\section{Spallation target assembly sub-division}
The most critical region in term of LBE activation is the \acrfull{STA} \cite{KeijersDesignAssembly} since most of the spallation products are produced there. In the STA the production rate of neutron activation products is also maximized because of the very high neutron fluxes. For this reason the rest of the LBE source term assessment concentrates on the analysis for this region, giving conservative results in term of activity and decay heat produced in the LBE.
The LBE source term study for the STA starts from a deeper analysis of the proton and neutron fluxes in the considered region.\\ 

Fig. \ref{pf_heatmap} depicts a 2D map (XZ-view on the left, XY-view on the right) of the proton fluxes in the STA. Protons are relevant for the LBE activation only within their stopping range, i.e.,  in the region right below the beam window. The plot also shows that proton flux radial spread is limited to the STA wrapper. 
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{images/plots/LBE/heatmap_proton.png}
\caption{Proton flux distribution in the spallation target assembly. xz view on the left, xy cross section 5 cm below the beam window on the right.}\label{pf_heatmap}
\end{figure}
The left plot in Fig. \ref{pf_heatmap} shows a strong proton flux axial gradient (with a significant axial zone without protons). This is an indication that using assembly-averaged proton fluxes in depletion analysis might not produce correct source term estimates. A refinement of the depletion model with a sub-division of the STA into three axial regions is proposed as a correction.\\
% Fig. \ref{pf_cum} depicts the axial boundaries of the three considered regions. They are retrieved looking at the cumulative distribution of proton flux in axial direction .
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{images/plots/LBE/proton_cum.png}
\caption{Cumulative proton flux integrated in axial direction.}\label{pf_cum}
\end{figure}
% The mathematical expression of the curve drawn in Fig. \ref{pf_cum} is presented in eq. \ref{pf_cum_eq}. It represents the proton flux integrated up to the height z as function of z itself. It is then divided by the the flux integrated all over the spallation target assembly. 
% \begin{equation}
%     Cum (z) = \frac{\int_{z_b}^{z} dz \int dx \int dy \phi_p (x,y,z)}{\int_{z_b}^{z_t} dz \int dx \int dy \phi_p (x,y,z)}
%     \label{pf_cum_eq}
% \end{equation}
The axial distribution of the proton flux is presented in the form of cumulative distribution in Fig. \ref{pf_cum}. It suggests to divide the assembly into three regions:
\begin{itemize}
\item the (axially)-central region consisting in the actual spallation target as it covers the entire proton stopping range in LBE. The LBE source term in this region is generated by nuclide interactions with protons and high energy neutrons;
\item the upper region where the neutron fluxes are still high but the presence of the protons is negligible;
\item the lower region, with neutron and proton flux characteristics similar to the upper region, but including a lower LBE volume.
\end{itemize}

The neutron and proton fluxes in the STA were already reported in Fig. \ref{sta_flux}, above.\\

In this work, the main objective for the analysis of the LBE source term in the spallation target assembly is to assess the deviations arising from a zone division approach, as well as to evaluate the actual inventory of safety relevant nuclides during irradiation and after shutdown.\\

LBE depletion simulations are carried out for each of the three STA axial regions previously identified. The spectral shape and the intensity of the neutron and proton fluxes were calculated for the entire STA volume and for the central axial region. The latter were used for the LBE depletion of the STA central region. The neutron flux (not the proton flux, which was assumed negligible) calculated in the entire STA volume was used for the depletion of the STA lower and upper regions, after correctly re-scaling the flux intensity to account for the different LBE volumes.

The volume-weighted sum of the nuclide inventories calculated in the three regions was compared to the nuclide inventory predicted for the entire STA using a single depletion calculations with volume-average fluxes. The latter will be later referred to as the \textit{homogeneous approach}. The discrepancy between the two approaches is assessed and the outcome for the relevant nuclides, discussed in Sect. \ref{AccCondObservables}, is presented in Fig. \ref{discr_3z}.
\begin{figure}[h]
\centering
\includegraphics[width=0.88\textwidth]{images/plots/LBE/activity_sta/discrepancy.png}
\caption{Variation in nuclide activity in STA due to the modeling assumptions. Impact of zone division is exploited.}\label{discr_3z}
\end{figure}
The impact of a zone division approach at the \acrshort{STA} level is not negligible. All the spallation products and $^{208}$Po, $^{209}$Po are underestimated by the homogeneous approach because of their non-linear dependence on the proton flux intensity. $^{210}$Po is mainly produced by neutron interactions ( (n,$\gamma$) in $^{210}$Bi) and it is slightly overestimated by the homogeneous approach. This information is valuable since $^{210}$Po is the most safety concerning nuclides and one of the main "showstopper" for the design of MYRRHA and analogous facilities because of its radiotoxicity.
\begin{center}
\fbox{\begin{minipage}{0.98\textwidth}
The production of spallation products in the spallation target assembly is underestimated by a homogeneous depletion of the entire LBE volume in the \acrshort{STA}. On the contrary $^{210}$Po is overestimated by this approach.
\end{minipage}
}
\end{center}
