Experimental analyses of bismuth sample reactivity worth at Kyoto University Critical Assembly

ABSTRACT

Sample reactivity worth experiments are carried out by substituting aluminum (Al) plates for bismuth (Bi) ones at the Kyoto University Critical Assembly. At the beginning, uncertainty quantification of bismuth isotope is conducted by deterministic calculations with nuclear data library JENDL-4.0, with the use of experimental results of sample reactivity worth. Then, with the combined use of current (Bi) and previous (Pb) experimental results that demonstrate the comparative difference in the sensitivity and uncertainty of Bi and Pb isotopes, experimental results of cross-section uncertainties of Bi isotope are available for examination of neutron characteristics of Pb–Bi coolant material in the accelerator-driven system. From the experimental analyses, further uncertainty analyses by neutron transport calculations are needed for several reactions of Bi isotope, especially with the use of the covariance data of capture, elastic scattering and inelastic scattering reactions in another nuclear data library.

KEYWORDS:

• KUCA
• Bi
• sample reactivity worth
• criticality
• sensitivity
• uncertainty
• JENDL-4.0 covariance data

1. Introduction

The accelerator-driven system (ADS) is a potentially promising solution to the achievement of nuclear transmutation of minor actinides and long-lived fission products in a fast neutron spectrum core, and to the use of spallation neutrons generated by the injection of high-energy protons onto a heavy metal target. At the Kyoto University Critical Assembly (KUCA), a feasibility study [1–5] on ADS has been carried out by coupling the solid-moderated and -reflected core, with 100 MeV protons from the fixed-field alternating gradient (FFAG) accelerator. Also, at the VENUS-F facility [6–8] in SCK/CEN, ADS experimental studies have been conducted in a lead (Pb)-reflected core with 14 MeV neutrons for implementation of an experimental facility, such as the Multi-purpose hYbrid Research Reactor for High-tech Applications (MYRRHA) [9].

In upcoming experimental facilities [9,10], liquid lead–bismuth (Pb–Bi) has been selected as not only target material for the generation of spallation neutrons but also coolant material in a fast neutron spectrum core. Before engineering studies on ADS, at the KUCA and the VENUS-F facilities, experimental analyses of reactor physics parameters have been conducted in a subcritical system to establish techniques for accurate measurement of subcriticality and reaction rates, and precise calculation methodologies of ADS kinetic parameters by both stochastic and deterministic approaches. Moreover, sensitivity and uncertainty analyses of reactor physics parameters have been quantitatively evaluated with the use of covariance data of nuclear data libraries, including criticality of actual ADS reactors [11,12], and precision of cross sections of Pb isotopes [13–15].

The main characteristics of coolant material Pb–Bi eutectic (LBE) in ADS are recognized as follows: chemically inactive; high boiling point mechanically; excellent neutron economy caused by large scattering cross sections. From the viewpoint of neutronics, LBE exerts considerable impact on nuclear design parameters for numerical simulations of neutron interactions of Pb and Bi isotopes. To validate their cross sections, uncertainty of Pb isotopes has been analyzed with the use of experimental data obtained at the KUCA facility [13,14] and at the National Criticality Experiments Research Center (NCERC) in the Nevada National Security site [15]. In experimental studies at KUCA and NCERC, Pb sample reactivity worth has been measured with the combined use of Pb and highly-enriched uranium (HEU) plates. Furthermore, numerical simulations have been conducted through both stochastic and deterministic approaches with nuclear data libraries of JENDL-4.0 [16] and ENDF/B-VII.1 [17], in order to examine the uncertainty of criticality induced by the cross sections of Pb isotopes. In a subsequent study on Pb–Bi cross-section uncertainty, the first attempt was made of experimental analyses of the quantification of Bi isotope cross-section uncertainty by conducting Bi sample reactivity worth experiments in a critical state at KUCA, with the consideration of neutron spectrum in the actual ADS core [10], as shown in Figure S1. As in, and with the use of the results of, previous Pb sample reactivity worth experiments, reference samples (aluminum: Al) were substituted for test ones (Bi), and current Bi sample reactivity worth was abundantly obtained to qualitatively interpret the Pb–Bi neutron characteristics.

The main objective of this study was to examine the neutron characteristics of Pb–Bi coolant material in ADS, through experimental studies on uncertainty of criticality induced by the JENDL-4.0 covariance data of Bi isotope cross sections and comparative studies on Bi and Pb sample reactivity worth. The details of sample reactivity worth experiments and evaluation methods are described in Section 2. The results of numerical analyses by the stochastic and deterministic calculations with JENDL-4.0 are presented in Section 3, the comparative study on Bi and Pb sample reactivity worth is detailed in Section 4, and the conclusions are summarized in Section 5.

2. Sample reactivity worth experiments

2.1. Core configuration

KUCA has three cores designated A, B and C: A and B are polyethylene solid-moderated and -reflected cores, and C is a light water-moderated and -reflected one. The three cores are operated at a zero power in the normal operating state. Sample reactivity worth experiments were carried out in the A-core (Figure 1), which has polyethylene moderator and reflector rods (Tables S1 and S2, respectively), and four different fuel assemblies, including HEU, polyethylene moderator (p), polyethylene reflector (PE), graphite (Gr) and Al plate: normal ‘F,’ partials ‘40’ and ‘14,’ and test ‘f’ (Figure 2(a–d), respectively). Normal fuel assembly ‘F’ is composed of 60 unit cells, and upper and lower polyethylene blocks about 600 and 550 mm long, respectively, in an Al sheath (54 × 54 × 1520 mm; Table S3). For the normal and partial fuel assemblies, a unit cell in the fuel region is composed of two HEU (Table S4) fuel plates 2 × 2 in square and 1/8 in (3.18 mm; 1/16 in× 2) thick, polyethylene plates 2 × 2 in square and 1/8 in thick. Numerals of 40 and 14 correspond to the number of unit cells in the partial fuel assemblies used for reaching the criticality mass. The test (reference) fuel assembly ‘f’ is composed of 40 unit cells with two HEU fuel plates 1/8 in (1/16 in× 2) thick and two Al plates 2 × 2 in square and 1/8 in (1/16 in× 2) thick, 20 unit cells of HEU and polyethylene plates as in the normal fuel assembly (Figure 2(a)).

Figure 1. Top view of the KUCA A-core in Bi sample reactivity worth experiments (Reference core).

Figure 2. Schematic drawing of fuel assemblies (Figure 1) in the A-core. (a). 1/8”p60EUEU fuel rod; (b). ‘40’ partial fuel rod; (c). ‘14’ partial fuel rod; and (d). Reference fuel rod ‘f’ containing Al plates.

2.2. Experimental settings

Sample reactivity worth was measured by substituting Al plates for Bi ones (Table S5) in the test fuel region at the critical state. In the test fuel region, five test fuel assemblies were set around the center of the core (14, M), (15, L), (15, M), (15, O) and (16, M), as shown in Figure 1. The sample reactivity worth experiments ranged between three and five fuel assemblies, as shown in Figure 3, by substituting the reference fuel rods for Bi fuel ones. The substitution was made for a total of 40 unit cells in the central region of the core, by changing Al plates in Figure 2(d) for Bi ones in Figure 4. The sample reactivity worth induced by the substitution was experimentally obtained through the difference between the excess reactivities of the Al reference core (Figure 1) and the Bi test core (Figure 3). The critical state was adjusted by maintaining control rods (C1, C2 and C3: Table S6; Figure 1) at the certain positions shown in Table 1; excess reactivity was then deduced from the difference between the critical and super-critical states in the core. The experimental excess reactivity was obtained by combining the reactivity worth of each control rod evaluated by the rod drop method with its integral calibration curve by the positive period method. The experimental error of excess reactivity measurement was estimated at less than 3%, demonstrating a statistical error obtained by the measurements several times.

Table 1. Control rod positions at the critical state in reference (Al) and test (Bi) cores shown in Figures 1 and 2, respectively

	Rod position [mm]
Core	C1	C2	C3	S4, S5, S6
Reference (Al)	1200.00	715.57	1200.00	1200.00
Case 1	1200.00	676.77	1200.00	1200.00
Case 2	1200.00	662.37	1200.00	1200.00
Case 3	1200.00	663.32	1200.00	1200.00
Case 4	1200.00	658.28	1200.00	1200.00

1200.00 [mm]: Upper limit.

Figure 3. Patterns of Bi sample reactivity worth experiments (b: test fuel rod; Figure 4). (a) Case 1; (b) Case 2; (c) Case 3; and (d) Case 4.

Figure 4. Schematic drawing of test fuel rod containing Bi.

2.3. Evaluation method

2.3.1. Criticality bias

The experimental sample reactivity worth $\mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}$ was deduced from the difference between two excess reactivities $\mathrm{\Delta }{\rho }_{Excess}^{Exp,Al}$ and $\mathrm{\Delta }{\rho }_{Excess}^{Exp,Bi}$ obtained by the positive period method in the reference and test cores, respectively, when the Al plates were substituted for the Bi ones, as follows:

(1) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}={\rho }_{Excess}^{Exp,Bi}-{\rho }_{Excess}^{Exp,Al}\\ & =\left(1-\frac{1}{k_{Clean}^{Exp,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Exp,Al}}\right)\\ & =\frac{1}{k_{Clean}^{Exp,Al}}-\frac{1}{k_{Clean}^{Exp,Bi}}\end{array}$(1)

where $k_{Clean}^{Exp,Al}$ and $k_{Clean}^{Exp,Bi}$ indicate the effective multiplication factors deduced by the experimental excess reactivities obtained in super-critical cores (clean core) before (Al) and after (Bi) substituting Al plates for Bi ones, respectively, when all the control and safety rods were withdrawn.

In MCNP [18] analyses, numerical sample reactivity worth $\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}$ was deduced from the difference between two excess reactivities $\mathrm{\Delta }{\rho }_{Excess}^{MCNP,Al}$ and $\mathrm{\Delta }{\rho }_{Excess}^{MCNP,Bi}$ in the reference and test cores, respectively, by the same method as that of the experimental sample reactivity worth, as follows:

(2) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\end{array}$(2)

where $k_{Clean}^{MCNP,Al}$ and $k_{Clean}^{MCNP,Bi}$ indicate the effective multiplication factors in super-critical cores before and after substituting Al plates for Bi ones, respectively. Here, $k_{Critical}^{MCNP,Al}$ and $k_{Critical}^{MCNP,Bi}$ are requisite for defining the values of effective multiplication factors in critical cores before and after substituting Al plates for Bi ones, since these numerical values are not always units.

On the basis of the experimental methodology shown in Equation (1)(1) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}={\rho }_{Excess}^{Exp,Bi}-{\rho }_{Excess}^{Exp,Al}\\ & =\left(1-\frac{1}{k_{Clean}^{Exp,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Exp,Al}}\right)\\ & =\frac{1}{k_{Clean}^{Exp,Al}}-\frac{1}{k_{Clean}^{Exp,Bi}}\end{array}$(1) , where Al plates were substituted for Bi ones, a numerical approach of sample reactivity worth $\mathrm{\Delta }{\rho }_{Al\to Bi}^{Cal}$ can be generally expressed as follows:

(3) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{Cal}={\rho }_{Excess}^{Cal,Bi}-{\rho }_{Excess}^{Cal,Al}=\left(1-\frac{1}{k_{Clean}^{Cal,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Cal,Al}}\right)\\ =\frac{1}{k_{Clean}^{Cal,Al}}-\frac{1}{k_{Clean}^{Cal,Bi}}\end{array}$(3)

where $k_{Clean}^{Cal,Al}$ and $k_{Clean}^{Cal,Bi}$ indicate the effective multiplication factors in super-critical cores.

Numerical sample reactivity worth $\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}$ in Equation (2)(2) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\end{array}$(2) can be rewritten with the use of the concept of Equation (3)(3) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{Cal}={\rho }_{Excess}^{Cal,Bi}-{\rho }_{Excess}^{Cal,Al}=\left(1-\frac{1}{k_{Clean}^{Cal,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Cal,Al}}\right)\\ =\frac{1}{k_{Clean}^{Cal,Al}}-\frac{1}{k_{Clean}^{Cal,Bi}}\end{array}$(3) , as follows:

(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4)

where ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}$ indicates the difference between inverse values of eigenvalue calculations in the two critical states evaluated by MCNP6.1. Here, the first term in Equation (4)(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4) is defined as ‘criticality bias’ [13] as follows:

(5) ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}=\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Critical}^{MCNP,Al}}$(5)

2.3.2. Sensitivity coefficient

Sensitivity coefficient S of the integral reactor physics parameter (effective multiplication factor) R is defined by the ratio of the rate of change in R and a certain parameter x as follows:

(6) $S=\frac{dR}{R}/\frac{dx}{x}$(6)

The effective multiplication factor k_eff is expressed by neutron balance equation as follows:

(7) $\mathbf{A}\varphi =\frac{1}{k_{eff}}\mathbf{F}\varphi$(7)

where A and F indicate operators of transport and fission terms, respectively, and ϕ the forward neutron flux. Multiplying Equation (7)(7) $\mathbf{A}\varphi =\frac{1}{k_{eff}}\mathbf{F}\varphi$(7) by adjoint neutron flux ϕ* and integrating over whole volume and energy, the following equation is obtained:

(8) $\frac{1}{k_{eff}}=\frac{⟨{\varphi }^{\ast }\mathbf{A}\varphi ⟩}{⟨{\varphi }^{\ast }\mathbf{F}\varphi ⟩}$(8)

where brackets < > indicate integration over the whole volume and energy.

With the use of an operator B, Equation (7)(7) $\mathbf{A}\varphi =\frac{1}{k_{eff}}\mathbf{F}\varphi$(7) is expressed as follows:

(9) $\left(\mathbf{A}-\frac{1}{k_{eff}}\mathbf{F}\right)\varphi =\mathbf{B}\varphi =0$(9)

Considering the theoretical background [19] and using adjoint neutron flux ϕ *, the sensitivity coefficient in Equation (6)(6) $S=\frac{dR}{R}/\frac{dx}{x}$(6) is finally expressed by applying the first-order perturbation approximation [20], as follows:

(10) $S=\frac{⟨{\varphi }^{\ast }\left(-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }x}\right)\varphi ⟩}{⟨{\varphi }^{\ast }A\varphi ⟩}x$(10)

Finally, in a previous Pb study [14], sensitivity analyses of k_eff were conducted by the SAGEP code [21].

2.3.3. Uncertainty

For the cross-section uncertainty analyses of nuclear data [22], the uncertainty of reactor physics parameter ν is expressed as follows:

(11) $\begin{array}{rl}& \nu ={\mathbf{G}}_{tar}\mathbf{M}({\mathbf{G}}_{tar}{)}^{\mathbf{t}}\\ & =\sum _i\hspace{0.17em}\sum _j{s}_i{c}_{i,j}{s}_j\\ & \equiv \sum _i\hspace{0.17em}\sum _j{\upsilon }_{i,j}(1\le i,j\le p)\end{array}$(11)

where G_tar (1 × p) indicates the sensitivity vector of reactor physics parameters, M (p × p) the covariance matrix of nuclear reaction parameters, s_i the sensitivity coefficient, $c_{i,j}$ the covariance, ${\upsilon }_{i,j}$ the factor of uncertainty and p the number of nuclear reactions including the nuclides. Thus, the contribution of uncertainty u_i in each nuclear reaction is defined as follows:

(12) $u_i\equiv \sum _i{\upsilon }_{i,j}$(12)

Generally, since sensitivity coefficient s_i and covariance c_{i, j} are dominant in the energy group, the factor of uncertainty is finally expressed with the use of the maximum number of energy group G as follows:

(13) ${\upsilon }_{i,j}=\sum _g\sum _{g{}^{\prime }}{s}_g^i{c}_{g,g{}^{\prime }}^{i,j}{s}_{g{}^{\prime }}^j\left(1\le g,g{}^{\prime }\le G\right)$(13)

where g and g’ indicate the energy groups.

3. Numerical analyses

3.1. Eigenvalue calculations

Numerical analyses were conducted with the use of the Monte Carlo code MCNP6.1 together with the JENDL-4.0 nuclear data library for transport. For actual experimental analyses, the capability of eigenvalue calculations by MCNP6.1 was useful in the discussion with the use of JENDL-4.0 for processing important data analyses, and JENDL-4.0 has already been compared with other nuclear data libraries in a previous study [13], while demonstrating a reference library. Notably, the numerical analyses system was the mostly same as previous study in Ref [14].

In the reference core shown in Figure 1, criticality was reached by adjusting the position of control rod C2 and withdrawing control rods C1 and C3, and safety rods S4, S5 and S6 from the core; excess reactivity was then deduced from the combined use of control rod worth of C2 by the rod drop method and its calibration curve by the positive period method. The measured excess reactivity was attained within an uncertainty of 3%, and compared with the numerical one as shown in Table 2. Here, effective delayed neutron fraction (β_eff) was attained by MCNP6.1 with JENDL-4.0 in both reference (Al: 798 ± 3 pcm) and test (Bi: 801 ± 3 pcm) cores. Since two values were almost same within statistical errors, the β_eff in reference core was used, when converting measured values in dollar units into ones in pcm units. The numerical excess reactivity was obtained by the MCNP6.1 eigenvalue calculations with JENDL-4.0 within a statistical error of 6 pcm through 2,000 active cycles of 25,000 histories. By comparing the measured and calculated results shown in Table 2, the C/E (calculation/experiment) value revealed good agreement within a relative difference of 4%, except in Case 1.

Table 2. Comparison between the results of measured and calculated (Equation (2)(2) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\end{array}$(2) ; MCNP6.1) excess reactivities in reference (Al) and test (Bi) cores shown in Figures 1 and 2, respectively

Core	Calculation [pcm]	Experiment [pcm]	C/E
Reference (Al)	88 ± 6	87 ± 1	1.01 ± 0.07
Case 1	129 ± 6	143 ± 3	0.90 ± 0.05
Case 2	164 ± 6	165 ± 3	0.99 ± 0.04
Case 3	156 ± 6	163 ± 3	0.96 ± 0.04
Case 4	166 ± 6	171 ± 3	0.97 ± 0.04

β_eff = 798 ± 3 [pcm] and ʌϕ = (3.39 ± 0.01)E-05 [s] by MCNP6.1 with JENDL-4.0.

C/E: Calculation/Experiment.

Furthermore, the measured control rod worth in the reference core was compared with the calculated one by the same method used for excess reactivity, as shown in Table 3. As shown in Tables 2 and 3, the MCNP eigenvalue calculations with JENDL-4.0 revealed accurate reproduction of the experimental results of excess reactivity and control rod worth, respectively, with the C/E values within 6%, ranging widely between 87 and 945 pcm.

Table 3. Comparison between the results of measured and calculated (MCNP6.1) control rod worth in reference core (Al reference core) shown in Figure 1

Rod	Calculation [pcm]	Experiment [pcm]	C/E
C1	1005 ± 10	945 ± 32	1.06 ± 0.04
C2	454 ± 10	438 ± 1	1.04 ± 0.02
C3	360 ± 11	350 ± 8	1.03 ± 0.04

β_eff = 798 ± 3 [pcm] and ʌϕ = (3.39 ± 0.01)E-05 [s] by MCNP6.1 with JENDL-4.0.

C/E: Calculation/Experiment.

3.2. Criticality bias

As discussed in Section 3.1, the accuracy of MCNP6.1 calculations was confirmed in terms of the general definition of sample reactivity worth by the MCNP approach. Here, the actual objective of this subsection was to compare the difference between experimental and numerical sample reactivity worth defined in Equations (1)(1) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}={\rho }_{Excess}^{Exp,Bi}-{\rho }_{Excess}^{Exp,Al}\\ & =\left(1-\frac{1}{k_{Clean}^{Exp,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Exp,Al}}\right)\\ & =\frac{1}{k_{Clean}^{Exp,Al}}-\frac{1}{k_{Clean}^{Exp,Bi}}\end{array}$(1) and (2(2) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\end{array}$(2) ), respectively, and to confirm the precision of MCNP calculations of the Bi sample reactivity worth experiments.

Special mention is made of the accuracy of numerical results by MCNP, especially of the absolute values shown in Table 4, and additional investigation was requisite to find the reason for the discrepancy between the results of experiments and calculations in Equation (4)(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4) shown in Table 4. Then, as suggested in Ref [13], ‘criticality bias’ of sample reactivity worth was useful in the investigation of the discrepancy, when a formulation of sample reactivity worth by MCNP6.1 defined in Equation (2)(2) $\begin{array}{c}\mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\end{array}$(2) is changed into that by Equation (4)(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4) . The criticality bias ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}$ defined in Equation (5)(5) ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}=\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Critical}^{MCNP,Al}}$(5) was around 30 pcm, as shown in Table 4 and Figure 5. From these results, the criticality bias of 37 pcm at most was confirmed as being included in the sample reactivity worth about 80 pcm, even in the analyses of MCNP calculations, although the absolute value of sample reactivity worth was small.

Table 4. Comparison of measured and calculated Bi sample reactivity worth in Equations (1)(1) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}={\rho }_{Excess}^{Exp,Bi}-{\rho }_{Excess}^{Exp,Al}\\ & =\left(1-\frac{1}{k_{Clean}^{Exp,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Exp,Al}}\right)\\ & =\frac{1}{k_{Clean}^{Exp,Al}}-\frac{1}{k_{Clean}^{Exp,Bi}}\end{array}$(1) and (4(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4) ), respectively, and criticality bias in Equation (5)(5) ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}=\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Critical}^{MCNP,Al}}$(5)

Core	Experiment in Equation (1)(1) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{Exp}={\rho }_{Excess}^{Exp,Bi}-{\rho }_{Excess}^{Exp,Al}\\ & =\left(1-\frac{1}{k_{Clean}^{Exp,Bi}}\right)-\left(1-\frac{1}{k_{Clean}^{Exp,Al}}\right)\\ & =\frac{1}{k_{Clean}^{Exp,Al}}-\frac{1}{k_{Clean}^{Exp,Bi}}\end{array}$(1) [pcm]	Calculation in Equation (4)(4) $\begin{array}{rl}& \mathrm{\Delta }{\rho }_{Al\to Bi}^{MCNP}={\rho }_{Excess}^{MCNP,Bi}-{\rho }_{Excess}^{MCNP,Al}\\ & =\left(\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)-\left(\frac{1}{k_{Critical}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Al}}\right)\\ & ={\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}+\left(\frac{1}{k_{Clean}^{MCNP,Al}}-\frac{1}{k_{Clean}^{MCNP,Bi}}\right)\end{array}$(4) [pcm]	Criticality bias in Equation (5)(5) ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}=\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Critical}^{MCNP,Al}}$(5) [pcm]
Case 1	56 ± 3	41 ± 8	24 ± 6
Case 2	78 ± 3	76 ± 8	6 ± 6
Case 3	76 ± 3	68 ± 8	33 ± 6
Case 4	84 ± 3	78 ± 8	37 ± 6

Figure 5. Comparison between the values of criticality bias at critical state evaluated by Equation (5)(5) ${\mathrm{\Delta }}_{Critical,Al\to Bi}^{MCNP}=\frac{1}{k_{Critical}^{MCNP,Bi}}-\frac{1}{k_{Critical}^{MCNP,Al}}$(5) .

3.3. Sensitivity

Sensitivity coefficients of k_eff (Case 4) in Equation (10)(10) $S=\frac{⟨{\varphi }^{\ast }\left(-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }x}\right)\varphi ⟩}{⟨{\varphi }^{\ast }A\varphi ⟩}x$(10) were analyzed by the SAGEP code, for cross-section data of inelastic scattering, elastic scattering and capture reactions in Bi isotope (²⁰⁹Bi) shown in Figures 6–8, respectively, compared as ²⁷Al, ²³⁵U and ²³⁸U that are mainly core components. The sensitivity coefficients of inelastic scattering reactions (Figure 6(a)) of ²⁰⁹Bi were found to be dominant over the high-energy (MeV) region, like those of ²⁷Al shown in Figure 6(b). The sensitivity coefficients of ²⁰⁹Bi elastic scattering reactions revealed an increasing tendency between epi-thermal and fast neutron energy regions shown in Figure 7(a), although the coefficients of ²⁰⁹Bi and ²⁷Al were compared with the mostly same distribution around 1 MeV of the elastic scattering reactions (Figure 7(a,b), respectively). For the sensitivity coefficients of capture reactions, ²⁰⁹Bi was found at a highly-negative peak around 10³ eV regions shown in Figure 8(a), whereas ²⁷Al and ²³⁵U showed a locally strong depression around the thermal neutron region, as shown in Figure 8(b). Nonetheless, from all the results of sensitivity coefficients, absolute values of ²⁰⁹Bi were markedly very small in vertical axes shown in Figures 6–8 as compared with the values of ²⁷Al and ²³⁵U, demonstrating that the impact of ²⁰⁹Bi cross sections was considered minor in the sensitivity coefficient analyses of k_eff in the Bi sample reactivity worth experiments at KUCA.

Figure 6. Sensitivity coefficients of inelastic scattering reactions of ²⁰⁹Bi and ²⁷Al in JENDL-4.0. (a) ²⁰⁹Bi and (b)²⁷Al.

Figure 7. Sensitivity coefficients of elastic scattering reactions of ²⁰⁹Bi and ²⁷Al in JENDL-4.0. (a) ²⁰⁹Bi and (b)²⁷Al.

Figure 8. Sensitivity coefficients of capture reactions of ²⁰⁹Bi, ²⁷Al, ²³⁵U and ²³⁸U in JENDL-4.0. (a) ²⁰⁹Bi and (b) ²⁷Al, ²³⁵U and ²³⁸U.

3.4. Uncertainty

Uncertainty analyses by the UNCERTAINTY [23] code of the MARBLE system [24] were conducted with the use of JENDL-4.0 covariance data (107-energy-group). Since the covariance data of H, C and Al nuclides consisted mainly of core components are not provided in JENDL-4.0, the uncertainty was analyzed for several reactions of ²⁰⁹Bi, ²³⁵U and ²³⁸U, including capture, elastic scattering, inelastic scattering, fission and (n, 2n) reactions, comprising reference and test fuel assemblies of the KUCA A-core. Nonetheless, among the covariance data of ²⁰⁹Bi, inelastic scattering reactions were only prepared in JENDL-4.0. As shown in Table 5, the results of uncertainty induced by covariance data were large, with total reactivity of 24.4 pcm, compared with the experimental error around 3 pcm of sample reactivity worth (Table 2). The value of total uncertainty was acquired by the square root of the sum of squares of reaction-wise contributions, disregarding the covariance between different nuclides in the sum of squares. Among the nuclides, the reaction-wise contribution was dominant mainly over the capture (19.4 pcm) and fission (9.7 pcm) reactions of ²³⁵U, and reasonable in the inelastic scattering reactions (10.0 pcm) of ²⁰⁹Bi. In other words, non-negligible contribution of ²⁰⁹Bi inelastic scattering reactions was observed in the uncertainty analyses of Bi sample reactivity worth.

Table 5. Reaction-wise contribution [pcm] to changes in Bi sample reactivity worth (Case 4) induced by covariance data of JENDL-4.0

	Reactions
Isotopes	Capture	Elastic	Inelastic	Fission	(n, 2n)	Total
^235U	19.4	1.9	4.1	9.7	0.1	22.2
^238U	2.6	0.0	0.3	0.1	0.0	2.6
^209Bi	-	-	10.0	-	-	10.0
					Total	24.4

4. Comparative study on Bi and Pb

Bi sample reactivity worth experiments were considered successfully carried out from the viewpoint of the reproducibility of previous Pb sample reactivity worth experiments, since the measured excess reactivity of the Al reference core was compared with the Bi and Pb experiments under the same condition, as shown in Table 6. With the combined use of experimental and numerical results, a comparative study on Bi and Pb sample reactivity worth was instrumental in examining the neutron characteristics of Pb–Bi coolant material in the actual ADS experimental facility.

Table 6. Comparison of the results of measured excess reactivities between Bi and Pb test cores

Core	Bi sample [pcm]	Pb sample [pcm] (Ref. [13])
Reference (Al)	87 ± 1	92 ± 5
Case 1	143 ± 3	186 ± 7
Case 2	165 ± 3	202 ± 8
Case 3	163 ± 3	237 ± 9
Case 4	171 ± 3	248 ± 9

In terms of the absolute values of sample reactivity worth shown in Table 6, the difference between Bi and Pb sample reactivity worth clearly emphasized the significance of the characteristics of the actual ADS facility attributed to the reactivity effect. Interestingly, on the basis of the neutron characteristics of Pb–Bi, an ADS with a Pb–Bi coolant core could exactly be analyzed by nuclear design calculations. Additionally, from the results of the uncertainty of Bi and Pb isotopes shown in Table 7, the impact of Bi induced by nuclear covariance data was considered small compared with that of total contribution of Pb isotopes, and invaluable in understanding the reason for choosing Pb–Bi as coolant material in ADS.

Table 7. Comparison of reaction-wise contributions [pcm] between Bi and Pb (Ref [14].) sample reactivity worth (Case 4) induced by covariance data of JENDL-4.0

	Reactions
Isotopes	Capture	Elastic	Inelastic	Fission	(n, 2n)	Total
^204Pb	0.1	−0.4	1.6	-	0.0	1.7
^206Pb	−1.0	−4.9	20.0	-	−0.8	19.4
^207Pb	0.9	−2.6	9.0	-	1.5	8.8
^208Pb	−0.6	2.2	11.3	-	3.1	11.9
^209Bi	-	-	10.0	-	-	10.0

5. Conclusions

The Bi sample reactivity worth experiments were carried out at KUCA, and the uncertainties of criticality induced by the covariance data of Bi isotope were obtained by numerical calculations. The comparison between the experiments and the calculations by MCNP6.1 with JENDL-4.0 revealed the importance of the effect of criticality bias on the precision of numerical simulations. Additionally, sensitivity coefficients of the Bi isotope were relatively small with the comparison of ²⁷Al, ²³⁵U and ²³⁸U comprising of fuel plates and core components, and uncertainty induced by its cross sections demonstrated a reasonable result of the Bi sample reactivity worth. From the results of Bi isotope uncertainty, the comparative study on Bi and Pb sample reactivity worth was instrumental in emphasizing the neutron characteristics and the impact of Pb–Bi coolant material in ADS.

Finally, further uncertainty analyses by neutron transport calculations are needed in several reactions of the Bi isotope, especially with the use of the covariance data of capture, elastic scattering and inelastic scattering reactions in another nuclear data library. Thus, precise uncertainty analyses are expected to help in understanding of the neutronics of Pb–Bi coolant material for ADS in nuclear design calculations.

Acknowledgments

This study was supported by “Research and Development for Engineering Resolution of Nuclear Transmutation Cycle by Accelerator-Driven System” (131101) carried out under the Innovative Nuclear Research and Development Program by the Ministry of Education, Culture, Sports, Science and Technology of Japan. The authors are grateful to all the technical staff at KUCA for their assistance during the experiments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary material

Supplemental data can be accessed here.

Additional information

Funding

This study was supported by “Research and Development for Engineering Resolution of Nuclear Transmutation Cycle by Accelerator-Driven System” (131101) carried out under the Innovative Nuclear Research and Development Program by the Ministry of Education, Culture, Sports, Science and Technology of Japan.