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

Technical Summary: Mapping Parametric Error Profiles onto Nuclear Structure Configurations in Deformed Proton Radioactivity

Problem Statement
Theoretical descriptions of proton radioactivity, particularly near the proton drip line in the transitional mass region between proton magic numbers $Z=50$ and $Z=82$, are frequently hindered by parametric uncertainties in nuclear potential models. Traditional implementations of the deformed Woods–Saxon potential framework typically rely on deterministic global fits or parameters adopted from macro-microscopic mass models. These approaches often obscure strong parameter correlations and degeneracies, particularly in soft transitional regimes where multiple deformation configurations can yield similar half-life predictions. Furthermore, standard deterministic calculations provide only single point-value predictions, lacking the capacity to evaluate statistical confidence bounds or propagate epistemic uncertainties from experimental inputs to final half-life outputs.

Methodology
To address these limitations, the authors establish a robust Bayesian uncertainty quantification (UQ) framework. The methodology integrates the following components:

• Theoretical Model: A deformed Woods–Saxon potential coupled with a semi-classical Wentzel–Kramers–Brillouin (WKB) approximation. The model accounts for orientation-dependent barrier geometry modulated by multipole deformations ($\beta_2, \beta_4$) and includes spin-orbit and Coulomb terms.
• Bayesian Calibration: The framework employs Markov-chain Monte Carlo (MCMC) sampling (using the emcee package) to calibrate model parameters against experimental half-lives for odd-A proton emitters across the $Z=50–82$ region.
• Parameter Space: The calibration extracts joint and marginal posterior distributions for potential geometry parameters (depth $V_0$, radius $r_0$, diffuseness $a_0$), multipole deformations ($\beta_2, \beta_4$), spectroscopic factors ($P_0$), and spin-orbit parameters.
• Sensitivity Analysis: A signed relative sensitivity index ($S_{signed}$) is introduced to dynamically re-order parameter hierarchies. This metric quantifies the percentage variation in calculated half-lives resulting from shifting specific parameters to their 16th and 84th percentiles of the posterior distribution, normalized by the baseline median prediction.

Key Results

• Parameter Correlations and Degeneracies: The analysis reveals that, contrary to concerns about "sloppiness" in certain transitional nuclei (e.g., $^{113}$Cs), the majority of the investigated region exhibits a lack of significant parameter degeneracies. The quadrupole ($\beta_2$) and hexadecapole ($\beta_4$) deformations are statistically decoupled from each other and from core potential parameters, operating as independent physical mechanisms.
• Systematic Evolution of Parameters:
  • Quadrupole Deformation ($\beta_2$): Exhibits a transition around $Z=71$. The lighter sector ($Z=53–69$) shows stable prolate shapes with narrow uncertainty bands, while the heavier sector ($Z \ge 71$) approaches the $Z=82$ shell closure, leading to a softening of the potential and a widening of posterior uncertainty bands.
  • Hexadecapole Deformation ($\beta_4$): Shows a distinct evolutionary pathway, shifting from positive values in light and heavy sectors to negative values in the intermediate region ($Z=65–73$). Uncertainty bands for $\beta_4$ systematically expand near the $Z=82$ closure.
• Global Sensitivity Hierarchy: The sensitivity analysis uncovers a dynamic, mass-dependent interplay:
  • Nuclear radius ($r_0$) and quadrupole deformation ($\beta_2$) are the primary drivers of half-life uncertainty.
  • $r_0$ sensitivity peaks near $Z=55$ ($\approx 50\%$) due to the exponential dependence of tunneling on barrier width.
  • $\beta_2$ sensitivity is highest in the mid-shell region ($Z \le 59$), peaking at $\approx 70\%$ for $^{113}$Cs, before declining and resurging in the heavy transitional region.
  • The hexadecapole parameter $\beta_4$ exhibits a bimodal sensitivity landscape, with sharp peaks at $Z=59$ and $Z=75–79$, but is quenched to near zero in the intermediate mass interval.
• Uncertainty Quantification: Propagating parametric uncertainties yields posterior predictive medians and differentiated $\sigma$ bands that tightly encompass experimental half-lives across multiple orders of magnitude without systematic bias. The width of these uncertainty bands is non-uniform, reflecting localized structural anomalies and intrinsic parametric sensitivities. Notably, near the $Z=82$ shell closure, the structural softening of the nuclear potential manifests as a systematic expansion and volatility of the $\sigma$ bands.

Significance and Claims
The paper claims that the Bayesian UQ methodology serves as a sensitive probe to extract reliable nuclear structure information directly from deformed proton decay systematics. A primary contribution is the demonstration that macroscopic behaviors of parametric error distributions can be mapped onto potential energy configurations. Specifically, the systematic expansion of uncertainty bands near shell closures is physically interpreted as a direct consequence of the structural softening of the nuclear potential, which triggers enhanced parametric degeneracies and shape coexistence.

The authors assert that this framework moves beyond superficial point estimation to provide a rigorous, multi-dimensional diagnostic of theoretical models. By successfully differentiating between regions of high sensitivity and those of structural softening, the methodology offers a more robust assessment of the reliability and intrinsic limitations of theoretical predictions for proton radioactivity. The work concludes that while the current framework is effective, future extensions could incorporate more comprehensive microscopic frameworks, such as advanced coupled-channels calculations or triaxial degrees of freedom, to further solidify predictive power.