Application of the extended bias factor method for highly reliable benchmark suite

. It was tried to get useful information feedback on cross section and covariance by application of extended bias factor method with experimental benchmarks' C/E, sensitivity coefficient and covariance data. As a result, it was succeeded to distinguish “the case unrecommended by specialists” as outlier case. And it was succeeded to get the hints that there is room for improvement in cross section or covariance of some nuclides by investigating outlier cases judged as recommended case by specialists.


Introduction
Differences in the keff values between continuous energy Monte Carlo calculations (C) and critical experiments (E) potentially contain hidden information for better making adjustment of cross sections and their covariance data. In this study, a possible approach, which is formed by mediating the bias factor corrected E (for short E') and combining the specialist judgement with this, is explained.

Used benchmarks and covariance data 2.1 Overview of used benchmarks
238 cases (UO 2 :173, MOX:65) in ICSBEP [1] /IRPhEP [2] that were investigated in the JAEA report [3], edited by Reactor Integral Test Working Group (WG), JENDL Committee, were used. In the report, benchmarks similar to LWR were selected from ICSBEP/IRPhEP, and those keff were calculated by MVP [4] (continuous energy Monte Carlo code developed by JAEA). On the basis of the calculation results, recommended case and unrecommended case for JENDL development were judged by specialists in the WG. Table 1 is the list of the benchmarks.
In this study, 185 cases (UO 2 :131, MOX:54) whose differences of keff between MVP and MCNP are less than about 0.1%dk were used. It is expected that the * Corresponding author: tokasiki@nfi.co.jp ** Present affiliation: Toshiba Energy Systems & Solutions Corporation input files of 185 cases do not contain significant input error, because keff of MVP and MCNP were consistent though each of input file was made independently. Therefore, the input files of these cases are expected to be reliable. These cases are shown in Table 1 with " ".
In the following, JENDL-4.0u were used consistently for cross section and covariance data.

Used multi-group covariance data, sensitivity coefficients and tools
Multi-group covariance data, sensitivity coefficients and tools were made by aforementioned WG, as an activity of the WG.
Sensitivity coefficients of 212 cases (UO 2 :147, MOX:65) are available that are part of benchmarks investigated in the JAEA report [3]. Additionally, sensitivity coefficients of undisclosed 28 experiments (UO 2 :23, MOX:5), that were submitted by the WG members (These 28 cases were not used in this study), are available. Those file formats are Sensitivity Data File (SDF) of SCALE, and available group structures are aforementioned.
The tools, file format changer (NJOY2016 output to COVERX and MCNP output to SDF) and analysis tool of extended bias factor (PE method type. In the following, "extended bias factor" means "PE method type") [12], were available and used in this study.
The above mentioned database and tools will be opened as JAEA report in FY 2022. Table 2 shows available covariance data in JENDL-4.0u for used benchmarks in this study.    In Fig. 1, useful information feedback to cross section is not clear due to large uncertainty from C and E. Therefore, extended bias factor method was applied to reduce uncertainty of cross section.

Extended bias factor method
Bias factor method [12] evaluates prediction value as (calculated value of system k) corrected by multiplying with , where is bias factor ⁄ of experiment i. As a premise, each sensitivity coefficient of k and i is required to be similar.
In this evaluation process, uncertainty of cross section can be reduced as following equations [12].
where Variance of ⁄ can be written as following [12].
⁄ ∆ ∆ ∆ (2) where is covariance data of cross section. If each sensitivity coefficient of k and i is similar, uncertainty of cross section becomes small.
In the extended bias factor method, uncertainty of cross section can be reduced further by using various experiments [12].
where w i are obtained by solving the following simultaneous linear equation determined to minimize variance of ⁄ .
where "cov" means covariance of each term. Variance of ⁄ can be written as following for extended bias factor method [12].  (5), and covariance between ∆ and ∆ was not considered for convinience). † In this study, the errors were assumed zero except for statistical uncertainty of Monte Carlo calculation, because MVP and MCNP calculation models are sufficiently precise.

Outlier cause of LCT-007
Benchmark LCT-007 (LEU-COMP-THERM-007) in ICSBEP is the experiment of water-reflected 4.738wt.%-enriched uranium dioxide fuel-rod arrays [1]. Table 3 and Fig. 4 show array geometry of each case in the experiment. Figure 5(b) shows E'/E-1 of keff with X-axis as pin pitch. Figure 5(b) implies dependency on pin pitch (= dependency on H/U). It was considered that the dependency implies that there is room for improvement in H-1 and S(α,β) of H 2 O. Incidentally, Fig.5(a), that shows C/E -1 before reduction of cross section's uncertainty, does not show clear dependency on pin pitch.   As a trial, E' of cases 1 and 4 in Fig. 6 were estimated when cross section of H-1 and S(α,β) of H 2 O were replaced to ENDF/B-VII.1 from JENDL-4.0u (The reactivities were the value evaluated in the JAEA report [3]. See gray color plots in Fig.6). Figure 6 shows that dependency on pin pitch is expected to be weaken after replacing cross section of H-1 and S(α,β) of H 2 O to ENDF/B-VII.1 (After the replacing cross sections, absolute value of E'/E-1 of case 1 whose pin pitch is 1.26cm is expected to decrease rather than case 4 whose pin pitch is 2.52cm). This result implies that there is room for improvement in H-1 and S(α,β) of H 2 O in JENDL-4.0u. *1: The data are referred from JAEA report [3].

Outlier cause of LCT-010 cases 1 -4
Benchmark LCT-010 (LEU-COMP-THERM-010) in ICSBEP is the experiment of water-moderated U(4.31%U-235)O 2 fuel rods reflected by lead, uranium, or steel walls [1]. The reflector of cases 1 -4 in the experiment were lead. Table 4 and Fig.7 show array geometry of each case in the experiment. LCT-010 cases 1 -3 are judged as outliers, and it is considered that the influence of the reflecting wall of lead is large in those cases. Therefore, Fig.8 shows elastic scattering cross sections (n,el) in Pb-206, 207 and 208. Figure 8 shows that there are about 7% relative difference in Pb-207 and about 4% in Pb-206 between ENDF/B-VIII.0 and JENDL-4.0u. On the other hand, covariance data of JENDL-4.0u for lead's nuclides in low energy are about 4E-4 that means about 2% = √(4E-4) as standard deviation. Aforementioned relative differences 7% and 4% are ~3 times of 2% in the covariance data. As a trial, E'/E-1 were evaluated again with lead nuclides' covariance data (for elastic scattering cross sections) multiplied by 9 over the entire energy range, where 9 means 3 times as standard deviation aforementioned. That is, the original standard deviation 2% in covariance data was changed to 6% to cover relative differences between ENDF/B-VIII.0 and JENDL-4.0u in Fig.8. Figure 9 shows the E'/E-1.
Dependency of E'/E-1 on distance from lead reflector was weaken after lead's covariance data were multiplied by 9. This result implies that there is room for improvement in lead's cross section or covariance in JENDL-4.0u.

Conclusion
It was tried to get useful information feedback on cross section and covariance by application of extended bias factor method with experimental benchmarks' C/E, sensitivity coefficient and covariance data. As a result, it was succeeded to distinguish "the case unrecommended by specialists" as outlier case. And it was succeeded to get the hints that there is room for improvement in "the cross section of H-1, S(α,β) of H 2 O, elastic scattering cross section or covariance of lead's nuclides" by investigating outlier cases judged as recommended case by specialists.