Trends on Major Actinides from an Integral Data Assimilation

: Nuclear data evaluations on major actinides can be improved by integral data assimilation. Appropriate integral measurements with reliable experimental techniques have been selected such as ICSBEP, IRPhE and MASURCA critical masses, PROFIL irradiation experiments and the FCA-IX experimental programme (critical masses and spectral indices). Highly reliable analyses are possible with the use of as-built geometries calculated with the TRIPOLI4 Monte Carlo code. The C/E values have been used in an integral data assimilation solving the Bayes equation. The trends on the JEFF3.1.1 235 U capture cross section are quite consistent with recent differential measurements. Assimilation results suggest up to a 2.5% decrease for 238 U capture from 3 keV to 60 keV, and a 4-5% decrease for 238 U inelastic in the plateau region. For this energy range, uncertainties are respectively reduced from 3-4 to 1-2% and from 6-9% to 2-2.5% for 238U capture and 238 U inelastic. Results on 239 Pu fission cross sections are included in posterior uncertainties. The increase trends on 239 Pu capture cross-section is of around 3% in the [2 keV-100 keV] energy range. For 240 Pu capture cross section, the increase is of around 4% in the [3 keV-100 keV] energy range and goes in the same direction as a recent evaluation.


INTRODUCTION
Five out the six Generation IV concepts are Fast Reactors with breeding capability (they produce Pu out of depleted Uranium) and with ability to use possibility Pu and MA, that would otherwise be a waste (Pu from PWR-MOX not suitable for PWR).
In France, 2006 June 28 act requires the design of a generation of sodium fast reactors, those being the most mature Generation IV concept.It is the reason for which CEA with its industrial partners started the design of ASTRID 1,2 (Advanced Sodium Technological Reactor for Industrial Demonstration), a 600 MWe sodium-cooled fast reactor concept.Its construction should demonstrate the feasibility of a SFR design with enhanced safety and its ability to achieve a zero breeding gain hence enabling to use the entire Uranium ore and not only 235  Sodium Fast Reactors require the development and validation of scientific calculation tools as a proof of their characteristics used in their safety dossiers.Nuclear data, the input parameters of the neutronic codes, constitute the main source of uncertainty in neutronic calculations 3 .For instance, uncertainty (in pcm) on critical mass for the ASTRID core (using JEFF3.1.1 and its associated COMAC-V1 nuclear data covariances) amounts up to 1558 pcm.Major actinides are contributing significantly to this uncertainty as Table 1 illustrates.Existing fertile blankets with depleted U induces great sensitivities to 238 U cross-sections, notably inelastic.The use MOx fuel with Pu retrieved from PWR-MOx spent fuel induces high sensitivities not only to 239 Pu cross sections but also to other Pu isotopes cross sections (notably 240 Pu).

STRATEGY USED
Integral Data Assimilation (IDA) can contribute to nuclear data improvement if attention is being given to ways that minimize the possibility of creating compensating errors.This approach has been done in the past and it is important to learn from past integral data assimilation works to develop a strategy that can avoid important compensating errors.

Previous integral data assimilation works
Since the early seventies, several integral experiments assimilation work for fast reactor applications have been conducted.
Among them, the statistical adjustment method (called BARACCA) was used to produce two CARNAVAL libraries (III and IV) 4 with the objectives of answering needs for PHENIX and SUPER-PHENIX respectively.For the adjustment, for each nuclear data, equiprobable probabilities were associated to an interval of possible values.This choice was done because assuming that the discrepancy in C/E values for integral experiments came mainly from normalization issues when performing microscopic measurements.Because, there were no constraints in the a priori multigroup cross sections other than the boundary of the initial interval, work resulted in compensating errors between 238 U capture cross sections and 239 Pu fission cross section.
ERALIB1 5 is a library resulting from an integral data adjustment whose experimental database includes 355 integral data (critical masses, spectral indices, buckling etc.) coming from 71 distinct experimental programs.Prior variances-covariances matrices associated to nuclear data were produced from expert judgements.The improvements made on the assimilation process include:  The use of Gaussian distributions for multigroup nuclear data to perform adjustments based on Bayes Inference using Generalized Least Square Methods. The use of statistical indicators (χ 2 criteria) to identify inconsistent integral data, which need to be eliminated from the experimental database.However, the experimental database included many unreliable measurements and some of covariance were lacking physical meaning (especially for 23 Na).This led in particular to discrepant sodium void worth.

Current Strategy
To avoid deficiencies identified in previous Integral Data Assimilation (IDA) work, a more reliable nuclear data library was used JEFF3.1.1 6 for all isotopes with the exception of 23 Na for which JEFF3.2 was used 7 .The associated COMAC V1 covariance 8 were used as they mimic the evaluation process itself.
Only reliable integral experiments have been selected.They are ICSBEP, IRPhE and MASURCA critical masses, PROFIL irradiation experiments and the FCA-IX experimental programme (critical masses and spectral indices).Highly reliable experiment analyses are now possible avoiding method approximation and using as-built geometries as it is possible with the TRIPOLI4 Monte Carlo code 9 .
Marginalization technique has been used for light and structural isotopes for which approximations (anisotropy, secondary neutron energy distribution) in the integral data assimilation technique are rather high.A test on whether Prompt Fission Neutron Spectrum (PFNS) needs to be fitted or not (such as in ERALIB1).It appears from this test that PFNS is a major source of uncertainty and requires to be fitted 12 .
Attention has been given to ways that minimize compensating errors.Compensating errors between 238 U capture and 239 Pu fission (such as in CARNAVAL IV) has been eliminated by using first only U-fueled experiments then adding Pu-fueled experiments.
The IDA has been using Bayesian inference 10 with a minimization of a cost function (1) in which is the covariance matrix associated to nuclear data and the one associated to integral experiments. ( Solutions for nuclear data are given by equation (2) where S is the sensitivity matrix calculated with the ERANOS code system 11 while associated uncertainties are given by equation (3):

Trends for 235 U Capture
As seen on Figure 1, the trends were a -27 to -33 % decrease at 1-2 keV (end of RRR) which is in agreement with Danon measurements (2017) 13 and a +8 to +10% increase in the 10-100keV region (URR) which is in agreement with Jandel measurements (2012) 14 .Uncertainties are significantly reduced.The consistency of integral data assimilation results on 235 U capture cross section with recent differential measurements encourages us to rely on the simultaneous use of PROFIL C/E and various U235 enriched critical masses to reassess capture cross sections.

Trends for 238 U Capture and Inelastic
For 238 U inelastic and capture cross sections, a comparison of 235 U-238 U assimilation results and U-Pu assimilation results allows us to conclude that the trends proposed for these cross sections are not the result of compensating errors with Pu nuclear data.Assimilation results suggest up to a 2.5% decrease for 238 U capture from 3 keV to 60 keV, and a 4-5% decrease for 238 U inelastic in the plateau region.For these energy range, uncertainties are respectively reduced from 3-4 to 1-2% and from 6-9% to 2-2.5% for 238 U capture and 238 U inelastic.The simultaneous use of GODIVA and FLATTOP-235U and JEZEBEL 239Pu and FLATTOP-239Pu is relevant to reassess 238 U inelastic cross sections.Their critical mass C/E are greatly affected by the reflecting properties of the FLATTOP core depleted Uranium blankets very sensitive to this cross section.
Assimilation results for 238 U capture are compared with differential measurements 15 and with the "a priori" JEFF-3.1.1,CIELO and JEFF-3.3evaluation (Figure 2).Posterior uncertainties for assimilation results are in dotted line.In the plateau region (~1 MeV to 6 MeV), a 5-6% decrease is suggested for 238 U inelastic.This decrease trend is in agreement with recent evaluations (see suggested value compared to CIELO and JEFF-3.3ones on Figure 3; Posterior uncertainties for assimilation results are in dotted line).Notably, the simultaneous use of FLATTOP-U235/GODIVA and FLATTOP-Pu239/JEZEBEL enables us to discriminate contributions from U238 and fissile isotopes.Indeed, GODIVA is a bare U235 sphere while FLATTOP-U235 is surrounded by a U depleted blanket.The same occurs for JEZEBEL and FLATTOP-Pu.

Trends for Pu isotopes
Summary of trends on Pu isotope data, which can a significant impact for the ASTRID core, are given in Table 2.
Table 2. Trends on Pu isotope cross sections Results on 239 Pu fission cross sections are included in posterior uncertainties which have been significantly reduced through the IDA process (from 3.5-2% to 2.5-0.5%) .There is no differential measurement able to reach that type of uncertainty.This means that IDA process will be needed in a way or another.The increase trends on 239 Pu capture cross-section is of around 3% in the [2 keV-100 keV] energy range.From 4keV up to 30 keV, the ENDF/B-VIII evaluation corresponds to JEFF-3.1.1.For 240 Pu capture cross section, the increase is of around 4% in the [3 keV-100 keV] energy range and goes in the same direction as the recent ENDF/B.VIII evaluation though at a much lower level.

IMPACT ON THE ASTRID CFV CORE
There is a significant impact of 238 U capture and inelastic on the critical mass of the ASTRID core but also 239 Pu, 240 Pu and 242 Pu fission cross-sections.With a prior value for the critical mass of 1.02688 (1.02908 with JEFF-3.1.1 for all isotopes except 23 Na for which JEFF3.2 is used: Impact of using 23 Na evaluation from JEFF-3.2 instead of JEFF-3.1.1 on the ASTRID core reactivity: -220 pcm.) the IDA conducted led to a posterior value of 1.02255 (-435 pcm).The Integral Data Assimilation reduces uncertainties associated to nuclear data significantly with, in particular, a significant reduction on 239  Table 3. Impact of the nuclear data trends (in pcm) on the critical mass of the ASTRID core and associated uncertainty (a priori uncertainties using COMAC-V1 nuclear data covariance are also given).

CONCLUSION AND PERSPECTIVES
Integral Data Assimilation (IDA) in this work has proved to be efficient in identifying the sources of possible normalization problems in the differential measurements.This achievement is mainly due to the progresses in identifying the different sources of uncertainties whether they are from nuclear data evaluations themselves (through covariance) or from integral experiments (either for their set up or their modelling).
The trends on the JEFF3.1.1 235 U capture cross section are quite consistent with recent differential measurements.
IDA results suggest also a 2.5% decrease for 238 U capture from 3 keV to 60 keV, and a 4-5% decrease for 238 U inelastic in the plateau region.For this energy range, uncertainties are respectively reduced from 3-4 to 1-2% and from 6-9% to 2-2.5% for 238 U capture and 238 U inelastic.
The increase trend on 239 Pu capture cross section is of around 3% in the [2 keV-100 keV] energy range.For 240 Pu capture cross section, the increase is of around 4% in the [3 keV-100 keV] energy range and goes in the same direction as the recent ENDF/B.VIII evaluation though at a much lower level.
As perspectives for future works, there is a need of more differential measurements and more reliable nuclear data covariance.
For instance, IDA has identified the lack of differential measurements, in particular for prompt fission neutron spectrum or 238  nuclear data covariance data with better reliability are required (for PFNS and some capture cross sections).
There is also a request to have covariance associated to anisotropy of scattering and distribution of secondary energy neutrons in particular for light and structural isotopes.This will enable to improve the IDA by incorporating all sources of uncertainties.Also adding more integral experiments will help increasing the reliability of the IDA, specifically if these are defined to target a given spectrum or nuclide.

Figure 2 :
Figure 2: Assimilation Results compared to major evaluations for 238 U captureBecause Prompt Fission Neutron Spectrum (PFNS) lacks some differential measurements, variances are large.Since integral experiments available cannot distinguish between  and  (PFNS) the possibility of having compensating errors is high.Hence, a parametric study has been conducted with 235 U and 238 U PFNS fitted or not through assimilation.The magnitude of trends on 238 U capture cross sections differ on whether 235 U and 238 U PFNS is fitted or not through assimilation.In the plateau region (~1 MeV to 6 MeV), a 5-6% decrease is suggested for 238 U inelastic.This decrease trend is in agreement with recent evaluations (see suggested value compared to CIELO and JEFF-3.3ones on Figure3; Posterior uncertainties for assimilation results are in dotted line).

Figure 3 :
Figure 3: Assimilation Results compared to major evaluations for 238 U inelastic U.

Table 1 .
Uncertainty breakdown (in pcm) on critical mass for the ASTRID core (using COMAC-V1 nuclear data covariances).
U inelastic and parametric studies have shown that