\section{Nuclide concentrations}
The time-evolution of the nuclide inventory of irradiated fuel is modeled taking the \acrshort{EoC} concentrations of each FA, in both the  critical and sub-critical configurations, as initial conditions of the Bateman equations.
%The nuclide vector as a function of the burnup of each FA bundle can be estimated. It mimics the evolution of the nuclide concentrations in an individual fuel assembly being reshuffled in all the available core positions. This phenomenon is a direct consequence of the equilibrium condition since the composition and the burnup level of a FA in a certain position is defined and it stays the same from cycle to cycle if equilibrium is achieved.
The nuclides buildup is reported as a function of the burnup level of a FA. 
Analysing the buildup of the various atomic species certain common patterns emerge. 
In the following  the investigated nuclides are grouped as actinides and fission products.\\

\textit{Actindes buildup}\\
Generally, the actinides have a similar buildup in the critical and sub-critical configurations.
The buildup of some nuclides is maximized in the sub-critical case: for instance $^{242}$Cm ( $t_{\frac{1}{2}} = $ 162.93 y) and $^{243}$Cm ( $t_{\frac{1}{2}} = $ 28.90 y) have a final concentration 135\% larger than in the critical case.\\
The plutonium isotopes show different trends. 
The production of $^{238}$Pu ( $t_{\frac{1}{2}} = $ 87.70 y) is maximized in the sub-critical configuration as a result of overproduction of its parent nuclide $^{242}$Cm.
Conversely the concentration of $^{239}$Pu ( $t_{\frac{1}{2}} = $ 24129.50 y) is lower in the sub-critical core, indicating a higher burn rate in this configuration due to the larger neutron flux.\\
% The concentration deviations between the critical and sub-critical modes for the plutonium isotopes are approximately 5\% since most of the final concentration depend on the initial composition of the fuel, that is common  to both models.\\
The larger production of $^{241}$Am and $^{242m}$Am ( $t_{\frac{1}{2}} = $ 141.00 y) in the sub-critical mode is not explained by the $^{241}$Pu buildup, since the latter is under-produced in the sub-critical configuration.
Concerning $^{242m}$Am, its neutron capture is lower in the harder spectrum of the sub-critical core, which results in a larger  $^{242m}$Am buildup and a lower  production of $^{243}$Am ( $t_{\frac{1}{2}} = $ 7364.98 y).\\
Tab. \ref{tab_act_scCr} compares the actinide concentrations of the most irradiated FA (that with the largest burnup) for the \acrfull{EoC} configurations of the critical and sub-critical cores. The comparison is carried out on FAs that reached the discharge burnup, higher in the critical configuration. 
Under the equilibrium assumption, the irradiation condition of a fuel assembly with largest burnup level, placed in the core periphery corresponds to the condition of a FA loaded in the core after it has undergone the whole in-core reshuffling procedure.
Therefore the composition in the fuel assembly with highest burnup level is equal to the composition of a FA at \acrfull{EoI}, before being unloaded from the core. 
\begin{table}[h]
    \centering
    \caption{\acrshort{EoI} concentration (FA with largest burnup) ratio between sub-critical end critical mode for selected actinides.}
\scalebox{0.92}{
\begin{tabular}{lrrrrrrrrr}
\toprule
\textbf{Nuclide} &       $^{238}$Pu &       $^{239}$Pu &       $^{240}$Pu &       $^{241}$Pu &       $^{241}$Am &      $^{242m}$Am &       $^{242}$Cm &       $^{244}$Cm &       $^{246}$Cm \\
\midrule
\textbf{Buildup} & 1.03 & 0.96 & 1 & 0.94 & 1.08 & 1.15 & 1.36 & 0.88 & 0.82 \\
\bottomrule
\end{tabular}
}

    \label{tab_act_scCr}

\end{table}

\textit{Fission products buildup}\\
The fission products considered in this chapter are those reported in Sect. \ref{AccCondObservables} and they are split into two categories depending on how their concentration builds up, as shown in Fig. \ref{fig:volatile_buildup} for the sub-critical configuration. Forthe sake of comparison, the nuclide concentration buildup curves are normalized to the corresponding nuclide concentration at \acrshort{EoC} for the critical core.\\
\begin{figure}
    \centering
    \begin{minipage}{\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/ref_buildup/shortLived.png} 
    \end{minipage}

        \begin{minipage}{0.95\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/ref_buildup/longLived.png}
    \end{minipage}
\caption{Buildup of volatile fission products in sub-critical configuration, divided in two categories depending to their evolution. Normalized on the \acrshort{EoC} concentration in critical configuration.}
\label{fig:volatile_buildup}
\end{figure}
The first category of fission products includes all the considered nuclides with relatively short half-life. The buildup of these nuclides is driven by the cycle power, and it reaches its maximum in the first irradiation cycle --- e.g., when the FA is in the core center. Because of the IN-to-OUT in-core FA shuffling scheme, the concentrations of these nuclides decrease as the FA moves away from the core center and the FA power decreases. The sub-critical configuration yields higher peak concentrations for these nuclides because of the larger power generate in the central FAs compared to the critical core.\\

The second category of fission products includes the relatively long-lived nuclides. The concentration buildup of these nuclides is proportional to the FA burnup and it attains its maximum at \acrshort{EoI}. These nuclides show different behaviour and evolution depending on the specific production path.\\
Because of its long half-life and its negligible capture cross section, $^{137}$Cs ( $t_{\frac{1}{2}} = $ 30.05 y) has a linear dependence on burnup. Because of this, it builds up more in the critical configuration where higher discharge burnup are achieved at \acrshort{EoI}.\\
$^3$H ( $t_{\frac{1}{2}} = $ 12.34 y) and $^{85}$Kr ( $t_{\frac{1}{2}} = $ 10.76 y) show a similar evolution but the slope tends to decrease since they slightly decay while building up. This behaviour is emphasized in $^{125}$Sb ( $t_{\frac{1}{2}} = $ 2.76 y).
% $^3$H production in sub-critical mode is enhanced. In particular the slope, that represents the production rate, is higher during the first cycles. It creates an offset that stays constant in the successive irradiation cycles.
$^3$H production in sub-critical mode is enhanced. During the irradiation cycle in the core central position its production is greater in sub-critical configuration. The threshold reactions above 20 MeV become important, in particular $^3$H is largely produced from the oxigen contained in the MOX fuel. The mechanism is not relevant when the FA is irradiated further for the center since the high energy tail of the spectrum is negligible. Further discussion on this is presented in appendix \ref{h3comments}.\\
The non-linear evolution of $^{134}$Cs ( $t_{\frac{1}{2}} = $ 2.07 y) is explained by the fact that it is not a direct fission product but it is produced by neutron capture in $^{133}$Cs (stable nuclide), the end product of A=133 isobaric $\beta^-$ decay chain.\\
$^{127m}$Te ( $t_{\frac{1}{2}} = $ 106.1 d) is the only one among the identified volatile nuclides that is not shown in the plots. It mostly comes from fission and $\beta_{-}$ decay of $^{127}$Sb ( $t_{\frac{1}{2}} = $ 3.85 d) and it reaches the peak buildup during the third irradiation cycle.\\ 
The indication on the maximum buildup of each radionuclide can be employed in a safety evaluation to obtain conservative results when evaluating the concentration and the associated dose in case of release during accidental sequences.
\begin{center}
\fbox{\begin{minipage}{0.98\textwidth}
The volatile nuclides generated in irradiated MOX fuel and that are considered relevant for a safety assessment can be divided into two categories depending on their half-life. Short-lived nuclides reach their peak concentration in the first irradiation cycle, when the FA generates a larger power. For the same reason we observed that they are produced in larger quantities in the sub-critical mode rather than in the critical one.
Long-lived nuclides build up during irradiation and they reach their peak concentration at \acrshort{EoI}.
Their concentration is proportional to the FA burnup. At discharge, the FA burnup is higher in the critical configuration. 
\end{minipage}
}
\end{center}
