinks at N = 28, 82, and 126 and predicts shell quenching at N = 50 in nickel and tin isotopes.","lead":"Adding a pairing-based correction to standard Skyrme nuclear calculations makes them reproduce the sudden jumps in nuclear charge radii at magic neutron numbers in calcium, nickel, tin, and lead. If the correction transfers reliably, it gives a cheap way to predict nuclear sizes for exotic isotopes without a new microscopic theory.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The untested transfer of the fitted constant a0 from RMF to Skyrme HFB is load-bearing: without a per-chain/per-force refit or quantitative validation, the reported kinks may be artifacts of a single coefficient.","rationale":"The paper is transparent about its scope and the phenomenological nature of the added term, and the figures do show real changes of slope after adding the correction. The mechanism is plausible: a positive correction proportional to sqrt(A)|D_n - D_p| naturally suppresses the correction at neutron magic numbers and inflates it between shells, which can generate the inverted parabolic shapes and kinks described in Section IV. That the same functional form works qualitatively across four chains with two Skyrme forces and one relativistic force is a genuine, falsifiable observation. However, the central claim as stated in the abstract and summary is stronger than the evidence presented. The only free parameter, a0 = 0.561, was fitted in a different theoretical framework and to different nuclei, and no uncertainty estimate or statistical comparison is supplied. Since D_{n,p} is not a direct experimental observable but depends on the pairing treatment, the same coefficient may not carry over to Skyrme HFB. The paper's own statements about overestimated Ca radii and deviating Pb slopes indicate that the amplitude is not universally correct. A per-chain refit of a0 is the minimal decisive check: if the fitted values cluster near 0.561, the transferability concern is retired; if they scatter widely, the claimed universality is unsupported. The reader already issued CONDITIONAL based on the same weakest assumption, so my stress-test leaves that verdict unchanged.","tokens_in":22495,"tokens_out":6066,"duration_ms":68244,"concrete_test":"Refit a0 separately for each even-even chain (Ca, Ni, Sn, Pb) and each force (SLy5, SkM*) using the same HFB solver and Eq. (3), minimizing the rms deviation of differential charge radii delta R^2(N) = R^2(N) - R^2(ref) against the experimental data used in Figs. 1-4. Report the fitted a0 with its uncertainty for each chain and force. If the optimal values are mutually inconsistent or scatter by more than about 20% around a0 = 0.561, the universal-transferability assumption fails and the kinks cannot be attributed to a single physics-motivated constant. As a secondary check, compare |D_n - D_p| for a fixed nucleus, e.g., 44Ca or 132Sn, computed with HFB(SLy5) and RHB(NL3); large framework-dependent differences would explain any refit drift.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Eq. (3): R_ch^2 = <r_p^2> + 0.7056 + a0 sqrt(A) |D_n - D_p|, with a0 = 0.561 taken directly from Ref. [69], a relativistic calculation fitted to K and Ca charge radii. The scale of D_{n,p} = sum_{k>0} u_k v_k is not framework-independent: it depends on the pairing interaction, the quasiparticle cutoffs, and the mean-field model. In this work, the Skyrme HFB uses mixed-type pairing with V0 refitted per chain (Table I), so |D_n - D_p| for a given nucleus need not match the RMF value that fixed a0. No quantitative metric, such as chi-squared or residual analysis, is provided, so the visually reported kinks could reflect a coefficient that is accidentally adequate for one chain but too large or too small for others; indeed, the text itself reports overestimated 46-50Ca radii and deviating slopes in Pb. Because the correction is positive and largest for mid-shell nuclei when the proton species is magic (D_p ~ 0), it inflates mid-shell radii and produces an inverted parabola almost by construction, independent of whether the underlying Skyrme charge radii are physical. The load-bearing assumption is that a0 and the sqrt(A)|D_n - D_p| form transfer across frameworks and isotopic chains; this is asserted but not tested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends a phenomenological correction to the nuclear charge radius formula, R_ch^2 = <r_p^2> + 0.7056 + a0 sqrt(A) |D_n - D_p| with a0 = 0.561, to Skyrme Hartree-Fock-Bogoliubov calculations with the SLy5 and SkM* functionals. The correction term, taken from Ref. [69], is intended to account for neutron and proton Cooper-pair condensation around the Fermi surface. The authors compute differential mean-square charge radii and the three-point indicator Delta_r^(2N) for even-even Ca, Ni, Sn, and Pb isotopic chains, compare with experimental data and with relativistic RHB(NL3) results, and conclude that the modified model reproduces the shell-closure kinks at N = 28, 82, and 126 and predicts shell quenching at N = 50 in Ni and Sn.","tokens_in":22858,"tokens_out":5329,"duration_ms":61661,"significance":"If the claimed result is robust, the paper provides a useful extension of a phenomenological pairing-condensation correction to non-relativistic energy density functionals, a class of models that has difficulty describing kinks and odd-even effects in charge radii. The manuscript is honest about several limitations, uses two Skyrme forces and a relativistic comparison, and explicitly identifies that a0 was fitted in Ref. [69]. The main value is as a systematic test; however, the central claim rests on an untested transfer of a fitted constant across frameworks and chains, and the evidence is presented only through visual comparison. No quantitative validation metric, no benchmark of the Delta D scales, and no code or reproducibility details are provided, so the improvement, while plausible, is not yet demonstrated. The paper would be significantly strengthened by adding transferability checks and error measures.","major_comments":[{"comment":"The load-bearing assumption is that a0 = 0.561, fitted in Ref. [69] to potassium and calcium charge radii within a relativistic mean-field framework, transfers without renormalization to Skyrme HFB calculations that use chain-dependent pairing strengths V0. The quantity D_{n,p} = sum_{k>0} u_k v_k depends on the pairing interaction, the quasiparticle cutoff, and the mean-field model, so the scale of |D_n - D_p| in Skyrme HFB need not match the RMF scale that fixed a0. The manuscript does not compare D_n and D_p between HFB and RHB for common nuclei, does not refit a0 within Skyrme HFB, and does not test the sensitivity of the predicted kinks to the value of a0. Please add a transferability test, for example by varying a0 by +/-20% or by fitting a0 to one chain and checking the others, and report representative D_n and D_p values for the four chains.","section":"Section II, Eq. (3) and Table I"},{"comment":"The central claims that HFB* 'reproduces' the trends, that deviations are 'slight', and that SkM* behaves better than SLy5 for tin are based entirely on visual inspection. No RMS deviation, mean absolute error, chi-square, or other quantitative metric is given for the differential charge radii or for Delta_r^(2N). Without such metrics the reader cannot judge whether the improvement over HFB is statistically meaningful, whether the overestimates in 46-50Ca and the Pb slopes are acceptable, or which force is more successful. Please add per-chain error measures for HFB, HFB*, RHB, and RHB* relative to the experimental data, and report the numerical slopes around N = 28, 50, 82, and 126.","section":"Section III and Figs. 1-4"},{"comment":"The calcium chain is presented as a validation of the model, but this is partly circular because a0 = 0.561 was adjusted in Ref. [69] to reproduce the inverted parabolic shape and the odd-even staggering in potassium and calcium charge radii. The authors state this in Section II and then in Section III interpret the Ca agreement as support for the model. The Ca results should be framed as a consistency check of the cross-framework transfer rather than as an independent test. The independent validation should rest on the Ni, Sn, and Pb chains, or the authors should refit a0 using Skyrme HFB on calcium and then judge the Ca comparison on that basis.","section":"Section III, calcium discussion"},{"comment":"The numerical implementation lacks the details needed to reproduce D_{n,p}. The value of D_{n,p} depends on the quasiparticle energy cutoff, the spatial box or basis size, and the pairing strengths, but only the V0 values are listed in Table I. Since the correction term and its framework dependence are the core of the paper, please specify the HFB numerical parameters (cutoff, box size, number of basis states, and the neutron/proton pairing gaps used to fit V0). Reporting D_n, D_p, and Delta D for representative nuclei such as 40Ca, 48Ca, 68Ca, 100Sn, 132Sn, and 208Pb would also help the reader assess the transferability concern raised above.","section":"Section II, Eq. (4) and Section III, Table I"}],"minor_comments":[{"comment":"In Figs. 1 and 2, HFB* and RHB* are both denoted by 'open diamond', so the two model families cannot be distinguished in black-and-white print; please use different markers or line styles and define them clearly in each caption.","section":"Figs. 1 and 2 captions"},{"comment":"The reference nucleus for the nickel chain is inconsistent: Fig. 1 says 'relative to the references 40Ca and 56Ni', while the text says 'with respect to reference nuclei 40Ca and 58Ni'. Please unify the definition.","section":"Fig. 1 caption and Section III"},{"comment":"The constant 0.7056 is the square of the proton charge radius, presumably r_p = 0.84 fm. Please state this explicitly so that the value is not a magic number, and give the relevant references at that point.","section":"Eq. (3)"},{"comment":"There are numerous grammatical and typographical errors, such as 'gives raise to', 'the influence of new term', 'the theoretical results obtained by Eq. (3) are labeled by HFB*', and 'combining the existing literatures'. A careful language edit is needed.","section":"Throughout"},{"comment":"The summary states that shell closure effects are 'slightly distorted' in the Ca and Pb isotopes, but the body text describes overestimated 46-50Ca radii and deviating Pb slopes. Please quantify these deviations with the metrics requested in the major comments so that 'slightly' is meaningful.","section":"Section IV"},{"comment":"The text and captions refer to the lead reference nucleus as '182Pb'; please verify whether this is the intended experimental reference and state it consistently with the data compilation used.","section":"Figs. 2 and 4"}],"recommendation":"major_revision","confidential_remarks":"The paper is closely connected to the authors' earlier work, including Ref. [69] where a0 was fitted and Ref. [115] where Eq. (3) was already applied in RHB calculations. The new element here is the extension to Skyrme HFB with two forces, which is a legitimate contribution, but the overlap with the preceding papers should be made explicit. I would ask the editor to require the transferability test for a0 and the quantitative error analysis before publication; without those, the visual agreement could be fortuitous and the central claim is not yet supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nTwo things to know. This is the first Skyrme-HFB application of the pair-condensation charge-radius correction that An and collaborators built in RMF. That is a real but narrow novelty. The paper does what it claims visually: with the extra term, even-even Ca, Ni, Sn, and Pb show the kinks at N=28, 82, 126 and the N=50 quenching in Ni and Sn for both SLy5 and SkM*. Seeing the same qualitative correction under two forces is the best evidence that the effect is not a single-functional accident. The authors are also transparent about failures—overestimated 46-50Ca and Pb, missing deformation in neutron-deficient Pb, and they explicitly say the term is phenomenological.\n\nThe soft spot is a0=0.561. It is carried over from the earlier RMF fit with no re-fit and no sensitivity study. The scale of D_n and D_p depends on the pairing interaction, the cutoff window, and the mean field; the authors refit V0 per chain but never show whether the Skyrme D values are in the same range as the RMF values that fixed a0. Without that, the kinks could be an artifact of an accidentally adequate constant. Also, Ca is not an independent test because a0 was fitted to Ca/K data—the Ni, Sn, and Pb chains are the real check, and they are the strongest part of the paper. The evidence is otherwise visual: no residuals, no chi-square, no numerical tables. Minor issues: inconsistent figure captions (56Ni vs 58Ni) and no code or data released.\n\nMy bottom line: plausible physics, honest limitations, and falsifiable N=50 predictions. It deserves a serious referee, but the referee should request a sensitivity scan of a0, a comparison of D values between Skyrme and RMF, and a table of computed radii. Then it becomes a useful reference. I wouldn't cite it in my own work yet; I'd consider it for a reading group after revision.","headline":"First Skyrme-HFB test of the pair-condensation charge-radius correction shows real kinks, but the unfitted constant a0 makes the result vulnerable; worth refereeing with a demand for sensitivity analysis.","tokens_in":23378,"tokens_out":4252,"would_cite":false,"duration_ms":49258,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81V35"],"pacs":["21.10.Ft","21.60.Jz","21.30.Fe"],"model":"deepseek-v4-flash","headline":"A single Cooper-pair-condensation correction to the Skyrme HFB charge-radius formula reproduces the measured shell-closure kinks at N=28, 82, and 126 and the shell quenching at N=50 in even-even Ca, Ni, Sn, and Pb isotopes.","keywords":["nuclear charge radii","Skyrme density functionals","Hartree-Fock-Bogoliubov","Cooper pair condensation","shell closure","shell quenching","differential charge radii","isotope shifts"],"falsifier":"Refit $a_0$ independently for each of the four isotopic chains and for each Skyrme force; if the best-fit values scatter well outside the uncertainty of 0.561, the universal transfer premise is false. A complementary experiment is a high-precision charge radius of $^{100}$Sn, where HFB$^*$ predicts a kink that plain HFB does not.","tokens_in":22277,"feed_emoji":"⚛️","tokens_out":11792,"duration_ms":118352,"temperature":0.7,"pith_summary":"Skyrme density functionals have long failed to reproduce the abrupt kinks in nuclear charge radii at shell closures. This paper argues that the failure is cured by adding one term, $a_0\\sqrt{A}|D_n-D_p|$, to the charge-radius formula, where $D_{n,p}=\\sum_{k>0}u_kv_k$ measures the Cooper-pair condensation of neutrons or protons near the Fermi surface. With the coefficient $a_0=0.561$ taken unchanged from earlier relativistic calculations, Skyrme HFB with the SLy5 and SkM$^*$ forces reproduces the trend of differential charge radii in even-even Ca, Ni, Sn, and Pb isotopes, including kinks at $N=28$, 82, and 126 and shell quenching at $N=50$ in Ni and Sn. If correct, this identifies the neutron-proton pairing asymmetry, rather than an artifact of one relativistic model, as the common origin of these shell effects in nuclear size.","feed_headline":"Pair-condensation term fixes nuclear radius shell kinks","feed_subtitle":"Skyrme HFB with one Cooper-pair term lands Ca, Ni, Sn, and Pb kinks at N=28, 82, and 126.","key_machinery":"The load-bearing object is the correction term $a_0\\sqrt{A}\\,|D_n-D_p|$ appended to the mean-square charge radius formula. Here $D_{n,p}=\\sum_{k>0}u_kv_k$ is a sum over quasiparticle amplitudes in the canonical HFB basis, so it measures the occupancy spread, or Cooper-pair condensation, of neutrons and protons near the Fermi surface, and $|D_n-D_p|$ is the neutron-proton asymmetry of that pairing condensate. The $D$ values come directly from the HFB self-consistent solution, while the overall strength $a_0=0.561$ is a fixed phenomenological coefficient fitted earlier to potassium and calcium charge radii and transferred without adjustment to Skyrme HFB. The term adds extra charge radius where neutron and proton pairing differ most, which is exactly at and beyond shell closures, turning the smooth Skyrme HFB curves into curves with kinks and quenching.","core_discovery":"The central claim is that the modified charge-radius formula $R_{\\rm ch}^2=\\langle r_p^2\\rangle+0.7056+a_0\\sqrt{A}\\,|D_n-D_p|$ with $a_0=0.561$ makes non-relativistic Skyrme HFB calculations reproduce the measured differential charge radii of even-even Ca, Ni, Sn, and Pb isotopes, whereas the same calculations without the extra term give smooth, monotonic curves that miss the shell closures. In the modified calculations, the kinks at $N=28$, 82, and 126 appear, the $N=50$ shell quenching appears in Ni and Sn, and the inverted parabolic-like shapes between filled shells are captured and weaken from Ca toward Pb. The authors take this as evidence that the difference between neutron and proton Cooper-pair condensation around the Fermi surface, computed self-consistently from the HFB wave functions, is the mechanism behind the discontinuous behaviour of nuclear charge radii.","pith_inferences":["Editorial inference: the natural next test is to apply the same $|D_n-D_p|$ term to odd-A and odd-odd nuclei, where the parent formula produces odd-even staggering in Ca and K; matching the Ni, Sn, and Pb staggering data would test whether the same mechanism covers both shell kinks and odd-even effects.","Editorial inference: because $a_0$ is fixed and universal in the paper, the formula makes concrete predictions for other magic-adjacent chains, for example Kr, Sr, or Zr near $N=50$ and 82, before those radii are precisely measured.","Editorial inference: the Pb failure toward neutron-deficient isotopes suggests a direct extension in which the same pair-condensation term is included on top of deformed Skyrme HFB; if the $N=126$ kink survives deformation, the mechanism is robust, and if not, the missing physics lies in the interplay of pairing and shape.","Editorial inference: the self-consistent $D_{n,p}$ values could be used as a cheap diagnostic of pairing saturation in functional fits, separate from their role in charge radii."],"forward_implications":["The same formula and the same $a_0$ work in both relativistic and non-relativistic mean-field frameworks, so the correction offers a unified way to add shell effects to mean-field charge radii.","Skyrme HFB acquires the ability to describe fine structure in charge radii, including the kinks at $N=28$, 82, and 126 and the $N=50$ quenching, without changing the energy functional.","The inverted parabolic-like radii between filled shells are reproduced and are predicted to weaken progressively from Ca to Pb isotopic chains.","A shell-closure kink at $^{100}$Sn is predicted by both HFB$^*$ and RHB$^*$, giving a falsifiable target for future precision measurements.","The residual deviations in Ca and Pb show that the correction is not complete by itself; the authors attribute part of the Pb mismatch to the neglect of shape deformation toward neutron-deficient regions."],"supporting_citations":[{"why":"Supplies the modified charge-radius formula and the fitted coefficient $a_0=0.561$ that the paper transfers to Skyrme HFB.","marker":"[69]"},{"why":"Introduced the neutron/proton pair-condensation ansatz behind $D_n$ and $D_p$ and the differential radius analysis.","marker":"[67]"},{"why":"Earlier application of the $D_{n,p}$ diagnostic to differential charge radii, grounding the shell and odd-even interpretation used here.","marker":"[34]"},{"why":"Spherical HFB solver in which all Skyrme calculations in the paper are carried out.","marker":"[76]"},{"why":"Defines the SLy5 Skyrme interaction, one of the two effective forces tested.","marker":"[89]"},{"why":"Defines the SkM* Skyrme interaction, the other effective force tested.","marker":"[106]"},{"why":"Defines the NL3 relativistic Lagrangian used for the RHB and RHB* comparison.","marker":"[116]"},{"why":"Compiled experimental charge radii tables used as the reference data for Ca, Ni, Sn, and Pb.","marker":"[20, 21]"},{"why":"Collinear laser spectroscopy charge radii of nickel isotopes used as experimental reference points.","marker":"[74]"},{"why":"Additional nickel isotope charge radius data used in the differential comparisons.","marker":"[118]"}],"fun_headline_variants":["Cooper-pair term captures nuclear charge radii kinks","Pair condensation term reproduces shell effects in nuclear radii","Skyrme with pair term fixes charge radii discontinuities","Neutron pair condensation term heals charge radii shell gaps","One pair term replicates nuclear charge radii shell closures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on one fitted number: $a_0=0.561$, obtained from potassium and calcium charge radii in a relativistic model, must stay valid in Skyrme HFB and across Ca, Ni, Sn, and Pb.","fun_headline_variants_meta":{"raw":{"variants":["Cooper-pair term captures nuclear charge radii kinks","Pair condensation term reproduces shell effects in nuclear radii","Skyrme with pair term fixes charge radii discontinuities","Neutron pair condensation term heals charge radii shell gaps","One pair term replicates nuclear charge radii shell closures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000302,"raw_usage":{"total_tokens":1767,"prompt_tokens":1001,"completion_tokens":766,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":688}},"tokens_in":617,"tokens_out":766,"duration_ms":8729,"temperature":1.0,"reasoning_tokens":688,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:48:47.984402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Refit $a_0$ independently for each of the four isotopic chains and for each Skyrme force; if the best-fit values scatter well outside the uncertainty of 0.561, the universal transfer premise is false. A complementary experiment is a high-precision charge radius of $^{100}$Sn, where HFB$^*$ predicts a kink that plain HFB does not.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the modified charge-radius formula and the fitted coefficient $a_0=0.561$ that the paper transfers to Skyrme HFB."},{"cited_title":"An, L.-S","cited_arxiv_id":null,"evidence_quote":"Introduced the neutron/proton pair-condensation ansatz behind $D_n$ and $D_p$ and the differential radius analysis."},{"cited_title":"Bennaceur and J","cited_arxiv_id":null,"evidence_quote":"Spherical HFB solver in which all Skyrme calculations in the paper are carried out."},{"cited_title":"Chabanat, P","cited_arxiv_id":null,"evidence_quote":"Defines the SLy5 Skyrme interaction, one of the two effective forces tested."},{"cited_title":"Bartel, P","cited_arxiv_id":null,"evidence_quote":"Defines the SkM* Skyrme interaction, the other effective force tested."},{"cited_title":"Malbrunot-Ettenauer, S","cited_arxiv_id":null,"evidence_quote":"Collinear laser spectroscopy charge radii of nickel isotopes used as experimental reference points."},{"cited_title":"Sommer, K","cited_arxiv_id":null,"evidence_quote":"Additional nickel isotope charge radius data used in the differential comparisons."}],"review_version":1}