- ホーム
- > 電子洋書
Description
The third, revised edition of this popular textbook and reference, which has been translated into Russian and Chinese, expands the comprehensive and balanced coverage of nuclear reactor physics to include recent advances in understanding of this topic.
The first part of the book covers basic reactor physics, including, but not limited to nuclear reaction data, neutron diffusion theory, reactor criticality and dynamics, neutron energy distribution, fuel burnup, reactor types and reactor safety.
The second part then deals with such physically and mathematically more advanced topics as neutron transport theory, neutron slowing down, resonance absorption, neutron thermalization, perturbation and variational methods, homogenization, nodal and synthesis methods, and space-time neutron dynamics.
For ease of reference, the detailed appendices contain nuclear data, useful mathematical formulas, an overview of special functions as well as introductions to matrix algebra and Laplace transforms.
With its focus on conveying the in-depth knowledge needed by advanced student and professional nuclear engineers, this text is ideal for use in numerous courses and for self-study by professionals in basic nuclear reactor physics, advanced nuclear reactor physics, neutron transport theory, nuclear reactor dynamics and stability, nuclear reactor fuel cycle physics and other important topics in the field of nuclear reactor physics.
Table of Contents
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
