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TRANSCRIPT
ACTAUNIVERSITATIS
UPSALIENSISUPPSALA
2015
Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1315
Nuclear data uncertaintyquantification and data assimilationfor a lead-cooled fast reactor
Using integral experiments for improved accuracy
ERWIN ALHASSAN
ISSN 1651-6214ISBN 978-91-554-9407-0urn:nbn:se:uu:diva-265502
Dissertation presented at Uppsala University to be publicly examined in polhemsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Thursday, 17 December 2015 at 09:15for the degree of Doctor of Philosophy. The examination will be conducted in English.Faculty examiner: Dr. Oscar Cabellos (OECD Nuclear Energy Agency (NEA)).
AbstractAlhassan, E. 2015. Nuclear data uncertainty quantification and data assimilation for a lead-cooled fast reactor. Using integral experiments for improved accuracy. Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology 1315. 85 pp.Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9407-0.
For the successful deployment of advanced nuclear systems and optimization of current reactordesigns, high quality nuclear data are required. Before nuclear data can be used in applicationsthey must first be evaluated, tested and validated against a set of integral experiments, andthen converted into formats usable for applications. The evaluation process in the past wasusually done by using differential experimental data which was then complemented with nuclearmodel calculations. This trend is fast changing due to the increase in computational powerand tremendous improvements in nuclear reaction models over the last decade. Since thesemodels have uncertain inputs, they are normally calibrated using experimental data. However,these experiments are themselves not exact. Therefore, the calculated quantities of model codessuch as cross sections and angular distributions contain uncertainties. Since nuclear data areused in reactor transport codes as input for simulations, the output of transport codes containuncertainties due to these data as well. Quantifying these uncertainties is important for settingsafety margins; for providing confidence in the interpretation of results; and for deciding whereadditional efforts are needed to reduce these uncertainties. Also, regulatory bodies are nowmoving away from conservative evaluations to best estimate calculations that are accompaniedby uncertainty evaluations.
In this work, the Total Monte Carlo (TMC) method was applied to study the impact of nucleardata uncertainties from basic physics to macroscopic reactor parameters for the EuropeanLead Cooled Training Reactor (ELECTRA). As part of the work, nuclear data uncertaintiesof actinides in the fuel, lead isotopes within the coolant, and some structural materials havebeen investigated. In the case of the lead coolant it was observed that the uncertainty in the keff
and the coolant void worth (except in the case of 204Pb), were large, with the most significantcontribution coming from 208Pb. New 208Pb and 206Pb random nuclear data libraries with realisticcentral values have been produced as part of this work. Also, a correlation based sensitivitymethod was used in this work, to determine parameter - cross section correlations for differentisotopes and energy groups.
Furthermore, an accept/reject method and a method of assigning file weights based onthe likelihood function are proposed for uncertainty reduction using criticality benchmarkexperiments within the TMC method. It was observed from the study that a significant reductionin nuclear data uncertainty was obtained for some isotopes for ELECTRA after incorporatingintegral benchmark information. As a further objective of this thesis, a method for selectingbenchmark for code validation for specific reactor applications was developed and applied tothe ELECTRA reactor. Finally, a method for combining differential experiments and integralbenchmark data for nuclear data adjustments is proposed and applied for the adjustment ofneutron induced 208Pb nuclear data in the fast energy region.
Keywords: Total Monte Carlo, ELECTRA, nuclear data, uncertainty propagation, integralexperiments, nuclear data adjustment, uncertainty reduction
Erwin Alhassan, Department of Physics and Astronomy, Applied Nuclear Physics, Box 516,Uppsala University, SE-751 20 Uppsala, Sweden.
© Erwin Alhassan 2015
ISSN 1651-6214ISBN 978-91-554-9407-0urn:nbn:se:uu:diva-265502 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-265502)
Dedicated to Mabel & Emily
List of papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I E. Alhassan, H. Sjöstrand, J. Duan, C. Gustavsson, A.J. Koning, S.
Pomp, D. Rochman, M. Österlund. Combining Total Monte Carloand Benchmarks for nuclear data uncertainty propagation on aLead Fast Reactor’s Safety. Nuclear Data Sheets (2014) 118,542-544.
My contribution: I wrote the scripts and I performed the simulations,
the analyses and interpretation of results. I also wrote the paper.
II E. Alhassan, H. Sjöstrand, P. Helgesson, A.J. Koning, M. Österlund,
S. Pomp, D. Rochman. Uncertainty and correlation analysis of leadnuclear data on reactor parameters for the European Lead CooledTraining Reactor. Annals of Nuclear Energy (2015) 75, 26-37.
My contribution: I wrote most of the scripts and I performed the
simulations, the analyses and interpretation of results. I also wrote the
paper.
III H. Sjöstrand, E. Alhassan, J. Duan, C. Gustavsson, A.J. Koning, S.
Pomp, D. Rochman, M. Österlund. Propagation of nuclear datauncertainties for ELECTRA burn-up calculations. Nuclear DataSheets (2014) 118, 527-530.
My contribution: I wrote part of the scripts, performed part of the
simulations and took part in the writing of the paper.
IV E. Alhassan, H. Sjöstrand, P. Helgesson, M. Österlund, S. Pomp, A.J.
Koning, D. Rochman. On the use of integral experiments foruncertainty reduction of reactor macroscopic parameters withinthe TMC methodology. In review process after revision, Progress ofNuclear Energy, 2015.
My contribution: I developed the method; wrote most of the scripts;
and also performed the simulations, the analyses and the interpretation
of the results. I also wrote the paper.
v
V E. Alhassan, H. Sjöstrand, P. Helgesson, M. Österlund, S. Pomp, A.J.
Koning, D. Rochman. Selecting benchmarks for reactorsimulations: an application to a Lead Fast Reactor. Submitted toAnnals of Nuclear Energy, 2015.
My contribution: I developed the method; wrote the scripts; and also
performed the simulations, the analyses and the interpretation of the
results. I also wrote the paper.
Reprints were made with permission from the publishers.
Other papers not included in this thesis
List of papers related to this thesis but not included in the comprehensive sum-
mary. I am first author or co-author of all listed papers.
1. E. Alhassan, H. Sjöstrand, J. Duan, P. Helgesson, S. Pomp, M. Öster-
lund, D. Rochman, A.J. Koning. Selecting benchmarks for reactorcalculations. In proc. PHYSOR 2014 International Conference, Kyoto,Japan. 28 September - 3 October 2014.
2. H. Sjöstrand, E. Alhassan, S. Conroy, J. Duan, C. Hellesen, S. Pomp, M.
Österlund, A.J. Koning, D. Rochman. Total Monte Carlo evaluationfor dose calculations. Radiation Protection Dosimetry (2013) 161 (1-4), 312-315.
3. R. Della, E. Alhassan, N.A. Adoo, C.Y. Bansah, E.H.K. Akaho, B.J.B.
Nyarko. Stability analysis of the Ghana Research Reactor-1 (GHARR-1). Energy Conversion and Management (2013) 74, 587-593.
4. C.Y. Bansah, E.H.K. Akaho, A. Ayensu, N.A. Adoo, V.Y. Agbodemegbe,
E. Alhassan, R. Della. Theoretical model for predicting the relativetimings of potential failures in steam generator tubes of a PWR dur-ing a severe accident. Annals of Nuclear Energy (2013) 59, 10-15.
5. J. Duan, S. Pomp, H. Sjöstrand, E. Alhassan, C. Gustavsson, M. Öster-
lund, D. Rochman, A.J. Koning. Uncertainty Study of Nuclear ModelParameters for the n+56Fe Reactions in the Fast Neutron Region Be-low 20 MeV. Nuclear Data Sheets (2014) 118, 346-348.
6. P. Helgesson, D. Rochman, H. Sjöstrand, E. Alhassan, A.J. Koning.
UO2 vs MOX: propagated nuclear data uncertainty for keff, withburnup. Nuclear Science and Engineering (2014) 3, 321-336.
7. E. Alhassan, H. Sjöstrand, J. Duan, C. Gustavsson, A.J. Koning, S.
Pomp, D. Rochman, M. Österlund. Uncertainty analysis of Lead crosssections on reactor safety for ELECTRA. In proc. SNA + MC 2013,02401 (2014), EDP Sciences.
8. E.K. Boafo, E. Alhassan, E.H.K. Akaho. Utilizing the burnup capa-bility in MCNPX to perform depletion analysis of an MNSR fuel.Annals of Nuclear Energy (2014) 73, 478-483.
9. P. Helgesson, H. Sjöstrand, A.J. Koning, D. Rochman, E. Alhassan, S.
Pomp. Incorporating experimental information in the TMC method-ology using file weights. Nuclear Data Sheets (2015) 123, 214-219.
vii
10. S. Pomp, A. Al-Adili, E. Alhassan, C. Gustavsson, P. Helgesson, C.
Hellesen, A.J. Koning, M. Lantz, M Österlund, D. Rochman, V. Simutkin,
H. Sjöstrand, A. Solders. Experiments and theoretical data for study-ing the impact of fission yield uncertainties on the nuclear fuel cyclewith TALYS/GEF and the TMC method. Nuclear Data Sheets (2015)123, 220-224.
11. A. Al-Adili, E. Alhassan, C. Gustavsson, P. Helgesson, K. Jansson, A.J.
Koning, M. Lantz, A. Mattera, A.V. Prokofiev, V. Rakopoulos, H. Sjos-
trand, A. Solders, D. Tarrio, M. Osterlund, S. Pomp. Fission activitiesof the nuclear reactions group in Uppsala. In proc., Scientific Work-shop on Nuclear Fission dynamics and the Emission of Prompt Neutronsand Gamma Rays, THEORY-3. Physics Procedia (2015) 64, 145-149.
12. P. Helgesson, H. Sjöstrand, A.J. Koning, J. Ryden, D. Rochman, E. Al-hassan, S. Pomp. Sampling of systematic errors to compute like-lihood weights in nuclear data uncertainty propagation. In Press,Nuclear Instruments and Methods A, 2015.
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Nuclear data needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Differential data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Integral data (benchmarks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Nuclear data evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Nuclear data libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Nuclear data definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Resonance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Covariance data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Sources of uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Statistical estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Uncertainty quantification approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Deterministic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.2 Stochastic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Simulation tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 TALYS based code system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 Processing codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.3 Neutron transport codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Model calculations with TALYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Nuclear data uncertainty calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.1 Global uncertainty analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.2 Local uncertainty analyses - Partial TMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Reactor Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Reactor description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.2 Reactor neutronic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.3 Convergence of the first 4 moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Nuclear data uncertainty reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.1 Binary Accept/Reject method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5.2 Reducing uncertainty using file weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5.3 Combined benchmark uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Benchmark selection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.6.1 Benchmark cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.7 Correlation based sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.8 Burnup calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.9 Nuclear data Adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.9.1 Production of random nuclear data libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.9.2 Nuclear data adjustment of 208Pb in the fast region . . . . . . . . . . . . . . . . 47
5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1 Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Global uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Partial variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 Nuclear data uncertainty reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.5 Similarity Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6 Correlation based sensitivity measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 Nuclear data adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7 Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1. Introduction
’If you would be a real seeker after truth, it is necessary that at leastonce in your life you doubt, as far as possible, all things.’
- Renè Descartes
1.1 Background
Today, about 1.4 billion people globally still have no access to electricity,
and an additional one billion more only have access to unreliable supply of
electricity [1]. As a result of population growth and economic development
especially in developing countries, the global energy demands are projected
to rise [2]. With increasing energy consumption worldwide, nuclear power
is expected to play an increasing role in providing the energy needs for the
future [3]. Due to public acceptance issues, the next generations of nuclear
power reactors must not only be economically competitive with other energy
sources but must also address waste management, proliferation and safety con-
cerns. The GEN-IV International Forum (GIF) was therefore initiated with
very challenging technology goals, which include sustainability, economics,
safety, reliability, proliferation resistance and physical protection as reported
in the GEN-IV Technology Roadmap [4]. The six reactor concepts identified
by the GEN-IV International Forum as most promising advanced reactor sys-
tems are [4]: the gas-cooled fast reactor (GFR), the lead-cooled fast reactor
(LFR), the molten salt reactor (MSR), the sodium fast reactor (SFR), the very-
high-temperature reactor (VHTR) and the supercritical water-cooled reactor
(SCWR).
The Lead Fast Reactor concept, was ranked top in sustainability by the GIF
because it uses a closed fuel cycle for the conversion of fertile isotopes, and in
proliferation resistance and physical protection because of its long-life core [5].
Its safety features are enhanced by the choice of a relatively inert coolant
which has the capability of retaining hazardous radionuclides such as iodine
and cesium even in the event of a severe accident. As part of GEN-IV devel-
opment in Sweden, the GENIUS project which was a collaboration between
Chalmers University of Technology, Royal Institute of Technology and Up-
psala University was initiated for the development of the GEN-IV concept in
1
Sweden [6]. The development of a lead-cooled Fast Reactor called ELECTRA
(European Lead-Cooled Training Reactor), which will permit full recycling of
plutonium and americium in the core, was proposed within the project.
For the design and successful implementation of GEN-IV reactor concepts,
high quality and accurate nuclear data are required. For several decades, re-
actor design has been supported by computer simulations for the investiga-
tion of reactor behavior under both steady state and transient conditions. The
physical models implemented in these simulations codes are dependent on the
underlying nuclear data used, implying that uncertainties in nuclear data are
propagated to the outputs of these codes. Before nuclear data can be used
in applications, they are first evaluated, benchmarked against integral experi-
ments and then converted into formats usable for applications. The evaluation
of neutron induced reactions usually involves the combination of nuclear re-
action models with experimental data (nuclear reaction models are adjusted
to reproduce experimental data). However, these experiments are themselves
not exact and therefore, the calculated quantities of model codes such as cross
sections and angular distributions, contain uncertainties.
In the past, nuclear data uncertainties within the Reactor Physics community
were mostly propagated using deterministic methods. With this approach, the
local sensitivities of a particular response parameter, such as the keff, to the
variations in input parameters (nuclear data in our case), can be determined by
using the generalized perturbation theory [7]. Once the sensitivity coefficient
matrix has been determined, it is combined with the covariance matrix to ob-
tain the corresponding uncertainty on any response parameter of interest. For
example, sensitivity profiles obtained by using the so-called perturbation card
in MCNP [8] are combined with covariance data using the SUSD code [9] to
obtain nuclear data uncertainty on the reactor response parameter of interest.
This approach relies on the assumption that the sensitivity of the output pa-
rameter depends linearly on the variation in each input parameter [10]. With
the increase in computational power, however, Monte Carlo methods are now
possible. In the Nuclear Research and Consultancy Group (NRG), Petten, The
Netherlands, a method called ’Total Monte Carlo (TMC)’ was developed for
nuclear data evaluation and uncertainty propagation. An advantage of this ap-
proach is that it eliminates the use of covariances and the assumption of linear-
ity used in the perturbation approach. Quantifying nuclear data uncertainties,
is important for reactor safety assessment, reliability and risk assessment, and
for deciding where additional efforts need to be taken to reduce these uncer-
tainties.
In this work, the TMC method was applied to study the impact of nuclear data
uncertainties from basic nuclear physics to macroscopic reactor parameters
for the ELECTRA reactor in respect of major and minor actinides, structural
materials and the coolant. This work is important because, if all uncertainties
2
(including nuclear data uncertainties) are not taken into account in reactor de-
sign safety margins could be under assigned to key reactor safety parameters.
This may lead to severe accidents. As part of the work, the impact of nuclear
data uncertainties of some of the actinides in the fuel (PAPER I and IV), lead
isotopes in the coolant (PAPER II) and some structural materials at beginning
of life (BOL) have been estimated. In addition, the propagation of 239Pu trans-
port data uncertainties in reactor burnup calculations was carried out in PA-PER III. A further objective has been to develop methodologies for selecting
benchmarks that can help in reactor code validation (PAPER V). Also, meth-
ods for reducing nuclear data uncertainties using integral benchmarks have
been developed and presented in more detail in PAPER I and IV. Finally, a
method for combining differential experiments and integral benchmark data
for data assimilation and nuclear data adjustments using file weights based on
the likelihood function is proposed in section 4.9. The proposed method is
applied for the adjustment of neutron induced reactions of 208Pb in the fast
energy region and the preliminary results are presented in Section 5.7.
1.2 Nuclear data needs
For the successful deployment of advanced nuclear systems and for the op-
timization of current reactor designs, accurate information about the nuclear
reactions taking place within the reactor core are required. Using sensitivity
and uncertainty analyses, a list of priorities for improvement of nuclear data
has been determined [11, 12] and this includes both lead and plutonium iso-
topes.
From the application side, a preliminary attempt to assign target uncertain-
ties for some GEN-IV systems have been carried out by the Organisation for
Economic Co-operation and Development (OECD)/Nuclear Energy Agency
(NEA) expert group [11]. From Ref. [11], target uncertainties within 1-sigma
have been identified for the following parameters (target uncertainties in brack-
ets) for fast reactors: multiplication factor at Beginning of Life (BOL) (0.3%),
power peaking factor (2%), reactivity coefficients at BOL (7%) while nuclide
densities at the end of life should be within 10% uncertainty. To fulfil these
targets the scientific community must develop both theoretical nuclear physics
models as well as uncertainty quantification and uncertainty reduction meth-
ods. In parallel, the initiation of high-quality differential experimental mea-
surements are necessary.
To achieve the goal of further reducing nuclear data uncertainties in macro-
scopic reactor parameters, the reduction of nuclear data uncertainties using
integral benchmark data, and nuclear data adjustments using both differential
and integral data is presented as part of this work.
3
1.3 Outline of thesis
The thesis is structured as follows; the background, nuclear data needs for
advanced nuclear systems and the outline of the thesis are presented in Chap-
ter 1. In Chapter 2, nuclear data definitions, differential and integral bench-
mark experimental data, the nuclear data evaluation process which involves
the combination of differential experimental data and model calculations, as
well as nuclear covariance data are described. Sources of uncertainties, statis-
tical estimations, uncertainty quantification approaches and uncertainty anal-
ysis in Reactor Physics modeling are given in Chapter 3. In Chapter 4, the
simulation tools used in this work and the application of the TMC method for
uncertainty quantification of macroscopic reactor parameters are presented for
global and local (partial) variations of nuclear data. Also, methods developed
for benchmark selection, for nuclear data uncertainty reduction, and nuclear
data adjustments in the fast energy region are described. Furthermore, model
calculations and generation of random nuclear data using the TALYS based
code system (T6) are presented. In chapter 5, the results obtained are pre-
sented and discussed. Chapter 6 contains the conclusion and the outlook for
future work and finally, the summary of the thesis in Swedish is presented in
Chapter 7.
4
2. Nuclear data
’It is nice to know that the computer understands the problem.But I would like to understand it too.’
- Eugene P. Wigner
Nuclear data are physical parameters that describe the properties of the atomic
nuclei and the fundamental physical relationship that governs their interac-
tions [13]. These include atomic data, nuclear reaction data, thermal scatter-
ing, radioactive decay data and fission yields data.The data are important for
theoretical nuclear models development and for applications involving radia-
tion and nuclear technology [14]. Because of the wide variety of applications
nuclear data can be divided into three types [15]. The first being transport
data which describes the interactions of various projectiles such as neutrons
and protons with a target nucleus. Transport data are usually associated with,
e.g., cross sections and angular distributions [15]. These data are utilized for
both transport and depletion calculations. The second is fission yield data
which are, e.g., used for the calculation of waste disposal inventories and de-
cay heat, for depletion calculations and in the calculation of beta and gamma
ray spectra of fission product inventories [16]. The third is decay data which
describes, among others, the nuclear levels, half-lives, Q-values and decay
schemes [13, 15]. These data are used for, e.g., dosimetry calculations and for
estimating decay heat in nuclear repositories.
This chapter presents a description of experimental data which include both
differential and integral data; model calculations; nuclear data definitions; as
well as resonance parameters. Also, the nuclear data evaluation process, nu-
clear data libraries containing large sets of nuclear data, and covariance data
in the ENDF formatted libraries are presented.
2.1 Experimental dataExperimental data can be divided into differential and integral data. Differen-
tial data are microscopic quantities that describes the properties of nuclei and
their interactions with particles while integral data mostly take into account the
global behavior of a macroscopic system. Experimental data are needed for
fine tuning nuclear reaction models and for nuclear data assimilation. In the
following subsections, differential experimental data and integral benchmarks
used for nuclear data evaluation are presented.
5
2.1.1 Differential data
Microscopic quantities such as cross sections, fission yields and angular dis-
tributions are measured at a large number of experimental facilities, such as
accelerators, world-wide. Typically, these data are measured as a function of
the energy of an incoming particle, e.g., neutron. These data are collected,
compiled and stored in the EXFOR database (Exchange among nuclear re-
action data centers) [17]. The EXFOR database is maintained by the Interna-
tional Network of Nuclear Reaction Data Centres (NRDC) and coordinated by
the Nuclear Data Section of the International Atomic Energy Agency (IAEA).
The database contains experimental and bibliographic information on exper-
iments for neutron, charged particle and photon-induced reactions on a wide
range of isotopes and incident energies [18]. In Fig. 2.1, an example of dif-
ferential experimental data for the 208Pb(n,2n) cross section as a function of
incident neutron energy is presented. Usually, when there exist discrepan-
cies between similar experiments for the same reaction, comparisons of the
original publications containing the measurements is carried out. The mea-
surements are sometimes compared with theoretical model calculations [18].
Accurate experimental data are important for nuclear data evaluation and the
Figure 2.1. An example of experimental data for 208Pb(n,2n) cross section as a
function of incident neutron energy. The data were obtained from the EXFOR
database [17].
calibration of nuclear reaction model codes. For these reasons, a careful as-
sessment of possible systematics and statistical experimental uncertainties are
needed [19]. The sources of experimental uncertainties are presented in sec-
tion 3.2. Users of nuclear data, e.g., the nuclear reactor community, usually
give feedback which helps in prioritizing measurements of particular isotopes
and reactions.
6
2.1.2 Integral data (benchmarks)
Integral experiments are used to measure macroscopic quantities such as flux
or keff. Indeed, an integral experiment is not used for measuring microscopic
quantities such as cross-sections. However, integral data obtained from these
experiments, are used for testing and validating nuclear data libraries. An
example of extensive testing of nuclear data libraries with a large set of crit-
icality safety and shielding benchmarks is presented in Ref. [20]. Integral
benchmarks (only referred to as benchmarks in this thesis), are integral exper-
iments which have gone through a strict validation process. For example, the
ICSBEP Handbook contains models (normally in MCNP) of the integral ex-
periment with an evaluated benchmark value. The evaluated benchmark value
is the obtained experimental observable from the benchmark experiment (e.g.
keff) corrected for the simplifications done in the modeling of the experiment.
The evaluated benchmark value is also associated with an evaluated uncer-
tainty.
In Fig. 2.2, a diagram showing the cylindrical benchmark model for the hmf57
(case 3) benchmark is presented. From the diagram, the following can be
observed: the central cylinder made of HEU (yellow), surrounded by a lead
reflector (blue), with a source placed in the middle of the HEU core. The eval-
uated benchmark value of the hmf57 case 3, is keff = 1.0000±0.0032 pcm.
Figure 2.2. The cylindrical benchmark model for hmf57 (cases 3) showing the lead
reflector (in blue) the HEU fuel (yellow) and the source (red). The figure was taken
from the ICSBEP Handbook [21]. Note, the diagram is not to scale.
In the past, benchmark testing of nuclear data was done after the evaluation
process, however, in most modern evaluations the benchmark testing step is
carried out as an integral part of the evaluation process [22]. There are a
number of international efforts geared towards providing the nuclear commu-
nity with qualified benchmark data. One of such projects is the International
Criticality Safety Benchmark Evaluation Project (ICSBEP) mentioned above
7
which contains criticality safety benchmarks derived from experiments that
were performed at various nuclear critical facilities around the world [21].
These benchmarks are categorized according to their fissile media (Plutonium,
HEU, LEU, etc.), their physical form (metal, compound, solution etc.), their
neutron energy spectrum (thermal, intermediate, fast and mixed spectra) and a
three digit reference number. Other benchmarks used for nuclear data and re-
actor applications are the Evaluated Reactor Physics Benchmark Experiments
(IRPHE) [23] which contains a set of reactor physics-related integral data and
the Radiation shielding experiments database (SINBAD) [24] which contains
a compilation of reactor shielding benchmarks, fusion neutronics and accel-
erator shielding experiments. For existing reactor technology, these bench-
marks can be used to validate computer codes, test and validate nuclear data
libraries, nuclear data adjustments and uncertainty reduction [21, 25]. In this
work, benchmark data are used for nuclear data testing, data adjustment and
nuclear data uncertainty reduction.
2.2 Model calculationsModern nuclear data evaluation in the fast region is performed using nuclear
reaction models [26]. These models are used for providing data where experi-
mental data are scarce or unavailable [27]. The use of nuclear reaction models
also has an added advantage that various partial cross sections can be automat-
ically summed up to the total cross section. This leads to internal consistency
within evaluated files [27, 28]. Where experimental data are available, they
are used for constraining and fine-tuning model parameters used as inputs to
nuclear model codes.
Nuclear reaction theories have been implemented in well-known nuclear re-
action codes such as TALYS [26, 29] and EMPIRE [30] for theoretical model
calculations. In this work nuclear model calculations have been performed
using the TALYS code. More details on the TALYS code can be found in
section 4.1.1.
2.3 Nuclear data evaluationSeveral approaches exist for nuclear data evaluation. These methods include:
experimental data interpolation, Bayesian methods, re-normalization of exist-
ing evaluations, copy and paste from other nuclear data evaluations and nu-
clear reaction modeling [31].
The basic steps involved in nuclear data evaluation process involving nuclear
reaction modeling are:
1. The selection and careful analysis of differential experimental data mostly
obtained from the EXFOR database.
8
2. Analysis of the low energy region which includes thermal energy, and
the resolved and unresolved resonance regions. Analyses of neutron res-
onances are usually performed using R-matrix codes for light nucleus
reactions and for heavy nuclides in the low incident energy regions [32].
These codes are complemented with data from the Atlas of neutron reso-
nances [33] which contains a compilation of resonance parameters, ther-
mal neutron cross sections, and other quantities. In Ref. [33], the reso-
nance parameters are analyzed in terms of the multilevel Breit-Wigner
(MLBW) formalism. In new evaluations, however, the use of the Reich-
Moore approximation is encouraged [34].
3. Theoretical model calculations using nuclear reaction codes such as EM-
PIRE [30] or the TALYS code [26, 29] for the fast energy region. Where
available, parameters and their uncertainties as recommended in the Ref-
erence Input Parameter Library (RIPL) [27] are used as inputs for these
nuclear reaction codes. The RIPL database contains Reference Input Pa-
rameters for nuclear model calculations. In the fast region, the Hauser-
Feshbach, pre-equilibrium and fission models, are used for modeling
medium and heavy nucleus reactions [22, 32]. Here model parameters
in these codes are adjusted and fine tuned to fit selected experimental
data. These model adjustments are important because current models
are deficient and hence are not able to reproduce available experimental
data. Also, even if models do reproduce differential experimental data,
final adjustments are sometimes needed to reproduce integral data. By
adding resonance information as explained in (2) above, and other quan-
tities such as the average number of fission neutrons, fission neutron
spectra, (n,f) cross section, for actinides, a complete ENDF file which
covers from thermal to high energies can be produced.
4. The data produced is then checked and tested using utility codes such as
CHECKR, FIZCON and PSYCHE [35] to verify that the data conforms
to current formats and procedures. The data once checked, are then
validated against a large set of integral benchmark experiments. The end
product is a nuclear data library which contains information on a large
number of incident particles for a large set of isotopes and materials.
Feedback from model adjustments to fit both differential and integral
data and from applications are needed for the improvement of theoretical
models and for identifying energy regions where additional experimental
efforts are needed.
2.4 Nuclear data libraries
After successful evaluation, nuclear data are compiled and stored in nuclear
data libraries. The ENDF format is the accepted format for data storage. The
9
content of an evaluated nuclear data file include the following: general infor-
mation such as the number of fission neutrons (nubar) and delayed neutron
data; resonance parameters; reaction cross sections; decay schemes; fission
neutron multiplicities and nuclear data covariance information which are rep-
resented by the so-called MF-numbers. These MF-numbered files are sub-
divided into different reaction channels such as (n,tot), (n,el), (n,inl), which
are represented by MT-numbers [34]. There are some national and interna-
tional evaluated nuclear data libraries currently available from various nu-
clear data centers. These are, the Japanese Evaluated Nuclear Data Library
(JENDL) [36], Evaluated Nuclear Data Library (ENDF/B) [31] from the USA,
the TALYS Evaluated Nuclear Data Library (TENDL) [37], Joint Evaluated
Fission and Fusion File (JEFF) [38] from the OECD/NEA data bank, the Chi-
nese Nuclear Data Library (CENDL) [39] and the Russian Nuclear Evaluated
Data library (BROND) [40]. Other libraries for use in certain applications are
the International Reactor Dosimetry File (IRDF) for reactor dosimetry appli-
cations [41], the European Activation File (EAF) [42] and the Fusion Eval-
uated Nuclear Data Library (FENDL) from the International Atomic Energy
Commission (IAEA) [43].
2.5 Nuclear data definitions
The relationships between different reaction cross sections in the fast energy
region in the evaluated nuclear data files (ENDF) are presented in this section.
The total cross section (σtot) is given as a summation of the elastic scattering
cross section (σel) and non-elastic (σnon−el) cross section:
σtot(MT = 1) = σel(MT = 2)+σnon−el(MT = 3) (2.1)
where MT=1, 2 and 3 are the total, elastic and non-elastic cross sections in
ENDF nomenclature. The non-elastic cross section (σnon−el) which repre-
sents all other cross sections except the elastic scattering cross section can be
expressed as:
σnon−el(MT = 3) = σinl(MT = 4)+σ2n(MT = 16)+σ3n(MT = 17)
+σ f ission(MT = 18)+σnα(MT = 22)+σnp(MT = 28) (2.2)
+σγ(MT = 102)+σchargeparticle(MT = 103−107)
where σinl is the (n,inl), σ2n is the (n,2n), σ3n is the (n,3n), σ f ission is the (n,f),
σnα is the (n,nα), σnp is the (n,np), σγ is the (n,γ) and σchargeparticle represents
the cross sections of producing charge particles. The MT numbers as presented
in Eq. 2.2, are the corresponding MT numbers of the cross sections in the
ENDF nomenclature. The inelastic cross section (MT=4) is given as the sum
of the cross sections for the 1st-40th excited states. In the situation where the
10
elastic channel is fed by the compound nucleus decay besides the shape elastic
scattering, the elastic cross section is expressed as [26, 29]:
σel(MT = 2) = σshape−el +σcomp−el (2.3)
where σshape−el and σcomp−el are the shape elastic and compound elastic cross
sections. The shape elastic part comes from the optical model while the com-
pound elastic part comes from compound nucleus theory. The total fission
cross section (σ f ission) can be found on MT=18 in the ENDF formatted file
and is given as the summation of the first, second, third and fourth chance
fission cross sections:
σ f ission(MT = 18) = σn, f (MT = 19)+σn,n f (MT = 20)
+σn,2n f (MT = 21)+σn,3n f (MT = 38) (2.4)
where σn, f is the first chance fission cross section and σn,n f , σn,2n f and σn,3n fare the second, third and fourth chance fission cross sections respectively. The
cross sections discussed above can be found on MF=3.
The angular distributions, which is made up of elastic and inelastic angular dis-
tributions, can be found on MF=4. The elastic angular distribution
[dσ el
dΩ
]has
two components - the direct and compound parts. The direct (shape-elastic)
part comes directly from the optical model while the compound part comes
from compound nucleus theory [26, 29]:
dσ el
dΩ=
dσ shape−el
dΩ+
dσ comp−el
dΩ(2.5)
Similarly, the inelastic angular distribution to a single discrete state i
[dσ i
n,n′
dΩ
]can be given as the addition of the direct and compound parts [26, 29]:
dσ in,n′
dΩ=
dσ i,directn,n′
dΩ+
dσ i,compoundn,n′
dΩ(2.6)
2.6 Resonance parameters
The resolved and unresolved resonance parameters are found in File 2 (MF2-
MT151) in the evaluated nuclear data file. These parameters include Eλ , Γ,
Γn, Γγ , and Γ f , where Eλ is the resonance energy, Γ,Γn,Γγ ,Γ f are the total,
neutron, radiative and fission widths respectively. These parameters can not
be predicted from theory but can be determine from experiments. In Fig. 2.3,
11
E
0
Figure 2.3. An example of 239Pu (n,γ) cross section at Eλ = 35.5 eV. The plot was
generated from the ENDF/B-VII.1 nuclear data library using the JANIS nuclear data
tool [44].
an example of a single resonance of 239Pu capture cross section at the reso-
nance energy of Eλ = 35.5 eV is presented. The plot was generated from the
ENDF/B-VII.1 nuclear data library using the JANIS nuclear data tool [44].
Where σ0 is the cross section for the formation of the compound nucleus,
σ0Γγ
Γis the height of the resonance peak for the (n,γ) cross section,
Γγ
Γis the
probability of decay via γ-emissions capture and, Γ, the total width, given as
the sum of the partial widths is expressed as:
Γ = Γn +Γγ +Γ f (2.7)
The resonance parameters are needed for the calculation of cross sections at
reactor operating temperatures due to Doppler broadening of neutron reso-
nances.
2.7 Covariance data
Covariance data specifies nuclear data uncertainties and their correlations.
These data are needed for the assessment of uncertainties of design and safety
parameters in nuclear technology applications [22]. Several methods are avail-
able for the computation of nuclear data covariances. These methods can be
classified under three categories: deterministic, Monte Carlo, and Hybrid ap-
proaches [45]. In this work, the Monte Carlo method was used to implicitly
determine the covariance of nuclear data.
Covariance data are usually stored in MF31-35 and MF40 in the evaluated
nuclear data file. Once these covariance data are available in a nuclear data
12
evaluated file, uncertainty propagation for current and advanced reactor appli-
cations can be performed as discussed in more detail in section 3.3.
13
3. Uncertainty Quantification
’It is in the admission of ignorance and the admission of uncertainty thatthere is a hope for the continuous motion of human beings in some direction
that doesn’t get confined, permanently blocked, as it has so many timesbefore in various periods in the history of man.’
- Richard P. Feynman
This chapter presents uncertainty quantification approaches made up of deter-
ministic and stochastic methods currently utilized for nuclear data sensitivity
and uncertainty analysis. Also, sources of uncertainties and a brief discussion
of statistical estimations used to make statistical deductions are discussed.
Uncertainty propagation involves the quantification of the effects of uncertain
input parameters on model outputs. This involves studying the impact of vari-
ability of the input parameters on model outputs. A closely related field known
as sensitivity analysis involves the evaluation of the impacts of the variation of
each input parameter in contributing to the output uncertainty. With sensitivity
analysis, model inputs that cause significant uncertainty in the output and the
relationships between the input and output parameters are identified. Fig. 3.1
illustrates the relation between the input, the model and the model output. If
Model
g = f(x1,x2,…,xn)
g(X)
x1
x2
xn
Figure 3.1. A flowchart showing the propagation of uncertainty in input parame-
ters through a model. The figure shows the input X = (x1,x2, ...,xn), the model
g = f (x1,x2, ...,xn) and the model output = g(X).
a model, g = f (x1,x2, ...,xn), then the uncertainties in the input parameters,
i.e., a collection of random variables X = (x1,x2, ...,xn), can be propagated
through the model to obtain a distribution of g(X). From the distribution, the
first four moments can be determined as presented later in section 3.2. g(X)could be the neutron cross sections - in which case, the input variable X are
14
the distributions of the nuclear reaction model parameters and the model rep-
resents the nuclear reaction models such as the optical and Hauser-Feshbach
models. g(X) could also represent any reactor quantity of interest such as the
keff; in this case, the input parameter X represents the distributions of the neu-
tron cross sections. The model represents the geometrical and physical model
of a neutron transport code such as presented in section 4.1.3.
In the TMC method discussed in more detailed in section 3.3.2, the model in-
cludes both the nuclear physics and the neutron transport codes, consequently
linking the reaction model parameter distributions, X, all the way to the macro-
scopic reactor parameter distribution, g(X), of interest.
3.1 Sources of uncertainties
As mention earlier, nuclear data evaluation benefits from both experimental
data and nuclear reaction modeling. Since the models have uncertain inputs,
they are normally calibrated using experimental data. However, since exper-
iments are not exact, measured quantities contain uncertainties. The errors
in measurements come from a variety of sources such as from the method of
measurement, errors due to the measuring instrument used and errors coming
from the person performing the experiment [46]. If an error is defined as the
difference between the measured and the true value, the uncertainty is defined
as the estimate of the magnitude of the error. Measurement errors are usu-
ally classified as random or systematic errors. Random errors are caused by
unknown and unpredictable changes in the experiment, e.g., detector noise.
These errors can be estimated by repeating a particular measurement under
the same conditions; the mean value of the measurements then becomes the
best estimate of the measured quantity. Systematic errors on the other hand are
reproducible inaccuracies that shift all measurements within an experiment in
a systematic way. An example of systematic uncertainty is the detector ef-
ficiency. These errors can be identified by using a more accurate measuring
instrument or by comparing a given result with a measurement of the same
quantity performed with a different method [47]. In many cases, the errors
can not be directly inferred from measurements but their magnitude is rather
inferred from uncertainty propagation.
Similar to experimental uncertainties, the sources of uncertainties in modeling
can generally be classified into two categories:
1. Aleatory or random uncertainty, which arises from the random nature of
the system under investigation or by the random process in the model-
ing, e.g., Monte Carlo modeling.
15
2. The epistemic uncertainty [46], which results in a systematic shift be-
tween the modeled response parameter and the response parameters true
value eg. model defect.
In Reactor Physics, a response parameter can be defined as any quantity of
interest such as the keff or reaction rates, that can be measured or predicted in
a simulation [48]. Normally, epistemic uncertainties can arise from one of the
following sources [46, 10]:
1. From the data and the parameters used in the model, e.g., cross sections,
boundary conditions.
2. Uncertainties in the physics of modeled processes as a result of ’incom-
plete knowledge’.
3. Assumptions and simplifications of the methods used for solving model
equations.
4. From uncertainties arising from our inability to model complex geome-
tries accurately.
3.2 Statistical estimationSince evaluated nuclear data are obtained from models in combination with
experiments which come with their corresponding uncertainties, statistical de-
ductions are used to extract valuable information. Statistical information is
normally provided by moments of the respective probability distributions [47].
The mean is commonly used to describe the best estimate of a parameter in a
normal distribution while the spread is best described using the standard devi-
ation and the variance. The uncertainties in physical observations are usually
interpreted as standard deviations.
If we consider a collection of random variables X = (x1, ...,xn) and g is a func-
tion of X given as g = f (x1,x2,x3, ...,xn), then the expectation of g(X), which
is the first central moment, denoted by E(g(X)) can be defined as [47]:
E(g(X))≡∫SX
g(X)p(X)dX (3.1)
where p(X) is the probability density function and SX is the n-dimensional
space formed by all the possible values of X .
The second central moment, the variance of g(X), is the dispersion of a ran-
dom variable around its mean, and can be expressed as:
V (g(X))≡ E[(g(X)−E(g(X)))2
](3.2)
From Eq. 3.2, the standard deviation which is the positive square root of the
variance denoted by σ can be expressed as:
σ ≡ [V (g(X))]1/2 ≡ [E[(g(X)−E(g(X)))2
]]1/2(3.3)
16
Other higher moments are the skewness and the kurtosis. The skewness or the
third central moment, S(g(X)), is a measure of the degree of asymmetry of a
particular distribution and can be given as:
S(g(X))≡E([g(X)−E(g(X))]3
)σ(g(X))3
(3.4)
A perfectly symmetric (Gaussian) distribution has a skewness value of zero
while a negative skewness value implies a long tail of data more to the left of
the mean. A positive skewness value indicates a long tail of data towards the
right of the mean. The kurtosis which is the fourth central moment, can be
given as:
K(g(X))≡E([g(X)−E(g(X))]4
)σ(g(X))4
(3.5)
The kurtosis measures the ’peakedness’ of a probability distribution of ran-
dom variables. It determines whether the data are peaked or flat, relative to a
normal distribution.
For a multivariate probability distribution X = (x1,x2...,xn), the second or-
der central moments comprise not only the variances but also the covariance.
Covariance can be defined as a measure of how much two random variables
change together; it measures the strength of the correlation between two vari-
ables. The covariances between different input variables can be described by
the so-called covariance matrix. Covariance matrices are symmetric compris-
ing of off-diagonal covariances and variances along the diagonal. The sample
covariance of N observations between two random variables x j and xk, can be
expressed as:
cov(x j,xk) =1
N −1
N
∑i=1
(xi j − x j)(xik − xk) (3.6)
where x j and xk are the mean values of the x j and xk variables respectively.
If all the variables in X are uncorrelated the covariance matrix is made up of
only the diagonal elements.
Using Eq. 3.6, the correlation coefficient (ρx jxk ), which is the measure of
strength of the linear relationship between two variables can be given as:
ρx jxk =cov(x j,xk)
σx j σxk
(3.7)
where σx j and σxk are the standard deviation of x j and xk respectively. A
perfect negative correlation is represented by the value -1, while a 0 indicates
no correlation and a +1 indicates a perfect positive correlation.
17
3.3 Uncertainty quantification approachesMethods for nuclear data uncertainty quantification in best-estimate model
predictions are usually either stochastic such as the TMC method used in
this work or deterministic such as the perturbation approach utilized in, e.g.,
Ref. [49]. There are recently also hybrid methods that combine the strengths
of the Monte Carlo and the deterministic approaches for uncertainty analy-
sis [45].
3.3.1 Deterministic methods
With the deterministic approach, local sensitivities of particular response pa-
rameters such as keff to variations in input parameters (nuclear data in our
case), can be determined by using the generalized perturbation theory, e.g., [49].
The local sensitivities of the model’s response to input parameter variation are
accomplished by computing the response with input parameter values per-
turbed usually within 1% from the nominal parameter values, i.e., by varying
x1,x2, ...,xn, individually by 1%, the response of g(x) is computed. The sensi-
tivity vector (Sx = (Sx1, ...,Sxn)), can be computed using:
Sx1=
�g(X)
�x1(3.8)
The variable, X, is typically comprised of cross sections in multi-group for-
mat. This method relies on the assumption that the sensitivity of the output
parameter depends linearly on the variation in each input parameter [10].
Given that Vx denotes the covariance matrix for the parameters (x1, ...,xn), then
by using the ’sandwich rule’, the variance of the response can be expressed
as [46]:
var(g(X)) = SxVxSTx (3.9)
where the superscript ’T’ indicates a transposed matrix and the covariance
matrix (Vx) contains both diagonal and off-diagonal elements for correlated
parameters. In the case where the parameters are uncorrelated, Eq. 3.9 be-
comes:
var(g(X)) =n
∑i=1
S2xi
var(xi) (3.10)
This approach has been extensively used to evaluate the impact of neutron
cross-section uncertainties on some significant reactor response parameters
related to the reactor fuel cycle [49].
3.3.2 Stochastic methods
The focus in this thesis is on Monte Carlo or stochastic methods used for
nuclear data uncertainty propagation. These methods are based on random
18
sampling of input parameters, X. For each random set, m, of input parame-
ters, Xm = (x1,m, ...,xn,m) a set of output responses, g(Xm) of interest are pro-
duced [10]. Stochastic methods used for nuclear data uncertainty propagation
can generally be classified into two types:
1. Propagation of nuclear data uncertainties from basic nuclear physics.
2. Uncertainty propagation from existing covariance information that come
with modern nuclear data evaluations
Among (1), the Total Monte Carlo (TMC) method is used in this work and
described in detail in the next subsection.
Total Monte Carlo (TMC)The TMC methodology (here called ’original TMC’) was first proposed by
Koning and Rochman in 2008 [50] for nuclear data uncertainty propagation.
With the method, inputs to a nuclear reaction code such as TALYS [26, 29],
are created after sampling from nuclear model parameter distributions to cre-
ate random nuclear data files [26]. Experimental data and their uncertainties
are accounted for by using an accept/reject approach where calculations that
fall within an acceptance band determined from comparison with experimen-
tal data, are accepted and those that do not fulfil this criterion are rejected.
The acceptance band was obtained by visual evaluation methods [51]. This
approach has been criticised for not including experimental data in a more rig-
orous way. There are, however, on-going work with a goal of incorporating
both differential [51, 52] and integral experiments (PAPER IV) into the TMC
methodology in a more rigorous way. The inclusion of integral experimental
data in a rigorous way is also one of the main objectives of this thesis and is
described in detail in section 4.5 and in PAPER IV.
To create a complete ENDF file covering from thermal to fast neutron ener-
gies, non-TALYS data such as the neutron resonance data, total (n,tot), elastic
(n,el), capture (n,γ) or fission (n,f) cross sections at low neutron energies, av-
erage number of fission neutrons, and fission neutron spectra (for fissionable
nuclei) are added to the results obtained from the TALYS code using other
auxiliary codes [26].
A summary of the TMC method is depicted in a flow chart in Fig. 3.2. From
the figure, parameters in both phenomenological and microscopic models im-
plemented in nuclear reaction codes such as the optical model, pre-equilibrium,
compound nucleus models are adjusted to reproduce experimental data. These
parameters could be the real central radius (rv) or the real central diffuseness
(av) of the optical model for example. The output of the codes such as cross
sections, fission yields and angular distributions are compared with differen-
tial experimental data by defining an uncertainty band which covers most of
19
Compare with Experimental
data
model parameters
Physical models
A large set of accepted random ENDF files
Applications:Reactor calculations;Depletion studies,Transient analysis
Observables: cross sections, fission yields, angular distributions
Simulations
1.006 1.008 1.01 1.012 1.014 1.0160
5
10
15
20
25
keff values
Num
ber
of c
ount
s/bi
n
01 1 012 1
obs
Figure 3.2. A flowchart depicting the Total Monte Carlo approach for uncertainty
analysis. Random files generated using the TALYS based code system [26] are pro-
cessed and used to propagate nuclear data uncertainties in reactor calculations.
the experimental data available. Data that falls within this uncertainty band are
accepted while those that do not fulfil this criterion are rejected. The accepted
files are then processed into the ENDF format using the TEFAL code [53].
These ENDF formatted files are translated into usable formats and fed into
neutron transport codes to obtain distributions in reactor parameters of inter-
est. From these distributions, statistical information such as the moments pre-
sented in subsection 3.2 can be inferred.
In Fig. 3.3, the (n,el) and (n,γ) of 50 random 208Pb files are plotted as a func-
tion of incident neutron energy. A spread in data can be observed for the
entire energy region as presented in Fig. 3.3. This is expected as each file con-
tains a unique set of nuclear data obtained from the distribution of the nuclear
model parameters. Due to the variation of nuclear data, different distributions
with their corresponding mean values and standard deviations can be obtained
for different response parameters such as keff, temperature feedback coeffi-
cients and kinetic parameters [54]. By varying nuclear data using the TMC
method for a particular response parameter, the observed total variance of the
response parameter (σ 2obs) in the case of Monte Carlo neutron transport codes,
e.g., MCNP or SERPENT, can be expressed as:
σ2obs = σ2
ND +σ2stat (3.11)
where σ2ND is the variance of the response parameter under study due to nu-
clear data uncertainties and, σ2stat is the variance due to statistics from the
Monte Carlo code. In the case of deterministic codes, there are no statistical
20
10−2 10−1 100 101102
103
104
105208Pb (n,el)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
10−2 10−1 100 101
10−2
100
102
104
208Pb (n,γ)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
Figure 3.3. 50 random 208Pb cross sections plotted as a function of incident neutron
energy. Left: 208Pb(n,el) and right: 208Pb(n,γ). Note that each random file contains a
unique set of nuclear data.
uncertainties and Eq. 3.11 becomes:
σ2obs = σ2
ND (3.12)
With the ’original TMC’ described above, the time taken for a single calcula-
tion is increased by a factor of n, where n (the number of samples or random
files) ≥ 500 making it not suitable for some applications. As a solution, a faster
method called the ’Fast TMC’ was developed [55]. By changing the seed of
the random number generator within the Monte Carlo code and changing nu-
clear data at the same time, a spread in the data that is due to both statistics
and nuclear data is obtained (same as in Eq. 3.11). However, by using different
seeds for each simulation, a more accurate estimate of the spread due to statis-
tics is obtained and, therefore, the statistical requirement on each run could be
lowered, thereby reducing the computational time involved for each calcula-
tion. The usual rule of the thumb used for original TMC is: σstat � 0.05σobs.
However, for fast TMC, σstat � 0.5σobs [55]. A detailed presentation of fast
TMC methodology is found in Refs. [55, 56, 57]. In this work, the fast TMC
was used and from here on, only referred to as the TMC method.
Other Monte Carlo methods from basic nuclear physicsOther Monte Carlo uncertainty propagation methods from basic nuclear physics
include, the Unified Monte Carlo (UMC) approach proposed by D.L. Smith [58]
and the Backward-Forward Monte Carlo (BFMC) proposed in Ref. [59]. The
UMC method is based on the applications of Bayes theorem and the principle
of maximum entropy as well as on fundamental definitions from probability
theory [58]. The method seeks to incorporate experimental data into model
calculations in a more rigorous and consistent manner [45].
The Backward-Forward Monte Carlo (BFMC) method involves the Backward
21
and the Forward Monte Carlo steps. In the Backward step, a covariance matrix
of model parameters is obtained by using a generalized χ2 with differential
data as constrains leading to observables consistent with experimental data.
The generalized χ2 is used to quantify the likelihood of model calculation
with respect to a set of experimental constraints. Starting from the covariance
matrix of model parameters obtained from the backward step, a sampling of
the Backward Monte Carlo parameter distribution is performed. The result-
ing distribution of ND observables hence includes experimental uncertainty
information [45].
Sampling from the evaluated co-variance matrixSeveral other approaches are based on Monte Carlo sampling of nuclear data
inputs based on covariance information that come with new nuclear data eval-
uations. The disadvantage of these approaches is that they rely on the assump-
tion of normal distributions of the input parameters. The covariance data avail-
able are usually not comprehensive and complete too [26]. One such method
has been implemented in the AREVA GmbH code NUDUNA (NUclear Data
UNcertainty Analysis) [60]. With this method, nuclear input parameters are
first randomly sampled according to a multivariate distribution model based
on covariance data. A large set of random data is generated and used for
the computation of different response parameters. This method can include
uncertainties of multiplicities, resonance parameters, fast neutron cross sec-
tions and angular distributions. Another method is the GRS (Gesellschaft für
Anlagen-und Reaktorsicherheit, Germany) method implemented in the SUSA
(Software for Uncertainty and Sensitivity Analysis) code which depends on
randomly grouped cross sections generated from existing covariance files [61]
which are propagated to macroscopic reactor parameters.
Similarly, in Ref. [62], a stochastic sampling method for quantifying nuclear
data uncertainties is accomplished by utilizing perturbed ACE formatted nu-
clear data generated using multigroup nuclear data covariance information.
This approach has been implemented successfully in the NUSS tool [62]. In
another study, the SharkX tool [63] under development at the PSI, is used
in combination with the CASMO-5 code [64] for uncertainty quantification
and sensitivity analysis. Cross sections, fission spectrum, neutron multiplic-
ities, decay constants as well as fission yields are perturbed based on statis-
tical sampling methods. Also, uncertainty and sensitivity analyses applied to
lattice calculations using perturbed multigroup cross section libraries with the
DRAGON lattice code [65] have been presented in Refs. [66, 67]. In this work,
however, the random nuclear data were produced using the TMC method.
22
4. Methodology
’With the current and future computer technology, and the accumulatedamount of knowledge in the nuclear data community, we believe that this
methodology is technologically condemned to succeed.’- D. Rochman and A.J. Koning
In this chapter, the methodology used is presented. First, in section 4.1, the
simulation tools used in this work are briefly presented. In section 4.2, model
calculations performed with the TALYS code is presented. Methods for com-
puting uncertainties due to both global and partial (local) variations of nu-
clear data on neutronic parameters are discussed in 4.3. In section 4.4, the
description of the ELECTRA reactor together with the propagation of nuclear
data uncertainties for some macroscopic reactor parameters using the TMC
method is presented. Section 4.5 describes methods developed as part of this
work for nuclear data uncertainty reduction using integral experiments. This
includes an accept/reject method and a method of assigning file weights based
on the likelihood function. In addition, a methodology developed for select-
ing benchmark experiments for reactor simulations, is presented in section 4.6.
Also, in section 4.7, a correlation based sensitivity method is used to determine
the sensitivity of benchmarks and application cases to different cross sections
for particular isotopes and energy groups. Finally, a method for uncertainty
propagation of nuclear data uncertainties in burnup calculations as well as a
method for combining differential and integral experimental data for nuclear
data adjustments are explained in sections 4.8 and 4.9.
4.1 Simulation tools
The work in this thesis was done using a number of computer codes. These
codes include the TALYS based code system used for nuclear reaction cal-
culations and the production of random nuclear data files. The NJOY [68]
and PREPRO [69] codes for nuclear data processing, and MCNP5/X [8] and
SERPENT [70] codes for reactor core calculations. Brief descriptions of these
codes are presented in the subsequent subsections.
23
4.1.1 TALYS based code system
The TALYS based code system is made up of a group of codes coupled to-
gether with a script called AutoTALYS [26]. This code system is used for
nuclear data evaluation and the production of random nuclear data libraries.
An example is the TALYS Evaluated Nuclear Data Library (TENDL) [37]
which contains complete ENDF formatted nuclear data libraries including co-
variance matrices for many isotopes, particles, energies, reaction channels and
secondary quantities. The codes included in the TALYS based code system
are the TALYS code [26, 29], TASMAN [71], TAFIS [72], TANES [73],
TARES [74] and TEFAL [53] codes. These codes are sometimes referred
to as the T6 code package.
The TALYS code which forms the main basis for the TMC methodology is a
state of the art nuclear physics code used for the predictions and analysis of
nuclear reactions [26, 29]. In the TMC methodology, the TALYS code is used
to generate nuclear data for all open channels in the fast neutron energy re-
gion, i.e., beyond the resonance region. This is achieved by fine tuning model
parameters of various nuclear reaction models within the code so that model
calculations reproduce differential experimental data. In situations where ex-
perimental data are unavailable TALYS is used for the prediction and extrapo-
lation of data [26]. The output of TALYS include total, elastic, inelastic cross
sections, elastic and inelastic angular distributions and other reaction channels
such as (n,2n), (n,np). To create a complete ENDF file covering from ther-
mal to fast neutron energies, non-TALYS data such as the neutron resonance
data, cross sections at low neutron energies, average number of fission neu-
trons, and fission neutron spectra are added to the results obtained from the
TALYS code. This is achieved by using other auxiliary codes [26] such as
the TARES code [74] for resonance parameters, the TAFIS and TANES codes
for the average number of fission neutrons and the fission neutron spectrum
respectively [72, 73]. The TASMAN code [71], is used to create input files to
TALYS and the other codes by generating random distributions of input pa-
rameters by randomly sampling each input parameter from a distribution with
a specific width for each parameter.
The uncertainty distribution in nuclear model parameters is often assumed to
be either Gaussian or uniform shaped [26]. The different input files created
by the TASMAN code are then run multiple times with the TALYS code, each
time with a different set of model parameters, to obtain distributions in calcu-
lated quantities. From the distributions obtained, statistical information such
as the mean, standard deviations and variances and a full covariance matrix
which includes both diagonal and off-diagonal elements can be obtained. Fi-
nally, the TEFAL code is used to translate the nuclear reaction results obtained
from all the different modules within the T6 code package into ENDF format-
ted nuclear data libraries [53]. It must be noted that even though these codes
24
are coupled together, each code can work as a standalone simulation tool. Be-
cause of the huge amount of time involved in the production of these random
nuclear data files, most of the random files used in this work were obtained
from the TENDL project [75]. However, random 208Pb and 206Pb files were
produced as part of this work as further described in section 4.2.
4.1.2 Processing codes
Between the ENDF formatted evaluated nuclear data and the users of nuclear
data are a set of data-processing codes. These codes prepare nuclear data from
the ENDF format to a variety of usable formats used in application codes [51].
Even though these codes are often overlooked, the accuracy of transport and
reactor calculations depend to a large extent on the assumptions and approxi-
mations introduced by these processing codes [51]. One widely used code is
the NJOY nuclear data processing code [52] which is used to convert ENDF
format files into useful forms for practical applications. To reflect the tem-
peratures in real systems, for instance, energy dependent cross sections have
to be reconstructed from resonance parameters, and then Doppler broadened
to defined temperatures. In this subsection the processing of the nuclear data
using the NJOY and PREPRO codes are presented.
NJOY processing codeThe NJOY processing code [68] is used for preparing ENDF formatted nuclear
data into usable formats for use in deterministic and Monte Carlo transport
codes which are used for reactor calculations and analyses. In this work, the
NJOY code was used to process random ENDF files into the ACE format at
defined temperatures using the following module sequence:
MODER −→ RECONR −→ BROADR −→UNRESR−→ HEAT R −→ PURR −→ ACER (4.1)
The MODER module was used to convert ENDF input data into NJOY blocked
binary mode. These data were then reconstructed into pointwise cross sec-
tions which are Doppler broaden using the BROADR module. The UNRESR
module is used to calculate effective self-shielded pointwise cross sections in
the unresolved resonance region while the HEATR module is used to gener-
ate pointwise heat production and radiation damage production cross sections.
PURR is used to prepare unresolved region probability tables mostly used by
MCNP and finally the ACER module converts the libraries into ACE format.
PREPRO codeSimilar to the NJOY code, PREPRO is a collection of modular codes designed
to prepare data in the ENDF format into usable formats for applications [69].
25
In this work, the following module sequence was used:
LINEAR −→ RECENT −→ SIGMA1 −→ GROUPIE −→ FIXUP (4.2)
The LINEAR module was used to convert ENDF random nuclear data files
into a linear-linear interpolable form. The RECENT module was used to
reconstruct the resonance contributions into a linear interpolable form. The
SIGMA1 module was used to Doppler broaden cross sections to defined tem-
peratures for use in applications while the GROUPIE module can be used to
calculate self-shielded cross sections and for collapsing pointwise ENDF files
into multigroup cross sections. In this work, the GROUPIE module was used
to calculate multigroup cross sections for random ENDF formatted nuclear
data for use in cross section-parameter correlation calculations discussed in
section 4.7. The FIXUP module was used for testing data formats from the
different modules for consistency.
4.1.3 Neutron transport codes
Neutron transport codes are used to simulate the transport of neutrons in mate-
rials. Some of these codes, such as SERPENT and MCNP, can also calculate
the criticality. Two transport codes were used in this work: the SERPENT
code version 1.1.17 [70] was used for all simulations involving the ELECTRA
reactor. For the benchmark cases, criticality calculations were performed us-
ing the MCNPX code version 2.5 [8]. These codes are briefly presented in this
subsection.
SERPENT Monte Carlo codeThe 3-D continuous-energy Reactor Physics code Serpent (version 1.1.17) [70]
developed at VTT Technical Research Centre in Finland was used for simula-
tions in this work. Serpent is specialized in 2-D lattice physics calculations but
has the capability of modeling complicated 3-D geometries as well. It also has
a built-in burnup capability for reactor analyses. SERPENT uses the universe-
based geometrical modeling for describing two or three dimensional fuel and
reactor core configurations [70].
SERPENT utilizes following types of data for simulations:
1. Continuous energy neutron data which are made up of reaction cross sec-
tions, energy and angular distributions, fission yields and delayed neu-
tron data used for transport calculations. For this thesis, the uncertainties
in these quantities are considered.
2. Thermal scattering data for moderator materials such as hydrogen bound
in light water (H2O), deuterium bound in heavy water (D2O) and carbon
in graphite, which are used to account for the chemical binding of the
26
moderator nuclei [70]. This data was not used in this work since only
fast systems were considered.
3. For burnup calculations, radioactive decay and fission yield data are used
in combination with data from (1). These data were only used in PAPERIII, and were taken from JEFF 3.1 nuclear data library. The uncertainties
in these data were not considered.
The SERPENT code was used in this work because the geometry input file for
the ELECTRA reactor was created using the SERPENT code. Furthermore,
SERPENT is an open source code and therefore access to the nuclear data
inputs used in the code was straight forward. The code is also well validated
with an in-built burnup capability.
MCNPX codeMCNPX, which stands for Monte Carlo N-Particle eXtended, is a continuous-
energy, general-purpose Monte Carlo radiation transport code used for track-
ing many particle types over a broad range of energies [8]. MCNPX is fully
three-dimensional. MCNPX was used for simulations for all the criticality
benchmarks used in this work. The MCNP code was used because it is a
robust and widely used code. Furthermore, the benchmarks obtained from
Ref. [21] came with corresponding MCNP geometry input files for criticality
calculations.
4.2 Model calculations with TALYS
A successful TMC uncertainty propagation relies on realistic central values
from the output of the TALYS based code system. These central values are
complete evaluations with their corresponding ’best’ input parameter sets. The
selection of these ’central values’ is guided by high-quality experimental data,
complemented by the nuclear model codes such as TALYS and the experience
of the evaluator [28, 75]. Once the central value is obtained, an uncertainty
band is defined for the model parameters using experimental data as a guide.
In Table 4.1, typical uncertainties of some of the model parameters for 208Pb
used in this work, are given as a fraction (%) of their absolute values.
The central value used in this work was provided by the TENDL team. The
newer versions of TALYS come with a directory known as ’best’ which con-
tains the ’best’ set of model input parameters per nuclide. This directory
contains, for example, adjusted optical model and level density parameters
and, therefore, saves the inexperience user from conducting an entire evalua-
tion in the fast region. By adding best y to the TALYS input file, TALYS is
prompted to use the ’best’ parameter sets for calculation instead of the default
27
parameters. The uncertainties in Table 4.1 were derived taken differential ex-
perimental data into consideration. As mentioned earlier, this approach has
been criticized because experimental information was not included in a statis-
tically rigorous way. There is, however, on-going work with the objective of
incorporating experimental information including their correlations in a more
rigorous way [51, 76, 77]. The approach defined in [51] has been applied for
the production of nuclear data libraries in TENDL-2015 [78].
Table 4.1. Uncertainty of some nuclear model parameters of TALYS for 208Pb, givenas a fraction(%) of the absolute value. A complete list of all the model parameters canbe found in Ref. [26].
Parameter Uncertainty(%) Parameter Uncertainty(%)
rnV 1.5 an
V 2.0
vn1 1.9 vn
2 3.0
vn3 3.1 vn
4 5.0
wn1 9.7 wn
2 10.0
dn1 9.4 dn
2 10.0
dn3 9.4 rn
D 3.5
anD 4.0 rn
SO 9.7
anSO 10.0 vn
so1 5.0
vnso2 10.0 wn
so1 20.0
wnso2 20.0 Γγ 5.0
a(207Pb) 4.5 a(206Pb) 6.5
a(208Pb) 5.0 a(205Pb) 6.5
σ2 19.0 M2 21.0
gπ (207Pb) 6.5 gν (207Pb) 6.5
In this thesis, methods for the inclusion of integral experimental information
into the TMC methodology are presented. In PAPER IV, uncertainty reduc-
tion methods using benchmark information are presented. Also, a method for
combining differential and integral experimental data for nuclear data adjust-
ments is presented in sections 4.9 and 5.7.
A large set of random 208Pb and 206Pb nuclear data libraries were produced
using the TALYS based code system. The inputs to the TALYS based code
system were obtained by random sampling of nuclear model parameters from
a uniform distribution. The resonance parameters used were adopted from the
JEFF-3.1 library together with their corresponding background cross sections
found on MF3 using the TARES code. Uncertainties in the resonance region
used in this work are default uncertainties within the TARES code that repre-
sent the best effort of the code developer at the time [74]. As a final step, the
TEFAL code was used to translate the output from the TALYS and TARES
codes into the ENDF format. These files, after data checking, have been ac-
cepted by the TENDL team and can be obtained from TENDL-2014 [37]. The
28
random ENDF nuclear data produced were processed into the ACE format
using the NJOY processing code and used for reactor core calculations.
4.3 Nuclear data uncertainty calculation
In the following subsections, the application of the TMC methodology for
the computation of isotope specific global and reaction specific local (partial
TMC) uncertainties due to nuclear data is presented. For the variation of nu-
clear data, the one-at-a-time approach [79] was used. This was done by vary-
ing one isotope at a time while keeping all other isotopes as either JEFF-3.1
or ENDF/B-VII.0 nuclear data libraries. The one-isotope-at-a-time was used
because the goal is to quantify and identify which nuclear data inputs that have
significant impact on the uncertainty of the output.
4.3.1 Global uncertainty analyses
The TMC approach was used to assess the global effect of nuclear data un-
certainty (per isotope) on reactor parameters as discussed later in section 4.4.
Random files in ENDF format, produced using the TALYS based code system
are processed into the ACE format with the NJOY processing code for reactor
calculations. Each random nuclear data file contains a unique set of resonance
parameters (MF2), reaction cross-sections (MF3), angular and energy distri-
butions (MF4 and MF5 respectively), double differential distributions (MF6),
among others. A bash script is used to run the SERPENT code multiple times,
each time with a different nuclear data input file and distributions in any reac-
tor quantity of interest can be obtained. The uncertainties inferred from these
distributions are therefore due to the global variation of nuclear data which
comes from the variation in model parameters.
In Fig. 4.1, a diagram showing the practical implementation of the Total Monte
Carlo method is presented. As can be seen in the diagram, the process can be
divided into three major sections:
1. Model calculations using the TALYS based code system which as men-
tioned earlier, involves comparing model calculations with differential
experimental data to obtain a specific uncertainty for each nuclear model
parameter. Subsequently, by running the TALYS based code system a
large number of times, each time with a different unique set of model
parameters, a set of random nuclear data files are obtained.
2. The nuclear data files generated from (1) are processed into usable for-
mats for neutron transport codes using the NJOY code. A bash script
29
Output from model calculations by TALYS based system
Nuclear data processing
Reactor core simulations
A large set of random ENDF files
(TENDL project)
Rand n
Rand 2
Rand 1
NJOY
ACE n
ACE 2
ACE 1
SIM 2
SIM 1
SERPENT
JEFF-3.1 Library
SIM n
T6 code package
+ EXFOR
+ Resonance parameters
Feedback to model calculations
Figure 4.1. Practical implementation of the TMC methodology for nuclear data un-
certainty propagation. The method is composed of model calculations, nuclear data
processing, reactor core simulations and statistical analyses.
is used to handle the interaction between the SERPENT code, the ran-
dom nuclear data files database and the base library. The base library is
the library used for the isotopes not under investigation. In the case of
the serpent calculations of ELECTRA, JEFF-3.1 was used as the base
library while in the case of criticality benchmarks, ENDF/B-VII.0 was
used.
3. Reactor calculations are performed using the processed random files
from (2) to obtained distributions in any macroscopic reactor parame-
ter of interest. Statistical information are inferred from the probability
distributions for the reactor parameters under consideration using statis-
tical deductions. The information obtained are important, not only for
uncertainty and sensitivity analysis, but can be used to prioritize which
cross section data should be focused on for improvement of both model
calculations and experiments as shown in the feedback loop in Fig. 4.1.
The outputs from the different codes are handled automatically by a suite
of bash scripts.
As seen from the figure, uncertainties from basic nuclear physics are propa-
gated all the way to reactor parameters. Nuclear data uncertainties on some
neutronic parameters are presented in more detail in Paper I, II and IV.
30
4.3.2 Local uncertainty analyses - Partial TMC
In section 4.3.1, the methods for computing isotope specific global uncertain-
ties due to nuclear data for some reactor safety parameters were presented.
However, for the purpose of giving informed feedback to model calculations
and differential measurements, it is of interest to quantify the contributions of
different partial cross sections and parts of the ENDF file to the global uncer-
tainties obtained. To achieve this goal, perturbed random files were produced
by varying specific parts of the ENDF file while keeping other parts constant
for a large number of random nuclear data. To investigate the impact of only
resonance parameters on reactor parameters, for example, only MF2 was per-
turbed. This implies that each complete ENDF file after perturbation con-
tains a unique set of resonance parameters such as the scattering radius, the
average level spacing and the average reduced neutron width. For practical
implementation, the first file or central file (i.e., random file zero files pro-
duced or obtained from the TENDL-2012 [75]) was kept as the unperturbed
file while different sections of the random ENDF files are perturbed and a
unique set of random files produced. By using this method of partial variation
(also called ’partial TMC’), the impact of uncertainties due to angular (MF4)
and energy (MF5) distributions, and double-differential (MF6) distributions
which are normally difficult to propagate using other uncertainty quantifica-
tion methods, can be quantified.
In Fig. 4.2, perturbed random ACE 208Pb cross sections are plotted as a func-
tion of incident neutron energy. In the top left and top right, the (n,el) and
(n,γ) cross sections are presented respectively, after perturbing only resonance
parameter data. As can be observed, the partial variation of only resonance pa-
rameters, affect both 208Pb(n,el) (top left) and 208Pb(n,γ) (top right) cross sec-
tions from thermal up to about 5 MeV and 1 MeV for the (n,el) and the (n,γ)
cross sections respectively. In the bottom left and bottom right of Fig. 4.2, the208Pb(n,el) and 208Pb(n,γ) are presented for the partial variation of the (n,el)
cross section in the fast energy range respectively. A spread is observed in the
fast region for the partial variation of 208Pb(n,el) cross section (bottom left) as
can be observed from Fig. 4.2. Since the results in the fast energy region were
obtained with the TALYS code, the spread can be attributed to the variation
of model parameters within the TALYS code. The lack of spread observed for
the (n,γ) is expected as the variation of the (n,el) cross section has no impact
on the (n,γ) cross section.
All the perturbed random files were processed into ACE files with the NJOY
processing code at 600 K and used in the SERPENT code for reactor core
calculations to obtain distributions in any reactor quantity of interest. The
variance of the response parameter (reactor quantity of interest) due to the
partial variation (σ2(n,xn),obs) can be expressed as:
σ2(n,xn),obs = σ2
(n,xn),ND +σ2stat (4.3)
31
where σ2(n,xn),obs is the variance of the observable due to partial variation, σ2
stat
is the mean value of the variance due to statistics and σ2(n,xn),ND is the variance
due to nuclear data as a result of partial variation and (n,xn) = (n,γ), (n,el),(n, inl), (n,2n), resonance parameters or angular distributions. In this way,
nuclear data uncertainties due to specific reaction channels or specific parts of
the ENDF files were studied and quantified. A detailed presentation of this
method and its application to lead isotopes has been presented in PAPER II.
In principle, the impact of energy groups on macroscopic reactor parameters
for different cross sections can be studied.
10−2 10−1 100 101102
103
104
105208Pb, MF2, (n,el)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
10−2 10−1 100 101
10−4
10−2
100
102
104
208Pb, MF2, (n,γ)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
10−2 10−1 100 101103
104
105208Pb, MF3−MT2, (n,el)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
10−2 10−1 100 101
10−4
10−2
100
102
104
208Pb, MF3−MT2, (n,γ)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
Figure 4.2. Random ACE 208Pb cross sections are plotted as a function of incident
neutron energy for varying only resonance parameter data (2 top plots) and then only
elastic scattering cross sections (2 bottom plots). For the top left: 208Pb(n,el) and top
right: 208Pb(n,γ), only MF2 (resonance parameters) were varied while for bottom left:208Pb(n,el) and bottom right: 208Pb(n,γ), only the elastic scattering cross sections in
the fast energy range were varied.
4.4 Reactor Physics
In following subsections, the description of the ELECTRA reactor is presented
together with the propagation of nuclear data uncertainties for some macro-
scopic reactor parameters using the TMC method. The impact of nuclear data
32
uncertainties on the multiplication factor and the coolant temperature coeffi-
cient is presented.
4.4.1 Reactor description
The ELECTRA - European Lead-Cooled Training Reactor, is a conceptual
0.5 MW lead cooled reactor fuelled with (Pu,Zr)N with an estimated average
neutron flux at beginning of life (BOL) of 6.3×1013n/cm2s [6]. The fuel com-
position was chosen such that the Pu vector resembles a typical spent fuel of
a pressurized water reactor UOX fuel with a burnup of 43 GWd/tonne, which
was allowed to cool for four years before reprocessing with an additional two
years storage before loading into the ELECTRA core.
Figure 4.3. Radial view of the ELECTRA core showing the hexagonal fuel assembly
in the center made up of 397 fuel rods, the lead coolant (pink), the control assembly
showing the six rotating control drums around the fuel assembly with the control rods
fully inserted.
The extra storage time after reprocessing gives the initial fuel vector realistic
levels of Am, which is a product from beta decay of 241Pu [6]. The fuel com-
position is made up of 60% mol of ZrN and 40% mol of PuN. ELECTRA is
cooled by pure lead. Fig. 4.3 shows the radial configurations of the ELECTRA
core. The objective is to achieve a 100 % heat removal via natural convection
while ensuring enough power density to keep the coolant in a liquid state. The
core is hexagonally shaped with an active core height of 30 cm and consists of
397 fuel rods. Reactivity compensation is achieved by the rotation of absorb-
ing drums made up of B4C enriched to 90% in 10B. [6]. Because of the hard
33
spectrum, ELECTRA has a relatively small negative Doppler constant, how-
ever, the presence of a large negative coolant temperature coefficient makes
temperature control a possible way of managing reactivity transients.
In Fig. 4.4, the neutron flux spectrum in the fuel as a function of neutron en-
ergy using the SERPENT code is presented. The neutron flux in the fuel was
estimated by defining a detector within the fuel material with user defined en-
ergy boundaries from 1e-5 to 20 MeV. As seen in the figure, the peak of the
spectrum occurs at about 700 keV. The relatively hard spectrum allows for an
efficient use of both the fissile and fertile isotopes within the ELECTRA core.
A detailed description of the reactor has been presented in Ref [6].
10−2 10−1 100 1010.000
0.005
0.010
0.015
0.020
0.025
Neutron energy (MeV)
Flux
per
leth
argy
Figure 4.4. Neutron flux per lethargy in the fuel against neutron energy. The flux was
normalized with the total flux. The peak of the spectrum occurs at about 700 keV.
4.4.2 Reactor neutronic parameters
The propagation of nuclear data uncertainties to reactor neutronic parameters
such as the keff and the coolant temperature coefficient is presented in this
subsection. Each quantity of interest is evaluated by simulating the reference
core or perturbed configurations and the criticality change was determined
while varying nuclear data for the isotope of interest. The JEFF-3.1 nuclear
data library was used as the base library. The reference temperature of the
fuel and coolant were 1200 K and 600 K respectively. More details have been
presented in PAPER II.
Neutron multiplication factor (keff)To quantify the uncertainty in keff due to uncertainties in nuclear data (ND),
ND for specific isotopes were varied while computing the keff each time,
and given probability distributions were obtained. From these distributions,
corresponding mean values and standard deviations were determined. Using
Eq. 3.11, the uncertainty due to nuclear data was extracted. In Paper II, the
34
results for the variation of 204,206,207,208Pb nuclear data for the keff and other
safety parameters are presented in more detail. More results on the nuclear
data uncertainty of some actinides and structural materials are presented in
PAPER I, IV and in Chapter 5.
Coolant Temperature Coefficient (CTC)The impact of nuclear data uncertainty on the CTC was quantified by per-
forming criticality calculations with the SERPENT Monte Carlo code (version
1.1.17) [80] at two different coolant densities corresponding to the tempera-
tures T1 = 600 K and T2 = 1800 K and varying lead nuclear data each time
while keeping all other isotopes as the JEFF-3.1 library. Since the density ef-
fect is dominant in the CTC, all lead cross sections used in the calculation of
the CTC were processed with the NJOY99.336 code at 600 K. Since the keff is
≈ 1 for both configurations, the CTC for a temperature change from T1 to T2
can be expressed as:
CTC =ke f f (T1)− ke f f (T2)
T1 −T2(4.4)
The nuclear data uncertainty in the CTC is propagated here similar to equa-
tion 3.11. If the statistical uncertainty on the keff at T1 and T2 are σstat,T1
and σstat,T2respectively, the combined statistical uncertainty (σstat,comb) for
the computation of CTC assuming that the statistical errors at T1 and T2 are
uncorrelated, was computed using:
σ2stat,comb = σ2
stat,T1+σ2
stat,T2(4.5)
From the square of the total spread (σobs) of the CTC distribution which is
equal to the quadratic sum of the nuclear data uncertainty (σND) and the com-
bined statistical uncertainty (σstat,comb), the uncertainty due to nuclear data can
be extracted as follows:
σND = [σ2obs −σ2
stat,comb]1/2 (4.6)
Since the difference between ke f f (T1) and ke f f (T2) is usually small, the CTCdistribution can easily be dominated by statistics and hence longer computer
time is needed in the Monte Carlo simulations to obtain statistical uncer-
tainty as small as possible; the usual rule of the thumb used for fast TMC
is: σstat � 0.5×σobs [55].
More information on uncertainty propagation for other reactor parameters
such as the coolant void worth and the delayed neutron fraction, can be found
in PAPER II.
4.4.3 Convergence of the first 4 moments
Convergence and consistency verifications of cross sections probability distri-
butions are important in the TMC method to ensure that the final keff converges
35
and that enough random nuclear data libraries are used. Therefore, the con-
vergence of the keff distributions obtained were determined by computing the
first four moments of the distribution as a function of the random sampling of
nuclear data. In Fig. 4.5, an illustration of the convergence of the keff obtained
in the case of varying 239Pu nuclear data is presented. From the figure, small
fluctuations can be observed in the convergence toward the final keff values
(top left), the associated standard deviation σ(keff) (top right), the skewness
(bottom left) and the kurtosis (bottom right) as a function of sample numbers.
50 100 150 200 2501.0000
1.0005
1.0010
1.0015
1.0020
Number of random files
Ave
rage
kef
f
239Pu
50 100 150 200 250150
200
250
300
350
400
Number of random files
σ k eff[p
cm]
239Pu
50 100 150 200 250−1.5
−1.0
−0.5
0.0
0.5
Number of random files
Ske
wne
ss
239Pu
50 100 150 200 2501
2
3
4
5
6
Number of random files
Kur
tosi
s
239Pu
Figure 4.5. An illustration of convergence for the keff in the case of varying 239Pu
nuclear data for the pmf5c1 (case 1) benchmark. The first four moments of the dis-
tribution are presented: mean (top left), the standard deviation σ(keff) (top right), the
skewness (bottom left) and the kurtosis (bottom right).
Another approach is to compute the uncertainty of the uncertainty in nuclear
data for each isotope and for each reactor response parameter as presented in
Ref. [56]. This approach has also been used in PAPERS II and IV.
4.5 Nuclear data uncertainty reduction
Even though information on differential measurements together with their un-
certainties are included (implicitly) in the production of random files in the
TMC methodology, wide spreads have been observed in parameter distribu-
tions (known here as our ’prior distribution’) leading to large uncertainties in
36
reactor parameters for some nuclides for the European Lead-Cooled Training
Reactor (PAPER I and II). To meet integral uncertainty requirements, the
present nuclear data uncertainties for reactor systems must be reduced signif-
icantly [7, 11]. This can be achieved by comparing the random files against
the results from integral experiments (benchmarks). Using benchmark infor-
mation as a constraint, weights are assigned to the nuclear data libraries de-
pending on their agreement with the benchmarks observable (e.g. keff). The
benchmark observable could vary depending on the type of benchmark. In
this work, benchmarks from the ICSBEP Handbook [21] and consequently
only the keff have been used as the benchmark observable. The weighted nu-
clear data libraries are used for a specific reactor system, to reduce the nuclear
data uncertainty in a reactor response parameter such as the keff. A binary
accept/reject method and a method of assigning file weights based on the like-
lihood function for nuclear data uncertainty reduction are presented in this
section, with more details in PAPER IV.
4.5.1 Binary Accept/Reject method
It was demonstrated earlier in PAPER I that, by setting a more stringent cri-
terion for accepting random files based on integral benchmark information,
nuclear data uncertainty could be reduced further. In PAPER I, however,
arbitrary limits were set on accepting random files using criticality bench-
marks without including the evaluated benchmark uncertainty information.
The method was improved by including benchmark uncertainty information in
the computation of an acceptance interval which is proportional to the bench-
mark uncertainty and presented in PAPER IV. In the paper, an acceptance in-
terval (FE) directly proportional to the combined benchmark uncertainty (σB, j)
was used:
FE ∝ σB, j (4.7)
The combined benchmark uncertainty is presented and discussed in more de-
tail in section 4.5.3. By introducing a proportionality constant κ which defines
the magnitude of the spread and given as the inverse of the Pearson correla-
tion coefficient (R) computed between the benchmark and the application case,
Eq. 4.7 becomes:
FE = κσB, j (4.8)
where κ is expressed as:
κ =1
|R| (4.9)
In theory, κ could have been set to 1 for all benchmarks. However, by letting
κ > 1, more conservative and in practice, less weights are given to bench-
marks with weak correlation to the application case. In Fig. 4.6, a correlation
plot example between the application case (ELECTRA) and the 239Pu Jezebel
37
benchmark is presented showing the evaluated benchmark keff value and the
corresponding acceptance band (FE). For the practical implementation of the
0.98 0.99 1 1.01 1.020.98
0.99
1
1.01
1.02
1.03
keff values for 239Pu Jezebel benchmark
k eff v
alue
s fo
r ELE
CTR
A FE
Figure 4.6. keff correlation plot between ELECTRA and the 239Pu Jezebel benchmark
showing the acceptance band (FE ). A correlation coefficient of R=0.84 was obtained,
signifying a strong linear relationship between the two systems.
binary accept/reject method, the following is considered: given that i denotes
the random files (random nuclear data) and keff(i), is the calculated bench-
mark keff for the random file i. Then the maximum value of keff(i) is kMaxeff =
kBeff,exp +FE and the minimum value, kMin
eff = kBeff,exp −FE , where kB
eff,exp is the
evaluated experimental benchmark value. Hence an acceptance range is de-
fined as kMineff ≤ keff(i) ≤ kMax
eff , and any random file i that falls within this range
is accepted and therefore assigned a binary value of one while those that do
not meet this criteria take binary values of zero and are therefore rejected.
Using only the accepted files for an application, a posterior distribution in a pa-
rameter of interest, e.g., keff can be obtained. The posterior distribution should
normally be narrower in spread than the prior distribution.
4.5.2 Reducing uncertainty using file weights
A more rigorous method is to base the uncertainty reduction on the likeli-
hood function using Bayesian theory. According to Baye’s rule, the posterior
distribution which is the distribution of parameters after taking into account
new data, is proportional to the likelihood function multiplied by the prior
distribution. The prior represents the distribution of parameters before the in-
clusion of new information [32]. Calibration of nuclear data using differential
information has been performed by many authors (Refs. [28, 52, 51, 81]). In
Ref. [52], file weights proportional to the likelihood function were assigned
38
to the TENDL random nuclear data files depending on how well they agreed
with differential cross-sections:
wi =e−
12 χ2
i
e−12 χ2
min(4.10)
where i is the random file number, wi is the weight for the random file i. Ex-
perimental uncertainties and their correlations were included by computing
a generalized χ2i which takes into consideration the differential experimental
covariance matrix and their correlations for the random file i. The method was
also used in this thesis, as discussed in section 4.9.2.
A similar approach is applied to nuclear data uncertainty reduction for reactor
safety parameters by introducing integral benchmark experimental informa-
tion as an additional constraint in the Total Monte Carlo chain. Using TENDL
random nuclear data libraries as our prior, file weights are assigned to each
random file depending on their quality with respect to a benchmark value us-
ing a modified likelihood function. Similar to the binary accept/reject case,
the correlation between the benchmark and the application case is taken into
account by introducing the Pearson correlation (R) as presented in Eq. 4.11:
wi, j =e−
12 χ2
i |R|
e−12 χ2
min|R|(4.11)
where χ2 is expressed by:
χ2i, j =
(kBeff(i)− kB
eff,exp)2
σ2B, j
(4.12)
where, kBeff(i) is the calculated benchmark value for the ith random file and
kBeff is the evaluated experimental benchmark value; σ2
B, j which is the com-
bined benchmark uncertainty for the benchmark B for the jth isotope whose
nuclear data uncertainty is being reduced is described in the next section in
Eq. 4.13. Introducing the correlation coefficient (R) into Eq. 4.11 is a com-
promise between acquiring good accuracy and still preserve random files in
the case where there is a weak correlation between the systems. After the
computation of the weights, the weighted moments of the distributions can be
calculated. More details on uncertainty reductions using file weights is pre-
sented in PAPER IV.
4.5.3 Combined benchmark uncertainty
The evaluated benchmark uncertainty normally contains information on uncer-
tainties in geometry, material compositions and experimental setup. However,
39
nuclear data uncertainties are not taken into account in the evaluated bench-
mark uncertainty. Uncertainties from geometrical modeling of the benchmark
in, e.g., MCNPX, the calculation bias, the uncertainties from statistics (in the
case of a Monte Carlo code) and the uncertainties in nuclear data have an
impact on the calculation of the response parameter of the benchmark. To
use a benchmark to reduce uncertainties, all these uncertainties can be taken
into account by computing a combined benchmark uncertainty for a particular
benchmark (B) and isotope (j), given as:
σ2B, j = σ2
E +σ2C, j (4.13)
where σC, the uncertainty in the calculation which takes into account the un-
certainties in nuclear data of all isotopes within the benchmark other than the
isotope whose uncertainty is being reduced, the uncertainties from geometri-
cal modeling, the computational bias and the uncertainties from statistics, is
expressed as:
σ2C, j = ∑
over all p,where p= j
σ2ND,p +σ2
calc,bias +σ2geo,mod +σ2
stat (4.14)
where p is the index for the different isotopes contained in the benchmark,
σND,p is the nuclear data uncertainty of the benchmark for the pth isotope,
and j is the isotope whose nuclear data uncertainty is being reduced, σcalc,biaswhich is the computational bias, takes into account the uncertainty from the
numerical methods used to solve the transport equation, σgeo,mod is the geo-
metrical modeling uncertainties, σstat is the statistical uncertainty in the case
where a Monte Carlo code is used. More details have been presented in PA-PER IV. It can be noted that in this work, the nuclear data uncertainties for the
benchmarks, for the different isotopes, is computed based on the uncertainties
contained in the TENDL library, using the TMC method.
4.6 Benchmark selection method
In PAPER I, a method for selecting benchmarks for reactor code validation,
based on the Total Monte Carlo method is proposed. Such a method is useful
for the code user to choose which benchmark could be used when validating
the reactor code for a particular application. This method has been described
in more detailed in PAPER V.
A flow diagram that illustrates the benchmark selection method is presented
in Fig. 4.7. The method involves first, the production of random nuclear data
libraries. It must be noted that, even though the random nuclear data used in
this work were obtained based on the TMC method [26], other approaches also
exist for random nuclear data production as presented earlier in section 3.3.2.
40
(4b) Reactor calculation and code validation with
benchmark B
(1) Random Nuclear Data (ND) libraries for isotope j (TENDL project)
(2) Process random ND with, e.g., NJOY
(3a) Perform transport calculation with random ND files for reactor application (application case)
(3b) Perform transport calculation with random ND files for benchmark, B (benchmark case)
Yes
1BB(4a) SB > 0.3?
No
Figure 4.7. Flow chart diagram for the benchmark selection process. Random nuclear
data files obtained from the TENDL project for the isotope j, are processed into the
ACE format and used for calculations with the benchmark and the application case.
Similarities between benchmarks and application cases are quantified using a similar-
ity index (SB). As a rule of the thumb, a limit of SB > 0.3 is set for the similarity index
(the limit has been chosen arbitrarily).
Using the same processed random files, reactor simulations are performed for
the benchmark under consideration and an application case. The application
case is defined as the reactor system under consideration - for this, a model
of the system with full geometry, concentrations, isotopic compositions is
required. The benchmark case can be either a criticality, reactor physics or
shielding benchmark which are available in various handbooks such as the
ICSBEP [21], IRPHE [23] and SINBAD [24]. In this work, however, only
criticality benchmarks were used. From this, the Pearson correlation coeffi-
cient can be computed between the application case and one or several bench-
marks. The computation of the correlation coefficient is the first step towards
the computation of the similarity index.
The goal is to identify benchmarks with similarity in neutron spectrum, iso-
topic composition and consequently a similarity in reaction rates with a reactor
application system. To include the contribution from both the isotopic compo-
41
sition and neutron spectrum, a Similarity Index (SB) expressed as the product
of the Pearson correlation coefficient (R) and the Variance Ratio (VR) com-
puted as the ratio between the variance of the benchmark to the variance of the
application case for a particular varied isotope is proposed.
Given two sequences of reactor calculations expressed as: X = (xi : i= 1, ...,xn)
and Y = (yi : i = 1, ...,yn), where X and Y are a collection of reactor response
parameter variables computed for the application case and the benchmark re-
spectively with n random files. The similarity index (SB) which is a measure
that quantifies the relationship between X and Y is expressed as:
SB = R×(
Var(Y )Var(X)
)for Var(Y )≤Var(X) (4.15)
where Var(Y), the variance of the benchmark due to the variation of nuclear
data of the isotope under consideration, is used here as a measure of the impor-
tance of that isotope in contributing to the overall uncertainty in the response
parameter such as the keff for example. The ratio between Var(Y) and Var(X)
is used for comparing the sensitivities between the application and the bench-
mark to the variation of the isotope of interest. A high value signifies similar
sensitivity to the variation of a particular isotope for both the application and
the benchmark case. Eq. 4.15 is valid for Var(Y) ≤ Var(X). In a case where
Var(Y) ≥ Var(X), Eq. 4.15 is modified as follows:
SB = R×(
Var(X)
Var(Y )
)for Var(Y )≥Var(X) (4.16)
The similarity index is interpreted as a measure that quantifies the similarity
between the two systems and its value is given between +1 and -1. For inter-
preting SB, we propose the following:
1. Very strong similarity: 0.7 ≤ SB ≤ 1.00
2. Strong similarity: 0.5 ≤ SB ≤ 0.69
3. weak similarity: 0.2 ≤ SB ≤ 0.49
4. Very weak similarity: SB ≤ 0.19
It should be noted that the ranges presented have been chosen arbitrarily. A
high SB computed between the application case and the benchmark case sig-
nifies a strong similarity between the two systems for the particular isotope of
interest and thus, the benchmark under consideration can be selected as a good
representation of the reactor system under investigation for the validation of
reactor codes using the nuclear data of that particular isotope as input.
4.6.1 Benchmark cases
Criticality safety benchmarks used in this work were taken from the Inter-
national Handbook of Evaluated Criticality Safety Benchmark Experiments
42
(ICSBEP) [21]. The benchmarks used in this work covers four categories: the
Plutonium Metallic Fast (pmf), Plutonium Metallic Intermediate (pmi), Highly
enriched uranium Metallic Fast (hmf) and Low enriched uranium Compound
Thermal (lct) systems. Each benchmark, comes with an evaluated benchmark
keff with a corresponding evaluated benchmark uncertainty.
4.7 Correlation based sensitivity analysis
To determine the sensitivity of benchmarks to various partial cross sections
for particular isotopes and energy groups, a correlation based sensitivity anal-
ysis was performed. This is also important for understanding the underlying
physics implemented in reactor simulation codes and for nuclear data adjust-
ments. In Fig. 4.8, the correlation based Monte Carlo sensitivity method is
presented in a flowchart.
Random files from TENDL project p jp j
(cross section (xs), Parameter) correlation computations
Monte Carlo sensitivity analysis
Linearize cross sections (xs) Module: Linear
Doppler broaden xs Module: Sigma1
Cal. group averaged xs Module: Groupie
Linearize cross sections (xs)Module: Linear
Doppler broaden xsModule: Sigma1
Cal. group averaged xsModule: Groupie
PREPRO code
Reconstructs cross sections from resonance parameters
Module: RECENT
Figure 4.8. A flowchart depicting the correlation based Monte Carlo sensitivity
method. Correlation coefficients are computed between random cross sections av-
eraged over an energy group and any reactor response parameter of interest.
Given a set of n random files, Pearson correlation coefficients are computed
between a reactor response parameter such as the keff and a particular par-
tial cross section averaged over a specific energy group. Random files ob-
tained from the TENDL project were first linearized and reconstructed from
43
resonance parameters using the LINEAR and RECENT modules of the PRE-
PRO processing code respectively [69]. The cross sections were then Doppler
broadened and collapsed into 187 energy groups (from thermal energy to 20
MeV) using the SIGMA1 and the GROUPIE modules of the PREPRO code
respectively. Correlation coefficients were computed between the cross sec-
tion for each energy group and the reactor response parameter between the 0
- 20 MeV energy range for different reaction channels. These correlations are
interpreted as representing the relationship between a particular partial cross
section and a reactor response parameter of interest. A high positive correla-
tion coefficient signifies a strong sensitivity/importance of a particular partial
cross section to the variance of a particular response parameter for a designated
incident energy group. The method was utilized and presented in PAPERS IIand section 5.6, for correlation based sensitivity analysis of ELECTRA.
4.8 Burnup calculationsIn steady state calculations as presented in section 4.4, the total uncertainty
in a physical observable was estimated as the sum of the statistical uncer-
tainty from the Monte Carlo code and the uncertainty due to nuclear data.
However, statistical uncertainties are not propagated for all quantities in bur-
nup calculations. Therefore, to propagate nuclear data uncertainties in bur-
nup calculations, we performed two sets of transport calculations with the
SERPENT code: (1) SERPENT calculations with constant nuclear data and
random seeds and (2) SERPENT calculations with random nuclear data and
random seeds. In Fig. 4.9, keff distributions for simulations with constant nu-
clear data (light bars) and simulations with random 239Pu nuclear data (blue)
are presented.
0.89 0.9 0.91 0.92 0.930
10
20
30
40
50
60
70
Keff
# of
rand
om fi
les
740 random Pu−239740 random Pu−239Constat NDConstant ND
Figure 4.9. keff distributions for two sets of simulations: The blue bars represent the
740 random files, where each simulation has unique 239Pu nuclear data and the light
bars represents 700 simulations with unique seeds but constant nuclear data.
From (1) above, since it was only the pseudo random numbers of the code
44
that were changed, the uncertainty obtained is as a result of statistics. In this
work, the TMC method was used to estimate decay heat and inventory uncer-
tainties due to the variation of 239Pu transport data as a function of time and
presented in more detailed in PAPER III. It was observed that, by using criti-
cality benchmarks as presented in PAPER I to select acceptable nuclear data
files for reactor burnup calculations, inventory uncertainties could be reduced
further.
4.9 Nuclear data Adjustments
As mentioned earlier, the evaluation process in the past was done by using dif-
ferential experimental data which was then complemented with nuclear model
calculations [26]; post adjustments to fit a set of integral measurements was
treated as a separate entity from the evaluation process [82]. In Ref. [83], a
method to couple differential and integral data for the evaluation and adjust-
ment of resonance parameters was proposed and utilized. Feedbacks from
integral benchmark experiments were used to adjust resonance parameters in
the resolved resonance region. In Refs. [82, 84], a method that recombines
the evaluation using implicitly differential data, with a posteriori adjustments
to integral measurements referred to as the ’Petten method’ was developed
and utilized. A random search for the best possible nuclear data file which
contains the ’best’ possible combination of nuclear model parameters is per-
formed based on distance minimization. In this way, a file which gives a good
description of integral benchmark data while implicitly taken into consider-
ation differential data, can be identified. In the Ref. [82], similar to what is
proposed in section 4.9.2, both differential and Integral data were combined
for the evaluation and adjustment of neutron induced reactions of 63,65Cu,. In
Refs. [82, 84], however, experimental data were not handled in a statistically
rigorous way.
In this work, an approach for combining differential and integral experimental
data for nuclear data adjustments using file weights based on the likelihood
function is proposed. Once the ’best’ file is identified, it follows the normal
course of events from data checking to data testing and validation to the final
product (an evaluated nuclear data file usually in ENDF format). A flowchart
showing the proposed method of combining differential and integral bench-
mark data in nuclear data evaluation is presented in Fig. 4.10. As can be seen
from the figure, model parameters in a nuclear reaction code such as TALYS,
are adjusted to reproduce differential (A) and integral (B) experimental data.
Starting from TALYS database of model parameters which also contains the
RIPL database (Reference Input Parameter Library) [27], uncertainties are as-
signed to model parameters in the nuclear reaction code, TALYS. The TALYS
code is run multiple times, each time with a different set of model parameters
45
(2)
Model parameters
Nuclear reaction code e.g.
TALYS
Model adjustments
to fit exp. data
Data checking
Adjustment to reproduce integral data
Evaluated nuclear data
Application e.g. transport calculations
(1)
Differential experiments
TALYS database of model parameters
Integral experiments
Physical observables
e.g. cross sections,
fission yields etc.
Data processing
A B
Data testing
and validation
BNL codes are used. Eg. CHECKR, FIZCON, PSYCHE
(3)
Figure 4.10. Flowchart showing the proposed method of combining differential and
integral benchmark data in nuclear data evaluation based on nuclear reaction model-
ing. Models are adjusted to reproduce both differential (A) and integral (B) experi-
mental data by assigning weights based on the likelihood function.
to obtain physical observables such as cross sections and angular distributions.
These observables are adjusted to fit differential data by computing weights
proportional to the likelihood function as given in Eq. 4.18, using differential
experimental information for each reaction channel. The files are again ad-
justed to fit integral benchmark experiments by computing weights based on
the likelihood function using integral benchmark information using Eq. 4.19.
The methodology for combining the weights from differential and integral
data is presented in section 4.9.2. These weights are then combined and the
’best’ file selected. To verify that the selected file conforms to ENDF format,
the file is tested using Brookhaven National Laboratory (BNL) utility codes:
CHECKR, FIZCON and PSYCHE codes [35]. Once this is done, the data
is validated against a large set of integral benchmarks and the final file is an
evaluated nuclear data file. For the file to be used in applications, it must be
processed into usable formats using processing codes such as NJOY. Several
feedbacks are possible as shown in Fig. 4.10:
1. Feedback from model adjustment to fit differential experimental data can
be given to model calculations for the adjustment of model parameters
and their uncertainties.
2. Feedback from integral data adjustments, can be given to differential ex-
perimentalist, to identify energy regions where discrepancies exist and
46
consequently new data is needed. Furthermore, feedback to model cal-
culations can be provided, to enable the adjustment of a priori model
parameter and their uncertainties.
3. By identifying subsets of the nuclear data, which contributes most sig-
nificantly to the nuclear data uncertainty in the application, one can se-
lect which experimental nuclear data is most needed. In addition, this
process can also guide which part of the model calculations that should
be prioritized, i.e., feedback from the application to both model calcula-
tions and experimental measurements.
This study is, therefore, an attempt to present an automatic approach for an
optimal search for the best file based on differential and integral data, and
nuclear reaction models using the likelihood function.
4.9.1 Production of random nuclear data libraries
The first step is to identify the optimal set of parameters (nominal file) which
reproduces differential experimental data. A good starting point is to use ref-
erence input parameters from the RIPL database [27] for nuclear model cal-
culations as shown in Fig. 4.10. The nominal file (with its parameter set) used
in this work was obtained from the TENDL-2014 [37] evaluation. To obtain a
non-informative prior as much as possible, the model parameter uncertainties
presented earlier in Table 4.1 were all multiplied by a factor of three. The
parameters were also sampled from a uniform distribution.
The elastic angular distributions (MF4-MT2) were adopted from ENDF/B-
VII.1 nuclear data library while the resonance parameters were adopted from
the JEFF-3.1 library together with their corresponding background cross sec-
tions using the TARES code. All the parameters were randomly varied within
these uncertainties, and a total of 3000 random nuclear data files were pro-
duced. These files were processed into ACE files at 293 K using NJOY99
version 336 and then processed into XY tables by means of a bash script.
Experimental information was included by computing file weights using the
likelihood function. In this work, the weights were computed based on ex-
perimental data from four cross sections: (n,tot), (n,el), (n,inl) and (n,γ) using
Eq. 4.10 and integral experiments using Eq. 4.19.
4.9.2 Nuclear data adjustment of 208Pb in the fast region
The following were used for the adjustment procedure: differential experi-
mental data from the EXFOR database (Exchange among nuclear reaction
data centers) [17], criticality benchmarks from the ICSBEP Handbook [21],
the state of-the-art reaction code TALYS, the TARES code [74], the MCNPX
47
code and a Monte Carlo adjustment procedure based on the likelihood func-
tion. Since the adjustment was done for the fast incident neutron energy re-
gion, only lead sensitive benchmarks that exhibit fast spectra, were used. In
order to include assumptions on the magnitude of the uncertainties and the
correlation between the different differential experiments, the default EXFOR
interpretation rules as presented in Ref. [77], were used in this work. In Ta-
ble 4.2, the benchmark experiments used in this work, showing the evalu-
ated benchmark keff, the evaluated benchmark uncertainty, and the combined
benchmark uncertainty are presented.
Table 4.2. Criticality safety benchmarks used in this work with their case numbers,the evaluated benchmark keff and the evaluated benchmark uncertainty. These bench-marks were obtained from the International Handbook of Evaluated Criticality SafetyBenchmark Experiments (ICSBEP) [21]. HEU-MET-FAST stands for Highly EnrichedUranium (HEU) Metallic Fast and LEU-COMP-THERM for Low Enriched Uranium(LEU) Compound Thermal benchmarks. The combined benchmark uncertainty givenin column 5 were computed using Eq. 4.13.
Benchmark category caseEvaluated
Benchmark keff
Evaluated Benchmark
uncertainty [pcm]
Combined benchmark
uncertainty [pcm]
PMF035 1 1.0000 160 1183
HMF027 1 1.0000 250 1163
HMF057 1 1.0000 200 1110
HMF057 2 1.0000 230 1149
HMF057 3 1.0000 320 1122
HMF057 4 1.0000 400 1183
HMF057 5 1.0000 190 1087
HMF064 1 0.9996 80 635
More details on how weights are assigned to random nuclear data libraries,
based on differential experimental data using the likelihood function can be
found in Refs. [52, 77]. The methods presented in Refs. [52, 77], were used in
this work for nuclear data adjustments.
Given that CE is the differential covariance matrix which is made up of both
diagonal and off-diagonal elements, a generalized χ2k(E) can be computed us-
ing:
χ2k(E) = (x− τ(p(k)))TC−1
E (x− τ(p(k))) (4.17)
where τ(p(k)) is a vector of the calculated observables found in the kth random
file, p(k) is the parameter set of the kth random file and x is a vector of exper-
imental observables. The χ2k(E) estimator is used to quantify the likelihood of
model calculations with respect to a set of experimental constraints using the
likelihood function:
wk(E) = e− 1
2 χ2k(E) (4.18)
48
where wk(E) is the weight assigned to the kth random file based on differential
experimental data (E) (the weights are computed based on (n,tot), (n,el), (n,inl)
and (n,γ) cross sections). More details on how differential data are incorpo-
rated into the TMC methodology and how outliers are treated is presented in
Refs. [52, 77]
Once these weights have been computed, they can be combined with weights
computed from integral benchmark data:
wk(B) = e− 1
2 χ2k(B) (4.19)
where the χ2 of the benchmark for the kth random file, assuming that the
benchmarks are uncorrelated (since we use one benchmark at a time), χ2k(B), is
given by:
χ2k(B) =
(kBeff(k)− kB
eff,exp)2
σ2B, j
(4.20)
where, kBeff(k) is the calculated benchmark value for the kth random file, kB
eff,exp
is the evaluated experimental benchmark value and σ2B, j is the combined bench-
mark uncertainty given in Eq. 4.13. Since the adjustment is made for a general
purpose file, the correlation coefficient (R) as utilized in Eq. 4.11 was not in-
cluded in the χ2 computation.
If we assume that the differential and integral experimental data are uncorre-
lated, a combined weight can be computed from:
wT,k = wk(E)×wk(B) (4.21)
where wT,k is the total weight (combined weight) for the kth random file, wk(E)
and wk(B) are the weights assigned to the kth random file, using differential and
integral benchmark data respectively. To compare wk(E) and wk(B), the weights
were both normalized with the maximum weights computed for the differen-
tial and the integral experimental data respectively. Since experimental covari-
ance data for benchmarks were not readily available, the ’one-benchmark-at-
a-time’ approach was used.
In the case where covariance data including their correlations are available
for the integral experiments, the generalized χ2 as given in Eq. 4.17 can be
used for χ2k(B). The inclusion of multiple benchmarks with their correlations
is recommended for future work. By combining these weights, a new ’best’
file (random file with the largest weight) is selected as the file which com-
pares best with both differential and integral benchmark data. This file was
compared with the major nuclear data libraries as presented in Table 5.7 and
validated by comparing its performance against integral benchmarks results.
It should be noted that the same benchmarks used for adjustment should not
to be used for validation.
49
5. Results and Discussions
’With four parameters I can fit an elephant, andwith five I can make him wiggle his trunk.’
- John von Neuman
In this chapter, a summary of the results obtained are presented and discussed.
First, results from model calculations using the TALYS code are presented.
Subsequently, nuclear data uncertainties and keff distributions for global and
partial variations of nuclear data are presented. Furthermore, results from the
benchmark selection method, nuclear data uncertainty reduction and correla-
tion based sensitivity measure are also discussed. Finally, results from the
adjustment of 208Pb nuclear data in the fast region using both differential and
integral benchmark information are discussed.
5.1 Model calculations
Model calculations were carried out with the TALYS code in combination with
other auxiliary codes as presented earlier in section 4.2 to generate 208Pb and206Pb random nuclear data. A total of 550 each of 208Pb and 206Pb nuclear
data libraries were produced. The random files generated were processed into
the ACE format with the NJOY code at 600 K. Fig. 5.1 shows 50 random ACE208Pb cross sections as a function of energy for the following neutron induced
reactions: (n,tot), (n,el), (n,2n) and (n,γ). These cross sections were compared
with the 208Pb nuclear data from the ENDF/B-VII.1 nuclear data library. The
ENDF/B-VII.1 data were processed at the same temperature and with the same
version of the NJOY code as the random files.
From Fig 5.1, a spread can be observed for the entire energy region for all the
studied reactions. This is not surprising as each line represents a random nu-
clear data file with a unique set of model parameters. Using the TMC method
described earlier in section 3.3.2, the spread in nuclear data observed here is
propagated all the way through to macroscopic reactor parameters (PAPERII). A detailed and extensive presentation of choice of parameters and models
used for the TALYS calculations is, however, beyond the scope of this work.
50
10−2 10−1 100 101102
103
104
105208Pb (n,tot)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)Random filesENDF/B−VII.1
10−2 10−1 100 101102
103
104
105208Pb (n,el)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
Random filesENDF/B−VII.1
5 10 15 200
1
2
3
208Pb (n,2n)
Incident Energy (MeV)
Cro
ss s
ectio
n (b
)
Random filesENDF/B−VII.1
10−2 10−1 100 101
10−2
100
102
104
208Pb (n,γ)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
Random filesENDF/B−VII.1
Figure 5.1. Random ACE cross sections as a function of energy for some neutron-
induced reactions on 208Pb. Results are compared with ENDF/B-VII.1 nuclear data
library. Top left: (n,tot), top right: (n,el), bottom left: (n,2n) and bottom left: (n,γ).
5.2 Global uncertainties
In Fig. 5.2, the keff distributions for varying the nuclear data of some of the ac-
tinides contained within the fuel: 238Pu, 239Pu, 240Pu and 241Am are presented.
Gaussian distributions are observed for all the actinides under consideration
except for 238Pu and 241Am which exhibited asymmetric behaviors with tails
in the high keff region; skewness values of 0.73 and 0.28 were recorded for241Am and 238Pu respectively. The skewness has been described in more de-
tail in section 3.2. Large deviations between the JEFF-3.1 and TENDL-2012
keff values were obtained for the variation of 241Pu and 242Pu nuclear data.
The reason for the deviation has to be investigated further. Delayed neutron
data were not varied in the random actinide data. Hence no computations were
performed to assess the impact of actinide data uncertainties on the βeff.
In Table 5.1, the global nuclear data uncertainty in the keff is presented for238,239,240Pu, 241Am, 56Fe and 58Ni. Large uncertainties were obtained for239,240Pu, however, the global uncertainties of 238Pu and 241Am were observed
to be relatively small but significant. Since partial variations were not per-
formed for these isotopes, it is difficult to ascertain the contributions from
51
0.99 1 1.010
10
20
30
40
50
keff values
Num
ber o
f cou
nts/
bin 239Pu
0.98 0.99 1 1.01 1.020
5
10
15
20
25
keff values
Num
ber o
f cou
nts/
bin 240Pu
0.999 1 1.001 1.0020
5
10
15
20
keff values
Num
ber o
f cou
nts/
bin 241Am
0.995 1 1.0050
5
10
15
20
keff values
Num
ber o
f cou
nts/
bin 238Pu
Figure 5.2. keff distribution for ELECTRA due to the variation of actinide nuclear
data. Top left: 239Pu, top right: 240Pu, bottom left: 241Am and bottom right: 238Pu.
partial channels to the global uncertainty observed. Small uncertainties were
observed for most structural materials on the keff except for 56Fe and 58Ni,
which had significant impact and are therefore presented in Table 5.1.
Table 5.1. The impact of nuclear data on the keff for the variation of actinide nucleardata and some structural materials for ELECTRA. The uncertainty of uncertainty ofthe nuclear data uncertainties computed are also presented.
Isotope σND(keff) [pcm]
238Pu 295±12239Pu 748±19240Pu 1015±32
241Am 70±356Fe 185±658Ni 20±1
The probability distributions of the keff for the variation of the lead nuclear
data: 204,206,207,208Pb is depicted in Fig. 5.3. The random nuclear data files for207Pb and 204Pb were obtained from TENDL-2012 [75] while 208Pb and 206Pb
random files were produced as part of this work and can be obtained from
the TENDL-2014 nuclear data library [37]. In Table 5.2, the global nuclear
data uncertainties together with their uncertainties for 204,206,207,208Pb are pre-
sented for keff, coolant temperature coefficient, and the coolant void worth.
52
0.98 0.99 1 1.01 1.020
10
20
30
40
50
keff values
Num
ber o
f cou
nts/
bin
208Pb
1.0021.0031.0041.0051.0061.0070
10
20
30
40
keff values
Num
ber o
f cou
nts/
bin
207Pb
1 1.002 1.004 1.0060
10
20
30
40
keff values
Num
ber o
f cou
nts/
bin
206Pb
0.9995 1 1.0005 1.0010
10
20
30
40
keff values
Num
ber o
f cou
nts/
bin
204Pb
Figure 5.3. keff distribution for ELECTRA for varying lead nuclear data at 600 K
coolant temperature. Top left: 208Pb, top right: 207Pb, bottom left: 206Pb and bottom
right: 204Pb. The keff distribution for 208Pb, 207Pb and 206Pb slightly deviate from
Gaussian distribution with tails in the high keff region.
Table 5.2. Nuclear data uncertainty (global) in some macroscopic reactor parametersfor ELECTRA due to the variation of 204,206,207,208Pb nuclear data. The results areall given in pcm. The values quoted in the sixth row are the quadratic sum of the NDuncertainties coming from 204,206,207,208Pb (σND,Pb,tot ). The uncertainty of uncertaintyof the nuclear data uncertainties computed are also presented.
Isotopes σND(ke f f ) σND(CTC) σND(CVW )208Pb 896±28 61±2 890±28207Pb 118±4 - 117±4206Pb 136±5 - 136±5204Pb 12±2 - 12±2
Total(σND,Pb,tot)[pcm] 914 61 907
Relative uncertainties (%) 0.9 2.6 3.3
It was assumed in Table 5.1 that the uncertainties between the lead isotopes
were uncorrelated. It should be noted, that to obtain the ND uncertainty for the
CTC in pcm/K the values in column three must be divided by the difference
in temperature (1200 K) between the two perturbed states.
Large uncertainties in the keff were observed for 208Pb indicating that the
ELECTRA core is highly sensitive to 208Pb nuclear data variation and hence
its uncertainties. Relatively large uncertainties in the keff were recorded for206Pb and 207Pb. The uncertainty from the 204Pb was, however, small. The ob-
53
served spread in the CTC for 204,206,207Pb was dominated by statistics. Except
for 204Pb, the impact of nuclear data uncertainty for all lead isotopes on the
CVW were relatively high.
5.3 Partial variations
In Fig. 5.4, we present the distribution in keff due to the variation of elastic
scattering, resonance parameters, angular distributions and neutron capture
cross sections of 208Pb. Non Gaussian shapes are observed for the variation
1 1.01 1.02 1.030
5
10
15
20
keff values
Num
ber o
f cou
nts/
bin
resonance parameters
1.006 1.008 1.01 1.012 1.014 1.0160
5
10
15
20
25
keff values
Num
ber o
f cou
nts/
bin elastic
1.006 1.0065 1.0070
5
10
15
20
keff values
Num
ber o
f cou
nts/
bin
capture
1 1.002 1.004 1.006 1.008 1.010
5
10
15
20
keff values
Num
ber o
f cou
nts/
bin
angular distribution
Figure 5.4. keff distribution for ELECTRA for varying resonance parameters only (top
left), the elastic scattering only (top right) and neutron capture only (bottom left) in
the fast region, and angular distributions only (bottom right) of 208Pb.
in the elastic scattering, resonance parameters and the angular distributions as
can be seen from the figure. Their skewness values are presented in Table 5.4.
High tails are observed in the high keff regions for the elastic scattering cross
section and the resonance parameter variations. A tail in the low keff region
was, however, observed for the angular distributions. Nuclear data uncertainty
in keff due to partial variations of 206,207,208Pb nuclear data are presented in
Table 5.3. The major contribution to the global uncertainty comes from the
resonance parameters, however, the (n,el) and angular distributions also had
significant impact as can be seen in the Table 5.3.
54
Table 5.3. Nuclear data uncertainty in keff due to partial variations of 208Pb,207Pb and 206Pb nuclear data. All lead files used here were obtained from TENDL-2012 [75]. The uncertainty of uncertainty of the nuclear data uncertainties computedare also presented.
208Pb 207Pb 206Pb
Nuclear data varied σND(ke f f )[pcm] σND(ke f f )[pcm] σND(ke f f )[pcm]
n,el 289±12 58±3 50±2
n,2n 7±3 4±6 5±4
n,γ 83±4 10±2 10±2
n, inl 8±3 30±2 23±2
Resonance parameters 862±35 55±3 145±6
Angular distributions 226±9 101±4 107±5
The bulk contribution to the nuclear data uncertainty on the keff and the CVWcome from uncertainties in the resonance parameters, the elastic scattering
cross section and from the angular distributions. Uncertainties due to (n,2n)
and (n,inl) were found to be small for all isotopes. The impact from the (n,γ)
on the keff was also observed to be small as expected since fast reactors like
ELECTRA have a small fraction of capture reactions in the core. The small
impact of the 208Pb (n,inl) cross section observed can be attributed to the small
fraction of inelastic reactions in the ELECTRA core (the inelastic channel of208Pb opens at about 2.7 MeV which is well above the peak of the ELECTRA
spectrum, 700 keV).
In Table 5.4, the skewness values of keff distributions for the partial variation
of 208,207,206Pb are presented. High skewness values are recorded for 207Pb
and 206Pb (n,el) cross sections as can be observed from the table.
Table 5.4. Table showing the skewness values for the keff distribution due to partialvariation of 208Pb, 207Pb and 206Pb nuclear data.
Skewness values
keff208Pb 207Pb 206Pb
Resonance parameters 0.75 0.12 -0.31
(n,el) cross section 0.98 0.86 0.73
Angular distributions -0.48 -0.18 -0.16
(n.γ) cross section 0.08 -0.12 0.04
Global keff 0.58 0.37 0.33
5.4 Nuclear data uncertainty reduction
In this section the results from the uncertainty reduction methods presented
in section 4.5 and PAPER IV are presented. For a benchmark to have a high
55
ability to reduce the ND uncertainty for a particular application, the bench-
mark should have: (1) high sensitivity to the isotope under investigation, (2)
a low combined benchmark uncertainty and (3) a high absolute value of the
correlation coefficient computed between the benchmark observable and the
application response parameter.
In Fig. 5.5, keff distributions due to the variation of 239Pu nuclear data after
implementing the method of assigning file weights based on the likelihood
function is presented using benchmarks for the ELECTRA reactor. It can be
observed from the figure that the posterior distributions have narrower spreads
compared to the prior distributions. This can be attributed to the high sensi-
tivity to the variation of 239Pu nuclear data for the pmf benchmarks and the
strong correlation obtained between the pmf benchmarks and ELECTRA due
to 239Pu nuclear data variation. The distributions of weights assigned to ran-
0.985 0.99 0.995 1 1.005 1.01 1.0150
50
100
150
200
keff values
Nor
mal
ized
cou
nts/
bin
pmf1c1 Posterior distributionPrior distribution
0.985 0.99 0.995 1 1.005 1.01 1.0150
50
100
150
200
keff values
Nor
mal
ized
cou
nts/
bin
pmf9c1 Posterior distributionPrior distribution
0.985 0.99 0.995 1 1.005 1.01 1.0150
50
100
150
200
keff values
Nor
mal
ized
cou
nts/
bin
pmf10c1 Posterior distributionPrior distribution
0.985 0.99 0.995 1 1.005 1.01 1.0150
50
100
150
200
keff values
Nor
mal
ized
cou
nts/
bin
pmf11c1 Posterior distributionPrior distribution
Figure 5.5. keff distributions due to the variation of 239Pu nuclear data after combining
prior information with integral benchmark information from pmf1c1 (top left), pmf9c1
(top right), pmf10c1 (bottom left) and pmf11c1 (bottom right) benchmarks for the
ELECTRA reactor using the method of assigning file weights based on the likelihood
function. ’c’ denotes the case of the benchmark.
dom 239Pu and 240Pu nuclear data after combining the prior information with
integral benchmark information from pmf1c1 (top left), pmf8c1 (top right),
pmf9c1 (bottom left) and pmf10c1 (bottom right) benchmarks for the ELEC-
TRA reactor are presented in Fig. 5.6. It can be observed from the figure that
a large number of random nuclear data files were assigned with low weights
and, therefore, contributed less to the final uncertainty obtained. This resulted
in narrower posterior distributions.
56
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
Weights
Num
ber o
f cou
nts/
bin pmf1c1
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Weights
Num
ber o
f cou
nts/
bin pmf8c1
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
Weights
Num
ber o
f cou
nts/
bin pmf9c1
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
Weights
Num
ber o
f cou
nts/
bin pmf10c1
Figure 5.6. Distributions of weights assigned to random 239Pu nuclear data after com-
bining prior information with integral benchmark information from pmf1c1 (top left),
pmf8c1 (top right), pmf9c1 (bottom left) and pmf10c1 (bottom right) benchmarks for
the ELECTRA reactor. ’c’ denotes the case of the benchmark.
5.5 Similarity Index
As proof of concept, similarities between different benchmarks calculated
with the method presented in section 4.6 are presented. These similarities are
important for classifying benchmarks for reactor calculations and code vali-
dation purposes. In Fig. 5.7, a similarity matrix computed between different
plutonium metallic fast benchmarks denoted by "pmf" due to the variation of239Pu nuclear data are presented. The values displayed in the matrices are sim-
ilarity indices computed. Very strong positive similarity indices are recorded
in Fig. 5.7 for all benchmarks. This signifies a strong similarity in neutron
spectrum and isotopic concentration between the two benchmarks.
The highest similarity index of 0.97 was recorded between pmf5c1 vs. pmf8c1
benchmarks. The main factors that contributed to the high similarity index ob-
tained are the high correlation coefficient (R=0.997) and high variance ratio
of 0.968 computed between the two systems. As can be observed, the pmf5c1
benchmark with the following composition: (239−241Pu: 94.79/ 4.90 / 0.31
[at. % ]) has similar plutonium composition with the pmf8c1 benchmark:
(239−241Pu: 93.59/ 5.10 / 0.30 [at. %]); both benchmarks have fast spectrum
too.
In Table 5.5, a summary of similarity indices computed between ELECTRA
and a set of lead sensitive benchmarks using 208,207,206Pb nuclear data are pre-
57
0.69
0.87
0.92
0.93
0.96
0.74
1.00
0.94
0.82
0.79
0.76
0.73
1.00
0.74
0.71
0.90
0.93
0.97
1.00
0.73
0.96
0.73
0.93
0.96
1.00
0.97
0.76
0.93
0.76
0.95
1.00
0.96
0.93
0.79
0.92
0.78
1.00
0.95
0.93
0.90
0.82
0.87
1.00
0.78
0.76
0.73
0.71
0.94
0.69
Benchmarkspmf1c1 pmf2c1 pmf5c1 pmf8c1 pmf9c1 pmf10c1 pmf11c1
Ben
chm
arks
pmf11c1
pmf10c1
pmf9c1
pmf8c1
pmf5c1
pmf2c1
pmf1c1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5.7. Similarity matrix between selected plutonium sensitive benchmarks due
to the variation of 239Pu nuclear data. (pmf) denotes plutonium metallic fast bench-
marks. ’c’ denotes the case of the benchmark. The values displayed in the matrix are
similarity indices.
Table 5.5. Table showing similarity indices computed between ELECTRA and a setof lead sensitive benchmarks due to the variation of 208,207,206Pb nuclear data. 300random nuclear data files were used for each isotope.
Benchmark category Case SB(208Pb) SB(
207Pb) SB(206Pb)
PU-MET-FAST-035 (case 1) 0.23 (3) -0.06 (4) -0.03 (4)
HEU-MET-FAST-027 (case 1) 0.25 (3) -0.01 (4) -0.02 (4)
HEU-MET-FAST-064 (case 1) 0.64 (2) -0.03 (4) -0.001 (4)
HEU-MET-FAST-057 (case 1) 0.80 (1) -0.06 (4) -0.04 (4)
HEU-MET-FAST-057 (case 2) 0.76 (1) -0.05 (4) -0.02 (4)
HEU-MET-FAST-057 (case 3) 0.68 (2) -0.03 (4) -0.02 (4)
HEU-MET-FAST-057 (case 4) 0.67 (2) -0.08 (4) 0.0004 (4)
HEU-MET-FAST-057 (case 5) 0.59 (2) -0.03 (4) -0.005 (4)
LEU-COMP-THERM-010 (case 1) 0.17 (4) 0.04 (4) 0.01 (4)
LEU-COMP-THERM-017 (case 1) 0.04 (4) 0.06 (4) 0.01 (4)
sented. Results in brackets represent the strength of similarity between the ap-
plication and the benchmarks (1, 2, 3 and 4 represent very strong, strong, weak
and very weak similarities respectively). Strong similarity indices are recorded
between ELECTRA and the hmf57c1 (SB=0.80), hmf57c2 (SB=0.76) and the
hmf64c1 (SB=0.64) benchmarks due to the variation of 208Pb nuclear data.
This is not surprising since high variance ratios of 0.8, 0.77 and 0.64 were
obtained between ELECTRA and the hmf57c1, hmf57c2 and the hmf64c1
benchmarks respectively. High variance ratios (close to 1) signifies that the
58
systems exhibit similar sensitivity to the variation of nuclear data for the iso-
tope of interest. Since the lct10c1 and lct17c1 are thermal benchmarks, very
weak similarity indices were obtained between these benchmarks and ELEC-
TRA. Since the highest similarity indices were recorded between ELECTRA
and the hmf57c1 and hmf57c2 benchmarks, these benchmarks can be consid-
ered as good candidates for reactor code validation using 208Pb nuclear data
as input. Very weak similarities were, however, obtained between ELECTRA
and the benchmarks in the case of 207Pb and 206Pb nuclear data variation as
can be seen from the table. This is expected since all the benchmarks have
weak sensitivities to the variation of 207Pb and 206Pb nuclear data.
5.6 Correlation based sensitivity measure
In this section, results from the correlation based sensitivity method presented
in section 4.7 and PAPER II are presented and discussed. In Fig. 5.8, an
example of the cross section-keff correlations (not included in the papers) is
presented as a function of incident neutron energy for ELECTRA, using a
187 energy group structure. Four partial cross sections: (n,f), (n,el), (n,γ) and
(n,inl) cross sections are presented for 241Am nuclear data.
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.001 0.01 0.1 1 10
Co
rre
latio
n
Incident Energy (MeV)
241Am(n,g)
(n,el)(n,inl)
(n,f)
Figure 5.8. Cross section-keff correlation against incident neutron energy for 241Am.
The high correlations are observed for the 241Am (n,f) cross section in the high
energy region at about 1 MeV. This could be attributed to the relatively large241Am (n,f) cross sections in the high energy region. Also, the ELECTRA
reactor exhibits a hard spectrum. Weak correlations were, however, obtained
for the (n,γ), (n,inl) and (n,el) cross sections. The correlation based sensitivity
measure has been applied to lead isotopes and presented in PAPER II.
59
5.7 Nuclear data adjustments
In section 4.9, a new method for nuclear data adjustment based on combining
information from both integral and differential nuclear data is proposed. In
this section, the results from the method are reported.
The weights for the differential data were computed based on the (n,tot) and
(n,el) cross sections and the (n,tot), (n,el), (n,inl) and (n,g), cross sections, in
the energy region of 1 to 20 MeV. From the 3000 prior random files, a new
best file (the file with the highest weight) based on the differential data was
selected. In Fig. 5.9, this new best file, together with 50 random files and
the prior central file is compared with the ENDF/B-VII.1 evaluation for the208Pb(n,el) cross section. Similar comparisons have been made for the other
reaction channels. A comparison of file weights computed between different
5 10 15 20
2000
4000
6000
8000
208Pb (n,el)
Incident Energy (MeV)
Cro
ss s
ectio
n (m
b)
Random filesENDF/B−VII.1Central filebest file (differential data only)
Figure 5.9. Comparing the prior adjustments and central file with ENDF/B-VII.1
nuclear data library for 208Pb(n,el) cross section in the energy range of 5 MeV to 20
MeV. The central file represents the evaluators best effort before the adjustment and
the ’best file’ is the file selected after adjustment with differential data.
evaluations and this work are presented in Table 5.7. The weights are com-
pared with ENDF/B-VII.1, JEFF-3.1, JENDL-4.0, CENDL-3.1, the central
file and this work (the new best file). From the table, the weights (based on
differential experimental data), show that our best file performed better com-
pared to the other nuclear data libraries. This is expected since our best file
is chosen based on the agreement with these experiments. It should also be
noted that other channels, such as (n,α), (n,p), and angular distributions, were
not included in the computation of the weights. Furthermore, since the outliers
were identified by considering the deviation of experiments to model calcula-
tions [77], this could be a possible reason why high weights were obtained for
our best files.
60
Table 5.6. Comparison of file weights between different nuclear data libraries andthis work for prior adjustments with only differential data. The weight for the newbest file, is the maximum weight computed using only differential experimental data.
Weights based on:
Nuclear data [(n,tot) and (n,el)] [(n,tot), (n,el), (n,inl), (n,γ)]
ENDF/B-VII.1 1.00e-08 5.15e-56
JEFF-3.1 1.02e-08 1.30e-50
JENDL-4.0 7.33e-08 2.42e-60
CENDL-3.1 9.93e-09 3.48e-49
Central file 2.72e-12 9.85e-111
New best file 1.40e-7 6.83e-42
In Table 5.7, ratios of calculated to experimental values (C/E) for different lead
sensitive benchmarks for adjustments using differential data only, the hmf57
case 1 benchmark only, and a combination of differential and hmf57 case 1
benchmark data are compared with the ENDF/BVII.0 evaluation.
Table 5.7. The ratios of calculated to experimental values (C/E) for different leadsensitive benchmarks for adjustments using differential data only, the hmf57 case 1benchmark only, and a combination of differential and hmf57 case 1 benchmark data.For the benchmark adjustment, the ENDF/B-VII.0 was used as the reference libraryfor all isotopes, except for 208Pb which was varied. In the case of ENDF/B-VII.0, allisotopes were maintained as the ENDF/B-VII.0 nuclear data library. In the case ofthe benchmarks, only the HMF057-01 was used for adjustment, the other benchmarkswere used for testing and validation. The statistical uncertainties obtained range from39 to 44 pcm.
BenchmarksDifferential
data only
Benchmark
(HMF057-01)Differential data
+ HMF057-01ENDF/B-VII.0
PMF035-01 0.99873 1.00635 0.99873 0.99856
HMF027-01 1.00222 1.00676 1.00222 1.00182
HMF057-02 1.00109 1.00851 1.00109 0.99888
HMF057-03 1.02020 1.03023 1.02020 1.01726
HMF057-04 0.99049 0.99793 0.99049 0.98784
HMF057-05 1.02470 1.03580 1.02470 1.02188
HMF057-06 0.99992 1.00909 0.99992 0.99713
HMF064-01 0.99835 1.00896 0.99796 0.99460
HMF064-02 0.99970 1.01225 0.99970 0.99610
As outlined in section 4.9, the 3000 208Pb files adjusted with differential data,
were further adjusted by using the hmf57 case 1 benchmark. The adjustment
was carried out in the fast region, above the unresolved resonance region. The
reason to choose this benchmark is that the hmf57 case 1 benchmark is lead
sensitive and also exhibits a fast neutron spectrum. Consequently, three differ-
61
ent files were obtained: one from adjustment with differential data, one from
adjustment with the hmf57 case 1 benchmark, and one from the combination
of the weights obtained from differential and benchmark adjustments. These
files were validated against nine different benchmarks using the MCNPX ver-
sion 2.5 code and their performance are compared to the ENDF/BVII.0 eval-
uation as presented in Table 5.7. The benchmarks, as shown in the table,
were used for testing and validation. This was done in order not to use the
same benchmarks for both calibration and validation. For example, the best
file obtained after adjustment with hmf57 case 1 (random file 0534 with a keff
= 1.00000±0.00038), was validated against benchmarks as presented in Ta-
ble 5.7.
In Fig. 5.10, a ratio of the best file obtained from the adjustments with com-
bined differential and integral experimental data to the ENDF/B-VII.1 eval-
uation for the 208Pb(n,el) cross section is presented. From the figure, a non-
Figure 5.10. Comparing adjustments with combined differential and integral experi-
mental data with ENDF/B-VII.1. The ratio between the two evaluations against inci-
dent energy is presented below the plot.
smooth curve is observed for the ENDF/B-VII.1 evaluation at about 10 MeV,
which is difficult to explain from reaction theory. The evaluation (best file)
from this work, however, gave a smoother curve as can be seen from Fig. 5.10.
However, for a complete evaluation of a general purpose file more channels,
such as (n,α) and (n,p), and a large set of benchmarks with different spectra
and composition will have to be used.
In the case of adjustment with only differential data, very few files were as-
signed with significant weights. This accounts for the reason why the adjusted
value based on only differential data gave the same results (the same file was
62
selected) with the adjustments with combined differential and benchmark data,
as can be observed from Table 5.7. The few significant weights obtained are
in agreement with the observations made earlier in Ref. [77]. This could be
as a result of the sampling of model parameters from a rather wide param-
eter space, or could be attributed to model defects. This, however, must be
investigated further in future research. Also, to take full advantage of integral
information, multiple benchmarks together with their uncertainties and corre-
lations must be used. Multiple benchmarks have, however, not been used in
this work.
63
6. Conclusion and outlook
’I think; therefore I am.’- Renè Descartes
The Total Monte Carlo method was used to study the impact of nuclear data
uncertainties of 238,239,240Pu, 241Am, 204,206,207,208Pb and some structural ma-
terials of importance to the conceptual ELECTRA reactor. Electra is a plu-
tonium fuelled and lead cooled fast reactor. For the actinides studied, large
uncertainties were observed in the keff due to 239,240Pu nuclear data uncer-
tainties. Relatively small uncertainties were, however, recorded for 238Pu and241Am nuclear data. In the case of the lead coolant, it was observed that the
uncertainty in the keff and the coolant void worth were large with the most
significant contribution coming from 208Pb nuclear data uncertainties. The
dominant contributions to the uncertainty in the keff came from uncertainties
in the resonance parameters in the case of 208Pb; however, elastic scattering
cross section and the angular distributions also had significant impacts. New208Pb and 206Pb random files with realistic central values have also been pro-
duced as part of this work.
In this thesis, methods for the inclusion of integral experiments for nuclear
data uncertainty reduction of reactor parameters within the TMC method have
been developed. It was observed from the study that significant reductions of
uncertainties due to nuclear data were obtained for some isotopes for ELEC-
TRA after incorporating integral benchmark information.
In addition, a method for combining differential experiments and integral bench-
mark data for data assimilation and nuclear data adjustments using file weights
based on the likelihood function is proposed. The proposed method was ap-
plied for the adjustment of neutron induced 208Pb nuclear data in the fast en-
ergy region.
Furthermore, a method is proposed for computing the similarities between
benchmarks and a specific application. This method can be used to select
benchmarks for code validation for a specific application. Also, a correlation
based sensitivity method was used to study cross section-parameter correla-
tions for different energy groups and reactions.
As an outlook, the following are recommended for future research:
64
1. Uncertainty quantification for specific isotopes in this work were per-
formed using the ’one-isotope-at-a-time’ approach where each isotope
was varied one after the other. This approach was used because the main
goal was to identify the nuclear data inputs with significant impact on
the uncertainty of the output. This approach, however, does not take
into consideration the interaction between different nuclear data files.
To take these interactions into consideration, the random nuclear data
libraries for all isotopes could be varied simultaneously; this gives the
global effect. Also, cross-correlations between isotopes were not taken
into account in this work. This is recommended for future work.
2. A more detailed work in the resonance region and the angular distribu-
tions of lead isotopes is needed, taking into consideration differential ex-
periments as well as integral benchmark data. This is important because
it was observed that the resonance region and the angular distributions
had significant impact, especially in the case of 208Pb. Furthermore,
significant amount of work is needed to obtain better angular distribu-
tions [85].
3. Also, the ’one-benchmark-at-a-time’ approach was used for the reduc-
tion of nuclear data uncertainties. However, to benefit fully from all the
benchmark information available, multiple benchmarks taking into con-
sideration their covariance data and correlations must be used. The use
of multiple benchmarks is also recommended for use in the combined
adjustment methodology based on differential and integral experiments
proposed in this work.
4. The uncertainty quantification carried out in this thesis were performed
at steady state for the ELECTRA reactor and, to some extent, burnup.
A comprehensive study to assess the impact of nuclear data uncertain-
ties on reactor burnup calculations by varying both transport and fission
yield data is recommended.
5. Feedback from nuclear data adjustments can be used to inform the dis-
tributions of model parameters and their uncertainties. This will help
improve nuclear data as well as nuclear reaction models. For example,
from the few files with significant weights obtained from the adjustment
with differential data (as observed in section 5.7), feedback to model
calculations could be provided for the adjustment of model parameters
and their uncertainties.
65
7. Sammanfattning
"Human progress is neither automatic nor inevitable ..."- Martin Luther King, Jr.
Idag saknar cirka 1,4 miljarder människor världen över fortfarande tillgång
till elektricitet. Ytterligare en miljard har endast sporadisk tillgång till elek-
tricitet [1]. Som ett resultat av befolkningsökningen och den ekonomiska
utvecklingen, i synnerhet i utvecklingsländer, förväntas de globala energibeho-
ven öka kraftigt [2]. Energianvändningen världen över ökar kraftigt varvid
kärnkraften förväntas spela en allt större roll vad gäller att möta energibehovet.
För att få allmänhetens acceptans måste nästa generations kärnreaktorer dels
kunna konkurrera ekonomiskt med andra energikällor och dels måste samhäl-
lets krav avseende avfallshantering, kärnämneskontroll, och säkerhet uppfyl-
las. GEN-IV International Forum (GIF) startades därmed med mycket ut-
manande tekniska mål, som omfattar hållbarhet, ekonomi, säkerhet, tillför-
litlighet, kärnämneskontroll och fysiskt skydd. De sex reaktorkoncept som
identifierats som de mest lovande fjärde generationens (GEN-IV) reaktorsys-
tem är [4]: den gaskylda snabbreaktorn, den blykylda snabbreaktorn, den na-
triumkylda snabbreaktorn, saltsmältereaktorn, högtemperaturreaktorn och den
superkritiska vatttenkylda reaktorn.
GIF har rankat konceptet med den blykylda snabbreaktorn högst vad gäller
hållbarhet, eftersom det använder en sluten bränslecykel för omvandling av
fertila isotoper. Den rankas också högt vad gäller icke-spridningsaspekter
och fysiskt skydd, bl.a. beroende på att härden har en lång livslängd [5].
Reaktorns säkerhetsfunktioner förstärks av att kylmediet är relativt inert och
har förmåga att hålla kvar farliga radionuklider, som exempelvis jod och ce-
sium, även om en allvarlig olycka skulle inträffa. Som en del av GEN-IV-
utvecklingen i Sverige lanserades 2009 GENIUS-projektet för utveckling av
GEN-IV-konceptet i ett svenskt sammanhang [6]. Projektet var ett samar-
bete mellan Chalmers tekniska högskola, Kungliga Tekniska högskolan och
Uppsala universitet. Inom ramen för projektet föreslogs utvecklingen av en
blykyld snabbreaktor som kallas ELECTRA, European Lead-Cooled Training
Reactor, som ska klara av en fullständig återvinning av plutonium och ameri-
cium i härden.
För ett framgångsrikt införande av GEN-IV-reaktorer krävs kärndata av hög
66
kvalitet. Kärnadata kan t.ex. vara sannolikheten för en kärnreaktion. Denna
sannolikhet benämns tvärsnitt. Ett exempel på ett tvärsnitt är fissionstvärsnit-
tet, d.v.s. sannolikheten för att en atomkärna ska klyvas.
Innan man kan använda kärndata i tillämpningar måste de evalueras, valid-
eras mot integrala experiment och konverteras till format som kan användas
för olika tillämpning. Evalueringsprocessen har tidigare skett med hjälp av
differentiella experimentella data som sedan kompletterades med kärnmodell-
beräkningar. Detta förfaringssätt håller på att förändras tack vare avsevärda
förbättringar av kärnreaktionsteorier och tillgången till ökad beräkningska-
pacitet. Eftersom kärnmodellkoderna inte är perfekta valideras de vanligtvis
mot en omfattande uppsättning experimentella data. Dessa data är dock inte
exakta och därför innehåller även storheter beräknade med hjälp av model-
lkoder, t.ex. tvärsnitt, osäkerheter. Eftersom kärndata används i reaktorkoder
som indata för simuleringar, innehåller resultaten från reaktorkoderna osäk-
erheter på grund av dessa data. För att kunna bedöma reaktorsäkerheten är
det därför viktigt att kunna kvantifiera dessa osäkerheter. Vi vill också kunna
bestämma var det krävs ytterligare ansträngning för att minska dessa osäker-
heter.
Tills nyligen skedde fortplantning av osäkerheter till största delen med hjälp
av deterministiska metoder. Med detta tillvägagångssätt kan man avgöra o-
säkerheter för specifika reaktorparametrar, som t.ex. reaktorkriticiteten, med
hjälp av störningsberäkningar [7]. Denna metod innehåller en rad antagan-
den och förenklingar [10]. Med tillgång till ökad datorkraft kan mer exakta
metoder baserade på Monte Carlo-beräkningar användas. I Nuclear Research
and Consultancy Group (NRG), Petten, Nederländerna, har man utvecklat en
metod som kallas ’Total Monte Carlo (TMC)’ för evaluering av kärndata och
fortplantning av osäkerheter.
I mitt arbete har jag tillämpat TMC-metoden för att studera osäkerheter i
reaktorsäkerhetsparametrar för ELECTRA-reaktorn på grund av osäkerheter i
kärndata. Kärndatan kan då också kopplats till olika kärnfysikparametrar. Jag
har studerat betydelsen av aktinider i bränslet, strukturmaterial och kylmedel.
Detta arbete är viktigt eftersom säkerhetsmarginaler för viktiga reaktorsäk-
erhetsparametrar kan bli felaktiga om inte alla osäkerheter, inklusive kärn-
dataosäkerheter, beaktas vid utformningen av reaktorn. Denna avhandling om-
fattar även utveckling inom TMC-metoden. En sammanfattning av de artiklar
som ingår i denna avhandling presenteras nedan.
I ARTIKEL I kombinerar jag Total Monte Carlo-metoden med integrala ex-
periment (benchmarks) för fortplantning av osäkerheter från kärndata. Jag ob-
serverade att osäkerheter i kärndata kan minskas avsevärt genom att introduc-
era ett kriterium för acceptans/avvisning baserat på de integrala experimenten.
I ARTIKEL II använde jag TMC-metoden för att studera påverkan av kärn-
67
data osäkerheterna i blyisotoperna på reaktorsäkerhetsparametrar för ELEC-
TRA. Som en del av arbetet studerade jag både osäkerheter på isotopnivå
samt vilka delar av kärndata som bidrog till osäkerheten. Jag studerade även
korrelationer mellan tvärsnitts och reaktorparametrar med hjälp av en korrela-
tionsbaserad känslighetsmetod. Både voidkoefficient och kriticitet studerades.
Stora osäkerheter observerades i kriticiteten för alla isotoper (med undantag av204Pb), med ett betydande bidrag från 208Pb-data. Det dominerande bidraget
till osäkerheten i reaktorparametrarna härstammade från osäkerheter i reso-
nansparametrar.
I ARTIKEL III användes TMC-metoden för att fortplanta osäkerheter i 239Pu-
kärndata till osäkerheter i sammansättningen, dvs. isotopinnehållet, av ELEC-
TRAs bränsle under reaktorns livstid. Resultaten visar att osäkerheterna för en
del minoritets-aktinider var tillräckligt stora för att man vid framtida forskning
bör fortsätta undersöka deras påverkan på säkerhetsparametrar vid upparbet-
ning. Det observerades även att integrala experiment kan användas för att
minska osäkerheterna.
I ARTIKEL IV föreslår jag två olika metoder för att minska osäkerheten i
kärndata med hjälp av en uppsättning integrala experiment. Efter att jag tilläm-
pat de två metoderna observerade jag en avsevärd reduktion av osäkerheten i
kriticitet p.g.a. osäkerheter i 239Pu och 208Pb-kärndata.
I ARTIKEL V föreslår jag en metod för att välja integrala experiment för
reaktorberäkningar som kommer att hjälpa till att validera reaktorkoder genom
att kvantifiera likheter mellan en reaktortillämpning och ett eller flera inte-
grala experiment. Studien har särskild betydelse eftersom man i allmänhet vill
ersätta kostsamma fullskaliga experiment med enklare integrala experiment.
Metoden tillämpades på ELECTRA med hjälp av en uppsättning integrala krit-
icitetsexperiment.
Som en del av denna avhandling föreslår jag även en metod för att kombin-
era differentiella experiment och integrala experiment för korrigering av kärn-
data med hjälp av filvikter baserat på Maximum likelihood-funktionen. Den
föreslagna metoden tillämpades för korrigering av neutroninducerad 208Pb-
kärndata i det snabba energiområdet. Denna metod öppnar upp för flera nya
möjligheter för hur man kan använda både differentiella och integrala experi-
ment i kärndataevaluering.
68
AcknowledgmentI would like to express my sincere gratitude and appreciation to my team of
supervisors made up of Henrik Sjöstrand, Dimitri Rochman, Michael Öster-
lund and Stephan Pomp. Without your careful guidance, patience and sup-
port, this thesis would not have been possible. To Henrik, thanks for your
great interest in my work and your infectious enthusiasm; it would have been
difficult without your continuous stream of ideas. Thanks also for the late
nights ... To Dimitri, thanks for your generosity with your knowledge, and for
your patience to listen to my crazy ideas that sometimes didn’t make sense.
Thanks also for teaching me ’Total Monte Carlo’; Michael, I am grateful for
the several discussions and also for your help with my visa and resident per-
mit matters. To the man who hired me Stephan, the ’supreme leader’ (I should
reference Cecilia G. here, I think), thanks for believing in me.
I would also like to acknowledge financial support from the Swedish Research
Council through the GENIUS project.
Special thanks to Arjan J. Koning from the IAEA, Vienna and Steven van der
Marck from NRG, Petten, for the great discussions and regular feedbacks that
this thesis has benefited from.
It is also my pleasure to have made acquaintance with the following people
with whom I have had discussions on various interesting subjects and issues:
Temitope Taiwo, Jan Blomgren, Andreas Pautz, Hakim Ferroukhi, Ernst van
Groningen and Mattias Klintenberg.
I wish also, to express my gratitude to the staff, colleagues and friends at
the Division of Applied Nuclear Physics who have made my stay in Uppsala
fun: Mattias L., Cecilia, Ali, Tom, Augusto, Petter, Andrea, Iwona, Federico,
Matilda, Peter, Bill, Andreas, Vasily, Kaj, Diego, Eric, Carl, Sophie,..., and
also to Inger and Marja from the department administration. Ali, Andrea, Kaj
and Bill; it was fun to have you guys around at some conferences and summer
schools, you should visit me in Ghana :-). To my beer buddies Tom and Au-
gusto, thanks for the time out at O’learys.
To my African team here in Uppsala: Sidi from Mauritania, Francisco from
Mozambique, Baba from Senegal and Gerald from Uganda, thanks to y’all for
the beers and exciting night outs; Francisco, keep the tradition alive.
To the doctoral board of the Faculty of Science and Technology, Uppsala Uni-
versity (TNDR) for the 2014/2015, you guys were awesome; you made my
tenure as chair much fun. Good-luck in your future careers.
Family now: To my parents, and siblings: Emelia, Elliot, Elvis and Eva;
thanks for your support and encouragement. I am also grateful to our ’Vienese
family’ (The Scherhak family) for being a wonderful family to me and my
wife here in Europe.
Finally, to Mabel & Emily, thanks for making my life complete.
69
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