Applying the continuous-energy Monte Carlo calculation code, MVP3, to analysis of kinetic parameters measured for light-water moderated UO₂ and MOX cores of the TCA and EOLE critical facilities

ABSTRACT

In order to evaluate the accuracies of the kinetics parameters predicted with the continuous-energy Monte Carlo calculation code, MVP3, core analysis was performed for light-water moderated UO₂ and mixed oxide (MOX) fuel cores for which the ratios of effective delayed-neutron fraction (β_eff) to prompt-neutron life time (ℓ) – represented hereafter by β_eff/ℓ – and β_eff values were measured. The results obtained with the JENDL-4.0-based neutron library showed that (the calculation values (C)/the measurement values (E)−1) ranged from −1.0% to 4.0% for the β_eff/ℓ values with an average of 1.0% and a standard deviation of 1.1% for the 17 cores (the UO₂, and UO₂-MOX mixed cores) tested in the Tank-Type Critical Assembly (TCA). With respect to the β_eff of one UO₂ core in TCA, C/E − 1 was 1.2%. For the β_eff values of the UO₂ and MOX cores tested in the EOLE critical facility, C/E − 1 was −1.5% for the former and −4.6% for the latter. The calculated β_eff value of the MOX core using the JEFF-3.2-based neutron library was larger by 4% than that calculated using the JENDL-4.0-based neutron library and showed better agreement to the measurement.

KEYWORDS:

• TCA
• eole
• UO₂ core
• MOX core
• effective delayed-neutron fraction
• prompt-neutron life time
• MVP3
• JENDL-4.0
• ENDF/B-VII.1
• JEFF3.2

1. Introduction

Effective delayed-neutron fractions (β_eff) and prompt-neutron life times (ℓ) are basic kinetic parameters in controlling the nuclear behavior of reactor cores when they depart from the static state by insertion of reactivity. Accordingly, these kinetic parameters are involved in the safety analysis of nuclear power plants. While they can be directly measured, they are usually predicted with nuclear analysis codes. Therefore, it is important to evaluate the accuracies of the predicted results.

While deterministic analysis codes have traditionally been used for predicting kinetic parameters, stochastic analysis methods have also been developed since they are expected to provide more accurate results. Nagaya et al. [1,2] developed the eigenvalue method using the differential-operator sampling technique associated with the Monte Carlo perturbation techniques, and installed it in the continuous-energy Monte Carlo calculation code, MVP3 [3]. Using the methods coupled with the neutron library based on JENDL-4.0 [4], Nagaya [2] validated both the methods and the library by analyzing the measured kinetic parameters for the experimental cores with various neutron-energy spectra. While these experimental cores widely covered bare spheres, spherical cores with a fertile reflector, heterogeneous fast cores, light-water moderated solution cores and light-water moderated lattice cores, the number of the light-water moderated lattice cores were limited to two cores.

The aim of this study is to evaluate the accuracy of kinetic parameters calculated with MVP3 by analyzing the measured kinetic parameters for light-water moderated lattice cores. The measured parameters included β_eff/ℓ [5] and β_eff [6,7], which were obtained for the critical cores tested in the Tank-Type Critical Assembly (TCA) of the Japan Atomic Energy Agency and the EOLE critical facility (EOLE) of the French Atomic and Alternative Energies Commission. As a neutron library of MVP3, the library based on JENDL-4.0 [4] was mainly used; the other libraries based on JENDL-3.3 [8], ENDF/B-VII.1 [9] and JEFF-3.2 [10] were also examined.

This study adopted the calculation methods which are recommended by Nagaya et al. [1,2] to provide most precise results of the kinetic parameters for comparison with the measured results. In addition to these results, the calculation results with some other methods [3] were also examined to discuss the accuracy of the methods installed in MVP3. Furthermore, the calculated effective neutron multiplication factors of the cores were also summarized in order to provide the comprehensive information relating to the reactivity calculations of MVP3.

Section 2 highlights the characteristics of the experimental cores which the kinetic parameters were measured for, Section 3 explains the analysis models and calculation conditions of MVP3, Section 4 presents the analysis results and discussions, and Section 5 mentions the conclusions. Appendix 1 is also provided to present the analysis results with the other methods in MVP3 for the kinetic parameters, and Appendix 2 shows the analysis results of the effective neutron multiplication factors.

2. Major characteristics of experimental cores for measuring kinetic parameters

Table 1 summarizes the major characteristics of the experimental cores in TCA and EOLE for the measurements of the kinetic parameters. In Table 1, ‘Core ID’ represents the identifications of the cores in this study.

Table 1. Summary of the major characteristics of the experimental cores for the measurements of the kinetic parameters [5–7].

Critical facility		TCA																					EOLE
Core ID		TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7		TCA8	TCA9	TCA10	TCA11	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17		TCA18		MISTRAL1	MISTRAL2
Core configuration		One-region core								Two-region core											One-region core
		Core region
M/F ratio		1.50	1.83	1.83	1.83	1.83	1.83	2.48		1.76	2.00	2.38	2.95	1.76	2.00	2.38	2.95	2.40	2.96		1.83		1.75	1.76
Fuel pitch (cm)		1.85	1.96	1.96	1.96	1.96	1.96	2.15		1.66	1.73	1.82	1.96	1.66	1.73	1.82	1.96	1.48	1.58		1.96		1.32	1.32
Enrichment of UO2 fuel (wt%)		2.60	2.60	2.60	2.60	2.60	2.60	2.60		2.59	2.59	2.59	2.59								2.60		3.7
Number of UO2 fuel rods		361	309	342	361	484	625	225		49	49	49	49								292		744
Pu content of MOX fuel (wt%Pu)														3.4	3.4	3.4	3.4	4.9	4.9					7.0/8.7
Number of MOX fuel rods														49	49	49	49	100	100					1572/88
Water gap between core and driver regions (cm)										0.06	1.79	1.45	0.00	0.06	1.79	1.45	0.00	0.03	0.42
		Driver region
M/F ratio										1.83	1.83	1.83	1.83	1.83	1.83	1.83	1.83	1.50	1.50
Fuel pitch (cm)										1.96	1.96	1.96	1.96	1.96	1.96	1.96	1.96	1.85	1.85
Enrichment of UO2 fuel (wt%)										2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60	2.60
Number of UO2 fuel rods										296	284	276	272	288	260	236	244	224	164
Core total fuel rods		361	309	342	361	484	625	225		345	333	325	321	337	309	285	293	324	264		292		744	1660
Fraction of MOX fuel rods														0.15	0.16	0.17	0.17	0.31	0.38					1.00
Water temperature (℃)		22.7	23.0	22.9	23.9	23.0	24.8	20^a		20.5	20.8	20.9	13.7	18.9	20.6	16.2	20.4	21.0	21.0		20^a		20.0	20.0
Boron content in water (ppm)				72.34	146.6	345	554																～ 300
Critical water height (cm)		99.20	81.28	85.78	96.85	84.99	85.79	148.18		69.28	68.80	68.32	67.84	68.72	68.15	69.28	69.26	83.66	87.30		92.24		80.00	80.00
Measured kinetic parameter		βeff/ℓ																			βeff

2.1. TCA cores

Tsuruta and Kitamoto [5] performed the measurements of the β_eff/ℓ values for the 17 light-water moderated cores in TCA with the pulsed-neutron technique. The cores were composed of UO₂ fuel rods with enrichments of 2.6 wt% and 2.59 wt%, and MOX fuel rods with Pu contents of 3.4 wt% and 4.9 wt%. The major specifications of the fuel rods are listed in Table 2. It is evident that the isotopic compositions of ²³⁹Pu of the MOX fuel rods are roughly 90 wt%; moreover, the density of the pellets for the 3.4 MOX fuel rods is lower than those of the other pellets, since green pellets without sintering were used for the 3.4 MOX fuel rods.

Table 2. Major specifications of the UO₂ and MOX fuel rods loaded in the TCA cores [5].

Fuel rod type	2.60 UO2	2.59 UO2	3.4 MOX	4.9 MOX
Pellet diameter (cm)	1.25	1.07	1.07	0.86
Pellet density (g/cm^3)	10.4	10.4	6.8	10.1
Enrichment (wt%)	2.60	2.59	0.71(Nat. U)	0.71(Nat. U)
Pu content (wt%)			3.4	4.9
Pu isotope ratio (Pu-238/239/240/241/242) (wt%)			0.01/90.64/8.46/0.84/0.05	0.05/90.24/8.48/1.15/0.09
Cladding inner dia. (cm)	1.27	1.08	1.08	0.87
Cladding outer dia. (cm)	1.42	1.22	1.22	1.00
Cladding material	Al	Al	Zry-2	Zry-2

The core configurations of TCA1 through TCA7 are the one-region cores of the UO₂ fuel rods. Those of TCA8 through TCA17 are the two-region cores composed of a central core region of either the UO₂ or MOX fuel rods and a driver region of the UO₂ fuel rods. The one-region cores and the central and driver regions of the two-region cores were loaded with fuel rods in a homogeneous square lattice. The moderation (M/F) ratio of the fuel cells, which is defined as a moderator-to-fuel-pellet volume ratio, ranged from 1.50 to 2.96, which is consistent with typical M/F ratios of the fuel cells and fuel assemblies at the cold and hot operating conditions of commercial light-water reactor cores.

The series of the cores, TCA1, TCA2 and TCA7, were used to investigate the influence of the M/F ratios of the UO₂ cores, and that fromTCA2 through TCA5 to study the influence of the boron concentrations in the moderator. The cores from TCA8 through TCA11 were the references for the cores from TCA12 through TCA15, which had the central MOX core regions (7 × 7 lattice) with the M/F ratios ranging from 1.66 to 1.96. The driver regions of the cores from TCA8 through TCA15 were composed of the 2.60 UO₂ fuel rods with a M/F ratio of 1.82. TCA 16 and TCA 17 were used to examine the influence of the extended central MOX core regions (10 × 10 lattice) composed of the 4.9 MOX fuel rods. The core radial configurations of TCA1 through TCA17 [5] are schematically shown in Figure S1 in the online supplemental material accompanying this manuscript.

The relative uncertainties (1σ) in the measured results ranged from 0.19% to 2.62% depending on the cores. They were obtained as the errors in the least-square fitting of the measured prompt neutron decay constants for the subcritical cores in order to determine that corresponding to the critical core [5]. The measured results were compared with the calculated results, and the results of the comparison are shown in the later section.

The measured result of β_eff for one UO₂ core tested in TCA was originally obtained by Haga and Kobayashi [11] using the substitution technique; it was later reevaluated by Nakajima [6]. The major characteristics of the core is shown as TCA18 in Table 1. The specifications of TCA18 are similar to those of TCA2; however, the numbers of fuel rods, the critical water heights and the radial core shapes are slightly different. The core radial configuration of TCA18 is shown in Figure S2 in the online supplemental material accompanying this manuscript. The measured β_eff is 7.71 × 10⁻³, and the relative uncertainty (1σ) is 2.2% [6].

2.2 EOLE cores

The measured results of β_eff for one UO₂ core (MISTRAL1) and one MOX core (MISTRAL2) tested in EOLE were analyzed in this study. The measurements were performed with the noise-analysis technique [7] as a part of the MISTRAL experimental program [12]. MISTRAL1 was composed of 3.7 wt% UO₂ fuel rods, and MISTRAL2 was composed of MOX fuel rods with Pu contents of 7.0 wt% and 8.7 wt%. The fuel rods were PWR-type test fuel rods with an effective fuel length of 80 cm. They were covered by over-claddings of aluminum alloy (AG3), which were used to adjust the M/F ratio of the cores to about 1.8. The enrichment of the UO₂ fuel rods and the Pu contents of the MOX fuel rods are relatively higher than those of the TCA cores; moreover, they are closer to those used in current commercial light-water reactors. The core radial configurations of MISTRAL1 and MISTRAL2 are shown in Figure S2 in the online supplemental material accompanying this manuscript. The relative uncertainties (1σ) in the measured results were 1.5% for MISTRAL1 and 1.6% for MISTRAL2. They were obtained considering the uncertainties in the Diven factor, the fission integral of the core, the spectral density of interaction power, and the average voltages of fission chamber amplifiers [7]. The measured results were compared with the calculated results and the comparison results are shown in the later section.

3. Analysis models and calculation conditions of MVP3

Based on the specifications of the fuel rods and the structure materials of TCA [5,13–15], the three-dimensional geometrical models of the TCA cores were composed of the fuel rods, water, core grids, structure materials and a core tank (an outer diameter of 184 cm and a height of 208 cm). The boundaries of the outside of the core tank were assumed to be in a vacuum condition. The modes of the vertical cross sections of the TCA cores are schematically shown in Figure S3 in the online supplemental material accompanying this manuscript.

The three-dimensional geometrical models of MISTRAL1 and MISTRAL2 were composed according to the procedure in Reference [16], where the core radial geometrical model included the core and water reflector inside the core tank (an inner diameter of 94 cm). The core axial geometrical model consists of six regions. For the fuel rods, in addition to the effective fuel part, the model included a fuel spacer and a spring in the top part of the fuel rod, as well as a fuel spacer in the bottom part of the fuel rod. Water reflectors (a length of 5 cm) were placed at the outsides of the fuel rods. The two intermediate grids, which were located outside of the effective fuel section in the axial direction, and the top and end grids were not included in the model. The modes of the MISTRAL cores [16] are schematically shown in Figure S4 in the online supplemental material accompanying this manuscript. Some studies [17,18] demonstrated that the differences in neutron multiplication factors (k_eff) between the models with and without the grids were comparable to the statistical uncertainty in the calculations, 0.02%Δk_eff, and negligibly small.

The calculations with MVP3 for the TCA and MISTRAL cores were performed using the neutron library based on JENDL-4.0 [4] for 100 inactive and 5,000 active batches with 40,000 histories per batch. Accordingly, the calculations provided the results of β_eff, ℓ and β_eff /ℓ calculated with the eigenvalue method using the differential-operator-sampling technique associated with the Monte Carlo perturbation technique [1,2]. The adopted values in this study were those obtained by taking into account the indirect-perturbed-source effect [1]. Under the neutron history conditions, the statistical uncertainties (1σ) of β_eff /ℓ and β_eff ranged from 0.3% to 0.4%, which are comparable to the measurement uncertainties or lower than them, while those of ℓ ranged from 0.03% to 0.07%, which were considerably smaller than those of β_eff/ℓ and β_eff . The statistical uncertainties of k_eff ranged from 0.005%Δk_eff to 0.006%Δk_eff. The neutron libraries based on JENDL-3.3 [8], ENDF/B-VII.1 [9], and JEFF-3.2 [10] were also partly used for examining the sensitivity of β_eff /ℓ and β_eff to nuclear-data libraries.

Since the MVP3 calculations provide the results of kinetic parameters with some other methods [3] in addition to the above-mentioned methods recommended by Nagaya [1,2] for the most accurate calculation, the calculation results with the other methods were also examined and the results are presented in Appendix 1.

4. Analysis results and discussions

4.1. Calculated results of β_eff and ℓ

The results of β_eff and ℓ in the MVP3 calculations coupled with the JENDL-4.0-based neutron library are shown in Figure 1 for all the cores.

Figure 1. Calculated results of β_eff and ℓ for the TCA and EOLE cores.

The following paragraphs involve a brief discussion on the trends in the kinetic parameters relating to the core characteristics.

Since the delayed neutron fractions in the total numbers of fission neutrons (β) for major nuclides contributing to fissions in nuclear fuel in light-water reactor cores are considerably different. The β values are 6.507 × 10⁻³ (at 0.0253 eV) for the thermal fission of ²³⁵U, 2.161 × 10⁻³ (at 0.0253 eV) for the thermal fission of ²³⁹Pu and 1.762 × 10⁻² (at 2.0 MeV) for the fast fission of ²³⁸U, according to JENDL-4.0 [4]. Therefore, the values of β_eff mainly depend on the fuel compositions within the cores. While the fractional contributions of ²³⁸U to fissions are roughly 5% for the TCA cores, and while their differences between the cores are relatively small, the fractional contributions of ²³⁹Pu vary according to the fractions of the MOX fuel rods in the cores. Moreover, because the β value of ²³⁹Pu is smaller than that of ²³⁵U, the β_eff values of the cores from TCA12 through to TCA17 are smaller than those of the other UO₂ cores. Since MISTRAL2 was fully loaded with MOX fuel rods, the β_eff value is the smallest among all the cores.

The ratio of the leakage of prompt neutrons from the cores to that of delayed neutrons also has effect on β_eff, and therefore, the core volumes are also major factors to affect the β_eff values. The systematic trend of slightly decreasing β_eff values from TCA2 through to TCA6 is the result of the core volume increasing in order to compensate for the negative reactivity caused by the boron-concentration increase in the moderator [5].

The ℓ values depend on macroscopic absorption cross sections of the cores which are sensitive to fuel and moderator conditions Comparison of the calculated ℓ values of TCA1, TCA2 and TCA7 indicates that the values increase with the M/F ratios. The calculated ℓ values of the cores from TCA2 through TCA6 decrease with the content of boron in the moderator. The differences in the core configurations of TCA8 through TCA11 are related to the M/F ratios in the central core regions. In addition, there are the effect of the water-gap widths between the central core regions and the driver regions as indicated by Tsuruta and Kitamoto [5]. The gaps varied: 0 cm for TCA8, 1.787 cm for TCA9, 1.454 cm for TCA10 and 0 cm for TCA11. Since the central core regions consisted of the small 7 × 7 lattice, the effect of changing the water-gap width was relatively large with respect to the ℓ values. The ℓ values of TCA12 through TCA15, which were partially loaded with MOX fuel rods, are smaller than those of the cores from TCA8 through TCA11. This is because the MOX fuel rods have relatively large thermal-neutron-absorption cross sections. The effect of the changes in the water-gap widths was similar to that of TCA8 through TCA11. For TCA16 and TCA17, the central core regions of the MOX fuel rods (the 10 × 10 lattice) are larger than those of TCA12 through TCA15. Moreover, the water-gap width of TCA17 is larger than that of TCA16; accordingly, the ℓ values for those cores are smaller than those of the cores TCA12 through TCA15; moreover, the ℓ value of TCA17 is slightly larger than that of TCA16. The ℓ value of MISTRAL1 is smaller than that of TCA18 since the enrichment (3.7 wt%) of the UO₂ fuel for MISTRAL1 is larger than that of TCA18 (2.6 wt%). Since the absorption cross sections of the MOX fuel are larger than those of the UO₂ fuel, the ℓ value of MISTRAL2 is smaller than that of MISTRAL1.

4.1.1. Comparison of calculated and measured β_ef /ℓ values

The calculated results of β_eff/ℓ with respect to the TCA cores were compared with the measured results. The comparison results are shown in Table 3 and Figure 2, represented by (the calculation values (C)/the measurement values (E)−1)%. Accordingly, the uncertainties (1σ) in C/E-1 are composed of the associated uncertainties in the calculated and measured results. The error bars in Figure 2 with the MVP3 calculation results show the range of uncertainties in C/E-1. While the uncertainties in the calculations range from 0.30% to 0.36% and their deviations are relatively small, those in the measured results ranged from 0.19% to 2.62% and their deviations are relatively large. The calculated results of this study generally agreed to the measurements even though the characteristics of the cores vary in terms of M/F ratios, boron concentrations, and fractions of MOX fuel rods in the cores. They might have slightly overestimated the measurements. After all, the average value of C/E-1 was 1.0% with a standard deviation of 1.1%. The largest overestimation was seen for TCA6 which is the core with the largest boron content (554 ppm) in the moderator and the largest underestimation was seen for TCA7 which is the UO₂ core with the largest M/F ratio. In the scope of this study, correlations of C/E-1 with the boron concentrations and the M/F ratios were not confirmed.

Table 3. Comparison of the calculated and measured β_eff/ℓ values in C/E-1% for the TCA cores.

Core	TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7	TCA8	TCA9	TCA10	TCA11	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17
This study (MVP3+ JENDL-4.0)	1.38	1.72	1.37	1.53	1.51	4.04	−1.08	0.98	0.35	0.71	0.49	1.46	−0.93	0.76	1.16	1.54	0.30
(Uncertainty (%) in C/E − 1 for this study ^a)	(0.68)	(1.70)	(2.68)	(0.47)	(0.80)	(0.72)	(1.95)	(0.36)	(1.00)	(0.33)	(1.18)	(1.56)	(0.66)	(0.58)	(0.95)	(0.83)	(0.95)
Tsuruta & Kitamoto [5]	−0.30	0.00	−0.70	−0.50	0.00	3.40	−3.90									−2.10	−3.90

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

Figure 2. Comparison of the calculated and measured β_eff/ℓ values in C/E-1% for the TCA cores. This figure shows the results of this study (MVP3 + JENDL-4.0) and those of Tsuruta and Kitamoto [5]. The error bars for this study show the range of the uncertainties (1σ) in C/E-1 composed of those in the calculated and measured results.

The calculated results of Tsuruta and Kitamoto [5], which were reported for some of the cores, are also shown in Table 3 and Figure 2. Their calculation results were obtained using deterministic calculations in two-dimensional geometry and four neutron-energy groups. The used codes were UGMG42 for the fast-energy group constants, THERMOS for the thermal-energy group constants, KAK for the diffusion calculations and 2DPERT for the β_eff and ℓ calculations. The neutron leakage in the axial direction was taken into account by using the axial buckling. Their calculation results were slightly lower than those obtained in this study; however, the general trend of C/E-1 is similar to this study. Their largest discrepancies were also observed for TCA6 and TCA7.

4.1.2. Comparison of calculated and measured β_eff values

The comparison of the calculated results obtained with the MVP3 calculations in this study and the measured results is shown in Table 4. As before, the uncertainties (1σ) in C/E-1 are composed of the uncertainties from the calculated and measured results. The calculation of β_eff for TCA18 has previously been performed by Nakajima [6] with the deterministic code TWODANT in the 107 neutron-energy groups in combination with the neutron library based on JENDL-3.2 [19], as well as Nagaya [2] with MVP-2 [20] modified to include the eigenvalue method using the differential-operator-sampling technique. The calculated results of Nakajima [6] overestimated the measurement by 2.9%, thereby exceeding the measurement uncertainty of 2.2%. Nagaya [2] reported 7.80 × 10⁻³ for β_eff, which agreed to 7.803 × 10⁻³ of this study as expected. The deviations of the results from the measurement were within the uncertainty in C/E-1.

Table 4. Comparison of the calculated and measured β_eff values in C/E-1%.

Core	TCA18	MISTRAL1	MISTRAL2
This study (MVP3+ JENDL-4.0)	1.21	−1.45	−4.61
(Uncertainty (%) in C/E-1 for this study ^a)	(2.25)	(1.51)	(1.59)
Nakajima [6]	2.9
Nagaya [2]	1.2
Litaize et al. [7]		2.4	0.1
Hibi et al. [12]		−0.8	−2.5
NUPEC [17]		−2	−3

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

The calculated result of MISTRAL1 in this study showed a slight underestimation; however, the deviation from the measurement was within the uncertainty in C/E-1. The value of C/E-1 for MISTRAL2 was −4.61%; the deviation of the calculated result exceeded the uncertainty in C/E-1. Table 4 also shows the C/E-1 values for the MISTRAL cores in the references [7,12,17]. Litaize et al. [7] analyzed MISTRAL1 and MISTRAL2 with the deterministic code, APOLLO2.5, in combination with the CEA93.V6 library based on the nuclear-data library, JEF2.2. Hibi et al. [12] analyzed the same core with the deterministic codes, SRAC-CITATION [21], in combination with the neutron library based on JENDL-3.2 [19] using a three-dimensional geometrical model and the 21 neutron-energy groups. The results of the Nuclear Power Engineering Corporation (NUPEC) [17] were obtained with the same methods as Hibi et al. [12], with JENDL-3.2 replaced by JENDL-3.3 [8]. While the results of Hibi et al. [12] and NUPEC [17] are similar to those of this study with respect to MISTRAL1 and MISTRAL2, those of Litaize et al. [7] are slightly larger for MISTRAL1; however, the results of Litaize et al. shows better agreement with the measurement of MISTRAL2 than the any other analysis.

4.1.3 Sensitivity of kinetics parameters to nuclear-data libraries

The differences in C/E-1 for the measured kinetic parameters are the result of the uncertainties in the analysis codes and the nuclear-data libraries on which the neutron libraries of the codes are based. Since the calculations of the kinetic parameters with MVP3 adopt the detailed-physics models in order to exclude uncertainties in the physics models of the deterministic calculation codes, they are expected to provide information on the uncertainties in the nuclear-data libraries. Therefore, the results of C/E − 1 of this study can be attributed to biases in the nuclear data, such as the β values and microscopic-absorption cross sections in JENDL-4.0 for major contributing nuclides. For the future updating of the nuclear data in JENDL-4.0, it would be useful to provide the analysis results of β_eff and ℓ for the experimental cores obtained with the neutron libraries based on JENDL-3.3 [8], ENDF/B-VII.1 [9], and JEFF-3.2 [10]. The calculations were performed for TCA1, TCA6 and TCA17 for which the measured β_eff/ℓ values were available, and for TCA18, MISTRAL1 and MISTRAL2 for which the measured β_eff values were available. The calculation models and conditions were the same as those mentioned in Section 3.

Table 5 shows the calculated results of β_eff, ℓ and β_eff/ℓ of the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries and the comparison results to the values of the JENDL-4.0-based neutron library, which are shown as (β_eff /JENDL-4.0 − 1), (ℓ/JENDL-4.0 − 1) and (β_eff/ℓ/JENDL-4.0 − 1): (the calculated result with the other neutron library/that with the JENDL-4.0-based neutron library−1)%.

Table 5. Calculated results of β_eff, ℓ and β_eff /ℓ with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries and the ratios to those obtained with the JENDL-4.0-based neutron library.

Core	Library	βeff	(βeff/JENDL-4.0–1)%	ℓ (s)	(ℓ/JENDL-4.0–1)%	βeff/ℓ (1/s)	(βeff/ℓ/JENDL-4.0–1)%
TCA1	JENDL-4.0	7.749E-03		3.494E-05		2.218E+02
	JENDL-3.3	7.788E-03	0.50	3.500E-05	0.18	2.225E+02	0.32
	ENDF/B-VII.1	7.702E-03	−0.61	3.497E-05	0.09	2.201E+02	−0.75
	JEFF-3.2	7.968E-03	2.82	3.487E-05	−0.20	2.285E+02	3.02
TCA6	JENDL-4.0	7.524E-03		2.839E-05		2.650E+02
	JENDL-3.3	7.539E-03	0.21	2.841E-05	0.07	2.653E+02	0.14
	ENDF/B-VII.1	7.484E-03	−0.54	2.836E-05	−0.12	2.639E+02	−0.41
	JEFF-3.2	7.736E-03	2.82	2.837E-05	−0.08	2.727E+02	2.90
TCA17	JENDL-4.0	5.378E-03		3.041E-05		1.769E+02
	JENDL-3.3	5.387E-03	0.17	3.049E-05	0.28	1.767E+02	−0.11
	ENDF/B-VII.1	5.437E-03	1.09	3.060E-05	0.62	1.777E+02	0.46
	JEFF-3.2	5.573E-03	3.61	3.042E-05	0.05	1.832E+02	3.56
TCA18	JENDL-4.0	7.803E-03		3.787E-05		2.060E+02
	JENDL-3.3	7.757E-03	−0.59	3.797E-05	0.25	2.043E+02	−0.84
	ENDF/B-VII.1	7.721E-03	−1.05	3.792E-05	0.12	2.036E+02	−1.17
	JEFF-3.2	8.003E-03	2.56	3.784E-05	−0.08	2.115E+02	2.65
MISTRAL1	JENDL-4.0	7.768E-03		2.727E-05		2.849E+02
	JENDL-3.3	7.785E-03	0.22	2.734E-05	0.27	2.847E+02	−0.07
	ENDF/B-VII.1	7.681E-03	−1.12	2.730E-05	0.10	2.814E+02	−1.24
	JEFF-3.2	7.934E-03	2.14	2.717E-05	−0.37	2.920E+02	2.50
MISTRAL2	JENDL-4.0	3.596E-03		1.387E-05		2.593E+02
	JENDL-3.3	3.595E-03	−0.04	1.389E-05	0.13	2.588E+02	−0.18
	ENDF/B-VII.1	3.612E-03	0.43	1.397E-05	0.71	2.585E+02	−0.28
	JEFF-3.2	3.722E-03	3.51	1.382E-05	−0.34	2.693E+02	3.86

The differences in the calculated results of β_eff between the JENDL-3.3-based neutron library and the JENDL-4.0-based neutron library are less than 0.6%, and, for the ENDF/B-VII.1-based neutron library, the differences are less than 1.1%, while the calculated results of the JEFF-3.2-based neutron library are larger than those of JENDL-4.0, and the differences range from 2.1 to 3.6%. The differences in the calculated results of ℓ between the JENDL-3.3-based neutron library and the JENDL-4.0-based neutron library are less than 0.3%; for the ENDF/B-VII.1-based neutron library, the differences are less than 0.7%; finally, for the JEFF-3.2-based neutron library, the differences are less than 0.4%. The differences in the calculated results of β_eff/ℓ between the JENDL-3.3-based neutron library and the JENDL-4.0-based neutron library are less than 0.8%, and, for the ENDF/B-VII.1-based neutron library, the differences are less than 1.2%, while the calculated results of the JEFF-3.2-based neutron library are larger than those of JENDL-4.0, and the differences range from 2.5 to 3.9%. The differences in the calculated results of β_eff/ℓ between the nuclear-data libraries are mainly caused by the differences in the β_eff values.

For further study, the β values of ²³⁵U, ²³⁹Pu and ²³⁸U for the thermal fission or the fast fission in ENDF/B-VII.1 [9] and JEFF-3.2 [10] were examined since the β_eff values obtained with the neutron libraries based on them showed noticeable differences compared with those of JENDL-4.0 [4]. They are shown in Table 6 together with the ratios to those of JENDL-4.0 [4]. As seen in Table 6, the β value of ²³⁵U in ENDF/B-VII.1 is close to that of JENDL-4.0, and that of ²³⁹Pu is 4% larger than that of JENDL-4.0, while that of ²³⁸U in ENDF/B-VII.1 are 5% smaller than that of JENDL-4.0. The smaller β value of ²³⁸U of ENDF/B-VII.1 explains that the β_eff and β_eff/ℓ values of the UO₂ cores (TCA1, TCA6, TCA18 and MISTRAL1) for the ENDF/B-VII.1-based library are slightly smaller than those for the JENDL-4.0-based library. The trend that while the β_eff and β_eff/ℓ values of the MOX cores (TCA17 and MISTRAL2) for the ENDF/B-VII.1-based library are slightly larger than those for the JENDL-4.0 is caused by the combination of the larger β value of ²³⁹Pu and the smaller β value of ²³⁸U.

Table 6. βvalues of ²³⁵U, ²³⁹Pu, and ²³⁸U for JENDL-4.0 [4], ENDF/B-VII.1 [9], JEFF-3.2 [10], and JEFF-3.3 [22].

Nuclide (neutron energy)	Nuclear data library	β	Ratio to JENDL-4.0
^235U (0.0253 eV)	JENDL-4.0	6.507 × 10^−3	-
	ENDF/B-VII.1	6.505 × 10^−3	1.000
	JEFF-3.2	6.650 × 10^−3	1.022
	JEFF-3.3	6.721 × 10^−3	1.033
^239Pu (0.0253 eV)	JENDL-4.0	2.161 × 10^−3	-
	ENDF/B-VII.1	2.240 × 10^−3	1.037
	JEFF-3.2	2.261 × 10^−3	1.046
	JEFF-3.3	2.268 × 10^−3	1.050
^238U (2.0 MeV)	JENDL-4.0	1.762 × 10^−2	-
	ENDF/B-VII.1	1.670 × 10^−2	0.948
	JEFF-3.2	1.858 × 10^−2	1.055
	JEFF-3.3	1.818 × 10^−2	1.030

Table 6 shows that the β values in JEFF-3.2 are larger than those of JENDL-4.0 by 2% for ²³⁵U, 5% for ²³⁹Pu, and 6% for ²³⁸U. This explains the differences in the β_eff and β_eff/ℓ values between the calculations with the JEFF-3.2- and JENDL-4.0-based neutron libraries for the TCA and MISTRAL cores. In Table 6, the β values in JEFF-3.3 [22] are also shown for comparison with those of JEFF-3.2. The β value of ²³⁵U for JEFF-3.3 are 1% larger that of JEFF-3.2, and that of ²³⁹Pu is close to that for JEFF-3.2 while that of ²³⁸U is 2% smaller than that for JEFF-3.2. Accordingly, the JEFF-3.3-based neutron library is expected to provide similar calculation results of the β_eff values to those of JEFF-3.2.

Table 7 shows the values of (C/E − 1) when the JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries are adopted for comparison with the measured values. As for TCA1, the (C/E − 1) value of the ENDF/B-VII.1-based library is smaller than those of the JENDL-4.0- and JENDL-3.3-based libraries. This is attributed to the smaller β value of ²³⁸U for ENDF/B-VII.1. The (C/E − 1) value for the JEFF-3.2-based library is larger than those for the other neutron libraries. This is caused by the larger β values of ²³⁵U, and ²³⁸U for JEFF-3.2. As for the relatively larger (C/E − 1) values of TCA6, further sturdy including on the measured value is necessary. Concerning TCA17 (a partially-MOX-loaded core), the (C/E − 1) value of the ENDF/B-VII.1-based library is slightly larger than those of the JENDL-4.0- and JENDL-3.3-based libraries. This is caused by the larger β value of ²³⁹Pu in ENDF/B-VII.1.

Table 7. Comparison of the calculated and measured β_eff/ℓ and β_eff values in C/E − 1% for the MVP3 calculations with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries.

	βeff/ℓ			βeff
Library/core	TCA1	TCA6	TCA17	TCA18	MISTRAL1	MISTRAL2
JENDL-4.0 (Uncertainty (%) in C/E-1 for this study ^a)	1.38 (0.68)	4.04 (0.72)	0.30 (0.95)	1.21 (2.25)	−1.45 (1.51)	−4.61 (1.59)
JENDL-3.3	1.70	4.18	0.19	0.61	−1.23	−4.65
ENDF/B-VII.1	0.61	3.61	0.76	0.14	−2.55	−4.20
JEFF-3.2	4.44	7.06	3.87	3.80	0.66	−1.26

^aUncertainty (1σ) in C/E-1 was composed of those in the calculated and measured results.

With respect to TCA18, the trend in the (C/E − 1) values are similar to those of TCA1. The (C/E − 1) values for ENDF/B-VII.1, JENDL-4.0, and JEFF-3.1.1 [23] have reported for TCA18 by van der Marck [24], who analyzed the core with the continuous-energy Monte Carlo calculation code, MCNP6 [25]. The (C/E − 1) values for JENDL-4.0, and JEFF-3.2 in this study accord with those for JENDL-4.0, and JEFF-3.1.1 by van der Marck [23], respectively. The (C/E − 1) value for ENDF/B-VII.1 in this study agrees to that by van der Marck [23] considering the statistical uncertainties of the calculations. For the measured β_eff values of MISTRAL1, the calculation result of the JEFF-3.2-based neutron library is close to the measurement while those of the JENDL-4.0- and JENDL-3.3-based neutron libraries show underestimation of 1%, and that of the ENDF/B-VII.1-based neutron library also shows underestimation of 3%. This trend is explained by the differences in the β values of ²³⁵U and ²³⁸U for the libraries as mentioned before. For the measurement of MISTRAL2, the calculation result of JEFF-3.2-based neutron library shows better agreement to the measurement than those of the other neutron libraries. This trend is explained by the larger β values of ²³⁵U, ²³⁹Pu and ²³⁸U for JEFF-3.2 compared with the other libraries.

5. Conclusions

The analysis of the measured β_eff/ℓ values for the 17 TCA experimental cores indicated that the calculated results of MVP3 coupled with the JENDL-4.0-based neutron library are, generally speaking, in agreement with the associated measurements, and the average value of (C/E-1) are 1.0% with a the standard deviation of 1.1%. According to the sensitivity study with respect to the other nuclear-data libraries, the analysis results coupled with the JENDL-3.3 and ENDF/B-VII.1-based neutron libraries demonstrated similar accuracies to those of JENDL-4.0. Moreover, the analysis results coupled with the JEFF-3.2-based neutron library overestimated the measurements by more than 4%.

The calculated results of MVP3 coupled with the JENDL-4.0-based neutron library agreed to the measured β_eff values of TCA18 (UO₂ core) and MISTRAL1 (UO₂ core), considering the uncertainties composed of the measurement and calculation uncertainties, while they underestimated by 5% the measured one of the MISTRAL2 (MOX core). The analysis results coupled with the JENDL-3.3- and ENDF/B-VII.1-based neutron libraries showed comparable trend as those with JENDL-4.0 for TCA18 and MISTRAL1. The analysis results with JEFF-3.2 overestimated the measurement of TCA18, while they agreed to the measurements of MISTRAL1 UO₂ and MISTRAL2. The sensitivity study on the nuclear-data libraries indicated that these differences in the (C/E − 1) values between the neutron libraries were attributed to the differences in the β values of ²³⁵U, ²³⁹Pu, and ²³⁸U of the nuclear-data libraries on which the neutron libraries of MVP3 based.

The calculations of the kinetic parameters with MVP3 were performed with the methods recommended by Nagaya et al. for the most accurate calculation: the differential-operator-sampling technique taking into account the indirect-perturbed-source effect. In addition, the comparison of the calculated results with the other methods available with MVP3 to those with the recommended methods were carried out, and the differences in the kinetic parameters between the other methods and the recommended method were summarized.

Calculation results of the continuous-energy Monte Carlo calculation codes have been used as reference analysis results to verify physics models and neutron libraries adopted by deterministic analysis codes, which have been applied to the nuclear design and the safety analysis of commercial light-water reactors. This study provides information on uncertainties in the kinetic parameters calculated with MVP3. These uncertainties will be taken into account when MVP3 will be used in the verification of the deterministic analysis codes.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary Material:

Supplemental data for this article can be accessed here.

Appendix 1

Calculation Results with the Other Methods Available with MVP3

The MVP3 calculations provided the results of the kinetic parameters, β_eff and ℓ, with the methods [3] other than the differential-operator-sampling technique with the indirect-perturbed-source effect [1,2], which were adopted to obtain the calculation results for the comparison with the measured results. The other methods for β_eff and ℓ are 1) the differential-operator-sampling technique without the indirect-perturbed-source effect, 2) the correlated-sampling technique with the indirect-perturbed-source effect, 3) the correlated-sampling technique without the indirect-perturbed-source effect [3]. The other methods only for β_eff are 4) the Meulekamp’s method, 5) the importance function weight method (the Nauchi’s method), and 6) the constant weight methods (the non-adjoint weighted method) as the next fission probability methods [3]. Table A1 lists some of numerical results of the other methods compared with those of the differential-operator-sampling technique with the indirect-perturbed-source effect (hereafter, they are referred to as ‘the references.’). Figures A1 and A2 illustrate some of comparison results for the calculated kinetic parameters obtained with the other methods as the ratios to the references for all the cores.

Figure A1. Ratios of the calculated β_eff obtained with the other methods in MVP3 [3] to the reference (the results with the differential-operator-sampling technique with the indirect-perturbed-source effect) for all the cores.

Figure A2. Ratios of the calculated ℓ obtained without the indirect-perturbed-source effect for the differential-operator-sampling technique to the reference (with the indirect-perturbed-source effect for the differential-operator-sampling technique) for all the cores.

Table A1. Calculated results of the kinetic parameters, β_eff and ℓ, with the methods [3] other than the differential-operator-sampling technique as the eigenvalue method taking into account the indirect-perturbed-source effect [1,2].

		Perturbation method								Next fission probability method
		Differential-operator-sampling				Correlated sampling
Core		With indirect-perturbed-source effect		Without indirect-perturbed-source effect		With indirect-perturbed-source effect		Without indirect-perturbed-source effect		Meulekamp's method
TCA1	βeff	7.749E-03	(0.30)	7.692E-03	(0.12)	7.748E-03	(0.31)	7.692E-03	(0.12)	7.76E-03	(0.21)
	Ratio to Reference^a	1.000		0.993		1.000		0.993		1.002
	ℓ (s^−1)	3.494E-05	(0.05)	3.944E-05	(0.02)
	Ratio to Reference^a	1.000		1.129
TCA6	βeff	7.524E-03	(0.29)	7.436E-03	(0.12)	7.531E-03	(0.30)	7.436E-03	(0.12)	7.51E-03	(0.21)
	Ratio to Reference^a	1.000		0.988		1.001		0.988		0.999
	ℓ (s^−1)	2.839E-05	(0.03)	2.985E-05	(0.01)
	Ratio to Reference^a	1.000		1.051
TCA17	βeff	5.378E-03	(0.36)	5.559E-03	(0.14)	5.380E-03	(0.36)	5.559E-03	(0.14)	5.63E-03	(0.25)
	Ratio to Reference^a	1.000		1.034		1.000		1.034		1.047
	ℓ (s^−1)	3.041E-05	(0.05)	3.634E-05	(0.02)
	Ratio to Reference^a	1.000		1.195
TCA18	βeff	7.803E-03	(0.30)	7.698E-03	(0.12)	7.810E-03	(0.31)	7.698E-03	(0.12)	7.75E-03	(0.21)
	Ratio to Reference^a	1.000		0.987		1.001		0.987		0.993
	ℓ (s^−1)	3.79E-05	(0.04)	4.20E-05	(0.02)
	Ratio to Reference^a	1.000		1.110
MISTRAL1	βeff	7.77E-03	(0.31)	7.67E-03	(0.13)	7.77E-03	(0.32)	7.67E-03	(0.13)	7.74E-03	(0.41)
	Ratio to Reference^a	1.000		0.988		1.000		0.988		0.997
	ℓ (s^−1)	2.726E-05	(0.04)	3.052E-05	(0.01)
	Ratio to Reference^a	1.000		1.119
MISTRAL2	βeff	3.596E-03	(0.37)	3.561E-03	(0.16)	3.599E-03	(0.38)	3.561E-03	(0.15)	3.567E-03	(0.52)
	Ratio to Reference^a	1.000		0.990		1.001		0.990		0.992
	ℓ (s^−1)	1.387E-05	(0.07)	1.866E-05	(0.03)
	Ratio to Reference^a	1.000		1.345

(): Statistical uncertainties in the calculations (1σ) (%)

^aThe result with the differential-operator-sampling with the indirect-perturbed souce effect for the perturbation method.

Table A2. Calculated results of β_eff with the Meulekamp’s method, the importance function method, and the constant weight method in the next fission probability methods for MISTRAL1.

	Perturbation method		Next fission probability method
	Differential-operator-sampling with indirect-perturbed-source effect		Meulekamp’s method		Importance function weight method (Nauchi’s method)		Contant weight method (Non-adjointweighted method)
βeff	7.748E-03	(1.01)	7.741E-03	(0.43)	7.659E-03	(0.43)	7.021E-03	(0.0028)
Ratio to Reference^a	1.000		0.999		1.000		0.906

(): Statistical uncertainties in the calculations (1σ) (%)’

^aThe result with the differential-operator-sampling with the indirect-perturbed souce effect for the perturbation method.

Concerning β_eff calculated with the perturbation method, the results of the differential-operator-sampling technique and the correlated-sampling technique are almost identical as previously indicated by Nagaya et al. [1]. As for the indirect-perturbed-source effect for the perturbation methods, the effects on the calculated β_eff are less than 2% for the most cores, and the trends of the effects are different between the cores of from TCA12 through TCA17, which are the two-region cores composed of the central core region of the MOX fuel rods and the driver region of the UO₂ fuel rods, and the other cores, as shown in Figure A1. As for the indirect-perturbed-source effect on the calculated ℓ, the effects are larger than 10% for the most cores, and they are larger for the MOX cores, TCA16, TCA17 and MISTRAL2, which were loaded with the high Pu content MOX fuel rods. It is noted that the number of propagation batches of 10 for the calculation of the indirect-perturbed-source effect were quite enough for all the cores, which accords with the results previously reported by Nagaya et al. [1].

The results of β_eff calculated with Meulekamp’s method are compared with the references, and the comparison results are shown in Table A1 and Figure A1. The differences are less than 1% for the most cores. The exceptions are for those of from TCA12 through TCA17, which are the two-region cores composed of the central core region of the MOX fuel and the driver region of the UO₂ fuel rods. Table A2 shows the comparison results of the importance function method, and the constant weight method in the next fission probability methods, which were obtained from a preliminary calculation for MISTRAL1 with smaller neutron histories. The results with the importance function method are close to the reference and that with Meulekamp’s method. The result with the constant weight method are 10% lower than the reference.

Appendix 2

Calculated Effective Neutron Multiplication Factors

The calculated results of k_eff for the TCA and EOLE cores obtained with the JENDL-4.0-based neutron library are shown in Table A3 and Figure A3. The k_eff values with the JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based libraries are also shown for a part of the cores. The k_eff values with the JENDL-4.0-, ENDF/B-VII.1- and JEFF-3.2-based libraries were close to 1.0 for the majority of cores. For the UO₂ cores of TCA without boron in the moderator, the average value obtained with the JENDL-4.0-based neutron library was 1.00015 with a standard deviation of 0.00038 Δk_eff. This suggests that these libraries have a good performance for light-water reactor cores loaded with low-enriched UO₂ fuel, which is in-line with the results of Chiba et al. [26].

Figure A3. K_eff values for the critical cores obtained with the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1-, and JEFF-3.2-based neutron libraries.

Table A3. Effective neutron multiplication factors for the critical cores with respect to the JENDL-4.0-, JENDL-3.3-, ENDF/B-VII.1- and JEFF-3.2-based neutron libraries. The statistical uncertainties were less than 0.005 to 0.006%Δk_eff.

Core	TCA1	TCA2	TCA3	TCA4	TCA5	TCA6	TCA7	TCA8	TCA9	TCA10	TCA11
JENDL4.0	1.00026	1.00076	1.00067	1.00009	1.00258	1.00316	1.00072	0.99994	1.00006	0.99975	0.99995
JENDLE-3.3	0.99748					1.00072
ENDF/B-VII.1	1.00107					1.00335
JEFF-3.2	1.00046					1.00295
Core	TCA12	TCA13	TCA14	TCA15	TCA16	TCA17	TCA18	MISTRAL1	MISTRAL2
JENDL4.0	1.00016	0.99994	1.00023	1.00023	0.99934	1.00018	1.00057	1.00191	1.00684
JENDLE-3.3						0.99837	0.99795	0.99998	1.00717
ENDF/B-VII.1						0.99996	1.00105	1.00193	1.00711
JEFF-3.2						0.99947	1.00043	1.00283	1.00194

The cores showing k_eff values of 1.002 to 1.003 were TCA5, TCA6 and MISTRAL1; they were the UO₂ cores with boron concentrations of 300 ppm to 500 ppm. Even in the results with the JENDL-3.3-based library for these cores, these results are higher than those of the other UO₂ cores. Tsuruta and Kitamoto [5] reported the core characteristics of a derivative core of TCA6 which had the same number of UO₂ fuel rods as TCA6. The fuel pitch of the cores was 2.15 cm (M/F ratio: 2.48), the boron concentration was 554 ppm, the critical water height was 86.89 cm, and the water temperature was 25.9 °C. The kinetic parameters were not measured for it. The additional MVP3 calculations were performed for the core in order to add the information pertaining to the correlation between boron concentrations and the k_eff values. The k_eff of the derivative core was 1.00243 ± 0.00005(1σ), which is larger than that of the TCA UO₂ cores without boron in the moderator. Including the additional result, this study confirmed a correlation between boron concentrations and k_eff for the light-water moderated UO₂ cores. It may indicate potential biases in the thermal-neutron-absorption cross sections of ¹⁰B with respect to the nuclear-data libraries.

As shown in Table A and Figure A, the results for JENDL-3.3 were systematically lower than those for JENDL-4.0, ENDF/B-VII and JEFF-3.2 (except for MISTRAL2). The nuclear-data libraries for JENDL-4.0, ENDF/B-VII and JEFF-3.2 had a lower thermal-neutron-capture cross section 1.3% lower than that of JENDL-3.3 with respect to ²³⁸U. This results in the k_eff values for light-water reactor cores to be 0.2 to 0.3% Δk_eff lower than those of JENDL-3.3 [27,28].

While the k_eff value for MISTRAL2 with respect to JEFF-3.2 were 1.00194, k_eff was approximately 1.007 for JENDL-4.0, JENDL-3.3, and ENDF/B-VII. Relatively large discrepancies in the k_eff values for MISTRAL2 were correlated to the uncertainties in the cross sections in the nuclear-data libraries [23]. The MOX fuel rods loaded in MISTRAL2 and the other MOX experimental cores of the MISTRAL program had high contents of ²⁴¹Am. This was because roughly ten years had elapsed from the fuel fabrication when the fuel rods were used in the MISTRAL experiments performed from 1995 to 2000, resulting in the ²⁴¹Pu isotopes decaying to ²⁴¹Am isotopes. The reactivity calculations of the MOX cores of the MISTRAL program were influenced by the capture cross sections of ²⁴¹Am in the thermal- and epithermal- energy regions in addition to the relevant cross sections of the other major uranium and plutonium isotopes. The capture cross sections of ²⁴¹Am still have large uncertainties. According to Yamamoto et al. [28], the thermal-neutron-capture cross section of ²⁴¹Am for JENDL-4.0 was 7% larger than that of JENDL-3.3 and 9% lower than that of JEFF-3.2. That of ENDF/B-VII.1 [9] adopted the same value as that of JENDL-4.0. Terada et al. [29] recently reported a new measurement result for the thermal-neutron-capture cross section of ²⁴¹Am which is 3% larger than that of JENDL-4.0 and 6% lower than that of JEFF-3.2.

The MOX fuel loaded into the TCA cores had high ²³⁹Pu contents and low contents of ²⁴⁰Pu and ²⁴¹Pu; the contents of ²⁴¹Am were not reported [5]. The uncertainties in the neutron-capture cross sections of ²⁴¹Am did not affect the k_eff values for the cores, TCA12 through TCA 17, all of which were partially loaded with MOX fuel rods.