Sensitivity and uncertainty analysis of beff for MYRRHA using a Monte Carlo technique

This paper presents a nuclear data sensitivity and uncertainty analysis of the effective delayed neutron fraction beff for critical and subcritical cores of the MYRRHA reactor using the continuous-energy Monte Carlo N-Particle transport code MCNP. The beff sensitivities are calculated by the modified k-ratio method proposed by Chiba. Comparing the beff sensitivities obtained with different scaling factors a introduced by Chiba shows that a value of a=20 is the most suitable for the uncertainty quantification of beff. Using the calculated beff sensitivities and the JENDL-4.0u covariance data, the beff uncertainties for the critical and subcritical cores are determined to be 2.2±0.2% and 2.0±0.2%, respectively, which are dominated by delayed neutron yield of Pu and U.


Introduction
To promote research and development of nuclear technology for various applications such as accelerator-driven systems, the Generation-IV reactors, and production of medical radioisotopes, the Belgian Nuclear Research Centre (SCK·CEN) has proposed a cutting-edge research reactor combined with a proton accelerator, MYRRHA [1,2]. From the viewpoint of ensuring safety margins and reducing uncertainty in the MYRRHA design parameters, uncertainty quantification of reactor physics parameters is one of the most important tasks. To this end, nuclear data sensitivity and uncertainty (S/U) analyses have been extensively conducted for various MYRRHA core configurations using different calculation tools, geometric models, and nuclear data libraries [3][4][5]; these works have focused on the effective neutron multiplication factor k eff as the primary neutronic safety parameter. The effective delayed neutron fraction b eff can be ranked second in the list of neutronic safety parameters, because, besides of reactor kinetics, it is used to determine other design and safety parameters such as control rod worth and Doppler coefficient.
Continuous-energy Monte Carlo transport codes such as the Monte Carlo N-Particle transport code MCNP [6] have been widely used in calculating not only k eff but also its nuclear data sensitivities and kinetic parameters including b eff . Although these codes have no capability to directly calculate the b eff sensitivities owing to technical cumbersomeness, it is approximately expressed as a function of two different k eff sensitivities by the so-called "k-ratio method [7]"; this indicates that uncertainty in b eff can be quantified by the sensitivity method from the approximate b eff sensitivities and evaluated covariance data of the nuclear data library. Although this method has been applied to the b eff S/U analysis with a deterministic code SUSD3D for MYRRHA in the studies of Kodeli [8], within the seventh framework programme solving CHAllenges in Nuclear DAta (CHANDA) project [9], the analysis using the Monte Carlo transport code has not yet been tackled.
The k-ratio method itself is currently subdivided into two techniques: the prompt k-ratio method [7] and the modified k-ratio method proposed by Chiba [10,11]. Our previous study [12] by MCNP for a critical configuration of the VENUS-F zero-power reactor at the SCK·CEN site [13] demonstrated that the prompt k-ratio method involves large statistical uncertainty in the calculated b eff sensitivities, and it would be currently difficult to reduce it only by increasing the number of neutron source histories. On the other hand, we also demonstrated that Chiba's modified kratio method can alleviate this kind of problem.
In this study, we conducted the S/U analysis of b eff for two types of MYRRHA configurations (i.e. critical mode and subcritical mode) using Chiba's modified k-ratio method. b eff and its sensitivities were calculated using the MCNP version 6.1.1, and JENDL-4.0u [14,15] was used as the nuclear data library since it contains covariance data of delayed neutron yield n d .

MYRRHA core models
MYRRHA is designed to operate both in critical mode as a lead-bismuth cooled fast reactor and in subcritical mode driven by 600-MeV linear proton accelerator. Figures 1 and  2 show horizontal sectional views of the MYRRHA critical and subcritical configurations, respectively, which are homogenized on assembly level. The analyses were carried out for the critical and subcritical core configurations at beginning of cycle using the assembly-based homogenized models [16]. As illustrated, the critical and subcritical cores consist of 78 and 58 fuel assemblies (FAs), respectively; these are loaded with MOX with high Pu content. Besides FAs, the cores contains in-pile sections for material testing, irradiation rigs for medical isotope production, and subassemblies containing safety and control rods. More details of the MYRRHA core design are given in Van den Eynde et al. [2].

The b eff sensitivities
where a(≠ 0) is a scaling factor; k is the effective multiplication factor calculated using the nuclear data library JENDL-4.0u; k is the effective multiplication factor calculated using the library in which n d is multiplied with (a + 1) (Appendix A). If a = À 1, then Chiba's modified k-ratio method reduces to the prompt k-ratio method. The statistical uncertainty in b eff propagated from dk and dk is expressed as follows: Here correlations were disregarded for the sake of simplicity. Using the definition of the sensitivity and equation (1), the b eff sensitivity to parameter x is expressed as follows: where S k x ð ≡ ð∂k=∂xÞ=ðx=kÞÞ and S k x ð ≡ ð∂k=∂xÞ=ðx=kÞÞ are the sensitivities of k and k to the parameter x, respectively. The statistical uncertainty in S b eff x is expressed as where dS k x and dS k x are the statistical uncertainties in S k x and S k x , respectively.

The b eff uncertainties
Using the sensitivity profile obtained by the abovementioned methods, the b eff uncertainty (standard deviation) due to nuclear data was evaluated by the uncertainty propagation law considering covariance, which is expressed as where cov(z, g ; z 0 , g 0 ) denotes a (g, g 0 ; z, z 0 ) component of variance-covariance matrix (g, g 0 : energy group, z, z 0 : reaction), which was obtained by processing covariance data stored in JENDL-4.0u with NJOY [17] and ERRORJ [18]. In this analysis, we employed eight parameters: fission, neutron capture, elastic scattering, inelastic scattering, and (n, 2n) reaction cross sections for major constituent materials: U, Pu, 241 Am, 16 O, 56 Fe, Pb, and 209 Bi, as well as prompt and delayed neutron yields and prompt neutron spectra for U and Pu isotopes. Here correlations between the reactions were also considered. Absence of covariance data for delayed neutron spectrum in JENDL-4.0u did not allow us to estimate the contribution to the b eff and confirm the conclusion made by Kodeli [19] on its significance.  Here the value calculated directly by the adjoint weighting method [20] for the same number of histories for critical and subcritical cores were estimated to be 323 ± 4 and 320 ± 4 pcm, respectively. It can be seen that the calculated statistical uncertainty decreases with increase in a, which was about |a| times smaller than that by the prompt k ratio method (a = À 1). In addition, we see that the calculated b eff values exceeding their 1s statistical uncertainties decrease as a increases. The similar trend can be seen in the previous study conducted for the VENUS-F reactor using MCNP [12] and the benchmark analysis for fast neutron systems conducted by Chiba using the deterministic transport code CBG [10,21]; this reason is linked to the approximation used in equation (1).

Sensitivity
Figures 4 and 5 show comparisons of the b eff sensitivity profiles of 239 Pu fission and elastic scattering cross sections and 239 Pu n p and n d for the critical core with different scaling factors, respectively. As demonstrated in Iwamoto et al. [12], the statistical uncertainty at a = 1 is very large for all parameters except n d ; this is caused by the small difference between S k x and S k x (Appendix A). In contrast, the statistical uncertainty for n d is negligibly small for all the selected a values; this is owing to an approximation of S k n d ≃ ða þ 1ÞS k n d (Appendix A). In addition, as with the b eff values, dS b eff x decrease as a increases. However, it should be noted that the b eff sensitivities change within about 1s statistical uncertainties, while the nominal values of b eff tend to exceed their 1s statistical uncertainties. This indicates that the influence of the change in the b eff sensitivities that results from increasing a on the uncertainty quantification of b eff is expected to be small. Figures 6 and 7 show the b eff sensitivity profiles with respect to 239 Pu and 238 U reaction parameters for the critical core with a = 20, respectively, in which the statistical uncertainties appear to be sufficiently small. Table 1 summarizes the major b eff sensitivities together with 1s statistical uncertainties which were calculated with a = 20. As expected from the definition of b eff , n d of major   fuel materials (i.e. 239 Pu and 238 U) shows positive high sensitivities; in contrast, n p demonstrates negative sensitivities. Table 2 lists the b eff uncertainties due to nuclear data with different scaling factors for the critical and subcritical cores. Small scaling factors produce large both uncertainty values and their statistical uncertainties. This arises from the large sensitivities with large statistical uncertainties in the sensitivity profile as mentioned above; these statistical uncertainties can be reduced by increasing a. In addition, we see from Table 2 that, as a increases, the calculated total b eff uncertainties for the critical and subcritical cores approach values of 2.2% and 2.0%, respectively. Table 3 summarizes top 15 contributors to the b eff uncertainty together with 1s statistical uncertainties. Overall, the statistical uncertainties are small enough to identify the main contributors; namely, it can be concluded that the b eff uncertainty is dominated by n d , followed by the elastic and inelastic scattering cross sections, for both cores.

Conclusion
We have conducted the nuclear data S/U analysis of b eff for critical and subcritical cores of the MYRRHA reactor using the MCNP code. The b eff sensitivities were calculated by Chiba's modified k-ratio method. Although the nominal b eff values appear to worsen as a increases, comparing the b eff sensitivities and their statistical uncertainties calculated with different scaling factors shows that the b eff sensitivities are more stable to the a change than the nominal b eff values, and that a value of a = 20 is the most suitable for the uncertainty quantification. Using the calculated b eff sensitivities and the JENDL-4.0u covariance data, the b eff uncertainties for the critical and subcritical cores have been determined to be 2.2 ± 0.2% and 2.0 ± 0.2%, respectively, which are dominated by n d of 239 Pu and 238 U.
To account for the optimal a value more clearly, further investigation, especially for non-linear effects on c a and c † a , is needed. Moreover, it would be of interest to compare our results with those previously performed using the deterministic code within the ongoing CHANDA project [8].