In this work we present a scheme for computing temperature-dependent unresolved resonance region cross sections in Monte Carlo neutron transport simulations. This approach relies on the generation of equiprobable cross section magnitude bands on an energy-temperature mesh. The bands are then interpolated in energy and temperature to obtain a cross section value. This is in contrast to the typical procedure of pre-generating probability tables at all temperatures present in the
simulation. As part of this work, a flexible probability table generation capability is integrated into the continuous-energy neutron transport code OpenMC [1]. Both single-level and multi-level Breit-Wigner formalisms are supported, as is modeling of the resonance structure of competitive reactions. A user-specified cross section band tolerance is enabled with
batch statistics. Probability tables are generated for all 268 ENDF/B-VII.1 [2] isotopes that have an unresolved resonance region evaluation. Integral benchmark simulations of the Big Ten critical assembly show that, for a system that is sensitive to the unresolved resonance region, a temperature interval of ∼200 K around 293.6 K is sufficient to reproduce the keff
value that is obtained with probability tables generated exactly at room temperature. A finer mesh of < 50 K is required to reproduce some cross section values at the common target relative difference of 0.1

Brown, Forrest B.

Forget, Benoit Robert Yves

Smith, Kord S.

Walsh, Jonathan Alan

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PHYSOR 2016 - Unifying Theory and Experiments in the 21st CenturySun Valley Resort, Sun Valley, Idaho, USA, May 1 - 5, 2016, on CD-ROM (2016)ON-THE-FLY DOPPLER BROADENING OF UNRESOLVED RESONANCE REGIONCROSS SECTIONS VIA PROBABILITY BAND INTERPOLATIONJonathan A. Walsh, Benoit Forget, and Kord S. SmithDepartment of Nuclear Science & EngineeringMassachusetts Institute of Technology77 Massachusetts Avenue, 24-107Cambridge, MA, USA 02139walshjon@mit.edu; bforget@mit.edu; kord@mit.eduForrest B. BrownXCP-3, Monte Carlo CodesLos Alamos National LaboratoryLos Alamos, NM, USA 87545fbrown@lanl.govABSTRACTIn this work we present a scheme for computing temperature-dependent unresolved resonance region cross sections inMonte Carlo neutron transport simulations. This approach relies on the generation of equiprobable cross section magnitudebands on an energy-temperature mesh. The bands are then interpolated in energy and temperature to obtain a cross sectionvalue. This is in contrast to the typical procedure of pre-generating probability tables at all temperatures present in thesimulation. As part of this work, a flexible probability table generation capability is integrated into the continuous-energyneutron transport code OpenMC [1]. Both single-level and multi-level Breit-Wigner formalisms are supported, as is mod-eling of the resonance structure of competitive reactions. A user-specified cross section band tolerance is enabled withbatch statistics. Probability tables are generated for all 268 ENDF/B-VII.1 [2] isotopes that have an unresolved resonanceregion evaluation. Integral benchmark simulations of the Big Ten critical assembly show that, for a system that is sensitiveto the unresolved resonance region, a temperature interval of ∼200 K around 293.6 K is sufficient to reproduce the keffvalue that is obtained with probability tables generated exactly at room temperature. A finer mesh of < 50 K is requiredto reproduce some cross section values at the common target relative difference of 0.1%.Key Words: unresolved resonance region, Doppler broadening, probability tables, cross sections, MonteCarlo1. INTRODUCTIONAt intermediate neutron energies, between the highly-structured cross sections of the resolved resonance region and therelatively smooth cross sections of the fast energy region, many isotopes have resonances which are so tightly spaced inenergy that they are unresolvable with current experimental techniques [3]. The energies over which this phenomenon isobserved comprise the unresolved resonance region (URR). The upper and lower energies of the URR vary from isotope toisotope with heavier nuclei typically exhibiting unresolved resonance structure at lower energies and over narrower energyintervals.While the precise structure of cross sections in the URR is unknown, sometimes by orders of magnitude, energy-averagedcross section values are available. However, it is generally not advisable to use these averaged cross sections in transportsimulations, especially for systems with an appreciable flux at URR energies. Energy self-shielding effects are effectivelyneglected when resonance structure is not taken into account. Without resonance structure, there are no flux depressions atresonance energies and reaction rates are over-predicted. The result is usually an under-prediction of keff due to increasedURR absorption and increased scattering to energies below the URR where the capture-to-fission ratio is typically higher.Walsh, et al.Further, because of the structure in URR cross sections, Doppler broadening effects are also observed. Therefore, high-fidelity simulations of fast reactor systems demand methods for modeling temperature-dependent resonance structure in theURR. In Sec. 2 we describe the mostly widely used method for generating temperature-dependent, structured URR crosssections, and how it has been adapted in this work to compute these cross sections on-the-fly, instead of as a pre-processingstep. The results of integral benchmark simulations are presented in Sec. 3. Additional results and analysis to be includedin the full paper are listed in Sec. 4.2. PROBABILITY TABLES AND ON-THE-FLY DOPPLER BROADENINGThe most common way of incorporating URR cross section resonance structure into Monte Carlo neutron transport simu-lations is with the probability table method [4]. The method, which relies on randomly sampling cross section magnitudes,has been implemented in many established nuclear data processing codes such as NJOY [5] and CALENDF [6] for pre-generating data to be used in particle transport codes (e.g MCNP [7], TRIPOLI [8]). Details of various probability tablegeneration implementations are documented elsewhere [9–11]. In Sec. 2.1 we describe the adapted implementation, whichis the focus of this work, for on-the-fly calculations of temperature-dependent cross sections. The results of both differentialcross section and integral benchmark testing are given in Sec. 2.22.1. ImplementationThough precise cross section values in the URR are unknown, average resonance parameters can be deduced through acombined examination of the parameters of resolved resonances and fitting of gross structure in the unresolved region. Theseparameters are provided in ENDF-6 format [12] nuclear data evaluations. Additionally, theoretical statistical distributions ofURR resonance parameters are well-known. For example, the distance in energy between adjacent resonances is describedby the Wigner distribution for level spacings. Partial reaction widths can be obtained by sampling a χ2x-distribution with anumber of degrees of freedom that depends on the reaction, x. An ensemble of URR resonances can be randomly generatedby sampling these resonance parameter distributions.For a single temperature, T , the probability table method proceeds by first establishing a grid in both the incident neutronenergy,En, and total cross section magnitude, σt, variables. Then, one realization of URR resonances is generated and crosssection values are calculated. Though any formalism can be used, a summation over single-level Breit-Wigner (SLBW)resonances is the standard prescribed by the ENDF-6 format. A schematic showing an En − σt mesh imposed on onerealization of URR cross sections is displayed in Fig. 1a. At each energy, the band containing the cross section magnitudeis recorded, as is the value itself. The tallied values are indicated with circular markers in the schematic. In addition, reactioncross section values conditional on the band of the total cross section magnitude are recorded. After many independentrealizations of cross section structure are tallied, at each energy, the probability of having a total cross section in a givenband is simply the number of realizations that fall within that band divided by the total number of realizations. The meantotal cross section magnitude for each band can also be determined by dividing the sum of cross section magnitudes thatfall within a band by the number of times that the magnitude is within that band. Mean reaction cross sections that areconditional on the band of the total cross section can be computed in an analogous manner. Transport codes can thensample these probability-cross section pairs each time that a cross section value is needed in order to capture the effects ofURR resonance structure.Probability table method implementations typically require that this process be repeated for each temperature that is presentin the model of the system being simulated. For models with continuous or highly-detailed temperature distributions, thiswould result in a massive increase in probability table data memory requirements. In order to mitigate this problem, an on-the-fly method for calculating temperature-dependent cross section values directly from resonance parameters at each eventwithin the simulation was investigated [13]. This approach is very memory efficient because it relies only on temperature-2/5 PHYSOR 2016 - Unifying Theory and Experiments in the 21st CenturySun Valley, Idaho, USA, May 1 - 5, 2016On-the-Fly URR Doppler Broadeningindependent average resonance parameters. With respect to simulation runtime for simple criticality calculations, 1 though,it can be quite inefficient due to the sampling of resonance parameters and calculation of cross sections at each event. Forthis reason, we develop an interpolation scheme which allows for the calculation of cross sections at all intermediate pointson a coarse temperature grid. Consistent interpolation is enabled through the generation of equiprobable cross sectionmagnitude surfaces on En − σt meshes. A schematic illustrating these surfaces can be found in Fig. 1b. In the transportsimulation a surface is randomly sampled and interpolation2 is performed in both the En and T variables.104 105En [eV]10−310−210−1100101102103104105σt[b]E0 E1 E2 E3 E4 E5 E6 E7P0, 〈σ0〉P1, 〈σ1〉P2, 〈σ2〉P3, 〈σ3〉P4, 〈σ4〉P5, 〈σ5〉P6, 〈σ6〉P7, 〈σ7〉(a) URR realization with an En − σt meshEn [keV]20406080100120140160log 10(T[K])−101234567σt[b]5101520253035404581216202428323640(b) Equiprobable cross section surfacesFigure 1: Probability table schematics10−4101σn[b]Reference Cross Section (ENDF71x)Calculated Cross Section10−4En [MeV]10−510−410−310−210−1100Rel.Diff. Rel. Diff. Magnitude(a) 239U elastic scattering cross sections10−4101102σn[b]Reference Cross Section (ENDF71x)Calculated Cross Section10−4En [MeV]10−510−410−310−210−1100Rel.Diff. Rel. Diff. Magnitude(b) 243Pu elastic scattering cross sectionFigure 2: Code-to-code comparison of SLBW cross section calculations1In transient fast reactor analysis simulations with multiphysics feedback, the runtime penalty will not be severe due to other overhead associated withdetailed tallies, communication between physics packages, etc.2Thus far, linear-linear and log-log interpolation have been explored.PHYSOR 2016 - Unifying Theory and Experiments in the 21st CenturySun Valley, Idaho, USA, May 1 - 5, 20163/5Walsh, et al.2.2. TestingIn order to test the implementation of the probability table generation capability, code-to-code comparisons with NJOYare performed. SLBW cross sections computed with both codes, using the same set of resonance parameters are shown inFigs. 2a and 2b.Table I: Code-to-code comparison of Big Ten keff eigenvaluesProbability Tables keff 1σNJOY99.393 (ENDF71x ACE Library) 1.00467 0.00010OpenMC Implementation 1.00466 0.00010The results of keff calculations using the room temperature improved Big Ten integral benchmark model are given in Table I.Excellent agreement is observed between eigenvalues computed using the probability tables generated with the two codes.3. RESULTSWith the satisfactory performance of the probability table generation capability demonstrated in the results of the testingdescribed in Sec. 2.2, we now seek to investigate the extent to which interpolation of equiprobable cross section magnitudebands to intermediate temperatures is able to reproduce results obtained with cross section data that is generated exactly atthose intermediate temperatures. The results of integral benchmark simulations are given in Sec. 3.1.3.1. Integral BenchmarksThe results of keff eigenvalue simulations of the improved Big Ten critical assembly model are shown in Table II. Eachsimulation is performed with probability table data generated at a different room temperature-bounding interval. Theseresults indicate that integral tallies are reproducible with a fairly wide temperature interpolation interval. Interpolation tothe Big Ten model temperature of 293.6 K using data generated at bounding temperatures of 200 K and 400 K gives akeff value that is statistically indistinguishable from the exact 293.6 K result at a relatively tight 1σ uncertainty of 10 pcm.It is expected that intervals even wider than this 200 K will produce acceptable results for the smoother cross sectionsencountered at elevated model temperatures.Table II: Big Ten keff eigenvalues for varying temperature interpolation intervals∆T [K] keff 1σ0 1.00466 0.00010100 1.00463 0.00010200 1.00468 0.00010400 1.00533 0.000104/5 PHYSOR 2016 - Unifying Theory and Experiments in the 21st CenturySun Valley, Idaho, USA, May 1 - 5, 2016On-the-Fly URR Doppler Broadening4. FUTUREWORKIn addition to the material presented in this summary, the full paper will contain:• Expanded integral benchmark analyses including reaction rate spectra• Comparisons of interpolated and directly-calculated differential cross sections• Discussion of temperature correlation effects which are accounted for in this methodology• Comments on competitive reaction resonance structure and the use of a multi-level resonance formalisms, as relatedto temperature effects• Recommendations for the generation of equiprobable cross section surfaces for use in fast reactor simulations withtemperature feedbackREFERENCES[1] P. K. Romano and B. Forget. “The OpenMC Monte Carlo particle transport code.” Ann. Nucl. Energy, 51: pp.274–281 (2013).[2] M. Chadwick et al. “ENDF/B-VII.1 nuclear data for science and technology: Cross sections, covariances, fissionproduct yields and decay data.” Nucl. Data Sheets, 112: pp. 2887–2996 (2011).[3] A. Foderaro. The Elements of Neutron Interaction Theory. MIT Press (1971).[4] L. Levitt. “The probability table method for treating unresolved neutron resonances in Monte Carlo calculations.”Nucl. Sci. Eng., 49: pp. 450–457 (1972).[5] R. Macfarlane and D. Muir. The NJOY Nuclear Data Processing System, Version 91. Technical Report LA-12740-M,Los Alamos National Laboratory (1994).[6] J. C. Sublet, P. Ribon, and M. Coste-Delclaux. CALENDF-2010: User Manual. Technical Report CEA-R-6277,French Alternative Energies and Atomic Energy Commission (2011).[7] T. Goorley et al. “Initial MCNP6 release overview.” Nucl. Technol., 180: pp. 298–315 (2012).[8] TRIPOLI-4 Project Team, 2008. TRIPOLI-4 User Guide. Technical Report CEA-R-6169, French Alternative Energiesand Atomic Energy Commission (2008).[9] T. M. Sutton and F. B. Brown. “Implementation of the probability table method in a continuous-energy Monte Carlocode system.” In: International Conference on the Physics of Nuclear Science and Technology. Vol. 2, p. 891, LongIsland, New York, October 5-8, 1998.[10] M. Dunn and L. Leal. “Calculating probability tables for the unresolved-resonance region using Monte Carlo meth-ods.” Nucl. Sci. Eng., 148: pp. 30–42 (2004).[11] R. MacFarlane and A. Kahler. “Methods for processing ENDF/B-VII with NJOY.” Nucl. Data Sheets, 111: pp.2739–2890 (2010).[12] A. Trkov, M. Herman, and D. Brown. ENDF-6 Formats Manual. Technical Report BNL-90365-2009 Rev.2, NationalNuclear Data Center, Brookhaven National Laboratory (2012).[13] J. A. Walsh et al. “Direct, on-the-fly calculation of unresolved resonance region cross sections in Monte Carlo simula-tions.” In: Proc. Joint Int. Conf. on Mathematics and Computation (M&C), Supercomputing in Nuclear Applications(SNA) and the Monte Carlo (MC) Method (2015).PHYSOR 2016 - Unifying Theory and Experiments in the 21st CenturySun Valley, Idaho, USA, May 1 - 5, 20165/5

On-the-fly doppler broadening of unresolved resonance region cross sections via probability band interpolation

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On-the-fly doppler broadening of unresolved resonance region cross sections via probability band interpolation

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