Mapping parametric error profiles onto nuclear structure configurations in deformed proton radioactivity

Jizheng Bo

REVIEW 5 major objections 4 minor 60 references

Mapping parametric error profiles onto nuclear structure configurations in deformed proton radioactivity

T0 review · 5 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Bayesian uncertainty bands in proton-decay half-lives widen near the Z=82 shell closure, and the paper reads this as a map of nuclear potential softness.

desk verdict Useful Bayesian application to deformed proton decay, but the central mapping claim rests on unidentified priors and one data point per nucleus. read the letter →

arxiv 2607.18911 v1 pith:2NKEUDZT submitted 2026-07-21 nucl-th

Jizheng Bo This is my paper · ORCID

classification nucl-th

keywords protonradioactivityBayesianuncertaintyquantificationdeformedWoods–SaxonpotentialWKBapproximationparametersensitivityshellstructurehalf-lifepredictionMCMC

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 the widths and volatility of Bayesian uncertainty bands in calculated proton-decay half-lives carry physical information about nuclear shell structure. Calibrating a deformed Woods–Saxon potential plus WKB tunneling model to one experimental half-life per odd-A emitter in the Z=50–82 region, the author extracts ten-parameter posteriors for fourteen nuclei. The study reports that posterior predictive medians reproduce experimental half-lives across orders of magnitude, that sensitivity hierarchies reorder with mass, and that near the Z=82 shell closure the σ bands of deformation parameters systematically expand. The proposed reading is that this expansion reflects a softened nuclear potential with shape coexistence, so parametric error profiles might be mapped onto potential energy configurations. If correct, Bayesian UQ becomes a spectroscopic tool that reads shell structure from decay systematics.

What carries the argument

The machinery is a deformed Woods–Saxon potential (radius and diffuseness with β2, β4 angular dependence), a multipole-expanded Coulomb field, an orientation-averaged WKB tunneling width, and a Bayesian calibration using MCMC with independent Gaussian priors. The signed relative sensitivity index (half-life shift from 16th to 84th percentile of each marginal posterior) converts posterior widths into fractional half-life sensitivities. The central object doing the interpretive work is the Z-dependence of the 1σ/2σ credible-interval widths of the deformation parameters.

What would settle it

Generate synthetic half-lives from the same WKB model with fixed deformation parameters (no shell-dependent softening), run the same MCMC calibration with identical priors, and check whether the σ bands still widen near Z=82; if they do, the widening is an inference artifact. Alternatively, repeat the fit with prior widths varied by a factor of 10 and see whether the band expansion persists.

Watch

Extended reading notes

Core claim

Constrained by experimental half-lives, the posterior distributions of the deformed potential parameters show little pairwise correlation for most nuclei, with the exception of 113Cs where sloppiness appears. The posterior predictive half-lives fall within the 1σ and 2σ bands for every nucleus without systematic bias. The signed relative sensitivity indices reveal that radius r0 and quadrupole deformation β2 dominate, with β4 active only in two windows near Z=59 and Z=75–79. As Z approaches 82, the 1σ and 2σ bands of β2 and β4 expand and become volatile, which the author interprets as evidence that the nuclear potential softens near the shell closure; the mapping between error-profile struct

Load-bearing premise

The assumption that a single experimental half-life per nucleus, combined with independent Gaussian priors whose widths are not stated, is enough to identify ten model parameters per nucleus, so the posterior widths and their Z-dependence reflect nuclear structure rather than prior choices or unidentifiability.

Editorial extensions

If this is right

If the error-profile mapping holds, posterior σ-band widths become an observable that can be compared across models and used to locate shell closures.
Bayesian calibration turns a single half-life per nucleus into ten parameter posteriors, and the posterior predictive distributions provide half-life uncertainties that can be propagated to other decay observables.
Mass-dependent re-ordering of sensitivities means no fixed parameter ranking can be assumed; future fits must allow hierarchy changes along the Z sequence.
Detection of sloppiness in 113Cs warns that some nuclei admit degenerate parameter combinations that point values cannot reveal.

Reading between the lines

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

The widening of σ bands near Z=82 could be an identifiability artefact of fitting ten parameters to one datum; an independent check would re-fit with synthetic data from a flat potential and see whether similar band structure emerges.
If the mapping is physical, the same pipeline applied to alpha decay or cluster emission should show analogous band expansion at other magic numbers (Z=50, N=82, N=126).
A hierarchical Bayesian model that pools information across the Z sequence could tighten the posteriors and directly test whether the per-nucleus widths reflect structure rather than prior spread.
Sensitivity indices computed at posterior medians may miss nonlinear compensation; a global variance-based sensitivity analysis on the posterior could refine the hierarchy claims.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take: this paper does something new and useful—first MCMC calibration of the deformed Woods–Saxon WKB model for odd-A proton emitters, with posteriors, correlation matrices, and a sensitivity ranking. That part is competently executed. But the headline claim, that σ-band widening near Z=82 maps onto structural softening, is not supported by the evidence. The fit is 10 parameters per nucleus against a single half-life, the prior hyperparameters in Eq. (3) are never stated, and the 'posterior predictive' agreement in Fig. 9 is evaluated on the same data used in the likelihood. Any widening of posterior bands under those conditions is exactly what an unidentifiable, prior-dominated fit produces; it does not require the potential to soften.

What is genuinely good: the author is transparent about sloppiness in 113Cs, the correlation matrices are shown in full, and the sensitivity index is clearly defined. The Bayesian methodology itself is standard, and the application to this decay mode is a legitimate extension of ref. [47]. The paper also honestly notes use of AI tools for polishing; no issue there.

The soft spots are not tiny. The sensitivity index in Eq. (21) sets perturbations from posterior percentiles, so a parameter with a wide prior automatically looks important—it cannot separate physical sensitivity from prior width. The claimed degeneracy-driven band inflation near Z=82 is contradicted by the paper's own correlation matrices: most off-diagonal elements are near zero, so there is no evidence that β2–β4 or β2–r0 compensation inflates the marginalized bands. The text asserts this in Sec. III C but the figures don't show it.

Would I recommend engaging with it? Yes, for a referee. The application is a first, the code appears reproducible, and the issues are fixable in revision: specify prior hyperparameters, run a prior-sensitivity analysis, include a held-out nucleus or cross-validation, and temper the conclusion to 'posterior widths are currently dominated by prior assumptions' rather than a structural probe. As it stands, the central claim is circular and should not be published as is.

Referee Report

5 major / 4 minor

Summary. The manuscript develops a Bayesian uncertainty-quantification framework for the deformed Woods–Saxon potential with a semi-classical WKB description of odd-A proton radioactivity in the Z = 50–82 region. For each of the 14 proton emitters, ten model parameters (V0, a0, r0, P0, Rc0, β2, β4, Vso0, Rso, aso) are calibrated to the single experimental half-life through MCMC (emcee), using an independent Gaussian prior per parameter (Eq. 3) and a likelihood containing only experimental uncertainties (Eq. 2). The paper reports posterior medians and credible intervals, Pearson correlation matrices, a signed relative sensitivity index (Eq. 21), and posterior predictive half-life bands (Fig. 9). The central claim, stated in the abstract and conclusion, is that the systematic expansion and volatility of the σ bands near the Z = 82 shell closure reflect a structural softening of the nuclear potential, so that parametric error profiles can be mapped onto potential energy configurations.

Significance. The methodological program—full Bayesian posterior sampling for a decay-model calibration, with covariance diagnostics and a quantitative sensitivity index—is timely and appropriate for a field that has largely relied on deterministic point fits, and the figures are informative. If the central claim were established, it would constitute a new spectroscopic use of proton-decay systematics. Credit is due for providing MCMC diagnostics and 68% credible intervals for all nuclei, and for flagging sloppiness in 113Cs. However, the paper's significance rests entirely on the proposition that the posterior widths carry physical information about the nuclear potential. That proposition is not established: the priors are unspecified, the model is underdetermined with one datum per nucleus, the predictive check is circular, and the reported correlation matrices contradict the accompanying narrative. As presented, the mapping claim is unsupported.

major comments (5)

[Sec. II A (Eq. 3); Tables I–II; Fig. 7] The prior hyperparameters {ᾱ_i, σ_{α_i}} of Eq. (3) are never given. Each nucleus contributes exactly one experimental half-life to the likelihood (Eq. 2), so ten parameters are constrained by a single scalar; any direction in which the half-life is insensitive has posterior equal to prior, and no prior-sensitivity analysis is supplied. Tables I–II indicate this limit is reached: in Table II the spin-orbit parameters are nearly identical for all 14 nuclei (V_so0 ≈ 6.2 ± 1.0 MeV, a_so = 0.75 ± 0.05 fm), and several β2/β4 credibles in Table I span the full plausible range (e.g., 171Au: β2 = 0.14(+0.21/−0.44); 121Pr: β4 = 0.08(+0.33/−0.35)). The Z ≈ 82 widening of the deformation bands in Fig. 7 could therefore be an identifiability/prior artifact rather than a physical softening—precisely the distinction the central claim relies on.
[Sec. III D, Fig. 9] Fig. 9 is presented as validation: the posterior predictive median and 1σ/2σ bands 'tightly encompass' the experimental half-lives. But those half-lives are the same data that enter the likelihood (Eq. (2) via Eq. (20)); for a flexible model with wide posteriors, containing the fitted points is unsurprising. The further statement in Sec. III D that this 'demonstrates that the framework does not suffer from parameter overfitting' does not follow and is not tested: there is no held-out nucleus, no leave-one-out analysis, and no comparison with an independent observable (e.g., a second decay branch or a spectroscopic factor from a different reaction). A genuine out-of-sample check is required before the predictive bands can be claimed to carry structural information.
[Sec. III C, Eq. (21)] The sensitivity index S_i in Eq. (21) is the half-life response to shunting the parameter between the 16th and 84th marginals of the posterior. When the marginal is prior-dominated, those percentiles are fixed by the prior, not by the data; S then cannot separate 'the parameter matters for the half-life' from 'the prior is broad'. The Sec. III C account of the near-Z=82 regime—that degeneracy 'geometrically inflates the marginalized posterior uncertainty bands'—is asserted, but the reported heavy-nucleus correlations do not show the compensating channels: for 171Au (Fig. 6) β2–β4 = 0.038 and β2–r0 = 0.069. The hierarchy in Fig. 8 should be recomputed with explicit priors and with a data-driven perturbation scale (e.g., credible intervals of a profile likelihood).
[Sec. III A vs Figs. 4–6] The text in Sec. III A claims that the off-diagonal correlation elements 'remain consistently small for almost all parameter pairs' and that deformation–geometry correlations are 'near-zero'. The figures contradict this: for 109I (Fig. 4) β2–r0 = 0.798 and β2–a0 = 0.611; for 145Tm (Fig. 5) a0–r0 = −0.519 and r0–P0 = −0.439. These are substantial correlations, and the same manuscript uses strong degeneracy in 113Cs as a central example. The presence or absence of parameter compensation is used both to exclude (Sec. III A) and to invoke (Sec. III C) degeneracy; this internal inconsistency must be resolved.
[Sec. III (dataset)] The experimental input—half-lives, Q_p values, parent/daughter spins and parities, and the assigned uncertainties σ_i—is never tabulated; Fig. 9 displays only the ratio log10(T_Bayes/T_exp). The likelihood of Eq. (2) is therefore non-reproducible, and the claimed agreement cannot be independently checked. The data table (or a direct reference to it) is required.

minor comments (4)

[Sec. II A] Typo in the first sentence: 'poseterior' should be 'posterior'.
[Eq. (13)] The radial kernel Kλ is typeset with garbled exponents ('rλ(λ−2) −1 Rλ−2c'); rewrite in standard notation and check dimensions.
[Eq. (21), Fig. 8] Eq. (21) defines a 'signed' index, but Fig. 8 plots |S_i|; state explicitly whether the sign information is discarded or retained.
[Abstract] The abstract and text refer to the 'Z = 50–82 region', but the fitted sequence is Z = 53–81 (no Z = 50, 52, or 82 nucleus appears); the abstract should match the data.

Circularity Check

2 steps flagged · score 6.0 of 10

In-sample half-life agreement is a fit loop; Z≈82 σ-band expansion is confounded with prior width.

fitted input called prediction [Abstract; Sec. II A Eq. (2); Sec. II C Eq. (20); Sec. III D Fig. 9]
"Constrained by experimental half-lives, the joint and marginal posteriors and covariance topologies are extracted ... propagating these parametric uncertainties yields posterior predictive medians and differentiated σ bands that tightly encompass experimental half-lives across multiple orders of magnitude without systematic bias."

The posterior (Eq. 1) is built from a likelihood (Eq. 2) whose y_i are the experimental half-lives, and Eq. (20) computes T1/2 from parameters sampled from that same posterior. With 10 free parameters per nucleus and only one half-life constraint, the model can essentially reproduce each y_i, so log10(T_Bayes/T_exp) ≈ 0 in Fig. 9 is enforced by the calibration, not independently predicted. Calling this in-sample agreement a 'posterior predictive' success is a fitted-input-called-prediction step.
other [Sec. II A Eq. (3); Sec. III C (Z≈82 interpretation); Sec. IV Conclusion]
"Independent priors are assigned to the model parameters p(α)∝exp[-1/2 Σ_i (α_i-ᾱ_i)^2/σ_{αi}^2], where the hyperparameters {σ_αi} and {ᾱ_i} describe the variance and mean ... Near the Z=82 major shell closure, the systematic widening of the half-life σ uncertainty bands are physically interpreted as a direct consequence of the structural softening of the nuclear potential, which triggers enhanced parametric degeneracies and shape coexistence."

The hyperparameters in Eq. (3) are never reported. Each nucleus contributes one experimental half-life in Eq. (2), while 10 parameters are sampled, so the likelihood is a single scalar constraint; in directions where T1/2 is insensitive, p(α_i|D) ≈ p(α_i). The paper's own Fig. 8 shows |S_β4| ≈ 0 over Z=63–73 and |S_Rc0| < 1.1% universally, so those widths are not data-determined. The Z≈82 widening of the β2/β4 posterior bands is therefore consistent with prior spread or unidentifiability, and the claimed 'structural softening' is a re-description of the fitted/prior widths rather than an independently derived potential-energy signature.

full rationale

Score is 6 rather than 0 because the central numerical validation reduces to the fit. The posterior predictive half-lives in Sec. III D / Fig. 9 are computed with parameters sampled from a posterior whose likelihood (Eq. 2) is constructed from the same experimental T1/2 values; with ten parameters per nucleus and one observable, the median agreement log10(T_Bayes/T_exp) ≈ 0 is a property of the calibration loop, not an out-of-sample check. The novel structural claim, that Z≈82 σ-band broadening maps onto potential softening, further requires the posterior widths to be data-dominated rather than prior-dominated. Since Eq. (3)'s hyperparameters are not given and one datum cannot identify ten parameters, the widening is equally consistent with an unidentifiability/prior artifact; the mapping is therefore a re-description of posterior/prior widths rather than an established nuclear-structure inference. The only self-citation, [47] (J.Z. Bo, Phys. Rev. C 113, 064303 (2026)), appears in a generic list of Bayesian UQ applications and is not load-bearing; no uniqueness theorem or ansatz is smuggled in via the author's prior work. The WKB and deformed-Coulomb machinery is standard and parameter-free given the potential, so the circularity is confined to the calibration/validation loop and the structural interpretation, not to the whole formal derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central statistical output rests on a heavily underdetermined calibration (10 parameters per nucleus from 1 half-life) and on unspecified priors; the physics interpretation additionally assumes the WKB/deformed-potential model is a faithful description with no unmodeled systematic error. The listed axioms are the unstated premises that must hold for the posterior widths and their Z-trend to carry the structural meaning the paper assigns to them.

free parameters (2)

Per-nucleus potential parameters (V0, a0, r0, P0, Rc0, β2, β4, Vso0, Rso, aso) = Tables I–II for 14 nuclei; e.g., 109I: V0=54.42, a0=0.6481, r0=5.666, P0=0.6708, Rc0=5.764, β2=0.3357, β4=0.05649
Each nucleus is calibrated to a single experimental half-life; all posteriors, sensitivity rankings, and the σ-band widening derive from these fitted values.
Prior hyperparameters {ᾱ_i, σ_{α_i}} = Not stated
Eq. (3) defines independent Gaussian priors; with 10 parameters per datum, the prior is the dominant constraint. Without these values the results are unreproducible and the posterior widths cannot be interpreted physically.

assumptions (4)

domain assumption The deformed Woods–Saxon potential with the angular-dependent diffuseness of Eq. (8) and multipoles β2, β4 is an adequate description of the proton-daughter interaction.
Invoked in Sec. II B; the entire half-life calculation and all parameter posteriors depend on this potential form, adopted from ref. [43].
domain assumption The WKB approximation and orientation-averaged single-particle width of Eqs. (16)–(19) accurately compute the proton decay width.
Sec. II C; WKB is a semi-classical approximation; no comparison to coupled-channels or exact tunneling is provided.
ad hoc to paper The likelihood includes only experimental uncertainties; there is no model-discrepancy term.
Eq. (2) defines σ_i as experimental uncertainty only; this choice makes posterior widths appear as pure parameter uncertainty, which is central to the 'error profile maps structure' interpretation.
ad hoc to paper Independent Gaussian priors with unspecified hyperparameters (Eq. 3) adequately encode prior knowledge, and the posterior width is not dominated by the prior.
Needed for the 10-parameter/1-datum calibration; without prior values, the claim that posterior σ bands reflect physical softening is untestable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Mapping parametric error profiles onto nuclear structure configurations in deformed proton radioactivity." pith.science (2026). https://pith.science/paper/2NKEUDZT

@misc{pith2026260718911,
  author       = {Pith},
  title        = {Pith review of: Mapping parametric error profiles onto nuclear structure configurations in deformed proton radioactivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2NKEUDZT}},
  note         = {Machine review of arXiv:2607.18911}
}

abstract

Theoretical descriptions of proton radioactivity near the drip lines are often challenged by parametric uncertainties in nuclear potential models. To address this, a robust Bayesian uncertainty quantification (UQ) framework is established to calibrate the deformed Woods--Saxon potential coupled with the semi-classical WKB approximation for odd-$A$ proton emitters across the $Z=50$--$82$ region. Constrained by experimental half-lives, the joint and marginal posteriors and covariance topologies are extracted for the potential geometry, multipole deformations, and spectroscopic factors. Using a signed relative sensitivity index, a dynamic, mass-dependent re-ordering of parameter hierarchies is unraveled. Crucially, propagating these parametric uncertainties yields posterior predictive medians and differentiated $\sigma$ bands that tightly encompass experimental half-lives across multiple orders of magnitude without systematic bias. Furthermore, near the $Z=82$ shell closure, the structural softening of the nuclear potential manifests as a systematic expansion and volatility of the $\sigma$ bands in the deformation profiles. This correlation suggests that the macroscopic behavior of parametric error distributions can potentially be mapped onto potential energy configurations, underscoring that the Bayesian UQ methodology may serve as a sensitive probe to extract reliable nuclear structure information directly from deformed proton decay systematics.

Figures

Figures reproduced from arXiv: 2607.18911 by the authors.

**Figure 4.** FIG. 4. Posterior Pearson correlation matrix for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (15 more)

**Figure 6.** Figure 6: FIG. 6. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. Systematic evolution of the quadrupole deformation [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: FIG. 8. Evolution of parameter sensitivities across the in [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9. Relative posterior predictive deviation for proton [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: FIG. 12. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 13.** Figure 13: FIG. 13. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]

**Figure 14.** Figure 14: FIG. 14. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 15.** Figure 15: FIG. 15. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]

**Figure 16.** Figure 16: FIG. 16. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

**Figure 17.** Figure 17: FIG. 17. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p014_17.png]

**Figure 18.** Figure 18: FIG. 18. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p015_18.png]

**Figure 19.** Figure 19: FIG. 19. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]

**Figure 20.** Figure 20: FIG. 20. Supplementary posterior diagnostics for [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 3 linked inside Pith

[47]

J. Z. Bo, Phys. Rev. C113, 064303 (2026)

2026
[1]

Pf¨ utzner, M

M. Pf¨ utzner, M. Karny, L. V. Grigorenko, and K. Riis- ager, Rev. Mod. Phys.84, 567 (2012)

2012
[2]

J. G. Cubiss, A. N. Andreyev, A. E. Barzakh, P. Van Duppen, S. Hilaire, S. P´ eru, S. Goriely, M. Al Monthery, N. A. Althubiti, B. Andel, S. An- talic, D. Atanasov, K. Blaum, T. E. Cocolios, T. Day Goodacre, A. de Roubin, G. J. Farooq-Smith, D. V. Fedorov, V. N. Fedosseev, D. A. Fink, L. P. Gaffney, L. Ghys, R. D. Harding, M. Huyse, N. Imai, D. T. Joss, S...

2023
[3]

Jackson, C

K. Jackson, C. Cardinal, H. Evans, N. Jelley, and J. Cerny, Physics Letters B33, 281 (1970)

1970
[4]

Delion, R

D. Delion, R. Liotta, and R. Wyss, Physics Reports424, 113 (2006)

2006
[5]

H. T. Fortune, Phys. Rev. C93, 034325 (2016)

2016
[6]

Caurier, G

E. Caurier, G. Mart ´ ınez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys.77, 427 (2005)

2005
[7]

M. G. Mayer, Phys. Rev.75, 1969 (1949)

1969

Show all 60 references

[8]

Haxel, J

O. Haxel, J. H. D. Jensen, and H. E. Suess, Phys. Rev. 75, 1766 (1949)

1949
[9]

Hinke, M

C. Hinke, M. B¨ ohmer, P. Boutachkov, T. Faestermann, H. Geissel, J. Gerl, R. Gernh¨ auser, M. G´ orska, A. Got- tardo, H. Grawe,et al., Nature486, 341 (2012)

2012
[10]

H. Wang, Z. Wang, D. Bai, D. Ni, and Z. Ren, Chinese Physics C49, 044111 (2025)

2025
[11]

Mahaux, P

C. Mahaux, P. Bortignon, R. Broglia, and C. Dasso, Physics Reports120, 1 (1985)

1985
[12]

Otsuka, T

T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, and Y. Akaishi, Phys. Rev. Lett.95, 232502 (2005)

2005
[13]

S. G. Nilsson, Dan. Mat. Fys. Medd.29, 1 (1955)

1955
[14]

Hagen, T

G. Hagen, T. Papenbrock, A. Ekstr¨ om, K. A. Wendt, G. Baardsen, S. Gandolfi, M. Hjorth-Jensen, and C. J. Horowitz, Phys. Rev. C89, 014319 (2014)

2014
[15]

Bender, P.-H

M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys.75, 121 (2003)

2003
[16]

Benjedi and P

R. Benjedi and P. Buganu, Phys. Rev. C114, 014322 (2026)

2026
[17]

S. W. Ødeg ˚ ard, G. B. Hagemann, D. R. Jensen, M. Bergstr¨ om, B. Herskind, G. Sletten, S. T¨ orm¨ anen, J. N. Wilson, P. O. Tjøm, I. Hamamoto, K. Spohr, H. H¨ ubel, A. G¨ orgen, G. Sch¨ onwasser, A. Bracco, S. Leoni, A. Maj, C. M. Petrache, P. Bednarczyk, and D. Curien, Phys...

2001
[18]

Cejnar, J

P. Cejnar, J. Jolie, and R. F. Casten, Rev. Mod. Phys. 82, 2155 (2010)

2010
[19]

C. Qi, R. Liotta, and R. Wyss, Progress in Particle and Nuclear Physics105, 214 (2019)

2019
[20]

D. S. Delion, R. J. Liotta, and R. Wyss, Phys. Rev. Lett. 96, 072501 (2006)

2006
[21]

A. Zdeb, M. Warda, and K. Pomorski, Phys. Rev. C87, 024308 (2013)

2013
[22]

Ring, Progress in Particle and Nuclear Physics37, 193 (1996)

P. Ring, Progress in Particle and Nuclear Physics37, 193 (1996)

1996
[23]

J. Meng, H. Toki, S. Zhou, S. Zhang, W. Long, and L. Geng, Progress in Particle and Nuclear Physics57, 470 (2006)

2006
[24]

Vautherin and D

D. Vautherin and D. M. Brink, Phys. Rev. C5, 626 (1972)

1972
[25]

Decharg´ e and D

J. Decharg´ e and D. Gogny, Phys. Rev. C21, 1568 (1980)

1980
[26]

Descouvemont and D

P. Descouvemont and D. Baye, Reports on Progress in Physics73, 036301 (2010)

2010
[27]

Descouvemont, Computer Physics Communications 200, 199 (2016)

P. Descouvemont, Computer Physics Communications 200, 199 (2016)

2016
[28]

Freer, H

M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meißner, Rev. Mod. Phys.90, 035004 (2018)

2018
[29]

Y. Qian, Z. Ren, and D. Ni, Phys. Rev. C94, 024315 (2016)

2016
[30]

von Oertzen, M

W. von Oertzen, M. Freer, and Y. Kanada-En’yo, Physics Reports432, 43 (2006). 10 V0 55+2 1 0.590 0.775 0.960a0 a0 0.64+0.08 0.07 5.40 5.95 6.50 r0 r0 5.7+0.2 0.3 0.150 0.255 0.360P0 P0 0.25+0.08 0.07 5.40 5.75 6.10 Rc0 Rc0 5.8+0.1 0.2 0.07 0.31 0.55 2 2 0.40+0.08 0.25 0.020 0....

2006
[31]

J. M. Dong, H. F. Zhang, and G. Royer, Phys. Rev. C 79, 054330 (2009)

2009
[32]

Qian and Z

Y. Qian and Z. Ren, Phys. Rev. C84, 064307 (2011)

2011
[33]

Z. Wang, D. Bai, and Z. Ren, Phys. Rev. C105, 024327 (2022)

2022
[34]

Ni and Z

D. Ni and Z. Ren, Phys. Rev. C83, 014310 (2011)

2011
[35]

I. I. Gontchar, M. V. Chushnyakova, and O. M. Sukhareva, Phys. Rev. C105, 014612 (2022)

2022
[36]

Bhattacharya and G

M. Bhattacharya and G. Gangopadhyay, Physics Letters B651, 263 (2007)

2007
[37]

D. N. Basu, P. R. Chowdhury, and C. Samanta, Phys. Rev. C72, 051601(R) (2005)

2005
[38]

Ni and Z

D. Ni and Z. Ren, Phys. Rev. C81, 024315 (2010)

2010
[39]

D. S. Delion, Z. Ren, A. Dumitrescu, and D. Ni, Journal of Physics G: Nuclear and Particle Physics45, 053001 (2018)

2018
[40]

Z. Wang, Z. Ren, and D. Bai, Phys. Rev. C101, 054310 (2020)

2020
[41]

Xu and Z

C. Xu and Z. Ren, Phys. Rev. C76, 027303 (2007)

2007
[42]

Xu and Z

C. Xu and Z. Ren, Phys. Rev. C68, 034319 (2003)

2003
[43]

Scamps, D

G. Scamps, D. Lacroix, G. G. Adamian, and N. V. An- tonenko, Phys. Rev. C88, 064327 (2013)

2013
[44]

Esbensen and C

H. Esbensen and C. N. Davids, Phys. Rev. C63, 014315 (2000)

2000
[45]

Heyde and J

K. Heyde and J. L. Wood, Rev. Mod. Phys.83, 1467 (2011)

2011
[46]

D. R. Phillips, R. J. Furnstahl, U. Heinz, T. Maiti, W. Nazarewicz, F. M. Nunes, M. Plumlee, M. T. Pratola, S. Pratt, F. G. Viens, and S. M. Wild, Journal of Physics G: Nuclear and Particle Physics48, 072001 (2021)

2021
[48]

J. M. Bub, M. Piarulli, R. J. Furnstahl, S. Pastore, and D. R. Phillips, Phys. Rev. C111, 034005 (2025)

2025
[49]

Jaiswal, Physics Letters B874, 140243 (2026)

S. Jaiswal, Physics Letters B874, 140243 (2026)

2026
[50]

Van Ravenzwaaij, P

D. Van Ravenzwaaij, P. Cassey, and S. D. Brown, Psy- chonomic bulletin & review25, 143 (2018)

2018
[51]

Brooks, A

S. Brooks, A. Gelman, G. Jones, and X. Meng,Handbook of Markov Chain Monte Carlo, Chapman & Hall/CRC Handbooks of Modern Statistical Methods (CRC Press, 2011)

2011
[52]

B. A. Berg, Introduction to markov chain monte carlo simulations and their statistical analysis (2004), arXiv:cond-mat/0410490 [cond-mat.stat-mech]

2004 arXiv
[53]

J. J. Waterfall, F. P. Casey, R. N. Gutenkunst, K. S. Brown, C. R. Myers, P. W. Brouwer, V. Elser, and J. P. Sethna, Phys. Rev. Lett.97, 150601 (2006)

2006
[54]

M. K. Transtrum, B. B. Machta, K. S. Brown, B. C. Daniels, C. R. Myers, and J. P. Sethna, The Journal of chemical physics143, 10.1063/1.4923066 (2015)

2015 doi
[55]

Talou, D

P. Talou, D. Strottman, and N. Carjan, Phys. Rev. C60, 054318 (1999)

1999
[56]

G. I. Bykhalo, V. N. Orlin, and K. A. Stopani, Descrip- tion of the shape of medium and heavy nuclei using a finite deformed one-particle potential with deformation- dependent diffuseness (2021), arXiv:2107.08245 [nucl-th]

2021 arXiv
[57]

˚Aberg, P

S. ˚Aberg, P. B. Semmes, and W. Nazarewicz, Phys. Rev. C56, 1762 (1997)

1997
[58]

Zhang, L.-J

D.-M. Zhang, L.-J. Qi, H.-F. Gui, S. Luo, B. He, X.-J. Wu, and X.-H. Li, Analytic formula for the proton ra- dioactivity spectroscopic factor (2024), arXiv:2403.04249 [nucl-th]

2024 arXiv
[59]

Goodman and J

J. Goodman and J. Weare, Communications in applied mathematics and computational science5, 65 (2010)

2010
[60]

Foreman-Mackey, D

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, Publications of the Astronomical Society of the Pa- cific125, 306 (2013)

2013

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [47]

J. Z. Bo, Phys. Rev. C113, 064303 (2026)

2026

[2] [1]

Pf¨ utzner, M

M. Pf¨ utzner, M. Karny, L. V. Grigorenko, and K. Riis- ager, Rev. Mod. Phys.84, 567 (2012)

2012

[3] [2]

J. G. Cubiss, A. N. Andreyev, A. E. Barzakh, P. Van Duppen, S. Hilaire, S. P´ eru, S. Goriely, M. Al Monthery, N. A. Althubiti, B. Andel, S. An- talic, D. Atanasov, K. Blaum, T. E. Cocolios, T. Day Goodacre, A. de Roubin, G. J. Farooq-Smith, D. V. Fedorov, V. N. Fedosseev, D. A. Fink, L. P. Gaffney, L. Ghys, R. D. Harding, M. Huyse, N. Imai, D. T. Joss, S...

2023

[4] [3]

Jackson, C

K. Jackson, C. Cardinal, H. Evans, N. Jelley, and J. Cerny, Physics Letters B33, 281 (1970)

1970

[5] [4]

Delion, R

D. Delion, R. Liotta, and R. Wyss, Physics Reports424, 113 (2006)

2006

[6] [5]

H. T. Fortune, Phys. Rev. C93, 034325 (2016)

2016

[7] [6]

Caurier, G

E. Caurier, G. Mart ´ ınez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys.77, 427 (2005)

2005

[8] [7]

M. G. Mayer, Phys. Rev.75, 1969 (1949)

1969