Weston M. Stacey

Nuclear Reactor Physics

Second Edition, Completely Revised and Enlarged

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Contents

Preface xxiii

Preface 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 20

1.4 Summary of Cross-Section Data 24

Low-Energy Cross Sections 24

Spectrum-Averaged Cross Sections 24

1.5 Evaluated Nuclear Data Files 24

1.6 Elastic Scattering Kinematics 27

Correlation of Scattering Angle and Energy Loss 28

Average Energy Loss 29

2 Neutron Chain Fission Reactors 33

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

Nuclear Reactor Physics. Weston M. StaceyCopyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3.527-40679-1

viii I 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 38

Prompt Fission Neutron Time Dependence 38

Source Multiplication 39Effect of Delayed Neutrons 39

2.4 Classification of Nuclear Reactors 40

Physics Classification by Neutron Spectrum 40

Engineering Classification by Coolant 41

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

Partial and Net Currents 43

Diffusion Theory 45Interface Conditions 46Boundary Conditions 46Applicability of Diffusion Theory 47

3.2 Solutions of the Neutron Diffusion Equation in NonmultiplyingMedia 48

Plane Isotropic Source in an Infinite Homogeneous Medium 48Plane Isotropic Source in a Finite Homogeneous Medium 48

Line Source in an Infinite Homogeneous Medium 49

Homogeneous 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 52

3.5 Neutron Diffusion and Migration Lengths 53Thermal Diffusion-Length Experiment 53Migration Length 55

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

Contents lix

Interpretation of Criticality Condition 60Optimum Geometries 61

3.7 Reflected Reactor 62

Reflected 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 72

Local Power Peaking Factor 733.9 Control Rods 73

Effective 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 82

Power Outer Iteration on Fission Source 82

Limitations on Mesh Spacing 83

3.11 Nodal Approximation 83

3.12 Transport Methods 85

Transmission and Absorption in a Purely Absorbing Slab ControlPlate 87

Escape Probability in a Slab 87Integral Transport Formulation 87

Collision Probability Method 88

Differential 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 102

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

XI

Contents

Slowing-Down Density 105Slowing Down with Weak Absorption 106

Fermi Age Neutron Slowing Down 107

Neutron Energy Distribution in the Thermal Range 108

Summary 111

4.2 Multigroup Calculation of Neutron Energy Distribution in an InfiniteMedium 111

Derivation of Multigroup Equations 111

Mathematical Properties of the Multigroup Equations 113

Solution of Multigroup Equations 114

Preparation of Multigroup Cross-Section Sets 115

4.3 Resonance Absorption 117

Resonance Cross Sections 117Doppler Broadening 119

Resonance Integral 122

Resonance Escape Probability 122

Multigroup Resonance Cross Section 122

Practical Width 122

Neutron Flux in Resonance 123

Narrow Resonance Approximation 123

Wide Resonance Approximation 124

Resonance Absorption Calculations 124

Temperature Dependence of Resonance Absorption 127

4.4 Multigroup Diffusion Theory 127

Multigroup Diffusion Equations 127Two-Group Theory 128

Two-Group Bare Reactor 129

One-and-One-Half-Group Theory 129

Two-Group Theory of Two-Region Reactors 130

Two-Group Theory of Reflected Reactors 133

Numerical Solutions for Multigroup Diffusion Theory 137

5 Nuclear Reactor Dynamits 1435.1 Delayed Fission Neutrons 143

Neutrons Emitted in Fission Product Decay 143

Effective 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 ixi

5.5 Delayed Neutron Kernel and Zero-Power Transfer Function 155

Delayed Neutron Kernel 155Zero-Power Transfer Function 155

5.6 Experimental Determination of Neutron Kinetics Parameters 156Asymptotic Period Measurement 156Rod Drop Method 157

Source Jerk Method 157Pulsed Neutron Methods 157Rod Oscillator Measurements 158

Zero-Power Transfer Function Measurements 159Rossi-a Measurement 159

5.7 Reactivity Feedback 161Temperature Coefficients of Reactivity 162

Doppler Effect 162Fuel and Moderator Expansion Effect an Resonance EscapeProbability 164Thermal Utilization 165Nonleakage Probability 166

Representative Thermal Reactor Reactivity Coefficients 166Startup Temperature Defect 167

5.8 Perturbation Theory Evaluation of Reactivity TemperatureCoefficients 168

Perturbation Theory 168

Sodium 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 171

Stability 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 179

Rod Oscillator Method 179Correlation Methods 179Reactor Noise Method 181

5.11 Reactor Transients with Feedback 183

Step Reactivity Insertion (pex < ß): Prompt Jump 184

Step Reactivity Insertion (pex < ß): Post-Prompt-Jump Transient 185

5.12 Reactor Fast Excursions 186

Step Reactivity Input: Feedback Proportional to Fission Energy 186

Ramp Reactivity Input: Feedback Proportional to Fission Energy 187

xii I Contents

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

5.13 Numerical Methods 190

6 Fuel Burnup 197

6.1 Changes in Fuel Composition 197Fuel 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 an 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 lxiii

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 I 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 lxv

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 340

PI 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-Po 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 354

PL and D-PL Ordinates 355Spatial Differencing and Iterative Solution 357Limitations an 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 I Contents

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

10 Neutron Slowing Down 385

10.1 Elastic Scattering Transfer Function 385Lethargy 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 393B 1 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 Pl Approximation 405Extended Age Approximation 405Grueling—Goertzel Approximation 406Summary of PI Continuous Slowing-Down Theory 407Inclusion of Anisotropic Scattering 407Inclusion of Scattering Resonances 409P1 Continuous Slowing-Down Equations 410

10.5 Multigroup Discrete Ordinates Transport Theory 411

11 Resonance Absorption 415

11.1 Resonance Cross Sections 415

Contents lxvii

11.2 Widely Spaced Single-Level Resonances in a HeterogeneousFuel–Moderator Lattice 415Neutron Balance in Heterogeneous Fuel–Moderator Gell 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 1-120 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 I Co ntents

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 lxix

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-P,1 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 Pi 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: 9 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 lxxi

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 642,u,-Mode Approximation 644?-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 S tudies 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 8-Function 678

A. Alternative Representations 678B. Properties 678

C. Derivatives 679

D Some Properties of Special Functions 681

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

E.2 Matrix Algebra 689

xxii I Contents

F Introduction to Laplace Transforms 691F.1 Motivation 691F.2 "Cookbook" Laplace transforrns 694

Index 697