The fission mechanism of Actinide elements

It is important to review the basic phenomena of the fragment yield distribution discovered in experiments before discussing the predicted results of the BNN models to better understand the evaluation of the fragment yields by BNN in γ-induced thorium fission reactions. In this subsection, we list the main experiments and their conclusions. The phenomenological Brosa model [36, 37], which is based on the mass distribution of fission products, postulates the existence of two asymmetric fission modes. The origin of these distinct asymmetric fission modes is attributable to the influence of fission barriers. The Brosa model further describes the broadening of fission observables through stochastic neck rupture, where longer necks correspond to a broader distribution. With the advancement of experimental research, three primary fission modes have been proposed for the nearly stable actinide region: two mass asymmetric modes, standard I (ST1) and standard II (ST2), and a superlong symmetric mode (SL) [38-40].

Schmidt et al. [40, 41] investigated fission reactions of 70 short-lived radioactive isotopes in 2000, using the secondary beam facility at GSI Darmstadt. The primary beam of 1A GeV ²³⁸U was irradiated on a beryllium target. The mass and charge of the produced fragments were analyzed using a Fragment Separator (FRS). These products were utilized as a secondary beam, which was excited by electromagnetic interaction with the secondary lead target, leading to fission of the nucleus within an excitation energy range of approximately 11 MeV, primarily by the giant dipole resonance. This study provides the first experimental characterization of the transition from asymmetric fission in actinides to symmetric fission in pre-actinides at the atomic level. Chatillon et al. [42] conducted a similar experiment at GSI in 2019, employing the R3B/SOFIA (Reactions with Relativistic Radioactive Beams/Studies on FIssion with Aladin) setup [43-45]. The mass and nuclear charge of the fission fragments were measured in coincidence with the total prompt neutron multiplicity, which provides evidence for the existence of a new compact symmetric fission mode in the light Th isotopic chain [46]. Based on the predictions of the BNN, a detailed discussion of the photofission mechanism of the Th isotopes is presented in Sect. 3.2 and Sect. 3.3.

Total element yield of fission products

The BNN-CY model was constructed to predict the PNF fragment charge yields based on learning the experimental charge yield data in the EXFOR database, including 1437 experimental data points from the PNF products of 14 nuclides (^217~230Th). All measured data were obtained from Refs. [40, 42]. The input dataset of BNN-CY consists of , which refers to the number of charges of the fission fragments, the number of charges and neutrons of the fission nucleus, and the excitation energy (Ei) of the compound nucleus, respectively. The output dataset ti represents the charge yield of fission fragments. The fission charge yields of the compound nucleus ²¹⁷Th predicted by the BNN-CY model are illustrated in Fig. 2. For clarity, the prediction results of the neural network without the odd-even parameters only displayed the central values and did not include confidence intervals. The predictive results from single-layer neural networks, excluding odd-even parameters, demonstrate two notable anomalies: negative values emerge in the vicinity of the fission fragment charge Z_f = 35, while an abnormal upward tail becomes apparent for Z_f > 60. Moreover, the double-layer neural network without odd-even parameters generated aberrant negative predictions near Z_f = 54. The production of fission fragments with even numbers of protons is usually enhanced because the fully paired proton configuration tends to survive scission with a high probability. To describe the odd-even effect in the reaction system, we performed the methods described in Ref. [16, 32], an additional input δ = ± 0.1 was added to the BNN-CY learning collection to represent the number of atoms of even and odd fragments, respectively. The input parameter set becomes . As can be seen in Fig. 2, BNN-CY evaluations without the odd-even parameter δ failed to capture the odd-even effect in the charge yield distribution of the PNF fragments, while the inclusion of the odd-even parameter δ allowed the BNN-CY to accurately reproduce the effect. Based on the values of in Eq. (6), to screen the number of neurons in the hidden layers, a double-layer network with 30 neurons was adopted for the study.

Fig. 2Full-screen View

(Color online) The photonuclear fission charge yields of γ + ²¹⁷Th at excitation energy of 11 MeV predicted by BNN-CY model constructed based on the EXFOR data without ²¹⁷Th. Comparison of the charge yields for the ²¹⁷Th nuclide in this work (solid lines) and the experimental data from Ref. [42] (solid squares). The shadow region corresponds to the confidence interval (CI) at 80%

Generally, a double-layer network significantly enhances the learning performance of data points owing to its larger number of connection parameters compared to a single-layer network, even when both have the same number of neurons. To verify this and better describe the odd-even effect in the reaction system, we conducted a more detailed comparison of the evaluation results of single-layer neural networks with odd-even parameters and those of double-layer neural networks with odd-even parameters. In particular, a single-layer network with 30 neurons and a double-layer network with 15–15 neurons were compared, with iterations of the BNN-CY sampling carried out 10⁵. For both single-layer and double-layer networks, the standard deviations obtained using Eq. (6) were 1.7 × 10² and 1.3 × 10², indicating similar uncertainties. In Figs. 3 and 4, as same as the double-layer network, the single-layer network can effectively reproduce the odd-even effect in the charge distribution of the fission fragments. However, the confidence interval of the double-layer network was slightly larger than that of the single-layer network.

Fig. 3Full-screen View

(Color online) The charge yield distribution of fragments from γ-induced fission of thorium isotopes at excitation energy of 11 MeV. Comparison of one-layer (green lines) and two-layer (pink lines) BNN-CY learning results of the fragment charge yields from Ref. [40]. Open circles represent the experimental data. The shadow region corresponds to CI at 80%

Fig. 4Full-screen View

(Color online) The charge yield distribution of fragments from γ-induced fission of thorium isotopes at different excitation energies. Comparison of one-layer (green lines) and two-layer (pink lines) BNN-CY learning results for the fragment charge yields from Ref. [42]. Open circles represent the experimental data. The shadow region corresponds to the CI at 80%

Figure 3 presents a comparison between the results of the BNN-CY evaluation and the experimental charge yield data for the PNF fragments of the targets ^217~229Th at an average excitation energy of 11 MeV, as reported by Schmidt et al. [40]. The BNN-CY results were in good agreement with the experimental data obtained. Notably, neural networks incorporating odd-even parameters successfully reproduced the characteristic odd-even staggering in the charge distribution of the reaction system. Furthermore, these calculations accurately captured the transition from asymmetric to symmetric fission in the PNF mechanism across the thorium isotopic chains. The number of peaks in the fission fragment charge distribution evolves with increasing neutron number of the target nuclei: single-peak distributions for neutron-deficient thorium targets transition through triple-peak patterns in intermediate-mass thorium nuclei, eventually becoming double-peak distributions for heavier thorium isotope targets. This systematic evolution demonstrates the strong dependence of the fission mechanisms on the number of neutrons along the Th isotopic chain. The experimental data on the charge yield of ^221~230Th fission fragments at an average excitation energy of approximately 14 MeV, obtained by Chatillon et al. [42] are shown in Fig. 4 to exhibit a similar phenomenon.

The results predicted by the double-layer network with the smallest value were selected for further evaluation and prediction, focusing on the charge yield values associated with the three prominent peaks observed in the fission fragment charge distribution across the Th isotopic chain. Fragments of Z_f = 36 (Kr), 45 (Rh), and 54 (Xe) were analyzed because of their critical role in quantifying the competition between the symmetric and asymmetric fission modes. These yields serve as direct indicators of the evolving fission mechanism. Furthermore, the fission product ¹³⁵Xe has a substantial neutron absorption cross section, acting as a reactor “poison” that significantly reduces the operational power, and the evaluation of its production is of great importance. Figure 5 presents the evolution of the yields of the fission fragments for Z_f = 45 (Rh) as a function of the target mass. The observed systematic decrease in the Rh fragment yields with increasing target neutron number clearly demonstrates the diminishing contribution of the symmetric fission channels. This trend culminates in near-zero fragment-charge yields for ^230~232Th, establishing asymmetric fission as the dominant mechanism in heavier thorium isotopes, whereas symmetric fission prevails in lighter Th isotopes. In particular, higher charge yields were observed at an excitation energy of 11 MeV in heavier Th isotopes than in the 13~15 MeV range. Figure 6 presents the evolution of the fission fragment yields for Z_f = 36 (Kr) and 54 (Xe) as a function of the target mass number. The BNN-CY results successfully reproduced the symmetry characteristics of the charge yield distributions. The experimental data revealed a perfectly symmetric charge yield distribution, where the fragment Kr and Xe yields were identical, which was accurately reproduced by the BNN-CY predictions. Moreover, the charge yields of the fragments representing asymmetric fission mechanisms showed a systematic increase with the neutron number of the target nucleus, with a more pronounced growth rate observed at an excitation energy of 14 MeV. These results provide clear evidence of the transition from symmetric fission in neutron-deficient thorium isotopes to asymmetric fission in heavier counterparts.

Fig. 5Full-screen View

(Color online) The BNN-CY predicted fission charge yields of Z_f = 45 of γ + ^216~232Th at excitation energy of 11 MeV (full line) and 14 MeV (dotted line). Comparison of the charge yields for the Th isotopic chain in this study (lines) and the experimental data from Refs. [40, 42] (Solid dots). The squares represent the experimental data of the charge yield at an excitation energy of 11 MeV, while the circular dots represent the experimental data of the charge yield in the excitation energy range of 13–15 MeV. The shadow region corresponds to the CI at 80%

Fig. 6Full-screen View

(Color online) The BNN-CY predicted fission charge yields of Z_f = 36 and 54 of γ + ^216~232Th at excitation energies of 11 MeV (red full line and dotted line) and 14 MeV (blue full line and dotted line). Comparison of the charge yields for the Th isotopic chain in this work (full lines) and the experimental data (solid dots) from Refs. [40, 42]. The squares represent the experimental data of the charge yield at an excitation energy of 11 MeV, while the circular dots represent the experimental data of the charge yield in the excitation energy range of 13–15 MeV. For the sake of clarity, the CI are not shown in this BNN-CY results

Mass chain yield distribution

The BNN-CY prediction in Sect. 3.2 indicates that the fission mechanism of ²³²Th will be dominated by asymmetric fission. In this section, to explore the energy dependence of the mass yield distribution of fission fragments, the BNN-MY model was constructed based on learning experimental mass yield data in the EXFOR database, which includes 771 data points within an energy range of 8 ~ 80 MeV for the PNF products of ²³²Th [47-54]. The mass yields of the fission fragments of ²³²Th were measured by Naik et al. [49-54] using the activation method. To avoid systematic inconsistencies between different experimental datasets, which could lead to artificially widened confidence intervals in the model, we selected data exclusively from the same group (Naik et al.) for the training set, while using data from other groups for validation. The charge distribution exhibited a pronounced odd-even effect, which appeared to be obscured in the mass distribution. Consequently, there is no need to introduce the δ parameters as input corrections for the mass yield data. The network consists of a set of input variables , which denote the mass number of the fission fragments, the number of charges and neutrons of the fission nucleus, and the incident γ energy (Ei), respectively. The output data ti represent the mass yield of the fission fragments. We compared the single-layer network with 38 neurons and the double-layer network with 18–20 neurons, performing 10⁵ BNN-MY sampling iterations. The standard deviations obtained for the single- and double-layer networks were 6.4 × 10³ and 3.8 × 10³, respectively. As shown in Fig. 7, the CI of the double-layer network is significantly narrower than that of the single-layer network. The double-layer network successfully reproduced the peak structures, whereas the results from the single-layer network appeared to be smoother. Based on these considerations, we selected a double-layer network as the more suitable model for further data evaluation in this study.

Fig. 7Full-screen View

(Color online) The mass yield distribution of fragments from γ-induced fission of ²³²Th at different energies. Comparison of one-layer (blue lines) and two-layer (pink lines) BNN-MY learning results of fragment mass yields of γ + ²³²Th in γ energy range of 8–80 MeV from Naik et al. [49-54]. The shadow region corresponds to the CI at 80%

Figure 7 presents the BNN evaluation for regions with sparse experimental data points. In the mass region Af < 90, experimental measurements are particularly scarce, demonstrating that the trained BNN model can predict the physical behaviors in regions where experimental data are currently scarce or inaccessible. The results show that the fission yield exhibits a monotonic increase with energy in the valley region (fragment mass number A_f ≈ 100 ~ 130), whereas the twin peaks corresponding to the asymmetric fission modes diminish in prominence. In Fig. 8, A_f = 91 and 140 were selected for BNN prediction because the fragments with these mass numbers were respectively located in the light peak region and the heavy peak region of the asymmetric fission mode of ²³²Th, and their yields were relatively high. Most importantly, these two fragments have sufficient experimental data that can be compared with the BNN prediction results. In general, the fragment yield of Af=91, 140 representing asymmetric fission increases gradually within the incident γ energy range of 5 ~ 40 MeV. However, when the γ energy exceeds 40 MeV, the yield decreases significantly rather than continuing to increase. In 2010, Demekhina et al. [38] compared the PNF fragment mass yields of ²³²Th at γ energies of 50 and 3500 MeV. They observed that symmetric fission dominated as the energy increased, and its contribution increased by almost an order of magnitude. Moreover, owing to the additional emission of neutrons from the fission nucleus at higher excitation energies, the mass numbers of the fission fragments decrease, leading to a leftward shift in the mass-yield distribution. Figure 9 displays the mass yield distributions evaluated by BNN-MY of ²³²Th PNF fragments at incident γ energies of 8, 25, and 80 MeV. It can be seen that the yield of symmetric fission (A_f = 100 ~ 130) increases with energy. In contrast, the contributions of the light fragment peak (A_f = 80 ~ 100) and heavy fragment peak (A_f = 130 ~ 150) decrease progressively, which represents the symmetric fission mechanism. This suggests that with an increase in the excitation energy of the nucleus, particle evaporation (predominant neutron emission) occurs from the excited nucleus, which opens a new decay channel leading to the formation of neutron-deficient fission fragments (primarily through symmetric fission) [55, 56]. However, no obvious phenomenon of the light-fragment peak moving towards a lower mass number was observed. This indicates that the PNF mechanism of ²³²Th changes with the incident γ energy. In the low-energy region, the PNF mechanism of ²³²Th mainly involves asymmetric fission, whereas in the high-energy region, it involves symmetric fission. Figure 10 shows the predictive capability of BNN-MY using a test set that excludes the PNF fragment mass yield data of ²³² at the γ energies of 7.64 MeV and 17.5 MeV. A comparison between the predicted results and experimental measurements [57, 58] shows that BNN-MY can effectively predict the mass yields of PNF fragments for unstudied reactions, and the trained model successfully reproduces the results reported by other groups.

Fig. 8Full-screen View

(Color online) The excitation function of the mass yield of photonuclear fission fragments within the incident γ energy range of 5 MeV to 80 MeV. The predicted results of BNN-MY (A_f = 91 as red line and A_f = 140 as blue line) are compared with the experimental data of the light asymmetric peak (Af = 91 as square points) and the heavy asymmetric peak (A_f = 140 as circle points) [47-54]. The shadow region corresponds to the CI at 80%

Fig. 9Full-screen View

(Color online) The BNN-MY predicted fission mass yield of γ + ²³²Th at γ energy of 8 MeV, 25 MeV, and 80 MeV. For clarity, the CI are not shown in this BNN-MY results

Fig. 10Full-screen View

(Color online) The BNN-MY predicted fission mass yield of γ + ²³²Th at γ energy of 7.64 MeV and 17.5 MeV, where the experimental data for these specific reactions were not included in the training dataset of BNN-MY. Comparison of the mass yields for the ²³²Th nuclide in this work (solid lines) and the experimental data (solid points) from Ref. [57, 58]. The shadow region corresponds to the CI at 80%