Calculation of $\beta$-decay half-lives within a Skyrme-Hartree-Fock-Bogoliubov energy density functional with the proton-neutron quasiparticle random-phase approximation and isoscalar pairing strengths optimized by a Bayesian method

Background: For data on radioactive nuclei, $\beta$ decay provides some of the most important information, applicable to various fields. However, some $\beta$-decay data are not available due to experimental difficulties. For this reason, theoretically calculated results have been embedded in the data on radioactive nuclear $\beta$ decay to compensate for the missing information.

Purpose: It is necessary to treat various nuclear correlations as precisely as possible for theoretical $\beta$-decay calculations. In particular, the pairing correlation is one of the most important factors for reproducing $\beta$-decay half-lives correctly. Therefore, we first study the effect of zero- and finite-range isovector pairings on half-lives. Second, we investigate the isoscalar pairing strengths, which are determined through experimental data of half-lives. Finally, we predict the isoscalar pairing strengths and half-lives of neutron-rich nuclei.

Methods: To calculate the $\beta$-decay half-lives, a proton-neutron quasiparticle random-phase approximation on top of a Skryme energy density functional is applied with an assumption of spherical symmetry. The half-lives are calculated by including the allowed and first-forbidden transitions. The isoscalar pairing strength is estimated by a Bayesian neural network (BNN). We verify the predicted isoscalar pairing strengths by preparing the training data and test data.

Results: It was confirmed that the finite-range isovector pairing ensures that the $\beta$-decay half-lives are insensitive to the model space, while the zero-range one is largely dependent on it. The half-lives calculated with the BNN isoscalar pairing strengths reproduced most of the experimental data, although those of highly deformed nuclei were underestimated. We also studied the predictive performance on new experimental data that were not used for the BNN training and found that they were reproduced well.

Conclusions: Our study demonstrates that the isoscalar pairing strengths determined by the BNN can reproduce experimental data with the same accuracy as other theoretical works. To achieve a more precise prediction, the nuclear deformation is important.

Figure 1

Difference between the calculated and experimental $Q_{\beta }$ [42] for the even-even (E-E), even-odd (E-O), odd-even (O-E), and odd-odd (O-O) nuclei. The mean values of $\delta Q=Q_{\beta ,\mathrm{calc}}-Q_{\beta ,\exp }$ (defined as $\delta \overline{Q}$) and the standard deviations ($\sigma$) are shown by the solid and dotted lines, respectively. The dashed line represents $\delta Q=0$.

Figure 2

(a) Average pairing gaps of proton ($p$) and neutron ($n$) and (b) $\beta$-decay half-life of ${}^{128}\mathrm{Cd}$ as a function of the cutoff energy ${\varepsilon }_{\mathrm{cut}}$.

Figure 3

Half-lives of Cd isotopes calculated with different model spaces ${\varepsilon }_{\mathrm{cut}}$ for (a) the zero-range and (b) the finite-range isovector pairing interactions.

Figure 4

(a) Mean deviation $\overline{r}$ from the experimental data of 950 nuclei and (b) standard deviation $s$ as a function of the isoscalar pairing strength $V$. The results of different ranges of $T$ are plotted. The dotted lines of $\overline{r}=\pm 0.3$ are also depicted.

Figure 5

Same as Fig. 4, but for the $Z\ge 20$ nuclei with the quadrupole deformation parameter ${\beta }_2<0.15$ and all nuclei with $Z<20$. The total number of target nuclei is 485.

Figure 6

Optimized isoscalar pairing strengths $V_{\mathrm{opt}}$ in the $N\text{−}Z$ plane determined so as to reproduce the $\beta$-decay half-lives of NUBASE2016 [51], and the corresponding projections to the $N$ and $Z$ axes. The calculation is performed by the $\mathrm{HFB}+pn\mathrm{QRPA}$. Neutron and proton magic numbers are shown by the double and dashed lines.

Figure 7

Isoscalar pairing strengths in the Cd isotopes estimated by the BNN ($V_{\mathrm{BNN}}$). The top, middle, and bottom panels are the results of setting 1 ($T_{\mathrm{div}}=1\mathrm{s}$), 3 ($T_{\mathrm{div}}=0.1\mathrm{s}$) of Table 1, and that using all data for training (i.e., $T_{\mathrm{div}}=0\mathrm{s}$), respectively. The mean values of $V_{\mathrm{BNN}}$ are shown by the dashed lines and the shaded areas are the uncertainties of $1\sigma$. The data set of training $V_{\mathrm{opt}}$ (filled circle) and test $V_{\mathrm{opt}}$ (open circle) are plotted together. Evolution of $V_{\mathrm{BNN}}$ with increasing number of training data can be seen through the panels (a)–(c).

Figure 8

(a) Distribution of $r_i$ as a function of $Q_{\beta }$. (b) The histogram of the statistics of $r_i$ with a bin $\mathrm{\Delta }{r}_i=0.2$. The mean deviation and the standard deviation of the histogram are $\overline{r}=-0.155$ and $s=1.153$, respectively.

Figure 9

(a) Number of nuclei, (b) mean deviation $\overline{r}$, and (c) standard deviation $s$ within the upper limit of half-life $T_{\exp }^{\max }$. The results of Skyrme HFB $+ pn\mathrm{QRPA}$ with $V_{\mathrm{BNN}}$ (this work), ${\mathrm{D}3\mathrm{C}}^{*}$ [6], and pnFAM [18] are plotted together for comparison.

Figure 10

Calculated $\beta$-decay half-lives of this work, ${\mathrm{D}3\mathrm{C}}^{*}$ [6], and $pn\mathrm{FAM}$ [18] for Kr, Rb, Cd, and In isotopes. The solid circles are taken from NUBASE2016 [51].

Figure 11

Ratios between the calculated and experimental half-lives in the $N\text{−}Z$ plane.

Figure 12

Same as Fig. 10, but for Mo, Tc, Sm, and Eu isotopes.

Figure 13

Percentages of the contributions from the first-forbidden transitions to the total $\beta$-decay rates.

Figure 14

Ratios between the calculated $T_{\mathrm{calc}}$ by the Skyrme $\mathrm{HFB}+pn\mathrm{QRPA}$ with BNN and the newly measured $T_{\exp }$ at RIKEN [4]. Data points of 14 nuclei that are not listed in the NUBASE2016 [51] are highlighted by the solid circles.

Figure 15

(a) Isoscalar pairing strengths $V$ and (b) their uncertainties $\mathrm{\Delta }V$ for Te isotopes estimated by the BNN with NUBASE2016 [51] (prior $V_{\mathrm{BNN}}$) and with NUBASE2016 and new measurements [4] (new $V_{\mathrm{BNN}}$). The optimized strengths $V_{\mathrm{opt}}$ estimated with NUBASE2016 and new data are also shown.

Figure 16

(a) $\beta$-decay half-lives and (b) their uncertainties of Te isotopes calculated with the isoscalar pairing strengths estimated by the BNN with the prior $V_{\mathrm{BNN}}$ and new $V_{\mathrm{BNN}}$. The experimental data of NUBASE2016 [51] and new measurements [4] are also plotted for comparison.