EXAMENSARBETE INOM TEKNISK FYSIK, AVANCERAD NIVÅ, 60 HP STOCKHOLM, SVERIGE 2018

Quantification of the uncertainty on the secondary sodium activation due to uncertainties on nuclear data

Master thesis report

COPPERE BENJAMIN

KTH SKOLAN FÖR ARKITEKTUR OCH SAMHÄLLSBYGGNAD Résumé

L’activation du sodium secondaire du cœur ASTRID est une problématique majeure du projet car cette activation nécessite la mise en place de protections neutroniques au niveau du cœur et de l’échangeur intermédiaire. Actuellement, le schéma de calcul fourni des résultats « best estimate », c'est-à-dire sans incertitudes. Pour pouvoir justifier l’utilisation de ces résultats dans les différents projets relatifs à ASTRID, il est nécessaire de connaître l’incertitude sur l’activation du sodium secondaire. Le logiciel NUDUNA a permis d’appliquer une méthode Total Monte-Carlo à notre problème pour déterminer l’incertitude sur l’activation du sodium secondaire. Cette méthode consiste à faire varier les paramètres importants de l’étude de manière aléatoire grâce à des tirages sur les matrices de covariance. Ces tirages aléatoires servent ensuite de données d’entrée au code stochastique MCNP. Après avoir effectué un très grand nombre de calculs MCNP, le principe de Wilks permet de déterminer l’incertitude sur l’activation du sodium secondaire due à l’incertitude sur les données nucléaires. L’application de cette méthode sur ASTRID et Superphénix permet d’aboutir à une valeur d’incertitude d’activation du sodium secondaire convergée. Cette incertitude est de 100% pour le réacteur ASTRID alors que l’incertitude sur l’activation du sodium secondaire est plus faible pour Superphénix avec 66%. L’incertitude due au spectre des neutrons est 9%, valeur plus faible comparée à l’impact des sections efficaces. L’origine de l’incertitude sur l’activation du sodium secondaire provient de l’incertitude sur la section efficace de diffusion élastique du 23Na. La comparaison entre le calcul et la mesure sur le réacteur Superphénix a prouvé que la méthode dans son ensemble est conservative, ce qui est confortant en termes de sureté.

Abstract

The activation of the sodium secondary circuit in the ASTRID core is a major concern because this activation leads to the setting up of protections on the core and the intermediate heat exchangers. Nowadays, the calculation scheme gives the best estimate values, that is to say, without uncertainties. To justify the use of these values on the different part of the ASTRID project, it is mandatory to evaluate the uncertainty on the activation of the secondary sodium. The software NUDUNA enables to apply a Total Monte-Carlo method which allows determining the uncertainty on the secondary sodium activation. The method consists of varying important parameters of the study by doing random samples on the covariance matrices. These random draws are then used as input to the stochastic code MCNP. After performing many MCNP calculations, the Wilks’s principle enables to determine the uncertainty on the activation of the secondary sodium due to uncertainties on nuclear data. The method is applied on the ASTRID and Superphénix reactors to obtain a converged value of the uncertainty on the activation of the secondary sodium. This uncertainty is 100% for the ASTRID reactor whereas the uncertainty is 66% for Superphénix which is a smaller value. The uncertainty due to the neutron spectrum on the ASTRID activation is 9%. This value is smaller compared to the uncertainty due to neutron cross sections. The origin of the uncertainty on the sodium activation comes from the inelastic scattering cross section of the 23Na nuclide. The comparison between calculations and measurements on the Superphénix reactors proves that the method applies conservatism, which is good in term of safety.

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Acknowledgement

My master thesis for the KTH Diploma in Nuclear Engineering took place in the Framatome Company and more precisely the "Neutron transport theory, Radioprotection and Criticality" section in Lyon for a period of 6 months.

First of all, I would like to thank Jean-Michel Perrois as Head of the Safety & Processes Department.

I would also like to acknowledge Amélie Hee-Duval, head of the "Neutron transport theory, Radioprotection and Criticality" section, for her welcome and her help throughout this internship.

I would like to thank my work placement mentors, Guillaume Nolin and Pierre-Marie Demy, for their availability, their help and their sympathies that they gave me during these 6 months.

I am grateful to Dr. Oliver Buss for his cooperation and commitment during my master thesis, for his availability and his presence to help me with the NUDUNA software.

I have a gratitude for Anne-Claire Scholer, Guillaume Vandermoere, Matthieu Culioli, Florent Beck, Pierre Boisseau, François Mollier and Denis Verrier for their wise advice that allowed me to complete my studies.

I especially thank Guillaume Testard with whom I shared my office during these 6 months for his good mood and his precious advice.

Finally, I would like to warmly thank all the members of the section for their welcome and good humor on a daily basis.

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Table of contents Résumé ...... 2 Abstract ...... 2 Acknowledgement ...... 3 1. Introduction ...... 8 1.1. Context ...... 8 1.2. Choice of tools based on current methods ...... 9 2. Problematic of the secondary sodium activation on SFR reactors ...... 9 2.1. Issues due to the activation of the sodium secondary circuit ...... 9 2.2. Calculation of the secondary sodium activation ...... 10 2.3. ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) ...... 10 2.4. Superphénix ...... 12 3. Methodology and tools used for the evaluation of the uncertainties ...... 13 3.1. Procedures to determine uncertainties due to nuclear data ...... 13 3.2. Uncertainty propagation theory ...... 14 3.3. Methodology to determine uncertainties due to nuclear data ...... 17 3.3.1. Creation of a nuclide database ...... 17 3.3.2. Variance and covariance information from the ENDF6 file ...... 19 3.3.3. Random draws on the input parameters ...... 19 3.3.4. The sum rules ...... 21 3.3.5. Creation of the random libraries ...... 24 3.3.6. Stochastic calculations with a transport code ...... 25 3.3.7. Analysis of the stochastic code results ...... 25 3.4. Uncertainty due to the neutron source spectrum ...... 27 3.5. Different tools for the study of sodium fast reactors ...... 28 3.5.1. Stochastic calculation tool: MCNP ...... 28 3.5.2. ADVANTG ...... 29 3.5.3. NUDUNA: NUclear Data UNcertainty Analysis ...... 29 4. Results for the quantification of the uncertainty on the secondary sodium activation ...... 30 4.1. Procedure to determine uncertainties on the secondary sodium activation ...... 30 4.2. ASTRID ...... 31 4.2.1. Reference calculation ...... 31

4.2.2. Convergence of the estimator I95/95 ...... 31 4.2.3. Uncertainty on the activation of the secondary sodium ...... 32 4.2.4. Origin of the uncertainty on the secondary sodium activation ...... 34

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4.2.5. Impact of all nuclides on the secondary sodium activation uncertainty ...... 35 4.2.6. The behaviour of the neutrons in the core ...... 36 4.2.7. Neutron source spectrum result...... 39 4.3. Superphénix ...... 40 4.3.1. Reference calculation ...... 40 4.3.2. Preliminary calculations on the Superphénix reactor ...... 41 4.3.3. Uncertainty on the activation of the secondary sodium ...... 43 4.3.4. Impact of all nuclides on the secondary sodium activation ...... 44 4.3.5. Difference between calculations and measurements ...... 45 Conclusion ...... 47 Perspective ...... 47 References ...... 48

List of figures Figure 1: Description of the ASTRID core [6] ...... 11 Figure 2: Description of the ASTRID vessel ...... 12 Figure 3: MCNP calculation of neutrons streaming from core to IHX [7] ...... 12 Figure 4: Description of the Superphénix vessel [7] ...... 13 Figure 5: Core lattice of Superphénix, vertical view [7] ...... 13 Figure 6: 23Na correlation matrix for the capture and the elastic scattering cross sections ...... 16 Figure 7: Different steps for the quantification of uncertainties due to nuclear data ...... 17 Figure 8: Structure of an ENDF6 file ...... 19 Figure 9: Different values of the parameters MF ...... 19 Figure 10: Interface of the nuclear data sampling function in NUDUNA ...... 21 Figure 11: Summary of the important sum rules ...... 23 Figure 12: Creation of the MCNP input files in NUDUNA ...... 24 Figure 13: Illustration of modifications done by NUDUNA on MCNP input files ...... 24 Figure 14: Illustration of the 1st 휷 quantile ...... 26 Figure 15: Illustration of the 95% quantile with a 95% confidence level on the normal distribution ...... 26 Figure 16: Neutron energy spectrum for a 0.2MeV incident neutron ...... 28 Figure 17: Normalized activation of the secondary sodium in the IHX 2 for 181 MCNP calculations ...... 34

Figure 18: Activation of the secondary sodium in the IHX 2 as a function of 23Na elastic scattering cross-section .. 35 Figure 19: Activation of the secondary sodium in the IHX 2 as a function of 23Na capture cross-section ...... 35 Figure 20: Elastic and inelastic cross-section for 23Na as a function of the energy of the neutron ...... 38

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Figure 21: Energy of the neutron as a function of the number of collisions ...... 39 Figure 22: Normalized activation of the secondary sodium in the IHX 2 for 181 MCNP calculations ...... 39 Figure 23: 23Na elastic scattering cross-section as a function of the energy of the incident neutron ...... 42 Figure 24: Comparison of neutron streaming between ASTRID (left) and Superphénix (right) ...... 43 Figure 25: Activation of the secondary sodium in the loop NE for 181 MCNP calculations ...... 43

List of tables Table 1: List of all nuclides of the ASTRID and Superphénix core ...... 18 Table 2: List of important nuclides of the ASTRID core ...... 19 Table 3: Illustration of the order of the Wilks’s formula ...... 27 Table 4: Activation of the secondary sodium in different intermediate heat exchangers ...... 31 Table 5: Values for the activation of the secondary sodium by modifying 23Na cross-sections ...... 32 Table 6: Values for the activation of the secondary sodium by modifying all nuclides cross-sections ...... 33 Table 7: Values for the activation of the secondary sodium by modifying all nuclides cross-sections ...... 36 Table 8: Average number of collisions on different nuclides using the PTRAC card ...... 37 Table 9: Activation values after sampling on the neutron spectrum ...... 40 Table 10: List of important nuclides of the Superphénix core ...... 40 Table 11: Reference calculations in different loops for the core Superphénix ...... 41 Table 12: Summary of all calculation tests for Superphénix in the loop NE ...... 41 Table 13: Values for the activation of the loop NE for 181 calculations ...... 44 Table 14: Values for the activation of the secondary sodium for 181 calculations ...... 45 Table 15: Comparison of confidence interval between the calculation and the measurements ...... 46

Glossary ADVANTG An Automated Variance Reduction Parameter Generator ASTRID Advanced Sodium Technological Reactor for Industrial Demonstration Bq Becquerel BWR Boiling Water Reactor CANDU CANada Deuterium Uranium CEA French Alternative Energies and Atomic Energy Commission COMAC COvariance MAtrices of Cadarache DR Dose Rate

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ENDF Evaluated Nuclear Data File GUI Graphical User Interface IHX Intermediate Heat Exchanger JEFF the Joint Evaluated Fission and Fusion File MCNP Monte Carlo N Particle NE Northeast NUDUNA NUclear Data UNcertainty Analysis NW Northwest PWR Pressurized Water Reactor RRR Resolved Resonance Regions SCC Scientific Calculation Code SE Southeast SFR Sodium Fast Reactor SVD Singular Value Decomposition SW Southwest TMC Total Monte-Carlo URR Unresolved Resonance Regions

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1. Introduction

1.1. Context The world energy demand has been constantly increasing over the last decades and drives the need for a reliable and affordable energy supply. The International Energy Agency predicts a growth of 0.83% per year in the world population over the period 2015-2050 bringing it from 7.3 billion to 9.7 billion. Accordingly, and with the increasing digitalization of the world economy, the world electricity demand will rise by 2.1% per year on average over the same time period. This rise will lead to an increase of 80% in the installed capacity. According to the World Energy Outlook (WEO) [1], “broad policy commitments and plans have been announced by countries, including national pledges to reduce greenhouse-gas emissions and plans to phase out fossil-energy subsidies, even if the measures to implement these commitments have yet to be identified or announced” [1]. To cope with this trend, a new energy mix between oil, gas, coal and low-carbon sources has to be found, while being in accordance with the concerns on climate change. Thus, in the new energy scenarios, the share of fossil fuels which produces large amounts of greenhouse gases is dropping. On the contrary, the share of nuclear and gas is increasing, the share of renewables seeing the largest growth [1].

Not only the supply but as well the sustainability of electricity are a matter of increasing concern. Indeed, the effects of different conflicts on oil prices and the potential threats to oil and gas supplies have stressed the importance of being energetically independent [1]. Nuclear energy provides a safe, almost CO2 free energy source and can be used for electricity base load. In 2016, about 450 nuclear reactors are operating worldwide and represent 378 GW installed capacity providing more than 11% of the world electricity [1]. 68 new reactors are currently under construction in the world.

However, a strong public concern is perceived with regard to safety, especially after the 2011 Fukushima Daiichi accident in Japan. To reduce the risk of accident, quantifying uncertainties in calculations is essential. It is the reason why the demand of the regulatory authorities to evaluate uncertainties has increased recently to prove that nuclear power plants are functioning with a high level of safety. Methods for quantifying uncertainties in the calculations must be developed to improve the knowledge of this field. There are four types of uncertainties in the calculations [2]: - Simplified modeling of physics. Modeling is leading to a simplification of the reality which implies a bias in the calculations, for example when the transport equation is simplified by the approximation of the diffusion. - Imperfect numerical schemes to solve the equations: multigroup approximation, discretization in space and energy. - Description of the model: the uncertainty on the dimension of the components, their density, their isotopic composition, etc. - Numerical data of the calculations: in the nuclear field it corresponds typically to the neutron cross-sections. The origin of the uncertainties on the cross-sections comes from the error of measurements as well as empirical models used for the reconstruction of the cross-sections.

The purpose of this report is, therefore, the evaluation of the uncertainties on the secondary sodium activation due to uncertainties on nuclear data. This evaluation is one of the mandatory demands so that the ASN, the French authority, could ensure a safe and reliable operation of the ASTRID reactor. The uncertainties must be known with a high level of confidence because the calculated values with uncertainties must not exceed values imposed by the project due to a restricted zoning in the buildings. In this report, only uncertainties due to nuclear data are considered. In a first part, the issues due to the sodium activation are explained. Then, the methodology

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to evaluate uncertainties on the activation of the secondary sodium is described. In the last part, the results are presented for the two SFR reactors studied.

1.2. Choice of tools based on current methods The evaluation of uncertainties in the radiation protection field is a crucial step of a reactor core design because it provides information on how parameters comply with safety margins. In recent years, two factors have enabled improvements in uncertainties quantification for neutron transport theory calculations [3]: an increase of the computation power and the addition of covariance data in the nuclear data libraries. Some parameters are missing in the libraries so improvements are still possible. It gives insights to the nuclear data expert groups about where future efforts should be spent to improve weaknesses in the existing data. Regarding the different methods used in the nuclear field to determine the uncertainties due to nuclear data [4], a stochastic code is chosen to apply the Total Monte-Carlo method. Indeed, the uncertainties on the secondary sodium activation are impossible to determine with a deterministic code because the geometry is too complex. To determine these uncertainties, a Monte-Carlo code is needed. MCNP is used because this code is the reference code in the neutron transport theory. The analysis of past studies done illustrates that the Monte-Carlo method has shown good results for the quantification of uncertainties with few simplifications of the problem. After performing all the stochastic calculations, results have to be analysed to determine the uncertainties on the activation of the sodium secondary circuit. Lots of methods exist to perform statistical analysis [5] [6] [7]. Regarding all different ones used in the nuclear field, the Wilks’s principle seems to be the most common [5]. The assumptions of this principle are little penalizing with a manageable number of calculations, that is why the Wilks’s principle is the most appropriate method for our problem.

2. Problematic of the secondary sodium activation on SFR reactors

2.1. Issues due to the activation of the sodium secondary circuit Secondary sodium, which continually flows through the secondary loop, is irradiated and activated as it crosses the intermediate heat exchanger (IHX) in the vessel. The activation of the secondary sodium is considerably low compared to the primary sodium because it is further away from the core and because its volume subjected to irradiation is lower. The separation of primary and secondary sodium allows confining the fission products resulting from a possible failure of pipes. During its circulation in the secondary loop, the 23Na nucleus absorbs a neutron and becomes an excited 24Na nucleus. It then decays by an electron emission with a half-life of 15 hours, then by photon emissions. The decay of the 24Na is one of the two major causes of the activation of the secondary loop. The second cause is the passage of tritium. Its production and diffusion in the reactor are not taken into account in this report.

The activation of the sodium secondary circuit leads to operating and construction constraints that have repercussions on safety (collective dose, etc.) and on the cost of the power plant. The secondary sodium activation on ASTRID is one of the uncertainty sources that must be quantified in particular for the sizing of some components or the need to put neutron protections. The secondary sodium flows in different rooms so a dose rate is mapped along the secondary loop. This area zoning leads to operational constraints. The goal is to get an activation value including uncertainties less than the value required to be in a guarded blue zone [8]. To achieve the required value for the activation of the secondary sodium, the ASTRID project teams propose to protect the intermediate heat exchangers with borated steel plates and reinforce the neutron protections around the core. These are heavy technologies, expensive and difficult to implement. It is, therefore, necessary to dimension these

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protections as accurately as possible. The margin for the sizing of the protections is small so it is necessary to take into account the uncertainties on the nuclear data in order to have a reliable estimate of the activation of the secondary sodium. To sum up, the main concern related to the activation of the secondary sodium is to evaluate the design constraints on the equipment constituting the secondary loop and the buildings which shelter them.

2.2. Calculation of the secondary sodium activation A calculation code is used to directly calculate the activation of the secondary sodium in the intermediate heat exchanger (IHX). obtain an estimate of the flux in each volume of the intermediate heat exchanger. Thanks to 23 the flux, the code determines the average reaction rate Ai of the Na activation reaction in each volume i of the exchanger according to equation 1. By applying this formula, it supposes that the equilibrium is reached: the consumption of neutrons is equal to the production by decay.

퐸2 23 23 ( ) (1) 퐴푖 푁푎 = 푁푖 ∫ 𝜎푎푐푡푖푣푎푡푖표푛 푁푎 (퐸) × 휙푖 퐸 푑퐸 퐸1

 Φi: the average neutron flux in volume i  E1 and E2: the boundaries of the considered energy domain. In our case, the whole energy spectrum is considered so E1=0 and E2=+∞ 23  Ni: the Na concentration in volume i  𝜎: microscopic capture cross-section of the 23Na nuclide The steady-state activity of the sodium in the secondary loop is calculated thanks to the following equation:

푛 23 퐴푖 푁푎 퐴푡표푡 = ∑ × 푉푁푎 (2) 푉푖 푖=1

 Vi: volume of the sodium in each volume i of the heat exchanger

 VNa: volume of sodium in the total secondary loop

2.3. ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) The Generation IV International Forum [9] created in 2001 at the initiative of the US Department of Energy (DoE) brings together 12 countries for the development of the future nuclear reactors on the horizon 2030-2035. The sustainable development criteria imposed for these reactors are as follows: • The improvement of safety with respect to the current reactors. • The economic competitiveness of these reactors. • The reduction of induced waste. • Resistance to external aggressions. During the forum [9], six promising reactor concepts were selected including fast neutron reactors. The ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) project is a French SFR project led by the CEA and supported by industrialists such as EDF or Framatome to meet Generation IV forum requirements. The design studies for the 600 MWe (1500 MWth) ASTRID reactor began in 2010. France has made the strategic choice to focus mainly on sodium fast neutron reactors (SFR) because such reactors have already existed in France: Rapsodie, Phénix, and Superphénix.

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The sodium fast reactors have specific features. These reactors are said to be fast because the neutrons used to maintain the chain reaction have a high energy (greater than 1 MeV, wherein a pressurized water reactor the neutrons used have an energy less than 0.025eV on average). The thermal neutron reactors in the current power plant burn mainly 235U, the only fissile nuclide of natural uranium. The energy potential of this raw material is not fully used because uranium is mainly composed of 238U (99.3 % of natural uranium is 238U). Fast neutron reactors convert 238U into a fissile nucleus 239Pu when it is irradiated by a rapid flux. 238U is thus said to be fertile. As it consumes its initial fuel, a sodium fast reactor (SFR) produces a new fuel. A fraction of this fuel is burned in the reactor and contributes to the energy production of the reactor. By exploiting such reactors, it is possible to reach the breed generation, that is to say, to have more fissile material at the output of the reactor than the input.

The technology chosen for this project is the integrated vessel because of feedbacks from the French authorities with Phénix and Superphénix. With an integrated vessel, the entire primary circuit (pumps, filters, exchangers) is in the vessel, which also contains the primary sodium. This primary sodium, whose role is to cool the core, is subjected to irradiation and is strongly activated. It is confined to the vessel to avoid the emission of out-of-vessel dose rates from pipes and to reduce the risk of radioactive sodium leakage. This is one of the great safety advantages of this technology. The configuration of the core [10] is presented Figure 1 and the vessel is described Figure 2. The core has the following specifications: a very low burn-up reactivity swing (due to a small cycle reactivity loss) and a reduced sodium void effect with regard to past designs such as the EFR (around -2$).

Figure 1: Description of the ASTRID core [6]

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Figure 2: Description of the ASTRID vessel

The knowledge of the major activation zones in the IHX is essential to understand the behavior of the reactor. A flux calculation over a mesh was implemented on the whole ASTRID model [11]. Figure 3 illustrates that the neutron flux arriving at the IHX comes from leakage from the top and the base of the assemblies. The radial contribution is minor compared to the axial contribution. This calculation illustrates the phenomenon of axial leakages. One consequence is that neutrons are bypassing the lateral protections to reach the intermediate heat exchanger. The neutrons cross a high quantity of sodium compared to materials structures during their lives.

Figure 3: MCNP calculation of neutrons streaming from core to IHX [7]

2.4. Superphénix Superphénix (SPX) was a nuclear power station prototype on the Rhône river at Creys-Malville in France. Superphénix was a 1242 MWe fast breeder reactor. This reactor had two main goals: reprocessing nuclear fuel from France's PWR nuclear reactors and being an economic generator of power on its own. Compared to ASTRID, the Superphénix core is composed of 4 pumps and 8 intermediate heat exchangers. Two intermediate heat

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exchangers are assembled as loops: NE, NW, SE, and SW as it is shown Figure 4 below [11]. The vertical view of the core lattice and main equipment is presented Figure 5 [11]. Construction began in 1974 and finished in 1983. After many tests, the plant was connected to the grid in December 1986. In operation, Superphénix had 3 incidents in the sodium part during its life. The number of incidents has been extremely low for a prototype reactor of this size. In December 1998, the Prime Minister made a ministerial decree to close permanently Superphénix. With the dismantling of Superphénix, the French fast neutron reactor sector gradually stopped until the Generation IV forum.

Figure 5: Core lattice of Superphénix, vertical view [7] Figure 4: Description of the Superphénix vessel [7]

In this report, the study of the Superphénix core is done because it is one sodium fast reactors where measurements were performed under-functioning. The global methodology applied on ASTRID is implemented on Superphénix to obtain a comparison between the calculation and measurements. The goal is to determine if the theory applied on ASTRID is accurate. The measurements were made on the second semester of 1996 during a steady state operation phase. This measurement campaign took benefits from the feedback of a first campaign done during the first semester of 1996. The uncertainties of these measurements are assessed between 14% and 27% at 90% of the nominal power. A gamma spectrometry was used to measure the dose due to 24Na activity.

3. Methodology and tools used for the evaluation of the uncertainties

3.1. Procedures to determine uncertainties due to nuclear data There are four methods in the nuclear field to determine uncertainties due to nuclear data [3]:  The simplest method is to know a penalizing value for each parameter. It is possible in some cases to determine a value that encompasses uncertainties. In general, this method is not applied because the distributions of the random parameters of the problem are unknown, so it is difficult to determine a penalizing value.  One approach for estimating systematic uncertainties is gauging a code with the help of benchmark experiments [4]. The gauging procedure yields a bias on the calculation model, which has to be compared

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to experimental values. The differences between the calculations and experiments enable to determine the uncertainties on the calculations. This bias depends on the chosen nuclear data set. The most important matter is to consider an experiment representative of the parameters studied.  The perturbation theory consists in studying the effects of the modification of the nuclear data, for example, the cross-sections. The goal is to analyze the consequence of the X% variation of the cross- section values on the observable. The perturbed cross sections values are then combined with covariance matrices to evaluate the uncertainties. Uncertainty quantification methods which rely on perturbation theory permit calculating sensitivity coefficients of output parameters with respect to input perturbations [3].  The Monte-Carlo method is based on random draws on covariance matrices to obtain modified values of the parameters studied. The manipulations of the parameters take place exclusively on the nuclear data so any neutron transport theory solvers may be used to run the model with the modified inputs. There are two different cases: - Creation of random input libraries by random draws then used in a deterministic code. In this case, the Monte-Carlo technique is used only once for the creation of the dataset; - It is possible to apply Monte-Carlo techniques both for the creation of the dataset and for the transport equation. In this case, the method is called Total Monte-Carlo (TMC). The solution proposed to take into account uncertainties on our observable is the Total Monte-Carlo method. It consists of generating random libraries of nuclear data for each nuclide considered as relevant. A random library is composed of all the random draws for all the nuclides for specific reactions. In this method, the parameters are considered as random variables so it is possible to define a mean value, a variance, a covariance. It means that the parameters of the problem are varied randomly (or pseudo-randomly, since the probability distributions of each of these parameters are known) from the covariance matrices which are nothing more than probability density functions. Monte Carlo methods are often referred as “model-free” because they do not require alterations on the problem model: any solver can be used to process the sampled input on any kind of system [3].

3.2. Uncertainty propagation theory Before explaining the method, it is important to define the concept of the uncertainty propagation theory. First, the variance and the covariance are defined [12]. The variance of a random variable describes, to first order, the amount of dispersion of the quantity around its own expected value. By considering one random variables x with its expectation value E:

푉푎푟(푥) = 퐸{[푥 − 퐸(푥)]2} (3)

By considering two random variables x and y, the definition of the covariance is: 퐶표푣(푥, 푦) = 퐸{[푥 − 퐸(푥)][푦 − 퐸(푦)]} 푎푛푑 퐶표푣(푥, 푥) = 푉푎푟(푥) (4)

The interpretation of this definition is as follows [12]. The quantities [푥 − 퐸(푥)] and [푦 − 퐸(푦)] represent the deviation from the mean value for the single points of the dataset. If the two variables have a positive relationship, either higher or lower values than the expectation value, then the two deviations show a concordant sign, and then a positive product. The result is a positive average value of these products, i.e. a positive covariance. Similarly, if two variables vary together via a negative relationship, then the two deviations from the mean show a discordant sign, and then a negative product. The result is a negative average value of these products, i.e. a negative covariance. If the two variables are poorly correlated, then the two deviations show a concordant sign in some items and a discordant sign in other items. The products are then in part positive and in part negative. This finally results in a relatively small average value of these products, i.e. a relatively small

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covariance. The extreme case occurs when the two variables are independent. In this case, the covariance is zero. The covariance matrix generalizes the notion of variance to multiple dimensions.

In a simulation which uses input containing uncertainties, it is of interest how the uncertainties of the input 푇 propagate to the results. Modeling the input as an m-dimension random vector 푿 = (X1, X1, … , Xm) , a resulting quantity g(푿) is a random variable which follows a distribution depending on 푿. This section describes two methods to estimate the uncertainty 푉푎푟(푔(푿)) in g(푿) given information on 푿 [13].

Linear uncertainty propagation The uncertainties can be described as the zeroth and the first orders in a Taylor expansion of g(푿) about the expected value of 푿, assuming g(푿) has continuous first partial derivatives on its domain. By only considering the first order of the Taylor expansion [13]:

푚 푚 휕푔 휕푔 푉푎푟(푔(푿)) ≈ ∑ ∑ ( ) ( ) 퐶표푣(푋푖, 푋푗) 휕푋푖 휕푋푗 (5) 푖=1 푗=1

Where 퐶표푣(푋푖, 푋푗) is the covariance between Xi and Xj. The covariance describes the uncertainty of Xi and Xj but also their linear correlation because the covariance can be written as [13]:

퐶표푣(푋푖, 푋푗) = 𝜎(푋푖)𝜎(푋푗)𝜌(푋푖, 푋푗) (6)

2 Where 𝜎(푋푖) is the standard deviation: 𝜎 (푋푖) = 푉푎푟(푋푖) and 𝜌 is the correlation coefficient, which can be shown to satisfy −1 ≤ 𝜌 ≤ 1 [16]. The closer |𝜌| is to 1, the greater is the linear dependence between Xi and Xj. 𝜌 is positive if Xi and Xj tend to “vary in the same direction” and negative in the opposite case. If all different random variables are uncorrelated, thanks to the relation 퐶표푣(푋푖, 푋푖) = 푉푎푟(푋푖), equation 5 simplifies to [13]: 푚 휕푔 2 푉푎푟(푔(푿)) ≈ ∑ ( ) 푉푎푟(푋푖) (7) 휕푋푖 푖=1

Quite intuitively, the uncertainty of g(푿) thus depends on the uncertainty of the different arguments 푋푖 and on how strongly g depends on the arguments. Equation 5 can be interpreted similarly, but it also takes the linear correlation into account.

In the nuclear field, the covariance matrices are not represented because it is difficult to perform an analysis of those matrices. Nuclear researchers prefer to represent the correlation matrix [13]. Figure 6 illustrates the example of the 23Na correlation matrix for the MT number 102 (capture) and number 2 (elastic scattering). This matrix enables to determine the relation between the two reactions for each energy bin. In this matrix, 175 energy bins are considered so the dimension of the correlation matrix is 175x175: the correlation coefficient between the energy bins Ei and Ej is 𝜌(퐸푖, 퐸푗). In the nuclear field, researchers prefer to represent colour for the correlation coefficient for a better reading of the matrix. It is, therefore, easy to see the relation between variables. In a large part of the energy spectrum, the two cross-sections are varying in the same way because the correlation factor between them is closed to one (green color).

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Figure 6: 23Na correlation matrix for the capture and the elastic scattering cross sections

Monte Carlo estimation A more direct way to estimate the propagated uncertainty is to generate a random sample 푥(1), 푥(2), … , 푥(푚) from 푿 and to evaluate g for all observations of this random sample. In this way, one has obtained a random sample from g(푿) namely g(푥(1)), g(푥(2)), … , g(푥(푚)). It is then straightforward to estimate the standard deviation 𝜎푔(푋) = √푉푎푟(푔(푿)) of g(푿) by considering m samples of the distribution g(푿):

푚 1 𝜎 = √ ∑( g(푥(푖)) − 퐸(푔(푿))2) 푔(푋) 푚 − 1 (8) 푖=1

The major disadvantage is that more evaluations of g(푿) are necessary compared to the linear method, which becomes a problem if g(푿) is computationally expensive to evaluate (which often can be considered the case in applied nuclear physics). It may be argued that it is poorly invested time to do such accurate uncertainty propagation since the knowledge of the distribution of 푿 may be a larger limitation than the approximation in linear uncertainty propagation. In nuclear data uncertainty propagation, it is often the case that the only information on 푿 consists of the expected value and covariance, then the distribution has to be assumed. It is the reason why a normal distribution is considered. In this methodology, the only approximation is the finite number of samples. Thanks to this method, the g(푿) standard deviation is obtained without simplification compared to the linear uncertainty propagation. In this report, the Monte Carlo estimation is used but the equation for the determination of the standard deviation of g(푿) is different. Instead of using the equation 8, the Wilks’s principle is used because this principle gives a more accurate value of 𝜎푔(푋).

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3.3. Methodology to determine uncertainties due to nuclear data The Total Monte-Carlo method was developed by Mr. Dimitri Rochman [14] and enables to carry out a propagation of uncertainties of any type: technological (geometry, the composition of materials) or related to the calculation code. To propagate the uncertainties due to the nuclear data, the knowledge of the variance/ covariance matrices is mandatory. In the rest of this report, a nuclear data is characterized by several physical parameters named Mfi. In the following, six different physical parameters Mfi are considered:  Resonance parameters,  Non-resonant effective cross-sections,  Multiplicity of secondary particles emitted, especially neutron production,  Energetic distribution of the final state of the particles,  Angular distribution of the final state of the particles,  Data on radioactive decay and fission products. Each of these parameters is a random variable defined with a mean value and a variance. The point is that Mfi parameters are not independent, so the covariance to have the links between them is mandatory. Two quantities must be distinguished here: each parameter has a specific uncertainty defined by the variance but there is an uncertainty due to the fact that Mfi parameters are not independent: it is the covariance. The methodology using the Total Monte-Carlo technique is explained to help the reader to understand the logical analysis. Figure 7 below presents the whole method to evaluate uncertainties due to nuclear data. The next paragraphs present in more details each block of the figure.

Creation of a nuclide database

Variance and covariance information from the ENDF6 file

Random draws on the input parameters

The sum rules

Creation of the random libraries

Stochastic calculations with a transport code

Analysis of the stochastic code results

Figure 7: Different steps for the quantification of uncertainties due to nuclear data

3.3.1. Creation of a nuclide database The first step is to select a number of nuclides which are important for the problem. In the calculation input files of the ASTRID and Superphénix model, many nuclides are considered to represent materials which constitute the primary vessel. The main hypothesis is that the nuclides of the core are excluded. Nuclides constituting the core capture most of the fission neutrons so neutrons which reach the IHX come from the external part of the core where leakages occur. It is considered that nuclides constituting the core have a low impact on the activation of

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the secondary sodium because few interactions between those nuclides and neutrons occur. This hypothesis is verified a posteriori part 4. The list of all nuclides for the two reactors is presented Table 1.

Fe54 Stainless steel Ni58 Stainless steel Mo97 Stainless steel

Fe56 Stainless steel Ni60 Stainless steel Mo98 Stainless steel

Fe57 Stainless steel Ni61 Stainless steel Mo100 Stainless steel

Fe58 Stainless steel Ni62 Stainless steel Mn55 Stainless steel

Cr50 Stainless steel Ni64 Stainless steel Cu63 Stainless steel

Cr52 Stainless steel Mo92 Stainless steel Cu65 Stainless steel

Cr53 Stainless steel Mo94 Stainless steel N14 Stainless steel

Cr54 Stainless steel Mo95 Stainless steel N15 Stainless steel

S32 Stainless steel Mo96 Stainless steel P31 Stainless steel

S33 Stainless steel Si28 Stainless steel Si29 Stainless steel

S34 Stainless steel He3 Helium Si30 Stainless steel

S36 Stainless steel He4 Helium Ti46 Stainless steel

Na23 Liquid sodium Co59 Borated steel Ti47 Stainless steel

Al27 Stainless steel C B4C Ti48 Stainless steel

B C + Borated Stainless steel Nb93 Stainless steel B10 4 Ti49 steel

B C + Borated Stainless steel Ar36 Argon B11 4 Ti50 steel

Ar38 Argon W182 Tungsten Mg24 MgO

Ar40 Argon W183 Tungsten Mg25 MgO

H1 Stainless steel W184 Tungsten Mg26 MgO

W186 Tungsten O16 MgO Table 1: List of all nuclides of the ASTRID and Superphénix core

In the choice of the important nuclides, this one is done according to the COMAC base [15]. This database from the CEA gives recommendations on neutron libraries where information on variance and covariance can be found. For nuclides where there is information in the COMAC database, the library recommended by COMAC is taken and this nuclide is considered as important. If there is no information for a nuclide, the reference library JEFF-

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3.1.1 is taken and no draws are made on these nuclides. Nuclides of our problem are the nuclides located between the core (excluded) and the intermediate heat exchangers. The first job was an important bibliographic task and data gathering on the ENDF6 files. All nuclide libraries (JEFF, JENDL, ENDFB-VII) were downloaded to find information on nuclides which composed ASTRID and Superphénix. The important nuclides for ASTRID are presented Table 2.

Fe54 Ni58 Mo97 Cr50 Mg24 Mo92

Fe56 Ni60 Mo98 Cr52 Mg25 Mo94

Fe57 Ni61 Mo100 Cr53 Mg26 Mo95

Fe58 Ni62 Mn55 Cr54 Na23 Mo96

O16 Ni64 B10 C Al27 B11 Table 2: List of important nuclides of the ASTRID core

3.3.2. Variance and covariance information from the ENDF6 file The ENDF6 file defines a precise format for the storage of nuclear data and covariance matrices for all reactions. These files are well-structured to find all information needed. Figure 8 presents the structure of the ENDF6 file. For each material [16], several sections are written corresponding to a different MF number which corresponds to the MFi parameters. Figure 9 presents all value for the MF parameters. In each Mf file, there are several reactions corresponding to an MT number. The most important MT numbers are 1 for the total number cross-section, 2 for elastic scattering, and 102 for the capture cross-section. There are more than 200 types of reactions. In addition, this file contains the covariance matrices for all parameters. In the nuclear field, different libraries for the storage of the information can be found. The most common libraries are JEFF, ENDBF, TENDL, and JENDL. Each nuclide has an ENDF6 file in each of these libraries. The data is different between libraries so one task of my internship was to study all the different format to find the information needed to solve the problem.

Figure 8: Structure of an ENDF6 file Figure 9: Different values of the parameters MF

3.3.3. Random draws on the input parameters The primordial task of the methodology to evaluate uncertainties is the random draws on the input parameters. 푇 For our study, an observable 휃 = 푔((X1, X2, … , Xm) ) and the covariance matrix 퐶표푣(푋푖, 푋푗) is considered. The expectation value E and the covariance matrix are empirical estimates of the real

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푇 datasets 푿 = (X1, X2, … , Xm) . The variables 푋푖 follow a normal or log-normal distribution with the density probability P(푋푖). In our problem, the distribution of all the 푋푖 are standard Gaussian distributions. To perform a random sample of a random variable 푋푖, the first step is to decompose 퐶표푣(푋푖, 푋푗) with the help of the Cholesky decomposition and the Singular Value Decomposition (SVD) [4] into:

푡 2 퐶표푣(푋푖, 푋푗) = 푈 퐷 푈 (9)

With U orthogonal and D diagonal. A random draw of a coordinate 푋푖 is noted 푋푖,푟푎푛푑표푚 and given by the following equation [4]:

푡 푋푖,푟푎푛푑표푚 = 푈 퐷풁 + 퐸(푋푖) (10)

푇 Where 풁 = (Z1, Z2, … , Zm) is a vector of a standard Gaussian random number. Each coordinate of 풁 is the result of a random draw on the standard Gaussian distribution illustrated Figure 15. The value 푋푖,푟푎푛푑표푚 is varying 푡 around its expectation value 퐸(푋푖) by the value 푈 퐷풛. This quantity through the covariance matrix compiles the uncertainty of the parameter 푋푖. Between two different samples, the coordinates Zi and Zj are different so the outcome 푋푖,푟푎푛푑표푚and 푋푗,푟푎푛푑표푚 are different. In our case, the observable 휃 is the value of the secondary sodium activation. The different 푋푖 are the different Mfi parameters. The function g is performed by the stochastic code: all the input parameters enable to determine the value of the secondary sodium activation thanks to the stochastic code.

푇 퐴푐푡푖푣푎푡푖표푛푟푒푓푒푟푒푛푐푒 = 푔((푀푓푖1, 푀푓푖2, … , 푀푓푖m) ) for the reference calculation { 푇 퐴푐푡푖푣푎푡푖표푛푎푓푡푒푟 푟푎푛푑표푚 푠푎푚푝푙푖푛푔 = 푔((푀푓푖1,푟푎푛푑표푚, 푀푓푖2,푟푎푛푑표푚, … , 푀푓푖m,푟푎푛푑표푚) ) (11)

The purpose of this method is to obtain N random samples of the variable Zi with the method described above and for each sample, a value 푀푓푖푖,푟푎푛푑표푚 is obtained. When all the modified parameters (푀푓푖1,푟푎푛푑표푚, 푀푓푖2,푟푎푛푑표푚 … , 푀푓푖푚,푟푎푛푑표푚) are drawn, a stochastic calculation is performed to obtain a value of the sodium activation corresponding to this specific input data. Then a different series (푀푓푖′ , 푀푓푖′ … , 푀푓푖′ ) is created and a new value of the secondary sodium activation is 1,푟푎푛푑표푚 2,푟푎푛푑표푚 푚,푟푎푛푑표푚 calculated. After performing many calculations, a statistical process enables to determine the value of the uncertainty on the secondary sodium activation.

The software NUDUNA performs the random draws on the different parameters. As illustrated Figure 10, a function in NUDUNA enables to choose the nuclides for which the uncertainties propagation is performed. The choice of the temperature is only done for cross-sections because it is the only parameters that vary significantly with the temperature. There is also a choice of Mfi parameters, the distribution for each parameter and the type of simulation. It is possible to randomly draw all parameters, only some parameters or make no draws and keep the mean value for each parameter. As a reminder, all Mfi parameters are [4]: - Resonance parameters - Non-resonant effective cross-sections - Multiplicity of secondary particles emitted, especially neutron production - Energetic distribution of the final state of the particles - Angular distribution of the final state of the particles - Data on radioactive decay and fission products

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Figure 10: Interface of the nuclear data sampling function in NUDUNA

3.3.4. The sum rules The ENDF6 files are very large and contain a huge quantity of information but unfortunately, some parameters are missing. Indeed, even considering 6 parameters among more than 200, there is a margin of progress to obtain all the data needed for the problem. There are therefore laws to manage the lack of information. To perform random draws on parameters, it is important to determine which information is missing and how it is possible to compensate the lack: it is the laws of conservation on the draws. For the six different Mfi parameters, the different sum rules for the lack or the excess of information in the ENDF6 file [4] are presented below.

1. The neutron multiplicity information is encoded in the file 1 of the ENDF6 file and contains three different sections for the total, prompt and delayed neutron multiplicity yields. First of all, NUDUNA determines a common energy grid for all parameters. Indeed in the ENDF6 file, the energy grid can be different between the total and the prompt neutron multiplicity for example. By interpolation, a common energy grid for all points is generated. Then all points are indexed and a common covariance table is established according to file 31 information. The multiplicity matrix has the same dimension as the covariance matrix associated. For the lack of information, the equation that must be verified is the equation 12 where m is the neutron multiplicity:

푚푡표푡푎푙(퐸) = 푚푑푒푙푎푦푒푑(퐸) + 푚푝푟표푚푝푡(퐸) (12)

After the random sampling on variance and covariance matrices, the sum rule for the different multiplicity contributions mi must be conserved. Five different cases exist [4]:

o If there is covariance given for 푚푑푒푙푎푦푒푑(퐸) and/or 푚푝푟표푚푝푡(퐸), then 푚푡표푡푎푙(퐸) is evaluated according to equation 12;

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o If there is covariance given for 푚푡표푡푎푙(퐸) and 푚푑푒푙푎푦푒푑(퐸), the prompt one is the difference between the random draws of the two other contributions;

o If there is covariance given for 푚푡표푡푎푙(퐸), the equation 12 is conserved by adjusting solely the prompt contribution because we approximate that the prompt one dominates the delayed multiplicity;

o If there is covariance given for 푚푝푟표푚푝푡(퐸) and 푚푡표푡푎푙(퐸), then 푚푑푒푙푎푦푒푑(퐸) is adjusted such that the sum rules gets fulfilled. Thus, the assumption of perfect correlation between

푚푝푟표푚푝푡(퐸) and 푚푡표푡푎푙(퐸) is done. The covariance of the total multiplicity points is ignored in favor of the covariance of the prompt one; o If there is covariance given for all 3 contributions in the sum rule and no correlation, the code is stopped because possible ambiguities can occur.

2. The resonance contributions can be divided into the resolved resonance regions (RRR) and unresolved resonance regions (URR). The parameters and uncertainties are encoded in the files 2 and 32 respectively of the ENDF6 file. The covariance information for the URR is fully implemented and for the RRR, the most important formalisms are considered. As for the neutron multiplicity, the first step is to find a common energy grid to create one large covariance matrix. In the RRR regime, NUDUNA supports the most important formalisms [4]: Reich-Moore, Single-level Breit- Wigner and Multilevel Breit-Wigner. NUDUNA randomizes all parameters of the formalism according to their covariance and enforces the positivity bounds for the widths and energy parameters.

3. The cross-section information is encoded in the file 3 and its uncertainty is encoded in file 33 of the ENDF6 file. Usually, file 3 gives only the fast neutron and background cross-sections, the full cross-section is given by the sum of the file 3 tabulations and the resonance contributions in file 2. As the previous case, a common energy grid is created for one nuclide because all matrices for all reactions must have the same size to perform matrix operations. However, up to now, covariance data are incomplete and the sum rule must be restored by hand. For this, three rules are followed [4]: o If a cross-section, which is the outcome of a sum, has no uncertainty information encoded in file 33, then this cross-section is set according to its sum rule. o If there is uncertainty information encoded in file 33 for a cross-section 𝜎 being the outcome of a

sum rule and also for at least one cross-section 𝜎푎 corresponding to one of the addends, then we ignore the random draw of 𝜎 and set it according to its sum rule. Hereby a perfect correlation

between 𝜎 and 𝜎푎 is assumed and uncertainty contributions of addends for which there is no information given in file 33 are neglected. o If there is uncertainty information encoded in file 33 for the sum but not for any of the addends, then we re-scale the addends with a common factor in order to conserve the sum rule.

This ad-hoc restoration will become obsolete as soon as complete covariance information will be available. Some evaluations include also correlations between different reaction types, such that the covariance matrix does not only consist of small diagonal blocks. The dimension can easily exceed 1000: it creates major numerical challenges.

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4. The angular distribution of the final state particles is encoded in the file 4 of the ENDF6 file. The angular distributions are expressed as normalized probability distributions, which are mostly given in Legendre representation. Legendre coefficients are chosen randomly according to their covariance table. However, the resulting Legendre sum represents a probability distribution and must, consequently, be larger than zero for all angles. Each random sample of the Legendre coefficients is checked whether it fulfills the positivity bound in each cross-section channel. If this criterion is not met, then the random draw is rejected and new ones are generated until the criterion is met.

5. The energy distribution of neutron-induced reactions is encoded in the file 5 of the ENDF6 tape. The covariance in file 35, however, is not suited to set up a random model for the file 5 information. One goal of my internship was to implement in NUDUNA the random draw of the energy distribution. The spectrum considered is the 239Pu neutron spectrum due to prompt neutrons. All information needed for the 239Pu spectrum is present in the ENDF6 file JENDL-4.0-up1 to implement the random draws. The result is a probability density function. The sum rule applied is a normalization of the spectrum after random draws: the sum over all energy bins is equal to one.

6. Decay data are stored in File 8. The current ENDF6 format supports neither covariance matrices for branching ratios nor correlations between data of different nuclides. The numerical treatment is simple

because NUDUNA only samples both half-life values and branching ratios. For the branching ratios 훽푖, only the constraint given equation 13 must be enforced:

∑ 훽푖 = 1 (13) 푖

The summary of the sum rules is presented Figure 11.

Figure 11: Summary of the important sum rules

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3.3.5. Creation of the random libraries After the sampling of the parameters, random libraries are created to gather all the sampling information. A random library is composed of all the random draws for all the important nuclides, all the reactions, and all the temperatures. From an original ENDF6 file for a given nuclide, NUDUNA reads the file and then modify each of the Mfi parameters based on the random draws and then generates N random ENDF6 libraries. Then, another function in NUDUNA generates the N input files for the transport code corresponding to the N random libraries created. The GUI interface for this function is presented Figure 12. During random draws, the cross-sections are modified so it is necessary to modify the absolute path of the xsdir file. This file is used to match the materials in MCNP to the cross-section files. In the MCNP file, the materials are defined with an .XXc extension, which makes possible to take the effective cross-section of a material thanks to the extension. In our problem, the extensions are changed so that MCNP uses the files containing modified cross sections as input, illustrated Figure 13.

Figure 12: Creation of the MCNP input files in NUDUNA

Figure 13: Illustration of modifications done by NUDUNA on MCNP input files

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In the case of Figure 13, random draws are performed on the nuclides 23Na (11023.) and 10B (5010.). For nuclides where there are no random draws on the variance/covariance matrices for the cross-section, the extension remains unchanged. For 23Na and 10B, the extension corresponds to the number of the random sampling.

3.3.6. Stochastic calculations with a transport code The random libraries are created to be the input of the stochastic code. This code calculates the overall value of the activation of the secondary sodium for all the different inputs. The input files are different so the results gather all the information needed for the evaluation of the uncertainty due to nuclear data. This step requests a lot of times because the number of calculations is important (approximately 500 calculations). In this study, a code for reduction variance techniques is needed in parallel of the stochastic code. Neutrons are crossing several meters of sodium so weight-windows are created to reduce drastically the computing time. This variance reduction technique is coupled to the stochastic code.

3.3.7. Analysis of the stochastic code results After obtaining all the results, it is necessary to perform a statistical processing to determine the uncertainty on the activation of the secondary sodium. There are many methods for analyzing the results:  The Wilks’s principle is explained the next paragraph [5].  The Bootstrap method [17]. It is the practice of estimating properties of an estimator (such as its variance) by doing samples from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data.

 The Tukey method [6]. The Tukey's test compares the mean values of all pairwise samples {휇푖 − 휇푗} and identifies any difference between two means that is greater than the expected standard error. The Tukey method is conservative when there are unequal sample sizes. As part of my internship, only the Wilks’s principle is used. This method is frequently employed in the nuclear field because it contains little-penalizing assumptions. This method is independent of the number of parameters and the principle is applicable for whatever distributions of all parameters [5] [7]. Given the number of parameters in our problem, this assumption is very useful. This method is efficient but asks for a big computational power. The Wilks’s principle can be applied to the SFR reactors because variance reduction techniques are used to reduce the time to perform calculations. To be applied, the random draws must be independent and follow the same law. This hypothesis is easily verified because by definition the random draws to create the input files are independent. Then, an assumption of the study part 3.3.3 is that the distributions of all the Mfi are standard Gaussian distributions so they follow the same law. This principle enables to determine the first 훽% quantile with an α% confidence level [18]. The one-sided tolerance is taken for the activation of the secondary sodium because an upper limit is needed and not an interval. First, the two concepts of quantile and confidence level are defined [19]:  In statistics, quantiles are cut points dividing the range of a probability distribution into contiguous intervals with known probabilities. There is one less quantile than the number of groups created. Then, 훽 quantiles are values that partition a finite sample into values higher or lower than the value of 훽. In our study, the 95-quantile is considered. This notion is easily understood Figure 14. In this example, the 95- quantile is noted Q95 so 95% of the values of the integral are lower than Q95 and 5% of the values are higher. 푇  Let suppose a random variable 푿 = (X1, X2, … , Xm) from a population having a probability density 푇 function 푃(푿) and an observable 휃 = 푔((X1, X2, … , Xm) ) where 휃 is the parameter to be estimated.

Further, let’s suppose that L1(X1, X2, … , Xm) and L2(X1, X2, … , Xm) are two statistics such that L1< L2

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with probability 1. The interval (L1, L2) is called a [100 × (1 − 훼)]% confidence level for 휃 if L1 and L2 can be selected such that [19]: (14) 푃(퐿1 < 휃 < 퐿2) = 1 − 훼

Limits L1 and L2 are called the confidence limits for 휃, and 1 − α is called the confidence coefficient. The value 1 − α is generally taken as 0.90, 0.95, 0.99, and 0.999. For a given sample, confidence limits are not

unique. In other words, many pairs of statistics L1 and L2 exist. For example, let consider the Gaussian normal distribution illustrated Figure 15 [19]. To obtain 95% of all the values for the normal distribution, the regular interval is the value [-1.96, 1.96]. In addition to this pair, there are other many pairs of values (not symmetric about zero) that could give the probability 0.95 in the equation 14. However, this particular pair [-1.96, 1.96] gives the minimum width interval. In this study, the 95% quantile with a 95% confidence level is studied. In the nuclear field, the common values are 95% because it is good enough to quantify uncertainties with a good accuracy.

Figure 14: Illustration of the 1st 휷 quantile Figure 15: Illustration of the 95% quantile with a 95% confidence level on the normal distribution

The Wilks’s principle states that if there is a sample of N independent runs of a random variable, then the pth maximum of this sample is greater than the 95% quantile with a 95% confidence level noted I95/95. The I95/95 is an estimator that is larger than the true Q95 in 95% of the cases (95% of the draws). I95/95 is a random variable so this estimator is different between each sample. In terms of probability, it corresponds to:

Prob (Q95% ≤ I95/95) ≥ 95% Table 3 shows the minimum number of simulations to determine the pth maximum corresponding to the Wilks order for one observable [7]:

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Table 3: Illustration of the order of the Wilks’s formula

rd For example, if 124 simulations of one random variable are performed, the 3 highest value is our estimator I95/95 and this value is greater than the true Q95 in 95% of the cases. By increasing the number of simulations, the estimator I95/95 converges to the true value of the 95% quantile with a 95% confidence level. This principle makes it possible to minimize the number of simulations but guarantees a strong conservatism if the number of simulations is not sufficient.

All the values of the activation given by the stochastic code combined with the Wilks’s principle enable to evaluate the uncertainties on the sodium secondary circuit. The first step is to perform a reference calculation without random sampling. Then all the calculations with the random libraries are performed. Thanks to the Wilks’s principle, the estimator I95/95 is known. The difference between the reference calculation and the estimator I95/95 enables to evaluate uncertainties on the observable due to modifications on the input parameters.

3.4. Uncertainty due to the neutron source spectrum One objective of my study is to quantify the uncertainty on the sodium activation due to the uncertainty on the neutron source spectrum. Different types of uncertainties on the neutron spectrum exist: the intensity, the angular distribution, and the energetic distribution. The intensity of the source related to the neutron multiplicity is considered as an input data so this value is fixed. The angular distribution is considered as isotropic in the problem. The uncertainty on the source spectrum is determined by studying the uncertainty on the energetic distribution of the spectrum. This uncertainty corresponds to the error on the energy of the neutrons after the fission. This uncertainty must be propagated during each calculation. The neutron spectrum is encoded in the file 5 of the ENDF6 files and its covariance matrix associated is in file 35. The same method applied to nuclear data is used. Random draws on the values of the spectrum are done before each transport calculations. These draws generate random modifications on the input data. The modifications represent variations of the input data in the range of their uncertainties. This source of uncertainty is determined separately from other nuclear data because we expect that this uncertainty is low compared to the uncertainty due to the neutron cross sections.

For the ASTRID and Superphénix cores which are SFR reactors, only the 239Pu is used for the definition of the neutron source spectrum. Indeed, in such reactors, most of the fissions come from the 239Pu so other nuclides are neglected for the neutron spectrum. Moreover, the spectrum due to delayed neutrons is neglected so only the neutron spectrum due to prompt neutrons is taken.

A bibliographic task was to find an ENDF library where information on the spectrum and the covariance associated are present. After studying many libraries, the JENDL-4.0 updated 1 library was taken. In this library, neutron spectrums are available for 23 different incident neutron energy: 10-5 eV, 0.1 MeV, 0.2MeV, 1MeV, 2.5 MeV, 5.5 MeV… 20 MeV. In a sodium fast reactor, the average energy of neutrons which induce fission is closed to 0.2MeV so this energy is chosen for the neutron spectrum. In an ENDF6 file, the spectrum is tabulated for different energies. Figure 16 represents the spectrum from JENDL-4.0 updated 1 used in the calculations.

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Figure 16: Neutron energy spectrum for a 0.2MeV incident neutron In the part 3.3, the methodology to determine uncertainties on the secondary sodium activation due to nuclear data is explained. First, a nuclide database must be defined to choose on which parameters and nuclides the random draws are performed. Nuclides are chosen thanks to the recommendations of the COMAC database from the CEA. Among the four methods to determine uncertainties due to nuclear data, the Total Monte-Carlo method is retained. This method consists of performing random draws on parameters thanks to the variance and covariance information. These parameters are then used as input to a stochastic code and many calculations are done with modified input data. In this report, the uncertainty on the neutron source spectrum, which is often negligible, is studied. After performing all the calculations, the results are post-processing thanks to the Wilks’s principle to determine the uncertainty on the secondary sodium activation. This method requires an important computational power but with a sufficient number of calculations, the observable is determined with a good accuracy. Now that the procedure is described, the next part presents all the tools to put the methodology into practice.

3.5. Different tools for the study of sodium fast reactors To apply the Total Monte-Carlo method for the quantification of the uncertainties on the secondary sodium activation, three different codes are needed: - A code to prepare all the inputs for the stochastic code by doing random draws on the input parameters (block in parenthesis A Figure 7). - A stochastic code and more precisely a transport code to perform the high number of calculations to evaluate the observable (block in parenthesis B Figure 7). - A code to perform the variance reduction technique so the time to perform a calculation is reduced. This code is not used if the transport code contains directly variance reduction techniques (block in parenthesis B Figure 7).

3.5.1. Stochastic calculation tool: MCNP Monte Carlo N-Particle Transport Code (MCNP) is a numerical simulation code that uses the Monte Carlo method to model nuclear physics processes [20]. This is the reference code in the neutron transport theory field. The Monte Carlo method consists in following the history of a very large number of particles in a system, from their

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"birth" (external source or fission neutrons) to their "death" (captured by a nucleus or leaking out of the system). By the statistical analysis of all these simulated histories, the natural behavior of the system at a given moment can be deduced. As part of this report, MCNP5 is used to determine the uncertainty on ASTRID and SUPERPHENIX secondary sodium activation due to the uncertainties on nuclear data. It should be noted that the source of neutrons of the core to be propagated with MCNP comes from an ERANOS2.3 / PARISV1 calculation using the JEFF-3.1 reference library. The MCNP models of the two cores, as well as the methodology for the calculation of the secondary sodium activation, are validated as it is used in studies carried out within Framatome.

3.5.2. ADVANTG ADVANTG is a deterministic code which performs reduction variance techniques by creating weight windows [21]. In the path from the core to the IHX, neutrons are propagated over more than 9 orders of magnitude of neutron flux attenuation which is very low. The neutrons from the source have a very low probability of reaching the IHX and activate the 23Na. Too many neutrons are necessary to reach an acceptable level of convergence in light of the computing capacity. Therefore, variance reduction methods have to be implemented to obtain sufficiently convergent results (uncertainty MCNP ≈ 5%) with a reasonable number of simulated particles (hundred thousand of source neutrons). It is the reason why the ADVANTG code is used to produce weight windows [21]. In short, these weight windows constitute a mesh where each zone has a weight following deterministic calculations of neutron flux and importance with the solver Denovo. These weights make it possible to control the neutron population in the different calculation zones to improve the convergence of a given calculation. For this, particles are "killed" with the Russian roulette method in certain areas of little interest for obtaining the desired result (high window weight). In return, the particles are multiplied with the split method in the areas that contribute to the desired result and thus improve the statistic (weak window weights). These operations are carried out while maintaining the total statistical weight of the particles. It should be noted that ADVANTG generates 27 weight windows corresponding to the division of the spectrum into 27 energy groups and a division of the model into a spatial mesh chosen by the user. The weight windows are a discretization in the phase space (energy + space). The weight windows are powerful but a very complex tool, so it is necessary to supplement these very succinct explanations with those of the reference [21].

3.5.3. NUDUNA: NUclear Data UNcertainty Analysis NUDUNA is a GUI which allow for a full Monte-Carlo sampling of the nuclear data inputs. The method generates many numbers of input files and for each input files, a transport calculation is performed. The difference in all input files leads to a different result for the observable. The random drawing on the Mfi input parameters is based on covariance tapes for nuclear data distributed in ENDF6 format. The NUDUNA code enables to perform four blocks Figure 7. NUDUNA takes uncertainties from files 1 to 5 of the ENDF6 files and covariance data from files 31 to 35. Two functions handle the reading part of the ENDF6 file and random sampling. The output file is a new ENDF6 file "modified" but not readable by MCNP. A module in NUDUNA verifies that the physics is preserved with the function "sum rules" described paragraph 3.3.4. As explained before, the creation of random libraries and creation of input files for MCNP are the two tasks performed by NUDUNA.

To determine the uncertainty on the neutron source spectrum, the NUDUNA source code must be changed. NUDUNA is encoded in FORTRAN. The first task was to understand the whole code and learn a new language: the FORTRAN. Secondly, functions were created to read the file 5 and 35 of the ENDF6 file and write information in variables. At the end, a function implemented in NUDUNA performs the random draws on the values of the spectrum.

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In the part 3.5, the three different codes needed to apply the Total Monte-Carlo method are presented. To summarize, the NUDUNA code creates all the random input data thanks to sampling on the variance and covariance matrices. Two different functions read information from the ENDF6 files, perform random sampling on the initial ENDF6 file, write a new ENDF6 file “modified” and transform ENDF6 tapes to obtain an input used by the stochastic code. The stochastic code MCNP5 calculates the secondary sodium activation for each input file. To reduce the computing times of MCNP calculations, the code ADVANTG is used to perform variance reduction techniques by creating weight windows on the model. The last part presents the results obtained on the ASTRID and Superphénix reactors.

4. Results for the quantification of the uncertainty on the secondary sodium activation

4.1. Procedure to determine uncertainties on the secondary sodium activation Before presenting all the results, the description of the procedure to obtain results helps the reader to understand the progression and the logic of the study. The results are dissociated in two parts, one for ASTRID and one for Superphénix. Even if two parts are written, similitudes can be found:  First, a reference calculation is performed and the definition is the same for both cases. The reference calculation corresponds to all the important nuclides (different for ASTRID and Superphénix) chosen with the COMAC database without performing random draws.

 In a second part, the convergence of the estimator I95/95 is determined iteratively. For each Wilks’s rank Table 3, two different series of calculations are done. When the two results of the series are approximatively the same, the minimum number of calculations to have a converged value of the estimator I95/95 is fixed by the Wilks’s order. First, two series of 59 calculations are done (Wilks’s order 1) and the same values of the activation have to be obtained, otherwise two series of 93 calculations (Wilks’s order 2) are done to compare results. If the two activation values are different for the two series of 93 calculations, 124 calculations are performed, etc. A simplified case studying only the impact of the sodium on the ASTRID reactor is considered to have a time- saving and the verification that this case is representative of the problem is done the next part.  In the third part, the quantification of the uncertainty on the secondary sodium activation due to the uncertainties on nuclear data is done. The number of MCNP calculations is equal to the number of simulations corresponding to the Wilks’s order found previously. The comparison between the hypothesis (preponderance of the 23Na impact) and the impact of all nuclides is studied.  After verifying the impact of the 23Na on the uncertainty of the secondary sodium (supposed preponderant), the impact of other nuclides except the sodium is studied to classify nuclides in order of importance.  In a fifth paragraph, the purpose is to understand the behaviour of the ASTRID neutrons source. The goal is to verify the hypothesis with physical arguments: for example the hypothesis that the nuclides of the core are neglected for the activation of the secondary sodium.  Then, the uncertainty of the source spectrum on the activation of the secondary sodium is studied on ASTRID.  Thanks to Superphénix, the results allow comparing the whole method to measurements done on the reactor under-functioning. This last part enables criticisms of the model applied on ASTRID.

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4.2. ASTRID

4.2.1. Reference calculation The reference calculation corresponds to all the important nuclides with the COMAC database without performing random draws. This is the most up-to-date data without any disturbance; the mean value is taken for the parameter. For other nuclides, the JEFF-3.1.1 library is chosen. The results are calculated with the code MCNP5_v160 using weight-windows generated by ADVANTG. The IHX 2 secondary sodium is the most activated because of its proximity to a ramp. This ramp is used for loading or unloading the fresh or spent fuel. To place this ramp, some neutron lateral protections closed to IHX 2 and 3 are removed from the core. The removal of lateral protection constitutes a privileged way for the neutrons to reach the intermediate heat exchangers. Angularly, the IHX 2 is closer to the axis of the ramp so this intermediate heat exchanger is the most activated one as shown Table 4. In the rest of this part, only the intermediate heat exchanger 2 is considered.

IHX Normalized activation

Reference calculation in the IHX 1 0.58

Reference calculation in the IHX 2 1

Reference calculation in the IHX 3 0.68

Reference calculation in the IHX 4 0.47

Table 4: Activation of the secondary sodium in different intermediate heat exchangers

4.2.2. Convergence of the estimator I95/95 The goal of this paragraph is to determine the minimum number of MCNP calculations needed to obtain a good convergence of the I95/95 estimator. The method applied is iterative: calculations are done progressively by following the Wilks’s order Table 3. First of all, the minimum of 59 calculations is done, and then the Wilks’s principle to determine the statistic is applied. If the statistic is not good, then 93 calculations are performed. The calculations are done by only sampling the 23Na cross-sections because of two reasons: - The time to create random libraries for 23Na is low compared to other nuclides - This case is supposed representative of the problem. After studying the Figure 3, the leakages are on the top and the base of the core. As a consequence, the neutron path between the core and the IHX crosses a huge volume of sodium. One hypothesis is considered: most of the neutrons interactions are with the 23Na so the case studied is representative of the problem. After performing many calculations illustrated Table 5, the good number of calculations to apply the Wilks’s principle is 181, corresponding to the 5Th order. Two series of 181 calculations are performed and results converge to the same value. The value of the estimator I95/95 related to the activation of the sodium secondary

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circuit converges to approximately 2.00 if the reference calculation is set to 1. The MCNP uncertainty related to the convergence of the calculations is 2% for all the calculations. If the MCNP uncertainty is taken into account, the two series of 181 calculations have the same result. The number of MCNP calculations for one series is now set to 181 because the convergence of the estimator is achieved. To be sure that the asymptote is obtained, the test on 396 calculations is made. Thus, the number of random libraries created for the 23Na is 396. The results for 59 and 93 calculations must not be taken into account because the statistic is not good, the difference with the reference calculation is not the same.

Calculation Value of the estimator Difference compared to the I95/95 reference calculation (%)

Reference calculation in the IHX 2 1 -

3.37 1st series of 59 calculations Wilks’s order 1 237% (1st maximum)

2.95 2nd series of 59 calculations Wilks’s order 1 195% (1st maximum)

2.02 1st series of 93 calculations Wilks’s order 2 102% (2nd maximum)

2.15 2nd series of 93 calculations Wilks’s order 2 115% (2nd maximum)

1st series of 181 calculations Wilks’s order 5 2.02 th 102% (5 maximum)

2nd series of 181 calculations Wilks’s order 5 2.00 th 100% (5 maximum)

Wilks’s principle on 386 Wilks’s order 1.97 calculations 13 th 97% (13 maximum) Table 5: Values for the activation of the secondary sodium by modifying 23Na cross-sections

4.2.3. Uncertainty on the activation of the secondary sodium In this part, the impact of the 23Na nuclear data uncertainties is studied. For the calculations, the number of random libraries created for all nuclides considered as important is 181. As explained paragraph 3.3.1, core materials are excluded. The selected nuclides considered as important are shown Table 2. In this report, some Mfi parameters are useless and are not considered in the study of the uncertainty on the activation of the secondary

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sodium. Nuclides from the core are excluded so the Mfi parameters multiplicity of secondary particles and Data on radioactive decay are not considered. The energetic distribution is studied paragraph 4.2.7. The impact of the angular distribution is considered as minor because the distribution is isotropic. The study focus on the Mfi parameter related to the random draw of the neutron cross-sections. From the Wilks’s test on 181 iterations th which gives the 95% quantile with a 95% confidence level, the 5 maximum of the 181 iterations is our I95/95 estimator. Two different series of calculations are made to take into account the impact of 23Na. These two series are confronted with the reference calculation in the IHX 2: - The cross-sections of all the nuclides Table 2 are modified by random draws on the covariance matrices; - The 23Na cross-sections are modified by random draws on the covariance matrices;

The goal is to determine the impact of the 23Na cross-section uncertainties on the total uncertainty due to nuclear data.

Difference compared to Value of the estimator Calculation the reference calculation Mean value I 95/95 (%)

Reference calculation in the IHX 2 1 - -

Drawing on cross-sections for all 2.04 nuclides th 104% 1.08 (5 maximum)

Drawing on 23Na cross-sections 2.02 th 102% 1.06 (5 maximum)

Table 6: Values for the activation of the secondary sodium by modifying all nuclides cross-sections

Table 6 illustrates that the impact of the 23Na nuclide is preponderant on the nuclear data uncertainty. Table 6 shows that the difference with the reference calculation is in the same range for the "Drawing on cross-sections for all nuclides" and "Drawing on 23Na cross-sections" series. The mean MCNP uncertainty is 2% for the two series of calculation. The value of the column “Difference compared to the reference calculation (%)” is directly the uncertainty due to the drawing studied because the difference between the reference calculation and I95/95 is the uncertainty due to perturbed input data. As a conclusion, the uncertainty due to nuclear data is overwhelmingly by the 23Na nuclide. Then, if the mean values are considered, this value is almost the same for all calculations. It shows that the Wilks’s principle applies conservatism. By considering the I95/95 estimator, a margin of safety is applied because this value is higher than the mean value. For all calculations, the mean value and the reference value are different because the distributions of the parameters are not perfectly Gaussian.

The set of calculations carried out makes it possible to provide a converged value for the uncertainty on the activation of the secondary sodium. This uncertainty is approximately 100% for the ASTRID reactor. The corresponding value of the 95% quantile with a 95% confidence without uncertainties is 2.02 if the reference calculation is 1. As a conclusion, the uncertainty on ASTRID secondary sodium activation due to nuclear data is mainly due to the 23Na cross-section uncertainty and the value is 100%. The hypothesis for the reference calculation is verified.

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4.2.4. Origin of the uncertainty on the secondary sodium activation To reduce the uncertainty on the secondary sodium activation, the origin of this uncertainty must be known. Figure 17 shows 181 MCNP calculations yielding to 181 different secondary sodium activation values. The only modified quantities are the 23Na cross-sections by making drawing on the variance/covariance matrices. As a reference, all activation values are divided by the reference calculation in the IHX 2. This figure illustrates a very significant dispersion of the activation, values ranging from 0.4 to 2.8 times the value of the reference. Due to the uncertainties, calculation values are higher and lower than the reference value 1. One uncertainty must be added to each calculation: the convergence of the MCNP calculation. The average error due to the convergence is 2%. As a conclusion, this figure illustrates a very strong link between the uncertainty on the activation of the sodium (due to the dispersion of values) and the value of the total cross-section of 23Na (different at each calculation). The purpose of the next paragraph is to define the origin of this strong link.

Figure 17: Normalized activation of the secondary sodium in the IHX 2 for 181 MCNP calculations

The origin of the uncertainty on the secondary sodium activation comes from the uncertainty on the 23Na elastic scattering cross-section. Figure 18 and Figure 19 present the values of the 23Na capture and elastic scattering cross-sections as a function of the secondary sodium activation of previous MCNP calculations. Figure 18 demonstrates the correlation between the value of the elastic scattering cross-section and the value of the secondary sodium activation because the coefficient R2 is close to one. This value of R2 is not sufficient for a linear regression but the presence of a correlation is certain. On the other hand, Figure 19 shows the non-correlation between the value of the 23Na capture cross-section and the value of the secondary sodium activation. It is therefore assumed that the dominant parameter for the uncertainty on the secondary sodium activation is the elastic scattering cross-section of 23Na. The dominant parameter is proved thanks to these figures but some correlations might be hidden by this parameter that is why calculations are done paragraph 4.2.3. The value of the elastic scattering cross-section uncertainty given by the CEA is low, ~3% in the energy interval considered. This value of 3% is difficult to reduce due to technological limits. The reduction of the uncertainty on the activation of the secondary sodium seems to be arduous. A high uncertainty on the activation will always remain unless the measurement of the 23Na elastic scattering cross-section improves.

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Figure 18: Activation of the secondary sodium in the IHX 2 as a function of 23Na elastic scattering cross-section

Figure 19: Activation of the secondary sodium in the IHX 2 as a function of 23Na capture cross-section

4.2.5. Impact of all nuclides on the secondary sodium activation uncertainty In this part, the impact of all nuclides except 23Na is studied. For the calculations, the number of random libraries created for all nuclides considered as important is 181. The goal is to determine the uncertainty on the secondary sodium activation if the uncertainty on 23Na cross-sections is excluded. The study focus on the Mfi parameter related to the random draw of the neutron cross-sections. From the Wilks test on 181 iterations, the 5th maximum of the 181 iterations is our I95/95 estimator. Two different series of calculations are performed to take into account the impact of the nuclides: - The 23Na cross-sections are modified by random draw on the covariance matrices; - The cross-sections of all nuclides except 23Na are modified by random draw on the covariance matrices;

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Difference compared to Value of the Calculation the reference calculation Mean value estimator I 95/95 (%)

Reference calculation in the IHX 2 1 - -

Drawing on 23Na cross-sections 2.02 th 102% 1.06 (5 maximum)

Drawing on cross-sections of all 1.30 23 nuclides except Na th 30% 1.05 (5 maximum)

Table 7: Values for the activation of the secondary sodium by modifying all nuclides cross-sections

Table 7 proves that the impact of all nuclides except 23Na is low compared to this nuclide. The hypothesis that the uncertainty on the secondary sodium activation is mainly due to the uncertainty on the 23Na nuclide is verified. This assumption is explained by the axial leakage due to the good efficiency of the lateral neutron protections. It must be noticed that the value of 30% is not totally negligible but this value is encompassed by the value of 102% found previously. The uncertainty on the activation of the secondary sodium remains about 100%.

4.2.6. The behaviour of the neutrons in the core A module named PTRAC in MNCP enables to understand the behavior of the core. Thanks to the PTRAC card, it is possible to track all neutrons that terminate their lives in the IHX 2. The PTRAC card describes the interactions of a parent particle and the particles which it in turn creates. The team that created the PTRAC output faced the issue of writing events of a particle and its progeny in an output file. The data structure of a particle and all the progeny which recreate other progeny is a tree data structure with branches. The event lines in PTRAC describe the life of a particle of known energy moving in a specified direction [22]. Tracks are defined by the particle’s action between two points in time and space. When the parent particle moves between n “events”, (n -1) tracks are defined. When the particle splits its original energy, one of the resulting particles is the daughter particle and the subsequent progeny get written in a separate block. The other particle becomes the continuation of the parent. The use of this card enables to: - Determine the average behavior of the neutrons which are involved in the activation of the secondary sodium. - Determine the average number of collisions with nuclides per history to verify the hypothesis that only the external part of the core is important on the activation of the secondary sodium. - Determine an average energy at which neutrons are captured in the IHX 2 and thus contribute to the activation.

The number of histories studied that terminate to a capture in the IHX 2 is 421 over 300 000 neutrons from the source. One history gathers many neutrons because, during a history, the particle splits due to weight windows so progenies are created. At each collision between a neutron and one nuclide, many parameters are known [22]: - The position (x,y,z) and the direction (u,v,w) of the neutron;

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- The energy of the neutron; - The weight of the neutron. This parameter is very important because it determines if a neutron has a high or low contribution to the activation of the secondary sodium. Two neutrons captured in the IHX 2 can be compared: neutron 1 and 2. If the neutron 1 has a weight equal to two times the neutron 2, the contribution to the activation of the neutron 1 is two times higher than the contribution of the neutron 2; - The nuclide where the collision occurred; - The time; - The type of reaction (scattering, capture, fission); - The creation of a new particle or not.

Based on 421 histories, the Table 8 first column presents the average number of collisions on different nuclides by considering the weight of the neutrons. The second column gives the chance to obtain a collision with special nuclides by considering one collision.

Average number of collisions in one Percentage to obtain a collision on a Nuclide history by considering the weight of the nuclide by considering the weight of the neutrons neutrons (%)

Sodium 23 133 63.5

Iron 56 42 19.9

Nickel 58 11 5.4

Magnesium 24 10 4.8

Oxygen 16 5 2.4

Uranium 238 4 1.9

Chromium 52 4 2.1

Boron 10 0.1 0.05

Table 8: Average number of collisions on different nuclides using the PTRAC card

The physical behavior of the neutron in the ASTRID core explains the origin of the uncertainty on the secondary sodium activation. The Table 8 illustrates that the average number of collision in the sodium is the most important one. The uncertainty on the value of the 23Na elastic scattering cross-section is low but many collisions with the sodium occur so the global uncertainty which is related to the number of collisions is high. Then there are also many collisions in alloys because there are nuclides such as iron, chromium, and nickel. In average, there are only 4 collisions with the uranium. This result shows that the nuclides from the core have a low impact on the

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activation of the secondary sodium. The hypothesis excluding the nuclides from the core in the list of the important nuclides is verified. The weight of the neutrons has to be considered to keep the physics. If a history has 20 collisions with the uranium but a weight very low, the contribution on the 420 other histories has to be low. The second column presents the percentage to obtain a collision with a nuclide if one collision is considered. As previously, if one collision is considered, the chance to have a collision with 23Na is high. The weight of the final collision (capture in the IHX 2) varies from three orders of magnitude so there are unequal histories between neutrons. If collisions with 23Na are considered, there are three different cases of interactions: - Capture: this interaction is excluded because when a capture occurs, the history of the neutron is stopped so this kind of interaction is not taken into account in the count; - Elastic and inelastic scattering: by studying each type of reaction for each collision, nearly 99% of the reactions are elastic scattering. By considering the elastic (red) and inelastic scattering (green) cross- sections illustrated Figure 20, it is easily understood that the probability to have an elastic scattering is very high for all energies because the red curve is higher than the green one.

Figure 20: Elastic and inelastic cross-section for 23Na as a function of the energy of the neutron

A second study is the energy of the neutron after each collision. It is important to determine how the neutron losses its energy. The result expected is a small decrease of the neutron energy. If the energy decreases quickly, the thermal energy is reached and the neutron is absorbed before arriving into the IHX. One goal is also to find a “cutting energy”: below this value, it is impossible to find a neutron in the IHX, all neutrons are captured. Figure 21 presents the evolution of the neutron energy during its history. Each point corresponds to a collision. A neutron with a high weight is considered to be representative of the problem. In this case, the energy decreases slowly at the beginning but after 100 collisions, the energy of the neutron loses 5 orders of magnitude. At the end, the energy is 180meV when it is captured in the IHX 2. The thermal energy for the neutron is reached in the IHX 2. Based on 421 histories, the mean energy at which the neutron is absorbed in the IHX is 341meV by taking into account the weight of the neutrons. This value corresponds to slow neutrons so the probability to be captured by a nuclide is really high.

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Figure 21: Energy of the neutron as a function of the number of collisions

4.2.7. Neutron source spectrum result Studying the uncertainty on the secondary sodium activation due to the neutron source spectrum enables to compare different sources of nuclear data uncertainties. In this part, the random parameter is a vector containing all values of the spectrum for each energy bin. Thanks to the covariance matrix associated, random draws are performed on the values of the spectrum. The number of random draws is 181, the same as random sampling on neutrons cross-sections. The reference calculation corresponds to all the important nuclides with the COMAC database by choosing the latest versions at our disposal without performing random draws and by choosing the JENDL spectrum without performing random draws. In the MCNP input files, the only modified parameter is the spectrum. Figure 22 shows 181 MCNP calculations yielding to 181 secondary sodium activation values. As a reference, all values of the activation are divided by the value of the reference calculation. The figure shows that the dispersion is low compared to random draws on neutron cross-sections because values are ranging from 0.8 to 1.1 times the reference. The average value of the MCNP error due to convergence is 3.6%.

Figure 22: Normalized activation of the secondary sodium in the IHX 2 for 181 MCNP calculations

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Table 9 presents the Wilks’s principle results to determine the impact of the neutron spectrum on the activation of the secondary sodium. As expected, the uncertainty is low compared to neutron cross-sections because this uncertainty is approximately 9%. This study was done apart from sampling on the neutron cross-sections because the uncertainty due to cross-sections would have erased the uncertainty due to the neutron source spectrum. As a conclusion, the uncertainty on ASTRID secondary sodium activation due to nuclear data is mainly due to the uncertainty on the 23Na cross-section values and remains 100%.

Difference compared to Value of the estimator Calculation the reference calculation Mean value I 95/95 (%)

Reference calculation in the IHX 2 1 - -

Drawing on the values of the neutron 1.09 source spectrum th 9% 0.98 (5 maximum)

Table 9: Activation values after sampling on the neutron spectrum

4.3. Superphénix

4.3.1. Reference calculation The reference calculation corresponds to all the important nuclides with the COMAC database by choosing the latest versions at our disposal without performing random draws. For other nuclides, the JEFF-3.1.1 library is chosen. The only difference with ASTRID is the number of nuclides considered as important because the materials in the MCNP input files are different. Some nuclides are removed (in red Table 10) and 1H is added. Except for the change in the nuclides, the reference calculations in the two reactors are the same.

Fe54 Ni58 Mo97 Cr50 Mg24 Mo92

Fe56 Ni60 Mo98 Cr52 Mg25 Mo94

Fe57 Ni61 Mo100 Cr53 Mg26 Mo95

Na23 Fe58 Ni62 Mn55 Cr54 Mo96 (JEFF-3.2)

O16 Ni64 B10 C Al27 B11

H1 Table 10: List of important nuclides of the Superphénix core In this part, only the NE loop is considered because it is the most activated loop as shown Table 11. As a reminder, each loop contains two intermediate heat exchangers.

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Loop Activation of the sodium secondary circuit (Bq/cm3)

NE 7.93

SE 4.50

SW 0.72

NW 0.90

Table 11: Reference calculations in different loops for the core Superphénix

4.3.2. Preliminary calculations on the Superphénix reactor Many test calculations were performed by Framatome on Superphénix contrary to ASTRID. The first task of this part is to compare the new method to previous ones. Indeed, the method to determine the activation of the sodium secondary circuit changes with the use of ADVANTG. The first set of calculations is to determine the impact of this new method. Table 12 illustrates all the calculation results. The original calculation is the Superphénix input file from older studies without modifications (older nuclides libraries). Then the updated libraries are taken into account. First, only the new 23Na library (JEFF-3.2) is taken into account and then new libraries are considered for all nuclides. Between the original calculation and the two other calculations, the updated spectrum from the JEFF-3.2 library is taken to have a neutron spectrum with the latest value available. The new spectrum increases the activation of the secondary sodium by 3%. The calculated activation represents the result given by MCNP and the measured activation is obtained directly from the measurement. Then the C/M factor is the ratio between the calculated and measured values.

Calculated activation Measured Calculation 3 C/M Factor (Bq/cm3) activation (Bq/cm )

Original calculation 7.93 6.23 1.27 (without modifications)

Only the new source neutron spectrum 8.17 6.23 1.31 (JEFF3.2)

Only the new nuclide library for 23Na without 9.55 6.23 1.53 sampling + JEFF3.2 spectrum

New library for all nuclides without sampling + the JEFF3.2 spectrum = 10.53 6.23 1.69 reference case for the calculations on 4.3.3

Table 12: Summary of all calculation tests for Superphénix in the loop NE

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The use of the new libraries increases the value of the secondary sodium activation. The table shows that if the new 23Na library is taken into account, the calculated activation of the secondary sodium with MCNP increases by 14%. The origin of this difference is presented Figure 23. In the part of the spectrum corresponding to fast neutrons, values of the new library are slightly lower than to the older library. By considering the Figure 18, the activation of the secondary sodium is high when the value of the elastic scattering cross-section is low. It means that if the cross-section decreases, the mean free path is higher so neutrons travel a larger distance between collisions. As a consequence, fewer collisions occur with the new libraries so the thermal energy is reached later. The chance to reach the IHX and be captured by 23Na is higher.

Figure 23: 23Na elastic scattering cross-section as a function of the energy of the incident neutron

The C/M factors for each calculation are varying between 1.27 and 1.69. The calculations are slightly above the measurements due to the changes in the cross-sections libraries. The value 1.69 is in the upper part of the C/M acceptable values but results are still representative if the uncertainties on measurement are considered (between 14% and 27%). This factor is more useful when the Wilks’s principle is applied. The C/M factor is presented here to have an order of magnitude of this parameter and to see the evolution of this factor in different cases.

The impact of the new libraries is different between ASTRID and Superphénix. First, a comparison between “Only the new nuclide library for 23Na without sampling” and “New library for all nuclides without sampling” must be done. The table shows an increase of 10% of the activation value. By comparison, a MCNP calculation shows that for ASTRID, the increase is 2%. The difference comes from the different core configuration between ASTRID and Superphénix [11]. Figure 24 shows that in ASTRID the leakages are on the top and the base of the core, showing the good efficiency of the lateral neutron protections. In this case, the neutrons path between the core and the IHX contains a high quantity of sodium. With Superphénix, lateral neutron protections are less efficient so a lateral leakage occurs. This is the main leakage so the neutron path contains more stainless steel than ASTRID. As a consequence, the stainless steel cross-sections are more solicited in the calculation of the activation, hence the difference between 10% and 2%. After studies on the cross-sections, the nuclide responsible for the increase of the activation is the 56Fe. Like 23Na, the values of the cross-sections for the updated library are lower compared to the older ones. It is the reason why the activation of the secondary sodium is increasing. The other nuclides of the stainless steel have also an impact but lower than the 56Fe.

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Figure 24: Comparison of neutron streaming between ASTRID (left) and Superphénix (right)

4.3.3. Uncertainty on the activation of the secondary sodium The results obtained on Superphénix with our model are confronted with the reality because, under functioning, activation measures were made. Such as ASTRID, some Mfi parameters are useless and are not considered in the study. The study focus on the Mfi parameter related to the random draws of the neutron cross-sections. The number of random libraries created is equivalent to the calculation performed on ASTRID: 198 for each nuclide considered as important. The same libraries as the ASTRID case are used. The calculations performed are the same as ASTRID to make a comparison between the two reactors. The ASTRID case shows that the statistics of the Wilks’s principle is good for 181 calculations so the number of iterations is set to 181. The 5th maximum of the 181 iterations is our I95/95 estimator.

Figure 25: Activation of the secondary sodium in the loop NE for 181 MCNP calculations

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The random draws on the neutron cross sections prove the existence of an uncertainty on the sodium activation. Like ASTRID, Figure 25 proves a dispersion of the results for the Superphénix reactor. The reference value is set to 1. The only modified quantity is the 23Na cross-section by making drawing on the variance/covariance matrices. Values are ranging from 0.6 to 1.9 times the value of the reference so the impact of the 23Na cross-sections on the uncertainty is high. The same conclusion as ASTRID is made: there is a very strong link between the uncertainty on the activation of the sodium (due to the dispersion of values) and the value of the 23Na total cross-section (different at each calculation). After studying the values of the 23Na cross-sections, the activation of the secondary sodium is mainly due to the uncertainty on the 23Na elastic scattering cross-section. Two different series of calculations are made to justify this hypothesis. These two series are confronted with the reference calculation on the loop NE: - The cross-sections of all nuclides are modified by random draw on the covariance matrices - The 23Na cross-sections are modified by random draw on the covariance matrices

Value of the activation Difference compared

Calculation of the secondary circuit to the reference I95/95 /M Mean value (Bq/cm3) calculation (%)

Reference calculation in the 10.53 - - - loop NE

Drawing on cross-sections for 17.49 all nuclides 66% 2.8 10.21 (5th maximum)

Drawing on 23Na cross- 17.55 sections 66% 2.8 10.64 (5th maximum)

Table 13: Values for the activation of the loop NE for 181 calculations

The uncertainty on the sodium activation is mainly due to the 23Na nuclide. First, Table 13 shows that the convergence of the statistic is very good because the difference between “Drawing on cross-sections for all nuclides” and “Drawing on 23Na cross-sections” is the same. The mean MCNP convergence is 3% for both. Then, if the mean value is considered, this value is almost the same for all calculations. It is fortifying because on average the same physical effects appear between the different cases so the difference between the mean values has to be minor. The uncertainty on the Supeprhénix secondary sodium activation due to nuclear data is mainly due to the 23Na cross-section uncertainty and the value is 66%. As a remark, the value for “Drawing on 23Na cross- sections” is higher than the value with drawing on all nuclides. It can be considered as wrong values because normally the highest value is the drawing on all nuclides. The difference comes from the MCNP convergence because if this uncertainty is taken into account, the values are in the same range so the activation for the “Drawing on cross-sections for all nuclides” can be the highest value.

4.3.4. Impact of all nuclides on the secondary sodium activation uncertainty The impact of all nuclides except 23Na is studied for the Superphénix core. Like ASTRID, the number of random libraries created for all nuclides considered as important is 181. The goal is to determine the uncertainty on the

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sodium secondary circuit if the uncertainty on 23Na cross-sections is excluded. Two different series of calculations are performed to take into account the impact of 23Na: - The cross-sections of 23Na are modified by random draw on the covariance matrices; - The cross-sections of all nuclides except 23Na are modified by random draw on the covariance matrices;

Difference Value of the compared to the Calculation Mean value estimator I95/95 reference calculation (%)

Reference calculation in the loop NE 10.53 - -

Drawing on 23Na cross-sections 17.55 66% 10.64 (5th maximum)

Drawing on cross-sections of all 14.54 23 nuclides except Na th 38% 10.47 (5 maximum)

Table 14: Values for the activation of the secondary sodium for 181 calculations

Table 14 proves that the impact of all nuclides except 23Na is low but higher compared to ASTRID. The hypothesis that the uncertainty of the secondary sodium activation is mainly due to the uncertainty on the 23Na elastic scattering cross-section is verified. The difference with ASTRID is that the impact of other nuclides is more important because the difference with the reference is 38% compared to 30% with ASTRID. The origin comes from the lateral leakage on Superphénix explained Figure 24. The neutron main path in Superphénix goes through more stainless steel than ASTRID; this is the reason why the impact of other nuclides is higher on the activation of the secondary sodium. The mean MCNP convergence is 2% for the two series of calculation. It must be noticed that the value of 38% is not totally negligible but this value is encompassed by the value of 66% found previously. The uncertainty on the Superphénix secondary sodium activation remains 66%.

4.3.5. Difference between calculations and measurements

The two-sided tolerance interval is chosen to apply the Wilks’s principle. The I95/95/M factor is used with the Wilks’s principle to compare calculations and measurements. Since the beginning, only the one-sided tolerance interval is considered because only one value for the estimator is needed. In this case, it is relevant to use the two-sided tolerance to compare interval. Indeed, values are given with an uncertainty so compare the coverage of the intervals between cases is the good method. If the experimental values are in the calculated interval, our calculations are acceptable. The Wilks’s principle enables to compare confidence intervals between series of calculations. The calculations are made on 181 iterations. The reference [5] demonstrates that the closest Wilks’s rank compared to the number of calculations on the two-sided tolerance is obtained with 176 calculations for the 95% quantile with a 95% confidence level. Table 15 presents the confidence interval for three cases. For the calculations, the value comes from the Wilks’s principle on the two-sided tolerance. For the experimental value, the uncertainty of 27% on the measurements is taken into account. This table shows that with the Superphénix core, our model is conservative

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because values of the interval are higher for the computed values compared to the measurement. For the lower values of the intervals, the conservatism is so high with the computed values that the experimental value is not included in the calculated intervals. This conservatism is good for safety but too important values prove that the method could be improved. The conservatism applied is a factor 2 which is significant compared to the uncertainties on the measurements. These results raise some questions regarding the model applied on ASTRID. If the model applied on Superphénix is extrapolated on ASTRID, one problem is raised on the nuclear data. Maybe some data are missing in the model or some data are inaccurate and lead to an important conservatism. Another possibility is that the methodology is an overall conservatism method. The values found with ASTRID could be verified with other methods less conservative if possible.

Confidence interval Calculation (Bq/cm3)

Drawing on cross-sections [ퟓ. ퟑퟑ, ퟏퟖ. ퟏퟕ] for all nuclides

23 Drawing on Na cross- [ퟔ. ퟎퟐ, ퟏퟗ. ퟖퟒ] sections

Measurements with the [ퟐ. ퟗퟑ, ퟗ. ퟓퟑ] uncertainty of 27%

Table 15: Comparison of confidence interval between the calculation and the measurements

The relevance of the calculation scheme has been tested by benchmarking the Superphénix measurements with the calculated activations on the secondary loop. The ratio between calculated and measured values shows a discrepancy of 2.8 which is very not bad for a neutron attenuation of 9 orders of magnitude through several meters of structures. Nevertheless, the last set of calculations shows an important conservatism applied to the two models ASTRID and Superphénix. The last part presents all the results for the quantification of the uncertainties on the activation of the sodium secondary circuit. In a first part, the results of the Total Monte-Carlo method for the ASTRID reactor states that the uncertainty on the secondary sodium activation due to uncertainties on nuclear data is 100%. This uncertainty is mainly due to the uncertainty on the 23Na elastic scattering cross-section and the impact of other nuclides except 23Na is neglected. The PTRAC card in MCNP5 enables to validate the hypothesis that the neutrons of the core are excluded in the activation of the secondary sodium. The quantification of the uncertainty on the neutron spectrum demonstrates that its impact is lower compared to neutrons cross-sections because the uncertainty on the sodium activation is 9%. In a second part, the same scheme was applied on Superphénix to compare values obtained with the calculation and measurements. In this reactor, the uncertainty on the activation of the secondary sodium due to nuclear data is 66%. The impact of other nuclides than 23Na is higher compared to ASTRID due to the presence of axial leakage. The comparison with measurements shows that the method applies conservatism on the results obtained.

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Conclusion

This document presents a method for quantifying uncertainties on the secondary sodium activation due to uncertainties on nuclear data. The goal is to determine a reliable estimate of these uncertainties so that the protections on the facility are dimensioned correctly. The consequence is a good sizing of components which avoid heavy and expensive technologies. The use of the software NUDUNA enables to perform a full Monte-Carlo Method. The knowledge of the variance and covariance information from the ENDF6 files enables to create the input data for the stochastic code MCNP. Thanks to many calculations using nuclide random libraries, the uncertainty on the secondary sodium activation is determined with a good accuracy. For ASTRID and Superphénix, the uncertainty on the secondary sodium activation is mainly due to the uncertainty on the 23Na elastic scattering cross-section. The origin comes from the high quantity of sodium present on the neutron path between the core and the intermediate heat exchanger. The uncertainty on the ASTRID secondary sodium activation due to uncertainties on nuclear data is 100%. This value is not negligible and difficult to reduce because the uncertainty on the elastic scattering cross-section of the 23Na nuclide is already low (3%). For this reactor, the uncertainty due to the neutron spectrum is 9% which is negligible compared to the uncertainty due to neutron cross-sections. The uncertainty on the Superphénix secondary sodium activation due to uncertainties on nuclear data is 66%. The impact of other nuclides than 23Na is higher compared to ASTRID because more stainless steel is present on the path of the neutrons.

The method applied to quantify uncertainties on the secondary sodium activation seems promising but some improvements are necessary so that the method could be more reliable. Indeed, the last part shows that the method applies a large conservatism so further studies have to be carried out. The methodology studied in this report enables to begin a nuclear safety demonstration to the French authority. Future studies in the same field will increase the knowledge on the nuclear data uncertainties thus the demand of the ASN to quantify these uncertainties will be satisfied.

Perspective

During the master thesis, an important work was done on the method applied but there is a room for improvement. The last part shows that the method applies conservatism so future modifications are needed to find new tools to reduce the degree of conservatism. Then improvements to reduce the calculation time would be very useful. The time to create random libraries is very variable, that is the reason why nuclides of the core are excluded. If a method to reduce the computed time is found, the user must take into account nuclides of the core to improve the reliability of the method.

To reduce the uncertainties on the secondary sodium activation, methods can be applied such as Bayesian procedures for example. In this report, the uncertainty on the 23Na elastic scattering cross-section is almost impossible to reduce because the value is already low (3%). Nevertheless, reduction of uncertainties on the Mfi parameters could be done thanks to MOCABA [23] for example. One task of a possible future internship would be to create a big database which gathers all the nuclear data available. The user makes an inventory of all the nuclides of the problem and the goal is to know where variance and covariance information for Mfi parameters are available for: o Neutron cross-sections

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. All nuclides . All reactions . All temperatures o Multiplicity of secondary particles emitted, especially neutron production o Energetic distribution of the final state of the particles to know the neutron spectrum for example o Angular distribution of the final state of the particles o Data on radioactive decay and fission products

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