The method of characteristics applied to solving slowing down equation to estimate the self-shielded resonance cross sections with an explicit geometrical effect

The conventional resonance treatment in the transport lattice codes requires resonance integral tables in which resonance integrals are tabulated as a function of the background cross sections to be a measure of dilution. Typically self-shielded resonance cross sections in the resonance integral table are generated by performing slowing down calculations with point-wise cross sections defined on an ultra fine energy grid for one-dimensional cylindrical pin cells. Collision probability, interface current method and discrete ordinate method have been used for the one-dimensional cylindrical slowing down calculations. These resonance integral tables are to be used in estimating the self-shielded resonance cross sections for the rectangular or hexagonal pin cells, which results in a reactivity difference due to the geometrical effect on the effective resonance cross sections. In order to improve this problem, the method of characteristics has been applied to the slowing down calculations for two-dimensional square pin cells. The geometrical effect on the reactivity has been quantitatively analyzed by using the Monte Carlo code MCNP and the transport lattice code KARMA. The method of characteristics has been implemented into the MERIT code developed at KAERI for slowing down calculations. The computation results show that the reactivity differences and the discrepancies of the effective resonance cross sections due to the geometrical inconsistency could be significantly improved by using the method of characteristics.     

Iterative resonance self-shielding methods using resonance integral table in heterogeneous transport lattice calculations

This paper describes the iteration methods using resonance integral tables to estimate the effective resonance cross sections in heterogeneous transport lattice calculations. Basically, these methods have been devised to reduce an effort to convert resonance integral table into subgroup data to be used in the physical subgroup method. Since these methods do not use subgroup data but only use resonance integral tables directly, these methods do not include an error in converting resonance integral into subgroup data. The effective resonance cross sections are estimated iteratively for each resonance nuclide through the heterogeneous fixed source calculations for the whole problem domain to obtain the background cross sections. These methods have been implemented in the transport lattice code KARMA which uses the method of characteristics (MOC) to solve the transport equation. The computational results show that these iteration methods are quite promising in the practical transport lattice calculations.     

精确处理共振干涉效应的子群参数制作方法

传统子群参数制作方法在单共振核素的条件下计算子群参数。然而，该方法中的单共振核素假设与燃料成分中多种核素并存的情况不相符，不能精确处理多核素之间的共振干涉效应。针对此问题，提出了一种新的子群参数制作方法，使用多核素等效截面制作共振积分表，在子群参数制作过程中考虑共振干涉效应。计算结果表明，在不改变子群计算方法的情况下，新的子群参数可更精确地处理共振干涉效应。     

The treatment of resonance interference effects in the subgroup method

This paper describes the development of a method to treat resonance interference effects within the framework of the subgroup method. The new procedure provides for the treatment of multiple resonance absorbers in which the subgroup weights are determined using a least squares technique and based on the cross sections generated from a mixture of multiple resonance isotopes and a suitably wide range of background cross sections. The method was implemented in the Method of Characteristics code DeCART and validated using MCNP. In order to implement the new method, the NJOY code was used for the calculation of neutron spectra and resonance parameters in for each representative LWR mixture. The resonance parameters, lambda, of the scattering isotopes are computed not just with U-238 as the resonance isotope as in previous applications of the subgroup method, but also with U-235 as resonance isotope for the energy groups in which U-238 has no significant resonances. After developing a procedure for generating lambda factors for scattering isotopes, a method is then described for generating subgroup parameters in a homogeneous system. Again NJOY is used for resonance calculations of a set of mixtures for each resonance isotope at each selected temperature. The group average cross sections instead of the resonance integrals of these mixtures are used to generate subgroup parameters using an optimization algorithm. The generated library is then verified by comparing the solution from DeCART with the solution from MCNP. The method is then extended to a heterogeneous system. The code RMET21 is used for transport calculations for the heterogeneous system. The interference effect from the most important resonance isotopes is treated by generating subgroup weights with resonance cross sections for the cases with two resonance isotopes. The results indicate that the subgroup method can accurately represent resonance interference effects within the framework of the subgroup method.     

Effects of Mutual Shielding on Resonance Integrals of 235U, 238U, 239Pu and 240pu below 150 eV

The results of some quantitative studies on resonance interference are presented. The calculations were performed on a FORTRAN IV program RICM2, which solves numerically the slowing down of neutrons over many resonance levels in a two region lattice, and gives reaction rates, average cross sections and effective resonance integrals of the nuclides concerned.

Three combinations of resonant nuclides, ²³⁵U-²³⁸U, ²³⁰Pu-²³⁸U and ²³⁹Pu-²¹⁰Pu, were considered, in conjunction with three oxide fuel rod radii, 0.2, 0.5 and 2.0 cm, the moderator (light water) to fuel volume ratio being maintained constant at 2.0. An energy range below 150eV has been covered by the present calculations. The effects of resonance interference have been found to be appreciable in this energy range.     

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 238 U 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.     

An advanced approach to calculation of neutron resonance self-shielding

Based on the combination of subgroup method and characteristics method, a resonance self-shielding calculation code SGMOC is programmed. SGMOC code can handle the complex (both in geometry and resonant components) resonance problems. The numerical results are in good agreement with those of MCNP. In order to improve the SGMOC calculation accuracy, two techniques are utilized, i.e., the resonance interference effects between resonant nuclides are considered, and on the other hand, the elastic scattering resonance is taken into account. These two techniques can enhance the accuracy remarkably.     

Analysis of Plate Lattice Heterogeneity Effects of Fast Reactor with Coarse Group Constants

Abstract

An advanced method has been developed to analyze the heterogeneity effect of a multi-region plate lattice system accurately enough with the coarse group constants. This treatment is characterized by the modification brought to effective admixture cross sections and correction on elastic removal cross sections. The mutual interference among different plates are taken into account in determining the admixture cross sections. The effective cross sections thus obtained agree well with the results obtained by ultra fine spectrum codes. The elastic removal cross sections of light nuclides are corrected near the sodium resonance by means of a simple analytical expression for the flux depression. With this correction remarkable improvement was observed in the flux heterogeneity. In solving the multi-group transport equation for flux heterogeneity, the fission source iteration was found to be dispensable under most conditions.     

Interference Effect of Strong Scattering Resonances on Elastic Removal Cross Sections

The interference effect from the strong scattering resonance on the elastic removal cross sections was investigated for the core composition of fast reactors. To find the relation of the group elastic removal cross sections of background nuclides to the scattering property of the mixture, they were numerically calculated in the energy ranges of typical resonances. It was shown that the interference effect could be taken into account in the conventional group constant set by using two different methods.

In the first method, the shielding foctors of the background nuclides were characterized by the parameters ξ and σ₀ for the resonance nuclide. The interpolation scheme was similar to that adopted in the conventional set. In the second method, the shielding factors of the imaginary background nuclide were linearly fitted to those of the resonance nuclide. The composition dependence of macroscopic cross sections could be easily obtained with use of the linear relation of which coefficients were determined, beforehand, for the typical core composition. The accuracy of the second method was examined by comparing with the exact values. The present method could predict the macroscopic elastic removal cross sections within the errors of several percents.     

Development of a Fine and Ultra-Fine Group Cell Calculation Code SLAROM-UF for Fast Reactor Analyses

A cell calculation code SLAROM-UF has been developed for fast reactor analyses to produce effective cross sections with high accuracy in practical computing time, taking full advantage of fine and ultra-fine group calculation schemes.

The fine group calculation covers the whole energy range in a maximum of 900-group structure. The structure is finer above 52.5 keV with a minimum lethargy width of 0.008. The ultra-fine group calculation solves the slowing down equation below 52.5 keV to treat resonance structures directly and precisely including resonance interference effects. Effective cross sections obtained in the two calculations are combined to produce effective cross sections over the entire energy range.

Calculation accuracy and improvements from conventional 70-group cell calculation results were investigated through comparisons with reference values obtained with continuous energy Monte Carlo calculations. It was confirmed that SLAROM-UF reduces the difference in k-infinity from 0.15 to 0.01% for a JOYO MK-I fuel subassembly lattice cell calculation, and from ?0.21 % to less than a statistical uncertainty of the reference calculation of 0.03% for a ZPPR-10A core criticality calculation.     

The pseudo-resonant-nuclide subgroup method based global–local self-shielding calculation scheme

The pseudo-resonant-nuclide subgroup method (PRNSM) based global–local self-shielding calculation scheme is proposed to simultaneously resolve the local self-shielding effects (including spatial self-shielding effect and the resonance interference effect) for large-scale problems in reactor physics calculations. This method splits self-shielding calculation into global calculations and local calculations. The global calculations obtain the Dancoff correction factor for each pin cell by neutron current method. Then an equivalent one-dimensional (1D) cylindrical problem for each pin cell is isolated from the lattice system by preserving Dancoff correction factor. The local calculation is to perform self-shielding calculations of the equivalent 1D cylindrical problem by the PRNSM. The numerical results show that PRNSM obtains accurate spatial dependent self-shielded cross sections and improves the accuracy of dealing with the resonance interference over the conventional Bondarenko iteration method and the resonance interference factor method. Furthermore, because both global and local calculation is linearly proportional to the size of problems, the global–local calculation scheme could be applied to large-scale problems.     

Multi-Level Correction to Breit-Wigner Single-Level Formula

A formula for neutron cross sections of heavy nuclei is derived on the basis of the Wigner-Eisenbud theory. The derived formula is the same as the single-level one introduced by Breit and Wigner except for the inclusion of additional interference parameters (u, v) which represent the contribution of the interference effect between resonances. The present formula is therefore applicable to reactor calculations without much modifications of the existing resonance integral codes.

The present formula has been applied to the analyses of the ²³⁵U and ²³³U fission cross sections and the ²³⁸U total cross sections in the resolved resonance region. By the use of least squares fits of the experimental data, the interference parameters (u, v) are obtained for resonance levels of these nuclei in their typical resonance regions. It is shown that the present formula well represents the experimental data.

In the unresolved resonance region, both the single-level resonance parameters and the present interference parameters are generated by using a random sampling method. The contributions of the interference between the resonances, to the Doppler effect are also evaluated for the fission cross sections of ²³⁵U and ²³⁹Pu in the unresolved resonance energy region of 1 to 2 keV.     

A hybrid method to generate few-group cross sections for fast reactor analysis

The accuracy of fast reactor core calculation is usually determined by the accuracy of self-shielded few-group cross sections. To further improve the accuracy of cross section generation, a hybrid method is proposed. In the hybrid method, the Monte-Carlo method is used to deal with the resonance effect in both the resolved and unresolved resonance range. The self-shielded ultrafine-group total, fission and elastic scattering cross sections are tallied by the Monte-Carlo method. The scattering transfer matrices are then generated in a synthesis way by using the tallied elastic scattering cross sections and a problem-independent elastic scattering function. The angular flux moments for the group condensation are calculated in an explicit deterministic way. Several tests are done to verify the hybrid method. The results show that the hybrid method avoids the disadvantages of both the traditional deterministic method and the pure Monte-Carlo method. It is a more accurate method to generate the few-group cross sections for fast reactor cores.     

Elastic Removal Self-Shielding Factors for Light and Medium Nuclides with Strong-Resonance Scattering

The self-shielding factors for elastic removal cross sections of light and medium weight nuclides were calculated for the parameter, σ_o within the conventional concept of the group constant sets. The numerical study were performed for obtaining a simple and accurate method. The present results were compared with the exact values and the conventional ones, and shown to be remarkably improved. It became apparent that the an-isotropy of the elastic scattering did not affect to the self-shielding factors though it did to the infinite dilution cross sections. With use of the present revised set, the neutron flux were calculated in an iron medium and in a prototype FBR and compared with those by the fine spectrum calculations and the conventional set. The present set showed the considerable improvement in the vicinity of the large resonance regions of sodium, iron and oxygen.     

Development of a physics analysis procedure for the prismatic very high temperature gas-cooled reactors

A new physics analysis procedure has been developed for a prismatic very high temperature gas-cooled reactor based on a conventional two-step procedure for the PWR physics analysis. The HELIOS and MASTER codes were employed to generate the coarse group cross sections through a transport lattice calculation, and to perform the 3-dimensional core physics analysis by a nodal diffusion calculation, respectively. Physics analysis of the prismatic VHTRs involves particular modeling issues such as a double heterogeneity of the coated fuel particles, a neutron streaming in the coolant channels, a strong core-reflector interaction, and large spectrum shifts due to changes of the surrounding environment and state parameters. Double heterogeneity effect was considered by using a recently developed reactivity-equivalent physical transformation method. Neutron streaming effect was quantified through 3-dimensional Monte Carlo transport calculations by using the MCNP code. Strong core-reflector interaction could be handled by applying an equivalence theory to the generation of the reflector cross sections. The effects of a spectrum shift could be covered by optimizing the coarse energy group structure. A two-step analysis procedure was established for the prismatic VHTR physics analysis by combining all the methodologies described above. The applicability of our code system was tested against core benchmark problems. The results of these benchmark tests show that our code system is very accurate and practical for a prismatic VHTR physics analysis.     

Anomalous Isotope Fractionation in Uranium Enrichment Process

A benchmark calculation for a deep penetration problem of 14 MeV neutrons through a 3m thick iron slab was carried out by using a vectorized continuous energy Monte Carlo code MVP with the JENDL-3 and ENDF/B-IV cross sections. Reference solutions for neutron spectra and averaged cross sections were obtained at various locations through the iron slab with good statistics owing to a high computation speed of the code. The accuracy of multigroup calculations with the JSSTDL/J3 library was investigated by comparison with the obtained reference solutions.

Both calculations with JENDL-3 and ENDF/B-IV showed a similar attenuation of total fluxes from thermal to 14 MeV through the slab, while differences of one order at the maximum were observed in the calculated fluxes in the resonance energy region. The multigroup calculations with the JSSTDL/J3 295- and 125-group libraries underestimate the streaming effect through the cross section minima above the well-known 24 keV window, which resulted in the underestimation of fluxes above this window by more than two decades at 3 m penetration compared with the continuous energy method. Taking into account the spatial dependence of averaged cross sections, the underestimation was reduced to about one decade. It was found, however, that an accurate prediction of streaming effect is fairly difficult by the multigroup method.     

Doppler effect experiment of resonance materials for rock like oxide fuels

A Doppler effect experiment of resonance materials such as erbium, tungsten, thorium and uranium was carried out in the Fast Critical Assembly of Japan Atomic Energy Research Institute. Cylindrical shaped samples of 150 mm in stack length and 23 mm in diameter were fabricated. The sample was contained in capsules and placed at the center of the core. Temperature of the sample was raised up to 1073 K. Measured Doppler reactivities of erbium, thorium and tungsten are comparable with those of uranium. Analysis was performed using the SRAC code system. Effective absorption cross sections of the samples were generated by two different methods. One is the f-table method based on the NR approximation and the other is the PEACO method that performs direct calculations of resonance absorption with an ultra fine energy group structure. Calculated results were compared with the measured values. For all samples except the tungsten, the f-Table method gives 17 % smaller reactivity than the PEACO method. Both methods predict the measurements within the error of 6 %. For the tungsten sample, the calculations underestimate the measurement by about 10 %.     

Japanese Evaluated Nuclear Data Library Version 3 Revision-2: JENDL-3.2

The revision work of JENDL-3 has been made by considering feedback information of various benchmark tests. The main revised quantities are the resonance parameters, capture and inelastic scattering cross sections, and fission spectra of main actinide nuclides, the total and inelastic scattering cross sections of structural materials, the resonance parameters the capture and inelastic scattering cross sections of fission products, and the γr-ray production data. The revised data were released as JENDL-3.2 in June 1994. The preliminary benchmark tests indicate that JENDL-3.2 predicts various reactor characteristics more successfully than the previous version of JENDL-3.1.     

Radially and azimuthally dependent resonance self-shielding treatment for general multi-region geometry based on a unified theory

A unified resonance self-shielding method, which can treat general sub-divided fuel regions, is developed for lattice physics calculations in reactor physics field. In a past study, a hybrid resonance treatment has been developed by theoretically integrating equivalence theory and ultra-fine-group slowing-down calculation. It can be applied to a wide range of neutron spectrum conditions including low moderator density ranges in severe accident states, as long as each fuel region is not sub-divided. In order to extend the method for radially and azimuthally sub-divided multi-region geometry, a new resonance treatment is established by incorporating the essence of sub-group method. The present method is composed of two-step flux calculation, i.e. ‘coarse geometry + fine energy’ (first step) and ‘fine geometry + coarse energy’ (second step) calculations. The first step corresponds to a hybrid model of the equivalence theory and the ultra-fine-group calculation, and the second step corresponds to the sub-group method. From the verification results, effective cross-sections by the new method show good agreement with the continuous energy Monte-Carlo results for various multi-region geometries including non-uniform fuel compositions and temperature distributions. The present method can accurately generate effective cross-sections with short computation time in general lattice physics calculations.     

Representation of the neutron cross sections of several fertile and fissile nuclei in the resonance regions

In this paper we review several problems related to the measurement, analysis and evaluation of the neutron cross sections of the main fertile and fissile nuclides in the resonance region. In particular we discuss the ENDF/B-V representation of these cross sections.In recent years little progress has been made in improving our knowledge of the resolved resonance parameters of the fertile nuclei. We suggest that this absence of progress is due to a lack of adequate methodologies to deal with the systematic errors arising from uncertainties in the analysis of the measurements.We comment on the ENDF/B treatment of the unresolved resonance region and recommend the validation of the unresolved resonance range evaluations with appropriate transmission and self-indication measurements.