Discussion on variety of cross-section candidate obtained by cross-section adjustment method

. The cross-section adjustment method gives only one best-estimated cross-section set based on Bayesian theory. In this study, cross-section sets other than the best-estimated set were discussed. The cross-section sets were obtained by the random sampling method using a covariance matrix adjusted by the cross-section adjustment method. 1,000,000 cross-section sets were obtained by the random sampling method. From these cross-section sets, significant cross-section sets other than the best-estimated set were considered. 593,340 cross-section sets that accurately evaluate nuclear characteristics were obtained. The result of the cross-section adjustment method shows that the change due to U235 χ is the largest positive. However, k eff is not changed to the positive side by U235 χ in 179,994 of the 593,340 cross-section sets obtained by the random sampling method. Although the best-estimated cross-section set is obtained by the cross-section adjustment method, the result suggests there are various candidates for the adjusted cross-section set.


Introduction
Accurate evaluation of nuclear characteristics is essential for the safe and efficient operation of nuclear reactors. Reduction of the uncertainties is important for higher evaluation accuracy. Among the uncertainties that lead to the uncertainty of nuclear characteristics, the uncertainty arising from the nuclear data is a major factor.
The cross-section adjustment method [1,2] is one of the methods for reduction of the uncertainty arising from nuclear data. The cross-section adjustment method provides a best-estimate cross-section set whose calculated values reproduce experimental values well. However, it is pointed out that some of the results of the cross-section adjustment method have no nuclear physics justification [3]. In addition, it has been reported that there are various cross-section sets whose C/E (calculation/experiment) values are consistent with the results of the cross-section adjustment method due to the compensation effect [4,5]. Some of these various crosssection sets may be more reliable than the result of the cross-section adjustment method. However, the significance of these various cross-section sets has not been assessed. Therefore, in this study, the significance of the various cross-section sets was discussed.
The random sampling method can generate crosssection sets based on covariance data and is used in the community of reactor physics [6][7][8]. In this study, the random sampling method was used to generate crosssection sets that accurately evaluate nuclear characteristics. Then, the significance of the crosssection set was assessed by comparison. The random sampling method is summarized in Sec. 2. A * Corresponding author: s-odani@ne.see.eng.osaka-u.ac.jp comparison of the cross-section sets obtained by the random sampling method with that by the cross-section adjustment method is shown in Sec. 3. Finally, the conclusion is summarized in Sec. 4.

Random sampling using crosssection adjustment method
For random sampling, an adjusted cross-section set and an adjusted covariance matrix obtained by the crosssection adjustment method were used. Assuming that experimental error, calculation error, and cross-section error are independent of each other, the adjusted crosssection set is obtained by = 0 + T ( T + e + m ) −1 ( e − c ( 0 )), (1) where is the adjusted cross-section set, 0 is the original cross-section set, is the covariance matrix, is the sensitivity coefficient of the cross-section set, e is the covariance matrix of the experimental error, m is the covariance matrix of the calculation error, e is the experimental value and c is the calculated value. The adjusted covariance matrix adj is obtained by The multivariate normal random number according to the adjusted covariance matrix is obtained by where is the vector whose elements are multivariate normal random numbers, is the vector whose elements are standard normal random numbers, and is the mean vector. When the adjusted covariance matrix is a positive semidefinite matrix, the adjusted covariance matrix is expanded using as To satisfy equation (4), adj must be a positive semidefinite matrix. However, it has been reported that the covariance matrix may not be positive semidefinite [8][9][10]. This problem is caused by the distortion of eigenvalues due to rounding. In this study, when the covariance matrix is not positive semidefinite, the nearest covariance matrix, which is positive semidefinite matrix obtained by Higham's method [11], was used. In this case, the nearest covariance matrix near is expanded using as When using the nearest covariance matrix, the random number is obtained by 3 Random sampling using two uranium benchmarks

Calculation condition
The cross-section adjustment method was performed using the criticality of light water-moderated uranium benchmarks summarized in ICSBEP [12]. The benchmark problems are HEU-SOL-THERM-001-001 (HST1-1) and LEU-COMP-THERM-001-001 (LCT1-1). HST1-1 is high-enriched uranium fuel system and LCT1-1 is low-enriched uranium fuel system. In addition, both systems do not contain plutonium. Sensitivity coefficients and covariance data were used from a package included in the MCNP6.2 [13] packages, WHISPER [14]. Sensitivity coefficients included in WHISPER were calculated with ENDF/B-VII.1 [15]. The calculations used 100,000 neutrons per cycle, 100 skip cycles, and 600 active cycles. Covariance data in WHISPER is the same data that is distributed with SCALE 6.1 and is 44-group energy structure [16]. Then, the two benchmark problems are assumed to be independent, and have zero covariance elements, and the diagonal elements are each variance.

Selection of target nuclide reactions
Due to computer memory utilization issues, not all cross-sections of nuclides included in the benchmark problem were adjusted using the cross-section adjustment method. The adjusted cross-section was selected by the contribution to the uncertainty of criticality. The uncertainty of criticality due to a particular reaction is obtained by where is the standard deviation of criticality due to a particular reaction, α is the sensitivity coefficient of the reaction, and α is the covariance matrix of the reaction.
In this study, the threshold was set at 1/100 of the uncertainty of criticality due to all reactions. Therefore, the cross-section with > 0.005 % were adjusted by the cross-section adjustment method. The target nuclides for the cross-section adjustment method are summarized in Table 1.

Results of cross-section adjustment method
The calculated and experimental values before the cross-section adjustment method are summarized in Table 2. The changes in criticality for the two benchmark problems due to the cross-section adjustment method are summarized in Table 3. The amount of change in criticality due to each nuclide reaction in the adjusted cross-section set is summarized in Figure 1. Then, the amount of change in criticality due to each nuclide reaction ⁄ is obtained by where g is a certain energy group, g is the sensitivity coefficient at a certain energy group, and Δ g g ⁄ is the change of cross-section at a certain energy group. As shown in Table 3, the difference between the adjusted and experimental values is small. Figure 1 shows that the C/E value is improved mainly by the adjustment of U235 χ.  As shown in Figure 1, the negative adjustments in U238 (n,γ) and (n,n') compensate the positive adjustments in H1 (n,n) and U235 χ. The compensation effects occurred in U238 (n,γ) and (n,n') because these reactions are sensitive only in LCT1-1, as shown in Figure 2.

Effect of using the nearest covariance matrix
As mentioned in Sec. 2, the nearest covariance matrix was used to generate multivariate normal random number. To evaluate the effect of using the nearest covariance matrix, the uncertainty of criticality arising from cross-section was compared.
First, the nearest covariance matrix was compared to the original covariance matrix by the uncertainty of the criticality of each benchmark problem. The difference in standard deviations of the criticality in each benchmark problem ∆ is obtained by where org is the original covariance matrix. The differences in each benchmark problem are summarized in Table 4. As shown in Table 4, the difference between the nearest covariance matrix and the original covariance matrix is small, so we concluded the effect of using the nearest covariance matrix is small. Even if each matrix element in the nearest covariance matrix is significantly different from the original covariance matrix, the differences shown in Table 4 could be underestimated due to cancellation of positive and negative values. Therefore, the focus was on the variance of the each cross-section in all energy groups for all nuclide reactions. The difference due to each cross-section ∆ is obtained by where i is the nuclide reaction, g is the energy group, ig near is the variance of the nearest covariance matrix, and ig org is the original variance. The maximum difference due to each cross-section in each benchmark problem is summarized in Table 5. As shown in Table 5, the difference due to each cross-section is also small. Considering the above, the effect of using nearest covariance matrices was determined to be small.

Result of random sampling and discussion
The nearest covariance matrix was used to generate 1,000,000 random cross-section sets. From these random cross-section sets, the focus was on crosssection sets that accurately evaluate criticality. In this study, accurate evaluation of criticality means that the adjusted criticality is in the range shown in Table 6. This range was determined by the uncertainty of the experimental and calculated values, as shown in Table  2. At this time, statistical uncertainty and experimental uncertainty are assumed to be independent. As a result, 593,340 cross-section sets were in this range. The 593,340 cross-section sets were compared to the adjusted cross-section set obtained by the cross-section adjustment method. As shown in Figure 1, the crosssection adjustment method shows the largest positive change in criticality by U235 χ. The frequency of the change in criticality by U235 χ in the 593,340 crosssection sets in HST1-1 is shown in Figure 3. Figure 3 shows that the results of the cross-section adjustment method are included in the most frequent bin. However, in 179,994 of the 593,340 cross-section sets, criticality is not changed to the positive side by U235 χ. This result indicates that there are significant candidates for the cross-section set that accurately evaluate criticality, other than the results of the cross-section adjustment method. The 179,994 cross-section sets whose criticality is negatively adjusted by U235 χ were investigated. Focusing on the fourth most frequent bin in Figure 3, the amount of adjustment due to reactions other than U235 χ was investigated. The average amount of change in criticality due to all reactions is shown in Figure 4. The fourth most frequent bin is in the range of −0.35 % to −0.14 % of criticality change due to U235 χ. Then, the results shown in Figure 4 are the average of the criticality adjustments for the 84,120 cross-section sets in this range. Figure 4 shows that the negative adjustment due to U235 χ is mainly compensated by H1(n,n). The standard deviations of criticality obtained by substituting the nearest covariance matrix into equation (7) are shown in Table 7. As shown in Table 7, the standard deviation arising from U235 χ is larger than 0.09 % (the amount of change in criticality by the crosssection adjustment method). Therefore, the negative adjustment by U235 χ is consistent with the covariance data. Furthermore, it is also consistent with the covariance data that criticality is adjusted by H1 (n,n), which is the second largest when criticality is not adjusted by U235 χ. In the cross-section adjustment method, the bestestimated cross-section set is obtained, where criticality is adjusted mainly by U235 χ due to the largest standard deviation. However, the results obtained by using the random sampling method indicate that the cross-section sets that adjust criticality mainly in H1(n,n) is also significant.

Conclusion
In this study, the significance of various cross-section sets that accurately evaluate nuclear characteristics was assessed. The random sampling method was used to obtain various cross-section sets. The random crosssection sets, which accurately evaluate nuclear characteristics, were compared to the cross-section set obtained by the cross-section adjustment method. The comparison indicated that there are other significant candidates that are consistent with the covariance data. By using the random sampling method, significant candidates can be obtained without changing the benchmark problems used for the cross-section adjustment method. Therefore, it may be useful when the result of the cross-section adjustment method is unreliable.