Nanosecond-Scale Proton Emission from Triaxially Deformed Lu-148 Predicted with High Accuracy Qp Value via Novel Bayesian Evaluation

Abstract

The half-life of the odd-odd deformed proton emitter $^{148}$Lu is predicted to be $196_{-129}^{+420}$ ns via the Wentzel-Kramers-Brillouin (WKB) approximation, in which the potential is extracted from the triaxial relativistic Hartree-Bogoliubov theory in continuum (TRHBc) and the proton decay energy $Q_{\rm p}$ is computed as 2.015(89) MeV by the Bayesian Neural Network - Beihang (BNN-BH) model for the first time. As a decisive factor, the uncertainty of $S_{\rm p}$ has been improved from 411 keV (Bayesian Machine Learning, BML) to 89 keV (BNN-BH) by taking the ensemble uncertainty into account and confining the error estimation to the neighboring nuclei. In consequence, the magnitude of the half-life's uncertainty can be reduced from 4 orders to 1 order, compared to that ($5.5_{-5.3}^{+636}$ ns) with $S_{\rm p}$ from the BML model. We also found that the range of half-life predicted by the TRHBc + WKB approach is consistent with those from the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) + WKB approach, and with those from an empirical formalism with the $S_{\rm p}$ obtained with the BNN-BH model. Furthermore, the means from the above 3 ways agree well with the experimental data for $^{149}$Lu, which gives us confidence to recommend a measurement of the half-life of proton emitter $^{148}$Lu.

Figures (3)

• Figure 1: Potential energy surfaces for $^{148}$Lu and $^{147}$Yb in the $\beta$-$\gamma$ plane from the TRHBc theory. The energy interval between each contour line is 0.4 MeV. All energies are normalized with respect to the energy of absolute minimum (triaxial ground state) indicated by a red ball. The ground state deformation predicted by the DRHBc theory is denoted by a black ball.
• Figure 2: Deviations of $S_{\rm p}$ in $^{148-186}$Lu isotopes with the BNN-BH and the BML niu_nuclear_2022 models, in comparison with the available data. The solid red (blue) lines with uncertainties show the deviations of the predictions from the BNN-BH (BML) models. Shaded regions represent the $1\sigma$ uncertainties. The solid pink line corresponds to the proton drip line ($^{155}$Lu). Experimental data are taken from Ref. PhysRevLett.128.112501 ($^{149}$Lu) and NNDC noauthor_nndc_nodate ($^{150-186}$Lu).
• Figure 3: Half-lives of proton emitters $^{148}$Lu (cyan region) and $^{149}$Lu (yellow region) in WKB approximation. The red solid circles with uncertainties refer to the predictions with $S_{\rm p}$ from the BNN-BH model, and the blue ones refer to those from the BML model niu_nuclear_2022. Experimental data are taken from Ref. PhysRevLett.128.112501.