\section{Modeling assumption}\label{lbeModeling}
The LBE depletion is calculated for the MYRRHA sub-critical configuration assuming  a equilibrium core.
What emerges from previous source term studies \cite{Stan12}, \cite{Stan16} is that the sub-critical mode of operation generates the largest LBE activity. Besides, most of the radioactive nuclides responsible for the LBE activity are produced directly/indirectly by proton-induced spallation reactions, i.e.,  a mechanism not present in a critical reactor.\\

\textit{LBE static simulation}\\
The LBE is considered stagnant, that is, the coolant forced circulation in the reactor and the mixing phenomena are not modelled. Recycling effects not taken into account can have an impact on the production/depletion of certain nuclides.\\

\textit{No replacement of filtering of the LBE during the reactor lifetime}\\
All radionuclides produced from LBE activation during reactor operation are retained in the LBE for the entire lifetime of the reactor, i.e., 40 years. This assumption neglects the migration of volatile activation products to the cover gas \cite{Aerts2020BehaviorLBE} (as postulated in one of the \acrshort{DBA}s), and results in a conservative estimation of the source term where the concentration of volatile radionuclides in LBE is overestimated.\\  

\textit{Transport/depletion decoupling}\\
%The main assumption on the model is that \acrshort{BoC} neutronic conditions, i.e. the neutron and are restored after every irradiation cycle and the only variation is in the LBE nuclide inventory. This assumption comes from the definition of equilibrium discussed Sect. \ref{eqCycleDef}. The neutronic behaviour evolves during the irradiation cycle because of in-cycle fuel depletion and it brings to \acrshort{EoC} condition, showing a slight spectral variation in the neutron flux.\\
Proton and neutron fluxes and spectra are calculated only once at \acrshort{BoC}, and then they are adopted for the LBE depletion throughout the entire reactor lifetime. This assumptions follows the definition of equilibrium cycle discussed in Sect. \ref{eqCycleDef}. Also, the in-cycle spectral variations are neglected. The neutron and proton spectra calculated in the STA are displayed in Fig. \ref{sta_flux}.\\

%A strong model simplification is introduced assuming the equilibrium condition is achieved and that there is negligible spectral variations during the irradiation cycle. Following this assumptions constant particle spectra can be employed throughout the transient since the fluxes stays the same from cycle to cycle and they undergo only a slight variation moving from \acrshort{BoC} to \acrshort{EoC}.
%The proton flux undergoes no significant variation since it comes from a external source while the neutron spectrum tends to harden a bit during the cycle: the hardening is quite small and it is considered negligible.\\
%This assumption was already exploited in the previous works on the same topic and it gives advantages in term of computational cost since the transport simulation of the core at equilibrium is carried out only one time to retrieve the neutron/proton flux at \acrshort{BoC} and \acrshort{EoC}. 
%This approach aims to decouple transport and depletion; if the fluxes do not vary all the reaction rates are pre-calculated and the simulation to perform is purely depletion. The model is built in ALEPH2.8: the spectra are fed from external file, then ALEPH performs the convolution with the cross sections to generate the reactor rates. An advanced depletion solver (RADAU7) carries out the depletion calculation itself solving the Bateman system of equations.\\

%\textit{Particle spectra}\\
%Neutron and proton spectra at \acrshort{BoC} are used for depletion calculations. The output is the spectrum normalized on the source particle, in this case the proton beam current and a unitary volume is considered. To obtain the actual value of the spectrum, the volume of LBE contained in a certain region must be evaluated. Both analytical and statistical methods are employed for this purpose.
%Eq. \eqref{f4tally} summarizes the procedure to obtain the actual flux from MCNPX tally outputs:
%\begin{equation}
%    \phi (E) = \frac{1}{V} \underbrace{\int_{dV}\; d \Vec{r}\; n(\Vec{r}, E)}_\text{MCNP F4 tally} v
    \label{f4tally}
%\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=0.99\textwidth]{images/plots/LBE/fluxes.png}
\caption{Neutron (left) and proton (right) energy-dependent flux in the spallation target at BoC. The neutron flux in the whole spallation target assembly is depicted in blue.}\label{sta_flux}
\end{figure}

%The velocity $v$, expressed in particles/s, is retrieved from the proton source, i.e. the proton current. In order to obtain conservative results the beam current at \acrshort{EoC} is taken; the current increases during the cycle to counterbalance the negative reactivity induced by fuel depletion. A current of 4 mA is considered providing a particle source of $2.5e+16$ protons/s.\\
%The neutron and proton spectra in the STA are presented in Fig. \ref{sta_flux}.
%The proton flux has a peak at the energy it exits from the beam tube (600 MeV) and then it reduces down to 1 MeV. At lower energies the flux is almost zero.
%The neutron fluxes has a high energy tail both considering only the actual spallation target and the whole \acrfull{STA} but there is a larger epithermal region averaging on the STA.
%Moving further from the target neutrons interact with the LBE and structures around and they lose their energy generating the epithermal tail shown in the plot.\\

\textit{LBE discrete regions depending on irradiation conditions}\\
The depletion of the LBE coolant was simulated independently for three volumetric regions of the reactor characterized by different irradiation conditions, e.g., particle fluxes and spectra:
\begin{itemize}
    \item Spallation target assembly (STA);
    \item Reactor core (without the STA);
    \item Reactor pool.
\end{itemize}

For each region, volume-average particle fluxes were used for the LBE depletion calculation. This approach is in line with the previous analysis carried out for the LBE source term assessment of the MYRRHA Design Revision 1.4. For the MYRRHA Design Revision 1.6 only one volumetric region was adopted, for which volume-average particle fluxes were used. The region included the STA, the reactor core and the reactor pool.
In this work the two approaches are compared for the LBE source term of the MYRRHA Design Revision 1.8.\\

The \textit{spallation target assembly} is characterized by a 600 MeV-proton beam impinging on the LBE spallation target and inducing spallation reactions that yield  high energy neutrons into the system. The LBE source term in this region is dominated by spallation products and high energy neutron activation products.\\

Because of the proton short stopping range in LBE and because of the direction of the proton beam, only a negligible proton flux is available in the \textit{reactor core}. Compared to the \textit{spallation target assembly} the neutron spectrum in the reactor core is softer as it mostly generates from neutron-induced fission in the driver MOX fuel. Still, a high-energy tail of spallation neutrons is still present. In the \textit{reactor core} region most of the source terms radionuclides produced in LBE are produced via neutron activation.\\

In the \textit{reactor pool} the neutron fluxes and spectra, respectively, have a lower intensity and are softer  (because of the multiple neutron-scattering reactions in LBE) with respect to those in the core. The main production mechanism for the source terms radionuclides in LBE is neutron activation.



