Use of integral data assimilation and differential measurements as a contribution to improve 235 U and U cross sections evaluations in the fast and epithermal energy range

. Critical mass calculations of various HEU-fueled fast reactors result in large discrepancies in C/E values, depending on the nuclear data library used and the con ﬁ guration modeled. Thus, it seems relevant to use integral experiments to try to reassess cross sections that might be responsible for such a dispersion in critical mass results. This work makes use of the Generalized Least Square method to solve Bayes equation, as implemented in the CONRAD code. Experimental database used includes ICSBEP Uranium based critical experiments and bene ﬁ ts from recent re-analyses of MASURCA and FCA-IX criticality experiments (with Monte-Carlo calculations) and of PROFIL irradiation experiments. These last ones provide very speci ﬁ c information on 235 U and 238 U capture cross sections. Due to high experimental uncertainties associated to ﬁ ssion spectra, we chose to consider either ﬁ tting these data or set them to JEFF-3.1.1 evaluations. The work focused on JEFF-3.1.1 235 U and 238 U evaluations and results presented in this paper for 235 U capture and 238 U capture, and inelastic cross sections are compared to recent differential experiment or recent evaluations. Our integral experimentassimilationworknotablysuggestsa30%decreasefor 235 Ucapturearound1 – 2.25keV,a10%increasein theunresolvedresonancerangewhenusingJEFF-3.1.1as “ apriori ” data.Theseresultsareinagreementwithrecent microscopic measurements from Danon et al. [Nucl. Sci. Eng. 187 , 291 (2017)] and Jandel et al. [Phys. Rev. Lett. 109 , 202506 (2012)]. For 238 U cross sections, results are highly dependent on ﬁ ssion spectra.


Introduction
Critical mass calculations of various HEU-fueled fast reactors result in large discrepancies in C/E values, depending on flux spectra, fuel enrichment, structural materials present and so on. These C/E values, calculated with the Monte-Carlo code TRIPOLI-4 [1], are shown in Figure 1. Table 1 gives some specifications about fuel and structural materials present in each configuration. Figure 1 underlines that critical mass C/E values for Uranium-fueled configurations of Fast Reactors calculated with JEFF-3.2 library are systematically overestimated (except for BIGTEN and GODIVA) and are larger than those calculated with JEFF-3.1.1. Discrepancy between the two sets of calculations goes from ∼250 to ∼630 pcm for BIGTEN. Moreover, large C/E values are observed for FCA-IX configurations (overestimation up to ∼800 pcm), BIGTEN and GODIVA when using the JEFF-3.1.1 library. Except for FLATTOP-235 U, all critical masses for configurations with HEU fuel exceed experimental uncertainties when using JEFF-3.1.1. Although MASURCA 1B and FCA-IX configurations [2] have similar spectra (as they both contain graphite) but significantly different Uranium enrichments and geometries, the large discrepancy observed in their C/E values (using either JEFF-3.1.1 or JEFF-3.2) rise concerns of possible compensating errors between 235 U and 238 U evaluations in the JEFF libraries in the fast and epithermal energy range.

Integral experiments assimilation
Considering the very large C/E values presented in Section 1, it seems relevant to use integral data assimilation to identify which nuclear data are responsible for these discrepancies. This was performed using the CONRAD code from CEA [3], which can solve analytically Bayes' theorem.

Bayesian inference
As a reminder, Bayes' theorem [4] generalized to continuous probability densities is given: where the vector x contains the parameters to be reassessed (in our case, the 33-group cross sections) in the view of new observations enclosed in the vector y. U gathers all the "background" information, that is, hypotheses or approximations made to obtain the values for x and y.
In practice, probability densities associated to each multigroup cross-section are assumed to be Gaussian distributions, as this choice maximizes the entropy [5]. Using Laplace approximation [6], we then assume that the posterior probability density function solution of equation (1) can be well-approximated by a Gaussian distribution: where E is a vector containing integral measurements values, C is a vector containing associated calculated values, M s is the covariance matrix associated to nuclear data s, M E is the covariance matrix associated to C/E values. For a Gaussian distribution, the central value is associated to its maximum. Thus, optimal solutions for s and associated covariances, M s are determined by minimizing a cost function (using Generalized Least Square method): 2.2 Integral data assimilation strategy and results for posterior C/E The JEFF-3.1.1 library was chosen as the a priori as it gives more satisfying results than JEFF-3.2 for Uranium configurations sensitive to the fast energy range, as seen in Figure 1. For our assimilation work, we used critical mass C/E of MASURCA 1B, FCA-IX cores 1-7, FLATTOP-235 U and GODIVA, as well as variations of concentration ratios C/E from PROFIL-2A irradiation experiments [7,8]. Experimental correlation matrix for FCA-IX configurations is provided in reference [2]. Experimental correlation matrix for PROFIL experiments is given in C/E used in the assimilation work were calculated using the Monte-Carlo code TRIPOLI-4 (except for PROFIL's variation of concentrations ratios, calculated with ECCO/ ERANOS) and 33-group sensitivity coefficients to nuclear data were calculated using the ECCO/ERANOS code [9]. For nuclear data covariance matrices, we used COMACV1.0 [10], except for 235 U n for which we used the COMMARA-2.0 matrix [11].
Critical mass C/E values for these configurations provide a great variety of sensitivity profiles to 235 U capture and 238 U capture and inelastic cross sections (this is shown for 235 U capture in Fig. 2). Using all these C/E values with their associated sensitivity coefficients in a single assimilation calculation allows us to make the most of both the redundant or complementary information they provide for the whole fast energy range.
Notably, the simultaneous use of GODIVA and FLATTOP-235 U critical masses can help avoiding compensations between 235 U and 238 U cross sections, as these fast spectrum critical configurations are similar, except for the presence of natural Uranium reflector in FLATTOP-235 U. Critical mass sensitivities to 238 U inelastic and capture and 235 U capture cross sections for these two configurations are shown in Figure 3. One can see that critical mass sensitivities to 235 U cross sections are similar whereas sensitivity coefficients to 238 U cross sections are important for FLATTOP-235 U and low for GODIVA.
The nuclear data fitted through assimilation are 235 U and 238 U capture, elastic, inelastic 33-groups cross sections as well as their fission spectrum x (unless specified otherwise) and multiplicity n. 235 U and 238 U fission were not fitted, as JEFF-3.1.1 evaluations are in good agreement with Neutron Standard from IAEA [12] for these cross sections. Also, it should be noted that assimilation work does not take into account sensitivities to angular distributions as no covariance matrices are currently available for these data. Taking into account these approximations through marginalization is the topic of future works.
In this integral data assimilation work, an effort was made to try to reduce risks of compensating errors by relying on the Neutron Cross-section Standards [12] for 235 U and 238 U fission cross sections and by using PROFIL-2A C/E (which add a specific constraint on 235 U or 238 U capture cross sections). Nevertheless, as this will be shown in the following sections, high uncertainties associated to fission spectra can have a significant impact on assimilation result. Also, as differences in JEFF-3.1.1 and JENDL-4.0 carbon evaluations were found to have a non-negligible impact for some critical masses used in this work (Tab. 3), we ran CONRAD calculations for both of these options. For these reasons, the results presented in Section 3 are sets of trends that include the four alternatives considered: fission spectra fitted or not and carbon evaluation either from JEFF-3.1.1 or JENDL-4.0. Assimilation trends are presented in this manner to stress that the variability in the results due to these choices can be seen as additional uncertainties.
Experimental correlations between FCA-IX critical mass C/E were taken into account using the matrix provided by JAEA [2]. Also, correlations between PROFIL irradiation experiments were calculated. Figure 4 displays post-assimilation C/E for critical masses compared with prior JEFF-3.1.1C/E values for the case where fission spectra are set to JEFF-3.1.1 and JEFF-3.1.1 graphite evaluation is used. A priori and a posteriori C/E values for the PROFIL irradiation experiment are given in Table 4, along with experimental uncertainties.
Post-assimilation C/E values are well-included in 1s experimental uncertainties, except for MASURCA 1B and FCA-IX 6, which however remain in 2s total uncertainties. This means there exists an optimal set of cross sections for the experimental database taken into account, and no inconsistency between C/E had been found.

Comparison of assimilation trends with differential measurements
To discuss the reliability of the trends on cross sections suggested through the integral data assimilation, we compared them to recent differential measurements from the EXFOR database [13] when they are available or recent evaluations otherwise. In this section, trends are given relative to JEFF-3.1.1.

235 U capture cross section
Assimilation results suggest a significant modification for 235 U capture: a ∼30% decrease around 1-2 keV and a ∼10% increase in the unresolved resonance range (URR) when using JEFF3.1.1 as "a priori" data. This is shown in Figure 5, along with prior and posterior uncertainties. One can notice that from 1 to 500 keV, posterior uncertainties are sufficiently low to consider assimilation trends as possible recommendations for a change in 235 U capture cross section. As mentioned earlier, the two curves displayed in Figure 5 represent an envelope, in which the assimilation results for the following four cases are included: uncertainties on graphite evaluation choice (JEFF-3.1.1 or JENDL-4.0) and fission spectra (fitted or set to JEFF-3.1.1). For 235 U capture cross sections, differences in posterior uncertainties for these four cases do not exceed 0.5% in the energy range of interest. Thus, only one curve is displayed in Figure 5.   Impact on critical mass À260 pcm À420 pcm À280 pcm À230 pcm Focusing on the end of the resolved resonances range (RRR) from 1 to 2.25 keV, we compared our assimilation trends in this energy range with recent differential measurements made at RPI. Figure 6 displays results of these measurements as published in reference [14] (as they are not currently available in the EXFOR database) with a comparison to ENDF/B-VII and JENDL-4.0. One has to note that for 235 U capture cross section, JEFF-3.1.1 and ENDF/B-VII.1 evaluations are identical. This graph of Figure 6 shows that our assimilation results are in good agreement with Danon measurements at RPI as they suggest a ∼33% decrease of 235 U capture cross section from JEFF-3.1.1 at around 2 keV. This issue on 235 U capture was already addressed in WPEC Subgroup 29 [15], which underlined an overestimation of this cross section in the end of the RRR in the JEFF-3.1 evaluation.
In the URR, from 10 to 100 keV, most recent measurements performed by Jandel et al. [16] at LANSCE with the DANCE detector are consistent with assimilation trends from 3 keV to 1 MeV (Fig. 7).
Comparing now assimilation results to JEFF-3.3t3 [17] (in Fig. 8), one can see that they agree well in the end of the RRR (considering that assimilation results uncertainties in this range is around 9%). In the URR, from 10 to 100 keV, JEFF-3.3t3 evaluation suggests a higher increase from JEFF-3.1.1 (around 20%) than our assimilation results. Figure 9 shows a comparison between Jandel et al. [16] measurements, JEFF-3.3t3 [17] and JEFF-3.1.1 evaluations. Compared to Jandel measurements, it seems that JEFF-3.3t3 235 U capture cross section evaluation is slightly higher whereas JEFF-3.1.1 appears to underestimate this cross section in the 10-100 keV energy range.

238 U capture cross section
Unlike 235 U capture, trends for 238 U capture are highly dependent on fission spectra values. As it can be seen in Figure 10, in the case where fission spectra are fitted through assimilation, resulting trends on 238 U capture are included in posterior uncertainties. When fission spectra  5. Trends from assimilation work for 235 U capture (relative to JEFF-3.1.1) compared with a priori and a posteriori nuclear data uncertainties. The two red dotted curves represent an envelope gathering all the trends suggested by assimilation results (that includes cases with fission spectra fitted or not, and with graphite evaluation from JEFF-3.1.1 or JENDL-4.0). are not fitted and set to JEFF-3.1.1, trends suggested (À4% up to À7% from JEFF-3.1.1) by the assimilation work are higher than posterior uncertainties from 10 keV to 3 MeV. Dependency of the results on fission spectra values is also reflected by the differences in posterior uncertainties for the two cases (Fig. 10) are at the same level as critical masses sensitivity coefficients for this cross section. Moreover, from 100 keV to 1 MeV, these sensitivity coefficients are noticeably lower than those of some critical masses. This is not the case for D 236 U 235 U whose sensitivity profile dominates all the critical mass sensitivity profiles to 235 U capture. The constraint brought by PROFIL-2A C/E on 238 U capture is thus less important than for 235 U capture. This is shown in Figure 11. Also, a priori correlations between 238 U cross sections might amplify the impact of fission spectra on assimilation results.
In the end, the great impact of fission spectra on 238 U capture results suggests possible compensations between 238 U capture and 238 U and 235 U fission spectra in our assimilation work. This assimilation results for 238 U capture cross section are all the more questioning as these can have a significant impact on fast reactor calculations. For instance, the trend suggested by the assimilation (for the case where fission spectra are set to JEFF-3.1.1) has an impact of around +500 pcm on the reactivity of the SFR ASTRID. Details of this impact per energy group (for a 33group sensitivity calculation) are given in Table 5. Thus, considering the high sensitivity of some fast reactors critical masses to this cross section, assimilation results should be clarified, for instance by using a wider experimental database for the assimilation.     10. Trends from assimilation work for 238 U capture (relative to JEFF-3.1.1) compared with a priori and a posteriori nuclear data uncertainties. For both cases (fission spectra fitted or not), the two dotted lines have to be seen as an additional uncertainty associated to the choice of graphite evaluation.

238 U inelastic cross section
As for 238 U capture cross section, trends for 238 U inelastic depend on whether fission spectra are fitted through assimilation or set to JEFF-3.1.1. Indeed, some of the critical configurations that are the most sensitive to 238 U inelastic cross are also the most sensitive to 238 U fission spectrum (FCA-IX 6, FCA-IX 7 and FLATTOP-235 U). Besides, all critical configurations are highly sensitive to 235 U capture.
All sets of trends for 238 U inelastic are shown in Figure 12, along with associated uncertainties. A posteriori uncertainties are sufficiently low in the plateau region (∼1 to 6 MeV) to consider assimilation trends as possible recommendations. For this energy range, assimilation results propose a 4%-8% decrease (from JEFF-3.1.1 238 U inelastic cross section) depending on whether fission spectra are fitted or not. For 238 U inelastic cross sections, differences in posterior uncertainties for these four cases do not exceed 0.5% in the energy range of interest. Thus, only one curve is displayed in Figure 12.
Assimilation results are compared to CIELO [18] (evaluation version of September the 29th, 2017), JEFF-3.1.1 and JEFF-3.3t3 [17] evaluations in Figure 13. Focusing on the plateau region, we observe that CIELO and JEFF-3.3t3 evaluations are both lower than JEFF-3.1.1 in this region, but the level of decrease is different.
Once again, the dependency of assimilation results for 238 U inelastic cross sections on fission spectra is a hint of possible compensation errors in the results. Assimilation work can be improved with the use of a wider database including more C/Es sensitive to 238 U inelastic cross sections.

Conclusion
C/E values from several critical masses calculations and from PROFIL irradiation experiments were used in a Bayesian inference approach as implemented in the CONRAD code to investigate cross sections that might need reassessment. These C/E values provide a great variety of sensitivity profiles to 235 U and 238 U cross sections, including capture and inelastic. Trends suggested for 235 U capture, which are in agreement with recent differential measurements made at RPI and LANSCE, confirm that significant modifications are needed for this cross section in JEFF-3.1.1 (∼30% decrease around 1-2.25 keV and ∼10% increase in the 10-100 keV energy range). This issue was already addressed in WPEC Subgroup 29, which underlined an overestimation of this cross section in the end of the RRR [15]. JEFF-3.3t3 seems to go in the right direction with a decrease of around 25% from JEFF-3.1.1 in the end of RRR and an increase up to 20% in the URR. Comparisons of integral data assimilation results with recent differential measurements constitute a key step in our study as sources of uncertainties are different. For 238 U cross sections, results are highly dependent on whether fission spectra are fitted or not. For 238 U capture cross section, the integral data assimilation suggests a 4%-7% decrease of the cross section from 10 keV to 3 MeV in the case where fission spectra are set to JEFF-3.1.1 evaluations. Such modifications on 238 U capture can have a significant impact on critical mass calculations of Fast Reactors. Thus, these results should be further confirmed by assimilation results using a wider experimental database.
For 238 U inelastic cross sections, integral data assimilation suggests a 4% to 8% decrease in the plateau region (from around 1 to 6 MeV), depending on whether fission spectra are fitted or not. JEFF-3.3t3 and CIELO evaluations also point toward a decrease from JEFF-3.1.1 in this energy region but at different levels. Previous work from Santamarina [19], using the RDN code and targeted on integral measurements with a strong sensitivity to 238 U inelastic cross section (including Pu-fueled systems), suggested a reduction trend of À11% ± 3% (in a case where 238 U fission spectra were not re-estimated).
In the end, this assimilation work focusing on 235 U and 238 U nuclear data with a reduced database enables us to deduce possible trends on these data independently from Pu isotopes nuclear data. Results presented in this work have to be confirmed by the addition of other integral experiments. Notably, trends on 238 U capture and inelastic cross sections might possibly exhibit compensating errors. Besides, posterior uncertainties from this work are probably underestimated: indeed, we did not take into account uncertainty from nuclear data which are not fitted (structural material, fission cross sections, etc.). An attempt to take into account these approximations through marginalization is under study.