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The study of unstable nuclei beyond the nucleon drip line is an important method to study the nuclear interaction and structure in the extremely neutron- or proton-rich system, and various nuclides beyond the proton drip line mainly decay by spontaneous one-proton emission. Using the deformed Woods-Saxon potential, spin-orbit potential and the expanded Coulomb potential to construct the daughter-proton potential, based on the quantum tunneling model and the microscopic Gamow state theory, the half-lives data of various proton emitters are systematically calculated. By using nuclear data from different source and comparing to experiments, the dependence of proton emission on decay energy and spectroscopic factors is evaluated. Additionally, based on previous observations, the half-life of the possibly lighter proton emitter in the fpg-shell below has been theoretically predicted. Our results are compiled into a comprehensive dataset of half-lives for both experimentally confirmed emitters (50 < Z < 84) and theoretically predicted emitters (30 < Z < 50), providing a useful reference for future experimental investigations related to the proton drip line. The datasets presented in this paper, including our results of calculation, are openly available at
https://www.doi.org/10.57760/sciencedb.27551 .-
Keywords:
- Decay by proton emission /
- Lifetimes and widths
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图 1 球形情况下151Lu质子发射核中的质子-子核相互作用势与Gamow态质子波函数. 实线代表势能的不同部分, 波函数实部(绿色虚线)和虚部(紫色虚线)对应右侧坐标轴, 其能量的实部如黄色水平虚线所示
Figure 1. The daughter-proton interaction potential in 151Lu in the spherical case and the Gamow state wave function of the emitted proton. The solid lines denotes different parts of potential, while the real part (green dashed line) and the imaginary part (purple dashed line) of the wave function are corresponding to the right axis with the real part of the energy as the yellow dotted horizontal line.
图 2 151Lu核中子核Woods-Saxon半径的形变(虚线) 与母核质子放射性衰变宽度的角分布(实线)
Figure 2. The angular distrbution of the deformed radius (dashed line) and the decay width (solid line).
图 3 本文的理论模型在实验衰变能和FRDM衰变能参数下对半衰期的计算误差 (a) 使用拟合得到的谱因子; (b) 使用RMF理论得到的谱因子. 阴影部分代表对数下的计算值与实验值相差在1以内
Figure 3. Errors of the theoretical model in experimental and FRDM decay energies: (a) using the spectroscopic factor by fitting; (b) using the spectroscopic factor of RMF theory. The shaded region indicates that the difference between our calculated value and the experimental value under logarithmic conditions is smaller than 1.
图 4 本文的计算结果和其他理论的结果(包括解析的半经典计算[11]与拓展的Geiger-Nuttall定律[44])与实验值的对比. 横轴代表母核的核子数, 纵轴代表各组半衰期计算结果与实验值的对数的差值
Figure 4. The difference between this work and other theoretical methods (including an analytic semiclassical solution[11] and the new Geiger-Nuttall law[44]). The horizontal axis denotes the nucleon number of the parent nucleus, while the vertical axis illustrates the difference between the logarithms of the calculated and experimental half-lives.
图 5 本文理论预言的质子数30到50的一系列质子放射核的半衰期, 及其与(a)母核质子数和(b)实验衰变能之间的关系, 其中与衰变能的关系符合Geiger-Nuttal定律, 如图中虚线所示
Figure 5. The predicted half-lives for serial proton emitters in the range from $ Z = 30 $ to $ Z = 50 $ in this work, and the dependence on (a) the proton number of parent nuclei and (b) the decay energy, which is consistent to the Geiger-Nuttal law as the dashed lines.
表 1 一些质子放射核的相关实验和理论数据, 其中质子轨道为球形极限下发射质子占据的轨道, $ Q_{\rm{exp}} $和$ Q_{\rm{FRDM}} $分别为实验(除表中的特别标注外, 由NNDC和AME 2020中列出的实验值得到[3,40,41]) 和FRDM理论[39]得到的衰变能, $ \beta_{2, 4} $为形变参数, $ T_{1/2}^{\rm{exp}} $为实验半衰期以10为底的对数, 最后两列为使用RMF和形变依赖的拟合两种方法得到的谱因子
Table 1. Experimental data of various proton emitters, where the orbital is occupied in the spherical limit, $ Q_{\rm{exp}} $ and $ Q_{\rm{FRDM}} $ are the decay energy by experiments (obtained from NNDC and AME 2020 except for those specifically marked values) and FRDM theory, $ \beta_{2, 4} $ are deformation parameters, $ T_{1/2}^{\rm{exp}} $ is the experimental half-life, and the last two columns denote the spectroscopic factors calculated by RMF and deformation-dependent fitting.
母核 $ n\ell_j $ $ Q_{\rm{exp}} $/MeV $ Q_{\rm{FRDM}} $/MeV $ \beta_2 $ $ \beta_4 $ $ \log_{10}T_{1/2}^{\rm{exp}} $ $ S_p^{\rm{RMF}} $ $ S_p^{\rm{fit}} $ 109I $ 2 d_{5/2} $ 0.820 0.821 0.139 0.056 –4.029 0.726 0.099 112Cs $ 2 d_{5/2} $ 0.816 0.901 0.185 0.052 –3.310 0.369 0.063 113Cs $ 2 d_{3/2} $ 0.973 0.681 0.195 0.054 –4.752 0.373 0.057 117La $ 2 d_{3/2} $ 0.820 0.581 0.282 0.106 –1.602 0.311 0.024 121Pr $ 2 d_{3/2} $ 0.890 0.671 0.304 0.087 –2.000 0.122 0.019 130Eu $ 2 d_{3/2} $ 1.028[45] 1.111 0.331 0.018 –3.046 0.816 0.014 131Eu $ 2 d_{3/2} $ 0.947 0.961 0.331 0.018 –1.699 0.029 0.014 135Tb $ 2 f_{7/2} $ 1.188 1.131 0.322 –0.037 –3.027 0.028 0.016 140Ho $ 2 f_{7/2} $ 1.094 0.881 0.276 –0.047 –2.222 0.952 0.025 141Ho $ 2 f_{7/2} $ 1.177 0.761 0.253 –0.039 –2.387 0.008 0.032 141Hom $ 3 s_{1/2} $ 1.243 0.827 0.253 –0.039 –5.137 0.048 0.032 144Tm $ 1 h_{11/2} $ 1.712 1.201 0.254 –0.064 –5.721 0.558[43] 0.031 145Tm $ 1 h_{11/2} $ 1.736 1.141 0.231 –0.068 –5.499 0.580 0.039 146Tm $ 1 h_{11/2} $ 1.196 0.951 0.219 –0.057 –1.125 0.962 0.045 146Tmm $ 1 h_{11/2} $ 1.120 0.833 0.219 –0.057 –0.703 0.962 0.045 147Tm $ 1 h_{11/2} $ 1.059 0.661 –0.187 –0.022 0.587 0.581 0.187 147Tmm $ 2 d_{3/2} $ 1.127 0.729 –0.187 –0.022 –3.444 0.953 0.187 150Lu $ 1 h_{11/2} $ 1.270 1.001 –0.167 –0.035 –1.197 0.497 0.199 151Lu $ 1 h_{11/2} $ 1.241 1.001 –0.167 –0.035 –0.896 0.490 0.199 151Lum $ 2 d_{3/2} $ 1.319 1.079 –0.167 –0.035 –4.796 0.858 0.199 155Ta $ 1 h_{11/2} $ 1.453 1.191 0.021 0.000 –2.538 0.422 0.324 156Ta $ 2 d_{3/2} $ 1.020 0.591 –0.063 0.001 –0.842 0.761 0.274 156Tam $ 1 h_{11/2} $ 1.122 0.693 –0.063 0.001 0.933 0.493 0.274 157Ta $ 3 s_{1/2} $ 0.935 0.771 –0.084 0.014 –0.527 0.797 0.257 159Rem $ 1 h_{11/2} $ 1.720 1.301 0.085 0.003 –4.665 0.387[43] 0.171 160Re $ 2 d_{3/2} $ 1.267 0.821 0.107 0.004 –3.045 0.507 0.137 161Re $ 3 s_{1/2} $ 1.197 0.761 0.128 0.018 –3.357 0.892 0.111 161Rem $ 1 h_{11/2} $ 1.321 0.885 0.128 0.018 –0.678 0.290 0.111 164Irm $ 1 h_{11/2} $ 1.814 1.928 0.118 –0.007 –4.155 0.339[43] 0.123 165Irm $ 1 h_{11/2} $ 1.740 1.311 0.129 0.006 –3.462 0.187 0.110 166Ir $ 2 d_{3/2} $ 1.152 0.711 0.140 –0.005 –0.824 0.415 0.098 166Irm $ 1 h_{11/2} $ 1.324 0.883 0.140 –0.005 –0.076 0.188 0.098 167Ir $ 3 s_{1/2} $ 1.070 0.731 0.151 –0.004 –1.028 0.912 0.088 167Irm $ 1 h_{11/2} $ 1.245 0.906 0.151 –0.004 0.848 0.183 0.088 170Au $ 2 d_{3/2} $ 1.472 0.981 0.129 0.007 –3.493 0.511 0.110 170Aum $ 1 h_{11/2} $ 1.757 1.266 0.129 0.007 –2.973 0.137[43] 0.110 171Au $ 3 s_{1/2} $ 1.448 0.961 0.129 –0.006 –4.770 0.848 0.110 171Aum $ 1 h_{11/2} $ 1.707 1.220 0.129 –0.006 –2.654 0.087 0.110 176Tl $ 3 s_{1/2} $ 1.265 1.031 –0.115 –0.03 –2.284 0.926 0.233 177Tl $ 3 s_{1/2} $ 1.156 0.981 –0.115 –0.03 –1.176 0.733 0.233 177Tlm $ 1 h_{11/2} $ 1.963 1.788 –0.115 –0.03 –3.346 0.022 0.233 185Bi $ 3 s_{1/2} $ 1.598[46] 1.611 0.000 0.012 –4.191 0.011 0.400 
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表 2 本文的理论模型对一些质子放射核的半衰期的计算值, 使用表 1中的两组谱因子和衰变能. 第二、三列半衰期使用实验衰变能[3,40,41], 第四、五列的半衰期使用由FRDM理论[39]得到的衰变能$ Q_{\rm{FRDM}} $计算, 最后两列展示了其他理论计算得到的结果
Table 2. Data calculated by our theoretical model, using different spectroscopic factors and decay energies in Table 1. The second and third columns correspond to the experimental decay energy[3,40,41], while the fourth and fifth columns correspond to the decay energy obtained by FRDM[39] as $ Q_{\rm{FRDM}} $. The last two columns are the results of other theoretical works.
母核 基于实验衰变能 基于FRDM衰变能 其他理论研究结果 $ \log_{10}T_{1/2}^{\rm{RMF}} $ $ \log_{10}T_{1/2}^{\rm{fit}} $ $ \log_{10}T_{1/2}^{\rm{RMF}} $ $ \log_{10}T_{1/2}^{\rm{fit}} $ Ref. [11] Ref. [44] 109I –4.673 –3.809 –4.688 –3.824 –4.098 –3.507 112Cs –3.587 –2.816 –4.823 –4.053 –3.261 –2.844 113Cs –5.746 –4.928 –1.171 –0.353 –5.599 –4.796 117La –2.884 –1.765 2.101 3.219 –2.699 –2.072 121Pr –2.840 –2.032 1.147 1.955 –3.020 –2.552 130Eu –4.218 –2.468 –5.211 –3.461 –3.470 –3.121 131Eu –1.680 –1.379 –1.876 –1.575 –2.354 –2.141 135Tb –3.279 –3.033 –2.652 –2.405 –3.647 –3.380 140Ho –3.123 –1.545 –0.031 1.547 –1.909 –1.902 141Ho –2.013 –2.611 4.341 3.743 –2.878 –2.811 141Hom –5.188 –5.007 0.549 0.729 –5.588 –5.783 144Tm –5.435 –4.185 –1.044 0.206 –4.686 –5.216 145Tm –5.603 –4.436 –0.355 0.812 –4.835 –5.401 146Tm –1.214 0.120 2.058 3.393 –0.204 –1.272 146Tmm –0.313 1.022 4.129 5.463 0.702 –0.999 147Tm 0.725 1.218 8.322 8.814 0.408 0.681 147Tmm –3.591 –2.884 3.079 3.786 –3.728 –2.455 150Lu –1.148 –0.750 2.250 2.647 –1.530 –1.199 151Lu –0.844 –0.452 2.248 2.639 –1.211 –0.911 151Lum –5.053 –4.418 –2.304 –1.669 –5.213 –3.899 155Ta –2.318 –2.204 1.191 0.509 –2.764 –2.397 156Ta –0.986 –0.542 8.836 9.280 –0.901 –0.180 156Tam 1.177 1.433 9.178 9.434 0.839 1.101 157Ta –0.299 0.193 2.923 3.416 –0.505 –0.797 159Rem –3.966 –3.610 –0.290 0.066 –4.399 –4.586 160Re –3.136 –2.568 3.681 4.250 –3.438 –2.450 161Re –3.439 –2.533 3.901 4.807 –3.575 –3.277 161Rem –0.415 0.003 5.771 6.189 –0.910 –0.729 164Irm –4.152 –3.711 –4.890 –4.448 –4.577 –4.247 165Irm –3.392 –3.161 0.413 0.645 –4.051 –3.550 166Ir –1.080 –0.454 7.189 7.814 –1.386 –0.801 166Irm 0.255 0.537 6.672 6.953 –0.370 –0.344 167Ir –1.142 –0.126 5.450 6.465 –1.186 –1.347 167Irm 1.142 1.460 6.213 6.530 0.532 0.546 170Au –4.130 –3.462 2.053 2.720 –4.331 –3.254 170Aum –2.954 –2.857 1.592 1.688 –3.693 –3.330 171Au –4.954 –4.066 1.332 2.220 –5.028 –4.460 171Aum –2.390 –2.491 2.344 2.243 –3.313 –2.992 176Tl –2.470 –1.871 0.799 1.397 –2.451 –2.361 177Tl –0.978 –0.481 1.742 2.239 –1.035 –1.263 177Tlm –3.149 –4.174 –1.962 –2.988 –4.611 –3.543 185Bi –3.383 –4.944 –3.495 –5.055 –5.214 –4.730
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表 3 对fpg壳层中可能的质子放射核的半衰期的理论预言. 其中衰变能Q根据NNDC数据库和AME 2020质量表中的单质子分离能实验值计算得到[3,40,41] (#上标表示该数值为AME 2020中的评价值), 形变参数$ \beta_{2, 4} $和谱因子$ S_p^{\rm{RHB}} $则分别由FRDM[39]和RHB[48]的计算结果得到. 最后一列为本文预言的质子放射性半衰期(以秒为单位) 的对数
Table 3. The theoretical prediction on the half-lives of fpg-shell possible proton emitters. The decay energy Q is taken from the proton separation energy in NNDC and AME 2020[3,40,41] (the superscript # denotes values that are not obtained from purely experimental data in AME 2020), and the deformation parameters $ \beta_{2, 4} $ and spectroscopic factor $ S_p^{\rm{RHB}} $ are obtained by the results of FRDM[39] and RHB[48], respectively. The predicted half-lives using our model are listed in the last column.
母核 质子轨道 Q / MeV $ \beta_2 $ $ \beta_4 $ $ S_p^{\rm{RHB}} $ $ \log_{10}T_{1/2}^{\rm{pred}} $ 60Ga $ 2 p_{3/2} $ 0.340# 0.106 0.041 0.53 –5.140 63As $ 2 p_{3/2} $ 1.350# 0.184 0.014 0.61 –15.627 68Br $ 1 f_{5/2} $ 0.500# 0.220 –0.082 0.80 –5.071 69Br $ 1 f_{5/2} $ 0.640 0.233 –0.106 0.78 –7.489 72Rb $ 1 f_{5/2} $ 0.710# –0.357 0.022 0.83 –7.713 73Rb $ 1 f_{5/2} $ 0.640 –0.366 0.025 0.83 –6.751 75Y $ 1 g_{9/2} $ 1.720# 0.401 0.001 0.92 –12.882 76Y $ 1 g_{9/2} $ 1.080# 0.402 –0.012 0.84 –9.505 81Nb $ 1 g_{9/2} $ 1.110# 0.430 –0.047 0.12 –8.391 84Tc $ 1 f_{5/2} $ 1.350# 0.492 –0.021 0.90 –11.544 88Rh $ 1 g_{9/2} $ 1.580# –0.243 –0.101 0.73 –10.976 92Ag $ 1 g_{9/2} $ 1.350# –0.011 0.000 0.56 –9.074 93Ag $ 1 g_{9/2} $ 1.090# 0.000 0.000 0.39 –6.967 96In $ 1 g_{9/2} $ 1.680# 0.053 0.001 0.49 –10.374 97In $ 1 g_{9/2} $ 0.890# –0.021 0.000 0.19 –3.965
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