Multigroup Cross Sections Research Articles

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Verification and Analysis of the Problem-Dependent Multigroup Macroscopic Cross-Sections for Shielding Calculations

Multigroup constants are the foundation of neutron and photon transport problems, and the accuracy of multigroup cross-sections has a significant impact on shielding calculation. Challenges have arisen in generating accurate multigroup macroscopic cross-sections for some problems using the widely used cross-section processing code TRANSX 2.15. We developed the multigroup neutron-photon macroscopic cross-section processing module in the shielding transport code ARES. The module is capable of handling the neutron-photon coupled master libraries in MATXS format and providing the problem-dependent multigroup macroscopic cross-sections tailored to each specific shielding physics problem. The self-designed problems with a single nuclide, as well as the iron and OKTAVIAN experiments, are used to verify and analyze the accuracy of neutron and photon macroscopic cross-sections. Results indicate that the macroscopic cross-sections, neutron flux and neutron-photon leakage spectrum obtained by the MGMXS module are consistent with corresponding reference values. As for the JANUS Phase I fixed source shielding benchmark, the relative differences in the reaction rate between the calculation results and experimental data are within 20%. The module provides better problem-dependent multigroup macroscopic cross-sections for neutron and photon shielding calculations that satisfy the accuracy requirements of engineering applications.

Developing a Reactivity-Equivalent Physical Transformation to Simulate an Axially Heterogeneous Boiling Water Reactor

Hitachi is advancing their designs for a conceptual reactor called the resource-renewable boiling water reactor (RBWR), a concept reactor similar to the advanced boiling water reactor with a harder neutron spectrum. This design aims to minimise construction costs and waste production as well as to utilise separated plutonium and minor actinide fuel. However, the axial heterogeneity of the design poses calculation difficulties. The aim of this work is to use a known method, reactivity-equivalent physical transformation (RPT), for calculating fuel with double heterogeneity and apply it to a BWR-type fuel pin. This could reduce the calculation time needed for optimisation of the design of the RBWR. The objective of the study is to use SCALE 6.2 to produce an equivalent axial pin model by comparison with the burnup and neutron spectra of a radial model of the fuel. This model can then be used for 2D burnup calculations, and in future work will be used for the generation of two-group and multigroup cross-sections for further deterministic calculations as part of a two-step approach for analysis of the RBWR. The RPT method has been extensively tested on spherical fuel, and SCALE is a standard industry code. The initial radial model is a hexagonal assembly with 20% enriched UO2 fuel in a zircaloy cladding, surrounded by light water moderator. The derived axial model has a water density distribution taken from Hitachi’s RBWR designs. Criticality over 70 GWd/tU burnup is estimated using the model. The application of the RPT to the BWR pin was shown to be possible, but to have limitations with the introduction of additional radial complexity. For a single pin, excellent agreement between the radial and axial models could be found across a range of water densities, but in the case of an assembly level calculation distinct equivalence models were required for each water density. In addition, the produced RPT model is validated using SCALE’s 3D Monte Carlo module, KENO.

Multigroup cross-sections generated using Monte-Carlo method with flux-moment homogenization technique for fast reactor analysis

The development of fast reactors with complex designs and operation status requires more accurate and effective simulation. The Monte-Carlo method can generate multi-group cross-sections in arbitrary geometry without approximation on resonances treatment and leads to good results in combination with diffusion codes. However, in previous studies, the coupling of Monte-Carlo generated multi-group cross-sections (MC-MGXS) and transport solvers has shown relatively large biases in fast reactor problems. In this paper, the main contribution to the biases is proved to be the neglect of the angle-dependence of the total cross-sections. The flux-moment homogenization technique (MHT) is proposed to take into account this dependence. In this method, the angular dependence is attributed to the transfer cross-sections, keeping an independent form for the total sections. For the MET-1000 benchmark, the multi-group transport simulation results with MC-MGXS generated with MHT are improved by 700 pcm and an additional 120 pcm with higher order scattering. The factors that cause the residual bias are discussed. The core power distribution bias is also significantly reduced when MHT is used. It proves that the MC-MGXS with MHT can be applicable with transport solvers in fast reactor analysis.

Annular fuel neutronic analysis

The annular fuel was proposed to increase the power density while keeping or improving the safety margins, given the considerably lower surface heat flux and lower fuel temperature than conventional solid fuel. In this paper, neutronic characteristics of this new fuel design were analysed using the new transport code of the Westinghouse, called PARAGON2, using Ultra-Fine Energy Mesh Library for multi-group cross section calculations. The bibliographical review showed there is a lack of deterministic transport codes capable to model such fuel with acceptable accuracy, because the equivalence relations for heterogeneous resonance integrals adopted in the standard codes were formulated only for solid cylindrical rods. Therefore, the main objective of this paper is to perform neutronic analysis for annular fuel using PARAGON2, evaluating the use of Ultra-Fine Energy Mesh Library to eliminate resonance self-shielding calculation. The results of this paper analysed the power distributions, the ratio of U-238 capture rate to U-235 fission rate and behaviour of the infinite multiplication factor for annular fuel and compared with the Monte Carlo code MCNP.

Development and verification of a Monte Carlo two-step method for lead-based fast reactor neutronics analysis

With the rise of economic and safety standards for nuclear reactors, new concepts of Gen-IV reactors and modular reactors showed more complex designs that challenge current tools for reactor physics analysis. A Monte Carlo (MC) two-step method was proposed in this work. This calculation scheme uses the continuous-energy MC method to generate multi-group cross-sections from heterogeneous models. The multi-group MC method, which can adapt locally-heterogeneous models, is used in the core calculation step. This calculation scheme is verified using a Gen-IV modular lead-based fast reactor (LFR) benchmark case. The influence of homogenized patterns, scatter approximations, flux separable approximation, and local heterogeneity in core calculation on simulation results are investigated. Results showed that the cross-sections generated using the 3D assembly model with a locally heterogeneous representation of control rods lead to an accurate estimation with less than 270 pcm bias in core reactivity, 0.5% bias in control rod worth, and 1.5% bias on power distribution. The study verified the applicability of multi-group cross-sections generated with the MC method for LFR analysis. The study also proved the feasibility of multi-group MC in core calculation with local heterogeneity, which saves 85% time compared to the continuous-energy MC.

Establishment of DeCART/MIG stochastic sampling code system and Application to UAM and BEAVRS benchmarks

In this study, a DeCART/MIG uncertainty quantification (UQ) analysis code system with a multi-correlated cross section stochastic sampling (S.S.) module was established and verified through the UAM (Uncertainty Analysis in Modeling) and the BEAVRS (Benchmark for Evaluation And Validation of Reactor Simulations) benchmark calculations. For the S.S. calculations, a sample of 500 DeCART multi-group cross section sets for two major actinides, i.e., 235U and 238U, were generated by the MIG code and covariance data from the ENDF/B-VII.1 evaluated nuclear data library. In the three pin problems (i.e. TMI-1, PB2, and Koz-6) from the UAM benchmark, the uncertainties in kinf by the DeCART/MIG S.S. calculations agreed very well with the sensitivity and uncertainty (S/U) perturbation results by DeCART/MUSAD and the S/U direct subtraction (S/U-DS) results by the DeCART/MIG. From these results, it was concluded that the multi-group cross section sampling module of the MIG code works correctly and accurately. In the BEAVRS whole benchmark problems, the uncertainties in the control rod bank worth, isothermal temperature coefficient, power distribution, and critical boron concentration due to cross section uncertainties were calculated by the DeCART/MIG code system. Overall, the uncertainties in these design parameters were less than the general design review criteria of a typical pressurized water reactor start-up case. This newly-developed DeCART/MIG UQ analysis code system by the S.S. method can be widely utilized as uncertainty analysis and margin estimation tools for developing and designing new advanced nuclear reactors.

Deep Learning for Multigroup Cross-Section Representation in Two-Step Core Calculations

We investigate using deep learning, a type of machine-learning algorithm employing multiple layers of artificial neurons, for the mathematical representation of multigroup cross sections for use in the Griffin reactor multiphysics code for two-step deterministic neutronics calculations. A three-dimensional fuel element typical of a high-temperature gas reactor as well as a two-dimensional sodium-cooled fast reactor lattice are modeled using the Serpent Monte Carlo code, and multigroup macroscopic cross sections are generated for various state parameters to produce a training data set and a separate validation data set. A fully connected, feedforward neural network is trained using the open-source PyTorch machine-learning framework, and its accuracy is compared against the standard piecewise linear interpolation model. Additionally, we provide in this work a generic technique for propagating the cross-section model errors up to the keff using sensitivity coefficients with the first-order uncertainty propagation rule. Quantifying the eigenvalue error due to the cross-section regression errors is especially practical for appropriately selecting the mathematical representation of the cross sections. We demonstrate that the artificial neural network model produces lower errors and therefore enables better accuracy relative to the piecewise linear model when the cross sections exhibit nonlinear dependencies; especially when a coarse grid is employed, where the errors can be halved by the artificial neural network. However, for linearly dependent multigroup cross sections as found for the sodium-cooled fast reactor case, a simpler linear regression outperforms deeper networks.

An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering: Part III. Uncertainty Propagation and Multigroup Covariance Matrices

An analytic benchmark for nuclear data uncertainty propagation in k-eigenvalue calculations is demonstrated. Flat-flux-weighted cross-section covariance matrices are available in the ENDF/B library for many isotopes. For application-specific purposes, flux-weighted multigroup cross sections with carefully constructed energy group boundaries are desired. In this paper, we use the covariance information from ENDF/B-VII.1 for the defined continuous-energy cross section and an artificially inflated variance version of the same covariance matrix for first-order and Monte Carlo propagation of uncertainty calculations. A flat-flux weighting function is used for the continuous-energy cross-section uncertainty collapse resulting in a higher propagated uncertainty on the k-eigenvalue as the group structure becomes coarser. The results of this analytic benchmark suggest that the reporting of flat-flux-weighted multigroup cross-section covariance matrices at the ENDF level may lead to inaccurate predictions of the uncertainty on the k-eigenvalue for certain applications. This work implies that not only should the resonance parameter uncertainties that go into the calculation of the continuous-energy cross sections be published, but the parameter uncertainties should also be processed into continuous-energy cross-section uncertainties that can be collapsed to application-specific multigroup cross-section covariance matrices.

Assimilation of integral experiments on high-energy nuclear parameters

Current assimilation of integral experiments often consists in adjusting multi-group cross sections with feedbacks from critical reference benchmarks. In order to maintain the constraints coming from nuclear models, we present here a method to achieve assimilation of integral experiment on nuclear parameters, from which nuclear data are evaluated. This method, based on Bayesian inference, uses continuous energy reactivity sensitivities to nuclear parameters, throughout all the nuclear data types (cross section, angular distribution, energy distribution, fission multiplicity and spectrum). This improvement was made possible by coupling a stochastic transport code and a nuclear data evaluation code. The study of a test case – the assimilation of Jezebel ICSBEP benchmark on a plutonium-239 toy evaluation – shows that angular and energy distributions have a non-negligible impact on the assimilation process and results.

GIDI+: A GNDS 2.0 suite of C++ APIs to access nuclear and atomic data for use in radiation transport codes

GIDI+ is a C++ package that reads GNDS formatted nuclear reaction and structure data, and photo-atomic reaction data as needed by radiation transport codes. As of version 3.25, GIDI+ supports reading GNDS 2.0 formatted data. GNDS is a new extensible hierarchy that has been internationally adopted as the new standard for storing nuclear and photo-atomic data, replacing ENDF-6 which has been the standard since the 1960’s. GIDI+ 3.25 supports the GNDS 2.0 map file which can contain a list of all GNDS data needed to make a complete nuclear reaction data library as needed by radiation transport codes. GIDI+ has functions that make it easy to obtain multi-group data from a GNDS file, including summed data (e.g., summing multi-group reaction cross sections to obtain the total multi-group cross section). In addition, the MCGIDI sub-package of GIDI+ can be used by Monte Carlo transport codes to look up data (e.g., cross section, deposition energy) as a function of projectile energy and material temperature. The MCGIDI sub-package can use either GNDS multi-groupor continuous energy data. MCGIDI also has functions for sampling a reaction from a GNDS reactionSuite and functions for sampling a reaction’s outgoing particle energy and angular data. Recent work on GIDI+ includes: supporting GNDS 2.0, speeding up load times and adding support for URR probability tables. GIDI+ is currently implemented in LLNL’s Mercury [4] (Monte Carlo) and Ardra [5] (deterministic) radiation transport codes. An effort to update GEANT4 to use the latest GIDI+ is underway and should be completed by March 2023. GIDI+ is released under the MIT license and can be found at github.com/LLNL/gidiplus.

Preservation of kinetics parameters generated by Monte Carlo calculations in two-step deterministic calculations

The generation of accurate kinetic parameters such as mean generation time Λ and effective delayed neutron fraction βeff via Monte Carlo codes is established. Employing these in downstream deterministic codes warrants another step to ensure no additional error is introduced by the low-order transport operator when computing forward and adjoint fluxes for bilinear weighting of these parameters. Another complexity stems from applying superhomogenization (SPH) equivalence in non-fundamental mode approximations, where reference and low-order calculations rely on a 3D full core model. In these cases, SPH factors can optionally be computed for only part of the geometry while preserving reaction rates and K-effective, but the impact of such approximations on kinetics parameters has not been thoroughly studied. This paper aims at studying the preservation of bilinearly-weighted quantities in the Serpent–Griffin calculation procedure. Diffusion and transport evaluations of IPEN/MB-01, Godiva, and Flattop were carried out with the Griffin reactor physics code, testing available modeling options using Serpent-generated multigroup cross sections and equivalence data. Verifying Griffin against Serpent indicates sensitivities to multigroup energy grid selection and regional application of SPH equivalence, introducing significant errors; these were demonstrated to be reduced through the use of a transport method together with a finer energy grid.

A Study of the Covariance Data in ENDF/B VIII.0 for Low Z Isotopes

Thirty group covariance data have been produced from ENDF/B VIII.0 MF 33 for H1, H2, He4, Li6, Li7, Be9, B10, B11, C12, C13, N15, O16, the JENDL O16 and F19. Multi-group cross section covariance data was produced by NJOY routine ERRORR. Negative eigenvalues were found only for F19, but the covariance constraints required for the condition of the sum of partial cross sections being equal to the resultant total cross section were found to be incorrect for H2, Li6, B10, C12, C13, and O16. An easy correction is suggested for Li6 and C13. An expansion of the ENDF-6 formats manual is suggested for the MF 33 chapter and a new proposed covariance normalization scheme is proposed.

Preliminary verification of multi-group cross-sections generation and locally heterogeneous transport calculation using OpenMC with CEFR start-up tests benchmark

The multi-group (MG) Monte-Carlo (MC) scheme is a hybrid method based on MC solvers and deterministic strategies. This paper focuses on the study of MG schemes for the analysis of fast reactors. The MG scheme in this paper is composed of two steps. In the cross-section generation step, the OpenMC continuous-energy (CE) mode is used with 2D or 3D high-fidelity cluster geometry. In the core calculation step, both homogeneous and locally heterogeneous core models are computed using OpenMC in multi-group mode. The MG schemes are applied and validated in the China Experimental Fast Reactor (CEFR) start-up tests and compared to the CE scheme. The results show an overprediction of core criticality. The locally heterogeneous scheme improves significantly the accuracy in predicting control rod worth. Moreover, synthesizing the results from deterministic code, the locally heterogeneous scheme exhibits a stable performance across different codes, core sizes, and control rod designs. The locally heterogeneous core model with cross-sections generated from the 3D cluster has a good ability in the prediction of feedback coefficients including the sodium void reactivity, temperature reactivity, and subassembly swap reactivity. The computation speed of the locally heterogeneous MG scheme is almost the same as the homogenous MG scheme and its core calculation uses only 7.2% computation time of the reference CE scheme.

Feasibility of deterministic self-shielding models for Molten Salt Fast Reactor analysis using DRAGON5 code with ENDF/B-VIII.0 data library

Due to its complex geometry and neutron spectrum, the GenIV Molten Salt Fast Reactor was largely studied using the “one-step” calculation scheme, based on the Monte Carlo method. However, for whole-core simulations, this method requires a long computing time to get accurate results. A deterministic “two-step” calculation scheme must be considered and applied to minimize the computational resources. In general, the first step is to obtain multi-group homogenized and condensed cross-sections, while the second step is using those multi-group cross-sections to perform the whole-core simulations. The treatment of the resonance cross-section behavior is one of the biggest problems that affect the accuracy of the multi-group cross-section and handling such a problem is required before proceeding to the second step calculation. So, it is essential to choose a proper and accurate self-shielding model. In this paper, the different self-shielding models are performed with the deterministic code DRAGON5, using ENDF/B-VIII.0 with various energy group libraries on which the self-shielding model is implemented. The infinite multiplication factor Kinf and the radial distribution of absorption, nu-fission, and capture rates were calculated at the low temperature to demonstrate the rim effect. To guarantee accurate predictions of the results revealed by the self-shielding models, the temperature was increased in the range [893 K–1123 K], and a series of different parameters, such as Kinf, average percent error (AVG), root mean square (RMS), and mean relative error (MRE), were calculated as functions of temperature. The calculation’s accuracy is determined by comparing the results with the stochastic transport code OpenMC. It was found that the deterministic transport code DRAGON5 can accurately treat the effect of self-shielding behavior in the MSFR; hence, DRAGON5 code can be utilized as a tool to generate the multi-group homogenized cross-sections of MSFR for whole-core calculations.

Development and verification of a new nuclear data processing code AXSP

The advanced reactor design needs an accurate cross-section generation code. In this study, a new nuclear data processing code AXSP is developed, and the method and performance of which are described. Compared with the NJOY program, the precision of the unresolved resonance processing module UnresXS has been significantly improved due to the adoption of a more accurate solution method and the consideration of in-sequence overlap integrals. The time consumption of PUnresXS has been decreased significantly due to an optimized sorting algorithm. At the same time, other modules of AXSP are relatively comprehensive. The function of resolved resonance cross-section reconstruction and linearization is the ReconXS module. The Doppler broadening module is BroadXS by using Gauss–Hermite quadrature and Gauss–Legendre quadrature from 0 K temperature pointwise cross section to any temperature which is defined by the user. The shielding factor in the unresolved resonance energy region is calculated by the UnresXS or the PUnresXS module, which are developed based on the Bondarenko method and the probability table method, respectively. The ACE formatted cross sections for the Monte Carlo code is processed by the ACEXS module, and the multigroup cross sections are generated by the GroupXS module. The cross sections processed by different modules were verified by the NJOY2016 code, and the multigroup cross sections were also verified by using the critical benchmarks. The multiplication factor difference between AXSP and NJOY2016 is less than 20 pcm. In addition to this, the ZPR6/7 fast reactor is used for ACE format library verification. The results show that the criticality calculated by AXSP has a good agreement with that of NJOY2016.

Monte Carlo Source Convergence Acceleration by Hybrid Multigroup and Continuous Energy Neutron Transport

A novel source convergence acceleration method for Monte Carlo eigenvalue calculations is proposed in this paper. The method consists of simulating the bulk of the inactive cycles with online-generated multigroup cross sections. Then the active cycles are simulated with continuous-energy cross sections to preserve full fidelity. The method was implemented in the Monte Carlo code SCONE and tested on several three-dimensional full-length assembly models. In some cases, the same multigroup cross sections were used for several spatially separated materials in order to limit statistical uncertainties. The method was shown to accelerate calculations by a factor of 2.5 to 5 at the cost of a slightly increased standard deviation in the flux distribution estimated across several independent simulations. The memory usage due to storing multigroup cross sections does not seem to be prohibitive for practical applications.

Implementation of Resonance Upscattering Treatment in FRENDY Nuclear Data Processing System

The resonance upscattering effect (the thermal agitation effect) is implemented in the generation capability of multigroup neutron cross sections of the FRENDY nuclear data processing system. The resonance upscattering effect is considered by (1) the variation of self-shielding factors (effective cross sections) due to the change in the ultra-fine group spectrum and (2) the variation of group-to-group elastic scattering cross sections. Since the upscattering effect is considered in the ultra-fine group spectrum calculation, an iteration calculation is necessary to consider the upscattering. The impacts of the iteration strategy (Jacobi or Gauss-Seidel), as well as the number of iterations, are discussed. In the verification calculations, impacts on the ultra-fine group spectrum, effective cross sections, and neutronics characteristics (the Doppler effect) are confirmed. The effect of energy group structure and the impact of resonance upscattering treatments on the Doppler effect through the variation of effective cross sections and the elastic scattering matrix are investigated. The results indicate that FRENDY can provide appropriate multigroup cross sections considering the resonance upscattering effect.

New approach to find the equilibrium cycle in PWR reactors using genetic algorithms

This work presents a methodology applied for performing in-core fuel management PWR (ICFM), using genetic algorithms (GAs). A system was developed in which the GAs interact directly with a reactor physics code created for this purpose. The nodal code, called NOD2ABC, is responsible for providing the neutron characteristics of the reactor that one wants to carry out the ICFM and sending them to the GAs to optimize them, based on some criteria. This code was developed in 2-D and uses the Nodal Expansion Method (NEM). The reference reactor of the Central Nuclear Almaraz II, made available by the International Atomic Energy Agency (IAEA), was chosen and used in this work with the objective of qualifying and validating all the methodology used in the ICFM. The Assemblies of Almaraz II were modeled using the SCALE 6.0 system, developed by the Reactor and Nuclear Systems Division (RNSD), of the Oak Ridge National Laboratory (ORNL), with the purpose of generating the homogenized multigroup cross sections applied in the implemented nodal code. A new ICFM approach, using GAs, was structured to find a possible equilibrium cycle of the modeled reactor. The equilibrium cycle found reached all the physical conditions provided, lasts 240 days with the reactor operating at full power and has an average discharge burnup of 28.93 MWd/kgHM.

ACE-FRENDY-CBZ: a new neutronics analysis sequence using multi-group neutron transport calculations

ABSTARCT We propose a new neutronics analysis sequence using multi-group neutron transport calculations named ACE-FRENDY-CBZ. This sequence is free from uses of any application libraries; with the ACE files as the starting point, multi-group cross sections of media comprising a target system are calculated with the FRENDY code, and multi-group neutron transport calculations are performed with modules of the CBZ code system. The ACE-FRENDY-CBZ sequence was tested against the eight fast neutron systems, and good agreement in the neutron multiplication factors with the reference Monte Carlo results was obtained within 20 pcm differences in the bare systems and within 60 pcm differences in the reflected systems. It was also found that the adoption of the consistent P approximation increases the errors. In order to investigate this issue, we adopted the sub-group method to calculate spatially-dependent current-weighted total cross sections in the reflector regions, and it was suggested that the uses of the spatially-dependent cross sections with the consistent P approximation has a possibility to further improve the numerical accuracy.