Nuclear Reactor Physics - Gamma Explorer · Nuclear Reactor Physics. Weston M. Stacey. Second...

Nuclear Reactor Physics

Weston M. Stacey

Second Edition, Completely Revised and Enlarged

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Weston M. StaceyNuclear Reactor Physics

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Nuclear Reactor Physics

Weston M. Stacey

Second Edition, Completely Revised and Enlarged

The Author

Prof. Weston M. StaceyGeorgia Institute of TechnologyNuclear & Radiological Engineering900 Atlantic Drive, NWAtlanta, GA 30332-0425USA

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vii

Contents

Preface xxiiiPreface to 2nd Edition xxvii

PART 1 BASIC REACTOR PHYSICS

1 Neutron Nuclear Reactions 31.1 Neutron-Induced Nuclear Fission 3

Stable Nuclides 3Binding Energy 3Threshold External Energy for Fission 4Neutron-Induced Fission 5Neutron Fission Cross Sections 5Products of the Fission Reaction 8Energy Release 10

1.2 Neutron Capture 13Radiative Capture 13Neutron Emission 19

1.3 Neutron Elastic Scattering 201.4 Summary of Cross-Section Data 24

Low-Energy Cross Sections 24Spectrum-Averaged Cross Sections 24

1.5 Evaluated Nuclear Data Files 241.6 Elastic Scattering Kinematics 27

Correlation of Scattering Angle and Energy Loss 28Average Energy Loss 29

2 Neutron Chain Fission Reactors 332.1 Neutron Chain Fission Reactions 33

Capture-to-Fission Ratio 33Number of Fission Neutrons per Neutron Absorbed in Fuel 33

viii Contents

Neutron Utilization 34Fast Fission 34Resonance Escape 36

2.2 Criticality 37Effective Multiplication Constant 37Effect of Fuel Lumping 37Leakage Reduction 38

2.3 Time Dependence of a Neutron Fission Chain Assembly 38Prompt Fission Neutron Time Dependence 38Source Multiplication 39Effect of Delayed Neutrons 39

2.4 Classification of Nuclear Reactors 40Physics Classification by Neutron Spectrum 40Engineering Classification by Coolant 41

3 Neutron Diffusion Theory 433.1 Derivation of One-Speed Diffusion Theory 43

Partial and Net Currents 43Diffusion Theory 45Interface Conditions 46Boundary Conditions 46Applicability of Diffusion Theory 47

3.2 Solutions of the Neutron Diffusion Equation in NonmultiplyingMedia 48Plane Isotropic Source in an Infinite Homogeneous Medium 48Plane Isotropic Source in a Finite Homogeneous Medium 48Line Source in an Infinite Homogeneous Medium 49Homogeneous Cylinder of Infinite Axial Extent with Axial LineSource 49Point Source in an Infinite Homogeneous Medium 49Point Source at the Center of a Finite Homogeneous Sphere 50

3.3 Diffusion Kernels and Distributed Sources in a HomogeneousMedium 50Infinite-Medium Diffusion Kernels 50Finite-Slab Diffusion Kernel 51Finite Slab with Incident Neutron Beam 52

3.4 Albedo Boundary Condition 523.5 Neutron Diffusion and Migration Lengths 53

Thermal Diffusion-Length Experiment 53Migration Length 55

3.6 Bare Homogeneous Reactor 57Slab Reactor 57Right Circular Cylinder Reactor 59

Contents ix

Interpretation of Criticality Condition 60Optimum Geometries 61

3.7 Reflected Reactor 62Reflected Slab Reactor 62Reflector Savings 64Reflected Spherical, Cylindrical, and Rectangular ParallelepipedCores 65

3.8 Homogenization of a Heterogeneous Fuel–Moderator Assembly 65Spatial Self-Shielding and Thermal Disadvantage Factor 65Effective Homogeneous Cross Sections 69Thermal Utilization 71Measurement of Thermal Utilization 72Local Power Peaking Factor 73

3.9 Control Rods 73Effective Diffusion Theory Cross Sections for Control Rods 73Windowshade Treatment of Control Rods 76

3.10 Numerical Solution of Diffusion Equation 77Finite Difference Equations in One Dimension 78Forward Elimination/Backward Substitution Spatial SolutionProcedure 79Power Iteration on Fission Source 79Finite-Difference Equations in Two Dimensions 80Successive Relaxation Solution of Two-DimensionalFinite-Difference Equations 82Power Outer Iteration on Fission Source 82Limitations on Mesh Spacing 83

3.11 Nodal Approximation 833.12 Transport Methods 85

Transmission and Absorption in a Purely Absorbing Slab ControlPlate 87Escape Probability in a Slab 87Integral Transport Formulation 87Collision Probability Method 88Differential Transport Formulation 89Spherical Harmonics Methods 90Discrete Ordinates Method 94

4 Neutron Energy Distribution 1014.1 Analytical Solutions in an Infinite Medium 101

Fission Source Energy Range 102Slowing-Down Energy Range 102Moderation by Hydrogen Only 103Energy Self-Shielding 103Slowing Down by Nonhydrogenic Moderators with No Absorption 104

x Contents

Slowing-Down Density 105Slowing Down with Weak Absorption 106Fermi Age Neutron Slowing Down 107Neutron Energy Distribution in the Thermal Range 108Summary 111

4.2 Multigroup Calculation of Neutron Energy Distribution in an InfiniteMedium 111Derivation of Multigroup Equations 111Mathematical Properties of the Multigroup Equations 113Solution of Multigroup Equations 114Preparation of Multigroup Cross-Section Sets 115

4.3 Resonance Absorption 117Resonance Cross Sections 117Doppler Broadening 119Resonance Integral 122Resonance Escape Probability 122Multigroup Resonance Cross Section 122Practical Width 122Neutron Flux in Resonance 123Narrow Resonance Approximation 123Wide Resonance Approximation 124Resonance Absorption Calculations 124Temperature Dependence of Resonance Absorption 127

4.4 Multigroup Diffusion Theory 127Multigroup Diffusion Equations 127Two-Group Theory 128Two-Group Bare Reactor 129One-and-One-Half-Group Theory 129Two-Group Theory of Two-Region Reactors 130Two-Group Theory of Reflected Reactors 133Numerical Solutions for Multigroup Diffusion Theory 137

5 Nuclear Reactor Dynamics 1435.1 Delayed Fission Neutrons 143

Neutrons Emitted in Fission Product Decay 143Effective Delayed Neutron Parameters for Composite Mixtures 145Photoneutrons 146

5.2 Point Kinetics Equations 1475.3 Period–Reactivity Relations 1485.4 Approximate Solutions of the Point Neutron Kinetics Equations 150

One Delayed Neutron Group Approximation 150Prompt-Jump Approximation 153Reactor Shutdown 154

Contents xi

5.5 Delayed Neutron Kernel and Zero-Power Transfer Function 155Delayed Neutron Kernel 155Zero-Power Transfer Function 155

5.6 Experimental Determination of Neutron Kinetics Parameters 156Asymptotic Period Measurement 156Rod Drop Method 157Source Jerk Method 157Pulsed Neutron Methods 157Rod Oscillator Measurements 158Zero-Power Transfer Function Measurements 159Rossi-α Measurement 159

5.7 Reactivity Feedback 161Temperature Coefficients of Reactivity 162Doppler Effect 162Fuel and Moderator Expansion Effect on Resonance EscapeProbability 164Thermal Utilization 165Nonleakage Probability 166Representative Thermal Reactor Reactivity Coefficients 166Startup Temperature Defect 167

5.8 Perturbation Theory Evaluation of Reactivity TemperatureCoefficients 168Perturbation Theory 168Sodium Void Effect in Fast Reactors 169Doppler Effect in Fast Reactors 169Fuel and Structure Motion in Fast Reactors 170Fuel Bowing 171Representative Fast Reactor Reactivity Coefficients 171

5.9 Reactor Stability 171Reactor Transfer Function with Reactivity Feedback 171Stability Analysis for a Simple Feedback Model 172Threshold Power Level for Reactor Stability 174More General Stability Conditions 175Power Coefficients and Feedback Delay Time Constants 178

5.10 Measurement of Reactor Transfer Functions 179Rod Oscillator Method 179Correlation Methods 179Reactor Noise Method 181

5.11 Reactor Transients with Feedback 183Step Reactivity Insertion (ρex < β): Prompt Jump 184Step Reactivity Insertion (ρex < β): Post-Prompt-Jump Transient 185

5.12 Reactor Fast Excursions 186Step Reactivity Input: Feedback Proportional to Fission Energy 186Ramp Reactivity Input: Feedback Proportional to Fission Energy 187

xii Contents

Step Reactivity Input: Nonlinear Feedback Proportional toCumulative Energy Release 187Bethe–Tait Model 188

5.13 Numerical Methods 190

6 Fuel Burnup 1976.1 Changes in Fuel Composition 197

Fuel Transmutation–Decay Chains 198Fuel Depletion–Transmutation–Decay Equations 199Fission Products 203Solution of the Depletion Equations 204Measure of Fuel Burnup 205Fuel Composition Changes with Burnup 205Reactivity Effects of Fuel Composition Changes 206Compensating for Fuel-Depletion Reactivity Effects 208Reactivity Penalty 208Effects of Fuel Depletion on the Power Distribution 209In-Core Fuel Management 210

6.2 Samarium and Xenon 211Samarium Poisoning 211Xenon Poisoning 213Peak Xenon 215Effect of Power-Level Changes 216

6.3 Fertile-to-Fissile Conversion and Breeding 217Availability of Neutrons 217Conversion and Breeding Ratios 219

6.4 Simple Model of Fuel Depletion 2196.5 Fuel Reprocessing and Recycling 221

Composition of Recycled LWR Fuel 221Physics Differences of MOX Cores 222Physics Considerations with Uranium Recycle 224Physics Considerations with Plutonium Recycle 225Reactor Fueling Characteristics 225

6.6 Radioactive Waste 226Radioactivity 226Hazard Potential 226Risk Factor 226

6.7 Burning Surplus Weapons-Grade Uranium and Plutonium 233Composition of Weapons-Grade Uranium and Plutonium 233Physics Differences Between Weapons- and Reactor-GradePlutonium-Fueled Reactors 234

6.8 Utilization of Uranium Energy Content 2356.9 Transmutation of Spent Nuclear Fuel 2376.10 Closing the Nuclear Fuel Cycle 244

Contents xiii

7 Nuclear Power Reactors 2497.1 Pressurized Water Reactors 2497.2 Boiling Water Reactors 2507.3 Pressure Tube Heavy Water–Moderated Reactors 2557.4 Pressure Tube Graphite-Moderated Reactors 2587.5 Graphite-Moderated Gas-Cooled Reactors 2607.6 Liquid-Metal Fast Breeder Reactors 2617.7 Other Power Reactors 2657.8 Characteristics of Power Reactors 2657.9 Advanced Generation-III Reactors 265

Advanced Boiling Water Reactors (ABWR) 266Advanced Pressurized Water Reactors (APWR) 267Advanced Pressure Tube Reactor 268Modular High-Temperature Gas-Cooled Reactors (GT-MHR) 268

7.10 Advanced Generation-IV Reactors 269Gas-Cooled Fast Reactors (GFR) 270Lead-Cooled Fast Reactors (LFR) 271Molten Salt Reactors (MSR) 271Super-Critical Water Reactors (SCWR) 272Sodium-Cooled Fast Reactors (SFR) 272Very High Temperature Reactors (VHTR) 272

7.11 Advanced Sub-critical Reactors 2737.12 Nuclear Reactor Analysis 275

Construction of Homogenized Multigroup Cross Sections 275Criticality and Flux Distribution Calculations 276Fuel Cycle Analyses 277Transient Analyses 278Core Operating Data 279Criticality Safety Analysis 279

7.13 Interaction of Reactor Physics and Reactor Thermal Hydraulics 280Power Distribution 280Temperature Reactivity Effects 281Coupled Reactor Physics and Thermal-Hydraulics Calculations 281

8 Reactor Safety 2838.1 Elements of Reactor Safety 283

Radionuclides of Greatest Concern 283Multiple Barriers to Radionuclide Release 283Defense in Depth 285Energy Sources 285

8.2 Reactor Safety Analysis 285Loss of Flow or Loss of Coolant 287Loss of Heat Sink 287

xiv Contents

Reactivity Insertion 287Anticipated Transients without Scram 288

8.3 Quantitative Risk Assessment 288Probabilistic Risk Assessment 288Radiological Assessment 291Reactor Risks 291

8.4 Reactor Accidents 293Three Mile Island 294Chernobyl 297

8.5 Passive Safety 299Pressurized Water Reactors 299Boiling Water Reactors 299Integral Fast Reactors 300Passive Safety Demonstration 300

PART 2 ADVANCED REACTOR PHYSICS

9 Neutron Transport Theory 3059.1 Neutron Transport Equation 305

Boundary Conditions 310Scalar Flux and Current 310Partial Currents 310

9.2 Integral Transport Theory 310Isotropic Point Source 311Isotropic Plane Source 311Anisotropic Plane Source 312Transmission and Absorption Probabilities 314Escape Probability 314First-Collision Source for Diffusion Theory 315Inclusion of Isotropic Scattering and Fission 315Distributed Volumetric Sources in Arbitrary Geometry 316Flux from a Line Isotropic Source of Neutrons 317Bickley Functions 318Probability of Reaching a Distance t from a Line Isotropic Sourcewithout a Collision 318

9.3 Collision Probability Methods 319Reciprocity Among Transmission and Collision Probabilities 320Collision Probabilities for Slab Geometry 320Collision Probabilities in Two-Dimensional Geometry 321Collision Probabilities for Annular Geometry 322

9.4 Interface Current Methods in Slab Geometry 323Emergent Currents and Reaction Rates Due to Incident Currents 323Emergent Currents and Reaction Rates Due to Internal Sources 326

Contents xv

Total Reaction Rates and Emergent Currents 327Boundary Conditions 329Response Matrix 329

9.5 Multidimensional Interface Current Methods 330Extension to Multidimension 330Evaluation of Transmission and Escape Probabilities 332Transmission Probabilities in Two-Dimensional Geometries 333Escape Probabilities in Two-Dimensional Geometries 335Simple Approximations for the Escape Probability 337

9.6 Spherical Harmonics (PL) Methods in One-DimensionalGeometries 338Legendre Polynomials 338Neutron Transport Equation in Slab Geometry 339PL Equations 339Boundary and Interface Conditions 340P1 Equations and Diffusion Theory 342Simplified PL or Extended Diffusion Theory 343PL Equations in Spherical and Cylindrical Geometries 344Diffusion Equations in One-Dimensional Geometry 347Half-Angle Legendre Polynomials 347Double-PL Theory 348D-P0 Equations 349

9.7 Multidimensional Spherical Harmonics (PL) Transport Theory 350Spherical Harmonics 350Spherical Harmonics Transport Equations in CartesianCoordinates 351P1 Equations in Cartesian Geometry 352Diffusion Theory 353

9.8 Discrete Ordinates Methods in One-Dimensional Slab Geometry 354PL and D-PL Ordinates 355Spatial Differencing and Iterative Solution 357Limitations on Spatial Mesh Size 358

9.9 Discrete Ordinates Methods in One-Dimensional SphericalGeometry 359Representation of Angular Derivative 360Iterative Solution Procedure 360Acceleration of Convergence 362Calculation of Criticality 362

9.10 Multidimensional Discrete Ordinates Methods 363Ordinates and Quadrature Sets 363SN Method in Two-Dimensional x–y Geometry 366Further Discussion 369

9.11 Even-Parity Transport Formulation 3699.12 Monte Carlo Methods 371

Probability Distribution Functions 371

xvi Contents

Analog Simulation of Neutron Transport 372Statistical Estimation 373Variance Reduction 375Tallying 377Criticality Problems 378Source Problems 379Random Numbers 380

10 Neutron Slowing Down 38510.1 Elastic Scattering Transfer Function 385

Lethargy 385Elastic Scattering Kinematics 385Elastic Scattering Kernel 386Isotropic Scattering in Center-of-Mass System 388Linearly Anisotropic Scattering in Center-of-Mass System 389

10.2 P1 and B1 Slowing-Down Equations 390Derivation 390Solution in Finite Uniform Medium 393B1 Equations 394Few-Group Constants 395

10.3 Diffusion Theory 396Lethargy-Dependent Diffusion Theory 396Directional Diffusion Theory 397Multigroup Diffusion Theory 398Boundary and Interface Conditions 399

10.4 Continuous Slowing-Down Theory 400P1 Equations in Slowing-Down Density Formulation 400Slowing-Down Density in Hydrogen 403Heavy Mass Scatterers 404Age Approximation 404Selengut–Goertzel Approximation 405Consistent P1 Approximation 405Extended Age Approximation 405Grueling–Goertzel Approximation 406Summary of Pl Continuous Slowing-Down Theory 407Inclusion of Anisotropic Scattering 407Inclusion of Scattering Resonances 409Pl Continuous Slowing-Down Equations 410

10.5 Multigroup Discrete Ordinates Transport Theory 411

11 Resonance Absorption 41511.1 Resonance Cross Sections 415

Contents xvii

11.2 Widely Spaced Single-Level Resonances in a HeterogeneousFuel–Moderator Lattice 415Neutron Balance in Heterogeneous Fuel–Moderator Cell 415Reciprocity Relation 418Narrow Resonance Approximation 419Wide Resonance Approximation 420Evaluation of Resonance Integrals 420Infinite Dilution Resonance Integral 422Equivalence Relations 422Heterogeneous Resonance Escape Probability 423Homogenized Multigroup Resonance Cross Section 423Improved and Intermediate Resonance Approximations 424

11.3 Calculation of First-Flight Escape Probabilities 424Escape Probability for an Isolated Fuel Rod 425Closely Packed Lattices 427

11.4 Unresolved Resonances 428Multigroup Cross Sections for Isolated Resonances 430Self-Overlap Effects 431Overlap Effects for Different Sequences 432

11.5 Multiband Treatment of Spatially Dependent Self-Shielding 433Spatially Dependent Self-Shielding 433Multiband Theory 434Evaluation of Multiband Parameters 436Calculation of Multiband Parameters 437Interface Conditions 439

11.6 Resonance Cross-Section Representations 439R-Matrix Representation 439Practical Formulations 441Generalization of the Pole Representation 445Doppler Broadening of the Generalized Pole Representation 448

12 Neutron Thermalization 45312.1 Double Differential Scattering Cross Section for Thermal

Neutrons 45312.2 Neutron Scattering from a Monatomic Maxwellian Gas 454

Differential Scattering Cross Section 454Cold Target Limit 455Free-Hydrogen (Proton) Gas Model 455Radkowsky Model for Scattering from H2O 455Heavy Gas Model 456

12.3 Thermal Neutron Scattering from Bound Nuclei 457Pair Distribution Functions and Scattering Functions 457Intermediate Scattering Functions 458Incoherent Approximation 459

xviii Contents

Gaussian Representation of Scattering 459Measurement of the Scattering Function 460Applications to Neutron Moderating Media 460

12.4 Calculation of the Thermal Neutron Spectra in HomogeneousMedia 462Wigner–Wilkins Proton Gas Model 463Heavy Gas Model 466Numerical Solution 468Moments Expansion Solution 470Multigroup Calculation 473Applications to Moderators 474

12.5 Calculation of Thermal Neutron Energy Spectra in HeterogeneousLattices 474

12.6 Pulsed Neutron Thermalization 477Spatial Eigenfunction Expansion 477Energy Eigenfunctions of the Scattering Operator 477Expansion in Energy Eigenfunctions of the Scattering Operator 479

13 Perturbation and Variational Methods 48313.1 Perturbation Theory Reactivity Estimate 483

Multigroup Diffusion Perturbation Theory 48313.2 Adjoint Operators and Importance Function 486

Adjoint Operators 486Importance Interpretation of the Adjoint Function 487Eigenvalues of the Adjoint Equation 489

13.3 Variational/Generalized Perturbation Reactivity Estimate 489One-Speed Diffusion Theory 490Other Transport Models 493Reactivity Worth of Localized Perturbations in a Large PWR CoreModel 494Higher-Order Variational Estimates 495

13.4 Variational/Generalized Perturbation Theory Estimates of ReactionRate Ratios in Critical Reactors 495

13.5 Variational/Generalized Perturbation Theory Estimates of ReactionRates 497

13.6 Variational Theory 498Stationarity 498Roussopolos Variational Functional 498Schwinger Variational Functional 499Rayleigh Quotient 499Construction of Variational Functionals 500

13.7 Variational Estimate of Intermediate Resonance Integral 50013.8 Heterogeneity Reactivity Effects 50213.9 Variational Derivation of Approximate Equations 503

Contents xix

13.10 Variational Even-Parity Transport Approximations 505Variational Principle for the Even-Parity Transport Equation 505Ritz Procedure 506Diffusion Approximation 507One-Dimensional Slab Transport Equation 508

13.11 Boundary Perturbation Theory 508

14 Homogenization 51514.1 Equivalent Homogenized Cross Sections 51614.2 ABH Collision Probability Method 51714.3 Blackness Theory 52014.4 Fuel Assembly Transport Calculations 522

Pin Cells 522Wigner–Seitz Approximation 523Collision Probability Pin-Cell Model 524Interface Current Formulation 527Multigroup Pin-Cell Collision Probabilities Model 528Resonance Cross Sections 529Full Assembly Transport Calculation 529

14.5 Homogenization Theory 529Homogenization Considerations 530Conventional Homogenization Theory 531

14.6 Equivalence Homogenization Theory 53114.7 Multiscale Expansion Homogenization Theory 53514.8 Flux Detail Reconstruction 538

15 Nodal and Synthesis Methods 54115.1 General Nodal Formalism 54215.2 Conventional Nodal Methods 54415.3 Transverse Integrated Nodal Diffusion Theory Methods 547

Transverse Integrated Equations 547Polynomial Expansion Methods 549Analytical Methods 553Heterogeneous Flux Reconstruction 554

15.4 Transverse Integrated Nodal Integral Transport Theory Models 554Transverse Integrated Integral Transport Equations 554Polynomial Expansion of Scalar Flux 557Isotropic Component of Transverse Leakage 558Double-Pn Expansion of Surface Fluxes 558Angular Moments of Outgoing Surface Fluxes 559Nodal Transport Equations 561

15.5 Transverse Integrated Nodal Discrete Ordinates Method 56115.6 Finite-Element Coarse Mesh Methods 563

xx Contents

Variational Functional for the P1 Equations 563One-Dimensional Finite-Difference Approximation 564Diffusion Theory Variational Functional 566Linear Finite-Element Diffusion Approximation in OneDimension 567Higher-Order Cubic Hermite Coarse-Mesh DiffusionApproximation 569Multidimensional Finite-Element Coarse-Mesh Methods 570

15.7 Variational Discrete Ordinates Nodal Method 571Variational Principle 571Application of the Method 579

15.8 Variational Principle for Multigroup Diffusion Theory 58015.9 Single-Channel Spatial Synthesis 58315.10 Multichannel Spatial Synthesis 58915.11 Spectral Synthesis 591

16 Space–Time Neutron Kinetics 59916.1 Flux Tilts and Delayed Neutron Holdback 599

Modal Eigenfunction Expansion 600Flux Tilts 601Delayed Neutron Holdback 602

16.2 Spatially Dependent Point Kinetics 602Derivation of Point Kinetics Equations 604Adiabatic and Quasistatic Methods 605Variational Principle for Static Reactivity 606Variational Principle for Dynamic Reactivity 607

16.3 Time Integration of the Spatial Neutron Flux Distribution 609Explicit Integration: Forward-Difference Method 610Implicit Integration: Backward-Difference Method 611Implicit Integration: θ Method 612Implicit Integration: Time-Integrated Method 615Implicit Integration: GAKIN Method 616Alternating Direction Implicit Method 619Stiffness Confinement Method 622Symmetric Successive Overrelaxation Method 623Generalized Runge–Kutta Methods 624

16.4 Stability 625Classical Linear Stability Analysis 625Lyapunov’s Method 627Lyapunov’s Method for Distributed Parameter Systems 629Control 631Variational Methods of Control Theory 631Dynamic Programming 633Pontryagin’s Maximum Principle 634

Contents xxi

Variational Methods for Spatially Dependent Control Problems 636Dynamic Programming for Spatially Continuous Systems 638Pontryagin’s Maximum Principle for a Spatially ContinuousSystem 639

16.5 Xenon Spatial Oscillations 641Linear Stability Analysis 642μ-Mode Approximation 644λ-Mode Approximation 645Nonlinear Stability Criterion 649Control of Xenon Spatial Power Oscillations 650Variational Control Theory of Xenon Spatial Oscillations 650

16.6 Stochastic Kinetics 652Forward Stochastic Model 653Means, Variances, and Covariances 656Correlation Functions 658Physical Interpretation, Applications, and Initial and BoundaryConditions 659Numerical Studies 660Startup Analysis 663

APPENDICES

A Physical Constants and Nuclear Data 669

B Some Useful Mathematical Formulas 675

C Step Functions, Delta Functions, and Other Functions 677C.1 Introduction 677C.2 Properties of the Dirac δ-Function 678

A. Alternative Representations 678B. Properties 678C. Derivatives 679

D Some Properties of Special Functions 681

E Introduction to Matrices and Matrix Algebra 687E.1 Some Definitions 687E.2 Matrix Algebra 689

xxii Contents

F Introduction to Laplace Transforms 691F.1 Motivation 691F.2 “Cookbook” Laplace transforms 694

Index 697

xxiii

Preface

Nuclear reactor physics is the physics of neutron fission chain reacting systems.It encompasses those applications of nuclear physics and radiation transport andinteraction with matter that determine the behavior of nuclear reactors. As such, itis both an applied physics discipline and the core discipline of the field of nuclearengineering.

As a distinct applied physics discipline, nuclear reactor physics originated inthe middle of the twentieth century in the wartime convergence of internationalphysics efforts in the Manhattan Project. It developed vigorously for roughly thenext third of the century in various government, industrial, and university R&Dand design efforts worldwide. Nuclear reactor physics is now a relatively maturediscipline, in that the basic physical principles governing the behavior of nuclearreactors are well understood, most of the basic nuclear data needed for nuclear re-actor analysis have been measured and evaluated, and the computational method-ology is highly developed and validated. It is now possible to accurately predictthe physics behavior of existing nuclear reactor types under normal operating con-ditions. Moreover, the basic physical concepts, nuclear data, and computationalmethodology needed to develop an understanding of new variants of existing reac-tor types or of new reactor types exist for the most part.

As the core discipline of nuclear engineering, nuclear reactor physics is funda-mental to the major international nuclear power undertaking. As of 2000, there are434 central station nuclear power reactors operating worldwide to produce 350,442MWe of electrical power. This is a substantial fraction of the world’s electrical power(e.g., more than 80% of the electricity produced in France and more than 20% ofthe electricity produced in the United States). The world’s electrical power require-ments will continue to increase, particularly as the less developed countries striveto modernize, and nuclear power is the only proven technology for meeting thesegrowing electricity requirements without dramatically increasing the already unac-ceptable levels of greenhouse gas emission into the atmosphere.

Nuclear reactors have additional uses other than central station electricity pro-duction. There are more than 100 naval propulsion reactors in the U.S. fleet (plusothers in foreign fleets). Nuclear reactors are also employed for basic neutronphysics research, for materials testing, for radiation therapy, for the productionof radio-isotopes for medical, industrial, and national security applications, and

xxiv Preface

as mobile power sources for remote stations. In the future, nuclear reactors maypower deep space missions. Thus nuclear reactor physics is a discipline importantto the present and future well-being of the world.

This book is intended as both a textbook and a comprehensive reference on nu-clear reactor physics. The basic physical principles, nuclear data, and computa-tional methodology needed to understand the physics of nuclear reactors are devel-oped and applied to explain the static and dynamic behavior of nuclear reactors inPart 1. This development is at a level that should be accessible to seniors in physicsor engineering (i.e., requiring a mathematical knowledge only through ordinaryand partial differential equations and Laplace transforms and an undergraduate-level knowledge of atomic and nuclear physics). Mastery of the material presentedin Part 1 provides an understanding of the physics of nuclear reactors sufficient fornuclear engineering graduates at the B.S. and M.S. levels, for most practicing nu-clear engineers and for others interested in acquiring a broad working knowledgeof nuclear reactor physics.

The material in Part 1 was developed in the process of teaching undergradu-ate and first-year graduate courses in nuclear reactor physics at Georgia Tech fora number of years. The emphasis in the presentation is on conveying the basicphysical concepts and their application to explain nuclear reactor behavior, usingthe simplest mathematical description that will suffice to illustrate the physics.Numerous examples are included to illustrate the step-by-step procedures for car-rying out the calculations discussed in the text. Problems at the end of each chapterhave been chosen to provide physical insight and to extend the material discussedin the text, while providing practice in making calculations; they are intended asan integral part of the textbook. Part 1 is suitable for an undergraduate semester-length course in nuclear reactor physics; the material in Part 1 is also suitable fora semester-length first-year graduate course, perhaps with selective augmentationfrom Part 2.

The purpose of Part 2 is to augment Part 1 to provide a comprehensive, detailed,and advanced development of the principal topics of nuclear reactor physics. Thereis an emphasis in Part 2 on the theoretical bases for the advanced computationalmethods of reactor physics. This material provides a comprehensive, though nec-essarily abridged, reference work on advanced nuclear reactor physics and the the-oretical bases for its computational methods. Although the material stops shortof descriptions of specific reactor physics codes, it provides the basis for under-standing the code manuals. There is more than enough material in Part 2 for asemester-length advanced graduate course in nuclear reactor physics. The treat-ment is necessarily somewhat more mathematically intense than in Part 1.

Part 2 is intended primarily for those who are or would become specialists innuclear reactor physics and reactor physics computations. Mastery of this materialprovides the background for creating the new physics concepts necessary for devel-oping new reactor types and for understanding and extending the computationalmethods in existing reactor physics codes (i.e., the stock-in-trade for the profes-sional reactor physicist). Moreover, the extensive treatment of neutron transportcomputational methods also provides an important component of the background

Preface xxv

necessary for specialists in radiation shielding, for specialists in the applicationsof neutrons and photons in medicine and industry, and for specialists in neutron,photon, and neutral atom transport in industrial, astrophysical, and thermonuclearplasmas.

Any book of this scope owes much to many people besides the author, and thisone is no exception. The elements of the subject of reactor physics were developedby many talented people over the past half-century, and the references can only be-gin to recognize their contributions. In this regard, I note the special contributionof R.N. Hwang, who helped prepare certain sections on resonance theory. The se-lection and organization of material has benefited from the example of previousauthors of textbooks on reactor physics. The feedback from a generation of stu-dents has assisted in shaping the organization and presentation. Several people(C. Nickens, B. Crumbly, S. Bennett-Boyd) supported the evolution of the manu-script through at least three drafts, and several other people at Wiley transformedthe manuscript into a book. I am grateful to all of these people, for without themthere would be no book.

Atlanta, Georgia Weston M. StaceyOctober 2000

xxvii

Preface to 2nd Edition

This second edition differs from the original in two important ways. First, a sectionon neutron transport methods has been added in Chapter 3 to provide an introduc-tion to that subject in the first section of the book on basic reactor physics that isintended as the text for an advanced undergraduate course. My original intentionwas to use diffusion theory to introduce reactor physics, without getting into themathematical complexities of transport theory. I think this works reasonably wellfrom a pedagogical point of view, but it has the disadvantage of sending BS gradu-ates into the workplace without an exposure to transport theory. So, a short sectionon transport methods in slab geometry was added at the end of the diffusion theorychapter to provide an introduction.

Second, there has been a resurgence in interest and activity in the improvementof reactor designs and in the development of new reactor concepts that are more in-herently safe, better utilize the uranium resources, discharge less long-lived wasteand are more resistant to the diversion of fuels to other uses. A section has beenadded in Chapter 7 on the improved Generation-III designs that will be comingonline over the next decade or so, and a few sections have been added in Chap-ters 6 and 7 on the new reactor concepts being developed under the Generation-IVand Advanced Fuel Cycle Initiatives with the objective of closing the nuclear fuelcycle.

The text was amplified for the sake of explication in a few places, some addi-tional homework problems were included, and numerous typos, omissions andother errors that slipped through the final proof-reading of the first edition werecorrected. I am grateful to colleagues, students and particularly the translatorspreparing a Russian edition of the book for calling several such mistakes to myattention.

Otherwise, the structure and context of the book remains unchanged. The firsteight chapters on basic reactor physics provide the text for a first course in reactorphysics at the advanced undergraduate or graduate level. The second eight chapterson advanced reactor physics provide a text suitable for graduate courses on neutrontransport theory and reactor physics.

I hope that this second edition will serve to introduce to the field the new genera-tion of scientists and engineers who will carry forward the emerging resurgence of

xxviii Preface to 2nd Edition

nuclear power to meet the growing energy needs of mankind in a safe, economical,environmentally sustainable and proliferation-resistant way.

Weston M. StaceyAtlanta, GeorgiaMay 2006

Part 1: Basic Reactor Physics

3

1Neutron Nuclear Reactions

The physics of nuclear reactors is determined by the transport of neutrons andtheir interaction with matter within a reactor. The basic neutron nucleus reactionsof importance in nuclear reactors and the nuclear data used in reactor physics cal-culations are described in this chapter.

1.1Neutron-Induced Nuclear Fission

Stable Nuclides

Short-range attractive nuclear forces acting among nucleons (neutrons and pro-tons) are stronger than the Coulomb repulsive forces acting among protons at dis-tances on the order of the nuclear radius (R ≈ 1.25 × 10−13A1/3 cm) in a stablenucleus. These forces are such that the ratio of the atomic mass A (the number ofneutrons plus protons) to the atomic number Z (the number of protons) increaseswith Z; in other words, the stable nuclides become increasingly neutron-rich withincreasing Z, as illustrated in Fig. 1.1. The various nuclear species are referred toas nuclides, and nuclides with the same atomic number are referred to as isotopes ofthe element corresponding to Z. We use the notation AXZ (e.g., 235U92) to identifynuclides.

Binding Energy

The actual mass of an atomic nucleus is not the sum of the masses (mp) of the Zprotons and the masses (mn) of A−Z neutrons of which it is composed. The stablenuclides have a mass defect

�= [Zmp + (A−Z)mn] − Amz (1.1)

This mass defect is conceptually thought of as having been converted to energy(E = �c2) at the time that the nucleus was formed, putting the nucleus into anegative energy state. The amount of externally supplied energy that would have

4 1 Neutron Nuclear Reactions

Fig. 1.1 Nuclear stability curve. (From Ref. 1; used with permission of McGraw-Hill.)

to be converted to mass in disassembling a nucleus into its separate nucleons isknown as the binding energy of the nucleus, BE = �c2. The binding energy pernucleon (BE/A) is shown in Fig. 1.2.

Any process that results in nuclides being converted to other nuclides with morebinding energy per nucleon will result in the conversion of mass into energy. Thecombination of lowA nuclides to form higherA nuclides with a higher BE/A valueis the basis for the fusion process for the release of nuclear energy. The splitting ofvery high A nuclides to form intermediate-A nuclides with a higher BE/A value isthe basis of the fission process for the release of nuclear energy.

Threshold External Energy for Fission

The probability of any nuclide undergoing fission (reconfiguring its A nucleonsinto two nuclides of lower A) can become quite large if a sufficient amount of ex-ternal energy is supplied to excite the nucleus. The minimum, or threshold, amountof such excitation energy required to cause fission with high probability depends onthe nuclear structure and is quite large for nuclides with Z < 90. For nuclides withZ > 90, the threshold energy is about 4 to 6 MeV for even-A nuclides, and gener-ally is much lower for odd-A nuclides. Certain of the heavier nuclides (e.g., 240Pu94and 252Cf98) exhibit significant spontaneous fission even in the absence of any ex-ternally supplied excitation energy.

1.1 Neutron-Induced Nuclear Fission 5

Fig. 1.2 Binding energy per nucleon. (From Ref. 1; used with permission of McGraw-Hill.)

Neutron-Induced Fission

When a neutron is absorbed into a heavy nucleus (A,Z) to form a compound nu-cleus (A+ 1,Z), the BE/A value is lower for the compound nucleus than for theoriginal nucleus. For some nuclides (e.g., 233U92, 235U92, 239Pu94, 241Pu94), thisreduction in BE/A value is sufficient that the compound nucleus will undergo fis-sion, with high probability, even if the neutron has very low energy. Such nuclidesare referred to as fissile; that is, they can be caused to undergo fission by the ab-sorption of a low-energy neutron. If the neutron had kinetic energy prior to beingabsorbed into a nucleus, this energy is transformed into additional excitation en-ergy of the compound nucleus. All nuclides with Z > 90 will undergo fission withhigh probability when a neutron with kinetic energy in excess of about 1 MeV isabsorbed. Nuclides such as 232Th90, 238U92, and 240Pu94 will undergo fission withneutrons with energy of about 1 MeV or higher, with high probability.

Neutron Fission Cross Sections

The probability of a nuclear reaction, in this case fission, taking place can be ex-pressed in terms of a quantity σ which expresses the probable reaction rate for nneutrons traveling with speed v a distance dx in a material with N nuclides perunit volume:

σ ≡ reaction ratenvN dx

(1.2)

The units of σ are area, which gives rise to the concept of σ as a cross-sectionalarea presented to the neutron by the nucleus, for a particular reaction process, and


Fig. 1.3 Fission cross sections for 233U92. (From http://www.nndc.bnl.gov/.)

to the designation of σ as a cross section. Cross sections are usually on the order of10−24 cm2, and this unit is referred to as a barn, for historical reasons.

The fission cross section, σf , is a measure of the probability that a neutron anda nucleus interact to form a compound nucleus which then undergoes fission.The probability that a compound nucleus will be formed is greatly enhanced ifthe relative energy of the neutron and the original nucleus, plus the reduction inthe nuclear binding energy, corresponds to the difference in energy of the groundstate and an excited state of the compound nucleus, so that the energetics are justright for formation of a compound nucleus in an excited state. The first excitedstates of the compound nuclei resulting from neutron absorption by odd-A fissilenuclides are generally lower lying (nearer to the ground state) than are the firstexcited states of the compound nuclei resulting from neutron absorption by theheavy even-A nuclides, which accounts for the odd-A nuclides having much largerabsorption and fission cross sections for low-energy neutrons than do the even-Anuclides.

Fission cross sections for some of the principal fissile nuclides of interest fornuclear reactors are shown in Figs. 1.3 to 1.5. The resonance structure correspondsto the formation of excited states of the compound nuclei, the lowest lying of whichare at less than 1 eV. The nature of the resonance cross section can be shown to giverise to a 1/E1/2 or 1/υ dependence of the cross section at off-resonance neutronenergies below and above the resonance range, as is evident in these figures. Thefission cross sections are largest in the thermal energy region E



Fig. 1.5 Fission cross sections for 239Pu94. (From http://www.nndc.bnl.gov/.)



Fission cross sections for 238U92 and 240Pu94 are shown in Figs. 1.6 and 1.7.Except for resonances, the fission cross section is insignificant below about 1 MeV,above which it is about 1 barn. The fission cross sections for these and other even-Aheavy mass nuclides are compared in Fig. 1.8, without the resonance structure.

Products of the Fission Reaction

A wide range of nuclides are formed by the fission of heavy mass nuclides, but thedistribution of these fission fragments is sharply peaked in the mass ranges 90<A< 100 and 135


Fig. 1.7 Fission cross sections for 240Pu94. (From http://www.nndc.bnl.gov/.)

Fig. 1.8 Fission cross sections for principal nonfissile heavymass nuclides. (From Ref. 15; used with permission of ArgonneNational Laboratory.)


Fig. 1.9 Yield versus mass number for 235U92 fission. (From Ref. 15.)

Energy Release

The majority of the nuclear energy created by the conversion of mass to energyin the fission event (207 MeV for 233U92) is in the form of the kinetic energy(168 MeV) of the recoiling fission fragments. The range of these massive, highlycharged particles in the fuel element is a fraction of a millimeter, so that the recoilenergy is effectively deposited as heat at the point of fission. Another 5 MeV is inthe form of kinetic energy of prompt neutrons released in the fission event, dis-tributed in energy as shown in Fig. 1.11, with a most likely energy of 0.7 MeV (for235U92). This energy is deposited in the surrounding material within 10 to 100 cmas the neutron diffuses, slows down by scattering collisions with nuclei, and isfinally absorbed. A fraction of these neutron absorption events result in neutroncapture followed by gamma emission, producing on average about 7 MeV in theform of energetic capture gammas per fission. This secondary capture gamma en-


Fig. 1.10 Average number of neutrons emitted per fission.(From Ref. 12; used with permission of Wiley.)

Fig. 1.11 Fission spectrum for thermal neutron-induced fissionin 235U92. (From Ref. 12; used with permission of Wiley.)


Table 1.1 235U92 Fission Energy Release

Form Energy (MeV) Range

Kinetic energy fission products 168

1.2 Neutron Capture 13

Thus, in total, about 200 MeV per fission of heat energy is produced. One Watt ofheat energy then corresponds to the fission of 3.1× 1010 nuclei per second. Since1 g of any fissile nuclide contains about 2.5 × 1021 nuclei, the fissioning of 1 gof fissile material produces about 1 megawatt-day (MWd) of heat energy. Becausesome fissile nuclei will also be transmuted by neutron capture, the amount of fissilematerial destroyed is greater than the amount fissioned.

1.2Neutron Capture

Radiative Capture

When a neutron is absorbed by a nucleus to form a compound nucleus, a num-ber of reactions are possible, in addition to fission, in the heavy nuclides. We havealready mentioned radiative capture, in which the compound nucleus decays bythe emission of a gamma ray, and we now consider this process in more detail.An energy-level diagram for the compound nucleus formation and decay associ-ated with the prominent 238U92 resonance for incident neutron energies of about6.67 eV is shown in Fig. 1.12. The energy in the center-of-mass (CM) system of anincident neutron with energy EL in the lab system is Ec = [A/(1 + A)]EL. Thereduction in binding energy due to the absorbed neutron is �EB . If Ec +�EB isclose to an excited energy level of the compound nucleus, the probability for com-

Fig. 1.12 Energy-level diagram for compound nucleusformation. (From Ref. 12; used with permission of Wiley.)


Fig. 1.13 Radiative capture cross section for 232Th90. (From http://www.nndc.bnl.gov/.)

pound nucleus formation is greatly enhanced. The excited compound nucleus willgenerally decay by emission of one or more gamma rays, the combined energy ofwhich is equal to the difference in the excited- and ground-state energy levels of thecompound nucleus.

Radiative capture cross sections, denoted σγ , for some nuclei of interest for nu-clear reactors are shown in Figs. 1.13 to 1.21. The resonance nature of the cross sec-tions over certain ranges correspond to the discrete excited states of the compoundnucleus that is formed upon neutron capture. These excited states correspond toneutron energies in the range of a fraction of an eV to 103 eV for the fissile nu-clides, generally correspond to neutron energies of 10 to 104 eV for even-A heavymass nuclides (with the notable exception of thermal 240Pu94 resonance), and cor-respond to much higher neutron energies for the lower mass nuclides. The 1/ν“off-resonance” cross-section dependence is apparent.

The Breit–Wigner single-level resonance formula for the neutron capture crosssection is

σγ (Ec)= σ0γ

(E0

Ec

)1/2 11+ y2 , y =

2

(Ec −E0) (1.3)

where E0 is the energy (in the CM) system at which the resonance peak occurs (i.e.,Ec +�EB matches the energy of an excited state of the compound nucleus), thefull width at half-maximum of the resonance, σ0 the maximum value of the totalcross section (at E0), and γ the radiative capture width (γ / is the probabilitythat the compound nucleus, once formed, will decay by gamma emission). The


Fig. 1.14 Radiative capture cross section for 233U92. (From http://www.nndc.bnl.gov/.)



Fig. 1.16 Radiative capture cross section for 239Pu94. (From http://www.nndc.bnl.gov/.)



Fig. 1.18 Radiative capture cross section for 240Pu94. (From http://www.nndc.bnl.gov/.)

Fig. 1.19 Radiative capture cross section for 56Fe26. (From http://www.nndc.bnl.gov/.)


Fig. 1.20 Radiative capture cross section for 23Na11. (From http://www.nndc.bnl.gov/.)

Fig. 1.21 Radiative capture cross section for 1H1. (From http://www.nndc.bnl.gov/.)


fission resonance cross section can be represented by a similar expression with thefission width f , defined such that f / is the probability that the compoundnucleus, once formed, will decay by fission.

Equation (1.3) represents the cross section describing the interaction of a neu-tron and nucleus with relative (CM) energy Ec . However, the nuclei in a materialare distributed in energy (approximately a Maxwellian distribution characterized bythe temperature of the material). What is needed is a cross section averaged overthe motion of the nuclei:

σ̄ (E,T )= 1υ(E)

∫dE ′ |v(E)− v(E ′)|σ(Ec)fmax(E ′, T ) (1.4)

whereE andE ′ are the neutron and nuclei energies, respectively, in the lab system,and fmax(E ′) is the Maxwellian energy distribution:

fmax(E′)= 2π

(πkT )3/2

√E ′e−E ′/kT (1.5)

Using Eqs. (1.3) and (1.5), Eq. (1.4) becomes

σ̄γ (E,T )= σ0γ

(E0

E

)1/2�(ξ, x) (1.6)

where

x = 2

(E −E0), ξ =

(4E0kT /A)1/2(1.7)

A is the atomic mass (amu) of the nuclei, and

�(ξ, x)= ξ2√π

∫ ∞−∞e−(1/4)(x−y)2ξ2 dy

1+ y2 (1.8)

Neutron Emission

When the compound nucleus formed by neutron capture decays by the emission ofone neutron, leaving the nucleus in an excited state which subsequently undergoesfurther decays, the event is referred to as inelastic scattering and the cross section isdenoted σin. Since the nucleus is left in an excited state, the energy of the emittedneutron can be considerably less than the energy of the incident neutron. If thecompound nucleus decays by the emission of two or more neutrons, the eventsare referred to as n − 2n, n − 3n, and so on, events, and the cross sections aredenoted σn,2n, σn,3n, on so on. Increasingly higher incident neutron energies arerequired to provide enough excitation energy for single, double, triple, and so on,neutron emission. Inelastic scattering is the most important of these events innuclear reactors, but it is most important for neutrons 1 MeV and higher in energy.


1.3Neutron Elastic Scattering

Elastic scattering may take place via compound nucleus formation followed by theemission of a neutron that returns the compound nucleus to the ground state ofthe original nucleus. In such a resonance elastic scattering event the kinetic energyof the original neutron–nuclear system is conserved. The neutron and the nucleusmay also interact without neutron absorption and the formation of a compoundnucleus, which is referred to as potential scattering. Although quantum mechani-cal (s-wave) in nature, the latter event may be visualized and treated as a classicalhard-sphere scattering event, away from resonance energies. Near resonance en-ergies, there is quantum mechanical interference between the potential and reso-nance scattering, which is constructive just above and destructive just below theresonance energy.

The single-level Breit–Wigner form of the scattering cross section, modified toinclude potential and interference scattering, is

σs(Ec)= σ0n

(E0

Ec

)1/2 11+ y2 +

σ02R

λ0

y

1+ y2 + 4πR2 (1.9)

where (n/) is the probability that, once formed, the compound nucleus de-cays to the ground state of the original nucleus by neutron emission, R � 1.25×10−13A1/3 centimeters is the nuclear radius, and λ0 is the reduced neutron wave-length.

Averaging over a Maxwellian distribution of nuclear motion yields the scatteringcross section for neutron lab energy E and material temperature T :

σ̄s (E,T )= σ0n

ψ(ξ, x)+ σ0R

λ0χ(ξ, x)+ 4πR2 (1.10)

where

χ(ξ, x)= ξ√π

∫ ∞−∞

ye−(1/4)(x−y)2ξ2

1+ y2 dy (1.11)

The elastic scattering cross sections for a number of nuclides of interest in nu-clear reactors are shown in Figs. 1.22 to 1.26. In general, the elastic scattering crosssection is almost constant in energy below the neutron energies corresponding tothe excited states of the compound nucleus. The destructive interference effectsjust below the resonance energy are very evident in Fig. 1.26.

The energy dependence of the carbon scattering cross section is extended to verylow neutron energies in Fig. 1.27 to illustrate another phenomenon. At sufficientlysmall neutron energy, the neutron wavelength

λ0 = hp= h√

2mE= 2.86× 10

−9√E(eV)

cm (1.12)

1.3 Neutron Elastic Scattering 21

Fig. 1.22 Elastic scattering cross section for 1H1. (From http://www.nndc.bnl.gov/.)

Fig. 1.23 Elastic scattering cross section for 16O8. (From http://www.nndc.bnl.gov/.)


Fig. 1.24 Elastic scattering cross section for 23Na11. (From http://www.nndc.bnl.gov/.)

Fig. 1.25 Elastic scattering cross section for 56Fe26. (From http://www.nndc.bnl.gov/.)

1.3 Neutron Elastic Scattering 23

Fig. 1.26 Elastic scattering cross section for 238U92. (From http://www.nndc.bnl.gov/.)

Fig. 1.27 Total scattering cross section of 12C6. (From Ref. 12; used with permission of Wiley.)


becomes comparable to the interatomic spacing, and the neutron interacts not witha single nucleus but with an aggregate of bound nuclei. If the material has a regu-lar structure, as graphite does, the neutron will be diffracted and the energy depen-dence of the cross section will reflect the neutron energies corresponding to mul-tiples of interatomic spacing. For sufficiently small energies, diffraction becomesimpossible and the cross section is once again insensitive to neutron energy.

1.4Summary of Cross-Section Data

Low-Energy Cross Sections

The low-energy total cross sections for several nuclides of interest in nuclear reac-tors are plotted in Fig. 1.28. Gadolinium is sometimes used as a “burnable poison,”and xenon and samarium are fission products with large thermal cross sections.

Spectrum-Averaged Cross Sections

Table 1.3 summarizes the cross-section data for a number of important nuclidesin nuclear reactors. The first three columns give fission, radiative capture, andelastic scattering cross sections averaged over a Maxwellian distribution with T =0.0253 eV, corresponding to a representative thermal energy spectrum. The nexttwo columns give the infinite dilution fission and radiative capture resonance in-tegrals, which are averages of the respective resonance cross sections over a 1/Espectrum typical of the resonance energy region in the limit of an infinitely diluteconcentration of the resonance absorber. The final five columns give cross sectionsaveraged over the fission spectrum.

Example 1.1: Calculation of Macroscopic Cross Section. The macroscopic cross sec-tion � = Nσ , where N is the number density. The number density is related tothe density ρ and atomic number A by N = (ρ/A)N0, where N0 = 6.022 × 1023is Avogadro’s number, the number of atoms in a mole. For a mixture of isotopeswith volume fractions υi , the macroscopic cross section is � = �iυi(ρ/A)iN0σi ;for example, for a 50:50 vol % mixture of carbon and 238U, the macroscopic ther-mal absorption cross section is �a = 0.5(ρC/AC)N0σaC + 0.5(ρU/AU)N0σaU =0.5(1.60 g/cm3 per 12 g/mol)(6.022 × 1023 atom/mol)(0.003 × 10−24 cm2) +0.5(18.9 g/cm3 per 238 g/mol)(6.022 × 1023 atom/mol)(2.4 × 10−24 cm2) =0.0575 cm−1.

1.5Evaluated Nuclear Data Files

Published experimental and theoretical results on neutron–nuclear reactions arecollected by several collaborating nuclear data agencies worldwide. Perhaps the

1.5 Evaluated Nuclear Data Files 25

Tabl

e1.

3Sp

ectr

um-A

vera

ged

Ther

mal

,Res

onan

ce,a

ndFa

stN

eutr

onCr

oss

Sect

ions

(bar

ns)

The

rmal

Cro

ssSe

ctio

nR

eson

ance

Cro

ssSe

ctio

nFi

ssio

nSp

ectr

umC

ross

Sect

ion

Nuc

lide

σf

σγ

σel

σf

σγ

σf

σγ

σel

σin

σn,2

n

233U

9246

941

11.9

774

138

1.9

0.07

4.4

1.2

4×

10−3

235U

9250

787

15.0

278

133

1.2

0.09

4.6

1.8

12×

10−3

239Pu

9469

827

47.

830

318

21.

80.

054.

41.

54×

10−3

241Pu

9493

832

611.1

573

180

1.6

0.12

5.2

0.9

21×

10−3

232T

h 90

–6.

513.7

–84

0.08

0.09

4.6

2.9

14×

10−3

238U

92–

2.4

9.4

227

80.

310.

074.

82.

612×

10−3

240Pu

940.

0526

41.

58.

981

031.

40.

094.

32.

04×

10−3

242Pu

94–

16.8

8.3

5.6

1130

1.1

0.09

4.8

1.9

7×

10−3

1H

1–

0.29

20.5

–0.

15–

4×

10−5

3.9

––

2H

1–

5×

10−4

3.4

–3×

10−4

–7×

10−6

2.5

––

10B

5–

443

2.1

–0.

22–

8×

10−5

2.1

0.07

–12

C6

–0.

003

4.7

–0.

002

–2×

10−5

2.3

0.01

–16

O8

–2×

10−4

3.8

–6×

10−4

–9×

10−5

2.7

––

23N

a 11

–0.

473.

0–

0.31

–2×

10−4

2.7

0.5

–56

Fe26

–2.

512.5

–1.

4–

3×

10−3

3.0

0.7

–91

Zr 4

0–

1.1

10.6

–6.

9–

0.01

5.0

0.7

–13

5X

e 54

–2.

7×

106

3.8×

105

–7.

6×

103

–0.

014.

91.

0–

149Sm

62–

6.0×

104

373

–3.

5×

103

–0.

224.

62.

2–

157G

d 64

–1.

9×

103

819

–76

1–

0.11

4.7

2.2

11×

10−3

Sour

ce:D

ata

from

http

://w

ww.

nndc

.bnl

.gov

/.


Fig. 1.28 Low-energy absorption (fission + capture) crosssections for several important nuclides. (From Ref. 12; usedwith permission of Wiley.)

most comprehensive computerized compilation of experimental data is the EXFORcomputer library (Ref. 11). The computerized card index file CINDA (Ref. 8), whichcontains comprehensive information on measurements, calculations, and evalua-tions of neutron–nuclear data, is updated annually. The plethora of sometimes con-tradictory nuclear data must be evaluated before it can be used confidently in reac-tor physics calculations. Such evaluation consists of intercomparison of data, use ofdata to calculate benchmark experiments, critical assessment of statistical and sys-tematic errors, checks for internal consistency and consistency with standard neu-tron cross sections, and the derivation of consistent preferred values by appropri-ate averaging procedures. Several large evaluated nuclear data files are maintained:

1.6 Elastic Scattering Kinematics 27

Fig. 1.29 Scattering event in lab and CM systems. (FromRef. 12; used with permission of Wiley.)

(1) United States Evaluated Nuclear Data File (ENDF/B), (2) Evaluated Nuclear DataLibrary of the Lawrence Livermore National Laboratory (ENDL), (3) United King-dom Nuclear Data Library (UKNDL), (4) Japanese Evaluated Nuclear Data Library(JENDL), (5) Karlsruhe Nuclear Data File (KEDAK), (6) Russian (formerly Soviet)Evaluated Nuclear Data File (BROND), and (7) Joint Evaluated File of NEA Coun-tries (JEF). Processing codes are used to convert these data to a form that can beused in reactor physic calculations, as discussed in subsequent chapters.

1.6Elastic Scattering Kinematics

Consider a neutron with energy EL = 12mv2L in the laboratory (L) system incidentupon a stationary nucleus of massM . Since only the relative masses are importantin the kinematics, we set m = 1 and M = A. It is convenient to convert to thecenter-of-mass (CM) system, as indicated in Fig. 1.29, because the elastic scatteringevent is usually isotropic in the CM system.


The velocity of the CM system in the L system is

vcm = 11+A(vL +AVL)=

vL1+A (1.13)

and the velocities of the neutron and the nucleus in the CM system are

vc = vL − vcm = AA+ 1 vL

(1.14)

Vc =−vcm = −1A+ 1 vL

The energy of the neutron in the CM system, Ec , is related to the energy of theneutron in the lab, EL, by

Ec = 12

v2c +1

2AV 2c =

A

A+ 11

2v2L =

A

A+ 1EL (1.15)

Correlation of Scattering Angle and Energy Loss

From consideration of conservation of momentum and kinetic energy, it can beshown that the speeds of the neutron and the nucleus in the center-of-mass systemdo not change during the scattering event:

v ′c = vc =A

A+ 1 vL (1.16)

V ′c = Vc =−1A+ 1 vL

With reference to Fig. 1.30, the scattering angles in the lab and CM systems arerelated by

tan θL = v′c sin θc

vcm + v ′c cos θc= sin θc(1/A)+ cos θc (1.17)

The law of cosines yields

cos(π − θc)= (v′c)

2 + (vcm)2 − (v ′L)22vcmv ′c

(1.18)

Fig. 1.30 Relation between lab and CM scattering angles.(From Ref. 12; used with permission of Wiley.)

1.6 Elastic Scattering Kinematics 29

which may be combined with Eqs. (1.13) and (1.16) to obtain a relationship betweenthe incident and final energies of the neutron in the lab system and the scatteringangle in the CM system:

12m(v

′L)

2

12m(vL)

2≡ E

′L

EL= A

2 + 1+ 2A cos θc(A+ 1)2 =

(1+ α)+ (1− α) cos θc2

(1.19)

where α ≡ (A− 1)2/(A+ 1)2.

Average Energy Loss

Equation (1.19) states that the ratio of final to incident energies in an elastic scat-tering event is correlated to the scattering angle in the CM system, which in turnis correlated via Eq. (1.17) to the scattering angle in the lab system. The maximumenergy loss (minimum value of E ′L/EL) occurs for θc = π (i.e., backward scatter-ing in the CM system), in which case E ′L = αEL. For hydrogen (A= 1), α = 0 andall of the neutron energy can be lost in a single collision. For other nuclides, onlya fraction (1 − α) of the neutron energy can be lost in a single collision, and forheavy nuclides (α→ 1) this fraction becomes very small.

The probability that a neutron scatters from energy EL to within a differentialband of energies dE ′L about energy E ′L is equivalent to the probability that a neu-tron scatters into a cone 2π sin θcdθc about θc :

σs(EL)P (EL→E ′L)dE ′L =−σcm(EL, θc)2π sin θcdθc (1.20)

where the negative sign takes into account that an increase in angle correspondsto a decrease in energy, σs is the elastic scattering cross section, and σcm(θc) is thecross section for scattering through angle θc . Using Eq. (1.19) to evaluate dE ′L/dθc ,this becomes

P(EL→E ′L)={ 4πσcm(EL, θc)(1− α)ELσs(EL) , αEL ≤E

′L ≤EL

0, otherwise(1.21)

Except for very high energy neutrons scattering from heavy mass nuclides, elasticscattering in the CM is isotropic, σcm(θc)= σs/4π . In this case, Eq. (1.21) may bewritten

σs(EL→E ′L) ≡ σs(EL)P (EL→E ′L)=σs(EL)

(1− α)EL , αEL ≤E′L ≤EL

(1.22)

= 0, otherwise

The average energy loss in an elastic scattering event may be calculated from

〈�EL〉 ≡EL −∫ ELαEL

dE ′LE ′LP (EL→E ′L)=1

2(1− α)EL (1.23)


Table 1.4 Number of Collisions, on Average, to Moderate a Neutron from 2 MeV to 1 eV

Moderator ξ Number of Collisions ξ�s/�a

H 1.0 14 –D 0.725 20 –H2O 0.920 16 71D2O 0.509 29 5670He 0.425 43 83Be 0.209 69 143C 0.158 91 192Na 0.084 171 1134Fe 0.035 411 35238U 0.008 1730 0.0092

and the average logarithmic energy loss may be calculated from

ξ ≡∫ ELαEL

dE ′L ln(EL

E ′L

)P(EL→E ′L)

= 1+ α1− α lnα = 1−

(A− 1)22A

ln

(A+ 1A− 1

)(1.24)

The number of collisions, on average, required for a neutron of energy E0 to bemoderated to thermal energies, say 1 eV, can be estimated from

〈no. collisions〉 � ln[E0(eV )/1.0]ξ

(1.25)

The results are shown in Table 1.4 for E0 = 2 MeV.The parameter ξ , which is a measure of the moderating ability, decreases with

nuclide mass, with the result that the number of collisions that are needed to mod-erate a fast neutron increases with nuclide mass. However, the effectiveness of anuclide (or molecule) in moderating a neutron also depends on the relative prob-ability that a collision will result in a scattering reaction, not a capture reaction,which would remove the neutron. Thus the parameter ξ�s/�a , referred to as themoderating ratio, is a measure of the effectiveness of a moderating material. Eventhough H2O is the better moderator in terms of the number of collisions requiredto thermalize a fast neutron, D2O is the more effective moderator because the ab-sorption cross section for D is much less than that for H.

Example 1.2: Moderation by a Mixture. The moderating parameters for a mixtureof isotopes is constructed by weighting the moderating parameters of the indi-vidual isotopes by their concentrations in the mixture. For example, in a mixtureof 12C and 238U the average value of ξ�s = NCξCσsC + NUξUσsU = NC(0.158)(2.3×10−24 cm2)+NU(0.008)(4.8×10−24 cm2), where the fission spectrum rangeelastic scattering cross sections of Table 1.3 have been assumed to hold also in the

References 31

slowing-down range. The total absorption cross section is�a =NCσaC+NUσaU =NC(0.002×10−24 cm2)+NU(280×10−24 cm2) in the slowing-down range, wherethe resonance range cross sections from Table 1.3 have been used.

References

1 H. Cember, Introduction to HealthPhysics, 3rd ed., McGraw-Hill, NewYork (1996).

2 C. Nordborg and M. Salvatores,“Status of the JEF Evaluated NuclearData Library,” Proc. Int. Conf. Nu-clear Data for Science and Technology,Gatlinburg, TN, Vol. 2 (1994), p. 680.

3 R. W. Roussin, P. G. Young, andR. McKnight, “Current Status ofENDF/B-VI, Proc. Int. Conf. Nu-clear Data for Science and Technology,Gatlinburg, TN, Vol. 2 (1994), p. 692.

4 Y. Kikuchi, “JENDL-3 Revision 2:JENDL 3-2,” Proc. Int. Conf. Nu-clear Data for Science and Technology,Gatlinburg, TN, Vol. 2 (1994), p. 685.

5 R. A. Knief, Nuclear Engineering, Tay-lor & Francis, Washington, DC (1992).

6 J. J. Schmidt, “Nuclear Data: TheirImportance and Application in FissionReactor Physics Calculations,” in D. E.Cullen, R. Muranaka, and J. Schmidt,eds., Reactor Physics Calculations forApplications in Nuclear Technology,World Scientific, Singapore (1990).

7 A. Trkov, “Evaluated Nuclear DataProcessing and Nuclear Reactor Cal-culations,” in D. E. Cullen, R. Mu-ranaka, and J. Schmidt, eds., ReactorPhysics Calculations for Applications inNuclear Technology, World Scientific,Singapore (1990).

8 CINDA: An Index to the Literatureon Microscopic Neutron Data, In-ternational Atomic Energy Agency,

Vienna; CINDA-A, 1935–1976(1979); CINDA-B, 1977–1981 (1984);CINDA-89 (1989).

9 D. E. Cullen, “Nuclear Cross SectionPreparation,” in Y. Ronen, ed., CRCHandbook of Nuclear Reactor Calcu-lations I, CRC Press, Boca Raton, FL(1986).

10 J. L. Rowlands and N. Tubbs, “TheJoint Evaluated File: A New NuclearData Library for Reactor Calculations,”Proc. Int. Conf. Nuclear Data for Basicand Applied Science, Santa Fe, NM,Vol. 2 (1985), p. 1493.

11 A. Calamand and H. D. Lemmel,Short Guide to EXFOR, IAEA-NDS-1,Rev. 3, International Atomic EnergyAgency, Vienna (1981).

12 J. J. Duderstadt and L. G. Hamilton,Nuclear Reactor Analysis, Wiley, NewYork (1976), Chap. 2.

13 H. C. Honeck, ENDF/B: Specifica-tions for an Evaluated Data File forReactor Applications, USAEC reportBNL-50066, Brookhaven National Lab-oratory, Upton, NY (1966).

14 I. Kaplan, Nuclear Physics, 2nd ed.,Addison-Wesley, Reading, MA (1963).

15 L. J. Templin, ed., Reactor PhysicsConstants, 2nd ed., ANL-5800, Ar-gonne National Laboratory, Argonne,IL (1963).

Problems

1.1. Demonstrate that the speeds of the neutron and nucleus inthe CM system do not change in an elastic scattering event byusing conservation of momentum and kinetic energy.


1.2. Estimate the probability that a 1-MeV neutron will bemoderated to thermal without being captured in a mixture ofuranium and water with NH/NU = 1:1. Repeat for a 1:1mixture of uranium and carbon.

1.3. Neutrons are slowed down to thermal energies in a 1:1mixture of H2O and 4% enriched uranium (4%235U,96%238U). Estimate the thermal value of η= νσf /(σc + σf ).Repeat the calculation for a mixture of (2%235U, 2%239Pu,96%238U).

1.4. Estimate the probability that a fission neutron will have ascattering collision with H2O in the mixtures of Problem 1.3.

1.5. Calculate the average energy loss for neutrons at 1-MeV,100-keV, 10-keV, and 1-keV scattering from carbon. Repeatthe calculation for scattering from iron and from uranium.

1.6. Repeat Problem 1.5 for scattering from hydrogen andsodium.

1.7. Calculate the moderating ratio and the average number ofcollisions required to moderate a fission neutron to thermalfor a 1:1 mixture of 12C : 238U. Repeat for a 10:1 mixture.

1.8. Calculate the thermal absorption cross section for a 1:1 wt%mixture of carbon and 4% enriched uranium (e.g., 4%235U,96%238U).

1.9. Derive Eq. (1.21) from Eqs. (1.20) and (1.19).1.10. Calculate the average number of scattering events required to

moderate a neutron’s energy from above the resonance rangeto below the resonance range of 238U for carbon, H2O andD2O moderators.

33

2Neutron Chain Fission Reactors

2.1Neutron Chain Fission Reactions

Since two or three neutrons are released in every neutron-induced fission reac-tion, the possibility of a sustained neutron chain reaction is obvious, as illustratedin Fig. 2.1. To sustain a fission chain reaction, one or more of the neutrons pro-duced in the fission event must, on average, survive to produce another fissionevent. There is competition for the fission neutrons in any assembly—some willbe absorbed in fuel nuclides as radiative capture events rather than fission events,some will be absorbed by nonfuel nuclides, and some will leak out of the assembly.A scattering event does not compete for a neutron because the scattered neutronremains in the assembly and available for causing a fission event, but a scatteringevent does change a neutron’s energy and thus, because the various cross sectionsare energy dependent, does change the relative likelihood of the next collision be-ing a fission event.

Capture-to-Fission Ratio

The fission cross sections for the fissile nuclides increase approximately as 1/υwith decreasing neutron energy, but then so do the capture cross sections of thefissile nuclides. The probability that a neutron that is captured in a fissile nuclidecauses a fission is just σf /(σf + σγ ) = 1/(1 + σγ /σf ) = 1/(1 + α), where α ≡σγ /σf is referred to as the capture-to-fission ratio. The capture-to-fission ratio forthe principal fissile nuclides decreases as the neutron energy increases. For highneutron energies, the fission probability, which varies as (1 + α)−1, is larger for239Pu than for 235U or 233U, but the situation is reversed for low-energy thermalneutrons.

Number of Fission Neutrons per Neutron Absorbed in Fuel

The product of the fission probability for a neutron absorbed in the fuel and theaverage number of neutrons released per fission, η≡ νσf /(σf + σγ )= ν/(1+ α),provides a somewhat better characterization of the relative capabilities of the var-

34 2 Neutron Chain Fission Reactors

Fig. 2.1 Schematic of a fission chain reaction. (From Ref. 3; used with permission of Wiley.)

ious fissile nuclides to sustain a fission chain reaction. This quantity is plotted inFig. 2.2 for the principal fissile nuclides. For high neutron energies, η is larger for239Pu than for 235U or 233U, but the situation is reversed for low-energy thermalneutrons.

Neutron Utilization

The probability that a neutron is absorbed in a fissile nuclide instead of being ab-sorbed in another nuclide or leaking from the assembly is

absorb fissileabsorb fissile+ absorb nonfissile+ leak= absorb fissile

absorb total1

(1+ leak/absorb total) ≡ fPNL (2.1)

where f is the fraction of the absorbed neutrons which are absorbed in the fissilenuclides, or the utilization:

f = Nfisσfisa

Nfisσfisa +Notherσ othera

(2.2)

and PNL refers to the nonleakage probability. Since the absorption cross section,σa = σf + σγ , is much greater for thermal neutrons than for fast neutrons for thefissile nuclides, but comparable for fast and thermal neutrons for the nonfissilefuel nuclides and for structural nuclides, the utilization for a given composition ismuch greater for thermal neutrons than for fast neutrons (and, in fact, is usuallyreferred to as the thermal utilization).

Fast Fission

The product ηf is the number of neutrons produced, on average, from the fis-sion of fissile nuclides for each neutron absorbed in the assembly. There will also

2.1 Neutron Chain Fission Reactions 35

Fig. 2.2 η for the principal fissile nuclides. (From Ref. 9; usedwith permission of Electric Power Research Institute.)

be neutrons produced by the fission of the nonfissile fuel nuclides, mostly byfast neutrons. Defining the fast fission factor ε ≡ total fission neutron productionrate/fission neutron production rate in fissile nuclides, ηf ε is the total number offission neutrons produced for each neutron absorbed in the assembly, and ηf εPNLis the total number of fission neutrons produced, on average, for each neutronintroduced into the assembly by a previous fission event.


Fig. 2.3 Neutron balance in a thermal neutron fission assembly.(From Ref. 1; used with permission of Taylor & Francis.)

Resonance Escape

The parameters ηf ε must be evaluated by averaging over the energy of the neu-trons in the assembly, of course. When the neutron population consists predom-inantly of thermal neutrons, the thermal spectrum-averaged cross sections givenin Table 1.3 may be used to estimate η and f , and the cross sections averagedover the fission spectrum may be used in estimating ε, which should now alsoinclude fast fission in the fissile nuclides. In this case, it is necessary to take intoaccount separately the capture of fission neutrons while they are slowing downto the thermal energy range, predominantly by the capture resonances of thefuel nuclides. The probability that a neutron is not captured during the slowing-down process is referred to as the resonance escape probability and denoted p. Thiscompetition for neutrons is illustrated schematically in Fig. 2.3 (leakage is ne-glected).

2.2 Criticality 37

2.2Criticality

Effective Multiplication Constant

The product ηf εpPNL is the total number of fission neutrons produced, on average,by one fast neutron from a previous fission event. This quantity is referred to as theeffective multiplication constant of the assembly:

k = ηf εpPNL ≡ k∞PNL (2.3)

where k∞ refers to the multiplication constant of an infinite assembly with noleakage.

If exactly one neutron, on average, survives to cause another fission, a conditionreferred to as criticality (k = 1), the neutron population in the assembly will re-main constant. If less than one neutron, on average, survives to produce anotherfission event, a condition referred to as subcriticality (k < 1), the neutron populationin the assembly will decrease. If more than one fission neutron, on average, sur-vives to cause another fission, a condition referred to as supercriticality (k > 1), theneutron population in the assembly will increase. The effective multiplication con-stant depends on the composition (k∞) and size (PNL) of an assembly and on thearrangement of the materials within the assembly (f and p). The composition af-fects k both by the relative number of nuclides of different species that are presentand by the determination of the neutron energy distribution, which determines theaverage cross sections for each nuclide. The arrangement of materials determinesthe spatial neutron distribution and hence the relative number of neutrons at thelocations of the various nuclides.

The fissile nuclide 235U is only 0.72% of natural uranium. Fuel enrichment toachieve a higher fissile content, hence larger value of f , is a major means of in-creasing the multiplication constant. The number of fission neutrons produced foreach neutron absorbed in fissile material, η, is significantly larger for fast neutronsthan for thermal neutrons, because the capture-to-fission ratio is smaller and thenumber of neutrons per fission is larger. On the other hand, for a given fuel en-richment, the utilization, f , is greater for thermal neutrons than for fast neutronsbecause the absorption cross section is much greater for thermal neutrons thanfor fast neutrons for the fissile nuclides, but comparable for fast and thermal neu-trons for the nonfissile fuel nuclides and for structural nuclides. On the whole, theamount of fissile material necessary to achieve a given value of the multiplicationconstant is substantially less in a fast neutron spectrum than in a thermal neutronspectrum.

Effect of Fuel Lumping

Lumping the fuel rather than distributing it uniformly can have a significant ef-fect on the multiplication constant. For example, if natural uranium is distributed


uniformly in a graphite lattice, the values of the various parameters are η ≈ 1.33,f ≈ 0.9, ε ≈ 1.05, and p ≈ 0.7, yielding k∞ ≈ 0.88 (i.e., the assembly is subcriti-cal). If the fuel is lumped, the strong resonance absorption at the exterior of thefuel elements reduces the number of neutrons that reach the interior of the fuelelements, increasing the resonance escape probability to p ≈ 0.9. Lumping the fuelalso reduces the thermal utilization f , for the same reason, but the effect is not sosignificant. Lumping the fuel was the key to achieving criticality (k = 1) in the firstgraphite-moderated natural uranium reactors and is crucial in achieving criticalityin present-day D2O-moderated natural uranium reactors.

Leakage Reduction

The multiplication constant can be increased by reducing the leakage, most ofwhich is due to fast neutrons. This can be done simply by increasing the size.The leakage can also be reduced by choosing a composition that moderates theneutrons quickly before they can travel far or by surrounding the assembly witha material with a large scattering cross section (e.g., graphite), which will reflectleaking neutrons back into the assembly.

Example 2.1: Effective Multiplication Factor for a PWR. For a typical pressurized wa-ter reactor (PWR), the various parameters are η ≈ 1.65, f ≈ 0.71, ε ≈ 1.02, andp ≈ 0.87, yielding k∞ ≈ 1.04. The nonleakage factors for fast and thermal neu-trons are typically 0.97 and 0.99, yielding k ≈ 1.00.

2.3Time Dependence of a Neutron Fission Chain Assembly

Prompt Fission Neutron Time Dependence

If there are N0 fission neutrons introduced into an assembly at t = 0, and if l is theaverage time required for a fission neutron to slow down and be absorbed or leakout, l = 1

ν�f υ, the number of neutrons, on average, in the assembly at time t = l is

(k)N0. Continuing in this fashion, the number of neutrons in the assembly at timet =ml (m integer) is (k)mN0. The quantity l is typically ≈ 10−4 s for assemblies inwhich the neutrons slow down to thermal before causing another fission, and istypically ≈ 10−6 s for assemblies in which the fission is produced by fast neutrons.For example, a 12 % change in absorption cross section, which could be producedby control rod motion, causes an approximately 0.005 change in k. The neutronpopulation after 0.1 s in a thermal assembly (0.1 s = 103l) in which k = 1.005,would be N(0.1)= (1.005)1000N0 ≈ 150N0. In a thermal assembly with k = 0.995,the neutron population after 0.1 s would be N(0.1)= (0.995)1000N0 ≈ 0.0066N0.

An equation governing the neutron kinetics described above is

dN(t)

dt= k− 1

lN(t)+ S(t) (2.4)

2.3 Time Dependence of a Neutron Fission Chain Assembly 39

which simply states that the time rate of change of the neutron population is equalto the excess of neutron production (by fission) minus neutron loss by absorptionor leakage in a neutron lifetime plus any external source that is present. For aconstant source, Eq. (2.4) has the solution

N(t)=N(0)e(k−1)t/ l + Slk− 1 [e

(k−1)t/ l − 1] (2.5)

which displays an exponential time behavior. Using the same example as above,with the source set to zero, leads to N(0.1) = N(0) exp(5.0) = 148N(0) for k =1.005 and N(0.1)=N(0) exp(−5.0)= 0.00677N(0) for k = 0.995.

Source Multiplication

Equation (2.4) does not have a steady-state solution for k > 1 and does not have aunique steady-state solution for k = 1. However, for k < 1, the asymptotic solutionis

Nasymptotic = lS01− k (2.6)

This equation provides a method to measure the effective multiplication factor kwhen k < 1 by measuring the asymptotic neutron population which results fromplacing a source S0 in a multiplying medium.

Effect of Delayed Neutrons

It would be very difficult, if not impossible, to control a neutron fission chain as-sembly which responded so dramatically to a 12 % change in absorption cross sec-tion. Fortunately, a small fraction (β ≈ 0.0075 for 235U fueled reactors) of the fis-sion neutrons are delayed until the decay (λ≈ 0.08 s−1) of the fission fragments.For an assembly that was critical prior to t = 0, the equilibrium concentration ofsuch delayed neutron precursor fission fragments is found from the balance equa-tion:

dC0

dt= 0= βνNf σf υN0 − λC0 = β

lN0 − λC0 (2.7)

where Nf is the density of fuel nuclei, N0 the neutron population, and C0 thepopulation of delayed neutron precursor fission fragments.

When the 12 % change in cross section occurs at t = 0, the multiplication of theprompt neutrons after 0.1 s (1000l) is [(1− β)k]1000. During each multiplicationinterval l there is a source λlC of delayed neutrons from the decay of fission frag-ments. This source results in (1−β)kλlC neutrons in the following multiplicationinterval, [(1− β)k]2λlC neutrons in the second following multiplication interval,and so on. There is such a delayed neutron source in each of the 1000 multiplica-tion intervals in our example. To simplify the problem, we assume that the fission


fragment concentration does not change (i.e., C = C0). Thus the number of neu-trons after 0.1 s (1000l) is

N(1000l) = [(1− β)k]mN0 + λlC0[(1− β)k]m−1 + λlC0[(1− β)k]m−2

+ · · · + λlC0(1− β)k+ λlC0

= [(1− β)k]mN0 + λlC0 (1− k[1− β])m−1

1− k(1− β)

={[(1− β)k]m

[1− β

k(1− β)[1− k(1− β)]]+ β

1− k(1− β)}N0

(2.8)

where Eq. (2.7) has been used in the last step. Evaluating this expression for k =1.005 yields N(t = 0.1 s)= 3.03N0, instead of the 150N0 found without taking thedelayed neutrons into account. If we had taken into account the changing fissionfragment population, we would have found a slightly larger number. Nevertheless,the fact that some of the neutrons emitted in fission are delayed results in a ratherslow and hence controllable response of a neutron chain fission reacting assembly,provided that (1− β)k < 1.

2.4Classification of Nuclear Reactors

Physics Classification by Neutron Spectrum

From the physics viewpoint, the main differences among reactor types arise fromdifferences in the neutron energy distribution, or spectrum, which causes differ-ences in the neutron–nuclear reaction rates and the competition for neutrons. Thefirst level of physics classification categories are then thermal reactors and fastreactors, corresponding to the majority of the neutron–nuclear reactions involv-ing neutrons in the thermal energy range (E < 1 eV) and to the majority of theneutron–nuclear reactions involving neutrons in the fast energy range (E > 1 keV),respectively

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