Experimental analyses of βeff/Λ in accelerator-driven system at Kyoto University Critical Assembly

ABSTRACT The capability of βeff  /Λ obtained by λ-mode and ω-mode calculations is examined for target subcriticality in the accelerator-driven system through a comparison with that obtained in the pulsed-neutron source (PNS) experiments at the Kyoto University Critical Assembly. Directly measured results of βeff  /Λ, α and ρ$ in the PNS experiments are validated by varying the experimental conditions: the external neutron source, detector position, detector type and subcriticality ranging between 500 and 7500 pcm. The numerical analyses of βeff  /Λ are conducted by using MCNP6.1 together with ENDF/B-VII.1 for λ-mode calculations and PARTISN (SCALE6.2.2 with ENDF-VII.1 for the effective cross sections) for both the λ-mode and the ω-mode calculations. The comparison between calculated and measured βeff  /Λ with varying subcriticality shows good agreement between the experiments and the ω-mode calculations, although a difference is observed between the experiments and the λ-mode calculations.


Introduction
Basic research on accomplishing an operation of the accelerator-driven system (ADS) has been conducted continuously for aiming at the transmutation of minor actinide (MA) and long-lived fission products to reduce the burden on geological repositories. The operation at a subcritical state is highly conducive to the safety margin for the criticality accident even when loading large amounts of MA involving considerable uncertainty in the nuclear data. The progress of the reactivity accident in ADS operations was indicated to be sensitively attributable to subcritical settings characterized by a factor of the effective delayed neutron fraction (β eff ) [1]. Accuracy in the evaluation of β eff is affected importantlyfor ADS design, and monitoring of subcriticality is requisite. Also, since the value of β eff is very low (0.2~0.3%) in ADS with plutonium as the main fuel [2], an on-line monitoring technique of subcriticality has been proposed with the use of measured prompt neutron decay constant (α) and calculated kinetics parameters of β eff and generation time (Λ) [3]. For the assessment of ADS safety, β eff and Λ are emphasized quite important. In actual ADS design, the effective neutron multiplication factor (k eff ) has been evaluated by λ-mode (k-criticality) calculations [4,5].
At the Kyoto University Critical Assembly (KUCA), ADS feasibility studies have been conducted, approaching the actual ADS, by the combined use of a subcritical core and an external neutron source generated by the injection of 100 MeV protons onto a heavy metal target [6][7][8], and measurements of subcriticality ρ $ in dollar units and α have been carried out for examining the accuracy of the measurement methodology [9]. Moreover, the impact of an external neutron source on neutronic characteristics has been investigated with the use of 14 MeV neutrons and spallation neutrons, indicating that the neutron flux distribution was significantly distorted by inducing higher-mode components in the deep subcritical core (k eff = 0.85) [10]. In subcriticality measurement by the α-fitting method, α has been measured from time evolution of neutron flux distribution in the fundamental mode by setting a masking time in the fitting process so as to minimize the influence of highermode components [11]. Thus, for the spatial effect on subcriticality measurements at KUCA, the detector located diagonally to the core exerts experimentally less influence of the higher-mode components.
Here, in general, the λ-mode calculation is widely recognized as not being strictly correct for supercritical and subcritical cores, because the neutron spectra are evaluated as harder (in the subcritical core) and softer (in the supercritical core) than the actual spectra due to the manipulation of neutron number generated by fission reaction as a parameter of k eff . The bias in the neutron spectrum results in the acquisition of a different k eff . The difference between k eff values obtained by λ-mode and ω-mode (αcriticality) calculations has been explained through a comparison of neutron production spectra [12]. Nonetheless, since the difference was observed only in subcriticality (with very deep subcritical core (k eff = 0.7)) far from the target subcriticality of ADS ranging between k eff = 0.97 and k eff = 0.95, the remaining issue is whether the ω-mode calculation is requisite for the numerical analysis of ADS. The necessity of the ωmode calculation is easily revealed by a comparison between experimental results and experimental analyses (the λ-mode and the ω-mode calculations). Here, a high degree of independence between the calculations and the experiments is necessary because a calculation bias is easily induced by introducing a correction factor for such spatial effect into experimental results. Accordingly, examination of calculation validity is considered necessary through a comparison of directly measurable reactor physics parameters, including ρ $ , α and β eff /Λ, in pulsedneutron source (PNS) experiments [13].
The objective of this study was to experimentally validate measured β eff /Λ values by the comparison of ρ $ and α with different detectors and external neutron sources, demonstrating the capability of λ-mode and ω-mode calculations in the analyses of subcritical cores and the influence of subcriticality on kinetics parameters (β eff and Λ) in ADS experiments at KUCA. The experimental settings and the numerical analyses of ADS experiments are described in Sec. 2; the experimental results of ρ $ and α with different detectors and external neutron sources are shown in Sec. 3; the capability of λ-mode and ω-mode calculations is examined through the comparison with the measured β eff /Λ in Sec. 4; the conclusions are summarized in Sec. 5.

Experimental settings
At KUCA, ADS experiments were carried out in a uranium-lead (U-Pb) core with 14 MeV neutrons (Figure 1(a)) and spallation neutrons (Figure 1(b)). The core comprised normal fuel rods (1/8"p60EUEU) of highly-enriched uranium (HEU; 50.8 × 50.8 × 1.5875 mm) and a polyethylene moderator (p; 50.8 × 50.8 × 3.158 mm) in an aluminum sheath 54 × 54 × 1524 mm, as shown in Figure 2(a), and U-Pb fuel rods composed of an HEU plate and a Pb plate (Pb: 50.8 × 50.8 × 3.011 mm) in the center, as shown in Figure 2(b). The core spectrum was approximately hard in the central region for the spectrum of the actual ADS, and the driver (normal fuel) region has an H/U (hydrogen/uranium) ratio of approximately 50 in the thermal reactor.
Time evolution according to the injection of external neutrons was obtained from the signals of four BF 3 detectors set around the core. Furthermore, in case of spallation neutrons, an optical fiber type detector containing a Eu:LiCaAlF 6 scintillator [16] was additionally installed at location(10, I; Figure 1(b)) symmetrical to BF 3 #2 so as to examine the influence of the detector on measured subcriticality.
Subcriticality was obtained by full insertion of control and safety rods, and by the substitution of the fuel assembly for polyethylene rods, as shown in Table 1. In Cases I-1 to I-5, the subcriticality was experimentally deduced with the combined use of control rod worth and its calibration curve obtained by the positive period method. Moreover, in Cases II through VII, some of the fuel rods 'F' (Figure 1) were substituted for polyethylene reflectors and configured as shown in Figure 3. The subcriticality in dollar units was acquired experimentally by the extrapolated area ratio method [17]. Also, α was obtained by the α-fitting method in PNS experiments. The subcriticality level ranged between 500 and 7500 pcm.

Numerical analyses
Numerical analyses were conducted by using MCNP6.1 [18] together with ENDF/B-VII.1 [19] (total histories were 5×10 8 : 5×10 5 histories per cycle and 10 3 active cycles) and by PARTISN [20] (with mesh size less than 10 × 10 × 10 mm; transport cross section instead of P L scattering treatment; EO 16 quadrature for S N [21]; Figure  4). In PARTISN analyses, seven effective cross sections (7-energy group) were generated as described in Ref [22] with the SCALE6.2.2 code system [23], as shown in Figure 5. The validation of the numerical analyses was confirmed through comparison between measured reactivity and calculated reactivity. Here, excess reactivity and control rod worth (C1, C2, and C3) were measured by the positive period method and the rod drop method, respectively, and calculated as follows: (1) where ρ cal excess and ρ cal rod are the calculated excess reactivity and control rod worth, respectively, k critical eff the value of the effective multiplication factor at the critical state so as to       13 14 15 16 17 18 (e) Case VI (f) Case VII reduce the calculation bias induced by the nuclear data, k clean eff the value of effective multiplication factor at the withdrawal of all control rods, and k rod eff the value of effective multiplication factor at the insertion of control rod C1, C2 or C3 at the critical state. The difference was confirmed at less than 5% in control rod worth by MCNP6.1, as shown in Table 2, although large uncertainty was observed in excess reactivity attributed to estimating small reactivity, considering valid for subsequent analyses. For PARTISN calculations, while the C/E (calculation/experiment) ratio was 1.25 at most, the reproducibility was considered acceptable by comparison with experiments involving variations of subcriticality shown in Table 3.

Subcriticality in dollar units
The measured ρ $ compared with the calculated ρ MCNP $ by using MCNP6.1 and deduced as follows: where k case eff is the effective neutron multiplication factor in each of the cases shown in Table 3, and β MCNP eff the ↑ A ↑ A' Fig. 4(a)). Fig. 4(b)).  comparable until deep subcriticality. Also, notable is that no spatial effect of detector location was observed except for that of BF 3 #4 detector. The results by BF 3 #4 detector placed near the neutron source were not reliable since ρ $ values were significantly overestimated in deep subcriticality (over 6$) in both 14 MeV and spallation external neutron sources. The large error in deep subcriticality of 9$ in BF 3 #1 and BF 3 #2 was caused by the low count rate of delayed neutrons. As examination of detector type dependency on ρ $ , LiFCAF fiber detector indicated almost the same ρ $ value compared with that by BF 3 #2, validating the measurement results and capability of the λ-mode calculation for ρ $ .

Prompt neutron decay constant
Measured α was compared with calculated one (α MCNP ) by using MCNP6.1 and deduced as follows: where Λ MCNP is generation time obtained by MCNP6.1. In addition to α MCNP , the prompt neutron decay constant by the ω-mode calculation with PARTISN (α PARTISN ) was added to compare the difference between λ-mode and ω-mode calculations, as shown in Figures 8 and 9. Here, the error of the measurement was evaluated by the fitting error. The large error was obtained in the result by BF 3 #4 with spallation neutrons in Figure 9 since the PNS histogram was largely influenced by the decay of the neutron source. As in ρ $ , no dependence on any external neutron source was observed in the measurements, indicating that the decay of neutron flux in the fundamental mode was measured correctly. Also, the results from all detectors were equivalent, except from BF 3 #4. Furthermore, the difference in measurement methodology was around 5% between the α-fitting method and   the Feynman-α method as described in Reference [12], demonstrating that the measurement was valid. In comparing α MCNP with measured α, α MCNP was overestimated by 1100 1/s (k eff = 0.97). Conversely, α PARTISN agreed with the measured ones, demonstrating that the λ-mode calculation has possibility to be incapable for evaluating the α value even for target subcriticality in ADS operations through the comprehensive comparisons.

Results and discussion
4.1. Evaluation of β eff /Λ β eff /Λ as representative of the cores were experimentally deduced by combining α and ρ $ at BF 3 #2 detector, as follows: Also, calculated β eff /Λ values were obtained by combining adjoint-weighted effective delayed neutron fraction β PARTISN eff by PARTISN, the adjoint-weighted generation time Λ PARTISN was defined as follows: where ϕ and ϕ + are the forward and adjoint fluxes in λ-mode or ω-mode calculations, respectively, χ and χ d the total and delayed neutron fission spectra, respectively, ν and ν d the number of total and delayed neutrons by fission reaction, respectively, Σ f the effective fission cross section, v the neutron velocity, < > the integration over energy group and phase space, and g and g' the energy groups. Here, χ d (g') was obtained by the extraction of nuclear data of 235 U from ENDF/B-VII.1 by NJOY99 [24]. Also, ν d (g) was defined as follows: where ν p (g), which was obtained as in the procedure for χ d (g'), is the number of prompt neutrons by fission reaction. Furthermore, β eff /Λ values by using MCNP6.1 were obtained by combining kinetics parameters (evaluated with KOPT option in MCNP6.1). By varying subcriticality, the value of β eff /Λ tended to decrease with deep subcriticality as shown in Figure 10. Interestingly, β eff /Λ was differed even when subcriticality was the same, although the core size was different in Cases I-2 and III. Also, the nonsignificant difference of β eff /Λ values was observed between 14 MeV and spallation neutron sources in measurements. Thus, β eff /Λ was considered independent of any external neutron source, indicating that the measurement was conducted correctly to extract fundamental-mode components in the time evolution of neutron flux. In the comparison between calculations, the λ-mode calculation by PARTISN showed a bias compared with that by MCNP6.1, and, the results were almost the same as the measured ones at shallow subcritical states (Cases I-1 to I-5). At deep subcriticality, however, the λ-mode calculations showed a large difference as compared with the experiments. Conversely, ω-mode calculations correctly evaluated the experiments in the whole range except for Case III, indicating that the kinetics parameters were not correctly evaluated by the λ-mode calculation even for target subcriticality in ADS for Case IV (k eff = 0.97).
Here, ρ MCNP $ by λ-mode calculation showed agreement with measured ρ $ even in deep subcritical states, as shown in Figures 6 and 7. Conversely, as shown in Figures 8 and 9, the comparison of α values was observed the large discrepancy between experiments and calculations as subcriticality becomes deeper. The discrepancy in the comparison between the experiment and the calculation was considered attributable to Λ value for estimating calculated α value by Equation (4). The difference of Λ value between λ-mode and ω-mode calculations could be also explained by comparing between ρ $ and β eff /Λ since ρ $ values agreed between the experiment and the λ-mode calculation more accurately than the results of β eff /Λ ( Figure 10).

Discussion
The discussion is focused on the variation of each kinetics parameter by the calculations toward the subcriticality. β eff and Λ were compared in terms of the λ-mode or the ω-mode calculation as shown in Figure 11. The values of β eff were independent of subcriticality in ω-mode calculations; however, λ-mode calculations showed a slightly increased tendency toward subcriticality. Notably, Λ values revealed a different tendency compared with β eff values under λ-mode and ω-mode calculations. Furthermore, β eff values were equivalent in Cases I-5 and IV, by contrast, however, Λ values were different even at almost the same subcritical states, indicating that Λ is sensitive to the size of the core and to the insertion of control rods.   Figure 11. Relationship between calculated β eff and Λ for subcriticality variation.
In qualitative calculation property, the ω-mode calculation involves -α/v value in absorption term compared to the λ-mode calculation, indicating softer neutron spectrum by decreasing the total absorption cross section of -α/v (since lower the energy, the more increase -α/v value). Furthermore, the adjoint neutron spectrum is estimated softer by the λ-mode calculation to induce fission reactions, resulting in the overestimation of β eff value in the λ-mode calculation since the importance of delayed neutrons is emphasized by the adjoint neutron flux. Furthermore, the different estimation of adjoint neutron spectra is also considered to influence the value of Λ since observation showed overestimation of β eff and underestimation of Λ in the λ-mode calculation. Accordingly, calculated k eff by the λ-mode calculation is implied to be different from actual neutron multiplication factor (in the subcritical system) by combined use of α, β eff and Λ since the spectrum calculation was inadequate in the λ-mode calculations for supercritical and subcritical cores.
Investigation of the cause in the variation of Λ is considered limited since the number of energy groups is insufficient for characterizing the neutron spectra of ϕ and ϕ + . Thus, further investigation with a higher number of energy groups is needed.

Conclusion
ADS experiments were carried out to evaluate β eff /Λ values by varying detector type, detector position, external neutron source, and subcriticality. The capability of λ-mode and ω-mode calculations was examined by comparing directly measured ρ $ , α, and β eff /Λ in the PNS experiment with the target subcriticality of ADS ranging between 500 and 7500 pcm.
The measurement of ρ $ and α indicated slight dependence on an external neutron source but not on any spatial effect except for BF 3 detector located near the neutron source. Furthermore, the values of ρ $ and α between BF 3 and LiCAF detectors (installed at symmetric positions) were equivalent, demonstrating validity of the experiments by comprehensive comparison. In the experimental analyses, calculated ρ $ (λ-mode and ω-mode) showed good agreement with measured ρ $ within the whole range of subcriticality; however, the value of α by the λ-mode calculation showed a difference in the experiment at deep subcriticality. Conversely, α obtained by the ω-mode calculation agreed with the experiments.
The calculated results of β eff /Λ were compared with the measured ones to examine their capability under varying subcriticality variation; consequently, agreement was observed between the experiments and ω-mode calculations under a wide range of subcriticality. Notably, however, the λ-mode calculations differed from the experiments even under slight subcriticality, implying the necessity of introducing ω-mode calculations in ADS design for evaluating actual neutron multiplication factor in the subcritical system and kinetics parameters.

Disclosure statement
No potential conflict of interest was reported by the authors.