The Pennsylvania State University
The Graduate School
College of Engineering
EXAMINATION AND IMPROVEMENT OF THE SHEM ENERGY GROUP STRUCTURE
FOR HTR AND DEEP BURN HTR DESIGN AND ANALYSIS
A Dissertation in
Nuclear Engineering
by
Tholakele Prisca Ngeleka
© 2012 Tholakele Prisca Ngeleka
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2012
The dissertation of Tholakele Prisca Ngeleka was reviewed and approved* by the following:
Kostadin N. Ivanov
Distinguished Professor of Nuclear Engineering
Dissertation Advisor
Chair of Committee
Maria Avramova
Assistant Professor of Nuclear Engineering
Samuel Levine
Professor Emeritus of Nuclear Engineering
Chimay J. Anumba
Professor of Architectural Engineering
Arthur Motta
Chair of Nuclear Engineering Program
*Signatures are on file in the Graduate School.
ii
Abstract
The purpose of this study was to study and improve the SHEM energy group structures (281 and
361) and General Atomics-193 energy group structure utilizing the more systematic, consistent,
and sophisticated energy group selection method referred to as contributon and point-wise crosssection driven (CPXSD) method. The SHEM-281 and -361 energy group structures were
developed for LWR and General Atomics energy group structure was developed for the fast
reactors. Pebble bed and Prismatic hexagonal block type fuel are used for cell analysis.
DRAGON transport code was used for this task taking advantage of its capability to compute
adjoint fluxes for reactor analysis. MCNP5 was used for generation of the reference solution
selected due to its accuracy of neutron transport calculations. Comparisons with DRAGON
calculations are presented. Pebble fuel element and Prismatic hexagonal block models were
created for both codes. In the DRAGON code, analysis are conducted for the starting energy
group structure by computing both forward and adjoint fluxes as well as the reaction rates and keffective. Then forward and adjoint fluxes were used in computing the importance function of
the groups, and the groups with high importance function are subdivided accordingly. The whole
energy group interval of interest was divided into fast, epithermal and thermal regions. Firstly,
the improvement was done for fast region and a new library was created and applied in the fuel
cell analysis until the selected target criteria’s were met (10 pcm relative deviation of ∆k / k and
1 percent deviation of reaction rate of interest). Then similar procedure was repeated for
epithermal and thermal regions. The dominant parameters for each energy region were
considered as required such as the fission cross section for fast region, absorption and scattering
iii
cross sections for epithermal region and absorption cross section for thermal region and keffective applied for all energy regions. Pebble fuel element and the Prismatic hexagonal block
were analyzed for depletion based on the improved energy group structure SHEM_TPN-531.
iv
Table of Contents
List of Figures ........................................................................................................................... ix
List of Tables ............................................................................................................................ xi
Acknowledgements ................................................................................................................ xvii
Chapter 1 ....................................................................................................................................1
Introduction ................................................................................................................................1
1.1 Background .......................................................................................................................1
1.2 Motivation .........................................................................................................................5
1.3 Research Objectives ...........................................................................................................7
1.4 Scope of Research..............................................................................................................8
Chapter 2 ..................................................................................................................................10
Literature Review ......................................................................................................................10
2.1 Introduction .....................................................................................................................10
2.2 Nuclear Energy Group Structures.....................................................................................10
2.2.1 SHEM Energy Group Structure .................................................................................11
2.2.2 Ultra Fine Energy Group Structure ............................................................................16
2.2.3 Contributon and Point-Wise Cross Section Driven Method........................................17
2.2.4 HTR Energy Group Structure Selection Method ........................................................18
2.2.5 An Adaptive Energy Group Constructor ....................................................................19
Chapter 3 ..................................................................................................................................21
Nuclear Physics Theory.............................................................................................................21
v
3.1 Introduction .....................................................................................................................21
3.2 Transport Equation ..........................................................................................................21
3.2.1 Angular Discretization...............................................................................................25
3.2.2 Energy Discretization ................................................................................................28
3.3 Adjoint Transport Equation..............................................................................................30
3.4 Neutron Energy Regions ..................................................................................................31
3.4.1 Neutron Slowing Down .............................................................................................32
3.4.2 Resonance Absorption ...............................................................................................34
3.4.3 Neutron Thermalization .............................................................................................39
Chapter 4 ..................................................................................................................................40
Monte Carlo Reference Results .................................................................................................40
4.1 Introduction .....................................................................................................................40
4.2 MCNP5 Code Description ...............................................................................................40
4.3 HTR Fuel Specifications ..................................................................................................41
4.4 MCNP5 Results ...............................................................................................................46
Chapter 5 ..................................................................................................................................61
Multi-group Structure Analysis .................................................................................................61
5.1 Introduction .....................................................................................................................61
5.2 DRAGON Code Description ............................................................................................61
5.3 Cross Section Library Generation ....................................................................................67
5.4 Sensitivity Analysis Results .............................................................................................72
5.4.1 DRAGON Results used in the Sensitivity Study ........................................................72
5.4.2 Sensitivity Analysis Comparison with MCNP5 Results .............................................75
Chapter 6 ..................................................................................................................................82
Multi-group Energy Structure Improvement ..............................................................................82
vi
6.1 Introduction .....................................................................................................................82
6.2 Contributon and Point-Wise Cross Section Driven Method ..............................................83
6.3 Improvement of SHEM-281 Energy Group Structure.......................................................85
6.3.1 Fast Energy Region Improvement..............................................................................85
6.3.2 Epithermal Energy Region Improvement ...................................................................90
6.3.3 Thermal Energy Region Improvement .......................................................................93
6.3.4 Improved SHEM-281 Energy Group Structure for all Regions ..................................98
6.4 SHEM-361 Energy Group Structure Results .................................................................. 104
6.4.1 Fast Energy Region Improvement............................................................................ 105
6.4.2 Epithermal Energy Region Improvement ................................................................. 110
6.4.3 Thermal Energy Region Improvement ..................................................................... 114
6.4.4 Improved Energy Group Structure for all Regions (SHEM-361) .............................. 119
6.5 General Atomics-193 Energy Group Structure ............................................................... 125
6.5.1 Fast Energy Region Improvement............................................................................ 125
6.5.2 Epithermal Energy Region Improvement ................................................................. 130
6.5.3 Thermal Energy Region Improvement ..................................................................... 135
6.5.4 Improved GA-193 Energy Group Structure for all Regions...................................... 140
Chapter 7 ................................................................................................................................ 147
Comparative Analysis of ENDF Data Files ............................................................................. 147
7.1 Introduction ................................................................................................................... 147
7.2 Comparison Results ....................................................................................................... 147
7.3 Nuclear Data Advancements .......................................................................................... 153
Chapter 8 ................................................................................................................................ 156
Depletion Analysis .................................................................................................................. 156
8.1 Introduction ................................................................................................................... 156
8.2 Pebble Fuel Element ...................................................................................................... 158
8.3 Prismatic Hexagonal Block ............................................................................................ 163
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Chapter 9 ................................................................................................................................ 169
Conclusions and Recommendations ........................................................................................ 169
9.1 Conclusions ................................................................................................................... 169
9.2 Recommendation ........................................................................................................... 174
References .............................................................................................................................. 176
Appendices ............................................................................................................................. 181
A1 MCNP5 Input Decks ...................................................................................................... 181
A1.1 Pebble Fuel Element ................................................................................................ 181
A1.2 Prismatic Hexagonal Block...................................................................................... 187
A2 DRAGON Input Decks .................................................................................................. 200
A2.1 Pebble Fuel Element ................................................................................................ 200
A2.2 Prismatic Hexagonal Block...................................................................................... 203
A3 Energy Group Structures ................................................................................................ 208
A3.1 SHEM-361 .............................................................................................................. 208
A3.2 SHEM-281 .............................................................................................................. 211
A3.3 GA-193 ................................................................................................................... 213
A3.4 SHEM_TPN-407 ..................................................................................................... 215
A3.5 SHEM_TPN-531 ..................................................................................................... 218
A3.6 GA_TPN-537 .......................................................................................................... 222
A4 Depletion Data Analysis ................................................................................................ 226
A4.1 Pebble Fuel Element Nuclides Concentrations ......................................................... 226
A4.2 Pebble Fuel Element Fission Products Concentrations ............................................. 227
A4.3 Prismatic Hexagonal Block Nuclides Concentrations ............................................... 229
A4.4 Prismatic Hexagonal Block Fission Products Concentrations ................................... 230
A4.5 Pebble Fuel Element Criticality Data per Burnup Step ............................................. 232
A4.6 Prismatic Hexagonal Block Criticality Data per Burnup Step................................... 232
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List of Figures
Figure 3.1: The position and direction characterizing a neutron .................................................22
Figure 3.2: Low energy resonances for U-238 ...........................................................................34
Figure 3.3: Low energy resonances for Th-232..........................................................................35
Figure 3.4: Flux depreciation under the resonance region ..........................................................36
Figure 3.5: Unresolved resonances for U-235 (t2.lanl.gov) ........................................................37
Figure 3.6: Unresolved resonances for U-238 (t2.lanl.gov) ........................................................37
Figure 4.1: MCNP5 Pebble model .............................................................................................48
Figure 4.2: MCNP5 Prismatic block model ...............................................................................49
Figure 4.3: Pebble keffective and source convergence ...............................................................54
Figure 4.4: Prismatic keffective and source convergence ...........................................................54
Figure 5.1: DRAGON flow chart [11] .......................................................................................66
Figure 5.2: Flow chart for DRAGON library production ...........................................................71
Figure 6.1: Importance function for fast energy region for 281, 309 1nd 323 groups. ................ 87
Figure 6.2: Importance function for epithermal energy region for 281 and 333 groups .............. 91
Figure 6.3: Importance function for thermal energy region for 281, 297 and 313 groups ........... 95
Figure 6.4: Importance function for fast energy region for 361,389 and 403 groups................. 106
Figure 6.5: Importance function for epithermal energy region for 361 and 455 groups ............ 111
Figure 6.6: Importance function for thermal energy region for 361, 393 and 395 groups ......... 115
Figure 6.7: Importance function for fast energy region for 193 and 227 groups ....................... 127
Figure 6.8: Importance function for epithermal energy region ................................................. 132
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for 193, 235, 319, 355, 399 and 485 groups ............................................................................. 132
Figure 6.9: Importance function for thermal energy region for 193, 205 and 211 groups ......... 137
Figure 7.1: Neutron capture on natural carbon for ENDF/B-VII.1 ........................................... 149
and ENDF/B-VII.0 (Chadwick et al, 2011).............................................................................. 149
Figure 8.1: K-effective versus burnup ..................................................................................... 159
Figure 8.2: Neutron flux spectrum at the beginning of life ....................................................... 160
Figure 8.3: Neutron flux spectrum at the end of life ................................................................. 160
Figure 8.4: Nuclides concentration in atom/barm.cm ............................................................... 162
Figure 8.5: Fission products buildup in a Pebble fuel element ................................................. 163
Figure 8.6: Prismatic block K-effective versus burnup ............................................................ 164
Figure 8.7 Neutron flux spectrum at the beginning of life ........................................................ 164
Figure 8.8: Neutron flux spectrum at the end of life ................................................................. 165
Figure 8.9: Nuclides concentration in atom/barn.cm ................................................................ 167
Figure 8.10: Fission products buildup in a Prismatic hexagonal block ..................................... 168
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List of Tables
Table 4.1: Coated Particle specifications (common for all types of fuel) ....................................43
Table 4.2: Material specification (common for all fuel types) ....................................................44
Table 4.3: Pebbles representative of PBMR fuel ........................................................................45
Table 4.4: Prismatic fuel lattice data ..........................................................................................46
Table 4.5: Pebble fuel element results calculated using ENDF/B-VII.0 .....................................50
Table 4.6: Pebble fuel element results calculated using ENDF/B-VI.8.......................................50
Table 4.7: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8 ........... 51
Table 4.8: Prismatic block results calculated using ENDF/B-VII.0 ............................................52
Table 4.9: Prismatic block results calculated using ENDF/B-VI.8 .............................................52
Table 4.10 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8 .. 53
Table 4.11: Pebble fuel element results calculated using ENDF/B-VII.0 ...................................55
Table 4.12: Pebble fuel element results calculated using ENDF/B-VI.8 .....................................56
Table 4.13: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8 ......... 56
Table 4.14: Prismatic block results calculated using ENDF/B-VII.0 ..........................................57
Table 4.15: Prismatic block results calculated using ENDF/B-VI.8 ...........................................57
Table 4.16 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8 .. 58
Table 4.17: ENDF/B-VII.0 and ENDF/B-VI.8 data files improvement comparisons. .................60
Table 5.1: Pebble reaction rates and criticality for SHEM-281 energy group structure ............... 72
Table 5.2: Prismatic block reaction rates and criticality for SHEM-281 energy group structure . 73
Table 5.3: Pebble reaction rates and criticality for SHEM-361 energy group structure ............... 73
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Table 5.4: Prismatic block reaction rates and criticality for SHEM-361energy group structure .. 74
Table 5.5: Pebble reaction rates and criticality for GA-193 energy group structure ....................74
Table 5.6: Prismatic block reaction rates and criticality for GA-193 energy group structure ...... 75
Table 5.7: Criticality calculation comparisons for ENDF/B-VII.0..............................................76
Table 5.8: Criticality calculation comparisons for ENDF/B-VII.0..............................................76
Table 5.9: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0........................ 77
Table 5.10: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0.............. 78
Table 5.11: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0...................... 80
Table 5.12: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0.............. 81
Table 6.1: Fast group selected in the fast range..........................................................................86
Table 6.2: Eigen-value results for fast energy group structure improvement ..............................87
Table 6.4: Prismatic block results ..............................................................................................89
Table 6.5: Epithermal energy groups selected in the epithermal range .......................................90
Table 6.6: Eigen-value resulted for epithermal energy group structure improvement ................. 90
Table 6.7: Pebble FE results ......................................................................................................92
Table 6.8: Prismatic block results ..............................................................................................93
Table 6.9: Thermal groups selected in the thermal region ..........................................................94
Table 6.10: Eigen-values resulted for thermal energy group structure improvement .................. 94
Table 6.11: Pebble FE results ....................................................................................................96
Table 6.12: Prismatic block results ...........................................................................................97
Table 6.13: Energy group structure improved from SHEM-281 to 407 ......................................98
Table 6.14: Reaction for SHEM-281 and 407 energy group structures .......................................99
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Table 6.15: Reaction for SHEM-281 and 407 energy group structures ..................................... 100
Table 6.16: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the
pebble FE ................................................................................................................................ 101
Table 6.17: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the
prismatic block ........................................................................................................................ 101
Table 6.18: MCNP5 results for the pebble ............................................................................... 102
Table 6.19: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the
pebble FE ................................................................................................................................ 103
Table 6.20: MCNP5 results for the prismatic block ................................................................. 103
Table 6.21: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the
prismatic block ........................................................................................................................ 104
Table 6.22: Fast group selected in the fast range ...................................................................... 105
Table 6.23: Eigen-value results for fast energy group structure improvement .......................... 107
Table 6.24: Pebble FE results .................................................................................................. 108
Table 6.25: Prismatic block results .......................................................................................... 109
Table 6.26: Epithermal energy groups selected in the epithermal range ................................... 110
Table 6.27: Eigen-value resulted for epithermal energy group structure improvement ............. 111
Table 6.28: Pebble FE results .................................................................................................. 112
Table 6.29: Prismatic block results .......................................................................................... 113
Table 6.30: Thermal groups selected in the thermal region ...................................................... 114
Table 6.31: Eigen-values resulted for thermal energy group structure improvement ................ 116
Table 6.32: Pebble FE results .................................................................................................. 117
Table 6.33: Prismatic block results .......................................................................................... 118
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Table 6.34: Energy group structure improved from SHEM-361 to SHEM_TPN-531 ............... 119
Table 6.35: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures ....... 120
Table 6.36: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures ....... 121
Table 6.37: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the
pebble FE ................................................................................................................................ 122
Table 6.38: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the
prismatic block ........................................................................................................................ 122
Table 6.39: MCNP5 results for the pebble FE ......................................................................... 123
Table 6.40: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the
pebble FE ................................................................................................................................ 123
Table 6.41: MCNP5 results for the prismatic block ................................................................. 124
Table 6.42: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the
pebble FE ................................................................................................................................ 124
Table 6.43: Fast group selected in the fast range ...................................................................... 126
Table 6.44: Eigen-values resulted for fast energy group structure improvement ....................... 126
Table 6.45: Reaction rates for GA-193 and 227 energy group structures.................................. 128
Table 6.46: Reaction rates for GA-193 and 227 energy group structures.................................. 129
Table 6.47: Epithermal group selected in the epithermal range ................................................ 131
Table 6.48: Eigen-values resulted for epithermal energy group structure improvement ............ 131
Table 6.49: Reaction rates for GA-193, 235, 319, 355, 399, and 485 energy group structures .. 133
Table 6.50: Reaction rates for GA-193, 235, 319, 355, 399, 485 energy group structures ........ 134
Table 6.51: Thermal groups selected in the thermal region ...................................................... 135
Table 6.52: Eigen-values resulted for thermal energy group improvement ............................... 136
xiv
Table 6.53: Pebble FE results .................................................................................................. 138
Table 6.54: Prismatic block results .......................................................................................... 139
Table 6.55: Energy group structure improved from GA-193 to GA_TPN-537 ......................... 140
Table 6.56: Reaction rates for GA-193 and GA_TPN-537 energy group structures.................. 141
Table 6.57: Reaction rates for GA-193 and GA_TPN-537 energy group structures.................. 142
Table 6.58: Comparisons for 537 with GA-193 energy group structure for the pebble FE. ....... 143
Table 6.59: Comparisons for GA_TPN-537 with GA-193 energy group structure for the
prismatic block. ....................................................................................................................... 143
Table 6.60: MCNP5 results for the pebble FE ......................................................................... 144
Table 6.61: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the
pebble FE ................................................................................................................................ 145
Table 6.62: MCNP5 results for the prismatic block. ................................................................ 145
Table 6.63: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the
prismatic block. ....................................................................................................................... 146
Table 7.1: Pebble FE results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons ................ 150
Table 7.2: END/B-VII.0 and ENDF/B-VII.1 % deviations for the pebble reaction rates and keffective in pcm ..................................................................................................................... 151
Table 7.3: Prismatic block results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons........ 152
Table 7.4: END/B-VII.0 and ENDF/B-VII.1 % deviation for the prismatic block reaction rates
and k-effective in pcm ........................................................................................................... 153
Table 7.5: ENDF/B-VII.1, ENDF/B-VII.0 and ENDF/B-VI.8 data file advancements ............. 155
Table 8.1: Nuclides concentration in atom/barm.cm ................................................................ 161
Table 8.2: Fission products in atom/barn.cm ........................................................................... 162
Table 8.3: Nuclides concentration in atom/barn.cm ................................................................. 166
xv
Table 8.4: Fission products in atom/barn.cm ........................................................................... 168
Table 9.1: Pebble fuel element deviations ................................................................................ 170
Table 9.2: Prismatic hexagonal block deviations ..................................................................... 171
xvi
Acknowledgements
I give great honor to God Almighty for giving me the strength, confidence and patience for
undertaking the study. Thank you for providing a great team to work with and all opportunities
accompanied the study. A conducive environment and all the campus community and friendly
student organizations, worth mentioning is the International Christian Fellowship.
My sincere gratitude goes to Dr Kostadin N. Ivanov for his supervision and assistance
throughout the study. I thank him for giving me this opportunity to be part of his research group,
Fuel Dynamics and Management Research Group. I am thankful to Dr Samuel Levine, for his
contribution throughout the study.
My special thanks to the Department of Mechanical and Nuclear Engineering of the
Pennsylvania State University, and the members of the Reactor Dynamics and Fuel Management
Research Group for the assistance and support.
I greatly appreciate help rendered by the Idaho National Laboratory (INL) for funding this study.
Thank you to Abderrafi Ougouag, Hans Gougar for being part of this work.
xvii
Not forgetting the technical support obtained from Alan Hebert and Vincent Descotes of the
Ecole Polytechnique de Montreal, Michael Pope of INL, Volkan Seker of the University of
Michigan.
Thanking parents, family and friends for their support, encouragements and lot of prayers.
xviii
Chapter 1
Introduction
1.1 Background
Nuclear energy is presently contributing around 14 % of the world’s electric energy needs and
produces vast amount of energy from a small amount of fuel. Its advantages lie on the fact that it
does not have green gas emissions (CO 2 , SO 2 and NO 2 ) since it is a major problem with the
current fossil fuel power generation industries. Since the advent of nuclear reactor industry,
many power reactors have been designed and operated such as Light water reactors (Pressurized
and Boiling water reactors), Heavy water reactors (CANDU) and Magnox gas cooled reactor
(named after its cladding material magnesium non-oxidizing alloy) [34]. Gas cooled type
reactors were also designed and operated such as German Arbeitsgemeinschaft Versuchs Reaktor
(AVR) for testing of fuel and other reactor components; however, it was shut down after high
temperature fuel instabilities. A demonstration plant (Peach Bottom) and the fuel development
plants, the DRAGON reactors were built in the United States of America (USA) but were shut
down following technical challenges. Therefore, gas cooled nuclear reactor research and
development was not performed for quite some time.
1
In spite of the challenges that gas cooled reactors encountered, the inherent safety features and
high outlet temperatures have caused recent interest throughout the world for initiating research
to improve their design. They are currently among the proposed generation IV nuclear reactors
(NGNP) that are under development, namely gas cooled fast reactor (GFR), very high
temperature reactor (VHTR), all grouped under high temperature reactors (HTRs). These
reactors promise to provide a secure nuclear system using materials with improved management
of nuclear waste, effective utilization of fuel, economically sound characteristics, and a well
developed safety performance. Their high temperature outlet can be utilized in other processes
such as the hydrogen production and thus make these reactors more useful [9]. There is a need to
improve the accuracy of HTR analysis to make HTRs more economically viable. Improvement
can be made by reducing the computer analysis uncertainties and using more sophisticated
methods.
Due to recent interests on these concepts, research facilities were constructed. At this stage, the
research (experimental) facilities in operation are a 10 megawatt (MW) high temperature reactor
(HTR-10), a high temperature engineering test reactor (HTTR) and the ASTRA critical facility
[33] to obtain detailed understanding of HTRs. All these facilities use helium gas as a coolant
and are graphite moderated with high temperature outlet of approximately 850°C to 950°C.
HTR-10 is a pebble bed type reactor that has been in operation at the Institute of Nuclear Energy
in China (INET) since December 2002. HTTR is a 30 MW thermal power reactor using
hexagonal assemblies (pin in block) type of fuel, operated in Oarai research reactor of Japan
Atomic Research Institute.
2
Whereas ASTRA is a Russian zero power critical facility at the Kurchatov Institute built for the
investigation of neutronics of high temperature reactors such as the GT-MHR reactor under
development at General Atomics in USA, [30].
Major developments are now focusing on General Atomics design of the GT-MHR and on the
VHTR under development at the Idaho National Laboratory (INL). Both GT-MHR and VHTR
use a prismatic block fuel type. South Africa had major contributions into the HTR research and
developments through the Pebble Bed Modular Reactor that was designed to use pebbles, an
extension of the German AVR. The similarities of the Pebble fuel element and the prismatic
hexagonal block fuel types are that they are both embedded with tri-structural isotopic coated
particles. These coated particles consists of a kernel (UO 2 ), low density carbon (buffer layer),
pyrolytic carbon (inner and outer layers) and silicon carbide. Consequently, these coated
particles make an HTR design to be a double heterogeneous system, firstly between the particles
in a graphite matrix and then between the fuel element and other structural materials like
moderator and a reflector. This results in a significant change in the physics of neutron slowing
down, absorption and scattering processes in the high temperature reactors as compared to that in
the light water reactors. Hence, advanced ways of treating double heterogeneity in graphite
moderators and neutron behavior in these reactors is of utmost importance.
3
One of the tasks in meeting the research objectives of the development program is to obtain
excellent fine energy group structures that allow accurate calculation of neutron cross sections
for reactor analysis. Neutron cross section is the probability that the various types of nuclear
reactions (fission, scattering and absorption) will occur. The most important dependent
parameters during the reactions for the equilibrium neutron conditions are energy, direction, and
space position. Neutron group cross sections play a very important role in reactor core analysis
and design calculations [8]. Neutron group cross sections are generated from continuous cross
section data files that have been collected for the past decades by experimental measurements
and theoretical calculations (in the United States of America). These data files contain all of
various types of neutron cross sections, e.g. scattering, absorption etc. for all the important
nuclides resulting in a vast volume of data stored in standardized format in the Evaluated
Nuclear Data File (ENDF). The ENDF data files contain both neutron and photon cross sections.
Its latest version is ENDF/B, which contains the complete and evaluated data ready for use by
the nuclear designers and nuclear analysts. Some nuclear data files also exist in other countries
such as Evaluated Nuclear Data Library of the Lawrence Livermore National Laboratory
(ENDL), United Kingdom Nuclear Data Library (UKNDL), Japanese Evaluated Nuclear Data
Library (JENDL), Karlsruhe Nuclear Data File (KEDAK), Russian Evaluated Nuclear Data File
(BROND) and Joint Evaluated Fission File of NEA Countries (JEFF), etc.
4
1.2 Motivation
To advance the development of the accurate analysis of the HTR and ongoing relevant research,
INL, in collaboration with Pennsylvania State University (PSU), have been engaged on
sensitivity studies of energy group structures for the analysis of HTRs [10]. Their previous study
addressed the broad energy group structure used in analyzing graphite moderated high
temperature gas cooled reactors. It is likely that one of the largest sources of error encountered in
that study [10] stems from the method through which nodal leakage was incorporated into the
spectrum calculation.
In these analysis, tables of neutron cross sections were generated for a few selected values of fast
and thermal buckling. Neutron cross sections for the core simulation were interpolated from
these tables using two group internodal currents, omitting much of the information contained in
the fine group energy spectrum. Furthermore, the fine energy group structure hardwired within
INL’s COMBINE-6 code that was used to generate broad group constants is fixed for light water
reactors and has not been optimized for HTR analysis. If the fine energy group structure is not
sufficiently refined in energy regions of importance, such a group structure may prevent the
flexibility needed for more accurate cell level energy collapsing. COMBINE-6 has also some
limitations that must be addressed explicitly in their own right.
As a result, the examination of a new fine energy group structure that is optimized for the
compositions expected in the (Next Generation Nuclear Plant) NGNP was necessary. Hence the
5
goal was to study the currently available energy group structures that are in use like SHEM
energy group structures of Santamarina and Hfaiedh [17] that were developed for LWRs as well
as General Atomics energy group structure, which was developed for the fast reactors.
The SHEM energy group structure has been optimized for LWR applications, addressing their
fuel components as well as the structural materials expected to be present in them. It was verified
to be accurate for both uranium and mixed oxide fuels in LWRs. In addition, it does address
actinides extensively in that their resonant reactions are well covered by the structure. This may
imply that the SHEM energy group structure might be applicable to NGNP and deep burn (DB)
applications without further modifications. However, it is uncertain whether it addresses all the
actinides that arise in the case of very high burnup applications, such as those contemplated for
NGNP and DB-NGNP. Furthermore, the SHEM energy group structure does not give any
particular attention to graphite and its slowing down properties. Although the graphite neutron
cross sections appear very smooth (with no resonance structures) at all energies below about 2
MeV, the absence of special consideration of graphite may imply an inadequate coverage of the
NGNP and deep burn physical situations. Therefore it was necessary to examine this potential
shortcoming of the SHEM energy group structure, and be modified to cover all NGNP and deep
burn related physical phenomena.
General Atomics energy group structure developed for fast energy reactors has been repeatedly
used for the analysis of HTRs without consideration of the effects that may be caused by the
6
underlying purpose of its development. At this point there are no details on the nuclear reactions,
isotopes or any materials addressed for this energy group structure. Pelloni et al. [29] noted that
the GA-193 energy group structure has 92 fast energy groups below 14.92 MeV that are equally
spaced. GA-193 energy group structure is incorporated in MICROX-2 code system that creates
the broad group cross sections with resonance interference and self-shielding from fine group
and point-wise cross section. Therefore, it is important to give a thorough investigation on its
performance towards the thermal reactor physical phenomena.
1.3 Research Objectives
The purpose of this PhD research was to study and improve the SHEM energy group structures
(281 and 361) as well as GA-193 energy group structure utilizing the more systematic,
consistent, and sophisticated energy group selection method referred to as contributon and pointwise cross-section driven (CPXSD) [19], a method developed by PSU. The sensitivity analysis
on these energy group structures were conducted for the pebble fuel element and prismatic
hexagonal block based on PBR and VHTR respectively. Reference solution was developed with
the MCNP5 code, where three energy boundaries were selected for three regions that correspond
to the SHEM-281, SHEM-361 and GA-193 energy group structures. This study was based on
ENDF/B-VII.0 and ENDF/B-VI.8 nuclear data libraries. Then the energy group structures were
improved using the CPXSD method mentioned above. The best performing energy group
structure was selected for the depletion calculations on both pebble FE and Prismatic hexagonal
block.
7
1.4 Scope of Research
The outline is as follows:
 Chapter 1 presents an overview of the background, motivation of the study and the
description of the research objectives, as well as the research scope.
 Literature review is presented in chapter 2. The currently available energy group
structures and methods used during their development are summarized.
 In chapter 3, nuclear physics theory and neutron transport equation theory is
presented. Included is the Boltzmann transport equation accompanied by methods
that can be used in solving it. Basic information is provided on neutron slowing
down, resonance absorption and thermalization processes.
 Chapter 4 presents the description of the continuous energy Monte Carlo Neutron
Particle Transport (MCNP) code used to develop a reference solution for this study. A
detailed description of used HTR fuel specifications is given. Then, the results
(reference solution) along with the sensitivity analysis for the evaluated data files
(ENDF/B-VI.8 and ENDF/B-VII.0) are provided.
 A description of DRAGON, a multi-group deterministic transport code used for the
energy group structure improvement and burnup calculations is given in Chapter 5.
Methods utilized for cross section generation is also given in details. Multi-group
energy structure sensitivity analysis are shown.
8
 The energy group structure improvement for SHEM-281, SHEM-361 and GA-193
are shown in chapter 6. Each energy group structure was improved for three different
energy regions (fast, epithermal and thermal) separately. These were then compared
to the reference solution (MCNP5 results in chapter 4).
 Chapter 7 presents the sensitivity analysis for the evaluated data files (ENDF/B-VII.1
and NDF/B-VII.0) that were conducted in DRAGON for selected energy group
structures. The differences in both evaluated data files are also discussed.
 Depletion analysis and the results are discussed in chapter 8.
 Conclusions drawn from the study and recommendations made based on findings are
given in chapter 9.
9
Chapter 2
Literature Review
2.1 Introduction
This section gives an overview of the work conducted in developing energy group structures and
the method used by other researchers [1] [10] [17] [18] [24] [25]. It also highlights the
considerations made during those developments to account for important phenomena such as
self-shielding, resonance absorption and overlapping and other nuclear reactor physics
assumptions made.
2.2 Nuclear Energy Group Structures
The distribution of neutron production, slowing down and interaction in a core of a reactor
depends on the neutron energy. At high energy, neutron energy dependence is dominated by
fission spectrum. At intermediate energies, it is dominated by both neutron slowing down and
resonance absorption. While at the low energies it is dominated by thermalization process [8].
Since neutronic calculations form the basis of the research and development of the nuclear
industry, energy group structures have been studied to reduce the errors in these calculations.
However, physics related challenges such as self-shielding are the major concerns and many
researchers are involved in ongoing improvements of the currently used energy group structures
to eliminate all the errors experienced. These energy group structures are used to generate
10
neutron cross section libraries utilized by different transport and diffusion codes for reactor
simulations, analysis and design. Some codes e.g. Monte Carlo utilizes continuous energies and
some utilize multi-group energies e.g. DRAGON, COMBINE etc. Due to the current advances in
both computing and programming languages it has became feasible to develop and use finer
energy group structures for improved core calculations and simulations [18], so finer energy
group structures has been developed and in some places developments are still in progress to
reduce the errors in reactor calculations.
2.2.1 SHEM Energy Group Structure
One of the latest multi-group energy structure developed by Hfaiedh and Santamarina is the
optimized SHEM energy group structure that promised to eliminate simplified self-shielding
calculations of thermal resonances [17]. SHEM energy group structure determination was meant
to correct the setback that was encountered in the XMAS-172 neutron energy group structure
used and developed in CEA-UKEA. The intention was also to elude the use of self-shielding
model in U-238 first large resonances and also in those for LWR absorbers, such as Silver (Ag),
Indium (In), Cadmium (Cd) and Hafnium (Hf) claiming that the assumptions involved in XMAS
were to crude. Reason for elimination of self-shielding model was due to their hypotheses that
were made:
(i)
Pure slowing down,
(ii)
Asymptotic flux and fine structure uncoupling ( Φ ≅ ψ .ϕ )
(iii)
Wide resonance assumption,
11
P ij (inaccurate probability on a neutron interaction) from interface current method for
(iv)
Dancoff calculations
Work conducted by Mosca et al, [26] for fast reactors also showed similar problems related to
the self-shielding model approximations as applied in APPOLO-2. SHEM energy group
structure’s expectations were to account for the mutual-shielding effects (resonance
overlapping). The isotopes and resonances that were accounted for included mostly actinides,
fission products and absorbers, moderators/coolants and the structural materials. The resonances
of the isotopes were considered up to 23 eV . The actinides involved are U-234, U-235, U-236,
U-238, Np-237, Pu-238, Pu-239, Pu-240, Pu-241, Pu-242, Am-241, Am-243, Cm-243 and Cm244 whom resonances possibly appear at different energies within the range of interest. Whereas,
fission product resonances involved were for Xe-135, Sm-149, Rh-103, Xe-131, Cs-133, Pm147, Sm-152, Tc-99, Nd-145, Eu-153, Mo-95, Eu-155 and Eu-154. Absorber isotopes considered
were Ag-107, Ag-109, Cd-113, In-115, Hf-176, Hf-177, Hf-178 and Hf-179, and the burnable
poisons were Gd-155, Gd-157 and Er-167. For structural materials and coolants, the isotopes
considered were Mn-55, Ni-58, Fe-56, Al-27, Na-23 and O-16. The multi-group resonance
reaction rate (Equation 2.1) was compared to point-wise mesh reaction rate (Equation 2.2),
which was taken as a reference.
R mg = ∑ σ gφ g
2.1
g
R
ref
=
umax
∫
duσ ( u ) φ ( u )
2.2
umin
12
E 
u = log  0 
 E 
σg =
2.3
1
duσ ( u )
∆u ∫g
2.4
Where:
σ ( u ) - is the cross section at lethargy u
φ ( u ) - is the neutron flux at lethargy u
σ g - is the group g cross section
g - present a group number with flat slowing down flux assumption
φ g - is the group g flux
Resonant absorption approximation and fine structure is presented in equation 2.5 where
subscript c and m represents fuel and moderator respectively.
Vc Rcϕc Pcc + Vm Rmϕm Pmc =
Vc Rcϕc
2.5
Then a reciprocity theorem was applied, which details are found in Nasr and Roushdy [27]. The
flux was simplified to be ϕ ≅ ψ .φ , where ψ =
Rmϕm
- is the macroscopic slowing down flux. The
∑m
discretization technique of energy group structure was used with small group numbers
consequently reaching the required target of accuracy (about 1%) on reaction rate. Point-wise
13
resonances were calculated using Breit and Wigner multilevel approximation while the flux was
calculated based on heavy nuclide slowing down models such as wide resonance (WR) and
narrow resonance (NR) models. The NR model was used for high energy resonances (actinides)
and it assumes that the neutrons absorbed in the resonance have been scattered at energies higher
than that of the resonance. WR model was also used for low energy resonances with an
assumption that the resonance is very wide compared to the energy change of the scattered
nuclide. Though wide resonance assumptions has reported to cause problems for self-shielding
model, in this work it was used on resonant nuclide slowing down operator shown to give
accurate results. An algorithm was developed for the groups inside the resonance consisting of
discretization of energy variable for equal distribution of reaction rate errors between energy
groups of the resonance. The central peak group was determined and the central group
boundaries fixed as the nearest point-wise energy values around the peak energy value.
Then the group was extended from both ends of the resonance resulting in an increase in a group
error. The calculation was stopped when the total group error reached the target accuracy, which
was fixed in the external iteration. Group widths under the resonance were selected as wide as
possible without violating the established error.
The findings concluded that SHEM energy group structure (281 groups) has proven its accuracy
in calculating the resonance absorption up to 23 eV without using self-shielding approximation
method and also it managed to account for actinides, fission products and absorbers. It also
14
contributed in solving the mutual-shielding effects between resonant isotopes. The SHEM energy
group structure reproduced the structural material neutron cross sections and accounted for U238 threshold reactions, oxygen and sodium coolant resonances, which is well suited for fast
spectra calculations (e.g. LMFBR). The overall contribution of the SHEM energy group structure
optimization task was the introduction of an innovative energy scheme that will eliminate selfshielding model. Its contribution to calculation of reaction rates more accurately will be
advantageous for the nuclear industry, because it allows better design, safety, and control of the
reactor.
The SHEM energy group structure was later refined from 281 to 361 energy groups by Hebert
and Santamarina, [13] by expanding the number of groups in the ranges between 22.5 eV and
11.4 keV using subgroup projection method (SPM). This took advantage of the computing
resource of DRAGON code version 4.02. This work increased the number of groups in the above
mentioned energy ranges from 38 to 118 groups. The idea was to remove the slowing down
correlation model used before, and it is assumed that below 22.5 eV the effects were taken care
of in the previous optimization work and above 11.4 keV the correlation effects vanishes. SPM is
a subgroup approach based on the CALENDF type probability tables. Detailed description of this
method is published by Hebert and Santamarina [13].
15
2.2.2 Ultra Fine Energy Group Structure
Hurai and Ouislomen, 2008 [18] also developed an optimized ultra fine energy group structure
with the intention of avoiding the entire self-shielding problem, and also expected to eliminate
the errors that are caused by resonance interference and overlapping effects. This broadens their
applications to different fuel assemblies. Major concern was the simplifying approximations that
are used in PARAGON and WIMS for self-shielding treatment. The ultra fine energy group
structure has 6064 groups and it was developed based on SHEM energy group structure as
discussed above. It is also developed for light water reactors.
The group boundaries of the SHEM energy group structure were refined keeping in mind the
importance of the location and the practical widths of the resonances. A certain number of
iterations were performed before attaining the final ultra fine energy group structure. This work
used MCNP5 for comparisons of the group boundary selection method. The equation below
presents the iteration method used, wherein the cell iterates until equation 2.6 is satisfied.
Max σ Pg − σ Mg ≤ ε , ∀ g , ∀σ Mg ≥ ξ
2.6
Where ε was set to 3%, ξ was chosen to be equal to 10 barns, σ Pg is the multi-group cross
section, σ Mg is the Monte Carlo derived cross sections from the reaction rates and flux
calculations and ∀ represents all groups. Then, the resulting broad group distribution for their
proposed ultra fine energy group structure was: fast energy range is 20 MeV to 9.118 keV (about
64 groups), resonance energy range is 9.118 keV to 4.0 eV (5877 groups) and the thermal energy
16
range is 4.0 eV to 0.0 eV (123 groups). The proposed energy group structure was then tested and
compared to MCNP calculations where the results showed deviations in an acceptable range
resulting in confidence that their use will lead to the reduction of neutronic calculation errors.
2.2.3 Contributon and Point-Wise Cross Section Driven Method
Alpan and Haghighat, 2003 [1] developed a method for the selection of energy group structure
based on contributon theory (William, 1991) [35]. Contributon and Point-wise Cross Section
Driven (CPXSD) method focus on the self-shielding problem as well. CPXSD method refined
the selected initial arbitrary group structure by using the importance of energy groups in the
initial energy group structure and point-wise cross section of an important isotope/material in the
problem. Cross sections were processed for the initial energy group structures using NJOY. Then
self-shielding calculations were performed. Adjoint and forward fluxes were calculated and
further used in the calculation of the importance function. Where maximum importance is
identified, the relevant group structures are refined considering the point-wise cross section of
the important material. If there is a resonance structure, resonance and non-resonance parts are
determined and their areas are computed. Those resonances with larger areas are enclosed in a
sub-group and the remaining resonances and non-resonances are combined about the size of the
largest resonance area. Thus sub-groups are placed within the important groups. If that is not the
case, the group is partitioned evenly by the user’s judgement.
17
The number of subdivisions in other groups is set based on their importance compared to the
maximum importance. The iteration is performed until the subdivisions set for that group are
closely matched. Then a new multi-group energy structure is constructed and ready for use in
generation of the new library.
2.2.4 HTR Energy Group Structure Selection Method
For HTR analysis, Mkhabela et al, 2007 [24] developed a neutron energy group structure
selection method based on systematic survey of the group structures that can predict the
modeling objectives such as effective multiplication factor, reaction rates, and flux distributions.
COMBINE6 code was used for cross section generation, Nodal Expansion Method (NEM) for
core calculations and Monte Carlo Neutron Particle (MCNP) code for the reference solution.
The objectives produced by the group structure were compared to the group structure of the
reference solution. Later, Han, 2008 continued with similar work in determining the best broad
energy group structures consisting of 5, 6, 7, 8 or 9 groups. The starting group structure was
COMBINE6 fine group structure developed at INL. It consisted of 166 energy groups.
Mphahlele, 2008 [25] also conducted a broad group structure selection for PBR analysis. The
starting energy group structure was a recommended best energy group structures by Han, 2008.
It was improved by adding the suitable group boundaries for point-wise resonance calculations
using MICROX-2 code.
18
The new group structure boundaries were examined based on the boundaries of General
Atomics-193 energy group structure. K-effective for the pebble bed reactor was used as the
parameter for these tests and where the minimum deviation between k-effective was observed,
those boundaries were selected as good boundaries. The k-effective for the reference solution
was calculated using Monte Carlo methods. This selection was done for three different
temperatures (800K, 1000K and 1200K). The best boundaries from all three temperatures were
combined into one energy group structure. Also the effectiveness of the new energy group
structure was tested. The iterative method was repeated until 8-group, 10-group and 13-group
structures were obtained.
2.2.5 An Adaptive Energy Group Constructor
Mosca et al, 2011 [26] developed another method to construct a multi-group energy structure
referred to as an adaptive energy mesh constructor (AEMC) and integrated it into nuclear data
processing project GALILEE. AEMC is used to optimize the energy group over infinite
homogeneous medium neutron problems characterizing a given reactor, in this case a fast
reactor. Parameters that were considered are medium temperature, isotopic concentration, cross
section and slowing down of neutrons.
19
The optimization was carried out in two steps, namely, the investigation of the energy group
structure that have results close to the Monte Carlo solutions used as references and followed by
the selection of the appropriate self-shielding model. However, the use of a shielding model later
revealed that it underestimates the U-238 fission production rates as it was pointed out by
Hfaiedh and Santamarina, 2005 [17].
Nonetheless, three self-shielding models were used, that is sub-group self-shielding, LivolantJeanpierre approach and the extended Livolant-Jeanpierre approach. Then, the fine energy group
structure was collapsed into broad energy group structure. AEMC tool searches for the optimal
boundaries with minimum errors during multi-group transport calculations for a specified
problem using swarm algorithm. At the end, the optimization technique was tested with
heterogeneous medium for accuracy where the neutron flux is known to be anisotropic. Details
on the technique used and the outcomes for using different shielding models are published on
Mosca et al, 2011 [26].
20
Chapter 3
Nuclear Physics Theory
3.1 Introduction
This chapter gives a brief summary of forward and adjoint transport equation used to solve
neutron problems. Other neutron physics concepts such as neutron slowing down, resonance
absorption, and thermalization process are highlighted since they form part of the problem
statement of this work.
3.2 Transport Equation
The rate at which nuclear reactions occur is determined by the neutron flux distribution and the
corresponding material distribution in the reactor. Neutron’s behavior and population as occurs
in the reaction rates is the key element in the reactor and chain reaction control. Neutron
distribution has been studied by investigating their motion as they stream inside the reactor [8].
Diffusion theory has been used to analyze nuclear reactors using neutron multi-group theory.
However, this technique has limitations because it is not valid between regions of different
compositions. Therefore, the Boltzmann transport equation adopted from gas dynamics (equation
3.1) is used to study neutrons transport in the reactors in developing the neutron energy group
spectrum.
21
∂n
ˆ ⋅∇n r , Ω
ˆ , E , t +ν ∑ n r , Ω
ˆ , E, t =
+ νΩ
t
∂t
(
)
(
)
∫ π d Ωˆ ' ∫
4
where:
∞
0
(
) (
) (
ˆ '→Ω
ˆ n r , E ', Ω
ˆ ', t + s r , E , Ω
ˆ ,t
d E'ν ' ∑ s E ' → E , Ω
)
3.1
∂n
- is the rate of change of neutrons of neutron density
∂t
ˆ = iΩ
ˆ + jΩ
ˆ + kΩ
ˆ = angular distribution of neutrons.
The omega in equation 3.1 is given by Ω
x
y
z
ˆ =sin θ sin θ and Ω
ˆ =sin θ cos θ , Ω
ˆ =cos θ as shown in Figure 3.1
where Ω
y
z
x
Z
ΩZ
ΩX
θ
Ω
ΩY
Y
φ
X
Figure 3.1: The position and direction characterizing a neutron
22
Streaming term
(

)
ˆ ⋅∇n r , Ω
ˆ , E , t This term defines the flow of neutrons in a volume d 3 r about r at energies
νΩ
ˆ about Ω̂ . Angular flux is presented by
between E and E + dE traveling in a direction dΩ
(
)

ˆ and Ω̂ is the unit vector that gives a direction of a particle.
nv r , E , Ω
Collision term
(

)
ˆ , E , t - is the rate at which neutron suffer collisions at point r in a volume d 3 r at
ν ∑t n r , Ω
ˆ about Ω̂ . The collision types
energies between E and E + dE traveling in a direction dΩ

involved are elastic, fission, inelastic scattering, and absorption. ∑t ( r , E ) , is the total
macroscopic cross section at point r and energy E giving the probability that a neutron will
interact per unit length.
Scattering term
∫ π d Ωˆ ' ∫
4
∞
0
(
) (
ˆ '→Ω
ˆ n r , E ', Ω
ˆ ', t
dE 'ν ' ∑ s E ' → E , Ω
)
is the neutron gain at time t due to

scattering in a volume d 3 r about r at energies between E ' and E '+ dE to energies between E
and E + dE traveling in direction Ω '+ d Ω ' and scattered in direction Ω + d Ω .
23
Source term
^


s  r , E , Ω, t  This is the source of neutrons in a volume d 3 r . Rate of neutron are produced at


position r having energy E , moving in direction Ω at time t . The analysis in this program deal
with a reactor in equilibrium where
∂n
= 0 . Therefore the transport equation becomes:
∂t
ˆ ⋅∇n ( r , Ω, E ) + v ∑ n ( r , Ω, E ) =
vΩ
t
(
) (
) (
∞
∫ d Ω '∫ d Ev' ' ∑ s E ' → E, Ωˆ ' → Ωˆ n r , E, Ωˆ + s r , E, Ωˆ
4π
0
)
3.2
It is complicated to solve the transport equation due to space, energy and angular dependence of
neutrons. There are methods to simplify transport calculations to obtain solutions. In this
program the numerical discretization methods are applied to solve the transport equation. Here,
(
)
ˆ is expanded as explained in the next section.
φ r, E, Ω
24
3.2.1 Angular Discretization
The angular discretization method is most appropriate for obtaining solution in transport theory
codes where the energy spectrum is initially represented by many energy groups. PN method
uses a flux expansion into a series of Ω̂ (angle), which results in an accurate flux presentation at
position r and energy E in direction Ω . The reaction rate accuracy then depends on the accuracy
of the cross section as given in ENDF data files. Discrete Ordinates method results in S N
equations. The Discrete Ordinates divides the 4π
solid angle into segments ∆Ω n and
approximates each segment by a linear expression or weights defined by its value, w within the
segment. The value of n determines the order of the approximation. The S 2 approximation is
better than diffusion theory and the S 4 approximation is adequate for many practical purposes.
Normally, the ∆Ω n is expressed as units of 4π or that

=
φ (r , E )

N
=
w φ (r , E), n
∑
n =1
n m
∑w
n
= 1 and
1, N
The transport equation reduces to:
(
)
ˆ ⋅∇φ r, Ω
ˆ , E + ∑ φ ( r, E ) =
Ω
n
t n
(
)

∑ wn '∫ dE ' ∑ s E '− E , Ωˆ 'n → Ωˆ n φn ( r , E ') + sn ( r , E )
N
n =1
25
3.3
Thus a treatment of the neutron directional dependence is summarized. Equation 3.3 is solved to
determine neutron flux in the S N method where:

φ=
(r , E )
(
)
ˆ dΩ
ˆ
,Ω
∫ φ r , E=
4π

∑ w φ (r , E)
N
n =1
3.4
m m
(
)

ˆ in terms of spherical harmonics:
The first step in the PN method is to expand φ r , E , Ω
(
N
+l
l=0
m= −l
) ∑ ∑ϕ
ˆ
=
φ r, E, Ω
( )
ˆ
where: Ylm Ω
lm
( r.E ) Ylm ( Ωˆ )
3.5

represents the spherical harmonics. Here φ ( r , E ) =
∫π d ΩY ( Ωˆ ) φ ( r , E, Ω ) .

lm
4
Considering a one dimensional equation to simplify a problem,
r = z , x= y= 0
ˆ =∑ 2l + 1P ( µ )
Ylm Ω
l
l = 0 4π
( )
N
where µ = Cosθ
Pl ( µ ) is a Legendre polynomial
26
In one dimension, the flux expands into:
 2l + 1 
 φn ( z , E ) Pl ( µ )
l = 0  4π 
N
φ ( z, E , µ ) = ∑ 
3.6
Pl ( µ ) = Legendre polynomial
The general form of the equation becomes:
( l + 1) ∂ϕl +1 + l ∂ϕl −1 + ∑ ϕ z, E =
)
t l (
( 2l + 1) ∂z ( 2l + 1) ∂z
3.7
∞
∫ dE ' ∑ ( E ' → E ) ϕ ( z, E ) + s ( z, µ , E )
sl
l
0
+1
P du
∫ P=
l m
1=
if l m
−1
= 0 if l ≠ m
The angular components in the PN expansion of the differential scattering cross section is
+1
∑ sl ( E ' →
=
E ) 2π ∫ d µ0 ∑ s ( E ' → E , µ0 ) Pl ( µ0 )
−1
ˆ '⋅ Ω
ˆ
µ0 = Ω
27
3.8
l
( n − m )!P m µ ' P m µ cos m φ '− φ
Here Pl ( µ0 ) =
Pl ( µ ') Pl ( µ ) + 2∑
) l ( )
(
)
l (
m =1 ( n + m ) !
3.9
where µ0 = cos θ 0 , µ = cos θ and µ ' = cos θ ' . The orthorgonality of the Legendre equations
eliminates the last term in the above equation.
3.2.2 Energy Discretization
Both the spherical harmonic formulation ( PN ) of the transport equation as given in equation 3.7
and discrete ordinates ( S N ) formulation use the multi-group discretization of the energy variable
E . The discretization of energy in equation 3.7 will provide adequate results, particularly for the
higher PN equations and S N equations. Of importance in solving the multi-group equations are
the different reaction rates that occur over the overall energy interval from 10−3 eV to 107 eV in
a nuclear reactor. The neutron spectrum is quite different in each of the three broad energy
ranges, fast, epithermal and thermal. In the fast region, the energy spectrum is dominated by the
fission, in the epithermal region by the slowing down and resonance absorption, and in the
thermal region by thermalization. The solutions to PN and S N equations are handled in energy
groups of three broad regions. The energy spectrum is divided into many energy groups, maybe
up to several hundred groups, forming a large matrix. In each group g , the group cross sections
are determined as shown in equation 3.10.
28
∫
Eg −1
Eg
∑ xg =
dE ∑ x ( E ) ϕ ( E )
∫
Eg −1
Eg
3.10
dEϕ ( E )
where ∑ xg = macroscopic cross section for reaction x .
x = f (fission), a (absorption), etc.
For example, the actual S N equation then reduces to:
1 ∂ϕ ng ^
+ Ω n ⋅∇ϕng +=
∑tg ϕng
ν g ∂t
∑w ∑∑
n'
n'
g '→ g
sn '→n
ϕng'' + Sng
3.11
g'
Eg −1
n = 1,..............., N and g = 1................, G and ϕ g =
∫ ϕ ( E )dE
Eg
The equation 3.11 is the S N solved to obtain ϕ g .
29
3.3 Adjoint Transport Equation
The adjoint operator is defined as:
Eg −1
∫ψ
Eg −1
+
Mψ dψ =
Eg
∫ ψM ψ
+
+
dψ +
3.12
Eg
where, ψ + is the adjoint flux and M + is the adjoint operator matrix.
g g'
M = Ω ⋅∇ψ ng + ∑tg ψ ng − ∑ wg ' ∑ ∑ sgn''→
ψ n ' + S ng
→n
3.13
g'
M + = transposed of M when the matrix M = M ij and M + = M ji formed by interchanging rows
and columns. The Eigen - functions of ψ + are then orthogonal to those of M , so that
( Mψ ) = λψ
( M ψ ) = ηψ
+
+
+
The matrix multiplication of the S N multi-group equation is
ψ M +ψ + − ψ + Mψ =
0
(η − λ ) ψψ + =
3.14
since ψ and ψ + are orthogonal, thus
M +ψ ng = Ω ⋅∇ψ ng + + ∑tg ψ ng + − ∑ wg ' ∑ ∑ ggn'→'→ngψ ng' '+ + S gg +
30
3.15
These equations can be shown to define the adjoint function to be the importance function where
ψ g + is the probability that a neutron born in energy group g will fission. Therefore, the
importance of the group ( g ) is given by:
Cg=
∫
V
d 3 r ∫ d Ω ψ g ( r , Ω )ψ g + ( r , Ω )
3.16
4π
Adjoint function/flux is the response of a detector in the core to a unit point source inserted at
position ( r0 , Ω0 , E0 ) , such as it is a measure of the importance of neutron events contributing to
the response of a detector [8] [3]. So it does not perform the neutron flux calculation for neutron
source or energy. An important application of adjoint flux is in the perturbation theory where it is
used to analyze the small changes that occur in the reactor [3] [15] [21]. The neutron importance
and its properties are discussed thoroughly by Henry, 1975 [16].
3.4 Neutron Energy Regions
As stated previously, neutron energy regions in nuclear reactors are estimated to be in the range
of 10 -3 to 107 electron volts. This large energy range is conveniently subdivided into three
regions, fast, epithermal and thermal. The fast region is dominated by slowing down process
through elastic and inelastic scattering processes. Epithermal also known as 1/E region is
dominated by resonance absorption and scattering processes in which slowing down of neutrons
also occurs. In the thermal region both up scattering and down scattering occur as the neutrons
are thermalized, and subsequently are absorbed.
31
3.4.1 Neutron Slowing Down
Neutron slowing down is governed by two processes, elastic and inelastic scattering where
elastic scattering can easily occur with neutron of any energy while the inelastic scattering
requires a sufficiently high energy to excite the target nucleus to a higher energy level. It occurs
in the fast energy region. The slowing down process also depends on the atomic mass of a
moderator, the lighter the nucleus the better the moderating ability. However, a moderator must
exhibit low neutron absorption.
Graphite (carbon) has the highest atomic mass, M C =12 of any moderator whereas the hydrogen
in water has the lowest mass M H =1. Thus, during the slowing down of neutrons from the fast to
thermal energies due to mainly elastic scattering, neutrons lose less energy per collision with
graphite than hydrogen. During elastic scattering, the initial neutron energy E1 is related to the
scattered energy EL by:
E
=
2
E1
(1 + α ) + (1 − α ) Cosθ 
2 
3.17
 u −1 
where α = 
 and θ = scattered angle of the neutron
 u +1
2
For C , α = 0.716 whereas α = 0 for H .
32
Of importance is the average energy lost per collision in a moderator. It is convenient to
calculate the energy loss as a function of lethargy u . The average gain in lethargy per collision
is independent of the neutron energy.
dE
E 
u = ln  0  and du = −
E
 E 
where, (−) indicates a decrease in E , as u increases.
+1
 E1 
=
ξ ln=
 
 E2 
 E1 
 d ( Cosθ )
2 
∫ ln  E
−1
3.18
+1
∫ d ( Cosθ )
−1
ξ = 1+
α
ln α
1−α
3.19
Lethargy is the convenient variable used to perform fast spectrum calculations. The average
lethargy change per collision ξ is used to calculate the average number of collisions needed to
thermalize neutrons starting at 2 MeV . It takes 118 collisions for neutrons to thermalize in
graphite whereas only 18 collisions are needed in H . The large energy changes that take place in
light water reactor allow the neutron to escape the resonances relatively often while slowing
down in the epithermal energy range.
It also allows the total energy spectrum to be divided into two groups, fast and thermal, for
successful analysis of the light water reactors. This is not the case for graphite moderated
33
reactors. Due to the extra number of scattering required for a neutron to slow down through the
epithermal region, there is greater chance for the neutrons to be captured by the resonance cross
sections. For this reason, more energy groups are required in analyzing graphite moderated
reactors.
3.4.2 Resonance Absorption
Resonance absorption is dominant in the epithermal energy region of the nuclear reactor because
neutrons with low energies have a high possibility to collide during slowing down thereby have
high probability to be absorbed [8]. The significant resonance absorption in thermal reactors is in
the lower energy resonances (resolved) such as those resonances of heavy nuclides such as U238 and Th-232 (see Figure 3.2 and 3.3).
Figure 3.2: Low energy resonances for U-238
34
Figure 3.3: Low energy resonances for Th-232
This resonance absorption cannot be ignored since it affects the multiplication factor, fuel
burnup, breeding and reactor control characteristics (reactivity, control rods, period, prompt and
delayed neutrons) [8]. Generally, in strong resonance regions there will be a depression of
neutron flux (see figure 3.4), and this need to be treated thoroughly to ensure the accurate kinetic
behavior of a reactor. This is known as energy self-shielding. There is also a related spatial selfshielding encountered in some cases but it is not a major concern in this study.
35
Resonance
σ
φ
φ (E)
E
Figure 3.4: Flux depreciation under the resonance region
At energies of keV, the resonance structure becomes fine such that they cannot be treated as
individual resonances (unresolved resonances) and in this case they are solved by means of
models. Examples of unresolved resonances are given in Figure 3.5 and 3.6.
36
Figure 3.5: Unresolved resonances for U-235 (t2.lanl.gov)
Figure 3.6: Unresolved resonances for U-238 (t2.lanl.gov)
37
The models are narrow resonance (NR) and wide resonance (WR). Narrow resonance assumes
that the practical width of the resonance is small compared to the average energy lost in the
collision with a moderator. Wide resonance assumes that the practical width of the resonance is
very wide compared to the energy lost in the collision with a moderator. Details assumptions and
equations of these models are given in literature [8] [3] [31].
In heterogeneous cases, which is true for all nuclear reactors flux now depends on both lethargy
and position. So, they can be computed by considering average values per region by assuming
first collision probabilities. This considered that the probability that a neutron born in the fuel
region will undergo its first collision in the moderator region (equation 3.20). Resonant
absorption and Wigner and Bell-Wigner approximations are used for proper considerations of
heterogeneity.


Vc N 0 I eff
=
p exp  −

 (V ξ ∑ S ) ' f + (V ξ ∑ S ) 'm 
3.20
I eff = ∫ σ a ,eff ( u ) du
3.21
Subscript f and m represents the fuel and moderator regions respectively. V is the volume of the
fuel region and moderator, N is the neutron density and (ξ ∑ ) is the moderating power. Equation
3.21 is the effective resonance integral.
38
3.4.3 Neutron Thermalization
In this region, the thermal neutron cross section depends on temperature and the physical state of
the scattering medium. This complicates the nuclear reactor analysis in this region since there is
a neutron up-scattering during the collisions. The equivalence of the thermal neutron energy and
the binding energy of the scattering nucleus also make the analysis difficult when determining
the change in neutron energy and angle. In heterogeneous reactor cores such as HTGRs where
considerable absorption occurs at thermal energies a detail modeling is required. In pure
graphite, the thermalization process is described by the Maxwell-Boltzmann distribution
(equation 3.22).
Neutron energy ( ET ) and its speed ( vT ) as functions of temperature ( T ) are expressed in
equation 3.23 and 3.24 respectively. Subscript ( m ) is the neutron mass.
M (E)
=
2π
(π kT )
3
2
 E 
E exp  −

 kT 
3.22
E=
kT
= 8.62 ×10−5 T (eV )
T
=
vT
3.23
2kT
= 1.28 ×104 T ( cm / sec )
m
3.24
However in reality for an actual reactor, the thermal spectrum is modified by the presence of
absorption, in-scattering, leakages etc. There are methods available for use to model such
conditions [8] [20].
39
Chapter 4
Monte Carlo Reference Results
4.1 Introduction
This chapter presents the MCNP5 work as a reference solution to the study. An overview of
Monte Carlo code, MCNP (version 5.1.5) is given in section 4.2. MCNP5 is used to give a
reference detailed accurate solution to be compared with the DRAGON code results. The
DRAGON code is a deterministic code having inherent approximations resulting in some
inaccuracy in the solution; techniques are available to reduce these inaccuracies. The MCNP5
code gives accurate solutions to the same problem, which can be used to guide technique to
increase the accuracy of the DRAGON code. HTR fuel specifications that are used for the
analysis are tabulated in section 4.3
4.2 MCNP5 Code Description
Monte Carlo Neutron Particle Transport code system (MCNP) is a continuous energy statistical
code developed at Los Alamos National Laboratory [23]. For this study, the used MCNP5
version 5.1.5 was supplied by the Radiation Safety Information Computational Center (RSICC)
organization of USA that conducts and regulates the use of nuclear computational tools. The
MCNP5 code models exactly the geometry in the calculation and then tracks individual neutron
particles as they traverse nuclear region based on accurate reactor physics laws and stores all
40
reactions of the neutron as it travels through the medium as the history of that particle. The
results are therefore statistical and the more histories the more accurate the solution.
Running thousands of histories gives an accurate solution that is a reference solution to be
compared with other codes that are less accurate such as deterministic codes. MCNP5 can handle
both simple and complex geometries and uses continuous energy neutron cross section data. It is
the best choice for analysis that involves complex geometries since it statistically can evaluate
the system without geometric approximations. However it is time consuming, especially when
many histories are being sampled to obtain solutions having small uncertainties. A random
number generation method is used to determine the particle interactions (absorption, scattering,
fission) as well as energy losses, new scattered direction, and the number of neutrons created in a
fission process. For criticality calculations, k-effective cycle (multiplication factor is a ratio of
the number of neutrons generated at the end of the cycle to those created during previous cycle)
is estimated by averaging over the events in the life time cycle. The first few cycles are discarded
to eliminate bad initial source distributions. MCNP5 is chosen to give reference solution to this
study.
4.3 HTR Fuel Specifications
There are two forms of HTR fuel used in this research work. Pebbles (fuel element of 6 cm
diameter) are used for pebbled bed reactors and prismatic (hexagonal blocks) are used for
modular high temperature gas cooled reactors. Both fuel elements are composed of TRISO
41
coated particles (CP) that are dispersed or embedded in a graphite matrix. The current design for
pebble fuel element has 15000 of CP’s and each prismatic cylinder in the hexagonal block has
about 3000 CP’s each. The fuel specifications used in this study were taken from the benchmark
specification by DeHart [6] for the HTGR fuel element under the US Department of Energy
(DOE)’s NGNP development program.
Table 4.1 presents the CP’s specifications, and the lattice data and material specification is
presented in Table 4.2. Table 4.3 and Table 4.4 give the specifications for the pebble bed and
prismatic lattice data respectively.
42
Table 4.1: Coated Particle specifications (common for all types of fuel)
Coated Particle specifications
Item
Units
Value
UO 2 fuel density
g/cm3
10.4
Uranium enrichment (by
%
8.2
ppm
1
Outer coated particle radius
mm
0.455
Fuel kernel radius
mm
0.25
Coated material
-
C/C/SiC/C
Coated thickness
mm
0.09/0.04/0.035/0.04
Coated densities
g/cm3
1.05/1.9/3.18/1.9
mass 235U/(235U+238U)
Fuel natural boron impurity
by mass
Coated Particle lattice data
Item
Units
Value
Unit cell grain square array
cm
0.16341
cm
0.10137
Grain outer radius
cm
0.0455
Packing fraction of coated
%
9.043
Graphite matrix density
g/cm3
1.75
Graphite matrix natural
ppm
0.5
g
6.806e-04
pitch (cubical outer
boundary)
Unit cell grain outer radius
(spherical outer boundary
particles
boron impurity by mass
UO 2 fuel mass
43
Table 4.2: Material specification (common for all fuel types)
Material
Nuclide
Atoms per barn cm
( concentrations)
UO 2 fuel
U-238
2.12877e-02
U-235
1.92585e-03
O
4.64272e-02
B-10
1.14694e-07
B-11
4.64570e-07
C (natural)
5.26449e-02
C (natural)
9.52621e-02
C (natural)
4.77240e-02
S (natural)
4.77240e-02
Pebble/Compact carbon
C (natural)
8.77414e-02
matrix
B-10
9.64977e-09
B-11
3.90864e-08
Pebble outer
C (natural)
8.77414e-02
coating/Prismatic block (note:
B-10
9.64977e-09
fuel grain has the same
B-11
3.90864e-08
He-3
3.71220e-11
He-4
2.65156e-05
Inner low density carbon
kernel coating
Pyrolytic carbon kernel
coating (inner and outer)
Silicon carbide kernel coating
packing)
Helium Coolant
44
Table 4.3: Pebbles representative of PBMR fuel
Item
Units
Value
Unit cell coolant outer radius
cm
3.53735
Pebble radius
cm
3.0
Radius of fuel zone
cm
2.5
Pebble outer carbon coating
cm
0.5
ppm
0.5
-
15,000
%
9.043
Graphite matrix density
g/cm3
1.75
Graphite matrix natural boron
ppm
0.5
Pebble outer carbon density
g/cm3
1.75
UO 2 fuel mass per pebble
g
10.210
(spherical outer boundary)
thickness
Pebble outer carbon natural
boron impurity by mass
Number of coated particles per
pebble
Packing fraction of coated
particles
impurity by mass
45
Table 4.4: Prismatic fuel lattice data
Item
Units
Value
Triangular pitch (coolant channel – rod
cm
1.880
Fuel channel diameter
cm
1.270
Coolant channel diameter
cm
1.588
Fuel compact (centered in fuel channel)
cm
1.245
Compact height
cm
4.93
Number of coated particles per compact
-
3000
Packing fraction of coated particles
%
19.723
Graphite matrix density
g/cm3
1.75
Graphite matrix natural boron impurity by
ppm
0.5
g
2.042
channel and rod channel – rod channel)
diameter
mass
UO 2 fuel mass per compact
4.4 MCNP5 Results
The HTR models developed in MCNP5 for both the Pebble fuel element and Prismatic
hexagonal block fuel are shown in Figures 4.1 and 4.2 below. The corresponding results are
given in Table 4.5 to 4.16. The reaction rates were sampled for three energy regions, fast,
epithermal and thermal regions. The energy ranges were selected as 1.100 ×10−10 → 3.142 ×10−6
MeV for
thermal
region,
3.142 ×10−6 → 1.156 ×10−1 MeV for
epithermal
region
and
1.156 ×10−1 → 1.964 ×10+1 MeV for fast region and these values corresponds to the SHEM
energy group structures that form the foundation of this work. The pebble absorption rates, nu-
46
fission rates and the average flux for all energy regions are presented in Table 4.5 that were
calculated using the ENDF/B-VII.0 version of the evaluated Nuclear Data files and Table 4.6
presents results for ENDF/B-VI.8 which was succeeded by ENDF/B-VII.0. The percent
deviation of the pebble reaction rates is presented in Table 4.7 where thermal region is shown to
be higher than 1 percent, but epithermal and fast regions have percent deviations below 1
percent. The eigenvalue relative deviation in pcm of ∆k / k between the data files is below the
acceptable 500 pcm difference. The average flux from the ENDF/B-VI.8 calculations is higher
compared to the ENDF/B-VII.0 results. It is observed that the difference was in the helium
region of the pebble. This is thought to be due to the updates that are in ENDF/B-VII.0 for He-3
(see Table 4.17).
47
Figure 4.1: MCNP5 Pebble model
48
Figure 4.2: MCNP5 Prismatic block model
49
Table 4.5: Pebble fuel element results calculated using ENDF/B-VII.0
PEBBLE
ENDF/B-VII.0
Energy Range
Reaction Rates and criticality
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.24977E-01
2.51396E-01
5.33901E-03
Nu-Fission (fissions/cm3-s)
1.40447E+00
1.01377E-01
8.65859E-03
Average Flux (particles/cm2-s)
1.09878E+00
1.32725E+00
5.97252E-01
1.52881 ± 0.00046
K-effective and deviation
Table 4.6: Pebble fuel element results calculated using ENDF/B-VI.8
PEBBLE
ENDF/B-VI.8
Energy Range
Reaction Rates and criticality
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.34328E-01
2.50483E-01
5.35821E-03
Nu-Fission (fissions/cm3-s)
1.42250E+00
1.01017E-01
8.68439E-03
Average Flux (particles/cm2-s)
1.29381E+00
1.22106E+00
5.05711E-01
1.52858 ± 0.00047
K-effective and deviation
50
Table 4.7: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8
PEBBLE
ENDF/B-VII.0
Energy Range
Reaction Rates and
criticality
Thermal
Epithermal
Fast
Absorption
-1.28991
0.36327
-0.35949
Nu-Fission
-1.28361
0.35469
-0.29800
Average flux
-17.74941
8.00124
15.32708
K-effective- Relative
Deviation in pcm of ∆k / k
23
with previous version of
ENDF
Table 4.8 and 4.9 presents the reaction rates for the prismatic block fuel element in the same
energy ranges. In this case the percent deviation (Table 4.10) for all reactions is below 1 percent
except for the nu-fission in the thermal region. The average fluxes deviation percent is also
below 1. The criticality relative deviation in pcm of ∆k / k is also below the acceptable 500
pcm difference.
51
Table 4.8: Prismatic block results calculated using ENDF/B-VII.0
PRISMATIC
ENDF/B-VII.0
Energy Range
Reaction Rates and criticality
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.97577E-01
2.97869E-01
7.38072E-03
Nu-Fission (fissions/cm3-s)
1.37041E+00
1.30683E-01
1.08548E-02
Average Flux (particles/cm2-s) 1.45410E+00
2.19054E+00
9.41143E-01
1.46946 ± 0.00156
K-effective and deviation
Table 4.9: Prismatic block results calculated using ENDF/B-VI.8
PRISMATIC
ENDF/B-VI.8
Energy Range
Reaction Rates and criticality
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.93087E-01
2.97834E-01
7.45265E-03
Nu-Fission (fissions/cm3-s)
1.32850E+00
1.30331E-01
1.07793E-02
Average Flux (particles/cm2-s)
1.44680E+00
2.18889E+00
9.42938E-01
1.46923 ± 0.00160
K-effective and deviation
52
Table 4.10 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8
PRISMATIC
ENDF/B-VII.0 and ENDF/B-VI.8 % deviation
Energy Range
Reaction Rates and criticality
Thermal
Epithermal
Fast
Absorption
0.64366
0.011907
-0.97457
Nu-Fission
3.05850
0.26963
0.69606
Average Flux
0.50197
0.074997
-0.19075
K-effective- Relative Deviation in pcm
23
of ∆k / k with previous version of ENDF
Figure 4.3 and 4.2 below presents the keffective and source convergence for the Pebble and
Prismatic block respectively. For the pebble fuel element the runs had 2000 cycles (150 cycles
skipped) and 15000 histories per cycle. While for the Prismatic block the runs had 1000 cycles
(150 were skipped) and 200 histories per cycle. The results were normalized to the mean of the
active cycles.
53
1.015
Keffective and source_entropy
1.01
Source_entropy
Keffective
1.005
1
0.995
0.99
0.985
0.98
0
500
1000
1500
Number of cycles
2000
2500
Figure 4.3: Pebble keffective and source convergence
1.15
Keffective and source_entropy
Source_entropy
1.1
Keffective
1.05
1
0.95
0.9
0.85
0.8
0
200
400
600
800
Number of cycles
1000
1200
Figure 4.4: Prismatic keffective and source convergence
54
The General Atomics energy group structure was also studied in this work. The energy ranges
for GA-193 group structure were selected to be between 5.000 ×10−10 → 3.059 ×10−6 MeV for
thermal, 3.059 ×10−6 → 1.111×10−1 MeV for epithermal, and 1.111×10−1 → 1.49182 ×10+1 MeV
for fast region. This cutoff energies are the closest possible to the cutoffs used for the SHEM
energy group structures. GA-193 energy group structure was developed for fast reactors, the fast
region consists of 49 groups while the slowing down (epithermal) region consists of 42 groups
and the thermal region consists of 102 groups. The reference solution was computed for the
corresponding cutoff points in MCNP5 to ensure accuracy during DRAGON results
comparisons. Table 4.11 and 4.12 presents Pebble fuel elements reaction rates calculated using
ENDF/B-VII.0 and ENDF/B-VI.8 respectively. The comparisons are shown in Table 4.13.
Similarly, the prismatic hexagonal block results are shown in Table 4.14 and 4.15, with their
comparisons in Table 4.16.
Table 4.11: Pebble fuel element results calculated using ENDF/B-VII.0
PEBBLE
ENDF/B-VII.0
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
0.72463533
0.251696718
0.005378067
Nu-Fission (fissions/cm3-s)
1.403901224
0.101907034
0.008697482
Average Flux (particles/cm2-s)
1.085554073
1.321156429
0.616512002
55
Table 4.12: Pebble fuel element results calculated using ENDF/B-VI.8
PEBBLE
ENDF/B-VI.8
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
0.734000653
0.250770526
0.005395323
Nu-Fission (fissions/cm3-s)
1.421954896
0.101527706
0.008721427
Average Flux (particles/cm2-s)
1.291840555
1.218793081
0.509942205
Table 4.13: Percent deviation of pebble FE results for ENDF/B-VII.0 and ENDF/B-VI.8
PEBBLE
ENDF/B-VII.0 and ENDF/B-VI.8 results % deviation
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption
-1.29242
0.36798
-0.32085
Nu-Fission
-1.28596
0.37223
-0.27531
Average Flux
-19.00287
7.74801
17.28592
56
Table 4.14: Prismatic block results calculated using ENDF/B-VII.0
PRISMATIC
ENDF/B-VII.0
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
0.697164716
0.298228638
0.007429926
Nu-Fission (fissions/cm3-s)
1.336392733
0.131598539
0.01080447
Average Flux (particles/cm2-s)
1.440463559
2.185138181
0.950793997
Table 4.15: Prismatic block results calculated using ENDF/B-VI.8
PRISMATIC
ENDF/B-VI.8
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
0.693019287
0.298201228
0.007487775
Nu-Fission (fissions/cm3-s)
1.327794793
0.130979016
0.010827201
Average Flux (particles/cm2-s)
1.442456795
2.183731411
0.952400355
57
Table 4.16 Percent deviation of prismatic block results for ENDF/B-VII.0 and ENDF/B-VI.8
PRISMATIC
ENDF/B-VII.0 and ENDF/B-VI.8 results % deviation
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption
0.59461
0.00919
-0.77860
Nu-Fission
0.64337
0.47077
-0.21039
Average Flux
-0.13837
0.06438
-0.16895
The Evaluated Nuclear Data Files (ENDF/B-VII.0) version contains data with reactions with
incident neutrons, protons and photons of approximately 400 isotopes. Its advancements
compared to ENDF/B-VI.8 are that it has new cross sections for U , Pu , Th , Np and Am
actinides isotopes resulting in the improved performance in integral validation criticality and
neutron transmission [5]. It also has more precise standard cross sections for neutron reactions on
H , 6 Li ,
10
B , Au and for
235
U and 238U fission. The thermal neutron scattering was improved
including a set of neutron cross sections on fission products. Neutron and proton induced
evaluations were extended up to 150 MeV and new light nucleus neutron proton reactions were
added. Among these developments are large suites of photonuclear reactions, post fission beta
delayed photon decay spectra, new radioactive decay beta, new method for uncertainties and
covariance’s with covariance evaluations, new actinides fission energy deposition [4]. These
advances attribute to the reaction rates deviations for ENDF/B-VII.0 and ENDF/B-VI.8 results
shown above for the pebble fuel element and the prismatic hexagonal block fuel. Table 4.17
58
below presents the comparisons for ENDF/B-VII.0 and ENDF/B-VI.8 libraries. NSUB is the
sub-library identification number in ENDF/B-VI format. The last two columns present the
number of materials (or isotopes) for specified libraries.
59
Table 4.17: ENDF/B-VII.0 and ENDF/B-VI.8 data files improvement comparisons.
No.
NSUB
Sub-library name
Short name
ENDF/B-VII.0
ENDF/B-VI.8
1
0
Photonuclear
g
163
-
2
3
Photo-atomic
photo
100
100
3
4
Radioactive decay
decay
3838
979
4
5
Spontaneous
s/fpy
9
9
fission yields
5
6
Atomic relaxation
ard
100
100
6
10
Neutron
n
393
328
7
11
Neutron
31
31
fission n/fpy
yields
8
12
Thermal scattering
tsl
20
15
9
19
Standards
std
8
8
10
113
Electro-atomic
e
100
100
11
10010
Proton
p
48
35
12
10020
Deuteron
d
5
2
13
10030
Triton
t
3
1
14
20030
3
He-3
2
1
He
60
Chapter 5
Multi-group Structure Analysis
5.1 Introduction
Sensitivity analysis were peformed on the currently available energy group structures, SHEM281, SHEM-361 and GA-193. The DRAGON transport code is used for the analysis. Section 5.2
describes the DRAGON code. The method used for the library generation is discussed in section
5.3. DRAGON libraries were generated using ENDF/B-VII.0 data files and used to analyze the
pebble fuel element and prismatic hexagonal block models for reaction rates and criticality. The
results and the observations are discussed in section 5.4.
5.2 DRAGON Code Description
DRAGON code is a lattice physics code, which is divided into many different calculation
modules linked together using a GAN generalized driver. A GAN generalized driver is used to
call sequential series of modules sharing a common calling convention and template to build
FORTRAN applications through linking independent modules. The modules exchange the
information through defined data structures. These data structures are memory resident or
persistent [22]. A simplified flow chart in Figure 5.1 presents the data structure and sequence of
modules and its brief description is given below.

MACROLIB
61

MICROLIB

GEOMETRY

TRACKING

ASMPIJ

FLUXUNK

EDITION

BURNUP

DRAGLIB

MULTICOMPO
MACROLIB is a standard data structure used to transfer group ordered macroscopic cross
sections between its modules. It can either be used on its own or included into MICROLIB or
EDITION structures. It can be created by MAC (macro library generation), LIB (micro library
generation) and EDI (editing) modules or modified by SHI (perform self-shielding using
generalized Stammler’s method), USS (perform self-shielding using subgroup method) and EVO
(burnup) modules. It is required for a successful execution of assembly (ASM) and flux (FLU)
modules.
MICROLIB is used to transfer microscopic and macroscopic cross sections between modules. It
follows the MACROLIB substructure. It can be created by LIB and EDI modules or modified by
MAC, SHI, USS and EVO modules.
62
GEOMETRY is used to transfer the geometry between modules. It is created by the GEO
(generate a geometry) module. It is required for a successful execution of the tracking modules
SYBILT, EXCELT and MCCGT modules.
TRACKING is used to transfer the general tracking information between its modules. It is a
stand-alone structure. It is created by either: SYBILT, EXCELT or MCCGT modules. It is
required for a successful execution of the ASM (assembly) module.
ASMPIJ is used to transfer multi-group response and collision probability matrices between
modules. It is a stand-alone structure. It can be created by the ASM module. It is required for a
successful execution of the FLUX module.
FLUXUNK is used to transfer fluxes between modules. It is a stand-alone structure. It can be
created by the FLUX module. It is required for a successful execution of the EDI and EVO
modules.
EDITION is used to store condensed and merged microscopic and macroscopic cross sections. It
is a stand-alone structure but can contain MICROLIB and MACROLIB sub-structures. It can be
created by EDI module. It is required for a successful execution of the COMPO (database
construction) module.
63
BURNUP is used to store burnup information. It is created by the EVO module. It is required to
deplete the fuel assembly.
DRAGLIB is used to recover isotopic, dilution and temperature dependent information including
multi-group microscopic cross sections and burnup data. It is a stand-alone structure. It is created
by the DRAGR module which is in NJOY.
MULTICOMPO is used to store reactor related information and to classify it using immutable
list of local and global parameters. It is a stand-alone structure. It is created by the COMPO
module. For details in specific modules refer to [22].
The DRAGON code is designed to simulate the neutronic behavior of a unit cell, a fuel assembly
and multi-assembly arrays of the nuclear reactor. It has a number of lattice code functional
characteristics such as the interpolation of microscopic cross sections supplied by the standard
library. It computes the resonance self-shielding in multi-dimensional geometries and the multigroup and multi-dimensional neutron flux that can take into account the neutron leakage. It
considers transport-transport and transport-diffusion equivalence calculations and can edit the
condensed and homogenized properties of nuclear reactor calculations. It also performs the
depletions analysis. DRAGON has two main components, one solves the multi-group flux and
the other calculates the collision probabilities (CP). It contains a multi-group iterator to control
the different algorithms to solve the neutron transport equation. These algorithms are SYBIL
64
option that solves the integral transport equation using the collision probability method (for 1-D)
and the interface current method (for 2-D) Cartesian or hexagonal assemblies. EXCELL option
solves the integral transport equation using collision probability method for 2-D and 3-D
geometries. There is also the MCCG option that uses a long characteristics method for 2D and
3D geometries. An option is also available for treating specular boundary conditions in 2D
rectangular geometry. In solving of the integral transport equations, the results include reaction
rates, neutron flux and the multiplication factor (k-effective).
65
BIVACT
GEO
TRIVACT
SYBILT
Geometry
SNT
Track
ASM
EXCELT
SHI
Track
NXT
MCCGT
USS
Draglib
LIB
asmpij
FLU
flux
Microlib
(Macrolib)
2
Microlib(Macrolib)
1
EVO
Microlib(Macrolib)
2
EDI
burnup
edition
COMPO
compo
END
Figure 5.1: DRAGON flow chart [11]
66
5.3 Cross Section Library Generation
The DRAGON library is referred to as DRAGLIB and is generated using a Python Script [14].
The ENDF/B data files are processed using NJOY through an object oriented Python Script
named PyNjoy. NJOY is a nuclear data processing system that converts the evaluated nuclear
data files ENDF/B format into usable libraries for nuclear reactor analysis. It transforms
continuous cross section data stored in data files like ENDF/B format for use with Monte Carlo
codes or lattice physics transport codes. It handles all nuclear reactor parameters such as
resonances, Doppler broadening, heating, radiation damage, scattering, gas production, neutrons
and charged particles, photo-atomic interactions, self-shielding, probability tables, photon
production and high energy interactions. NJOY consists of a set of modules from which several
can be selected to perform specific calculations. For DRAGON library generation, the following
modules are used:

RECONR

BROADR

GROUPR

UNRESR

PURR

THERMR

GROUPR
67
The RECONR module reads ENDF/B data and produces a common energy grid for all reactions
to obtain all cross sections within specified tolerance by linear interpolation. It then reconstructs
resonance cross sections from resonance parameters and cross sections from ENDF/B using
nonlinear interpolation schemes. The output is written in point-wise- ENDF (PENDF) file with
all cross sections kept on a unionized energy grid where they are improved for accuracy and
usability.
BROADR generates Doppler broadened cross sections from the PENDF module, the output file
of RECONR. It uses the SIGMA1 module, also called the kernel broadening because it uses a
detailed integration of the integral transport equation defining the cross section. It is known to be
accurate since it treats all resonances and non-resonance cross section including multilevel
effects. The output is written on the PENDF file.
The GROUPR produces self-shielded multi-group cross sections, anisotropic group to group
scattering matrices as well as anisotropic photon production matrices. It presents fission as a
group to group matrix. If necessary, self-shielding for scattering matrices and the photon
production matrices may be modeled. It uses the Bondarenko narrow resonance weighting
scheme. It also allows an option for computing a weighted flux for various mixtures of heavy
absorbers with light moderators. GROUPR uses an accurate point-wise solution of the integral
slowing down equation to account for intermediate resonance effects in the epithermal range. It
uses the PENDF module for input.
68
The UNRESR module produces the effective self-shielded cross sections for resonance reactions
in the unresolved energy range. The unresolved resonance range begins at the energy where it is
difficult to measure individual resonances. It extends to energies where the effects of fluctuations
in the resonance cross section become unimportant for analysis. Resonance information is
averaged for resonance widths and spacing. This is then converted to the effective cross sections.
The UNRESR module uses PENDF module and the ENDF data from BROADR as input. The
effective cross sections are then written into the PENDF module.
The PURR module produces the probability tables that are used to treat unresolved resonance
self-shielding cross section. The unresolved self-shielding data generated by UNRESR is suitable
for use in multi-group calculation after processing by GROUPR. This is done by the generation
of resonances using statistical analysis. Then, cross sections sampling is random at selected
energies and are accumulated into probability tables and Bondarenko moments. These will not be
used for Monte Carlo codes where the probability tables are used. The probability tables are
written on an output file (PENDF module).
The THERMR module generates point-wise neutron scattering cross sections in the thermal
range. It generates elastic cross sections for crystalline materials and non crystalline materials. It
also generates inelastic cross sections and energy to energy transfer matrices for gas free atoms
or for a bound scatters. Since in thermal energy range, neutron scatterings are temperature
dependent, THERMR module considers this temperature. Therefore it expects the requested
69
temperature (T) to be of the temperature included in the ENDF/B thermal file or within few
degrees of the value such that it can pick the closest possible value if necessary. It uses ENDF
and PENDF module from the RECONR as input.
DRAGR is an interface module developed to perform advanced lattice code functions. These are
self-shielding models that have capabilities of presenting distributed and mutual resonance selfshielding effects, leakage models with space dependent isotropic or anisotropic streaming
effects, availability of the characteristics method and burnup calculations featuring energy
production reactions [12]. The capability of radiative transport of gamma energy produced by
decay or any nuclear reaction in the lattice is among the essential features. DRAGR produces a
DRAGLIB which is a direct access cross section library in a format compatible for DRAGON
code.
The list of modules discussed are the ones specifically used for the generation of library for this
study, Please note that NJOY has a various modules to select depending on the user’s need. For
NJOY details can review the NJOY, 99.90 manual [28].
70
ENDFB/VI or
VII
NJOY
RECONR
BROADR
PURR
THERMR
PENDF
GROUPR
GENDF
DRAGR
DRAGLIB
Figure 5.2: Flow chart for DRAGON library production
71
5.4 Sensitivity Analysis Results
5.4.1 DRAGON Results used in the Sensitivity Study
In this section, DRAGON analysis results are given, which are used in the sensitivity analysis for
optimization of multi-group libraries. The cross section libraries were generated for the available
group structures, SHEM-281, SHEM-361 and GA-193 using ENDF/B-VII.0. These libraries
were used to compute the reaction rates and criticality of pebble fuel element and prismatic
hexagonal block with specifications given in Chapter 4. The results are shown in Tables 5.1
through 5.6.
Table 5.1: Pebble reaction rates and criticality for SHEM-281 energy group structure
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
7.35093E-01
2.58643E-01
6.26975E-03
Nu-Fission (fissions/cm3-s)
1.40377E+00
1.04519E-01
8.63650E-03
Average Flux (particles/cm2-s)
1.07132E+00
1.32069E+00 5.97364E-01
K-effective
1.51692 (convergence = 2.79E-09)
72
Table 5.2: Prismatic block reaction rates and criticality for SHEM-281 energy group structure
PRISMASTIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
6.86963E-01
3.05376E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.46823E+00 2.25204E+00 1.03317E+00
K-effective
1.46094 (convergence =1.53E-07)
Table 5.3: Pebble reaction rates and criticality for SHEM-361 energy group structure
PEBBLE
Group
Structure
SHEM-361
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s) 7.350480E-01 2.586860E-01 6.261710E-03
Nu-Fission (fissions/cm3-s)
1.403680E+00 1.047250E-01 8.636350E-03
Average Flux (particles/cm2-s) 1.071270E+00 1.320840E+00 5.973630E+00
K-effective
1.517046 (convergence = 3.44E-08)
73
Table 5.4: Prismatic block reaction rates and criticality for SHEM-361energy group structure
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
SHEM-361
Absorption (collisions/cm3-s)
6.86881E-01
3.05456E-01 7.65884E-03
Nu-Fission (fissions/cm3-s)
1.31808E+00
1.31889E-01 1.11035E-02
Average Flux (particles/cm2-s)
1.46805E+00
2.25232E+00 1.03317E+00
K-effective
Epithermal
Fast
1.46107 (convergence = 3.44E-07)
Table 5.5: Pebble reaction rates and criticality for GA-193 energy group structure
PEBBLE
Energy Range
Group Structure
Reaction Rates
Thermal
Epithermal
Fast
GA-193
Absorption (collisions/cm3-s)
6.25219E-01
3.68398E-01
6.36363E-03
Nu-Fission (fissions/cm3-s)
1.11943E+00
1.04135E-01
8.71626E-03
Average Flux (particles/cm2-s)
8.98270E-01
1.30071E+00
6.10101E-01
K-effective
1.30713 (convergence = 4.22E-08)
74
Table 5.6: Prismatic block reaction rates and criticality for GA-193 energy group structure
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
GA-193
Absorption (collisions/cm3-s)
5.609600E-01 4.312260E-01 7.790330E-03
Nu-Fission (fissions/cm3-s)
1.076820E+00 1.307680E-01 1.120650E-02
Average Flux (particles/cm2-s)
1.177660E+00 2.220720E+00 1.055180E+00
K-effective
1.218799 (convergence = 5.02E-07)
Thermal
Epithermal
Fast
5.4.2 Sensitivity Analysis Comparison with MCNP5 Results
The reaction rates for Pebble FE and prismatic hexagonal block computed with DRAGON using
the SHEM-281, SHEM-361 and GA-193 generated libraries are compared to the reference
solution calculations from MCNP5 (chapter 4). K-effective relative percent deviation in pcm is
given in Table 5.7 and 5.8 for each model. Table 5.9 and 5.10 shows the relative percent
deviation for the reaction rates comparisons.
75
Table 5.7: Criticality calculation comparisons for ENDF/B-VII.0
PEBBLE
deviation in pcm
Group Structure
K-effective
MCNP5
-
1.52881
DRAGON
SHEM-281
1.51692
1189
SHEM-361
1.517046
1176.4
GA-193
1.307130
22168.5
Table 5.8: Criticality calculation comparisons for ENDF/B-VII.0
PRISMATIC
deviation in pcm
Codes
Group Structure
K-effective
MCNP5
-
1.46946
DRAGON
SHEM-281
1.46094E+00
852.4
SHEM-361
1.461067E+00
839.3
GA-193
1.21880E+00
25066.1
76
Table 5.9: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0
PEBBLE
Energy Range
MCNP5
Reaction rates
Thermal
Epithermal
Fast
2.51396E-01
5.33901E-03
1.40447E+00
1.01377E-01
8.65859E-03
Average Flux (particles/cm2-s) 1.09878E+00
1.32725E+00
5.97252E-01
3
SHEM-281 Absorption (collisions/cm -s) 7.350480E-01 2.586430E-01
6.26975E-03
Absorption (collisions/cm3-s) 7.24977E-01
Nu-Fission (fissions/cm3-s)
DRAGON
Nu-Fission (fissions/cm3-s)
% deviation
1.40377E+00
1.04519E-01
8.63650E-03
Average Flux (particles/cm2-s) 1.07132E+00
1.32069E+00
5.97364E-01
Absorption
-1.389212277
-2.882545451
-17.43271822
Nu-Fission
0.05013748
-3.099762817
0.255135027
Average Flux
2.499473073
0.494391026
-0.018703628
3
SHEM-361 Absorption (collisions/cm -s) 7.350480E-01 2.586860E-01
6.261710E-03
Nu-Fission (fissions/cm3-s) 1.403680E+00 1.047250E-01
8.636350E-03
Average Flux (particles/cm2-s) 1.071270E+00 1.320840E+00 5.973630E-01
% deviation
Absorption
-1.389212277
-2.899649913
-17.28212864
Nu-Fission
0.056545572
-3.302965595
0.25686741
Average Flux
2.504023558
0.483089478
-0.018536195
77
Table 5.10: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0
PRISMATIC
Energy Range
MCNP5
Reaction rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.97577E-01
2.97869E-01
7.38072E-03
Nu-Fission (fissions/cm3-s)
1.37041E+00
1.30683E-01
1.08548E-02
Average Flux (particles/cm2-s)
1.45410E+00
2.19054E+00 9.41143E-01
Absorption (collisions/cm3-s)
6.869630E-01 3.053760E-01 7.65777E-03
DRAGON
SHEM-281
% deviation
SHEM-361
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
Average Flux (particles/cm2-s)
1.46823E+00
2.25204E+00 1.03317E+00
Absorption
1.521522941
-2.520222601 -3.753748785
Nu-Fission
3.807588733
-0.699405992 -2.290771967
1.11035E-02
Average Flux
-0.972081048 -2.807681093 -9.778257997
Absorption (collisions/cm3-s)
6.868810E-01 3.054560E-01 7.658840E-03
Nu-Fission (fissions/cm3-s)
1.318080E+00 1.318890E-01 1.110350E-02
Average Flux (particles/cm2-s) 1.468050E+00 2.252320E+00 1.033170E+00
% deviation
Absorption
1.53327792
-2.547080042 -3.768246022
Nu-Fission
3.818534366
-0.922847458 -2.290771967
Average Flux
-0.959702214 -2.820463348 -9.778257997
78
Since the cutoff points for GA-193 energy group structure were slightly different from the
SHEM energy group structures, their reference solution was generated separately. Though there
might not be much difference in the results due to the fact that the cutoff points were selected to
be as close as possible to the SHEM energy group structures. The comparisons are therefore
presented in Table 5.11 and 5.12 for both models. The observation is that the reaction rates
percent deviation is very high in some cases for the GA-193 group structure, thus the energy
group structure should not be used or either trusted for simulation for other reactors other than
fast reactors for which it was developed. Note that the errors in k-effective are greater than 1000
pcm for the SHEM energy group structure for both the Pebble fuel element and the Prismatic
hexagonal block. This large difference in k-effective is probably due to the large deviations of 17
% in the fast absorption reaction rate in the Pebble fuel element. However, the large difference in
k-effective for the Prismatic hexagonal block appears to be due to the 3.8 % deviation in the
thermal nu-fission reaction rate. The k-effective deviation is more sensitive to deviations in
thermal nu-fission reaction rate than fast absorption rate reaction rates.
79
Table 5.11: Pebble FE reaction rates calculation comparisons for ENDF/B-VII.0
PEBBLE
Energy Range
MCNP5
Reaction rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.24635E-01
2.51697E-01
5.37807E-03
Nu-Fission (fissions/cm3-s)
1.40390E+00
1.01907E-01
8.69748E-03
Average Flux (particles/cm2-s) 1.08555E+00
1.32116E+00 6.16512E-01
Absorption (collisions/cm3-s)
6.25219E-01
3.68398E-01
6.36363E-03
Nu-Fission (fissions/cm3-s)
1.19427E+00
1.04135E-01
8.71626E-03
DRAGON
GA-193
Average Flux (particles/cm2-s) 8.98270E-01
% deviation
1.30708E+00 6.10101E-01
Absorption
13.719498
-46.365834
-18.325594
Nu-Fission
14.932049
-2.186273
-0.215900
Average Flux
17.252395
1.065463
1.039883
80
Table 5.12: Prismatic block reaction rates calculation comparisons for ENDF/B-VII.0
PRISMATIC
Energy Range
MCNP5
Reaction rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s) 6.97165E-01 2.98229E-01
7.42993E-03
Nu-Fission (fissions/cm3-s) 1.33639E+00 1.31599E-01
1.08045E-02
Average Flux (particles/cm2-s) 1.44046E+00 2.18514E+00
9.50794E-01
Absorption (collisions/cm3-s) 5.60960E-01 4.31226E-01
7.79033E-03
Nu-Fission (fissions/cm3-s)
1.07682E+00 1.30768E-01
1.12065E-02
Average Flux (particles/cm2-s) 1.17766E+00 2.22072E+00
1.05518E+00
DRAGON
GA-193
% deviation
Absorption
19.5369492
-44.5957717
-4.8507087
Nu-Fission
19.4233870
0.6311153
-3.7209620
Average Flux
18.2443740
-1.6283556
-10.9788243
81
Chapter 6
Multi-group Energy Structure Improvement
6.1 Introduction
The SHEM multi-group energy structure has been optimized for LWRs. The objective of this
dissertation was to optimize it for graphite moderated reactors, both the Pebble bed and the
Prismatic type reactors by altering the energy group structure. The DRAGON-4 code was used
for the optimization process where the currently available energy group structure (SHEM-281,
SHEM-361 and GA-193) are improved, and the obtained results are compared to the MCNP5
results wherein the MCNP5 results are assumed to be reference results. The advantages of using
the DRAGON code are its capabilities to calculate the flux and the adjoint flux for each of its
energy groups. The adjoint flux allows computing the neutron importance function of each
energy group, which is used to improve the energy group structure. The group that has higher
importance function was subdivided into two groups or more and a new energy group structure
was developed from which a new library was created. The new library is applied in the
DRAGON code to analyze a pebble fuel element and prismatic hexagonal block. Then, a pebble
fuel element and prismatic hexagonal block were analyzed for k-effective, flux spectrum and
reaction rates (absorption, scattering, production, fission cross sections). These parameters were
analyzed for three macro energy groups, fast, epithermal and thermal regions. A pebble fuel
element comprises of 15000 TRISO coated particles consisting of fuel kernel of UO 2 which is
surrounded by low density carbon, inner pyrolitic carbon, silicon carbide and outer pyrolitic
82
carbon. It has a diameter of 6 cm. Each prismatic cylinder in a hexagonal block has 3000 coated
particles of the same specifications. A fuel rod of 1.245 cm diameter is placed in a fuel channel
diameter of 1.270 cm and the compact height is 4.93 cm.
6.2 Contributon and Point-Wise Cross Section Driven Method
The improvement of the energy group structures was conducted using the contributon and pointwise cross section driven method (CPXSD) developed by Alpan and Haghighat in 2005 initially
for shielding problems [2]. Nateekol [19] in her PhD extended the method to TRIGA (Penn
State) reactor. In this study the method was applied for first time to HTR problems. In the former
two applications the method was applied using S N formulation. In this study the method was
applied in Collision Probability. The CPXSD method was derived based on the product of the
forward and adjoint angular fluxes, and the point-wise cross section of important
isotope/material of interest. It is an iterative method that selects effective fine and broad group
energy structures for a problem of interest. The energy dependent response flux referred to as
contributon is expressed by equation 6.1:
=
C (E)
∫ d r∫π d ΩΨ ( r , E, Ωˆ )Ψ ( r , E, Ωˆ )

v
+

6.1
4
(

ˆ
In equation 6.1, C ( E ) is the importance function, Ψ r , E , Ω
(
)
)
is the forward flux and

ˆ is the adjoint flux dependent on position ( r ), energy (E) and direction ( Ω̂ ). When
Ψ + r , E, Ω
83
considering spherical harmonics expansion of forward flux and its adjoint, and using
orthogonality, the group-dependent contributon C ( E ) is given by:
=
Cg
L
2l + 1 m
Ψ l , g , s Ψ lm, g,+, s
m =0 4π
l
∑V ∑ ∑
s∈D
s
l =0
6.2
In equation 6.2, s presents a uniform material region in D , Vs is the volume of the sub-domain.
Where l and m are polar and azimuthal indices for the spherical harmonic polynomial, and g is
the energy group [1]. Also, Ψ lm, g , s is the flux moment and Ψ lm, g,+, s is the adjoint flux moment.
Therefore, the fine energy group structure improvement followed in this study is as follows:
(i) An initial multi-group energy structure was selected (SHEM-281 and SHEM-361 groups)
and GA-193 group structures.
(ii) Cross sections were generated for the initial multi-group energy structure with the
established procedure of cross section generation relevant to DRAGON transport code as
discussed in chapter 5.
(iii) The forward and adjoint flux calculations were performed to determine the importance
function of the groups in the initial energy group structure of interest.
(iv) After identifying the energy groups with higher importance, this energy group structure is
improved by the resonance structure of a spectrum representing the unit cell (fuel) by
dividing the energy group into two or more energy groups.
84
(v) When the improvement process was complete for all energy groups, the new energy group
structure was used for cross section generation. The new cross section library was used to
calculate the reaction rates and k-effective of the problem of interest.
(vi) The reaction rates and k-effective calculated using the new library are compared with the
results obtained from the previous library analysis. If the results are within a specified
tolerance, the procedure ends; otherwise, steps (iii) through (v) were repeated.
6.3 Improvement of SHEM-281 Energy Group Structure
In this section the SHEM-281 energy group structure was improved for three different energy
regions (Fast, Epithermal and Thermal) separately. After each region has met the target criteria,
they are combined to make up a whole SHEM-281 improved energy group structure. This was
done to study the effect of each region behavior during the energy group structure improvement
and to ensure that there was no under estimations in the procedure.
6.3.1 Fast Energy Region Improvement
The fast energy region was selected to be between 1.156235 ×10−6 → 1.964030 ×10+1 MeV . The
starting energy group structure is SHEM-281. The two selected criteria for determining fine
group structure were 10 pcm relative deviation of ∆k / k and 1% relative deviation of the reaction
rates. The reaction rate used for the fast region was the total neutron production (Nu-fission)
(ν ∑ ) and
f
the k-effective. During the fast region energy group structure improvement the
85
epithermal (163 groups) and thermal (81 groups) regions were kept constant. Table 6.1 shows the
number of groups obtained during fast energy group structure improvement.
Table 6.1: Fast group selected in the fast range
SHEM 281-group structure improvement
Number of Groups in different energy ranges
Group structure number
Fast
Epithermal
Thermal
Total
1
37
163
81
281
2
65
163
81
309
3
79
163
81
323
The importance function for the starting and improved energy group structure is presented in
Figure 6.1. It was observed that as the energy groups with high importance function are split as
defined previously their importance was reduced. The relative deviation differences of ∆k / k
and the percent relative deviation for selected reaction rates are shown in Table 6.2. The 1%
relative deviation in the reaction rate was met in the 309 energy group structure, however, the
relative deviation for ∆k / k was slightly higher than that of the target criteria. Then further
improvement to 323 energy groups resulted in the achievement of both target criteria’s, thus, the
fast range of the energy derived from 323 energy group structure was selected to be used for
further improvement in the final energy group structure.
86
0.1
281-groups
0.08
309 groups
323 groups
Importance
0.06
0.04
0.02
0
1.00E-01
1.00E+00
1.00E+01
Energy (MeV)
Figure 6.1: Importance function for fast energy region for 281, 309 1nd 323 groups.
Table 6.2: Eigen-value results for fast energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Nu-fission rate for
Deviation in pcm
Fast region
Group
structure
K-effective
SHEM-281 1.516920E+00
of ∆k / k with
previous group
(ν ∑ )
% Relative
Deviation with
previous group
-
8.63650E-03
-
f
309
1.516816E+00
10.4
8.60509E-03
0.363688994
323
1.516797E+00
1.9
8.59872E-03
0.074025954
87
Table 6.3 and 6.4 gives the data for each energy group structure for the Pebble FE and Prismatic
hexagonal block, respectively. It was anticipated that the reaction rates in each energy group
would increase with the energy group structure improvement, however, this did not happen in the
groups wherein reaction rates decreased and the k-effective also decreased. Note that the changes
are very small.
Table 6.3: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
SHEM-281
Absorption (collisions/cm3-s)
7.35093E-01
2.58643E-01 6.26975E-03
Nu-Fission (fissions/cm3-s)
1.40377E+00
1.04519E-01 8.63650E-03
Average Flux (particles/cm2-s)
1.07132E+00
1.32069E+00 5.97364E-01
K-effective
309
Fast
1.51692 (convergence =2.79E-09)
Absorption (collisions/cm3-s)
7.35102E-01
2.58647E-01 6.24979E-03
Nu-Fission (fissions/cm3-s)
1.40369E+00
1.04522E-01 8.60509E-03
Average Flux (particles/cm2-s)
1.07423E+00
1.32072E+00 5.96607E-01
K-effective
323
Epithermal
1.516816 (convergence = 1.171E-08)
Absorption (collisions/cm3-s)
7.35095E-01
2.58656E-01 6.24692E-03
Nu-Fission (fissions/cm3-s)
1.40368E+00
1.04521E-01 8.59872E-03
Average Flux (particles/cm2-s)
1.07420E+00
1.32071E+00 5.96446E-01
K-effective
1.516797 (convergence = 2.92E-08)
88
Table 6.4: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
6.86963E-01
3.05376E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
1.11035E-02
309
323
Average Flux (particles/cm2-s)
1.46823E+00 2.25204E+00 1.03317E+00
K-effective
1.46094+1.53E-07
Absorption (collisions/cm3-s)
6.86974E-01
3.05382E-01
7.64310E-03
Nu-Fission (fissions/cm3-s)
1.31817E+00
1.31601E-01
1.10625E-02
Average Flux (particles/cm2-s)
1.47142E+00 2.25208E+00 1.03184E+00
K-effective
1.46083 (convergence = 1.39E-07)
Absorption (collisions/cm3-s)
6.86965E-01
3.05393E-01
7.63941E-03
Nu-Fission (fissions/cm3-s)
1.31815E+00
1.31599E-01
1.10542E-02
Average Flux (particles/cm2-s)
1.47137E+00 2.25207E+00 1.03156E+00
K-effective
1.46080 (convergence = 1.26E-07)
89
6.3.2 Epithermal Energy Region Improvement
Similarly, starting from the SHEM-281 energy group structure the epithermal energy region
( 3.14 ×10
−6
→ 1.156 ×10−1 ) MeV was improved. The absorption reaction rate was used for
comparisons in this energy range as well as the k-effective. The energy group numbers per
region are given in Table 6.5 where both fast (37 groups) and thermal (81 groups) regions were
kept constant.
Table 6.5: Epithermal energy groups selected in the epithermal range
SHEM 281-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
37
163
81
281
2
37
215
81
333
Table 6.6: Eigen-value resulted for epithermal energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Deviation in pcm
Group structure K-effective
of ∆k / k with
previous group
Absorption
% Relative
Deviation with
previous group
SHEM-281
1.51692E+00
-
2.58643E-01
-
333
1.51699E+00
6.6
2.58696E-01
0.02049157
90
The importance function plot is given in Figure 6.2 showing the reduction of energy group
importance as the group structure was increased. The target criteria of 10 pcm relative deviation
of ∆k / k and the 1 percent relative deviations of the objective function were met with the 333
energy group structure. The absorption reaction rates and k-effective comparisons results for
energy groups are presented in Table 6.6. The additional reaction rates data for all regions for
the energy group structure of interest are given in Table 6.7 and 6.8 for pebble FE and Prismatic
hexagonal block respectively.
0.08
281-groups
Importance
0.06
333-groups
0.04
0.02
0
1.00E-06
1.00E-04
1.00E-02
1.00E+00
Energy (MeV)
Figure 6.2: Importance function for epithermal energy region for 281 and 333 groups
91
Table 6.7: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
SHEM-281
Absorption (collisions/cm3-s)
7.35093E-01
2.58643E-01 6.26975E-03
Nu-Fission (fissions/cm3-s)
1.40377E+00
1.04519E-01 8.63650E-03
Average Flux (particles/cm2-s) 1.07132E+00
1.32069E+00 5.97364E-01
K-effective
333
Epithermal
Fast
1.51692 (convergence = 2.79E-09)
Absorption (collisions/cm3-s)
7.35160E-01
2.58696E-01 6.14524E-03
Nu-Fission (fissions/cm3-s)
1.40380E+00
1.04659E-01 8.52606E-03
Average Flux (particles/cm2-s) 1.07430E+00
1.33637E+00 5.82209E-01
K-effective
1.51699 (convergence = 1.76E-08)
92
Table 6.8: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
6.86963E-01
3.05376E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
1.11035E-02
Average Flux (particles/cm2-s) 1.46823E+00
2.25204E+00
1.03317E+00
K-effective
333
1.46094 (convergence = 1.53E-07)
Absorption (collisions/cm3-s)
6.87040E-01
3.05454E-01
7.50768E-03
Nu-Fission (fissions/cm3-s)
1.31829E+00
1.31793E-01
1.09601E-02
Average Flux (particles/cm2-s) 1.47153E+00
2.27932E+00
1.00693E+00
K-effective
1.46105 (convergence = 1.67E-07)
6.3.3 Thermal Energy Region Improvement
The thermal region energy range is between 1.10 ×10−10 − 3.14 ×10−6 MeV . The epithermal (163
groups) and fast (37 groups) region were kept constant as the starting energy group structure
(SHEM-281) was improved for the thermal region. The number of energy groups in each energy
range is presented in Table 6.9 and the related importance function plot in Figure 6.3. Table 6.10
presents the comparisons for the selected reaction rates (absorption, neutron production) and the
k-effective. The selected energy group structure for use in the final group structure for all regions
is 313.
93
Table 6.9: Thermal groups selected in the thermal region
SHEM 281-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
37
163
81
281
2
37
163
97
297
3
37
163
113
313
Table 6.10: Eigen-values resulted for thermal energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Deviation in
pcm of
∆k / k with
Group
structure
K-effective
SHEM-281 1.516920E+00
(ν ∑ )
previous
group
Absorption
% Relative
Deviation with
previous group
-
7.35093E-01
-
1.40377E+00
-
( ∑a )
Nu-Fission
% Relative
Deviation with
previous group
f
297
1.516752E+00
16.8
7.35875E-01
0.106381097
1.40488E+00
0.079072783
313
1.51675E+00
0.6
7.35578E-01
0.040360116
1.40438E+00
0.035590228
94
0.3
0.25
281-groups
297-groups
Importance
0.2
313-groups
0.15
0.1
0.05
0
1.00E-09
1.00E-08
1.00E-07
Energy (MeV)
1.00E-06
1.00E-05
Figure 6.3: Importance function for thermal energy region for 281, 297 and 313 groups
95
Table 6.11: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281 Absorption (collisions/cm3-s) 7.35093E-01
2.58643E-01
6.26975E-03
Nu-Fission (fissions/cm3-s)
1.40377E+00
1.04519E-01
8.63650E-03
Average Flux (particles/cm2-s) 1.07132E+00
1.32069E+00
5.97364E-01
K-effective
297
1.51692 (convergence = 2.79E-09)
Absorption (collisions/cm3-s) 7.35875E-01
2.58078E-01
6.04495E-03
1.40488E+00
1.03443E-01
8.42752E-03
Average Flux (particles/cm2-s) 1.08391E+00
1.34062E+00
5.68356E-01
Nu-Fission (fissions/cm3-s)
K-effective
313
1.516752 (convergence = 4.48E-08)
Absorption (collisions/cm3-s) 7.35578E-01
2.58160E-01
6.26099E-03
1.40438E+00
1.03727E-01
8.63651E-03
Average Flux (particles/cm2-s) 1.08155E+00
1.31388E+00
5.97365E-01
Nu-Fission (fissions/cm3-s)
K-effective
1.51675 (convergence = 3.97E-08)
96
Table 6.12: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
6.86963E-01
3.05376E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.46823E+00
2.25204E+00 1.03317E+00
K-effective
297
1.46094 (convergence = 1.53E-07)
Absorption (collisions/cm3-s)
6.87903E-01
3.04718E-01
7.37759E-03
Nu-Fission (fissions/cm3-s)
1.31962E+00
1.30306E-01
1.08321E-02
Average Flux (particles/cm2-s)
1.48700E+00
2.28755E+00
9.82949E-01
K-effective
313
1.46075 (convergence = 8.05E-08)
Absorption (collisions/cm3-s)
6.87543E-01
3.04796E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31901E+00
1.30634E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.48323E+00
2.24100E+00 1.03317E+00
K-effective
1.46075 (convergence = 1.40E-08)
97
6.3.4 Improved SHEM-281 Energy Group Structure for all Regions
This section combines all the individually improved fast, epithermal and thermal energy regions
to 407 energy group structure as shown in Table 6.13 and the new group structure is referred to
as SHEM_TPN-407. The neutron cross sections were generated for this energy group structure
and Pebble fuel element and Prismatic hexagonal block analysis were conducted as the same as
previous sections. Then, the objective functions were compared to the SHEM-281 as the starting
group structure, see Table 6.14 and Table 6.15.
Table 6.13: Energy group structure improved from SHEM-281 to 407
SHEM 281-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
37
163
81
281
2
79
215
113
407
98
Table 6.14: Results for SHEM-281 and 407 energy group structures
PEBBLE
Energy Range
Group Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
7.35093E-01
2.58643E-01
6.26975E-03
Nu-Fission (fissions/cm3-s)
1.40377E+00
1.04519E-01
8.63650E-03
Average Flux (particles/cm2-s)
1.07132E+00
1.32069E+00
5.97364E-01
K-effective
1.51692 (convergence = 2.79E-09)
SHEM_TPN-407 Absorption (collisions/cm3-s)
7.35662E-01
2.58207E-01
6.13159E-03
Nu-Fission (fissions/cm3-s)
1.40454E+00
1.03868E-01
8.48825E-03
Average Flux (particles/cm2-s)
1.08167E+00
1.32958E+00
5.81286E-01
K-effective
1.516901 (convergence = 9.15E-09)
99
Table 6.15: Results for SHEM-281 and 407 energy group structures
PRISMATIC
Energy Range
Group Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-281
Absorption (collisions/cm3-s)
6.86963E-01
3.05376E-01
7.65777E-03
Nu-Fission (fissions/cm3-s)
1.31823E+00
1.31597E-01
1.11035E-02
Average Flux (particles/cm2-s) 1.46823E+00
2.25204E+00
1.03317E+00
K-effective
1.46094 (convergence = 1.53E-07)
SHEM_TPN-407 Absorption (collisions/cm3-s)
6.87645E-01
3.04868E-01
7.48962E-03
1.31920E+00
1.30833E-01
1.09108E-02
Average Flux (particles/cm2-s) 1.48344E+00
2.25833E+00
1.00532E+00
Nu-Fission (fissions/cm3-s)
K-effective
1.46095 (convergence = 1.65E-07)
The Pebble fuel element results are compared in Table 6.16 below. The comparisons are based
on the relative deviation of ∆k / k in pcm for k-effective and percent relative deviation for
reaction rates. Similarly, Table 6.17 shows the comparisons for the prismatic hexagonal block. It
was observed that the fast region has higher reaction relative deviation in the fast region which is
believed to be the effect from other changes related to the improvement in epithermal and
thermal energy regions since during its improvement the relative deviations were very low.
Epithermal and thermal energy regions relative deviation are below the selected target criteria of
1 percent. The relative deviation of ∆k / k also met the 10 pcm target criteria.
100
Table 6.16: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the
pebble FE
% relative deviation for three regions
Relative
Deviation in
pcm of ∆k / k
Reaction
rates
Thermal
Epithermal
with previous
group
Fast
Absorption 0.077345303 -0.16885677 -2.253249157
Nu-Fission
0.054822219 -0.626757038 -1.746531971 k-effective
Average Flux 0.956853754
0.6686322
-1.9
-2.765936217
Table 6.17: Comparisons for SHEM_TPN-407 with SHEM-281 energy group structure for the
prismatic block
% relative deviation
Relative
Deviation in
pcm of ∆k / k
Reaction
rates
Thermal
Epithermal
Fast
Absorption
0.099179082 -0.166629492
-2.24511
Nu-Fission
0.073529412 -0.583950532
-1.76614
Average Flux 1.025319528
0.278524396
101
with previous
group
-2.77026
k-effective
1
Table 6.18 and 6.20 are the MCNP5 reference solution recalled from chapter 4 and are compared
with the results for a new energy group structure (407 groups) in table 6.19 and 6.21 for both fuel
types. The improvement is observed in the reaction rates relative deviations with no significant
trend.
Table 6.18: MCNP5 results for the pebble
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.24977E-01
2.51396E-01
5.33901E-03
Nu-Fission (fissions/cm3-s)
1.40447E+00
1.01377E-01
8.65859E-03
Average Flux (particles/cm2-s)
1.09878E+00
1.32725E+00
5.97252E-01
1.52881 ± 0.00046
K-effective and deviation
102
Table 6.19: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the
pebble FE
% relative deviation
criticality
Relative
Deviation in
pcm of ∆k / k
Reaction
Rates
Thermal
Epithermal
Fast
Absorption
1.452496263
2.637657019
12.9260987
Nu-Fission
0.00468709
2.398653219 -2.006787014 K-effective
Average Flux -1.582165109
with previous
group
-1190.9
0.175106886 -2.746718853
Table 6.20: MCNP5 results for the prismatic block
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.97577E-01
2.97869E-01
7.38072E-03
Nu-Fission (fissions/cm3-s)
1.37041E+00
1.30683E-01
1.08548E-02
Average Flux (particles/cm2-s)
1.45410E+00
2.19054E+00
9.41143E-01
1.46946 ± 0.00156
K-effective and deviation
103
Table 6.21: Comparisons of SHEM_TPN-407 energy group structure to MCNP5 results for the
prismatic block
% relative deviation
Relative
Deviation in
pcm of ∆k / k
Reaction Rates Thermal
Epithermal
Fast
Absorption
-1.44432
2.29573
1.45406
Nu-Fission
-3.88186
0.11465
0.51288
Average Flux
1.97817
3.00192
6.38377
with previous
group
K-effective
-851.4
6.4 SHEM-361 Energy Group Structure Results
Similarly, the SHEM-361 energy group structure was improved for fast, epithermal and thermal
regions separately. After each region improvements, the energy group structures selected for
different regions are combined to make a complete improved energy group structure. Thus the
effect of each region is not underestimated.
104
6.4.1 Fast Energy Region Improvement
The fast energy region as previously selected is 1.156 ×10−6 → 1.964 ×10+1 MeV . The starting
energy group structure is SHEM-361. The same target criteria of 10 pcm relative deviation of
∆k / k and 1 percent relative deviation of the objective function are used. The reaction rate used
for the fast energy region was the total neutron production (Nu-Fission) and the k-effective. As
the fast region was being improved, the epithermal (243 groups) and thermal (81 groups) regions
remained constant. The number of energy groups obtained during fast energy region
improvement is shown in Table 6.22. The importance function for all energy groups is presented
in figure 6.4 showing a significant reduction of the importance function as the energy groups are
being improved.
Table 6.22: Fast group selected in the fast range
SHEM 361-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
37
243
81
361
2
65
243
81
389
3
79
243
81
403
105
0.1
361 groups
389 groups
0.08
Importance
403 groups
0.06
0.04
0.02
0
1.00E-01
1.00E+00
1.00E+01
Energy (MeV)
Figure 6.4: Importance function for fast energy region for 361,389 and 403 groups
The relative deviation differences for ∆k / k and percent relative deviation for the selected
reaction rates are given in Table 6.23. Both target criteria’s were met by improving the SHEM361 to 389 groups, however 9.9 was relatively too close to the 10 pcm , so the decision was made
to continue with another improvement that led to 403 energy group structures. Therefore 403
energy group structure was selected for further use. Table 6.24 and 6.25 present additional
information of the three regions reaction rates obtained during the energy group structure
improvement for the pebble fuel element and prismatic hexagonal block respectively.
106
Table 6.23: Eigen-value results for fast energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Deviation in
pcm of ∆k / k
Nu-fission rate
for Fast region
(ν ∑ )
% Relative
Deviation with
previous group
Group
structure
K-effective
with previous
group
361
1.51705E+00
-
8.63635E-03
389
1.516947E+00
9.9
8.60508E-03
0.36207
403
1.516928E+00
1.9
8.598720E-03
0.07391
107
f
Table 6.24: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
361
Absorption (collisions/cm3-s)
7.350480E-01 2.586860E-01 6.261710E-03
Nu-Fission (fissions/cm3-s)
1.403680E+00 1.047250E-01 8.636350E-03
Thermal
Epithermal
Fast
Average Flux (particles/cm2-s) 1.071270E+00 1.320840E+00 5.973630E-01
389
K-effective
1.517046 (convergence = 3.44E-09)
Absorption (collisions/cm3-s)
7.350610E-01 2.586860E-01 6.249900E-03
Nu-Fission (fissions/cm3-s)
1.403610E+00 1.047300E-01 8.605080E-03
Average Flux (particles/cm2-s) 1.074170E+00 1.320870E+00 5.966070E-01
403
K-effective
1.516947 (convergence = 3.45E-08)
Absorption (collisions/cm3-s)
7.350540E-01 2.586940E-01 6.247040E-03
Nu-Fission (fissions/cm3-s)
1.403600E+00 1.047290E-01 8.598720E-03
Average Flux (particles/cm2-s) 1.074140E+00 1.320870E+00 5.964460E-01
K-effective
1.516928 (convergence = 3.08E-08)
108
Table 6.25: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
361
Absorption (collisions/cm3-s)
6.86881E-01
3.05456E-01
7.65884E-03
Nu-Fission (fissions/cm3-s)
1.31808E+00
1.31889E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.46805E+00
2.25232E+00 1.03317E+00
K-effective
389
1.46107 (convergence = 3.44E-07)
Absorption (collisions/cm3-s)
6.86902E-01
3.05450E-01
7.64322E-03
Nu-Fission (fissions/cm3-s)
1.31803E+00
1.31894E-01
1.10625E-02
Average Flux (particles/cm2-s)
1.47126E+00
2.25239E+00 1.03184E+00
K-effective
403
1.46099 (convergence = 2.99E-08)
Absorption (collisions/cm3-s)
6.86893E-01
3.05462E-01
7.63965E-03
Nu-Fission (fissions/cm3-s)
1.31801E+00
1.31892E-01
1.10542E-02
Average Flux (particles/cm2-s)
1.47122E+00
2.25237E+00 1.03156E+00
K-effective
1.46096 (convergence = 1.84E-07)
109
6.4.2 Epithermal Energy Region Improvement
(
)
During the improvement of epithermal energy region 3.142 ×10−6 → 1.156 ×10−1 MeV for
SHEM-361 energy group structure the target criteria’s were met at 455 energy group structure
(Table 6.26). The fast (37 groups) and thermal (81 groups) regions were kept constant during the
epithermal region improvement. Significant improvement in the importance function is shown in
Figure 6.5. The relative deviation in pcm of ∆k / k and percent relative deviation of the selected
reaction rates (absorption rate) are given in Table 6.27. Pebble fuel element and prismatic
hexagonal block results also given in Table 6.28 and Table 6.29 respectively.
Table 6.26: Epithermal energy groups selected in the epithermal range
SHEM 361-group structure improvement
Group structure number
Number of Groups in Different Energy Ranges
Fast
Epithermal
Thermal
Total
1
37
243
81
361
2
37
337
81
455
110
0.1
361 groups
0.08
Importance
455-groups
0.06
0.04
0.02
0
1.00E-06
1.00E-04
1.00E-02
1.00E+00
Energy (MeV)
Figure 6.5: Importance function for epithermal energy region for 361 and 455 groups
Table 6.27: Eigen-value resulted for epithermal energy group structure improvement
Eigen-values results of fine energy group structure
Group
structure
K-effective
Relative
Deviation in
pcm of ∆k / k
with previous
group
361
1.51705E+00
-
2.58686E-01
-
455
1.516965E+00
8.1
2.58849E-01
-0.06301
111
Absorption
( ∑a )
% Relative
Deviation with
previous group
Table 6.28: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
361
Absorption (collisions/cm3-s)
7.35048E-01
2.58686E-01 6.26171E-03
Nu-Fission (fissions/cm3-s)
1.40368E+00
1.04725E-01 8.63635E-03
Epithermal
Fast
Average Flux (particles/cm2-s) 1.07127E+00 1.32084E+00 5.97363E-01
K-effective
455
1.51705 (convergence = 3.44E-09)
Absorption (collisions/cm3-s)
7.35051E-01
2.58849E-01 6.09478E-03
Nu-Fission (fissions/cm3-s)
1.40359E+00
1.04896E-01 8.47538E-03
Average Flux (particles/cm2-s) 1.07414E+00 1.34333E+00 5.75134E-01
K-effective and precision
1.51697 (convergence = 5.78E-08)
112
Table 6.29: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
361
Absorption (collisions/cm3-s)
6.86881E-01
3.05456E-01
7.65884E-03
Nu-Fission (fissions/cm3-s)
1.31808E+00
1.31889E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.46805E+00
2.25232E+00 1.03317E+00
K-effective
455
1.46107 (convergence = 3.44E-08)
Absorption (collisions/cm3-s)
6.86888E-01
3.05664E-01
7.44161E-03
Nu-Fission (fissions/cm3-s)
1.31800E+00
1.32109E-01
1.08942E-02
Average Flux (particles/cm2-s)
1.47120E+00
2.29130E+00 9.94678E-01
K-effective
1.46101 (convergence = 1.59E-07)
113
6.4.3 Thermal Energy Region Improvement
(
)
In thermal region 0 → 3.14 ×10−6 MeV , the selected reaction rates for improving the energy
group structure were absorption and nu-fission and then k-effective. The fast and epithermal
energy regions were kept constant as well. The number of energy group structures obtained
during the improvement is given in Table 6.38. Improving the SHEM-361 to 393 did not meet
the 10 pcm target criteria of ∆k / k , while both absorption rate and nu-fission rate were met.
Then the energy group structure was further improved to 395 groups, it is also importance to
note that the difference between 393 and 395 groups is not just two groups, but some energy
group structures with less importance were taken out while some energy group structure were
subdivided into two or more energy groups leading to 395 energy groups. Then all the target
criteria’s were met and 395 energy group structures were selected for further use. The
importance function is shown in Figure 6.6.
Table 6.30: Thermal groups selected in the thermal region
SHEM 281-group structure improvement
Group structure number
Number of Groups in Different Energy Ranges
Fast
Epithermal
Thermal
Total
1
37
243
81
361
2
37
243
113
393
3
37
243
115
395
114
0.3
361 groups
393-groups
0.25
Importance
395 groups
0.2
0.15
0.1
0.05
0
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
Energy (MeV)
Figure 6.6: Importance function for thermal energy region for 361, 393 and 395 groups
The relative deviation in pcm of ∆k / k and percent relative deviation of selected reaction rates is
given in Table 6.31. The results for all regions of both pebble fuel element and prismatic
hexagonal block are given in Table 6.32 and Table 6.33.
115
Table 6.31: Eigen-values resulted for thermal energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Deviation
in pcm of
∆k / k
( ∑a )
% Relative
Deviation
with previous
group
7.35048E-01
-
Nu-Fission
Group
structure
K-effective
with
previous
group
361
1.517046E+00
-
393
1.516874E+00
17.2
7.35535E-01 -0.066254177 1.40430E+00
395
1.516877E+00
-0.3
7.355350E-01
Absorption
116
0
(ν ∑ )
% Relative
Deviation with
previous group
1.40368E+00
-
f
1.404300E+00
-0.04417
0
Table 6.32: Pebble FE results
PEBBLE
Energy Range
Group
Structure
361
Reaction Rates
Thermal
Absorption (collisions/cm3-s) 7.350480E-01
Nu-Fission (fissions/cm3-s)
Epithermal
Fast
2.586860E-01 6.261710E-03
1.403680E+00 1.047250E-01 8.636350E-03
Average Flux (particles/cm2-s) 1.071270E+00 1.320840E+00 5.973630E-01
K-effective
393
1.517046 (convergence = 3.44E-09)
Absorption (collisions/cm3-s) 7.355350E-01
Nu-Fission (fissions/cm3-s)
2.581980E-01 6.261940E-03
1.404300E+00 1.039360E-01 8.636490E-03
Average Flux (particles/cm2-s) 1.081480E+00 1.314040E+00 5.973640E-01
K-effective
395
1.516874 (convergence = 2.93E-08)
Absorption (collisions/cm3-s) 7.355350E-01
Nu-Fission (fissions/cm3-s)
2.581980E-01 6.262060E-03
1.404300E+00 1.039360E-01 8.636500E-03
Average Flux (particles/cm2-s) 1.081550E+00 1.314040E+00 5.973640E+00
K-effective
1.516877 (convergence = 7.45E-08)
117
Table 6.33: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
361
Absorption (collisions/cm3-s)
6.86881E-01
3.05456E-01
7.65884E-03
Nu-Fission (fissions/cm3-s)
1.31808E+00
1.31889E-01
1.11035E-02
393
395
Average Flux (particles/cm2-s)
1.46805E+00 2.25232E+00 1.03317E+00
K-effective
1.46107 (convergence = 3.44E-07)
Absorption (collisions/cm3-s)
6.87471E-01
3.04865E-01
7.65908E-03
Nu-Fission (fissions/cm3-s)
1.31887E+00
1.30926E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.48307E+00 2.24130E+00 1.03317E+00
K-effective
1.46090 (convergence = 1.06E-07)
Absorption (collisions/cm3-s)
6.87471E-01
3.04865E-01
7.65896E-03
Nu-Fission (fissions/cm3-s)
1.31887E+00
1.30926E-01
1.11035E-02
Average Flux (particles/cm2-s)
1.48317E+00 2.24130E+00 1.03317E+00
K-effective
1.46090 (convergence = 9.01E-08)
118
6.4.4 Improved Energy Group Structure for all Regions (SHEM-361)
The energy group structure selected during the specific regions improvement are combined
leading to 531 energy groups referred to as SHEM_TPN-531. The library was generated for this
group structure and used for the analysis of the pebble FE and prismatic hexagonal block
benchmark problems. Reaction rates and k-effetive were compared with the ones from the
original SHEM-361 energy group structure as shown in Table 6.37 and 6.38. The MCNP5 results
were again recalled (Table 6.39 and 6.41) and they were compared with the results obtained from
the new energy group structure analysis (see Table 6.40 and 6.42).
Table 6.34: Energy group structure improved from SHEM-361 to SHEM_TPN-531
SHEM 361-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
37
243
81
361
2
79
337
115
531
119
Table 6.35: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures
PEBBLE
Energy Range
Group Structure
SHEM-361
Reaction Rates
Thermal
Epithermal
Fast
2.58686E-01
6.26171E-03
1.40368E+00 1.04725E-01
8.63635E-03
Absorption (collisions/cm3-s) 7.35048E-01
Nu-Fission (fissions/cm3-s)
Average Flux (particles/cm2-s) 1.07127E+00 1.32084E+00 5.97363E-01
K-effective
1.51705 (convergence = 3.44E-08)
SHEM_TPN-531 Absorption (collisions/cm3-s) 7.35554E-01
Nu-Fission (fissions/cm3-s)
2.58361E-01
6.08033E-03
1.40434E+00 1.04105E-01
8.43757E-03
Average Flux (particles/cm2-s) 1.08158E+00 1.33654E+00 5.74212E-01
K-effective
1.51688 (convergence = 2.45E-08)
120
Table 6.36: Reaction rates for SHEM-361 and SHEM_TPN-531 energy group structures
PRISMATIC
Energy Range
Group Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-361
Absorption (collisions/cm3-s)
6.86881E-01
3.05456E-01
7.65884E-03
Nu-Fission (fissions/cm3-s)
1.31808E+00
1.31889E-01
1.11035E-02
Average Flux (particles/cm2-s) 1.46805E+00
K-effective
2.25232E+00 1.03317E+00
1.46107 (convergence = 3.44E-07)
SHEM_TPN-531 Absorption (collisions/cm3-s)
6.87493E-01
3.05078E-01
7.42278E-03
1.31892E+00
1.31148E-01
1.08449E-02
Average Flux (particles/cm2-s) 1.48322E+00
2.28032E+00
9.93066E-01
Nu-Fission (fissions/cm3-s)
K-effective
1.46091 (convergence = 9.21E-09)
121
Table 6.37: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the
pebble FE
% relative deviation
Relative
Deviation in pcm
Reaction rates
Thermal
Epithermal
Fast
Absorption
-0.06884
0.12563
2.89665
Nu-Fission
-0.04701
0.59203
2.30167
Average Flux
-0.96241
-1.18864
3.87553
of ∆k / k with
previous group
k-effective
16.3
Table 6.38: Comparisons for SHEM_TPN-531 with SHEM-361 energy group structure for the
prismatic block
% relative deviation
Reaction rates
Thermal
Epithermal
Fast
Absorption
-0.08910
0.12375
3.08219
Nu-Fission
-0.06373
0.56184
2.32899
Average Flux
-1.03334
-1.24316
3.88165
122
Relative Deviation
in pcm of ∆k / k
with previous group
k-effective
15.8
Table 6.39: MCNP5 results for the pebble FE
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.24977E-01
2.51396E-01
5.33901E-03
Nu-Fission (fissions/cm3-s)
1.40447E+00
1.01377E-01
8.65859E-03
Average Flux (particles/cm2-s)
1.09878E+00
1.32725E+00
5.97252E-01
1.52881 ± 0.00046
K-effective and deviation
Table 6.40: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the
pebble FE
% relative deviation
Criticality
Relative
Deviation in
pcm of ∆k / k
Reaction rates
Thermal
Epithermal
Fast
Absorption
1.43803
2.69569
12.19203
Nu-Fission
-0.00955
2.62085
-2.61949
Average Flux
-1.59062
0.69494
-4.01251
with previous
group
k-effective
-1192.7
It is important to compare the deviation in Table 6.37 with the corresponding deviation for the
Pebble fuel element in Table 6.40. Only the thermal nu-fission reaction rate deviation was
123
decreased by increasing the number of energy groups to 531. However, the fast absorption
reaction rate remained high at 12 %. The increase in energy groups to 531 showed no
improvement in the k-effective.
Table 6.41: MCNP5 results for the prismatic block
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.97577E-01
2.97869E-01
7.38072E-03
Nu-Fission (fissions/cm3-s)
1.37041E+00
1.30683E-01
1.08548E-02
Average Flux (particles/cm2-s)
1.45410E+00
2.19054E+00
9.41143E-01
1.46946 ± 0.00156
K-effective and deviation
Table 6.42: Comparisons of SHEM_TPN-531 energy group structures to MCNP5 results for the
pebble FE
% relative deviation
Criticality
Relative
Deviation in
pcm of ∆k / k
Reaction rates
Thermal
Epithermal
Fast
Absorption
-1.46675
2.36299
0.56668
Nu-Fission
-3.90392
0.35456
-0.09166
Average Flux
1.96363
3.93731
5.22859
124
with previous
group
k-effective
-855.1
Similar comparison can be made for the Prismatic hexagonal block by comparing Table 6.38 to
Table 6.42. Again the k-effective was not improved using the 531 energy group structure
whereas the nu-fission reaction rate remained relatively high at 3.9 %. In fact, the 531 energy
group structure increases the deviation.
6.5 General Atomics-193 Energy Group Structure
General Atomics energy group structure was divided into three energy regions (fast, epithermal
and thermal) and each region was improved separately. Similar target criteria’s are used for the
improvement of GA-193 energy group structure, which are 10 pcm relative deviation of ∆k / k
and 1 percent deviation of reaction rates.
6.5.1 Fast Energy Region Improvement
The fast energy region of GA-193 which is between 1.111×10−1 → 1.492 ×10+1 MeV has 49
groups as shown in Table 6.43. This was increased to 83 groups while keeping the epithermal
and thermal regions constant.
The relative deviations for k-effective and percent relative
deviation for nu-fission rate are presented in Table 6.44. These were met with the group
structure improvement to 227 energy groups.
125
Table 6.43: Fast group selected in the fast range
GA 193-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
49
42
102
193
2
83
42
102
227
Table 6.44: Eigen-values resulted for fast energy group structure improvement
Eigen-values results of fine energy group structure
Relative Deviation Nu-fission rate for
Fast region
in pcm of ∆k / k
Group
structure
K-effective
with previous
group
(ν ∑ )
193
1.30713E+00
-
8.71626E-03
227
1.30712E+00
0.5
8.70836E-03
f
% Relative
Deviation with
previous group
0.090635
The importance function plotted in Figure 6.13. The data for the pebble fuel element and
prismatic hexagonal block results for this region are given in Table 6.45 and 6.46, where a slight
decrease in both k-effective and nu-fission rate was observed.
126
0.016
0.014
GA-193
0.012
Importance
GA-227
0.01
0.008
0.006
0.004
0.002
0
1.00E-01
1.00E+00
1.00E+01
Energy (MeV)
1.00E+02
Figure 6.7: Importance function for fast energy region for 193 and 227 groups
127
Table 6.45: Reaction rates for GA-193 and 227 energy group structures
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
GA-193
Absorption (collisions/cm3-s)
6.25219E-01
3.68398E-01
6.36363E-03
Nu-Fission (fissions/cm3-s)
1.11943E+00
1.04135E-01
8.71626E-03
Average Flux (particles/cm2-s)
8.98270E-01
1.30071E+00
6.10101E-01
K-effective
227
1.30713 (convergence = 4.22E-08)
Absorption (collisions/cm3-s)
6.25221E-01
3.68399E-01
6.36101E-03
Nu-Fission (fissions/cm3-s)
1.19428E+00
1.04135E-01
8.70836E-03
Average Flux (particles/cm2-s)
8.98254E-01
1.30708E+00
6.09821E-01
K-effective
1.30712 (convergence = 8.42E-08)
128
Table 6.46: Reaction rates for GA-193 and 227 energy group structures
PRISMATIC
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
193
Absorption (collisions/cm3-s)
5.609600E-01
4.312260E-01
7.790330E-03
Nu-Fission (fissions/cm3-s)
1.076820E+00
1.307680E-01
1.120650E-02
Average Flux (particles/cm2-s) 1.177660E+00
K-effective
227
2.220720E+00 1.055180E+00
1.218799 (convergence = 5.02E-07)
Absorption (collisions/cm3-s)
5.60962E-01
4.31226E-01
7.78699E-03
Nu-Fission (fissions/cm3-s)
1.07683E+00
1.30769E-01
1.11963E-02
Average Flux (particles/cm2-s)
1.17764E+00
2.22072E+00
1.05469E+00
K-effective
1.21879 (convergence = 7.80E-08)
129
6.5.2 Epithermal Energy Region Improvement
The epithermal energy region of 3.059 ×10−6 → 1.111×10−1 MeV was improved from 42 groups
to 334 groups. The energy group structures selected in this region is given in Table 6.47, while
the relative deviation for the k-effective and the percent relative deviation for the absorption
reaction rate are given in Table 6.56. The epithermal region improvement of GA-193 was
challenging due to the instability of the reaction rates behavior. The reaction rates vary with no
trend as the energy group structures are increased. The k-effective showed a high sensitivity on
the energy group structure changes. This is attributed to the fact that GA-193 was developed for
fast energy reactors, which use thorium fuel cycle therefore it is obvious that the resonances and
other consideration made were based on the thorium fuel cycle physics and also there are
breeding properties involved in such reactors. The target criteria of 1 percent deviation for
reaction rates was met at 235 groups, however the k-effective was higher than the targeted value.
Therefore the improvement continued with intention to reach the target 10 pcm . But, the keffective high sensitivity observed as the work progresses leading into accepting the closest
possible value to 10 pcm target criteria, which is 17.9 pcm . Therefore 485 energy groups were
selected for this region.
130
Table 6.47: Epithermal group selected in the epithermal range
GA-193 group structure improvement
Group structure
number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
49
42
102
193
2
49
84
102
235
3
49
168
102
319
4
49
204
102
355
5
49
334
102
485
Table 6.48: Eigen-values resulted for epithermal energy group structure improvement
Eigen-values results of fine energy group structure
Relative
Deviation in pcm
Absorption
∆k / k with
previous group
( ∑a )
% Relative
Deviation with
previous group
Group
structure
K-effective
193
1.30713E+00
235
1.30675E+00
37.8
3.68842E-01
0.120376747
319
1.32884E+00
2208.9
3.56242E-01
3.416096865
355
1.32166E+00
717.4
3.59880E-01
1.01089252
399
1.36102E+00
3935.4
3.39782E-01
5.584639324
485
1.36120E+00
17.9
3.39685E-01
0.028547716
3.68398E-01
131
0.03
0.025
Importance
0.02
GA-193
GA-235
GA-319
GA-355
GA-399
GA-485
0.015
0.01
0.005
0
1.00E-06
1.00E-04
1.00E-02
Energy (MeV)
1.00E+00
Figure 6.8: Importance function for epithermal energy region
for 193, 235, 319, 355, 399 and 485 groups
132
Table 6.49: Reaction rates for GA-193, 235, 319, 355, 399, and 485 energy group structures
PEBBLE
Energy Range
Group Structure
193
Reaction Rates
3
Absorption (collisions/cm -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
Thermal
Epithermal
Fast
6.25219E-01
3.68398E-01
6.36363E-03
1.11943E+00
1.04135E-01
8.71626E-03
8.98270E-01
1.30071E+00
6.10101E-01
K-effective and precision
235
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
1.30713E+00 (precision = 4.22E-08)
6.24948E-01
3.68842E-01
6.20890E-03
1.19375E+00
1.04416E-01
8.57722E-03
8.97862E-01
1.32643E+00
5.91044E-01
K-effective and precision
319
Absorption (collisions/cm3 -s)
3
1.30675+1.36E-07
6.37583E-01
Nu-Fission (fissions/cm -s)
1.21789E+00
1.02432E-01
8.51358E-03
9.16016E-01
1.32989E+00
5.82311E-01
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
1.32884+6.56E-08
6.33980E-01
3.59880E-01
6.14268E-03
1.21101E+00
1.02142E-01
8.51358E-03
9.10839E-01
1.32941E+00
5.82311E-01
K-effective and precision
399
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
1.32166+8.73E-08
6.54072E-01
3.39782E-01
6.14288E-03
1.24939E+00
1.03116E-01
8.51360E-03
9.39705E-01
1.33678E+00
5.82312E-01
K-effective and precision
485
6.14309E-03
Average Flux (particles/cm2-s)
K-effective and precision
355
3.56242E-01
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
1.36102+4.11E-08
5.64199E-01
3.39685E-01
6.11164E-03
1.24963E+00
1.03082E-01
8.48345E-03
1.34103E+00
5.78120E-01
9.39888E-01
K-effective and precision
1.36120+4.81E-08
133
Table 6.50: Reaction rates for GA-193, 235, 319, 355, 399, 485 energy group structures
PRISMATIC
Energy Range
Group Structure
193
Reaction Rates
Thermal
3
5.609600E-01
4.312260E-01 7.790330E-03
Nu-Fission (fissions/cm3-s)
1.076820E+00
1.307680E-01 1.120650E-02
Average Flux (particles/cm2-s)
1.177660E+00
2.220720E+00 1.055180E+00
1.218799+5.02E-07
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
5.60578E-01
4.31830E-01
7.59023E-03
1.07609E+00
1.31155E-01
1.10260E-02
1.17684E+00
2.25431E+00
1.02219E+00
K-effective and precision
319
1.21827+3.86E-07
Absorption (collisions/cm3 -s)
5.76707E-01
Nu-Fission (fissions/cm3-s)
1.10705E+00
1.28131E-01
1.09433E-02
Average Flux (particles/cm2-s)
1.21070E+00
2.25899E+00
1.00706E+00
K-effective and precision
355
7.50419E-03
5.75205E-01
4.17293E-01
7.50414E-03
Nu-Fission (fissions/cm -s)
1.10417E+00
1.27779E-01
1.09433E-02
Average Flux (particles/cm2-s)
1.20755E+00
2.25882E+00
1.00707E+00
K-effective and precision
1.24892+1.19E-07
Absorption (collisions/cm3 -s)
3
Nu-Fission (fissions/cm -s)
2
Average Flux (particles/cm -s)
5.99303E-01
3.93188E-01
7.50431E-03
1.15043E+00
1.29395E-01
1.09434E-02
1.25814E+00
2.27443E+00
1.00707E+00
K-effective and precision
485
4.15752E-01
1.24612+1.15E-07
Absorption (collisions/cm3 -s)
3
399
Fast
Absorption (collisions/cm -s)
K-effective and precision
235
Epithermal
1.29077+3.67E-07
Absorption (collisions/cm3 -s)
5.99483E-01
3.93048E-01
7.46356E-03
Nu-Fission (fissions/cm3-s)
1.15077E+00
1.29341E-01
1.09042E-02
Average Flux (particles/cm2-s)
1.25851E+00
2.28181E+00
9.99809E-01
K-effective and precision
1.29102+1.23E-07
134
6.5.3 Thermal Energy Region Improvement
(
)
Thermal energy region 5.000 ×10−10 → 3.059 ×10−6 MeV has 102 energy groups. The fast (49
groups) and epithermal (42 groups) are kept constant during thermal region improvement.
During the improvement of the GA-193 to 205 groups, the selected reaction rates (absorption)
and the k-effective met the target criteria of 1 percent deviation and the 10 pcm relative
deviation of ∆k / k . However, the nu-fission did not meet the target criteria. Therefore the 205
energy group structure was further improved to 211 where all reaction rates and the k-effective
target criteria’s were met. A slight increase in the absorption rate and nu-fission rate was
observed from GA-193 energy group to 205 energy group, which is a positive improvement
when compared to the reference solution from the MCNP5. The k-effective is decreasing, which
is different from what was expected (Table 6.52).
Table 6.51: Thermal groups selected in the thermal region
GA193-group structure improvement
Group structure number
Number of Groups in Different energy ranges
Fast
Epithermal
Thermal
Total
1
49
42
102
193
2
49
42
114
205
3
49
42
120
211
135
Table 6.52: Eigen-values resulted for thermal energy group improvement
Eigen-values results of fine energy group structure
Relative
Deviation in
pcm of
% Relative
% Relative
∆k / k with
Deviation
Deviation
ν ∑f
( ∑ a ) with previous
previous
with previous
group
Absorption
group
Nu-Fission
group
(
)
Group
structure
K-effective
193
1.307125E+00
-
205
1.307042E+00
8.3
6.25886E-01 -0.1066826 1.19502E+00 -6.7528298
211
1.30703E+00
0.9
6.25885E-01 0.0001598 1.19501E+00 0.0008368
6.25219E-01
-
1.11943E+00
-
The importance function plotted in Figure 6.9 shows a high importance in the energy ranges of
5.50 ×10−8 → 2.18 ×10−7 MeV and that of 2.38 ×10−6 → 3.06 ×10−6 MeV . This was reduced as the
energy group structure was divided into two or more groups thus improving the reaction rates
results. Table 6.53 and 6.54 presents the reaction rates of all regions for the Pebble fuel element
and prismatic hexagonal block respectively. The changing of one region results in the changes in
the reaction rates for all regions showing that all regions energy group structure has effect from
any changes that are happening in the neighboring regions.
136
0.08
GA-193
importance
0.06
GA-205
GA-211
0.04
0.02
0
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
Energy (MeV)
Figure 6.9: Importance function for thermal energy region for 193, 205 and 211 groups
137
Table 6.53: Pebble FE results
PEBBLE
Energy Range
Group
Structure
Reaction Rates
Thermal
Epithermal
Fast
193
Absorption (collisions/cm3-s)
6.25219E-01
3.68398E-01
6.36363E-03
Nu-Fission (fissions/cm3-s)
1.11943E+00
1.04135E-01
8.71626E-03
Average Flux (particles/cm2-s) 8.98270E-01
1.30071E+00
6.10101E-01
K-effective
205
1.30713 (convergence = 4.22E-08)
Absorption (collisions/cm3-s) 6.258860E-01 3.677320E-01 6.363510E-03
Nu-Fission (fissions/cm3-s)
1.195020E+00 1.033030E-01 8.716260E-03
Average Flux (particles/cm2-s) 9.159530E-01 1.290220E+00 6.101010E-01
K-effective
211
1.307042 (convergence = 1.38E-07)
Absorption (collisions/cm3-s)
6.25885E-01
3.67732E-01
6.36363E-03
Nu-Fission (fissions/cm3-s)
1.19501E+00
1.03303E-01
8.71629E-03
Average Flux (particles/cm2-s) 9.16074E-01
1.29022E+00
6.10103E-01
K-effective
1.30703 (convergence = 2.13E-07)
138
Table 6.54: Prismatic block results
PRISMATIC
Energy Range
Group
Structure
193
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s) 5.609600E-01 4.312260E-01 7.790330E-03
Nu-Fission (fissions/cm3-s)
1.076820E+00 1.307680E-01 1.120650E-02
Average Flux (particles/cm2-s) 1.177660E+00 2.220720E+00 1.055180E+00
K-effective
205
1.218799 (convergence = 5.02E-07)
Absorption (collisions/cm3-s) 5.617200E-01 4.304660E-01 7.790330E-03
Nu-Fission (fissions/cm3-s)
1.077700E+00 1.297970E-01 1.120660E-02
Average Flux (particles/cm2-s) 1.205090E+00 2.194490E+00 1.055180E+00
K-effective
211
1.218705 (convergence = 5.83E-08)
Absorption (collisions/cm3-s)
5.61720E-01
4.30465E-01
7.78985E-03
Nu-Fission (fissions/cm3-s)
1.07769E+00
1.29796E-01
1.12065E-02
Average Flux (particles/cm2-s) 1.20526E+00
2.19448E+00
1.05518E+00
K-effective
1.21870 (convergence = 3.86E-07)
139
6.5.4 Improved GA-193 Energy Group Structure for all Regions
Three energy group structures selected during each region’s improvement are combined and
totaling to 537 and is referred to as GA_TPN-537. The library was generated for this final energy
group structure and used for the analysis of the pebble FE and prismatic hexagonal block. Table
6.55 gives the number of the starting and final energy groups. The reaction rates for the initial
GA-193 and 537 are given in Table 6.56 and 6.57 for the pebble FE and prismatic hexagonal
block. The reaction rates were compared with the ones from the original GA-193 energy group
structure as shown in Table 6.58 and 6.59.
Table 6.55: Energy group structure improved from GA-193 to GA_TPN-537
GA 193-group structure improvement
Group structure number
Number of Groups in different energy ranges
Fast
Epithermal
Thermal
Total
1
49
42
102
193
2
83
334
120
537
140
Table 6.56: Reaction rates for GA-193 and GA_TPN-537 energy group structures
PEBBLE
Energy Range
Group Structure
193
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s) 6.25219E-01 3.68398E-01 6.36363E-03
Nu-Fission (fissions/cm3-s) 1.11943E+00 1.04135E-01 8.71626E-03
Average Flux (particles/cm2-s) 8.98270E-01 1.30071E+00 6.10101E-01
K-effective
537
1.30713 (convergence = 4.22E-08)
Absorption (collisions/cm3-s) 6.549040E-01 3.389820E-01 6.109270E-03
Nu-Fission (fissions/cm3-s) 1.250420E+00 1.022130E-01 8.475520E-03
Average Flux (particles/cm2-s) 9.585390E-01 1.323410E+005.778350E-01
K-effective
1.361109 (convergence = 2.06E-08)
141
Table 6.57: Reaction rates for GA-193 and GA_TPN-537 energy group structures
PRISMATIC
Group
Structure
193
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s) 5.609600E-01 4.312260E-01 7.790330E-03
Nu-Fission (fissions/cm3-s)
1.076820E+00 1.307680E-01 1.120650E-02
Average Flux (particles/cm2-s) 1.177660E+00 2.220720E+00 1.055180E+00
K-effective
537
1.218799 (convergence = 5.02E-07)
Absorption (collisions/cm3-s) 6.003060E-01 3.922280E-01 7.460500E+00
Nu-Fission (fissions/cm3-s)
1.151720E+00 1.283040E-01 1.089400E-02
Average Flux (particles/cm2-s) 1.288040E+00 2.253810E+00 9.993210E-01
K-effective
1.290921 (convergence = 7.71E-08)
142
Table 6.58: Comparisons for 537 with GA-193 energy group structure for the pebble FE.
% relative deviation
Relative
Deviation in
pcm of
∆k / k with
previous
group
Reaction rates
Thermal
Epithermal
Fast
Absorption
4.53273
-8.67775
-4.16351
Nu-Fission
10.47592
-1.88039
-2.84042
Average Flux
6.28759
1.71542
-5.58395
k-effective
5398.4
Table 6.59: Comparisons for GA_TPN-537 with GA-193 energy group structure for the
prismatic block.
% relative deviation
Relative
Deviation in
pcm of ∆k / k
Reaction rates
Thermal
Epithermal
Fast
Absorption
6.55432
-9.94269
99.89558
Nu-Fission
6.50332
-1.92044
-2.86855
Average Flux
8.56961
1.46818
-5.58969
143
with previous
group
k-effective
7212.2
The MCNP5 results for both pebble fuel element and prismatic hexagonal block are given in
Table 6.60 and 6.62. They are then compared with the improved energy group structure results in
Table 6.61 and 6.63. A significant change in the reaction rates deviations is observed. However,
it is believed that there are some underlying issues about the energy group structure. The keffective was improved significantly about 5000 pcm from initial deviations.
Table 6.60: MCNP5 results for the pebble FE
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
7.24977E-01
2.51396E-01
5.33901E-03
Nu-Fission (fissions/cm3-s)
1.40447E+00
1.01377E-01
8.65859E-03
Average Flux (particles/cm2-s)
1.09878E+00
1.32725E+00
5.97252E-01
1.52881 ± 0.00046
K-effective and deviation
144
Table 6.61: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the
pebble FE
% relative deviation
Relative
Deviation in
pcm of
Reaction
rates
Thermal
Epithermal
Fast
Absorption
9.66549
-34.83965
-14.42692
Nu-Fission
10.96881
-0.82508
2.11433
Average Flux
12.76364
0.28946
3.25110
∆k / k with
previous
group
k-effective
-16770.1
Table 6.62: MCNP5 results for the prismatic block.
Revise to use MCNP5 values given in Chapter 4
Energy Range
Reaction Rates
Thermal
Epithermal
Fast
Absorption (collisions/cm3-s)
6.97577E-01
2.97869E-01
7.38072E-03
Nu-Fission (fissions/cm3-s)
1.37041E+00
1.30683E-01
1.08548E-02
Average Flux (particles/cm2-s)
1.45410E+00
2.19054E+00
9.41143E-01
1.47467 ± 0.00156
K-effective and deviation
145
Table 6.63: Comparisons of GA_TPN-537 energy group structures to MCNP5 results for the
prismatic block.
% relative deviation
Relative
Deviation in
pcm of ∆k / k
Reaction
rates
Thermal
Epithermal
Fast
Absorption
13.94410
-31.67800
-100980.973
Nu-Fission
15.95797
1.82043
-0.36076
Average Flux
11.41982
-2.88848
-6.18167
146
with previous
group
k-effective
17853.9
Chapter 7
Comparative Analysis of ENDF Data Files
7.1 Introduction
The purpose of this section was to analyze the latest released ENDF/B-VII.1. Two multi-group
energy structures (SHEM-361 and SHEM_TPN-531) were selected for the analysis. Section 7.2
compares the Pebble fuel element and Prismatic hexagonal block analysis based on the ENDF/BVII.0 and ENDF/B-VII.1. The discussion on the differences of the evaluation data files is given
in section 7.3.
7.2 Comparison Results
Table 7.1 presents the Pebble fuel element results for ENDF/B-VII.0 and ENDF/B-VII.1. This
was computed for SHEM-361 and the improved energy group structure SHEM_TPN-531. Their
percent relative deviations are shown in Table 7.2. The fission reaction rate percent relative
deviation is below 1 percent. The absorption reaction rate for the thermal and epithermal energy
regions for both energy groups are also below 1 percent, however it is observed that the
absorption reaction rate at fast energy region is higher. The relative percent deviation of ∆k / k is
higher than the accepted 500 pcm for the pebble FE. Similar behavior is observed for the
prismatic hexagonal block results with the higher percent relative deviation at the fast energy
region (Table 7.3 and 7.4). As seen that the reaction rate percent relative deviations between the
147
ENDF/B-VII.1 and ENDF/B-VII.0 data files is insignificant (Table 7.2 and 7.4), this may be due
to the fact important nuclides like U-235, U-238 were not changed during data advancements,
noting their importance (fuel kernel). The relative deviation for ∆k / k is below 500 pcm for the
prismatic hexagonal block. It is noticeable that the k-effective sensitivity for the pebble fuel
element is higher compared to that of the prismatic hexagonal block, which can be attributed to
the geometry. This is in agreement with the MCNP5 results (Chapter 4) when comparing
ENDF/B-VI.8 and ENDF/B-VII.0.
It is pointed out in the Nuclear Data Sheets by Chadwick et al, 2011[4] that the validation testing
for ENDF/B-VII.1 maintained a good performance compared to that of ENDF/B-VII.0 for
nuclear criticality of which the improved performance is attributed to the new structural
materials evaluations. Figure 7.1 shows the neutron cross section for natural carbon as an
example. Concerns were raised on the shortfall of major actinides (U-235, U-238 and Pu-239)
for the previous ENDF/B-VII.0 data files, for example poor performance of Pu-239 in thermal
ranges and the cross section capture for U-235 that is reported to be 25% higher in the 1 keV
region. Therefore, U-235, U-238, Pu-239 were changed back to the ENDF/B-VI.8 data neutron
parameters since there are no new advancements on these nuclides as yet.
148
Figure 7.1: Neutron capture on natural carbon for ENDF/B-VII.1
and ENDF/B-VII.0 (Chadwick et al, 2011)
149
Table 7.1: Pebble FE results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons
PEBBLE
ENDF/B-VII.0
Energy Range
Group Structure
Reaction Rates
Thermal
Epithermal
Fast
SHEM-361
Absorption (collisions/cm3-s)
7.35048E-01
2.58686E-01
6.26171E-03
Nu-Fission (fissions/cm3-s)
1.40368E+00
1.04725E-01
8.63635E-03
Average Flux (particles/cm2-s) 1.07127E+00
1.32084E+00
5.97363E-01
1.51705 (convergence = 3.44E-08)
K-effective
SHEM_TPN-531 Absorption (collisions/cm3-s)
7.35554E-01
2.58361E-01
6.08033E-03
1.40434E+00
1.04105E-01
8.43757E-03
Average Flux (particles/cm2-s) 1.08158E+00
1.33654E+00
5.74212E-01
Nu-Fission (fissions/cm3-s)
1.51688 (convergence = 2.45E-08)
K-effective
ENDF/B-VII.1
SHEM-361
Absorption (collisions/cm3-s)
7.34864E-01
2.58774E-01
6.35707E-03
Nu-Fission (fissions/cm3-s)
1.39795E+00
1.04702E-01
8.63426E-03
Average Flux (particles/cm2-s) 1.07119E+00
1.32065E+00
5.97300E-01
1.51129 (convergence = 7.49E-08)
K-effective
SHEM_TPN-531 Absorption (collisions/cm3-s)
7.35374E-01
2.58449E-01
6.17179E-03
1.39987E+00
1.04081E-01
8.43539E-03
Average Flux (particles/cm2-s) 1.07863E+00
1.33634E+00
5.74150E-01
Nu-Fission (fissions/cm3-s)
1.51123 (convergence = 2.40E-08)
K-effective
150
Table 7.2: END/B-VII.0 and ENDF/B-VII.1 % deviations for the pebble reaction rates and keffective in pcm
% Deviation
SHEM-361
Reaction Rates
Thermal
Epithermal
Fast
Absorption
0.025032
-0.034018
-1.522907
Nu-Fission
0.408213
0.021962
0.024200
Average Flux
0.007468
0.014385
0.010546
K-effective ( pcm )
SHEM_TPN-531
575.6
Absorption
0.024471
-0.034061
-1.504195
Nu-Fission
0.318228
0.023054
0.025837
Average Flux
0.272749
0.014964
0.010797
K-effective ( pcm )
565.6
151
Table 7.3: Prismatic block results using ENDF/B-VII.0 and ENDF.B-VII.1 comparisons
PRISMATIC
ENDF/B-VII.0
Energy Range
Group Structure
SHEM-361
Reaction Rates
Thermal
Epithermal
Fast
3.05456E-01
7.65884E-03
1.31808E+00
1.31889E-01
1.11035E-02
Average Flux (particles/cm2-s) 1.46805E+00
2.25232E+00
1.03317E+00
Absorption (collisions/cm3-s) 6.86881E-01
Nu-Fission (fissions/cm3-s)
1.46107 (convergence = 3.44E-07)
K-effective
3
SHEM_TPN-531 Absorption (collisions/cm -s) 6.87493E-01
3.05078E-01
7.42278E-03
1.31892E+00
1.31148E-01
1.08449E-02
Average Flux (particles/cm2-s) 1.48322E+00
2.28032E+00
9.93066E-01
Nu-Fission (fissions/cm3-s)
1.46091 (convergence = 9.21E-09)
K-effective
ENDF/B-VII.1
SHEM-361
Absorption (collisions/cm3-s) 6.86708E-01
3.05532E-01
7.75397E-03
1.31376E+00
1.31859E-01
1.11008E-02
Average Flux (particles/cm2-s) 1.46806E+00
2.25199E+00
1.03306E+00
Nu-Fission (fissions/cm3-s)
1.45672 (convergence = 1.94E-07)
K-effective
3
SHEM_TPN-531 Absorption (collisions/cm -s) 6.87323E-01
3.05155E-01
7.51480E-03
1.31469E+00
1.31118E-01
1.08421E-02
Average Flux (particles/cm2-s) 1.48009E+00
2.27997E+00
9.92958E-01
Nu-Fission (fissions/cm3-s)
1.45665 (convergence = 5.16E-08)
K-effective
152
Table 7.4: END/B-VII.0 and ENDF/B-VII.1 % deviation for the prismatic block reaction rates
and k-effective in pcm
% Deviation
SHEM-361
Reaction Rates
Thermal
Epithermal
Fast
Absorption
0.025186
-0.024881
-1.242094
Nu-Fission
0.327749
0.022746
0.024317
Average Flux
-0.000681
0.014652
0.010647
K-effective (pcm)
SHEM_TPN-531 Reaction Rates
434.5
Thermal
Epithermal
Fast
Absorption
0.024728
-0.025239
-1.239697
Nu-Fission
0.320717
0.022875
0.025819
Average Flux
0.211027
0.015349
0.010875
K-effective (pcm)
425.8
7.3 Nuclear Data Advancements
The most important advances of the new evaluated data files (ENDF/B-VII.1) include an
increase in neutron reaction cross section coverage from 393 to 423 nuclides, and the covariance
uncertainty data for 190 of the most important nuclides. New R-matrix evaluations on light
nuclides (He, Li, and Be) are give as well as resonance parameter analysis at lower energies and
statistical high energy reactions for isotopes (Cl, K, Ti, V, Mn, Cr, Ni, Zr and W). Also, changes
153
on thermal neutron reactions on fission products (Mo, Tc, Rh, Ag, Cs, Nd, Sm, Eu) and neutron
absorbers (Cd, Gd). Minor actinides evaluations for isotopes (U, Np, Pu, and Am) were
improved, but there were no improvements on the major actinides (U-235, U-238, and Pu-239)
as discussed above. They then adopted JENDL-4.0 evaluations for Cm, Bk, Cf, Es, Fm, isotopes
and other minor actinides. New data for fission energy release evaluations and a new decay data
sub-library was created. Fission product yield advances for fission spectrum neutrons and 14
MeV neutrons incident on Pu-239.
Table 7.5 presents the comparisons for the ENDF/B-VII.0, ENDF/B-VII.1 libraries and
ENDF/B-VI.8. NSUB is the sub-library number in ENDF/B-VI format. The last two columns
give the materials (or isotopes) for the libraries.
154
Table 7.5: ENDF/B-VII.1, ENDF/B-VII.0 and ENDF/B-VI.8 data file advancements
No.
NSUB
Sub-library name
Short name
ENDF/B-
ENDF/B-
ENDF/B-
VII.1
VII.0
VI.8
1
0
Photonuclear
g
163
163
-
2
3
Photo-atomic
photo
100
100
100
3
4
Radioactive decay
decay
3817
3838
979
4
5
Spontaneous
s/fpy
9
9
9
fission yields
5
6
Atomic relaxation
ard
100
100
100
6
10
Neutron
n
423
393
328
7
11
Neutron fission
n/fpy
31
31
31
yields
8
12
Thermal scattering
tsl
20
20
15
9
19
Standards
std
8
8
8
10
113
Electro-atomic
e
100
100
100
11
10010
Proton
p
48
48
35
12
10020
Deuteron
d
5
5
2
13
10030
Triton
t
3
3
1
14
20030
he3
2
2
1
3
He
155
Chapter 8
Depletion Analysis
8.1 Introduction
Depletion is an important aspect of reactor analysis for the reactor safety and for the prediction
of the economic performance of the reactor. It includes various nuclear reactions, and is
described by the isotopic depletion rate equations where isotopic concentrations are solved as a
function of time and position. Secondly, the multi-group transport/diffusion equations are solved
for the neutron flux. The depletion calculations in this study were performed using DRAGON
code. It uses an EVO module wherein depletion equations for different isotopes in the library are
solved using burnup chains as available in the generated library. Depletion can be performed at
constant flux or constant power in MW/tonne of initial uranium using burnup time steps (in
Days), assuming linear flux/power changes. The burnup mixtures of the unit cell are solved using
the rate equations describing the isotopic changes in the core composition during the reactor
operation. The rate equations are expressed as shown below, Marleau, et al, 2010 [22].
dN k
+ Nk (t ) Λk (t ) =
Sk ( t ) , k = 1, K
dt
8.1
Λ k ( t ) =λk + σ a ,k ( t ) φ ( t ) ,
8.2
Sk ( t )
L
∑Y
σ
(t )φ (t )
K
N ( t ) + ∑ mkl ( t ) N l ( t ) ,
kl
f ,l
l
l 1 =l 1
156
8.3
∞
σ x ,l ( t ) φ ( t ) = ∫ σ x ,l ( u )φ ( t , u ) du
8.4
0
σ x , k ( t f , u ) φ ( t f , u ) − σ x , k ( t0 , u ) φ ( t0 , u )
σ x ,k ( t , u ) φ ( t , u ) =
σ x , k ( t0 , u ) φ ( t0 , u ) +
( t − t0 )
t f − t0
8.5
Where
K = number of depleting isotopes
L = number of fissile isotopes
N k ( t ) = time dependent number density for k th isotope
λk = radioactive decay constant for k th isotope
σ x ,k ( t , u ) = time and lethargy dependent microscopic cross section for nuclear reaction x on k th
isotope, x can be absorption fission and radiative capture cross sections
φ ( t , u ) = time and lethargy dependent neutron flux
Ykl = fission yield for production of fission product k by fissile l
mkl = radioactive decay constant or σ x ,l ( t ) φ ( t ) term for the production of isotope k by isotope
l
The advantage of the DRAGON code in depletion analysis is that it allows the calculations of
rate equations for as many isotopes as required per type of the nuclear reactor of interest. Thus,
157
there is sufficient capability for monitoring the nuclides concentrations, fission products buildup,
and burnable poison concentrations. Similar HTGR models as in previous chapters are used
(Pebble fuel element and Prismatic hexagonal blocks) for depletion analysis. The depletion
analysis were performed using input multi-group cross section library based on selected
SHEM_TPN-531 multi-group structure at a constant power of 62 MW/tonne and for burnup
steps of 0.5, 5, 10, 20, 30, 40, 50, 60, 70, 80, 100 and 120 GWD/tonne of initial uranium.
8.2 Pebble Fuel Element
Figure 8.1 presents the criticality behavior of the Pebble fuel element during the depletion
process per burnup steps in GWD/tonne. The effect of the xenon fission product buildup and keffective is observed at the beginning of the depletion process. The xenon causes a sharp drop in
the k-effective during the first few days of core depletion. The overall criticality curve
presentation produce good results as expected for assumed constant power. The k-effective
results are in agreement with the work reported by Dehart and Goluoglu [7], where depletion
calculations were conducted using Serpent and TRITON/NEWT and TRITON/KENO codes.
The neutron flux spectrum for the beginning of life and end of life are given in Figures 8.2 and
8.3 respectively, showing an increase in neutron flux peak for thermal regions for the end of life.
The neutron flux spectrum increase observed in the end of life (Figure 8.3) as compared to the
beginning of life (Figure 8.2) is due to the number of nuclides and fission products with thermal
energy that are produced during burnup process. These nuclides and fission products then
158
compete with the control rods resulting in a need to remove control rods from the system in order
to keep the reactor at constant power. The nuclide concentration changes, for example of U-235
as it fissions has been reduced by 98 percent from the beginning of life to the end of life. More
nuclides and fission products were produced (Table 8.1) and their concentration increased with
an increase in burnup time step. Pu-239 increased by 98 percent from its initial production
(beginning of life) to the end of life. The summary of the nuclides concentrations in atom/barncm are given in Table 8.1 and Figure 8.4 for selected burnup steps. Fission products are shown in
Table 8.2 and Figure 8.5. The xenon-135 buildup is about 66 percent and that of Sm149 is 10
percent from beginning of life to the end of life.
1.6
1.5
1.4
K-effective
K-effective
1.3
1.2
1.1
1
0.9
0.8
0.7
0
20
40
60
80
Burnup (GWD/t)
100
Figure 8.1: K-effective versus burnup
159
120
140
0.5
Flux/lethargy width
0.4
Pebble_BOL
0.3
0.2
0.1
0
1.00E-09
1.00E-07
1.00E-05
1.00E-03
1.00E-01
1.00E+01
Energy (MeV)
Figure 8.2: Neutron flux spectrum at the beginning of life
0.8
Flux/lethargy width
0.6
Pebble_EOL
0.4
0.2
0
1.00E-09
1.00E-07
1.00E-05 1.00E-03
Energy (MeV)
1.00E-01
1.00E+01
Figure 8.3: Neutron flux spectrum at the end of life
160
Table 8.1: Nuclides concentration in atom/barm.cm
Burnup (GWD/t)
Nuclides
10
20
40
80
120
U-235
1.64E-03
1.38E-03
9.27E-04
2.98E-04
4.27E-05
U-236
4.84E-05
9.15E-05
1.63E-04
2.48E-04
2.58E-04
U-238
2.12E-02
2.11E-02
2.09E-02
2.03E-02
1.96E-02
Np-237
3.69E-07
1.47E-06
5.53E-06
1.81E-05
2.95E-05
Pu-238
8.61E-09
7.39E-08
6.21E-07
4.87E-06
1.19E-05
Pu-239
7.61E-05
1.30E-04
1.88E-04
1.98E-04
1.70E-04
Pu-240
5.02E-06
1.61E-05
4.15E-05
7.46E-05
7.90E-05
Pu-241
7.40E-07
4.68E-06
2.14E-05
5.10E-05
5.06E-05
Pu-242
2.12E-08
2.95E-07
3.40E-06
2.78E-05
6.95E-05
Am-241
3.89E-09
4.90E-08
4.41E-07
1.75E-06
1.58E-06
Am-242m
1.71E-11
3.51E-10
4.42E-09
1.98E-08
1.74E-08
Am-243
2.61E-10
7.80E-09
2.02E-07
4.02E-06
1.77E-05
Cm-242
1.43E-10
3.58E-09
6.57E-08
6.41E-07
1.20E-06
Cm-244
4.47E-12
2.86E-10
1.69E-08
8.86E-07
8.07E-06
Cm-245
2.11E-14
2.72E-12
3.13E-10
2.73E-08
2.74E-07
161
1.0E+00
1.0E-02
Concentration (atom/barm.cm)
1.0E-04
1.0E-06
1.0E-08
1.0E-10
1.0E-12
1.0E-14
1.0E-16
1.0E-18
1.0E-20
U235
U236
U238
Np237
Pu238
Pu239
Pu240
Pu241
Pu242
Am241
Am242m
Am243
Cm242
Cm244
Cm245
1.0E-22
1.0E-24
0
20
40
60
80
Burnup (GWD/t)
100
120
140
Figure 8.4: Nuclides concentration in atom/barm.cm
Table 8.2: Fission products in atom/barn.cm
Burnup (GWD/t)
Fission
Products
10
20
40
80
120
Kr-85
6.22E-07
1.19E-06
2.18E-06
3.62E-06
4.37E-06
Sr-90
1.40E-05
2.70E-05
5.03E-05
8.66E-05
1.09E-04
Ag-110m 2.71E-10
1.46E-09
8.30E-09
5.15E-08
1.58E-07
Cs-137
1.55E-05
3.08E-05
6.09E-05
1.19E-04
1.73E-04
Xe-135
1.62E-08
1.54E-08
1.34E-08
8.74E-09
5.57E-09
Sm-149
1.43E-07
1.41E-07
1.32E-07
9.65E-08
6.76E-08
Sm-151
4.12E-07
4.72E-07
5.04E-07
4.98E-07
4.77E-07
162
1.0E-03
Concentration (atom/barm.cm)
1.0E-04
1.0E-05
1.0E-06
1.0E-07
1.0E-08
1.0E-09
1.0E-10
Kr85
Sr90
Ag110m
1.0E-11
Cs137
Xe135
Sm149
1.0E-12
Sm151
1.0E-13
0
20
40
100
60
80
Burnup (GWD/t)
120
140
Figure 8.5: Fission products buildup in a Pebble fuel element
8.3 Prismatic Hexagonal Block
Similar depletion analysis were conducted for the prismatic hexagonal block and the criticality
behavior is presented in Figure 8.6. The neutron flux spectrum at the beginning and end of life
are shown in Figure 8.7 and 8.8 with significant neutron flux spectrum peak in thermal region as
a result more nuclides and fission products are produced and the U-235 is reduced. A significant
decrease in U-235 concentration of about 95 percent is observed. Pu-239 concentration had
increased by 98 percent during the burnup process. The xenon-135 buildup change is about 51
percent. This is in agreement with prismatic hexagonal blocks depletion analysis results reported
by Rohde et al, 2011[32], where BGCore and Helios codes were used. The nuclides
163
concentration per selected burnup steps is given in Table 8.3 and Figure 8.9. The fission products
are given in Table 8.4 and Figure 8.10.
1.5
1.4
K-effective
1.3
K-effective
1.2
1.1
1
0.9
0.8
0.7
0
20
40
60
80
Burnup (GWD/t)
100
120
140
Figure 8.6: Prismatic block K-effective versus burnup
0.6
Flux/lethargy width
0.5
Prismatic_BOL
0.4
0.3
0.2
0.1
0
1.00E-09
1.00E-07
1.00E-05
1.00E-03
Energy (MeV)
1.00E-01
1.00E+01
Figure 8.7 Neutron flux spectrum at the beginning of life
164
0.7
0.6
Flux/lethargy width
Prismatic_EOL
0.5
0.4
0.3
0.2
0.1
0
1.00E-09
1.00E-07
1.00E-05 1.00E-03
Energy (MeV)
1.00E-01
1.00E+01
Figure 8.8: Neutron flux spectrum at the end of life
165
Table 8.3: Nuclides concentration in atom/barn.cm
Burnup (GWD/t)
Nuclides
10
20
40
80
120
U-235
1.64E-03
1.39E-03
9.55E-04
3.67E-04 9.32E-05
U-236
5.02E-05
9.39E-05
1.64E-04
2.45E-04 2.58E-04
U-238
2.12E-02
2.11E-02
2.08E-02
2.02E-02 1.94E-02
Np-237
4.78E-07
1.88E-06
6.93E-06
2.19E-05 3.56E-05
Pu-238
1.26E-08
1.07E-07
8.74E-07
6.51E-06 1.64E-05
Pu-239
8.74E-05
1.48E-04
2.16E-04
2.45E-04 2.27E-04
Pu-240
6.09E-06
1.87E-05
4.52E-05
7.77E-05 8.48E-05
Pu-241
1.09E-06
6.60E-06
2.83E-05
6.65E-05 7.24E-05
Pu-242
3.21E-08
4.24E-07
4.40E-06
3.05E-05 6.99E-05
Am-241
5.77E-09
6.98E-08
5.91E-07
2.38E-06 2.71E-06
Am-242m 2.81E-11
5.52E-10
6.60E-09
3.06E-08 3.45E-08
Am-243
4.92E-10
1.39E-08
3.28E-07
5.34E-06 2.00E-05
Cm-242
2.32E-10
5.55E-09
9.28E-08
7.91E-07 1.53E-06
Cm-244
1.03E-11
6.22E-10
3.35E-08
1.40E-06 9.88E-06
Cm-245
6.13E-14
7.51E-12
8.14E-10
6.11E-08 5.23E-07
166
3.0E-01
3.0E-03
3.0E-05
Concentration (atom/barm.cm)
3.0E-07
3.0E-09
3.0E-11
3.0E-13
3.0E-15
3.0E-17
3.0E-19
3.0E-21
U235
U236
U238
Np237
Pu238
Pu239
Pu240
Pu241
Pu242
Am241
Am243
Cm242
Cm244
Cm245
3.0E-23
0
20
40
60
80
Burnup (GWD/t)
100
120
Figure 8.9: Nuclides concentration in atom/barn.cm
167
140
Table 8.4: Fission products in atom/barn.cm
Burnup (GWD/t)
Fission
Products
10
20
40
80
120
Kr-85
6.17E-07
1.17E-06
2.13E-06
3.51E-06
4.30E-06
Sr-90
1.39E-05
2.66E-05
4.91E-05
8.38E-05
1.07E-04
Ag-110m
3.56E-10
1.95E-09
1.10E-08
6.25E-08
1.68E-07
Cs-137
1.54E-05
3.07E-05
6.07E-05
1.19E-04
1.74E-04
Xe-135
1.69E-08
1.64E-08
1.50E-08
1.13E-08
8.37E-09
Sm-149
1.48E-07
1.51E-07
1.49E-07
1.25E-07
9.83E-08
Sm-151
4.40E-07
5.28E-07
6.00E-07
6.60E-07
6.74E-07
1.0E-04
Concentration (atom/barm.cm)
1.0E-05
1.0E-06
1.0E-07
1.0E-08
1.0E-09
1.0E-10
1.0E-11
Kr85
Sr90
Ag110m
Cs137
Xe135
Sm149
Sm151
1.0E-12
1.0E-13
0
20
40
60
80
Burnup (GWD/t)
100
120
140
Figure 8.10: Fission products buildup in a Prismatic hexagonal block
168
Chapter 9
Conclusions and Recommendations
9.1 Conclusions
The objective of this study was to investigate the applicability of the SHEM energy group
structures (281 and 361) and GA-193 for HTR applications and improve them utilizing the more
systematic, consistent, and sophisticated energy group selection method referred to as
contributon and point-wise cross-section driven (CPXSD). The DRAGON code was used for the
energy group structure improvement and the MCNP5 as reference for comparing the results to
determine the magnitude of the improvement.
MCNP5 code was selected to provide a reference solution to the study. Therefore both the pebble
fuel element and prismatic hexagonal block are the basic components of the PBR and VHTR
analyzed by DRAGON and MCNP5 for comparing results. Then, energy boundaries were
selected to divide the energy spectrum into three regions (fast, epithermal and thermal) that
correspond to the SHEM-281, SHEM-361 and GA-193 energy group structures. These analysis
were based on ENDF/B-VII.0 and ENDF/B-VI.8 data files. Additionally, the sensitivity caused
by the data files was studied. When comparing reaction rates from the three regions, the thermal
region had higher percent deviations and the epithermal and fast regions had their percent
deviations below 1 percent. The major changes were observed to be due to helium advancement
169
from ENDF/B -VI.8 to ENDF/B -VII.0 data files. Also neutron sub-library was increased from
328 to 393 as well as radioactive decay (from 979 to 3838), and a new 163 photonuclear sublibrary was added. Similar Pebble fuel element and Prismatic hexagonal block models were
created for the DRAGON code analysis. The neutron cross sections were generated using
SHEM-361, SHEM-281 and GA-193 energy group structures. Then, the pebble fuel element and
prismatic hexagonal block fuel analysis were subjected to these libraries and the results were
compared to the MCNP5 (reference solution) as shown in Table 9.1 and 9.2.
Table 9.1: Pebble fuel element deviations
Group Structure
SHEM-281
SHEM-361
GA-193
K-effective
1.528750
1.516920
1.517046
1.307130
deviation in pcm
% deviation
SHEM-281
Reaction rates
Absorption
Nu-Fission
Average Flux
Thermal
-1.389212277
0.05013748
2.499473073
Epithermal
-2.882545451
-3.099762817
0.494391026
Fast
-17.43271822
0.255135027
-0.018703628
% deviation
SHEM-361
Reaction rates
Absorption
Nu-Fission
Average Flux
Thermal
-1.389212277
0.056545572
2.504023558
Epithermal
-2.899649913
-3.302965595
0.483089478
Fast
-17.28212864
0.25686741
-0.018536195
K-effective
deviation
MCNP5
DRAGON
170
1189
1176.4
22168.5
Table 9.2: Prismatic hexagonal block deviations
K-effective
deviation
MCNP5
DRAGON
% deviation
SHEM-281
Absorption
Nu-Fission
Average Flux
% deviation
SHEM-361
Absorption
Nu-Fission
Average Flux
Group Structure
K-effective
1.46946
SHEM-281
1.46094E+00
SHEM-361
1.461067E+00
GA-193
1.21880E+00
Reaction rates
Thermal
Epithermal
1.521522941
-2.520222601
3.807588733
-0.699405992
-0.972081048
-2.807681093
Thermal
1.53327792
3.818534366
-0.959702214
Epithermal
-2.547080042
-0.922847458
-2.820463348
deviation in pcm
852.4
839.3
25066.1
Fast
-3.753748785
-2.290771967
-9.778257997
Fast
-3.768246022
-2.290771967
-9.778257997
The k-effective difference between the MCNP5 calculation and DRAGON are approximately
1.19 % for the pebble fuel element and 0.85 % for the prismatic hexagonal fuel block. For the
DRAGON code to become reliable for analyzing Pebble bed and Prismatic reactors, the keffective deviation should be less than 500 pcm . Further examination of Table 9.1 and 9.2 show
that for the Pebble fuel element the major reaction rates deviation is -17 % for the fast absorption
rate and for the Prismatic block is 3.8 % for the thermal nu-fission reaction rates. This could
explain the major errors that occur in the DRAGON calculation. A small error of 3.83 in the
thermal nu-fission reaction rate for the Pebble fuel element could propagate into an error greater
than 1 % in k-effective whereas a -17 % error in fast absorption rate for the Prismatic hexagonal
block would propagate also into an error greater than 1 % for k-effective because k-effective is
less sensitive to errors in the fast absorption rate. Note that an increase in the MCNP5 thermal
171
nu-fission and a decrease in MCNP5 fast absorption rate both cause the MCNP5 k-effective to be
greater than the corresponding DRAGON k-effective. It is interesting to note that the errors are
different for reaction rates for the Pebble fuel and Prismatic block. This might indicate that
geometry plays some part in the way DRAGON calculates reaction rates (see Table 5.7, 5.8 and
5.9 and 5.10).
Optimizing the SHEM-281 and SHEM-361multi-group structures to 407 and 531 groups caused
relatively small changes in the k-effective and reaction rates. The small changes in the keffective and reaction rates due to the increase in the multi-groups (from 281 to 407 and from
361 to 531) groups were insignificant compared to the significant deviation in k-effective caused
by the 17 % deviation in the fast absorption reaction rate in the Pebble fuel element and the 3.8
% deviation in the nu-fission reaction rate for the Prismatic hexagonal block. Thus, using
SHEM_TPN-407 and SHEM_TPN-531 group structures in the DRAGON calculation does not
correct for the relatively large deviation in the k-effective for either Pebble fuel element or
Prismatic hexagonal block. Nor, will it correct the deviation in the fast absorption reaction rate in
the Pebble fuel element or the nu-fission reaction rate in the Prismatic hexagonal block. The
DRAGON code calculation method of the Pebble cell fast absorption reaction rate and the
Prismatic block nu-fission reaction rate should be corrected so that the DRAGON code
calculates these reaction rates more accurately.
172
It is also observed that SHEM-281 energy group structure development had solved sufficient
nuclear physics problems that were introduced by the assumptions considered for XMAS-172
energy group structure. This was done by eliminating the self-shielding calculation of thermal
resonances, removing self-shielding models for U-238 resonances, Er = 6.7 and 20.9 eV .
Sufficient nuclides and fission products were taken in to account. Then, SHEM-281 was
improved to SHEM-361 using a subgroup model in the regions of 22.5 eV and 11.14 keV
(epithermal region). However, there is little effect on the reaction rates and k-effective results as
shown in chapter 5 when comparing the two energy group structures. This is in agreement with
the results obtained in this study, the improvement of SHEM energy group structures had little
effect on the reaction rates and k-effective as the pebble fuel element and prismatic hexagonal
block models were analyzed.
GA-193 was improved to GA_TPN-537 energy group structure. The fast energy region was
improved from 49 to 83 groups, epithermal region from 42 to 334 and thermal region from 102
to 120. When comparing the results obtained using GA-193 and GA_TPN-537 energy group
structures, the relative percent deviation for k-effective was 5398.4 pcm for the pebble fuel
element and 7212.2 pcm for the prismatic hexagonal block. By improving GA-193 to GA_TPN537, the relative percent deviation is reduced from 22168.5 pcm to 16770.1 pcm to the MCNP5
results for the pebble fuel element. For the prismatic hexagonal block, the relative percent
deviation is reduced from 25066.1 pcm to 17853.9 pcm to the MCNP5 results. There was a
significant improvement in reaction rates as well but the deviations remain very large.
173
There was also recently published ENDF/B-VII.1 that was decided to be tested in this study and
be compared to ENDF/B-VII.0. The neutron cross sections were generated for both data files and
then pebble fuel element and prismatic hexagonal block models were analyzed using these
libraries. There were no significant changes in the reaction rates, the reason being that there were
no significant changes in the important nuclides (U-235, U-238). The observed changes were in
k-effective, which were about 500 pcm in difference. This is attributed to the advancement of
structural materials, for example natural carbon cross sections (Figure 7.1), which is important
for HTR since graphite is the moderator.
Lastly, the SHEM_TPN-531 energy group structure was used for the depletion analysis. The
objective was to test its ability for the HTR deep burn analysis. It is interesting to note that the
change in k-effective with increase in burnup, Figure 8.1 is a straight line once the sharp drop in
k-effective due to xenon buildup is passed. No such straight line occurs for the change in keffective as a function of burnup in light water reactors. Then, the conclusion is that the energy
group structure is applicable for HTR deep burn analysis as the results are showing a good
agreement with other researchers [7] [32].
9.2 Recommendation
The main recommendation based on the results of this study is to determine why there are large
deviations, -17 % in the fast absorption reaction rate for the Pebble fuel element and 3.8 % in the
thermal nu-fission reaction rate, when DRAGON calculates the reaction rates. Once these
174
problems are fixed DRAGON should be an excellent code to generate multi-group cross section
for graphite moderated reactors. In addition, although DRAGON does create broad group cross
sections, these cross sections cannot be reprocessed within DRAGON for further cross section
improvement. This should be corrected.
Next step will be to utilize the SHEM_TPN-531 multi-group structure with DRAGON and
develop an optimized broad group cross-section structure for core analysis using again the
contributon method. Such study would require coupling DRAGON with core analysis code,
which has also forward and adjoint flux calculation capability.
175
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180
Appendices
A1 MCNP5 Input Decks
A1.1 Pebble Fuel Element
Pebble
1 1 -10.4 -13
u=5 imp:n=1 $ kernel UO2
2 2 -1.05 13 -14 u=5 imp:n=1 $ inner low density carbon
3 3 -1.9 14 -15 u=5 imp:n=1 $ inner pyro carbon layer
4 4 -3.18 15 -16 u=5 imp:n=1 $ Silicon carbide layer
5 3 -1.90 16 -17 u=5 imp:n=1 $ Outer pyro carbon layer
6 5 -1.75 17
u=5 imp:n=1 $ graphite matrix
7 5 -1.75 -18 19 -20 21 -22 23 lat=1 u=8 imp:n=1 $ cube
fill=-14:14 -14:14 -14:14
c (121)
8 260r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r
8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r
8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 260r
c (169)
8 231r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 231r
c (285)
8 173r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 6r 5 14r 8 6r 8 173r
c (401)
8 115r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r
8 6r 8 7r 5 12r 8 7r 8 115r
c (469)
8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r
8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r
c (469)
8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r
8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r
c (549)
8 57r 8 7r 5 12r 8 7r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 8 5 24r 8 8
181
8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r
8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8
5 24r 8 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r
8 4r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 7r 5 12r 8 7r 8 57r
c (625)
8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r
8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8
8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r
8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r
c (625)
8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8
8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r
8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
182
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (697)
8 8r 5 10r 8 8r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 3r 5 20r 8 3r
8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8 5 26r
8 5 318r 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 2r
5 22r 8 2r 8 3r 5 20r 8 3r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 8r
5 10r 8 8r
c (625)
8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r
8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 8 5 24r 8 8
8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r
8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r
c (625)
8 28r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 4r 5 18r 8 4r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8
5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8 5 26r 8 8
8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r
8 4r 5 18r 8 4r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 28r
c (549)
8 57r 8 7r 5 12r 8 7r 8 5r 5 16r 8 5r 8 4r 5 18r 8 4r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 8 5 24r 8 8 8 8 5 24r 8 8
8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r
8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8 5 24r 8 8 8 8
5 24r 8 8 8 8 5 24r 8 8 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r
5 18r 8 4r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 7r 5 12r 8 7r 8 57r
c (469)
8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r
8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r
c (469)
8 86r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
183
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 2r 5 22r
8 2r 8 2r 5 22r 8 2r 8 2r 5 22r 8 2r 8 3r 5 20r 8 3r 8 4r 5 18r
8 4r 8 5r 5 16r 8 5r 8 6r 5 14r 8 6r 8 7r 5 12r 8 7r 8 86r
c (401)
8 115r 8 7r 5 12r 8 7r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 4r 5 18r
8 4r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r 8 3r 8 3r 5 20r
8 3r 8 3r 5 20r 8 3r 8 4r 5 18r 8 4r 8 5r 5 16r 8 5r 8 6r 5 14r
8 6r 8 7r 5 12r 8 7r 8 115r
c (285)
8 173r 8 6r 5 14r 8 6r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r 8 5r 8 5r 5 16r
8 5r 8 6r 5 14r 8 6r 8 173r
c (169)
8 231r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r 8 7r 8 7r 5 12r
8 7r 8 7r 5 12r 8 7r 8 231r
c (121)
8 260r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r
8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r
8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 8r 5 10r 8 8r 8 260r
8 0 -40 41 -42 43 -44 45 u=52 fill=8 imp:n=1
9 5 -1.75 40:-41:42:-43:44:-45 u=52 imp:n=1
10 0 -24 u=53 fill=52 imp:n=1
11 5 -1.75 24 u=53 imp:n=1
12 0
-25 u=54 fill=53 imp:n=1
13 13 -0.000176 25 u=54 imp:n=1
14 0 -26 fill=54 imp:n=1
15 0 26 imp:n=0
c Surfaces
13 so 0.025
14 so 0.034
15 so 0.038
16 so 0.0415
17 so 0.0455
18 px 0.081705
19 px -0.081705
20 py 0.081705
21 py -0.081705
22 pz 0.081705
23 pz -0.081705
24 so 2.5
25 so 3.0
*26 so 3.53735
40 px 2.369445
41 px -2.369445
42 py 2.369445
43 py -2.369445
44 pz 2.369445
45 pz -2.369445
c imp:n 1 13r 0
c
184
c Material data
mode n
m1 92238.66c 2.12877E-02 $ U-238
293.6 1993
92235.66c 1.92585E-03
$ U-235
293.6K 1997
8016.62c 4.64272E-02
$ Oxygen
293.6K 2000
5010.66c 1.14694E-07
$ Boron-10
5011.66c 4.64570E-07
$ Boron-11
m2 6000.24c 5.26449E-02 $ Natural Carbon (buffer)
mt2 grph.60t
m3 6000.24c 9.52621E-02 $ Natural Carbon
mt3 grph.60t
m4 6000.24c 4.77240E-02 $ Natural Carbon
14000.60c 4.77240E-02 $ Silicon
293.6K 1976
mt4 grph.60t
m5 6000.24c 8.77414E-02 $ Natural Carbon
5010.66c 9.64977E-09
$ Boron-10
5011.66c 3.90864E-08
$ Boron-11Fuel
mt5 grph.60t
m13 2003.66c 3.71220E-11 $ Helium gas outside the FE 293.6K 1990
2004.60c 2.65156E-05
$ Helium gas outside the FE 293.6K 1973
ksrc 0 0 0
kcode 2000 1.0 200 1200
prdmp j 20 0 2
c
c Tallies: Fluxes and Reaction rates
c Energy bins
c E0 3.14200E-06 1.15600E-01 1.96400E+01 $ 3 broad groups
E0 3.05902E-06 1.11090E-01 1.49182E+01
c
c Thermal, epithermal and fast fluxes
F4:N 9
$ graphite cube
11
$ graphite layer
13
$ heliumlayer
SD4 65.44984695
47.64748858
72.30820487
c
c Coated Particle flux
F14:N (1 < (7 [-14:14 -14:14 -14:14])<10) $ fuel kernel
(2 < (7 [-14:14 -14:14 -14:14])<10) $ buffer
(3 < (7 [-14:14 -14:14 -14:14])<10) $ IPyC
(4 < (7 [-14:14 -14:14 -14:14])<10) $ SiC
(5 < (7 [-14:14 -14:14 -14:14])<10) $ OPyC
SD14 0.987835
1.49702
0.984231
1.049562
1.43658
c
c reaction rates
c
F24:N 9
FM24 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7)
SD24 0.044085965
c
F44:N 11
FM44 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7)
SD44 47.64748858
185
c
F54:N 13
FM54 -1.0 13 (-1) (-2) (-6) (-7) (-6 -7)
SD54 72.30820487
c
F64:N (1 < (7 [-14:14 -14:14 -14:14])<10)
FM64 -1.0 1 (-1) (-2) (-6) (-7) (-6 -7)
SD64 0.987835
c
F74:N (2 < (7 [-14:14 -14:14 -14:14])<10)
FM74 -1.0 2 (-1) (-2) (-6) (-7) (-6 -7)
SD74 1.49702
c
F84:N (3 < (7 [-14:14 -14:14 -14:14])<10)
FM84 -1.0 3 (-1) (-2) (-6) (-7) (-6 -7)
SD84 0.984231
c
F94:N (4 < (7 [-14:14 -14:14 -14:14])<10)
FM94 -1.0 4 (-1) (-2) (-6) (-7) (-6 -7)
SD94 1.049562
c
F104:N (5 < (7 [-14:14 -14:14 -14:14])<10)
FM104 -1.0 5 (-1) (-2) (-6) (-7) (-6 -7)
SD104 1.43658
c
186
A1.2 Prismatic Hexagonal Block
Prismatic Assembly lattice pattern
c Hexagonal Lattice
c -------------------------------- note --------------------------------------c Preserve the coated particle packing fraction
c Removed the particles sitting at the boundary of the fuel compact
c ----------------------------------------------------------------------------c ----------------------------------------------------------------------------c Cells
10 10 -0.000176 -10 u=10 imp:n=1 $ coolant channel
20 0 -20 fill=1(1) u=10 imp:n=1
$ fuel channel
30 0 -30 fill=1(2) u=10 imp:n=1
40 0 -40 fill=1(3) u=10 imp:n=1
50 0 -50 fill=1(4) u=10 imp:n=1
60 0 -60 fill=1(5) u=10 imp:n=1
70 0 -70 fill=1(6) u=10 imp:n=1
80 80 -1.75 10 20 30 40 50 60 70 u=10 imp:n=1
c lattice
90 21 -10.4 -12
u=20 imp:n=1 $ kernel
91 22 -1.05 12 -13 u=20 imp:n=1 $ Inner low density kernel coating layer
92 23 -1.90 13 -14 u=20 imp:n=1 $ Inner Pyro Carbon kernel coating layer
93 24 -3.18 14 -15 u=20 imp:n=1 $ Silicon carbide layer
94 25 -1.90 15 -16 u=20 imp:n=1 $ Outer pyro Carbon coating layer
95 26 -1.75 16
u=20 imp:n=1 $ outside of the coating layer
85 26 -1.75 -16
u=50 imp:n=1 $ graphite medium interior
86 26 -1.75 16
u=50 imp:n=1 $ graphite medium exterior
96 0
-90 91 -92 93 -94 95 lat=1 u=21 imp:n=1 $ cube
fill=-6:6 -6:6 -20:20
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191
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50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
192
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
193
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
194
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
195
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 20 20 20 20 20 20 20 20 20 20 20 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 20 20 20 20 20 20 20 20 20 50 50
50 50 50 20 20 20 20 20 20 20 50 50 50
50 50 50 50 20 20 20 20 20 50 50 50 50
50 50 50 50 50 50 20 50 50 50 50 50 50
50 50 50 50 50 50 50 50 50 50 50 50 50
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row ***
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
$row
c
87 0
90 -91 92 -93 94 -95 u=21 imp:n=1 $ fuel cube exterior
97 0
-17 fill=21
u=1 imp:n=1 $ inside He gap
98 27 -0.000176 17
u=1 imp:n=1 $ outside
c Hexagonal cell
100 0 -100 200 -300 400 -500 600 u=30 fill=10 lat=2 imp:n=1 $ Hexagon
c
c Sample cell
200 0 -700 710 -720 730 -740 750 fill=30 imp:n=1
250 0 (700:-710:720:-730:740:-750)
imp:n=0
c
c
300 0 -700 710 -720 730 -740 750 20 30 40 50 60 70 imp:n=1
c ----------------------------------------------------------------------------c ----------------------------------------------------------------------------c SURFACE CARDS
c Cylinder surfaces
10 cz
0.79400
20 c/z 1.628127759 0.94 0.63500
196
30 c/z 0 1.88
0.63500
40 c/z -1.628127759 0.94 0.63500
50 c/z -1.628127759 -0.94 0.63500
60 c/z 0
-1.88 0.63500
70 c/z 1.628127759 -0.94 0.63500
c
c Hex cell surfaces
100 px 1.628127759
200 px -1.628127759
300 p 0.577350269 1 0 1.88
400 p 0.577350269 1 0 -1.88
500 p -0.577350269 1 0 1.88
600 p -0.577350269 1 0 -1.88
c
c Boundary Cell (Assembly)
*700 px 3.256
*710 px -3.256
*720 pz 2.465
*730 pz -2.465
*740 py 2.820
*750 py -2.820
c surface
17 cz 0.6225
12 so 0.025
13 so 0.034
14 so 0.038
15 so 0.0415
16 so 0.0455
90 px 0.0560
91 px -0.0560
92 py 0.0560
93 py -0.0560
94 pz 0.0601220
95 pz -0.0601220
c ----------------------------------------------------------------------------c ----------------------------------------------------------------------------c DATA CARDS
tr1 1.628127759 0.94 0
tr2 0 1.88
0
tr3 -1.628127759 0.94 0
tr4 -1.628127759 -0.94 0
tr5 0
-1.88 0
tr6 1.628127759 -0.94 0
mode n
c MATERIAL
m10 2003.66c 1.40000e-06 $ Helium gas outside the FE 293.6K 1990
2004.60c 9.99999e-01
$ Helium gas outside the FE 293.6K 1973
m21 92238.66c 3.05676e-01 $ U-238
293.6 1993
92235.66c 2.76538e-02
$ U-235
293.6K 1997
8016.62c 6.66662e-01
$ Oxygen
293.6K 2000
5010.66c 1.64692e-06
$ Boron-10
5011.66c 6.67090e-06
$ Boron-11
m22 6000.24c 1.00000e+00 $ Natural Carbon (buffer)
mt22 grph.60t
m23 6000.24c 1.00000e+00 $ Natural Carbon
mt23 grph.60t
m24 6000.24c 5.00000e-01 $ Natural Carbon
14000.60c 5.00000e-01 $ Silicon
293.6K 1976
197
mt24 grph.60t
m25 6000.24c 1.00000e+00 $ Natural Carbon
mt25 grph.60t
m26 6000.24c 9.99999e-01 $ Natural Carbon
5010.66c 1.09980e-07
$ Boron-10
5011.66c 4.45472e-07
$ Boron-11
mt26 grph.60t
m27 2003.66c 1.40000e-06 $ Helium gas outside the FE 293.6K 1990
2004.60c 9.99999e-01
$ Helium gas outside the FE 293.6K 1973
m80 6000.24c 9.99999e-01 $ Natural Carbon
5010.66c 1.09980e-07
$ Boron-10
5011.66c 4.45472e-07 $ Boron-11
mt80 grph.60t
c
c CRITICALITY CONTROL CARDS
kcode 200 1.0 200 1200
ksrc 0.000 0.000 0.000
c
c TALLY CARDS
c E0
3.14200E-06 1.15600E-01 1.96400E+01
E0 3.05902E-06 1.11090E-01 1.49182E+01
c -------------------- FLUX IN SYSTEM ------------------------FC4 Broad-group FLUX in CELLS (6-group)
F4:n
10
$ Coolant channel
98
$ Helium gap
80
$ Moderator
SD4
9.7642
2.4345e-01
23.0159
c
FC14
Broad-group FLUX in FUEL (6-group)
F14:N (90 < (96 [-6:6 -6:6 -20:20]) < 97) $ Fuel kernel
(91 < (96 [-6:6 -6:6 -20:20]) < 97)
$ 1 layer coating
(92 < (96 [-6:6 -6:6 -20:20]) < 97)
$ 2 layer coating
(93 < (96 [-6:6 -6:6 -20:20]) < 97)
$ 3 layer coating
(94 < (96 [-6:6 -6:6 -20:20]) < 97)
$ 4 layer coating
SD14 0.1963
0.2976
0.1956
0.2086
0.2855
FC24
Broad-group FLUX outside FUEL (6-group)
F24:N ( (95 85 86) < (96 [-6:6 -6:6 -20:20]) < 97) $ graphite matrix
SD24 4.8180
c -------------------------------------------------------------c
c ---------------- REACTION RATE IN SYSTEM --------------------F34:N 10
$ Coolant channel
FM34 -1.0 10 (-1) (-2) (-6) (-7)
SD34 9.7642
c
F44:N 98
$ Helium gap
FM44 -1.0 27 (-1) (-2) (-6) (-6 -7)
SD44 2.4345e-01
c
F54:N 80
$ Moderator
FM54 -1.0 80 (-1) (-2) (-6) (-6 -7)
SD54 23.0159
198
c
F64:N (90 < (96 [-6:6 -6:6 -20:20]) < 97) $ Fuel kernel
FM64 -1.0 21 (-1) (-2) (-6) (-6 -7)
SD64 0.1963
c
F74:N (91 < (96 [-6:6 -6:6 -20:20]) < 97) $ 1 layer coating
FM74 -1.0 22 (-1) (-2) (-6) (-6 -7)
SD74 0.2976
c
F84:N (92 < (96 [-6:6 -6:6 -20:20]) < 97) $ 2 layer coating
FM84 -1.0 23 (-1) (-2) (-6) (-6 -7)
SD84 0.1956
c
F94:N (93 < (96 [-6:6 -6:6 -20:20]) < 97) $ 3 layer coating
FM94 -1.0 24 (-1) (-2) (-6) (-6 -7)
SD94 0.2086
c
F104:N (94 < (96 [-6:6 -6:6 -20:20]) < 97) $ 4 layer coating
FM104 -1.0 25 (-1) (-2) (-6) (-6 -7)
SD104 0.2855
c
F114:N ( (95 85 86) < (96 [-6:6 -6:6 -20:20]) < 97) $ graphite matrix
FM114 -1.0 26 (-1) (-2) (-6) (-6 -7)
SD114 4.8180
c --------------------------------------------------------------
199
A2 DRAGON Input Decks
A2.1 Pebble Fuel Element
*----* HTR FUEL MODELING
* Pebble1
* Surrounded by Helium
* ENDF7_361
* Author: Tholakele Prisca Ngeleka
*----* Define STRUCTURES and MODULES
*-----LINKED_LIST
PEBBLE DISCR DISCR1 LIBRARY LIBRARY1 LIBRARY2
MACRO1 MACRO2
CP CPAM CALC CALCB OUT OUTA OUTB OUTC OUTD OUTE
OUTF OUTG OUTH OUT1 OUT2 OUT3 OUT4 OUT5
OUT6 OUT7 OUT8 COMPO FLUX DATABASE DATABASE1 ;
SEQ_ASCII calcFl1 calcFl2 calcFl3 calcFl4 calcFl5
calcFl6 calcFl7 calcFl8 calcFlA calcFlB
calcFlC calcFlD calcFlE calcFlF calcFlG
calcFlH multicompo ;
MODULE
LIB: GEO: SYBILT: USS: SHI: ASM:
FLU: EDI: T: COMPO: END: ;
*
*-----* Microscopic cross section
*-----*
LIBRARY := LIB: ::
NMIX 8 CTRA NONE
SUBG
MACR
MIXS LIB: DRAGON FIL: ENDF7_361
*
* Mixtures
* 1 = fuel kernel
* 2 = Inner low density carbon kernel coating (Buffer)
* 3 = Inner Pyro carbon kernel coating
* 4 = Silicon Carbide kernel coating
* 5 = Outer Pyro carbon kernel coating
* 6 = Compact carbon matrix
* 7 = Helium coolant
* 8 = Pebble outer coating
*
* Fuel kernel
MIX 1 293.6
U238 = 'U238' 2.12877E-02 1
U235 = 'U235' 1.92585E-03 1
O16 = 'O16' 4.64272E-02
B10 = 'B10' 1.14694E-07
B11 = 'B11' 4.64570E-07
* Inner low density carbon kernel (buffer)
MIX 2 293.6
200
Cnat = 'C12_GRA' 5.26449E-02
* Inner Pyro carbon kernel coating
MIX 3 293.6
Cnat = 'C12_GRA' 9.52621E-02
* Silicon carbide kernel coating
MIX 4 293.6
Si28 = 'Si28' 4.402E-02
Si29 = 'Si29' 2.235E-03
Si30 = 'Si30' 1.473E-03
C0
= 'C12' 4.772E-02
* Outer Pyro carbon kernel coating
MIX 5 293.6
Cnat = 'C12_GRA' 9.52621E-02
* Compact carbon matrix
MIX 6 293.6
Cnat = 'C12_GRA' 8.77414E-02
B10 = 'B10'
9.64977E-09
B11 = 'B11'
3.90864E-08
* Helium coolant
MIX 7 293.6
He3 = 'He3' 3.71220E-11
He4 = 'He4' 2.65156E-05
* Pebble outer coating
MIX 8 293.6
Cnat = 'C12_GRA' 8.77414E-02
B10 = 'B10' 9.64977E-09
B11 = 'B11' 3.90864E-08 ;
*------* GEOMETRY PEBBLE :
PEBBLE := GEO: :: SPHERE 3
EDIT 10000
R+ REFL RADIUS 0.0 2.5 3.0 3.53735
MIX 9 8 7
* Coated Particles
* NMISTR = Number of microstructures/coated particles types in region
* NMILG = Number of microstructure/coated particles regions
* NS = ARRAY OF SUB REGIONS IN THPebble1.x2mE COATED PATICLE/MICROSTRUCTURES; LEN=NMILG
* RS = RADIUS OF COATED PARTICLES/MICROSTRUCTURES, LEN=NS(I); I=1; NMISTR;
* milie = COMPOSITION OF EACH COATED PARTICLE/MICROSTRUCTURE, LEN=NMISTR;
* "NOTE: MILIE NO'S ARE GREATER THAN MIX NO'S"
* mixdil = Base composition of each region, LEN=NMILG
* fract = Microstructure type volume fraction in region LEN=NMILG
* mixgr = LIBRARY MIXTURES FOR EACH COATED PARTICLE/MICROSTRUCTURE LAYER; LEN=NS(I)
*NMISTR, NMILG
BIHET SPHE 1 1
(* NS *) 5
(* RS *) 0.0 0.025 0.034 0.038 0.0415 0.0455
(* milie *) 9
(* mixdil *) 6
(* fract *) 0.09042852
(* mixgr *) 1 2 3 4 5 ;
*----*
* TRACKING MODULES
* SYBILT -used for 1D geometries (plane, cyl or spherical)
* EXCELT-perform full cell collision probability tracking with isotropic/specular surface current
201
* NXT -extension of excelt for more complex geometries
* MCCGT-implementation of open characteristics method
* SNT -implementation of the discrete ordinates method (1D/2D/3D geom)
* BIVACT-perform a finite element (Diff/SPn) 1D/2D tracking
* TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D trackikng
*
*SYBILT (CASE 1)
DISCR := SYBILT: PEBBLE ::
TITLE 'Pebble1: HTR FUEL MODELING (SYBIL /SYBIL)'
MAXR 1000 ;
LIBRARY1 := USS: LIBRARY DISCR :: EDIT 1 ;
CP := ASM: LIBRARY1 DISCR :: EDIT 0 ;
CALC := FLU: CP LIBRARY1 DISCR :: EDIT 1
TYPE K ;
OUTA := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERG COMP
COND 38 281 361
MICR ALL
SAVE ;
*SYBILT (CASE 1)
DISCR1 := SYBILT: PEBBLE ::
TITLE 'Pebble1: HTR FUEL MODELING (SYBIL /SYBIL)'
MAXR 1000 ;
LIBRARY2 := USS: LIBRARY DISCR1 :: EDIT 1 ;
MACRO1 := LIBRARY2 :: STEP UP MACROLIB ;
MACRO2 := T: MACRO1 ;
CPAM := ASM: MACRO2 DISCR1 :: EDIT 0 ;
CALCB := FLU: CPAM MACRO2 DISCR1 :: EDIT 4
TYPE K ;
OUTB := EDI: CALCB MACRO2 DISCR1 ::
EDIT 2
MERGE COMP
MICR ALL
SAVE ;
END: ;
202
A2.2 Prismatic Hexagonal Block
*----* HTR FUEL MODELING
* Prismatic block
* Surrounded by Helium
* ENDF7 SHEM 361
* Author:Tholakele Prisca Ngeleka
*----* Define STRUCTURES and MODULES
*-----LINKED_LIST
PRISM DISCR DISCR1 LIBRARY LIBRARY1 LIBRARY2
CPAM OUTK OUTL OUTM OUTN OUTP OUTQ OUTR OUTS
COMPO CALC FLUX DATABASE3 ;
* SEQ_BINARY TRACK_F1 TRACK_F2 ;
SEQ_ASCII calcFl1 calcFl2 calcFl3 calcFl4
calcFl5 calcFl6 calcFl7 calcFl8
multicompo ;
MODULE
LIB: GEO: SYBILT: USS: ASM: SHI: FLU:
EDI: COMPO: END: ;
*
* Microscopic cross section
*-----LIBRARY := LIB: ::
NMIX 8 CTRA NONE
SUBG
MIXS LIB: DRAGON FIL: ENDF7_361
*
* Mixtures
* 1 = fuel kernel
* 2 = Inner low density carbon kernel coating (Buffer)
* 3 = Inner Pyro carbon kernel coating
* 4 = Silicon Carbide kernel coating
* 5 = Outer Pyro carbon kernel coating
* 6 = Compact carbon matrix
* 7 = Helium coolant
* 8 = Compact carbon matrix
*
* Fuel kernel
MIX 1 293.6
U238 = 'U238' 2.12877E-02 1
U235 = 'U235' 1.92585E-03 1
O16 = 'O16' 4.64272E-02
B10 = 'B10' 1.14694E-07
B11 = 'B11' 4.64570E-07
*
* Inner low density carbon kernel (buffer)
MIX 2 293.6
Cnat = 'C12_GRA' 5.26449E-02
*
* Inner Pyro carbon kernel coating
MIX 3 293.6
Cnat = 'C12_GRA' 9.52621E-02
203
*
* Silicon carbide kernel coating
MIX 4 293.6
Si28 = 'Si28' 4.402E-02
Si29 = 'Si29' 2.235E-03
Si30 = 'Si30' 1.473E-03
C0
= 'C12' 4.772E-02
*
* Outer Pyro carbon kernel coating
MIX 5 293.6
Cnat = 'C12_GRA' 9.52621E-02
*
* Compact carbon matrix
MIX 6 293.6
Cnat = 'C12_GRA' 8.77414E-02
B10 = 'B10'
9.64977E-09
B11 = 'B11'
3.90864E-08
*
* Helium coolant
MIX 7 293.6
He3 = 'He3' 3.71220E-11
He4 = 'He4' 2.65156E-05
*
* Pebble outer coating
MIX 8 293.6
Cnat = 'C12_GRA' 8.77414E-02
B10 = 'B10'
9.64977E-09
B11 = 'B11'
3.90864E-08 ;
*------*
* GEOMETRY PRISMATIC :
*
PRISM := GEO: :: HEX 36
HBC S30 REFL
CELL
* Ring 0 (center cell)
C
* Ring 1
F
* Ring 2
CF
* Ring 3
FC
* Ring 4
CFF
* Ring 5
FCF
* Ring 6
CFFC
* Ring 7
FCFF
* Ring 8
CFFCF
* Ring 9
FCFFC
* Ring 10
CFFCFF
*
204
* Coated Particles
* NMISTR = Number of microstructures/coated particles types in region
* NMILG = Number of microstructure/coated particles regions
* NS = ARRAY OF SUB REGIONS IN THE COATED PATICLE/MICROSTRUCTURES; LEN=NMILG
* RS = RADIUS OF COATED PARTICLES/MICROSTRUCTURES, LEN=NS(I); I=1; NMISTR;
* milie = COMPOSITION OF EACH COATED PARTICLE/MICROSTRUCTURE, LEN=NMISTR;
* "NOTE: MILIE NO'S ARE GREATER THAN MIX NO'S"
* misdil = Base composition of each region, LEN=NMILG
* fract = Microstructure type volume fraction in region LEN=NMILG
* mixgr = LIBRARY MIXTURES FOR EACH COATED PARTICLE/MICROSTRUCTURE LAYER; LEN=NS(I)
*
*NMISTR, NMILG
BIHET SPHE 1 1
(* NS *) 5
(* RS *) 0.0 0.025 0.034 0.038 0.0415 0.0455
(* milie *) 9
(* mixdil *) 6
(* fract *) 0.19723
(* mixgr *) 1 2 3 4 5
::: C := GEO: HEXCEL 1
EDIT 1000
SIDE 1.085
RADIUS 0.0 0.794
MIX 7 8
;
::: F := GEO: HEXCEL 2
EDIT 1000
SIDE 1.085
RADIUS 0.0 0.6225 0.635
MIX 9 7 8 ; ;
*
*PSPPLOT (Graphical presentation)
*----* TRACKING MODULES
* SYBILT -used for 1D geometries (plane, cyl or spherical)
* EXCELT-perform full cell collision probability tracking with isotropic/specular surface
* current
* NXT -extension of excelt for more complex geometries
* MCCGT-implementation of open characteristics method
* SNT -implementation of the discrete ordinates method (1D/2D/3D geom)
* BIVACT-perform a finite element (Diff/SPn) 1D/2D trackikng
* TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D tracking
*
*SYBILT (CASE 1)
* TRACKING MODULES
* SYBILT -used for 1D geometries (plane, cyl or spherical)
* EXCELT-perform full cell collision probability tracking with isotropic/specular surface current
* NXT -extension of excelt for more complex geometries
* MCCGT-implementation of open characteristics method
* SNT -implementation of the discrete ordinates method (1D/2D/3D geom)
* BIVACT-perform a finite element (Diff/SPn) 1D/2D tracking
* TRIVACT-perform a finite element (Diff/SPn) 1D/2D/3D trackikng
*
* SYBILT
*---------------------------------------------* Tracking calculation for self-shielding
*----------------------------------------------
205
* TITLE 'Prismatic: HTR FUEL MODELING (SYBILT)'
DISCR := SYBILT: PRISM ::
EDIT 1
MAXR 1000
;
LIBRARY1 := USS: LIBRARY DISCR ::
EDIT 1
;
CPAM := ASM: LIBRARY1 DISCR ::
EDIT 1
;
CALC := FLU: CPAM LIBRARY1 DISCR ::
EDIT 2
TYPE K ;
* 1 = fuel kernel
OUTK := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 1 0 0 0 0 0 0 0
MICR ALL
SAVE ;
calcFl1 := OUTK ;
* 2 = Inner low density carbon kernel coating (Buffer)
OUTL := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 1 0 0 0 0 0 0
MICR ALL
SAVE ;
calcFl2 := OUTL ;
* 3 = Inner Pyro carbon kernel coating
OUTM := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 1 0 0 0 0 0
MICR ALL
SAVE ;
calcFl3 := OUTM ;
* 4 = Silicon Carbide kernel coating
OUTN := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 0 1 0 0 0 0
MICR ALL
SAVE ;
calcFl4 := OUTN ;
* 5 = Outer Pyro carbon kernel coating
OUTP := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 0 0 1 0 0 0
MICR ALL
SAVE ;
206
calcFl5 := OUTP ;
* 6 = Compact carbon matrix
OUTQ := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 0 0 0 1 0 0
MICR ALL
SAVE ;
calcFl6 := OUTQ ;
* 7 = Helium coolant
OUTR := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 0 0 0 0 1 0
MICR ALL
SAVE ;
calcFl7 := OUTR ;
* 8 = Pebble outer coating
OUTS := EDI: CALC LIBRARY1 DISCR ::
EDIT 2
MERGE MIX 0 0 0 0 0 0 0 1
MICR ALL
SAVE ;
calcFl8 := OUTS ;
* COMPO (FLUX SAMPLING)
DATABASE3 := COMPO: ::
EDIT 5
COMM 'Multi-parameter reactor database' ENDC
INIT
;
DATABASE3 := COMPO: DATABASE3 OUTK OUTL OUTM OUTN
OUTP OUTQ OUTR OUTS ::
EDIT 3
STEP UP 'default'
;
multicompo := DATABASE3 ;
END: ;
207
A3 Energy Group Structures
A3.1 SHEM-361
Group number
Energy (eV)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
1.96403E+07
1.49182E+07
1.38403E+07
1.16183E+07
9.99999E+06
9.04836E+06
8.18730E+06
7.40817E+06
6.70319E+06
6.06530E+06
4.96585E+06
4.06569E+06
3.32871E+06
2.72531E+06
2.23130E+06
1.90139E+06
1.63654E+06
1.40577E+06
1.33694E+06
1.28696E+06
1.16205E+06
1.05115E+06
9.51119E+05
8.60006E+05
7.06511E+05
5.78443E+05
4.94002E+05
4.56021E+05
4.12501E+05
3.83884E+05
3.20646E+05
2.67826E+05
2.30014E+05
1.95008E+05
1.64999E+05
1.40000E+05
1.22773E+05
1.15624E+05
9.46645E+04
8.22974E+04
6.73794E+04
5.51656E+04
4.99159E+04
4.08677E+04
3.69786E+04
3.34596E+04
2.92810E+04
2.73944E+04
2.61001E+04
2.49991E+04
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
208
2.26994E+04
1.85847E+04
1.62005E+04
1.48997E+04
1.36037E+04
1.11377E+04
9.11881E+03
7.46585E+03
6.11252E+03
5.00451E+03
4.09735E+03
3.48107E+03
2.99618E+03
2.70024E+03
2.39729E+03
2.08410E+03
1.81183E+03
1.58620E+03
1.34358E+03
1.13467E+03
1.06432E+03
9.82494E+02
9.09681E+02
8.32218E+02
7.48517E+02
6.77287E+02
6.46837E+02
6.12834E+02
6.00099E+02
5.92941E+02
5.77146E+02
5.39204E+02
5.01746E+02
4.53999E+02
4.19094E+02
3.90760E+02
3.71703E+02
3.53575E+02
3.35323E+02
3.19928E+02
2.95922E+02
2.88327E+02
2.84888E+02
2.76468E+02
2.68297E+02
2.56748E+02
2.41796E+02
2.35590E+02
2.24325E+02
2.12108E+02
Group number
Energy (eV)
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
2.00958E+02
1.95996E+02
1.93078E+02
1.90204E+02
1.88877E+02
1.87559E+02
1.86251E+02
1.84952E+02
1.83295E+02
1.75229E+02
1.67519E+02
1.63056E+02
1.54176E+02
1.46657E+02
1.39504E+02
1.32701E+02
1.26229E+02
1.20554E+02
1.17577E+02
1.16524E+02
1.15480E+02
1.12854E+02
1.10288E+02
1.05646E+02
1.03038E+02
1.02115E+02
1.01605E+02
1.01098E+02
1.00594E+02
9.73287E+01
9.33256E+01
8.87741E+01
8.39393E+01
7.93679E+01
7.63322E+01
7.35595E+01
7.18869E+01
6.90682E+01
6.68261E+01
6.64929E+01
6.61612E+01
6.58312E+01
6.55029E+01
6.50460E+01
6.45923E+01
6.36306E+01
6.23083E+01
5.99250E+01
5.70595E+01
5.40600E+01
Group number
Energy (eV)
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
5.29895E+01
5.17847E+01
4.92591E+01
4.75173E+01
4.62053E+01
4.52904E+01
4.41721E+01
4.31246E+01
4.21441E+01
4.12270E+01
3.97295E+01
3.87874E+01
3.77919E+01
3.73038E+01
3.68588E+01
3.64191E+01
3.60568E+01
3.56980E+01
3.45392E+01
3.30855E+01
3.16930E+01
2.78852E+01
2.46578E+01
2.25356E+01
2.23788E+01
2.21557E+01
2.20011E+01
2.17018E+01
2.14859E+01
2.13360E+01
2.12296E+01
2.11448E+01
2.10604E+01
2.09763E+01
2.07676E+01
2.06847E+01
2.06021E+01
2.05199E+01
2.04175E+01
2.02751E+01
2.00734E+01
1.95974E+01
1.93927E+01
1.91997E+01
1.90848E+01
1.79591E+01
1.77590E+01
1.75648E+01
1.74457E+01
1.68305E+01
Group number
Energy (eV)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
209
1.65501E+01
1.60498E+01
1.57792E+01
1.48662E+01
1.47301E+01
1.45952E+01
1.44702E+01
1.42505E+01
1.40496E+01
1.35460E+01
1.33297E+01
1.26000E+01
1.24721E+01
1.23086E+01
1.21302E+01
1.19795E+01
1.18153E+01
1.17094E+01
1.15894E+01
1.12694E+01
1.10529E+01
1.08038E+01
1.05793E+01
9.50002E+00
9.14031E+00
8.97995E+00
8.80038E+00
8.67369E+00
8.52407E+00
8.30032E+00
8.13027E+00
7.97008E+00
7.83965E+00
7.73994E+00
7.60035E+00
7.38015E+00
7.13987E+00
6.99429E+00
6.91778E+00
6.87021E+00
6.83526E+00
6.81070E+00
6.79165E+00
6.77605E+00
6.75981E+00
6.74225E+00
6.71668E+00
6.63126E+00
6.60611E+00
6.58829E+00
Group number
Energy (eV)
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
6.57184E+00
6.55609E+00
6.53907E+00
6.51492E+00
6.48178E+00
6.43206E+00
6.35978E+00
6.28016E+00
6.16011E+00
6.05991E+00
5.96014E+00
5.80021E+00
5.72015E+00
5.61979E+00
5.53004E+00
5.48817E+00
5.41025E+00
5.38003E+00
5.32011E+00
5.21008E+00
5.10997E+00
4.93323E+00
4.76785E+00
4.41980E+00
4.30981E+00
4.21983E+00
4.00000E+00
3.88217E+00
3.71209E+00
3.54307E+00
3.14211E+00
2.88405E+00
2.77512E+00
2.74092E+00
2.71990E+00
2.70012E+00
2.64004E+00
2.62005E+00
2.59009E+00
2.55000E+00
2.46994E+00
2.33006E+00
2.27299E+00
2.21709E+00
2.15695E+00
2.07010E+00
1.98992E+00
1.90008E+00
1.77997E+00
1.66895E+00
Group number
Energy (eV)
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
1.58803E+00
1.51998E+00
1.44397E+00
1.41001E+00
1.38098E+00
1.33095E+00
1.29304E+00
1.25094E+00
1.21397E+00
1.16999E+00
1.14797E+00
1.12997E+00
1.11605E+00
1.10395E+00
1.09198E+00
1.07799E+00
1.03499E+00
1.02101E+00
1.00904E+00
9.96501E-01
9.81959E-01
9.63960E-01
9.44022E-01
9.19978E-01
8.80024E-01
8.00371E-01
7.19999E-01
6.24999E-01
5.94993E-01
5.54990E-01
5.20011E-01
4.75017E-01
4.31579E-01
3.90001E-01
3.52994E-01
3.25008E-01
3.05012E-01
2.79989E-01
2.54997E-01
2.31192E-01
2.09610E-01
1.90005E-01
1.61895E-01
1.37999E-01
1.19995E-01
1.04298E-01
8.97968E-02
7.64969E-02
6.51999E-02
5.54982E-02
Group number
Energy (eV)
351
352
353
354
355
356
357
358
359
360
361
210
4.73019E-02
4.02999E-02
3.43998E-02
2.92989E-02
2.49394E-02
2.00104E-02
1.48300E-02
1.04505E-02
7.14526E-03
4.55602E-03
2.49990E-03
A3.2 SHEM-281
Group number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Energy (eV)
1.96403E+07
1.49182E+07
1.38403E+07
1.16183E+07
9.99999E+06
9.04836E+06
8.18730E+06
7.40817E+06
6.70319E+06
6.06530E+06
4.96585E+06
4.06569E+06
3.32871E+06
2.72531E+06
2.23130E+06
1.90139E+06
1.63654E+06
1.40577E+06
1.33694E+06
1.28696E+06
1.16205E+06
1.05115E+06
9.51119E+05
8.60006E+05
7.06511E+05
5.78443E+05
4.94002E+05
4.56021E+05
4.12501E+05
3.83884E+05
3.20646E+05
2.67826E+05
2.30014E+05
1.95008E+05
1.64999E+05
1.40000E+05
1.22773E+05
1.15624E+05
9.46645E+04
8.22974E+04
6.73794E+04
5.51656E+04
4.99159E+04
4.08677E+04
3.69786E+04
3.34596E+04
2.92810E+04
2.73944E+04
2.61001E+04
2.49991E+04
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
2.26994E+04
1.85847E+04
1.62005E+04
1.48997E+04
1.36037E+04
1.11377E+04
9.11881E+03
7.46585E+03
6.11252E+03
5.00451E+03
4.09735E+03
3.48107E+03
2.99618E+03
2.57884E+03
2.21963E+03
1.91045E+03
1.61404E+03
1.34506E+03
1.13501E+03
1.06496E+03
9.07501E+02
7.48517E+02
6.12834E+02
5.01746E+02
4.10795E+02
3.53575E+02
3.19928E+02
2.83750E+02
2.41796E+02
1.97966E+02
1.62081E+02
1.32701E+02
1.08646E+02
8.89518E+01
7.50455E+01
6.14420E+01
5.26726E+01
4.57913E+01
4.39958E+01
4.01690E+01
3.37201E+01
2.76077E+01
2.46086E+01
2.25356E+01
2.23784E+01
2.21557E+01
2.20011E+01
2.17018E+01
2.14859E+01
2.13360E+01
211
Group number
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Energy (eV)
2.12296E+01
2.11448E+01
2.10604E+01
2.09763E+01
2.07676E+01
2.06847E+01
2.06021E+01
2.05199E+01
2.04175E+01
2.02751E+01
2.00734E+01
1.95974E+01
1.93927E+01
1.91997E+01
1.90848E+01
1.79591E+01
1.77590E+01
1.75648E+01
1.74457E+01
1.68305E+01
1.65501E+01
1.60498E+01
1.57792E+01
1.48663E+01
1.47301E+01
1.45952E+01
1.44702E+01
1.42505E+01
1.40496E+01
1.35460E+01
1.33297E+01
1.26000E+01
1.24721E+01
1.23086E+01
1.21302E+01
1.19795E+01
1.18153E+01
1.17094E+01
1.15894E+01
1.12694E+01
1.10529E+01
1.08038E+01
1.05793E+01
9.50002E+00
9.14031E+00
8.97995E+00
8.80038E+00
8.67369E+00
8.52407E+00
8.30032E+00
Group number
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Energy (eV)
8.13027E+00
7.97008E+00
7.83965E+00
7.73994E+00
7.60035E+00
7.38015E+00
7.13987E+00
6.99429E+00
6.91778E+00
6.87021E+00
6.83526E+00
6.81070E+00
6.79165E+00
6.77605E+00
6.75981E+00
6.74225E+00
6.71668E+00
6.63126E+00
6.60611E+00
6.58829E+00
6.57184E+00
6.55609E+00
6.53907E+00
6.51492E+00
6.48178E+00
6.43206E+00
6.35978E+00
6.28015E+00
6.16011E+00
6.05991E+00
5.96014E+00
5.80021E+00
5.72015E+00
5.61979E+00
5.53004E+00
5.48817E+00
5.41025E+00
5.38003E+00
5.32011E+00
5.21008E+00
5.10997E+00
4.93323E+00
4.76785E+00
4.41980E+00
4.30981E+00
4.21983E+00
4.00000E+00
3.88217E+00
3.71209E+00
3.54307E+00
Group number
Energy (eV)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
3.14211E+00
2.88405E+00
2.77512E+00
2.74092E+00
2.71990E+00
2.70012E+00
2.64004E+00
2.62005E+00
2.59009E+00
2.55000E+00
2.46994E+00
2.33006E+00
2.27299E+00
2.21709E+00
2.15695E+00
2.07010E+00
1.98992E+00
1.90008E+00
1.77997E+00
1.66895E+00
1.58803E+00
1.51998E+00
1.44397E+00
1.41001E+00
1.38098E+00
1.33095E+00
1.29304E+00
1.25094E+00
1.21397E+00
1.16999E+00
1.14797E+00
1.12997E+00
1.11605E+00
1.10395E+00
1.09198E+00
1.07799E+00
1.03499E+00
1.02101E+00
1.00904E+00
9.96501E-01
9.81959E-01
9.63960E-01
9.44022E-01
9.19978E-01
8.80024E-01
8.00371E-01
7.19999E-01
6.24999E-01
5.94993E-01
5.54990E-01
212
Group number
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
Energy (eV)
5.20011E-01
4.75017E-01
4.31579E-01
3.90001E-01
3.52994E-01
3.25008E-01
3.05012E-01
2.79989E-01
2.54997E-01
2.31192E-01
2.09610E-01
1.90005E-01
1.61895E-01
1.37999E-01
1.19995E-01
1.04298E-01
8.97968E-02
7.64969E-02
6.51999E-02
5.54982E-02
4.73019E-02
4.02999E-02
3.43998E-02
2.92989E-02
2.49394E-02
2.00104E-02
1.48300E-02
1.04505E-02
7.14526E-03
4.55602E-03
2.49990E-03
A3.3 GA-193
Group number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Energy (eV)
1.491820E+07
1.349860E+07
1.221400E+07
1.105170E+07
1.000000E+07
9.048370E+06
8.187310E+06
7.408180E+06
6.703200E+06
6.065310E+06
5.488120E+06
4.965850E+06
4.493300E+06
4.065700E+06
3.678800E+06
3.328700E+06
3.011900E+06
2.725300E+06
2.466000E+06
2.231300E+06
2.019000E+06
1.826800E+06
1.653000E+06
1.495700E+06
1.353400E+06
1.224600E+06
1.108000E+06
1.002600E+06
9.071800E+05
8.208500E+05
7.427400E+05
6.720600E+05
6.081000E+05
5.502300E+05
4.978700E+05
4.504900E+05
4.076200E+05
3.688300E+05
3.337300E+05
3.019700E+05
2.732400E+05
2.472400E+05
2.237100E+05
2.024200E+05
1.831600E+05
1.657300E+05
1.499600E+05
1.356900E+05
1.227700E+05
1.110900E+05
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
8.651700E+04
6.738000E+04
5.247500E+04
4.086800E+04
3.182800E+04
2.478800E+04
1.930500E+04
1.503400E+04
1.170900E+04
9.118800E+03
7.101700E+03
5.530800E+03
4.307400E+03
3.354600E+03
2.612600E+03
2.034700E+03
1.584600E+03
1.234100E+03
9.611200E+02
7.485200E+02
5.829500E+02
4.540000E+02
3.535800E+02
2.753600E+02
2.144500E+02
1.670200E+02
1.300700E+02
1.013000E+02
7.889300E+01
6.144200E+01
4.785100E+01
3.726600E+01
2.902300E+01
2.260300E+01
1.760400E+01
1.371000E+01
1.067700E+01
8.315300E+00
6.475900E+00
5.043500E+00
3.927900E+00
3.059000E+00
2.382400E+00
2.355000E+00
2.310000E+00
2.245000E+00
2.150000E+00
2.050000E+00
1.950000E+00
1.880000E+00
213
Group number
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Energy (eV)
1.820000E+00
1.740000E+00
1.650000E+00
1.550000E+00
1.470000E+00
1.395000E+00
1.325000E+00
1.275000E+00
1.225000E+00
1.175000E+00
1.140000E+00
1.127500E+00
1.117500E+00
1.100000E+00
1.085000E+00
1.075000E+00
1.065000E+00
1.055000E+00
1.037500E+00
1.012500E+00
9.950000E-01
9.850000E-01
9.750000E-01
9.600000E-01
9.400000E-01
9.200000E-01
9.000000E-01
8.830000E-01
8.630000E-01
8.250000E-01
7.750000E-01
7.250000E-01
6.915000E-01
6.665000E-01
6.375000E-01
6.125000E-01
5.950000E-01
5.825000E-01
5.625000E-01
5.410000E-01
5.160000E-01
4.950000E-01
4.850000E-01
4.775000E-01
4.725000E-01
4.650000E-01
4.550000E-01
4.400000E-01
4.250000E-01
4.170000E-01
Group number
Energy (eV)
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
3.970000E-01
3.700000E-01
3.550000E-01
3.450000E-01
3.350000E-01
3.250000E-01
3.150000E-01
3.050000E-01
2.950000E-01
2.850000E-01
2.750000E-01
2.650000E-01
2.550000E-01
2.450000E-01
2.350000E-01
2.250000E-01
2.100000E-01
1.900000E-01
1.700000E-01
1.500000E-01
1.300000E-01
1.100000E-01
9.750000E-02
9.250000E-02
8.750000E-02
8.250000E-02
7.750000E-02
7.250000E-02
6.750000E-02
6.250000E-02
5.500000E-02
4.500000E-02
3.500000E-02
2.765000E-02
2.265000E-02
1.750000E-02
1.250000E-02
9.000000E-03
7.500000E-03
6.000000E-03
4.500000E-03
3.000000E-03
1.500000E-03
214
A3.4 SHEM_TPN-407
Group number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Energy (eV)
1.96403E+07
1.49182E+07
1.38403E+07
1.27293E+07
1.16183E+07
9.99999E+06
9.04836E+06
8.18730E+06
7.40817E+06
6.70319E+06
6.06530E+06
5.51557E+06
4.96585E+06
4.51577E+06
4.29073E+06
4.06569E+06
3.69720E+06
3.32871E+06
3.02701E+06
2.72531E+06
2.66356E+06
2.60181E+06
2.47831E+06
2.35480E+06
2.23130E+06
2.06634E+06
1.98387E+06
1.90139E+06
1.83518E+06
1.76896E+06
1.70275E+06
1.63654E+06
1.57885E+06
1.52115E+06
1.46346E+06
1.40577E+06
1.34008E+06
1.28696E+06
1.22450E+06
1.16205E+06
1.10660E+06
1.05115E+06
1.00113E+06
9.51119E+05
9.05562E+05
8.60006E+05
8.21632E+05
7.83259E+05
7.44885E+05
7.06511E+05
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
6.74494E+05
6.42477E+05
6.10460E+05
5.78443E+05
5.57332E+05
5.36222E+05
5.15112E+05
4.84507E+05
4.56021E+05
4.34261E+05
4.05347E+05
3.83884E+05
3.52265E+05
3.20646E+05
2.94236E+05
2.81031E+05
2.67826E+05
2.48920E+05
2.30014E+05
2.12511E+05
1.95008E+05
1.80003E+05
1.64999E+05
1.52499E+05
1.46249E+05
1.40000E+05
1.31386E+05
1.27080E+05
1.22773E+05
1.15624E+05
1.05144E+05
9.99043E+04
9.46645E+04
8.84809E+04
8.22974E+04
7.48384E+04
6.73794E+04
6.12725E+04
5.51656E+04
5.25407E+04
4.99159E+04
4.53918E+04
4.08677E+04
3.69786E+04
3.34596E+04
3.13703E+04
2.92810E+04
2.73944E+04
2.61001E+04
2.49991E+04
215
Group number
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Energy (eV)
2.38492E+04
2.26994E+04
2.06421E+04
1.85847E+04
1.73926E+04
1.62005E+04
1.48997E+04
1.36037E+04
1.23707E+04
1.11377E+04
1.01283E+04
9.11881E+03
8.29233E+03
7.46585E+03
6.78918E+03
6.11252E+03
5.55851E+03
5.00451E+03
4.55093E+03
4.09735E+03
3.78921E+03
3.48107E+03
3.23863E+03
2.99618E+03
2.78751E+03
2.57884E+03
2.39923E+03
2.21963E+03
2.06504E+03
1.91045E+03
1.76224E+03
1.61404E+03
1.47955E+03
1.34506E+03
1.24003E+03
1.13501E+03
1.06496E+03
9.86231E+02
9.07501E+02
8.28009E+02
7.48517E+02
6.80676E+02
6.12834E+02
5.57290E+02
5.01746E+02
4.56271E+02
4.10795E+02
3.82185E+02
3.53575E+02
3.19928E+02
Group number
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Energy (eV)
3.01839E+02
2.83750E+02
2.62773E+02
2.41796E+02
2.19881E+02
1.97966E+02
1.80023E+02
1.62081E+02
1.47391E+02
1.32701E+02
1.20673E+02
1.08646E+02
9.87988E+01
8.89518E+01
8.19986E+01
7.50455E+01
6.82438E+01
6.14420E+01
5.70573E+01
5.26726E+01
4.92319E+01
4.57913E+01
4.39958E+01
4.20824E+01
4.01690E+01
3.69445E+01
3.37201E+01
3.21920E+01
3.06639E+01
2.91358E+01
2.76077E+01
2.61081E+01
2.46086E+01
2.35721E+01
2.25356E+01
2.23784E+01
2.21557E+01
2.20011E+01
2.17018E+01
2.14859E+01
2.13360E+01
2.12296E+01
2.11448E+01
2.10604E+01
2.09763E+01
2.07676E+01
2.06847E+01
2.06021E+01
2.05199E+01
2.04175E+01
Group number
Energy (eV)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
216
2.02751E+01
2.00734E+01
1.95974E+01
1.93927E+01
1.91997E+01
1.90848E+01
1.79591E+01
1.77590E+01
1.75648E+01
1.74457E+01
1.68305E+01
1.65501E+01
1.60498E+01
1.57792E+01
1.48663E+01
1.47301E+01
1.45952E+01
1.44702E+01
1.42505E+01
1.40496E+01
1.35460E+01
1.33297E+01
1.25997E+01
1.24721E+01
1.23086E+01
1.21302E+01
1.19795E+01
1.18153E+01
1.17094E+01
1.15894E+01
1.12694E+01
1.10529E+01
1.08038E+01
1.05793E+01
1.00396E+01
9.50002E+00
9.14031E+00
8.97995E+00
8.80038E+00
8.67369E+00
8.52407E+00
8.30032E+00
8.13027E+00
7.97008E+00
7.83965E+00
7.73994E+00
7.60035E+00
7.38015E+00
7.13987E+00
6.99429E+00
Group number
Energy (eV)
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
6.91778E+00
6.87021E+00
6.83526E+00
6.81070E+00
6.79165E+00
6.77605E+00
6.75981E+00
6.74225E+00
6.71668E+00
6.63126E+00
6.60611E+00
6.58829E+00
6.57184E+00
6.55609E+00
6.53907E+00
6.51492E+00
6.48178E+00
6.43206E+00
6.35978E+00
6.28015E+00
6.16011E+00
6.05991E+00
5.96014E+00
5.80021E+00
5.72015E+00
5.61979E+00
5.53004E+00
5.48817E+00
5.41025E+00
5.38003E+00
5.32011E+00
5.21008E+00
5.10997E+00
4.93323E+00
4.76785E+00
4.59382E+00
4.41980E+00
4.30981E+00
4.21983E+00
4.00000E+00
3.88217E+00
3.71209E+00
3.54307E+00
3.34250E+00
3.14211E+00
3.07759E+00
3.01308E+00
2.94856E+00
2.88405E+00
2.82958E+00
Group number
Energy (eV)
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
2.77512E+00
2.74092E+00
2.71990E+00
2.70012E+00
2.64004E+00
2.62005E+00
2.59009E+00
2.55000E+00
2.46994E+00
2.40000E+00
2.33006E+00
2.27299E+00
2.21709E+00
2.15695E+00
2.07010E+00
1.98992E+00
1.90008E+00
1.84002E+00
1.77997E+00
1.72446E+00
1.66895E+00
1.62849E+00
1.58803E+00
1.51998E+00
1.44397E+00
1.41001E+00
1.38098E+00
1.33095E+00
1.29304E+00
1.25094E+00
1.21397E+00
1.16999E+00
1.14797E+00
1.12997E+00
1.11605E+00
1.10395E+00
1.09198E+00
1.07799E+00
1.05649E+00
1.03499E+00
1.02101E+00
1.00904E+00
9.96501E-01
9.81959E-01
9.63960E-01
9.44022E-01
9.19978E-01
8.80024E-01
8.50031E-01
8.20037E-01
Group number
Energy (eV)
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
217
7.70018E-01
7.45008E-01
7.19999E-01
6.96249E-01
6.72499E-01
6.48749E-01
6.24999E-01
6.09996E-01
5.94993E-01
5.74991E-01
5.54990E-01
5.37500E-01
5.20011E-01
4.97514E-01
4.75017E-01
4.53298E-01
4.31579E-01
4.10790E-01
3.90001E-01
3.71497E-01
3.52994E-01
3.39001E-01
3.32004E-01
3.25008E-01
3.15010E-01
3.05012E-01
2.92500E-01
2.79989E-01
2.67493E-01
2.54997E-01
2.43094E-01
2.31192E-01
2.20401E-01
2.09610E-01
1.99808E-01
1.90005E-01
1.75950E-01
1.61895E-01
1.49947E-01
1.37999E-01
1.19995E-01
1.04298E-01
8.97968E-02
7.64969E-02
6.51994E-02
5.54982E-02
4.73019E-02
4.02999E-02
3.43998E-02
2.95460E-02
Group number
Energy (eV)
401
402
403
404
405
406
407
2.49394E-02
2.00104E-02
1.48300E-02
1.04505E-02
7.14526E-03
4.55602E-03
2.49990E-03
A3.5 SHEM_TPN-531
Group number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Energy (eV)
1.96403E+07
1.49182E+07
1.38403E+07
1.27293E+07
1.16183E+07
9.99999E+06
9.04836E+06
8.18730E+06
7.40817E+06
6.70319E+06
6.06530E+06
5.51557E+06
4.96585E+06
4.51577E+06
4.29073E+06
4.06569E+06
3.69720E+06
3.32871E+06
3.02701E+06
2.72531E+06
2.66356E+06
2.60181E+06
2.47831E+06
2.35480E+06
2.23130E+06
2.06634E+06
1.98387E+06
1.90139E+06
1.83518E+06
1.76896E+06
1.70275E+06
1.63654E+06
1.57885E+06
1.52115E+06
1.46346E+06
1.40577E+06
1.34008E+06
1.28696E+06
1.22450E+06
1.16205E+06
1.10660E+06
1.05115E+06
1.00113E+06
9.51119E+05
9.05562E+05
8.60006E+05
8.21632E+05
7.83259E+05
7.44885E+05
7.06511E+05
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
6.74494E+05
6.42477E+05
6.10460E+05
5.78443E+05
5.57332E+05
5.36222E+05
5.15112E+05
4.84507E+05
4.56021E+05
4.34261E+05
4.05347E+05
3.83884E+05
3.52265E+05
3.20646E+05
2.94236E+05
2.81031E+05
2.67826E+05
2.48920E+05
2.30014E+05
2.12511E+05
1.95008E+05
1.80003E+05
1.64999E+05
1.52499E+05
1.46249E+05
1.40000E+05
1.31386E+05
1.27080E+05
1.22773E+05
1.15624E+05
1.10384E+05
1.05144E+05
9.99043E+04
9.46645E+04
9.15727E+04
8.84809E+04
8.53891E+04
8.22974E+04
7.85679E+04
7.48384E+04
7.11089E+04
6.73794E+04
6.43259E+04
6.12725E+04
5.82190E+04
5.51656E+04
5.25407E+04
4.99159E+04
4.76538E+04
4.53918E+04
218
Group number
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Energy (eV)
4.31297E+04
4.08677E+04
3.89231E+04
3.69786E+04
3.52191E+04
3.34596E+04
3.13703E+04
3.03257E+04
2.92810E+04
2.83377E+04
2.73944E+04
2.67473E+04
2.61001E+04
2.55496E+04
2.49991E+04
2.38492E+04
2.26994E+04
2.06421E+04
1.96134E+04
1.85847E+04
1.73926E+04
1.62005E+04
1.55501E+04
1.48997E+04
1.42517E+04
1.36037E+04
1.29872E+04
1.23707E+04
1.17542E+04
1.11377E+04
1.06330E+04
1.01283E+04
9.62354E+03
9.11881E+03
8.70557E+03
8.29233E+03
7.87909E+03
7.46585E+03
7.12752E+03
6.78918E+03
6.45085E+03
6.11252E+03
5.83552E+03
5.55851E+03
5.28151E+03
5.00451E+03
4.55093E+03
4.32414E+03
4.09735E+03
3.78921E+03
Group number
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Energy (eV)
6.74494E+05
6.42477E+05
6.10460E+05
5.78443E+05
5.57332E+05
5.36222E+05
5.15112E+05
4.84507E+05
4.56021E+05
4.34261E+05
4.05347E+05
3.83884E+05
3.52265E+05
3.20646E+05
2.94236E+05
2.81031E+05
2.67826E+05
2.48920E+05
2.30014E+05
2.12511E+05
1.95008E+05
1.80003E+05
1.64999E+05
1.52499E+05
1.46249E+05
1.40000E+05
1.31386E+05
1.27080E+05
1.22773E+05
1.15624E+05
1.10384E+05
1.05144E+05
9.99043E+04
9.46645E+04
9.15727E+04
8.84809E+04
8.53891E+04
8.22974E+04
7.85679E+04
7.48384E+04
7.11089E+04
6.73794E+04
6.43259E+04
6.12725E+04
5.82190E+04
5.51656E+04
5.25407E+04
4.99159E+04
4.76538E+04
4.53918E+04
Group number
Energy (eV)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
3.53575E+02
3.44449E+02
3.35323E+02
3.27625E+02
3.19928E+02
3.07925E+02
2.95922E+02
2.88327E+02
2.84888E+02
2.76468E+02
2.68297E+02
2.56748E+02
2.49272E+02
2.41796E+02
2.35590E+02
2.29958E+02
2.24325E+02
2.18216E+02
2.12108E+02
2.06533E+02
2.00958E+02
1.95996E+02
1.93078E+02
1.90204E+02
1.88877E+02
1.87559E+02
1.86251E+02
1.84952E+02
1.83295E+02
1.75229E+02
1.67519E+02
1.63056E+02
1.54176E+02
1.46657E+02
1.43080E+02
1.39504E+02
1.32701E+02
1.26229E+02
1.20554E+02
1.17577E+02
1.16524E+02
1.15480E+02
1.12854E+02
1.10288E+02
1.05646E+02
1.03038E+02
1.02115E+02
1.01605E+02
1.01098E+02
1.00594E+02
219
Group number
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
Energy (eV)
9.73287E+01
9.33256E+01
9.10498E+01
8.87741E+01
8.63567E+01
8.39393E+01
8.16536E+01
7.93679E+01
7.63322E+01
7.35595E+01
7.18869E+01
6.90682E+01
6.68261E+01
6.64929E+01
6.61612E+01
6.58312E+01
6.55029E+01
6.50460E+01
6.45923E+01
6.36306E+01
6.23083E+01
5.99250E+01
5.84923E+01
5.70595E+01
5.55597E+01
5.40600E+01
5.29895E+01
5.17847E+01
5.05219E+01
4.92591E+01
4.75173E+01
4.62053E+01
4.52904E+01
4.41721E+01
4.31246E+01
4.21441E+01
4.12270E+01
3.97295E+01
3.87874E+01
3.77919E+01
3.73038E+01
3.68588E+01
3.64191E+01
3.60568E+01
3.56980E+01
3.45392E+01
3.30855E+01
3.16930E+01
2.97891E+01
2.78852E+01
Group number
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
Energy (eV)
2.62715E+01
2.46578E+01
2.35967E+01
2.25356E+01
2.23788E+01
2.21557E+01
2.20011E+01
2.17018E+01
2.14859E+01
2.13360E+01
2.12296E+01
2.11448E+01
2.10604E+01
2.09763E+01
2.07676E+01
2.06847E+01
2.06021E+01
2.05199E+01
2.04175E+01
2.02751E+01
2.00734E+01
1.95974E+01
1.93927E+01
1.91997E+01
1.90848E+01
1.85219E+01
1.79591E+01
1.77590E+01
1.75648E+01
1.74457E+01
1.68305E+01
1.65501E+01
1.60498E+01
1.57792E+01
1.53227E+01
1.48662E+01
1.47301E+01
1.45952E+01
1.44702E+01
1.42505E+01
1.40496E+01
1.35460E+01
1.33297E+01
1.29648E+01
1.26000E+01
1.24721E+01
1.23086E+01
1.21302E+01
1.19795E+01
1.18153E+01
Group number
Energy (eV)
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
1.17094E+01
1.15894E+01
1.12694E+01
1.10529E+01
1.08038E+01
1.05793E+01
1.00396E+01
9.50002E+00
9.14031E+00
8.97995E+00
8.80038E+00
8.67369E+00
8.52407E+00
8.30032E+00
8.13027E+00
7.97008E+00
7.83965E+00
7.73994E+00
7.60035E+00
7.38015E+00
7.13987E+00
6.99429E+00
6.91778E+00
6.87021E+00
6.83526E+00
6.81070E+00
6.79165E+00
6.77605E+00
6.75981E+00
6.74225E+00
6.71668E+00
6.63126E+00
6.60611E+00
6.58829E+00
6.57184E+00
6.55609E+00
6.53907E+00
6.51492E+00
6.48178E+00
6.43206E+00
6.35978E+00
6.28016E+00
6.16011E+00
6.05991E+00
5.96014E+00
5.80021E+00
5.72015E+00
5.61979E+00
5.53004E+00
5.48817E+00
220
Group number
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
Energy (eV)
5.41025E+00
5.38003E+00
5.32011E+00
5.21008E+00
5.10997E+00
4.93323E+00
4.76785E+00
4.59382E+00
4.41980E+00
4.30981E+00
4.21983E+00
4.00000E+00
3.88217E+00
3.71209E+00
3.54307E+00
3.34259E+00
3.14211E+00
3.07759E+00
3.01308E+00
2.94856E+00
2.88405E+00
2.82958E+00
2.77512E+00
2.74092E+00
2.71990E+00
2.70012E+00
2.64004E+00
2.62005E+00
2.59009E+00
2.55000E+00
2.46994E+00
2.40000E+00
2.33006E+00
2.27299E+00
2.21709E+00
2.15695E+00
2.07010E+00
1.98992E+00
1.90008E+00
1.84002E+00
1.77997E+00
1.72446E+00
1.66895E+00
1.62849E+00
1.58803E+00
1.51998E+00
1.44397E+00
1.41001E+00
1.38098E+00
1.33095E+00
Group number
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
Energy (eV)
1.29304E+00
1.25094E+00
1.21397E+00
1.16999E+00
1.14797E+00
1.12997E+00
1.11605E+00
1.10395E+00
1.09198E+00
1.07799E+00
1.05649E+00
1.03499E+00
1.02101E+00
1.00904E+00
9.96501E-01
9.81959E-01
9.63960E-01
9.44022E-01
9.19978E-01
9.00001E-01
8.80024E-01
8.50031E-01
8.35034E-01
8.20037E-01
7.95028E-01
7.70018E-01
7.45008E-01
7.19999E-01
6.96249E-01
6.72499E-01
6.48749E-01
6.24999E-01
6.09996E-01
5.94993E-01
5.74991E-01
5.54990E-01
5.37500E-01
5.20011E-01
4.97514E-01
4.75017E-01
4.53298E-01
4.31579E-01
4.10790E-01
3.90001E-01
3.71497E-01
3.52994E-01
3.39001E-01
3.32004E-01
3.20009E-01
3.05012E-01
Group number
Energy (eV)
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
2.92500E-01
2.79989E-01
2.67493E-01
2.54997E-01
2.43094E-01
2.31192E-01
2.20401E-01
2.09610E-01
1.99808E-01
1.82978E-01
1.61895E-01
1.49947E-01
1.37999E-01
1.28997E-01
1.19995E-01
1.04298E-01
8.97968E-02
7.64969E-02
6.51994E-02
5.54982E-02
4.73019E-02
4.02999E-02
3.43998E-02
2.95460E-02
2.49394E-02
2.00104E-02
1.48300E-02
1.04505E-02
7.14526E-03
4.55602E-03
2.49990E-03
221
A3.6 GA_TPN-537
Group number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Energy (eV)
1.49182E+07
1.34986E+07
1.22140E+07
1.10517E+07
1.00000E+07
9.04837E+06
8.18731E+06
7.40818E+06
6.70320E+06
6.06531E+06
5.48812E+06
4.96585E+06
4.49330E+06
4.06570E+06
3.67880E+06
3.32870E+06
3.17030E+06
3.01190E+06
2.86860E+06
2.72530E+06
2.59565E+06
2.46600E+06
2.34865E+06
2.23130E+06
2.12515E+06
2.01900E+06
1.92290E+06
1.82680E+06
1.73990E+06
1.65300E+06
1.57435E+06
1.49570E+06
1.42455E+06
1.35340E+06
1.28900E+06
1.22460E+06
1.16630E+06
1.10800E+06
1.05530E+06
1.00260E+06
9.54890E+05
9.07180E+05
8.64015E+05
8.20850E+05
7.81795E+05
7.42740E+05
7.07400E+05
6.72060E+05
6.40080E+05
6.08100E+05
Group number
Energy (eV)
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
5.79165E+05
5.50230E+05
5.24050E+05
4.97870E+05
4.74180E+05
4.50490E+05
4.29055E+05
4.07620E+05
3.88225E+05
3.68830E+05
3.51280E+05
3.33730E+05
3.17850E+05
3.01970E+05
2.87605E+05
2.73240E+05
2.60240E+05
2.47240E+05
2.35475E+05
2.23710E+05
2.13065E+05
2.02420E+05
1.92790E+05
1.83160E+05
1.74445E+05
1.65730E+05
1.57845E+05
1.49960E+05
1.42825E+05
1.35690E+05
1.29230E+05
1.22770E+05
1.16930E+05
1.11090E+05
1.08018E+05
1.04947E+05
1.01875E+05
9.88035E+04
9.57319E+04
9.26603E+04
8.95886E+04
8.65170E+04
8.41249E+04
8.17328E+04
7.93406E+04
7.69485E+04
7.45564E+04
7.21643E+04
6.97721E+04
6.73800E+04
222
Group number
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
Energy (eV)
6.55169E+04
6.36538E+04
6.17906E+04
5.99275E+04
5.80644E+04
5.62013E+04
5.43381E+04
5.24750E+04
5.10241E+04
4.95733E+04
4.81224E+04
4.66715E+04
4.52206E+04
4.37698E+04
4.23189E+04
4.08680E+04
3.97380E+04
3.86080E+04
3.74780E+04
3.63480E+04
3.52180E+04
3.40880E+04
3.29580E+04
3.18280E+04
3.09480E+04
3.00680E+04
2.91880E+04
2.83080E+04
2.74280E+04
2.65480E+04
2.56680E+04
2.47880E+04
2.41026E+04
2.34173E+04
2.27319E+04
2.20465E+04
2.13611E+04
2.06758E+04
1.99904E+04
1.93050E+04
1.87711E+04
1.82373E+04
1.77034E+04
1.71695E+04
1.66356E+04
1.61018E+04
1.55679E+04
1.50340E+04
1.46184E+04
1.42028E+04
Group number
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
Energy (eV)
1.37871E+04
1.33715E+04
1.29559E+04
1.25403E+04
1.21246E+04
1.17090E+04
1.13852E+04
1.10615E+04
1.07377E+04
1.04139E+04
1.00901E+04
9.76635E+03
9.44257E+03
9.11880E+03
8.86666E+03
8.61452E+03
8.36239E+03
8.11025E+03
7.85811E+03
7.60597E+03
7.35384E+03
7.10170E+03
6.90534E+03
6.70897E+03
6.51261E+03
6.31625E+03
6.11989E+03
5.92353E+03
5.72716E+03
5.53080E+03
5.37788E+03
5.22495E+03
5.07203E+03
4.91910E+03
4.76617E+03
4.61325E+03
4.46032E+03
4.30740E+03
4.18830E+03
4.06920E+03
3.95010E+03
3.83100E+03
3.71190E+03
3.59280E+03
3.47370E+03
3.35460E+03
3.26185E+03
3.16910E+03
3.07635E+03
2.98360E+03
Group number
Energy (eV)
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
2.89085E+03
2.79810E+03
2.70535E+03
2.61260E+03
2.54036E+03
2.46812E+03
2.39589E+03
2.32365E+03
2.25141E+03
2.17918E+03
2.10694E+03
2.03470E+03
1.97844E+03
1.92218E+03
1.86591E+03
1.80965E+03
1.75339E+03
1.69712E+03
1.64086E+03
1.58460E+03
1.54079E+03
1.49697E+03
1.45316E+03
1.40935E+03
1.36554E+03
1.32172E+03
1.27791E+03
1.23410E+03
1.19998E+03
1.16585E+03
1.13173E+03
1.09761E+03
1.06349E+03
1.02937E+03
9.95242E+02
9.61120E+02
9.34545E+02
9.07970E+02
8.81395E+02
8.54820E+02
8.28245E+02
8.01670E+02
7.75095E+02
7.48520E+02
7.27824E+02
7.07127E+02
6.86431E+02
6.65735E+02
6.45039E+02
6.24343E+02
223
Group number
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
Energy (eV)
6.03646E+02
5.82950E+02
5.66831E+02
5.50713E+02
5.34594E+02
5.18475E+02
5.02356E+02
4.86238E+02
4.70119E+02
4.54000E+02
4.41448E+02
4.28895E+02
4.16343E+02
4.03790E+02
3.91237E+02
3.78685E+02
3.66132E+02
3.53580E+02
3.43802E+02
3.34025E+02
3.24247E+02
3.14470E+02
3.04693E+02
2.94915E+02
2.85138E+02
2.75360E+02
2.67746E+02
2.60133E+02
2.52519E+02
2.44905E+02
2.37291E+02
2.29677E+02
2.22064E+02
2.14450E+02
2.08521E+02
2.02592E+02
1.96664E+02
1.90735E+02
1.84806E+02
1.78878E+02
1.72949E+02
1.67020E+02
1.62401E+02
1.57783E+02
1.53164E+02
1.48545E+02
1.43926E+02
1.39307E+02
1.34689E+02
1.30070E+02
Group number
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
Energy (eV)
1.26474E+02
1.22877E+02
1.19281E+02
1.15685E+02
1.12089E+02
1.08492E+02
1.04896E+02
1.01300E+02
9.84991E+01
9.56982E+01
9.28974E+01
9.00965E+01
8.72956E+01
8.44947E+01
8.16939E+01
7.88930E+01
7.67116E+01
7.45303E+01
7.23489E+01
7.01675E+01
6.79861E+01
6.58048E+01
6.36234E+01
6.14420E+01
5.97431E+01
5.80443E+01
5.63454E+01
5.46465E+01
5.29476E+01
5.12487E+01
4.95499E+01
4.78510E+01
4.65279E+01
4.52047E+01
4.38816E+01
4.25585E+01
4.12354E+01
3.99122E+01
3.82996E+01
3.66870E+01
3.49157E+01
3.40301E+01
3.31445E+01
3.21141E+01
3.10838E+01
3.00534E+01
2.90230E+01
2.82205E+01
2.74180E+01
2.66155E+01
Group number
Energy (eV)
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
2.58130E+01
2.50105E+01
2.42080E+01
2.33426E+01
2.24771E+01
2.18837E+01
2.12903E+01
2.01035E+01
1.94786E+01
1.88537E+01
1.82289E+01
1.76040E+01
1.71172E+01
1.66305E+01
1.61437E+01
1.56570E+01
1.51703E+01
1.46835E+01
1.41968E+01
1.37100E+01
1.33309E+01
1.29518E+01
1.25726E+01
1.21935E+01
1.18144E+01
1.14353E+01
1.10561E+01
1.06770E+01
1.03818E+01
1.00866E+01
9.79136E+00
9.49615E+00
9.20094E+00
8.90573E+00
8.61051E+00
8.31530E+00
8.08538E+00
7.85545E+00
7.65427E+00
7.45308E+00
7.35249E+00
7.25190E+00
7.05071E+00
6.47590E+00
6.29685E+00
6.11780E+00
5.93875E+00
5.75970E+00
5.58065E+00
5.40160E+00
224
Group number
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
Energy (eV)
5.22255E+00
5.04350E+00
4.90405E+00
4.76460E+00
4.62515E+00
4.48570E+00
4.34625E+00
4.20680E+00
4.06735E+00
3.92790E+00
3.81929E+00
3.71068E+00
3.60206E+00
3.49345E+00
3.38484E+00
3.27623E+00
3.16761E+00
3.05900E+00
2.88985E+00
2.72070E+00
2.55155E+00
2.38240E+00
2.35500E+00
2.31000E+00
2.24500E+00
2.15000E+00
2.05000E+00
1.95000E+00
1.88000E+00
1.82000E+00
1.74000E+00
1.65000E+00
1.55000E+00
1.47000E+00
1.39500E+00
1.32500E+00
1.27500E+00
1.22500E+00
1.17500E+00
1.14000E+00
1.12750E+00
1.11750E+00
1.10000E+00
1.08500E+00
1.07500E+00
1.06500E+00
1.05500E+00
1.03750E+00
1.01250E+00
9.95000E-01
Group number
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
Energy (eV)
9.85000E-01
9.75000E-01
9.60000E-01
9.40000E-01
9.20000E-01
9.00000E-01
8.83000E-01
8.63000E-01
8.25000E-01
7.75000E-01
7.50000E-01
7.25000E-01
6.91500E-01
6.66500E-01
6.37500E-01
6.12500E-01
5.95000E-01
5.82500E-01
5.62500E-01
5.41000E-01
5.16000E-01
4.95000E-01
4.85000E-01
4.77500E-01
4.72500E-01
4.65000E-01
4.55000E-01
4.40000E-01
4.25000E-01
4.17000E-01
3.97000E-01
3.83500E-01
3.70000E-01
3.55000E-01
3.45000E-01
3.35000E-01
3.25000E-01
3.15000E-01
3.05000E-01
2.95000E-01
2.85000E-01
2.75000E-01
2.65000E-01
2.55000E-01
2.45000E-01
2.35000E-01
2.25000E-01
2.17500E-01
2.10000E-01
2.00000E-01
Group number
Energy (eV)
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
1.90000E-01
1.80000E-01
1.70000E-01
1.60000E-01
1.55000E-01
1.50000E-01
1.45000E-01
1.40000E-01
1.35000E-01
1.30000E-01
1.25000E-01
1.20000E-01
1.15000E-01
1.10000E-01
1.06094E-01
1.02188E-01
9.50000E-02
9.25000E-02
8.75000E-02
8.25000E-02
7.75000E-02
7.25000E-02
6.75000E-02
6.25000E-02
5.50000E-02
4.50000E-02
3.50000E-02
2.76500E-02
2.26500E-02
1.75000E-02
1.25000E-02
9.00000E-03
7.50000E-03
6.00000E-03
4.50000E-03
3.00000E-03
1.50000E-03
225
A4 Depletion Data Analysis
A4.1 Pebble Fuel Element Nuclides Concentrations
Burnup steps (GWD/t)
Nuclides
0.5
5
10
20
30
40
50
60
70
80
100
120
U-233
4.85E-14
4.72E-13
9.02E-13
1.67E-12
2.40E-12
3.14E-12
3.92E-12
4.74E-12
5.57E-12
6.34E-12
7.55E-12
8.13E-12
U-234
8.23E-11
8.21E-10
1.60E-09
3.08E-09
4.73E-09
7.03E-09
1.06E-08
1.63E-08
2.48E-08
3.69E-08
7.25E-08
1.20E-07
U-235
1.91E-03
1.78E-03
1.64E-03
1.38E-03
1.14E-03
9.27E-04
7.36E-04
5.67E-04
4.21E-04
2.98E-04
1.28E-04
4.27E-05
U-236
2.55E-06
2.49E-05
4.84E-05
9.15E-05
1.30E-04
1.63E-04
1.91E-04
2.15E-04
2.34E-04
2.48E-04
2.61E-04
2.58E-04
U-237
1.51E-09
2.29E-08
4.70E-08
9.57E-08
1.43E-07
1.91E-07
2.38E-07
2.84E-07
3.31E-07
3.76E-07
4.77E-07
5.22E-07
U-238
2.13E-02
2.12E-02
2.12E-02
2.11E-02
2.10E-02
2.09E-02
2.07E-02
2.06E-02
2.05E-02
2.03E-02
2.00E-02
1.96E-02
Np-237
5.79E-10
8.87E-08
3.69E-07
1.47E-06
3.22E-06
5.53E-06
8.29E-06
1.14E-05
1.47E-05
1.81E-05
2.48E-05
2.95E-05
Np-238
4.05E-13
1.18E-10
5.29E-10
2.31E-09
5.46E-09
1.02E-08
1.66E-08
2.51E-08
3.60E-08
4.94E-08
8.44E-08
1.20E-07
Np-239
1.72E-06
2.00E-06
2.06E-06
2.20E-06
2.35E-06
2.51E-06
2.68E-06
2.88E-06
3.11E-06
3.36E-06
4.04E-06
4.50E-06
Pu-238
3.02E-13
9.47E-10
8.61E-09
7.39E-08
2.57E-07
6.21E-07
1.23E-06
2.13E-06
3.35E-06
4.87E-06
8.57E-06
1.19E-05
Pu-239
2.76E-06
4.06E-05
7.61E-05
1.30E-04
1.66E-04
1.88E-04
2.00E-04
2.05E-04
2.03E-04
1.98E-04
1.86E-04
1.70E-04
Pu-240
9.64E-09
1.37E-06
5.02E-06
1.61E-05
2.90E-05
4.15E-05
5.27E-05
6.20E-05
6.93E-05
7.46E-05
7.87E-05
7.90E-05
Pu-241
6.38E-11
1.00E-07
7.40E-07
4.68E-06
1.20E-05
2.14E-05
3.12E-05
4.00E-05
4.67E-05
5.10E-05
5.43E-05
5.06E-05
Pu-242
7.86E-14
1.36E-09
2.12E-08
2.95E-07
1.27E-06
3.40E-06
7.00E-06
1.23E-05
1.93E-05
2.78E-05
4.81E-05
6.95E-05
Pu-243
7.10E-18
1.51E-13
2.43E-12
3.63E-11
1.67E-10
4.75E-10
1.05E-09
1.96E-09
3.29E-09
5.10E-09
1.06E-08
1.67E-08
Am-241
1.58E-14
2.61E-10
3.89E-09
4.90E-08
1.88E-07
4.41E-07
7.84E-07
1.16E-06
1.51E-06
1.75E-06
1.86E-06
1.58E-06
Am-242m
4.13E-18
6.41E-13
1.71E-11
3.51E-10
1.68E-09
4.42E-09
8.39E-09
1.29E-08
1.70E-08
1.98E-08
2.11E-08
1.74E-08
Am-242
1.38E-17
3.53E-13
5.67E-12
7.63E-11
3.14E-10
7.95E-10
1.54E-09
2.52E-09
3.64E-09
4.76E-09
6.36E-09
6.55E-09
Am-243
3.50E-17
8.00E-12
2.61E-10
7.80E-09
5.36E-08
2.02E-07
5.50E-07
1.22E-06
2.33E-06
4.02E-06
9.61E-06
1.77E-05
Cm-242
1.66E-17
4.52E-12
1.43E-10
3.58E-09
2.07E-08
6.57E-08
1.50E-07
2.79E-07
4.47E-07
6.41E-07
1.01E-06
1.20E-06
Cm-243
1.27E-21
3.92E-15
2.59E-13
1.39E-11
1.28E-10
5.72E-10
1.72E-09
4.02E-09
7.84E-09
1.33E-08
2.79E-08
4.00E-08
Cm-244
2.59E-20
6.59E-14
4.47E-12
2.86E-10
3.15E-09
1.69E-08
6.14E-08
1.74E-07
4.18E-07
8.86E-07
3.11E-06
8.07E-06
Cm-245
5.66E-24
1.54E-16
2.11E-14
2.72E-12
4.45E-11
3.13E-10
1.38E-09
4.54E-09
1.20E-08
2.73E-08
1.06E-07
2.74E-07
226
A4.2 Pebble Fuel Element Fission Products Concentrations
Burnup (GWD/t)
Fission
Products
0.5
5
10
20
30
40
50
60
70
80
100
120
Kr-85
3.26E-08
3.19E-07
6.22E-07
1.19E-06
1.71E-06
2.18E-06
2.61E-06
2.99E-06
3.33E-06
3.62E-06
4.07E-06
4.37E-06
Sr-90
7.27E-07
7.13E-06
1.40E-05
2.70E-05
3.91E-05
5.03E-05
6.07E-05
7.02E-05
7.88E-05
8.66E-05
9.93E-05
1.09E-04
Ag-109
4.01E-09
6.33E-08
1.76E-07
5.32E-07
1.05E-06
1.72E-06
2.54E-06
3.49E-06
4.58E-06
5.78E-06
8.36E-06
1.10E-05
Ag-110m
3.99E-13
5.40E-11
2.71E-10
1.46E-09
4.01E-09
8.30E-09
1.47E-08
2.37E-08
3.58E-08
5.15E-08
9.74E-08
1.58E-07
Cs-137
7.79E-07
7.75E-06
1.55E-05
3.08E-05
4.59E-05
6.09E-05
7.57E-05
9.03E-05
1.05E-04
1.19E-04
1.47E-04
1.73E-04
Xe-131
1.02E-07
3.08E-06
6.61E-06
1.34E-05
1.99E-05
2.58E-05
3.13E-05
3.61E-05
4.03E-05
4.38E-05
4.81E-05
4.94E-05
Xe-133
5.18E-07
7.87E-07
7.86E-07
7.84E-07
7.82E-07
7.79E-07
7.76E-07
7.72E-07
7.68E-07
7.64E-07
7.54E-07
7.45E-07
Xe-135
1.67E-08
1.65E-08
1.62E-08
1.54E-08
1.45E-08
1.34E-08
1.23E-08
1.11E-08
9.91E-09
8.74E-09
6.94E-09
5.57E-09
Xe-136
1.34E-06
1.37E-05
2.75E-05
5.55E-05
8.38E-05
1.13E-04
1.42E-04
1.71E-04
2.01E-04
2.32E-04
2.93E-04
3.55E-04
Nd-143
8.60E-08
5.36E-06
1.23E-05
2.50E-05
3.61E-05
4.56E-05
5.34E-05
5.92E-05
6.30E-05
6.46E-05
6.18E-05
5.31E-05
Nd-144
7.12E-09
7.33E-07
2.83E-06
1.03E-05
2.13E-05
3.50E-05
5.10E-05
6.89E-05
8.86E-05
1.10E-04
1.57E-04
2.07E-04
Nd-145
4.95E-07
4.88E-06
9.64E-06
1.88E-05
2.75E-05
3.57E-05
4.35E-05
5.06E-05
5.72E-05
6.32E-05
7.28E-05
7.94E-05
Nd-146
3.78E-07
3.77E-06
7.56E-06
1.52E-05
2.29E-05
3.08E-05
3.89E-05
4.73E-05
5.59E-05
6.48E-05
8.40E-05
1.05E-04
Pm-147
6.09E-08
2.13E-06
4.47E-06
8.28E-06
1.11E-05
1.31E-05
1.44E-05
1.51E-05
1.54E-05
1.53E-05
1.41E-05
1.24E-05
Pm-148
6.77E-11
7.68E-09
1.78E-08
3.61E-08
5.20E-08
6.59E-08
7.82E-08
8.91E-08
9.88E-08
1.07E-07
1.21E-07
1.22E-07
Pm-148m
4.05E-11
7.81E-09
1.98E-08
4.01E-08
5.58E-08
6.72E-08
7.50E-08
7.95E-08
8.12E-08
8.04E-08
7.49E-08
6.45E-08
Pm-149
4.94E-08
5.46E-08
5.65E-08
6.01E-08
6.34E-08
6.67E-08
6.99E-08
7.32E-08
7.64E-08
7.96E-08
8.58E-08
8.91E-08
Pm-151
1.11E-08
1.15E-08
1.18E-08
1.25E-08
1.31E-08
1.37E-08
1.43E-08
1.50E-08
1.57E-08
1.64E-08
1.78E-08
1.88E-08
Sm-147
1.24E-10
5.35E-08
2.44E-07
9.68E-07
2.02E-06
3.27E-06
4.62E-06
5.96E-06
7.23E-06
8.35E-06
9.94E-06
1.06E-05
Sm-148
2.05E-11
3.52E-08
1.90E-07
8.62E-07
1.98E-06
3.52E-06
5.43E-06
7.71E-06
1.03E-05
1.33E-05
2.01E-05
2.76E-05
Sm-149
6.12E-08
1.42E-07
1.43E-07
1.41E-07
1.38E-07
1.32E-07
1.24E-07
1.16E-07
1.06E-07
9.65E-08
8.09E-08
6.76E-08
Sm-151
3.97E-08
3.07E-07
4.12E-07
4.72E-07
4.93E-07
5.04E-07
5.09E-07
5.10E-07
5.05E-07
4.98E-07
4.96E-07
4.77E-07
227
Eu-153
1.35E-08
2.11E-07
4.86E-07
1.23E-06
2.23E-06
3.45E-06
4.83E-06
6.35E-06
7.94E-06
9.54E-06
1.26E-05
1.49E-05
Eu-154
3.36E-11
5.83E-09
2.43E-08
1.03E-07
2.43E-07
4.46E-07
7.08E-07
1.02E-06
1.36E-06
1.72E-06
2.43E-06
2.96E-06
Eu-155
3.93E-09
2.98E-08
4.85E-08
8.20E-08
1.28E-07
1.93E-07
2.80E-07
3.87E-07
5.10E-07
6.44E-07
9.12E-07
1.14E-06
Gd-155
5.57E-12
1.77E-10
3.16E-10
5.16E-10
7.44E-10
1.03E-09
1.36E-09
1.68E-09
1.98E-09
2.22E-09
2.64E-09
2.84E-09
Gd-156
3.31E-10
2.49E-08
8.26E-08
2.68E-07
5.58E-07
9.97E-07
1.66E-06
2.62E-06
4.01E-06
5.94E-06
1.20E-05
2.16E-05
Gd-157
3.46E-10
6.37E-10
8.32E-10
1.16E-09
1.42E-09
1.65E-09
1.86E-09
2.07E-09
2.29E-09
2.51E-09
3.26E-09
3.95E-09
228
A4.3 Prismatic Hexagonal Block Nuclides Concentrations
Burnup (GWD/t)
Nuclides
0.5
5
10
20
30
40
50
60
70
80
100
120
U-233
6.39E-14
6.19E-13
1.19E-12
2.20E-12
3.18E-12
4.18E-12
5.24E-12
6.37E-12
7.55E-12
8.73E-12
1.09E-11
1.26E-11
U-234
1.09E-10
1.08E-09
2.10E-09
4.07E-09
6.31E-09
9.49E-09
1.45E-08
2.23E-08
3.39E-08
5.02E-08
9.89E-08
1.67E-07
U-235
1.91E-03
1.78E-03
1.64E-03
1.39E-03
1.16E-03
9.55E-04
7.75E-04
6.18E-04
4.83E-04
3.67E-04
1.97E-04
9.32E-05
U-236
2.67E-06
2.60E-05
5.02E-05
9.39E-05
1.32E-04
1.64E-04
1.92E-04
2.14E-04
2.32E-04
2.45E-04
2.58E-04
2.58E-04
U-237
1.99E-09
2.96E-08
6.06E-08
1.21E-07
1.80E-07
2.36E-07
2.89E-07
3.40E-07
3.87E-07
4.30E-07
5.15E-07
5.57E-07
U-238
2.13E-02
2.12E-02
2.12E-02
2.11E-02
2.09E-02
2.08E-02
2.07E-02
2.05E-02
2.04E-02
2.02E-02
1.98E-02
1.94E-02
Np-237
7.62E-10
1.15E-07
4.78E-07
1.88E-06
4.08E-06
6.93E-06
1.03E-05
1.40E-05
1.79E-05
2.19E-05
2.96E-05
3.56E-05
Np-238
6.04E-13
1.73E-10
7.67E-10
3.26E-09
7.55E-09
1.37E-08
2.17E-08
3.16E-08
4.33E-08
5.68E-08
8.86E-08
1.20E-07
Np-239
2.01E-06
2.31E-06
2.38E-06
2.52E-06
2.69E-06
2.86E-06
3.03E-06
3.21E-06
3.39E-06
3.58E-06
4.04E-06
4.38E-06
Pu-238
4.51E-13
1.40E-09
1.26E-08
1.07E-07
3.66E-07
8.74E-07
1.71E-06
2.91E-06
4.52E-06
6.51E-06
1.14E-05
1.64E-05
Pu-239
3.22E-06
4.68E-05
8.74E-05
1.48E-04
1.89E-04
2.16E-04
2.33E-04
2.42E-04
2.45E-04
2.45E-04
2.38E-04
2.27E-04
Pu-240
1.24E-08
1.71E-06
6.09E-06
1.87E-05
3.24E-05
4.52E-05
5.62E-05
6.53E-05
7.24E-05
7.77E-05
8.29E-05
8.48E-05
Pu-241
1.00E-10
1.52E-07
1.09E-06
6.60E-06
1.63E-05
2.83E-05
4.05E-05
5.14E-05
6.02E-05
6.65E-05
7.33E-05
7.24E-05
Pu-242
1.28E-13
2.13E-09
3.21E-08
4.24E-07
1.73E-06
4.40E-06
8.66E-06
1.46E-05
2.19E-05
3.05E-05
5.00E-05
6.99E-05
Pu-243
1.44E-17
2.91E-13
4.55E-12
6.37E-11
2.77E-10
7.50E-10
1.56E-09
2.77E-09
4.39E-09
6.40E-09
1.18E-08
1.76E-08
Am-241
2.48E-14
3.96E-10
5.77E-09
6.98E-08
2.59E-07
5.91E-07
1.03E-06
1.52E-06
1.99E-06
2.38E-06
2.77E-06
2.71E-06
Am-242m
7.20E-18
1.07E-12
2.81E-11
5.52E-10
2.56E-09
6.60E-09
1.24E-08
1.90E-08
2.54E-08
3.06E-08
3.61E-08
3.45E-08
Am-242
2.40E-17
5.87E-13
9.15E-12
1.16E-10
4.53E-10
1.09E-09
2.03E-09
3.20E-09
4.48E-09
5.75E-09
7.80E-09
8.69E-09
Am-243
7.12E-17
1.55E-11
4.92E-10
1.39E-08
9.12E-08
3.28E-07
8.49E-07
1.79E-06
3.26E-06
5.34E-06
1.16E-05
2.00E-05
Cm-242
2.89E-17
7.56E-12
2.32E-10
5.55E-09
3.06E-08
9.28E-08
2.04E-07
3.64E-07
5.65E-07
7.91E-07
1.23E-06
1.53E-06
Cm-243
2.61E-21
7.70E-15
4.96E-13
2.53E-11
2.21E-10
9.44E-10
2.71E-09
6.04E-09
1.13E-08
1.85E-08
3.72E-08
5.53E-08
Cm-244
6.40E-20
1.55E-13
1.03E-11
6.22E-10
6.55E-09
3.35E-08
1.16E-07
3.11E-07
7.02E-07
1.40E-06
4.26E-06
9.88E-06
Cm-245
1.76E-23
4.53E-16
6.13E-14
7.51E-12
1.19E-10
8.14E-10
3.47E-09
1.10E-08
2.80E-08
6.11E-08
2.14E-07
5.23E-07
229
A4.4 Prismatic Hexagonal Block Fission Products Concentrations
Burnup steps (GWD/t)
Fission
Products
0.5
5
10
20
30
40
50
60
70
80
100
120
Kr-85
3.25E-08
3.17E-07
6.17E-07
1.17E-06
1.68E-06
2.13E-06
2.54E-06
2.90E-06
3.23E-06
3.51E-06
3.97E-06
4.30E-06
Sr-90
7.25E-07
7.09E-06
1.39E-05
2.66E-05
3.83E-05
4.91E-05
5.90E-05
6.81E-05
7.63E-05
8.38E-05
9.66E-05
1.07E-04
Ag-109
4.06E-09
6.90E-08
1.97E-07
6.06E-07
1.19E-06
1.95E-06
2.85E-06
3.89E-06
5.04E-06
6.27E-06
8.89E-06
1.16E-05
Ag-110m
4.80E-13
6.86E-11
3.56E-10
1.95E-09
5.37E-09
1.10E-08
1.93E-08
3.05E-08
4.48E-08
6.25E-08
1.09E-07
1.68E-07
Cs-137
7.78E-07
7.74E-06
1.54E-05
3.07E-05
4.58E-05
6.07E-05
7.55E-05
9.01E-05
1.04E-04
1.19E-04
1.47E-04
1.74E-04
Xe-131
1.02E-07
3.07E-06
6.58E-06
1.33E-05
1.96E-05
2.54E-05
3.06E-05
3.52E-05
3.92E-05
4.26E-05
4.74E-05
4.99E-05
Xe-133
5.18E-07
7.85E-07
7.84E-07
7.82E-07
7.80E-07
7.77E-07
7.74E-07
7.71E-07
7.67E-07
7.64E-07
7.57E-07
7.51E-07
Xe-135
1.71E-08
1.71E-08
1.69E-08
1.64E-08
1.58E-08
1.50E-08
1.41E-08
1.32E-08
1.23E-08
1.13E-08
9.76E-09
8.37E-09
Xe-136
1.33E-06
1.36E-05
2.73E-05
5.50E-05
8.30E-05
1.11E-04
1.40E-04
1.69E-04
1.98E-04
2.28E-04
2.88E-04
3.48E-04
Nd-143
8.58E-08
5.35E-06
1.22E-05
2.49E-05
3.60E-05
4.56E-05
5.37E-05
6.03E-05
6.53E-05
6.87E-05
7.10E-05
6.82E-05
Nd-144
7.12E-09
7.30E-07
2.81E-06
1.02E-05
2.09E-05
3.41E-05
4.93E-05
6.61E-05
8.42E-05
1.04E-04
1.45E-04
1.90E-04
Nd-145
4.94E-07
4.86E-06
9.58E-06
1.86E-05
2.72E-05
3.52E-05
4.27E-05
4.97E-05
5.62E-05
6.22E-05
7.23E-05
8.03E-05
Nd-146
3.77E-07
3.77E-06
7.54E-06
1.52E-05
2.29E-05
3.08E-05
3.88E-05
4.71E-05
5.56E-05
6.44E-05
8.29E-05
1.03E-04
Pm-147
6.08E-08
2.12E-06
4.41E-06
8.07E-06
1.07E-05
1.25E-05
1.36E-05
1.43E-05
1.46E-05
1.45E-05
1.38E-05
1.28E-05
Pm-148
7.99E-11
8.95E-09
2.06E-08
4.08E-08
5.78E-08
7.19E-08
8.34E-08
9.28E-08
1.00E-07
1.06E-07
1.16E-07
1.18E-07
Pm-148m
4.77E-11
8.91E-09
2.24E-08
4.50E-08
6.25E-08
7.52E-08
8.39E-08
8.90E-08
9.13E-08
9.14E-08
8.80E-08
8.03E-08
Pm-149
4.93E-08
5.48E-08
5.69E-08
6.10E-08
6.46E-08
6.79E-08
7.09E-08
7.36E-08
7.62E-08
7.85E-08
8.27E-08
8.54E-08
Pm-151
1.11E-08
1.16E-08
1.20E-08
1.27E-08
1.34E-08
1.41E-08
1.47E-08
1.53E-08
1.60E-08
1.66E-08
1.77E-08
1.86E-08
Sm-147
1.23E-10
5.31E-08
2.41E-07
9.44E-07
1.95E-06
3.12E-06
4.36E-06
5.58E-06
6.72E-06
7.73E-06
9.26E-06
1.01E-05
Sm-148
2.42E-11
4.12E-08
2.20E-07
9.90E-07
2.26E-06
3.96E-06
6.06E-06
8.49E-06
1.12E-05
1.42E-05
2.09E-05
2.81E-05
Sm-149
6.13E-08
1.45E-07
1.48E-07
1.51E-07
1.52E-07
1.49E-07
1.45E-07
1.39E-07
1.32E-07
1.25E-07
1.11E-07
9.83E-08
Sm-151
3.97E-08
3.18E-07
4.40E-07
5.28E-07
5.69E-07
6.00E-07
6.25E-07
6.42E-07
6.54E-07
6.60E-07
6.80E-07
6.74E-07
230
Eu-153
1.35E-08
2.15E-07
5.04E-07
1.31E-06
2.40E-06
3.71E-06
5.20E-06
6.79E-06
8.44E-06
1.01E-05
1.32E-05
1.57E-05
Eu-154
3.76E-11
6.54E-09
2.76E-08
1.19E-07
2.83E-07
5.21E-07
8.26E-07
1.18E-06
1.58E-06
1.98E-06
2.78E-06
3.43E-06
Eu-155
3.92E-09
2.94E-08
4.80E-08
8.38E-08
1.34E-07
2.06E-07
3.00E-07
4.13E-07
5.40E-07
6.77E-07
9.50E-07
1.19E-06
Gd-155
5.62E-12
1.87E-10
3.44E-10
5.96E-10
9.12E-10
1.33E-09
1.82E-09
2.35E-09
2.87E-09
3.34E-09
4.23E-09
4.79E-09
Gd-156
3.34E-10
2.61E-08
8.80E-08
2.90E-07
6.10E-07
1.10E-06
1.82E-06
2.85E-06
4.27E-06
6.18E-06
1.17E-05
1.99E-05
Gd-157
3.66E-10
7.55E-10
1.03E-09
1.50E-09
1.91E-09
2.29E-09
2.68E-09
3.07E-09
3.48E-09
3.91E-09
5.13E-09
6.29E-09
231
A4.5 Pebble Fuel Element Criticality Data per Burnup Step
Burnup (GWD/t)
K-effective
0
1.516883
0.5
1.456028
5
1.428048
10
1.401606
20
1.344026
30
1.285172
40
1.227301
50
1.170394
60
1.11E+00
70
1.06E+00
80
9.98E-01
100
8.91E-01
120
8.03E-01
A4.6 Prismatic Hexagonal Block Criticality Data per Burnup Step
Burnup (GWD/t)
K-effective
0
1.460909
0.5
1.402415
5
1.372111
10
1.342032
20
1.279117
30
1.219249
40
1.164353
50
1.113787
60
1.07E+00
70
1.02E+00
80
9.78E-01
100
9.01E-01
120
8.34E-01
232
VITA
Tholakele Prisca Ngeleka
Education
2000
National Diploma in Chemical Engineering
Mangosuthu Technikon, Durban, South Africa
2001
Baccalaureus Technologiae in Chemical Engineering
Mangosuthu Technikon, Durban, South Africa
2006
Master of Science in Chemical Engineering
North West University, Potchefstroom, South Africa
2009
Master of Science in Nuclear Engineering
North West University, Potchefstroom, South Africa
2012
Doctor of Philosophy in Nuclear Engineering
Pennsylvania State University, Pennsylvania, United States of America