\section{Fuel assembly model limitations}
The main limitation of FA models is introduced by the boundary conditions and it implies the neutron flux in the core is not reproduced by the FA model. In radial direction the neutron flux is almost flat while the actual shape in the core has peak in the core center and it slopes down toward the external region. It implies the differential depletion in radial direction cannot be caught by any FA model. This phenomenon, noticed in the full core model, is counterbalanced by the radial dependant depletion itself, since the fuel will be more depleted where neutron flux is originally higher leading to a reduction of fissions when burnup is higher. Moreover, the 2D FA model is not able to simulate the neutron leakage through the axial boundary, bringing to a flat neutron flux profile in axial direction. This issue is overcome by 3D models, that are investigated to assess the impact of axial burnup.\\

\subsection{Spectral differences between core and fuel assembly simulations}\label{sp_shift}
Spectral differences between the core and FA simulation induced by modeling assumptions are explored in this section. The neutron spectrum of the fuel assembly irradiated in the core central position (101) is compared with the spectrum obtained in a FA simulation. The neutron spectra are depicted in Fig. \ref{fig:FAcoreSpectra}.\\
\begin{figure}
    \centering
    \begin{minipage}{0.85\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/fa_spectra/spectrum.png} 
    \end{minipage}
\caption{Spectral comparison among core and 2D/3D FA model.}
\label{fig:FAcoreSpectra}
\end{figure}

Spectra are similar in the high energy region while differences emerge in the lower energy tail that is not present in the 2D FA model. However this region of the spectrum is nearly negligible, being two orders of magnitude lower. On the contrary, the small spectral differences in the fast region induce largest effect on the reaction rates. The integrated reaction rates for two relevant reactions are shown in Tab. \ref{tab:rr} to emphasize the impact of spectral shift in the nuclide vector.
\begin{table}
\begin{minipage}{\textwidth}
\caption{Reaction rates of fission and neutron capture respectively in $^{239}$Pu and $^{240}$Pu}\label{tab:rr}
\begin{center}

\begin{tabular}{lrr}

\toprule
\textbf{Reaction Rate}       &   \makecell{$(n,f)$   $^{239}$Pu}     &    \makecell{$(n,\gamma)$  $^{240}$Pu}\\
\midrule
\textit{3D core model} &   2.79e15 & 7.14e14\\
\textit{2D fuel assembly model} &   2.88e15 & 7.82e14\\
\textit{3D fuel assembly model} &   2.86e15 & 7.94e14\\



\bottomrule
\end{tabular}
\end{center}

\end{minipage}
\end{table}
The two reactions depicted are the fission of $^{239}$Pu and the neutron capture in $^{240}$Pu. A good agreement is seen between the two FA models on the total numbers of fissions. In the core $^{239}$Pu fission reaction rate is slightly lower, but it is compensated by more fissions in $^{238}$U and $^{240}$Pu, that is an indication of a harder spectrum. The neutron capture in $^{240}$Pu is the first capture step bringing to the production of all minor actinides. 3D FA simulation provides the largest reaction rate, 1.5\% higher than 2D and 12\% higher than in the core simulation. From this analysis an increase in the production of minor actinides in the FA models is expected.\\
The spectral variations are marked between FA and core model as they are less relevant among 2D and 3D. They are only one aspect of the differences between the core and FA models. Another one to be investigated is the discrepancy in the power profile.

\subsection{Power profile discrepancies}
A large discrepancy in the 3D fuel assembly model with respect to the core simulation was noticed in the axial power profile. The first shows tails in the top and bottom regions, that are present in the core simulation only in minor entity in assemblies in specific position, near control/safety rods or in the external position. This non-physical behaviour is related to the assumption of perfect reflectivity in the FA models and it is connected to the discussion on the spectral shift.
%The leakage phenomenon is more effective in the axial boundary regions since the probability of a particle re-entering in the active part of the core once it reaches the boundary is low in general but it is amplified by the assumption of total reflectivity.
In the FA models the neutron spectrum is softer, neutrons re-enter in the system with lower energy and they produce fissions; this phenomenon is more visible in the edge regions since in that part of the system the neutron flux is lower. The consequence is a local peak of power generated and, in turn a higher burnup in the edges. The effect on the power is therefore produced by combination of spatial and spectral effects.\\

\textit{Partial reflectivity impact on power profile}\\
A method to take into account one aspect of the phenomenon is to set a fixed value of particles leakage through the boundaries. 
\begin{comment}
"The albedo boundary condition iteration is an attempt to simulate the effects of neutron
leakage in an infinite-lattice geometry. The method works by sampling leakage (k > 1) or
multiplication reactions (k < 1) each time a neutron crosses a repeated or periodic boundary.
It should be noted that this method is highly experimental, and does not have physical
foundation similar to deterministic leakage models."

\end{comment}

It tackles part of the issue connected to the number of neutrons in the system but it does not consider the spatial and spectral effects.\\
A parametric study has been carried out to investigate the influence of neutron leakage parameter (albedo) on the power profile, that is the quantity more effected by the modeling assumption. The results of the analysis are summarized in Fig. \ref{fig:rr_alb}.
\begin{figure}[h]
\centering
\includegraphics[width=0.94\textwidth]{images/plots/FA/albedo/parametric_rr.png}
\caption{Axial profile of the normalized fission reaction rate. The result of the core model central assembly is compared with the one of FA model. Green and purple curves show the effect of albedo.}
\label{fig:rr_alb}
\end{figure}
The fission reaction rate, being the count of the number of fissions in the system is proportional to the power with a scaling factor equal to $\kappa$, the energy release per fission. The plot is normalized and the total number of fissions is set to one. The reduction of albedo affects the power (fission reaction rate) profile. The impact is evident passing from an albedo 98.9\% to 95\%. The power profile tends to be more peaked in the center and more depressed in the edge reducing albedo.\\
% The phenomenon is supposed to be a spectral effect, since the number of neutrons leaking from the fuel assembly is uniform along the axial position. The neutrons in the edges are more prone to undergo fission since the spectrum is softer than in the center due to increased interaction with the coolant and other structures. The leakage of those epithermal neutrons affects more the shape then the neutrons lost in the center of fuel assembly since the fission reaction rates increase in the edges.\\

\textit{Partial reflectivity impact on nuclide vector}\\
The impact of partial reflectivity boundary condition on the nuclide vector is assessed and the discrepancies for a subset of nuclides are presented; a similar analysis on a \acrshort{PWR} system was carried out in \cite{alb_AS} to estimate the sensitivity of \acrshort{SNF} observables on the neutron leakage parameter itself. The neutron leakage rate have been chosen arbitrarily: it has been calculated in a reference fuel assembly criticality calculation imposing a target $k_{eff}$ corresponding to the multiplication factor of the core at equilibrium. The albedo obtained is of 95.94 and 97.49\% for the 2D and 3D model, respectively. It is used to run transport/depletion simulations to be compared with the results of models with pure reflective BCs, to investigate the impact of partial reflectivity and if it allows to target the core concentrations.\\ 

The comparison is carried out on a 2D model and the nuclide vectors at end of irradiation are compared. The deviations that arise are presented in Fig. \ref{fig:nuc_alb}.
\begin{figure}[h]
\centering
\includegraphics[width=0.84\textwidth]{images/plots/FA/albedo/dev_nuc.png}
\caption{Deviation on the EoI nuclide inventory of the 2D FA with partial reflective boundary conditions (albedo=0.959) with respect to perfect reflective boundary conditions.}
\label{fig:nuc_alb}
\end{figure}
The concentration of most of volatile FP is not affected, with deviation up to 2\%. An slight underproduction of xenon isotopes and overproduction of kripton is osbserved. The most discrepant among volatiles is $^{134}$Cs that is underproduced of 25\% in case of albedo BCs. This is consistent with the hypothesis that reflective boundary conditions soften the spectrum because $^{134}$Cs is produced by capture and the reaction rate increases if the spectrum gets softer. The same trend is observed on minor actinides, americium and curium isotopes.\\
The main conclusion from this preliminary study is that applying partial reflective boundary condition can modulate the spectrum to match the one of the core and attain more agreement with core result in term of nuclides inventory. The issue lies in the methodology employed to get the leakage fraction that should be more deeply investigated. Moreover the physical relationship between increase of the leakage rate and spectral hardening is not fully established.

%The main reason for FA model development is the possibility to perform uncertainty analysis. The accurate 2D model was further simplified with the purpose of performing statistical sampling lowering the computational cost in term of time and resources. The simplifications doesn't concern the geometry but only the simulation parameters: the number of particles simulated is reduced of a factor 10 and the predictor corrector was switched off. The discrepancies introduced have a very limited impact on the nuclides analyzed. Largest deviation of around 3\% occurs in $^{246}$Cu, meaning that the simplified model is suitable to perform uncertainty analysis being representative of the accurate one.\\
\begin{comment}
To take into account the bias introduced by perfect radial boundary condition the approach followed consists of comparing the FA model with one that employs albedo condition to determine the discrepancies in term of concerning nuclides concentrations, Fig. \ref{fig:nuc_alb}.
The leakage rate have been chosen arbitrarily: it has been calculated in a reference FA simulation imposing a target $k_{eff}$. The multiplication factor of the core at equilibrium is used. The necessary albedo is of 95.94 and 97.49\% for the 2D and 3D model, respectively.

Nuclide vector at end of irradiation is taken for comparison. The deviation that arise are presented in Fig. \ref{fig:nuc_alb}. Nuclides are divided in three categories according to their relevance to the analysis.
\begin{figure}[h]
\centering
\includegraphics[width=0.55\textwidth]{images/plots/FA/albedo/dev_nuc.png}
\caption{Deviation in EoI inventory comparing 2D FA model with and without leakage.}
\label{fig:nuc_alb}
\end{figure}
Among volatiles most of them are underestimated in the model with purely reflective BCs, even if the most discrepant, up to 30\%, is $^{134}$Cs that, on the contrary is produced more in that case.
The differences are of order of 1\% for fission products contributing to decay heat, while bigger deviations are retrieved in actinides. All of them but $^{237}$Np are underestimated with albedo BCs.
The take home message from the analysis is that there's an impact on the production of radionuclides beyond the power profile. The more affected nuclides are those produced not directly by fission since they are not linear with the power, this brings the shape to have an impact in the total production.\\
Once assessed the differences the approach in the following is to take as a reference for the fuel assembly the model with fully reflective boundary conditions since it gives more conservative results for most of the nuclides of main concern. 
\end{comment}