Decoding $\beta$-decay systematics: A global statistical model for ${\beta }^{-}$ half-lives^*

Statistical modeling of nuclear data provides a novel approach to nuclear systematics complementary to established theoretical and phenomenological approaches based on quantum theory. Continuing previous studies in which global statistical modeling is pursued within the general framework of machine learning theory, we implement advances in training algorithms designed to improve generalization, in application to the problem of reproducing and predicting the half-lives of nuclear ground states that decay $100\%$ by the ${\beta }^{-}$ mode. More specifically, fully connected, multilayer feed-forward artificial neural network models are developed using the Levenberg-Marquardt optimization algorithm together with Bayesian regularization and cross-validation. The predictive performance of models emerging from extensive computer experiments is compared with that of traditional microscopic and phenomenological models as well as with the performance of other learning systems, including earlier neural network models as well as the support vector machines recently applied to the same problem. In discussing the results, emphasis is placed on predictions for nuclei that are far from the stability line, and especially those involved in $r$-process nucleosynthesis. It is found that the new statistical models can match or even surpass the predictive performance of conventional models for $\beta$-decay systematics and accordingly should provide a valuable additional tool for exploring the expanding nuclear landscape.

Figure 1

Architecture of a typical fully connected feed-forward network having an input layer with three units, two hidden layers each, containing five units, and a single output unit, thus of structure $[3-5-5-1|56]$.

Figure 2

Partitioning of the nuclides of the data set NuSet-B into learning, validation, and test sets, as viewed in the $N\text{−}Z$ plane. Stable nuclides are also indicated.

Figure 3

Distribution of half-lives over time scale for NuSet-A nuclides. NuSet-B nuclides lie to the right of the vertical gray rectangle.

Figure 4

Plot showing calculated and experimental ${\beta }^{-}$-decay half-lives for the ${}^{28}\mathrm{Ni}$ isotopic chain. (Solid dots) Experimental data points. (Unfilled dots) New and more precise experimental half-lives recently deduced by Hosmer et al. [45]. Pluses: results generated by the $[2-5-5-5-5-1|111]$ ANN model with inputs $(Z,N)$. Solid lines trace the calculated values of the overall mode (learning, validation, and test sets), while dotted lines trace extrapolated values produced by the model.

Figure 5

Ratios of calculated to experimental half-life values for nuclides in the learning (black), validation (gray), and test (white) sets selected from NuSet-B, plotted versus half-life $T_{\beta ,\exp }$. Calculated values are generated by the standard ANN model of this work with architecture $[3-5-5-5-5-1|116]$. Total error equals ${\Sigma }^{(10)}$ [see Eq. (32)].

Figure 6

Same as Fig. 5, but ratios of calculated to experimental half-lives are plotted against the atomic number $Z$. The dashed lines indicate the magic numbers.

Figure 7

Absolute errors, relative to experiment, of the calculated $\beta$-decay half-lives of all nuclides $(p)$ in the full NuSet-B database plotted versus proton and neutron numbers $Z$ and $N$ for the $[3-5-5-5-5-1|116]$ network model. The bar on the right indicates the mapping from the absolute error values $\left|e_p\right|=|{\log }_{10}{T}_{\beta ,\exp }^p-{\log }_{10}{T}_{\beta ,\mathrm{calc}}^p|$ to the gray scale. Test nuclides are indicated as squares.

Figure 8

Values of ${\sigma }_{\mathrm{rms}}$ obtained in each isotopic chain with the favored $[3-5-5-5-5-1|116]$ network model, respectively, for the learning, validation, and test sets, and the full NuSet-B database, plotted against atomic number $Z$.

Figure 9

Regression analysis for the (i) learning, (ii) validation, (iii) test sets (prediction mode) and for the (iv) full database (overall mode). Solid lines represent exact agreement with the data (i.e., $\log {}_{10}{T}_{\beta ,\mathrm{calc}}=\log {}_{10}{T}_{\beta ,\exp })$, while dashed lines indicate the corresponding best linear fits. The dataset used is NuSet-B. The units used for $T_{\beta }$ are ms. The corresponding values of the parameters $a$ and $b$ of Eq. (22) and the correlation coefficient $R$ of [Eq. (23)] are given in each panel.

Figure 10

Half-lives given by the indicated models for the isotopic chain of ${}^{26}\mathrm{Fe}$, compared with experimental data. ANN refers to the standard network model developed based on the NuSet-B data and $\mathit{pn}\mathrm{QRPA}+\mathit{ff}\mathrm{GT}$ and ${\mathrm{GT}}^{*}$ to models of Ref. [20]. ANN points connected by solid lines are results for nuclides in the learning, validation, and test sets (NuSet-B); those connected by dotted lines are predictions for unknown nuclides absent from NuSet-B.

Figure 11

Same as in Fig.10 but for the isotopic chain of ${}^{47}\mathrm{Ag}$.

Figure 12

Same as in Fig.10 but for the isotopic chain of ${}^{50}\mathrm{Sn}$.

Figure 13

Same as in Fig.10 but for the isotopic chain of ${}^{28}\mathrm{Ni}$.

Figure 14

Same as in Fig.10 but for the isotopic chain of ${}^{48}\mathrm{Cd}$.

Figure 15

Same as in Fig.10 but for the isotopic chain of ${}^{83}\mathrm{Bi}$.

Figure 16

Same as in Fig.10 but for the $r$-ladder isotonic chain of $N=50$. Here ANN refers to the standard network model developed on the basis of the NuSet-B data and $\mathit{pn}\mathrm{QRPA}+\mathit{ff}\mathrm{GT}$ and ${\mathrm{GT}}^{*}$ to models of Ref. [20]. ANN points connected by solid lines are results for nuclides in the learning, validation, and test sets (NuSet-B); those connected by dotted lines are predictions for unknown nuclides absent from NuSet-B.

Figure 17

Same as in Fig.16 but for the $r$-ladder istonic chain of $N=82$.

Figure 18

Same as in Fig.16 but for the $r$-ladder isotonic chain of $N=126$.

Figure 19

Half-lives for the isotopic chain of ${}^{26}\mathrm{Fe}$, as given by the $\mathit{pn}\mathrm{QRPA}+\mathit{ff}\mathrm{GT}$ model of Ref. [20] and by the new ANN model (namely ANN-II) designed to fit and predict data generated by this “theory-thick” model rather than the experimental data. ANN-II points connected by solid lines are results for nuclides in the corresponding learning, validation, and test sets; those connected by dotted lines are predictions for nuclides outside these sets.

Figure 20

Same as in Fig.19 but for the isotopic chain of ${}^{50}\mathrm{Bi}$.

Figure 21

Same as in Fig.19 but for the isotonic chain of $N=82$.

Figure 22

Same as in Fig.19 but for the isotonic chain of $N=126$.

Figure 23

Half-lives for ${\beta }^{-}$-decaying nuclides that are found on or near a typical $r$-process path, with neutron separation energies up to 3 MeV. ANN refers to the standard model of this work and $\mathit{pn}\mathrm{QRPA}+\mathit{ff}\mathrm{GT}$ and ${\mathrm{GT}}^{*}$ to models of Ref. [20].