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核素β–衰变释放的β粒子与γ射线平均能量是计算反应堆衰变热的核心参数, 对核设施安全与工程应用至关重要. 然而, 许多核素的实验数据匮乏, 现有理论模型精度难以满足需求. 本文基于ENSDF数据库中543个实验数据准确的β–衰变核素(选自1136个β–衰变核素), 采用神经网络方法对核素衰变发射的β粒子、γ射线及中微子的平均能量进行预测, 对比了三种特征组(分别含特殊特征值$T_{1/2}$, $({1}/{T_{1/2}})^{1/5}$, $Q/3$)的模型性能. 结果表明: 相比特征组含$T_{1/2}$以及$({1}/{T_{1/2}} )^{1/5}$的模型, 特征组含$Q/3$的模型综合表现最佳, 其β粒子与中微子预测误差分别为28.11%/56.9%和35.33%/37.76%, 并且利用该特征组训练的机器学习模型成功补充了裂变产物区(质量数66—172) 291个核素的缺失数据. 核素图对比显示, 神经网络对规律性较强的β粒子及中微子能量预测与实验符合较好, 但对γ射线(训练误差76.9%)以及奇奇核、幻数附近核素的预测偏差显著. 本文证实经验特征值$Q/3$可有效提升模型性能, 同时揭示了数据规律性与模型泛化能力的关联, 为后续融合物理机理优化机器学习模型提供了依据.
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关键词:
- 衰变热 /
- β粒子与γ射线平均能量 /
- 机器学习 /
- 神经网络
The average β energy data and average γ energy data of the β–-decay nuclei play an important role in many fields of nuclear technology and scientific research, such as the decay heat and antineutrino spectrum calculation for different kinds of reactors. However, the reliable experimental measurements of the average energies for many nuclei are lacking, and the theoretical calculation needs to be improved to meet the requirements for accuracy in the technical applications. In this study, the average β, γ and neutrino energies of the β–-decay nuclei are investigated by the neural network method based on the newly evaluated experimental data of 543 nuclei that are selected from a total of 1136 β–-decay nuclei. In the neural network approach, three different feature sets are used for model training. Each feature set contains a feature characteristic value (one of the $T_{1/2}$, $\left( {1}/{T_{1/2}} \right)^{1/5}$, and$Q/3$), along with five identical feature values (Z, N, parity of Z, parity of N, and $\Delta Z$). The three feature values are selected based on the physical mechanism below. 1) The average energy is obviously related to Q value and approximately taken as $Q/3$ in the reactor industry. Therefore, the $Q/3$ is chosen as one feature value. 2) The half-live is related to the Q value of β–-decay, and $T_{1/2}$ is considered. 3) According to the Sargent’s law, $\left( {1}/{T_{1/2}} \right)^{1/5} \propto Q$, a more accurate $\left( {1}/{T_{1/2}} \right)^{1/5}$ value is selected. As a result, for the feature set of $T_{1/2}$, the training results for all three types of average energies are unsatisfactory. For the other groups, the relative errors of the average β energy data, are 19.32% and 28.11% for $\left( {1}/{T_{1/2}} \right)^{1/5}$ and $Q/3$ feature groups in the training set, and 82% and 56.9% in the validation set; the relative errors of the average γ energy are 28.9% and 76.9% for $\left( {1}/{T_{1/2}} \right)^{1/5}$ and $Q/3$ feature sets, respectively, and they are both >100% in the validation set; for the average neutrino energy, the relative errors in the training set are 27.82% and 35.33% for $\left( {1}/{T_{1/2}} \right)^{1/5}$ and $Q/3$ feature group, and 76.32% and 37.76% in the validation set, respectively. Considering the accuracy comparison of the three groups, the $Q/3$ feature set is chosen to predict the average energy data of nuclei in the fission product region (mass numbers range from 66 to 172), which lacks reliable experimental data. As a result, the average energy data with predicted values for 291 nuclei are supplemented. Besides, a comparison is made between the calculated data and the evaluated experimental data through the nuclide chart. It is found that the neural network accurately predicts the experimental data for the average β and neutrino energies which exhibit relatively strong regularity. However, it shows significant deviations in predictions for average gamma energy (relative error in the training set is 76.9%). Large deviation also emerges in the odd-odd nuclei and nuclei near magic numbers. This study confirms that integrating empirical relationships and physical principles can effectively improve the performance of the neural network, and simultaneously reveals the relationship between data regularity and model generalization capability. These findings provide a basis for using physical mechanisms to optimize machine learning models in the future. 
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Keywords:
- decay heat /
- average β and γ energies /
- machine learning /
- neural networks
[1] Algora A, Tain J, Rubio B, Fallot M, Gelletly W 2021 Eur. Phys. J. A 57 85
Google Scholar
[2] Nichols A 2015 J. Nucl. Sci. Technol. 52 17
Google Scholar
[3] Rana M A 2008 Nucl. Sci. Tech. 19 117
Google Scholar
[4] Song Q F, Zhu L, Guo H, Su J 2023 Nucl. Sci. Tech. 34 32
Google Scholar
[5] Xiao K, Li P C, Wang Y J, Liu F H, Li Q F 2023 Nucl. Sci. Tech. 34 62
Google Scholar
[6] Greenwood R, Helmer R, Putnam M, Watts K 1997 Nucl. Instrum. Methods Phys. Res. A 390 95
Google Scholar
[7] Valencia E, Tain J, Algora A, et al. 2017 Phys. Rev. C 95 024320
Google Scholar
[8] Tengblad O, Aleklett K, Von Dincklage R, Lund E, Nyman G, Rudstam G 1989 Nucl. Phys. A 503 136
Google Scholar
[9] Nichols A, Dimitriou P, Algora A, et al. 2023 Eur. Phys. J. A 59 78
Google Scholar
[10] Yoshida T, Nakasima R 1981 J. Nucl. Sci. Technol. 18 393
Google Scholar
[11] Fang J, Zhang X, Shi M, Niu Z 2025 Eur. Phys. J. A 61 1
Google Scholar
[12] Fang J, Chen J, Niu Z 2022 Phys. Rev. C 106 054318
Google Scholar
[13] Azevedo M, Ferreira R, Dimarco A, Barbero C A, Samana A R, Possidonio D 2020 Brazil. J. Phys. 50 57
Google Scholar
[14] Nakata H, Tachibana T, Yamada M 1995 Nucl. Phys. A 594 27
Google Scholar
[15] Nakata H, Tachibana T, Yamada M 1997 Nucl. Phys. A 625 521
Google Scholar
[16] Tachibana T, Yamada M, Yoshida Y 1990 Prog. Theor. Phys. 84 641
Google Scholar
[17] Borzov I 2006 Nucl. Phys. A 777 645
Google Scholar
[18] Kumar V, Srivastava P C 2023 Eur. Phys. J. A 59 237
Google Scholar
[19] Ni D, Ren Z 2012 J. Phys. G 39 125105
Google Scholar
[20] Nabi J U, Ishfaq M 2020 New Astron. 78 101356
Google Scholar
[21] Engel J, Bender M, Dobaczewski J, Nazarewicz W, Surman R 1999 Phys. Rev. C 60 014302
Google Scholar
[22] Minato F, Bai C 2013 Phys. Rev. Lett. 110 122501
Google Scholar
[23] Moller P, Nix J, Myers W, Swiatecki W 1995 At. Data Nucl. Data Tables 59 185
Google Scholar
[24] Möller P, Randrup J 1990 Nucl. Phys. A 514 1
Google Scholar
[25] Audi G, Bersillon O, Blachot J, Wapstra A 2003 Nucl. Phys. A 729 3
Google Scholar
[26] Moller P, Nix J, Kratz K 1997 At. Data Nucl. Data Tables 66 131
Google Scholar
[27] Möller P, Pfeiffer B, Kratz K L 2003 Phys. Rev. C 67 055802
Google Scholar
[28] Soloviev V G 2020 Theory of Atomic Nuclei, Quasi-particle and Phonons. (Boca Raton, FL: CRC Press) p226
[29] Kuz-Min V, Soloviev V 1988 Nucl. Phys. A 486 118
Google Scholar
[30] Soloviev V, Sushkov A 1989 Phys. Lett. B 216 259
Google Scholar
[31] Luo D, Clark B K 2019 Phys. Rev. Lett. 122 226401
Google Scholar
[32] Yoshida S 2020 Phys. Rev. C 102 024305
Google Scholar
[33] Chen Y, Wu X 2024 Int. J. Mod. Phys. E 33 2450012
Google Scholar
[34] Hizawa N, Hagino K, Yoshida K 2023 Phys. Rev. C 108 034311
Google Scholar
[35] Wu X, Ren Z, Zhao P 2022 Phys. Rev. C 105 L031303
Google Scholar
[36] Yang Z X, Fan X H, Li Z P, Liang H 2023 Phys. Lett. B 840 137870
Google Scholar
[37] Utama R, Chen W C, Piekarewicz J 2016 J. Phys. G 43 114002
Google Scholar
[38] Wu D, Bai C, Sagawa H, Zhang H 2020 Phys. Rev. C 102 054323
Google Scholar
[39] Xian Z Y, Ya Y, An R 2025 Phys. Lett. B 868 139662
Google Scholar
[40] Jiao B B 2024 Int. J. Mod. Phys. E 33 2450019
Google Scholar
[41] Zhang X, He H, Qu G, Liu X, Zheng H, Lin W, Han J, Ren P, Wada R 2024 Phys. Rev. C 110 014316
Google Scholar
[42] Zhang X, Liu X, Zheng H, et al. 2025 IEEE Trans. Nucl. Sci. 72 795
Google Scholar
[43] Rodríguez U B, Vargas C Z, Gonçalves M, Duarte S B, Guzmán F 2019 J. Phys. G 46 115109
Google Scholar
[44] Freitas P S, Clark J W 2019 arXiv: 1910.12345 [nucl-th]
[45] Ma N N, Bao X J, Zhang H F 2021 Chin. Phys. C 45 024105
Google Scholar
[46] Yuan Z, Bai D, Ren Z, Wang Z 2022 Chin. Phys. C 46 024101
Google Scholar
[47] Li W, Zhang X, Niu Y, Niu Z 2023 J. Phys. G 51 015103
Google Scholar
[48] Luo J, Xu Y, Li X, Wang J, Zhang Y, Deng J, Zhang F, Ma N 2025 Phys. Rev. C 111 034330
Google Scholar
[49] Akkoyun S 2020 Nucl. Instrum. Methods Phys. Res. B 462 51
Google Scholar
[50] Ma C W, Peng D, Wei H L, Niu Z M, Wang Y T, Wada R 2020 Chin. Phys. C 44 014104
Google Scholar
[51] Sun Q K, Zhang Y, Hao Z R, et al. 2025 Nucl. Sci. Tech. 36 52
Google Scholar
[52] Li W, Liu L, Niu Z, Niu Y, Huang X 2024 Phys. Rev. C 109 044616
Google Scholar
[53] Özdoğan H, Üncü Y A, Şekerci M, Kaplan A 2024 Appl. Radiat. Isot. 204 111115
Google Scholar
[54] Lay D, Flynn E, Giuliani S A, Nazarewicz W, Neufcourt L 2024 Phys. Rev. C 109 044305
Google Scholar
[55] 郭粤颖, 唐湘琪, 刘辉鑫, 吴鑫辉 2025 核技术 48 050003
Google Scholar
Guo Y Y, Tang X Q, Liu H X, Wu X H 2025 Nucl. Tech. 48 050003
Google Scholar
[56] Utama R, Piekarewicz J 2017 Phys. Rev. C 96 044308
Google Scholar
[57] Neufcourt L, Cao Y, Nazarewicz W, Viens F 2018 Phys. Rev. C 98 034318
Google Scholar
[58] Utama R, Piekarewicz J 2018 Phys. Rev. C 97 014306
Google Scholar
[59] Neufcourt L, Cao Y, Nazarewicz W, Olsen E, Viens F 2019 Phys. Rev. Lett. 122 062502
Google Scholar
[60] Neufcourt L, Cao Y, Giuliani S, Nazarewicz W, Olsen E, Tarasov O B 2020 Phys. Rev. C 101 014319
Google Scholar
[61] Lu Y, Shang T, Du P, Li J, Liang H, Niu Z 2025 Phys. Rev. C 111 014325
Google Scholar
[62] Zeng L X, Yin Y Y, Dong X X, Geng L S 2024 Phys. Rev. C 109 034318
Google Scholar
[63] Yiu T C, Liang H, Lee J 2024 Chin. Phys. C 48 024102
Google Scholar
[64] Yüksel E, Soydaner D, Bahtiyar H 2024 Phys. Rev. C 109 064322
Google Scholar
[65] 谭凯中, 高琬晴, 刘健 2025 核技术 48 050010
Google Scholar
Tan K Z, Gao W Q, Liu J 2025 Nucl. Tech. 48 050010
Google Scholar
[66] ENSDF: Evaluated Nuclear Structure Data File, National Nuclear Data Center https://www.nndc.bnl.gov/ensdf/ [2025-07-07]
[67] Wang M, Huang W J, Kondev F G, Audi G, Naimi S 2021 Chin. Phys. C 45 030003
Google Scholar
[68] Sargent B 1933 Proc. R. Soc. London, Ser. A 139 659
Google Scholar
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图 1 实验(543)与神经网络(834)所得β粒子平均能量系统变化的核素图
Fig. 1. Systematic variation of average β energy values from experiment (543) and neural network (834) in the nuclear chart.
图 2 实验(543)与神经网络(834)所得γ射线平均能量系统变化的核素图
Fig. 2. Systematic variation of average γ energy values from experiment (543) and neural network (834) in the nuclear chart.
图 3 实验(543)与神经网络(834)所得中微子平均能量系统变化的核素图
Fig. 3. Systematic variation of average neutrino energy values from experiment (543) and neural network (834) in the nuclear chart.
表 1 神经网络的超参数集
Table 1. Hyper parameters of neural network
层数 名称 神经元数目 激活函数 0 输入层 6 1 隐藏层 60 Tanh 2 隐藏层 60 Tanh 3 隐藏层 60 Tanh 4 隐藏层 60 Tanh 5 隐藏层 60 Tanh 6 输出层 1 
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表 2 其他超参数
Table 2. Other hyper parameters
其他超参数 值和特征 批大小 16 损失函数 均方差 优化器 RMSprop 学习速率 $ 1\times10^{-5} $
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表 3 平均能量计算结果
Table 3. Calculation results of average energies
平均能量类型 误差类型 特征组1 特征组2 特征组3 β粒子平均能 训练误差 $> 100{\text{%}} $ 19.32% 28.11% 验证误差 $> 100{\text{%}} $ 82% 56.9% γ射线平均能 训练误差 $> 100{\text{%}} $ 28.9% 76.9% 验证误差 $> 100{\text{%}} $ $> 100{\text{%}} $ $> 100{\text{%}} $ 中微子平均能 训练误差 $> 100{\text{%}} $ 27.82% 35.33% 验证误差 $> 100{\text{%}} $ 76.32% 37.76%
下载: 导出CSV
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[1] Algora A, Tain J, Rubio B, Fallot M, Gelletly W 2021 Eur. Phys. J. A 57 85
Google Scholar
[2] Nichols A 2015 J. Nucl. Sci. Technol. 52 17
Google Scholar
[3] Rana M A 2008 Nucl. Sci. Tech. 19 117
Google Scholar
[4] Song Q F, Zhu L, Guo H, Su J 2023 Nucl. Sci. Tech. 34 32
Google Scholar
[5] Xiao K, Li P C, Wang Y J, Liu F H, Li Q F 2023 Nucl. Sci. Tech. 34 62
Google Scholar
[6] Greenwood R, Helmer R, Putnam M, Watts K 1997 Nucl. Instrum. Methods Phys. Res. A 390 95
Google Scholar
[7] Valencia E, Tain J, Algora A, et al. 2017 Phys. Rev. C 95 024320
Google Scholar
[8] Tengblad O, Aleklett K, Von Dincklage R, Lund E, Nyman G, Rudstam G 1989 Nucl. Phys. A 503 136
Google Scholar
[9] Nichols A, Dimitriou P, Algora A, et al. 2023 Eur. Phys. J. A 59 78
Google Scholar
[10] Yoshida T, Nakasima R 1981 J. Nucl. Sci. Technol. 18 393
Google Scholar
[11] Fang J, Zhang X, Shi M, Niu Z 2025 Eur. Phys. J. A 61 1
Google Scholar
[12] Fang J, Chen J, Niu Z 2022 Phys. Rev. C 106 054318
Google Scholar
[13] Azevedo M, Ferreira R, Dimarco A, Barbero C A, Samana A R, Possidonio D 2020 Brazil. J. Phys. 50 57
Google Scholar
[14] Nakata H, Tachibana T, Yamada M 1995 Nucl. Phys. A 594 27
Google Scholar
[15] Nakata H, Tachibana T, Yamada M 1997 Nucl. Phys. A 625 521
Google Scholar
[16] Tachibana T, Yamada M, Yoshida Y 1990 Prog. Theor. Phys. 84 641
Google Scholar
[17] Borzov I 2006 Nucl. Phys. A 777 645
Google Scholar
[18] Kumar V, Srivastava P C 2023 Eur. Phys. J. A 59 237
Google Scholar
[19] Ni D, Ren Z 2012 J. Phys. G 39 125105
Google Scholar
[20] Nabi J U, Ishfaq M 2020 New Astron. 78 101356
Google Scholar
[21] Engel J, Bender M, Dobaczewski J, Nazarewicz W, Surman R 1999 Phys. Rev. C 60 014302
Google Scholar
[22] Minato F, Bai C 2013 Phys. Rev. Lett. 110 122501
Google Scholar
[23] Moller P, Nix J, Myers W, Swiatecki W 1995 At. Data Nucl. Data Tables 59 185
Google Scholar
[24] Möller P, Randrup J 1990 Nucl. Phys. A 514 1
Google Scholar
[25] Audi G, Bersillon O, Blachot J, Wapstra A 2003 Nucl. Phys. A 729 3
Google Scholar
[26] Moller P, Nix J, Kratz K 1997 At. Data Nucl. Data Tables 66 131
Google Scholar
[27] Möller P, Pfeiffer B, Kratz K L 2003 Phys. Rev. C 67 055802
Google Scholar
[28] Soloviev V G 2020 Theory of Atomic Nuclei, Quasi-particle and Phonons. (Boca Raton, FL: CRC Press) p226
[29] Kuz-Min V, Soloviev V 1988 Nucl. Phys. A 486 118
Google Scholar
[30] Soloviev V, Sushkov A 1989 Phys. Lett. B 216 259
Google Scholar
[31] Luo D, Clark B K 2019 Phys. Rev. Lett. 122 226401
Google Scholar
[32] Yoshida S 2020 Phys. Rev. C 102 024305
Google Scholar
[33] Chen Y, Wu X 2024 Int. J. Mod. Phys. E 33 2450012
Google Scholar
[34] Hizawa N, Hagino K, Yoshida K 2023 Phys. Rev. C 108 034311
Google Scholar
[35] Wu X, Ren Z, Zhao P 2022 Phys. Rev. C 105 L031303
Google Scholar
[36] Yang Z X, Fan X H, Li Z P, Liang H 2023 Phys. Lett. B 840 137870
Google Scholar
[37] Utama R, Chen W C, Piekarewicz J 2016 J. Phys. G 43 114002
Google Scholar
[38] Wu D, Bai C, Sagawa H, Zhang H 2020 Phys. Rev. C 102 054323
Google Scholar
[39] Xian Z Y, Ya Y, An R 2025 Phys. Lett. B 868 139662
Google Scholar
[40] Jiao B B 2024 Int. J. Mod. Phys. E 33 2450019
Google Scholar
[41] Zhang X, He H, Qu G, Liu X, Zheng H, Lin W, Han J, Ren P, Wada R 2024 Phys. Rev. C 110 014316
Google Scholar
[42] Zhang X, Liu X, Zheng H, et al. 2025 IEEE Trans. Nucl. Sci. 72 795
Google Scholar
[43] Rodríguez U B, Vargas C Z, Gonçalves M, Duarte S B, Guzmán F 2019 J. Phys. G 46 115109
Google Scholar
[44] Freitas P S, Clark J W 2019 arXiv: 1910.12345 [nucl-th]
[45] Ma N N, Bao X J, Zhang H F 2021 Chin. Phys. C 45 024105
Google Scholar
[46] Yuan Z, Bai D, Ren Z, Wang Z 2022 Chin. Phys. C 46 024101
Google Scholar
[47] Li W, Zhang X, Niu Y, Niu Z 2023 J. Phys. G 51 015103
Google Scholar
[48] Luo J, Xu Y, Li X, Wang J, Zhang Y, Deng J, Zhang F, Ma N 2025 Phys. Rev. C 111 034330
Google Scholar
[49] Akkoyun S 2020 Nucl. Instrum. Methods Phys. Res. B 462 51
Google Scholar
[50] Ma C W, Peng D, Wei H L, Niu Z M, Wang Y T, Wada R 2020 Chin. Phys. C 44 014104
Google Scholar
[51] Sun Q K, Zhang Y, Hao Z R, et al. 2025 Nucl. Sci. Tech. 36 52
Google Scholar
[52] Li W, Liu L, Niu Z, Niu Y, Huang X 2024 Phys. Rev. C 109 044616
Google Scholar
[53] Özdoğan H, Üncü Y A, Şekerci M, Kaplan A 2024 Appl. Radiat. Isot. 204 111115
Google Scholar
[54] Lay D, Flynn E, Giuliani S A, Nazarewicz W, Neufcourt L 2024 Phys. Rev. C 109 044305
Google Scholar
[55] 郭粤颖, 唐湘琪, 刘辉鑫, 吴鑫辉 2025 核技术 48 050003
Google Scholar
Guo Y Y, Tang X Q, Liu H X, Wu X H 2025 Nucl. Tech. 48 050003
Google Scholar
[56] Utama R, Piekarewicz J 2017 Phys. Rev. C 96 044308
Google Scholar
[57] Neufcourt L, Cao Y, Nazarewicz W, Viens F 2018 Phys. Rev. C 98 034318
Google Scholar
[58] Utama R, Piekarewicz J 2018 Phys. Rev. C 97 014306
Google Scholar
[59] Neufcourt L, Cao Y, Nazarewicz W, Olsen E, Viens F 2019 Phys. Rev. Lett. 122 062502
Google Scholar
[60] Neufcourt L, Cao Y, Giuliani S, Nazarewicz W, Olsen E, Tarasov O B 2020 Phys. Rev. C 101 014319
Google Scholar
[61] Lu Y, Shang T, Du P, Li J, Liang H, Niu Z 2025 Phys. Rev. C 111 014325
Google Scholar
[62] Zeng L X, Yin Y Y, Dong X X, Geng L S 2024 Phys. Rev. C 109 034318
Google Scholar
[63] Yiu T C, Liang H, Lee J 2024 Chin. Phys. C 48 024102
Google Scholar
[64] Yüksel E, Soydaner D, Bahtiyar H 2024 Phys. Rev. C 109 064322
Google Scholar
[65] 谭凯中, 高琬晴, 刘健 2025 核技术 48 050010
Google Scholar
Tan K Z, Gao W Q, Liu J 2025 Nucl. Tech. 48 050010
Google Scholar
[66] ENSDF: Evaluated Nuclear Structure Data File, National Nuclear Data Center https://www.nndc.bnl.gov/ensdf/ [2025-07-07]
[67] Wang M, Huang W J, Kondev F G, Audi G, Naimi S 2021 Chin. Phys. C 45 030003
Google Scholar
[68] Sargent B 1933 Proc. R. Soc. London, Ser. A 139 659
Google Scholar
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