Benchmark models for criticalities of FCA-IX assemblies with systematically changed neutron spectra

Abstract

New benchmark models with respect to criticality data are established on the basis of seven uranium-fueled assemblies constructed in the ninth experimental series at the fast critical assembly (FCA) facility. By virtue of these FCA-IX assemblies, where the simple combinations of uranium fuel and diluent (graphite and stainless steel) in their core regions were systematically varied, the neutron spectra of these benchmark models cover those of various reactor types, from fast to sub-moderated reactors. The sample calculations of the benchmark models by a continuous-energy Monte Carlo (MC) code showed obvious differences between even the latest versions of two major nuclear data libraries, JENDL-4.0 and ENDF/B-VII.1. The present benchmark models would be well suited for the assessment and improvement of the nuclear data for ²³⁵U, ²³⁸U, graphite, and stainless steel. In addition, the verification of the deterministic method was performed on the benchmark models by comparison with the MC calculations. The present benchmark models are also available to users of deterministic calculation codes for the assessment and improvement of nuclear data.

Keywords:

• FCA-IX assemblies
• uranium
• criticality
• benchmark
• Monte Carlo
• deterministic method
• JENDL-4.0
• ENDF/B-VII.1

1. Introduction

The fast critical assembly (FCA) facility of the Japan Atomic Energy Agency has played an important role in studies on the assessment and improvement of nuclear data and calculation methods in the reactor physics field. In the 1980s, seven uranium-fueled assemblies, the so-called FCA-IX assemblies, were constructed to evaluate and adjust the cross-section data for minor actinides at FCA, where criticalities, fission rate ratios, and small-sample reactivity worths for minor actinides were measured [1]. Each assembly is identified by a number (1–7) after the “IX,” e.g., IX-1 for the first one. FCA-IX assemblies are simple, both in geometry and composition. In the first six assemblies, IX-1–IX-6, each core region was composed of a simple combination of high-enriched uranium (HEU) fuel plates and diluent plates of graphite or stainless steel. Their neutron spectra covered those of various reactor types, from fast to sub-moderated reactors, by varying the combinations of fuel and diluent plates. In addition, assembly IX-7 was constructed as a low-enriched uranium (LEU) assembly without diluent.

The main purpose of this paper is to provide new benchmark models of FCA-IX assemblies with respect to criticalities (k_eff) as one of the most basic reactor physics parameters. Even simple assemblies, such as those of FCA-IX, include too many geometric details in exact experimental configurations. For the user's convenience, because the benchmark models should be as simple as possible, simplifying the benchmark models without increasing the total uncertainty far beyond that of the original experiment is important. Therefore, we have established three-dimensional heterogeneous benchmark models by simplifying the geometric details, except for the plate-wise heterogeneity, to a practical level. In the future, we will open the heterogeneous benchmark models, which include detailed data of FCA's fuel plates. However, in this paper, we provide the three-dimensional homogeneous benchmark models where the plate-wise heterogeneity is homogenized, together with correction factors for the heterogeneous effect. Adopting the correction factors on the results of the homogeneous benchmark models, users of Monte Carlo (MC) codes can obtain equivalent results to those of heterogeneous benchmark models. Furthermore, the present benchmark models provide available correction factors to users of deterministic calculation codes.

The following section describes the measurement data and uncertainties of the experimental configurations of FCA-IX assemblies. Section 3 describes the establishment of benchmark models and provides the specifications of homogeneous benchmark models together with several correction factors. Section 4 describes the sample benchmark calculations performed by the continuous-energy MC code MVP [2] with two major nuclear data libraries, JENDL-4.0 [3] and ENDF/B-VII.1 [4]. The verification of the deterministic calculation method performed by comparing with the MC results is also described. Finally, conclusions are presented in Section 5.

2. Experiments

2.1. General description of the FCA facility

FCA is a horizontal split-table type comprising two half-assemblies: one is constructed on a fixed table and the other is on a movable one [5]. Figure 1 shows a schematic of the FCA facility. Square tubes of stainless steel are horizontally stacked on each table to form a 51-row and 51-column square matrix. Each square tube is 5.52 cm square (outside dimension) and 132.4 cm long. As shown in Figure 1, the two half-assemblies are separated by 200 cm when the facility is not operating.

Figure 1. Schematic of the FCA facility.

Figure 2 illustrates the construction of the experimental assemblies at the FCA facility. Each assembly is realized by loading drawers, where rectangular plates or blocks of different kinds of material, such as enriched uranium, depleted uranium, graphite, and stainless steel, are packed, into the square tube matrices. There are two types of drawers: one is an ordinary drawer and the other is a special one for the safety and control rods (SCRs) that are connected to the SCR drive mechanisms. The ordinary drawers are made of 0.0771 cm thick stainless steel. The 5.08 cm cubic material unit, which is composed of stacking the fuel and diluent plates, and the depleted uranium blocks (DUBs), can be packed into the ordinary drawers. Figure 3 illustrates the XY-cross-sectional view of the matrix tube loading an ordinary drawer, where fuel and diluent plates are packed. For startup operation, a neutron source (NS) has to be present in each half of the uranium-fueled assemblies because there is no inherent source. Hence, the uranium-fueled assemblies include drawers with a hole for NS insertion (NS-hole drawer), where plates or blocks with a 2.0 cm diameter hole are partially stacked to form an X-directed hole. Since the NS-hole drawers are loaded in a horizontal row, NS can be inserted into the core from the outside along the hole.

Figure 2. Construction of the experimental assemblies at the FCA facility.

Figure 3. XY-cross-sectional view of matrix tube loading an ordinary drawer.

In FCA, fuel plates are used in the SCR drawers. The structure and contents of the SCR drawers are similar to the ordinary drawers, but they have some special features. The SCR drawer is composed of stainless steel with twice the side and bottom wall thickness (0.15 cm) of the ordinary drawer walls, because it has to be strong enough to undergo rapid acceleration and deceleration in scram operations. Hence, the units for SCRs have to be 1/16 inch (0.159 cm) thinner than those for the ordinary drawers along the X-direction owing to the thicker walls of the SCR drawers. All the safety rods are fully inserted when the facility is operating. Critical state is achieved by bringing the movable table to the fixed one and adjusting the insertion depth of the control rods.

Figure 4 illustrates an XZ-cross-sectional view near the interface between the fixed and movable half-assemblies when operating. Owing to a mechanical clearance, there is a small air gap between the drawers loaded in the fixed half and those loaded in the movable one, even at the closest position of the movable table to the fixed one. As a result, the interface gap is 0.21 cm in total owing to the mechanical clearance gap and the drawer front walls, which are filled with the air and the stainless steel, respectively. The front wall (3 mm) of the SCR drawer is also thicker than that of the ordinary drawer. Therefore, the region boundaries along the Z-direction in the SCR drawers are slightly different from those in the ordinary drawers owing to the difference between their front wall thicknesses.

Figure 4. XZ-cross-sectional view near interface between fixed and movable half-assemblies when operating.

2.2. Specific description of FCA-IX assemblies

The core specifications of the experimental configurations are summarized in Table 1. Figure 5 shows the XY-cross-sectional views of the plate-stacking patterns of core units. Each pattern was symmetric with respect to the YZ mid-plane within the core unit. In assemblies IX-1–IX-6, HEU plates were used as fuel. Note that LEU plates were partially used instead of HEU plates in assembly IX-4. In assembly IX-7, only LEU plates were stacked in core units. Graphite plates were used as diluent in assemblies IX-1–IX-3 and stainless steel plates in assemblies IX-4–IX-6.

Table 1. Core specifications of the experimental configurations.

						Atomic number density^b
						(×10^22 atoms/cm^3)
Assembly	Fuel	Diluent	Radius^a(cm)	Height(cm)	Volume (liter)	^12C	SS^c	^235U	^238U
IX-1	HEU	Graphite	30.43	60.96	177.4	6.578	0.919	0.226	0.017
IX-2	HEU	Graphite	23.20	40.64	68.7	6.135	0.920	0.452	0.033
IX-3	HEU	Graphite	18.03	35.56	36.3	5.716	0.920	0.678	0.050
IX-4	HEU	Stainless steel	27.59	50.80	121.5	0.035	6.976	0.452	0.033
IX-5	HEU	Stainless steel	20.42	40.64	53.2	0.038	6.549	0.678	0.050
IX-6	HEU	Stainless steel	22.89	40.64	66.9	0.019	2.378	0.678	0.050
IX-7	LEU	–	17.34	35.56	33.6	0.032	0.919	0.784	3.112

^aEquivalent radius of the experimental configuration. ^bStainless steel of the matrix tube and the drawer is included. ^cSummation of the atomic number densities of Cr, Mn, Fe, and Ni.

Figure 5. XY-cross-sectional views of plate-stacking patterns of core units.

In FCA assemblies, the total amount of fuel is divided into the fixed and movable half-assemblies owing to the feature of the horizontal split-table type. Therefore, in each core drawer of FCA-IX assemblies, the core units with the half of the core height given in Table 1 were packed and the blanket units of the DUBs with 15.24 cm height were packed behind the core units. In each blanket drawer, the blanket units of DUBs were packed to the same height as the core drawer.

In each of the FCA-IX assemblies, the plate-stacking pattern in SCR units was designed to be similar to that in core units. However, SCR units have to be 1/16 inch (0.159 cm) thinner than the ordinary core units owing to the thicker walls of the SCR drawers, as mentioned in Section 2.1. Therefore, a 1/16 inch thick diluent plate was removed from the respective core unit to design each unit for SCRs in assemblies IX-1–IX-6. In the SCR unit of assembly IX-7, a 1/16 inch thick LEU plate was removed because there was no diluent.

In the NS-hole drawers of assembly IX-1, the unit with the NS-hole was located 22.86 cm from the drawer front wall along the Z-direction. As for the other assemblies, units with an NS hole were located in a suitable position depending on the assembly.

In each assembly, the core, blanket, SCR drawers, and NS-hole drawers were loaded into the matrix tubes in accordance with the drawer-loading map. The loading position in the matrix is identified by the initial letters of the half (Fixed or Movable), row number, and column number. The row numbers (101–151) start from the top. The column numbers (1–51) increase along the X-direction. For example, the central position in the fixed half is F126/26. Figures 6 and 7 show the central portion of the drawer-loading maps at the fixed and movable halves of assembly IX-1 as an example.

Figure 6. Drawer-loading map at fixed half of assembly IX-1.

Figure 7. Drawer-loading map at movable half of assembly IX-1.

In the fixed half of assembly IX-1, there were 88 core and 289 blanket drawers, as shown in Figure 6. NS-hole drawers were loaded into 10 matrix tubes from F126/15 to F126/24. The loading numbers of the core and blanket drawers with NS-holes were 4 and 6, respectively. In each assembly, four SCR drawers were loaded at F124/24, F128/24, F124/28, and F128/28. In the movable half of assembly IX-1, there were 87 core and 290 blanket drawers, as shown in Figure 7. The core drawer at F131/28 and the blanket drawer at M131/28 were asymmetrically loaded in the experimental configuration, whereas other drawers were symmetric between both the halves, because the loading number of core drawers was adjusted to achieve criticality under the operational regulation of the FCA facility. Such asymmetric drawer-loadings occurred in the experimental configurations of assemblies IX-1–IX-4.

2.3. Criticalities of FCA-IX assemblies

In each of the FCA-IX assemblies, the excess reactivity of the experimental configuration was measured with the calibrated control rod. The excess reactivity is the system reactivity when all SCRs are fully inserted. The calibrations were performed using the rod compensation and positive period methods. The excess reactivity in units of dollars thus measured was converted into that in units of Δk/k by the effective delayed neutron fraction, which was re-evaluated with the delayed neutron data in the latest nuclear data library of JENDL-4.0. The uncertainties in the excess reactivity were conventionally evaluated by considering the mechanical reproducibility of the SCR positions, that of closeness between the two tables, the measurement errors of the doubling time, and the core temperature [5]. In this study, we also considered the uncertainties due to the delayed neutron data of decay constants (λ), fractional group yields (a), and the total number of delayed neutrons per fission (ν_d). Table 2 summarizes the excess reactivity and uncertainties due to the measurement techniques of FCA-IX assemblies. The uncertainties due to the measurement techniques are within 0.02% Δk/k, as shown in Table 2.

Table 2. Excess reactivity and uncertainties due to the measurement techniques (% Δk/k).

		Uncertainties
Assembly	Excess reactivity	λ, a	νd	Others^a	Subtotal^b
IX-1	0.336	±0.010	±0.008	±0.002	±0.013
IX-2	0.528	±0.014	±0.012	±0.003	±0.019
IX-3	0.424	±0.011	±0.010	±0.003	±0.016
IX-4	0.274	±0.008	±0.008	±0.002	±0.012
IX-5	0.461	±0.012	±0.012	±0.003	±0.018
IX-6	0.246	±0.007	±0.006	±0.002	±0.010
IX-7	0.224	±0.005	±0.006	±0.002	±0.009

^a Due to the mechanical reproducibility of the SCR positions, that of closeness between the two tables, the measurement errors of the doubling time, and the core temperature. ^bSquare root summation of squares of the individual uncertainties due to the measurement techniques.

The uncertainties due to the compositions in k_eff values were also evaluated. The standard errors of the mean of the plate weights and uranium enrichments were considered as main composition uncertainties. The reactivity effects of these uncertainties were estimated by the MC calculations using MVP. A reference k_eff value was calculated with the nominal weights for all plates and blocks, and the nominal enrichments for all the uranium plates. To evaluate the uncertainty due to the weight of the enriched uranium plates, the perturbed k_eff′ value was calculated by changing the enriched uranium weight by its standard error of mean and fixing all other conditions. The uncertainty was estimated as (k_eff′ − k_eff)/k_eff using the reference k_eff value to the perturbed k_eff′ value. For the stainless steel plates and DUB blocks, uncertainties were estimated in the same manner.

FCA has two different lots of graphite plates. The standard error of the mean of the plate weights is expected to be small when graphite plates are selected from one of the lots. However, we should consider two extreme cases where each assembly was constructed by the respective lots because there is no information that either lot was selected in constructing assemblies IX-1–IX-3. Therefore, the nominal weight of the graphite plates was assumed to be the middle value of the averaged weights of the two lots by taking its error to be the difference between the nominal weight and one of the averaged weights. Note that the standard error of mean of each extreme case is negligibly small in comparison with the thus-defined error.

To evaluate the uncertainty due to the enrichment of uranium plates, the perturbed k_eff′ value was calculated by changing the enrichment by the standard error of mean and fixing all other conditions.

Table 3 summarizes the uncertainties in k_eff value due to the compositions. The number of neutron histories was 200 million for each calculation. The statistical uncertainties of MC calculations were less than 0.01% Δk/k. In comparisons between Tables 2 and 3, the overall composition uncertainties dominate the measurement uncertainties. In assemblies IX-1–IX-3, the uncertainties are mainly caused by that in the weight of the graphite plates.

Table 3. Uncertainties in k_eff value due to the compositions (% Δk/k).

	Weight^a
Assembly	Graphite^b	Enrichment^c	Subtotal^d
IX-1	±0.235	±0.034	±0.24
IX-2	±0.181	±0.041	±0.19
IX-3	±0.104	±0.034	±0.11
IX-4	–	±0.050	±0.05
IX-5	–	±0.035	±0.04
IX-6	–	±0.057	±0.06
IX-7	–	±0.070	±0.07

^aUncertainties due to the weights of stainless steel, HEU, and LEU plates, and DUBs are negligibly small. ^b Due to the weight of graphite plates. ^cDue to the enrichment of HEU or LEU plates. ^dSquare root summation of squares of the individual uncertainties due to the compositions.

Table 4 summarizes the experimental k_eff values. The total uncertainties were estimated as the square root summation of squares of the uncertainties due to the measurement techniques and compositions. The results show that the total uncertainties in the experimental k_eff values are within 0.24% Δk/k.

Table 4. Summary of the experimental k_eff values.

Assembly	Experimental keff value
IX-1	1.0034	±0.0024*
IX-2	1.0053	±0.0019*
IX-3	1.0042	±0.0011*
IX-4	1.0027	±0.0005*
IX-5	1.0046	±0.0004*
IX-6	1.0025	±0.0006*
IX-7	1.0022	±0.0007*

*Square root summation of squares of the uncertainties due to the measurement techniques and compositions given in Tables 2 and 3, respectively.

3. Benchmark models

As mentioned in the previous section, there were too many geometric details in the exact experimental configurations, even simple assemblies, such as those in FCA-IX. For the user's convenience, because the benchmark models should be as simple as possible, simplifying the benchmark models without increasing the total uncertainty far beyond that of the original experiment is important. Therefore, the three-dimensional heterogeneous benchmark models were established by simplifying the geometric details, except for the plate-wise heterogeneity, to a practical level. Section 3.1 provides the benchmark k_eff values for the heterogeneous models. In the future, we will open the heterogeneous benchmark models, which include detailed data of FCA's fuel plates. However, in this paper, we establish the three-dimensional homogeneous benchmark models where the plate-wise heterogeneity is homogenized, together with correction factors for the heterogeneous effect. Section 3.2 provides the specific configuration and composition data of the homogeneous benchmark models. Section 3.3 provides the correction factors for the heterogeneous effect, so that users of MC codes can obtain results comparable to the benchmark k_eff values from the homogeneous calculation results. Section 3.4 provides several correction factors for the deterministic scheme, so that users of deterministic calculation codes can obtain results comparable to the benchmark k_eff values using simpler homogeneous calculation results based on the diffusion method.

3.1. Benchmark k_eff values of the heterogeneous benchmark models

The heterogeneous models, which maintain the plate-wise heterogeneity, are based on the experimental configurations with the following simplifications:

• • replacement of the SCR drawers with the ordinary core drawers,

• • elimination of the clearance gap and the drawer front walls at the interface,

• • elimination of the NS-holes,

• • symmetrization of the drawer-loading between half-assemblies (only for assemblies IX-1–IX-4).

The heterogeneous models, which exclude the geometric details of SCR drawers, the interface gap, and NS-holes, are desirable so as to easily perform benchmark calculations. By taking advantage of the symmetrized drawer-loading, the half-assembly under the reflective condition at the mid-plane instead of the full assembly can be easily applied in the benchmark calculations. The relevant reactivity worths corresponding to these simplifications were measured in each of the FCA-IX assemblies. The descriptions of the measurement techniques for these reactivity worths at FCA were reported in [5]. Table 5 summarizes the measured reactivity worths.

Table 5. Reactivity worths for the simplifications (% Δk/k).

			MC calculation
Assembly	Measurement		JENDL-4.0	ENDF-B/VII.1
(a) Reactivity worth for replacement of the SCR drawers with the ordinary core drawers
IX-1	0.01	±0.01	0.01	0.00
IX-2	0.03	±0.02	0.00	0.00
IX-3	0.01	±0.01	0.00	0.00
IX-4	−0.01	±0.01	−0.01	−0.02
IX-5	−0.01	±0.01	−0.01	−0.01
IX-6	0.00	±0.01	−0.02	−0.01
IX-7	0.12	±0.02	0.12	0.12
(b) Reactivity worth for elimination of the clearance gap and drawer front walls at the interface
IX-1	0.17	±0.06	0.18	0.19
IX-2	0.23	±0.07	0.25	0.25
IX-3	0.34	±0.06	0.30	0.30
IX-4	0.21	±0.05	0.22	0.23
IX-5	0.27	±0.07	0.28	0.28
IX-6	0.21	±0.05	0.22	0.22
IX-7	0.27	±0.06	0.32	0.30
(c) Reactivity worth for elimination of holes for the NS
IX-1	0.11	±0.01	0.09	0.09
IX-2	0.02	±0.01	0.02	0.02
IX-3	0.02	±0.01	0.01	0.01
IX-4	0.01	±0.01	0.03	0.03
IX-5	0.01	±0.01	0.01	0.01
IX-6	0.02	±0.01	0.01	0.01
IX-7	0.00	±0.01	0.00	0.01
(d) Reactivity worth for symmetrization of the drawer-loading between half-assemblies
IX-1	−0.14	±0.01	−0.13	−0.14
IX-2	−0.23	±0.01	−0.23	−0.21
IX-3	−0.52	±0.02	−0.48	−0.48
IX-4	Not measured		0.15	0.15
(e) Total reactivity worth^a (ΔT)
IX-1	0.15	±0.06^c	0.15	0.14
IX-2	0.05	±0.07^c	0.04	0.06
IX-3	−0.15	±0.06^c	−0.17	−0.17
IX-4	0.36^b	±0.05^c	0.39	0.39
IX-5	0.27	±0.07^c	0.28	0.28
IX-6	0.23	±0.05^c	0.21	0.22
IX-7	0.39	±0.06^c	0.44	0.43

^aSummation of the individual reactivity worths. ^bIn assembly IX-4, the MC calculation was adopted for the reactivity worth for symmetrization of the drawer-loading because there was no measurement. ^cSquare root summation of squares of the individual uncertainties.

The reactivity worths for simplifications were verified by the MC code MVP with the latest libraries, i.e., JENDL-4.0 and ENDF/B-VII.1. For each of the FCA-IX assemblies, we constructed an as-built model, which includes the geometric details of the experimental configuration. Starting from this as-built model, simplified models were constructed stepwise by each simplification, such as SCR drawers, the interface, NS holes, and the asymmetric drawer-loading. Each reactivity worth was estimated using the k_eff values of the two models before and after the relevant simplification. The calculation results are summarized in Table 5. The number of neutron histories was 200 million for each calculation. The statistical uncertainties of the MC calculations were 0.01% Δk/k on the reactivity worth. Each calculation value is consistent with the measured value within two times of the experimental error. From these results, the validity of the measured results for simplifications was confirmed using the geometric detailed MC calculations.

Table 6 summarizes the k_eff values of the heterogeneous benchmark models. The benchmark k_eff values were estimated by adopting the measured results of total reactivity worth on the experimental k_eff values. Note that the MC calculation for the reactivity worth to symmetrize the drawer-loading was adopted in assembly IX-4 because there was no measurement. The total uncertainties in the benchmark k_eff values are within 0.24% Δk/k. In Appendix 1, we prepared a correlation matrix between the benchmark k_eff values of assemblies IX-1–IX-7.

Table 6. Summary of k_eff values of the heterogeneous benchmark models.

	Heterogeneous benchmark
Assembly	keff value
IX-1	1.0049^a	±0.0024^b
IX-2	1.0058^a	±0.0020^b
IX-3	1.0027^a	±0.0013^b
IX-4	1.0063^a	±0.0007^b
IX-5	1.0073^a	±0.0008^b
IX-6	1.0048^a	±0.0008^b
IX-7	1.0061^a	±0.0009^b

^aSummation of the experimental k_eff value in Table 4 and the measured result of the total transformation Δ_T in Table 5(e). ^bSquare root summation of squares of their uncertainties.

3.2. Specifications of homogeneous benchmark models

For the homogeneous benchmark models, the composition data were provided by homogenizing the constituents of the packed plates, the drawer walls, the matrix tube, and the air gaps per cell. Here, the cell corresponds to product of the XY-cross-sectional area of 5.52^X×5.52^Y cm² (outer boundary of the square matrix tube) and the unit length along the Z-direction. Table 7 provides the homogenized atomic number densities of the core, blanket, and the empty matrix tube (MTX) cells of FCA-IX assemblies. Here, amount of uranium isotopes other than ²³⁵U was assumed to be that of ²³⁸U because there is only measurement data of the total uranium and ²³⁵U amounts for HEU and LEU plates.

Table 7. Homogenized atomic number densities of cells of FCA-IX assemblies (×10²⁴ atoms/cm³).

	IX-1	IX-2	IX-3	IX-4		IX-5	IX-6		IX-7	DUB^a	MTX^b
Nuclide	CH2	CH4	CH6	SH4	SH4-LEU	SH6	VH6	VH6V	LEU
H	7.696E−05	1.356E−04	1.917E−04	1.173E−04	2.355E−04	1.760E−04	1.760E−04	1.760E−04	3.767E−04	–	–
C	6.578E−02	6.135E−02	5.716E−02	3.522E−04	2.634E−04	3.832E−04	1.896E−04	1.896E−04	3.182E−04	–	–
O	3.462E−05	6.011E−05	8.429E−05	5.097E−05	1.026E−04	7.645E−05	7.645E−05	7.645E−05	1.642E−04	–	–
F	1.944E−05	3.888E−05	5.831E−05	3.888E−05	–	5.831E−05	5.831E−05	5.831E−05	–	–	–
Si	3.372E−06	6.588E−06	9.782E−06	8.290E−04	2.350E−04	7.147E−04	9.648E−06	9.648E−06	–	–	–
P	–	–	–	2.233E−05	6.381E−06	1.914E−05	–	–	–	–	–
S	–	–	–	5.690E−06	1.626E−06	4.877E−06	–	–	–	–	–
Ti	–	–	–	8.930E−06	2.551E−06	7.654E−06	–	–	–	–	–
Cr	1.810E−03	1.810E−03	1.810E−03	1.413E−02	7.059E−03	1.324E−02	4.721E−03	4.760E−03	1.810E−03	1.810E−03	1.229E−03
Mn	1.203E−04	1.203E−04	1.203E−04	7.212E−04	4.239E−04	7.013E−04	2.848E−04	3.437E−04	1.203E−04	1.203E−04	8.167E−05
Fe	6.475E−03	6.478E−03	6.481E−03	4.873E−02	2.469E−02	4.577E−02	1.678E−02	1.649E−02	6.472E−03	6.472E−03	4.393E−03
Co	–	–	–	1.399E−04	3.997E−05	1.199E−04	–	–	–	–	–
Ni	7.893E−04	7.893E−04	7.893E−04	6.181E−03	3.061E−03	5.776E−03	1.996E−03	2.121E−03	7.893E−04	7.893E−04	5.357E−04
Cu	–	–	–	2.529E−05	7.224E−06	2.167E−05	–	–	–	–	–
Nb	–	–	–	4.162E−06	1.189E−06	3.568E−06	–	–	–	–	–
Mo	–	–	–	1.956E−05	5.588E−06	1.676E−05	–	–	–	–	–
^235U	2.260E−03	4.519E−03	6.779E−03	4.519E−03	4.900E−03	6.779E−03	6.779E−03	6.779E−03	7.840E−03	8.442E−05	–
^238U^c	1.673E−04	3.347E−04	5.020E−04	3.347E−04	1.945E−02	5.020E−04	5.020E−04	5.020E−04	3.112E−02	4.017E−02	–

^aBlanket. ^bEmpty matrix tube. ^cAmount of uranium isotopes other than ²³⁵U was assumed to be that of ²³⁸U because there is only measurement data of the total uranium and ²³⁵U amounts for HEU and LEU plates.

Figures 8–14 describe the configurations of the homogeneous benchmark models. In these figures, mutually facing drawers in respective half-assemblies are given as “combined” drawers by virtue of the elimination of the clearance gap and the drawer front walls at the interface in the benchmark models. Using the combined drawers, the packing maps of the cells are described for the full assembly from the back end of the movable half to that of the fixed half along the Z-direction. Here, the maps are symmetric with respect to the mid-plane owing to the symmetrized drawer-loadings in the benchmark models. Simultaneously, the loading maps of the combined drawers are described in perspective seen towards the fixed half from the back end of the movable half.

Figure 8. Homogeneous benchmark model of assembly IX-1.

Figure 9. Homogeneous benchmark model of assembly IX-2.

Figure 10. Homogeneous benchmark model of assembly IX-3.

In assembly IX-4, extra core drawers were partially loaded, as shown in Figure 11. In the extra core drawers, the SH4-LEU cells, composed of LEU and stainless steel, were packed instead of the SH4 cells. In assembly IX-6, two types of core cells, VH6 and VH6V, were packed at the center and the edges of the core region, respectively, as shown in Figure 13. Note that the number densities of the stainless steel materials in the VH6 and VH6V cells were slightly different, as shown in Table 7.

Figure 11. Homogeneous benchmark model of assembly IX-4.

Figure 12. Homogeneous benchmark model of assembly IX-5.

Figure 13. Homogeneous benchmark model of assembly IX-6.

Figure 14. Homogeneous benchmark model of assembly IX-7.

Using these specific configuration and composition data, one can easily obtain the k_eff values of the homogeneous benchmark models by MC calculation codes or deterministic codes. The following sections provide correction factors from which users of MC calculation codes or deterministic codes can obtain results equivalent to those of the heterogeneous benchmark models.

3.3. Correction factor for Monte Carlo calculations

The heterogeneous effect should be considered to obtain results equivalent to those of the heterogeneous benchmark models. The correction factor for the heterogeneous effect was obtained as the ratio of the k^hetero_eff value for the heterogeneous benchmark model to the k^homo_eff value for the homogeneous one using MVP. In the heterogeneous benchmark model, the geometries of the plates, drawer walls, matrix tubes, and air gaps were modeled to maintain the plate-wise heterogeneity. The homogeneous benchmark model was equivalent to that provided in Section 3.2. The number of neutron histories was 200 million for each calculation. The MC statistical uncertainties were smaller than 0.0001 for all the correction factors. Table 8 provides the correction factors for the heterogeneous effect for JENDL-4.0 and ENDF/B-VII.1.

Table 8. Correction factor for the heterogeneous effect.

Assembly	JENDL-4.0	ENDF/B-VII.1
IX-1	1.0044	1.0047
IX-2	1.0095	1.0095
IX-3	1.0077	1.0073
IX-4	1.0123	1.0120
IX-5	1.0093	1.0090
IX-6	1.0045	1.0043
IX-7	1.0001	1.0000

Note: The correction factor was defined as f = k^hetero_eff/k_eff^homo with the k^hetero_eff value for the heterogeneous benchmark model and the k^homo_eff value for the homogeneous one by MC calculations.

The heterogeneous correction factors range from 1.0001 to 1.0123 for JENDL-4.0. In assembly IX-7, the correction is small because of its homogeneous plate-stacking. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.04%.

3.4. Correction factors for deterministic calculations

For users of deterministic calculation codes, we provided several correction factors, such as the ultra-fine (UF) energy group effect, anisotropic effect, mesh effect, heterogeneous cell effect, transport effect, S_N- and P_L-order effects, and effect of detailed-energy treatment on the core calculation. The deterministic calculation conditions and the definitions of the correction factors are summarized in Tables 9 and 10, respectively.

Table 9. Deterministic calculation conditions.

	Cell calculation		Core calculation
Method	Media	Energy group	Energy group	Mesh	Theory	Diffusion coefficient	SN order	PL order
0^a	Homo.	70Gr	70Gr	2	Diffusion	D = 1/(3Σtr)	–	–
1	Homo.	UF-70Gr^c	70Gr	2	Diffusion	D = 1/(3Σtr)	–	–
2	Hetero.^b	UF-70Gr^c	70Gr	2	Diffusion	D = 1/(3Σtr)	–	–
3	Hetero.^b	UF-70Gr^c	70Gr	2	Diffusion	Benoist's D//,D⊥	–	–
4	Homo.	UF-70Gr^c	70Gr	4	Diffusion	D = 1/(3Σtr)	–	–
5	Homo.	UF-70Gr^c	70Gr	6	Diffusion	D = 1/(3Σtr)	–	–
6	Homo.	UF-70Gr^c	70Gr	Infinite^d	Diffusion	D = 1/(3Σtr)	–	–
7	Homo.	UF-70Gr^c	70Gr	2	Transport	–	S8	P0 − Σtr
8	Hetero.^b	UF-70Gr^c	70Gr	2	Transport	–	S8	P0 − Σtr
9	Homo.	UF-70Gr^c	70Gr	6	Transport	–	S8	P0 − Σtr
10	Homo.	UF-70Gr^c	70Gr	2	Transport	–	S16	P0 − Σtr
11	Homo.	UF-70Gr^c	70Gr	2	Transport	–	S8	P4 extended
12	Homo.	UF-900Gr^c	900Gr	2	Transport	–	S8	P0 − Σtr

^aMethod-0 corresponds to the base calculation. ^bThe plate-stretch model was adopted for the heterogeneous media. ^cThe UF energy group calculation scheme was adopted. ^dThe k_eff value at the infinite mesh was evaluated by extrapolating with respect to the square of the mesh interval using the k_eff values of Method-1, -4, and -5.

Table 10. Definitions of correction factors for deterministic calculations.

Correction factor		Definition
Ultra-fine (UF)	fultra-fine	Method-1/Method-0
Anisotropic	fanisotropic	Method-3/Method-2
Mesh	fmesh	Method-6/Method-1
Heterogeneous cell	fhetero	Method-8/Method-7
Transport	ftrans	Method-9/Method-5
SN order	fsn	Method-10/Method-7
PL order	fpl	Method-11/Method-7
900 group	f900gr	Method-12/Method-7

The base calculation was defined as a simple one that was based on the diffusion method as follows. The effective macroscopic cross sections were prepared to consider the resonance self-shielding effect in the homogeneous media with the Bondarenko type of the 70-energy group constant set called JAERI Fast Set. Cell calculations were performed using the SLAROM-UF code [6]. Transport cross sections were conventionally defined as Σ^C_tr = Σ_t^C − Σ^C_s, with the current-weighted total cross sections (Σ^C_t) and the P₁ component of the scattering cross sections (Σ^C_s). Isotropic diffusion coefficients were defined as D = 1/(3Σ^C_tr). Using the 70-energy group effective cross sections, core calculations were performed by the CITATION diffusion calculation code [7]. Mesh intervals were conventionally set to 2.76 (=5.52/2) cm in the X- and Y-directions and 2.54 (=5.08/2) cm in the Z-direction. Hereafter, this condition of mesh intervals is called the 2-mesh condition.

The most accurate values by the deterministic calculation method can be obtained by adopting the correction factors. Table 11 provides the correction factors for JENDL-4.0 and ENDF-B/VII.1.

Table 11. Correction factors for deterministic calculations.

	Ultra-fine	Anisotropic	Mesh	Hetero.	Transport	SN-order	PL-order	900-group	Total^a
Assembly	fultra-fine	fanisotropic	fmesh	cell fhetero	ftrans	fsn	fpl	f900gr	ftotal
(a) JENDL-4.0
IX-1	1.0009	1.0000	0.9970	1.0040	1.0085	1.0000	0.9991	1.0003	1.0098^b
IX-2	1.0003	1.0000	0.9971	1.0087	1.0131	1.0000	0.9985	1.0007	1.0184^b
IX-3	1.0002	1.0000	0.9968	1.0068	1.0182	1.0000	0.9980	1.0009	1.0209^b
IX-4	0.9995	1.0000	0.9983	1.0102	1.0107	1.0000	0.9986	0.9912	1.0084^b
IX-5	0.9996	1.0000	0.9980	1.0078	1.0157	1.0000	0.9981	0.9935	1.0126^b
IX-6	1.0000	1.0000	0.9984	1.0047	1.0219	0.9999	0.9978	0.9973	1.0199^b
IX-7	1.0000	1.0000	0.9967	1.0000	1.0161	1.0000	0.9985	1.0013	1.0125^b
(b) ENDF-B/VII.1
IX-1	1.0006	1.0000	0.9971	1.0044	1.0083	1.0000	0.9990	1.0003	1.0097^b
IX-2	1.0003	1.0000	0.9972	1.0086	1.0129	1.0000	0.9984	1.0007	1.0181^b
IX-3	1.0002	1.0000	0.9969	1.0066	1.0180	1.0000	0.9978	1.0009	1.0204^b
IX-4	0.9998	1.0000	0.9983	1.0100	1.0110	1.0000	0.9988	0.9909	1.0087^b
IX-5	0.9998	1.0000	0.9981	1.0077	1.0159	1.0000	0.9982	0.9932	1.0128^b
IX-6	1.0002	1.0000	0.9985	1.0046	1.0220	0.9999	0.9980	0.9972	1.0203^b
IX-7	1.0001	1.0000	0.9967	1.0000	1.0160	1.0000	0.9981	1.0011	1.0119^b

^aThe total correction factor was evaluated as f_total = f_ultra-fine × f_anisotropic × f_mesh×f_hetero×f_trans×f_sn×f_pl×f_900gr. ^bAs described later, adoption of the weighting factor given in Table 14 is recommended to obtain the value consistent with the MC calculation.

3.4.1. Ultra-fine energy group effect

The effective cross sections were calculated in the UF energy group of about 100,000 below 50 keV by SLAROM-UF and were collapsed into 70 groups for core calculations. The correction factor for the effect of the UF calculation scheme was estimated as the ratio of the k_eff value by the UF calculation scheme to that by the base calculation. Here, the other conditions were fixed and the core calculations were commonly performed in the 70-energy group structure. The correction factors range from 0.9995 to 1.0009 for JENDL-4.0. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.03%; hence, they are not significant.

3.4.2. Anisotropic effect

In general, the diffusion coefficient of the vertical direction with respect to plates is different from that of the horizontal directions. The effect due to the anisotropic diffusion coefficients should be considered because of the use of the plate-type of fuel and diluent in all FCA assemblies. The anisotropic diffusion coefficients were defined as Benoist's formula [8]. The correction factor for the anisotropic effect was estimated as the ratio of the k_eff value by the anisotropic diffusion coefficient to that by the isotropic one, while the other conditions were fixed. In both calculations, a plate-stretch model was adopted. Figure 15 illustrates the XY-cross-sectional view of the plate-stretch model. Here, the stainless steel of the top and bottom walls of the matrix tube and that of the bottom wall of the drawer were counted in stretched areas without fuel to conserve the total amounts. In all FCA-IX assemblies, the correction factors are negligibly small for both libraries.

Figure 15. XY-cross-sectional view of the plate-stretch model.

3.4.3. Mesh size effect

The correction factor for the mesh size effect was estimated as the ratio of the k_eff value under the infinite-mesh condition to that under the 2-mesh condition by the diffusion calculations. Here, the k_eff value under the infinite-mesh condition was evaluated by extrapolating with respect to the square of the mesh interval. In the extrapolation, we used the results of the 2-, 4-, and 6-mesh conditions. Here, the intervals of the 4- and 6-mesh conditions corresponded to half and one-third of the 2-mesh condition, respectively. The correction factors range from 0.9967 to 0.9984 for JENDL-4.0. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.01%; hence, they are not significant.

3.4.4. Heterogeneous cell effect

The heterogeneous cell effect was conventionally considered for the plate-wise heterogeneity on the basis of the deterministic method. The correction factor was estimated as the ratio of the k_eff value by the plate-stretch model to that by the homogeneous model. The correction factors range from 1.0000 to 1.0102 for JENDL-4.0. In assembly IX-7, the effect is negligibly small because of its homogeneous plate-stacking. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.04%; hence, they are not significant.

It is noted that there are some differences between the heterogeneous correction factors by the MC and deterministic methods given in Tables 8 and 11, respectively. In the MC method, the geometric details of the plates, drawer walls, matrix tubes, and air gaps were modeled for the heterogeneous calculations as mentioned in Section 3.3. In contrast, the geometric details were one dimensionally simplified into the plate stretch models used for the heterogeneous calculations in the deterministic method. From the point of view of accuracy on the heterogeneous effect, adoption of the correction factors by MC calculations given in Table 8 is one of the useful options also for users of deterministic calculation codes.

3.4.5. Transport effect

The correction factor for the transport effect was estimated as the ratio of the k_eff value by the transport calculation to that by the diffusion one. Here, both the k_eff values were commonly calculated under the 6-mesh condition as the finest one. The diffusion calculation was performed with the isotropic diffusion coefficients defined as D = 1/(3Σ^C_tr). The transport calculation was performed under the P₀-order diagonal transport approximation (P₀ − Σ_tr) and the S₈ order. The transport core calculation was performed using the DANTSYS S_N neutron transport code [9]. The correction factors range from 1.0085 to 1.0219 for JENDL-4.0. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.03%; hence, they are not significant.

3.4.6. S_N- and P_L-order effects

The correction factor for the higher S_N-order effect was estimated as the ratio of the k_eff value under the S₁₆ order to that under the S₈ order. In all FCA-IX assemblies, the S_N-order effects are negligibly small for both libraries. The correction factor for the higher P_L-order effect was estimated as the ratio of the k_eff value under the P₄-extended transport approximation to that under the P₀ − Σ_tr one. The correction factors for P_L-order effect range from 0.9978 to 0.9991 for JENDL-4.0. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.04%; hence, they are not significant.

3.4.7. 900-group effect

Since many stainless steel plates were used in the core cells for the two assemblies IX-4 and IX-5, the core calculations with a fine energy group structure would be required to treat resonances in detail for structural materials in the higher energy regions. The 900-energy group constant set provided in the SLAROM-UF code was employed. The correction factor for the energy group effect on core calculations was estimated as the ratio of k_eff values of the 900-energy group to the 70-energy group. In both calculations, the other conditions were fixed and the UF calculation scheme was commonly adopted. The correction factors range from 0.9912 to 1.0013 for JENDL-4.0. For assemblies IX-4 and IX-5 with large amounts of stainless steel, the energy group effects are significant and increase with the stainless steel content. The discrepancies between the factors for JENDL-4.0 and ENDF-B/VII.1 are within 0.03%; hence, they are not significant.

4. Sample calculations and discussion

4.1. Results by Monte Carlo calculations

Sample calculations were performed using MVP to discuss the usefulness of the present benchmark models for the assessment of the nuclear data. Two major libraries, JENDL-4.0 and ENDF-B/VII.1, were employed. The number of neutron histories was 200 million for each calculation. The statistical uncertainties of the MC calculations were less than 0.01%. The ratios of the MC calculations to the experimental results (C/E) are shown in Figure 16 and Table 12. Here, the correction factor for the heterogeneous effect was adopted for the MC calculation result. The results are discussed by dividing FCA-IX assemblies into the following three categories 4.1.1, 4.1.2, and 4.1.3. The sensitivity analyses are additionally described in the Appendix 2.

Figure 16. Ratio of the MC calculation result to the experimental result (C/E).

Table 12. Ratio of the MC calculation result to the experimental result.

	C/E value^a
Assembly	JENDL-4.0	ENDF/B-VII.1	Difference^b
IX-1	1.0058	1.0042	−0.16%
IX-2	1.0038	1.0072	0.34%
IX-3	1.0014	1.0071	0.57%
IX-4	1.0026	1.0040	0.14%
IX-5	1.0017	1.0038	0.21%
IX-6	0.9982	1.0016	0.34%
IX-7	0.9971	1.0045	0.74%

^aThe correction factor for the heterogeneous effect given in Table 8 was adopted for the MC calculation result. ^bDifference between C/E values by JENDL-4.0 and ENDF/B-VII.1.

4.1.1. Comparison among assemblies IX-1, IX-2, and IX-3

In assemblies IX-1–IX-3, HEU and graphite were the main components of the core regions. As shown in Table 1, the graphite content per core cell decreases from assembly IX-1 to assembly IX-3, whereas the ²³⁵U content increases. The ratios of the atomic number density of ¹²C to that of ²³⁵U range from 29.1 to 8.4. The neutron spectrum of the core region becomes harder as the graphite content decreases from assembly IX-1 to assembly IX-3, as shown in Figure 17.

Figure 17. Neutron spectra at the core centers of FCA-IX assemblies.

For assembly IX-1, the calculation value by JENDL-4.0 overestimates the experimental value by 0.58%, as shown in Table 12. As the neutron spectrum becomes harder, the C/E value by JENDL-4.0 decreases from 1.0058 to 1.0014. The calculation value for assembly IX-3 agrees with the experimental value. In contrast, the calculation values by ENDF-B/VII.1 overestimate by 0.42%, 0.72%, and 0.71% for assemblies IX-1, IX-2, and IX-3, respectively. The assembly dependences of the C/E values are quite different between JENDL-4.0 and ENDF-B/VII.1. The difference between the C/E values of these two libraries obviously increases from −0.16% in assembly IX-1 to 0.57% in assembly IX-3 as the spectrum becomes harder. Assemblies IX-1–IX-3 would be effective in the systematic assessment of ²³⁵U and ¹²C cross sections at the softer neutron spectra.

4.1.2. Comparison among assemblies IX-4, IX-5, and IX-6

In assemblies IX-4 and IX-5, HEU and stainless steel were the main components of the core regions. Their spectra are harder than those in assemblies IX-1–IX-3 with graphite. As shown in Table 12, the calculation values by JENDL-4.0 overestimate the experimental value by 0.26% and 0.17% in assemblies IX-4 and IX-5, respectively. The calculation values by ENDF-B/VII.1 also overestimate the experimental value by 0.40% and 0.38% in assemblies IX-4 and IX-5, respectively. The overestimations by ENDF-B/VII.1 are larger than those by JENDL-4.0. Assemblies IX-4 and IX-5, together with the previous three assemblies, would be effective in the systematic assessment of ²³⁵U cross sections in a wide range of neutron energy.

In assembly IX-6, voided stainless steel plates were used instead of the stainless steel plates in the core region. As shown in Table 1, the stainless steel content of assembly IX-6 is smaller than that of assembly IX-5 by a factor of approximately 0.36, whereas their fuel contents are the same. From these composition features, the change in the C/E value of k_eff from assembly IX-6 to assembly IX-5 can be roughly regarded as the C/E value of the stainless steel reactivity worth. As shown in Table 12, the C/E value by JENDL-4.0 increases from 0.9982 in assembly IX-6 to 1.0017 in assembly IX-5, which means 0.35% Δk/k overestimation of the stainless steel reactivity worth. On the other hand, the C/E value by ENDF-B/VII.1 increases from 1.0016 in assembly IX-6 to 1.0038 in assembly IX-5. The stainless steel reactivity worth by ENDF-B/VII.1 also overestimates by 0.22% Δk/k. The stainless steel reactivity worth by ENDF/B-VII.1 is better than JENDL-4.0 by 0.13% Δk/k, although both still overestimate. The comparison between assemblies IX-5 and IX-6 would mainly be useful in the assessment of cross sections of stainless steel materials.

4.1.3. Comparison between assemblies IX-6 and IX-7

In assembly IX-7, LEU was the only component of the core region. One finds a significant difference of 0.74% between the C/E values using the two libraries for assembly IX-7. As shown in Table 1, the ²³⁸U content of assembly IX-7 is larger than that of assembly IX-6 by a factor of approximately 62. In contrast, the contents of ²³⁵U and stainless steel of assembly IX-7 are almost the same as those of assembly IX-6. From these composition features, the change in the C/E value of k_eff from assembly IX-6 to assembly IX-7 can be roughly regarded as the C/E value of the ²³⁸U reactivity worth. As shown in Table 12, the C/E value by JENDL-4.0 decreases from 0.9982 in assembly IX-6 to 0.9971 in assembly IX-7, which means a slight underestimation of the ²³⁸U reactivity worth. In contrast, the C/E value by ENDF/B-VII.1 increases from 1.0016 in assembly IX-6 to 1.0045 in assembly IX-7. The ²³⁸U reactivity worth by ENDF-B/VII.1 overestimates by 0.29% Δk/k. There is an obvious difference between the C/E values of ²³⁸U reactivity worth by JENDL-4.0 and ENDF/B-VII.1. The comparison between assemblies IX-6 and IX-7 would be useful in the assessment of ²³⁸U cross sections.

4.2. Results by deterministic calculations

For users of deterministic codes, it is worth noting that the assessment and improvement of nuclear data can be achieved by simpler and more efficient calculations on the basis of the diffusion method. Therefore, the verification of the deterministic calculation method was also performed through comparison with the MC calculation results. Here, JENDL-4.0 and ENDF/B-VII.1 were employed. The most accurate deterministic result (C_det) was estimated as C_det = f_total×C_base by multiplying the base calculation result (C_base) by the total correction factor (f_total) given in Table 11. The base calculation corresponds to Method-0 in Table 9. Table 13 compares the most accurate deterministic result and the MC calculation result. Here, the correction factor for the heterogeneous effect was adopted for the MC calculation result (C_MC). The ratio of C_det/C_MC by JENDL-4.0 decreases as the neutron spectrum becomes softer. One finds that the most accurate deterministic results underestimate the MC calculations by approximately 0.6% for the assemblies with the softer neutron spectra. The degrees of underestimations are almost equal to the accuracy of the nuclear data. The results by ENDF-B/VII.1 show a similar tendency to those by JENDL-4.0.

Table 13. Comparison between the most accurate deterministic result and the MC calculation result.

	Base calc.	Most accurate calc.
Assembly	Cbase	Cdet = ftotal×Cbase	Cdet/CMC*
(a) JENDL-4.0
IX-1	0.9952	1.0050	0.9944
IX-2	0.9856	1.0037	0.9941
IX-3	0.9786	0.9991	0.9950
IX-4	0.9982	1.0066	0.9977
IX-5	0.9943	1.0068	0.9978
IX-6	0.9829	1.0025	0.9995
IX-7	0.9913	1.0037	1.0005
(b) ENDF-B/VII.1
IX-1	0.9942	1.0038	0.9948
IX-2	0.9900	1.0079	0.9949
IX-3	0.9854	1.0055	0.9957
IX-4	0.9999	1.0086	0.9983
IX-5	0.9969	1.0097	0.9986
IX-6	0.9868	1.0068	1.0004
IX-7	0.9993	1.0112	1.0006

*The correction factor for the heterogeneous effect given in Table 8 was adopted for C_MC.

As mentioned in Section 3.4 for the deterministic calculation method, the current-weighted total cross sections were conventionally adopted in the definition of transport cross sections. However, there is another option for the definition of transport cross sections as discussed in [10]. Since the current is inversely proportional to the square of the resonance peak, the total cross section condensed by current weighting is extremely small in the resonance region, especially in the blanket region where a large amount of ²³⁸U was loaded. Therefore, flux weighting instead of current weighting was reconsidered for the transport cross section in the blanket region to overcome the above-mentioned underestimations. Here, the weighting factor (f_weight) was introduced as the ratio of the k^flux-weighed_eff value by the flux-weighted transport cross section to the k^{current-weighed}_eff value by the current-weighted transport cross section. The revised deterministic result (C_rev) was estimated as C_rev = f_weight × C_det. Table 14 compares the revised deterministic result and the MC calculation result. For all assemblies, the k^{flux − weighed}_eff values are larger than the k^{current − weighed}_eff values. The weighting factor by JENDL-4.0 increases from 1.0015 in assembly IX-7 to 1.0060 in assembly IX-1 as the spectrum becomes softer. The revised deterministic results by the weighting factors are confirmed to be consistent with the MC calculations within 0.2%, as shown in Table 14. The results by ENDF-B/VII.1 show a similar tendency to those by JENDL-4.0. From these results, the adoption of the weighting factor given in Table 14 is recommended to obtain values consistent with the MC calculation.

Table 14. Comparison between revised deterministic result and the MC calculation result.

	Weighting	Revised calc.
Assembly	factor ^a fweight	Crev = fweight×Cdet	Crev/CMC^b
(a) JENDL-4.0
IX-1	1.0060	1.0110	1.0003
IX-2	1.0053	1.0090	0.9994
IX-3	1.0046	1.0037	0.9996
IX-4	1.0026	1.0092	1.0003
IX-5	1.0025	1.0093	1.0003
IX-6	1.0020	1.0045	1.0015
IX-7	1.0015	1.0052	1.0020
(b) ENDF-B/VII.1
IX-1	1.0055	1.0093	1.0002
IX-2	1.0047	1.0126	0.9995
IX-3	1.0040	1.0095	0.9996
IX-4	1.0022	1.0108	1.0005
IX-5	1.0022	1.0119	1.0008
IX-6	1.0017	1.0085	1.0021
IX-7	1.0014	1.0126	1.0020

^aThe weighting factor was defined as f_weight = k^flux-weighed_eff/k^{current-weighed}_eff with the k^flux-weighed_eff value by the flux-weighted transport cross section and the k^{current-weighed}_eff value by the current-weighted transport cross section. ^bThe correction factor for the heterogeneous effect given in Table 8 was adopted for C_MC.

5. Conclusions

New benchmark models for criticalities of FCA-IX assemblies with systematically changed neutron spectra were established. This paper provides the specifications of homogeneous benchmark models, which were established by simplifying the geometric details and homogenizing the plate-wise heterogeneity, with several correction factors.

The sample calculations of the benchmark models were performed using the continuous-energy MC code with the two major nuclear data libraries, JENDL-4.0 and ENDF-B/VII.1. It was found that the assembly dependences of the C/E values were obviously different between JENDL-4.0 and ENDF-B/VII.1. The present benchmark models would be well suited for the assessment and improvement of the nuclear data for ²³⁵U, ²³⁸U, graphite, and stainless steel because of their advantages of simplicity and systematically changed neutron spectra. The verification of the deterministic method was further performed on the benchmark models by comparison with the MC calculations. The present simplified benchmark models are also available to users of deterministic calculation codes for the assessment and improvement of nuclear data.

Acknowledgements

The authors express thanks to Mr. M. Ishikawa for valuable information and helpful discussion about the weighting functions of transport cross sections. The authors also wish to thank Mr. T. Sakino for performing the calculations and to express their gratitude to the FCA staff for maintaining the facilities.

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix 1.

We prepared a correlation matrix between the benchmark k_eff values of assemblies IX-1–IX-7 in Table A1. The matrix was evaluated by adopting the method mentioned in [11]. All uncertainties were classified to either uncorrelated or correlated components between these assembles. The correlation factor was zero or unity according to whether they were uncorrelated or correlated.

Table A1. Correlation matrix between the benchmark k_eff values.

	IX-1	IX-2	IX-3	IX-4	IX-5	IX-6	IX-7
IX-1	1.000	0.904	0.836	0.106	0.072	0.110	0.004
IX-2	0.904	1.000	0.818	0.159	0.110	0.166	0.008
IX-3	0.836	0.818	1.000	0.209	0.145	0.218	0.010
IX-4	0.106	0.159	0.209	1.000	0.341	0.542	0.013
IX-5	0.072	0.110	0.145	0.341	1.000	0.354	0.018
IX-6	0.110	0.166	0.218	0.542	0.354	1.000	0.010
IX-7	0.004	0.008	0.010	0.013	0.018	0.010	1.000

The statistical uncertainties were identified as uncorrelated components: uncertainties due to the mechanical reproducibility of the SCR positions, that of closeness between the two tables, the measurement errors of the doubling time, the core temperature, and reactivity worths for the simplifications. The uncertainties due to the delayed neutron data were identified as correlated components because the same date for delayed neutron was adopted in all the assemblies. The uncertainties due to the compositions were considered as follows. For instance, the HEU plates were commonly used in assemblies IX-1–IX-6. The uncertainties due to the weights and uranium enrichments of HEU plates were assumed to be correlated between assemblies IX-1–IX-6 except assembly IX-7. For the other plates and blocks, the correlation factors were determined in the same manner.

Appendix 2.

Sensitivity analyses were performed to investigate the differences between JENDL-4.0 and ENDF/B-VII.1. The sensitivity coefficients are defined as $$\begin{array}{c}\hfill {\mathrm{S}}_{m,x,g}^L=\frac{d{k}_{\mathrm{eff}}/k_{\mathrm{eff}}}{d{\sigma }_{m,x,g}^L/{\sigma }_{m,x,g}^L}.\end{array}$$Here, m, x, g, and L denote the nuclide, reaction type, energy group, and library, respectively. Sensitivity coefficients were obtained by the 70-group diffusion calculation using the SAGEP generalized perturbation code [12]. Here, the group constant set based on JENDL-4.0 was employed to prepare the effective cross sections by SLAROM-UF. Figure B1(a) and (b) show the energy-wise sensitivity coefficients of the k_eff values to the ²³⁵U capture and the ²³⁵U fission cross sections (σ_{U235, cap} and σ_{U235, fis}, hereafter), respectively. These sensitivity coefficients shift to the high-energy region as the neutron spectrum becomes harder from assembly IX-1 to assembly IX-6. Using the sensitivity coefficients and the changes of cross sections by replacing from JENDL-4.0 to ENDF/B-VII.1, the changes in the k_eff values were evaluated as follows: $$\begin{array}{c}{\left(\frac{d{k}_{\mathrm{eff}}}{k_{\mathrm{eff}}}\right)}_{m,x,g}^{\mathrm{JENDL}-4.0\to \mathrm{ENDF}/\mathrm{B}-\mathrm{VII}.1}\hfill \\ & & =S_{m,x,g}^{\mathrm{JENDL}-4.0}\left(\frac{{\sigma }_{m,x,g}^{\mathrm{ENDF}/\mathrm{B}-\mathrm{VII}.1}-{\sigma }_{m,x,g}^{\mathrm{JENDL}-4.0}}{{\sigma }_{m,x,g}^{\mathrm{JENDL}-4.0}}\right).\hfill \end{array}$$Figure B2(a) and (b) show the energy-wise changes in the k_eff values caused by the changes of σ_{U235, cap} and σ_{U235, fis}, respectively. Table B1 summarizes the nuclide-reaction-wise changes in the k_eff values, which were summed over all energy groups.

Figure B1. Energy-wise sensitivity coefficients of k_eff values.

Figure B2. Energy-wise changes in k_eff values by replacing from JENDL-4.0 to ENDF/B-VII.1.

Table B1. Nuclide-reaction-wise changes in k_eff values by replacing from JENDL-4.0 to ENDF/B-VII.1.

		Changes in keff value (% Δk/k)
Nuclide	Reaction type	IX-1	IX-2	IX-3
(a) Assemblies IX-1, IX-2, and IX-3
^235U	Capture	−0.94	−0.40	−0.11
	ν	0.13	0.06	0.02
	Fission	0.32	0.41	0.45
	Elastic	−0.01	−0.02	−0.02
	Inelastic	0.05	0.08	0.10
^238U	Capture	0.01	0.02	0.02
	Inelastic	−0.10	−0.09	−0.07
	μ-average	0.01	0.01	0.02
^12C	Elastic	0.22	0.15	0.11
	μ-average	0.19	0.15	0.12
^52Cr	μ-average	−0.03	−0.03	−0.03
^56Fe	Elastic	−0.02	−0.02	−0.02
	Inelastic	−0.02	−0.01	−0.01
^58Ni	Capture	0.02	0.01	0.01
Other		−0.01	−0.01	−0.01
Total		−0.17	0.33	0.59
		Changes in keff value (% Δk/k)
Nuclide	Reaction type	IX-4	IX-5	IX-6	IX-7
(b) Assemblies IX-4, IX-5, IX-6, and IX-7
^235U	Capture	0.09	0.14	0.14	0.14
	ν	−0.01	−0.03	−0.05	−0.04
	Fission	0.48	0.47	0.45	0.48
	Elastic	−0.01	−0.02	−0.03	−0.01
	Inelastic	0.03	0.06	0.09	0.03
	μ-average	−0.02	−0.02	−0.01	−0.01
^238U	Capture	0.02	0.03	0.04	0.10
	ν	0.00	0.00	−0.01	−0.04
	Fission	0.00	0.00	0.00	0.03
	Inelastic	−0.11	−0.10	−0.12	0.12
	μ-average	−0.01	0.00	0.03	0.01
^52Cr	Elastic	−0.02	−0.01	0.01	0.00
	μ-average	−0.19	−0.17	−0.08	−0.03
^53Cr	Elastic	−0.02	−0.02	−0.01	0.00
	Inelastic	−0.03	−0.02	−0.01	0.00
	μ-average	−0.04	−0.03	−0.01	0.00
^55Mn	Inelastic	0.05	0.03	0.01	0.00
^54Fe	Capture	0.02	0.01	0.01	0.00
^56Fe	Elastic	−0.10	−0.09	−0.04	−0.01
	Inelastic	−0.06	−0.05	−0.01	0.01
	μ-average	0.06	0.08	0.02	0.01
^57Fe	Elastic	−0.07	−0.05	−0.02	−0.01
^58Ni	Capture	0.09	0.06	0.02	0.01
	μ-average	0.03	0.02	0.01	0.00
^60Ni	Elastic	−0.04	−0.03	−0.01	−0.01
Other		−0.02	−0.02	−0.03	0.00
Total		0.11	0.24	0.38	0.79

B.1.  Comparison among assemblies IX-1, IX-2, and IX-3

In assembly IX-1, the changes of σ_{U235, cap} and σ_{U235, fis} by replacing from JENDL-4.0 to ENDF/B-VII.1 mainly cause the k_eff changes of −0.94% Δk/k and 0.32% Δk/k, respectively, as shown in Table B1(a). The total k_eff change is −0.17% Δk/k due to a large cancelation of the k_eff changes caused by σ_{U235, cap}, σ_{U235, fis}, and other cross sections, such as the ¹²C elastic and the μ-average, i.e., the average cosine of the ¹²C scattering angle. Figure B2(a) shows that the negative change in k_eff caused by σ_{U235, cap} of several keV obviously decreases as the neutron spectrum becomes harder from assembly IX-1 to assembly IX-3. In assembly IX-3, the k_eff change caused by σ_{U235, cap} is −0.11% Δk/k, as shown in Table B1(a). In contrast, Figure B2(b) shows that the positive changes in k_eff caused by σ_{U235, fis} in the energy region from 10 keV to 1 MeV are almost constant in assemblies IX-1–IX-3. In assembly IX-3, the k_eff change caused by σ_{U235, fis} is 0.45% Δk/k, as shown in Table B1(a). From the results for assemblies IX-1–IX-3, the larger σ_{U235, fis} of ENDF/B-VII.1 compared with that of JENDL-4.0 in the energy region from 10 keV to 1 MeV is one of the factors that causes the overestimations in k_eff values by ENDF/B-VII.1. Simultaneously, the smaller σ_{U235, cap} of JENDL-4.0 compared with that of ENDF/B-VII.1 of several keV is one of the factors that causes the assembly dependence of C/E values by JENDL-4.0.

B.2.  Comparison among assemblies IX-4, IX-5, and IX-6

In assemblies IX-4–IX-6, the changes of σ_{U235, fis} by replacing from JENDL-4.0 to ENDF/B-VII.1 mainly cause the k_eff changes from 0.45% Δk/k to 0.48% Δk/k, as shown in Table B1(b), and their k_eff changes are almost constant in the energy region from 10 keV to 1 MeV, as shown in Figure B2(b). Therefore, the larger σ_{U235, fis} of ENDF/B-VII.1 compared with that of JENDL-4.0 in the energy region from 10 keV to 1 MeV is one of the factors that causes the overestimations in k_eff values by ENDF/B-VII.1 also in assemblies IX-4–IX-6.

As mentioned in Section 4.1, the C/E change of the k_eff values from assembly IX-6 to assembly IX-5 can be roughly regarded as the C/E value of the stainless steel reactivity worth. Table 12 shows that the C/E values of k_eff by JENDL-4.0 are 0.9982 and 1.0017 for assembly IX-6 and assembly IX-5, respectively. From these results by JENDL-4.0, the C/E change from assembly IX-6 to assembly IX-5 is 0.35%. On the other hand, the C/E change by ENDF/B-VII.1 is 0.22% because the C/E values of k_eff are 1.0016 and 1.0038 for assembly IX-6 and assembly IX-5, respectively. These results mean that the stainless steel reactivity worth by ENDF/B-VII.1 provides a better result than that by JENDL-4.0 by 0.13% Δk/k. On the other hand, Table B1(b) shows that the k_eff change caused by the μ-average of ⁵²Cr by replacing from JENDL-4.0 to ENDF/B-VII.1 becomes more negative from −0.08% Δk/k in assembly IX-6 to −0.17% Δk/k in assembly IX-5. It is one of the contributions that causes the difference between the C/E values of stainless steel reactivity worths by JENDL-4.0 and ENDF/B-VII.1.

B.3.  Comparison between assemblies IX-6 and IX-7

In assemblies IX-6 and IX-7, the positive changes in k_eff caused by σ_{U235, fis} are almost the same, as shown in Table B1(b). In contrast, the k_eff changes caused by the ²³⁸U cross sections are quite different. As mentioned in Section 4.1, the C/E change of the k_eff values from assembly IX-6 to assembly IX-7 can be roughly regarded as the C/E value of the ²³⁸U reactivity worth. Table 12 shows that the C/E values of k_eff by JENDL-4.0 are 0.9982 and 0.9971 for assembly IX-6 and assembly IX-7, respectively. From these results by JENDL-4.0, the C/E change from assembly IX-6 to assembly IX-7 is −0.11%. On the other hand, the C/E change by ENDF/B-VII.1 is 0.29% because the C/E values of k_eff are 1.0016 and 1.0045 for assembly IX-6 and assembly IX-7, respectively. These results imply that the ²³⁸U reactivity worths are different by 0.40% Δk/k between the libraries. On the other hand, Table B1(b) shows that the k_eff change caused by the ²³⁸U inelastic cross section by replacing from JENDL-4.0 to ENDF/B-VII.1 obviously increases, from −0.12% Δk/k in assembly IX-6 to 0.12% Δk/k in assembly IX-7, by 0.24% Δk/k. It mainly contributes to the difference between the C/E values of ²³⁸U reactivity worths by JENDL-4.0 and ENDF/B-VII.1.