aa r X i v : . [ nu c l - t h ] F e b Bayesian evaluation of charge yields of fission fragments of U C.Y. Qiao, J.C. Pei, ∗ Z.A. Wang, and Y. Qiang
State Key Laboratory of Nuclear Physics and Technology,School of Physics, Peking University, Beijing 100871, China
Y.J. Chen, N.C. Shu, and Z.G. Ge
China Institute of Atomic Energy, Beijing 102413, China
Recent experiments [Phys. Rev. Lett. 123, 092503(2019); Phys. Rev. Lett. 118, 222501(2017)] have made remarkable progress in measurements of the isotopic fission-fragment yields ofthe compound nucleus
U, which is of great interests for fast-neutron reactors and for benchmarksof fission models. We apply the Bayesian neural network (BNN) approach to learn existing evaluatedcharge yields and infer the incomplete charge yields of
U. We found the two-layer BNN is improvedcompared to the single-layer BNN for the overall performance. Our results support the normal chargeyields of
U around Sn and Mo isotopes. The role of odd-even effects in charge yields has alsobeen studied.
PACS numbers: 21.10.Re, 21.60.Cs, 21.60.Ev
I. INTRODUCTION
Nuclear fission is a very complex non-equilibriumquantum many-body dynamic process and a deeper un-derstanding of fission presents a well-known challenge innuclear physics [1]. There are still strong motivations tostudy nuclear fission with increasing wide nuclear appli-cations [2] in productions of energies and rare isotopes,and in fundamental physics such as synthesizing super-heavy elements [3, 4] and constraints on r -process [5].In particular, the high-quality energy dependent fissiondata is very needed for fast-neutron reactors. However,fission measurements are very difficult and fission dataare generally incomplete and have large uncertainties. Inmajor nuclear data libraries [6–9], evaluated fission yieldsare only available at thermal neutron energies, 0.5 and14 MeV.Recently, the isotopic U fission products with closeexcitation energies have been measured experimentallyby different methods [10, 11]. Previously, the fissionfragment mass distributions or charge distributions canbe obtained. However, precise measurements of full iso-topic fission yields are only possible very recently withthe inverse kinematics and magnetic spectrometers [12–15]. The correlated fission observables are very crucialfor deeper understandings of fission process. It was re-ported that the charge yields around Sn and Mo are ex-ceptionally small in the
U(n, f) reaction [10] but itwas normal in the later experiment on
U via transferreactions [11]. It would be interesting to evaluate the twodiscrepant experimental results.The microscopic fission dynamical models based onpotential energy surfaces (PES) can obtain reasonablefission mass yields and charge yields [16]. In addition,the non-adiabatic time-dependent density functional the-ory [17] is promising to obtain various fission observables ∗ [email protected] such as fission yields, total kinetic energies and neutronmultiplicities, due to developments of supercomputingcapabilities. The macro-microscopic fission models basedon complex PES have been successfully used for descrip-tions of fission yields [18]. Generally fission models withmore predictive ability would have less precision. For pre-cise evaluations of fission data, phenomenology modelssuch as Brosa model [19] and the recent GEF model [20]are very successful and have been widely used.It is known that machine learning is powerful to learnand infer from complex big data, which is of great in-terests in interdisciplinary physics subjects. Recently, itwas shown that Bayesian neural network can be usedfor evaluations of incomplete fission mass yields with un-certainty quantifications [21]. The machine learning hasbeen used in nuclear physics such as the extrapolationof nuclear masses [22–24], fission yields [21, 25], variousnuclear structure [26–31] and reaction observables [32–34]. The machine learning has also been widely appliedin other physics subjects, such as the constrains of equa-tion of state of neutron stars from gravitational wavesignals [35] and for facilitating the lattice QCD calcula-tions [36]. We speculate that machine learning is promis-ing for developing new evaluation methods of nucleardata, in regarding to correlated fission observables andexisting large uncertainties.Previously, we have applied BNN to evaluate fissionmass yields [21]. In this work, we apply BNN to evalu-ate the fission charge yields, in particular the discrepantcharge yields of the compound nucleus U. The chargedistribution data is usually scare and is very useful innuclear applications. There could be different energy de-pendent behaviors of charge distributions and mass dis-tributions [37]. The charge distributions show distinctodd-even effects [38], while odd-even effects are ambigu-ous in mass distributions. Compared to our previouswork, we employ a multi-layer neural network in thiswork.
II. THE MODELS
The BNN approach [39] adopts probability distribu-tions as connection weights and is naturally suitable foruncertainty quantifications, in contrast to standard neu-ral networks which optimize definite values for connec-tion weights. The BNN approach to statistical inferenceis based on Bayes’ theorem, which provides a connectionbetween a given hypothesis(in terms of problem-specificbeliefs for a set of parameters ω ) and a set of data ( x, t )to a posterior probability p ( ω | x,t) that is used to makepredictions on new inputs, which is written as p ( ω | x, t ) = p ( x, t | ω ) p ( ω ) p ( x, t ) , (1)where p (x,t | ω )is the ‘likelihood’ that a given model de-scribes the data and p ( ω ) is the prior density of the pa-rameters ω ; x and t are input and output data; p ( ω | x,t)is the probability distribution of parameters ω after con-sidering the data ( x, t ), i.e., the posterior distribution; p ( x, t ) is a normalization factor which ensures the inte-gral of posterior distribution is one.We adopt a Gaussian distribution for the likelihoodbased on an objective function, which is written as p ( x, t | ω ) = exp( − χ / , (2)where the objective function χ ( ω ) reads: χ ( ω ) = N X i =1 ( t i − f ( x i , ω )∆ t i ) , (3)Here N is the number of data points, and ∆ t i is theassociated noise scale which is related to specific observ-ables. The function f (x i , ω ) depends on the input datax i and the model parameters ω . In this work, the inputsof the network are given by x i = { Z fi ,Z i ,A i ,E i } , whichinclude the charge number Z fi of the fission fragments,the charge number Z i and mass number A i of the fis-sion nuclei and the excitation energy of the compoundnucleus E i =e i +S i (e i and S i are incident neutron energyand neutron separation energy, respectively); t i are thefission charge yields.The posterior distributions are obtained by learningthe given data. With new data x n , we make predictionsby averaging the neural network over the posterior prob-ability density of the network parameters ω , h f n i = Z f ( x n , ω ) p ( ω | x, t ) dω, (4)The high-dimensional integral in Eq.4 is approximated byMonte Carlo integration in which the posterior probabil-ity p ( ω | x, t ) is sampled using the Markov Chain MonteCarlo method [39].In BNN we need to specify the form of the functions f (x, ω ) and p ( ω ). In this work, we use a feed-forward neural network model defined the function f (x, ω ). Thatis f ( x, ω ) = a + H X j =1 b j tanh( c j + I X i =1 d ji x i ) , (5)where H is the number of neurons in the hidden layer, I denotes the number of input variables and ω = { a , b j , c j , d ji } is the model parameters, a is bias of output lay-ers, b j are the weights of output layers, c j is bias ofhidden layers, and d ji are weights of hidden layers. Intotal, the number of parameters in this neural network is1+(2+ I ) × H . To study the odd-even effects, an additionalinput variable to identify the odd-even charge number isemployed. The confidential interval (CI) at 95% level isgiven for uncertainty quantifications in this work. Moredetails about BNN can be found in Ref. [39]. III. RESULTS AND DISCUSSIONS
CI-2L (a)
Th-14MeV (d)
Pu-14MeV F i ss i on Y i e l d (b) U-14MeV (e)
Cm-0.5MeV F i ss i on Y i e l d Atomic Number (c)
Pu-0.5MeV
Atomic Number (f)
Fm-0.025eV
FIG. 1. (Color online) Comparison of one-layer (blue lines)and two-layer (red lines) BNN learning results of charge yieldsfrom JENDL [7]. The shadow region corresponds to the con-fidence interval(CI) at 95%.
Firstly, we apply BNN to learn the existing evalu-ated distributions of charge yields from JENDL [7],which includes 2303 data points of the neutroninduced fissions of 29 nuclei ( , , Th,
Pa, , , , , , U, , Np, , , , , Pu, , Am, , , , , , Cm, , Cf,
Es,
Fm ). We adopt a single hidden layer networkof 32 neurons and a double hidden layer network of16-16 neurons for comparison. We adopt 10 BNNsampling iterations in all calculations in this work. Alarge number of sampling iterations are required forlarge data sets and large parameter sets. Note thatthe computing costs of BNN are very high comparedto standard neural networks. The obtained standarddeviations χ N = P i ( t i − f ( x i )) /N are 1.43 × − and6.36 × − for the single-layer and double-layer networksrespectively. The single-layer network has been used pre-viously for mass yields [21]. Generally the double-layernetwork has much improved the learning performanceof charge yields. In this work, the charge distributionsare trained independently and the normalization of thecharge distributions are examined, which is close to therequired 2.0 within 2%. Fig.1 displays the BNN learningresults of charge distributions of 6 nuclei. It is shownthat the single-layer network is less precise compared tothe double-layer network. In particular, the single-layernetwork is not satisfactory for descriptions of Fm.Similarly, it was shown previously that the single-layernetwork is not satisfactory for mass yields around
Thand
Fm where neighboring nuclei in the learning setare not sufficient [21]. The confidential interval (CI) at95% level are also shown in Fig.1. It is consistent thatCI of the double-layer network is much smaller thanthat of the single-layer network. In
Pu, the odd-eveneffects in charge yields are shown and the double-layernetwork can reproduce the peak structures while theresults of the single-layer network are rather smooth.Note that the double-layer network has more connec-tion parameters than that of the single-layer network,although they have the same number of neurons. Thereis also a risk of overfitting or training convergence issueswith more parameters. Both the parameter numbersand the architecture can affect the network performance.After above considerations, we choose the double-layernetwork which is suitable for the present study.Next we test the predictive ability of BNN with alearning set without
U, compared with JENDL eval-uation data. The predicted charge distributions of
Uare shown in Fig.2. It can be seen that single-layer re-sults can not describe the detailed peak structures of thecharge distributions at low incident neutron energies. Atenergies of 14 MeV, the single-layer results are better. Atlow energies, the charge distributions have obvious odd-even effects, which disappear at high excitations. We seethe double-layer predictions can describe the energy de-pendence of the odd-even effects. It is also shown fromCI that the double-layer predictions have smaller uncer-tainties than that of the single-layer predictions.It is known that odd-even effects are considerable incharge distributions while it is not obvious in mass dis-tributions. The charge yields of proton-even nuclei arelarger than that of proton-odd nuclei around peaks. Tosimulate the odd-even effects in BNN, we add an addi-tional input variable δ = ± . i = { Z fi ,Z i ,A i ,E i , δ } . Similar methodshave been used in BNN to account odd-even pairing cor- CI-2L (c) 0.5MeV F i ss i on Y i e l d (d) 20 25 30 35 40 45 50 55 60 65 700.000.040.080.120.160.20 Atomic Number (e) 14MeV F i ss i on Y i e l d
20 25 30 35 40 45 50 55 60 65 70 (f) Atomic Number
FIG. 2. (Color online) The BNN predicted fission chargeyields of n+
U at neutron energies of 0.025 eV((a)(b)), 0.5MeV((c)(d)) and 14 MeV((e)(f)), after learning the JENDLdata without
U. The results of one-layer(left) and two-layer(right) are compared. The shadow region correspondsto CI at 95%. relations and shell corrections in estimations of globalnuclear masses [24]. Fig.3 displays the training perfor-mance of charge distributions of n+
U at neutron en-ergy of 0.5 MeV. It can be seen that with the single-layernetwork, the influences of additional odd-even input δ is significant, while the odd-even effects is not shownwithout the δ input. The associated CI with δ inputis slightly reduced. For the double-layer network, theodd-even effects are shown with and without the δ in-put, with similar uncertainties. The additional δ inputdoesn’t gain much performance for the double-layer net-work. Our main motivation is to evaluate the chargedistributions of the compound nucleus U, so that wetrained BNN with resampled learning of the evaluatedn+
U data from JENDL, as shown in Fig.3(c). In thiscase, we reproduce learning data very precisely with thedouble-layer network plus odd-even input and resampledlearning of n+
U data.Finally, we evaluate the charge distributions of fissionfragments of the compound nucleus
U, based on therecently experimental data. In a recent experiment, Wil-son et al. [10] for the first time used a novel techniquewhich involves the coupling of a high-efficiency γ -rayspectrometer to an inverse-kinematics neutron source toextract charge yields of fission fragments via γ - γ coinci-dence spectroscopy of U(n,f). This experiment data iscompared with results of the GEF evaluations [20] andcharge yields around Sn and Mo isotopes are significantly CI (a) one-layer F i ss i on Y i e l d (b) two-layer F i ss i on Y i e l d Atomic Number(c) resample
FIG. 3. (Color online) The BNN results of fission chargedistribution of n+
U at neutron energy of 0.5 MeV. (a) dis-plays the comparison of one-layer results without and withodd-even input. (b) displays the comparison of two-layer re-sults. (c) displays results of with odd-even input and resam-pled learning. The shadow region corresponds to CI at 95%. small with a deviation by 600%. However, in anotherrecent experiment by Ramos et al. [11], direct measure-ments of isotopic fission yields of
U performed usingthe neutron-transfer Be( U, U) Be reaction don’tshow the abnormal deviation. The excitation energiesin
U in two experiments are 6.5 MeV and 8.3 MeV re-spectively, which should have more or less similar fissionyields. The significant discrepancy can impact fissionstudies and nuclear applications.Fig.4 displays the BNN evaluations of the two exper-imental data. In Fig.4, BNN adopts the double-layernetwork with and without the odd-even indication. Thelearning data set includes evaluated charge distributionsfrom JENDL and the two incomplete experimental data.In particular, the n+
U data from JENDL has beenresampled twice. The evaluations without odd-even in-put and resamplings would not be satisfied. It can beseen that the evaluations with odd-even input have sig-nificantly improved descriptions of the peak details. Theassociated CI with odd-even input is also much smallerthan that of evaluations without odd-even input. Thecontroversial charge yields around Sn and Mo isotopes intwo evaluations are not small by our approach. This isconfirmed even without resampling n+
U data. This
CI GEF
FIG. 4. (Color online) The BNN evaluations of fission chargeyields of the compound nucleus
U with two different exper-imental data. (a) the experiment corresponds to an excitationenergy of 6.5MeV [10]. (b) the experiment corresponds to anexcitation energy of 8.3MeV [11]. The blue lines and red linesare the two-layer BNN plus resampled learning results, with-out and with odd-even input, respectively. The olive linesare the GEF evaluations taken from [11]. The shadow regioncorresponds to CI at 95%. is consistent with the latest experiment that the abnor-mal deviation is not seen. In addition, we speculate thatthe experimental charge yields at peaks around Z =40and Z =52 could be too large based on BNN evaluations.The BNN evaluation is also very successful for the 2017data with clear odd-even effects, although which has veryfew data points of even atomic number. The evaluationsby GEF are also shown [11]. For this particular case,BNN evaluations are comparable to GEF evaluations. IV. SUMMARY
In summary, we applied the double-layer Bayesian neu-ral network to learn and predict charge yields of fissionfragments for the first time. We found the performancesof the double-layer network performance are significantlybetter than the single-layer network although they havethe same number of neurons. The double-layer networkcan describe the odd-even effects of charge yields at lowenergies, while odd-even effects are not obvious in massyields. We also add an additional input in BNN to in-dicate the odd-even atomic number which are very use-ful to improve evaluations. We apply these methods toevaluate the incomplete charge yields of two recent ex-periments of
U. Our BNN evaluations don’t obtainabnormal small charge yields around Sn and Mo isotopesas reported in Ref. [10]. This is consistent with the latestexperiment [11]. The BNN evaluations are comparable toGEF evaluations for this particular evaluations. Furtherimprovement of BNN is still underway and is promisingfor quantitative modeling fission data for practical nu-clear applications.
ACKNOWLEDGMENTS
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