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Title: Kernel reconstruction methods for Doppler broadening — Temperature interpolation by linear combination of reference cross sections at optimally chosen temperatures

Abstract

This paper establishes a new family of methods to perform temperature interpolation of nuclear interactions cross sections, reaction rates, or cross sections times the energy. One of these quantities at temperature T is approximated as a linear combination of quantities at reference temperatures (T j). The problem is formalized in a cross section independent fashion by considering the kernels of the different operators that convert cross section related quantities from a temperature T 0 to a higher temperature T — namely the Doppler broadening operation. Doppler broadening interpolation of nuclear cross sections is thus here performed by reconstructing the kernel of the operation at a given temperature T by means of linear combination of kernels at reference temperatures (T j). The choice of the L 2 metric yields optimal linear interpolation coefficients in the form of the solutions of a linear algebraic system inversion. The optimization of the choice of reference temperatures (T j) is then undertaken so as to best reconstruct, in the L∞ sense, the kernels over a given temperature range [T min,T max]. The performance of these kernel reconstruction methods is then assessed in light of previous temperature interpolation methods by testing them upon isotope 238U. Temperature-optimizedmore » free Doppler kernel reconstruction significantly outperforms all previous interpolation-based methods, achieving 0.1% relative error on temperature interpolation of 238U total cross section over the temperature range [300 K,3000 K] with only 9 reference temperatures.« less

Authors:
ORCiD logo [1];  [1];  [1];  [2];  [1];  [1]
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1344987
Alternate Identifier(s):
OSTI ID: 1397789
Grant/Contract Number:
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 335; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Doppler broadening; nuclear cross sections; temperature interpolation; kernel reconstruction

Citation Formats

Ducru, Pablo, Josey, Colin, Dibert, Karia, Sobes, Vladimir, Forget, Benoit, and Smith, Kord. Kernel reconstruction methods for Doppler broadening — Temperature interpolation by linear combination of reference cross sections at optimally chosen temperatures. United States: N. p., 2017. Web. doi:10.1016/j.jcp.2017.01.039.
Ducru, Pablo, Josey, Colin, Dibert, Karia, Sobes, Vladimir, Forget, Benoit, & Smith, Kord. Kernel reconstruction methods for Doppler broadening — Temperature interpolation by linear combination of reference cross sections at optimally chosen temperatures. United States. doi:10.1016/j.jcp.2017.01.039.
Ducru, Pablo, Josey, Colin, Dibert, Karia, Sobes, Vladimir, Forget, Benoit, and Smith, Kord. Wed . "Kernel reconstruction methods for Doppler broadening — Temperature interpolation by linear combination of reference cross sections at optimally chosen temperatures". United States. doi:10.1016/j.jcp.2017.01.039. https://www.osti.gov/servlets/purl/1344987.
@article{osti_1344987,
title = {Kernel reconstruction methods for Doppler broadening — Temperature interpolation by linear combination of reference cross sections at optimally chosen temperatures},
author = {Ducru, Pablo and Josey, Colin and Dibert, Karia and Sobes, Vladimir and Forget, Benoit and Smith, Kord},
abstractNote = {This paper establishes a new family of methods to perform temperature interpolation of nuclear interactions cross sections, reaction rates, or cross sections times the energy. One of these quantities at temperature T is approximated as a linear combination of quantities at reference temperatures (Tj). The problem is formalized in a cross section independent fashion by considering the kernels of the different operators that convert cross section related quantities from a temperature T0 to a higher temperature T — namely the Doppler broadening operation. Doppler broadening interpolation of nuclear cross sections is thus here performed by reconstructing the kernel of the operation at a given temperature T by means of linear combination of kernels at reference temperatures (Tj). The choice of the L2 metric yields optimal linear interpolation coefficients in the form of the solutions of a linear algebraic system inversion. The optimization of the choice of reference temperatures (Tj) is then undertaken so as to best reconstruct, in the L∞ sense, the kernels over a given temperature range [Tmin,Tmax]. The performance of these kernel reconstruction methods is then assessed in light of previous temperature interpolation methods by testing them upon isotope 238U. Temperature-optimized free Doppler kernel reconstruction significantly outperforms all previous interpolation-based methods, achieving 0.1% relative error on temperature interpolation of 238U total cross section over the temperature range [300 K,3000 K] with only 9 reference temperatures.},
doi = {10.1016/j.jcp.2017.01.039},
journal = {Journal of Computational Physics},
number = C,
volume = 335,
place = {United States},
year = {Wed Jan 25 00:00:00 EST 2017},
month = {Wed Jan 25 00:00:00 EST 2017}
}

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  • The SIGMA1 kernel broadening method is presented to Doppler broaden to any required accuracy a cross section that is described by a table of values and linear-linear interpolation in energy-cross section between tabulated values. The method is demonstrated to have no temperature or energy limitations and to be equally applicable to neutron or charged-particle cross sections. The method is qualitatively and quantitatively compared to contemporary approximate methods of Doppler broadening with particular emphasis on the effect of each approximation introduced.
  • The TREND code is developed for reconstruction of neutron cross sections in the region of resolved resonances from the data of the international evaluated nuclear data libraries using the Reich-Moore, Adler-Adler, and Breit-Wigner formalisms and for calculation of the Doppler broadening of the resulting cross sections within the classical approximation. The TREND code is incorporated into the MCU code package for Monte Carlo reactor calculations. The TREND and MCU codes have been used to evaluate the data of the ROSFOND data bank on uranium and plutonium isotopes in comparison with the data of the ICSBEP data bank.
  • In this paper, two methods for computing temperature-dependent unresolved resonance region cross sections on-the-fly within continuous-energy Monte Carlo neutron transport simulations are presented. The first method calculates Doppler broadened cross sections directly from zero-temperature average resonance parameters. In a simulation, at each event that requires cross section values, a realization of unresolved resonance parameters is generated about the desired energy and temperature-dependent single-level Breit-Wigner resonance cross sections are computed directly via the analytical Ψ-x Doppler integrals. The second method relies on the generation of equiprobable cross section magnitude bands on an energy-temperature mesh. Within a simulation, the bands are sampledmore » and interpolated in energy and temperature to obtain cross section values on-the-fly. Both of the methods, as well as their underlying calculation procedures, are verified numerically in extensive code-to-code comparisons. Energy-dependent pointwise cross sections calculated with the newly-implemented procedures are shown to be in excellent agreement with those calculated by a widely-used nuclear data processing code. Relative differences at or below 0.1% are observed. Integral criticality benchmark results computed with the proposed methods are shown to reproduce those computed with a state-of-the-art processed nuclear data library very well. In simulations of fast spectrum systems which are highly-sensitive to the representation of cross section data in the unresolved region, k-eigenvalue and neutron flux spectra differences of <10 pcm and <1.0% are observed, respectively. The direct method is demonstrated to be well-suited to the calculation of reference solutions — against which results obtained with a discretized representation may be assessed — as a result of its treatment of the energy, temperature, and cross section magnitude variables as continuous. Also, because there is no pre-processed data to store (only temperature-independent average resonance parameters) the direct method is very memory-efficient. Typically, only a few kB of memory are needed to store all required unresolved region data for a single nuclide. However, depending on the details of a particular simulation, performing URR cross section calculations on-the-fly can significantly increase simulation times. Alternatively, the method of interpolating equiprobable probability bands is demonstrated to produce results that are as accurate as the direct reference solutions, to within arbitrary precision, with high computational efficiency in terms of memory requirements and simulation time. Analyses of a fast spectrum system show that interpolation on a coarse energy-temperature mesh can be used to reproduce reference k-eigenvalue results obtained with cross sections calculated continuously in energy and directly at an exact temperature to within <10 pcm. Probability band data on a mesh encompassing the range of temperatures relevant to reactor analysis usually require around 100 kB of memory per nuclide. Finally, relative to the case in which probability table data generated at a single, desired temperature are used, minor increases in simulation times are observed when probability band interpolation is employed.« less