Creation of problem-dependent Doppler-broadened cross sections in the KENO Monte Carlo code☆
Introduction
Cross-section files are generally provided in Evaluated Nuclear Data Format (ENDF) formatted data files (Trkov et al., 2011) that contain all of the necessary data to create continuous energy (CE) data libraries for use in a Monte Carlo calculation. To be useful, these ENDF data files are generally processed by a cross-section processing code such as AMPX (Dunn and Greene, 2002) or NJOY (MacFarlane and Muir, 2000) for use in a radiation transport code such as CE-KENO (Hollenbach et al., 2011). For one-dimensional cross sections, the data are usually provided at one temperature (designated as 0 K), and need to be Doppler-broadened to various temperatures before they can be used at reactor-level temperatures.
Exact Doppler-broadened cross sections can be done by the nuclear data-processing codes using Doppler broadening equations (Cullen et al., 1973); however, producing exact cross sections at a large number of temperatures would consume a significant amount of time and space, both in memory and on a hard disk. Therefore, cross-section libraries are generally only created at several different temperatures; for KENO, part of the SCALE code suite (ORNL, 2011), there are generally six temperatures created. For KENO in CE mode, if the temperature desired is not one of the pregenerated temperatures, then the closest temperature is used. A case containing materials that are 50 K away from a library temperature can produce significantly different results when compared with a case that is using the temperature-corrected cross sections.
Two-dimensional cross sections are generally provided for thermal moderators in order to account for crystalline effects encounted when neutrons are traveling at thermal speeds. Unlike the one-dimensional cross sections, the ENDF files are usually provided at a variety of temperatures. However, no Doppler broadening is done on these temperatures, so the end result is the same: If a temperature desired by the user is sufficiently far from the library temperatures, errors in the eigenvalue estimates can occur. Some previous work has been done to provide for on-the-fly (OTF) Doppler broadening of one-dimensional neutron cross sections (Yesilyurt et al., 2009, Yesilyurt et al., 2012, Brown et al., 2012, Martin et al., 2013, Trumbull, 2006) in other Monte Carlo codes. For example, MCNP6 (X-5 Monte Carlo Team, 2003) ships with a utility to generate fits to cross-section data so that cross sections can be calculated on-the-fly for any temperature as desired. KENO previously had no such capability to Doppler broaden cross sections.
In this paper two methods are discussed to temperature-correct the provided cross sections. A finite difference method is employed for the one-dimensional cross sections. This method is much faster than the exact Doppler-broadening method developed by Cullen and can use the data libraries that have already been created. For two-dimensional thermal moderator data, a simple unit-base interpolation scheme is used on the probability distributions of the double differential cross sections. By combining the aforementioned methods with temperature interpolation on the probability tables covering the unresolved resonance range (such as in Walsh et al. (2015)), KENO will have temperature-corrected neutron cross sections for all energy regions of interest (Hart et al., 2014).
Section snippets
One-dimensional method
For one-dimensional cross sections the approach to be implemented into KENO utilizes a finite-difference method similar to that used by SAMMY (Larson, 2008), which is well suited for resonance analysis and light water reactor (LWR) applications. This approach is based on the Leal-Hwang scattering method (Leal and Hwang, 1987), in which the Doppler broadened cross sections satisfy a heat equation of the formwhere F is the function of interest (in this case the cross section), u is
One-dimensional results
A variety of test cases were run to test the impact of the one-dimensional problem-dependent Doppler broadening.
Two-dimensional method
After the one-dimensional cross sections have been Doppler-broaded, the next step is to Doppler-broaden the two-dimensional cross sections (or kinematics data) used for the thermal moderators.
Benchmarks
The International Handbook of Evaluated Reactor Physics Benchmark Experiments (IHECRPhBE) (Committee, 2006) was prepared by a working group of experienced reactor physics personnel. It contains reactor physics benchmark specifications that have been derived from experiments performed at various nuclear experimental facilities around the world. The benchmark specifications are intended for use to validate calculation techniques. The most recent edition of the handbook contains data from 53
Conclusions
Typically, reactor analysis tools ship with only a subset of temperatures that a reactor physics analyst needs in order to accurately model a problem. A one-dimensional method to Doppler-broaden cross secions was expanded and coded into the Monte Carlo code KENO. By using this method, cross sections can be Doppler-broadened to any temperature that the user selects. Doppler broadening is done before neutron transport begins. Therefore the method has been christened as “problem-dependent Doppler
Acknowledgements
The work documented in this paper was performed with support from the U.S. Department of Energy Nuclear Criticality Safety Program.
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