Theoretical derivation and formulas

In this study, we investigated the half-lives of the β^--decay and βn probabilities of neutron-rich nuclei. According to Fermi’s β-decay theory, the standard formula of the β^--decay half-life T_1/2 through the Fermi and GT transitions is as follows [44]: $T_{1/2}=\frac{\kappa }{f_0(B_{\text{F}}+B_{\text{GT}})}\mathrm{,}$ (1) where $\kappa \equiv \frac{2{\pi }^3{\hslash }^7\ln 2}{m_{\text{e}}^5{c}^4{G}_{\text{F}}^2}=6147$ s, B_F and B_GT are the reduced transition probability term for the Fermi and GT transitions, respectively, and f₀ is the phase-space factor, also known as the Fermi integral.

With the Primakoff–Rosen approximation [45] applied to f₀, Eq. (1) was simplified to a manipulable form, which also imposed a relatively high value of the β-decay energy Qβ for the constraint on the selection of experimental data. All nuclei in this study were selected with Qβ greater than 3 MeV, taken from a new compilation [43, 36]. All other branches were equated to one or two effective GT branches, and a logarithmic formula was derived from Eq. (1) with the same notations as in Eq. (2) in removing the term B_F. $\begin{array}{l}-\ln (B_{\text{GT}})=-\ln (\kappa )+\ln \left[\frac{2\pi \alpha Z_{\text{f}}}{1-\exp (-2\pi \alpha Z_{\text{f}})}\right]\mathrm{,}\\ +\ln (T_{1/2})+\ln \left(\frac{1}{30}\left(E_0^5-10E_0^2+15E_0-6\right)\right)\mathrm{,}\end{array}$ (2) where $E_0\equiv \frac{E_{\text{i}}-E_{\text{f}}}{m_{\text{e}}{c}^2}$, E_i and E_f are the energy levels of the initial and final states, respectively, whose difference (E_i-E_f) is consequently the decay energy Qβ, and Z_f is the number of protons of the daughter nucleus.

Subsequently, a linear formula describing -ln(B_GT) was established based on physical meanings. After β-decay, an odd-odd (oo) nucleus decays into an even-even (ee) nucleus, whereas an odd-A (oA) nucleus decays into another oA nucleus. Qβ, calculated from ground state to ground state, does not necessarily reflect the actuality of E₀ of the effective branch. Thus, multiplicative factors were introduced to include the odevity in E₀. Moreover, the oo nucleus more likely decays into the excited state of the ee nucleus for small effective decay energies because the angular momentum of the ground state of the oo parent nucleus tends to differ significantly from that of the ground state of the ee daughter nucleus. With a similar analysis for the case of oA and ee nuclei, there is a pronounced stratification of reduced transition intensities of the three types owing to the different magnitudes of the multiplicative factors.

Considering Eq. (2) and the differences in effective decay energy owing to different odevities, a new formula was accordingly constructed for -ln(B_GT) with the Dirac function, $-\ln (B_{\text{GT}})=a_0+a_1{\delta }_{\text{oA}}+a_2{\delta }_{\text{oo}}\mathrm{.}$ (3) The Dirac functions δ_oo and δ_oA can discern the odevity. The corresponding term for ee nuclei was set as a constant, a₀. Furthermore, the pronounced peaks of the -ln(B_GT) values observed in the distribution of experimental data are bound up with the periodic arrangement of the nuclei, namely, the shell effect of the nucleus. Therefore, a correction term, a₃x₃, for the shell effect was added, $-\ln (B_{\text{GT}})=a_0+a_1{\delta }_{\text{oA}}+a_2{\delta }_{\text{oo}}+a_3{x}_3\mathrm{,}$ (4) where a₀, a₁, a₂, and a₃ are fitting parameters, $x_3={\displaystyle {\sum }_{N_k\mathrm{,}{Z}_k}}\exp \left[-{\left(\frac{N-N_k}{5}\right)}^2-{\left(\frac{Z-Z_k}{5}\right)}^2\right]$, and Nk and Zk describe the shell and sub-shell structures and were chosen to be (50, 32) and (82, 50) according to the locations of distinct peak points.

Eqs. (3) and (4) can only describe a single decay branch without considering the βn branch. In the below content, βo denotes the branch that is not βn. The relations between β-decay half-life and βn probability are as follows: $P_{\beta n}+P_{\beta \text{o}}=\mathrm{1,}$ (5) $\frac{1}{T_{1/2}}=\frac{1}{T_{\beta n}}+\frac{1}{T_{\beta \text{o}}}\mathrm{,}$ (6) $-\ln P_{\beta n}=-\ln \left(\frac{T_{1/2}}{T_{\beta n}}\right)\mathrm{,}$ (7) $-\ln \left(\frac{P_{\beta n}}{1-P_{\beta n}}\right)=-\ln \left(\frac{T_{\beta \text{o}}}{T_{\beta n}}\right)\mathrm{,}$ (8) where Tβn (Tβo) and Pβn (Pβo) denote the partial half-life and probability of the βn (β_o) branch. Thus, an analogical formula could be proposed for -ln(P_βn) with a new optimizing term for the effect of the difference between Qβn and Qβ on the βn probability, $-\ln (P_{\beta n})=a_0+a_1{\delta }_{\text{oA}}+a_2{\delta }_{\text{oo}}+a_3{x}_3+a_4{x}_4\mathrm{,}$ (9) where $x_4=\ln \left(\frac{Q_{\beta n}}{m_{\text{e}}{c}^2}\right)-\ln \left(\frac{Q_{\beta \text{o}}}{m_{\text{e}}{c}^2}\right)$, and Qβn (Qβo) denotes the decay energy of the βn (β_o) branch from ground state to ground state. This makes use of a direct logarithmization of Pβn, yet it may obtain nonphysical results during fitting, that is, Pβn > 1. Pβn was replaced by $\frac{P_{\beta n}}{1-P_{\beta n}}$ to avoid such results, and the derived formula is expressed as $-\ln \left(\frac{P_{\beta n}}{1-P_{\beta n}}\right)=a_0+a_1{\delta }_{\text{oA}}+a_2{\delta }_{\text{oo}}+a_3{x}_3+a_4{x}_4\mathrm{.}$ (10)

Bootstrap method

This study evaluated the fitting parameters and uncertainties of β^--decay half-lives and βn probabilities by applying the bootstrap method. In applied statistics, the bootstrap method was proposed by Efron Bradley in 1985 [46] and used to determine the accuracy of estimating the unknown parameters of a chosen estimator through the basic idea of resampling [47].

After the first proposal of its application on alpha decay laws [47], it was successfully applied to the uncertainty determination of nuclear mass models [48, 49], proton decay stability [49], and the binary cluster model [50].

Similar to Monte Carlo events, a new dataset was obtained by resampling with replacements from a given experimental dataset. A group of parameters in the model was obtained by minimizing the root-mean-square of the residuals. Repeating this process M times, the total uncertainty was thus ${\sigma }_{\text{tot}}=\sqrt{\frac{1}{MK}{\displaystyle \sum _{m\mathrm{,}k}}{(r_{m\mathrm{,}k})}^2}\mathrm{,}$ (11) where m denotes the m^th resampling of the dataset, k denotes the k^th nucleus in the original dataset among the K nuclei, and rm, k is the residual between the observed value and its estimated value from the m^th replication of bootstrap for the k^th nucleus.

The total uncertainty was then decomposed into the statistical uncertainty σ_sys and systematic uncertainty σ_sys, which can be written as ${\sigma }_{\text{stat}}=\sqrt{\frac{1}{(M-1)K}{\displaystyle \sum _{m\mathrm{,}k}}{(r_{m\mathrm{,}k}-{\overline{r}}_k)}^2}\mathrm{,}$ (12) ${\sigma }_{\text{sys}}=\sqrt{\frac{1}{K}{\displaystyle \sum _k}{\left(\frac{1}{M}{\displaystyle \sum _m}{r}_{m\mathrm{,}k}\right)}^2}\mathrm{.}$ (13) In this study, this method was used to assess the uncertainty on Eqs. (3), (4), (9), and (10). The global systematic uncertainty of model deficiency, σ_sys, and the statistical uncertainty appropriate to the specific nucleus were combined to evaluate the confidence interval. ${\sigma }_{\text{pred}\mathrm{,}k}=\sqrt{{\sigma }_{\text{stat}\mathrm{,}k}^2+{\sigma }_{\text{sys}}^2}\mathrm{.}$ (14) Because the reduced transition probability term and the probabilities of the βn and βo branches were in the logarithmic form, the uncertainties obtained were propagated to be different positive and negative deviations when the half-lives and probabilities were further calculated.

One effective decay branch

Nuclei were carefully selected from the newest compilation and evaluation with experimental half-lives smaller than 2 s and Q greater than 3 MeV because forbidden transitions may dominate long-lived transitions and are not suitable for study as effective GT branches. Nuclei with βn basically meet these two conditions. Among 256 selected nuclei, two datasets were taken: one containing all 256, which have half-life measurements, and the other containing 133, which have βn probability measurements.

Based on the assumption of one effective decay, the bootstrap method was used to investigate the first dataset, as listed in Table1. The shell effect did contribute to the description of half-lives, as indicated by the decreasing systematic uncertainty. Fig. 1 shows the results of the difference between the observed data and calculation corresponding to Eqs. (3) and (4). In general, the latter with the shell effect reproduced better than the former, especially where the major shell and sub-shell closures were located.

Fig. 1Full-screen View

Distribution of the difference between the calculated values and experimental values of ln T with error bars. In the one-branch case, the red and black squares correspond to Eq. (3) with three parameters and Eq. (4) with four parameters, respectively.

Two effective decay branches

Then, we considered the case in which there are neutron emissions after β decay, that is, two decay branches with βn emitting neutrons and βo without emitting. It should be noted that there is no distinction between the number of neutrons emitted. Eq. (4) was applied to each branch with the second dataset. The corresponding results are listed in the last two lines of Table2 with the notations Eq. (4)βn and Eq. (4)βo.

Moreover, the one-branch result was re-fitted to compare the adaptability of Eq. (4) with the changed dataset. The values of βn probability largely varied from nuclide to nuclide, resulting in larger uncertainties in Eq. (4)βn and Eq. (4)βo compared with 1b4p.

The values of -ln(B_GT) were calculated to deduce the partial half-life of the two branches, Tβn and T_βo, then the T_1/2 appearing in Eqs. (6) and (7), and the branching ratios. The experimental half-lives were generally well reproduced. In particular, the shorter the half-life, the better the consistency.

The root-mean-square deviation (RMS), which characterizes the goodness-of-fit of one dataset, is defined as $\text{RMS}=\frac{1}{K}{\displaystyle \sum _{k=1}^K}{[Y_{k\mathrm{,}\text{cal}}-Y_{k\mathrm{,}\text{exp}}]}^2\mathrm{.}$ (15) The results obtained by direct fitting to the βn probability are listed in Table3. The three results and experimental values were generally consistent within approximately 10% RMS (10.389%, 10.002%, and 10.231%, respectively).

The value given by -ln(P_βn) was close to the experimental value in general. However, the $-\ln \left(\frac{P_{\beta n}}{1-P_{\beta n}}\right)$ results were closer to the experimental data in several cases with great undulations, such as in the regions of shell closure. The form of $-\ln \left(\frac{P_{\beta n}}{1-P_{\beta n}}\right)$ was determined in a more physical transformation, guaranteeing that the calculated probabilities always lay between 0 and 1.

The experimental and calculated values of the half-life and probability are shown in Table4 for the two-branch case. The calculated values of the half-life were obtained from the fitting results of Eq. (4)βn and Eq. (4)βo listed in Table2, whereas the probability corresponds to Eq. (9) listed in Table3.

Because the uncertainties were estimated on a logarithmic scale, the distances from the upper and lower bounds to the predicted values were different when converting, and thus there were differences in the positive and negative directions of the uncertainties.

For the sake of convenience, the half-lives are presented in natural logarithmic form with T in units of seconds so that the total uncertainty on the calculated values can be given in the same scale in the sixth column. Because the two formulas Eqs. (9) and (10) study different objects, the calculated values of probability were converted from the natural logarithm form.

Predictions

According to the presented method, the predictions of β-decay half-lives and βn probabilities Pβn could be given for nuclei without experimental data, and the uncertainties according to Eq. (14). The predictions for neutron-rich nuclei in the intermediate mass zone offer important nuclear input and relevant data for nuclear physics applications, such as fission product yields in nuclear reactors [16], and the half-lives of nuclei participating in the rapid neutron capture process (r process) in astrophysics [51].

In TABLE 5, 123 nuclei without experimental values of Pβn are listed, 18 of which also have no half-life (in the last eighteen lines of the table). The predictions corresponding to Eq. (10) and Eq. (4) are given for the probability and half-life, respectively.