\section{Production mechanisms of relevant nuclides in the STA}
The investigation of the production mechanisms of relevant nuclides is carried out in this section. The analysis of the reaction rates explains which are the main reaction chains for the production and disappearance of the nuclides of interest.
The reaction rate multiplied by the concentration parent nuclide concentration at every instant constitutes the amount produced by that channel, mathematically expressed by eq. \eqref{prod_channel}.

\begin{equation}
     N_{P\rightarrow D} (t) \ = <\sigma \phi>_{P\rightarrow D} \int_{t_0} ^t N_P(t') \ dt'.
     \label{prod_channel}
\end{equation}

In the equation above $N_{P\rightarrow D} (t)$ is the amount of the nuclide $D$ (daughter) produced by a certain nuclide  $P$ (parent), $<\sigma \phi>_{P\rightarrow D}$ is the corresponding reaction rate (assumed time-independent) and  $N_P$ is the time-dependent concentration of $P$.
The concentration of the nuclide $D$ at a certain time $t$ is given by the sum of all contributions from its parent nuclides, to which a disappearance term must be subtracted because of the radioactive decay or the nuclear reactions of $D$.\\

\textit{Production of  $^{210}$Po}\\
$^{210}$Po poses the greatest concern both in terms of LBE activity and decay heat.
Its production in the spallation target assembly is mainly driven by the radioactive $\beta$-decay of $^{210}$Bi ( $t_{1/2} \ = \ 5 $ days). The latter is produced by neutron capture in $^{209}$Bi.
The second production path is the neutron capture in $^{209}$Po and it accounts for the 5\% of the total production. This implies that the concentration of $^{210}$Po is mainly affected by neutron flux and by the capture cross section of $^{209}$Bi. The capture cross section increases in the resonance region below 1 MeV and consequently the production is enhanced if the spectrum is softer.
\begin{table}[h]
    \centering
\caption{Reaction rates of $^{209}$Bi $(n,\gamma)$ reaction in the spallation target assembly. }
\begin{tabular}{llr}
\toprule
&                 &  Reaction rate  (s$^{-1}$) \\
\midrule
\ & \textbf{Homogeneous STA} &                 4.13e-12 \\
   \textbf{$^{209}Bi \xrightarrow{n,\gamma} \; ^{210}Bi$}                                           & \textbf{Bottom} &                 1.21e-12 \\
                                              & \textbf{Center (ST)} &                 1.53e-12 \\
                                              & \textbf{Top} &                 7.35e-13 \\
\bottomrule
\end{tabular}
\label{rr_bi210}
\end{table}
This explains the overestimation of $^{210}$Po in the homogeneous approach where the reaction rate is higher because of an overestimation of the epithermal neutron flux.\\
The reaction rates of $^{210}$Bi production in the regions are reported in Tab. \ref{rr_bi210}. The difference in the reaction rate in the homogeneous approach with respect to the sum in the three regions confirms what seen in the plot \ref{vol_act}, i.e., conservative results are obtained with the homogeneous approach for the production of $^{210}$Po. \\

\textit{Production of other polonium isotopes}\\
Compared to the $^{210}$Po behaviour the other polonium isotopes show an opposite trend, since they are mainly generated by proton reactions. The relevance of the proton flux is emphasized when treating the spallation region as a separate zone. More than 99\% of $^{208}$Po and $^{209}$Po come from $^{209}$Bi that undergoes respectively $(p,n)$ and $(p,\gamma)$ reactions.\\

\textit{Production of tritium}\\
$^3$H, that provides largest activity in the STA during operation and after shut down is produced by spallation reactions in lead and bismuth. It decays in $^{3}$He that in turns undergoes $(n,p)$ reactions during reactor operation and it produces other tritium. When $^3$H starts to build-up its main production path is self-production by the daughter $^{3}$He.\\

\textit{Production of $^{16}$N}\\
$^{16}$N is almost completely produced by $(n,\alpha)$ reactions in $^{19}$F, that is a stable nuclide found as impurity in the fresh LBE. The reaction has a threshold at around 1 MeV so the production would be negligible in case of softer spectrum.\\ 

\textit{Production of mercury isotopes}\\
Mercury isotopes present in the activated LBE are of major concern in case of release in the environment. They are mainly produced via spallation reactions. They are produced by direct spallation reactions in lead and bismuth, $\beta^+$-decay of thallium isotopes and spallation reactions in other nuclides produced during irradiation.
Other production mechanisms are neutron- and proton-induced reactions such as:
\begin{itemize}
\item $^{194}$Hg produced by $(n,3n)$ reaction in $^{196}$Hg,
\item $^{195}$Hg produced by $(n,2n)$ reaction in $^{196}$Hg,
\item $^{197}$Hg produced by $(n,\gamma)$ reaction in $^{196}$Hg and $(n,2n)$ reaction in $^{198}$Hg,
\item $^{203}$Hg by $(n,\gamma)$ reaction in $^{202}$Hg. 
\end{itemize}

\textit{Production of $^{207}$Bi}\\
$^{207}$Bi is important for the LBE decay heat in the long term from 500 d to 70 years of decay. It is mainly produced by neutron-induced reactions and it is depleted by neutron capture and $\beta^{+}$-decay in the stable $^{207}$Pb.
The major production paths are $(n,2n)$ reaction in $^{208}$Bi and $(n,3n)$ reaction in $^{209}$Bi. The second mechanism is more relevant at the beginning of irradiation because of the bismuth natural isotopic composition (100\% $^{209}$Bi) but when $^{208}$Bi builds up the two mechanisms have the same relevance.

%The mechanisms involved in the production of all the nuclides produced in the LBE during irradiation is investigated but here only those interesting for their activity or decay heat contribution were presented.
