The licensing of new reactors implies the use of verified and validated neutronic codes. Numerical validation can rely on sensitivity and uncertainty studies, but they require repeated execution of time-consuming neutron flux and depletion calculations. The computational costs can be reduced by using perturbation theories. However, the uncoupled Depletion Perturbation Theory is restricted to single integral values such as nuclide density. Relying on reduced-basis approaches, which reconstruct all nuclide densities at once, is one way to get around this restriction. Furthermore, the adjoint-based reduced-order model uses the direct and adjoint equations for projection. For diffusion or transport calculations, the Exact-to-Precision Generalized Perturbation Theory was developed. Still, no models for depletion calculations are readily available. Therefore, this paper describes a novel adjoint-based reduced-order model for the Bateman Equation. It uses a range-finding algorithm to create the basis and the uncoupled Depletion Perturbation Theory for the reconstruction of the first order replaced by with a first order formulation. Our paper shows that for several perturbed cases, the depletion reduced-order model successfully reconstructs the nuclide densities. As a result, this serves as a proof of concept for our adjoint-based reduced-order model, which can perform sensitivity and uncertainty burn-up analysis in a shorter time.

Reactor design and uncertainties propagation request repeated execution of neutron diffusion calculations. To address this, one can rely on the Generalized Perturbation Theory which is restricted to single integral values such as reaction rate ratios. To overpass such limitations, the perturbed flux could be calculated with the modal expansion method. The difficulty to calculate more than one eigenvector has limited its use. Hence, modal expansion lacks a clear explanation of its strengths and weaknesses. Using recent advances in eigenvalues solvers, we have successfully applied modal expansion to the 2D-BIBLIS and IAEA-3D benchmarks. Our results are promising for fission and capture cross sections perturbations. Still, modal expansion is faulty for local perturbations or when it concerns the reflector. In terms of computational cost, modal expansion is more efficient than solving the direct problem with a calculation time divided by 8 for a basis of 20 eigenvectors.

Incorrect representations of flux variations in a reactor core can introduce significant errors in the resulting calculation. In cases with strong energetic flux variations, it becomes important to refine the deterministic calculation to have a better description of resonances. Refining the energy mesh for the whole core can become quite computationally costly. Domain decomposition methods and parallel architecture seem to offer a solution, by allowing the user to refine only part of the problem. The present work aims to provide a domain decomposition method in a deterministic solver with the capability of better describing local heterogeneities and strong transients by adapting the energy refinement. The goal is to accomplish this without sacrificing calculation time. This paper investigates the use of domain decomposition methods with cross sections of different energy meshes within the IDT deterministic solver. In this novel method, subdomains have different energy discretizations and no overlap. Boundary fluxes are condensed or reconstructed to account for spectral differences between subdomains to make the boundary condition fully consistent across the whole core. Multi-grid cases, where subdomains have different energy discretizations are compared to classic cases with the same energy discretizations over the entire configuration. This method has been tested on two cases with different reflectors, to investigate the potential for this method to deal with heavy-material reflector design such as iron reflectors in modern PWR. Two boundary flux expressions have been examined. The gains in precision and computation time using this methodology have been estimated on a core configuration inspired by the well-known C5G7-benchmark. Preliminary results are promising both in terms of time and precision.(c) 2022 Elsevier Ltd. All rights reserved.

The CONRAD code is an object-oriented software tool developed at CEA since 2005. It aims at providing nuclear reaction model calculations, data assimilation procedures based on Bayesian inference and a proper framework to treat all uncertainties involved in the nuclear data evaluation process: experimental uncertainties (statistical and systematic) as well as model parameter uncertainties. This paper will present the status of CONRAD-V1 developments concerning the theoretical and evaluation aspects. Each development is illustrated with examples and calculations were validated by comparison with existing codes (SAMMY, REFIT, ECIS, TALYS) or by comparison with experiment. At the end of this paper, a general perspective for CONRAD (concerning the evaluation and theoretical modules) and actual developments will be presented.

In 2012 CEA produced a entire new evaluation of sodium nuclear data for the release of the JEFF-3.2 evaluated nuclear data library. During the evaluation process performed with the CONRAD code, several differential measurements (total and discrete inelastic cross-sections) have been used. However double differential data (elastic angular distribution) that were yet available in the EXFOR database were not incorporated in the analysis at that time. The experimental elastic angular distribution were discarded because of it was impossible to obtain a good agreement for both angle-integrated cross-sections and double differential ones. The underlying cause of this disagreement is expected to be due to the attribution of quantum numbers to resonance and related channel amplitudes. Indeed these numbers are imposed during the analysis but impact differently angular distributions and angle-integrated cross-sections. An automated search for an accurate set of quantum numbers has been implemented in order to produce a reliable quantum numbers set. In this paper we present a new evaluation of Na-23 taking into account both differential and double differential measurements. The analysis performed with the CONRAD code reached the level of agreement with experimental data for the total and inelastic cross-sections but this time with a significant improvement for the elastic angular distributions. This new evaluation produced in the ENDF-6 format has then been tested and validated on critical facilities calculation (MASURCA and ZPPR) in different configurations (nominal and voided) in order to assess its performances.

Due to the importance of radiation damage of materials in nuclear facilities, it is of interest to accurately calculate the damage and the corresponding uncertainty. The CONRAD code is developed at CEA Cadarache for evaluating nuclear data and corresponding uncertainties using various methods. Different Displacement per Atom (DPA) models have been recently implemented in CONRAD to perform the calculation, the sensitivity analysis, and the uncertainty propagation of radiation damage cross sections. The current version of CONRAD is able to compute damage cross section and the associated covariance matrix from nuclear model parameters and from DPA model parameters. In addition to the sensitivities of damage cross section of 56Fe to different nuclear model parameters and DPA model parameters, the present work illustrates the importance of a complete covariance matrix between different nuclear data, while these correlations are normally not given in the current evaluated nuclear data libraries.

Within the framework of the V&V of the new APOLLO3 French deterministic code, this paper focusses on its ability of analyzing integral mock-up experiments through IDT flux solver. In the present case, the UH1.2 experiments involving voiding effects in standard PWR UOX lattices are investigated through straightforward 281 groups 3D transport calculations using recent capabilities of IDT solver based on Sn short-characteristics method. Instead of using standard 2D calculations with imposed experimental Bzwe performed directly 3D 281 groups IDT calculations and checked the results against reference continuous-energy Monte-Carlo (TRIPOLI4) results in a simplified axial configuration. In both cases the IDT solver (with dedicated self-shielding model) shows an excellent agreement with Monte-Carlo results regarding both reactivity and fission rate reference parameters. In particular it is shown that linear short characteristics and refined spatial meshes (3x3x1) are necessary to find the best agreement (less than 1% fission rate and void effect errors). The next step of this work will be devoted to the analysis of the 3D TDTMOC solver in the same configurations with ad-hoc axial reflector treatment. To remove the limitations in terms of CPU time and memory resources of this kind of calculations, ongoing developments devoted to hybrid shared/distributed parallelization (OpenMP/MPI) are also pointed out.

Correlations between neutron inelastic scatterings angular distributions are not included in the Joint Evaluated Fission and Fusion (JEFF) nuclear data library, while they are key quantities for uncertainty propagation of nuclear data. By reproducing the angle-integrated cross sections and uncertainties of JEFF-3.1.1, the present work obtains covariance matrix between high energy model parameters using the least square method implemented in the CONRAD code. With this matrix, it is possible to generate correlations between angle-integrated cross sections and angular distributions, which are usually presented by Legendre coefficients. As expected, strong correlations are found, for example, between the Legendre coefficients of elastic and first-level-inelastic scatterings and the angle-integrated total, elastic, total inelastic cross sections.

Design of modern sodium fast reactors such as the French SFR project ASTRID are axially heterogeneous in order to maximize neutron leakage and therefore improve their natural behavior during transients. This feature is however challenging for neutronic calculations that usually rely on a first step based upon the generation of averaged cross sections over planar geometries. It is in particular the case of the APOLLO3 (R) reference calculation route that is currently considered for SFR applications. This paper investigates the corresponding biases in the case of a 3D fissile assembly calculation and highlights the consequences of the fundamental mode assumption. It is shown that even though an acceptable global accuracy can be achieved, the flux distribution presents a slight axial drift. The homogenization and energy collapsing of anisotropic cross sections over radial geometries is pointed responsible for it. Several proposals are made to correct this inconsistency, leading to new calculation schemes and enhanced performances. (C) 2018 Elsevier Ltd. All rights reserved.

This paper presents a comparison of homogenization techniques implemented in the APOLLO3 platform for transport core calculations: standard scalar flux weighting and new flux–moment homogenization, in different combinations with (or without) leakage models. Besides the historical B1-homogeneous model, a new B-heterogeneous one has indeed been implemented recently in the two/three-dimensional-transport solver using the method of characteristics. First analyses have been performed on a very simple Sodium Fast Reactor core with a regular hexagonal lattice. They show that using the heterogeneous leakage model in association with flux-moment homogenization strongly improves the prediction of keff and void reactivity effects. These good results are confirmed when the application is done to the fissile assemblies of the more complex CFV (Low Void Effect) core of the ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) project of sodium-cooled fast breeder reactor (Generation IV).

A consistent description of the neutron cross sections from thermal energy up to the MeV region is challenging. One of the first steps consists in optimizing the optical model parameters using average resonance parameters, such as the neutron strength functions. They can be derived from a statistical analysis of the resolved resonance parameters, or calculated with the generalized form of the SPRT method by using scattering matrix elements provided by optical model calculations. One of the difficulties is to establish the contributions of the direct and compound nucleus reactions. This problem was solved by using a slightly modified average R-Matrix formula with an equivalent hard sphere radius deduced from the phase shift originating from the potential. The performances of the proposed formalism are illustrated with results obtained for the 238U+n nuclear systems.

Fission product yields (FY) are fundamental nuclear data for several applications, including decay heat, shielding, dosimetry, burn-up calculations. To be safe and sustainable, modern and future nuclear systems require accurate knowledge on reactor parameters, with reduced margins of uncertainty. Present nuclear data libraries for FY do not provide consistent and complete uncertainty information which are limited, in many cases, to only variances. In the present work we propose a methodology to evaluate covariance matrices for thermal and fast neutron induced fission yields. The semi-empirical models adopted to evaluate the JEFF-3.1.1 FY library have been used in the Generalized Least Square Method available in CONRAD (COde for Nuclear Reaction Analysis and Data assimilation) to generate covariance matrices for several fissioning systems such as the thermal fission of U235, Pu239 and Pu241 and the fast fission of U238, Pu239 and Pu240. The impact of such covariances on nuclear applications has been estimated using deterministic and Monte Carlo uncertainty propagation techniques. We studied the effects on decay heat and reactivity loss uncertainty estimation for simplified test case geometries, such as PWR and SFR pin-cells. The impact on existing nuclear reactors, such as the Jules Horowitz Reactor under construction at CEA-Cadarache, has also been considered.

As model parameters, necessary ingredients of theoretical models, are not always predicted by theory, a formal mathematical framework associated to the evaluation work is needed to obtain the best set of parameters (resonance parameters, optical models, fission barrier, average width, multigroup cross sections) with Bayesian statistical inference by comparing theory to experiment. The formal rule related to this methodology is to estimate the posterior density probability function of a set of parameters by solving an equation of the following type: pdf(posterior) ∼ pdf(prior) × a likelihood function. A fitting procedure can be seen as an estimation of the posterior density probability of a set of parameters (referred as x→) knowing a prior information on these parameters and a likelihood which gives the probability density function of observing a data set knowing x→. To solve this problem, two major paths could be taken: add approximations and hypothesis and obtain an equation to be solved numerically (minimum of a cost function or Generalized least Square method, referred as GLS) or use Monte-Carlo sampling of all prior distributions and estimate the final posterior distribution. Monte Carlo methods are natural solution for Bayesian inference problems. They avoid approximations (existing in traditional adjustment procedure based on chi-square minimization) and propose alternative in the choice of probability density distribution for priors and likelihoods. This paper will propose the use of what we are calling Bayesian Monte Carlo (referred as BMC in the rest of the manuscript) in the whole energy range from thermal, resonance and continuum range for all nuclear reaction models at these energies. Algorithms will be presented based on Monte-Carlo sampling and Markov chain. The objectives of BMC are to propose a reference calculation for validating the GLS calculations and approximations, to test probability density distributions effects and to provide the framework of finding global minimum if several local minimums exist. Application to resolved resonance, unresolved resonance and continuum evaluation as well as multigroup cross section data assimilation will be presented.

As the need for precise handling of nuclear data covariances grows ever stronger, no information about covariances of prompt fission neutron spectra (PFNS) are available in the evaluated library JEFF-3.2, although present in ENDF/B-VII.1 and JENDL-4.0 libraries for the main fissile isotopes. The aim of this work is to provide an estimation of covariance matrices related to PFNS, in the frame of some commonly used models for the evaluated files, such as the Maxwellian spectrum, the Watt spectrum, or the Madland-Nix spectrum. The evaluation of PFNS through these models involves an adjustment of model parameters to available experimental data, and the calculation of the spectrum variance-covariance matrix arising from experimental uncertainties. We present the results for thermal neutron induced fission of 235 U. The systematic experimental uncertainties are propagated via the marginalization technique available in the CONRAD code. They are of great influence on the final covariance matrix, and therefore, on the spectrum uncertainty band width. In addition to this covariance estimation work, we have also investigated the importance on a reactor calculation of the fission spectrum model choice. A study of the vessel fluence depending on the PFNS model is presented. This is done through the propagation of neutrons emitted from a fission source in a simplified PWR using the TRIPOLI-4 ® code. This last study includes thermal fission spectra from the FIFRELIN Monte-Carlo code dedicated to the simulation of prompt particles emission during fission.

Benchmark measurements of irradiated and un-irradiated fuel samples were performed in the framework of the CERES collaborative program between AEA (UK Atomic Energy Agency) and CEA (French Atomic Energy Commission). These experiments provide relevant data for the validation of fuel burn-up and criticality-safety calculations for the whole fuel cycle. As part of this program, pile-oscillation measurements were carried out on a range of mixed-oxide samples with plutonium of various mass and isotopic contents, both in the MINERVE and DIMPLE reactors. Four core configurations, two over-thermalized situations and two pressurized water reactor (PWR)-type situations, were constituted with different forward and adjoint flux spectra, emphasizing fission and/or capture contributions. The experiments were analyzed using reference TRIPOLI4 calculations with the JEFF-3.1.1 library, using exact three-dimensional models of the core configurations. In a first step, calculations of each DIMPLE configuration were performed and compared with the experiment, showing very good agreements with a maximum C-E of −230 pcm. In the second step, reactivity worth experiments were analyzed, using recently developed exact-perturbation capabilities in TRIPOLI4. A consistent assimilation of calculation over experiment discrepancies was performed with the CONRAD code, using the integral data assimilation method. Covariance matrix on multigroup neutron cross sections and multiplicities were generated and significant trends were identified, especially on the 239Pu and 240Pu capture cross sections in the thermal energy range (E < 0.1 eV). Further investigations should be required to confirm these conclusions, due to the strong dependence of these trends and of posterior covariances to prior covariances.

This paper deals with the validation of the multiple scattering corrections developed in the CONRAD code for the capture yield calculations in the Resolved Resonance energy Range (RRR). In order to calculate the capture yields, analytic and stochastic calculation schemes implemented in CONRAD are described and compared with the analysis code SAMMY/SAMSMC. The results are in excellent agreement for a variety of samples. We concentrate the discussion here on 238U, 197Au and 55Mn.

The JEFF-3.1.1 Nuclear Data Library is the latest version of the Joint Evaluated Fission and Fusion Library. We present the status of the validation of this library using the Monte Carlo Code TRIPOLI 4.5 and the deterministic code package ERANOS 2.2 for fast reactor calculations. For this purpose, we reanalyze a selected set of integral experiments performed in the MASURCA mock-up (CEA/CADARACHE), in the ZPPR mock-up at ANL (USA), and in the SUPERPHENIX Power Reactor. Furthermore, we also present the analysis of pure sample irradiation experiments PROFIL and PROFIL-2 performed in the PHENIX reactor, as this kind of experiment provides a direct feedback on nuclear capture data. We observe good performances of these calculation tools for criticality calculations and fuel inventory prediction. From this validation work, some required improvements on nuclear data are highlighted, as well as the need for new specific integral experiments. The main trends observed are the following: —Reactivity of clean and fresh cores: the results obtained with JEFF-3.1.1 are consistent with those obtained using JEFF-3.1 within 80 pcm, but there is an overestimation of the calculated reactivity of all the experiments between 40 and 800 pcm depending on the spectrum hardness (Pu content) and fuel composition, the discrepancy being larger in hard spectra. Additional investigations are in progress to understand this behaviour. —Integral capture cross sections (PROFIL and PROFIL-2):/C/E - 1/≤ 3%, except for 241Pu (C/E ≈ 1:08), 242Pu (C/E ≈ 1:18), 237Np (C/E ≈ 0:92), 243Am (C/E ≈ 0:93), 244Cm (C/E ≈ 1:35); impact of the trends observed on the individual fission products put in PROFIL and PROFIL-2 is ≈ 4% on the part of Δρburnup due to fission products.