Research on stability and decay properties of superheavy nuclei based on neural network method

CHEN Haijun; SHENG Haowen; HUANG Wenhao; WU Binqi; ZHAO Tianliang; BAO Xiaojun

doi:10.7498/aps.74.20250720

Abstract
This study aims to develop a highly accurate method of predicting α-decay energy (Q_α) of superheavy nuclei (SHN) and to identify the region of enhanced stability (the “island of stability”) based on α-decay properties. Improving the accuracy of Q_α calculations is crucial for reliably predicting α-decay half-lives, which are essential for identifying newly synthesized superheavy elements.A modified liquid-drop model (LDM) formula for calculating Q_α is proposed, eliminating explicit dependence on magic numbers to improve universality. However, the initial LDM formula alone yields a high root-mean-square deviation (RMSD) of 663.5 keV compared with experimental Q_α values from the AME2016 database for 369 nuclei with Z ≥ 82. In order to significantly improve accuracy, a neural network (NN) method is combined with the LDM formula. For a feedforward backpropagation (BP) neural network with a 2-21-1 architecture (2 input neurons: proton number Z and mass number A; 21 hidden neurons; 1 output neuron), the correction term $ \text{δ}{\text{Q}}_{\text{α}} $ is developed. The network is trained using the Levenberg-Marquardt algorithm on a dataset of 369 nuclei (319 training, 50 validation). The final Q_α prediction is given by $ Q_\alpha ^{{\text{NN}}} = Q_\alpha ^{{\text{Eq}}{\text{. (2)}}} + \delta Q_\alpha ^{} $. The unified decay law (UDL) formula is then used to calculate α-decay half-lives (T_1/2), with and without NN correction (denoted as UDL and UDL^NN). The main results obtained are listed below.1) Improved Q_α accuracy: The NN correction dramatically reduces the RMSD between calculated and experimental Q_α values from 663.5 keV (LDM alone) to 89.2 keV.2) Capturing shell effects: Remarkably, although there is no explicit input of nuclear shell information, the NN-corrected Q_α predictions clearly reproduce known shell structures, including the expected shell closure near N = 184 for superheavy nuclei. This is evident in the systematic lowering of predicted Q_α values (implying increased stability) around the predicted doubly magic nucleus ²⁹⁸Fl (Z = 114, N = 184) and other known shell closures (e.g., N = 152, N = 162).3) Half-life predictions: Using the NN-corrected Q_α in the UDL formula (UDL^NN) further refines T_1/2 predictions, reducing the RMSD from 0.631 (UDL alone) to 0.423. The method reliably reproduces experimental half-lives and shell-related features, such as a significant increase in T_1/2 near shell closure (e.g. N = 126) and odd-A/odd-odd nuclei due to blocking effects.4) Validation: Predictions for recently synthesized neutron-deficient uranium isotopes ^214,216,218U agree well with new experimental data of Q_α and T_1/2. Predictions for Fl isotopic chains also show good agreement with experimental trends.5) Stability island prediction: Maps of predicted Q_α and T_1/2 in the superheavy region consistently identify minimum value (indicating maximum stability) near the theoretically predicted doube magic nucleus ²⁹⁸Fl. A potential secondary stability center near Z = 126 and N = 228 is suggested, but further verification is needed. The longest predicted region of T_1/2 coincides with the N = 184 shell closure.The conclusions drawn from the above findings are as follows. Integrating a neural network with a modified liquid-drop model formula provides a powerful and accurate method for predicting α-decay energies (Q_α) of heavy and superheavy nuclei. The NN successfully learns and corrects complex shell effects implicitly, significantly improving prediction accuracy (RMSD reduced by ~85%). By combining the UDL formula, this method yields reliable α-decay half-lives. The results strongly confirm the existence and location of the predicted "island of stability" centered on the double magic nucleus ²⁹⁸Fl, providing valuable theoretical guidance for future experiments on the synthesis and identification of superheavy elements.

Keywords:
superheavy nuclei /

neural network /

nuclear masses /

α decay
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图 1 由两个输入神经元、一个隐藏层和一个输出神经元组成的前馈神经网络

Figure 1. Feedforward neural network consisting of two input neurons, a hidden layer and an output neuron.

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图 2 α衰变能量的实验值与理论值之差

Figure 2. Difference between experimental and theoretical values of α decay energy.

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图 3 Po, Th, Rf同位素α衰变能和半衰期的理论结果与实验值的比较

Figure 3. Comparison of theoretical results and experimental values for Po, Th, Rf isotope $ \mathrm{\alpha } $ decay energy and half-life.

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图 4 Fl同位素$ \alpha $衰变能和半衰期的理论结果与实验值比较

Figure 4. Comparison of theoretical results and experimental values for Fl isotope $ \alpha $ecay energy and half-life.

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图 5 利用(2)式(左)和神经网络模型修正(右)预测超重核$ \alpha $衰变能量

Figure 5. Prediction of $ \alpha $ decay energy of superheavy nuclei using Eq. (2)(left) and neural network model correction (right).

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图 6 利用UDL公式和UDL^NN计算$ \alpha $衰变半衰期

Figure 6. Calculated $ \alpha $ decay half-life using the UDL formula and the UDL^NN.

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表 1 其他文献$ \alpha $衰变能和半衰期与本文工作的计算结果进行了比较

Table 1. The $ \alpha $ decay energy and half-life of other reference are compared with the calculation results of the work in this paper.

$ {Q}_{\alpha }/\mathrm{M}\mathrm{e}\mathrm{V} $
同位素	文献[49]		(2)式	NN	文献[51]		文献[52]
²¹⁴U	8.696		10.054	8.865	9.008		8.45
²¹⁶U	8.532		10.047	8.587	8.532		8.36
²¹⁸U	8.773		9.897	8.795	8.801		8.51
$ {T}_{1/2}/\mathrm{m}\mathrm{s} $
同位素		文献[45]		UDL		UDL^NN
²¹⁴U		0.52+0.63 –0.21		0.77		0.40
²¹⁶U		2.25+0.95 –0.4		2.12		1.93
²¹⁸U		0.65+0.08 –0.07		0.34		0.40

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表 2 母核质子数为Z = 120的同位素$ \alpha $衰变链的预测. 第1列为母核, 第2列为文献[49]计算的$ \alpha $衰变能量, 第3列为(2)式计算的$ \alpha $衰变能量, 第4列为神经网络学习后的衰变能量, 第5列为使用UDL公式计算的$ \alpha $衰变半衰期, 第6列为使用神经网络学习后的$ \alpha $衰变半衰期

Table 2. Prediction of the $ \alpha $ decay chains of isotopes with proton number Z = 120 in the parent nucleus. The first column in the table shows the parent nucleus, the second column shows the $ \alpha $ decay energy calculated in Ref. [49], the third column shows the decay energy calculated in Eq. (2), the fourth column shows the decay energy after neural network learning, the fifth column shows the $ \alpha $ decay half-life calculated using the UDL formula, and the sixth column shows the $ \alpha $ decay half-life after learning using the neural network.

核素	$ {Q}_{\alpha }^{\mathrm{L}\mathrm{Z}\mathrm{U}} $	$ {Q}_{\alpha }^{\mathrm{E}\mathrm{q}. \left(2\right)} $	$ {Q}_{\alpha }^{\mathrm{N}\mathrm{N}} $	UDL	$ {\mathrm{U}\mathrm{D}\mathrm{L}}^{\mathrm{N}\mathrm{N}} $
³⁰⁰120	13.05	9.83	13.03	–5.45	–5.59
²⁹⁶118	11.89	9.85	11.75	–3.21	–3.35
²⁹²116	10.64	10.34	10.08	–1.44	–1.57
²⁸⁸114	9.82	10.92	10.03	0.03	–0.09
²⁸⁴112	10.18	11.14	9.61	0.59	0.47
²⁸⁰110	10.04	10.74	10.63	–2.88	–3.19
³⁰²120	12.81	9.63	13.31	–6.05	–6.43
²⁹⁸118	11.86	9.34	11.85	–3.48	–3.86
²⁹⁴116	10.59	9.62	10.66	–1.12	–1.49
²⁹⁰114	9.56	10.20	9.69	0.99	0.63
²⁸⁶112	9.23	10.64	9.01	2.45	2.10
²⁸²110	9.52	10.56	10.34	–2.16	–2.74
³⁰⁴120	12.66	9.57	13.64	–6.73	–7.25
³⁰⁰118	11.75	8.97	12.04	–3.94	–4.46
²⁹⁶116	10.56	8.98	10.62	–1.03	–1.54
²⁹²114	9.18	9.45	9.39	1.91	1.43
²⁸⁸112	8.95	10.01	8.43	4.43	3.99
²⁸⁴110	8.48	10.21	9.94	–1.09	–1.84
³⁰⁶120	13.29	9.61	13.99	–7.41	–8.01
³⁰²118	11.65	8.76	12.29	–4.54	–5.13
²⁹⁸116	10.63	8.45	10.66	–1.17	–1.74
²⁹⁴114	8.85	8.72	9.14	2.67	2.16
²⁹⁰112	8.49	9.28	7.91	6.41	6.00
²⁸⁶110	8.12	9.71	9.43	0.35	–0.51
³⁰⁸120	13.07	9.73	14.32	–8.05	–8.68
³⁰⁴118	12.36	8.69	12.59	–5.20	–5.82
³⁰⁰116	10.57	8.08	10.77	–1.49	–2.06
²⁹⁶114	8.53	8.07	8.97	3.21	2.77
²⁹²112	8.30	8.52	7.49	8.17	7.87
²⁸⁸110	7.68	9.06	8.86	2.16	1.12
³¹⁰120	11.61	9.85	14.60	–8.57	–9.21
³⁰⁶118	12.61	8.73	12.89	–5.86	–6.46
³⁰²116	11.37	7.85	10.93	–1.94	–2.44
²⁹⁸114	8.32	7.53	8.89	3.47	3.18
²⁹⁴112	7.96	7.78	7.20	9.44	9.26
²⁹⁰110	7.56	8.33	8.26	4.24	2.91

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