Quantification of the uncertaintyon the secondary sodiumactivation due to uncertainties onnuclear data

INOM

EXAMENSARBETE 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

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

Na. 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

Na nuclide. The

comparison between calculations and measurements on the Superphénix reactors proves that the method

applies conservatism, which is good in term of safety.

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.

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 I

... 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

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:

Na 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 𝜷 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

Na elastic scattering cross-section .. 35

Figure 19: Activation of the secondary sodium in the IHX 2 as a function of

Na capture cross-section ... 35

Figure 20: Elastic and inelastic cross-section for

Na as a function of the energy of the neutron ... 38

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:

Na 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

Na 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

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

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 CO

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

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

Na nucleus absorbs a neutron and becomes an excited

Na nucleus. It then decays by an electron emission with a half-life of 15 hours, then by photon emissions. The decay of the

Na 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

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 the flux, the code determines the average reaction rate A

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.

𝐴

= 𝑁

∫ 𝜎

(𝐸) × 𝜙

(𝐸)𝑑𝐸

𝐸1

(1)

 Φ

: 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=+∞

 N

: the

Na concentration in volume i

 𝜎: microscopic capture cross-section of the

Na nuclide

The steady-state activity of the sodium in the secondary loop is calculated thanks to the following equation:

𝐴

= ∑ 𝐴

𝑉

× 𝑉

𝑛

𝑖=1

(2)

 V

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

 V

: 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.

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

U, the only fissile nuclide of natural uranium. The energy potential of this raw material is not fully used because uranium is mainly composed of

U (99.3 % of natural uranium is

U). Fast neutron reactors convert

U into a fissile nucleus

Pu when it is irradiated by a rapid flux.

U 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]

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

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 4: Description of the Superphénix vessel [7]

Figure 5: Core lattice of Superphénix, vertical view [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

Na 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

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:

𝑉𝑎𝑟(𝑥) = 𝐸{[𝑥 − 𝐸(𝑥)]

} (3)

By considering two random variables x and y, the definition of the covariance is:

𝐶𝑜𝑣(𝑥, 𝑦) = 𝐸{[𝑥 − 𝐸(𝑥)][𝑦 − 𝐸(𝑦)]} 𝑎𝑛𝑑 𝐶𝑜𝑣(𝑥, 𝑥) = 𝑉𝑎𝑟(𝑥) ⁽⁴⁾

The interpretation of this definition is as follows [12]. The quantities [𝑥 − 𝐸(𝑥)] and [𝑦 − 𝐸(𝑦)] 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

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 propagate to the results. Modeling the input as an m-dimension random vector 𝑿 = (X

, X

, … , X

)

, a resulting quantity g( 𝑿 ) is a random variable which follows a distribution depending on 𝑿 . This section describes two methods to estimate the uncertainty 𝑉𝑎𝑟(𝑔(𝑿)) in g( 𝑿 ) given information on 𝑿 [13].

Linear uncertainty propagation

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

𝑉𝑎𝑟(𝑔(𝑿)) ≈ ∑ ∑ ( 𝜕𝑔

𝜕𝑋

) ( 𝜕𝑔

𝜕𝑋

)

𝑚

𝑗=1 𝑚

𝑖=1

𝐶𝑜𝑣(𝑋

, 𝑋

)

(5)

Where 𝐶𝑜𝑣(𝑋

, 𝑋

) is the covariance between X

and X

. The covariance describes the uncertainty of X

and X

but also their linear correlation because the covariance can be written as [13]:

𝐶𝑜𝑣(𝑋

, 𝑋

) = 𝜎(𝑋

)𝜎(𝑋

)𝜌(𝑋

, 𝑋

) (6) Where 𝜎(𝑋

) is the standard deviation: 𝜎

(𝑋

) = 𝑉𝑎𝑟(𝑋

) 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 X

and X

. 𝜌 is positive if X

and X

tend to “vary in the same direction” and negative in the opposite case. If all different random variables are uncorrelated, thanks to the relation 𝐶𝑜𝑣(𝑋

, 𝑋

) = 𝑉𝑎𝑟(𝑋

), equation 5 simplifies to [13]:

𝑉𝑎𝑟(𝑔(𝑿)) ≈ ∑ ( 𝜕𝑔

𝜕𝑋

)

𝑚 2 𝑖=1

𝑉𝑎𝑟(𝑋

) (7)

Quite intuitively, the uncertainty of g( 𝑿 ) thus depends on the uncertainty of the different arguments 𝑋

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

Na 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 E

and E

is 𝜌(𝐸

, 𝐸

). 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).

Figure 6:

Na 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 𝑥

, 𝑥

, … , 𝑥

from 𝑿 and to evaluate g for all observations of this random sample. In this way, one has obtained a random sample from g( 𝑿 ) namely g(𝑥

), g(𝑥

), … , g(𝑥

). It is then straightforward to estimate the standard deviation 𝜎

= √𝑉𝑎𝑟(𝑔(𝑿)) of g( 𝑿 ) by considering m samples of the distribution g( 𝑿 ):

𝜎

= √ 1

𝑚 − 1 ∑(

𝑚

𝑖=1

g(𝑥

) − 𝐸(𝑔(𝑿))

) ₍₈₎

The major disadvantage is that more evaluations of g( 𝑿 ) are necessary compared to the linear method, which becomes a problem if g( 𝑿 ) 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 𝑿 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 𝑿 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( 𝑿 ) 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( 𝑿 ) is different. Instead of using the equation 8, the Wilks’s principle is used because this principle

gives a more accurate value of 𝜎

.

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.

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

Analysis of the stochastic code results Stochastic calculations with a transport code

Creation of the random libraries The sum rules

Random draws on the input parameters

Variance and covariance information from the ENDF6 file Creation of a nuclide database

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

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 B

C Ti48 Stainless steel

Nb93 Stainless steel B10 B

C + Borated

steel Ti49 Stainless steel

Ar36 Argon B11 B

C + Borated

steel Ti50 Stainless 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-

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.

For our study, an observable 𝜃 = 𝑔((X

, X

, … , X

)

) and the covariance matrix 𝐶𝑜𝑣(𝑋

, 𝑋

) is considered.

The expectation value E and the covariance matrix are empirical estimates of the real

datasets 𝑿 = (X

, X

, … , X

)

. The variables 𝑋

follow a normal or log-normal distribution with the density probability P(𝑋

). In our problem, the distribution of all the 𝑋

are standard Gaussian distributions. To perform a random sample of a random variable 𝑋

, the first step is to decompose 𝐶𝑜𝑣(𝑋

, 𝑋

) with the help of the Cholesky decomposition and the Singular Value Decomposition (SVD) [4] into:

𝐶𝑜𝑣(𝑋

, 𝑋

) = 𝑈

𝐷

𝑈 (9)

With U orthogonal and D diagonal. A random draw of a coordinate 𝑋

is noted 𝑋

and given by the following equation [4]:

𝑋

= 𝑈

𝐷𝒁 + 𝐸(𝑋

)

(10) Where 𝒁 = (Z

, Z

, … , Z

)

is a vector of a standard Gaussian random number. Each coordinate of 𝒁 is the result of a random draw on the standard Gaussian distribution illustrated Figure 15. The value 𝑋

is varying around its expectation value 𝐸( 𝑋

) by the value 𝑈

𝐷𝒛. This quantity through the covariance matrix compiles the uncertainty of the parameter 𝑋

. Between two different samples, the coordinates Z

and Z

are different so the outcome 𝑋

and 𝑋

are different. In our case, the observable 𝜃 is the value of the secondary sodium activation. The different 𝑋

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.

{ 𝐴𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛

= 𝑔((𝑀𝑓𝑖

, 𝑀𝑓𝑖

, … , 𝑀𝑓𝑖

)

) for the reference calculation

𝐴𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛

= 𝑔((𝑀𝑓𝑖

, 𝑀𝑓𝑖

, … , 𝑀𝑓𝑖

)

) ⁽¹¹⁾

The purpose of this method is to obtain N random samples of the variable Z

with the method described above and for each sample, a value 𝑀𝑓𝑖

is obtained. When all the modified parameters (𝑀𝑓𝑖

, 𝑀𝑓𝑖

… , 𝑀𝑓𝑖

) 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 (𝑀𝑓𝑖

, 𝑀𝑓𝑖

… , 𝑀𝑓𝑖

) is created and a new value of the secondary sodium activation is 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

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:

𝑚

(𝐸) = 𝑚

(𝐸) + 𝑚

(𝐸) (12)

After the random sampling on variance and covariance matrices, the sum rule for the different multiplicity contributions m

must be conserved. Five different cases exist [4]:

o If there is covariance given for 𝑚

(𝐸) and/or 𝑚

(𝐸) , then 𝑚

(𝐸) is evaluated

according to equation 12;

o If there is covariance given for 𝑚

(𝐸) and 𝑚

(𝐸) , the prompt one is the difference between the random draws of the two other contributions;

o If there is covariance given for 𝑚

(𝐸) , 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 𝑚

(𝐸) and 𝑚

(𝐸) , then 𝑚

(𝐸) is adjusted such that the sum rules gets fulfilled. Thus, the assumption of perfect correlation between 𝑚

(𝐸) and 𝑚

(𝐸) 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 𝜎

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 𝜎

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.

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

Pu neutron spectrum due to prompt neutrons. All information needed for the

Pu 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 𝛽

, only the constraint given equation 13 must be enforced:

∑ 𝛽

𝑖

= 1 ₍₁₃₎

The summary of the sum rules is presented Figure 11.

Figure 11: Summary of the important sum rules

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

In the case of Figure 13, random draws are performed on the nuclides

Na (11023.) and

B (5010.). For nuclides where there are no random draws on the variance/covariance matrices for the cross-section, the extension remains unchanged. For

Na and

B, 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 {𝜇

− 𝜇

} 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 𝛽% 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, 𝛽 quantiles are values that partition a finite sample into values higher or lower than the value of 𝛽 . 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.

 Let suppose a random variable 𝑿 = (X

, X

, … , X

)

from a population having a probability density function 𝑃( 𝑿 ) and an observable 𝜃 = 𝑔((X

, X

, … , X

)

) where 𝜃 is the parameter to be estimated.

Further, let’s suppose that L

(X

, X

, … , X

) and L

(X

, X

, … , X

) are two statistics such that L

< L