\section{Uncertainty on nuclide vector}
The nuclide vector and its evolution as function of irradiation/decay time is the main output of the burnup calculations performed in this work. 
\subsection{Cross sections uncertainty propagation}\label{unc_act_xs_sect}
The propagation of cross section uncertainty is performed considering all available nuclides which uncertainty is present in the three previously discussed libraries. A set of 200 samples is generated to perform statistical sampling. The determination of the sample size is based on observing the convergence of the inputs and of the outputs. The exhaustive list of the nuclides perturbed and the output convergence analysis are presented in appendix \ref{app_inc}, while the perturbation coefficients convergence is shown in \ref{stSMPS}. The output presented in this work is the uncertainty associated to the nuclide vector with main focus of the relevant nuclides already discussed.
\subsubsection{Impact of cross section uncertainty on actinides concentrations}
The first category of nuclides analysed comprises the actinides important for spent fuel management , and their concentration uncertainties at \acrshort{EoI} are depicted in Fig. \ref{unc_act_xs}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{images/plots/FA/uncertainty/act_xs.png}
\caption{Actinides concentration uncertainty at EoI coming from cross section uncertainty propagation.}\label{unc_act_xs}
\end{figure}

\textit{Uncertainty of uranium isotopes}\\
$^{235}$U ( $t_{\frac{1}{2}} = $ 7.038e8 y) and $^{238}$U ( $t_{\frac{1}{2}} = $ 4.468e9 y) show limited uncertainty. $^{235}$U neutron capture cross section uncertainty brings to a progressive increase of its uncertainty with burnup. On the other side, it constitutes the major source of uncertainty for $^{236}$U ( $t_{\frac{1}{2}} = $ 7.370e7 y), that constantly builds up in the system. The discrepancies of the libraries are explained looking at the relative standard deviation in the fast region where the uncertainty given by JENDL-4.0u \acrshort{NDL} is lower with respect to the others. $^{238}$U shows negligible uncertainty. The sensitivity to any production/depletion mechanism is low because it is present in large amount in the fresh fuel and its concentration varies less then 1\% during irradiation.\\

\textit{Uncertainty of plutonium isotopes}\\
Pu isotopes are initially present in the fresh fuel and depleted by means of fission, neutron capture and decay. Their uncertainty is in the range 0.25 to 2.5\% at \acrshort{EoI}. The uncertainty evolution of plutonium isotopes is shown in Fig. \ref{unc_pu}, it tends to increase with burnup and it evolves in the opposite way respect to their buildup, displayed in Fig. \ref{fig:actinides_buildup}. 
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{images/plots/FA/uncertainty/pu_unc_evolution.png}
\caption{Plutonium isotopes uncertainty evolution as function of the FA burnup.}\label{unc_pu}
\end{figure}

$^{238}$Pu production is mainly bond to $^{242}$Cm that is ahead in the capture chain while its depletion happens by neutron capture; since its concentration varies of 15\% during irradiation, the global effect of production/depletion has a limited impact. JEFF-3.3 and JENDL-40u give a similar evaluation of the relative standard deviation of $^{238}$Pu total cross section, larger than ENDF/B-VIII.0 and this is consistent with the corresponding bars in Fig. \ref{unc_act_xs}. The uncertainty concentration of $^{242}$Cm affects the $^{238}$Pu uncertainty in lower amount being its depletion greater than the production.\\
$^{239}$Pu has very low uncertainty, coming from its own total cross section; $^{240}$Pu is produced and depleted by neutron capture and the two mechanisms are of the same order of magnitude. So its uncertainty is both due to capture cross section of $^{239}$Pu and its own cross section.\\
$^{241}$Pu has relatively low half-life and it decays into $^{241}$Am other than undergoing fission. Its uncertainty is the highest amongst plutonium isotopes since its concentration has a largest variation during irradiation and it is more sensitive to the production/depletion during irradiation. $^{241}$Pu mainly comes from  neutron capture in $^{240}$Pu. The uncertainty given by JEFF-3.3 for this \acrshort{XS} is greater than the other libraries in the region from 0.01 to 1 MeV and this explains why the output uncertainty obtained with this \acrshort{NDL} is higher.\\
$^{242}$Pu ( $t_{\frac{1}{2}} = $ 3.735e5 y) concentration undergoes a limited variation, its uncertainty comes from $\sigma_{(n,\gamma)}$ of $^{241}$Pu and its own $\sigma_t$. Reduced uncertainty is observable with JEFF-3.3 since covariance matrices of $^{242}$Pu are not available in this library.\\

\textit{Uncertainty of americium isotopes}\\
The americium isotopes are produced by successive captures from $^{241}$Am that in turn comes from $\beta_-$ decay of $^{241}$Pu; the source of uncertainty for these nuclides is primarily attributed to the capture cross sections of their respective parent. JEFF-3.3 systematically underestimates their uncertainty since covariance matrices of $^{241}$Am, $^{242}$Am and $^{242m}$Am are not available.\\
$^{241}$Am is initially present in the fresh fuel, which results in a lower uncertainty, primarily influenced by the uncertainty on the concentration of $^{241}$Pu and its $\sigma_{(n,\gamma)}$ (neutron capture cross section). The uncertainty on the concentration of this nuclide has a strong impact on the decay heat rate at 70 years of decay since it is one of the main contributors. JENDL-4.0u \acrshort{NDL} gives the largest estimate of $^{242m}$Am and $^{243}$Am uncertainty since the it provides greater cross section uncertainties in the region from 0.01 to 1 MeV.\\

\textit{Uncertainty of curium isotopes}\\
$^{242}$Cm is an important contributor to the decay heat of \acrshort{SNF} in the first years of cooling and it shows very similar uncertainty of $^{242m}$Am, which is its major production path. The uncertainty associated to $^{242}$Cm capture cross section is of the order of 20 to 40\% in the energy range of interest in JEFF-3.3 but it has negligible impact on its concentration uncertainty. On the other hand, this same source of uncertainty has a significant effect on the product of the reaction, $^{243}$Cm. The uncertainty for $^{243}$Cm is consistently around 25 to 30\%, and it is primarily attributed to its production mechanism since it is less sensitive to its total cross section. The uncertainty associated to $^{243}$Cm is not propagated to $^{244}$Cm because the latter is only produced in limited amount through neutron capture reactions. The primary source of production of $^{244}$Cm is the $\beta_-$ decay of $^{244m}$Am. Therefore, the uncertainty in $^{244}$Cm arises from the uncertainties in the americium isotopes involved in this decay process.\\
The uncertainty on the concentrations of heavier curium isotopes increases up to 40\% and 50\% and is mainly driven by the uncertainties in their parents capture cross sections. $^{245}$Cm concentration uncertainty is overestimated by JEFF-3.3 since it gives greater standard deviations with respect to the other libraries. ENDF/B-VII.0 and JENDL-4.0u have the same covariance matrix for the capture cross section of $^{244}$Cm so the mismatch in the $^{245}$Cu concentration uncertainty is due to the concentration uncertainty of the parent nuclide. The opposite happens when considering $^{246}$Cu: even in this case JENDL-4.0u and ENDF/B-VIII.0 share the covariance matrix evaluations but they give much larger uncertainties than JEFF-3.3 and this translates in larger concentration uncertainties in $^{246}$Cm as depicted in the plot.\\

\subsubsection{Impact of cross section uncertainty on fission products concentrations: volatiles \& decay heat contributors}
The second category analyzed includes volatile fission products conveying the largest dose in case of release  identified in Sect. \ref{obervables} and the FPs contributing to decay heat. Almost all of them are direct fission products, so their production is proportional to the number of fissions in the system that is determined by the input power. The low uncertainty observed comes from the small fraction produced by neutron capture reactions and the statistical error inherent in Monte Carlo. The \acrshort{EoI} uncertainty concentration of both categories is depicted in Fig. \ref{unc_fp_xs} and the only non-negligible contribution comes from $^{134}$Cs and $^{3}$H.\\
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{images/plots/FA/uncertainty/fp_xs.png}
\caption{Fission products concentration uncertainty at EoI coming from cross section uncertainty propagation. Divided in volatiles and decay heat contributors.}\label{unc_fp_xs}
\end{figure}
$^{134}$Cs is almost 100\% produced by neutron capture in $^{133}$Cs and its uncertainty directly comes from the capture cross section of the parent. The uncertainty obtained with JEFF-3.3 and ENDF/B-VIII.0 is around 10\% in agreement with the relative standard deviation given on the $(n,\gamma)$ cross section that ranges from 4\% in the thermal up to 40\% in the fast region in both libraries. Lower uncertainty is obtained with JENDL-4.0u \acrshort{NDL} which does not provide uncertainty measures for $^{133}$Cs and fission products in general.\\
The uncertainty associated with $^3$H has a distinct source, which deserves further examination. In critical configurations, its primary production path is through ternary fission, a process in which three products are generated from a fission reaction. While this is a rare event with a low yield, it remains significant because it contributes to the accumulation of tritium within the fuel assembly. The uncertainty obtained from ENDF/B-VIII.0 library arises from the fact that the fission yield data provided in the library does not account for ternary processes, as demonstrated in Fig. \ref{fy_h3}, with JEFF-3.3 used as a reference.
\begin{figure}[h]
\centering
\includegraphics[width=0.89\textwidth]{images/plots/FA/uncertainty/fy_h3.png}
\caption{Independent fission yields as a function of product mass number for $^{239}$Pu in the fast region ($E = 400\;keV$).}\label{fy_h3}
\end{figure}
If tritium production by fission is not taken into account its concentration is underestimated of five orders of magnitude and in this case the amount produced is so small that the uncertainty observed comes from statistical fluctuations as shown by the last bar in Fig. \ref{unc_fp_xs} and it is irrelevant as the tritium produced is negligible. The assumption of neglecting ternary fission gives a strong underestimation of the production of one of the most important nuclides in the analysis.\\
A second issue connected to tritium is that the production mechanism is not the same in sub-critical configuration, where its production is maximized, so part of its uncertainty is not considered in a critical system. A deeper analysis on this regard is carried out in appendix \ref{tritium}.\\
The uncertainty of the other fission products coming from the \acrshort{XS} uncertainty is almost negligible, therefore to give a more reliable estimation of the uncertainty of the other relevant \acrshort{FP}s is necessary to broaden the analysis to the neutron fission product yields that are expected to be the main source of uncertainty for fission products.\\
\begin{center}
\fbox{
\begin{minipage}{0.98\textwidth}
The propagation of cross sections uncertainty by means of statistical sampling provides the uncertainty estimation of the nuclide concentrations. The impact on plutonium isotopes is limited to 2.5\% and it increases for minor actinides up to 60 \% in $^{246}$Cm. The uncertainty of fission products concentrations is negligible since their production is not affected by cross sections variation; $^{134}$Cs is the exception, being produced by neutron capture in $^{133}$Cs. The uncertainties vary depending on the \acrshort{NDL} employed since each of them provides different covariance matrices.
\end{minipage}}
\end{center}

\subsection{Nuclear fission product yields uncertainty propagation}
The need for uncertainty values of fission product concentrations drove to conduct the propagation of uncertainty for the \acrfull{NFPY} alongside the neutron cross-section data.
Covariance matrices are not provided for neutron fission product yields, the uncertainty measure is given in term of variance and therefore correlations among fission yields are not taken into account. This brings to quite large output uncertainties associated since $\sigma^2$ is of the order of 10\% for most of \acrfull{IFY}. Methods to reduce independent fission yields uncertainty were developed, such as introducing conservation equations and taking advantage of the uncertainty evaluations on the cumulative FY to generate covariance matrices from the available data as reported in \cite{Fiorito2016GenerationLibraries}. These methods are not applied in this work and the raw uncertainty data of \acrshort{IFY} are propagated. The only normalization procedure applied was on the sum of all the independent ﬁssion yields that must be equal to the fragments of the ﬁssion, that is slightly greater than two if ternary fission events are accounted.\\
The output uncertainties of the \acrshort{EoI} nuclide concentrations are presented in Fig. \ref{unc_fp_fy} and the perturbed \acrshort{IFY} comes from JEFF-3.3 \acrshort{NDL}\
\begin{figure}[h]
\centering
\includegraphics[width=0.99\textwidth]{images/plots/FA/uncertainty/fy_nucs.png}
\caption{Fission products concentration uncertainty at EoI coming from neutron fission product yields uncertainty propagation. Divided in volatiles and decay heat contributors.}\label{unc_fp_fy}
\end{figure}
The uncertainties on actinides are negligible since \acrshort{NFPY} data do not interact with their production/depletion, while they have a higher impact on the other observables. The production of a nuclide through fission is linearly dependent on its fission yield when there are no other production mechanisms involved, such as the decay of other fission products or neutron captures. On the contrary most of the nuclides investigated are both produced by fission and decay of other short lived fission products, which implies the uncertainty of a target nuclide is sensitive to the uncertainty on its own fission yields and all the fission products decaying in it.\\
The uncertainty on $^{134}$Cs, that is not a direct fission product, comes from the uncertainty of his parent $^{133}$Cs. The latter has a very small independent fission yield but it is the stable end of chain of the A=133 $\beta_-$ decay line so its uncertainty is determined by all the nuclides in the chain and it explains why $^{134}$Cs, $^{133}$Xe ( $t_{\frac{1}{2}} = $ 5.247 d) and $^{133}$I ( $t_{\frac{1}{2}} = $ 20.87 h) show the same level of uncertainty. The interaction among the \acrshort{FP} involved in the chain is clarified by Fig. \ref{corr_fp}, that shows the correlation among the fission products concentrations. 
\begin{figure}[h!]
\centering
\includegraphics[width=0.85\textwidth]{images/plots/FA/uncertainty/correlation_fp.png}
\caption{Correlation matrix among nuclides contributing to the uncertainty of $^{134}$Cs}
\label{corr_fp}
\end{figure}
The correlation matrix measures how much the population of an item is explained by another one in the list. The reaction chain analyzed in the following: $$^{133}Sb \xrightarrow{\beta_-} \ ^{133}Te \xrightarrow{\beta_-} \ ^{133}I \xrightarrow{\beta_-} \ ^{133}Xe \xrightarrow{\beta_-} \ ^{133}Cs \xrightarrow{(n,\gamma)} \ ^{134}Cs.$$
$^{132}$I ( $t_{\frac{1}{2}} = $ 2.295 h) and $^{132}$Xe (stable) are added to demonstrate they are not relevant in the production of $^{133}$I and $^{133}$Xe by capture and in fact they show null correlation. On the contrary there is full correlation in the chain between $^{134}$Cs and $^{133}$I, i.e. all the uncertainty comes from this fission product. In  turn $^{133}$I is both directly produced by fission and partly comes form $^{133}$Te ( $t_{\frac{1}{2}} = $ 12.45 min) decay as witnessed by their cross-correlation of $0.8$. The same conclusion, in a qualitative way can be drawn looking at the mismatch between independent and cumulative fission yield of each couple of nuclides.\\
$^{137}$Cs is a concerning volatile nuclide and it is bond to $^{137m}$Ba, contributing to decay heat at 3 y of decay; its uncertainty is due to its own fission yield and the one of $^{137}$Xe ( $t_{\frac{1}{2}} = $ 3.818 min) and $^{137}$I ( $t_{\frac{1}{2}} = $ 24.51 s), which contribute to its production. $^{3}$H shows the largest uncertainty among the subset of fission products considered. It is around 10\% and it is only induced by the uncertainty of its fission yield. Amongst krypton isotopes the one showing largest uncertainty is $^{91}$Kr and it mainly comes from its \acrshort{NFPY} uncertainty since there is a low mismatch between the independent and cumulative fission yield.\\
The uncertainty of $^{133}$Xe and its origin was already discussed while commenting Fig. \ref{corr_fp}; it is the xenon isotope with lower standard deviation around 4\% while $^{135m}$Xe ( $t_{\frac{1}{2}} = $ 15.3 min), $^{139}$Xe ( $t_{\frac{1}{2}} = $ 39.69 s) and $^{140}$Xe ( $t_{\frac{1}{2}} = $ 13.6 s) have the largest uncertainty around 8\%. The first mainly comes from $\beta$ decay of other fission products while the last two have quite high yield and are direct fission product, so their uncertainty derives from their own \acrshort{NFPY} data.\\

\begin{center}
\fbox{
\begin{minipage}{0.98\textwidth}
The \acrshort{NFPY} uncertainties in form of standard deviation are propagated and they produce a non-negligible uncertainty on the concentrations of the fission products. The uncertainty associated to the volatile nuclides and decay heat contributors is between 5 and 10\%. 
\end{minipage}}
\end{center}\\
Even though all the contributors to decay heat exhibit uncertainties of approximately 6\%, their impact on the overall uncertainty of decay heat is explored in the subsequent section.