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Generation of Fast Neutron Spectra using an Adaptive Gauss-Kronrod Quadrature Algorithm

Permanent Link: http://ufdc.ufl.edu/UFE0042859/00001

Material Information

Title: Generation of Fast Neutron Spectra using an Adaptive Gauss-Kronrod Quadrature Algorithm
Physical Description: 1 online resource (175 p.)
Language: english
Creator: TRIPLETT,BRIAN SCOTT
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: GAUSS -- INTEGRATION -- KRONROD -- LATTICE -- NUCLEAR -- NUMERICAL
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A lattice physics calculation is often the first step in analyzing a nuclear reactor. This calculation condenses regions of the reactor into average parameters (i.e., group constants) that can be used in coarser full core, time dependent calculations. This work presents a high fidelity deterministic method for calculating the neutron energy spectrum in an infinite medium. The spectrum resulting from this calculation can be used to generate accurate group constants. This method includes a numerical algorithm based on Gauss-Kronrod Quadrature to determine the neutron transfer source to a given energy while controlling numerical error. This algorithm was implemented in a pointwise transport solver program called Pointwise Fast Spectrum Generator (PWFSG). PWFSG was benchmarked against the Monte Carlo program MCNP and another pointwise spectrum generation program, CENTRM, for a set of fast reactor infinite medium example cases. PWFSG showed good agreement with MCNP, yielding coefficients of determination above 98% for all example cases. In addition, PWFSG had 6 to 8 times lower flux estimation error than CENTRM in the cases examined. With run-times comparable to CENTRM, PWFSG represents a robust set of methods for generation of fast neutron spectra with increased accuracy without increased computational cost.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by BRIAN SCOTT TRIPLETT.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Schubring, Duwayne Lee.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042859:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042859/00001

Material Information

Title: Generation of Fast Neutron Spectra using an Adaptive Gauss-Kronrod Quadrature Algorithm
Physical Description: 1 online resource (175 p.)
Language: english
Creator: TRIPLETT,BRIAN SCOTT
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: GAUSS -- INTEGRATION -- KRONROD -- LATTICE -- NUCLEAR -- NUMERICAL
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A lattice physics calculation is often the first step in analyzing a nuclear reactor. This calculation condenses regions of the reactor into average parameters (i.e., group constants) that can be used in coarser full core, time dependent calculations. This work presents a high fidelity deterministic method for calculating the neutron energy spectrum in an infinite medium. The spectrum resulting from this calculation can be used to generate accurate group constants. This method includes a numerical algorithm based on Gauss-Kronrod Quadrature to determine the neutron transfer source to a given energy while controlling numerical error. This algorithm was implemented in a pointwise transport solver program called Pointwise Fast Spectrum Generator (PWFSG). PWFSG was benchmarked against the Monte Carlo program MCNP and another pointwise spectrum generation program, CENTRM, for a set of fast reactor infinite medium example cases. PWFSG showed good agreement with MCNP, yielding coefficients of determination above 98% for all example cases. In addition, PWFSG had 6 to 8 times lower flux estimation error than CENTRM in the cases examined. With run-times comparable to CENTRM, PWFSG represents a robust set of methods for generation of fast neutron spectra with increased accuracy without increased computational cost.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by BRIAN SCOTT TRIPLETT.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Schubring, Duwayne Lee.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042859:00001


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1 GENERATION OF FAST NETURON SPECTRA USING AN ADAPTIVE GAUSS-KRONROD QUADRATURE ALGORITHM By BRIAN SCOTT TRIPLETT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

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2 2011 Brian Scott Triplett

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3 To the Creator, who orders the Universe accord ing to His purpose and enables our attempts to understand and describe that order

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4 ACKNOWLEDGMENTS I would first like to acknowledge Dr. Sami m Anghaie for his support and encouragement throughout my time at University of Florida. Without his guidance and support I would not be where I am today. I would also like to thank Dr Eric Loewen for his constant encouragement and not allowing me to slow down or quit. Brett Dooies and Andrew Caldwell were alwa ys available for disc ussing ideas and giving input into the content of this dissertation. I would like to ackno wledge Brett for peer-checking my work and assisting in the de velopment of the error quantifica tion procedure used to analyze the results. Finally, I would like to acknowle dge Kristen Triplett who provi ded support in the form of plot generation consulting and computer programming advice.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ................................................................................................................ ...........8 LIST OF FIGURES ............................................................................................................... ..........9 LIST OF ABBREVIATIONS ........................................................................................................1 1 ABSTRACT ...................................................................................................................... .............13 CHAPTER 1 INTRODUCTION ................................................................................................................ ..14 The Importance of Lattice Physics .........................................................................................14 Nuclear Cross Sections ........................................................................................................ ...15 Overview ...................................................................................................................... ...15 Cross Section Usage in Reactor Calculations .................................................................18 Asymptotic Flux Solution in Energy ...............................................................................20 Group Constants ............................................................................................................... ......22 2 CURRENT LATTICE PHYSICS/GROUP CONSTANT GENERATION METHODS ......30 Introduction to Group Constants ............................................................................................30 Weighting Spectrum Methods ................................................................................................31 Bondarenko Method ........................................................................................................31 Subgroup Method ............................................................................................................36 Direct Solution Methods ....................................................................................................... ..38 Fine Group Method .........................................................................................................38 Pointwise Method ............................................................................................................41 3 POINTWISE FLUX SPECTR A GENERATION WITH ADA PTIVE QUADRATURE .....48 Neutron Energy Transfer Mechanisms ...................................................................................48 Scattering Kinematics ......................................................................................................48 Scattering Kernel ............................................................................................................. 51 Uncorrelated Energy-Angle Distributions .......................................................................55 Legendre coefficients representation........................................................................56 Kalbach-Mann Systematics representation ..............................................................57 Quadrature Integration of the Transfer Source .......................................................................57 Overview of Gauss Quadrature Integration .....................................................................58 Limitations of Gauss Qu adrature Integration ..................................................................59 Gauss-Legendre Quadrature ............................................................................................59 Adjustment of Integration Limits ....................................................................................60

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6 Gauss-Kronrod Error Estimation .....................................................................................61 Neutron Transfer Source Calculati on Using Gauss-Kronrod Quadrature ..............................62 Bounds of Integration for Neutron Transfer ....................................................................63 Numerical Form of the Neut ron Slowing-Down Equation .............................................64 4 IMPLEMENTATION OF ADAP TIVE QUADRATURE METHODS .................................69 ENDF Input Data ............................................................................................................... .....69 Preprocessing with NJOY ....................................................................................................... 70 ENDF Data Processing .......................................................................................................... .71 Overview of EndfReader .................................................................................................71 Interpolation with in EndfReader .....................................................................................72 Binary Search Algorithm for Interpolation .....................................................................73 Required Input ................................................................................................................ ........73 Solution Algorithm ............................................................................................................ .....74 Gauss-Kronrod Transfer Treatment ................................................................................75 Energy Mesh Determination ............................................................................................75 Parallelization of the Isotope Transfer Source Calculation .............................................77 Flux Iteration Procedure ..................................................................................................77 Outputs ....................................................................................................................... .............78 5 PWFSG VALIDATION .........................................................................................................84 Overview ...................................................................................................................... ...........84 Spectrum Normalization ..................................................................................................86 Error Calculation ............................................................................................................. 87 U-238 Infinite Medium Case ..................................................................................................88 Model Description ...........................................................................................................88 Results ....................................................................................................................... ......89 SFR Fuel Cell Case ............................................................................................................ .....92 Model Description ...........................................................................................................92 Results ....................................................................................................................... ......93 UNF Fuel Cell Case ............................................................................................................ ....94 Model Description ...........................................................................................................94 Results ....................................................................................................................... ......95 6 CONCLUSIONS AND RECOMMENDATIO NS FOR FUTURE WORK ........................123 Conclusions ................................................................................................................... ........123 Future Work ................................................................................................................... .......124 APPENDIX A MCNP INPUT DATA ..........................................................................................................127 U-238 Infinite Medium ......................................................................................................... 127 SFR Fuel Cell ................................................................................................................. ......129 UNF Fuel Cell ................................................................................................................. ......131

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7 B PWFSG INPUT DATA ........................................................................................................133 U-238 Infinite Medium ......................................................................................................... 133 SFR Fuel Cell ................................................................................................................. ......134 UNF Fuel Cell ................................................................................................................. ......136 C CENTRM INPUT DATA .....................................................................................................139 U-238 Infinite Medium ......................................................................................................... 139 SFR Fuel Cell ................................................................................................................. ......146 UNF Fuel Cell ................................................................................................................. ......157 LIST OF REFERENCES ............................................................................................................ .170 BIOGRAPHICAL SKETCH .......................................................................................................175

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8 LIST OF TABLES Table page 3-1 Nodes and weights for Gauss-Lege ndre Quadrature of order 2 to 6 .................................67 3-2 Nodes and weights for the (G7, K15) Gauss-Kronrod pair .................................................67 5-1 Required execution times of the three programs for the three example cases ...................97 5-2 Error statistics for U-238 infinite medium case .................................................................98 5-3 Error Statistics for U-238 infinite medi um case with unresolved resonance treatment disabled in MCNP .............................................................................................................. 98 5-4 Homogenized number densities used in the SFR cell model .............................................99 5-5 Error statistics for SFR fuel cell case ...............................................................................100 5-6 Homogenized number densities used in the UNF cell model ..........................................101 5-7 Error statistics for UNF fuel cell case ..............................................................................102 6-1 Summary of improvement factors for test cases ..............................................................126

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9 LIST OF FIGURES Figure page 1-1 Diagram of neutron flux in space and angle ......................................................................26 1-2 Example cross section in nuclear analysis .........................................................................27 1-3 The flux self-shielding effect ............................................................................................ .28 1-4 Group constant generation process ....................................................................................29 2-1 Scattering band width for various isotopes ........................................................................45 2-2 Representation of the total cross sect ion probability density function for the Subgroup method. .............................................................................................................. 46 2-3 Regions of solution for the CENTRM program ................................................................47 3-1 Scattering kinematic vectors .............................................................................................. 68 4-1 Overall program/data flow for PWFSG program ..............................................................80 4-2 PWFSG solution algorithm ................................................................................................81 4-3 Example of how the GKQ algorithm br eaks up integration bo unds until error is acceptable .................................................................................................................... .......82 4-4 Example of flux estimation-iter ation procedure used in PWFSG .....................................83 5-1 Source distribution used for example cases .....................................................................103 5-2 Neutron energy spectra generated by MCNP, PWFSG, and CENTRM for the U-238 infinite medium case ........................................................................................................10 4 5-3 Neutron energy spectrum for the U-238 infin ite medium case in the resonance region between 18 and 19 keV ....................................................................................................105 5-4 Statistical relative error from MCNP U-238 infinite medium case .................................106 5-5 Residual flux error from difference in PWFSG or CENTRM and MCNP results for U-238 infinite medium case .............................................................................................107 5-6 Neutron energy spectra for the U-238 infinite medium case with CENTRM extending PW treatment to 10 MeV ................................................................................108 5-7 Various cross sections for U-238 between 10 keV and 10 MeV .....................................109 5-8 Angular scattering distribu tion versus energy for U-238 ................................................110

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10 5-9 Neutron energy spectra for the U-238 cas e with unresolved resonance treatment disabled in MCNP ............................................................................................................11 1 5-10 Standardized residuals ( z statistics) from PWFSG generated by comparison to MCNP with and without unres olved resonance treatment ..............................................112 5-11 Histogram of standardized residuals from PWFSG compared to MCNP with and without unresolved resonance treatment ..........................................................................113 5-12 Geometry used for SFR and UNF cell model ..................................................................114 5-13 Neutron energy spectra generated by MCNP, PWFSG, and CENTRM for the SFR fuel cell case ................................................................................................................ .....115 5-14 Neutron energy spectra for the SFR fuel cell case in the reso nance region between 18 and 19 keV .................................................................................................................... ...116 5-15 Statistical relative error from MCNP SFR fuel cell case .................................................117 5-16 Residual flux error from difference in PWFSG or CENTRM and MCNP results for SFR fuel cell case............................................................................................................. 118 5-17 Neutron energy spectra generated by MCNP, PWFSG, and CENTRM for the UNF fuel cell case ................................................................................................................ .....119 5-18 MCNP spectra for SFR fuel cell with U-10Zr fuel and UNF fuel cell with U-20TRU-10Zr fuel .........................................................................................................120 5-19 Statistical relative error from MCNP for the UNF fuel cell case ....................................121 5-20 Residual flux error from difference in PWFSG or CENTRM and MCNP results for UNF fuel cell case ............................................................................................................ 122

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11 LIST OF ABBREVIATIONS ACE A Compact ENDF ASCII American Standard Code for Information Interchange BONAMI Bondarenko AMPX Interpolator BWR Boiling water reactor CENTRM Continuous Ener gy Transport Module cm Centimeter COM Center-of-mass CRAWDAD Code to Read And Write Data for Discretized solution ENDF Evaluated nuclear data file eV Electronvolt GEMINEWTRN Group and ener gy-pointwise methodology implemented in NEWT for resonance neutronics GKQ Gauss-Kronrod Quadrature GLQ Gauss-Legendre Quadrature GQ Gauss Quadrature JEFF Joint Evaluated Fission and Fusion File JENDL Japanese Evaluate d Nuclear Data Library K Kelvin keV Kilo-electronvolt LHS Left hand side LWR Light water reactor MCNP Monte Carlo n-particle MeV Mega-electronvolt MG Multigroup

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12 MOX Mixed oxide MWD/MTHM Megawatt-days per me tric ton of heavy metal ORIGEN-S Oak Ridge Isotope Generation ORNL Oak Ridge National Laboratory PENDF Pointwise evaluated nuclear data file PMC Produce Multigroup Cross sections PWFSG Pointwise Fast Spectrum Generator PW Pointwise PWR Pressurized water reactor RHS Right hand side s Second SCALE Standardized Computer An alysis for Licensing Evaluation SFR Sodium fast reactor TRU Transuranic wt% Weight percent UNF Used nuclear fuel

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13 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GENERATION OF FAST NETURON SPECTRA USING AN ADAPTIVE GAUSS-KRONROD QUADRATURE ALGORITHM By Brian Triplett May 2011 Chair: DuWayne Schubring Major: Nuclear Engineering Sciences A lattice physics calculation is often the first step in analyzing a nuclear reactor. This calculation condenses regions of th e reactor into averag e parameters (i.e., group constants) that can be used in coarser full-core, time-dependent calculations. This work presents a high-fidelity deterministic method for calculating the neutron en ergy spectrum in an infinite medium. The spectrum resulting from this calculation can be us ed to generate accurate group constants. This method includes a numerical algorithm based on Gauss-Kronrod Quadrature to determine the neutron transfer source to a give n energy while controlling numerical error. This algorithm was implemented in a pointwise transport solver program called Pointwise Fast Spectrum Generator (PWFSG). PWFSG was benchmarked against the Monte Carlo program MCNP and another pointwise spectrum generation pr ogram, CENTRM, for a set of fa st reactor infinite medium example cases. PWFSG showed good agreemen t with MCNP, yielding coefficients of determination above 98% for all example cases. In addition, PWFSG had 6 to 8 times lower flux estimation error than CENTRM in the cases examined. With run-times comparable to CENTRM, PWFSG represents a robust set of methods for generation of fast neutron spectra with increased accuracy without increased computational cost.

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14 CHAPTER 1 INTRODUCTION The Importance of Lattice Physics A complete description of the neutron population in the reactor is necessary to accurately model a nuclear reactor. The neutron population determines th e distribution of the nuclear reactions in the reactor. These reactions govern the distribution of power, the utilization of fuel, and the safety margins in a nuclear system. It is impractical to completely model the ne utron population within a nuclear reactor as a function of time, space, direction, fuel exposur e, temperature, etc.; therefore, a coarser calculation must be performed. Th e coarser calculation typically discretizes one or more of the aforementioned variables. This process requires system parame ters that have been properly averaged over one or more of the neutron populat ion’s governing variables (e.g., space, energy). This averaging process is referred to as a lattice calculation or la ttice physics because it involves performing high-fidelity calculations on a small unit of a nuclear reactor, such as a single fuel pin or assembly. The average parameters resu lting from a lattice phys ics calculation are then used for core-wide calculations in a broade r time and space domain. Lattice physics computer programs include LANCER02 used by GE Hitachi Nuclear Energy,1 CASMO from Studsvik,2 and CENTRM/PMC from Oak Ridg e National Laboratory (ORNL).3 Since a lattice physics calculation is often the first step in reac tor analysis, errors in lattice physics can propagate into furthe r core-wide calculations. This can lead to misprediction of important reactor parameters such as neutron balance and reactor power distribution. A great effort has been made over the past few decades to improve multidimensional transport methods for core-wide solutions; however, much less pr ogress has been made on the lattice physics

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15 methods that inform these transport methods. R ecent efforts at ORNL have sought to increase the fidelity of lattice physics methods beyond the current state-of-the-art.4 The present work focuses on improving latti ce physics calculations for fast neutron spectrum reactors (i.e., reactors where neutrons re main at high energies and do not thermalize). A new approach to calculating the neutron tran sfer source based on Gauss-Kronrod Quadrature was implemented in a pointwise solution to the Neutron Slowing-Down Equation. This approach provided excellent agreement with high-f idelity Monte Carlo met hods and resulted in a factor of 6 to 8 reduction in spectrum estimati on error over the current fast spectrum generation methods for the cases studied. The following sections will provide the backgr ound of neutron transport as it relates to lattice physics. Chapter 2 will provide an ove rview of the common lattice physics methods, also known as group constant generation methods. Chapter 3 will introduce the new methods developed for this work. Chapter 4 will discu ss the implementation of these improved methods into a computer program called Pointwise Fast Spectrum Generator (PWFSG). Chapter 5 will include comparison of these improved methods ag ainst high-fidelity Monte Carlo methods and other lattice physics programs in a set of example cases. These example cases include a U-238 infinite medium, a sodium fast reactor (SFR) fuel cell, and a SFR fu el cell with recycled, transuranic-bearing fuel. Finally, Chapter 6 wi ll include conclusions and recommendations for future work. Nuclear Cross Sections Overview The distribution of neutrons in side a reactor core is the und erlying driver for many of the behaviors observed at a macroscopic scale. A complete description of this population requires complete knowledge of the every neutrons’ posi tion, energy (i.e., speed), and direction as a

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16 function of time. Such comprehensive knowledge of the neutron populati on is impractical to obtain; therefore, a number of assumptions ar e usually made about one or more of these independent variables to effici ently arrive at a solution. Although the underlying behavior of a reactor is governed by the neutron distribution, the final reactor performance ultimately depends on the rate of the reactions occurring inside of the reactor core. A reaction rate is a product of the neutron flux and the probability of a specific interaction occurring. It is possible to have a large number of neutrons with a specific position, energy, and direction that are no t reacting with the medium and not impacting reactor behavior. The neutron angular flux is given in Eq. (1-1) where n is the neutron distribution as a function of position, r ; energy, E ; direction, ˆ ; time, t ; and the neutron speed, v .5,6 ()()()ˆˆ ,,,,,, rEtvEnrEt = (1-1) Figure 1-1 gives a sketch of the neutron flux in sp ace and angle. At fast reactor energies (i.e., between a few eV and 20 MeV), the nuclear reac tion rates are independent of the incident neutron direction. This makes th e directionally-indepe ndent scalar flux an important quantity in reactor calculations. The definition of scalar flux is given in Eq. (1-2). ()()4ˆ ,,,,, rEtdrEt = (1-2) The differential solid angle, d in Eq. (1-2) is expa nded in Eq. (1-3) where is the azimuthal angle with respect to the x-axis and is the polar angle with respec t to the z-axis (see Fig. 1-1). ()sin ddd= (1-3) The differential polar angle is comm only condensed into the parameter, sin dd = (1-4) ddd = (1-5)

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17 The reaction rate for a specific reaction is a product of the sc alar flux, the atomic number density, and the reaction’s micros copic cross section. This micr oscopic cross section is the effective area presented by a nucleus to a neut ron for a given reaction. The microscopic cross section is often given in barns (1 b = 10-24 cm2). The relationship for reaction rate, Rx, is shown for nuclear reaction x in Eq. (1-6) where (,) Nrt is the atomic number density at position r and time t and ()x E is the microscopic cross section for reaction x at energy, E.6 ()()()(),,,,,xx R rEtNrtErEt = (1-6) The product of the microscopic cross section a nd number density can be combined into a single parameter known as the macroscopic cross s ection as shown in Eq. (1-7). The units of macroscopic cross section are area per volume (i.e., inverse length). ()()(),,,xxrEtNrtE = (1-7) The use of the macroscopic cross section leads to the definition of the reaction rate given in Eq. (1-8). ()()(),,,,,,xx R rEtrEtrEt = (1-8) Cross sections are estimated by a combination of experiment ation and modeling. In most cases, the cross section is not a smooth function of energy that is easily described by analytical functions. An example total cr oss section (i.e., sum of all reactions) for U-238 is given in Fig. 1-2. Below 4 eV and above 20 keV the cross section is relatively smooth; however, sharp peaks in the cross section occur between these energies. These cross section resonances occur at locations where the additional binding energy ad ded to the nucleus by the neutron plus the neutron kinetic energy corresponds to an excitati on level in the resulting compound nucleus (i.e., the combination of the nucleus plus the captured ne utron, U-239). This causes the probability of neutron interaction at th ese resonance energies to significantly increase.

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18 At higher energies, these res onances become more closely spaced eventually reaching a continuum. This behavior does not imply an absence of resonances but simply that the resonances are no longer resolvab le. Hence, this region is te rmed the unresolved resonance region. For the U-238 total cross section in Fig. 12, this region is between 20 keV and 149 keV. The cross section can also exhibit complicated behavior at much lo wer (<0.01 eV) or much higher (>20 MeV) energies. These energies are extremely rare in fission reactor analysis and are not considered in this work. While not shown explicitly, the reac tion rate of Eq. (1-8) is also a function of temperature. This temperature dependence is introdu ced by the Doppler-broadening phenomenon7 that causes the cross section resonances to broaden and flatte n at higher temperatures due to random atomic thermal motion. Cross section data are typically pr esented unbroadened, at a temperature of 0 K. Prior to use, the cross section data must be broadened to the appropria te temperature. Cross section data are collected from experi ments and models into libraries for use in nuclear physics calculations. These librarie s are promulgated by the laboratories and government organizations that produced them. Ex amples of cross section libraries include the ENDF/B series from the United States,8 the JEFF library from the European Union,9 and the JENDL library from Japan.10 These libraries may focus on di fferent isotopes and applications, but share many original experimental evaluations These libraries follow the standard ENDF-6 format11 that allows processing programs to manipulat e any set of library data, regardless of the source. Cross Section Usage in Reactor Calculations To determine the reaction rate s in a reactor system, the fl ux (i.e., neutron population) and cross sections (i.e., in teraction probabilities) must be know n. As the cross sections are known via experiment, the flux, with respect to all of its independent variables, is the parameter of

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19 interest. The equation for describing neutron be havior in a medium is the neutron transport equation. This equation describes the neutr on balance over differential position, energy, and angle. The neutron transport equati on is given in Eq. (1-9).5 () ()() ()()()041 ˆˆ ,,,,, ˆˆˆˆ ,,,,,,,,,t srEtrEt vEt dEdrEEtrEtSrEt =ŠŠ ++ (1-9) The negative terms on the right hand side (RHS) represent loss by streaming to another location in space (1st term) and loss by absorpti on and scatter out of the current phase space (2nd term). The two positive terms represent transfer into the phase space of interest from other phase spaces (3rd term) and neutron sources with in the current phase space (4th term). This source term can be a fixed neutron source or sources due to reactions such as fission or (n, xn). The cross section in the 3rd term is not a cross section in th e sense described in the previous section but a differential cross section. In this case the cross section is a double-differential scattering cross section in energy and solid angle, with units of inverse length, energy, and angle (e.g., 1/cm-eV-sr). This cross section describes th e probability per unit lengt h of neutron transfer from a specific energy E to a new energy E and from a direction ˆ to a new direction ˆ following a scattering event. When integrated ov er all possible resulti ng energies and angles, this parameter reduces to the scattering cross section at E Direct solution of the neutron tr ansport equation in its full form is not possible analytically; however, this is not necessary for most reactor ap plications. A number of simplifications can be applied to the neutron transport e quation to arrive at a form that is able to be implemented in a computer program. The neutron transport equati on is often solved at a steady-state conditions (i.e., / 0 t =) as in Eq. (1-10).

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20 ()() ()()()04ˆˆ ,,, ˆˆˆˆ ,,,,,,t srErE dEdrEErESrE += + (1-10) Discretization of space, energy, and angle is often performed as well. In the case of Diffusion Theory, the angular dependence is re moved completely thro ugh use of a diffusion coefficient applied to the streaming term.12 Diffusion theory is a pplicable when the angular dependence of the neutron flux in the system is w eak. This condition is met in the interior of many nuclear reactors. In regions of strong a ngular dependence (e.g., near reactor boundaries, areas of strong absorption), the diffusion assump tion yields inaccurate results. Diffusion theory has found wide application in many core simulation programs such as DIF3D,13 AETNA,14 and SIMULATE.15 Asymptotic Flux Solution in Energy A description of the asymptotic form of the neutron flux in energy will aid in determining the expected form of the neut ron energy behavior in a simplif ied case. The knowledge of this asymptotic solution will inform future solutions a nd can form a first estimate of the flux profile with respect to the energy variable. The first step in reducing the steady-state neut ron transport equation to its asymptotic form is assuming the neutrons are present in an infinite medium. This removes any spatial dependence of the neutron flux and el iminates the streaming term (i.e., 0 = ).16 ()()()()()04ˆˆˆˆˆ ,,,,tsEEdEdEEESE =+ (1-11) Equation (1-11) can be integrated over d to yield Eq. (1-12). ()() ()()()4 0444ˆ ˆˆˆˆ ,,,t sEdE dEdEdEEdSE = + (1-12)

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21 The next step is removing the angular dependence by integration over d The differential scattering cross section depends on the change in angle and not the specific incoming and outgoing neutron angles.12 This alters the scattering term to the form shown in Eq. (1-13). ()()1 00 421ˆˆ ,,ssdEEddEE+ Š = (1-13) The cosine of the scattering angle, 0 is given by the fo llowing relationship. ()00ˆˆ cos == (1-14) The angular dependence of the double-differential scattering cross secti on can now be removed because of the integration over all possible cosines from -1 to +1. This yields a single-differential scattering cross section in energy. ()()1 00 12,ssdEEEE+ Š = (1-15) Substitution of Eq. (1-15) into Eq. (1-12) gi ves the infinite medium transport equation. ()()()()()0 ts E EdEEEESE =+ (1-16) The neutron transport equation has now been simp lified to an integral equation in terms of energy only. If neutrons only lose energy in collisions, as is the case above thermal energies, the lower limit of integration of the scattering term is restricted to the current energy. This leads to the Neutron Slowing-Down Equation of Eq. (1-17).17 ()()()()()ts E EEdEEEESE =+ (1-17) In treating neutron slowing-down, the flux is often given in term s of the independent variable of lethargy. Lethargy is a measure of ne utron slow-down relative to a reference energy. As the neutron slows down its lethargy increases, as is defined in Eq. (1-18). Differential lethargy is defined in Eq. (1-19). lno E u E = (1-18)

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22 1 dudE E =Š (1-19) The parameter 0 E is a specified reference energy, which is typically a source energy or maximum reachable energy in the model. The neutron flux function in energy can be given in terms of lethargy, u as in Eq. (1-20). ()()uduEdE =Š (1-20) ()()uEE = (1-21) The solution to the slowing-down equation with isotropic scattering and no absorption in the center-of-mass frame of reference takes the form given in Eq. (1-22).12 () C E E = (1-22) The parameter C is a constant representing the source st rength or a normalization factor. If Eq. (1-22) is given with lethar gy as the independent variable, th en Eq. (1-22) reduces to the following form. ()uC = (1-23) This constant flux in lethargy is the asympto tic form of the flux which can serve as the starting point for more complicated spectru m calculations discussed in Chapter 2. Group Constants Spatial and energy dependence are often treated in a discre te fashion in the neutron transport equation. Without discretization of the neutron transport equation, the time and computational resources required to arrive at a solution would be impractical for engineering design and analysis. Discretization allows for a reactor system to be de scribed by a few spatial nodes or zones and energy to be defined by a few gr oups. The trade-off of this approach is that the flux dependence within the node, zone, or group is lost in the final solution.

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23 Reactor spatial dependence will often be discretized by homogenizing material properties over a volume and calculating an average flux in th is volume. This volume is often a repeating structure such as a fuel pin or fuel assembly. Care must be taken to preserve any important flux variation in energy or space within this volume if it will affect the final system solution. A similar discretization proce dure is performed for energy by calculating the average flux in an energy group. These groups are sections of the neutron energy spectrum with similar properties. For example, in LW R analysis it is common to have an energy group below a 1 eV assigned to the thermal group, neutrons in the resonance energy range (e.g., 1 eV to 100 keV) assigned to a slowing-down group and neutrons above 100 ke V assigned to a fast group.14 Since the energy dependence of the flux within the gr oup is lost during disc retization, corresponding average cross sections in energy must be de veloped for each group to be used with the energy-averaged flux. These average cross sections are known as group constants. The averaging of group constants is not as simp le as weighting the cr oss section value in space and energy. Instead, the flux at the specif ic energy and point in space must be used to weight a cross section at a given location and energy to pr eserve reaction rates. The flux weighted averaging of cross sections is complicated by self -shielding, which is where a large cross section depresses the flux at that specific energy and location. The energy self-shielding effect is often the result of strong resonances in th e cross section. As the neutron population approaches the resonan ce energy, the number of neutr ons available for interaction decreases due to neutron absorption or transfer aw ay from the resonance energy. Therefore, the sharp resonance peak shields itself from the ne utron interaction at the resonance energy. The result is a lower overall reacti on rate at the resonance energy than if the neutron population

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24 remained unshielded. Figure 1-3 is a sketch of how the flux is depressed in the vicinity of a resonance. The self-shielding effect can also have spatial dependence, such as in a heterogeneous zone with both a resonant absorber and non-resonant material. The non-resonant material would be shielded from the resonant neutron energy b ecause of the resonant absorber. Any spatial homogenization of this zone re quires consideration of this effect to product accurate group constants. This effect is of particular impor tance in LWRs, where neutrons are born in fuel material, slow-down in moderator ma terial, and then return to fuel material for reabsorption. A number of methods have been developed to account for spa tial self-shieldi ng in lattice physics, also called cell calculations.18,19,20 These methods are not disc ussed in Chapter 2. This work focuses on application to fast spectrum sy stems, where the energy self-shielding is of primary importance, and on a computer program developed for infinite medium calculations. The standard method for deterministically modeli ng reactor systems is shown in Fig. 1-4. The energy self-shielding calculation is typically performed first on a unit cell of the reactor. A simplified geometric model, such as one-dimensi on or infinite medium, is applied to the unit cell. During this step, the con tinuous cross section data is proce ssed into to a set of fine group constants, typically including tens, hundreds, or possibly thousands of en ergy groups. This work focuses on the process of generating the weighting spectrum for fine group constant generation. A method is presented in Chapter 3 for using co ntinuous nuclear data to generate a pointwise flux solution for use in group constant generation. The next step in reactor modeling is applying the fine group constants to a detailed cell calculation. This cell flux is then used to ge nerate a few broad group constants for use in the full-core simulation. This cell calculation can be repeated for diffe rent operating conditions

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25 (e.g., temperature, void, compos ition) resulting in a set of ce ll-averaged group constants as a function of energy and operating conditions. The re sult is a set of coar se group constants (e.g., fewer than 10) at a specific operating condition for each cell. The final coarse group constants generated from the cell calcul ation are then applied to a full-core reactor simulation with the appropriate average values used for each unit cell and energy group. The entire exercise of group constant genera tion would be trivial if the flux were known continuously as a function of space, energy, angl e, and time. If the flux were known in perfect detail, the cross sections could be averaged with zero bias. Ho wever, as knowledge of the flux profile is the goal of solving the neutron transp ort equation, it will not be known beforehand. Even if the flux were known completely in a r eactor system, any perturbation (e.g., temperature change, control rod insertion, fl ow or power change) could alte r the flux profile and invalidate the current set of group constants. Therefore, the difficulty of creating group constants arises from the requirement of the flux to be known fo r all potential operating conditions before the flux is actually known. Chapter 2 describes so me of the common approaches to handling the problem of group constant generation for bot h thermal and fast spectrum systems.

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26 Figure 1-1. Diagram of neutron flux in sp ace and angle. Diagram based on Figure 4-2 of Duderstadt and Hamilton.5 x y z r ˆ y eˆ x e ˆzeˆ d

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27 Figure 1-2. Example cross section in nuclear anal ysis. Example plot is total cross section of U-238. Total Cross Section (barns)

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28 Figure 1-3. The flux self-shi elding effect. The sharp resonance of the cross section, x shields the neutron flux, by removing neutrons from near resonance energies. The resulting reaction rate, Rx, is less than it would be if the flux had remained unshielded.5 E Rx x

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29 Figure 1-4. Group consta nt generation process Continuous Cross Section Data•Start with evaluated nuclear data library (ENDF/B, JENDL, JEFF, etc.) •Reconstruction of unresolved and resolved resonance parameters into continuous cross sections •Doppler broadening to a specific temperature Energy Averaging•Simplified Geometry (e.g., Infinite medium, one-dimensional) •Detailed treatment of energy (fine group or point-wise) •Accounts for energy self-shielding •Results in fine group constants (hundreds of groups) Spatial Averaging•Fine group constants used to solve for spatial flux within a unit cell •Accounts for spatial self-shielding within the cell •Detailed flux calculation (Discrete Ordinates, Integral transport) •Detailed flux in space used to collapse fine group constants to coarse group constants for specific cell at specific conditions (tens of groups or fewer) Whole System Model•Whole system modeled in space and time with appropriate coarse group constants for each cell •Pre-computed coarse group constants selected for cell based on reactor conditions in that cell

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30 CHAPTER 2 CURRENT LATTICE PHYSICS/GROUP C ONSTANT GENERATION METHODS Introduction to Group Constants The average value of a function over an interval is calculated as show n in Eqs. (2-1) and (2-2) where () x t is the function to be averaged over an independent variable t and () wt is the weighting function for averaging. ()() ()bb aadtxtwtxdtwt = (2-1) ()() () b a b adtxtwt x dtwt = (2-2) In the simple case of determining the mean va lue of a function, the weighting function of Eq. (2-2) is equal to one. In the case of gr oup constant generation wher e the reaction rates must be preserved, the weighting f unction is the scalar flux. () ()() () ,, ,b a b a E x E x E EdErErE r dErE = (2-3) The difficulty arising from solution of Eq. (2-3 ) is the requirement that flux is known as a function of energy at a specific location in space. The most straightforward approach to solving Eq. (2-3) is to assume an approximate energy dependence of the flux. This would require experiential basis of the reactor system in question. This me thod would also assume that the net errors associated with this approach would be ac ceptable. The next section describes some of the common weighting spectrum methods.

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31 Weighting Spectrum Methods Bondarenko Method In the original work on group constants, I. I. Bondarenko21 proposed flux spectrum of the form of Eq. (2-4) where 0() u is a smoothed flux spectrum without perturbation given in terms of lethargy. () () () 0 tu u u (2-4) Bondarenko proposed using the asymptotic flux so lution as described in Eq. (1-23) where 0() u is assumed to be constant. This reduces Eq. (2-3) to the form of Eq. (2-5). () () () 1b a b au x u t x u u tu du u du u = (2-5) Variations of Eq. (2-5) are made for the transport and elastic cross section but this basic form is the basis of the group constant gene ration method proposed by Bondarenko. While the Bondarenko method is valid for a wi de range of energies, the assumption of asymptotic flux is invalid near neutron sources, where neutrons have not yet reached asymptotic behavior. It is also inva lid at energies where neutrons have thermalized with the medium. In the group constant set presented by Bondarenko, the asymptotic flux spectrum was used for all groups lower than 2.5 MeV. For the highest en ergy groups an analytic expression representing the neutron fission spectrum was used.21 The group constant method of Bondarenko is an example of how to apply knowledge of the system through different forms of the flux sp ectrum. For light wate r reactors (LWRs), the unperturbed flux profile is often a combination of a fission energy distribution at high energies,

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32 the asymptotic form at slowing-down energies and a Maxwellian distribution at thermal energies. A noted weakness of a straightforward applic ation of Eq. (2-5) is that it considers a medium containing only a single resonance absorber. As is often the case of nuclear reactors, the medium in question could be a mixture of different resonant materials (e.g., fuel isotopes) with non-resonant materials (e.g., structure and cool ant isotopes). Bondare nko proposed a correction to Eq. (2-4) by including a background cross section, 0 () () () 0 0 tu u u = + (2-6) The 0 term represents the constant background sec tion from non-resonant isotopes. Spatial effects (leakage) can also be accounted for through 0 .7 Inclusion of the background cro ss section dilutes the effect of the resonance on the flux. As 0 goes to infinity, the flux perturbation goes to zero. This condition is known as infinite dilution. If this condition, coupled with the asymptotic form, is substituted into Eq. (2-3) the group constant is simply the mean value of the cross section over the interval. This can be interpreted as no self-s hielding taking place. () b au x u x baduu uu = Š (2-7) Recall from Eq. (2-4) that the slowing-dow n form assumes that the total neutron interaction rate, ()()tuu is relatively constant. This condition is true if a collision within a resonant material removes the neutron from the resonance either by absorption or by scatter out of the resonance. This condition is given in Eq. (2-8) where is the practical width of the resonance and is the average fractional energy loss per scatter in the absorber.

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33 E (2-8) The Bondarenko method is based upon this Narrow Resonance Approximation. If collisions occur inside a resonance and the neutron fails to leave the resonance, then the condition of Eq. (2-8) no longer holds. The constraint of a narrow resonance is we ll met if the scattering band (i.e., range of resultant energies) is wide compared to the res onance width. A plot of the scattering band width for various atomic weight isotopes is given in Fig. 2-1. At hi gh energies, the large scattering band and narrower resonances (recall Fig. 1-1) lead to the satisfaction of Eq. (2-8) and validity of the Narrow Resonance Approximation. At lower energies, the scattering band becomes logarithmically smaller and the resonances widen. These factors create difficulties for the Narrow Resonance Approach. At these energies the Wide or Intermediate Resonance Approximations7,12 are more appropriate. The Wide Resonance Approximation starts with the same fundamental equations as the Narrow Resonance method but assumes that a neutr on does not leave a resona nce upon collision. This condition is also expressed by the assumption that resonant isotope is infinitely absorbing (i.e., always absorbs neutrons which interact in the resonance rather th an transfer them). Therefore this approximation is also calle d the Wide Resonance Infinite Absorber Approximation. The Wide Resonance Approximati on arrives at the flux spectrum of Eq. (2-9).7 () () () 0 0 au u u = + (2-9) The only difference between Eq. (2-6) and Eq. (2-9) is the use of the abso rption cross section of the resonant isotope in the denominator rather than the total cross section. The Intermediate Resonance Approximation is a combination of the Narrow and Wide Resonant approximations. The narrow and wide resonance expressions are combined and

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34 weighted with the parameters and (1)Š, respectively.7 This approximation can be used for energies where neither the assumptions of th e narrow or wide resonance approximations are valid. The Bondarenko method based on the Narro w Resonance Approximation has been a standard approach for generation of group cons tants for three decades. The GROUPR module of the NJOY computer program is commonly used to produce group constants with the Narrow Resonance approach.22 The fundamental formula used in GROUPR is given in Eq. (2-10), where i is the isotope index, is the Legendre component of the angular flux (0 = for scalar flux), and () CE is an input wei ghting spectrum. () () () 1 0 i ii tCE E E += + (2-10) In the GROUPR module, () CE and 0 are input from the user result ing in cross sections at a set of temperatures and 0 values. A similar methodology is employed in the BONAMI module23 of the SCALE computer package.24 The strength of the Narrow Re sonance approach is its simplic ity and ability to generate reusable group constants. A library of group constants can be genera ted as a function of temperature and background cross section. Exam ples of such group cons tant libraries include the VITAMIN-B6 library25 processed by NJOY into the AMPX format used by SCALE. These group constant libraries are pse udo problem-independent because they have been processed with variable temperatures and background cross sect ions but a fixed weighting spectrum. These group constant libraries and are often the starting point for th e generation of problem-specific libraries. For example, the 199 group VITAMINB6 library was collapsed with a specific

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35 pressurized water reactor (PWR) and boiling wate r reactor (BWR) flux spectrum to create the 47 group BUGLE-96 library for pressure vessel shielding calculations.26 The Bondarenko method is often used in conjun ction with an f-factor approach. In this approach a set of group constant s are generated at zero temperat ure (no Doppler broadening) and infinite dilution conditions. This zero-temperature, infinite dilute set are scaled by an f-factor. ()()()00,,0,TfT = (2-11) In this scenario, a table of f-factors is genera ted for scaling of the zero-temperature, infinite dilute group constants. The LANCER02 lattice p hysics generates f-factors via NJOY from a set of infinitely dilute cross secti ons at 0 K and cross sections at other values of temperature and background cross section. Du ring LANCER02 execution, a loga rithmic interpolation is performed on the background cross section and a quadratic interpolati on is performed on the square root of the fuel te mperature. This allows the LANCER02 program to obtain group constants for a specific amount of moderation and reactor temp erature conditions without the need for recalculation of the group constants.1 The weakness of the Bondarenko method is th e assumption of narrow, widely-spaced resonances. Resonance overlap, e ither within or between resonant materials, will introduce error because the flux will not be shielded as 1 /t. Areas of particular concern are wide resonances at lower energies of fuel materials (e.g., U235, Pu-239) and the closely-spaced, overlapping resonance region of fuel materials. The former issue can be treated by di rect flux calculation in the thermal region (as in GROUPR ) or by a wide or intermediate resonance approximations. The latter issue is of great importance in fast reactor analysis and ofte n leads to the need to solve the flux with a higher degree of energy resolution.

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36 Subgroup Method The Subgroup method has found use in LWR la ttice physics applications such as CASMO,2 APOLLO2,27 and HELIOS.28 In newer lattice physics programs it has replaced the Bondarenko method because of its ability to create accurate group constants without resorting to direct solution of the transport equation. The Subgroup method offers improved accuracy over the Bondarenko method with simila r computational requirements. The Subgroup method, also known as the Multib and method or Probability Table method, was originally proposed as a me thod of self-shielding by Levitt29 and Nikolaev30 then advanced in the works of Cullen in the 1970s.31 The Subgroup method has been further refined in the works of Hrbert32 and incorporated into many modern lattice physics programs such as APOLLO2.32,33 The Subgroup method is based upo n subdivision of the total cross section into subgroups, also called bands, within an energy group. A proba bility density function is developed for the total cross section within the gr oup. Figure 2-2 demonstrates the translation from a cross section as a function of energy to a probability density function versus cross section value. The translation of a cross section versus lethargy or energy to a probability table is performed by taking a Riemann integral of the cross section an d replacing it with an equivalent Lebesgue integral34 as in Eq. (2-12). ()()()()1max 01g gu u g dufudf uŠ = (2-12) The LHS of Eq. (2-12) is the common defin ition of a cross section perturbed flux profile as in the Bondarenko approach. The RHS of Eq. (2 -12) contains the probability density function of where () d is the probability that the cro ss section value with lie between and

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37 d + This probability density f unction is then represented by a finite, weighted set of Dirac distributions at discrete va lues of the cross section, ()k Š. ()()1 K kk kw= =Š (2-13) Substituting Eq. (2-13) into Eq. (2-12) yields the relationship of Eq. (2-14) that forms the basis for Subgroup programs. ()()111g g K u kk u k g dufuwf uŠ= = (2-14) The weights and k values in Eq. (2-14) can be de termined by a number of methods.29,30,31 The Subgroup method is an improvement of the Bondarenko method for resonance self-shielding. Instead of adding a finer group structure, the Subgroup me thod adds a dimension of variation on the tota l cross section via probability tabl es. Some of the works of Cullen demonstrate that a 175 group, 2 band calculation (350 equations) can yield reaction rates with equal or less error than a 2020 group calculation for U-235 critical spheres.7 This improved efficiency has made the Subgroup method a popul ar choice for many LWR lattice applications. One drawback of the Subgroup approach is the lack of resonance interference treatment. A common example of this interference effect can be observed in LWRs between the 20.9 eV resonance of U-238 and the 20.45 eV resonance of Pu-240. When re sonances of multiple materials (e.g., a mixture of actinides in a fuel assembly) interfere with each other, the assumed shape function may be invalidated. Any calcu lation where the flux w ithin a region of the problem is assumed to follow a prescribed shap e dependent on the total cross section can be limited by this problem.

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38 Direct Solution Methods Fine Group Method The fine group approach involve s the solution of the discreti zed neutron transport equation with energy groups selected at energy (or leth argy) widths fine enough that the intra-group flux will be slowly varying (i.e., a linear variation or c onstant). The flux is then solved for directly, typically in simplified infinite medium or one -dimensional geometry. The resulting flux energy spectrum is then used as the weighting spectrum for group constant generation via Eq. (2-3). The multigroup (MG) form of the transpor t equation is obtained by integrating the transport equation over discrete energy ranges. The MG form of the steady-state, angularlyindependent neutron transport equation is shown in Eq. (2-15) for the energy range g E to 1 g E Š. ()()()()()11 1 11 00 10ˆˆ ,,, ˆˆ ,,,,,,gg gg g g g gEE t EE E E s E EdEdErErE dEddErEErEdESrE ŠŠ Š Š+ Š+= + (2-15) To simplify Eq. (2-15), group flux, group constants, and group source are defined in Eqs. (2-16), (2-17), and (2-18), respectively. ()()1ˆˆ ,,,g g E g ErdErE Š= (2-16) () ()() () 1 1,ˆ ,,, ˆ ,,g g g gE x E xg E EdErErE r dErE Š Š = (2-17) ()()1ˆˆ ,,,g gE g ESrdESrEŠ= (2-18) With the definitions above, Eq. (2-1 5) can be reduced to Eq. (2-19). ()()()()()1, 1 00 10ˆˆ ˆˆ ,,,,,g ggtgg E sg Err dEddErEErESr Š+ Š+= + (2-19)

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39 To accommodate the scattering te rm of Eq. (2-19), the total energy range considered is broken into G subintervals (i.e., energy groups) as s hown in Eq. (2-20). A group to group transfer cross section is th en defined in Eq. (2-21). ()()()()()11, 1 00 1 1ˆˆ ˆ ,,,,gg gggtgg G EE sg EE g rr dEddErEErESr ŠŠ + Š =+= + (2-20) () ()() () 11 10 ,0ˆ ,,,, ˆ ,,gg gg g gEE s EE sgg E EdEdErEErE r dErE ŠŠ Š = (2-21) Equation (2-20) can now be reduced to the fina l MG form of the tran sport equation shown in Eq. (2-22). ()()()()()1 ,0,0 1 1ˆˆˆˆ ,,,,G gtggsgggg g rrdrrSr+ Š =+=+ (2-22) The group fluxes in Eq. (2-22) are coupled vi a the scattering transfer term. This can become problematic as finer group structures ar e employed because the probability of transfer from every group to every other group must be calc ulated. In the worst, case this would require 2G calculations. Approximations can be made to Eq. (2-22) to reduce the computational burden of group-to-group scatter. A common approximation in fast neutron sy stems is to assume neutrons only down-scatter. This allows the scattering term of Eq. (2-22) to be summed from the current group to the highest energy group, ignoring all lower energy groups Another approximation is to space groups such that scatter will only occur to the next lowest group. This approximation is of great utility in coarse-group, whole-core simula tions because there is a fixed input to the lower groups and no summation loop required.

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40 Group constants must still be determined fo r application of the MG methods (see Eqs. (2-17) and (2-21)). The difference with the fi ne group approach is that the impact of the assumed weighting spectrum is minimized because th e energy group size is relatively fine. With the fine-group approach, the Bonda renko method can be used to develop initial group constants for the MG solution. These group constants can then be updated with the resulting flux from the fine group solution if desired. In the fine group approach the spatial de pendence is often treated with a course approximation. Infinite medium calculations are common when spatial dependence is of secondary importance, a zero-dimensional correc tion can be used to account for the leakage effect on the energy spectrum. In this zero-dime nsional approach the sp atial dependence of the flux approximated via a geometric buckling factor, B ()()ˆˆ ,,,iBrrEEe = (2-23) The lumped buckling parameter describes the loss of neutrons by leakage and can be given on a group-wise basis if necessary. The zero-dimensional transport equation can be derived by substitution of Eq. (2-23) into the neutron transport equation. ()()()()()1 00 1ˆˆˆˆ ,,,,ts E EiBEdEdEEESE+ Š +=+ (2-24) If the angular component of the flux and scatteri ng cross section are expanded into spherical harmonics the following relationship is obtained for planar geometry.35 ()() () ()()()()021 2ts EEiBEdEEEEPSE =+ +=+ (2-25)

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41 The zero-dimensional transport equation can be simplified through the recursion relationship for Legendre Polynomials given by Eq. (2-26) and the application of the orthogonality property of Legendre Polynomials. ()()()()()11210 PxxPxPx++Š++= (2-26) Use of these Legendre Polynomial prope rties results in Eq. (2-27) where is the Legendre order of the flux or differential cross section. () ()()() ()()()111 2121t s EiBEiBEEE dEEEESE +Š + ++= ++ + (2-27) The fast reactor group constant program MC2 solves Eq. (2-27) using a number of transport methods including ultra and hyper fine MG.35 Pointwise Method Another direct approach to developing problem dependent group constants is a pointwise (PW) method. In this approach, the flux is solv ed at a set of energy points that are spaced such that the flux variation between points can be assumed linear. Th e resulting PW flux is then a linear-interpolat able function in energy. At resonance energies this continuous energy, PW solution can be useful as it repres ents one of the highest fidelity methods currently available. This PW method was first introduced by Williams36 and has been implemented in the CENTRM module37 of the SCALE analysis package from Oak Ridge National Laboratory.24 The CENTRM method combines a classic MG approach with a PW flux solution. CENTRM applies a high-fidelity, PW soluti on where the flux has strong energy dependence (e.g., resonance energies) and a traditional MG so lution where the flux is slowly varying (e.g., high energies).

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42 Figure 2-3 shows the regions of solution fo r CENTRM. In the MG regions the group constants can be self-shielded by the Bonda renko method; the MG fluxes solved by the aforementioned MG solution methods. In the PW range the transport equation is written as Eq. (2-28) where n is the energy point where the flux will be balanced. ()()()()()(), 1 00 1ˆˆˆ ,, ˆˆ ,,,,,ntnn sn Errr ddErEErESr + Š+ =+ (2-28) Note that the form of the equation in Eq. (2-28) is very similar to the MG form of Eq. (2-22) with the exception of the cross section de finitions and the scattering source.36 The CENTRM program uses the same subroutines to solve the transport equation in the PW region as it does in the MG region. CENTRM uses sub-moment expansion38 to separate the final energy dependence, E out of the transfer source integrand of Eq (2-28) and into a series of sub-moments. The removal of the final energy dependence from the integrand allo ws for a continuous summation of the transfer source over initial energy, E as the calculation progresses from high energy to low energy. This avoids the recomputing of the tr ansfer source at each energy poin t and, therefore, makes the PW solution method in CENTRM more efficient. The primary difference between the PW and MG solutions is the definition of the resulting flux. The PW formula of Eq. (2-28) describes ne utron balance at a specifi c energy, while the MG form describes neutron balance ov er an energy interval. The units of flux in the PW solution are 1/eV-cm2-s and the MG solution flux has th e normal scalar flux units of 1/cm2-s. Conversion between the two quantities can be performed by dividi ng the MG flux by E or u This assumes the PW flux is an accurate representation of the average flux over the entire interval. The PW method does not conserve reaction rate s over an energy interval. By using a PW solution, the overall particle bala nce on an energy interval is not maintained. However, for the

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43 purposes of obtaining a weighting spectrum, a PW solution is typically acceptable because the overall flux shape is of primary importance and not the actual reaction rates occurring in the system. This effect is further mitigated by using a larger number of energy points. CENTRM uses enough points to assume lin ear flux dependence between the en ergy points. A typical PW solution for CENTRM includes 10,000 to 70, 000 points, depending on the isotopes present.37 The inclusion of spatial depende nce in a PW solution is dependent on the degree of fidelity desired. CENTRM contains opti ons ranging from infinite medium to one-dimensional geometry. Often in lattice physics methods for LWRs, the flux spectrum variation in a unit cell, particularly a large unit cell like a fuel assemb ly, can be significant enough to warrant inclusion of detailed spatial treatment. The PW solution can be solv ed in twoor three-dimensional space using the Discrete Ordinates (i.e., SN) method or the Method of Characteristics.6 An example of spatial and PW coupli ng is the GEMINEWTRN module of SCALE.4 GEMINENTRN couples a two-dimensional Discre te Ordinates method with the CENTRM PW flux module. This can yield highly accurate results for highly hete rogeneous fuel designs such as mixed oxide (MOX)-bearing BWR lattices. The strength of the fine group or PW approach is its high degree of fidelity. The neutron transport equation is solved directly for the materi al and temperature of in terest instead of using an assumed weighting spectrum perturbed by a func tion of the cross section or other factors. The drawback of this approach is that the resulting group consta nts are not reusable, because the detail of the system has been captured in th e higher accuracy group cons tants. Application of these group constants to temperatures and compos itions different from the original basis would introduce error. There is no stra ightforward means of scaling th e group constants to a different application with the fine-group approach.

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44 The PW solution method represents one of th e highest accuracy methods for calculating the neutron energy spectrum. The work presented here is based on the work of the PW solution method, particularly the methods employe d in the CENTRM computer package.

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45 Figure 2-1. Scattering band width for various isotopes Scattering Band Width (eV)

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F F igure 2-2. Re p en e pl o p resentation of t e rgy is shown o n o t from Figure 1 t he total cross s e n the left and th e of Hbert and C e ction probabili t e probability d e C oste.34 46 t y density funct e nsity versus cr o ion for the Sub g o ss section valu e g roup metho d e is shown rota t Total cross sec t t ed on the right. t ion versus Example

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F F igure 2-3. Re g g ions of solutio n n for the CENT R R M progra m D 47 D iagram from Fi gure F18.1.1 o f f Williams, Asg a a ri, and Hollen b b ach.37

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48 CHAPTER 3 POINTWISE FLUX SPECTRA GENERATI ON WITH ADAPTIVE QUADRATURE Neutron Energy Transfer Mechanisms The generation of accurate group constants is highly depende nt on adequately describing the energy dependence of the neutron flux. Collisions with nuclei in a system transfer neutrons from source energies to lower energies. In a thermal reactor, such as a light water reactor (LWR), these neutrons slow down until they th ermalize (reach energies comparable with the thermal motion of the reactor medium). In a fast reactor, the neutrons are usually absorbed or leak from the system before thermalizing and th us remain in the slow ing-down mode for their entire lifetime. The primary mode of neutron energy transfer within a nuclear reactor is scattering. Neutron scattering occurs when neutrons collide w ith the nucleus of an atom and impart some of their kinetic energy to the nucleus. If the nucleus is left in an excited state after the collision, the kinetic energy of the reaction is not conserved and this reaction is in elastic. In this scenario, the nucleus de-excites via some subsequent nuclear decay mechanism such as gamma ray emission. Otherwise, the scattering reaction is a two-body collision where kinetic energy is conserved and is considered elastic. As the scattering energy transfer mechanism is important to descri bing the neutron energy dependence and, by extension, determining the group constants, it is importan t to briefly describe the relationships that govern ener gy transfer in scattering reactions. Scattering Kinematics Figure 3-1a diagrams the neutro n-nucleus interaction before a nd after a scattering event. A neutron with mass, m and velocity, 0 v strikes a nucleus, assumed to be at rest, scatters into a new angle, 0 and imparts some of its kinetic energy to the nucleus. This assumption of the

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49 nucleus at rest is valid when the neutron energy is well above kT, where k is Boltzmann’s constant of 8.617x10-5 eV/K and T is the temperature of the medium in K. The conservation of energy before and afte r collision is given in Eq. (3-1) where Q is the excitation energy of the nucleus if the reaction is inelastic. For an elastic collision, Q is equal to zero. 222 00111 222omvmvMV Q =+Š (3-1) The conservation of momentum before and after collision is given in the x and y direction by Eqs. (3-2) and (3-3), respectively. ()()00000coscosmvmvMV =+ (3-2) ()()00000sinsin mvMV =+ (3-3) This system of coordinates is referred to as the laboratory system. To facilitate a full description of this scattering event, the system will be cast into a new frame of reference. In this new frame of reference, the entire system is movi ng at the same velocity as the center-of-mass of the neutron-nucleus pair. Therefore this syst em is referred to as the center-of-mass (COM) system. A diagram of the COM system is shown in Fig. 3-1b. Solution of the neutron ener gy and momentum equations in the COM system enables solution of the scattering angle in the laborator y system. The details of this procedure can be found in Ref. 12. Following this procedure and converting velocity to kinetic energy provides the following relationship for scatte ring angle in the laboratory system. () 011 cos 222 A QAEAE EEE +Š =ŠŠ (3-4)

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50 The variable E is used to denote initial energy and E to denote final energy. This convention is used to maintain consistency with the differentia l scattering cross section in the neutron transport equation. The cosine of the scattering angle on the left hand side (LHS) of Eq. (3-4) is commonly condensed into a single parameter, 0 The formulation for the scattering energy transfer in the lab system is then given by Eq. (3-5). () 012, EE EE EE =Š (3-5) The parameters 1 and 2 are defined in Eqs. (3-6) and (3-7), respectively. 11 22 A QA E + =Š (3-6) 21 2 A Š = (3-7) The above scattering angle formulation is used in many nuclear simulation programs such as CENTRM.37 It can also be useful to describe the energy tr ansfer in terms of the COM scattering angle. The energy-angle relationship in th e COM system can be developed using the veloci ty vectors as shown in Fig. 3-1c and the Law of Cosines. The result of this procedure is the following energy-angle relationship in the COM system. () () () ()211 1 2cE A E E EE E +Š =Š (3-8) () 1 1 A Q EA A E + =+ (3-9)

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51 Again the convention is used where E is the initial neutron energy before scatter and E is the final energy after scatter. Scattering Kernel The neutron scattering kernel de scribes the probability of ne utron energy transfer between two energies. The scattering kernel appear s in the neutron tran sport equation as a double-differential cross section, 00(,) s E EdEd This parameter gives the probability that a neutron will scatter from an energy, E to a new energy interval dE about E with a change of direction within 0d about 0 The value of 0d varies between -1 (full backscatter) to +1 (glancing collision). This allows the angular dependence of th e scattering kernel to be repres ented by an infinite series of Legendre Polynomials. Legendre Polynomials exis t in the domain of [-1, +1] and can be expanded to the necessary order to describe th e angular dependence of the scattering kernel. Application of these Legendre Polynomials to the double-differe ntial scattering cross section yields the relationship in Eq. (3-10), where 0() P is a Legendre Polynomial of order and () s E E is the associated polynomial coefficient. () () ()()00 021 2ssEEEEP = + = (3-10) Both sides of Eq. (3-10) can be multiplied by another Legendre Polynomial of order and integrated over the entire range of 0 resulting in Eq. (3-11). ()() () ()()()11 000000 11 021 2ssdEEPdEEPP ++ ŠŠ = + = (3-11) Legendre Polynomials are orthogonal functions. As such, Legendre Polynomials with the normalization of (21)/ 2 m + have the property of Eq. (3-12) where mn is the Kronecker delta function given in Eq. (3-13).

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52 ()() 1 12 21mn mndxPxPx m+ Š= + (3-12) { 1 if 0 if mnmn mn = = (3-13) Using the orthogonality property of Legendre Polynomials, Eq. (3 -11) reduces to Eq. (3-14). ()()()1 000 1,ssdEEPEE+ Š = (3-14) To determine the right hand side (RHS) of Eq. (3-14), another form of the scattering kernel is necessary. The scattering kernel is separate d into components and expressions are developed for each.39 ()()()()000000,,,ss E EdEdEpEdgEEdE = (3-15) The variable () s E is the total probability of scattering at E while 00(,) p E is the probability that, if scattered at E the neutron will scatter within the angle 0d about 0 Finally, 0(,) g EE is the probability that, on ce a neutron scatters through 0 the resulting energy will be within dE about E In both elastic and discrete inelastic scattering, the energy transfer is directly related to angle. These relationships are given via Eq. (3 -5) in the lab system or Eq. (3-8) in the COM system. This leads to the expression for 0(,) g EE given in Eq. (3-16) where the RHS is the Dirac delta function defined in Eq. (3-17). ()()0000,, g EEdEEEd =Š (3-16) ()()()00dxfxxxfx ŠŠ= (3-17) The usage of the Dirac delta function restrict s the energy probability density function to the specific energy resulting from the scatter. An alternate form of the scattering kernel is now given in Eq. (3-18).

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53 ()()()() 0 00000,,,ssd EEEpEEE dE =Š (3-18) The form of the scattering kern el in Eq. (3-18) can be insert ed into Eq. (3-14) yielding Eq. (3-19). ()()()() ()1 0 000000 1,,ssd EEdEpEEEP dE + Š =Š (3-19) The Dirac delta function property in Eq. (3-17) can be used to reduce Eq. (3-19) to Eq. (3-20). ()()() () ()0 000, ,,,ssdEE EEEpEEEPEE dE = (3-20) The result in Eq. (3-20) may be substituted back into Eq. (3-10) to yield an expression for the coefficient of the scattering kernel, where 0 is the specific scatte ring angle in the given coordinate system that results from an energy transfer of E to E () ()() ()()0 00000 021 ,, 2ssd EEEpEPP dE = + = (3-21) The primary focus of a slowing-down calcula tion is the scalar neutron flux energy dependence; therefore, the angul ar dependence of Eq. (3-21) can be removed via integration. Equation (3-22) is obtained by integrating Eq. (3-21) over angle and using the property of Legendre Polynomials given in Eq. (3-12). ()()() 0 00,ssd EEEpE dE = (3-22) The Jacobian in Eq. (3-22) accounts for the change in variable from a per unit energy to a per unit cosine basis. The Jacobian can be determ ined by taking the derivativ e of Eq. (3-5). This derivative is given fo r the laboratory system in Eq. (3-23). () () 1/2 01 1 1 4 QAAE d EEA dEEŠ +Š =++ (3-23)

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54 If a probability density function is directly supplied via experi ment or model, all terms of Eq. (3-22) are known and the scatte ring kernel is fully described. In some ENDF evaluations the function 00(,) p E is given as a tabulated function of prob ability versus cosine at a number of energies for a given isotope and reaction.11 Often the ENDF representation of the probabi lity density function is an expansion of Legendre Polynomials. The probability density f unction in this form would be given by a Legendre Polynomial expansion with 1 N + terms as in Eq. (3-24) where the ENDF libraries supply the Legendre coefficients, ()naE .11 ()() ()() 0 0 021 2N ssnn nd n EEEaEP dE =+ = (3-24) Elastic scattering is often isotro pic, or nearly isotropic, in the COM frame of reference. The closer the scattering is to isotropic, the fe wer terms required in the expansion in Eq. (3-24). For the latest ENDF/B library release, ENDF/B -VII.0, all elastic scattering is represented entirely in the COM system. A transition from the lab representation of Eqs. (3-22) and (3-24) to the COM system is performed by a change of variables for the pr obability density function of Eq. (3-22). ()()000,,ccc p EdpEd = (3-25) Substitution of Eq. (3-25) into Eq. (3 -22) yields the following result. ()()() ,c ssccd EEEpE dE = (3-26) The form of Eq. (3-26) is the same as the la b representation of Eq. (3-22). The probability density function is altered to a COM system representation and the Jacobian is that of COM angle to energy. This Jacobian can be determin ed by taking the derivative of Eq. (3-8), yielding Eq. (3-27).

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55 () () 21 1 2cA d dEEE + = (3-27) As with the lab system, it is common to represent the probability density function of Eq. (3-26) in the ENDF libraries as a finite se ries of Legendre Polynomials as demonstrated in Eq. (3-28), where the coefficients, ()naE are provided in the data libraries in the COM system. ()() ()() 021 2N c ssnc nd n EEEaEP dE =+ = (3-28) Equations (3-22), (3-24) (3-26), and (3-28) represent all the possible angular descriptions of scattering energy transfer in the lab and COM frames of refe rence present in File 4 of the current ENDF-6 standard.11 These equations are used to desc ribe the slowing-down of neutrons within a reactor system via elas tic and inelastic scatter. Uncorrelated Energy-Angle Distributions In elastic scatter, the neutron scattering a ngle has a unique value for a given incoming and outgoing energy. The same is true with discrete inelastic scatter (i.e., a scatter that leaves the nucleus at a specific excitation energy). For ot her reactions, this corr espondence does not exist and a full description of bot h the angular distribution, 000(,) p Ed and resulting energy distribution, 0(,) g EEdE are required. Two important examples of uncorrelated reactions are continuum inelastic and (n,xn) reactions. Continuum inelastic s catter occurs at energies where the discrete nuc lear excitation levels of an isotope reach a c ontinuum and are no longer resolvable. A (n,xn) reaction occurs when a neutron strikes a nucleus, freeing x-1 additional bound neutrons, which results in x neutrons from the scatter event.

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56 The ENDF-6 standard11 combines the energy and angle dependence into one distribution as described in Eq. (3-29), where ( ) mE is the multiplicity of the reaction at E The multiplicity is the number of neutrons that result fr om the reaction (e.g., 2 for (n,2n), for fission). ()()()(),,,, EEEmEfEE = (3-29) This differential cross sec tion would be given in units of barns per unit cosine. The ENDF-6 standard includes a number of ways for describing the energy-angle distribution following a reac tion for different types of incident and resultant particles. These data are included in File 6 of an ENDF library for a given isot ope. Below are two of the most common representations for neutron-nucleus reactions. Legendre coefficients representation The first common representation for energy-angle distributions of neut rons in the ENDF-6 standard11 is via Legendre Polynomials. () ()()021 ,,, 2 N AfEEfEEP =+ = (3-30) In this representation the angular component of the dist ribution is represented by an expansion of Legendre Polynomials, where the (,) f EE terms are the Legendre coefficients given for NA number of terms in the ENDF data files. The value of NA is dependent on the angular distribution at a given en ergy. If the angular distribution is complicated, more polynomial terms are necessary to describe the re sulting angular distribution. If only the total neutron energy transfer is desired, both sides of Eq. (3-30) can be multiplied by another Legendre Polynomial and integrated over angle. ()()()1 1,,, fEEdfEEP + Š = (3-31)

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57 The zero-order Legendre Polynomial coefficient, 0(,) f EE is the total probability of scattering from E to E The 0(,) f EE values would be equiva lent to the energy dist ribution data given in File 5 of the ENDF data. When determining only the probability of energy transfer from one energy to another, a full description of the resulting angular distribution is not nece ssary. Therefore the value of 0(,) f EE is sufficient for determining the probability of transfer from E to E. Kalbach-Mann Systematics representation Another means of describing an energy-a ngle distribution is through Kalbach-Mann Systematics.40,41 Kalbach and Mann proposed a gene ralized means of representing the energy-angle distribution of an incident particle on a nuc leus with the function shown in Eq. (3-32) where the parameters 0(,) f EE (,) aEE and (, ) rEE are stored in the ENDF data. () ()() ()() ()()()()()0,, ,,cosh,,sinh, 2sinh, aEEfEE fEEaEErEEaEE aEE =+ (3-32) As with the Legendre representati on, the energy transfer probability is desired, rather than the angular distribution. If Eq (3-32) is integrated ov er all possible values of Eq. (3-32) reduces to 0(,) f EE This parameter can be obtained from the ENDF tapes. Quadrature Integration of the Transfer Source One of the difficulties of representing the flux with a fine group structure is characterizing the group to group transfer cross sections. To obtain the flux in a specific group, the contribution from all other possible groups must be calcula ted. This requirement makes a fine group approach increasingly computati onally expensive as the number of groups and reaction types is increased. Methods such as sub-moment expansion38 in CENTRM or the RABANL35 module in MC2 have been developed to mitigate the issue of tr ansfer source calculation. This work presents and alternate form of accounting for the tran sfer source via a nume rical approximation.

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58 The transfer source term is represented by the in tegral of Eq. (3-22) or Eq. (3-26) over all possible neutron energies. Because the transfer source is given as a definite integral of continuous functions, numerical integration is used to obtain the value of th e scattering source at a given energy. One possible technique for nu merically integrating a function is the Gauss Quadrature (GQ) method.42 This section gives a brief ove rview of the GQ method and its refinements from the general form to applicati on to the neutron transf er source given in the neutron transport equation. Overview of Gauss Quadrature Integration Given a general function, () f x, to be integrated over an independent variable x, the integrand may be expressed by Eq. (3-33). ()b a I dxfx = (3-33) The GQ method seeks to numerically obtain an accurate estimate of the integral by selecting the optimum points (i.e., nodes) and evaluating the weighted sum of the function values at these points. The Gauss Quadrature Theorem st ates that “[the nodes] of the n -point Gaussian quadrature formula are precisely the roots of the orthogonal polynomial for th e same interval and weighting function.”43 This condition is stat ed in Eq. (3-34) where ()n P x is the function to be integrated represented in polynomial form, ()m P x is a polynomial which is orthogonal to the function on the interval [ a b ], and ( ) wx is a weighting function.44 ()()()b nmnm adxPxPxwx= (3-34) If the condition of Eq. (3-34) is met, then Eq. (3-35) is true according to the Gaussian Quadrature Theorem.

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59 ()1n ii i I w f x== (3-35) The values of xi are the roots of the orthogonal polynomial and the values of wi are determined by assuming the function can be represented by a polynomial of order 21 n Š and solving for the undetermined coefficients. An exam ple of this procedure for a 2nd order GQ estimate using Legendre Polynomials is given in Ref. 42. The utility of this numerical approximation is that it allows accurate estimation of complicated functions via a minimum number of function evaluations. Other numerical integration schemes such as the Trapezoid Rule or Simpson’s Rule45 typically require a greater number of function evaluations to obtain the same accuracy. Limitations of Gauss Quadrature Integration One potential drawback of GQ is the irregular sp acing of the nodes in the interval of [ a,b ]. This makes implementation of GQ for evenly spaced tabular data less straightforward. If the data are given in tabular form, interpol ation must also be performed. For a GQ estimate using n nodes (i.e., of order n ), the GQ method can exactly estimate a function of order 21 n Š If a function requires a very hi gh order polynomial to be represented accurately, the benefit of the GQ method is dimini shed because more function evaluations would be required. In some exceptional cases, the trapezoid rule with fi ne enough spacing is less expensive than the GQ method for the same accuracy.42 The GQ method also requires in tegration over a finite interv al. The GQ method cannot be applied to functions with singular ities or integrals with infinite or semi-infinite bounds. Gauss-Legendre Quadrature The GQ method relies on the appr opriate selection of weights and nodes. These values are tied to the choice of orthogonal pol ynomial in Eq. (3-34). The use of Legendre Polynomials with

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60 ()1 wx = is a common polynomial selection. Legendre Polynomials are orthogonal on the interval [-1,+1] and follow simple recursion rela tionships that facilitate implementation in a computer program. In an order n Gauss-Legendre Quadrature (GLQ) appr oach, the nodes are determined from the roots of the order n Legendre Polynomial. The corresponding weights can be determined by the properties of Legendre Polynomials at a given order by th e method described by Hildebrand.46 Using this method, the weight for order n and node xi can be calculated via Eq. (3-36). ()()() 2 2 2 121 1i i nix w nPx+Š = + (3-36) The nodes and weights do not change based on th e function to be integrated. Therefore, the node-weight pairs for a given GLQ order can be st ored in a computer pr ogram without need for recalculation during each execution. The GLQ nodes and weights for orders 2 thru 6 are given in Table 3-1. Adjustment of Integration Limits For GLQ integration to be practical, the rest riction of integration only over the interval [-1,+1] must be removed by a change in variable su ch that the new variable is equal to -1 at the lower limit of a and +1 at the upper limit of b as shown in Eq. (3-37). ()()1 1 b adxfxdtft+ Š= (3-37) The substitution vari able and differential is defined in Eq s. (3-38) and (3-39), respectively. 22 baba tx Š+ =+ (3-38) 2 ba dtdx Š = (3-39)

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61 The GLQ formulation can now be a pplied to the definite integrati on of any function that satisfies the aforementioned restrictions by changing the independent variab le and adjusting the value of dx Gauss-Kronrod Error Estimation Since the GLQ method is accurate to order 21 n Š any integral estimate of a function of greater than order 21 n Š will retain some numerical error. On e way to estimate this error is to compare a higher-order estimate to current estimate Necessary adjustments can then be made to step size, number of points, etc. to re duce numerical error to acceptable levels. Kronrod47 proposed a method for estimating the error in Gauss-Legendre Quadrature, known as Gauss-Kronrod Quadrature (GKQ) also known as Gauss-Legendre-Kronrod Quadrature. Kronrod st arted with an order n Gauss Estimate as given in Eq. (3-40) and inserted 1 n + additional nodes to obtain the higherorder Kronrod estimate of Eq. (3-41). ()1 n nii iGw f x== (3-40) ()()1 21 11 nn niijj ij K afxbfy+ + ===+ (3-41) Kronrod determined the additional weights, bj, and nodes, yj, which would yield an estimate that is order 3 1 n + accurate by inserting 1 n + additional nodes. The loworder, high-order estimate used in a GKQ estimate is known as a Gauss-Kronrod pair. The (G7, K15) pair, a common set used for GKQ, is given in Table 3-2. The GKQ method reuses the nodes from the GL Q estimate. This makes the GKQ more efficient than comparing a high-order and loworder GLQ estimate, where the nodes cannot be reused due to the irregular spaci ng. For a GKQ estimate, a total of 21 n + function evaluations would yield a 3 1 n + order integral estimate and an error estimate. The same number of function

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62 evaluations with GLQ integration would yield the more accurate 41 n + order integral estimate but without any error estimate. The error estimate from GKQ can be used to create adaptive algorithms, which reduce error below a limiting value. An error estimate is necessary for the numerical algorithm to adapt to the input functions and mainta in error below acceptable levels. Neutron Transfer Source Calculation Using Gauss-Kronrod Quadrature The Neutron Slowing-Down Equation in an in finite medium is given in Eq. (3-42). ()()()()()ts E E EdEEEESE =+ (3-42) The scattering source term on the RHS of Eq. (3-42) is the candidate for a GKQ integration treatment. While scattering is typically the largest cont ribution to energy transfer, the scattering source term can be generalized into a transfer ke rnel for any reaction causing an energy transfer. A general form of Eq. (3-42) is given in Eq. (3 -43) where the transfer source is generalized to any reaction, x with a multiplicity of m ()()()()()()0 all tx xEEdEmEEEESE =+ (3-43) All non-flux terms in Eq. (3-42) can be dete rmined from ENDF data based on given data and scattering kinematics. These data can be in the lab or COM frame of reference and given as Legendre expansion coefficients, as tabulated prob ability density functions, or in Kalbach-Mann Systematics.40,41 The transfer kernel in this genera l form can include many types of energy transfer reactions, including elastic scatte ring, inelastic scatte ring, or (n,xn). At the highest possible energy in the system, there is no transf er contribution from higher energies. Therefore the init ial value of the flux can be determined from Eq. (3-44).

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63 () () () max max maxtSE E E = (3-44) From this point the calculation can sweep down to lower energi es using the higher energy flux values in the transfer source. An exception to this general slowing-down tr eatment is fission. Fission reactions occurring at low energies yield neutrons at higher energies. Typically, th e implementation of a source and eigenvalue iteration procedure is necessary to describe the system Therefore, the source term in Eq. (3-43) is left as a general source term that can be either a fixed s ource, fission source, or a combination of both. Bounds of Integration for Neutron Transfer The appropriate bounds of integration must be de termined to calculate an accurate transfer source for a given reaction. These bounds can be input based on the kinematics of the reaction or the input ENDF data. For a ll slowing-down reactions, the bounds of integration will be from the present energy, at a minimum, to the highe st possible energy based on the reaction. For the case of elastic scatte ring, the minimum energy occurs during a glancing collision with no energy transfer; therefore, the minimum energy would be th e present neutron energy. In a glancing collision, the scattering cosine in both the laboratory and COM systems is +1. For the opposite case of maximum energy transfer (i.e., full backscatter), the scattering cosine is -1. For the case of elastic (i.e., 0 Q =) scattering, inserting a scattering cosine of -1 into Eqs. (3-5) or (3-8) yields the following result. maxE E = (3-45) 21 1A A Š=+ (3-46)

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64 The case of inelastic scattering follows the same procedure. Again the minimum and maximum energies are obtained by substitution of -1 and +1 into the kinematic equations. In this case the result is Eqs. (3-47) and (3-48). () () () 2 max 21 121 1 1QA E QA E AA AEA A Š =++ŠŠŠ Š (3-47) () () () 2 min 21 121 1 1 QA E QA E AA AEA A Š =+ŠŠŠ Š Š (3-48) For reactions where the kinematics are not easi ly described (e.g., continuum inelastic and (n,xn)) the available ENDF data must be used to determine the bounds of in tegration. In these cases, the ENDF evaluations typically include a secondary energy distribu tion that is either tabulated, given as parameters of a Legendre ex pansion, or as Kalbach-Mann parameters. The minimum and maximum energies are then determined by the in terpolated secondary energy distribution at a given in cident neutron energy. Numerical Form of the Neut ron Slowing-Down Equation With the bounds of integration determined and using the GKQ method for treating the transfer source, the Neutron Slowin g-Down Equation at a given energy E is expressed in Eq. (3-49). () () ()()()1 max,min, all111 ,, 2nn xx ixijxj xij tEE EafEEbfEESE E + == Š =++ (3-49) The i nodes and ia weights correspond to the Ga uss-Legendre estimate and the j nodes and j b weights to the higher-order Kronrod estim ate. The function evaluated by GKQ at i or j for reaction x is given in Eq. (3-50) with (,)xi g EE being the transfer function from i E to E determined either from scattering kinematics or directly from the given ENDF data. ()()()()(),,xixixixii f EEmEE g EEE = (3-50)

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65 The i E nodes are determined via Eq. (3-51), where xi are the corresponding nodes in the range of [-1,+1] given by the selected Gauss-Kronrod pair. maxminmaxmin22iiEEEE E x Š+ =+ (3-51) With the exception of the flux, the values in Eqs. (3-49) and (3-50) are all known. No numerical approximations other than those discu ssed in the GKQ approach have been used to this point. While a number of important assumpti ons were made to arrive at this slowing-down equation, this form is intended to describe th e energy behavior in a reactor system for the purposes of generating group constants and not fo r a full description of neutron behavior in energy, space, and time. The GKQ form of the slowing-down equation can be solved on a given pointwise (PW) mesh of energy points such that the resulting flux points are linea rly interpolatable to within a specified numerical error. Assuming no error is introduced by the re duction of the neutron transport equation to the slowing-down form, the following are the major sources of error in the resulting PW flux. 1. Error in the input nuclear data parameters. Wh ile potentially substantia l, this error cannot be controlled by solution of the slowing-dow n equation. Error can also arise from interpolation of parameters such as cross sect ions, transfer probabilities, and multiplicity. 2. Error from the GKQ integration technique. This error is controllable via adjustments in the bounds of integration or orde r of the GKQ estimate. 3. Error from the linearization of the PW flux. This error can be c ontrolled by applying the appropriate energy mesh. This mesh is often best informed by the underlying nuclear data, which are also linearly interpolatable to within a given error. 4. Any error resulting from numerical limitati ons of a computer. This could include floating-point precision error or truncation. The methods developed in this chapter have particular applicability to fast spectrum systems where the flux at each energy is a f unction of only higher energy fluxes. Chapter 4

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66 describes the implementation of this PW sp ectrum generation methodol ogy into a computer program. The algorithms used to control error in the transfer s ource, determine the energy mesh, process the ENDF data, etc. wi ll be described for the implem entation of these methods.

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67 Table 3-1. Nodes and weights fo r Gauss-Legendre Quadrature of order 2 to 6. Values from Table 22.1 of Chapra and Canale.42 Order Nodes Weights 2 0.577350269 1.0000000 3 0.774596669 0.000000000 0.5555556 0.8888889 4 0.861136312 0.339981044 0.3478548 0.6521452 5 0.906179846 0.538469310 0.000000000 0.2369269 0.4786287 0.5688889 6 0.932469514 0.661209386 0.238619186 0.1713245 0.3607616 0.4679139 Table 3-2. Nodes and weights for the (G7, K15) Gauss-Kronrod pair. Va lues from Table 5.3 of Kahaner, Moler, and Nash.48 Set Nodes Weights 7-point Gauss 0.9491079123 0.7415311856 0.4058451514 0.0000000000 0.1294849662 0.2797053915 0.3818300505 0.4179591837 15-Point Kronrod 0.9915453711 0.9491079123 0.8648644234 0.7415311856 0.5860672355 0.4058451514 0.0000000000 0.0229353220 0.0630920926 0.1047900103 0.1690047266 0.1903505781 0.2044329401 0.2094821411

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68 Figure 3-1. Scattering kinematic vectors. A) In the laboratory system. B) In the center-of-mass system. C) A vector diagram of resulta nt velocities and center-of-mass velocity. A B C 0 mv0, M V 0 mv 0 0 ,cmvc c ,cmv c M V ,c M V 0 c com v 0 v c v x y x y

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69 CHAPTER 4 IMPLEMENTATION OF ADAPTI VE QUADRATURE METHODS The methods developed in the previous chap ter have been implemented in a computer program called Pointwise Fast Spectrum Genera tor (PWFSG). This program is intended to demonstrate an adaptive GKQ integration transfer kernel treatment. PWFSG solves the Neutron Slowing-Down Equa tion in an infinite medium. The flux obtained by PWFSG is point wise (PW) and linearly interpolatab le. A high-level diagram of the inputs and outputs of PWFSG is shown in Fig. 4-1. ENDF Input Data Nuclear data, including cross sections and energy-angular di stributions, are collected and distributed by a number of institutions as Evaluated Nuclear Data Files (ENDF). PWFSG draws fundamental nuclear data from these ENDF “tapes ” which are ASCII text files that follow the ENDF-6 format.11 The ENDF-6 format specifies several types of 80 character lines of text, known as “cards,” that are structured to facilitate reading by FO RTRAN programs. The last 14 characters of each card are reserved for a four char acter material identifier (MAT), a two character file identifier (MF), a three character section id entifier (MT), and a five charac ter line number. These control characters are used by processing programs to determine their location in an ENDF tape. The first 66 non-control characters are the portion of the reco rd that contains the nuclear data. Most card types contain 6 entries of 11 char acter width followed by the control characters. The exception to this is the TEXT record, whic h contains one entry of 66 character width followed by the control characters. The mix of in teger and floating point data on each card is dependent on the type of record. Control (C ONT) records contain two floating point entries

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70 followed by four integer entries. The two fl oating point entries typi cally contain data; the remaining integer data are typically control flags or a count of the num ber of entries that follow. Tabulated data are represented in the ENDF -6 standard by the LIST, TAB1, and TAB2 record types. These are a series of multiple records which contain CONT records followed by floating point data. The CONT r ecords provide control data about the following tabulated data. The tabulated data are then gi ven as a simple list of values (LIST records), tabulated (), x y data (TAB1 records), or a collection of LIST or TAB1 records at differe nt values of an independent variable (TAB2 records). Preprocessing with NJOY In most instances, the ENDF-6 formatted data cannot be used directly in nuclear transport programs because the cross sections are given at one temperature (typically 0 K) and the resolved resonance energy region cr oss sections are given as res onance parameters. Therefore, the computer program NJOY22 is commonly used to perform preprocessing of the ENDF data. The first use of NJOY in PWFSG is processi ng the resonance data provided in the ENDF tapes. Isotopes that contain a large number of resonances (e.g., U-238) do not typically contain tabulated data for the entire res onance range. Instead the resonan ce data are stored as resonance parameters. These parameters can be in many fo rms such as Single or Multi-level Breit-Wigner, Reich-Moore, Alder-Alder, or Limited R-matrix.11 The purpose of resonance parameters is to keep the ENDF files more compact. A resonance parameter representation limits the required energy resolution to represent rapidly varying cross sections around a resonance. Resonance parameters are widely used in the more re cent ENDF evaluations, such as ENDF/B-VII.0, for medium and heavy mass isotopes that contain many resonances.8

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71 If a nuclear transport program, such as PWFSG, needs a PW representation of the cross section, the resonance parameter data must be converted into tabulated interpolatable cross sections. The resonances must also be broadened to the te mperature of the material to account for Doppler Broadening. PWFSG relies on NJOY to convert resona nce parameters into Doppler-broadened, tabulated cross sections. Th e NJOY modules RECONR and BROADR are used to reconstruct and broaden the resonances, respectively. The ou tput of these modules is a PENDF (pointwise ENDF) tape that follows the ENDF-6 standard an d contains linearly-interpolatable cross section data generated by reconstructing a nd broadening the resonances. The temperatures of the isotopes input into PWFSG are not required to be consistent. NJOY can process one material to a specific temperature and anot her material to a different temperature. The infinite medium calculation th en uses the cross sections and number densities provided by NJOY to determine the reaction prob abilities without the need to account for temperature effects. Therefore, a homogenous mixture of hot fu el could be mixed with cold coolant and simulated in PWFSG by pr oviding the appropriate PENDF tapes. ENDF Data Processing Overview of EndfReader A dynamic link library, EndfReader, was deve loped to facilitate the reading and interpolation of ENDF and PENDF data for PWFSG. The EndfReader library is based on accessing ENDF data using the standards provided in the ENDF-6 format specification. The EndfReader library read s ENDF formatted records such as: TEXT, CONT, LIST, TAB1, and TAB2. It scans ENDF tapes and breaks th em into materials, files, and section objects based on the MAT, MF, and MT tags, respectivel y. These components can then be accessed by a calling program such as PWFSG. The actual E NDF data are not processed at the time of the

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72 scan; instead, they are sorted into a data structure for later access to eliminate computational cost in processing large ENDF sections not used by the calling program. Due to the dynamic storage of ENDF records, the calling program is used to seek and rewind the ENDF tapes rather than EndfReader. A calling program will access a given section from a combination of MAT, MF, and MT numb er. It will then pro cess the section’s HEAD record and possibly the following CONT records to determine the number and type of records to follow. The calling program, based on the type of ENDF section being read, will make the appropriate calls to EndfReader instructing the types and numbe r of records to process next. These method calls will cause Endf Reader to seek forward in the ACSII formatted text contained in the ENDF section and create library objects representing record s of type CONT, TAB1, etc. Therefore, the EndfReader class library provides a layer of abstrac tion that removes the need for the calling program to process CONT, TAB1, etc. records, while the calling program directs the reading of the records an d interprets those record s into physical quantities. Interpolation within EndfReader The library objects made availa ble by EndfReader offer the ability to access ENDF data directly. There is also interpolation functiona lity for TAB and LIST records. TAB records, which contain tabulated two-dime nsional (TAB1) or three-dime nsional (TAB2) data, can be interpolated by their asso ciated interpolation law.11 LIST records, which c ontain lists of data at given independent variables (e.g., a list of Legendre coefficients at specific energies), can be interpolated to yield a new LIST record at an interpolat ed independent variable. EndfReader supports the five one-dimensi onal interpolation laws used by ENDF: 1. Constant (histogram) interpolation 2. Linear-linear interpolation 3. Linear-log interpolation 4. Log-linear interpolation 5. Log-log interpolation

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73 EndfReader interpolates depe ndent variables (e.g., cross se ction) using the appropriate interpolation law at a given inde pendent variable (e.g., energy). These dependent variables can be a single value (e.g., cross section at an energy) or a set of values (e.g., Legendre Coefficients at an energy). This functionality allows Endf Reader to present continuous data to the calling program. Binary Search Algorithm for Interpolation A PWFSG calculation spends a significant portion of its processing time on the interpolation of cross sections, tr ansfer probabilities, etc. To improve the efficiency of this interpolation, a bina ry search algorithm49 was implemented in EndfReader. A binary search algorithm operates on sorted lists of data by compar ing the middle of the list to the value sought. If the value sought is greater than th e middle value of th e list, then the lower half of the list is ignored and the mi ddle becomes the new lower bound and vice versa. This process is performed recursively until th e value sought is found. The binary search algorithm will converge logarithmically on the desire d value leading to sign ificant computational benefits when working with large sets of data. A small modification was made to the Endf Reader binary search algorithm. The continuous values sought by the calling PWFSG progr am are rarely the exact values tabulated in the ENDF data. Therefore, the EndfReader seeks th e interval where the value lies rather than the exact value itself. Once the algorithm has found th e desired interval, inte rpolation is performed on the ENDF data using the normal interpolation procedure. Required Input To perform the infinite medium slowingdown calculation, PWFSG requires continuous nuclear data (e.g., cross sections, transfer probabilities, etc.), the relative number densities of the

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74 isotopes in the system, and the source distributio n. Since the solution for an infinite medium, no geometrical information is requir ed. A diagram of the PWFSG input s is shown in Fig. 4-1. The continuous nuclear data for PWFSG are provided through the E ndfReader library. The source distribution and number densities must be specified in user input. When resonance parameters are used or the cross sections require broadening to a specific temperature, EndfReader requires a PENDF tape processed by NJOY in addition to the original ENDF tape. Solution Algorithm Figure 4-2 displays the solu tion algorithm used in PWFSG. PWFSG begins with ENDF and PENDF data and a given sour ce distribution. The PWFSG solu tion procedure begins at the highest source energy and solv es for the flux directly. () () () max max max tSE E E = (4-1) The flux can be solved directly at max E because no other flux values ar e needed (i.e., the transfer kernel is zero). PWFSG assumes that the source is not coupled to the flux as in an eigenvalue calculation. PWFSG is only able to handle fixed source calc ulations, although the input source can be a fission spectrum. This same consideration is true for thermal scattering treatment. Once the first flux point is determined, the flux solution algorithm progresses to lower energies. Each successive flux point is determ ined by the numerical fo rm of the slowing-down equation with GKQ integration developed in the pr evious chapter. The form of the slowingdown equation used in PWFSG is shown in Eqs. (4-2), (4-3), and (4-4), where the subscript k refers to isotope, subscript x to reaction, and subscript n to energy point. () () ()()()1 ,, allall11 ,1 ,, 2nn xk nixkinjxkjnn kxij tknE E afEEbfEESE E + == =++ (4-2)

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75 ( ) ( ) ( ) ( ) ( ),,,,,, x kinxkixkixkini f EEmEEgEEE = (4-3) ,max,,min,, x kxkxkEEE =Š (4-4) The total cross section at En is interpolated via EndfReader To calculate the transfer source, EndfReader interpolates the transfer cr oss section, multiplicity, and transfer probability for each reaction included in the ENDF eval uation to the appropriate node given by the Gauss-Kronrod pair. PWFSG uses the nodes of the (G7, K15) pair given in Table 3-2 to determine the energies, i E or j E for these evaluations. The flux term given in Eq. (4-3) is determined by interpolation of th e previously solved flux points. Gauss-Kronrod Transfer Treatment The GKQ method is used in Eq. (4-2) to calc ulate the transfer source. The function evaluations of Eq. (4-3) cons titute the node values with ia and j b being the weighting coefficients. For each reaction, the GKQ method is used to calculate the neutron transfer source to n E The GKQ method provides an error estimate for the integration of the transfer source. If the error is above a set tolerance, the integration ra nge for the GKQ treatment is divided in half and treated as two separate regions. The interval s are continually subdivided until the integration error is below the given tolera nce. This process is graphica lly depicted in Fig. 4-3. Energy Mesh Determination The energy mesh used for PWFSG is based on the total cross section of the system. First, a union of all the total cross s ection evaluation energies for every isotope in the system is generated. Following the creati on of a union mesh, the total cro ss section at each point in the union mesh is calculated by summing the macroscopic cross sections.

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76 After the creation of the union mesh, the me sh is thinned by removing any unnecessary points. During the mesh thinning process, an en ergy point is removed if the total cross section value at that point could be calcu lated via linear interpolation within a given error. This thinning can reduce the number of transport calculatio ns significantly. The energy mesh of U-238 contains 129,402 points when pr ocessed with the RECONR and BROADR modules of NJOY to an error tolerance of 0.1%. Th e thinning process of PWFSG usi ng an error tolerance of 0.1% yields an energy mesh of 43,247 poi nts (67% reduction). If a less stringent tolerance of 1% is used the number of mesh points is reduced even further to 21,924 points (83% reduction). The result of the thinning process in CENTRM yields a similar reduction.37 The choice of error tolerance is a balance of accuracy versus computational time and is left to the discretion of the user. One exception to the thinning is dete rmined by the following criterion. ()max1 nnEEE+Š (4-5) max max1 1 EE =Š (4-6) Equation (4-6) gives the maximu m possible energy change via an elastic scattering event. Elastic scattering reactions cause less energy change in the incident neutron than inelastic scatter and other reactions. The criterion of Eq. (4-5) states that the differe nce between the current energy point and the next energy point must be less than some fraction, of the maximum possible energy loss by elastic scat ter. This ensures that changes in the flux due to energy transfer are also included and not just t hose due to the total cross section.

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77 While PWFSG thins energy mesh, it tests the criteri on of Eq. (4-5). If the test fails, a new energy point is added at max() E In the example of the U-238 mesh, a total of 7158 points are added to the mesh. These points are include d in the aforementione d thinning results. Parallelization of the Isotope Transfer Source Calculation Each isotope in the summation term of Eq. (4-2) contributes separately to the overall transfer source. The summation of the neutron tr ansfer from each individual isotope results in the total transfer to the present energy, n E Due to the number of interpolations and GKQ evaluations, this transfer probability calculation is typically the most computationally intensive portion of the PWFSG calculation. In PWFSG, the transfer source summation term is broken into different process threads. For multi-core processing each core computes th e transfer source for a given isotope. The threads rejoin to sum the total transfer s ource which is used for the flux calculation. For cases where there are only a few isotopes, this parallelization does not yield any appreciable speedup. However, for cases with many isotopes, speedups of greater than 30% have been observed. Flux Iteration Procedure The GKQ procedure used during the PWFSG solution can require a flux at a point between n E and 1 n E +. Linear interpolation is normally used to obtain the flux for the GKQ procedure. In this case the flux is not known at En, so an interpolation cannot be performed between n E and 1 n E +. A flux extrapolation and iterati on procedure is used in PWFSG to predict fluxes between n E and 1 n E + (see Fig. 4-4). Prior to solving the flux at n E the flux at n E is predicted by a quadratic extrapolation. A quadratic fit to the last three solution point s is generated and the flux at n E is

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78 estimated via this quadratic fit. The actual flux at n E is then solved using the normal PWFSG procedure given in Eqs. (4-2) and (4-3), with the result compared to the estimate. If the difference is below a given error, the solution pro cedure continues to the next point. If the error is not acceptable, the estimate is replaced with the value calculated via Eqs. (4-2) and (4-3). This process is repeated until the flux at n E is converged. Typically, only one or two nodes lie within the interval of n E to 1 n E +. These points are also the lowest weight points from the GKQ algorithm (see Table 3-2). Therefore, typically only one or two iterations are needed for convergence. Wh en the solution is near the source energy, there are few higher energy flux values to draw from, so a greater than average number of iterations may be performed. In addition to the extrapolation/iteration pro cedure, an additional check is performed to ensure convergence. In the even t that the error starts to incr ease and the solution begins to diverge an extra energy point is inserted halfway between n E and 1 n E +. The solution procedure restarts with this new energy point as n E Outputs The output of PWFSG is a linearly-interpolatable PW flux. This flux represents a continuous function in energy and can therefore be used directly to produce group constants, according to Eq. (4-7) below. ()() () 1 1,g g g gE x E xg E EdEEE dEE Š Š= (4-7) An accurate flux energy spectrum will result in accurate energy dependence in the resulting group constants. PWFSG produces th is flux spectrum with high accuracy in energy

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79 using the GKQ integration procedure for the transf er source. PWFSG can then be coupled with another process to account for spatial effects as needed.

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80 Figure 4-1. Overall program/ data flow for PWFSG program ENDF/PENDF Data Files Source Distribution Number Densities EndfReader Library PWFSG Interpolated Data Parameters (cross sections, transfer probabilities, etc.) Pointwise Flux Distribution

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81 Figure 4-2. PWFSG solution algorithm Yes Process/check input Solve flux Write pointwise flux output Yes Yes Subdivide integration domain Integration error acceptable? Point converged? Update guess of Guess via quadratic extrapolation Determine flux at Emax Last energy point? No Advance to next energy point in mesh, En No No END Generate energy mesh Pointwise Flux START Source distribution Number densities ENDF/ PENDF tapes Thin energy mesh ()nE ()nE ()nE

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82 Figure 4-3. Example of how the GKQ algorithm breaks up inte gration bounds until error is acceptable. In this example, the GKQ algor ithm would attempt integration between Emin and Emax first. When that error is unaccep table it would attempt integration from Emin to E1 and E1 to Emax, where E1 is the halfway point between Emin and Emax. In this case, the error from the integration from E1 to Emax (region 4) would be acceptable and subdivision would cease. The lower half would continue subdividing into regions 1, 2, and 3 until the integration error in all regions is acceptable. 1 3 4E 2 EminE3E2E1Emax

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83 Figure 4-4. Example of flux estimation-iteration procedure used in PWFSG. The points En +1, En +2, and En +3 are used to estimate the value at En via a quadratic extrapolation, shown above as ()nest E When the GKQ points (denoted by x) are calculated, this estimate is used for the region where the fl ux is not known. Upon the first iteration the flux estimate is updated with the cal culated value which also updates the interpolated GKQ node points. This procedure is repeated until the valu e of the flux converges on the actual value, shown above as ()nact E EEn+ 1En En+ 2En+ 3 ()n estE ()n actE ()E

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84 CHAPTER 5 PWFSG VALIDATION Overview To quantify the utility of the GKQ algorithm, the PWFSG program was executed with three example cases and the results compared to those of other solution techniques. PWFSG results were compared to the high-f idelity Monte Carlo program, MCNP5.1.5150 and another deterministic spectrum generation program, C ontinuous Energy Transport Module (CENTRM).37 CENTRM uses a hybrid of pointwi se (PW) and multigroup (MG) flux solutions combined with a sub-moment expansion of the scattering source.36,38 The homogenous infinite medium PW flux solution algorithm used by CENTRM is similar to PWFSG with the exception the scattering source term (i.e., sub-moment expa nsion versus GKQ integration). The selected example problems are intended to represent a sodium fast reactor (SFR). The first example problem is an infinite medium of U-238. The second is a homogenized infinite medium with isotopic co mpositions representing a typical SFR pin. The third example is an infinite medium with isotopic compositions repres enting recycled nuclear fuel from a light water reactor (LWR) placed in a SFR. All MCNP cases simulated 5 billion particle histories using cust om weight-windows for particle population control. The weight window treatment was included to reduce MCNP stochastic error by contro lling particle population in the resonance region. The PWFSG and MCNP cases were executed on an 8-way Intel Xeon X5550 2.67 GHz processor that supported symmetric multi-proces sing and hyperthreading. This architecture allowed for up to 16 separate processes to be executed in parallel. MCNP utilized this architecture by running particle hi stories in parallel. PWFSG used this architecture in a more limited sense by calculating the transfer source for different isotopes on separate process threads;

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85 however, this speedup was only realized on cases with multiple isotopes. CENTRM did not support any parallelization and was run on Intel Core™ Duo 2.00 GHz architecture with SCALE utility scripts. PWFSG and CENTRM only support fixed sour ce distributions. To facilitate program-to-program comparison, every system was modeled with a fixed fission source distribution representing the ener gy distribution of fissi on neutrons from Pu-239. This spectrum is given by the Watt fi ssion distribution where E is given in units of MeV. ()()()1/2exp/sinh SECEabE =Š (5-1) For Pu-239 the parameters a and b are 0.966 and 2.842, respectively, with C as an arbitrary scaling factor. The fission neutron production in MCNP was disabled using the NONU card, with the above analytic expression specified on th e MCNP SDEF card. A plot of this source is shown in Fig. 5-1. CENTRM requires the neutron source to be given groupwise. Equation (5-1) was integrated over the 238 SCALE energy group struct ure and the integral of the source over each energy group was input into CENTRM as re quired in section F18.5 of Ref. 37. PWFSG was developed with a s ource input feature similar to MCNP. The type of fixed source spectrum and any spectrum-specific paramete rs are specified in the input. PWFSG then uses the analytic expression for the described fixed source spectrum. For this case, the Watt spectrum with the same a and b parameters were given in the input causing PWFSG to calculate the source at a given energy via in Eq. (5-1). All three programs used ENDF/B-VII.0 nuclear data in their calculation process. PWFSG used the native ENDF/B-VII.0 files and PENDF files produced by NJOY. ACE tables, also created by using NJOY from ENDF/B-VII.0 data files were used in MCNP. CENTRM uses the

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86 appropriate SCALE modules to pre-process the ENDF/B-VII.0 data into the required format. In particular the CENTRM process involved the execution of th e Bondarenko AMPX Interpolator (BONAMI)23 module, to generate the shielding MG cro ss sections as well as Code to Read And Write Data for Discretized solution (CRAWDAD)51 to generate the PW libraries used by CENTRM. The same energy mesh was used in all three programs so that quantitative comparison of the different spectra was possible. PWFSG and CENTRM use a similar algorithm for determine the appropriate energy mesh, while MCNP requires the energy mesh be input prior to execution. To obtain the same mesh for all three programs, PWFSG was run first to generate an energy mesh. This mesh was input into MCNP. Any differences in the CENTRM energy mesh were adapted to the PWFSG energy mesh via linear inte rpolation. This adaption is possible because “a linear variation of the flux between energy points is assumed” in CENTRM as noted in Section F18.1.1 of Ref 37. Spectrum Normalization Each program has a different internal pr ocedure for normalizing the flux spectrum; therefore, each output spectrum must be scaled so that program-to-progr am comparisons can be made. Once all three spectra where normalized appropriately, the residual errors between PWFSG and MCNP and between CENTRM and MCNP were calculated. The MCNP spectrum is assumed to be the true spectrum due to the high-fidelity of the Monte Carlo method, th e thorough verification an d validation of MCNP,52 and 5 billion particle histories tracked. The MCNP flux values were divided by thei r respective energy mesh interval size and scaled by an arbitrary value to place most of the flux values between 0 and 100, as shown in Eq. (5-2). Dividing by the ener gy mesh interval size was necessa ry because MCNP flux values

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87 are integrated over the energy mesh during tallying. This yields flux values in units of 1/cm2-s. This division converts the inte grated MCNP flux to its PW equivalent with units of 1/cm2-eV-s. () ,1,iMCNP iMCNP iMCNPiMCNPC EE Š = Š (5-2) The PWFSG and CENTRM results were then scal ed and shifted to obtain the closest fit to the MCNP results. The scali ng and shifting of flux energy spectrum is possible because the spectrum is simply a weighting function for the ge neration of group constants. This scaling and shifting will not have an imp act on the resulting group constants because the weighting spectrum is normalized during the group constant generation proce ss (See Eq. (4-7)). The flux points from PWFSG and CENTRM were s caled and shifted, as in Eq. (5-3), using constant values of A and B so that the spectral shape was preserved. ii A B =+ (5-3) The constants A and B were determined by minimizi ng the sums of the squares, Sr, of the residuals between MCNP and the respective program. ()2 ,,/ riMCNPiPWFSGCENTRM iS=Š (5-4) This minimization was performed using a non linear Generalized Reduced Gradient (GRG) optimization algorithm,53 subject to two constraints. First, the flux must remain positive over the region of comparison. Second, the mean of the residuals must be zero. Error Calculation The standard deviation of the residuals gives a measure of the variation of a program’s results about the MCNP data for the cases studied. For comparison between CENTRM and PWFSG results, a coefficients of determination (r2) were calculated as in Eq. (5-5).54

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88 2 tr tSS r S Š = (5-5) ()2 tiMCNPMCNP iS=Š (5-6) The St parameter is a measure of the total varia tion in the measured data. The purpose of the regression model is to then explain this va riation with a functional dependence (e.g., a linear function). In this application of regression analysis, the MCNP results take the role of the experimental data, with the model taken as the results from PWFSG or CENTRM. An r2 value of 100% would indi cate that a PWFSG or CENTRM model exactly matches the MCNP data and explains all of th e variation observed in MCNP. The r2 value is normalized by St allowing for comparison betw een the different cases. An improvement factor, F, was calculated via Eq. (5-7). () () 2 21 1CENTRM PWFSGr F r Š = Š (5-7) This improvement factor quantifies the improveme nt of the PWFSG match to the MCNP results compared to CENTRM. U-238 Infinite Medium Case Model Description The first example case is an infinite medium of U-238. The use of a single isotope in this case is intended to give a straightforward co mparison of PWFSG to CENTRM and MCNP. Only one isotope influences the cross section and energy transfer behavi or in the system, which allows for more direct inference on what is affecting the flux spectra calculated by the three programs. The infinite medium of U-238 is mode led at a temperature of 900 K with temperature-appropriate cross sectio ns. A number density of 1.0 wa s input into all the programs.

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89 For a single isotope in an infin ite medium, the number density onl y scales the final spectrum and does not affect its shape. Results The execution times required for the U-238 infin ite medium case by the three programs are summarized in Table 5-1. CENTRM and PWFSG executed in just a few minutes while MCNP required almost two days. The nor malized spectra are given in Fig. 5-2. Zoomed views of the resonance regions of the spect ra are given in Fig. 5-3. A plot of the MCNP relative error from th is simulation is shown in Fig. 5-4. The comparison of the flux spectra was truncated to between 1 keV and 10 MeV. This energy range includes the majority of the flux peak, as shown in Fig. 5-2, a nd had MCNP relative error below 1%. There is generally good agreement among the th ree programs. A plot of the PWFSG and CENTRM residuals (i.e., absolute error) compared to MCNP is shown in Fig. 5-5. The standard deviation of the residuals as well as the r2 values are shown in Table 5-2. Both programs have r2 values very close to 100% i ndicating good agreement with MCNP. The improvement factor for this case was 6.4, indicating that PWFSG pr ovides a significant impr ovement over CENTRM. The main region of discrepancy between the prog rams is in the unresolved resonance region (20 keV to 149 keV), where resonances exist but ar e unable to be measured experimentally. An upper PW cutoff of 25 keV was input into CENTRM to include the resolved resonance range of U-238. Above this point, CENTRM relies on the BONAMI module of SCALE to generate the self-shielded MG cr oss sections. The errors obser ved in CENTRM in this region are likely from the simplified Bondarenko approach in BONAMI. To test the contribution of the Bondarenko approach to the CENTRM results, the PW boundary was moved to the upper comparison boundary of 10 MeV. The new results are shown

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90 in Fig. 5-6. The standard deviation of the resi duals increased from 0.8240 to 4.20, the r2 value decreased from 99.65% to 90.86%, and the improveme nt factor increased to 166.3. There was also some non-physical variati on in the CENTRM spectrum at higher energies, which may be caused by a numerical issue. The agreement between CENTRM and MCNP for the CENTRM PW case was likely decreased due to the kinematic assumptions in CENTRM. From page F18.1.3 of the CENTRM manual it is noted that: Within the epithermal PW range, the slowing-do wn source due to elastic and discrete-level inelastic reactions is computed using the PW flux and the rigorous scatter kernel based upon the neutron kinematic relati ons for s-wave scattering. C ontinuum inelastic scatter is approximated by an analytical evaporation spec trum, assumed isotropic in the laboratory system.37 The assumptions of isotropic sc attering in the center-of-mass coordinate system for elastic and discrete-level inelastic scatte r (i.e., s-wave based scatter) and isotropic scattering in the laboratory system for continuum inelastic scatte r may cause large errors at higher energies. Scattering dominates in general fo r U-238 at high energies as show n in Fig. 5-7. Above 1 MeV, elastic and inelastic scattering cross sections cont ribute comparably to the total cross section. Above 2-3 MeV, almost half of the inelastic cont ribution is from continuu m inelastic. At these energies, the scattering can show strong angular dependence as shown in Fig. 5-8. These dependences are not well-represen ted with assumed isotropy. PWFSG has excellent agreement with MC NP throughout the comparison energy range with an r2 value of 99.95%. An examination of Fi g. 5-5 reveals that, as with CENTRM, the main discrepancy is in the unresolved resonanc e energy range. PWFSG currently has no method for including unresolved resonance contributions, where MCNP uses a probability table method. The infinite medium U-238 case was execute d again in MCNP w ith the unresolved resonance probability table method disabled. The spectra from this result are shown in Fig. 5-9

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91 and summarized in Table 5-3. The differen ce at the unresolved energies is reduced; however the r2 statistic and the standard de viation are changed only sligh tly. The improvement factor decreased slightly to a value of 5.4. For comparison of the differences between thes e two cases it is helpful to calculate the z statistic for the residuals. The z statistic for a given residual is the value of the residual minus the mean of the residuals and normalized by the standard deviation of the residuals as in Eq. (5-8). Due to the spectrum normalization the mean of the residuals, is zero for this analysis. i iz Š = (5-8) A plot of the z values versus energy for the unresolve d and no-unresolved cases is given in Fig. 5-10. A histogram of the z statistic values is given in Fig. 5-11. Even though the descriptive statistics do not show a distinct improvement for the no-unresolved case, it is clear from the histogram that the variation about the MCNP re sults is more evenly distributed for the case without unresolved resonances. The case with the unresolved res onances included has a stronger bias towards lower values that skews the histogr am. Despite the lack of improvement in the r2 statistic, the case wit hout unresolved resonances seems to exhibit less bias and a more random distribution of the residuals a nd is thus considered a better match to the MCNP data. Based on the results for the U-238 case as well as subsequent studies, it is concluded that both PWFSG and CENTRM provide good agre ement with MCNP results, yielding r2 values over 99% in their nominal usage. In this case, th e main source of CENTRM error is the scattering kinematics assumptions in the PW region. The main source of error for PWFSG is due to lack of unresolved resonance treatment. Other PWFSG error is likely due to statistical variation in the MCNP results and numerical error in PWFSG.

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92 SFR Fuel Cell Case Model Description The second example case is an SFR fuel cell in a hexagonal lattice as shown in Fig. 5-12. This example provides a more realistic scenario to test the effects of multiple isotopes with different resonance energies and energy transf er behavior. The entire fuel cell was modeled at a te mperature of 900 K, because CENTRM cannot model materials in the same zone at different temperatures. PWFSG and MCNP do not share this limitation, but were also modele d at 900 K for consistency. The fuel in the modeled SFR pin was a meta llic alloy of uranium and zirconium with 10 wt% zirconium (U-10Zr), with the uranium 15% enriched in U-235. The cladding was modeled as a simplified version of HT9, a co mmon choice for SFR cladding due to its corrosion and swelling resi stant properties.55 The composition, based on Table 1 of Ref. 55, was simplified as 12 wt% chromium, 1 wt% moly bdenum, and 87 wt% iron. The coolant was modeled as liquid sodium with a density of 0.874 g/cm3. The theoretical densities of the metal alloys were determined using Eq. (5-9) for a metal alloy with constituent elemental densities of i and weight fractions wi. 11i i iw = (5-9) The metal fuel alloy is assumed to be fabricated with this theoretical density with a gap between the fuel and cladding. Upon reaching operationa l temperatures, the fuel expands until making contact with the inner su rface of the cladding. This expansion lowers the fuel density. The fuel density used in the number dens ity calculations was adjusted to be 75% of the theoretical calculated from Eq. (5-9).56 While this adjustment does not change the overall number of the

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93 fuel atoms (i.e., density is lowered but area fr action increases proportionally to compensate), it does serve to remove the fuel-cladding gap. This condition is more characteristic of an actual SFR fuel pin at operational temperatures and also provides a simpler model. The fuel cell geometry and ma terials of Fig. 5-12 were hom ogenized into an infinite medium. The area fractions resulting from the ge ometry were used to scale the number densities for the fuel, cladding and coolant. The final number densities of the isotopes in the SFR fuel cell are given in Table 5-4. Results The execution times required for the SFR fuel cell case are summarized with the other cases in Table 5-1. The CENTRM time include d approximately 11 minutes of calculation time for CENTRM itself and the remainder in lib rary pre-processing by BONAMI, CRAWDAD, and other SCALE utility modules. Similar to th e U-238 infinite medium case, PWFSG and CENTRM required just over 10 mi nutes, while MCNP required al most a week of processing time. The normalized spectra for the SFR fuel cell are shown in Fig. 5-13 with a plot of the resonance region of the spectra in Fig. 5-14. A plot of the MCNP relative error for the SFR cell is shown in Fig. 515. For this case the comparison of the flux spectra was performed between 400 eV and 10 MeV. As with the U-238 infinite medium case, this truncation was based on selecting the region with MCNP relative error below 1% while including the majority of the flux peak. The comparison statistics for the SFR cell cas e are given in Table 5-5. The agreement between MCNP and PWFSG decreased slightly from Case 1 but the r2 statistic indicates a good fit with a value of 98.41%. The CENTRM results show more of a discrepancy for the SFR cell with an r2 of 87.54%. The improvement factor is 7. 8, which indicates a greater benefit of PWFSG over CENTRM when compared to the U-238 case.

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94 A plot of the CENTRM and PWFS G residuals compared to MCNP is shown in Fig. 5-16. The issues identified in the U-238 case again contribute to error in the SFR cell case. For this case, the CENTRM PW cutoff was set to 10 MeV to capture the resonance region of sodium and the structural materials. This PW cutoff cause d the inclusion of a si gnificant amount of high mass, anisotropic scattering isotopes in the PW solution between 100 keV to 10 MeV range. This inclusion caused an over-prediction by CE NTRM of the flux in this range and an under-prediction in the flux values below 100 keV. Figure 5-16 suggests the error in PWFSG again derives from the lack of an unresolved resonance treatment. The flux is under-estimated in the unresolved resonance region of the uranium isotopes between 10 keV to 200 keV. This causes the neut ron balance to be over-estimated, particularly in the lower energy ranges. W ithout an unresolved resonance treatment, PWFSG is redistributi ng the neutrons that would be lo st in the unresolved region to the lower resonance energies. Despite the error from the lack of unresolv ed resonance treatment, PWFSG still provides excellent agreement with MCNP. This exampl e case demonstrates that PWFSG is able to effectively model a multi-isot ope problem derived from a realistic SFR application. UNF Fuel Cell Case Model Description The final example case is a fuel cell with LWR used nuclear fuel (UNF). This example is to test programs’ abilities to simulate multiple, si milar resonant materials in a mixture. Such a condition violates many of the assumptions in traditional group constant generation methods such as the Bondarenko method and the Subgrou p method. The PW solution methods in PWFSG and CENTRM do not share these assumpti ons; MCNP relies on direct sampling of the

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95 cross sections. Therefore, thes e programs should be able to accurately model the overlapping resonances. The same area fractions and de nsity calculations from the SFR cell were applied to the UNF cell. The difference between the two models is in the composition of the fuel alloy. The fuel is assumed to be an alloy of 70 wt% de pleted uranium, 20 wt% transuranics (TRU), and 10 wt% zirconium (U-20TRU-10Zr). The detailed TRU composition was determin ed from the fuel depletion simulation program, ORIGEN-S.57 A pressurized water reactor (PWR ) fuel composition accumulated an exposure to 41,200 MWD/MTHM over three cycles. This fuel exposure was based on previous studies58 to determine fuel product compositions for advanced SFR designs. The UNF fuel cell model was simplified by discar ding isotopes in conc entrations less than 0.0001%. The final fuel number de nsities were determined using the alloy weight fractions and the isotopic composition of depl eted uranium with 0.2 wt% enrichment, the transuranic composition from ORIGEN-S, and the isotopic composition of zirconium. The cladding and coolant number densities fo r the UNF cell remained unchanged from the SFR cell model. The final number densities used in the UNF cell model are given in Table 5-6. Results The execution times required for the UNF fu el cell case by the three programs are summarized along with the ot her cases in Table 5-1. As with th e other cases, the majority of the time needed by CENTRM was for the spectrum calcu lation (18.5 minutes) with the remainder in library pre-processing. CENTRM and PWFSG both required less than 20 minutes to execute, while MCNP required almost 10 days. The normalized spectra for the UNF fuel cell are shown in Fig. 5-17. The spectra are similar to the SFR cell spectra as the total system cross section is still dominated by Na-23 and

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96 U-238. The SFR and UNF cell spectra from MCNP are shown for comparison in Fig. 5-18. The inclusion of the transuranic isotopes shifts the neutron spectra to slight ly lower energies and lessens the peaks and valleys in the resonances. The MCNP relative error for the UNF cell is show n in Fig. 5-19. As with the SFR cell, the program comparison was performed between 4 00 eV and 10 MeV with the UNF fuel cell comparison statistics given in Table 5-7. The agreement for both PWFSG and CENTRM is approximately the same as in the SFR cell case, with the r2 values decreasing slightly. The improvement factor is slightly less than the SFR cell case with a value of 7.4. The residual plot of Fig. 5-20 confirms that the same error tr ends are present as in the SFR cell case. The UNF cell case reveals that the inclusion of isotopes with overl apping resonances does not cause significant additional error in PWFSG or CENTRM. The PW treatment along with a rigorous scattering treatment avoids many of the deficiencies of traditional approaches when a high-fidelity neutron ener gy spectrum is desired.

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97 Table 5-1. Required execution times of the three programs for the three example cases Case PWFSG Execution Time (minutes) CENTRM Execution Time (minutes) MCNP Execution Time (hours) U-238 Infinite Medium 3.5<1.047 SFR Fuel Cell 12.012.4154 UNF Fuel Cell 17.719.3227

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98 Table 5-2. Error statistics for U-238 infinite medium case Program Standard Deviation of Residuals r2 1 r2 PWFSG 0.3257 99.95%0.0550% CENTRM 0.8240 99.65%0.3518% Table 5-3. Error Statistics fo r U-238 infinite medium case with unresolved resonance treatment disabled in MCNP Program Standard Deviation of Residuals r2 1 r2 PWFSG 0.3347 99.94%0.0596% CENTRM 0.7621 99.69%0.3087%

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99 Table 5-4. Homogenized number densities used in the SFR cell model Isotope Number Density (atoms/b-cm) U-235 1.8858619 x 10-3 U-238 1.0551567 x 10-2 Zr-90 1.8518384 x 10-3 Zr-91 4.0384114 x 10-4 Zr-92 6.1727947 x 10-4 Zr-94 6.2555785 x 10-4 Zr-96 1.0078032 x 10-4 Fe-54 7.5240415 x 10-4 Fe-56 1.1811136 x 10-2 Fe-57 2.7277064 x 10-4 Fe-58 3.6300765 x 10-5 Cr-50 8.2857689 x 10-5 Cr-52 1.5978281 x 10-3 Cr-53 1.8118088 x 10-4 Cr-54 4.5099755 x 10-5 Mo-92 1.2782226 x 10-5 Mo-94 7.9673577 x 10-6 Mo-95 1.3712469 x 10-5 Mo-96 1.4367084 x 10-5 Mo-97 8.2257585 x 10-6 Mo-98 2.0784037 x 10-5 Mo-100 8.2946654 x 10-6 Na-23 8.4013761 x 10-3

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100 Table 5-5. Error statistic s for SFR fuel cell case Program Standard Deviation of Residuals r2 1 r2 PWFSG 2.9460 98.41%1.59% CENTRM 8.2412 87.54%12.46%

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101 Table 5-6. Homogenized number densities used in the UNF cell model Isotope Number Density (atoms/b-cm) U-235 1.9614993 x 10-5 U-238 9.6642491 x 10-3 Am-241 1.3566604 x 10-4 Am-243 3.4663778 x 10-5 Cm-244 8.2528486 x 10-6 Cm-245 4.6175738 x 10-7 Np-237 1.5733084 x 10-4 Pu-238 5.5200518 x 10-5 Pu-239 1.3077054 x 10-3 Pu-240 6.5959724 x 10-4 Pu-241 2.3354193 x 10-4 Pu-242 1.5570373 x 10-4 Zr-90 1.8573215 x 10-3 Zr-91 4.0503688 x 10-4 Zr-92 6.1910718 x 10-4 Zr-94 6.2741007 x 10-4 Zr-96 1.0107872 x 10-4 Fe-54 7.5240415 x 10-4 Fe-56 1.1811136 x 10-2 Fe-57 2.7277064 x 10-4 Fe-58 3.6300765 x 10-5 Cr-50 8.2857689 x 10-5 Cr-52 1.5978281 x 10-3 Cr-53 1.8118088 x 10-4 Cr-54 4.5099755 x 10-5 Mo-92 1.2782226 x 10-5 Mo-94 7.9673577 x 10-6 Mo-95 1.3712469 x 10-5 Mo-96 1.4367084 x 10-5 Mo-97 8.2257585 x 10-6 Mo-98 2.0784037 x 10-5 Mo-100 8.2946654 x 10-6 Na-23 8.4013761 x 10-3

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102 Table 5-7. Error statistic s for UNF fuel cell case Program Standard Deviation of Residuals r2 1 r2 PWFSG 2.9204 98.24%1.76% CENTRM 7.9724 87.07%12.93%

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103 Figure 5-1. Source dist ribution used for example cases. Th e source distribution is generated from a Watt fission spectrum with fitting parameters intended to match the fission neutron distribution from Pu-239. Normalized Source

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104 Figure 5-2. Neutron energy sp ectra generated by MCNP, PWFSG, and CENT RM for the U-238 infinite medium case Normalized Flux (1/cm-eV-s)

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105 Figure 5-3. Neutron energy sp ectrum for the U-238 infinite medium case in the resonance region between 18 and 19 keV 18.0 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 Energy (keV) (10^3) 0 10 20 30 40 50 60 70 80 90 100Normalized Flux (1/cm-eV-s) MCNP PWFSG CENTRM

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106 Figure 5-4. Statistical re lative error from MCNP U-238 infinite medium case MCNP output truncates results below 0.01% result ing in a “flat” value of 0.01 between 20 keV and 300 keV above. Relative Error (%)

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107 Figure 5-5. Residual flux error from difference in PWFSG or CENTRM and MCNP results for U-238 infinite medium case

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108 Figure 5-6. Neutron energy spectra for the U-238 infinite medium case with CE NTRM extending PW treatment to 10 MeV Normalized Flux (1/cm-eV-s)

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109 Figure 5-7. Various cr oss sections for U-238 between 10 keV and 10 MeV Cross Section (barns)

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Figure 5 8. Angular B) For f centero A B scattering d f irst discrete o f-mass coo r d istribution v level inelas t r dinate syste m 110 v ersus energ y t ic scatterin g m y for U-238 g Distribut i A) For ela i ons are pre s stic scatteri n s ented in in n g.

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111 Figure 5-9. Neutron energy spectra fo r the U-238 case with unresolved res onance treatment disabled in MCNP Normalized Flux (1/cm-eV-s)

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112 Figure 5-10. Standa rdized residuals (z statistics) from PWFSG gene rated by comparison to MCNP with and without unresolved resonance treatment Z ( )

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113 Figure 5-11. Histogram of standardized re siduals from PWFSG compared to MCNP with and without unresolved resonance treatment 1 10 100 1000 10000 100000 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7Energy Point CountZ MCNP With Unresolved Resonances MCNP without Unresolved Resonances

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114 Figure 5-12. Geometry used for SFR and UNF cell model Fuel Coolant Cla d

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115 Figure 5-13. Neutron energy spectra generated by MCNP, PWFSG, and CENT RM for the SFR fuel cell case 102 103 104 105 106 107Energy (eV) 0 20 40 60 80 100 120Normalized Flux (1/cm-eV-s) MCNP PWFSG CENTRM

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116 Figure 5-14. Neutron energy spectra for the SFR fuel ce ll case in the resonance region between 18 and 19 keV

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117 Figure 5-15. Statistical relative error from MCNP SFR fuel cell case

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118 Figure 5-16. Residual flux error from difference in PWFSG or CENTRM and MCNP resu lts for SFR fuel cell case 102 103 104 105 106 107Energy (eV) -20 -10 0 10 20 PWFSG CENTRM

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119 Figure 5-17. Neutron energy spectra generated by MCNP, PWFSG, and CE NTRM for the UNF fuel cell case

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120 Figure 5-18. MCNP spectra for SFR fuel cell with U10Zr fuel and UNF fuel cell with U-20TRU-10Zr fuel

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121 Figure 5-19. Statistical re lative error from MCNP for the UNF fuel cell case

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122 Figure 5-20. Residual flux error from difference in PWFSG or CENTRM and MCNP results for UNF fuel cell case Residual Flux Error (1/cm-eV-s)

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123 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK Conclusions The example cases included in Chapter 5 de monstrate that the poi ntwise (PW) GaussKronrod Quadrature (GKQ) algorithm provides a r obust treatment of the tr ansfer source without a noticeable increase in the computational cost The PWFSG results have excellent agreement with the high-fidelity Monte Carlo results, wi th computation time comparable to CENTRM. PWFSG reduces the residual error a factor of 6 as compared to CENTRM. The improvement factors, which are a measure of the residual er ror decrease in PWFSG compared to CENTRM, are summarized in Table 6-1. Part of the computational efficiency in PW FSG is due to the scaling performance with increasing isotopes, due to parallel izing the transfer source calcul ation. This scaling would not be present on single-core architecture or in a model with only one isotope. The main cause of the error in CENTRM is the kinematic assumptions in the PW solution region. These assumptions may be acceptable fo r LWR applications where the resonance region lies at lower energies (e.g., below 100 keV). CENTRM would then rely on the traditional Bondarenko approach (i.e., BONAMI module) at higher energies (e.g., above 100 keV). However, for SFR applications where the PW region could extend into the MeV range, the kinematic assumptions are invalid and cause sign ificant error. The U238 infinite medium case demonstrates that both CENTRM and PWFS G offer good agreement with MCNP when the CENTRM PW treatment was limited to below 25 keV. The SFR and UNF fuel cell cases demonstrat e that CENTRM signifi cantly over-predicts the flux above 100 keV. Therefore, the GKQ al gorithm in PWFSG offers added robustness to the transfer source term calculation.

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124 Future Work The greatest discrepancy in the PWFSG results is from the lack of an unresolved resonance treatment. The effect of these unresolved resona nces is small compared to the CENTRM error, but still causes under-prediction of the flux in th e unresolved resonance range. The addition of an unresolved resonance treatmen t to PWFSG would be an importa nt next step in improving and demonstrating the GKQ transfer source approach. A possible method for implementing this is a stochastic approach that random ly adjusts the cross section in the unresolved resonance region based on probability tables as implemented in MCNP.59 The majority of the processing time required by PWFSG is for cross section and transfer probability interpolation. The development of a more efficient library lookup and interpolation scheme could make the GKQ algorit hm appealing for production use. The GKQ algorithm within PWFSG controls error by interval sizing rather than increasing the order of the node-weight set. Early implem entations of PWFSG foun d the interval-halving approach to be more efficient and robust th an increasing the order of the GKQ set; the (G7,K15) Gauss-Kronrod node-weight set was chosen to balance accuracy a nd computational cost. It is possible that a hybrid approach of interval sizing and quadr ature order may improve the performance of PWFSG. A parametric study would be beneficial to quantifying the merits of both approaches.

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125 Additional features could be added to PWFSG such as: € Internal resonance reco nstruction and temperat ure Doppler-broadening € Thermal scattering treatment € Eigenvalue calculation (i.e., fission source) capability Such features may be necessary for extens ion to other common r eactor applications, particularly LWR analysis. These consideratio ns have implementations in programs such as NJOY and CENTRM; integration into PWFSG would be straightforward. Finally, the GKQ algorithm in PWFSG provided only infinite medium spatial treatment to focus the comparison of the GKQ algorithm on the en ergy spectra effects, which are of primary importance in fast reactor applications. CENT RM provides methods to couple the PW solution with one-dimensional transport calculations. Extension of the PWFSG program to include spatial effects would extend the applicability of th is PW algorithm to applications where spatial variation within a unit cell is important.

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126 Table 6-1. Summary of improvement factors for test cases Case Improvement Factor, F U-238 Infinite Medium 6.4 SFR Fuel Cell 7.8 UNF Fuel Cell 7.4

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127 APPENDIX A MCNP INPUT DATA U-238 Infinite Medium Infinite Medium of U238 with Pu-239 Fission Source 1 1 0.01 -1 99 0 1 c 10000 cm sphere 1 so 10000 nps 5e9 c Uranium-238 at 900 K m1 92238.72c 1.0000 sdef erg=d1 sp1 -3 0.966 2.842 $ Watt fission spectrum of Pu-239 wwp:n 5 2 4 0 0 # wwe:n 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 4.00E-02 1.00E-01 2.00E-01 4.00E-01 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 1.00E+01 2.00E+01 wwn1:n 3.56E-04 0.0 wwn2:n 1.14E-03 0.0 wwn3:n 3.61E-03 0.0 wwn4:n 1.21E-02 0.0 wwn5:n 3.93E-02 0.0 wwn6:n 7.40E-02 0.0 wwn7:n 1.46E-01 0.0 wwn8:n 2.08E-01 0.0 wwn9:n 2.70E-01 0.0 wwn10:n 3.21E-01 0.0 wwn11:n 2.68E-01 0.0 wwn12:n 1.95E-01 0.0 wwn13:n 1.14E-01 0.0 wwn14:n 6.17E-02 0.0 wwn15:n 1.44E-02 0.0 wwn16:n 1.90E-04 0.0 nonu print prdmp 0 0 0 3 rand gen=2

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128 f4:n 1 e4 1.00085430009E-07 1.00256508946E-07 1.00427880384E-07 1.00599544322E-07 … Energy mesh truncated due to size. Mesh in cludes 43,283 points between 0.1 eV and 20 MeV.

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129 SFR Fuel Cell Infinite medium of homogenized SFR Fuel Cell at 900 K 1 1 0.01 -1 99 0 1 c 10000 cm sphere 1 so 10000 nps 5e9 c Homogenized fuel cell at 900 K m1 92235.72c 1.8858619E-03 92238.72c 1.0551567E-02 40090.72c 1.8518384E-03 40091.72c 4.0384114E-04 40092.72c 6.1727947E-04 40094.72c 6.2555785E-04 40096.72c 1.0078032E-04 26054.72c 7.5240415E-04 26056.72c 1.1811136E-02 26057.72c 2.7277064E-04 26058.72c 3.6300765E-05 24050.72c 8.2857689E-05 24052.72c 1.5978281E-03 24053.72c 1.8118088E-04 24054.72c 4.5099755E-05 42092.72c 1.2782226E-05 42094.72c 7.9673577E-06 42095.72c 1.3712469E-05 42096.72c 1.4367084E-05 42097.72c 8.2257585E-06 42098.72c 2.0784037E-05 42100.72c 8.2946654E-06 11023.72c 8.4013761E-03 sdef erg=d1 sp1 -3 0.966 2.842 $ Watt fission spectrum of Pu-239 wwp:n 5 2 4 0 0 # wwe:n 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 4.00E-02 1.00E-01 2.00E-01 4.00E-01 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 1.00E+01 2.00E+01 wwn1:n 3.56E-04 0.0 wwn2:n 1.14E-03 0.0

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130 wwn3:n 3.61E-03 0.0 wwn4:n 1.21E-02 0.0 wwn5:n 3.93E-02 0.0 wwn6:n 7.40E-02 0.0 wwn7:n 1.46E-01 0.0 wwn8:n 2.08E-01 0.0 wwn9:n 2.70E-01 0.0 wwn10:n 3.21E-01 0.0 wwn11:n 2.68E-01 0.0 wwn12:n 1.95E-01 0.0 wwn13:n 1.14E-01 0.0 wwn14:n 6.17E-02 0.0 wwn15:n 1.44E-02 0.0 wwn16:n 1.90E-04 0.0 nonu print prdmp 3j 3 rand gen=2 f4:n 1 e4 1.00085430009E-07 1.00256508946E-07 1.00427880384E-07 1.00599544322E-07 … Energy mesh truncated due to size. Mesh in cludes 49,114 points between 0.1 eV and 20 MeV.

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131 UNF Fuel Cell Infinite medium of homogenized SFR Fuel Cell with 20% TRU 1 1 0.01 -1 99 0 1 c 10000 cm sphere 1 so 10000 nps 5e9 c Homogenized fuel cell at 900 K m1 92235.72c 1.9614993E-05 92238.72c 9.6642491E-03 95241.72c 1.3566604E-04 95243.72c 3.4663778E-05 96244.72c 8.2528486E-06 96245.72c 4.6175738E-07 93237.72c 1.5733084E-04 94238.72c 5.5200518E-05 94239.72c 1.3077054E-03 94240.72c 6.5959724E-04 94241.72c 2.3354193E-04 94242.72c 1.5570373E-04 40090.72c 1.8573215E-03 40091.72c 4.0503688E-04 40092.72c 6.1910718E-04 40094.72c 6.2741007E-04 40096.72c 1.0107872E-04 26054.72c 7.5240415E-04 26056.72c 1.1811136E-02 26057.72c 2.7277064E-04 26058.72c 3.6300765E-05 24050.72c 8.2857689E-05 24052.72c 1.5978281E-03 24053.72c 1.8118088E-04 24054.72c 4.5099755E-05 42092.72c 1.2782226E-05 42094.72c 7.9673577E-06 42095.72c 1.3712469E-05 42096.72c 1.4367084E-05 42097.72c 8.2257585E-06 42098.72c 2.0784037E-05 42100.72c 8.2946654E-06 11023.72c 8.4013761E-03 sdef erg=d1 sp1 -3 0.966 2.842 $ Watt fission spectrum of Pu-239 wwp:n 5 2 4 0 0 # wwe:n 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 4.00E-02 1.00E-01 2.00E-01

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132 4.00E-01 1.50E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 1.00E+01 2.00E+01 wwn1:n 3.56E-04 0.0 wwn2:n 1.14E-03 0.0 wwn3:n 3.61E-03 0.0 wwn4:n 1.21E-02 0.0 wwn5:n 3.93E-02 0.0 wwn6:n 7.40E-02 0.0 wwn7:n 1.46E-01 0.0 wwn8:n 2.08E-01 0.0 wwn9:n 2.70E-01 0.0 wwn10:n 3.21E-01 0.0 wwn11:n 2.68E-01 0.0 wwn12:n 1.95E-01 0.0 wwn13:n 1.14E-01 0.0 wwn14:n 6.17E-02 0.0 wwn15:n 1.44E-02 0.0 wwn16:n 1.90E-04 0.0 nonu print prdmp 3j 3 rand gen=2 f4:n 1 e4 1.0008296558356E-07 1.0024910302666E-07 1.0041551646903E-07 1.0058220591232E-07 … Energy mesh truncated due to size. Mesh in cludes 49,481 points between 0.1 eV and 20 MeV.

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133 APPENDIX B PWFSG INPUT DATA U-238 Infinite Medium 0.001 0.5 0.001 0.1

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134 SFR Fuel Cell

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135
0.001 0.5 0.001 0.1


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136 UNF Fuel Cell

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137


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138
0.001 0.5 0.001 0.1


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139 APPENDIX C CENTRM INPUT DATA U-238 Infinite Medium =shell del C:\scale6\tmpdir\ft88f001 copy C:\scale6\data\xn238v7 ft88f001 end =ajax 0$$ 11 0 1$$ 1 T Input library / Number of isotopes to copy 2$$ 88 1 T Isotope ZAID 3$$ 92238 T end =bonami Input library / Scratch / Scratch / Output library 0$$ 11 0 18 1 Geometry flag / Number of zones / Mix Table length / XS edit flag / 1$$ 0 1 1 0 Bondarenko edit flag / Dancoff factor 0 0 T Mixture numbers by isotope 3$$ 1 Nuclide IDs 4$$ 92238 Nuclide concentrations (atom/b-cm) 5** 1.000 Mixtures by zone 6$$ 1 Temperatures by zone 8** 9.00000E+02 New Identifier in Mixing tables 10$$ 92238 Zone types (0 fuel, 1 mod, 2 clad) 11$$ 0 t end =worker Input master lib / Input working lib / Output lib 0$$ 1 0 4 Scratch / Scratch / Output Temp-Interpolated lib 18 19 0 Num nuclides to read from master lib / Num to read from working lib 1$$ 1 0 Output print option / Flag to copy master / Flag to copy working -2 -1 -1 Sequence # of master / Sequence # of working / Sequence # of output 1 1 1 T IDs from master to copy 2$$ 92238

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140 Thermal scat. kernel temps 6** 9.00000E+02 T end =crawdad Output CENTRM lib / Sc ratch / Scratch / CE-KENO lib 0$$ 81 17 18 77 Number of PW nuclides / CENTRM Identifier header / Print flag 1$$ 1 66666 1 Lib format 0/1= old/new / Temp interp method / Thermal lib out flag 0 0 1 Extra unused integer flags e T Data Blocks 2 and 3 repeated for each nuclide ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 92238 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T end =centrm CENTRM: U-238 Inf medium at 900 K Punch logical / Thermal kernel / MG working lib / PW lib 0$$ 7 0 4 81 MG flux restart / MG output / PW output / MG Angular output 0 25 15 16 Scratch for thermal matrices / Optional PW xs print / Scratch 17 18 19 Not used / Scratch 8 9 Geometry / Num Zones / Num Intervals / Left BC / Right BC 1$$ 0 1 1 0 0 Num Mixtures / Mix Table Length / Sn order / Max Pn Order 1 1 0 64 Source type / Inner iter max / Upscatter outer iter max 0 20 100 UMG range calc type / LMG range calc type / PW range calc type 2 2 3 MG source and XS linearization / Pn order for PW calc 3 64 PW Inelastic scat option 1 Mix XS print opt / Flux print opt / Balance Table print 2$$ 0 3 1 Vol source flag / Boundary sour ce flag / Group diffusion coeff 1 0 2 Density factors / Extra print opt / Mass scattering restriction 0 1 0 Restrictive order / correlate PW and MG flux / Not used 64 0 0 Upscatter converg / Point converge / Source normalization 3** 1.00000E-04 1.00000E-05 1.00000E+00

PAGE 141

141 Material Buckling (1/cm) / LMG range cutoff / UMG range cutoff 0.00000E+00 1.00000E+00 2.00000E+04 Thinning tolerance / Frac Leth argy gain allowance / Not used 1.00000E-03 1.00000E-01 0.0+00 T Mixture numbers 13$$ 1 Isotope identifiers 14$$ 92238 Isotope concentrations (atoms/barn -cm) 15** 1.0000 T Source number by interval 30$$ 1 Volumetric source by interval and group 31** 9.60817162E+00 3.17979593E+01 6.59123252E+01 8.56636328E+01 2.43855982E+02 3.77567052E+03 1.28597182E+04 4.60351593E+04 1.38805606E+05 8.15300416E+04 3.79836723E+05 2.48427549E+05 7.00361135E+04 3.27183503E+05 2.70261891E+05 8.33884439E+04 3.74944768E+04 3.36258630E+04 5.85884438E+04 4.43603989E+04 9.01939217E+04 8.26073314E+04 8.36331980E+04 1.86862972E+04 2.33946264E+04 1.30216633E+04 3.85419406E+04 6.56569747E+04 6.63587719E+04 8.37785121E+03 6.47534398E+04 2.47279973E+04 2.09256714E+04 4.53887848E+04 2.61351971E+04 2.61944753E+04 1.72324738E+04 1.70284606E+04 5.77192278E+04 4.66022640E+04 4.99461055E+04 3.18813210E+04

PAGE 142

142 1.26057183E+04 1.50973866E+04 7.29640582E+03 1.39403173E+03 3.16257286E+03 8.79634792E+02 5.44082929E+03 3.09154071E+03 7.39829777E+02 1.78832320E+03 4.78581570E+03 1.37584123E+03 1.92836547E+03 8.18316897E+02 6.21239675E+02 2.30800307E+02 2.85248739E+02 2.47910671E+02 1.66350391E+01 7.22464162E+01 3.73318571E+01 2.40880149E+01 7.17972519E+00 3.01104369E+01 1.72285861E+01 3.28882021E+00 2.14460394E+01 1.09147995E+01 1.28414744E+01 5.69800237E-01 4.99268036E+00 8.50801791E+00 5.78975626E-01 1.22854750E+00 7.58383952E-01 6.08854788E-02 3.57555982E-01 1.50727273E-01 1.33636800E+00 5.55129899E-02 7.29341847E-02 1.24595534E-01 1.37520215E-01 1.64285979E-01 1.25052132E-01 3.03432451E-02 5.95509100E-02 5.80038225E-02 6.33520693E-02 3.43022322E-02 5.35190864E-02 2.61154789E-02 7.07635424E-02 1.71328389E-02 1.69037426E-02 1.66714974E-02

PAGE 143

143 1.05932305E-02 1.51276633E-02 2.06023547E-02 1.35097265E-02 1.77278940E-02 1.52402758E-02 1.49822591E-02 5.28742327E-03 1.15134061E-02 1.03232229E-02 1.52244183E-02 8.98228896E-03 8.37672169E-03 4.87860058E-03 1.44154236E-02 4.73073359E-03 1.16612495E-02 2.25959045E-02 2.15908251E-02 2.05365695E-02 1.17925057E-02 7.63261220E-03 7.44410831E-03 3.64984888E-03 1.06528502E-02 6.84752080E-03 5.98340417E-03 4.53136990E-03 4.11089618E-03 5.23049636E-03 5.93595252E-03 2.30651781E-03 8.28927848E-03 4.68833258E-03 4.94308066E-03 4.42159243E-03 6.72609125E-04 1.10504696E-03 1.08476815E-03 1.06410289E-03 1.04302821E-03 2.41462206E-03 1.53760807E-03 9.30521181E-04 2.64379700E-03 8.94803183E-04 7.37182167E-04 1.07578364E-03 2.96845515E-04 1.46573523E-04 8.74074895E-05 2.88040554E-04 2.83093703E-04 2.78056296E-04 2.72842238E-04 2.67637866E-04

PAGE 144

144 2.36266798E-04 2.06301869E-04 2.27834294E-04 2.23216548E-04 2.90342781E-04 1.41968410E-04 1.85896124E-04 2.04400187E-04 1.99289297E-04 1.93998326E-04 1.88561965E-04 1.02369949E-04 1.00619824E-04 9.88582973E-05 9.70245353E-05 9.51573073E-05 4.69024254E-05 4.63891110E-05 4.59451129E-05 4.54225447E-05 1.80570699E-05 1.79600451E-05 1.78629090E-05 1.78011816E-05 1.77388567E-05 1.76407264E-05 1.75424699E-05 1.74789609E-05 1.74148298E-05 1.73155201E-05 1.72160676E-05 1.71506875E-05 1.70676326E-05 1.70011138E-05 1.68833560E-05 4.18810395E-05 4.13474874E-05 4.08069592E-05 4.02591739E-05 7.88444405E-05 7.65582839E-05 7.42017016E-05 7.17677517E-05 6.92482701E-05 3.36499964E-05 3.29835503E-05 6.39118808E-05 6.10689577E-05 5.80869496E-05 5.49431732E-05 2.62342689E-05 2.53737371E-05 2.44829653E-05 2.35585205E-05 2.25962658E-05 2.15911402E-05

PAGE 145

145 2.05368407E-05 1.94253475E-05 1.82461874E-05 1.69852318E-05 1.56226084E-05 1.41287680E-05 5.19545583E-06 4.91426775E-06 4.61596116E-06 4.29696175E-06 3.95224224E-06 3.57432714E-06 3.15119418E-06 1.31712363E-06 3.39885712E-06 3.93899278E-07 3.32630523E-07 1.13030151E-07 9.96495113E-08 4.41874128E-08 3.99621932E-08 3.52314226E-08 1.85727491E-08 1.11786295E-08 1.24561889E-08 1.05187008E-08 1.14415023E-08 1.08834188E-09 t Zone number by interval 36$$ 1 Mixture number by zone 39$$ 1 Temperature by zone 41** 9.00000E+02 t end

PAGE 146

146 SFR Fuel Cell =shell del C:\scale6\tmpdir\ft88f001 copy C:\scale6\data\xn238v7 ft88f001 end =ajax 0$$ 11 0 1$$ 1 T Input library / Number of isotopes to copy 2$$ 88 23 T Isotope ZAID 3$$ 92235 92238 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 T end =bonami Input library / Scratch / Scratch / Output library 0$$ 11 0 18 1 Geometry flag / Number of zones / Mix Table length / XS edit flag / 1$$ 0 1 23 1 Bondarenko edit flag / Dancoff factor 1 0 T Mixture numbers by isotope 3$$ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Nuclide IDs 4$$ 92235 92238 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Nuclide concentrations (atom/b-cm) 5** 1.8858619E-03 1.0551567E-02 1.8518384E-03 4.0384114E-04 6.1727947E-04 6.2555785E-04 1.0078032E-04 7.5240415E-04 1.1811136E-02 2.7277064E-04 3.6300765E-05 8.2857689E-05 1.5978281E-03 1.8118088E-04 4.5099755E-05 1.2782226E-05 7.9673577E-06 1.3712469E-05 1.4367084E-05 8.2257585E-06 2.0784037E-05 8.2946654E-06 8.4013761E-03 Mixtures by zone 6$$ 1 Temperatures by zone 8** 9.00000E+02 New Identifier in Mixing tables 10$$ 92235 92238 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Zone types (0 fuel, 1 mod, 2 clad) 11$$ 0 t end

PAGE 147

147 =worker Input master lib / Input working lib / Output lib 0$$ 1 0 4 Scratch / Scratch / Output Temp-Interpolated lib 18 19 0 Num nuclides to read from master lib / Num to read from working lib 1$$ 23 0 Output print option / Flag to copy master / Flag to copy working -2 -1 -1 Sequence # of master / Sequence # of working / Sequence # of output 1 1 1 T IDs from master to copy 2$$ 92235 92238 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Thermal scat. kernel temps 6** 9.00000E+02 9.00000E+02 9.000 00E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 T end =crawdad Output CENTRM lib / Sc ratch / Scratch / CE-KENO lib 0$$ 81 17 18 77 Number of PW nuclides / CENTRM Identifier header / Print flag 1$$ 23 66666 1 Lib format 0/1= old/new / Temp interp method / Thermal lib out flag 0 0 1 Extra unused integer flags e T Data Blocks 2 and 3 repeated for each nuclide ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 92235 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 92238 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40090 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV)

PAGE 148

148 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40091 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40092 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40094 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40096 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26054 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26056 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26057 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV)

PAGE 149

149 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26058 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24050 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24052 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24053 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24054 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42092 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42094 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV)

PAGE 150

150 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42095 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42096 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42097 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42098 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42100 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 11023 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 2.000E+07 T Nuclide temps to process 4** 9.00000E+02 T end =centrm CENTRM: Homogenized SFR Cell Punch logical / Thermal kernel / MG working lib / PW lib

PAGE 151

151 0$$ 7 0 4 81 MG flux restart / MG output / PW output / MG Angular output 0 25 15 16 Scratch for thermal matrices / Optional PW xs print / Scratch 17 18 19 Not used / Scratch 8 9 Geometry / Num Zones / Num Intervals / Left BC / Right BC 1$$ 0 1 1 0 0 Num Mixtures / Mix Table Length / Sn order / Max Pn Order 1 23 0 64 Source type / Inner iter max / Upscatter outer iter max 0 20 100 UMG range calc type / LMG range calc type / PW range calc type 2 2 3 MG source and XS linearization / Pn order for PW calc 3 64 PW Inelastic scat option 1 Mix XS print opt / Flux print opt / Balance Table print 2$$ 0 3 1 Vol source flag / Boundary sour ce flag / Group diffusion coeff 1 0 2 Density factors / Extra print opt / Mass scattering restriction 0 1 0 Restrictive order / correlate PW and MG flux / Not used 64 0 0 Upscatter converg / Point converge / Source normalization 3** 1.00000E-04 1.00000E-05 1.00000E+00 Material Buckling (1/cm) / LMG range cutoff / UMG range cutoff 0.00000E+00 1.00000E+00 5.00000E+06 Thinning tolerance / Frac Leth argy gain allowance / Not used 1.00000E-03 1.00000E-01 0.0+00 T Mixture numbers 13$$ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Isotope identifiers 14$$ 92235 92238 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Isotope concentrations (atoms/barn -cm) 15** 1.8858619E-03 1.0551567E-02 1.8518384E-03 4.0384114E-04 6.1727947E-04 6.2555785E-04 1.0078032E-04 7.5240415E-04 1.1811136E-02 2.7277064E-04 3.6300765E-05 8.2857689E-05 1.5978281E-03 1.8118088E-04 4.5099755E-05 1.2782226E-05 7.9673577E-06 1.3712469E-05 1.4367084E-05 8.2257585E-06 2.0784037E-05 8.2946654E-06 8.4013761E-03 T Source number by interval 30$$ 1 Volumetric source by interval and group 31** 9.60817162E+00

PAGE 152

152 3.17979593E+01 6.59123252E+01 8.56636328E+01 2.43855982E+02 3.77567052E+03 1.28597182E+04 4.60351593E+04 1.38805606E+05 8.15300416E+04 3.79836723E+05 2.48427549E+05 7.00361135E+04 3.27183503E+05 2.70261891E+05 8.33884439E+04 3.74944768E+04 3.36258630E+04 5.85884438E+04 4.43603989E+04 9.01939217E+04 8.26073314E+04 8.36331980E+04 1.86862972E+04 2.33946264E+04 1.30216633E+04 3.85419406E+04 6.56569747E+04 6.63587719E+04 8.37785121E+03 6.47534398E+04 2.47279973E+04 2.09256714E+04 4.53887848E+04 2.61351971E+04 2.61944753E+04 1.72324738E+04 1.70284606E+04 5.77192278E+04 4.66022640E+04 4.99461055E+04 3.18813210E+04 1.26057183E+04 1.50973866E+04 7.29640582E+03 1.39403173E+03 3.16257286E+03 8.79634792E+02 5.44082929E+03 3.09154071E+03 7.39829777E+02 1.78832320E+03 4.78581570E+03 1.37584123E+03 1.92836547E+03 8.18316897E+02 6.21239675E+02

PAGE 153

153 2.30800307E+02 2.85248739E+02 2.47910671E+02 1.66350391E+01 7.22464162E+01 3.73318571E+01 2.40880149E+01 7.17972519E+00 3.01104369E+01 1.72285861E+01 3.28882021E+00 2.14460394E+01 1.09147995E+01 1.28414744E+01 5.69800237E-01 4.99268036E+00 8.50801791E+00 5.78975626E-01 1.22854750E+00 7.58383952E-01 6.08854788E-02 3.57555982E-01 1.50727273E-01 1.33636800E+00 5.55129899E-02 7.29341847E-02 1.24595534E-01 1.37520215E-01 1.64285979E-01 1.25052132E-01 3.03432451E-02 5.95509100E-02 5.80038225E-02 6.33520693E-02 3.43022322E-02 5.35190864E-02 2.61154789E-02 7.07635424E-02 1.71328389E-02 1.69037426E-02 1.66714974E-02 1.05932305E-02 1.51276633E-02 2.06023547E-02 1.35097265E-02 1.77278940E-02 1.52402758E-02 1.49822591E-02 5.28742327E-03 1.15134061E-02 1.03232229E-02 1.52244183E-02 8.98228896E-03 8.37672169E-03 4.87860058E-03 1.44154236E-02

PAGE 154

154 4.73073359E-03 1.16612495E-02 2.25959045E-02 2.15908251E-02 2.05365695E-02 1.17925057E-02 7.63261220E-03 7.44410831E-03 3.64984888E-03 1.06528502E-02 6.84752080E-03 5.98340417E-03 4.53136990E-03 4.11089618E-03 5.23049636E-03 5.93595252E-03 2.30651781E-03 8.28927848E-03 4.68833258E-03 4.94308066E-03 4.42159243E-03 6.72609125E-04 1.10504696E-03 1.08476815E-03 1.06410289E-03 1.04302821E-03 2.41462206E-03 1.53760807E-03 9.30521181E-04 2.64379700E-03 8.94803183E-04 7.37182167E-04 1.07578364E-03 2.96845515E-04 1.46573523E-04 8.74074895E-05 2.88040554E-04 2.83093703E-04 2.78056296E-04 2.72842238E-04 2.67637866E-04 2.36266798E-04 2.06301869E-04 2.27834294E-04 2.23216548E-04 2.90342781E-04 1.41968410E-04 1.85896124E-04 2.04400187E-04 1.99289297E-04 1.93998326E-04 1.88561965E-04 1.02369949E-04 1.00619824E-04 9.88582973E-05 9.70245353E-05

PAGE 155

155 9.51573073E-05 4.69024254E-05 4.63891110E-05 4.59451129E-05 4.54225447E-05 1.80570699E-05 1.79600451E-05 1.78629090E-05 1.78011816E-05 1.77388567E-05 1.76407264E-05 1.75424699E-05 1.74789609E-05 1.74148298E-05 1.73155201E-05 1.72160676E-05 1.71506875E-05 1.70676326E-05 1.70011138E-05 1.68833560E-05 4.18810395E-05 4.13474874E-05 4.08069592E-05 4.02591739E-05 7.88444405E-05 7.65582839E-05 7.42017016E-05 7.17677517E-05 6.92482701E-05 3.36499964E-05 3.29835503E-05 6.39118808E-05 6.10689577E-05 5.80869496E-05 5.49431732E-05 2.62342689E-05 2.53737371E-05 2.44829653E-05 2.35585205E-05 2.25962658E-05 2.15911402E-05 2.05368407E-05 1.94253475E-05 1.82461874E-05 1.69852318E-05 1.56226084E-05 1.41287680E-05 5.19545583E-06 4.91426775E-06 4.61596116E-06 4.29696175E-06 3.95224224E-06 3.57432714E-06 3.15119418E-06 1.31712363E-06 3.39885712E-06

PAGE 156

156 3.93899278E-07 3.32630523E-07 1.13030151E-07 9.96495113E-08 4.41874128E-08 3.99621932E-08 3.52314226E-08 1.85727491E-08 1.11786295E-08 1.24561889E-08 1.05187008E-08 1.14415023E-08 1.08834188E-09 t Zone number by interval 36$$ 1 Mixture number by zone 39$$ 1 Temperature by zone 41** 9.00000E+02 t end

PAGE 157

157 UNF Fuel Cell =shell del C:\scale6\tmpdir\ft88f001 copy C:\scale6\data\xn238v7 ft88f001 end =ajax 0$$ 11 0 1$$ 1 T Input library / Number of isotopes to copy 2$$ 88 33 T Isotope ZAID 3$$ 92235 92238 95241 95243 96244 96245 93237 94238 94239 94240 94241 94242 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 T end =bonami Input library / Scratch / Scratch / Output library 0$$ 11 0 18 1 Geometry flag / Number of zones / Mix Table length / XS edit flag / 1$$ 0 1 33 1 Bondarenko edit flag / Dancoff factor 1 0 T Mixture numbers by isotope 3$$ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Nuclide IDs 4$$ 92235 92238 95241 95243 96244 96245 93237 94238 94239 94240 94241 94242 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Nuclide concentrations (atom/b-cm) 5** 1.9614993E-05 9.6642491E-03 1.3566604E-04 3.4663778E-05 8.2528486E-06 4.6175738E-07 1.5733084E-04 5.5200518E-05 1.3077054E-03 6.5959724E-04 2.3354193E-04 1.5570373E-04 1.8573215E-03 4.0503688E-04 6.1910718E-04 6.2741007E-04 1.0107872E-04 7.5240415E-04 1.1811136E-02 2.7277064E-04 3.6300765E-05 8.2857689E-05 1.5978281E-03 1.8118088E-04 4.5099755E-05 1.2782226E-05 7.9673577E-06 1.3712469E-05 1.4367084E-05 8.2257585E-06 2.0784037E-05 8.2946654E-06 8.4013761E-03 Mixtures by zone 6$$ 1 Temperatures by zone 8** 9.00000E+02

PAGE 158

158 New Identifier in Mixing tables 10$$ 92235 92238 95241 95243 96244 96245 93237 94238 94239 94240 94241 94242 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Zone types (0 fuel, 1 mod, 2 clad) 11$$ 0 t end =worker Input master lib / Input working lib / Output lib 0$$ 1 0 4 Scratch / Scratch / Output Temp-Interpolated lib 18 19 0 Num nuclides to read from master lib / Num to read from working lib 1$$ 33 0 Output print option / Flag to copy master / Flag to copy working -2 -1 -1 Sequence # of master / Sequence # of working / Sequence # of output 1 1 1 T IDs from master to copy 2$$ 92235 92238 95241 95243 96244 96245 93237 94238 94239 94240 94241 94242 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Thermal scat. kernel temps 6** 9.00000E+02 9.00000E+02 9.000 00E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9. 00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 9.00000E+02 T end =crawdad Output CENTRM lib / Sc ratch / Scratch / CE-KENO lib 0$$ 81 17 18 77 Number of PW nuclides / CENTRM Identifier header / Print flag 1$$ 33 66666 1 Lib format 0/1= old/new / Temp interp method / Thermal lib out flag 0 0 1 Extra unused integer flags e T Data Blocks 2 and 3 repeated for each nuclide ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 92235 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process

PAGE 159

159 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 92238 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 95241 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 95243 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 96244 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 96245 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 93237 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 94238 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process

PAGE 160

160 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 94239 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 94240 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 94241 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 94242 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40090 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40091 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40092 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process

PAGE 161

161 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40094 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 40096 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26054 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26056 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26057 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 26058 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24050 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process

PAGE 162

162 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24052 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24053 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 24054 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42092 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42094 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42095 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42096 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process

PAGE 163

163 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42097 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42098 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 42100 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T ZAID for PW mat / ENDF version / ENDF mod / Number of temps for iso 2$$ 11023 7 -1 1 MTs to be included / Mod number of therm data / Data source 0 -1 0 Lower PW energy limit (eV) / Upper PW energy limit (eV) 3** 1.00000E-04 1.000E+07 T Nuclide temps to process 4** 9.00000E+02 T end =centrm CENTRM: Homogenized Unf Cell Punch logical / Thermal kernel / MG working lib / PW lib 0$$ 7 0 4 81 MG flux restart / MG output / PW output / MG Angular output 0 25 15 16 Scratch for thermal matrices / Optional PW xs print / Scratch 17 18 19 Not used / Scratch 8 9 Geometry / Num Zones / Num Intervals / Left BC / Right BC 1$$ 0 1 1 0 0 Num Mixtures / Mix Table Length / Sn order / Max Pn Order 1 33 0 64 Source type / Inner iter max / Upscatter outer iter max 0 20 100 UMG range calc type / LMG range calc type / PW range calc type 2 2 3 MG source and XS linearization / Pn order for PW calc 3 64 PW Inelastic scat option

PAGE 164

164 1 Mix XS print opt / Flux print opt / Balance Table print 2$$ 0 3 1 Vol source flag / Boundary sour ce flag / Group diffusion coeff 1 0 2 Density factors / Extra print opt / Mass scattering restriction 0 1 0 Restrictive order / correlate PW and MG flux / Not used 64 0 0 Upscatter converg / Point converge / Source normalization 3** 1.00000E-04 1.00000E-05 1.00000E+00 Material Buckling (1/cm) / LMG range cutoff / UMG range cutoff 0.00000E+00 1.00000E+00 1.00000E+07 Thinning tolerance / Frac Leth argy gain allowance / Not used 1.00000E-03 1.00000E-01 0.0+00 T Mixture numbers 13$$ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Isotope identifiers 14$$ 92235 92238 95241 95243 96244 96245 93237 94238 94239 94240 94241 94242 40090 40091 40092 40094 40096 26054 26056 26057 26058 24050 24052 24053 24054 42092 42094 42095 42096 42097 42098 42100 11023 Isotope concentrations (atoms/barn -cm) 15** 1.9614993E-05 9.6642491E-03 1.3566604E-04 3.4663778E-05 8.2528486E-06 4.6175738E-07 1.5733084E-04 5.5200518E-05 1.3077054E-03 6.5959724E-04 2.3354193E-04 1.5570373E-04 1.8573215E-03 4.0503688E-04 6.1910718E-04 6.2741007E-04 1.0107872E-04 7.5240415E-04 1.1811136E-02 2.7277064E-04 3.6300765E-05 8.2857689E-05 1.5978281E-03 1.8118088E-04 4.5099755E-05 1.2782226E-05 7.9673577E-06 1.3712469E-05 1.4367084E-05 8.2257585E-06 2.0784037E-05 8.2946654E-06 8.4013761E-03 T Source number by interval 30$$ 1 Volumetric source by interval and group 31** 9.60817162E+00 3.17979593E+01 6.59123252E+01 8.56636328E+01 2.43855982E+02 3.77567052E+03 1.28597182E+04 4.60351593E+04 1.38805606E+05 8.15300416E+04 3.79836723E+05 2.48427549E+05 7.00361135E+04

PAGE 165

165 3.27183503E+05 2.70261891E+05 8.33884439E+04 3.74944768E+04 3.36258630E+04 5.85884438E+04 4.43603989E+04 9.01939217E+04 8.26073314E+04 8.36331980E+04 1.86862972E+04 2.33946264E+04 1.30216633E+04 3.85419406E+04 6.56569747E+04 6.63587719E+04 8.37785121E+03 6.47534398E+04 2.47279973E+04 2.09256714E+04 4.53887848E+04 2.61351971E+04 2.61944753E+04 1.72324738E+04 1.70284606E+04 5.77192278E+04 4.66022640E+04 4.99461055E+04 3.18813210E+04 1.26057183E+04 1.50973866E+04 7.29640582E+03 1.39403173E+03 3.16257286E+03 8.79634792E+02 5.44082929E+03 3.09154071E+03 7.39829777E+02 1.78832320E+03 4.78581570E+03 1.37584123E+03 1.92836547E+03 8.18316897E+02 6.21239675E+02 2.30800307E+02 2.85248739E+02 2.47910671E+02 1.66350391E+01 7.22464162E+01 3.73318571E+01 2.40880149E+01 7.17972519E+00 3.01104369E+01 1.72285861E+01 3.28882021E+00 2.14460394E+01

PAGE 166

166 1.09147995E+01 1.28414744E+01 5.69800237E-01 4.99268036E+00 8.50801791E+00 5.78975626E-01 1.22854750E+00 7.58383952E-01 6.08854788E-02 3.57555982E-01 1.50727273E-01 1.33636800E+00 5.55129899E-02 7.29341847E-02 1.24595534E-01 1.37520215E-01 1.64285979E-01 1.25052132E-01 3.03432451E-02 5.95509100E-02 5.80038225E-02 6.33520693E-02 3.43022322E-02 5.35190864E-02 2.61154789E-02 7.07635424E-02 1.71328389E-02 1.69037426E-02 1.66714974E-02 1.05932305E-02 1.51276633E-02 2.06023547E-02 1.35097265E-02 1.77278940E-02 1.52402758E-02 1.49822591E-02 5.28742327E-03 1.15134061E-02 1.03232229E-02 1.52244183E-02 8.98228896E-03 8.37672169E-03 4.87860058E-03 1.44154236E-02 4.73073359E-03 1.16612495E-02 2.25959045E-02 2.15908251E-02 2.05365695E-02 1.17925057E-02 7.63261220E-03 7.44410831E-03 3.64984888E-03 1.06528502E-02 6.84752080E-03 5.98340417E-03

PAGE 167

167 4.53136990E-03 4.11089618E-03 5.23049636E-03 5.93595252E-03 2.30651781E-03 8.28927848E-03 4.68833258E-03 4.94308066E-03 4.42159243E-03 6.72609125E-04 1.10504696E-03 1.08476815E-03 1.06410289E-03 1.04302821E-03 2.41462206E-03 1.53760807E-03 9.30521181E-04 2.64379700E-03 8.94803183E-04 7.37182167E-04 1.07578364E-03 2.96845515E-04 1.46573523E-04 8.74074895E-05 2.88040554E-04 2.83093703E-04 2.78056296E-04 2.72842238E-04 2.67637866E-04 2.36266798E-04 2.06301869E-04 2.27834294E-04 2.23216548E-04 2.90342781E-04 1.41968410E-04 1.85896124E-04 2.04400187E-04 1.99289297E-04 1.93998326E-04 1.88561965E-04 1.02369949E-04 1.00619824E-04 9.88582973E-05 9.70245353E-05 9.51573073E-05 4.69024254E-05 4.63891110E-05 4.59451129E-05 4.54225447E-05 1.80570699E-05 1.79600451E-05 1.78629090E-05 1.78011816E-05 1.77388567E-05 1.76407264E-05 1.75424699E-05

PAGE 168

168 1.74789609E-05 1.74148298E-05 1.73155201E-05 1.72160676E-05 1.71506875E-05 1.70676326E-05 1.70011138E-05 1.68833560E-05 4.18810395E-05 4.13474874E-05 4.08069592E-05 4.02591739E-05 7.88444405E-05 7.65582839E-05 7.42017016E-05 7.17677517E-05 6.92482701E-05 3.36499964E-05 3.29835503E-05 6.39118808E-05 6.10689577E-05 5.80869496E-05 5.49431732E-05 2.62342689E-05 2.53737371E-05 2.44829653E-05 2.35585205E-05 2.25962658E-05 2.15911402E-05 2.05368407E-05 1.94253475E-05 1.82461874E-05 1.69852318E-05 1.56226084E-05 1.41287680E-05 5.19545583E-06 4.91426775E-06 4.61596116E-06 4.29696175E-06 3.95224224E-06 3.57432714E-06 3.15119418E-06 1.31712363E-06 3.39885712E-06 3.93899278E-07 3.32630523E-07 1.13030151E-07 9.96495113E-08 4.41874128E-08 3.99621932E-08 3.52314226E-08 1.85727491E-08 1.11786295E-08 1.24561889E-08 1.05187008E-08 1.14415023E-08

PAGE 169

169 1.08834188E-09 t Zone number by interval 36$$ 1 Mixture number by zone 39$$ 1 Temperature by zone 41** 9.00000E+02 t end

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175 BIOGRAPHICAL SKETCH Brian Triplett holds a bachelor ’s degree in nuclear engineering and a master’s degree in nuclear engineering sciences from the Universi ty of Florida. His professional experience includes internships at the Crystal River Unit 3 nuclear power plant a nd Los Alamos National Laboratory. He currently works at GE Hitach i Nuclear Energy on the development of advanced neutronic methods in support of the design and licensing of boiling water reactors and sodium fast reactors.