Weston M. Stacey

Nuclear Reactor Physics

Second Edition, Completely Revised and Enlarged

I C E N T E N N I A L

B I C E N T E N N I A L

WILEY-VCH Verlag GmbH & Co. KGaA

vii

Contents

Preface xxiiiPreface to 2nd Edition xxvii

PART 1 BASIC REACTOR PHYSICS

1 Neutron Nuclear Reactions 3

1.1 Neutron-Induced Nuclear Fission 3Stable Nuclides 3

Binding 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 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

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 43

3.1 Derivation of One-Speed Diffusion Theory 43Partial 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 101

4.1 Analytical Solutions in an Infinite Medium 101Fission Source Energy Range 102Slowing-Down Energy Range 102Moderation by Hydrogen Only 103Energy Self-ShieKiing 203Slowing Down by Nonhydrogenic Moderators with No Absorption 104

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 215

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 143

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

5.2 Point Kinetics Equations 1475.3 Period-Reactivity Relations 2485.4 Approximate Solutions of the Point Neutron Kinetics Equations 250

One Delayed Neutron Group Approximation 250Prompt-Jump Approximation 153Reactor Shutdown 254

Contents xi

5.5 Delayed Neutron Kernel and Zero-Power Transfer Function 255Delayed Neutron Kernel 155

Zero-Power Transfer Function 2555.6 Experimental Determination of Neutron Kinetics Parameters 256

Asymptotic Period Measurement 256

Rod Drop Method 157Source Jerk Method 257Pulsed Neutron Methods 257Rod Oscillator Measurements 158Zero-Power Transfer Function Measurements 159Rossi-a Measurement 159

5.7 Reactivity Feedback 161Temperature Coefficients of Reactivity 2 62Doppler Effect 262

Fuel and Moderator Expansion Effect on Resonance EscapeProbability 264Thermal Utilization 265Nonleakage Probability 266Representative Thermal Reactor Reactivity Coefficients 166

Startup Temperature Defect 2675.8 Perturbation Theory Evaluation of Reactivity Temperature

Coefficients 268Perturbation Theory " 2 68Sodium Void Effect in Fast Reactors 269Doppler Effect in Fast Reactors 169Fuel and Structure Motion in Fast Reactors 270Fuel Bowing 172Representative Fast Reactor Reactivity Coefficients 2 72

5.9 Reactor Stability 272Reactor Transfer Function with Reactivity Feedback 172Stability Analysis for a Simple Feedback Model 2 72

Threshold Power Level for Reactor Stability 174More General Stability Conditions 275Power Coefficients and Feedback Delay Time Constants 2 78

5.10 Measurement of Reactor Transfer Functions 179Rod Oscillator Method 179

Correlation Methods 279Reactor Noise Method 182

5.11 Reactor Transients with Feedback 183

Step Reactivity Insertion (pex < fi): Prompt Jump 184Step Reactivity Insertion (pex < fi): 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 288

5.13 Numerical Methods 190

6 Fuel Burnup 197

6.1 Changes in Fuel Composition 297Fuel Transmutation-Decay Chains 2 98Fuel Depletion-Transmutation-Decay Equations 299Fission 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 220

6.2 Samarium and Xenon 211Samarium Poisoning 222Xenon Poisoning 223Peak Xenon 225Effect of Power-Level Changes 226

6.3 Fertile-to-Fissile Conversion and Breeding 22 7Availability of Neutrons 217Conversion and Breeding Ratios 229

6.4 Simple Model of Fuel Depletion 2296.5 Fuel Reprocessing and Recycling 221

Composition of Recycled LWR Fuel 222Physics 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 249

7.1 Pressurized Water Reactors 2497.2 Boiling Water Reactors 250

7.3 Pressure Tube Heavy Water-Moderated Reactors 255

7.4 Pressure Tube Graphite-Moderated Reactors 2587.5 Graphite-Moderated Gas-Cooled Reactors 2607.6 Liquid-Metal Fast Breeder Reactors 2627.7 Other Power Reactors 2657.8 Characteristics of Power Reactors 2657.9 Advanced Generation-Ill 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) 272Molten Salt Reactors (MSR) 272Super-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 276

Fuel Cycle Analyses 277Transient Analyses 278Core Operating Data 279Criticality Safety Analysis 279

7.13 Interaction of Reactor Physics and Reactor Thermal Hydraulics 280Power Distribution 280

Temperature Reactivity Effects 281Coupled Reactor Physics and Thermal-Hydraulics Calculations 281

8 Reactor Safety 283

8.1 Elements of Reactor Safety 283Radionuclides 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

Contents

Reactivity Insertion 287Anticipated Transients without Scram 288

8.3 Quantitative Risk Assessment 288Probabilistic Risk Assessment 288Radiological Assessment 291Reactor Risks 292

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 305

9.1 Neutron Transport Equation 305Boundary Conditions 320Scalar Flux and Current 310Partial Currents 310

9.2 Integral Transport Theory 320Isotropic Point Source 322Isotropic Plane Source 311Anisotropic Plane Source 322Transmission and Absorption Probabilities 314Escape Probability 314First-Collision Source for Diffusion Theory 315Inclusion of Isotropic Scattering and Fission 325Distributed Volumetric Sources in Arbitrary Geometry 326Flux from a Line Isotropic Source of Neutrons 327Bickley Functions 328Probability of Reaching a Distance t from a Line Isotropic Sourcewithout a Collision 328

9.3 Collision Probability Methods 329Reciprocity Among Transmission and Collision Probabilities 320Collision Probabilities for Slab Geometry 320Collision Probabilities in Two-Dimensional Geometry 322Collision 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 340Pi Equations and Diffusion Theory 342Simplified Pi or Extended Diffusion Theory 343Pi 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 352Pi 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 363SM 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 372

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 Pi and B\ Slowing-Down Equations 390Derivation 390Solution in Finite Uniform Medium 393B] 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 400P\ Equations in Slowing-Down Density Formulation 400Slowing-Down Density in Hydrogen 403Heavy Mass Scatterers 404Age Approximation 404Selengut-Goertzel Approximation 405Consistent Pi Approximation 405Extended Age Approximation 405Grueling-Goertzel Approximation 406Summary of P; Continuous Slowing-Down Theory 407Inclusion of Anisotropic Scattering 407Inclusion of Scattering Resonances 409Pi Continuous Slowing-Down Equations 410

10.5 Multigroup Discrete Ordinates Transport Theory 412

11 Resonance* Absorption 41511.1 Resonance Cross Sections 425

Contents xvii

11.2 Widely Spaced Single-Level Resonances in a HeterogeneousFuel-Moderator Lattice 425Neutron Balance in Heterogeneous Fuel-Moderator Cell 415

Reciprocity Relation 428Narrow 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 439^-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 Scatteririg Functions 458Incoherent Approximation 459

Contents

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

12 A 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 483

13.1 Perturbation Theory Reactivity Estimate 483Multigroup Diffusion Perturbation Theory 483

13.2 Adjoint Operators and Importance Function 486Adjoint 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 P WR 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 506

Diffusion Approximation 507One-Dimensional Slab Transport Equation 508

13.11 Boundary Perturbation Theory 508

14 Homogenization 51514.1 Equivalent Homogenized Cross Sections 52614.2 ABH Collision Probability Method 52714.3 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 53214.7 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 562

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

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 572Application 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 602Delayed 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 622Implicit Integration: 8 Method 622Implicit Integration: Time-Integrated Method 625Implicit Integration: GAKIN Method 626Alternating Direction Implicit Method 629Stiffness 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 632Variational Methods of Control Theory 632Dynamic Programming 633Pontryagin's Maximum Principle 634

Contents xxi

Variational Methods for Spatially Dependent Control Problems 636Dynamic Programming for Spatially Continuous Systems 638

Pontryagin's Maximum Principle for a Spatially ContinuousSystem 639

16.5 Xenon Spatial Oscillations 642Linear Stability Analysis 642/Lt-Mode Approximation 644

A-Mode Approximation 645Nonlinear Stability Criterion 649

Control of Xenon Spatial Power Oscillations 650

Variational Control Theory of Xenon Spatial Oscillations 650

16.6 Stochastic Kinetics 652Forward Stochastic Model 653

Means, Variances, and Covariances 656

Correlation Functions 658

Physical Interpretation, Applications, and Initial and BoundaryConditions 659

Numerical 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 677

C.I Introduction 677

C.2 Properties of the Dirac <5-Function 678

A. Alternative Representations 678

B. Properties 678

C. Derivatives 679

D Some Properties of Special Functions 681

E Introduction to Matrices and Matrix Algebra 687

E.I Some Definitions 687 ''

E.2 Matrix Algebra 689

Contents

F Introduction to Laplace Transforms 691F.I Motivation 691F.2 "Cookbook" Laplace transforms 694

Index 697