Nuclear Reactor Physics

Preface xxiii Preface to Second Edition xxvii Preface to Third Edition xxix Part 1 Basic Reactor Physics 1 1 Neutron–Nuclear Reactions 3 1.1 Neutron-Induced Nuclear Fission 3 Stable Nuclides 3 Binding Energy 3 Threshold External Energy for Fission 5 Neutron-Induced Fission 5 Neutron Fission Cross Sections 5 Products of the Fission Reaction 7 Energy Release 9 1.2 Neutron Capture 12 Radiative Capture 12 Neutron Emission 18 1.3 Neutron Elastic Scattering 19 1.4 Summary of Cross Section Data 23 Low-Energy Cross Sections 23 Spectrum-Averaged Cross Sections 24 1.5 Evaluated Nuclear Data Files 25 1.6 Elastic Scattering Kinematics 25 Correlation of Scattering Angle and Energy Loss 26 Average Energy Loss 27 2 Neutron Chain Fission Reactors 33 2.1 Neutron Chain Fission Reactions 33 Capture-to-Fission Ratio 33 Number of Fission Neutrons per Neutron Absorbed in Fuel 33 Neutron Utilization 34 Fast Fission 35 Resonance Escape 36 2.2 Criticality 37 Effective Multiplication Constant 37 Effect of Fuel Lumping 37 Leakage Reduction 38 2.3 Time Dependence of a Neutron Fission Chain Assembly 38 Prompt Fission Neutron Time Dependence 38 Source Multiplication 39 Effect 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 and Transport Theory 43 3.1 Derivation of One-Speed Diffusion Theory 43 Partial and Net Currents 43 Diffusion Theory 46 Interface Conditions 46 Boundary Conditions 46 Applicability of Diffusion Theory 47 3.2 Solutions of the Neutron Diffusion Equation in Nonmultiplying Media 48 Plane Isotropic Source in an Infinite Homogeneous Medium 48 Plane Isotropic Source in a Finite Homogeneous Medium 48 Line Source in an Infinite Homogeneous Medium 49 Homogeneous Cylinder of Infinite Axial Extent with Axial Line Source 49 Point Source in an Infinite Homogeneous Medium 49 Point Source at the Center of a Finite Homogeneous Sphere 50 3.3 Diffusion Kernels and Distributed Sources in a Homogeneous Medium 50 Infinite-Medium Diffusion Kernels 50 Finite-Slab Diffusion Kernel 51 Finite Slab with Incident Neutron Beam 52 3.4 Albedo Boundary Condition 52 3.5 Neutron Diffusion and Migration Lengths 53 Thermal Diffusion-Length Experiment 53 Migration Length 56 3.6 Bare Homogeneous Reactor 57 Slab Reactor 58 Right Circular Cylinder Reactor 59 Interpretation of Criticality Condition 61 Optimum Geometries 61 3.7 Reflected Reactor 62 Reflected Slab Reactor 63 Reflector Savings 65 Reflected Spherical, Cylindrical, and Rectangular Parallelepiped Cores 65 3.8 Homogenization of a Heterogeneous Fuel–Moderator Assembly 65 Spatial Self-Shielding and Thermal Disadvantage Factor 65 Effective Homogeneous Cross Sections 68 Thermal Utilization 70 Measurement of Thermal Utilization 70 Local Power Peaking Factor 71 3.9 Control Rods 72 Effective Diffusion Theory Cross Sections for Control Rods 72 Windowshade Treatment of Control Rods 74 3.10 Numerical Solution of Diffusion Equation 76 Finite-Difference Equations in One Dimension 76 Forward Elimination/Backward Substitution Spatial Solution Procedure 78 Power Iteration on Fission Source 78 Finite-Difference Equations in Two Dimensions 79 Successive Relaxation Solution of Two-Dimensional Finite-Difference Equations 81 Power Outer Iteration on Fission Source 81 Limitations on Mesh Spacing 82 3.11 Nodal Approximation 82 3.12 Transport Methods 84 Transmission and Absorption in a Purely Absorbing Slab Control Plate 86 Escape Probability in a Slab 86 Integral Transport Formulation 86 Collision Probability Method 88 Differential Transport Formulation 89 Spherical Harmonics Methods 89 Boundary and Interface Conditions 91 P 1 Equations and Diffusion Theory 92 Discrete Ordinates Method 93 4 Neutron Energy Distribution 101 4.1 Analytical Solutions in an Infinite Medium 101 Fission Source Energy Range 102 Slowing-Down Energy Range 102 Moderation by Hydrogen Only 103 Energy Self-Shielding 103 Slowing Down by Nonhydrogenic Moderators with No Absorption 104 Slowing-Down Density 105 Slowing 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 Infinite Medium 112 Derivation of Multigroup Equations 112 Mathematical Properties of the Multigroup Equations 114 Solution of Multigroup Equations 115 Preparation of Multigroup Cross-Section Sets 116 4.3 Resonance Absorption 118 Resonance Cross Sections 118 Doppler Broadening 120 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 126 Temperature Dependence of Resonance Absorption 126 4.4 Multigroup Diffusion Theory 127 Multigroup Diffusion Equations 127 Two-Group Theory 128 Two-Group Bare Reactor 128 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 135 5 Nuclear Reactor Dynamics 141 5.1 Delayed Fission Neutrons 141 Neutrons Emitted in Fission Product Decay 141 Effective Delayed Neutron Parameters for Composite Mixtures 143 Photoneutrons 144 5.2 Point Kinetics Equations 145 5.3 Period–Reactivity Relations 146 5.4 Approximate Solutions of the Point Neutron Kinetics Equations 148 One-Delayed Neutron Group Approximation 148 Prompt-Jump Approximation 151 Reactor Shutdown 153 5.5 Delayed Neutron Kernel and Zero-Power Transfer Function 153 Delayed Neutron Kernel 153 Zero-Power Transfer Function 154 5.6 Experimental Determination of Neutron Kinetics Parameters 155 Asymptotic Period Measurement 155 Rod Drop Method 155 Source Jerk Method 156 Pulsed Neutron Methods 156 Rod Oscillator Measurements 157 Zero-Power Transfer Function Measurements 158 Rossi-α Measurement 158 5.7 Reactivity Feedback 160 Temperature Coefficients of Reactivity 161 Doppler Effect 162 Fuel and Moderator Expansion Effect on Resonance Escape Probability 164 Thermal Utilization 165 Nonleakage Probability 165 Representative Thermal Reactor Reactivity Coefficients 166 Startup Temperature Defect 167 5.8 Perturbation Theory Evaluation of Reactivity Temperature Coefficients 168 Perturbation Theory 168 Sodium Void Effect in Fast Reactors 169 Doppler Effect in Fast Reactors 170 Fuel and Structure Motion in Fast Reactors 170 Fuel Bowing 171 Representative Fast Reactor Reactivity Coefficients 171 5.9 Reactor Stability 171 Reactor Transfer Function with Reactivity Feedback 171 Stability Analysis for a Simple Feedback Model 173 Threshold Power Level for Reactor Stability 174 More General Stability Conditions 176 Power Coefficients and Feedback Delay Time Constants 178 5.10 Measurement of Reactor Transfer Functions 179 Rod Oscillator Method 180 Correlation Methods 180 Reactor Noise Method 182 5.11 Reactor Transients with Feedback 184 Step Reactivity Insertion (ρ ex < β): Prompt Jump 185 Step Reactivity Insertion (ρ ex < β): Post-Prompt-Jump Transient 186 5.12 Reactor Fast Excursions 187 Step Reactivity Input: Feedback Proportional to Fission Energy 187 Ramp Reactivity Input: Feedback Proportional to Fission Energy 188 Step Reactivity Input: Nonlinear Feedback Proportional to Cumulative Energy Release 189 Bethe–Tait Model 190 5.13 Numerical Methods 192 6 Fuel Burnup 197 6.1 Changes in Fuel Composition 197 Fuel Transmutation–Decay Chains 198 Fuel Depletion–Transmutation–Decay Equations 199 Fission Products 203 Solution of the Depletion Equations 204 Measure of Fuel Burnup 205 Fuel Composition Changes with Burnup 205 Reactivity Effects of Fuel Composition Changes 206 Compensating for Fuel-Depletion Reactivity Effects 207 Reactivity Penalty 208 Effects of Fuel Depletion on the Power Distribution 209 In-Core Fuel Management 210 6.2 Samarium and Xenon 211 Samarium Poisoning 211 Xenon Poisoning 213 Peak Xenon 215 Effect of Power-Level Changes 215 6.3 Fertile-to-Fissile Conversion and Breeding 217 Availability of Neutrons 217 Conversion and Breeding Ratios 217 6.4 Simple Model of Fuel Depletion 219 6.5 Fuel Reprocessing and Recycling 221 Composition of Recycled LWR Fuel 221 Physics Differences of MOX Cores 222 Physics Considerations with Uranium Recycle 224 Physics Considerations with Plutonium Recycle 224 Reactor Fueling Characteristics 225 6.6 Radioactive Waste 225 Radioactivity 225 Hazard Potential 226 Risk Factor 226 6.7 Burning Surplus Weapons-Grade Uranium and Plutonium 232 Composition of Weapons-Grade Uranium and Plutonium 232 Physics Differences Between Weapons- and Reactor-Grade Plutonium-Fueled Reactors 232 6.8 Utilization of Uranium Energy Content 234 6.9 Transmutation of Spent Nuclear Fuel 236 6.10 Closing the Nuclear Fuel Cycle 242 7 Nuclear Power Reactors 247 7.1 Pressurized Water Reactors 247 7.2 Boiling Water Reactors 249 7.3 Pressure Tube Heavy Water–Moderated Reactors 253 7.4 Pressure Tube Graphite-Moderated Reactors 255 7.5 Graphite-Moderated Gas-Cooled Reactors 258 7.6 Liquid Metal Fast Reactors 260 7.7 Other Power Reactors 265 7.8 Characteristics of Power Reactors 266 7.9 Advanced Generation-III Reactors 267 Advanced Boiling Water Reactors (ABWR) 267 Advanced Pressurized Water Reactors (APWR) 267 Advanced Pressure Tube Reactor 269 Modular High-Temperature Gas-Cooled Reactors (gt-mhr) 269 7.10 Advanced Generation-IV Reactors 271 Gas-Cooled Fast Reactors (GFR) 271 Lead-Cooled Fast Reactors (LFR) 272 Molten Salt Reactors (MSR) 273 Supercritical Water Reactors (SCWR) 273 Sodium-Cooled Fast Reactors (SFR) 273 Very High Temperature Reactors (VHTR) 273 7.11 Advanced Subcritical Reactors 274 7.12 Nuclear Reactor Analysis 276 Construction of Homogenized Multigroup Cross Sections 276 Criticality and Flux Distribution Calculations 277 Fuel Cycle Analyses 278 Transient Analyses 279 Core Operating Data 280 Criticality Safety Analysis 280 7.13 Interaction of Reactor Physics and Reactor Thermal Hydraulics 281 Power Distribution 281 Temperature Reactivity Effects 282 Coupled Reactor Physics and Thermal Hydraulics Calculations 282 8 Reactor Safety 285 8.1 Elements of Reactor Safety 285 Radionuclides of Greatest Concern 285 Multiple Barriers to Radionuclide Release 285 Defense in Depth 287 Energy Sources 287 8.2 Reactor Safety Analysis 287 Loss of Flow or Loss of Coolant 288 Loss of Heat Sink 289 Reactivity Insertion 289 Anticipated Transients without Scram 289 8.3 Quantitative Risk Assessment 289 Probabilistic Risk Assessment 289 Radiological Assessment 290 Reactor Risks 293 8.4 Reactor Accidents 294 Three Mile Island 294 Chernobyl 298 Fukushima 300 8.5 Passive Safety 300 Pressurized Water Reactors 300 Boiling Water Reactors 301 Integral Fast Reactors 301 Passive Safety Demonstration 301 Part 2 Advanced Reactor Physics 305 9 Neutron Transport Theory 307 9.1 Neutron Transport Equation 307 Boundary Conditions 309 Scalar Flux and Current 310 Partial Currents 311 9.2 Integral Transport Theory 312 Isotropic Point Source 313 Isotropic Plane Source 313 Anisotropic Plane Source 315 Transmission and Absorption Probabilities 317 Escape Probability 317 First-Collision Source for Diffusion Theory 318 Inclusion of Isotropic Scattering and Fission 318 Distributed Volumetric Sources in Arbitrary Geometry 320 Flux from a Line Isotropic Source of Neutrons 320 Bickley Functions 321 Probability of Reaching a Distance t from a Line Isotropic Source without a Collision 322 9.3 Collision Probability Methods 323 Reciprocity Among Transmission and Collision Probabilities 323 Collision Probabilities for Slab Geometry 324 Collision Probabilities in Two-Dimensional Geometry 325 Collision Probabilities for Annular Geometry 326 9.4 Interface Current Methods in Slab Geometry 327 Emergent Currents and Reaction Rates Due to Incident Currents 327 Emergent Currents and Reaction Rates Due to Internal Sources 331 Total Reaction Rates and Emergent Currents 333 Boundary Conditions 334 Response Matrix 335 9.5 Multidimensional Interface Current Methods 336 Extension to Multidimension 336 Evaluation of Transmission and Escape Probabilities 338 Transmission Probabilities in Two-Dimensional Geometries 339 Escape Probabilities in Two-Dimensional Geometries 342 Simple Approximations for the Escape Probability 343 9.6 Spherical Harmonics (P L) Methods in One-Dimensional Geometries 344 Legendre Polynomials 344 Neutron Transport Equation in Slab Geometry 345 P L Equations 346 Boundary and Interface Conditions 347 P 1 Equations and Diffusion Theory 348 Simplified P L or Extended Diffusion Theory 350 P L Equations in Spherical and Cylindrical Geometries 351 Diffusion Equations in One-Dimensional Geometry 354 Half-Angle Legendre Polynomials 354 Double-P L Theory 355 D-P 0 Equations 357 9.7 Multidimensional Spherical Harmonics (P L) Transport Theory 357 Spherical Harmonics 357 Spherical Harmonics Transport Equations in Cartesian Coordinates 359 P l Equations in Cartesian Geometry 360 Diffusion Theory 361 9.8 Discrete Ordinates Methods in One-Dimensional Slab Geometry 362 P L and D-P L Ordinates 363 Spatial Differencing and Iterative Solution 366 Limitations on Spatial Mesh Size 367 9.9 Discrete Ordinates Methods in One-Dimensional Spherical Geometry 368 Representation of Angular Derivative 368 Iterative Solution Procedure 369 Acceleration of Convergence 371 Calculation of Criticality 372 9.10 Multidimensional Discrete Ordinates Methods 372 Ordinates and Quadrature Sets 372 S N Method in Two-Dimensional x–y Geometry 375 Further Discussion 378 9.11 Even-Parity Transport Formulation 379 9.12 Monte Carlo Methods 380 Probability Distribution Functions 380 Analog Simulation of Neutron Transport 381 Statistical Estimation 383 Variance Reduction 385 Tallying 387 Criticality Problems 389 Source Problems 390 Random Numbers 390 10 Neutron Slowing Down 395 10.1 Elastic Scattering Transfer Function 395 Lethargy 395 Elastic Scattering Kinematics 395 Elastic Scattering Kernel 396 Isotropic Scattering in Center-of-Mass System 398 Linearly Anisotropic Scattering in Center-of-Mass System 399 10.2 P 1 and B 1 Slowing-Down Equations 400 Derivation 400 Solution in Finite Uniform Medium 404 B 1 Equations 405 Few-Group Constants 407 10.3 Diffusion Theory 407 Lethargy-Dependent Diffusion Theory 407 Directional Diffusion Theory 408 Multigroup Diffusion Theory 409 Boundary and Interface Conditions 410 10.4 Continuous Slowing-Down Theory 411 P 1 Equations in Slowing-Down Density Formulation 411 Slowing-Down Density in Hydrogen 415 Heavy Mass Scatterers 415 Age Approximation 416 Selengut–Goertzel Approximation 416 Consistent P 1 Approximation 416 Extended Age Approximation 417 Grueling–Goertzel Approximation 418 Summary of P l Continuous Slowing-Down Theory 419 Inclusion of Anisotropic Scattering 419 Inclusion of Scattering Resonances 421 P l Continuous Slowing-Down Equations 422 10.5 Multigroup Discrete Ordinates Transport Theory 423 11 Resonance Absorption 429 11.1 Resonance Cross Sections 429 11.2 Widely Spaced Single-Level Resonances in a Heterogeneous Fuel–Moderator Lattice 429 Neutron Balance in Heterogeneous Fuel–Moderator Cell 429 Reciprocity Relation 432 Narrow Resonance Approximation 433 Wide Resonance Approximation 434 Evaluation of Resonance Integrals 434 Infinite Dilution Resonance Integral 436 Equivalence Relations 436 Heterogeneous Resonance Escape Probability 436 Homogenized Multigroup Resonance Cross Section 438 Improved and Intermediate Resonance Approximations 438 11.3 Calculation of First-Flight Escape Probabilities 439 Escape Probability for an Isolated Fuel Rod 439 Closely Packed Lattices 442 11.4 Unresolved Resonances 444 Multigroup Cross Sections for Isolated Resonances 446 Self-Overlap Effects 447 Overlap Effects for Different Sequences 448 11.5 Multiband Treatment of Spatially Dependent Self-Shielding 449 Spatially Dependent Self-Shielding 449 Multiband Theory 450 Evaluation of Multiband Parameters 453 Calculation of Multiband Parameters 454 Interface Conditions 455 11.6 Resonance Cross Section Representations 456 R-Matrix Representation 456 Practical Formulations 457 Generalization of the Pole Representation 461 Doppler Broadening of the Generalized Pole Representation 464 12 Neutron Thermalization 469 12.1 Double Differential Scattering Cross Section for Thermal Neutrons 469 12.2 Neutron Scattering from a Monatomic Maxwellian Gas 470 Differential Scattering Cross Section 470 Cold Target Limit 471 Free-Hydrogen (Proton) Gas Model 471 Radkowsky Model for Scattering from H 2 O 471 Heavy Gas Model 472 12.3 Thermal Neutron Scattering from Bound Nuclei 473 Pair Distribution Functions and Scattering Functions 473 Intermediate Scattering Functions 474 Incoherent Approximation 475 Gaussian Representation of Scattering 475 Measurement of the Scattering Function 476 Applications to Neutron Moderating Media 476 12.4 Calculation of the Thermal Neutron Spectra in Homogeneous Media 478 Wigner–Wilkins Proton Gas Model 480 Heavy Gas Model 483 Numerical Solution 486 Moments Expansion Solution 486 Multigroup Calculation 490 Applications to Moderators 491 12.5 Calculation of Thermal Neutron Energy Spectra in Heterogeneous Lattices 492 12.6 Pulsed Neutron Thermalization 494 Spatial Eigenfunction Expansion 494 Energy Eigenfunctions of the Scattering Operator 494 Expansion in Energy Eigenfunctions of the Scattering Operator 496 13 Perturbation and Variational Methods 501 13.1 Perturbation Theory Reactivity Estimate 501 Multigroup Diffusion Perturbation Theory 501 13.2 Adjoint Operators and Importance Function 504 Adjoint Operators 504 Importance Interpretation of the Adjoint Function 506 Eigenvalues of the Adjoint Equation 507 13.3 Variational/Generalized Perturbation Reactivity Estimate 508 One-Speed Diffusion Theory 508 Other Transport Models 511 Reactivity Worth of Localized Perturbations in a Large PWR Core Model 512 Higher Order Variational Estimates 512 13.4 Variational/Generalized Perturbation Theory Estimates of Reaction Rate Ratios in Critical Reactors 512 13.5 Variational/Generalized Perturbation Theory Estimates of Reaction Rates 515 13.6 Variational Theory 516 Stationarity 516 Roussopolos Variational Functional 517 Schwinger Variational Functional 517 Rayleigh Quotient 518 Construction of Variational Functionals 519 13.7 Variational Estimate of Intermediate Resonance Integral 519 13.8 Heterogeneity Reactivity Effects 521 13.9 Variational Derivation of Approximate Equations 522 Inclusion of Interface and Boundary Terms 523 13.10 Variational Even-Parity Transport Approximations 524 Variational Principle for the Even-Parity Transport Equation 524 Ritz Procedure 525 Diffusion Approximation 526 One-Dimensional Slab Transport Equation 527 13.11 Boundary Perturbation Theory 527 14 Homogenization 535 14.1 Equivalent Homogenized Cross Sections 536 14.2 ABH Collision Probability Method 537 14.3 Blackness Theory 541 14.4 Fuel Assembly Transport Calculations 543 Pin Cells 543 Wigner–Seitz Approximation 543 Collision Probability Pin-Cell Model 544 Interface Current Formulation 548 Multigroup Pin-Cell Collision Probabilities Model 549 Resonance Cross Sections 550 Full Assembly Transport Calculation 550 14.5 Homogenization Theory 551 Homogenization Considerations 551 Conventional Homogenization Theory 552 14.6 Equivalence Homogenization Theory 553 14.7 Multiscale Expansion Homogenization Theory 556 14.8 Flux Detail Reconstruction 560 15 Nodal and Synthesis Methods 563 15.1 General Nodal Formalism 564 15.2 Conventional Nodal Methods 567 15.3 Transverse Integrated Nodal Diffusion Theory Methods 570 Transverse Integrated Equations 570 Polynomial Expansion Methods 571 Analytical Methods 576 Heterogeneous Flux Reconstruction 577 15.4 Transverse Integrated Nodal Integral Transport Theory Models 577 Transverse Integrated Integral Transport Equations 577 Polynomial Expansion of Scalar Flux 581 Isotropic Component of Transverse Leakage 581 Double-P n Expansion of Surface Fluxes 582 Angular Moments of Outgoing Surface Fluxes 583 Nodal Transport Equations 584 15.5 Transverse Integrated Nodal Discrete Ordinates Method 585 15.6 Finite-Element Coarse-Mesh Methods 586 Variational Functional for the P 1 Equations 587 One-Dimensional Finite-Difference Approximation 588 Diffusion Theory Variational Functional 590 Linear Finite-Element Diffusion Approximation in One Dimension 591 Higher Order Cubic Hermite Coarse-Mesh Diffusion Approximation 593 Multidimensional Finite-Element Coarse-Mesh Methods 595 15.7 Variational Discrete Ordinates Nodal Method 595 Variational Principle 596 Application of the Method 604 15.8 Variational Principle for Multigroup Diffusion Theory 605 15.9 Single-Channel Spatial Synthesis 608 15.10 Multichannel Spatial Synthesis 614 15.11 Spectral Synthesis 616 16 Space–Time Neutron Kinetics 623 16.1 Flux Tilts and Delayed Neutron Holdback 623 Modal Eigenfunction Expansion 624 Flux Tilts 625 Delayed Neutron Holdback 626 16.2 Spatially Dependent Point Kinetics 626 Derivation of Point Kinetics Equations 628 Adiabatic and Quasistatic Methods 630 Variational Principle for Static Reactivity 631 Variational Principle for Dynamic Reactivity 632 16.3 Time Integration of the Spatial Neutron Flux Distribution 635 Explicit Integration: Forward-Difference Method 635 Implicit Integration: Backward-Difference Method 636 Implicit Integration: θ Method 637 Implicit Integration: Time-Integrated Method 640 Implicit Integration: GAKIN Method 642 Alternating Direction Implicit Method 645 Stiffness Confinement Method 648 Symmetric Successive Overrelaxation Method 648 Generalized Runge–Kutta Methods 649 16.4 Stability 651 Classical Linear Stability Analysis 651 Lyapunov’s Method 653 Lyapunov’s Method for Distributed Parameter Systems 655 Control 657 Variational Methods of Control Theory 657 Dynamic Programming 659 Pontryagin’s Maximum Principle 661 Variational Methods for Spatially Dependent Control Problems 662 Dynamic Programming for Spatially Continuous Systems 665 Pontryagin’s Maximum Principle for a Spatially Continuous System 666 16.5 Xenon Spatial Oscillations 667 Linear Stability Analysis 669 μ-Mode Approximation 671 λ-Mode Approximation 672 Nonlinear Stability Criterion 676 Control of Xenon Spatial Power Oscillations 677 Variational Control Theory of Xenon Spatial Oscillations 677 16.6 Stochastic Kinetics 680 Forward Stochastic Model 680 Means, Variances, and Covariances 684 Correlation Functions 685 Physical Interpretation, Applications, and Initial and Boundary Conditions 686 Numerical Studies 688 Startup Analysis 690 Appendices A Physical Constants and Nuclear Data 695 B Some Useful Mathematical Formulas 703 C Step Functions, Delta Functions, and Other Functions 705 C. 1 Introduction 705 C. 2 Properties of the Dirac δ-Function 706 Alternative Representations 706 Properties 706 Derivatives 707 D Some Properties of Special Functions 709 E Introduction to Matrices and Matrix Algebra 713 E. 1 Some Definitions 713 E. 2 Matrix Algebra 715 F Introduction to Laplace Transforms 717 F.1 Motivation 717 F.2 “Cookbook” Laplace Transforms 719 Index 723