\section{Particle-matter interaction}
Particle-matter interaction \cite{Tavernier2009} consists of the energy transfer between a projectile particle and a physical target. The nature of interaction depends mainly on the nature of the incoming particle and its energy. For the purpose of the work the discussion is limited to neutrons and protons. The latter are provided by the accelerator and they undergo spallation reactions.
\subsection{Spallation reactions}
A high energy proton beam is employed in MYRRHA as external driver. The beam is produced by a linear particle accelerator, a machine using alternating radio frequency (\acrshort{RF}) electromagnetic fields to accelerate charged particles in a straight line \cite{Seeman2020DesignColliders}.\\
The interactions of high-energy particles with the coolant just below the beam are called spallation reactions.
Spallation is a violent reaction in which a target is bombarded by very high energy particles \cite{Ahmed2015PropertiesRadiation}. The impinging particle, proton for example, breaks the target nucleus in several fragments: protons, neutrons, $\alpha$-particles and other particles are emitted. The general formulation for a spallation reaction is written as 
\begin{equation}
    p + ST \rightarrow SF_1 + SF_2 + ... + SF_m + (k)n,
\end{equation}
in which \acrshort{ST} is the spallation target and \acrshort{SF} are the fragments. The number of neutrons generated $k$ depends on the both the characters involved. The energy ranges from a few eV to several GeV with the energy of the projectile as upper bound. Spallation targets can be solid of liquid and are usually constituted by high-Z nuclei: an eutectic compound of lead and bismuth is used in MYRRHA.
\subsection{Neutron-matter interaction}
Neutrons are neutral particles and therefore they do not participate in the electromagnetic interaction and do not produce ionization of the atoms. They are generated by several kind of sources: we already discussed about spallation, other phenomena are spontaneus fission and $(\alpha,n)$ reactions.\\

The main process of neutron interaction \cite{Rinard1997NeutronIW} with matter is by nuclear forces. Unlike charged particles, neutrons collide rarely with atoms but in these interaction they can lose either all or a large part of their energy, which is caused by the short-range nature of nuclear forces. Since neutrons do not have an electric charge, they freely penetrate through the electron shells of atoms and are not repelled by the Coulomb field of the nucleus.\\
When considering possible energy transfer among neutrons and other particles it is useful to separate them in several groups depending on the neutron energy. A special group of neutrons are thermal neutron, those that are in thermal equilibrium and they present a Maxwellian distribution.\\
The main nuclear reactions the neutrons take part are:
\begin{itemize}
\item $(n, n)$ is the elastic or inelastic scattering of neutrons by nuclei;
\item $(n, \gamma)$ is the radiative capture reaction, at which a photons is released from the nucleus;
\item $(n, \alpha)$ is the reaction with the emission of an alpha particle;
\item $(n, p)$ is the reaction with a proton emission;
\item $(n, f)$ is the fission reaction of the nucleus, in which it is split in two (or rarely more) fission fragments
\end{itemize}
For the sake of simplicity only the most common reactions are listed but other interactions happen with lower probability and they are taken into account during neutronic evaluations.\\
The interaction probability is defined by the microscopic cross section $\sigma$, that represents the impact area of the neutron in the target and for this reason it is measured in barn $= 1e-24$ cm$^2$.
The microscopic cross section for the interaction $i$ is denoted $\sigma_i (E)$ and it depends on the target nuclide as well as the neutron energy.
The probability of absorption is the sum of all the interactions bringing to the disappearence of the incoming neutron; it is denoted as $\sigma_a$ and is split amongst capture and fission, eq.  \eqref{sigma_m}.
Neutron scattering changes the neutron position in the phase space, by modifying position, energy and angle. The phenomenon is divided in elastic ($\sigma_{es}$) and inelastic ($\sigma_{is}$), in which part of the neutron energy is transferred to the target nuclide. The total cross section ($\sigma_{t}$) represents the probability of a neutron to undergo an interaction of any kind.
\begin{equation}
    \sigma_a = \sigma_f + \sigma_c \;\;\;\;\;\; \sigma_s = \sigma_{es} + \sigma_{is} \;\;\;\;\;\; \sigma_t = \sigma_a + \sigma_s
    \label{sigma_m}
\end{equation}
The probability of interaction per unit of path is defined as
$$\Sigma_i(\Vec{r}, E) = \sigma_i(E) N_j(\Vec{r})$$
and it conveys the exponential reduction of neutron concentration as function of the free flight as shown in eq. \eqref{exponential}:
\begin{equation}
    N (x) = N (0) e^{-\Sigma x}.
    \label{exponential}
\end{equation}