\section{Codes and data employed}
In the following, the main tools and codes used to obtain the reported results are presented. The list shows the different topics addressed during the project.

\subsection{Monte Carlo codes for particle transport}
Monte Carlo codes are used to mimic the transport of particles inside a system by means of stochastic simulation of the phenomena involved \cite{Haghighat2020}. Their motion inside the system is traced and it allows to determine the average behavior of the particles simulated retrieving their energy and spatial distribution. Monte Carlo simulations provide the result of the general transport equation without directly solving it as opposed to deterministic codes based on the discretization in space, energy and angle, the three phase space parameters.\\
The main advantages of MC methods are possibility to handle complex geometries, materials and nuclear data, providing very accurate results with the statistical uncertainty associated. The drawback is that the convergence is slow since it is proportional to $\sqrt{N}$ for all statistical method with $N$ equal to the number of histories simulated.\\
Several Monte Carlo particle transport codes are available and provide different features. The following discussion is limited to the codes employed in this work. 
\subsubsection{MCNP6}
Monte-Carlo N-Particle (MCNP) \cite{WernerMCNP6.2} is one the most prominent codes available. It is commonly adopter for nuclear reactor modeling and it has been extensively validated and constantly maintained during its 40 years history.\\
The main requirement to fulfill for MYRRHA core modeling is the transport of protons, together with neutrons to mimic the proton source and the neutrons consequently generated from it by means of spallation reactions. MCNP fully complies to this requirement. However, MCNP is not able to perform materials depletion by itself and it has to be coupled by a depletion solver. This task is accomplished by ALEPH2 \cite{BaetenALEPH2-ACode}, that will be fully discussed in Sect. \ref{aleph}.
\subsubsection{Serpent-2}
Serpent \cite{Leppanen2015Serpent-aManual} is a multi-purpose continuous-energy neutron and photon transport code, developed since 2004. It is widely used for particle transport applications in particular reactor modeling. Serpent has built-in burnup calculation capability, performing depletion of burnable materials in the simulations. Depletion zone division can be implemented to discretize the fluxes into the same material to estimate spatial dependent depletion; this feature is used in the thesis to investigate the influence of axial burnup.\\
With respect to ALEPH2, Serpent is much faster but it needs larger hardware requirements since nuclear data are stored in the \acrshort{RAM}. The calculation speedup makes is very promising to perform calculations involving uncertainty propagation. The main limit of Serpent, related to this case study is that it is not able to transport protons; this implies only the MYRRHA critical model can be simulated in Serpent.   

\subsection{ALEPH2}\label{aleph}
The general-purpose burn-up code ALEPH2 wraps around MCNP/X Monte-Carlo radiation transport code and an advanced depletion code to solve the Bateman system of ordinary differential equations. It was developed in SCK CEN and recently upgraded to ALEPH v2.8. With respect to ALEPH1 some modification were introduced coming from user feedbacks and new necessities. The main upgrades involve the depletion code in use since ORIGEN-2.2 \cite{ORIGEN2} was replaced by the advance depletion solver based on RADAU5 implicit Runge-Kutta algorithm. Besides, new features were implemented such as the predictor corrector, heating calculation during irradiation and decay and the calculation of neutron sources during the decay.\\
The working principle of burn-up codes in general "and ALEPH2 in particular" is to take particles spectra and fluxes from the Monte Carlo simulation to generate the reaction rates to feed the depletion solver. Then, the output of the depletion calculation updates the input of the transport simulation for a new run. Briefly, burn-up codes take care of the coupling between transport and depletion, discussed from the theoretical point of view in the previous section.\\
The main advantages of ALEPH2 among other codes is the full nuclear data consistency, since the same energy grid is used both in transport and subsequent depletion calculation. Moreover, as already mentioned the code wraps around MCNP and the complexity of utilization is limited to the addition of a number of extra cards to the Monte Carlo transport simulation input. ALEPH2 generates the MCNP input and when  the transport simulation is completed it automatically extracts the fluxes in the materials/cells we are interested to burn. After depletion procedure is performed, the materials compositions are updated accordingly. The possibility to change materials is also implemented to allow considering system configuration variation, like the fuel assembly shuffling in a core simulations or the variation of boron concentration in PWRs coolant.\\
The main advantages of ALEPH2 for our specific case, being MYRRHA an ADS is that the code takes into account protons matter interactions. Moreover it gives the possibility to perform stand alone calculations, reading particle fluxes from external files and carrying out depletion calculations without the necessity of calling upon MCNP. This specific procedure is implemented for the LBE source term calculation as described in the following and it allows to speedup the calculation assuming the fluxes do not change from one irradiation cycle to the other.

\subsection{SANDY}
SANDY \cite{Fiorito2017NuclearSANDY} is a tool created to manage nuclear data file in ENDF-6 format. The code is able to read ENDF-6 file \cite{Herman2010} and to perform operations with them. It interfaces with NJOY \cite{Macfarlane2019The2016}, creating input files and reading the outputs to provide processed nuclear data, for example multigroup cross sections and covariance matrices. The main objective of SANDY is to perturb nuclear data starting from the processed data and generate perturbed files readable by Monte Carlo codes to carry out uncertainty propagation by statistical sampling. \\
The format used by some \acrshort{MC} codes is called \textit{ACE} and the processing, also in this case case, is performed through a NJOY module.
Perturbation of nuclear data is possible for cross sections, neutron fission product yields and $\Bar{\nu}$, but it has being implemented for other class of nuclear data contained in ENDF-6 files. The procedure to obtain perturbed data was breafly described in Sect. \ref{stSMPS}, starting from normally distributed random samples, to which the covariance matrix is applied to obtain relative perturbation coefficients.\\
The choice of the multivariate normal distribution is driven by the Central Limit Theorem \cite{Cacuci2003}, it establishes that the mean values of samples taken form a certain population converge to the normal one if a sufficient sample size is achieved. This implies whatever input distribution the output of the model will be normally distributed if the number of sample is large enough. So it seems the easiest choice to take normally distributed inputs. The other available option given by the code is the lognormal distribution that can be used to assess the bias given by the input distribution but it is out of the scope of this thesis.\\
The production of the samples to carry out uncertainty propagation of the models developed gave the opportunity to understand and to test different features of the sampling procedure implemented in SANDY; the contribution to some updates in the code, to fix bugs encountered during the procedure was part of the thesis work, that has been carried out under the supervision of the main developer of the code Luca Fiorito.

\subsection{NJOY}
NJOY is a comprehensive computer code developed to process nuclear data. It prduces pointwise and multigroup nuclear cross section and related quantities from evaluated nuclear data files in ENDF-6 format. It comprises many modules and each of them performs a certain task. As already mentioned SANDY code generates NJOY inputs requested and it reads and converts the outputs in a user friendly format on a python interface.
NJOY2016 includes a module called $ACER$ to translate ENDF-6 in $ACE$ format, the one requested by MCNP and Serpent MC codes.

\subsection{Nuclear data libraries}\label{nuclear_data}
Nuclear data are the main input data to feed to a neutronic simulation and it means their availability and accuracy is on main importance to obtain meaningful results. Nuclear data include energy dependant reaction cross sections, the energy and angular distributions of fission neutrons, fission product yields and the atomic and nuclear properties of excited states as well as their radioactive decay data.\\
Nuclear data are stored in libraries and different source are available coming from different evaluators. For this work some libraries are compared to assess the discrepancies and choose the best one tailored on this specific application, the one more complete in term of the nuclides we are interested in.\\
The three libraries used in this work are JEFF \cite{Koning2011StatusLibrary}, the joint evaluated fission and fusion nuclear data library, ENDF \cite{Brown2018ENDF/B-VIII.0Data}, the U.S.'s Evaluated Nuclear Data File and JENDL \cite{SHIBATA2011JENDL-4.0:Engineering}, the Japanese evaluated nuclear data library.
Among the several versions available, those chosen for performing the simulations are :
\begin{itemize}
    \item JEFF-3.1.2 \cite{OECD/NEA2012JEFFDVD}
    \item JEFF-3.3
    \item ENDFB-VIII.0
    \item JENDL-4.0u
\end{itemize}
The older release of JEFF is used in the reference neutronic model calculation and the LBE depletion simulations. The comparison with the other libraries listed above is performed to assess the discrepancies in term of nuclide concentrations and to select the evaluated data giving more conservative results. The largest prediction among the libraries is chosen to be conservative since is not possible to affirm a priori which library works better in a certain system and the reason for certain divergences in the outputs.\\
For fuel assembly simulations the results presented in the following are obtained using the newer release of JEFF, JEFF-3.3. Deviations in the nuclides concentrations coming from nuclear data variation are shown  and uncertainty propagation of neutronic cross section investigates the discrepancy due to the uncertainty of nuclear data themselves. The main focus is on the largest uncertainty for each nuclide giving the upper limit concentration for the nuclide examined.
Neutron fission product yield uncertainty analysis is only performed with data coming from JEFF-3.3.

\subsubsection{High energy incident neutron data}
Most of neutron libraries commonly have cross section data up to an energy of 20 MeV, that is the upper limit for neutrons coming from fission. However, in ADS applications the energies spans to much higher energies and cross sections for high energy reactions must be taken into account. For this purpose a tool from ALEPH2 provides custom libraries comprising cross sections data up to 20 MeV to which other libraries are appended.
In case of JEFF-3.1.2 library in the reference calculation the high energy data are provided by two additional sources, in particular TENDL-2013 \cite{Koning2019TENDL:Technology} to cover energy range 20-200 MeV and HEAD-2009 \cite{Korovin2010HighHEAD-2009} high-energy activation library to add data above 200 MeV.
\subsubsection{High energy incident proton data}
The proton reference library used for sub-critical configuration is produced in ALEPH2, merging nuclear data coming from several evaluators to provide the most complete information possible. JENDL and JEFF data are combined to TENDL to provide comprehensive data up to 200 MeV. HEAD-2009 is used to cover the higher energy range of the spectrum.
