Generation and analysis of independent fission yield covariances based on GEF model code

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Introduction
Fission yields are the basic nuclear data in nuclear fuel cycle calculation. As input data of reactor burnup calculation, fission yields play an important role in the prediction of inventories of fission product nuclides. The inventories of fission products determine the calculation accuracy of macro parameters of reactor physics, such as eigenvalue. The fission yield data are mainly obtained through theoretical model evaluation [1]. During the evaluation process, imperfect model parameters propagate uncertainties to fission yields. Therefore, it is necessary to further analyze the uncertainties of fission yields propagated to macro parameters. However, the most commonly used fission yield data in ENDF-6 format [2] only contain the mean value and uncertainty of each fission yield itself, lacking the correlations among different fission yields of fission products or fissioning systems. This affects the accuracy of the uncertainties of the macro parameters.
To provide the correlations among fission yield data and quantify the uncertainties more accurately, many studies have been carried out worldwide. Subgroup 37 (SG37) of the Working Party on International Nuclear Data Evaluation Co-operation (WPEC) was established with the goal of developing "Improved Fission Product Yield evaluation methodologies" [3]. Katakura proposed a Bayesian/general least-squares method to construct covariances among fission yields of fission products with the same mass number [4]. Fiorito et al. considered the different physical constraints of fission yields and generated the covariances of independent fission yields through an iterative generalized leastsquare method [5]. At the Oak Ridge National Laboratory (ORNL), Wahl's systematics and its random parameters were used to generate random fission yields, and a Bayesian approach based on constraints from the mass yields, the conservation of the mass and charge number was proposed to generate the covariances among different fission yields [6]. Recently, the covariance matrices of independent fission yields were determined based on the generalized least-square method by Kohsuke Tsubakihara, et al [7].
GEF (GEneral description of Fission observables) code [8,9] is a semi-empirical model designed to give a complete description of the fission process [10]. GEF code determines the corresponding fission products of different fission events through model parameters and determines the yields of different fission products through the Monte-Carlo process. Some important model parameters of GEF exist in the form of normal distribution, so the parameter samples are obtained by perturbation based on the probability distribution, and then fission yield samples are obtained by running GEF with the parameter samples. In this paper, the independent fission yield covariances of 235 U, 239 Pu, and 241 Pu thermal neutron-induced fissioning systems are generated respectively using the fission yield samples. These covariances are verified based on uncertainties of burnup-related responses. At the same time, the influence of the correlations among fissioning systems on the uncertainties of the burnup-related responses is analyzed. In addition, the model parameters of GEF are preliminarily adjusted through the Bayesian Monte Carlo method [10,11]. All uncertainty calculations in this paper are based on random sampling. This paper is organized as follows. Section 2 expounds on the detailed process of generating and verifying independent fission yield covariances for individual fissioning systems. Section 3 shows the influence of correlations among fissioning systems. Section 4 introduces the preliminary adjustment of GEF model parameters. Conclusions and perspectives are summarized in Section 5.

Generation of fission yield covariances
The process of generating independent fission yield covariances mainly includes three steps, which are summarized below.
(2) Calculate independent fission yield samples with the above parameter samples. (3) Calculate covariances among different fission products with the above fission yield samples. The GEF code version used in this paper is 2021.1.1 [12]. The GEF2021.1.1 determines the normal distribution of the 23 most important parameters for calculating fission yields, and these parameters are independent of each other. In the present work, we sample these parameters. The specific information on these model parameters is shown in Table 1. 1000 sets of model parameter samples are generated and based on these samples, independent fission yield samples of 235 U, 239 Pu, and 241 Pu thermal neutroninduced fissioning systems are calculated individually. The number of fission events of GEF is 2×10 5 . Based on these samples, the respective fission yield covariances of the three fissioning systems are calculated.

Verification of fission yield covariances
To verify the fission yield covariances, 1000 samples of fission yield are generated respectively for the three fissioning systems by resampling the above-obtained covariances. Besides, as mentioned above, there are model parameters. Based on the two types of samples, neutronics-burnup coupling calculation is conducted for the TMI-1 pin cell problem in UAM project [13]. The fuel material of this problem is UO 2 , and the 235 U enrichment is 4.85%. The calculation is based on the hot full power condition and the power density is 33.58 MW/tU. The burnup of the problem reaches 60 GWꞏd/tU. Figure 1 shows the geometry of the TMI-1 pin cell. The relative uncertainties of the responses, the infinite multiplication factor (k inf ), and the number densities of important fission product nuclides, are calculated based on the fission yield samples resampled based on the above-obtained covariances. The same uncertainties are also calculated based on the fission yield samples by sampling the GEF model parameters, which are taken as the reference results. The neutronicsburnup coupling calculation tool is the high-fidelity reactor physics code NECP-X [14]. Other burnup data and cross section data are extracted from ENDF/B-VII.1 evaluated nuclear data library [15].

Influence of correlations among fissioning systems
Generally, there are correlations among multiple fissioning systems. According to the analysis in Section 2, since the same set of model parameter sample of GEF can calculate one set of fission yield sample for each of the three fissioning systems ( 235 U+n th , 239 Pu+n th , and 241 Pu+n th ) simultaneously, the fission yield samples calculated by GEF actually include the correlations among fissioning systems. To analyze the influence of these correlations, two cases are calculated. Case 1 uses fission yield samples of three fissioning systems calculated by GEF in one random run of the TMI-1 problem. Case 2 uses fission yield samples of one fissioning system calculated by GEF in one random run of the TMI-1 problem and calculates the sum of the uncertainties of three fissioning systems with equation (1) : where  is relative uncertainties of burnup-related responses (k inf or nuclide number densities). In other words, Case 1 considers correlations among three fissioning systems and Case 2 assumes that the three fissioning systems are independent. Figure 5 shows the uncertainty results for the two cases. It can be seen that the uncertainties of Case 1 considering correlations are significantly higher than that of Case 2 assuming independence. The bias of relative uncertainty of k inf reaches about 100 pcm at the end of irradiation. For uncertainties of nuclide number densities, the two cases also differ greatly. Therefore, the influence of the correlations among fissioning systems on the quantification of uncertainty cannot be ignored. This section also adopts the method like Section 2 to calculate the joint covariances among independent fission yields of the three fissioning systems based on the fission yield samples calculated with GEF model parameter samples. The method to verify the joint covariances is also like Section 2. The fission yield samples of three fissioning systems are obtained based on the resampling with joint covariances, and the samples of three fissioning systems are used simultaneously in one random run of the TMI-1 problem to calculate the uncertainties. The reference uncertainty results are also based on fission yield samples calculated with GEF. The comparison of uncertainty results is shown in Figure 6. The uncertainty results of resampling show good consistency with the reference results, which proves that the GEF calculation process for the correlations among fissioning systems is also correct. Thus, when nuclear data users need the covariances of fission yield data, the GEF code can provide covariance data for individual fissioning systems and multiple fissioning systems at the same time.

Preliminary adjustment of GEF model parameters
This section adopts the Bayesian Monte Carlo method to adjust 23 parameters with normal distribution in GEF2021.1.1 which are shown in Table 1 and presents some preliminary numerical results.

Bayesian Monte Carlo method
The Bayesian Monte Carlo method combines Bayesian inference theory with Monte Carlo sampling and adjusts the prior nuclear data by calculating the weight of each sample.
Based on the work in this section, Bayesian inference theory is shown in equation (2): where σ is the prior GEF model parameter vector and  (3): where C i is the calculated fission product inventory sample vector based on prior parameter sample i; V E is the covariance matrix of experimental data. Then, i  , the weight of sample i is calculated with equation (4): We can obtain the posterior model parameters and the posterior parameter covariances through equation (5) and equation (6): , , 1,......, (6) where N is the number of samples; N  is the number of model parameters; i l  is the i-th prior sample of model parameter l.
The general steps for adjusting GEF model parameters are: (1) Sample GEF model parameters.
(2) Calculate fission yield samples with the above parameter samples.

Generation of pseudo-experimental data
The Bayesian Monte Carlo method has high requirements for experimental data, which requires that experimental data and corresponding calculated data have high consistency [10]. In addition, equation (3) shows that the covariances of experimental data are needed. Nevertheless, common integral experimental data, such as nuclide inventory data in SFCOMPO-2.0 spent fuel experimental database [16], usually have some outliers, and these experimental data do not contain covariances, which will seriously affect the effect of nuclear data adjustment. Therefore, this section generates pseudoexperimental data based on the fission yield data in the evaluated nuclear data files (ENDF). The accuracy of pseudo-experimental data is high, and covariance data are obtained. The specific steps of generation are: In the process of generating pseudo-experimental data for this work, fission yield data of 235 U thermal neutron-induced fission in JEFF-3.3 evaluated nuclear data library [17] are selected for sampling, and the sample size is 10000. The burnup problem is the TMI-1 pin cell and the calculation tool is NECP-X code. The nuclide number densities of 13 fission product nuclides at 60 GWꞏd/tU are taken as the final pseudoexperimental data. These nuclides are: 95 Mo, 99 Tc, 101 Ru, 103 Rh, 133 Cs, 143,145 Nd, 151,152 Sm, 151,153,155 Eu, and 155 Gd. These nuclides are important in the uncertainty analysis of the burnup credit system [18], so they are selected. The correlation matrix of the pseudo-experimental data of this work is shown in Figure 7. Correlations are shown for the number densities of different nuclides.

Numerical results
235 U thermal neutron-induced fission is selected as the fissioning system for adjustment of GEF model parameters. The number of model parameter samples is 10000 and each run of GEF uses 2×10 5 fission events. To maintain consistency with the pseudo-experimental data, the calculation of GEF-based prior fission yield samples also uses the burnup data of JEFF-3.3.
The relative adjustment of model parameters is shown in Figure 8. The results show that the adjustment range of all parameters is reasonable and within the prior uncertainties. The uncertainty results of prior and posterior parameters are shown in Figure 9. For most parameters, the posterior uncertainties are reduced, and the effect is more obvious for parameters with larger prior uncertainties.  The fission yield covariances of 235 U thermal neutron-induced fission calculated by GEF and the uncertainties of burnup-related responses calculated based on prior and posterior parameters are compared. The prior and posterior lower triangular correlation matrices of fission yields are shown in Figure 10. There are 67 fission product nuclides compared: 87 Ce. The fission yields of these nuclides are generally greater than 1%, so the calculation accuracy of GEF is higher. Figure  10 shows that the posterior correlations have decreased for most nuclides. Figure 11 shows the prior and posterior uncertainty results of the TMI-1 problem based on fission yield samples of 235 U thermal neutroninduced fission calculated by GEF. The basic setup for the calculation is the same as in Figure 2. The results show that the posterior uncertainties of k inf and nuclide number densities are significantly reduced. The reduction of uncertainty of k inf up to about 80 pcm throughout the process of irradiation. The above numerical results prove that the preliminary adjustment of GEF model parameters is effective.

Conclusions and perspectives
The independent fission yield covariances are generated based on the GEF model code in this paper. The samples of fission yield are calculated by sampling model parameters, and the covariances are calculated based on the samples of fission yield. The above process is carried out for 235 U, 239 Pu, and 241 Pu thermal neutron-induced fissioning systems, individually. To verify the correctness of the process, the fission yield samples calculated by GEF and the fission yield samples generated based on the covariances of this work are respectively applied to the random run of the TMI-1 pin cell burnup problem. The uncertainties of the infinite multiplication factor (k inf ) and nuclide number densities propagated by two types of fission yield samples are in good consistency for three fissioning systems separately. The correctness of the method of generating fission yield covariance is verified.
The influence of correlations among fissioning systems is also analyzed and quantified. The numerical results show that the influence cannot be ignored. The joint independent fission yield covariances of multiple fissioning systems are calculated with GEF and verified. Finally, the parameters of the GEF model code are preliminarily adjusted with the Bayesian Monte Carlo method based on the pseudo-experimental data. The results show that the adjustment was effective.
The following conclusions are summarized: (1) Independent fission yield covariances are obtained based on the sampling of GEF model parameters.
The influence of correlation among fissioning systems cannot be ignored. (2) When nuclear data users need the covariances of fission yield data, the GEF code can provide correct covariance data for individual fissioning systems and multiple fissioning systems simultaneously. (3) Preliminary adjustment of GEF model parameters is performed with the Bayesian Monte Carlo method and the effect is good.
In future work, we will do more numerical tests on the current nuclear data adjustment method and strive to improve the applicability of the method. We will try to apply the adjustment method to other nuclear data, not limited to fission yield-related data.