Expanded COG criticality validation suite for inter-laboratory benchmark data comparison

. The COG suite of criticality benchmarks has been formally expanded from 591 to 3,395 to cover the entire energy range from thermal to fast neutron spectra under a variety of reflector and moderator conditions and fissile materials. COG results have been compared with benchmark values from the International Criticality Safety Benchmark Evaluation Project Handbook for ENDF/B-VII.1, ENDF/B-VIII.0 and JEFF-3.3. COG results have been also compared with a MERCURY validation suite. Most of the results agreed with the benchmark values within ±3σ. Among the three cross section data, cases with ENDF/B-VIII.0 performed best with about 85% of the total cases within ±3σ range. A major inter-comparison project between COG, MCNP, MORET, and SCALE for ENDF/B-VIII.0 and JEFF-3.3 is in progress.


Introduction
COG [1] is a general purpose, multi-particle, highfidelity Monte Carlo code developed by LLNL. Since 2017, LLNL has focused on expanding COG benchmark cases as part of a collaborative effort of the benchmark inter-laboratory comparison study between the US Depart of Energy (DOE) Nuclear Criticality Safety Program (NCSP) and the French Institut de Radioprotection et de Sûreté Nucléaire (IRSN). The benchmark cases fully cover the entire range from thermal to fast neutron spectra for a wide variety of fissionable material forms in a variety of reflector and moderator conditions described in the International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook [2].
The original number of LLNL 591 benchmark cases (143 PU, 358 U-235, and 90 U-233) was expanded to 3,395 PU, HEU (Highly Enriched Uranium), IEU (Intermediate Enriched Uranium), LEU (Low Enriched Uranium), U-233, Mixed fuel, and SMF (Special Metal Fast) cases, providing valuable data for the interlaboratory benchmark data comparison. The number of benchmark cases in each of these six major categories is summarized in Table 1.
The cross section data libraries used are ENDF/B-VII.1, ENDF/B-VIII.0, and JEFF-3.3. Calculations for ENDF/B-VIII.0 were performed with: (a) continuousenergy cross sections based on ENDF/B-VIII.0 nuclear data as processed by the International Atomic Energy Agency (IAEA); (b) probability tables for the unresolved resonance region as processed by Brookhaven National Laboratory using NJOY within the ADVANCE system [3]; and (c) thermal scattering laws using algorithms developed by LLNL [4]. The most recent version, COG11.3, was used for all the benchmark cases.   To compare performance of each set with the benchmark data, the root mean square errors (RMSE) are also calculated. The RMSE is defined as, where N is the total number of cases, Kc,i and Kb,i are the calculated and benchmark k eff values, respectively. This represents a sample standard deviation of the differences between calculated and the benchmark values.
The RMSE for the six categories are compared in Fig. 1. ENDF/B-VIII.0 data performed better than others for PU, HEU, LEU, and Mixed categories.
The χ 2 (chi-squared) value is the indicator in determining the degree of difference between the calculated and the benchmark values. Here, χ 2 is defined as: where K c,i and K b,i are the calculated and benchmark k eff values, respectively. ν is the degree of freedom (DoF), and 2 is the variance. Fig. 2 shows the cumulative χ 2 values of the three different cross section data for six categories of the expanded COG criticality validation suite.

Comparison with MERCURY validation suite
To validate the newly expanded benchmark data set, COG input decks are translated into the MERCURY [5] input decks. From this effort, errors from the benchmark models, if any, cannot be identified. However, results from different (MERCURY) cross section data processing can be compared. The selected 3,350 COG benchmark cases matching the corresponding MERCURY validation suite based on ENDF/B-VIII.0 are compared.
For Mercury, calculations using ENDF/B-VIII.0 data were performed with: (a) continuous energy cross sections based on ENDF/B-VIII.0 nuclear data as processed by LLNL's FUDGE into the GNDS format; (b) thermal neutron scatter law data as processed with FUDGE; and (c) probability tables for the unresolved resonance region were not been included. The most recent version, Mercury 5.25.0-137 along with GIDI 3.19.65, was used for all benchmark cases. The COG to Mercury translation was done with software still in testing associated with Mercury 5.25 series. Table 5 summarizes categorized benchmark comparison cases.