Resonant Nuclides Research Articles

Extended R-matrix description of two-proton radioactivity

Two-proton (2p) radioactivity provides fundamental knowledge on the three-body decay mechanism and the residual nuclear interaction. In this work, we propose decay width formulae in the extended R-matrix framework for different decay mechanisms, including sequential 2p decay, diproton decay, tri-body decay, and sequential two-diproton decay. The diproton and tri-body formulae, combined with information on the two-nucleon transfer amplitude and Wigner single-particle reduced width, can reproduce well experimental 2p radioactivity half-lives. For the case of 67Kr, theoretical predictions for direct 2p decay give much larger half-lives than the recent measurement from RIKEN. A combination of direct and sequential 2p emission is analyzed by considering a small negative one-proton separation energy and a possible enhanced contribution from the p-wave component. The present method predicts that 71Sr and 74Zr may be the most promising candidates for future study on 2p radioactivity. Our model gives an upper limit of 55(4) keV for the decay width of 4p emission in recently found four-proton resonant nuclide, 18Mg, which agrees with the observed width of 115(100) keV.

Reactor physics study of square and hexagonal lattice cell using MANTRA (Method of ChAracteristics based Neutron TRAnsport Code) coupled with resonance treated multi energy group cross section library

Since last few years we have been developing a computer code MANTRA (Method of ChAracteristics based Neutron TRAnsport Code) in phased manner to solve neutron transport equation using Method of Characteristics (MOC) for lattice (i.e. assembly) as well as full core with a flexibility of choosing boundary condition out of vacuum/periodic/reflective options appropriate for the simulation. In literature, we have already reported about the development and performance of the code in triangular mesh network for mainly lattice level calculation with reflective and periodic boundary conditions and to some extent core level calculation with vacuum boundary condition using the processed cross section of benchmark problem. Later, we have explored the advantage of linear source assumption over the commonly used flat source assumption within a mesh in some selected transport calculations (where the variation in neutron flux is substantial) using MANTRA. We have also demonstrated the application of Krylov subspace iteration in a matrix-free framework to accelerate the solution of MANTRA for solving complex problems in effective manner. In continuation of the work, MANTRA is now coupled with WIMSD multi energy group cross section library. For the purpose of coupling, a module is developed in the code to process the cross section data and apply resonance treatment to account for the self-shielding effect of the resonant nuclides. The resonance treatment requires estimation of Dancoff factor, which is carried out using MANTRA for various arrangement of fuel elements. Finally, a number of benchmark problems based on PHWR, BWR, VVER lattices are studied with resonance treated library to compare the performance of MANTRA with other internationally recognized codes like HELIOS, DRAGON, WIMSD, MCNP etc.

Relationship between Tone’s method and embedded self-shielding method

The Tone’s and embedded self-shielding methods are based on the equivalence theory. Compared with the traditional equivalence theory, they solve a fixed-source problem for the whole problem domain to consider the shadowing effect in a lattice system rather than use the Dancoff factor. In this paper, we investigate the similarities and differences of these two methods and find the approximation that multi-group absorption cross sections of resonant nuclides are independent of regions is used in Tone’s method and not used in ESSM, which leads to the phenomena that these two methods produce the same numerical results in cases where there is only one resonant region and produce different results in cases where there are several resonant regions. The two methods are applied to the pressurized water reactor analysis and fast spectrum reactor analysis.

Improvements of subgroup method based on fine group slowing-down calculation for resonance self-shielding treatment

To combine the advantages of the accuracy of ultra-fine group method and efficiency of subgroup method for resonance self-shielding treatment, the method of fine-mesh subgroup method (FSM) has been raised in this paper. FSM adopts a fine energy mesh of 289 groups for resonance energy range and the subgroup parameters are generated for each group. The Coupling procedure ensures the elimination of extra resonance interference amendment as the group structure is sufficient enough. To improve the efficiency, the micro level optimization based on “one-group” approximation is proposed to solve the subgroup fixed source equations for only 8 subgroup levels for each resonant nuclide. Finally, a group condense procedure based on neutron slowing-down equation is employed for resonance range from 289 to 16 groups, and a 47-group structure library will be used for eigenvalue calculation. Benchmarks of pin cells and lattices in reactors are applied in numerical validation and the calculating results show that FSM has a good performance with both accuracy and efficiency for complicated resonance treatment.

Accurate resonance calculation method coupling simplified embedded self-shielding method and ultra-fine group method

An accurate resonance calculation method coupling embedded self-shielding method (ESSM) and ultra-fine group method is proposed in this paper. Based on the analysis on the ESSM theory, a simpler ESSM iteration is developed. Comparing with the original ESSM method, the simplified ESSM just performs the iteration between solving the fixed source transport equation and interpolating the resonance integral (RI) table for one representative resonant nuclide and energy group, which is helpful to increase the computation efficiency. The impact of heterogeneous RI table on the results is analyzed, because the accuracy of ESSM deeply relies on the heterogeneous RI table. The numerical results show that the heterogeneous RI tables generated by different manners will produce different results. A method coupling the simplified ESSM and ultra-fine group method is proposed, which is capable of treating the temperature distribution, spatial self-shielding effect and interference effect in the fuel rod, and providing exact spatially dependent self-shielded cross sections.

An Advanced Sodium-Cooled Fast Reactor Core Concept Using Uranium-Free Metallic Fuels for Maximizing TRU Burning Rate

In this paper, we designed and analyzed advanced sodium-cooled fast reactor cores using uranium-free metallic fuels for maximizing burning rate of transuranics (TRU) nuclides from PWR spent fuels. It is well known that the removal of fertile nuclides such as 238U from fuels in liquid metal cooled fast reactor leads to the degradation of important safety parameters such as the Doppler coefficient, coolant void worth, and delayed neutron fraction. To resolve the degradation of the Doppler coefficient, we considered adding resonant nuclides to the uranium-free metallic fuels. The analysis results showed that the cores using uranium-free fuels loaded with tungsten instead of uranium have a significantly lower burnup reactivity swing and more negative Doppler coefficients than the core using uranium-free fuels without resonant nuclides. In addition, we considered the use of axially central B4C absorber region and moderator rods to further improve safety parameters such as sodium void worth, burnup reactivity swing, and the Doppler coefficient. The results of the analysis showed that the final design core can consume ~353 kg per cycle and satisfies self-controllability under unprotected accidents. The fuel cycle analysis showed that the PWR–SFR coupling fuel cycle option drastically reduces the amount of waste going to repository and the SFR burner can consume the amount of TRUs discharged from 3.72 PWRs generating the same electricity.

Improvement and optimization of the pseudo-resonant-nuclide subgroup method in NECP-X

The pseudo-resonant-nuclide subgroup method is a newly emerged self-shielding calculation method to treat resonance interference effect. It has been implemented in the NECP-X code which is developed for the high-fidelity neutronics calculation. Though this method has high accuracy, its computation time occupies more than half of the total computation time (mainly including resonance self-shielding calculation, multi-group transport calculation and depletion calculation). To improve the efficiency of the resonance self-shielding calculation, some analysis and optimization are performed in several aspects: firstly, the maximum number of subgroups is reduced from 7 to 5; secondly, the number of dilution cross section points is reduced from 21 to 9; thirdly, the number of fuel meshes in solving SFSE is reduced from 15 to 5; finally, number of fuel meshes in solving one-group fixed-source equation for the SPH factor is reduced from 15 to 1. The acceleration ratio of the self-shielding calculation is 3.45 for assembly without poison and 3.49 for assembly with poison. The time proportion of self-shielding calculation is reduced to less than 30%. The accuracy loss is negligible in the accelerations. Besides, it is found that the subgroup condensation scheme can obtain better self-shielded cross sections of component resonant nuclides than the dilution interpolation scheme.

Resonance self-shielding treatment and analysis of resonance integral tables for Fully Ceramic Micro-encapsulated fuels with the Embedded Self-Shielding Method

A resonance calculation method for Fully Ceramic Micro-encapsulated (FCM) fuels containing stochastically dispersed tri-structural isotropic (TRISO) coated fuel particles has been proposed in this research. The purpose is to obtain effective homogenized resonance cross sections representing the effect of double heterogeneity (DH). The Monte Carlo code MVP is used to solve the heterogeneity in the isolated FCM fuels pin, and to couple with the method of characteristic (MOC) to generate heterogeneous resonance integral (RI) tables for the following Embedded Self-Shielding Method (ESSM) based resonance calculation. The ESSM method is employed to treat the heterogeneity in the lattice (Dancoff effect). Heterogeneous RI tables based on different packing fraction may differ in relationship between background and self-shielded cross sections. Analysis of this distinction’s impact on ESSM calculation has been conducted. Furthermore, resonance interference effect is neglected in ESSM because heterogeneous RI tables are developed for isolated resonant nuclide system. Therefore, Resonance Interference Factor (RIF) Method was applied to correct cross sections from ESSM. To obtain RIF, slowing-down equation need be solved twice for different homogeneous systems. Whereas, the cross section neglecting interference effect could be received by interpolating in homogeneous RI tables to improve efficiency. Evaluation of errors caused by interpolation in RIF method was performed with reference solutions from MVP. Numerical results show that the new method is proved effective in DH capability for resonance self-shielding calculation.

Practical resolution of angle dependency of multigroup resonance cross sections using parametrized spectral superhomogenization factors

Abstract Based on the observation that ignoring the angle dependency of multigroup resonance cross sections within a fuel pellet would result in nontrivial underestimation of the spatial self-shielding of flux, a parametrized spectral superhomogenization (SPH) factor library (PSSL) method is developed as a practical means of resolving the problem. Region-wise spectral SPH factors are calculated by the normal and transport corrected SPH iterations after ultrafine group slowing down calculations over various light water reactor pin-cell configurations. The parametrization is done with fuel temperature, U-238 number density, fuel radius, moderator source represented by Σ mod V mod , and the number density ratio of resonance nuclides to that of U-238 in a form of resonance interference correction factors. The parametrization is successful in that the root mean square errors of the interpolated SPH factors over the fuel regions of various pin-cells are within 0.1%. The improvement in reactivity error of the PSSL method is shown to be superior to that by the original SPH method in that the reactivity bias of −200 pcm to −300 pcm vanishes almost completely. It is demonstrated that the environment effect takes only about 4% in the reactivity improvement so that the pin-cell based PSSL method is effective in the assembly problems.

Compression of the Waveform of a Gamma Photon Into a Train of Short Pulses in an Optically Dense Oscillating Mössbauer Absorber

We study the potential of a method for manipulating the spectral-temporal characteristics of the Mossbauer γ photon on the basis of resonant interaction of the electromagnetic field of a single photon with an oscillating nuclear absorber. It is shown that the absorber enrichment by a resonant nuclide and an increase in its optical density allow one to considerably increase the peak probability of detecting the photon behind the absorber and improve the obtained-pulse shape. The estimates for the γ radiation of the Mossbauer nuclide 57Co with a photon energy of 14.4 keV and a lifetime of 141 ns of the excited state, which interacts with the oscillating resonant absorber enriched by the 57Fe nuclei, are given and the possibility of compressing the waveform of an incident photon into a train of pulses with durations of 10 ns and peak intensity two times exceeding that in the absence of the absorber is shown.

An improved DeCART library generation procedure with explicit resonance interference using continuous energy Monte Carlo calculation

DeCART multi-group cross section library generation procedure based on the KAERI library generation system was improved using the new alternative procedures based on the Monte Carlo slowing down calculations. In the new alternative procedures, the resonance integral tables were generated by solving the slowing down equation with continuous energy Monte Carlo calculations and the new procedures for the library generation were incorporated for the additional improvements. Most of fission product and actinide nuclides are processed as resonant nuclides to reduce the errors which are resulted from the approximate treatment of their resonance cross sections, and the one-group effective FPYs (Fission Product Yield) which are calculated by fission reaction rates are incorporated for improving the accuracy of the burnup calculations. Using the new procedure and ENDF/B-VII.1 evaluated nuclear data library, new DeCART cross section libraries are generated and they are tested for various benchmark calculations for pin-cell, FA, and core depletion problem against the McCARD reference solutions. The results confirmed that the new generated libraries can be used in the PWR SMR (Small Modular Reactor) core design.

Pseudo-resonant-nuclide subgroup method capable to precisely treat spatial self-shielding effect

Considering that the conventional Bondarenko-Iteration Method (BIM) would introduce much error in treating resonance interference effect, the Resonance-Interference-Factor Method (RIFM) and the Heterogeneous-Pseudo-Resonant-Isotope Method (HPRIM) was developed by researchers. Though these two methods can give relatively precise pin-averaged self-shielded cross sections by considering resonance interference effect, they cannot give exact spatial dependent self-shielded cross sections for coarsely modeling spatial self-shielding effect. In order to overcome this problem, the Pseudo-Resonant-Nuclide Subgroup Method (PRNSM) is proposed in this paper. The PRNSM takes into account the resonance interference effect by making resonance cross section table of the pseudo resonant nuclide on-line. The subgroup parameters of the pseudo resonant nuclide and the partial resonant nuclide are obtained by fitting method. The spatial dependent self-shielded cross sections are obtained by condensing the subgroup cross sections of the partial resonant nuclide with the subgroup flux which is obtained by solving subgroup fixed source problem of the pseudo resonant nuclide. The numerous numerical results show that the PRNSM can give both precise pin-averaged self-shielded cross sections and precise spatial dependent self-shielded cross sections for UO2 problems with different enrichments.

Heterogeneous Pseudo-Resonant Isotope Method for Resolved Resonance Interference Treatment in Resonance Self-Shielding Calculation

The theory of resonance interference factor (RIF) method is examined for thermal reactor problems, and the approximations and limitations are identified. To evaluate the interference effect between resonance isotopes, the RIF method establishes an approximate equivalent relationship between a heterogeneous system and a homogeneous system by introducing background cross sections, and the approximation is a source of deviation in self-shielding calculations. Furthermore, each resonance isotope is treated individually in the self-shielding procedure, which requires unnecessary calculation effort, especially for whole-core and burnup cases. Based on the analysis, a heterogeneous pseudo-resonant isotope method (HPRIM) is proposed to overcome these problems. The mixture of resonant nuclides is considered as a pseudo-resonant isotope, and the resonance integral is generated in a one-dimensional heterogeneous system. The numerical results show that HPRIM improves the accuracy of evaluating the resonance interference effect and improves the efficiency of the self-shielding procedure.

Temperature dependent subgroup formulation with number density adjustment for direct whole core power reactor calculation

The method and performance of the temperature dependent subgroup formulation based on the concept of number density adjustment are presented for the application to the direct whole core calculation of power reactors that involves substantial thermal feedback. The number density of each resonant nuclide is adjusted such that the resulting macroscopic cross section can carry proper temperature dependence even though the microscopic cross section is evaluated at the uniform average temperature. Specifically, the non-uniform temperature distribution is replaced by the corresponding temperature dependent number density distribution and by the uniform temperature for the microscopic cross section of the resonant nuclide. The number density adjustment is introduced not only in the subgroup fixed source problem, but also in the Bondarenko iteration to obtain the effective cross sections for the mixture. The subgroup level dependency of the number density adjustment factor is also introduced. In order to examine the effect of non-uniform temperature profiles on power reactor calculations, the OPR1000 fuel pin problems consisting of several cases having different power levels are constructed. The performance of the proposed method was verified by analyzing the OPR1000 fuel pin problems and by comparing with the corresponding Monte Carlo solutions in the aspect of group-wise cross sections as well as the reactivity. It is demonstrated that the proposed temperature dependent subgroup formulation can predict quite accurately the effects of non-uniform temperature distribution on the fuel temperature coefficients and the shielded cross sections.

Improvement on application of Dancoff factor and resonance interference table

An improvement for application of Dancoff factor is developed. It combines Stamm’ler’s two-term method for resonance integral calculation with neutron current method for Dancoff factor calculation. Stamm’ler’s formulation, which is originally derived for the infinite lattice geometry, can be easily revised to contain the Dancoff factor explicitly, while the neutron current method can easily calculate the Dancoff factor for general irregular assembly geometry. For the resonance interference effects, the resonance interference factor table is built in pairs of nuclides, only for the interference between 238U and other resonance nuclides, spanning over a range of background cross-section and number density ratio of the pairing nuclides. A series of verification calculations have been carried out for problems of infinite lattice and single assembly geometry, with two or multiple resonance absorbers. For these verification calculations, our improvement on Dancoff factor application and resonance interference give good results.

Improved resonance calculation of fluoride salt-cooled high-temperature reactor based on subgroup method

The subgroup method is improved in several aspects to address challenges brought on by design features of the Fluoride salt-cooled High-temperature Reactor (FHR). Firstly, the Dancoff correction is applied to resolve the double heterogeneity arising from embedding TRISO fuel particles in the matrix of pebbles. Secondly, a fast Resonance Interference Factor (RIF) scheme is proposed to treat the resonance interference effect in the FHR. In this scheme, the heterogeneous system is converted into a homogeneous one according to self-shielded cross section conservation of the dominant resonant nuclide. The resonance interference effect is considered in the equivalent homogenous system by correcting the non-interfered self-shielded cross sections with RIFs which are obtained by solving the slowing down equation in hyper-fine energy group (∼1M number of energy groups). Finally, the resonance elastic scattering effect becomes considerable due to high temperatures in the FHR. This effect is considered by substitution of the conventional Resonance Integral (RI) table with that generated by the Monte Carlo method. The Monte Carlo method is modified via the Doppler Broadening Correction Rejection (DBRC) method to implement the Doppler broadened scattering kernel. The numerical results show that the Dancoff correction can significantly reduce errors brought about by the double heterogeneity. The fast RIF scheme provides more accurate effective self-shielded cross sections than the conventional iteration scheme. In addition, the speedup ratio of the fast RIF scheme is ∼3.3 compared with the conventional on-the-fly RIF schemes for TRU TRISO. The scheme to generate RI table can resolve the resonance elastic scattering effect encountered by the conventional scheme.

Pointwise cross-section-based on-the-fly resonance interference treatment with intermediate resonance approximation

Abstract The effective cross sections (XSs) in the direct whole core calculation code nTRACER are evaluated by the equivalence theory-based resonance-integral-table method using the WIMS-based library as an alternative to the subgroup method. The background XSs, as well as the Dancoff correction factors, were evaluated by the enhanced neutron-current method. A method, with pointwise microscopic XSs on a union-lethargy grid, was used for the generation of resonance-interference factors (RIFs) for mixed resonant absorbers. This method was modified by the intermediate-resonance approximation by replacing the potential XSs for the non-absorbing moderator nuclides with the background XSs and neglecting the resonance-elastic scattering. The resonance-escape probability was implemented to incorporate the energy self-shielding effect in the spectrum. The XSs were improved using the proposed method as compared to the narrow resonance infinite mass-based method. The RIFs were improved by 1% in 235U, 7% in 239Pu, and >2% in 240Pu. To account for thermal feedback, a new feature was incorporated with the interpolation of pre-generated RIFs at the multigroup level and the results compared with the conventional resonance-interference model. This method provided adequate results in terms of XSs and k-eff. The results were verified first by the comparison of RIFs with the exact RIFs, and then comparing the XSs with the McCARD calculations for the homogeneous configurations, with burned fuel containing a mixture of resonant nuclides at different burnups and temperatures. The RIFs and XSs for the mixture showed good agreement, which verified the accuracy of the RIF evaluation using the proposed method. The method was then verified by comparing the XSs for the virtual environment for reactor application-benchmark pin-cell problem, as well as the heterogeneous pin cell containing burned fuel with McCARD. The method works well for homogeneous, as well as heterogeneous configurations.

Improvements of the subgroup resonance calculation code SUGAR

Due to its accuracy and geometric flexibility, the subgroup method is becoming a more and more attractive resonance calculation approach dedicated to obtaining resonance group macroscopic cross sections from multi-group libraries. In order to increase the efficiency of our subgroup code SUGAR, this paper contributes to the development of the code from four aspects. Firstly, subgroup parameters were proved to be problem-independent and the number of subgroups can be chosen automatically. This motivated us to produce a new multi-group library. Secondly, what subgroup method really needs is the relative subgroup flux within each multi-group instead of the relative multi-group flux between different groups. Thus, it is unnecessary to iteratively calculate in the whole energy range if the connections between different energy ranges can be approximated by a simple method. Thirdly, for problems with complex isotope compositions, resonant nuclides could be grouped according to their resonance characteristics. By this grouping, computational effort could be significantly reduced since nominal resonant nuclides turn out to be these nuclide groups rather than the actual nuclides. Finally, considering that most of the computational effort is spent on solving the subgroup neutron transport equation, an in-house matrix MOC solver is employed to replace the AutoMOC solver. In this way, the higher speed of the matrix MOC solver can be fully utilized by our subgroup code. To verify these theories and to prove the improvements, a series of benchmark problems were solved. It is demonstrated by these numerical results that these techniques can accelerate the code SUGAR by a factor of 5–32 with respect to overall computations without losing accuracy and geometric flexibility. It was also found that the more complex the resonant nuclide composition is, the sharper the acceleration effect appears to be.

A NOVEL APPROACH TO FIND OPTIMIZED NEUTRON ENERGY GROUP STRUCTURE IN MOX THERMAL LATTICES USING SWARM INTELLIGENCE

Energy group structure has a significant effect on the results of multigroup transport calculations. It is known that UO2–PUO2 (MOX) is a recently developed fuel which consumes recycled plutonium. For such fuel which contains various resonant nuclides, the selection of energy group structure is more crucial comparing to the UO2 fuels. In this paper, in order to improve the accuracy of the integral results in MOX thermal lattices calculated by WIMSD-5B code, a swarm intelligence method is employed to optimize the energy group structure of WIMS library. In this process, the NJOY code system is used to generate the 69 group cross sections of WIMS code for the specified energy structure. In addition, the multiplication factor and spectral indices are compared against the results of continuous energy MCNP-4C code for evaluating the energy group structure. Calculations performed in four different types of H2O moderated UO2–PuO2 (MOX) lattices show that the optimized energy structure obtains more accurate results in comparison with the WIMS original structure.

Improvements in Transport Calculations by the Energy-Dependent Fission Spectra and Subgroup Method for Mutual Self-Shielding

This paper shows two ways to improve the accuracy of the transport calculations. These improvements, implemented in the APOLLO2 code, concern the fission source calculation and the self-shielding models. The calculation of the fission source was generalized to fission spectra including an incident neutron energy dependence. The subgroup self-shielding model was updated for a mixture of resonant nuclides. Some tests on fast neutron systems like a critical sphere without reflector, a sodium-cooled cell, and a helium-cooled cell show that the use of four optimized incident macro groups for fission spectra guarantees a correct representation of the fission source.The tests on a critical sphere with a thick steel reflector and on a water-moderated mixed oxide cell prove that the subgroup self-shielding, accounting for the mutual shielding of several resonant nuclides, allows us to improve the accuracy of the neutron transport solution.