Similarity Coefficient Generation Using Adjoint-Based Sensitivity and Uncertainty Method and Stochastic Sampling Method

Abstract

In this study, a similarity coefficient generation code system was established using the Monte Carlo Code for Advanced Reactor Design (McCARD) transport code and the MIG multi-correlated input sampling code. We considered the adjoint-based sensitivity and uncertainty (S/U) and stochastic sampling (S.S.) approaches to the generation of the c_k similarity coefficient. To examine the code system, the c_k similarity coefficients of 23 relevant critical experiments and the System-Integrated Modular Advanced Reactor (SMART) small modular reactor (SMR) target system were generated using ENDF/B-VII.1 covariance data with the Los Alamos National Laboratory (LANL) 30-group energy group structure. Our results show that the similarity coefficients between the 16 LEU thermal-spectrum-based critical experiments and SMART are more than 0.90, which is the recommended criterion of the U.S. Nuclear Regulatory Commission (NRC). These results are very helpful for licensees and can be used to justify the determination of critical experiment benchmarks for computational bias estimations of the SMART target system. To examine the discrepancy in the similarity coefficient, c_k, due to the covariance data, similarity analyses for a 24 × 24 benchmark matrix were performed using ENDF-VIII.0, JENDL-5.0, and JEFF-3.3 covariance data. The results show that the selection of the covariance data used for c_k generation significantly impacts the similarity coefficient. Moreover, it was observed that the current results for the SCALE 6.1 covariance data show a consistent trend with the results reported in earlier studies.

1. Introduction

Recently, new nuclear reactor systems have been developed by various research institutes and industrial companies using advanced design methods and codes. In the design of new nuclear reactor systems, a designer should provide enough safety margins to ensure that the system is adequately subcritical under any condition. To perform safety analyses for a new system, uncertainties or biases in design codes and cross-section data should also be provided by comparing experimental and calculated safety parameters such as criticality. It is crucial to select the appropriate critical experiments when estimating safety margins and bias. In the selection of critical experiments, nuclear system designers and licensees should provide computational justifications to regulating bodies. According to the technical guidance in [1], a critical experiment employed for benchmarking codes or cross-sections should, to the greatest extent possible, encompass configurations exhibiting neutronic and geometrical characteristics closely resembling those envisaged for the proposed system. There are a few studies that have quantified the degree of similarity between critical experiments and target applications for justification [2,3,4,5]. In these studies, sensitivity techniques play a crucial role by quantifying first-order relative variations in k_eff to generate similarity coefficients, which are often referred to as similarity indices. The similarity coefficient of k_eff, c_k, gives information relative to the similarity in criticality between a critical experiment and a specific target system. These similarity coefficients or indices are comprehensive mathematical metrics, pivotal for assessing the relevance of both existing and new experiments within specific application conditions. A similarity coefficient is based on the definition of a correlation coefficient; hence, as the value approaches 1.0, it signifies an increasing similarity between two systems. B. L. Broadhead et al. [2] introduced the adjoint-based sensitivity and uncertainty (adjoint S/U) analysis method for validating benchmark datasets essential in critical safety applications. C. M. Perfetti and B. T. Rearden [3] utilized similarity coefficients to classify benchmark experiments into groups based on their neutronic similarity for criticality safety applications. In these two studies, the Tools for Sensitivity and Uncertainty Analysis Methodology Implementation (TSUNAMI) code in the Standardized Computer Analyses for Licensing Evaluation (SCALE) code package was used to calculate a correlation coefficient that was the numerical value of the degree of similarity between an experiment and a target system. The Monte Carlo N-Particle Transport (MCNP)-Whisper [4] simulation code is a Monte Carlo computer code package used to assist in nuclear criticality safety (NCS). It can provide bias and uncertainties to determine the upper subcritical limit (USL) and quantify similarity to select critical experiments. The American Nuclear Society (ANS) standards delineate the USL as “a limit on the computed k_eff set to guarantee that conditions determined to be subcritical will indeed remain subcritical [1]”. In determining the USL, we consider statistical uncertainty to be a 68% confidence level in the bias. Meanwhile, D. Huang et al. [5] proposed a simple non-intrusive method (i.e., the stochastic sampling method) to calculate similarity indices as an independent verification tool. In this study, similarity coefficients were estimated by the Sampler-KENO sequence using a different number of cross-section samples. It was observed that the adjoint S/U method-based Tools for Sensitivity and Uncertainty Analysis Methodology Implementation–Indices and Parameters (TSUNAMI-IP) results were verified by the stochastic sampling (SS) method. In these previous studies on similarity, a similarity threshold value for assuming the resemblance between two systems was determined by the evaluated nuclear covariance data in each used code. Furthermore, there is a noticeable absence of studies in which similarity calculations were conducted under the same calculation conditions for both the adjoint S/U method and the SS method.

In this study, a new Monte Carlo Code for Advanced Reactor Design–McCARD Input Generator (McCARD-MIG) coupling code system [6,7,8] for similarity quantification was introduced for both the adjoint S/U and S.S. methods. Section 2 explains the methodologies and code systems for similarity coefficient generation. In Section 3, to examine the similarity generation code system, an estimation of the degree of similarity between critical experiment benchmarks [9] and the System-Integrated Modular Advanced Reactor (SMART) [10] small modular reactor (SMR) target system is conducted using deterministic-based adjoint S/U methods. For verification, the similarity coefficients among some benchmarks determined with the adjoint S/U method are compared with those determined with the SS method. Furthermore, sensitivity analyses of the similarity coefficients were conducted to consider the influence of various evaluated nuclear covariance data. For this investigation, we utilized the latest cross-section covariance data: ENDF/B-VII.1 [11], ENDF/B-VIII.0 [12], JENDL-5.0 [13], and JEFF-3.3 [14]. The conclusions are given in Section 4.

2. Similarity Coefficient Generation

2.1. Similarity Coefficient Generation with the Adjoint-Based S/U Method

In the studies of [2,3,4,5], which concerned similarity tests between experiments and applications, the similarity coefficient between the multiplication factors of two systems, c_k, was defined as in Equation (1).

$$c_k=\frac{\mathrm{cov}\left[k_{eff}^I,k_{eff}^{II}\right]}{\sigma (k_{eff}^I)\cdot \sigma (k_{eff}^{II})},$$

(1)

The c_k is a Pearson correlation coefficient [15] and can describe the correlation relationship between two systems (i.e., system I and system II). If two systems contain the same materials, the c_k can help one identify how closely they are correlated to each other. In the same manner as the correlation coefficient, the similarity coefficient ranges from −1 to 1. If the similarity coefficient is close to 1, the two systems are strongly positively correlated. Conversely, when the correlation coefficient is −1, it indicates a strong negative correlation. In the common deterministic adjoint-based S/U method [2,3], the covariance term between k_eff values for two systems can be calculated as

$$\mathrm{cov}\left[k_{eff}^I,k_{eff}^{II}\right]={\displaystyle \sum _{j,\alpha ,g}{\displaystyle \sum _{j^{\prime },{\alpha }^{\prime },g^{\prime }}\mathrm{cov}\left[x_{\alpha ,g}^j,x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}\right]\left(\frac{\partial k_{eff}^I}{\partial x_{\alpha ,g}^j}\right)\left(\frac{\partial k_{eff}^{II}}{\partial x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}}\right)}},$$

(2)

where $x_{\alpha ,g}^j$ is the $\alpha$-type microscopic cross-section of isotope j for energy group g. The nuclear reaction cross-section covariance matrix, $\mathrm{cov}[x_{\alpha ,g}^i,x_{{\alpha }^{\prime },g^{\prime }}^{i^{\prime }}]$, can be taken from an evaluated nuclear data library, whereas the sensitivity coefficients can be obtained from adjoint analyses for the two systems.

However, if we try to obtain c_k using the Monte Carlo (MC) method or code, statistical uncertainty effects should also be considered. In general, the mean of an MC estimate on a multiplication factor k_eff and its variance can be expressed by

$$\overline{k_{eff}}=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N{k}_{eff,i}},$$

(3)

$${\sigma }^2\left[k_{eff}\right]=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(k_{eff,i}-\overline{k_{eff}}\right)}^2},$$

(4)

where i and N are the index number and total number of histories. If one assumes that the total uncertainty on k_eff is due to the statistical uncertainties of the MC calculations and the nuclear reaction cross-section uncertainties, Equation (4) can be rewritten as [16]

$${\sigma }^2\left(k_{eff}\right)=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(k_{eff,i}-<k_{eff}>+<k_{eff}>-\overline{k_{eff}}\right)}^2},$$

(5)

$$<k_{eff}>=\frac{1}{N}{\displaystyle \sum _{i=1}^N{k}_{eff,i}}.$$

(6)

The angular bracket in <k_eff> represents the operator, implying the expected value of a quantity on it. Using the first-order Taylor expansion for <k_eff> about the mean values of nuclear reaction cross-section, $<k_{eff}>-\overline{k_{eff}}$ can be expressed as

$$<k_{eff}>-\overline{k_{eff}}\approx {\displaystyle \sum _j{\displaystyle \sum _{\alpha }{\displaystyle \sum _g\left({\left(x_{\alpha ,g}^j\right)}_k-\overline{x_{\alpha ,g}^j}\right)}}\left(\frac{\partial k_{eff}}{\partial x_{\alpha ,g}^j}\right)}.$$

(7)

By substituting Equation (7) into Equation (5), one can obtain

$${\sigma }^2\left(k_{eff}\right)={\sigma }_{SS}^2\left(k_{eff}\right)+{\sigma }_{XX}^2\left(k_{eff}\right),$$

(8)

where

$${\sigma }_{SS}^2\left(k_{eff}\right)=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(k_{eff,i}-<k_{eff}>\right)}^2},$$

(9)

$$\begin{array}{ll}{\sigma }_{XX}^2\left(k_{eff}\right)& =\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(<k_{eff}>-\overline{k_{eff}}\right)}^2}\\ & ={\displaystyle \sum _{j,\alpha ,g}{\displaystyle \sum _{j^{\prime },{\alpha }^{\prime },g^{\prime }}\mathrm{cov}\left[x_{\alpha ,g}^j,x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}\right]\left(\frac{\partial k_{eff}}{\partial x_{\alpha ,g}^j}\right)\left(\frac{\partial k_{eff}}{\partial x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}}\right)}}\end{array}.$$

(10)

${\sigma }_{SS}^2\left(Q\right)$ is the statistical contribution to the variance of Q, whereas ${\sigma }_{XX}^2\left(Q\right)$ is the contribution from the nuclear reaction cross-section uncertainties, which is commonly known as the sandwich equation or rule [17] for common S/U analyses. In the same manner, the covariance term k_eff values for two systems can be derived as

$$\mathrm{cov}\left[k_{eff}^I,k_{eff}^{II}\right]={\mathrm{cov}}_{SS}\left[k_{eff}^I,k_{eff}^{II}\right]+{\mathrm{cov}}_{XX}\left[k_{eff}^I,k_{eff}^{II}\right],$$

(11)

where

$${\mathrm{cov}}_{SS}\left[k_{eff}^I,k_{eff}^{II}\right]=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N\left(k_{eff,i}^I-<k_{eff}^I>\right)\left(k_{eff,i}^{II}-<k_{eff}^{II}>\right)}\approx 0,$$

(12)

$$\begin{array}{c}{\mathrm{cov}}_{XX}\left[k_{eff}^I,k_{eff}^{II}\right]=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{i=1}^N\left(<k_{eff}^I>-\overline{k_{eff}^I}\right)\left(<k_{eff}^{II}>-\overline{k_{eff}^{II}}\right)}\\ ={\displaystyle \sum _{j,\alpha ,g}{\displaystyle \sum _{j^{\prime },{\alpha }^{\prime },g^{\prime }}\mathrm{cov}\left[x_{\alpha ,g}^j,x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}\right]\left(\frac{\partial k_{eff}^I}{\partial x_{\alpha ,g}^j}\right)\left(\frac{\partial k_{eff}^{II}}{\partial x_{{\alpha }^{\prime },g^{\prime }}^{j^{\prime }}}\right)}}\end{array}.$$

(13)

As shown in Equation (12), the statistical contribution to the covariance is 0 because it is independent. Finally, one can obtain c_k by substituting Equations (8) and (11) into Equation (1) in the MC method.

2.2. Similarity Coefficient Generation with the S.S. Method

In general, the similarity coefficients can be easily and directly generated using the S.S. method, which is similar to the Cholesky decomposition method employed in [5].

$$X^i=\left(\dots ,{\left(x_{\alpha ,g}^j\right)}^i,\dots \right),$$

(14)

where ${(x_{\alpha ,g}^i)}^k$ indicates the i-th microscopic cross-section of isotope j for energy group g sampled from a cross-section covariance matrix and Xⁱ is the i-th cross-section set. In general, one can generate N sets of cross-section samples from the standard normal distribution using the Cholesky decomposition method.

$${\sigma }^2(k_{eff}^I)=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(k_{eff}^I(X^i)-\overline{k_{eff}^I(X^i)}\right)}^2},$$

(15)

$${\sigma }^2(k_{eff}^{II})=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(k_{eff}^{II}(X^i)-\overline{k_{eff}^{II}(X^i)}\right)}^2},$$

(16)

$$\mathrm{cov}\left[k_{eff}^I,k_{eff}^{II}\right]=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left(k_{eff}^I(X^i)-\overline{k_{eff}^I(X^i)}\right)\left(k_{eff}^{II}(X^i)-\overline{k_{eff}^{II}(X^i)}\right)},$$

(17)

where $k_{eff}^I(X^i)$ and $k_{eff}^{II}(X^i)$ represent the k_eff s calculated by the i-th sampled cross-section set for systems I and II, respectively. Finally, c_k can be calculated using its definition and Equations (15)–(17). This c_k in the statistics has the same meaning and value as that in the adjoint S/U method from Equation (1).

2.3. McCARD/MIG Similarity Coefficient Generation Code System

In this study, the similarity coefficient generation code system was established using the McCARD MC transport code [6], the MIG multi-correlated input sampling code [7,8], and the SimTest editing utility. Figure 1 shows the flow chart for the similarity coefficient generation code system. The McCARD code already has the capability to perform the MC perturbation technique for sensitivity coefficient generation. The SimTest utility conducts the adjoint S/U analyses, as shown in Equation (10), and calculates the similarity coefficient, c_k, using the sensitivity coefficients and the covariance data. For the S.S. method, the MIG code can generate the nuclear reaction cross-section sample sets, as shown in Equation (14). A sequence of k_eff s and c_k can be calculated using McCARD with the sampled cross-section sets. The similarity coefficients can be easily and directly generated using the S.S. method.

3. Similarity Tests among Various Systems

3.1. Similarity Test for SMART SMR System

The similarity coefficients between the selected 23 critical experiment (CE) benchmarks [9] and the target application—the SMART SMR system [10]—were calculated with the adjoint S/U method. For the adjoint S/U analyses, a covariance data matrix with the LANL 30-group structure was used for two major actinide isotopes (i.e., ²³⁵U and ²³⁸U).

Table 1. Description of the selected critical experiment benchmarks and SMART.

ID	Short Name	Benchmark ID	Ref No.	Spectrum	235U Enrichments (w/o)	Pin Pitch (cm)	Pellet Radius (cm)
A	SMART		[10]	Thermal	2.82/4.88	1.26	0.4096
B	GODIVA	HEU-MET-FAST-001	[9]	Fast	94.0	-	-
C	FLATTOP25	HEU-MET-FAST-028		Fast	93.2	-	-
D	HMF-002c2	HEU-MET-FAST-002 case2		Fast	97.6	-	-
E	HMF-004	HEU-MET-FAST-004		Fast	97.7	-	-
F	HMF-018	HEU-MET-FAST-018		Fast	90.0	-	-
G	HMF-027	HEU-MET-FAST-027		Fast	90.0	-	-
H	HMF-032	HEU-MET-FAST-032		Fast	93.5	-	-
I	LCT-001c1	LEU-COMP-THERM-001 case1		Thermal	2.35	2.032	0.635
J	LCT-002c1	LEU-COMP-THERM-002 case1		Thermal	4.31	2.54	0.6325
K	LCT-003c1	LEU-COMP-THERM-003 case1		Thermal	2.35	1.684	0.5588
L	LCT-004c1	LEU-COMP-THERM-004 case1		Thermal	4.31	1.892	0.6325
M	LCT-005c1	LEU-COMP-THERM-005 case1		Thermal	4.31	2.398	0.6325
N	LCT-006c1	LEU-COMP-THERM-006 case1		Thermal	1.5	1.849	0.625
O	LCT-010c9	LEU-COMP-THERM-010 case9		Thermal	4.31	2.54	0.6325
P	LCT-017c13	LEU-COMP-THERM-017case13		Thermal	2.35	2.032	0.5588
Q	LMT-007c1	LEU-MET-THERM-007 case1		Thermal	4.95	1.30	0.38645
R	LMT-007c2	LEU-MET-THERM-007 case2		Thermal	4.95	1.53	0.38645
S	ORNL1	HEU-SOL-THERM-013 case1		Thermal	93.2	-	-
T	ORNL2	HEU-SOL-THERM-013 case2		Thermal	93.2	-	-
U	ORNL3	HEU-SOL-THERM-013 case3		Thermal	93.2	-	-
V	ORNL4	HEU-SOL-THERM-013 case4		Thermal	93.2	-	-
W	ORNL10	HEU-SOL-THERM-032		Thermal	93.2	-	-
X	AGN-201K		[18]	Thermal	19.5	-	-

shows the description of the selected 23 critical experiments and SMART. In Table 1, the benchmark ID ‘HMF’ stands for ‘HEU-MET-FAST’ and refers to a benchmark problem related to highly enriched uranium metal-type fuel-critical experiments characterized by a fast neutron spectrum. The benchmark IDs ‘LCT’ and ‘LMT’ stand for ‘LEU-COMP-THERM’ and ‘LEU-MET-THERM’, respectively. In the benchmark IDs, ‘COMP’ signifies a system with a mixture component, while ‘THERM’ indicates a thermal neutron spectrum system. Godiva, Flattop25, HMF-002 case2, HMF-004, HMF-018, HMF-027, and HMF-043 benchmarks are fast spectrum systems, whereas the others are thermal spectrum systems. Meanwhile, the SMART core has fuel assemblies that include 2.82 w/o- and 4.88 w/o-enriched UO₂ fuel rods, and its pin pitch is about 1.26 cm.

Table 2. Similarity coefficients among the 24 × 24 benchmark matrix problems with ENDF/B-VII.1 covariance.

	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X
A	1.00 *	0.32	0.34	0.32	0.40	0.34	0.33	0.32	0.98	0.98	0.99	0.99	0.99	0.99	0.98	0.98	0.99	0.99	0.97	0.97	0.97	0.97	0.97	0.99
B	0.32	1.00	0.95	0.99	0.97	0.96	0.97	0.99	0.20	0.22	0.23	0.28	0.24	0.25	0.22	0.20	0.34	0.28	0.17	0.17	0.18	0.18	0.17	0.19
C	0.34	0.95	1.00	0.95	0.98	0.99	0.99	0.96	0.22	0.24	0.24	0.30	0.26	0.26	0.24	0.22	0.35	0.30	0.19	0.20	0.20	0.20	0.19	0.21
D	0.32	0.99	0.95	1.00	0.97	0.96	0.97	0.99	0.20	0.22	0.23	0.28	0.24	0.25	0.22	0.20	0.34	0.28	0.17	0.18	0.18	0.18	0.17	0.19
E	0.40	0.97	0.98	0.97	1.00	0.98	0.99	0.98	0.28	0.30	0.31	0.37	0.32	0.33	0.31	0.28	0.42	0.36	0.26	0.26	0.26	0.26	0.25	0.27
F	0.34	0.96	0.99	0.96	0.98	1.00	0.99	0.97	0.22	0.24	0.24	0.30	0.26	0.26	0.24	0.22	0.35	0.30	0.19	0.19	0.20	0.20	0.19	0.21
G	0.33	0.97	0.99	0.97	0.99	0.99	1.00	0.98	0.22	0.23	0.24	0.30	0.25	0.26	0.24	0.22	0.35	0.30	0.19	0.19	0.19	0.19	0.18	0.20
H	0.32	0.99	0.96	0.99	0.98	0.97	0.98	1.00	0.20	0.22	0.23	0.29	0.24	0.25	0.22	0.20	0.34	0.28	0.17	0.18	0.18	0.18	0.17	0.19
I	0.98	0.20	0.22	0.20	0.28	0.22	0.22	0.20	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.99
J	0.98	0.22	0.24	0.22	0.30	0.24	0.23	0.22	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.99
K	0.99	0.23	0.24	0.23	0.31	0.24	0.24	0.23	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.99
L	0.99	0.28	0.30	0.28	0.37	0.30	0.30	0.29	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.98	0.98	0.98	0.98	0.99
M	0.99	0.24	0.26	0.24	0.32	0.26	0.25	0.24	0.99	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99
N	0.99	0.25	0.26	0.25	0.33	0.26	0.26	0.25	0.99	0.99	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99
O	0.98	0.22	0.24	0.22	0.31	0.24	0.24	0.22	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.99
P	0.98	0.20	0.22	0.20	0.28	0.22	0.22	0.20	0.99	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.99
Q	0.99	0.34	0.35	0.34	0.42	0.35	0.35	0.34	0.98	0.98	0.98	0.99	0.99	0.99	0.98	0.98	1.00	0.99	0.97	0.97	0.97	0.97	0.96	0.98
R	0.99	0.28	0.30	0.28	0.36	0.30	0.30	0.28	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.98	0.98	0.98	0.98	0.98	0.99
S	0.97	0.17	0.19	0.17	0.26	0.19	0.19	0.17	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.97	0.98	1.00	0.99	0.99	0.99	0.99	0.99
T	0.97	0.17	0.20	0.18	0.26	0.19	0.19	0.18	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.97	0.98	0.99	1.00	0.99	0.99	0.99	0.99
U	0.97	0.18	0.20	0.18	0.26	0.20	0.19	0.18	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.97	0.98	0.99	0.99	1.00	0.99	0.99	0.99
V	0.97	0.18	0.20	0.18	0.26	0.20	0.19	0.18	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.97	0.98	0.99	0.99	0.99	1.00	0.99	0.99
W	0.97	0.17	0.19	0.17	0.25	0.19	0.18	0.17	0.99	0.99	0.99	0.98	0.98	0.98	0.99	0.99	0.96	0.98	0.99	0.99	0.99	0.99	1.00	0.99
X	0.99	0.19	0.21	0.19	0.27	0.21	0.20	0.19	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.99	1.00

* The colors in the table represent a dark blue for similarity coefficients close to 1.0 and a dark red close to −1.0.

shows the similarity coefficients for a 24 × 24 benchmark matrix including the critical experiments and SMART, which were calculated using the McCARD/SimTest code sequence with ENDF/B-VII.1 covariance data [11]. The similarity coefficients between the high-enriched uranium (HEU) fast experiment benchmarks (B–H) and SMART ranged from 0.32 to 0.40, whereas those between the low-enriched uranium (LEU) thermal experiment benchmarks (I~W) and SMART (A) ranged from 0.97 to 0.99. The U.S. Nuclear Regulatory Commission (NRC) recommends that critical safety analyses should be conducted using critical experiments with c_k values in excess of 0.90. Studies have suggested that a target application should have more than 20 experiments with a c_k value greater than 0.80 [19]. Therefore, it is worth mentioning that all 16 LEU CE benchmarks (I~X) produced very high-level similarity coefficients with SMART in the ENDF/B-VII.1 covariance case.

To verify and validate the similarity coefficients with the adjoint S/U method, the S.S. method-based similarity coefficient generation was performed by the McCARD/MIG code sequence with the ENDF/B-VII.1 covariance data.

Table 3. Comparison between similarity coefficients of the adjoint S/U and S.S. methods.

S.S.	Godiva (B *)	Flattop25 (C)	LCT001c1 (I)	LCT002c1 (J)	ORNL1 (S)	ORNL2 (T)
Adjoint S/U
Godiva (B)	1.00	0.94 ± 0.01 **	0.19 ± 0.06	0.22 ± 0.06	0.20 ± 0.06	0.20 ± 0.06
		0.95	0.20	0.22	0.17	0.18
Flattop25 (C)	0.94 ± 0.01	1.00	0.21 ± 0.06	0.24 ± 0.07	0.16 ± 0.07	0.17 ± 0.07
	0.95		0.22	0.24	0.19	0.20
LCT001c1 (I)	0.19 ± 0.06	0.21 ± 0.06	1.00	0.99 ± 0.01	0.88 ± 0.01	0.88 ± 0.01
	0.20	0.22		0.99	0.99	0.99
LCT002c1 (J)	0.22 ± 0.06	0.26 ± 0.08	0.99 ± 0.01	1.00	0.91 ± 0.02	0.91 ± 0.02
	0.22	0.24	0.99		0.99	0.99
ORNL1 (S)	0.20 ± 0.06	0.26 ± 0.08	0.88 ± 0.01	0.91 ± 0.01	1.00	0.99 ± 0.01
	0.17	0.19	0.99	0.99		0.99
ORNL2 (T)	0.20 ± 0.06	0.26 ± 0.08	0.88 ± 0.01	0.91 ± 0.01	0.99 ± 0.01	1.00
	0.18	0.20	0.99	0.99	0.99

* Refers to the notations (B, C, I, J, S, T) from Table 1. ** In the S.S. results, the value following ± represents 2σ.

compares the similarity coefficients among six CEs (i.e., two fast systems and four thermal systems) using the S.S. method and the adjoint S/U method. In the S.S. method, the 95% confidence intervals of the uncertainties of the requested outputs were calculated by five repetitions of 100 McCARD runs with different sampled cross-section sets. The uncertainties of similarity coefficients from the 100 McCARD S.S. runs agree within 2~3 standard deviations with the McCARD adjoint S/U results. From the results, it was concluded that the S.S. method also works well, and we confirmed the effectiveness of the adjoint S/U method-based similarity coefficients.

3.2. Impact on the S.S. Method-Based Similarity Coefficients from MC Statistical Uncertainties

In this section, the impact of the MC statistical uncertainties on the similarity coefficients generated by the S.S. method was confirmed. By deriving the uncertainty as shown in Equation (5), the contributions to the covariance can be separated into two uncertainty terms on k_eff which come from the statistical uncertainties of the MC calculations and cross-section uncertainties as shown in Equation (11). Inserting Equations (5) and (11) into Equation (1) leads to

$$c_k^{SS}\simeq \frac{{\mathrm{cov}}_{SS}\left[k_{eff}^I,k_{eff}^{II}\right]+{\mathrm{cov}}_{XX}\left[k_{eff}^I,k_{eff}^{II}\right]}{\sqrt{{\sigma }_{SS}^2(k_{eff}^I)+{\sigma }_{XX}^2(k_{eff}^I)}\cdot \sqrt{{\sigma }_{SS}^2(k_{eff}^{II})+{\sigma }_{XX}^2(k_{eff}^{II})}}.$$

(18)

For the S.S. method-based similarity coefficients, considering that the uncertainty of k_eff is due to the statistical uncertainties of MC calculations, $c_k^{SS}$ will be changed by the γ ratio, as shown in Equation (19):

$$c_k^{SS}=\frac{1}{\sqrt{1+\frac{{\sigma }_{SS}^2(k_{eff}^I)}{{\sigma }_{XX}^2(k_{eff}^I)}}\cdot \sqrt{1+\frac{{\sigma }_{SS}^2(k_{eff}^{II})}{{\sigma }_{XX}^2(k_{eff}^{II})}}}{c}_k=\gamma c_k.$$

(19)

Figure 2 compares the $c_k^{SS}$ values between Flattop25 and the other three experiments (i.e., Godiva, LCT001c1, and LCT002c1) due to the change in the γ ratio that was calculated using ${\sigma }_{XX}(k_{eff})$ and ${\sigma }_{SS}(k_{eff})$ as shown in

Table 4. Uncertainties of k_efff from statistical uncertainties of MC calculations and cross-section uncertainties.

Uncertainties of keff (pcm)
Critical Experiment (ID *)		Godiva (B)	Flattop25 (C)	LCT001c1 (I)	LCT002c1 (J)
${\sigma }_{XX}(k_{eff})$		1266	1108	770	750
${\sigma }_{SS}(k_{eff})$	10,000 × 100 **	60	65	66	75
	10,000 × 10	225	233	225	262
	10,000 × 5	378	447	408	453
	1000 × 10	632	701	725	822

* Refers to the notations (B, C, I, J) from Table 1. ** Number of particles/cycle × number of active cycles in MC simulations.

. In Figure 2, dots indicate $c_k^{SS}$ from Equation (19), whereas cross points are represented by c_k, obtained from the repeated McCARD calculations with different sequences of random seeds. As the stochastic uncertainty of k_eff in a single MC calculation decreases or the γ ratio increases, $c_k^{SS}$ will approach the dotted line, which represents the c_k value obtained with the adjoint S/U method as shown in Figure 2. Therefore, it can be confirmed that an increase in statistical uncertainty leads to an increase in similarity coefficient errors. In common MC calculations, the stochastic uncertainties of k_eff are less than 50~100 pcm (percent milli). For example, a reactivity of 0.001 corresponds to 100 pcm in units. It was noted that the impact on the c_k value from statistical uncertainties in MC eigenvalue calculations was not significant. Moreover, it was observed that the c_k value estimated by Equation (19) agreed very well with the reference, directly calculated from repeated MC calculations.

3.3. Sensitivity of Similarity Coefficients due to Covariance Data

To examine discrepancies in the similarity coefficient, c_k, due to the covariance data, 24 × 24 benchmark matrix problem analyses were performed using the covariance data in each evaluated nuclear data library (i.e., ENDF/B-VIII.0 [12], JENDL-5.0 [13], and JEFF-3.3 [14]).

Table 5. Similarity coefficients among the 24 × 24 benchmark matrix problems with ENDF/B-VIII.0 covariance.

	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X
A	1.00 *	0.59	0.57	0.58	0.70	0.57	0.59	0.60	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.97	0.97	0.97	0.97	0.96	0.98
B	0.59	1.00	0.95	0.99	0.97	0.96	0.97	0.99	0.54	0.53	0.55	0.54	0.53	0.54	0.53	0.55	0.55	0.53	0.56	0.56	0.56	0.56	0.57	0.54
C	0.57	0.95	1.00	0.95	0.97	0.99	0.99	0.96	0.54	0.51	0.54	0.51	0.51	0.53	0.52	0.55	0.51	0.50	0.58	0.58	0.58	0.58	0.60	0.54
D	0.58	0.99	0.95	1.00	0.97	0.95	0.97	0.99	0.54	0.52	0.54	0.54	0.52	0.54	0.53	0.54	0.55	0.53	0.55	0.55	0.55	0.55	0.56	0.53
E	0.70	0.97	0.97	0.97	1.00	0.97	0.98	0.98	0.67	0.64	0.66	0.64	0.63	0.65	0.65	0.67	0.64	0.63	0.71	0.71	0.71	0.71	0.72	0.67
F	0.57	0.96	0.99	0.95	0.97	1.00	0.99	0.97	0.54	0.51	0.54	0.51	0.51	0.53	0.52	0.55	0.52	0.50	0.58	0.58	0.58	0.58	0.60	0.54
G	0.59	0.97	0.99	0.97	0.98	0.99	1.00	0.98	0.56	0.53	0.55	0.53	0.52	0.54	0.54	0.56	0.53	0.52	0.60	0.60	0.60	0.60	0.61	0.56
H	0.60	0.99	0.96	0.99	0.98	0.97	0.98	1.00	0.55	0.53	0.56	0.55	0.53	0.55	0.54	0.56	0.56	0.54	0.58	0.58	0.58	0.58	0.59	0.55
I	0.98	0.54	0.54	0.54	0.67	0.54	0.56	0.55	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.99	0.98	0.99
J	0.99	0.53	0.51	0.52	0.64	0.51	0.53	0.53	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.98	0.98	0.98	0.97	0.99
K	0.99	0.55	0.54	0.54	0.66	0.54	0.55	0.56	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.98	0.98	0.98	0.98	0.98	0.99
L	0.99	0.54	0.51	0.54	0.64	0.51	0.53	0.55	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.99	0.97	0.97	0.97	0.97	0.96	0.98
M	0.99	0.53	0.51	0.52	0.63	0.51	0.52	0.53	0.99	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.99	0.97	0.97	0.98	0.98	0.97	0.99
N	0.99	0.54	0.53	0.54	0.65	0.53	0.54	0.55	0.99	0.99	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.99	0.98	0.98	0.98	0.98	0.97	0.99
O	0.99	0.53	0.52	0.53	0.65	0.52	0.54	0.54	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.99	0.99	0.99	0.98	0.98	0.98	0.98	0.97	0.99
P	0.98	0.55	0.55	0.54	0.67	0.55	0.56	0.56	0.99	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.98	0.99	0.99	0.99	0.99	0.99	0.98	0.99
Q	0.99	0.55	0.51	0.55	0.64	0.52	0.53	0.56	0.98	0.99	0.99	0.99	0.99	0.99	0.99	0.98	1.00	0.99	0.96	0.96	0.96	0.96	0.95	0.98
R	0.99	0.53	0.50	0.53	0.63	0.50	0.52	0.54	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.99	1.00	0.97	0.97	0.97	0.97	0.96	0.98
S	0.97	0.56	0.58	0.55	0.71	0.58	0.60	0.58	0.99	0.98	0.98	0.97	0.97	0.98	0.98	0.99	0.96	0.97	1.00	0.99	0.99	0.99	0.99	0.99
T	0.97	0.56	0.58	0.55	0.71	0.58	0.60	0.58	0.99	0.98	0.98	0.97	0.97	0.98	0.98	0.99	0.96	0.97	0.99	1.00	0.99	0.99	0.99	0.99
U	0.97	0.56	0.58	0.55	0.71	0.58	0.60	0.58	0.99	0.98	0.98	0.97	0.98	0.98	0.98	0.99	0.96	0.97	0.99	0.99	1.00	1.00	0.99	0.99
V	0.97	0.56	0.58	0.55	0.71	0.58	0.60	0.58	0.99	0.98	0.98	0.97	0.98	0.98	0.98	0.99	0.96	0.97	0.99	0.99	1.00	1.00	0.99	0.99
W	0.96	0.57	0.60	0.56	0.72	0.60	0.61	0.59	0.98	0.97	0.98	0.96	0.97	0.97	0.97	0.98	0.95	0.96	0.99	0.99	0.99	0.99	1.00	0.98
X	0.98	0.54	0.54	0.53	0.67	0.54	0.56	0.55	0.99	0.99	0.99	0.98	0.99	0.99	0.99	0.99	0.98	0.98	0.99	0.99	0.99	0.99	0.98	1.00

* The colors in the table represent a dark blue for similarity coefficients close to 1.0 and a dark red close to −1.0.

,

Table 6. Similarity coefficients among the 24 × 24 benchmark matrix problems with JENDL-5.0 covariance.

	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X
A	1.00 *	0.11	0.03	0.11	0.09	0.03	0.03	0.10	0.92	0.90	0.91	0.81	0.85	0.89	0.91	0.94	0.73	0.77	0.83	0.83	0.83	0.83	0.82	0.89
B	0.11	1.00	0.44	0.99	0.39	0.46	0.49	0.91	−0.08	−0.10	−0.10	−0.17	−0.15	−0.12	−0.09	−0.06	−0.20	−0.18	0.00	0.00	0.00	0.00	0.00	−0.01
C	0.03	0.44	1.00	0.41	0.95	0.99	0.99	0.57	0.02	0.03	0.03	0.05	0.03	0.03	0.03	0.02	0.06	0.06	0.00	0.00	0.00	0.00	0.00	0.00
D	0.11	0.99	0.41	1.00	0.36	0.43	0.46	0.86	−0.10	−0.13	−0.13	−0.22	−0.19	−0.15	−0.11	−0.08	−0.26	−0.24	0.00	0.00	0.00	0.00	0.00	−0.02
E	0.09	0.39	0.95	0.36	1.00	0.94	0.95	0.51	0.08	0.09	0.08	0.09	0.09	0.08	0.09	0.08	0.09	0.10	0.07	0.07	0.07	0.07	0.07	0.08
F	0.03	0.46	0.99	0.43	0.94	1.00	0.99	0.59	0.02	0.03	0.03	0.05	0.04	0.04	0.03	0.02	0.07	0.07	0.00	0.00	0.00	0.00	0.00	0.01
G	0.03	0.49	0.99	0.46	0.95	0.99	1.00	0.62	0.02	0.03	0.03	0.05	0.04	0.04	0.03	0.02	0.07	0.07	0.00	0.00	0.00	0.00	0.00	0.00
H	0.10	0.91	0.57	0.86	0.51	0.59	0.62	1.00	0.01	0.01	0.02	0.02	0.01	0.02	0.00	0.01	0.04	0.04	0.00	0.00	0.00	0.00	0.00	0.00
I	0.92	−0.08	0.02	−0.10	0.08	0.02	0.02	0.01	1.00	0.98	0.98	0.94	0.97	0.97	0.98	0.99	0.88	0.91	0.86	0.87	0.87	0.87	0.86	0.92
J	0.90	−0.10	0.03	−0.13	0.09	0.03	0.03	0.01	0.98	1.00	0.97	0.96	0.98	0.97	0.99	0.97	0.91	0.95	0.87	0.87	0.87	0.87	0.86	0.93
K	0.91	−0.10	0.03	−0.13	0.08	0.03	0.03	0.02	0.98	0.97	1.00	0.96	0.97	0.99	0.97	0.98	0.92	0.94	0.79	0.79	0.79	0.79	0.78	0.86
L	0.81	−0.17	0.05	−0.22	0.09	0.05	0.05	0.02	0.94	0.96	0.96	1.00	0.99	0.98	0.94	0.92	0.98	0.99	0.73	0.73	0.73	0.73	0.71	0.81
M	0.85	−0.15	0.03	−0.19	0.09	0.04	0.04	0.01	0.97	0.98	0.97	0.99	1.00	0.98	0.97	0.95	0.96	0.98	0.81	0.81	0.81	0.81	0.80	0.88
N	0.89	−0.12	0.03	−0.15	0.08	0.04	0.04	0.02	0.97	0.97	0.99	0.98	0.98	1.00	0.96	0.96	0.94	0.96	0.76	0.76	0.76	0.76	0.75	0.84
O	0.91	−0.09	0.03	−0.11	0.09	0.03	0.03	0.00	0.98	0.99	0.97	0.94	0.97	0.96	1.00	0.98	0.89	0.93	0.89	0.89	0.89	0.89	0.88	0.94
P	0.94	−0.06	0.02	−0.08	0.08	0.02	0.02	0.01	0.99	0.97	0.98	0.92	0.95	0.96	0.98	1.00	0.85	0.89	0.88	0.88	0.88	0.88	0.87	0.93
Q	0.73	−0.20	0.06	−0.26	0.09	0.07	0.07	0.04	0.88	0.91	0.92	0.98	0.96	0.94	0.89	0.85	1.00	0.99	0.64	0.64	0.64	0.64	0.63	0.72
R	0.77	−0.18	0.06	−0.24	0.10	0.07	0.07	0.04	0.91	0.95	0.94	0.99	0.98	0.96	0.93	0.89	0.99	1.00	0.71	0.71	0.71	0.71	0.70	0.79
S	0.83	0.00	0.00	0.00	0.07	0.00	0.00	0.00	0.86	0.87	0.79	0.73	0.81	0.76	0.89	0.88	0.64	0.71	1.00	0.99	0.99	0.99	0.99	0.98
T	0.83	0.00	0.00	0.00	0.07	0.00	0.00	0.00	0.87	0.87	0.79	0.73	0.81	0.76	0.89	0.88	0.64	0.71	0.99	1.00	0.99	0.99	0.99	0.98
U	0.83	0.00	0.00	0.00	0.07	0.00	0.00	0.00	0.87	0.87	0.79	0.73	0.81	0.76	0.89	0.88	0.64	0.71	0.99	0.99	1.00	0.99	0.99	0.98
V	0.83	0.00	0.00	0.00	0.07	0.00	0.00	0.00	0.87	0.87	0.79	0.73	0.81	0.76	0.89	0.88	0.64	0.71	0.99	0.99	0.99	1.00	0.99	0.98
W	0.82	0.00	0.00	0.00	0.07	0.00	0.00	0.00	0.86	0.86	0.78	0.71	0.80	0.75	0.88	0.87	0.63	0.70	0.99	0.99	0.99	0.99	1.00	0.97
X	0.89	−0.01	0.00	−0.02	0.08	0.01	0.00	0.00	0.92	0.93	0.86	0.81	0.88	0.84	0.94	0.93	0.72	0.79	0.98	0.98	0.98	0.98	0.97	1.00

* The colors in the table represent a dark blue for similarity coefficients close to 1.0 and a dark red close to −1.0.

and

Table 7. Similarity coefficients among the 24 × 24 benchmark matrix problems with JEFF-3.3 covariance.

	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X
A	1.00 *	0.44	0.36	0.44	0.49	0.36	0.38	0.43	0.96	0.96	0.97	0.95	0.96	0.96	0.96	0.96	0.93	0.94	0.88	0.88	0.89	0.89	0.87	0.92
B	0.44	1.00	0.86	0.99	0.86	0.88	0.89	0.98	0.42	0.44	0.45	0.50	0.45	0.48	0.43	0.40	0.53	0.50	0.30	0.30	0.30	0.30	0.29	0.32
C	0.36	0.86	1.00	0.87	0.96	0.99	0.99	0.83	0.34	0.35	0.34	0.36	0.35	0.34	0.36	0.34	0.36	0.36	0.33	0.33	0.33	0.33	0.32	0.34
D	0.44	0.99	0.87	1.00	0.88	0.89	0.90	0.96	0.40	0.42	0.44	0.47	0.43	0.46	0.41	0.39	0.50	0.47	0.30	0.30	0.30	0.30	0.30	0.32
E	0.49	0.86	0.96	0.88	1.00	0.96	0.97	0.83	0.45	0.47	0.45	0.47	0.47	0.45	0.48	0.45	0.46	0.46	0.45	0.45	0.45	0.45	0.44	0.47
F	0.36	0.88	0.99	0.89	0.96	1.00	0.99	0.85	0.34	0.36	0.35	0.37	0.36	0.35	0.36	0.34	0.38	0.37	0.32	0.32	0.32	0.33	0.32	0.34
G	0.38	0.89	0.99	0.90	0.97	0.99	1.00	0.86	0.36	0.37	0.36	0.38	0.37	0.36	0.37	0.35	0.39	0.38	0.34	0.34	0.34	0.34	0.33	0.35
H	0.43	0.98	0.83	0.96	0.83	0.85	0.86	1.00	0.43	0.46	0.47	0.52	0.47	0.50	0.44	0.41	0.57	0.54	0.29	0.29	0.29	0.29	0.29	0.32
I	0.96	0.42	0.34	0.40	0.45	0.34	0.36	0.43	1.00	0.99	0.99	0.97	0.98	0.98	0.99	0.99	0.94	0.96	0.93	0.93	0.93	0.93	0.92	0.95
J	0.96	0.44	0.35	0.42	0.47	0.36	0.37	0.46	0.99	1.00	0.98	0.98	0.99	0.98	0.99	0.98	0.96	0.98	0.92	0.92	0.92	0.92	0.91	0.95
K	0.97	0.45	0.34	0.44	0.45	0.35	0.36	0.47	0.99	0.98	1.00	0.98	0.99	0.99	0.98	0.98	0.96	0.98	0.88	0.88	0.89	0.89	0.87	0.92
L	0.95	0.50	0.36	0.47	0.47	0.37	0.38	0.52	0.97	0.98	0.98	1.00	0.99	0.99	0.97	0.96	0.99	0.99	0.85	0.85	0.85	0.85	0.83	0.89
M	0.96	0.45	0.35	0.43	0.47	0.36	0.37	0.47	0.98	0.99	0.99	0.99	1.00	0.99	0.99	0.98	0.97	0.98	0.90	0.90	0.90	0.90	0.89	0.94
N	0.96	0.48	0.34	0.46	0.45	0.35	0.36	0.50	0.98	0.98	0.99	0.99	0.99	1.00	0.98	0.97	0.98	0.98	0.86	0.86	0.86	0.86	0.85	0.90
O	0.96	0.43	0.36	0.41	0.48	0.36	0.37	0.44	0.99	0.99	0.98	0.97	0.99	0.98	1.00	0.99	0.95	0.97	0.93	0.93	0.93	0.93	0.92	0.96
P	0.96	0.40	0.34	0.39	0.45	0.34	0.35	0.41	0.99	0.98	0.98	0.96	0.98	0.97	0.99	1.00	0.93	0.95	0.94	0.94	0.94	0.94	0.93	0.96
Q	0.93	0.53	0.36	0.50	0.46	0.38	0.39	0.57	0.94	0.96	0.96	0.99	0.97	0.98	0.95	0.93	1.00	0.99	0.80	0.80	0.80	0.80	0.78	0.84
R	0.94	0.50	0.36	0.47	0.46	0.37	0.38	0.54	0.96	0.98	0.98	0.99	0.98	0.98	0.97	0.95	0.99	1.00	0.84	0.84	0.84	0.84	0.83	0.88
S	0.88	0.30	0.33	0.30	0.45	0.32	0.34	0.29	0.93	0.92	0.88	0.85	0.90	0.86	0.93	0.94	0.80	0.84	1.00	0.99	0.99	0.99	0.99	0.99
T	0.88	0.30	0.33	0.30	0.45	0.32	0.34	0.29	0.93	0.92	0.88	0.85	0.90	0.86	0.93	0.94	0.80	0.84	0.99	1.00	0.99	0.99	0.99	0.99
U	0.89	0.30	0.33	0.30	0.45	0.32	0.34	0.29	0.93	0.92	0.89	0.85	0.90	0.86	0.93	0.94	0.80	0.84	0.99	0.99	1.00	0.99	0.99	0.99
V	0.89	0.30	0.33	0.30	0.45	0.33	0.34	0.29	0.93	0.92	0.89	0.85	0.90	0.86	0.93	0.94	0.80	0.84	0.99	0.99	0.99	1.00	0.99	0.99
W	0.87	0.29	0.32	0.30	0.44	0.32	0.33	0.29	0.92	0.91	0.87	0.83	0.89	0.85	0.92	0.93	0.78	0.83	0.99	0.99	0.99	0.99	1.00	0.98
X	0.92	0.32	0.34	0.32	0.47	0.34	0.35	0.32	0.95	0.95	0.92	0.89	0.94	0.90	0.96	0.96	0.84	0.88	0.99	0.99	0.99	0.99	0.98	1.00

* The colors in the table represent a dark blue for similarity coefficients close to 1.0 and a dark red close to −1.0.

show the similarity coefficients among the 24 × 24 benchmark matrix problems with each library’s covariance data.

Table 8. Comparison between similarity coefficients of various evaluated nuclear data library covariances.

Covariance		A	B	C	E	I	J	K	L	M	Q	R	W	X
ENDF/B-VII.1	A	1.00	0.33	0.34	0.41	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.97	0.99
ENDF/B-VIII.0		1.00	0.60	0.57	0.70	0.99	0.99	0.99	0.99	0.99	0.99	0.99	0.97	0.99
JENDL-5.0		1.00	0.11	0.03	0.10	0.93	0.90	0.92	0.82	0.85	0.74	0.78	0.83	0.89
JEFF-3.3		1.00	0.45	0.37	0.49	0.96	0.96	0.97	0.96	0.96	0.94	0.94	0.88	0.93
SCALE 6.1		1.00	0.23	0.19	0.23	0.85	0.74	0.82	0.68	0.71	0.58	0.57	0.75	0.82

summarizes the similarity coefficients between SMART and the representative 12 critical benchmarks with the covariance data from various evaluated nuclear data files (ENDFs): ENDF/B-VII.1 [11], ENDF/B-VIII.0 [12], JENDL-5.0 [13], and JEFF-3.3 [14]. In the thermal spectrum benchmark problems (I~X), which are anticipated to be similar to SMART, the similarity coefficients for ENDF/B-VII.1 are comparable to those of ENDF/B-VIII.0 and JEFF-3.3. However, there is a significant discrepancy when compared to the results from JENDL-5.0. In the fast spectrum benchmark problems, the similarity coefficients of JENDL-5.0 appear significantly lower than the others. In general, the selection of nuclear data libraries for evaluation did not significantly impact sensitivity coefficients in Equation (2). Therefore, the most influential factor affecting similarity coefficients was the covariance term between cross-sections, as shown in Equation (2). It was observed that JENDL-5.0 and SCALE 6.1 stood out among the various nuclear data libraries, as they provided significantly different covariance matrices compared to the other evaluation libraries, leading to differences in similarity coefficients. For the 16 LEU CE benchmarks (I~X) case, which produced very high-level similarity coefficients (i.e., ~0.90) to SMART in the ENDF/B-VII.1 covariance case, c_k ranged from 0.74 to 0.93 in JENDL-5.0. Overall, it was observed that the selection of the covariance data used for c_k generations significantly impacted the similarity coefficient. Moreover, because JENDL-5.0 covariance data gave a broad range of c_k similarity coefficients, they were expected to show effective discrimination in the similarity analyses.

Moreover, a comparison of similarity coefficients derived from the covariance data of the SCALE 6.1 code system [20] and the four ENDF covariance datasets is shown in Table 8. In the SCALE 6.1 case, the covariance data matrix with a 44-group structure was used for two major actinide isotopes. The SCALE 6.1 results were calculated using the covariance data employed in the SCALE 6.1 code by the McCARD/SimTest code systems with the ENDF/B-VII.1 library. In previous studies [2,3] which used the SCALE code package (i.e., SCALE 6.1 and 6.2) for similarity analyses, it was suggested that the c_k value should be greater than 0.70~0.80 for applications. The current results using the SCALE 6.1 covariance data show a consistent trend with the results reported in earlier studies. In the critical benchmarks (I~K) with fuel possessing a similar enrichment and geometry to that of SMART, similarity coefficients exceeding 0.7 were observed. From these comparative results, our new similarity coefficient generation code system was validated and verified.

4. Conclusions

In this study, a similarity coefficient generation code system based on the McCARD code was successfully established and verified. We considered the adjoint-based S/U approaches and the S.S. approaches for the generation of c_k similarity coefficients. In the adjoint S/U method, a formula was derived that describes the impact of statistical uncertainties arising from an MC code on the generation of the similarity coefficient. This formula is very valuable in that it facilitates the separation of the two influences: statistical uncertainties and nuclear reaction cross-sections.

To examine the code system, the c_k similarity coefficients among the relevant 23 critical experiments and the SMART SMR target system were generated using the ENDF/B-VII.1 covariance data with an LANL 30-group energy group structure. From the results, it can be seen that the c_k similarity coefficients between the 16 LEU critical experiments and SMART are over 0.90. These results are very helpful for licensees needing to justify the determination of critical experiment benchmarks for computational bias estimations of the SMART target system. Moreover, we confirmed the effects on the c_k similarity coefficient using the up-to-date cross-section libraries and their covariance data (i.e., ENDF/B-VIII.0, JENDL-5.0, and JEFF-3.3). Additionally, it was noted that the results of the SCALE 6.1 covariance data showed a consistent trend with the results reported in earlier studies. Overall, our observations indicate a notable influence of the selected covariance data on the generation of c_k similarity coefficient values, consequently exerting a significant impact on the resulting similarity coefficients.

Similarity coefficients play a crucial role in nuclear core design analyses by providing a quantitative measure of the likenesses between different core configurations. The newly developed similarity coefficient generation code systems for the adjoint S/U and S.S. approaches can be utilized for nuclear core and shielding design analyses for the development of next-generation nuclear reactors, and as tools to justify the application of existing critical facility or nuclear reactor results for a new system where experimental data are unavailable. In the near future, this similarity coefficient generation code system will be enhanced to generate discriminative similarity coefficients, as it currently tends to produce similar similarity coefficients for different fuel types and structures.