k_eff uncertainty quantification and analysis due to nuclear data during the full lifetime burnup calculation for a small-sized prismatic high temperature gas-cooled reactor

k eff uncertainty quantification and analysis due to nuclear data during the full lifetime burnup calculation for a small-sized prismatic high temperature gas-cooled reactor 1 first-author https://orcid.org/0000-0001-6978-1122 2 1 2 corresp haochen.heu@163.com haochen.heu@163.com haochen.heu@163.com 1 1 Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, College of Nuclear Science and Technology, Harbin Engineering University, Harbin 150001, China. Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, College of Nuclear Science and Technology, Harbin Engineering University, Harbin 150001, China. Department of Reactor Technology, Beijing Institute of Nuclear Engineering, China Nuclear Power Engineering Co., Ltd, Beijing 100089, China Department of Reactor Technology, Beijing Institute of Nuclear Engineering, China Nuclear Power Engineering Co., Ltd, Beijing 100089, China To benefit from recent advances in modeling and computational algorithms, as well as the availability of new covariance data, sensitivity and uncertainty analyses are needed to quantify the impact of uncertain sources on the design parameters of small prismatic high-temperature gas-cooled reactors (HTGRs). In particular, the contribution of nuclear data to the k eff uncertainty is an important part of the uncertainty analysis of small-sized HTGR physical calculations. In this study, a small-sized HTGR designed by China Nuclear Power Engineering Co., Ltd. was selected for k eff uncertainty analysis during full lifetime burnup calculations. Models of the cold zero power (CZP) condition and full lifetime burnup process were constructed using the Reactor Monte Carlo Code RMC for neutron transport calculation, depletion calculation, and sensitivity and uncertainty analysis. For the sensitivity analysis, the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance Characterization (CLUTH) method was applied to obtain sensitive information, and the "sandwich" method was used to quantify the k eff uncertainty. We also compared the k eff uncertainties to other typical reactors. Our results show that 235 U is the largest contributor to k eff uncertainty for both the CZP and depletion conditions, while the contribution of 239 Pu is not very significant because of the design of low discharge burnup. It is worth noting that the radioactive capture reaction of 28 Si significantly contributes to the k eff uncertainty owing to its specific fuel design. However, the k eff uncertainty during the full lifetime depletion process was relatively stable, only increasing by 1.12% owing to the low discharge burnup design of small-sized HTGRs. These numerical results are beneficial for neutronics design and core parameters optimization in further uncertainty propagation and quantification study for small-sized HTGR. Small-sized HTGR SU analysis Nuclear data Burnup Introduction 1 Owing to their significant inherent safety and applicability characteristics, high-temperature gas-cooled reactors (HTGRs) have gradually played indispensable roles in nuclear reactor development [ 1 1 - 3 3 1-3 ] . HTGRs can be split into two types based on their core design: pebble-bed HTGRs, such as the high-temperature reactor pebble-bed module (HTR-PM) developed in China [ 4 4 ] , and prismatic HTGRs, such as the modular high-temperature gas-cooled reactor (MHTGR-350), developed in the US [ 5 5 ] . Simultaneously, small reactors have become a hotspot in international nuclear energy research. HTGR technology is developing rapidly in China, and a new small-sized prismatic HTGR is under development by the China Nuclear Power Engineering Co., Ltd. The continued development of HTGRs requires verification of their designs with reliable high-fidelity physics models and efficient accurate codes. The predictive capability of codes for HTGR designs can be assessed using sensitivity and uncertainty (SU) analysis methods. Through advancements in computer modeling and computational algorithms, as well as the accessibility of new covariance data, SU analysis can quantify the impact of uncertainties on the design parameters of small prismatic HTGRs. In particular, the effective multiplication factor ( k eff ) is the most important parameter in reactor physical analysis, its uncertainty propagated by nuclear data is usually indicated as an interval of k eff value. Because the uncertainty of nuclear data exists naturally, the contributions of nuclear data on the k eff uncertainty are essential for the designer to optimize core lifetime and neutronics design. For SU analysis of HTGRs, a representative international project is the Coordinated Research Project (CRP) on the HTGR Uncertainty Analysis in Modeling (UAM), supported by the International Atomic Energy Agency (IAEA), which considers the peculiarities of HTGR designs and simulation requirements [ 6 6 , 7 7 ] . In the CRP, the coupled HTGR system calculations are divided into several steps, each of which can contribute to the total uncertainty. Simultaneously, the input, output, and assumptions for each step need to be identified. The resulting uncertainty in each step is calculated by considering all sources of uncertainties, including related uncertainties from previous steps [ 8 8 ] . Some in-depth studies have quantified the contribution of cross-section uncertainties to the eigenvalue uncertainty for some representative but simplified models for both the pebble-bed and prismatic HTGRs [ 6 6 - 10 10 6-10 ] . For SU analysis of the prismatic HTRG, local and global calculations have been performed, including steady-state and depletion calculations for the cell and core models. The k eff uncertainties due to the nuclear data for the fresh block core and mixed core of the MHTGR-350 have been quantified [ 11 11 ] . Although both the small-sized HTGR selected in this study and MHTGR-350 belong to prismatic HTGRs, there are some significant differences. Unlike the mixed core arrangement in MHTGR-350, the small-sized prismatic HTGR only has fresh fuel with burnable poison (BP) in the core at the beginning of life (BOL). At the same time, this study can enrich the content of the IAEA CRP in HTGRs. This study focuses on k eff uncertainty analysis due to the nuclear data during full lifetime burnup calculations of the small-sized prismatic HTGR. We will quantify the different cross sections contributions on the k eff uncertainty at the CZP condition and the full lifetime burnup process and analyze the mechanism in-depth. The following section describes the model details of the small-sized prismatic HTGR and SU analysis methods. Based on the first-order perturbation theory [ 12 12 ] , we selected the contribution-linked eigenvalue sensitivity/uncertainty estimation via track length importance characterization (CLUTCH) method [ 13 13 ] to perform the sensitivity analysis, and the "sandwich" rule [ 14 14 ] to quantify the k eff uncertainty by using the ENDF/B-VII.1 based covariance data [ 23 23 ] . The rest of this paper is structured as follows. In Section 2, we introduce the model and method applied in this study, especially the full core burnup model of the small-sized HTGR for neutron transport and depletion calculations. In Section 3, we present the SU analysis of k eff due to nuclear data in the full lifetime depletion calculation. In Section 4, we present the in-depth mechanism analysis of nuclear data contributions on k eff during the full lifetime burnup process. Finally, we present the numerical results and conclusions drawn from SU in Section 5. Models and Methodologies 1 Small-sized HTGR model 2 The small-sized HTGR, which is under development by the China Nuclear Power Engineering Co., Ltd., was selected as the research target in this study. This small-sized HTGR is a helium-cooled, graphite-moderated prismatic reactor and has some unique characteristics, such as fuel blocks and a burnable poison rod arrangement [ 15 15 ] . A representation of the core layout is shown in Fig.1 Fig.1 ; 30 hexagonal prism fuel blocks and 13 control rod blocks are closely arranged in the core. The seven control rod blocks are surrounded by fuel assemblies, including one center startup control block and six shutdown control blocks. The other six control blocks are on the six corners of the core beside the fuel blocks, and each fuel block contains 24 fuel rods and seven coolant channels within the graphite matrix. Each fuel rod or coolant channel has a hexagonal graphite cladding in the fuel blocks. The radial reflector around the core is also made of graphite, but the density is much lower than different from the hexagonal graphite cladding. The coolant gas and reflector material specifications are the same as those of the pebble bed reactor, which uses helium as the coolant and graphite as the moderator and reflector. In particular, several cylinder fuel pellets are added to the upper and bottom reflective layers to constitute a fuel rod and TRISO particles [ 16 16 ] are dispersed in the SiC matrix to form a fuel pellet. This is different from the pebble-bed HTGR, in which TRISO particles are dispersed in the graphite matrix. During the burnup calculation, every fuel kernel in the TRISO particles is a basic burnup unit. At the same time, as the depletion proceeds, fission nuclides are consumed and new fission products, such as Ce, Pr, Pu, and Np, are generated. Some of these will cause fission events again and introduce new uncertainties to the core. Moreover, owing to the arrangement of the reflective layers, fuel blocks, and control rod blocks presented in Fig.1 Fig.1 , the discrepancy in the burnup degree in different burnup areas will be gradually evident during the depletion process. Thus, a 24 burnup zone model (four zones in the radial direction and six zones in the axial direction) was established for the depletion calculation, as shown in Fig.2 Fig.2 . In this study, the reactor Monte Carlo code (RMC), developed by Tsinghua University, was used for calculating the neutron transport and depletion for the high-fidelity model of the small-sized prismatic HTGR [ 17 17 ] . The ENDF/B-VII.1 cross-section library [ 23 23 ] was chosen for the calculation. The setting parameters for the MC critical and burnup calculations are illustrated in Table 1 Table 1 . Based on these parameters, each MC critical calculation for this model can converge and fulfill the accuracy requirements of the calculated results. For the full core depletion calculation, the statistical error-based MC method was lower than 25 pcm in each burnup step. At the same time, fission poisons, such as 135 Xe and 149 Sm, can have a huge impact on reactivity at the BOL. Therefore, to study these nuclide reaction contributions to k eff uncertainty and its sensitivity variation during the burnup calculation, the time of the burnup steps must be set small at the BOL, as illustrated in Table 1 Table 1 . During the depletion calculation, the RMC produces a large amount of complicated nuclide information, including the nuclide densities for each burnup region in each burnup step. In addition, the predictor correction method was used for the RMC burnup calculation [ 18 18 ] . For the MC depletion calculation, the nuclide densities used for this burnup step were derived from the results of the previous burnup step. Therefore, the densities of the nuclides as the input parameters for the uncertainty calculation should be the average value of the predicted and corrected densities [ 19 19 ] . 1 1 Number of total generations 1 1 150 1 1 Inactive generations 1 1 50 1 1 Number of neutrons per generation 1 1 100,000 1 1 Burnup steps 1 1 15 1 1 Burnup sub-steps 1 1 10 1 1 Burnup step time (day) 1 1 0.5, 1.0, 3.5, 15, 30, 50×7, 100×2 Moreover, because all fresh fuels are input into the core simultaneously, there is a large amount of excess reactivity at the BOL. BP isotopes, such as 157 Gd, 10 B, or 167 Er, have large neutron absorption cross-section of themselves and little absorption cross-section of their products, which usually be chosen to balance the excessive reactivity at BOL to reduce the number of control rods as well as deepen the burnup and flatten the distribution of neutron fluence rate. For this burnup model, there are six center fuel blocks around the center control rod block, each containing three Gd 2 O 3 burnable poison rods, each of which is a unique burnup unit during the MC depletion calculation. SU analysis method 2 In this paper, the CLUTCH method [ 13 13 ] based on first-order perturbation theory was used for k eff sensitivity analysis during the Monte Carlo calculations. This method calculates the importance of events during a particle's lifetime by examining the number of fission neutrons created by that particle after those events occur. The aim is to produce an accurate and efficient method for calculating k eff sensitivity coefficients for nuclear cross-sections with a relatively small computational memory. The CLUTCH method only calculates the sensitivity information during the forward calculations. Therefore, a fine important weight function F *( r ) mesh number should be set to ensure accurate sensitivity. An interval of 1–2 cm mesh is typical for obtaining accurate F *( r ) estimates [ 13 13 ] . The F *( r ) mesh only needs to cover the fissionable regions in the core; therefore, in this study, the mesh only needed to cover the fuel block regions. The F *( r ) meshes are also calculated in the inactive generations, so at least 50 to 100 inactive histories should be simulated per mesh interval for sufficient F *( r ) convergence. In this paper, 1.8 cm length meshes in radial, 1.86 cm length meshes in axial, and 750 total histories and 600 inactive histories were set for the CLUTCH method calculations. After the sensitivity coefficients were measured by the CLUTCH method, the k eff uncertainty can be quantified using the "sandwich" rule [ 14 14 ] and the ENDF/B-VII.1 based covariance matrix [ 23 23 ] . SU analysis of k eff during the full lifetime depletion calculation 1 SU analysis of k eff at CZP condition 2 Based on the RMC depletion calculation, several burnup step results were chosen to investigate the uncertainty of k eff due to nuclear data. At the BOL, there is only fresh fuel in the core and no fission products, which is known as the cold zero power (CZP) condition. To observe the sensitivity and uncertainty contribution of key nuclide cross sections in the depletion process, SU analysis at the CZP condition should be considered. Through uncertainty quantification, the relative standard deviation of k eff due to nuclear data at the CZP condition was 0.6586%. The top 10 most crucial nuclide reaction covariance contributors to the k eff uncertainty for small-sized HTGRs under the CZP condition are presented in Table 2 Table 2 , where the numerical results were obtained by RMC. 1 1 Rank 2 1 Nuclide-reaction 2 1 Nuclide-reaction 1 1 Contributions to uncertainty in k eff (% Δ k / k ) 1 1 1 1 1 235 U 1 1 υ 1 1 235 U 1 1 υ 1 1 3.74×10 -1 ±2.90×10 -5 1 1 2 1 1 28 Si 1 1 n, γ 1 1 28 Si 1 1 n, γ 1 1 2.58×10 -1 ±3.59×10 -6 1 1 3 1 1 C-graphite 1 1 elastic 1 1 C-graphite 1 1 elastic 1 1 2.50×10 -1 ±4.64×10 -4 1 1 4 1 1 235 U 1 1 χ 1 1 235 U 1 1 χ 1 1 1.80×10 -1 ±2.61×10 -4 1 1 5 1 1 235 U 1 1 n, γ 1 1 235 U 1 1 n, γ 1 1 1.68×10 -1 ±2.22×10 -6 1 1 6 1 1 238 U 1 1 n, γ 1 1 238 U 1 1 n, γ 1 1 1.47×10 -1 ±4.89×10 -6 1 1 7 1 1 235 U 1 1 n, f 1 1 235 U 1 1 n, γ 1 1 1.32×10 -1 ±5.23×10 -6 1 1 8 1 1 235 U 1 1 n, f 1 1 235 U 1 1 n, f 1 1 1.28×10 -1 ±9.49×10 -6 1 1 9 1 1 C-graphite 1 1 n, γ 1 1 C-graphite 1 1 n, γ 1 1 9.14×10 -2 ±5.01×10 -7 1 1 10 1 1 157 Gd 1 1 n, γ 1 1 157 Gd 1 1 n, γ 1 1 6.49×10 -2 ±5.23×10 -6 The average number of neutrons emitted per fission event of 235 U is the main contributor to the k eff uncertainty and accounts for nearly 17.40% of the total uncertainty of k eff - based nuclear data. This phenomenon is similar to the results of a previously reported uncertainty analysis of HTR-10 [ 10 10 , 22 22 ] . However, the main k eff uncertainty contributors from the results of uncertainty analyses of typically pressurized water reactors (PWRs) and boiling water reactors (BWRs) [ 20 20 , 21 21 ] are different, which has detailed description in Sect. 3.2.2. Moreover, the radioactive capture reaction cross-section of 28 Si should be considered as a significant factor because it is the second contributor to k eff uncertainty. This reaction cross-section has an 11.98% contribution to the total uncertainty of k eff . This value is much higher in small-sized HTGRs than in other typical reactors [ 10 10 , 20 20 ] . In addition, the elastic scattering of C-graphite is the third contributor, and the fission spectrum of 235 U is the fourth contributor. It should be noted that the fuel pellet matrix materials are different between the pebble-bed and small-sized HTGRs: C-graphite is used for pebble-bed HTGRs and SiC for small-sized HTGRs. Furthermore, the volume ratio of Si in the small-sized HTGR fuel pellet was 55.87%, which was significantly higher than that in the pebble-bed reactor. For a more in-depth study on the effect of nuclear data in k eff uncertainty under the CZP condition, some necessary nuclide reaction energy sensitivity coefficient curves are presented in Fig. 3 Fig. 3 . As the first contributor to k eff uncertainty under CZP conditions for small-sized HTGRs, the average number of neutrons emitted per fission event of 235 U has a large sensitivity coefficient in the thermal neutron energy range (energy less than 1 eV). Based on the "sandwich" rule, the large uncertainty contribution of the average number of neutrons emitted per fission event of 235 U can be attributed to its large sensitivity coefficient. The large k eff uncertainty contribution of the radioactive capture reaction of 28 Si can be explained by the same reason. Additionally, C-graphite and 238 U as the resonance nuclides are mainly reflected in the resonance energy range (energy less than 0.1 MeV and more than 1 eV). Nevertheless, SU analysis in the CZP condition is just one stage of the depletion calculation. Next, the study focuses on the variation of the important nuclide reaction sensitivity coefficients and cross-section contributions on k eff uncertainty during the full lifetime depletion calculation. SU analysis of k eff in the depletion calculation 2 k eff sensitivity analysis 3 Generally, the k eff sensitivity to important nuclide reactions can be used to measure the degree of influence of these reactions on k eff . Here, we used RMC to evaluate the variations in the k eff value and calculated the k eff sensitivity to some important nuclide reactions during the full lifetime depletion process. The specific fuel burnup step times were set as 0, 0.5, 1.5, 5, 20, 50, 100, 150, 200, 250, 300, 350, 400, 500, and 600 days. The k eff results for these burnup steps are illustrated in Fig.4 Fig.4 . For the k eff uncertainty quantification, in a condition of keep k eff change trend during the full lifetime, select as few burnup step results as possible to reduce sensitivity coefficients calculation time. The 0, 5, 50, 150, 250, 400, and 600 days burnup step results were chosen for analysis. During the depletion process in the RMC, the nuclide density required by the transport calculation was obtained from the solution of the depletion equation. In this way, the nuclide density is updated in each burnup step. Therefore, the nuclide density variation of certain important nuclides and their effect on k eff uncertainty needs to be investigated. In general, some fission elements, fission products, and fission poison nuclides are considered, such as 235 U, 238 U, 239 Pu, 135 I, 135 Xe, 149 Sm, and 155 Gd. In the RMC Monte Carlo depletion calculation, nuclide density data are obtained from the results of the previous burnup step. Because the poison elements 135 Xe and 149 Sm are produced, as shown in Fig.5 Fig.5 , the large neutron absorption cross-section of these elements makes the k eff decline steeply from the BOL to 5 days. Owing to the space self-shielding effect of the burnable poison Gd, k eff exhibits an upward trend between 5 and 150 days. During the depletion, the burnable poison nuclide 155 Gd was consumed rapidly, and after nearly 250 days the amount in the reactor was very low, as illustrated in Fig.5a Fig.5a . Therefore, without the effect of the burnable poison Gd, the k eff value decreases with fuel depletion until the end of life. According to the sensitivity analysis results, the k eff sensitivity coefficients for some vital nuclide reactions were considerable during the depletion calculation. Table 3 Table 3 lists the 14 main nuclide reactions with average integrated sensitivity coefficients during the small-sized HTGR depletion calculation. It should be noted that the average integrated sensitivity coefficients are calculated by integrating all energy groups for all regions and the sensitivity of the mixture materials through seven burnup steps. Two illustrative line charts of the integrated sensitivity coefficients of these important nuclides and their reaction cross sections are presented in Fig.6 Fig.6 . In addition, Fig.7 Fig.7 shows the difference in integrated sensitivity coefficients from the BOL to the end-of-life (EOL). According to Table 3 Table 3 and Fig.6 Fig.6 , the integrated sensitivity coefficients of the average number of neutrons emitted per fission event of 235 U, elastic scattering of C-graphite, radioactive capture reaction of 239 Pu, and fission reaction of 239 Pu have more considerable variations than other nuclide reactions. 1 1 Nuclides 1 1 Nuclear reaction 1 1 Sensitivity coefficients 1 1 Spread (Max-Min) 1 1 STDEV 1 1 235 U 1 1 υ 1 1 9.36×10 -1 1 1 1.37×10 -1 1 1 5.16% 1 1 C-graphite 1 1 elastic 1 1 4.80×10 -1 1 1 1.32×10 -1 1 1 5.58% 1 1 235 U 1 1 n,f 1 1 3.63×10 -1 1 1 3.04×10 -2 1 1 2.08% 1 1 235 U 1 1 n,γ 1 1 -1.20×10 -1 1 1 6.87×10 -3 1 1 0.54% 1 1 238 U 1 1 n,γ 1 1 -1.20×10 -1 1 1 3.36×10 -3 1 1 0.11% 1 1 28 Si 1 1 n,γ 1 1 -5.54×10 -2 1 1 3.86×10 -3 1 1 0.15% 1 1 239 Pu 1 1 n,f 1 1 3.33×10 -2 1 1 8.02×10 -2 1 1 3.05% 1 1 239 Pu 1 1 n,γ 1 1 -2.23×10 -2 1 1 5.26×10 -2 1 1 2.02% 1 1 135 Xe 1 1 n,γ 1 1 -1.24×10 -2 1 1 1.03×10 -2 1 1 0.55% 1 1 10 B 1 1 n,γ 1 1 -7.72×10 -3 1 1 1.15×10 -4 1 1 0.02% 1 1 157 Gd 1 1 n,γ 1 1 -6.49×10 -3 1 1 1.61×10 -2 1 1 0.69% 1 1 149 Sm 1 1 n,γ 1 1 -4.68×10 -3 1 1 6.47×10 -3 1 1 0.29% 1 1 155 Gd 1 1 n,γ 1 1 -2.66×10 -3 1 1 3.55×10 -3 1 1 0.19% 1 1 235 U 1 1 χ 1 1 1.91×10 -10 1 1 1.57×10 -10 1 1 0.00% Moreover, the radioactive capture reaction of 28 Si has a relatively high sensitivity, and its sensitivity coefficient value essentially remains unchanged during the full lifetime, as illustrated in Fig.6 Fig.6 . At the same time, the integrated sensitivity coefficients of some poison isotopes, such as 135 Xe, 149 Sm, and 155 Gd, have no obvious variation, and these values are quite small during the full lifetime depletion. k eff uncertainty analysis 3 According to the previous analysis of the important nuclide reaction sensitivity coefficients, the uncertainty contributions of the reaction cross sections to k eff may have significantly different throughout the depletion calculation, which requires further study. The SU results calculated by RMC show that the rank of the top eight contributors during the full lifetime depletion calculation did not differ substantially, however the contribution values varied. There are 12 important nuclide reaction cross-section average contributions to the k eff uncertainty for the small-sized HTGR depletion calculation, which are illustrated in Table 4 Table 4 . These contributions to the k eff uncertainty are the average values of the results of the seven burnup steps. It is obvious that the elastic scattering of C-graphite, radioactive capture of 135 Xe, radioactive capture of 157 Gd, radioactive capture and fission reaction of 239 Pu, and the average number of neutrons emitted per fission event of 235 U change substantially throughout the depletion calculation through the standard deviation presented in Table 4 Table 4 . In addition, combined with the results summarized in Tables 3 Tables 3 and 4 4 , the radioactive capture reactions of poison isotopes 157 Gd and 135 Xe have relatively large variations in contribution values. However, their sensitivity coefficients were mostly stable during the lifetime depletion calculation. 1 1 Nuclides 2 1 Covariance Matrix 1 1 Average Contributions to Uncertainty in k eff (% Δ k / k ) 1 1 Spread (Max-Min) 1 1 STDEV 1 1   1 1 Nuclide reaction 1 1 Nuclide reaction 1 1   1 1   1 1   1 1 235 U 1 1 υ 1 1 υ 1 1 3.50×10 -1 1 1 6.98×10 -2 1 1 2.58% 1 1 28 Si 1 1 n,γ 1 1 n,γ 1 1 2.59×10 -1 1 1 1.33×10 -2 1 1 0.52% 1 1 C-graphite 1 1 elastic 1 1 elastic 1 1 2.57×10 -1 1 1 6.59×10 -2 1 1 2.90% 1 1 235 U 1 1 χ 1 1 χ 1 1 1.67×10 -1 1 1 2.49×10 -2 1 1 0.99% 1 1 235 U 1 1 n,γ 1 1 n,γ 1 1 1.56×10 -1 1 1 2.10×10 -2 1 1 1.00% 1 1 238 U 1 1 n,γ 1 1 n,γ 1 1 1.48×10 -1 1 1 4.43×10 -3 1 1 0.17% 1 1 235 U 1 1 n,f 1 1 n,γ 1 1 1.21×10 -1 1 1 1.02×10 -2 1 1 0.78% 1 1 135 Xe 1 1 n,γ 1 1 n,γ 1 1 5.13×10 -2 1 1 4.26×10 -2 1 1 2.48% 1 1 157 Gd 1 1 n,γ 1 1 n,γ 1 1 2.50×10 -2 1 1 6.45×10 -2 1 1 2.95% 1 1 239 Pu 1 1 n,f 1 1 n,f 1 1 2.41×10 -2 1 1 4.42×10 -2 1 1 2.26% 1 1 239 Pu 1 1 n,γ 1 1 n,γ 1 1 2.28×10 -2 1 1 4.97×10 -2 1 1 2.12% 1 1 149 Sm 1 1 n,γ 1 1 n,γ 1 1 7.23×10 -3 1 1 3.24×10 -3 1 1 0.40% The 12 important nuclide reaction cross-section contributions to k eff uncertainty in small-sized HTGR depletion calculations are shown in Fig.8 Fig.8 . The solid lines represent the uncertainty contribution variations of the nuclear reaction for each nuclide that exists at the BOL. The figure shows that the average number of neutrons emitted per fission event of 235 U contributions decreased with the depletion calculation, but it was still the most significant contributor to the uncertainty of k eff across the full lifetime. Furthermore, all cross sections other than the elastic scattering cross-section of C-graphite and the radioactive capture cross-section of 28 Si exhibited a downward trend. In particular, the radioactive capture reaction of 157 Gd showed a noticeable decline from BOL to nearly 150 days. This was mainly caused by the depletion of 157 Gd. Simultaneously, this also led to a reduction in the average number of neutrons emitted per fission event of 235 U, contributing to k eff uncertainty. The dotted lines in Fig.8 Fig.8 express the fission product reaction contribution to k eff uncertainty; all reaction contributions have an increasing trend with the nuclides produced during the burnup calculation. As an important fission product, 135 Xe is produced rapidly at the BOL and reaches equilibrium at 4 to 5 days. Simultaneously, the radioactive capture reaction of 135 Xe was the main contributor to fission products until nearly 450 days. After 450 days, the fission reaction and radioactive capture cross-section of 239 Pu became the main contributors. However, the contributions of the radioactive capture reaction of 149 Sm were low and barely changed. Figure 9 Figure 9 shows the total variation of the important nuclide reaction contributions to the k eff uncertainty. The phenomenon concluded with Tables 3 Tables 3 and 4 4 can be more intuitively seen in Fig.7 Fig.7 and Fig.9 Fig.9 that the radioactive capture of 135 Xe and 157 Gd has only tiny variations in sensitivity coefficients, but the contributions to k eff uncertainty differ significantly during the full lifetime. Considering the numerical results in Sect. 3.2.1, the fission spectrum of 235 U makes a large contribution to k eff uncertainty after the radioactive capture of 135 Xe and 157 Gd. However, the sensitivity coefficients do not noticeably increase during the depletion calculation. However, the nuclide reaction cross-section has uncertainty, which is presented by the covariance matrix based on the nuclear data library [ 23 23 ] . Because the “sandwich” rule is used to quantify uncertainty, although the integrated sensitivity of k eff to the 235 U fission spectrum is only -1.91×10 -10 , the large relative covariance explains why the 235 U fission spectrum is the fourth most significant contributor. In addition, the relative covariance of the radioactive capture reaction of 28 Si is not large in the ENDF/B-VII.1 covariance library [ 23 23 ] , but its contribution to the uncertainty of k eff is still significant in small-sized HTGR depletion calculations. The density of 28 Si remained almost unchanged throughout the lifetime. Thus, it can be concluded that the large volume ratio in the core and large average sensitivity coefficient are the main reasons that the radioactive capture of 28 Si is the second largest contributor to k eff uncertainty. After analyzing the contribution of some important nuclide reactions, the total uncertainty of k eff during the full lifetime depletion calculation was quantified, as shown in Fig.10 Fig.10 . According to the numerical results of the seven burnup steps, the uncertainty of k eff remained largely constant. Moreover, owing to the fission products constantly generated during the depletion process, the uncertainty has a slightly increasing trend. Since 2007, there have been many developments in reactor uncertainty analysis modeling, such as the OECD/NEA of Light Water Reactor (LWR) UAM, OECD/NEA of Boiling Water Reactor (BWR) UAM, and IAEA CRP UAM on HTGR [ 7 7 , 9 9 , 20 20 , 21 21 ] . The k eff uncertainty results of these reactor uncertainty analysis projects are presented in Table 5 Table 5 . The burnup value of the small-sized HTGR was only 10.789 GWd/tU at EOL. Nevertheless, the burnup of PWR and HTR-10 was much deeper. At the same time, the PWR and BWR all have an extensive increase in k eff uncertainty during the full lifetime depletion, and the uncertainty value increased by 28.98% and 40.22%, respectively. However, the uncertainty of k eff in small-sized HTGRs only increased by 1.12% in the full lifetime depletion calculation owing to the low discharge burnup design. 1 1 Reactor 1 1 EOL burnup (GWd/tU) 2 1 k eff uncertainty(%Δ k / k ) 1 1   1 1   1 1 BOL 1 1 EOL 1 1 PWR TMI-1 1 1 60 1 1 0.49 1 1 0.69 1 1 BWR PB-2 1 1 45 1 1 0.55 1 1 0.92 1 1 HTR-10 1 1 52.72 1 1 0.6609 1 1 — 1 1 Small-sized HTGR 1 1 10.789 1 1 0.6359 1 1 0.6431 After analyzing the k eff uncertainty due to the nuclear data during the burnup calculations for different types of reactors, the contributions of the top five most important nuclide reactions to k eff uncertainty in different typical reactors under CZP conditions were determined, as illustrated in Table 6 Table 6 . BWR, pebble-bed HTR-10, and small-sized HTGRs were included. It is clear that in BWR, the top contributor to k eff uncertainty is the radioactive capture reaction of 238 U. However, in the two HTGRs, the first contributor to k eff uncertainty was the average number of neutrons emitted per fission event of 235 U. Moreover, the radioactive capture reaction of 28 Si being the second contributor to k eff uncertainty is a novel finding in small-sized HTGRs. 1 1 Rank 2 1 BWR 1 1 %Δ k / k 2 1 HTR-10 1 1 %Δ k / k 2 1 Small-sized HTGR 1 1 %Δ k / k 1 1   2 1 Nuclide-reaction 1 1   2 1 Nuclide-reaction 1 1   2 1 Nuclide-reaction 1 1 1 1 1 238 U n, γ 1 1 238 U n, γ 1 1 0.30 1 1 235 U υ 1 1 235 U υ 1 1 0.38 1 1 235 U υ 1 1 235 U υ 1 1 0.37 1 1 2 1 1 235 U υ 1 1 235 U υ 1 1 0.28 1 1 C-gra elastic 1 1 C-gra elastic 1 1 0.31 1 1 28 Si n, γ 1 1 28 Si n, γ 1 1 0.26 1 1 3 1 1 235 U n, γ 1 1 235 U n, γ 1 1 0.14 1 1 235 U χ 1 1 235 U χ 1 1 0.25 1 1 C-gra elastic 1 1 C-gra elastic 1 1 0.25 1 1 4 1 1 235 U n, f 1 1 235 U n, f 1 1 0.14 1 1 C-gra n, γ 1 1 C-gra n, γ 1 1 0.19 1 1 235 U χ 1 1 235 U χ 1 1 0.18 1 1 5 1 1 235 U n, f 1 1 235 U n, γ 1 1 0.12 1 1 235 U n, γ 1 1 235 U n, γ 1 1 0.18 1 1 235 U n, γ 1 1 235 U n, γ 1 1 0.17 Mechanism analysis of k eff uncertainty 1 From the above SU analysis during the small HTGR depletion calculation, it was revealed that some cross-sections of 235 U, 28 Si, 157 Gd, C-graphite, 239 Pu, 135 Xe, and 149 Sm had high sensitivity coefficients or significant contributions to the uncertainty of k eff . The reason that the radioactive capture reaction cross-section of 28 Si has such a large contribution to the uncertainty of k eff was investigated in Sect. 3.2.2. Additionally, other changes in significant nuclide reaction sensitivity coefficients may directly affect the uncertainty of k eff during the full lifetime depletion process. Therefore, it is necessary to carry out further mechanistic analyses. Based on the results in Section 3, the average number of neutrons emitted per fission event, fission spectrum, radioactive capture reaction, and fission reaction of 235 U all have a significant contribution to k eff uncertainty during the depletion calculation. As one of the most important elements in fission reactors, the nuclear data of 235 U have a significant effect on k eff uncertainty and are valuable for further studies. According to the uncertainty quantification method introduced in Sect. 2, the uncertainty of k eff due to nuclear data depends on its covariance data and sensitivity coefficients. The four crucial reactions of the 235 U integrated sensitivity coefficients and their contribution to k eff uncertainty are presented in Fig.11 Fig.11 . From these two histograms, it is noteworthy that the high sensitivity of the average number of neutrons emitted per fission event of 235 U directly leads to a large 235 U contribution to k eff uncertainty. Although other crucial reactions of 235 U also significantly contribute to the total uncertainty of k eff , the k eff sensitivities to these reactions are not very significant and only slightly decrease during the full lifetime. In addition, the sensitivity coefficients of the 235 U fission spectrum are too small to be observed in this histogram, however the contribution to k eff uncertainty is still large due to the high relative covariance data [ 23 23 ] . However, the uncertainties of the average number of neutrons emitted per fission event and fission reaction, which are based on the covariance matrices, are much smaller [ 23 23 ] . Therefore, the average number of neutrons emitted per fission event and fission reaction of 235 U has a large amount of uncertainty, which can be attributed to their large sensitivity coefficients. Furthermore, the elastic scattering of C-graphite, the radioactive capture reaction, and the fission reaction of 239 Pu also have considerable impact and variation during the depletion process. The integrated sensitivity coefficients of these two important nuclides are shown in Fig.12 Fig.12 . In these two histograms, the elastic scattering of C-graphite has a large basal sensitivity during the full lifetime and even exhibits a slight growth at EOL. This trend also reflects the total uncertainty change in k eff to some extent. At the same time, the two important reactions of 239 Pu sensitivity coefficients have a significant growth at EOL, but their sensitivity coefficient values are still slight compared with those of C-graphite and 235 U. Based on this finding, the reason that the radioactive capture and fission reactions do not contribute significantly to k eff uncertainty at EOL can be explained clearly. Moreover, this phenomenon explains why the total k eff uncertainty does not increase significantly at EOL. Fission poison products, such as 135 Xe and 149 Sm, are generated during the full lifetime depletion process. These nuclides dramatically affect k eff due to their substantial absorption cross sections and therefore, the impact of these poison nuclide reaction cross-sections on k eff uncertainty should be studied. The integrated sensitivity coefficients of the radioactive capture reaction of 135 Xe, 149 Sm, and 157 Gd during the depletion calculation are shown in Fig.13 Fig.13 . Interestingly, based on the rapid production of 135 Xe at the BOL, the integrated sensitivity coefficients of 135 Xe increased significantly at 5 days and reached the highest value at 150 days. The integrated sensitivity coefficients of 149 Sm have slightly increased at 5 days and also get peak at 150 days, but its integrated sensitivity values are much lower than that of 135 Xe. The burnable poison material 157 Gd is input at the BOL. Its sensitivity coefficients show an obvious decrease after 150 days and nearly decrease to zero at EOL. After 150 days, the main contributor of poison elements was 135 Xe in the small-sized HTGR. However, the integrated sensitivity coefficients of these poison elements are still much smaller than those of C-graphite or 235 U. Therefore, based on the analysis results and the "sandwich" rule, the contributions to k eff uncertainty of these main poison nuclide reactions in small-sized HTGRs arise from their relative covariances. Conclusion 1 In this study, the k eff uncertainties due to the nuclear data in the CZP condition and full lifetime depletion calculation were quantified for a small-sized HTGR. RMC was used to generate the small-sized HTGR high-fidelity model and carry out critical calculations and for depletion and uncertainty calculations. In the depletion calculation, the predictor correction method was applied. In addition, the CLUTCH method was used for sensitivity analysis, and the "sandwich " method was utilized to quantify the k eff uncertainty through the ENDF/B-VII.1 covariance data. Our main findings are as follows: First, the uncertainty of k eff due to nuclear data at the CZP condition was considered. According to SU results in the CZP condition, the average number of neutrons emitted per fission event of 235 U is the most important contributor to the uncertainty of k eff . The radioactive capture reaction of 28 Si is the second largest contributor to the uncertainty in k eff because of its heavy volume ratio in the fuel pellet. This finding differs from the results of our study of the pebble-bed HTGR [ 10 10 ] . The total uncertainty of k eff due to the nuclear data at the CZP condition was approximately 636 pcm. Second, the 24 fuel zone model was used for the full lifetime depletion calculation. According to the results illustrated in Tables 2 Tables 2 and 4 4 , the top eight most important nuclide reaction contributors themselves to k eff uncertainty did not change across the full lifetime. However, the k eff uncertainty from the radioactive capture and fission reactions of 239 Pu increased significantly at EOL. Simultaneously, other types of reactors, such as PWR, BWR, and pebble bed HTGR, were compared with the small-sized HTGR in the full lifetime depletion k eff uncertainty quantification. The results showed that the small-sized HTGR had a lower burnup value at EOL, and its k eff uncertainty only changed slightly during the full lifetime depletion calculation. In the small-sized HTGR, the uncertainty of k eff during the full lifetime increased by 1.1196%, compared to 28.9855% for PWR and 40.2174% for BWR. Finally, the variation in k eff uncertainty due to nuclear data during the full lifetime depletion was analyzed. The average number of neutrons emitted per fission event of 235 U and elastic scattering of C-graphite significantly contribute to the uncertainty of k eff owing to their large sensitivity coefficients. However, this conclusion is contrary to the fission spectrum of 235 U, in which the significant contribution to the k eff uncertainty is due to the large covariance data of itself. Moreover, 239 Pu is one of the main fission products, and its important reaction cross-section contributions to k eff uncertainty increased at EOL, but did not surpass the contributions of 235 U or C-graphite owing to its small sensitivity coefficient. This is the key reason that the total uncertainty of k eff grew little at the EOL. In addition, the poison elements, 135 Xe, 149 Sm, and 157 Gd, were investigated in the depletion calculation. The k eff sensitivity coefficients for the poison element cross sections varied significantly during the full lifetime, as illustrated in Fig.13 Fig.13 . These changes have an obvious influence on the k eff value but did not significantly affect the k eff uncertainty. In general, the k eff uncertainty due to nuclear data was quantified, and some important nuclides and reactions were determined to contribute significantly to the k eff uncertainty. These findings are valuable for the design and optimization of new small-sized prismatic HTGRs. However, the nuclear data introduces non-negligible uncertainties to the nuclide density, which further contributes to the uncertainty of k eff during the depletion process. This work is now in progress, and the uncertainty results will be reported in the following papers. Reference Wu Z.X. , Safety Features of Modular High Temperature Gas-cooled Reactors (MHTGR) . Tsinghua Science and Technology , 1 , 8 - 11 March ( 1996 ). Safety Features of Modular High Temperature Gas-cooled Reactors (MHTGR) Zhang X.C. , She D. , Chen F.B. , et al . Temperature analysis of the HTR-10 after full load rejection . Nucl. Sci. Tech. 30 , 163 ( 2019 ). doi: 10.1007/s41365-019-0692-1 http://doi.org/10.1007/s41365-019-0692-1 Temperature analysis of the HTR-10 after full load rejection Gou F. , Liu Y. , Chen F.B. et al . Thermal behavior of the HTR-10 under combined PLOFC and ATWS condition initiated by unscrammed control rod withdrawal . Nucl. Sci. Tech. 29 , 123 ( 2018 ). doi: 10.1007/s41365-018-0472-3 http://doi.org/10.1007/s41365-018-0472-3 Thermal behavior of the HTR-10 under combined PLOFC and ATWS condition initiated by unscrammed control rod withdrawal She D. , Xia B. , Guo J. , et al. , Prediction calculations for the first criticality of the HTR-PM using the PANGU code . Nucl. Sci. Tech. 32 , 90 ( 2021 ). doi: 10.1007/s41365-021-00936-5 http://doi.org/10.1007/s41365-021-00936-5 Prediction calculations for the first criticality of the HTR-PM using the PANGU code Willianms P.M. , Silady F.A. , Dunn T.D. et al. , MHTGR development in the United States . Prog. Nucl. Energ. 28 , 265 - 346 ( 1994 ). doi: 10.1016/0149-1970(94)90002-7 http://doi.org/10.1016/0149-1970(94)90002-7 MHTGR development in the United States Reitsma F. , Kim W. , The contribution of cross section uncertainties to pebble bed reactor eigenvalues results: IAEA cooperative research project phase I stand alone neutronics . Paper presented at the PHYSOR2016 Conference ( Sun Valley, Idaho, USA 1 May. 2016 ). The contribution of cross section uncertainties to pebble bed reactor eigenvalues results: IAEA cooperative research project phase I stand alone neutronics Reitsma F. , Strydom G. , Tyobeka B. et al. , The IAEA coordinated research program on HTGR reactor physics, thermal-hydraulics and depletion uncertainty analysis: description of the benchmark test cases and phases . Paper presented at the HTR2012 Conference ( Tokyo, Japan 28 Oct - 1 Nov. 2012 ). The IAEA coordinated research program on HTGR reactor physics, thermal-hydraulics and depletion uncertainty analysis: description of the benchmark test cases and phases Bostelmann F. , Strydom G. , Reitsma F. et al. , The IAEA coordinated research program on HTGR uncertainty analysis: phase I status and Ex. I-1 prismatic reference results . Nucl. Eng. Des. 306 , 77 - 88 ( 2016 ). doi: 10.1016/j.nucengdes.2015.12.009 http://doi.org/10.1016/j.nucengdes.2015.12.009 The IAEA coordinated research program on HTGR uncertainty analysis: phase I status and Ex. I-1 prismatic reference results Strydom G. , IAEA Coordinated Research Project on HTGR Physics, Thermal-Hydraulics, and Depletion Uncertainty Analysis Prismatic HTGR Benchmark Specification: Phase II. Idaho National Laboratory, Technical Report INL/EXT-18-44815 ( 2018 ). doi: 10.2172/1467563 http://doi.org/10.2172/1467563 IAEA Coordinated Research Project on HTGR Physics, Thermal-Hydraulics, and Depletion Uncertainty Analysis Prismatic HTGR Benchmark Specification: Phase II. Idaho National Laboratory, Technical Report INL/EXT-18-44815 Hao C. , Cheng Y.Y. , Guo J. et al. , Mechanism analysis of the contribution of nuclear data to the keff uncertainty in the pebble bed HTR , Ann. Nucl. Energy 120 , 857 - 868 ( 2018 ). doi: 10.1016/j.anucene.2018.07.015 http://doi.org/10.1016/j.anucene.2018.07.015 Mechanism analysis of the contribution of nuclear data to the keff uncertainty in the pebble bed HTR Tae Y.H. , Lee H.C. , Jin Y.C. et al. , Improvement and application of DeCART/MUSAD for uncertainty analysis of HTGR neutronic parameters , Nucl. Eng. Tech. 52 , 461 - 468 ( 2020 ). doi: 10.1016/j.net.2019.08.006 http://doi.org/10.1016/j.net.2019.08.006 Improvement and application of DeCART/MUSAD for uncertainty analysis of HTGR neutronic parameters Williams M. L. , Perturbation Theory for Nuclear Reactor Analysis , ed. by Ronen Y. ( Boca Raton , Florida , 1986 ). Perturbation Theory for Nuclear Reactor Analysis Perfetti C.M. , Doctoral Dissertation (Advanced Monte Carlo Methods for Eigenvalue Sensitivity Coefficient Calculation . University of Michigan 2012 ). http://hdl.handle.net/2027.42/93973 http://hdl.handle.net/2027.42/93973 Doctoral Dissertation (Advanced Monte Carlo Methods for Eigenvalue Sensitivity Coefficient Calculation Pusa M. . Incorporating sensitivity and uncertainty analysis to a lattice physics code with application to CASMO-4 . Ann. Nucl. Energ. 40 , 153 - 162 ( 2012 ). doi: 10.1016/j.anucene.2011.10.013 http://doi.org/10.1016/j.anucene.2011.10.013 Incorporating sensitivity and uncertainty analysis to a lattice physics code with application to CASMO-4 Arzhannikov A. , Bedenko S. , Shmakov V. et al. , Gas-cooled thorium reactor at various fuel loadings and its modification by a plasma source of extra neutrons . Nucl. Sci. Tech. 30 , 181 ( 2019 ). doi: 10.1007/s41365-019-0707-y http://doi.org/10.1007/s41365-019-0707-y Gas-cooled thorium reactor at various fuel loadings and its modification by a plasma source of extra neutrons Vrinda Devi K.V. , Dubey J.N. , Gupta J. , et al . TRISO fuel volume fraction and homogeneity: a nondestructive characterization . Nucl. Sci. Tech. 30 , 41 ( 2019 ). doi: 10.1007/s41365-019-0573-7 http://doi.org/10.1007/s41365-019-0573-7 TRISO fuel volume fraction and homogeneity: a nondestructive characterization Wang K. , Li Z.G. , She D. et al. , RMC – A Monte Carlo code for reactor core analysis . Ann. Nucl. Energy 82 , 121 - 129 ( 2015 ). doi: 10.1016/j.anucene.2014.08.048 http://doi.org/10.1016/j.anucene.2014.08.048 RMC – A Monte Carlo code for reactor core analysis Li W.L. , Yu G.L. , Wang K. et al. , Comparative analysis of coupling schemes of Monte Carlo burnup calculation in RMC . Ann. Nucl. Energy 137 , 107024 ( 2020 ). doi: 10.1016/j.anucene.2019.107024 http://doi.org/10.1016/j.anucene.2019.107024 Comparative analysis of coupling schemes of Monte Carlo burnup calculation in RMC Shi G.L. , Guo Y.C. , Jia C.L. et al. , Improvement of sensitivity and uncertainty analysis capabilities of generalized response in Monte Carlo code RMC . Ann. Nucl. Energy 154 , 108099 ( 2021 ). doi: 10.1016/j.anucene.2020.108099 http://doi.org/10.1016/j.anucene.2020.108099 Improvement of sensitivity and uncertainty analysis capabilities of generalized response in Monte Carlo code RMC Zwermann W. , Aures A. , Gallner L. et al. , Nuclear data uncertainty and sensitivity analysis with XSUSA for fuel assembly depletion calculations . Nucl. Eng. Tech , 46 , 343 - 352 ( 2014 ). doi: 10.5516/NET.01.2014.711 http://doi.org/10.5516/NET.01.2014.711 Nuclear data uncertainty and sensitivity analysis with XSUSA for fuel assembly depletion calculations Ivanov K. , Avramova M. , Kodeli I. , Benchmark for Uncertainty Analysis in Modeling (UAM) for Design, Operation and Safety Analysis of LWRs.Volume 1- Specification and Supporting Data for the Neutronics Cases (Phase I) Version 1.0, Technical report, NEA/NSC/DOC23 ( 2007 ). https://www.oecdnea.org/science/docs/2007/nsc-doc2007-23 https://www.oecdnea.org/science/docs/2007/nsc-doc2007-23 Benchmark for Uncertainty Analysis in Modeling (UAM) for Design, Operation and Safety Analysis of LWRs.Volume 1- Specification and Supporting Data for the Neutronics Cases (Phase I) Version 1.0, Technical report, NEA/NSC/DOC23 Hao C. , Cheng Y.Y. , Teng Q.C. , Quantification and mechanism analysis of the kinf uncertainty propagated from nuclear data for the TRISO particle fuel pebble . Ann. Nucl. Energ. 127 , 248 - 256 ( 2019 ). doi: 10.1016/j.anucene.2018.12.011 http://doi.org/10.1016/j.anucene.2018.12.011 Quantification and mechanism analysis of the kinf uncertainty propagated from nuclear data for the TRISO particle fuel pebble Chadwick M.B. , Herman M. , Oblozinsky P. et al. , ENDF/B-VII.1 nuclear data for science and technology: cross sections, covariance, fission product yields and decay data . Nucl. Data Sheets , 112 , 2887 - 2996 ( 2011 ). doi: 10.1016/j.nds.2011.11.002 http://doi.org/10.1016/j.nds.2011.11.002 ENDF/B-VII.1 nuclear data for science and technology: cross sections, covariance, fission product yields and decay data All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Rong-Rui Yang, Yuan Yuan, Chen Hao, Ji Ma and Guang-Hao Liu. The first draft of the manuscript was written by Rong-Rui Yang and Chen Hao, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00969-w This work was supported by the National Natural Science Foundation of China (No. 12075067) and the National Key R&D Program of China (No. 2018YFE0180900) . 2021-07-19 2021-09-13 2021-09-15 2021-11-18 2021-11-01 2021-12-06 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00969-w http://dx.doi.org/10.1007/s41365-021-00969-w . 2021 Nuclear Science and Techniques 11 32