The β-decay half life is one of the most important weak interaction reaction rates. It is of great significance for not only nuclear structure research, but also for astrophysics.^[1–5] From astrophysical point of view, the β-decay half-lives are much important for nuclei synthesis (thus star elements abundance) and evolution of stars. It is the competition between β-decay rates and neutron capture rates that determines by which ways (rapid neutron capture process (r process) or the slow neutron capture process (s process)) the heavier nuclei in universe are produced. For r process, in which the β-decay rates is smaller than the neutron capture rates, seed nuclei can capture a neutron and hence make heavier nuclei before they decay by β-transition. It is believed that half of the neutron-rich atomic nuclei heavier than iron are made by the r process. Nucleus lighter than Fe is produced though thermonuclear reactions. Now, it is generally accepted that r process occurs in these environment, high temperature and high neutron density, although the actual site of r-process is hard to be determined.^[6–8] Under these conditions, neutron captures are dominant in comparison with β-decays, and the r-process path moves along a chain of extreme neutron-rich nuclei. Along these paths, the so called waiting point nuclei with magic number N = 50, 82, and 126 are of great interest because the large neutron separation energies of these nuclei make r-process matter flow slow down when it reaches these neutron magic nuclei. It waits for some β-decays to occur before further neutron captures are possible. The mass flow are carried to heavier nuclei and consequently the matter is gathered, which leads to the so called peaks in abundance distribution of r-process. Thus, the β-decay half-lives of the waiting point nuclei are very important because it is the key to understand the third r-process peak of heavy nuclei with mass numbers A∼200. From the point of view of nuclear science, β-decay half-lives of waiting point nuclei are also of great importance, because the decay rates are mainly determined by the Q values and the transition matrix elements, which are strongly affected by the nuclear structures and thus are very important quantity for nuclear structure study, especially the shell structure far from stability.

Despite their importance for astrophysics and nuclear physics, a few experimental data of half-lives are obtained only for waiting point nuclei with N = 50 and 82.^[3,9–11] For N = 126, experimental data are very absent. Thus, for astrophysical r-process simulations, the needed half-lives only rely on theoretical estimates. The situation of half-lives for nuclear structure study is also the same. Therefore, theoretical calculations of half-lives are very important. Since Fermi^[12] first proposed β-decays theory in the 1930s, many scientists have done a lot of theoretical research on the calculation of β-decay half-lives. Because the Q-values are much low, these studies have considered mainly on allowed transitions. The Quasiparticle Random Phase Approximation (QRPA) either on top of semi-empirical global models^[13–16] or the Hartree-Fock-Bogoliubov method or the shell model is applied to these calculations, which reproduces well the experimental half-lives of waiting point nuclei in r-process.^[17–19] The calculations can give a reasonable account of the few experimental data, however, some of these microscopic calculations are time-consuming and model dependent. Therefore, besides the microscopic calculations, there exist other phenomenological methods. Sargent^[20] proposed a law that the β-decay half-lives is proportional to the fifth power of β-decay energies . Recently, we analyzed the experimental data of nuclear β-decay half-lives and found an exponential law of β-decay half-lives for nuclei far from stability, which was similar to the relation between α-decay half-lives and α-decay energies.^[21–23] The results calculated with an empirical formula based on this exponential law reproduce the experimental data.

Very recently, the forbidden transitions are included in the QRPA and the shell model calculations. The first forbidden transitions are considered and half-lives are calculated for waiting point nuclei around neutron number N = 50, 82, and 126.^[24,25] It was pointed out that the forbidden transitions become very important and must be considered for waiting point nuclei around N = 126. As we know, the available experimental data around N = 126 are very poor. In addition, microscopic calculations including forbidden transitions are very time-consuming. Therefore, it is of great importance to give another theoretical calculations of the waiting point nuclei around N = 126. In this paper, we will re-exam the formulae in the literature^[22] by taking into account the forbidden transitions and the shell effects with the aim to give better description of β-decay half-lives for waiting point nuclei around N = 126, which are very useful for nuclear structures and the third peak elements production in stars.

The paper is organized in the following way. In Sec. 2, we propose a new set of parameters for empirical formula for calculating the -decay. In Sec. 3, we utilize the formula to calculate half-lives of -decays, and to predict the -decay half-lives of waiting point nuclei around N = 126. A summary is given in Sec. 4.

Zhang et al.^[21,22] systematically analyzed the experiment β-decay half-lives of nuclei and discovered an exponential relationship between -decay half-lives (in scale) and the nucleon numbers of parent nuclei. They proposed an exponential formula to calculate the β-decay half-lives of nuclei far from stability:
where is -decay half-life (in seconds) and Z is the proton number of parent nucleus. By a least-squares fit, the values of c₁, c₂, c₃ and c₄ are: , −0.2275, 0.3652, and −0.8852, respectively.^[22] If the shell effects are included, the formula has the form:
where S(Z, N) is a term of shell effect correction and can be written as:

Zhang et al.^[22] have tested the accuracy and reliability of the two formulae. For the chosen center of N = 131 in the Gaussian function in Eq. (3), Zhang et al. has given an explanation.^[22] On the one hand, -decay half-lives depend on the decay energies, which are affected by the shell effects near proton or neutron “magic number”. On the other hand, the -decay half-lives also depend on the energy level structure of the parent and daughter nuclei. According to the selection rule for β-transitions, forbidden transition plays an important role near the region of proton or neutron “magic number”. The superposition of these two effects causes the deviation of the Gaussian center from N = 126. It is interesting to see from the data that a bigger correction is needed near N = 131, instead of N = 126. As pointed out in Refs. [24,25] that the contributions of forbidden transitions, which were not fully considered by a global fit in Ref. [22], are large and must be considered separately for nuclei around N = 126. Therefore, we re-examine Eq. (2) and write it in the form:

By a least-squares fit to the available experimental data of 27 nuclei with Z = 73–83, N = 117–136, we obtain a new set of parameters: c₁=−0.002 87, c₂ = 0.041 88, c₃ = 0.785 26, c₄=−37.1312 and c₅=0.635 56, respectively. Here, the Gaussian center N=131.5 in Eq. (4) is a little bit different from the center of N = 131 in Eq. (3). The reason is that the fit for Eq. (3) is a global fit from light nuclei to heavy nuclei, while the parameters for Eq. (4) are obtained from a local fit around N = 126. Because the fit is mainly concentrated on nuclei around N = 126, the forbidden transitions are described more accurately in the new set of parameters. In the following, we will test the reliability of this new set of parameters, and use Eq. (4) with its new parameters to calculate and predict the half-lives of neutron-rich nuclei around N = 126, which is important for r process.

3. Results for -Decay Half-Lives of Nuclei around N = 126

In Table 1, we compared calculated results from Eq. (4) with experimental data. Theoretical results from Eqs. (1)–(2) and from FRDM model are also listed for comparison. From Table 1, one can see that the calculated results from Eqs. (1)–(2) with old parameters have larger deviations from experimental data. For example, the deviation of ¹⁹⁰Ta is about 5 times with old parameters for Eq. (1). However, calculated -decay half-lives from Eq. (4) with the new set of parameters agree with experimental data quite well. It can be seen in Table 1 that, the calculated results of Eq. (4) are much closer to the experimental data when compared with the FRDM model with or without forbidden transitions. These show that the influence of forbidden transitions are very important and must be considered for nuclei around N = 126.

Table 1

Table 1

Table 1 -decay half-lives (in seconds) for nuclei with Z = 73–83 and N = 117–136 from calculated and experimental results. The first column and second column represent the proton number and mass number of a parent nucleus, respectively. Calculated half-lives from Eq. (1) and Eq. (2) with old parameters[22] are represented as T1 and T2 in column 3 and column 4, respectively. T4 in column 5 represents the theoretical values from Eq. (4) with the new set of parameters. TFRDM I and TFRDM II in the sixth and seventh columns represent theoretical values from FRDM model without and with contributions of forbidden transitions, respectively.[15,16] The last column represents experimental data.[26] The symbol X represents unknown error bar. . Z A T1 T2 T4 TFRDM I TFRDM II Texp 73 190 26.3 22.1 4.0 1.5 1.4 5.3±0.07 73 192 10.1 7.6 1.8 0.6 0.5 2.2±0.07 75 195 39 34 9 3 3 6±1 75 196 24.1 20.3 6.4 4.0 3.6 2.4±1.5 76 200 14 11 5 187 7±4 77 200 58 54 21 124 44±6 77 201 36 32 14 130 21±5 77 202 22 19 9 68 13±2 78 203 56 52 25 564 22±4 78 204 34.7 30.9 16.2 321.8 10.3±0.14 79 204 139.6 145.9 65.8 455.3 38.2±0.14 79 205 86.8 86.1 43.0 222.0 32.5±0.14 79 206 54 51 28 21 30±X 80 207 135 143 74 40 174±1.2 80 209 52 85 39 34 37±8 81 208 338.50 399.28 196.12 99.62 183.18±0.000 24 81 209 210.63 259.07 131.15 91.28 129.66±0.000 42 81 210 131.1 238.9 101.6 57.5 78.0±0.018 81 211 82 362 107 71 100±40 81 212 51 386 126 29 100±40 81 213 32 126 89 32 60±40 82 213 127 1080 324 590 612±1.8 82 215 49 81 84 283 147±0.12 83 216 124 228 213 99 135±0.03 83 217 77.4 86.8 78.0 178.3 98.5±0.08 83 218 48 47 39 4 3 33±1 83 219 30 27 23 83 27 22±7

Table 1 -decay half-lives (in seconds) for nuclei with Z = 73–83 and N = 117–136 from calculated and experimental results. The first column and second column represent the proton number and mass number of a parent nucleus, respectively. Calculated half-lives from Eq. (1) and Eq. (2) with old parameters[22] are represented as T1 and T2 in column 3 and column 4, respectively. T4 in column 5 represents the theoretical values from Eq. (4) with the new set of parameters. TFRDM I and TFRDM II in the sixth and seventh columns represent theoretical values from FRDM model without and with contributions of forbidden transitions, respectively.[15,16] The last column represents experimental data.[26] The symbol X represents unknown error bar. .

In Figs. 1 and 2, the logarithms of the ratios between calculated values and experimental ones are plotted versus the proton and neutron numbers of parent nuclei for various models and formulae. In the two figure, it can be seen that theoretical value from Eqs. (1)–(2) are larger than experimental data for the smaller Z and N in general. Theoretical results T_{FRDM II} from FRDM model have the largest deviations from experimental data, while results T₄ from Eq. (4) with the new set of parameters have the smallest deviations. For example, the largest deviation is about 32 times for T_{FRDM II}, but the deviations of T₄ are less than 5 times, which shows that the new formula Eq. (4) can give good description of the -decay half-lives of nuclei around N = 126 region where the contribution of forbidden transitions are important.

Fig. 1.

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	Fig. 1. The logarithms of the ratios between calculated values from T1 (Eq. (1) with old parameters), T2 (Eq. (2) with old parameters), T4 (Eq. (4) with new parameters), and TFRDM II and experimental ones plotted versus the proton numbers of parent nuclei with Z = 73–83. Experimental data are taken from Ref. [26].

Fig. 2.

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	Fig. 2. The same as Fig. 1, but the results plotted versus N.

Based on the above formulae, we calculated the half-lives of 76 nuclei with Z = 73–82 and N = 121–139 for both nuclear and astrophysical purposes. The results together with theoretical calculations from FRDM models with and without forbidden transitions are listed in Table 2. It is clear seen that the calculated results with forbidden transitions are smaller than those without forbidden transitions, which shows the importance of forbidden transitions. It can bee seen in Table 2 that, in general, results from two models have small deviations and in general our results are smaller than the FRDM model results. More experimental data in future could be valuable to verify the predictions from different models.

Table 2.

Table 2.

Table 2. Predicted -decay half-lives (in seconds) for nuclei with Z = 73–82 and N = 121–139. The first column and second column represent the proton number and mass number of a parent nucleus, respectively. Calculated half-lives from Eq. (2) with old parameters[22] are represented as T2 in column 3. T4 in column 4 represents the theoretical values from Eq. (4) with the new set of parameters. TFRDM I and TFRDM II in the fifth and sixth columns represent theoretical values from FRDM model without and with contributions of forbidden transitions, respectively.[15,16] . Z A T2 T4 TFRDM I TFRDM II Z A T2 T4 TFRDM I TFRDM II 73 194 2.626 0.812 1.061 0.959 77 207 5.924 2.291 41.148 3.102 73 195 1.542 0.552 1.037 0.974 77 208 6.295 2.776 4.290 1.299 73 196 0.906 0.375 0.633 0.552 77 209 2.056 2.001 2.022 0.930 73 197 0.532 0.255 0.619 0.558 77 210 0.472 0.770 0.268 0.198 73 198 0.312 0.173 0.535 0.471 78 205 18.386 10.220 20.596 73 199 0.183 0.118 0.701 0.554 78 206 11.902 6.968 22.244 73 200 0.109 0.080 0.190 0.160 78 207 10.949 5.504 6.796 73 201 0.070 0.057 0.277 0.241 78 208 16.567 5.918 7.135 73 202 0.064 0.046 0.130 0.105 78 209 17.618 7.125 23.118 3.755 73 203 0.097 0.051 0.130 0.110 78 210 5.757 5.101 10.063 2.995 74 196 4.286 1.503 87.515 45.338 78 211 1.322 1.950 1.913 0.989 74 197 2.519 1.015 25.427 14.180 79 207 33.231 18.240 22.318 74 198 1.480 0.686 65.422 20.707 79 208 30.594 14.312 8.555 74 199 0.870 0.463 19.156 79 209 46.330 15.288 87.747 7.502 74 200 0.511 0.313 11.603 79 210 49.308 18.284 0.353 0.314 74 201 0.303 0.212 10.327 2.188 79 211 16.125 13.003 4.827 2.296 74 202 0.196 0.148 35.563 2.409 79 212 3.706 4.939 4.775 1.806 74 203 0.180 0.120 8.798 1.282 79 213 1.401 1.849 1.370 0.925 74 204 0.271 0.133 5.136 1.109 79 214 0.753 0.948 0.442 0.319 75 198 7.005 2.747 2.282 2.011 80 210 129.561 39.493 41.233 75 199 4.120 1.843 2.377 2.026 80 211 137.994 46.920 14.922 75 200 2.423 1.236 1.775 1.498 80 212 45.164 33.149 41.758 10.511 75 201 1.426 0.829 1.190 0.995 80 213 10.388 12.507 16.266 4.716 75 202 0.847 0.559 0.472 0.381 80 214 3.930 4.652 5.679 3.098 75 203 0.547 0.389 0.605 0.495 80 215 2.113 2.368 0.251 0.224 75 204 0.502 0.313 0.295 0.232 80 216 1.236 1.455 0.732 0.610 75 205 0.758 0.343 0.369 0.290 81 214 29.118 31.674 14.422 75 206 0.804 0.422 0.072 0.066 81 215 11.026 11.702 20.385 7.769 76 201 6.750 3.300 81.058 81 216 5.933 5.919 7.420 2.797 76 202 3.975 2.200 40.605 81 217 3.472 3.612 1.793 1.272 76 203 2.362 1.472 6.582 81 218 2.051 2.313 2.111 1.098 76 204 1.527 1.017 6.272 81 219 1.212 1.491 0.830 0.629 76 205 1.402 0.814 50.123 2.564 82 216 30.928 29.441 852.178 76 206 2.119 0.887 30.470 2.579 82 217 16.655 14.793 104.917 76 207 2.249 1.082 3.353 1.062 82 218 9.755 8.969 66.288 76 208 0.734 0.785 2.006 0.893 82 219 5.766 5.706 1.233 1.191 77 205 4.263 2.662 6.123 82 220 3.410 3.653 8.291 6.960 77 206 3.918 2.117 76.173 2.875 82 221 2.016 2.340 0.514 0.449

Table 2. Predicted -decay half-lives (in seconds) for nuclei with Z = 73–82 and N = 121–139. The first column and second column represent the proton number and mass number of a parent nucleus, respectively. Calculated half-lives from Eq. (2) with old parameters[22] are represented as T2 in column 3. T4 in column 4 represents the theoretical values from Eq. (4) with the new set of parameters. TFRDM I and TFRDM II in the fifth and sixth columns represent theoretical values from FRDM model without and with contributions of forbidden transitions, respectively.[15,16] .

In summary, by considering the contributions of forbidden transitions, the -decay half-lives for nuclei around N = 126 are studied and a new set of parameters for the exponential law have been proposed. With the new set of parameters, the -half-lives of 27 nuclei for nuclei with Z = 73-83 are calculated and compared with other theoretical results and experimental data. It is found that theoretical results considering the forbidden transitions agree well with experimental data. Meanwhile, we predict the -half-lives of some unknown neutron-rich nuclei around N = 126 for future nuclear and astrophysical study, especially for r process nucleosynthesis in stars.