## 6 MCNP6
### Uncertainty Propagation and Quantification
#### 3.2.2 Consistent Sampling Approach Verification  
Prior to this work, the capability of producing and using consistent sets of perturbed ND for both depletion (with Shark-X/CASMO5) and criticality (with NUSS/Serpent) calculations has been demonstrated. The Peach Bottom 2 pin-cell model of UAM in Hot Zero Power (HZP) conditions served as a test case [25]. The same Shark-X model was used while a consistent Serpent model was established based on the MCNP model used in [25]. The ENDF/B-VII.0 nuclear data library is used in both codes to calculate kinf of the pincell.  
Various sources of covariance information were considered for testing the consistent ND sampling, namely ENDF/B-VII.1, ENDF/B-VIII.0 and JEFF-3.3. 300 perturbed ACE formatted files were generated with the NUSS scripts for each of the covariance libraries. Covariance matrices with a 19 energy group structure were applied in the case of ENDF/B-VIII.0 and JEFF-3.3 while for ENDF/B-VII.1 they had a 44 energy group structure. Only 235U, 238U and 1H have been considered for this test albeit more nuclides are available. The following reaction channels were perturbed, when available: (_n_,_n_), (_n_,_n_\({}^{\prime}\)), (_n_,_2_n_), (_nf_), (_n_,_\(\gamma\)), total \(\overline{v}\) and \(\chi\).  
In order to provide the consistency of the ND sets employed in the CASMO5 and Serpent calculations, the perturbation factors produced by NUSS were used to generate the corresponding ND perturbations in Shark-X.  
All Serpent calculations were performed with 100,000 histories per cycle, 200 active cycles and 20 inactive cycles. This leads to a kinf uncertainty of about 15 pcm, e.g., much less than the ND uncertainty. Furthermore, the seed number of each perturbed calculation is set to that of the nominal calculation.  
Uncertainties in the kinf values, calculated with CASMO5 and Serpent on the basis of 300 sampled ND sets for each library, ENDF/B-VII.1, ENDF/B-VIII.0 and JEFF-3.3, are reported in Table 1. The kinf relative uncertainties are consistent, even though small discrepancies are observed, most likely due to the different mean values for kinf. The consistency is further demonstrated by ordering the Serpent ratio of the perturbed kinf with respect to the nominal value in increasing order and plotting it against the same quantity produced by Shark-X. This is shown in Figure 2(a) for ENDF/B-VII.1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **ENDF/B-VII.1** & **ENDF/B-VIII.0** & **JEFF-3.3** \\ \hline
**CASMO5** & 0.89\% & 0.57\% & 0.75\% \\
**Serpent** & 0.86\% & 0.55\% & 0.74\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of kinf uncertainties for various covariance sources using consistent samples in CASMO5 and Serpent.  
As expected, a slope of 1 is observed. Fluctuations around the unity slope are visible in Figure 3a. Further analysis on these fluctuations is carried by computing the \"distance\" between the Shark-X and Serpent perturbed kinf for a consistent perturbation; this distance is expressed as the ratio defined by Equation (1):  
\[d_{i}=\frac{k_{i}^{Shark-X}}{k_{nom}^{Shark-X}}-\frac{k_{i}^{Serpent}}{k_{nom}^{ Serpent}} \tag{1}\]  
The standard deviation of the distances is in the 50 pcm range and appears normally distributed, see Figure 3b. However, such fluctuations are somewhat larger than the inherent statistical spread expected from the ratio of two independent Serpent calculations, each calculation having a 15 pcm standard deviation (the resulting standard deviation, assuming that the correlation between both Serpent _k-eff_ values in Equation (1) is zero, should then be \(\sqrt{2}\)\(\sigma_{Serpent}\approx 20\) pcm). The larger fluctuations are most likely caused by the different neutron transport approaches of Serpent and CASMO5, MC and deterministic method of characteristics (MoC).