From fission yield measurements to evaluation: propositions of 235 U(n th ,f) Fission Yields for JEFF-4

. The study of fission yields has a major impact on the characterization and understanding of the fission process and its applications. The fission products have a direct influence on the amount of neutron poisons that limit the fuel burnup and on the evaluation of the decay heat of the reactor after shutdown. Fission yield evaluation represents the synthesis of experimental and theoretical knowledge in order to perform the best estimation of mass and independent fission product yields. Today, the lack of correlations between the different fission observables induces some inconsistencies in the evaluations. For instance, the mass yield uncertainties are drastically overestimated while this observable is the best known. In this last decade, different covariance matrices have been proposed but the experimental part of those are neglected. A consistent covariance matrix depends on the global process of evaluation from data up to the models used. The collaboration develops a new methodology in the field of fission products for the future version of the JEFF-library coupling the analyses of the CEA and NNL institutes.


Available fission yield covariance matrices
Several fission observables related to the fission process have been experimentally studied. Starting from the scission point up to the decay products, these include: -Fission fragment mass yields (preneutron) ( * ) -Fission fragment isotopic distributions ( | * ) (pre-neutron) -Neutron multiplicity ( * ) -Post-neutron emission mass yields ( ) (fission products) -Post-neutron emission isotopic distribution ( | ) -Independent yields ( , , ) -Isomeric ratio ( , , ) -Kinetic energy distributions ( | , ) -Cumulative yields ( , , ) -Chain yields ( ) Historically, JEFF evaluations are based on experimental data associated with phenomenological models (Brosa [1], Zp model [2], Madland England [3]) to extend the range of fission products up to isotopes of interest for the applications. Thus, we find the same classification in the sorting of the experimental data. Some data are also model-dependent as several chain yields for instance. Up to now, the evaluation of the independent fission product yields (post-neutron emission) and the cumulative yields respect the * Corresponding author: gregoire.kessedjian@cea.fr conservation laws but the uncertainty of each observable is based on the experimental data.
During the last decades, to fill the lack of a covariance matrix respecting the difference between the large independent yield uncertainties and the precise chain mass yields, several methodologies were proposed.
-C. Devillers [4] proposed in 1977 a sum method of covariance in order to respect the precise chain yield uncertainties. This method is applied by K. Tsubakihara et al. [5] for the JENDL library, L. Fiorito et al. [6,7] for the JEFF one [28] and T. Kawano et al. [8] followed by M.T. Pigni et al. [9] for the ENDF/B-VII.1 library. This method consists to determine the covariance elements between the independent yields ( , , ) to obtain the variance of cumulated yield evaluation ( , , ). This method assumes a null correlation between cumulative yields. This method only allows the filling of the lacks between independent and cumulative yields in order to respect the compatibility of the uncertainties of the two evaluations. Nevertheless, the hierarchy between the observables and the propagation of conservation laws is not properly defined. -In 2015, N. Terranova et al. [29] proposed a new approach considering the physical fission process to describe the covariance of the independent fission yields. Based on Brosa's model [1] and the Wahl parametrization of the Zp model [2] to describe the fission fragment distributions, the neutron emission yields ( * ) are adjusted in order to describe the JEFF-3.1.1 independent fission product yields. The covariance matrix is obtained using the generalized perturbation theory. The cumulative yields are deduced from the previous calculations. Nevertheless, the calculated uncertainties of the chain yields are overestimated in comparison to those of JEFF-3.1.1. -Another approach distributed by the NEA is the GEF modeling [10] based on a Brosa-like fission fragment description and many phenomenological parameters to describe a very large range of fissioning systems. The Monte Carlo code allows the generation of covariance of fission yields for one fissioning system and between fissioning systems [11] Each approach considers the independent and cumulative fission yields as a collection of values represented by random variables. These variables are independently defined by the experimental methods and the constraints of the analysis. Nevertheless, for thermal neutron-induced fission, the mass yields are often more precise than the independent (isotopic) yields due to the physical properties: at low energy ⁄ ≲ 1 the ionic charge shields the nuclear charge. Then the description of isotopic distribution is more difficult than mass yields [12]. It is exactly the opposite in certain inverse kinematic experiments where nuclear charge yields are nicely resolved [13]. Therefore, in the case of thermal neutron-induced fission, for the 235 U(nth,f) reaction, the mass yields are a complete ensemble whereas isotopic distributions are partially measured. Thus, the independent yields will be necessarily model-dependent to provide the complete and normalized distributions.
Where ( , | , ) is the probability to populate the metastable or ground state of the nucleus. According to eq. 1, the self-normalizations of isotopic and isomeric distributions result in the covariance matrix of independent yields being encapsulated in the covariance matrix of mass yields. In other words, the sum over nuclear charges of the covariance of the independent yields must provide the covariance of the mass yield evaluation. The covariance matrices of all observables are driven by the conservation laws of each observable, combined according to eq.1. The good precision of available mass yield measurements in the EXFOR database will govern the correlation of independent yields. Fig.2 estimated the mean anticorrelation requested between two isotopes of the same mass respecting the following equation: For a mass yield uncertainty of 5% and independent yield uncertainties of ~10% (according to JEFF3.3 evaluation) the mean anti-correlation expected is -0.55 to preserve the sum rule (red arrows in Fig. 2). Thus, the structure of the correlation matrix of the independent yields will be a superposition of the covariance matrix of evaluated mass yields and those from isotopic and isomeric distributions. If the evaluation of the independent yields includes models (e.g. isotopic distribution), the covariance due to the model parameters has to be embedded in the covariance structure driven by eq. 1. Up to now, no independent yield evaluation or model provides the constraints due to the hierarchy of the experimental data. Note that this logic is respected in the approach proposed by C. Devillers (and the following works) concerning the consistency of the independent yield summation to provide the chain yields. In the following work, we develop a complete definition of the covariance matrix of each observable up to cumulative yields in order to take into account these constraints.

New mass yield evaluation 2.1 Database
According to the mathematical description of independent yields (see eq.1), the common covariance term between two different yields comes at least from the covariance term of mass yields. Consequently, a complete evaluation of experimental data including assumptions on experimental correlations is required. For this, we select the measurement achieved with an absolute mass resolution in order to limit the possible bias on the mass structures. For this work, the datasets used are sorted according to the different kinds of experiments: -Cumulative yields by gamma spectrometry [14] -Cumulative yields using radiochemical separation [15,16], -Cumulative yields using magnetic separation criteria [17][18][19] -Independent yields using magnetic spectrometers (typically the HIAWATHA or LOHENGRIN spectrometers) [12,[20][21][22][23]

Re-normalization of data and ranking
The procedure of evaluation is driven by the generalized chi-squared test of compatibility of the a priori weighted mean values of data according to two axes: -Compatibility of measurements per mass (Fig. 4) -Compatibility of each dataset with the a priori weighted mean values (Fig. 5)   Fig. 4. Local 2 ( ) per mass as a function of the mass in comparison to the limit (black circles) for a 0.997 confidence level in two cases: using provided uncertainties (blue dots) and adding 2% systematic uncertainties (i.e. with associated correlations, red circles).  ii) If the generalized chi-squared test ² is rejected, the first solution corresponds to adding uncertainties to get a positive test. This solution is named the "conservative" solution considering that the mismatch between the data is due to an underestimation of uncertainties. The limit of additional relative uncertainty is 2.5%, using the limit of precision obtained with the Lohengrin spectrometer in the last study concerning the mass yields [24]. iii) If the generalized chi-squared test ² is rejected, the second solution is the sorting of each mass yield measurement using its contribution to the ² . While the test is negative, the main contributor per dataset is eliminated with an iterative process. This solution is called the "Strict" one. Using these different approaches, we ensure that the fission yield observables follow Gaussian distributions for the two proposed solutions. Thus, mean values, variances and correlation matrix fully describe the evaluated observables. Fig. 7 presents the multiplicity of available measurements as a function of mass for the different sorting of data. In order to correctly differentiate the mass yield measurements and the chain yield measurements, the analysis takes into account the isotopic distribution analysis (cf. §3) and the Q evolution matrix in order to correctly interpret the chain yields. Fig. 7. Multiplicity of mass yield measurements as a function of mass for the total data number (database), the conservative selection and the strict selection.  (Figs. 8 & 9), even considering additional uncertainty of 2.5%, for the masses 128, 134, 135 and 153, a choice has to be made in order to obtain a consistent Gaussian-like evaluation. For the three first masses, only one measurement of each mass can be considered as an outlier due to the comments in publications. The other measurements of masses 128, 134, and 135 are consistent. For mass 153, there are two inconsistent groups of two measurements considering the 2.5% additional uncertainty. Thus, we cannot settle the debate but we recommend the value which gives the lowest statistical ² value.    10. This is the same as Fig.9, but for the strict evaluation. Except for the symmetrical region, there is a good agreement with the JEFF 3.3 evaluation.

Conservative and strict solutions
The strict solution corresponds to data selection validating the ² tests. Therefore we eliminate the mass yield measurements starting from the greatest contribution per mass to the ² . Thus, with this method, we exclude 5 measurements to satisfy the 2 ( ) per dataset and 34 measurements to obtain an acceptable local 2 ( ) per mass. Fig. 10 presents the strict evaluation in comparison to the JEFF-3.3 evaluation. For the final data selection, based on the full correlation matrix of datasets (Fig. 11), the total 2 over all measurements, the local 2 ( ) per mass and the 2 ( ) per dataset have to satisfy the statistical test for a confidence level of 0.997 corresponding to ±3 for a Gaussian-like distribution. The P-value corresponds to the probability that the 2 random variable is greater than this the 2 value obtained [26].

11.
Correlations between renormalized measurements 〈 ( )〉. The red square shows the correlations of measurements for the same dataset, contributing to the 2 ( ) value. The orange arrows show the correlations of different measurements for the same mass A from different datasets, contributing to the local 2 ( ) per mass. Fig. 12 presents the uncertainties of each evaluation. According to the used datasets, the two proposed solutions represent the upper and lower limits of uncertainties for the mass yield evaluation. We note that Table 1 shows the statistical tests of compatibility of our propositions of evaluation with the mean values of JEFF-3.3 [28] or ENDF\B-VIII.0 [9] evaluations and the GEF-2021/1.1 model [10]. We note that only the JEFF-3.3 mass yields are in agreement with our "conservative" evaluation. All the tests are negative (P-Values < (1-CL), for 0.997 confidence level) except for the compatibility of the JEFF-3.3 mean values with our conservative evaluation.

Impact of experimental correlation in the evaluations
The force of the proposed method is the capability to test the impact of the experimental covariance matrices (Fig.  6). Then, for the datasets of H. Thierens et al. [14], M. Shima et al. [16], A. Bail et al. [21] and R. B. Strittmatter et al. [22], it is possible to build the experimental covariance matrices based on the information on the systematic uncertainties or the reference measurement uncertainties. Even if there are assumptions in these covariance estimations, we observe an important disparity between all datasets. In particular, the option of null covariance when there is no information is always a weakness.  Then, we tested the impact of correlations of experimental data in the statistical test (Fig.4) for the sorting of data and the uncertainty propagation. In this calculation, we arbitrarily add 1% to 3% of systematic uncertainties (filling in the covariance elements) and we compare the results for the same data selection. The analyzed values are the mean values, the relative uncertainties and the correlation matrix (Figs 13). 2 presents the Shannon entropy calculated using the eigenvalues of the different correlation matrices [25].
We note that the systematic uncertainties are a negligible impact on the mean values. Moreover, Table  2 shows that the information provided by the correlation matrix is similar for all data sorting or regularization methods (strict or conservative). The correlation matrix of fission yields is not sensitive to the filling of experimental correlations where there are some lacking data. Nevertheless, the additional systematic uncertainties induced an increase of evaluated standard deviations of evaluated mass yields according to Fig. 14. Typically, adding 3% of systematic uncertainties increases the final uncertainties of evaluated fission yields by around 1% after self-normalization without a significant difference in the correlation matrix.

Isotopic and isomeric distributions
In order to correctly interpret the chain mass yields, we need to consider the isotopic distribution per mass according to eq. 1. In this approach, we consider that the JEFF-3.3 evaluation represents a realistic synthesis of experimental knowledge of isotopic measurements. In this library, the existing measurements are completed with the GEF model in order to extrapolate distributions beyond the 3 or 4 most produced nuclear charges per mass. Thus, in this proposed approach for the JEFF 4 library, we combine our analysis of mass yields to the isotopic distribution per mass from JEFF-3.3: this is the "mixed method" based on the capitalisation of the maximum of knowledge available from datasets today. Then, to extract the self-normalized isotopic distribution from JEFF-3.3 and their correlation, we assume this library as a synthesis of knowledge about the N(A,Z) independent rates. Applying the conservations laws, we obtain: -The symmetry of nuclear charge yields between the light charge and heavy charge : With + = 92 -The isotopic rates respecting the nuclear charge conservation All these normalizations are already done in the JEFF-3.3 evaluation but we apply the generalized perturbation theory [26] in order to generate the covariance matrix at each step ( Fig. 15 & 16). In this work, the N(A,Z) independent rates are assumed without any correlation. Experimentally, it is consistent with what we observed in the previous part concerning the mass yield evaluation: the mix of many independent datasets generates a correlation matrix of experimental mean values close to the identity matrix. In future work, we will test this assumption with a complete description of the correlation matrix associated with isotopic rates [27]. The impact of our complete uncertainty propagation on independent yields Y(A,Z,m) induces a negligible change on the initial JEFF-3.3 uncertainties but ensures consistency between independent yields through the correlation matrix.

Cumulative yield evaluation
To take into account the chain yields in the (independent) mass yield analysis, we have to calculate the ( , , ) cumulative yields from the independent yields. This observable is calculated using the decay data defining the Q matrix: With: Where is the identity matrix and is the matrix of decay corresponding to the list of ( , , ) isotopes. The matrix is deduced from the JEFF-3.3 decay data file [28]. Using this equation, we can introduce a correction factor between the mass yield datasets and the chain yield datasets in our analysis process presented in Fig. 6. The convergence of the sorting of data is obtained after 2 iterations due to the low correction applied about 0% to 1%, close to the uncertainty, except for the 136 Cs with a difference of 7 %. Fig. 18. Comparison of cumulative yield uncertainties from our conservative process of evaluation (black circles), the JEFF-3.3 chain yield evaluation (red points) and the calculated cumulative yield uncertainties from JEFF-3.3 independent fission yields.
The last step corresponds to the uncertainty propagation from the covariance matrix of independent yields ( , , ) up to the covariance matrix of ( , , ) cumulative yields. Fig. 17 presents the cumulative yields of the strict analysis in comparison to the JEFF-3.3. According to the standard deviation of our evaluation, we observe a good agreement. Then, we calculated the propagation uncertainty up to chain yields using the covariance matrix developed. For the conservative analysis, we obtain a limit of precision of 2.5 % in the best case compared to about 1% accuracy in JEFF-3.3 (Fig. 18). Note that the independent yields and the chain yields of JEFF-3.3 are two different evaluations whereas in our analysis both observables are extracted from a single analysis. Fig. 19 presents the relative uncertainties of the chain yields for the strict evaluation. We find the same precision level as JEFF-3.3. The process used in this work allows us to claim that the strict evaluation is the best estimation of the mass yields assuming no bias in the chain yields. In Figs 20 & 21, we also calculated the chain yield uncertainty from the JEFF-3.3 independent yields. Then we note that without a correlation matrix, the level of uncertainties of the chain yields remains at the level of those of independent yields. Fig.20 represents the correlation matrix of the chain yields closest to the correlation matrix of the independent yields (Fig. 16).

Conclusion and Perspectives
The proposed analysis method of several fission observables allows us to determine a consistent evaluation of the independent and cumulative yields. In this global method, we conclude that the sorting of the data is an essential milestone to reach the best estimation of fission yields, limiting the bias and uncertainties. If there is no bias in the very accurate measurements of chain yields, then the consistency of our work with the JEFF-3.3 library allows us to conclude that the strict evaluation corresponds to the best assessment of the 235 U(nth,f) independent fission yields. Nevertheless, that requires us to exclude a part of the available measurements. On the other hand, to limit the potential bias of evaluation, the conservative method proposed a synthesis of the actual knowledge taking into account the inconsistency of the datasets used. These two evaluations represent the upper and lower limits on the precision of these observables.