A Bayesian-Neural-Network Prediction for Fragment Production in Proton Induced Spallation Reaction
aa r X i v : . [ nu c l - t h ] J u l A Bayesian-Neural-Network Prediction for Fragment Production in Proton InducedSpallation Reaction
Chun-Wang Ma , , ∗ Dan Peng , Hui-Ling Wei , Yu-Ting Wang , , and Jie Pu , Institute of Particle and Nuclear Physics,Henan Normal University, Xinxiang 453007, China School of Physics, Henan Normal University,Xinxiang 453007, China
Fragments productions in spallation reactions are key infrastructure data for various applications.Based on the empirical parameterizations spacs , a Bayesian-neural-network (BNN) approach isestablished to predict the fragment cross sections in the proton induced spallation reactions. Asystematic investigation have been performed for the measured proton induced spallation reactionsof systems ranging from the intermediate to the heavy nuclei and the incident energy ranging from168 MeV/u to 1500 MeV/u. By learning the residuals between the experimental measurements andthe spacs predictions, the BNN predicted results are in good agreement with the measured results.The established method is suggested to benefit the related researches in the nuclear astrophysics,nuclear radioactive beam source, accelerator driven systems, and proton therapy, etc.
Keywords: Bayesian neural network (BNN), spallation reaction, cross sections
I. INTRODUCTION
Spallation reaction is one of the violent nuclear reac-tions, which happens when a high energy light particlehits on a target nucleus. The spallation reaction can nat-urally happen in cosmos where the high energy cosmicray collides on nuclei [1], which results in the elementsvariation in universe. It can also artificially happen innuclear facility, such as the accelerator driven systems(ADS) for nuclear waste disposal, radioactive nucleusproduction and so on, or during the proton therapy pro-cess using the accelerated protons. The incident energyof spallation reaction is above tens of A MeV, which cov-ers the range of the intermediate energy, the relativisticenergy and even higher. As the result of spallation reac-tion, various of radioactive nuclei can be produced, theresearch of which has important applications in many dis-ciplines including nuclear physics, nuclear astrophysics,the isotopic-separation-online (ISOL) type radioactive-ion-beam facilities, and the incoming third generationof radioactive nuclear beams facilities, the ADS for nu-clear energy [2] and nuclear waste transmutation [3, 4],radioactive nuclei synthesis (especially for the extremenuclei and nuclear isomers [5]), accelerator material ra-diation [6], proton therapy [7, 8].For its extensive applications, both experimental andtheoretical interests have been attracted. In early time,the experiments were usually carried out using the ac-celerated proton on the synchrocyclotron or by the cos-mic rays. After the reverse kinetic technique has beenproposed, massive experiments have been performed tomeasure the spallation fragments above the Fe cover-ing the incident energy from a few hundreds of A MeV to ∗ Corresponding author. Email: [email protected] above A GeV. Prodigious data for fragments in spallationreactions have been assembled.On the theoretical side, many models have been de-veloped. The quantum molecular dynamics model hasbeen improved for spallation reactions, for example, [9–11]. The statistical multi-fragmentation model [12–14],the Li`ege intranuclear cascade ( incl++ ) model [15–17](which has been implanted in the openMC, GEANT4and FLUKA toolkits [18, 19]), can be used to simulatethe spallation reactions, which are usually followed bya secondary decaying simulation to reproduce the ex-perimental results (A review is also recommended [20]).Some semi-empirical parameterizations, including the epax [21] and the spacs [22], can globally predict theresidue fragments in the spallation reaction. The nuclearenergy agency (NEA) systematically compared the inter-national codes and models for the intermediate energyactivation yields to meet the needs in the ADS desig-nation, energy amplification and medical therapy [23].Difficulties still exist for the reasons that the spallationreaction involves a wide range of incident energy, as wellas a wide range of nucleus from the light to heavy one.Since the important applications of spallation reactionsand the resultant residues productions, it is importantto improve the theoretical predictions for the proton in-duced spallation reactions. The present models are lim-ited in predicting the light fragments, in particular forthe proton induced reaction at low energy. It is neces-sary to propose a new method for the exactly predictionof the fragments in spallation reactions.The neural network method, as one kind of machinelearning technologies, has found a rise of applications innuclear physics. In the standard neural network, it ishard or even impossible to control the complexity of themodel, which is likely to lead to overfitting problem andreduce the generalization ability of network [24]. TheBayesian neural network (BNN) provides a good mannerto avoid overfitting automatically by defining vague pri-ors for the hyperparameters that determines the modelcomplexity [25]. Prior distribution about the model pa-rameters can be incorporated in Bayesian inference andcombined with training data to control complexity of dif-ferent parts of the model [26]. Successful examples ofBNN applications in nuclear physics can be found in thepredictions of nuclear mass [29–31], nuclear charge radii[32], nuclear β − decay half-life [33], fissile fragments [34],and spallation reactions [35]. Based on the vast num-bers of measured fragments around the world-wide labo-ratories, it is quite promising to predict spallation crosssections accurately and give reasonable uncertainty eval-uations with the BNN approach.In this article, a new method is proposed to predictthe fragment cross sections in proton induced spallationreactions. The BNN method is described in Sec. II. Theresults and discussions are presented in Sec. III, and aconclusion is given in Sec. IV. II. BNN METHOD
The key principle of Bayesian learning is to deduce theposterior probability distributions through the prior dis-tribution. The process of Bayesian learning is startedby introducing the prior knowledge for model parame-ters. Based on the given training data D = ( x ( i ) , y ( i ) )and model assumptions, the prior distributions for allthe model parameters are updated to the posterior dis-tribution using the Bayes’ rule, p ( θ | D ) = p ( D | θ ) p ( θ ) /p ( D ) ∝ L ( θ ) p ( θ ) . (1)where θ denote the model parameters. The posterior dis-tribution p ( θ | D ) combines the likelihood function p ( D | θ )with the prior distribution p ( θ ), which contains the in-formation about θ derived from the observation and thebackground knowledge, respectively.The introduction of prior distribution is a crucial stepwhich allows the prediction to go from a likelihood func-tion to a probability distribution. The normalized quan-tity p ( D ) can be directly understood as the edge dis-tribution of the data, which can be obtained from theintegration of the selected model hypothesis and priordistribution p ( θ ), p ( D ) = Z θ p ( D | θ ) p ( θ ) dθ. (2)In this work, the prior distributions p ( θ ) are set as theGaussian distributions. The precisions (inverse of vari-ances) of these Gaussian distributions are set as gammadistributions [30], which automatically control the com-plexity of different parts of the model. The likelihoodfunction p ( D | θ ) and the objective function χ ( θ ) are given by p ( D | θ ) = exp ( − χ / ,χ = N X i [ y i − f ( x i ; θ )] / ∆ y i . where ∆ y i is the associated noise scale. The func-tion f ( x, θ ) is a multilayer perceptron (MLP) network,which is also known as the “back-propagation” or “feed-forward”. A typical MLP network consists of a set ofinput variables ( x i ), a certain hidden layers, and one ormore outputs ( f k ( x ; θ )). For an MLP network with onehidden layer and one output, the function is defined as, f ( x ; θ ) = a + H X j =1 b j tanh ( c j + I X i =1 d ji x i ) , (3)where H denotes the number of hidden unites, and I isthe number of input variables. θ = ( d ji , c j ) and θ =( b j , a ) are the weights and bias of the hidden layers andoutput layer, respectively.Based on the theoretical principles and prior knowl-edge, the posterior distribution be obtained from the datausing Eq. (1). A predictive distribution of output y new for a new input x new is obtained by integrating the pre-dictions of the model with respect to the posterior dis-tribution of the model parameters, p ( y new | x new , D ) = Z p ( y new | x new , θ ) p ( θ ) dθ. (4)In the MLP network, what is interested is to get a rea-sonable prediction with the new input x new rather thanthe posterior distribution for parameters. For a new in-put x new , the model prediction y new can be obtainedfrom the mathematical expectation of posterior distribu-tion, y new = E [ y new | x new , D ] = Z f ( x new , θ ) p ( θ | D ) dθ. (5)The integral of Eq. (5) is complex, and a numerical ap-proximation algorithm will reduce the complexity. TheMarkov chain Monte Carlo (MCMC) methods are appliedto optimize the model control parameters, and obtain thepredictive distribution. As one of the MCMC methods,the hybrid Monte Carlo (HMC) algorithm is firstly in-troduced by Neal [36] to deal with the model parametersand Gibbs sampling for hyperparameters. The HMC isa form of the metropolis algorithm, where the candidatestates are found by means of dynamical simulation. Itmakes the effective use of gradient information to reducerandom walk behavior. In concept, the Gibbs sampler isthe simplest Markov chain sampling methods, which isalso known as the heatbath algorithm. The hyperparam-eters are updated separately using the Gibbs sampling,which allows their values to be used in chasing good step-sizes for the discretized dynamics, and helps to minimizethe amount of tuning needed for a good performance inHMC. The integral of Eq. (5) is approximately calculatedas, y new = 1 /K K X k =1 f ( x new , θ t ) . (6)where K is the number of iteration samples. In a previouswork, the BNN approach has been adopted to learn andpredict the cross sections directly [35]. To provide somephysical guides, a recent empirical parameterizations forfragments prediction, which is named as the spacs [22],for spallation reaction has been adopted to obtain thefragment cross sections. In this work, the BNN approachis employed to reconstruct the residues between the ex-perimental data ( σ exp ) and the theoretical predictions( σ th ), i.e., y i = lg( σ exp ) − lg( σ th ) . (7)The cross sections predictions with BNN approach arethen given as, σ BNN + th = σ th × y new . (8)where σ th and y new denotes the spacs results and theBNN prediction, respectively. In this work, σ th refers tothe predictions by the spacs parameterizations, whichis proposed recently and gains its success in spallationreactions. TABLE I. A list of the adopted data for the measured frag-ments in the X + p spallation reactions. A X + p E(MeV/u) Numbers Z i Reference361 42 9-17 Ar + p
546 42 9-17 [37]765 38 9-17 Ar + p
352 45 9-17 [37]356 48 10-20 Ca + p
565 54 10-20 [38]763 54 10-20300 128 10-27500 136 10-27 Fe + p
750 148 8-27 [39]1000 152 8-261500 157 8-27168 73 48-55 [40]200 96 48-55 [41]
Xe + p
500 271 41-56 [42]1000 604 3-56 [43]
Au + p
800 352 60-80 [44]
Pb + p
500 249 69-83 [45]1000 458 61-82 [46]
U + p The inputs of neural network are the mass numbers A pi ( A i ), the charge numbers Z pi ( Z i ) of the projec-tile (fragment) nucleus, and the bombarding energy (in SPACS H=5 H=10 H=15 H=20 H=23 H=25 A - f a c t o r Model
FIG. 1. The A-factor for the predictions by the spacs and theBNN + spacs model with different hidden neurons denotedby H of the validation set. MeV/u) E i , i.e., x i = ( A pi , Z pi , E i , Z i , A i ). System-atic experiments have been performed at the LawernceBerkeley Laboratory (LBL), RI Beam Facility (RIBF)RIKEN, and FRagment Separator (FRS) GSI, whichcover a broad range of spallation nuclei from Ar to
U. The range of the incident energy changes from 168MeV/u to 1500 MeV/u, which is relevant for the ADSand proton therapy applications. As listed in Table I,3511 data in 20 different reactions will be used in thiswork. The entire data are divided into two different sets,which serve as the learning set and the validation set,respectively. The learning set is built by randomly se-lecting 3211 data set and the remaining 300 data as thevalidation set.
III. RESULTS AND DISCUSSIONS
The network is trained with different model structures,and 2000 iteration samples are taken in each training.Because the fragment cross sections may differs in sev-eral orders of magnitude, an A-factors method [48, 49] isintroduced to indicate the validation results of differentmodels, as shown in Fig. 1. The A-factor is defined as, A f = 1 /N N X i =1 | σ exp − σ pre | σ exp + σ pre , (9)where σ exp and σ pre denotes the measured data and pre-dicted data, respectively. Below we discuss the predic-tions by the BNN + spacs method, and compare themto the measured data and the spacs predictions.In Fig. 1, the A-factor for the spacs predictions andBNN method with different hidden neurons are com-pared. It is seen that even the BNN with five hiddenneurons can significantly improve the predictions. TheA-factor decreases with the increasing numbers of H .When H is increased to 23, the A-factor cannot be fur-ther minimized. A 5-23-1 structure is taken as the opti-mal network structure, which means that 5 inputs x i =( A pi , Z pi , E i , Z i , A i ), 1 output y i = lg( σ exp ) − lg( σ th ) andsingle hidden layer with 23 hidden neurons are included.The BNN + spacs predictions for fragment cross sec-tions in the 1 A GeV
Xe + p, 168 A MeV
Xe +p, 356 A MeV Ca + p and 1 A GeV
U + p reac-tions are shown in Fig. 2 to Fig. 5, and compared to theexperimental data, as well as the spacs predictions.In Fig. 2, the predicted and measured fragment crosssections in the 1 A GeV
Xe + p reaction are compared.It is seen that the BNN + spacs predictions agree quitewell with the measured data for fragments from Z = 3to 54. For the fragments with Z ≤
25, the underestima-tion of experimental data by spacs has been improvedsignificantly.Fig. 3 shows the BNN + spacs predictions for frag-ment cross sections in the 168 A MeV
Xe + p reaction,which has been measured at RIBF, RIKEN recently [40].In [40], only the cross sections for the Z ≥
48 fragmentsare reported. Compared to the measured fragments, theBNN + spacs predictions are very close to the spacs ones. For the light and the medium fragments ( Z ≤ spacs predictions are much higher than the spacs ones, which is similar to the results shown in Fig.2. In addition, the uncertainties are relatively large forthe Z ≤
11 isotopes, which may be caused by the insuf-ficient data in the training set in this incident energy.The spallation of intermediate nuclei are of interests inthe proton therapy and nuclear astrophysics. The com-position of interstellar matter, which are influenced bythe cosmic ray (mainly high energy proton) induced spal-lation reactions. The Ca + p at 356 A MeV have beenstudied, for which the predicted and measured results areshown in Fig. 4. Compared to the measured results, theBNN + spacs and spacs predictions both can reproducethe experimental data quiet well. The predicted fragmentcross sections by the BNN + spacs method are in linewith those by spacs except for the Z = 3 isotopes.The predictions to the fragment cross sections in the 1 A GeV
U + p reaction are compared in Fig. 5. Themeasured data cover the fragments from Z = 74 to 92[47]. It can be seen that the BNN + spacs method canpredict the results well, while the spacs highly overesti-mate the measured results for Z = 91 and 92. The BNN+ spacs method show the sign of larger than the spacs method for the fragments of smaller I . It seems that forthe Z = 20 isotopes, fragments For the spallation of aheavy system as U, the predictions by BNN + spacs become worse, which indicates that the BNN should befurther improved by incorporating more data for small Z fragments produced in the heavy systems.The BNN + spacs predictions is further verified byusing the correlation between the cross section and av-erage binding energy, which has been performed in Ref.[35]. It is generally believed that the isotopic cross sec- tion depends on the average binding energy in the formof σ = C e ( B ′ − /τ , where C and τ are free parame-ters, and B ′ = ( B − ǫ p ) /A (in which ǫ p = 0 . − N +( − Z ] ǫ A − / is the pairing energy for the fragment,and ǫ = 30 MeV). It is clearly seen that the predictedisotopic cross sections by BNN + spacs model for the Z = 17 and 41 obey the correlation very well, whichimproves both the previous BNN method and also the spacs method.From the above results, which cover the fragment crosssection predictions for proton induced reactions from theintermediate nuclei to the heavy nuclei, and for the inci-dent energy from 168 A MeV to 1 A GeV, it is seen thatthe BNN approach improves the quality of the empirical spacs parameterizations through the reconstruction ofthe residual cross sections between the spacs predictionsand measured data. But the BNN + spacs method canbe a new tool to predict the fragment cross section in thespallation reactions since it can work independently afterthe network is formed.If we revisit the foundation of the BNN approach, itis natural that the BNN + spacs model should have abetter prediction than the spacs parameterizations sincethe difference between the spacs and the measured datahas been minimized. This is why the BNN + spacs im-proves the prediction, and also avoids the nonphysicalphenomenon by forming a direct BNN learning networkfrom the measured data as shown in Ref. [35]. The phys-ical implantations of spacs play important roles to makethe BNN + spacs method reasonable in physics, and theleaning and predicting abilities of the BNN also improvethe predictions where the spacs parameterizations do notwork well.It is indicated from the results that the spacs tends tounderestimate the cross sections for fragments with rel-ative small Z , while overestimates the fragments with Z close to the heavy spallation nuclei. These shortcomingshave been overcome by the BNN + spacs model. Limita-tions still exist for the constructed BNN + spacs modelin this work since the absence of experimental data forreactions of incident energy below 100 A MeV, and thesmall spallation systems. For the applications in the pro-ton therapy, the incident energy maybe lower than 100 A MeV, and the mass of the spallation nuclei are smallerthan 30. The smallest spallation reaction adopted in thiswork is for Ar. If we consider the interstellar matters,most of the nuclei have mass numbers smaller than 56.In spallation reactions induced by the high energy cosmicrays and in the proton therapy process, we should im-prove the prediction model to cover the small spallationsystems, for which the spacs parameterizations do notwork well. It is important to improve the BNN + spacs predictions by introducing new data for the spallation re-actions of intermediate energy (for example below 100 A MeV) and intermediate/small systems (
A < -10 -5 -15 -10 -5 -15 -10 -5 BNN+SPACS SPACS EXP
Z=3 Z=7 Z=11 ( m b ) Z=25 Z=29 Z=33Z=45I Z=49I Z=53 I
FIG. 2. The BNN + spacs predictions of fragment cross sections of 1000 A MeV
Xe + p compared with spacs andexperimental data (taken from [43]). In the x axis I = N − Z denotes the neutron excess of fragment. The measured data, theBNN + spacs and the spacs predictions are plotted as the squares, circles and triangles, respectively. The experimental andBNN + spacs error bars are too small to be shown. IV. CONCLUSION
In this article, the BNN approach is proposed to pre-dict the fragment cross sections in proton induced spalla-tion reactions combined to the spacs parameterizations.Based on the 3,511 measured fragment cross sectionsin 20 spallation reaction systems, the optimal networkstructure has been established to be 5-23-1, which in-cludes 5 inputs, 1 output and a single hidden layer with23 hidden neurons. By reconstructing the residuals be-tween the measured data and the spacs predictions, theBNN + spacs method is verified to well reproduce the experimental data. It is also shown that the BNN + spacs method can yield a better global prediction com-pared to the spacs parameterizations. The establishedBNN + spacs method potentially can be applied into theresearches of nuclear physics, nuclear astrophysics, ADS,and proton therapy, etc.
ACKNOWLEDGEMENT
This work is supported by the National Natural Sci-ence Foundation of China (grant Nos. U1732135 and11975091). [1] Y. G´enolini et al ., Phys. Rev. C , 034611 (2018).[2] A. Letourneau et al ., Nucl. Instrum. Meth. B , 299(2000).[3] W. Gudowski, Nucl. Phys. A , C436 (1999).[4] J. Cugnon et al ., Adv. Space Research , 1332 (2007).[5] S.A. Karamian et al ., NIMA , 488 (2009).[6] M. Numajiri et al ., J. Rad. Nucl. Chem. , 481 (2003).[7] D. Schardt et al ., Rev. Mod. Phys. , 383 (2010). [8] H. A. WShih et al ., Inter. J. Radi. Oncology BiologyPhysics , S642 (2008).[9] J. Su, L. Zhu, C. Guo, Z. Zhen, Phys. Rev. C , 014602(2019).[10] S. Xu, G. Yang, M. Jin, J. Su, Phys. Rev. C , 024609(2020).[11] F. Zhang, J. Su, Chin. Phys. C , 024103 (2019).[12] J. P. Bondorf et al ., Phys. Rep. , 133 (1995). -15 -10 -5 -15 -10 -5 -15 -10 -5 SPACS BNN+SPACS EXP
Z=3
168 A MeV
Xe + p
Z=7 Z=11 ( m b ) Z=25 Z=29 Z=33Z=49 I Z=52 I Z=54 I FIG. 3. Similar to Fig. 2 but for the 168 A MeV
Xe + p reaction (experimental data taken from Ref. [40]). -10 -5 -15 -10 -5 -4 -2 0 2 4 6 8 1010 -10 -5 -4 -2 0 2 4 6 8 10 -4 -2 0 2 4 6 8 Z=3
356 A MeV Ca + p
Z=5 Z=8 ( m b ) Z=10 Z=13 Z=15Z=17 I Z=19 I Z=20
SPACS BNN+SPACS EXP I FIG. 4. Similar to Fig. 2 but for the 356 A MeV Ca + p reaction (experimental data taken from Ref. [38]). -15 -10 -5 -15 -10 -5 -15 -10 -5 Z=20 Z=30 Z=39 ( m b ) Z=48 Z=57 Z=66 I Z=75 I Z=84 I SPACS BNN+SPACS EXP
Z=91
FIG. 5. Similar to Fig. 2 but for the 1 A GeV
U + p reaction (experimental data taken from Ref. [47]). -10 -8 -6 -4 -2 BNN BNN+SPACS SPACS EXP Linear Fit of EXP ( m b ) B’=(B- p )/A Xe+p Z=17 -11 -9 -7 -5 -3 -1 BNN BNN+SPACS SPACS EXP Linear Fit of EXP ( m b ) B’=(B- p )/A Z=41 1 A GeV
Xe+p
FIG. 6. Isotopic cross section dependence on average binding energy for fragments of Z = 17 and 41 produced in the 1 A GeV
Xe + p reaction (experimental data taken from [43]). The open circles, solid circles, triangles and squares denote the databy the BNN predictions (see Ref. [35]), BNN + spacs in this work, spacs , and the measured data, respectively. The solid linedenote the fitting to the experimental data (see text for explanation).[13] A. S. Botvina and I. N. Mishustin, Phys. Rev. C ,061601 (2001).[14] N. Buyukcizmeci et al ., Eur. Phys. J. A , 57 (2005).[15] A. Boudard et al ., Phys. Rev. C , 014606 (2013).[16] D. Mancusi et al ., Phys. Rev. C , 054602 (2014).[17] D. Mancusi et al ., Phys. Rev. C , 034602 (2015). [18] Z.-L. Zhao, Y.W. Yang, S. Hong, Nucl. Sci. Tech. , 10(2019).[19] A. Lamrabet, A. Maghnouj, J. Tajmouati, M. Bencheikh,Nucl. Sci. Tech. , 54 (2019), and ibid, , 75 (2019).[20] J.-C. David, I. Leya, Prog. Part. Nucl. Phys. , 103711(2019). [21] K. Summerer and B. Blank, Phys. Rev. C , 034607(2000).[22] C. Schmitt et al ., Phys. Rev. C , 039901 (2016). Andthe Erratum: SPACS: A semi-empirical parameterizationfor isotopic spallation cross sections (Phys. Rev. C ,064605 (2014)).[23] R. Michel, P. Nagel, International codes and model in-tercomparison for intermediate energy activation yields,OECD/NEA, NSC/DOC(97)-1 (1997).[24] C. L. Fan. statistics and Computing , 13 (2005).[25] A. Vehtari, J. Lampinen, Pattern Recognition Letters, ,1183 (2000).[26] J. Lampinen, A. Vehtari, Neural Networks, , 257(2001).[27] D. J. C. MacKay. Neural Computation , 448 (1992).[28] R.M. Neal, Bayesian Learning for Neural Networks ,Springer, New York (1996).[29] R. Utama et al ., Phys. Rev. C , 014311 (2016).[30] Z. M. Niu, H. Z. Liang, Phys. Lett. B , 48, (2018).[31] L. Neufcourt et al ., Phys. Rev. C 98, 034318 (2018).[32] R. Utama et al ., J. Phys. G: Nucl. Part. Phys. , 114002(2016). [33] Z. M. Niu et al ., Phys. Rev. C , 064307 (2019).[34] Z. A. Wang et al ., Phys. Rev. Lett. , 122501 (2019).[35] C.W. Ma, D. Peng, H.L. Wei et al., Chin. Phys. C ,014104 (2020).[36] R. M. Neal,Technical Report CRG-TR-93-1 (1993).[37] C. N. Knott et al ., Phys. Rev. C , 398 (1997).[38] C.-X. Chen et al ., Phys. Rev. C , 1536 (1997).[39] C. Villagrasa-Canton et al ., Phys. Rev. C , 4603(2007).[40] X.H . Sun et al ., Phys. Rev. C , 064623 (2020).[41] C. Paradela et al ., Phys. Rev. C , 044606 (2017).[42] L. Giot et al ., Nucl. Phys. A , 116 (2013).[43] P. Napolitani et al ., Phys. Rev. C , 67 (2007).[44] F. Rejmund et al ., Nucl. Phys. A , 540 (2001).[45] L. Audouin et al ., Nucl. Phys. A , 1 (2006).[46] T. Enqvist et al ., Nucl. Phys. A , 481 (2001).[47] J. Taleb et al ., Nucl. Phys. , 413 (2003).[48] S.K. Sharma, B. Kamys, F. Goldenbaum et al. , Eur.Phys. J. A , 150 (2017).[49] Y. D. Song, H. L. Wei, C. W. Ma, Sci. China-Phys. Mech.Astron.62