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REVIEW 3 major objections 5 minor 92 references

Impact of Nuclear Deformation of Parent and Daughter Nuclei on One Proton Radioactivity Lifetimes

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read One-proton decay half-lives are reproduced most accurately when the empirical formula includes the deformations of both parent and daughter nuclei.

desk verdict A serviceable new empirical formula for 1p-decay half-lives that includes both parent and daughter deformation; the predictive-superiority claim is weaker than advertised because the head-to-head RMSE is in-sample. read the letter →

arxiv 2411.17112 v1 pith:KCP4KM6J submitted 2024-11-26 nucl-th

A. Jain , Pranali Parab , G. Saxena , Mamta Aggarwal This is my paper · ORCID
classification nucl-th
keywords one-protonradioactivityprotondecayhalf-livesnucleardeformationquadrupoleshapecoexistencesemi-empiricalformuladriplinebranchingratios
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

This paper argues that one-proton radioactivity half-lives are better predicted when the empirical decay formula includes the quadrupole deformation (prolate or oblate shape) of both the parent nucleus and the daughter nucleus, in addition to the usual decay energy and angular-momentum terms. The authors fit a five-parameter formula to 52 known proton-emitting states, treating the 18 half-life upper limits with a censored-data likelihood, and on the 27 heavy-emitter subset they report a root-mean-square error of $0.6998$ and a reduced $\chi^2$ of $0.6010$, lower than the nine formulas they compare. If the formula is correct, it provides a simple predictive tool for half-lives of proton emitters that have not been measured yet, and it indicates that the shapes of both partners matter for the decay rate. The paper further argues that shape coexistence—two nearly degenerate shapes in the same nucleus—can open alternative decay pathways and change branching ratios.

What carries the argument

The central object is Eq. (5), a five-parameter empirical formula for $\log_{10}T_{1/2}$: $$\log_{10}T_{1/2}=a+\frac{b\sqrt{\mu}}{Z_d $A^{{1/3}}$}+\frac{c\sqrt{\mu}\,Z_d}{\sqrt{Q}}+d\sqrt{l(l+1)}+e\left[(-1)^{\kappa_p+\kappa_p\kappa_d}(\kappa_p\beta_p)^{1/2}+(-1)^{\kappa_d+\kappa_p\kappa_d}(\kappa_d\beta_d)^{1/2}\right]\frac{Z_d}{\sqrt{Q}}.$$ The first three terms are the usual decay-law structure, the fourth is the centrifugal-barrier hindrance, and the last term carries the new physics: it adds the deformation-weighted contributions of parent and daughter, with the $\kappa$ convention ($2$ prolate, $-1$ oblate) and the sign factor enforcing constructive addition when both shapes match. The deformation values $\beta_p,\beta_d$ come from relativistic mean-field calculations, and the constants $a$--$e$ are determined by a least-squares fit with a maximum-likelihood loss that only penalizes predictions above the upper-limit data. This term is what lets the formula connect half-lives to the shapes of both partner nuclei.

What would settle it

Refit Eq. (5) on the same 52 data points using deformation values from a second independent mass model instead of the relativistic mean-field values, and see whether the two-deformation formula still beats the other eight formulas in Table 1; if the RMSE advantage disappears, the claim that both-nucleus deformation is what drives the improvement fails. A complementary check is to measure the half-life (and ideally the shape) of a newly predicted emitter such as $^{83}$Tc, $^{47}$Co, or $^{71}$Rb, and compare with the formula's extrapolation.

Watch

Extended reading notes

Core claim

The paper's central claim is that Eq. (5), a semi-empirical relation with five fitted constants, captures one-proton decay lifetimes across a wide mass range ($21\le A\le 185$) by adding deformation terms for both the emitting and the residual nucleus to the standard $\log_{10}T_{1/2}\sim Z_d/\sqrt{Q}$ systematics. For each nucleus the deformation enters as $(\kappa\beta)^{1/2} Z_d/\sqrt{Q}$, where $\beta$ is the quadrupole deformation and $\kappa=2$ for prolate, $-1$ for oblate shapes; a sign factor $(-1)^{\kappa_p+\kappa_p\kappa_d}$ makes same-shape parent-to-daughter transitions add constructively while damping shape-changing transitions. Fitted to 34 true experimental half-lives plus 18 upper limits by maximum likelihood, the formula yields RMSE $0.6998$ and $\chi^2=0.6010$ on the 27 heavy emitters used for Table 1, the lowest of the compared formulas, and its predicted half-lives for new candidates fall in an experimentally accessible range ($\log_{10}T_{1/2}\approx -8.9$ to $-6.3$ s). The authors read the negative fitted coefficient $e$ as deformed nuclei decaying faster than spherical ones, and they use potential-energy surfaces to show that in shape-coexisting nuclei transitions from a second minimum can sometimes agree with measured half-lives better than the ground-to-ground transition.

Load-bearing premise

The whole prediction stands on the assumption that the shapes (quadrupole deformations) of the parent and daughter nuclei computed by relativistic mean-field theory are the true shapes; for most drip-line nuclei there is no experimental deformation to check against, so if the calculation is wrong for a candidate nucleus, the predicted half-life inherits that error in a way that the five fitted constants cannot reveal.

Editorial extensions

If this is right

  • New proton emitters can be screened quickly: the formula turns measured Q-values and computed deformations into half-life predictions, as done for the 24 candidates in Table 3, without full barrier-penetration calculations.
  • The comparison in Table 1 implies that the deformation terms, not just extra parameters, improve the fit; formulas without both-nucleus deformation terms have RMSEs of about $0.89$--$2.38$, so shape effects are not a negligible correction.
  • Because the formula predicts $\log_{10}T_{1/2}$ between roughly $-8.9$ and $-6.3$ s for the candidates in Table 3, several of these nuclei should be detectable as proton emitters with existing techniques.
  • For shape-coexisting nuclei, the calculated ground-to-ground, ground-to-second-minimum, second-minimum-to-ground, and second-minimum-to-second-minimum transition half-lives differ; matching an observed lifetime to one of these variants can identify which shape state participates in the decay.

Reading between the lines

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

  • A cleaner test of the physical content of the deformation term would be to refit Eq. (5) with the deformation coefficients set to zero and compare RMSE; the paper does not report this ablation, so part of the improvement could in principle come from the two extra fit parameters rather than from the shapes themselves.
  • Because the fitted deformation values come from one theoretical model, the extrapolated half-lives are conditional on that model's shapes; an independent check would be to compare the formula's predictions for candidates whose deformations have since been measured or constrained by other observables.
  • The same same-shape sign structure could be exported to other decay modes; if it improves $\alpha$-decay or two-proton formulas as well, it would suggest a common empirical rule that tunneling lifetimes depend on whether parent and daughter shapes are similar.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a new semi-empirical formula, Eq. (5), for one-proton radioactivity half-lives that includes the quadrupole deformations of both the parent and daughter nuclei, along with the usual Q-value and angular-momentum terms. The formula is fitted to 52 proton-emitting states (34 with accurate half-lives and 18 with upper limits) using a censored-data maximum-likelihood loss. The authors report the lowest RMSE and chi-squared among nine formulas on 27 heavy-emitter points, show a k-fold cross-validation plot, and use the formula to predict half-lives of candidates. They also discuss shape coexistence in selected parent-daughter pairs, considering G-G, G-S, S-G, and S-S transitions and their effect on half-lives and branching ratios.

Significance. If the central comparison were reliable, the formula would be a practically useful addition to the 1p-decay toolbox: it is explicit, easily reusable, and treats parent and daughter deformations simultaneously, which existing empirical formulas do not. The authors also deserve credit for using a censored-data loss and for reporting a k-fold cross-validation exercise; both are steps in the right direction. However, the key superiority claim is not yet established, because the comparison in Table 1 is in-sample for Eq. (5) while the competitors are evaluated with their published coefficients, and the cross-validation is not applied to the competing formulas. The deformation inputs are entirely theoretical (RMF) for highly exotic nuclei, and the predicted half-lives carry no uncertainty estimates. The shape-coexistence analysis is qualitative and post hoc. For these reasons the paper requires revision before the central claims can be accepted.

major comments (3)
  1. [Table 1 and Theoretical Framework (Half-life calculations)] The headline comparison in Table 1 is not a predictive test. The coefficients a-e in Eq. (5) are least-squares fitted to the same 27 heavy-emitter points that are subsequently scored, whereas the competing formulas are evaluated with their published coefficients, which were obtained on other datasets. In-sample RMSE is therefore expected to favour the more flexible refitted formula, and the gap 0.6998 vs 0.8854 does not demonstrate predictive superiority. In addition, the text states that the deformation-term exponent p was scanned from 0 to 5 in steps of 0.5 and the best value selected on the same data; this model-selection step is an additional adaptive choice that is not counted in Np when computing chi-squared via Eq. (7). The k-fold CV in Fig. 1 is not tabulated and is not performed for any competing formula, so it does not repair the comparison. Please provide an out-of-sample comparison in which Eq. (5) and all formulas in Table 1 are fitted/selected on identical training folds and scored on identical test folds, or downgrade the claim to 'best in-sample fit on the 2020 dataset.'
  2. [Theoretical Framework (RMF deformations) and Table 3/Table 4] All fits and predictions use RMF values of beta_p and beta_d for nuclei at or beyond the proton drip line, where there are no experimental deformations. Any systematic error in these RMF shapes is absorbed into the fitted coefficients, so the good in-sample RMSE cannot validate the deformation inputs. The extrapolated half-lives in Tables 3 and 4 are quoted to two decimals in log10T without propagating uncertainties from beta, Q, or l; for the least-bound candidates this can correspond to factors of order 10 in half-life. The authors state that WS4, HFB, and FRDM deformation tables were checked qualitatively, but no quantitative sensitivity test is reported. Please add a sensitivity analysis using alternative deformation sets for the fit and for the predicted candidates, and report approximate uncertainties on the predicted log10T values.
  3. [Correlation between half-life and shape coexistence (Fig. 3, Table 5)] The four-transition scheme (G-G, G-S, S-G, S-S) is applied post hoc: for each nucleus the transition that gives better agreement with experiment is identified after inspecting the data, without any model-comparison statistic or penalty for the additional transition channels. No wave-function overlaps or mixing amplitudes are computed, so the half-lives for the secondary-minimum paths should be regarded as schematic. The text itself acknowledges that this part is 'preliminary' and 'premature,' yet the conclusions and abstract present shape coexistence as a key outcome. Please either add a quantitative selection criterion (e.g., a likelihood comparison with a degrees-of-freedom penalty) or explicitly and consistently frame the shape-coexistence section as exploratory rather than as a validated prediction.
minor comments (5)
  1. [Abstract and text] The abstract contains 'signicant' twice; the Introduction has 'performes'; the shape-coexistence section has 'inuenced'; Eq. (6) and the data-availability statement contain rendering artifacts ('vuut', 'left datasets'). The manuscript should be carefully proofread.
  2. [Eq. (5)] The formula is difficult to parse as typeset: the exponents such as (-1)^(kappa_p + kappa_p kappa_d) need unambiguous parentheses, and the term preceding sqrt{l(l+1)} appears garbled (the factor d is separated from the radical). Please rewrite Eq. (5) with clear mathematical formatting.
  3. [Table 5] The 'Decay Modes' column mixes probabilities and percentages (e.g., 'p=0.89; alpha=11'); please use a single convention throughout, preferably probabilities with stated uncertainties.
  4. [Figure 1 and k-fold CV] The text states that average RMSE values across folds are shown, but no numerical values are given in the text or tables, making the cross-validation claim hard to reproduce. Please tabulate the per-fold and average RMSE for training and test splits.
  5. [Table 2 caption] The caption refers to 'columns 8-27' for the compared models, but the printed table does not appear to contain that many model columns; please correct the column reference.

Circularity Check

2 steps flagged · score 6.0 of 10

Headline superiority claim rests on in-sample RMSE, and shape-coexistence agreement is post-hoc channel selection; actual extrapolations remain independent.

  1. fitted input called prediction [Results and Discussion, Table 1; fitting paragraph for Eq. (5)]
    "Finally, fitting the model to all 52 data points yields the following coefficient values: a= −9.5853, b= −0.0516, c= 0.1135, d= 0.0620, and e= −0.0058. ... It is evident from the Table 1 that the present formula shows the lowest RMSE and χ2 values than all the other formulas compared, demonstrating exceptional predictive accuracy."

    The five parameters a–e are obtained by fitting Eq. (5) to the full 52-point dataset, which includes the 27 heavy-emitter points whose RMSE is reported in Table 1. The Table 1 RMSE is therefore an in-sample residual, not an out-of-sample prediction. The competing formulas are quoted with their published fixed coefficients, so a five-parameter model refitted to the evaluation set is almost guaranteed to show lower in-sample RMSE. The k-fold cross-validation in Fig. 1 is applied only to the present formula, and no comparable cross-validated RMSE is given for the competitors. The claimed 'exceptional predictive accuracy' based on this comparison is thus partly forced by construction and does not establish predictive superiority.

  2. other [Correlation between half-life and shape coexistence, paragraph following Fig. 3]
    "In few nuclei, the S-G, S-S or G-S transitions are matching better with data demonstrating qualitatively how the nuclear shapes affect half-lives."

    For shape-coexisting parent–daughter pairs the paper computes all four possible transition channels (G-G, G-S, S-G, S-S) and then, after comparing with the experimental half-life, highlights the channel that matches best as the likely decay path ('the discrepancies in few nuclei suggest the transitions ... that might have inuenced the lifetimes'). The transition label is effectively a discrete free parameter selected post hoc to minimize disagreement with the data. The resulting agreement is therefore data-driven selection rather than an independent prediction of which shape transition occurs, so it does not provide independent confirmation of the shape-coexistence hypothesis.

full rationale

The core empirical formula is openly presented as a fit, and the genuine extrapolations to new candidates in Tables 3 and 4, using external Q-values and RMF/NSM deformations, are not circular. Self-citations [52] and [59] are mostly motivational or supportive of a sign convention, and the deformation inputs are compared with WS4, HFB and FRDM, so no load-bearing self-citation chain is established. The k-fold cross-validation is a legitimate internal generalization check for Eq. (5). Circularity enters in two localized places. First, the headline claim of lowest RMSE and 'exceptional predictive accuracy' relative to other formulas is based on residuals on data points used to fit Eq. (5), while the competitors keep fixed published coefficients; the in-sample comparison is statistically forced and no fair out-of-sample competitor comparison is supplied. Second, the shape-coexistence agreement is obtained by post hoc selection among the four computed G-G/G-S/S-G/S-S channels, so the 'matching better' transition is chosen after seeing the data rather than predicted. Because the central competitive claim is partly an in-sample fit but the formula does provide independent extrapolations, the overall circularity is partial, giving a score of 6.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central formula is an empirical fit with five free coefficients and a hand-selected functional form; the shape-coexistence analysis adds a hand-chosen energy cutoff and a post-hoc transition-path choice. The paper introduces no new particles, forces, or conserved quantities.

free parameters (7)
  • a = -9.5853
    Intercept in Eq. (5), fitted to the 52-point dataset.
  • b = -0.0516
    Coefficient of sqrt(mu)*Zd*A^{1/3} in Eq. (5).
  • c = 0.1135
    Coefficient of sqrt(mu)*Zd/sqrt(Q) in Eq. (5).
  • d = 0.0620
    Coefficient of the centrifugal barrier term sqrt(l(l+1)).
  • e = -0.0058
    Coefficient of the parent and daughter deformation terms in Eq. (5).
  • deformation-term exponent = 1/2
    Chosen after scanning p from 0 to 5 in 0.5 increments for |beta|^p and (kappa*beta)^p; the final form adopts the square-root variant from Ref. [48] because it yielded the lowest RMSE.
  • shape-coexistence energy cutoff = 1 MeV
    The condition Delta E <= 1 MeV between the two minima is used to select nuclei for the shape-coexistence half-life and branching-ratio analysis; it is chosen by hand in the section on shape coexistence.
assumptions (6)
  • domain assumption RMF theory with the NL3* parameter set yields reliable deformation parameters for proton-unbound drip-line nuclei.
    The fitted formula uses beta_p and beta_d from RMF calculations; any error in these deformation inputs propagates into the fitted coefficients. Invoked in the fitting paragraph of the Half-life calculations section.
  • domain assumption Nilsson-Strutinsky method gives reliable potential energy surfaces and secondary minima for shape-coexistence analysis.
    Used for the PES plots, for the alternative beta values in Table 4, and for the coexistence transition analysis.
  • domain assumption The experimental Qp, l, and half-lives from NUBASE2020 and AME2020 are correct.
    These values are the training data for the fit and the benchmarks for the RMSE comparison.
  • domain assumption A semi-classical WKB tunneling treatment justifies the linear terms in Eq. (5).
    The first three terms are stated to be based on the alpha-tunneling transmission coefficient (Refs. [66,67]); the formula is empirical, so this is a loose derivation assumption.
  • domain assumption The orbital angular momentum l is determined unambiguously by standard spin-parity selection rules.
    Used for the sqrt(l(l+1)) term; l values are taken from NUBASE2020 or Ref. [89].
  • ad hoc to paper Decays from or to secondary minima (G-S, S-G, S-S) are physically allowed and can be described by the same formula with Q values adjusted by the energy differences between minima.
    The paper computes half-lives for all four transition paths and then identifies agreement with experiment; no population model or mixing probability is provided, making this a paper-specific assumption.

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Cite this review

Pith. "Pith review of Impact of Nuclear Deformation of Parent and Daughter Nuclei on One Proton Radioactivity Lifetimes." pith.science (2026). https://pith.science/paper/KCP4KM6J

@misc{pith2026241117112,
  author       = {Pith},
  title        = {Pith review of: Impact of Nuclear Deformation of Parent and Daughter Nuclei on One Proton Radioactivity Lifetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCP4KM6J}},
  note         = {Machine review of arXiv:2411.17112}
}
read the original abstract

The influence of nuclear deformation on proton-decay half-lives has been systematically studied in microscopic theoretical frameworks for a wide range of nuclei with Z<82. Correlation between 1p-decay half-lives and the deformed nuclear shapes of both the parent and daughter nuclei has been investigated. Since the deformations of proton emitters and their residual nuclei impact the potential barrier and disintegration energy which are crucial for the accurate determination of half-lives, we incorporate the nuclear deformations of both the emitters and residues in a phenomenological manner and propose a new semi-empirical formula to estimate the 1p-decay half-lives. The robustness of this formula is demonstrated by the accurate predictions of the measured values while making it reliable for forecasting the properties of other potential proton emitters. The phenomenon of shape coexistence as observed in several proton emitters and their respective daughter nuclei, is particularly signicant in this context due to secondary minima in the potential energy surfaces of both the nuclei. Accounting for these factors signicantly affects the estimation of half-lives and branching ratios by introducing additional decay pathways and altering transition probabilities between different nuclear shapes.

Figures

Figures reproduced from arXiv: 2411.17112 by the authors.

Figure 1
Figure 1. RMSE for various data-sets of 34 true experimental data + 18 data with upper limit (37 ground states and 15 isomeric transitions) [69]. Each training and test set consist of 80% and 20% data split. The last bars are showing average of 5 data sets. crucial as it indicates a better fit when it converges to a lower value, allowing each formula to be compared on an equal basis in terms of their fitting parameters. It is… view at source ↗
Figure 2
Figure 2. Potential energy surfaces (PES) for the one-proton emitters 67Kr and 71Rb, along with their corresponding daughter nuclei 66Br and 70Kr. The energies are normalized to zero with respect to the lowest energy minima for each nucleus, allowing for a clear comparison of the deformation states and their relative stability. In order to establish the correlations between half-lives and shape coexistence, we selected only t… view at source ↗
Figure 3
Figure 3. 1p-decay half-lives related to various transitions for selected shape coexisting nuclei as found from NSM and RMF calculations. Here, G-G represents transition from ground state of parent nucleus to ground state of daughter nucleus. Similarly the others, where S refers to second minima state of the nucleus (see text for details). We have shown the half-life of 65Br with upper limit. we calculated the half-lives for … view at source ↗

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