Feedback from experimental isotopic compositions of used nuclear fuels on neutron cross sections and cumulative fission yields of the JEFF-3.1.1 library by using integral data assimilation

. Comparisons of calculated and experimental isotopic compositions of used nuclear fuels can provide valuable information on the quality of nuclear data involved in neutronic calculations. The experimental database used in the present study (cid:1) containing more than a thousand isotopic ratio measurements for UOX and MOX fuels with burnup ranging from 10 GWd/t up to 85 GWd/t (cid:1) allowed to investigate 45 isotopic ratios covering a large number of actinides (U, Np, Pu, Am and Cm) and ﬁ ssion products (Nd, Cs, Sm, Eu, Gd, Ru, Ce, Tc, Mo, Ag and Rh). The Integral Data Assimilation procedure implemented in the CONRAD code was used to provide nuclear data trends with realistic uncertainties for Pressurized Water Reactors (PWRs) applications. Results con ﬁ rm the quality of the 235 U, 239 Pu and 241 Pu neutron capture cross sections available in the JEFF-3.1.1 library; slight increases of +1.2 ±2.4%, +0.5 ±2.2% and +1.2 ±4.2% are respectively suggested, these all being within the limits of the quoted uncertainties. Additional trends on the capture cross sections were also obtained for other actinides ( 236 U, 238 Pu, 240 Pu, 242 Pu, 241 Am, 243 Am, 245 Cm) and ﬁ ssion products ( 103 Rh, 153 Eu, 154 Eu) as well as for the 238 U(n,2n) and 237 Np(n,2n) reactions. Meaningful trends for the cumulative ﬁ ssion yields of 144 Ce, 133 Cs, 137 Cs and 106 Ru for the 235 U(n th ,f) and 239 Pu(n th ,f) reactions are also reported.


Introduction
Numerous studies report comparisons of calculated (C) and experimental (E) isotopic compositions of used fuels for nuclear data validation purposes. In this work, we have used an experimental database that mainly contains proprietary data obtained from French PWRs fleet. Our database contains 1370 isotopic ratio measurements for UOX and MOX fuels with burnup ranging from 10 GWd/t up to 85 GWd/t. It allows to investigate 45 isotopic ratios for a large number of actinides (U, Np, Pu, Am and Cm) and fission products (Nd, Cs, Sm, Eu, Gd, Ru, Ce, Tc, Mo, Ag and Rh).
Interpretations of post-irradiation experiments (PIEs) of samples irradiated in nuclear power reactors can be performed with the deterministic calculation package DARWIN2.3 [1], which is a calculation tool designed for fuel cycle applications. It solves the Boltzmann and Bateman equations to compute fuel cycle parameters at any irradiation and cooling time. A flow chart of the DARWIN2.3 package for PWR calculations is presented in Figure 1 [1]. DARWIN2.3 includes the APOLLO2 deterministic transport code [2], which provides neutron data to the PEPIN2 depletion solver [3], namely selfshielded cross-sections libraries and multi-group neutron fluxes.
State-of-the-art calculated-to-experimental (C/E-1) ratios obtained with the JEFF-3.1.1 library [4] are summarized in reference [1]. The objectives of the present study are to include most of these C/E-1 ratios in the Integral Data Assimilation procedure of the CONRAD code [5,6] and to provide valuable trends on the nuclear data compiled in the JEFF-3.1.1 library. Guidelines to achieve these objectives and strategies to take into account uncertainties coming from experiments, nuclear data and numerical biases are discussed in references [7][8][9]. The originality of our approach lies in the use of the AGS code method [10] to generate covariances between the C/E-1 ratios and of the marginalization procedure [11] to propagate uncertainties of "nuisance" parameters. Final results indicate that reliable trends can be obtained for the capture cross sections of some important actinides ( 235 U, 236 U, 238 Pu, 239 Pu, 240 Pu, 241 Pu, 242 Pu, 241 Am,

Integral Data Assimilation procedure
The Integral Data Assimilation procedure implemented in the CONRAD code [5,6] is based on a two-step calculation scheme. The first step is a least-square fitting procedure, namely the analytic resolution of a standard generalized least-square equation [12], that consists in adjusting model parameters (i.e. nuclear data) on a given set of integral values. The second step is a marginalization procedure designed to propagate nuisance parameter uncertainties after the fitting procedure.

Governing equations
For a correct use of the Integral Data Assimilation procedure of the CONRAD code, two types of model parameters have to be defined, namely observable and nuisance parameters. The observable parameters are free variables whose values are adjusted during the fitting procedure. The nuisance parameters are fixed model parameters with known uncertainties. Nuisance parameter uncertainties are taken into account via the marginalization technique [11]. The algorithm consists in building a "full" covariance matrix S between the observable and nuisance parameters as follows [13]: where and with x being the set of adjusted parameters, M x being the posterior covariance matrix associated to x, u being the set of nuisance parameters and M u the corresponding covariance matrix. Matrices G x and G u contain the partial derivatives of the calculated values with respect to the observable and nuisance parameters:

Used nuclear fuel data and uncertainties
Integral values included in the fitting procedure are C/E-1 ratios obtained from the interpretation of PIEs with the   Figure 2. This burnup is representative of the burnup of PWR assemblies at discharge. More information can be found in references [1,14]. According to data reported in Figure 2, the concentrations of the major actinides 235 U, 239 Pu and 241 Pu at 40 GWd/t are accurately calculated for UOX fuels with the JEFF-3.1.1 library. The mean 〈C/Ei-1 ratios are À0.4 ± 3.7%, 0.0 ± 0.8% and À0.8 ± 1.8%, respectively. Average results obtained for MOX fuels remain consistent within the quoted uncertainties, except for the 241 Pu/ 238 U ratio. For both types of fuel, the largest C/E-1 ratios are obtained for the curium isotopes. These discrepancies, reaching about À50% for 247 Cm in UOX fuels, or discrepancies observed between UOX and MOX fuels, can be explained by using the Integral Data Assimilation procedure of the CONRAD code. The calculated-to-experimental ratios considered in this work share the same sources of systematic uncertainties (parameters p 1 to p 4 detailed in the next paragraph). Therefore, the AGS code method [10] has been used to estimate the covariance matrix M C/EÀ1 between the C/E-1 values. The originality of the AGS code method is to combine uncorrelated and correlated uncertainties as follows: in which D is a n Â n diagonal matrix containing the variances of the uncorrelated uncertainties and S is a n Â m rectangular matrix containing the correlated contributions for each of the j ∈ ⟦1, m⟧ sources of uncertainty. No correlation between the experimental isotopic ratios was provided alongside the results of the chemical analyses. Therefore, the covariance matrix D is a diagonal matrix that only contains the experimental uncertainties coming from the measurement process.
in which var (C i /E i À 1) represents the uncertainty associated with the ith experiment. For the S-matrix, four sources of systematic uncertainties are considered:   . . . with: Parameters p 1 and p 2 are related to the fuel and moderator temperatures, respectively. Their contributions D 1,i and D 2,i are calculated with DARWIN2.3 from direct perturbations of the parameters, by considering an uncertainty of ±50°C for the fuel temperature and ±2°C for the moderator temperature at 1s. Parameters p 3 and p 4 are numerical scheme-related uncertainties, depending on the use of DARWIN2.3. The former corresponds to the differences that could be obtained between deterministic and Monte-Carlo calculation schemes. It was estimated  from pin-cell depletion calculations performed with the deterministic code APOLLO2 and the Monte-Carlo code TRIPOLI-4 [15], with the same filiation chain. The latter corresponds to the differences due to resonant upscattering effects [16] on fuel inventory calculations. Examples of differences obtained on the calculated isotopic ratios for a UOX fuel (3.7 wt.% 235 U) at 40 GWd/t are given in Figure 3. Most of the obtained differences lie below 1%. The comparison of the deterministic and Monte-Carlo calculations schemes indicates that the differences are significant for the curium isotopes and non-negligible for 137 Cs. Among the fission products, 137 Cs seems to be an isolated case needing more extensive studies. As expected, in the case of the up-scattering effect, non-negligible increases of the concentration of all the actinides are observed. The final correlation matrix between the C/E-1 values reconstructed with the AGS code method (Eq. (6)) is shown in Figure 4. An interesting feature is the correlations between the UOX and MOX fuel data that introduces constraints on the variations of the observable parameters during the fitting procedure.

Observable parameters
Isotopic ratios depend on a wide number of nuclear data. Therefore, the CYRUS tool [17] was used, coupled with PEPIN2 depletion solver, to establish an exhaustive list of nuclear data involved in the build-up of actinides and fission products.
The selected set of observable parameters that will be adjusted with the CONRAD code is given in Table 2. It was chosen not to adjust fission cross sections, because it would imply modifying the burnup of the analysed samples. This is incompatible with the fact that the burnup is a fixed parameter for each sample with a known uncertainty of about ±2% (see next section). It was also decided not to fit the decay constants; in our study, decay constants of interest are well known, with quoted uncertainties lesser than 0.6% (see Tab. 3). Eventually, it was chosen not to fit some nuclear data (e.g. fission yields on samarium isotopes, or samarium cross sections) because the number of free parameters was too important with respect to the number of constraints, thus leading to an underdetermined system.
Parameter y c (X,A) indicates the thermal cumulative fission yield of nuclide X for actinide A. For reducing the number of free parameters, the (n,g) and (n,2n) cross sections are effective cross sections which are averaged over the neutron flux '(E): Prior uncertainties for neutron cross sections come from the COMAC-V2.0 covariance matrix database [18,19], developed at CEA Cadarache. For fission yields, they come from the JEFF-3.1.1 library.

Nuisance parameters
Nuisance parameters are fixed parameters whose uncertainties are propagated via the marginalization procedure of the CONRAD code (Sect. 2.1). The list of the nuisance parameters introduced in the CONRAD calculation is given in Table 3. It gathers all nuclear data involved in the buildup of nuclides considered, determined with the CYRUS tool, except the ones fitted (see previous section). The cross section uncertainties come from the COMAC-V2.0 database. Uncertainties for the decay data (periods, isomeric ratios and branching ratios) and fission yields are directly taken from the JEFF-3.1.1 library.
Nuisance parameters are not only the nuclear data involved in the build-up of actinides and fission products, but also experimental parameters such as the burnup scaling parameter. The relative uncertainty associated to the latter parameter is close to ±2%.

Nuclear data adjustment trends and uncertainties
Nuclear data adjustment trends provided by the Integral Data Assimilation of our used nuclear fuel database are reported in Table 4. Results are listed by ascending order of adjustment coefficient. In particular, those obtained for the europium capture cross sections and cumulative fission yield of 137 Cs confirm the results previously reported in references [8,9]. Table 4 also compares the relative uncertainties obtained before and after the marginalization of the nuisance parameters (Sect. 2.4). For most of the posterior values, the fitting uncertainties are lower than 1%. The marginalization provides more realistic uncertainties that allow discussing the quality of some cumulative fission yields and cross sections recommended in the JEFF-3.1.1 library and hence the content of the covariance database COMAC-V2.0.

Cumulative fission yields
Adjustment trends on a few number of cumulative fission yields were extracted from the Integral Data Assimilation of our used nuclear fuel data. This work provide information on the cumulative fission yields of 106 Ru, 144 Ce, 133 Cs and 137 Cs for both 235 U(n,f) and 239 Pu(n,f) reactions with relative uncertainties ranging from 1.3% to 5.7%. These trends can be directly applied to the  Table 4 provide the following results:

Neutron cross sections
Results obtained for the (n,g) and (n,2n) reactions can be divided in three groups.
-The first group concerns the reactions for which the adjustment trends remain within the uncertainties obtained after marginalization. This can be observed for six actinides and two fission products: 235 U(n,g), 236 U(n,g), 239 Pu(n,g), 241 Pu(n,g), 244 Cm(n,g), 246 Cm(n,g), 133 Cs(n,g) and 99 Tc(n,g). The case of technetium should be considered with caution because of the large uncertainty obtained (±7.8%). This first group refers to reactions that are correctly described in JEFF-3.1.1 or for which minor corrections are expected.
-The second group concerns the reactions for which the adjustment trends are higher than the uncertainty obtained after marginalization, while remaining lower than 5%. This is the case of three actinides: 238 U(n,2n), 240 Pu(n,g) and 242 Pu(n,g). -The third group concerns the reactions for which the adjustment trends are higher than the uncertainty obtained after marginalization and also higher than 5%. Six actinides and three fission products fulfil these conditions: 237 Np(n,2n), 241 Am(n,g), 243 Am(n,g), 242 Cm(n,g), 245 Cm(n,g), 247 Cm(n,g), 153 Eu(n,g), 154 Eu(n,g) and 103 Rh(n,g). Results obtained for the 247 Cm(n,g) and 237 Np(n,2n) reactions only indicate that such reactions are poorly known. Our used nuclear fuel data cannot provide accurate information to improve them. This latest group provides key information on reactions that have to be corrected in priority for improving significantly the DARWIN2.3 calculations for fuel cycle applications. However, adjustment trends reported in Table 4 need to be interpreted by examining the evolution with the burnup of the posterior calculated-to-experimental ratios provided by CONRAD at the end of the Integral Data Assimilation procedure. 241 Am and 243 Am are discussed below. Figures 5 and 6 show the prior and posterior C/E-1 values for the ratio 241 Am/ 238 U in UOX and MOX fuels respectively, from 20 GWd/t to 85 GWd/t. The review of the posterior results reveals a slight improvement in average for UOX fuels, while the average C/E discrepancy is slightly degraded for MOX fuels. This result might be explained by the non-negligible dispersion of the calculated-to-experimental values, especially for MOX fuels. Final uncertainties in Table 4 do not reflect the potential inconsistencies between the used nuclear fuel data.
The production chain of 244 Cm (Fig. 7) shows that the impact of 243 Am capture cross section can be quantified on the calculated-to-experimental ratios of 244 Cm/ 238 U. The prior and posterior C/E-1 values for UOX and MOX fuels are reported in Figures 8 and 9. The posterior values are closer to zero in average. However, a dependence of the results with the burnup can be observed. Below approximately 30 GWd/t, the 244 Cm/ 238 U ratio is still underestimated. This could be explained by the low amount of 244 Cm in the fuel at low burnup, suggesting that the experimental values at low burnup should be considered with caution. Hence, when considering only high burnup data on Figures 8 and 9, for which the 244 Cm content becomes more significant, one can observe a slight overestimation of the 244 Cm/ 238 U isotopic ratio. This indicates that the increase of the 243 Am capture cross section suggested in Table 4 is overcompensating the initial C/E-1 values. In that case, the obtained adjustment trend of +9.4 ± 2.2% should be considered as an upper limit.

Feedback on the COMAC-V2.0 library
Results reported in Table 4 can be used to assess the quality of the COMAC-V2.0 database: the mean posterior values obtained in this work should presumably remain within the limit of the prior uncertainties provided by COMAC-V2.0, otherwise the latter is underestimated. On the contrary, a prior uncertainty value which is significantly larger than the posterior trend may indicate that the prior uncertainty is overestimated. When comparing the posterior values with the prior uncertainties, it seems indeed that the prior uncertainties for the capture cross sections of 133 Cs, 236 U, 242 Pu, 244 Cm and 246 Cm are overestimated. As an example, the COMAC-V2.0 library provides a relative uncertainty for the 242 Pu(n,g) reaction which is close to 12%, while the present study suggests a slight increase of the capture cross section of +3.8 ± 2.8%. On the other hand, the COMAC-V2.0 database seems to underestimate the uncertainties of the 103 Rh(n,g), 241 Am(n,g) and 243 Am(n,g) reactions. This feedback constitutes guidelines for the improvement of the COMAC-V2.0 database.

Discussions of the results
Results provided by the Integral Data Assimilation procedure and presented in Section 3 were obtained using the JEFF-3.1.1 library. Since its official release in 2009, numerous validation studies were conducted on this library; furthermore new experimental data and evaluated nuclear data files are now available in the literature. This section compares results obtained in the present work with those reported in the literature and the international libraries in order to draw some recommendations for improving the JEFF library.  Tables 5 and 6. We have selected some works that provide nuclear data adjustment trends which are associated to the JEFF-3.1.1 library. From an integral point of view, many oscillations and activation experiments were carried out in the MINERVE Fig. 5. Evolution of the calculated-to-experimental discrepancies before and after integral data assimilation for the 241 Am/ 238 U isotopic ratio in UOX fuels.

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A. Rizzo et al.: EPJ Nuclear Sci. Technol. 5, 24 (2019) reactor of CEA Cadarache in order to obtain an insight on nuclear data adjustment trends for a large amount of isotopes. OSMOSE [20,21], CERES [22], BUC [23] and MAESTRO [24][25][26] are programs specifically dedicated to actinides, MOX fuels, fission products and structural materials respectively. PIEs of samples irradiated in thermal spectra (such as the ICARE and SHERWOOD programs carried out in the MELUSINE reactor of CEA Grenoble [27]) and in fast spectra (such as the PROFIL program carried out in the PHENIX reactor of CEA Marcoule [28]) provide integral information similar to used nuclear fuel data, namely isotopic ratios. One must pay attention that the reported C/E-1 values are not adjustment trends on nuclear data. This means that adjustment  [29], 243 Am [30] and 153 Eu [31] at the n_TOF and RPI facilities were considered. Resonance parameters extracted from these data provide useful information on the capture resonance integral. For the fission yields, results obtained at the ILL with the Lohengrin spectrometer [32][33][34] are listed.
From the evaluation point of view, it was chosen to focus on the case of 103 Rh, for which a new evaluation has been recently performed at IRSN (Institut de Radioprotection et de Sûreté Nucléaire) with the SAMMY code [35].
In addition to the above-mentioned works, several studies were conducted based on the analyses of integral experiments. Two recent Integral Data Assimilation studies were performed on actinides with the RDN code using criticality-safety benchmarks in association to used nuclear fuel data [36], and integral data provided by the CERES program [22]. Posterior results are reported in Table 5. Finally, a nuclear data trend for the 154 Eu(n,g) reaction has been also obtained from the analysis of used nuclear fuel data [37]. Results are reported in Table 6.
A detailed review of the results listed in Table 5 and 6 leads to three major conclusions.
-For the cumulative fission yields, trends suggested by our Integral Data Analysis are fully consistent with the ILL results. The agreement between the values remains within the limit of the reported uncertainties, which represents an encouraging result. An exhaustive comparison with a wider number of cumulative fission yields measured in various places could provide good indications about the robustness of our work. -For the capture cross sections of the fission products, most of the reported results are also consistent, excepted those related to the 103 Rh(n,g) reaction. The large inconsistencies between the BUC and MAESTRO experiments carried out in the MINERVE reactor justify the replacement of the 103 Rh evaluation by a new evaluated nuclear data file, such as the one produced at IRSN [35]. -The last conclusion concerns the actinides for which the numerous inconsistencies observed between the reported integral data can lead to erroneous conclusions. For some reactions, this confused situation can be clarified with microscopic data, such as recent time-of-flight experiments carried out in the RPI and n_TOF facilities. Examples given in Table 5 and 6 for the 242 Pu(n,g), 243 Am(n,g) and 153 Eu(n,g) reactions provide capture resonance integrals in favor of our Integral Data Assimilation results.
This comparative study indicates that our work provides consistent results for some actinides, fission products and cumulative fission yields; the present set of nuclear data adjustment trends represents a reliable guideline for improving the JEFF library.

Recommended evaluated nuclear data files and resonance parameters
The nuclear data adjustment trends suggested by our Integral Data Assimilation procedure can be used to propose cross section evaluations that could improve calculations related to the fuel cycle. Table 7 gives a list of recommended Evaluated Nuclear Data files or resonance parameters. We only consider capture reactions for which meaningful adjustment trends were obtained. Cumulative fission yields are not discussed because the evaluation process is more complex than the one for cross sections, since it generally relies on an experimental database containing a large number of independent and cumulative fission yield measurements. Adjustment trends obtained on fission yields can nevertheless be used for the validation of the next fission yield evaluation releases. Two types of recommendations can be distinguished. The first one consists in proposing existing Evaluated Nuclear Data files whose resonance parameters and neutron cross sections fulfil the nuclear data adjustment trends suggested by our study. This is the case for the 236 U, 238 U, 241 Am and 154 Eu nuclides. The proposed evaluations come from new or old versions of the JEFF library or from the Japanese library JENDL-4.0. Justifications of these choices are given in Table 7. One can precise that the recommended evaluation for 154 Eu capture cross-section relies on a work carried out in [40], which aims at assessing Fig. 9. Evolution of the calculated-to-experimental discrepancies before and after integral data assimilation for the 244 Cm/ 238 U isotopic ratio in MOX fuels.  [20,21] PROFIL program [28] ICARE and SHERWOOD program [27] Santamarina and Bernard [36] CERES program [22] n_TOF 242 Pu [29] n_TOF 243 Am [30] 238 Pu(n,g) -7.5 ± 7.0% -0.6 ± 2.6% +0.9 ± 5.6%   MAESTRO program [24][25][26] 1984 [32] 2009 [33] 2017 [34] 153 Eu [31] 103 Rh [35] 154 Eu [37] y

Conclusions
In this paper, comparisons of calculated and experimental isotopic compositions of used nuclear fuels are carried out to provide information on the accuracy of nuclear data from the JEFF-3.1.1 library for fuel cycle applications. The method applied relies on both a fitting algorithm and a marginalization procedure implemented in the CONRAD code.
Final results confirm the accuracy of 235 U, 239 Pu and 241 Pu neutron capture cross sections available in the JEFF-3.1.1 library. Slight increases of +1.2 ± 2.4%, +0.5 ± 2.2% and +1.2 ± 4.2% are respectively suggested, these all remaining within the limit of the quoted uncertainties. Results indicate the good quality of 236 U, 244 Cm, 246 Cm and 133 Cs capture cross sections as well, since the adjustment trends obtained remain within the quoted uncertainties. Our study also suggest to slightly increase 238 U(n,2n), 240 Pu(n,g) and 242 Pu(n,g), adjustment trends obtained being lower than +5%. Adjustment trends greater than +5% are obtained for 237 Np(n,2n), 241 Am(n,g), 243 Am (n,g), 242 Cm(n,g), 245 Cm(n,g), 247 Cm(n,g), 153 Eu(n,g), 154 Eu(n,g) and 103 Rh(n,g).
Meaningful adjustment trends for the cumulative fission yields of 144 Ce, 133 Cs, 137 Cs and 106 Ru for the 235 U(n th ,f) and 239 Pu(n th ,f) reactions, ranging from À8% to +9% approximately, are also reported.
Comparisons with data from the literature show a good agreement of the adjustment trends obtained on cumulative fission yields with experiments recently conducted at the ILL facility, and show a global agreement of the adjustment trends obtained on neutron cross sections with integral and microscopic experiments carried out in the past. Based on these results, several cross section evaluation are recommended to improve fuel cycle Table 7. Evaluated nuclear data files and new resonance parameters in agreement with the nuclear data adjustment trends provided by our Integral Data Assimilation study.

IDA results
Recommended evaluation or resonance parameters Comments 236 U(n,g) +2.2 ± 1.9% JENDL-4.0 JENDL-4.0 suggests an increase of +2% of the capture resonance integral. 242 Pu(n,g) +3.8 ± 2.8% Resonance parameters from reference [29] Recent time-of-flight measurements carried out at the n_TOF facility propose an increase of about +4% of the capture resonance integral. 240 Pu(n,g) +4.2 ± 2.8% Resonance parameters from reference [39] Recent resonance analysis suggests a slight increase of about +0.6% of the capture area of the 1st resonance. 238 U(n,2n) +4.8 ± 2.0% JEFF-3.3 [38] JEFF-3.3 suggests an increase of about +3%. 243 Am(n,g) +9.4 ± 2.2% Resonance parameters from reference [30] As discussed in Section 5, the obtained trend seems to be too high. 241 Am(n,g) +10.7 ± 4.7% JEFF-3.3 [38] (taken from JEFF-3. 2) The capture cross section was already improved in the JEFF-3.2 library. 153 Eu(n,g) +5.1 ± 2.4% Resonance parameters from reference [31] Resonance parameters extracted from RPI data are in agreement with our study. Possible improvements were already addressed in a previous study [9]. 103 Rh(n,g) +7.4 ± 4.7% Resonance parameters from reference [35] New resonance analysis, performed at IRSN Fontenay aux Roses, suggests an increase of +11% of the capture resonance integral, which is consistent with our results. 154 Eu(n,g) +8.4 ± 3.7% JEFF-3.0 The analysis from reference [40] suggests to use the resonance parameters of JEFF-3.0 calculations: these evaluations may come from other international libraries or be based on resonance parameters recently measured at time-of-flight facilities. The evaluations recommended in this paper represent a valuable feedback of the fuel cycle applications that can be used for the improvement of the future JEFF library releases and more generally for the improvement of nuclear data.