E-Book, Englisch, 1462 Seiten, eBook
Marguet The Physics of Nuclear Reactors
1. Auflage 2018
ISBN: 978-3-319-59560-3
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, 1462 Seiten, eBook
ISBN: 978-3-319-59560-3
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark
This comprehensive volume offers readers a progressive and highly detailed introduction to the complex behavior of neutrons in general, and in the context of nuclear power generation. A compendium and handbook for nuclear engineers, a source of teaching material for academic lecturers as well as a graduate text for advanced students and other non-experts wishing to enter this field, it is based on the author's teaching and research experience and his recognized expertise in nuclear safety.After recapping a number of points in nuclear physics, placing the theoretical notions in their historical context, the book successively reveals the latest quantitative theories concerning:• The slowing-down of neutrons in matter• The charged particles and electromagnetic rays• The calculation scheme, especially the simplification hypothesis• The concept of criticality based on chain reactions• The theory of homogeneous and heterogeneous reactors• The problem of self-shielding• The theory of the nuclear reflector, a subject largely ignored in literature• The computational methods in transport and diffusion theories Complemented by more than 400 bibliographical references, some of which are commented and annotated, and augmented by an appendix on the history of reactor physics at EDF (Electricité De France), this book is the most comprehensive and up-to-date introduction to and reference resource in neutronics and reactor theory.
Serge Marguet is an expert in reactor physics at Electricité De France (EDF), the French leading utility owning a fleet of 58 nuclear reactors. EDF is one of the major companies in the world nuclear field. He has taken part, for the last 30 years, in the development of the calculation scheme of the French nuclear reactors. At the beginning of the 2000, he headed the EDF research team on severe accidents, subject on which he was appointed as expert by the European Commission, in charge of the evaluation of the 6th European Framework Programme, related to severe accidents. Serge Marguet is also teaching reactor physics at the National Institute of Applied Sciences of Bourges (France) for 15 years, as well Neutronics in the EDF Institute of Technology Transfer.
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Weitere Infos & Material
1;Foreword to the 2011 Edition;6
2;Foreword to the 2017 Edition;8
3;Acknowledgements;10
4;Introduction;12
5;Contents;15
6;Contents for Volume 2;23
7;Part I: Neutronics;31
7.1;Chapter 1: Fundamentals of Nuclear Physics;32
7.1.1;1.1 Chemical Elements;32
7.1.2;1.2 Molecules;36
7.1.3;1.3 Isotopes;38
7.1.4;1.4 Atoms;41
7.1.5;1.5 Avogadro´s Number;43
7.1.6;1.6 Mass-Energy Equivalence;48
7.1.7;1.7 Neutrons;51
7.1.8;1.8 Electrons;54
7.1.9;1.9 Protons;57
7.1.10;1.10 The Electron Cloud;57
7.1.11;1.11 The Atomic Nucleus;69
7.1.12;1.12 Nuclear Spin;79
7.1.13;1.13 Radioactivity;80
7.1.13.1;1.13.1 Alpha Decay;90
7.1.13.2;1.13.2 ?- Decay;97
7.1.13.3;1.13.3 ?+ Radioactivity;100
7.1.13.4;1.13.4 Electron Capture;101
7.1.13.5;1.13.5 ? Radioactivity;102
7.1.13.6;1.13.6 Internal Conversion;103
7.1.13.7;1.13.7 (?-,n) Decay or Neutron Decay;104
7.1.13.8;1.13.8 Spontaneous Fission;105
7.1.14;1.14 Radioactive Decay Branches;105
7.1.15;1.15 Heavy Nucleus Chains;111
7.2;Chapter 2: Interaction Between Neutrons and Matter;117
7.2.1;2.1 Neutron Scattering;117
7.2.1.1;2.1.1 Elastic Scattering on a Fixed Target;118
7.2.1.2;2.1.2 Elastic Scattering on a Moving Target;125
7.2.1.3;2.1.3 Moderator;127
7.2.1.4;2.1.4 Inelastic Scattering;128
7.2.2;2.2 Transmutations;131
7.2.2.1;2.2.1 Absorption;132
7.2.2.2;2.2.2 (n,?) Neutron Capture or Radiative Capture;133
7.2.2.3;2.2.3 (n,?) Capture;134
7.2.2.4;2.2.4 Other Forms of Capture;134
7.2.2.5;2.2.5 High-Energy Reactions;135
7.2.2.6;2.2.6 Energy Balance;135
7.2.3;2.3 Fission;138
7.2.4;2.4 Fusion;138
7.2.5;2.5 Cross Sections;139
7.2.5.1;2.5.1 Basic Definitions;139
7.2.5.2;2.5.2 Measurement of Cross Sections;141
7.2.5.3;2.5.3 Notion of Flux and Reaction Rate;142
7.2.5.4;2.5.4 Resonance;144
7.2.6;2.6 Nuclear Fission;156
7.2.6.1;2.6.1 Fission Energy;159
7.2.6.2;2.6.2 Spontaneous Fission;161
7.2.6.3;2.6.3 Neutrons Produced by Fission;162
7.2.6.3.1;2.6.3.1 Theoretical Fission Spectrum;166
7.2.6.3.2;2.6.3.2 Average Energy of Fission Neutrons;169
7.2.6.4;2.6.4 Prompt Fission Photons;170
7.2.6.5;2.6.5 Delayed Fission Neutrons;171
7.2.7;2.7 Fission Products Resulting from Fission;175
7.2.7.1;2.7.1 Direct Yield of an Isotope;177
7.2.7.2;2.7.2 Total Chain Yield;178
7.2.7.3;2.7.3 Cumulative Yield of an Isotope;179
7.2.7.4;2.7.4 Slowing Down of Fission Products in Matter;179
7.3;Chapter 3: Interaction of Electromagnetic Radiation and Charged Particles with Matter;180
7.3.1;3.1 Electromagnetic Radiation;180
7.3.2;3.2 X-radiation;181
7.3.3;3.3 Interaction of Photons with Matter;184
7.3.3.1;3.3.1 Attenuation of a Photon Beam;185
7.3.3.2;3.3.2 Photon Transport;187
7.3.3.3;3.3.3 Rayleigh-Thomson Scattering;188
7.3.3.4;3.3.4 Photoelectric Effect;188
7.3.3.5;3.3.5 Compton Effect;193
7.3.3.6;3.3.6 Pair Production;198
7.3.3.7;3.3.7 Cumulative Effects;201
7.3.3.8;3.3.8 Scattered Radiation and Build-Up Factors;201
7.3.3.9;3.3.9 Application of Photon Attenuation in Matter;204
7.3.3.10;3.3.10 Photoneutrons;208
7.3.3.11;3.3.11 Photofission;209
7.3.4;3.4 Measuring Radiation;209
7.3.5;3.5 Interaction of Electrons with Matter;211
7.3.5.1;3.5.1 Ionization;213
7.3.5.2;3.5.2 Wilson Chamber;214
7.3.5.3;3.5.3 Excitation;216
7.3.5.4;3.5.4 Braking Radiation or Bremsstrahlung;216
7.3.5.5;3.5.5 Annihilation;217
7.3.6;3.6 Cherenkov-Mallet Effect;217
7.3.7;3.7 Charged Particles: Rutherford Diffusion;220
7.3.8;3.8 Transfer of Energy to Matter;225
7.3.9;3.9 Ion-Electron Pair Production by Ionization;231
7.3.10;3.10 Variation in Charge;232
7.3.11;3.11 Fission Products;232
7.3.12;3.12 Path Length in Matter;233
7.3.13;3.13 Biological Effects of Radiation;234
7.4;Chapter 4: Neutron Slowing-Down;238
7.4.1;4.1 Historical Background;238
7.4.2;4.2 Continuous-Energy Slowing-Down Theory;241
7.4.2.1;4.2.1 Elastic Collision with a Stationary Target;242
7.4.2.2;4.2.2 Collision Statistics;250
7.4.2.3;4.2.3 Effect of the Motion of the Target Nucleus;253
7.4.2.4;4.2.4 Transfer Probability as a Function of Angle;254
7.4.2.5;4.2.5 Isotropic Collision;257
7.4.3;4.3 Continuous Slowing-Down Theory;258
7.4.3.1;4.3.1 Slowing Down by Non-Absorbing Hydrogen;264
7.4.3.1.1;4.3.1.1 Neutronic Definition of the Napier´s Constant;271
7.4.3.2;4.3.2 Taking into Account Absorption by Hydrogen;273
7.4.3.3;4.3.3 Taking Account of a Spectral Source;274
7.4.3.4;4.3.4 Slowing Down by Targets Heavier Than Hydrogen;275
7.4.3.5;4.3.5 Influence of the Fast Fission Spectrum;283
7.4.3.6;4.3.6 Mixture of Moderators;286
7.4.4;4.4 Slowing Down in an Absorbing Medium;287
7.4.4.1;4.4.1 Slowly Varying Absorption: The Greuling-Goertzel Model;292
7.4.4.2;4.4.2 Slowing Down in a Medium with a Resonant Cross Section;295
7.4.4.3;4.4.3 Inelastic Slowing-Down;298
7.4.4.4;4.4.4 The Qn Slowing-Down Approximation;302
7.5;Chapter 5: Resonant Absorption;307
7.5.1;5.1 Cross Section Model;307
7.5.1.1;5.1.1 Historical Background;307
7.5.1.2;5.1.2 Intermediate Nucleus Theory;308
7.5.1.3;5.1.3 Principle of Reciprocity;311
7.5.2;5.2 Single-Level Breit-Wigner Formalism;312
7.5.2.1;5.2.1 Total Cross Section;313
7.5.2.2;5.2.2 Scattering Cross Section;314
7.5.2.3;5.2.3 Radiative Capture Cross Section;314
7.5.2.4;5.2.4 Fission Cross Section;316
7.5.2.5;5.2.5 Absorption Cross Section;316
7.5.2.6;5.2.6 Negative Resonances;317
7.5.2.7;5.2.7 Distribution of Resonances;317
7.5.2.8;5.2.8 Resonant Absorption;320
7.5.3;5.3 Self-Shielding;321
7.5.4;5.4 Slowing-Down Through Resonances;324
7.5.5;5.5 The Livolant-Jeanpierre Formalism;327
7.5.5.1;5.5.1 Homogeneous Medium;327
7.5.5.2;5.5.2 Fine Structure Equation;330
7.5.5.3;5.5.3 Tabulating Effective Cross Sections;332
7.5.6;5.6 Modeling the Slowing-Down Operator Using the Resonant Isotope;334
7.5.6.1;5.6.1 Narrow Resonance Approximation;334
7.5.6.2;5.6.2 Wide Resonance Approximation;335
7.5.6.3;5.6.3 Statistical Approach;336
7.5.6.4;5.6.4 All Resonance Model (TR);337
7.5.7;5.7 Heterogeneous Medium;339
7.5.7.1;5.7.1 Two-Media Problem;339
7.5.7.2;5.7.2 Accounting for Spatial Interaction;343
7.5.7.3;5.7.3 Generalization to Several Self-Shielding Regions;346
7.5.8;5.8 Accounting for Energy Interactions: Self-Shielding of Mixtures;348
7.5.9;5.9 Intermediate Resonance Model in Flux Calculations;349
7.5.10;5.10 The Probability Table Method;352
7.6;Chapter 6: Doppler Effect;358
7.6.1;6.1 An Intuitive Analysis of the Doppler Effect;358
7.6.2;6.2 Effective Interaction Cross Section with ``Hot´´ Matter;359
7.6.2.1;6.2.1 Distribution of the Target Nuclei Velocities in Matter: The Free Gas Model;360
7.6.2.2;6.2.2 Definition of the Effective Cross Section;361
7.6.2.3;6.2.3 Cross Section Inversely Proportional to Velocity;362
7.6.2.4;6.2.4 Constant Cross Section;362
7.6.3;6.3 Generalized Doppler Broadening: Bethe-Placzek Formula;366
7.6.4;6.4 Doppler Broadening of a Breit-Wigner Cross Section;370
7.6.4.1;6.4.1 Overview of the Breit-Wigner Formalism;370
7.6.4.2;6.4.2 Voigt´s Formula;372
7.6.4.3;6.4.3 Interference Function;378
7.6.5;6.5 Application to the Large Resonance of Uranium 238;379
7.6.6;6.6 Temperature Effect on Cross Sections;381
7.6.6.1;6.6.1 First Voigt Function ?;382
7.6.6.2;6.6.2 Interference Function;383
7.6.6.3;6.6.3 Asymptotic Numeric Evaluation;384
7.6.6.4;6.6.4 Derivatives of the Voigt Functions with Respect to Energy;387
7.6.6.5;6.6.5 Some Mathematical Properties of Voigt Profiles;388
7.6.7;6.7 Effective Resonance Integral;389
7.6.7.1;6.7.1 Homogeneous Medium;389
7.6.7.2;6.7.2 Heterogeneous Medium;392
7.6.7.3;6.7.3 Analytical Calculation of a Broadened Resonance: The Campos-Martinez Model;399
7.6.8;6.8 Effective Doppler Temperature;403
7.6.8.1;6.8.1 Lattice Bonding Effects;403
7.6.8.2;6.8.2 Heterogeneity Effects of the Temperature Field;405
7.7;Chapter 7: Thermalization of Neutrons;358
7.7.1;7.1 Historical Background;411
7.7.2;7.2 Boltzmann Theory of Gases;412
7.7.3;7.3 Application to Neutrons;416
7.7.4;7.4 Neutron Flux Spectrum;419
7.7.5;7.5 Neutron Thermalization Equation;421
7.7.6;7.6 Wigner-Wilkins Model: Free Proton Gas;425
7.7.7;7.7 Asymptotic Spectrum;428
7.7.8;7.8 Simplified Solution to Thermalization with Absorption;432
7.7.9;7.9 Horowitz-Tretiakoff Model;436
7.7.9.1;7.9.1 Principle;436
7.7.9.2;7.9.2 Case of Absorption Inversely Proportional to Speed;443
7.7.9.3;7.9.3 Case of a Finite Reactor (with Leakage);443
7.7.9.4;7.9.4 Thermalization Equation for a Homogeneous Medium;444
7.7.10;7.10 Heavy Gas Model;445
7.7.11;7.11 Cadilhac, Horowitz and Soulé Differential Model;446
7.7.12;7.12 Application of the Cadilhac Model to Heterogeneous Media;450
7.7.13;7.13 Graphical Representation of Flux over the Energy Spectrum;455
7.7.14;7.14 True Moderators;456
7.7.15;7.15 Heating and Cooling by Scattering;458
7.7.16;7.16 Thermalized Absorption;461
7.7.16.1;7.16.1 Calculation of Reaction Rate in a Pure Thermal Spectrum;464
7.7.16.2;7.16.2 Definition of the Westcott Coefficient g(T);465
7.7.17;7.17 Calculation of the Reaction Rate in a True Thermal Spectrum;470
7.7.17.1;7.17.1 Westcott Formalism: Introduction of the Coefficients r and s;473
7.7.17.2;7.17.2 Extension of the Model to Other Nuclides: The Linear Logarithmic Model;477
7.7.17.3;7.17.3 Progressive Junction at Epithermal Energy;480
7.7.17.4;7.17.4 Westcott Junction;481
7.7.17.5;7.17.5 Determination of Cut-Off Function;482
7.7.17.6;7.17.6 Limits of the Westcott Formalism;484
7.7.18;7.18 Application of the Westcott Formalism;485
7.8;Chapter 8: The Boltzmann Equation;488
7.8.1;8.1 Setting Up the Boltzmann Equation;488
7.8.1.1;8.1.1 Concept of Flux;491
7.8.2;8.2 The Integro-Differential Transport Equation;497
7.8.2.1;8.2.1 The Integro-Differential Transport Equation in Kinetics;497
7.8.2.2;8.2.2 The Integro-Differential Equation in Steady-State;498
7.8.2.2.1;8.2.2.1 Setting Up the Integro-Differential Form;498
7.8.2.2.2;8.2.2.2 The Eigenvalue Problem;502
7.8.2.2.3;8.2.2.3 Solutions of the Transport Equation for Simple Cases;505
7.8.2.2.4;8.2.2.4 Adjoint Transport Theory;510
7.8.2.2.4.1;8.2.2.4.1 The Adjoint Integro-Differential Equation;510
7.8.2.2.4.2;8.2.2.4.2 The Adjoint Equation for the Computation of Neutron Multiplication;514
7.8.2.2.4.3;8.2.2.4.3 Neutron Importance;516
7.8.2.2.4.4;8.2.2.4.4 Perturbation Theory Approach to the Subcritical Flux with Source;517
7.8.2.2.5;8.2.2.5 The Critical Reactor Eigenvalue Problem;520
7.8.2.2.6;8.2.2.6 Uncollided Flux;521
7.8.2.2.6.1;8.2.2.6.1 Green´s Functions, Uncollided Neutron Flux;521
7.8.2.2.6.2;8.2.2.6.2 Uncollided Flux from a Point Source;522
7.8.2.2.6.3;8.2.2.6.3 Uncollided Flux from a Plane Source;524
7.8.2.2.6.4;8.2.2.6.4 Uncollided Flux from an Isotropic Line Source;525
7.8.2.2.6.5;8.2.2.6.5 Using Homogeneous Green´s Functions;526
7.8.3;8.3 Integral Form of the Boltzmann Equation;530
7.8.3.1;8.3.1 Peierls Operator;530
7.8.3.2;8.3.2 The Volume Integral Form;533
7.8.3.3;8.3.3 The First Collision Probability;535
7.8.3.3.1;8.3.3.1 Definition of the First Collision Probability;535
7.8.3.3.2;8.3.3.2 Calculating First Collision Probabilities;539
7.8.3.3.3;8.3.3.3 Dirac´s Chord Method;543
7.8.3.4;8.3.4 1D Geometry;545
7.8.3.5;8.3.5 Escape Probabilities;547
7.8.3.5.1;8.3.5.1 Escape Probability from a Slab;547
7.8.3.5.2;8.3.5.2 Escape Probability from a Sphere;549
7.8.3.5.3;8.3.5.3 Internal Escape Probability from a Hollow Sphere;550
7.8.3.5.4;8.3.5.4 Escape Probability from a Cylinder;551
7.8.3.5.5;8.3.5.5 Concept of Opacity;553
7.8.3.5.6;8.3.5.6 Multiple Collision Escape Probability;553
7.8.3.5.7;8.3.5.7 Escape Probability in Transient States;555
7.8.3.5.8;8.3.5.8 Interface Current Method;557
7.8.3.6;8.3.6 The Integral Equation in 2D;561
7.8.3.7;8.3.7 Application to an Infinite Medium with a Fission Source;562
7.8.3.8;8.3.8 Graphical Solution to the Dispersion Equation;563
7.8.4;8.4 Third Form of the Transport Equation: the Surface-Integral Form;566
7.8.4.1;8.4.1 Placzek´s Lemma;567
7.8.4.2;8.4.2 Flux Equation at the Interface;569
7.8.4.3;8.4.3 Application to the Milne Problem;570
7.8.4.4;8.4.4 Second Complementarity Theorem;571
7.8.5;8.5 Concept of Characteristic Function;572
7.8.6;8.6 Fourier Transform of the Boltzmann Equation;576
7.8.6.1;8.6.1 Formalism;576
7.8.6.2;8.6.2 Resolution Using Green´s Function;578
7.8.7;8.7 The 1D Transport Equation;582
7.8.7.1;8.7.1 General Points;582
7.8.7.2;8.7.2 Lafore and Millot Method, Case Method;585
7.8.7.2.1;8.7.2.1 Historical Overview;585
7.8.7.2.2;8.7.2.2 Theory;590
7.8.7.3;8.7.3 Perovich Method;594
7.8.8;8.8 Asymptotic Solution for Diffusion;595
7.8.8.1;8.8.1 Exponential Relaxation of the Flux, Far from the Source;595
7.8.8.2;8.8.2 Finding the Dispersion Equation from the Asymptotic Flux;603
7.8.8.3;8.8.3 Critical Absorption Limiting the Asymptotic Solution;605
7.8.8.4;8.8.4 Definition of a Diffusion Coefficient from the Transport Equation;607
7.8.9;8.9 The 3D Transport Equations;612
7.9;Chapter 9: Computational Neutron Transport Methods;616
7.9.1;9.1 Discrete Ordinates Method Sn;616
7.9.2;9.2 Exact Sn Method;624
7.9.3;9.3 Legendre Polynomial Method;627
7.9.3.1;9.3.1 Theory and Application to 1D Transport;627
7.9.3.2;9.3.2 Multi-group 1D Transport and Diffusion Equivalence;642
7.9.4;9.4 SPn Method;646
7.9.5;9.5 Interfaces Between Different Media;651
7.9.6;9.6 Spherical Harmonics Method;653
7.9.6.1;9.6.1 Principle;653
7.9.6.2;9.6.2 P1 Approximation;661
7.9.7;9.7 Milne Problem;663
7.9.8;9.8 DPn Method;666
7.9.9;9.9 Semi-infinite Plane: Albedo Problem;669
7.9.9.1;9.9.1 Fundamentals of Discrete Eigenfunctions;669
7.9.9.2;9.9.2 Ganapol Method by Laplace Transform;675
7.9.10;9.10 Bn Method;680
7.9.11;9.11 Tn Method;690
7.9.12;9.12 Fn Method;693
7.9.13;9.13 Cn Method;693
7.9.14;9.14 The SKn Method;698
7.9.15;9.15 Method of Characteristics (MOC);700
7.9.15.1;9.15.1 Principle;700
7.9.15.2;9.15.2 Heterogeneous Geometries;702
7.9.15.3;9.15.3 Characteristic Direction Probabilities (CDP);707
7.9.16;9.16 Even-Odd Formulation of the Transport Equation;709
7.9.16.1;9.16.1 Even-Odd Flux Equation;710
7.9.16.2;9.16.2 Variational Nodal Method of the Even-Odd Formulation;714
7.9.16.3;9.16.3 Ritz Method;717
7.9.17;9.17 Variational Method for Time-Dependent Problems;720
7.9.18;9.18 Gauss-Seidel Method for Sources in Time-Dependent Problems;722
7.9.19;9.19 Probabilistic Approach: The Monte Carlo Method;723
7.9.19.1;9.19.1 Fundamental Concepts of the Monte Carlo Method;723
7.9.19.2;9.19.2 Application to Neutron Transport: A Simple 2D Case;728
7.9.19.3;9.19.3 Statistical Error;736
7.9.19.4;9.19.4 Calculation of Physical Quantities;736
7.9.19.5;9.19.5 Generalization, Biasing;737
7.9.19.6;9.19.6 Resonance Escape Probability Factor Calculation;739
7.9.19.7;9.19.7 Midway Monte Carlo;742
7.9.19.8;9.19.8 Quasi-Deterministic Approximation of the Importance Function;746
7.9.19.9;9.19.9 Example of a Monte Carlo Calculation;749
8;Part II: Reactor Physics;751
8.1;Chapter 10: Diffusion Approximation in Neutron Physics;616
8.1.1;10.1 Fick´s Law;752
8.1.1.1;10.1.1 Evaluation of the Neutron Diffusion Coefficient;752
8.1.1.2;10.1.2 Discussion of the Hypotheses;757
8.1.1.3;10.1.3 The Diffusion Equation in a Force Field;762
8.1.2;10.2 Boundary Conditions for a Medium Surrounded by a Vacuum in Diffusion Theory;764
8.1.2.1;10.2.1 P1 Approximation;765
8.1.2.2;10.2.2 Rulko´s Variational Approach;766
8.1.3;10.3 Boundary Conditions Between Any Two Media;770
8.1.3.1;10.3.1 Notion of a Reflector Albedo;771
8.1.4;10.4 Diffusion Equation in Energy;772
8.1.5;10.5 One-Group Diffusion Equation;774
8.1.6;10.6 ``Thermal Diffusion´´;776
8.1.6.1;10.6.1 ``Thermal´´ Diffusion Equation;776
8.1.6.2;10.6.2 Interpretation of the Thermal Scattering Path;778
8.1.6.3;10.6.3 Deriving the Four-Factor Formula;780
8.1.7;10.7 Scattering of an Isotropic Source in a Non-Multiplying Medium;780
8.1.7.1;10.7.1 Point Source in an Infinite Scattering Medium;781
8.1.7.2;10.7.2 Anisotropic Point Source in Spherical Geometry;784
8.1.7.3;10.7.3 Infinite Thin Rod Source in an Infinite Scattering Medium;790
8.1.7.4;10.7.4 Infinite Plane Source in an Infinite Scattering Medium;792
8.1.7.5;10.7.5 Infinite Plane Source in an Infinite Scattering Slab;794
8.1.7.6;10.7.6 Uniform Source in an Infinite Scattering Slab;796
8.1.7.7;10.7.7 Semi-infinite Slab Source;797
8.1.7.8;10.7.8 Extension to the Infinite Homogeneous Medium;799
8.1.7.9;10.7.9 Expansion on the Eigenfunctions of the Laplacian Operator;800
8.1.7.10;10.7.10 Superposition of Flux Induced by Point Sources;801
8.1.7.11;10.7.11 Absorbing Slab in an Infinite Source Medium;803
8.1.7.12;10.7.12 Thin Absorbing Slabs, the Galanin Method;804
8.1.7.13;10.7.13 Flux Transient;805
8.1.8;10.8 Measurement of the Scattering Path of a Moderator by Attenuation;808
8.1.9;10.9 Pulsed Neutron Method;812
8.1.10;10.10 Diffusion in a Homogeneous Slab;818
8.1.11;10.11 Source Thermalization Transient in Diffusion Theory;823
8.1.11.1;10.11.1 Infinite Medium;823
8.1.11.2;10.11.2 Finite Medium;824
8.1.11.3;10.11.3 Expansion on Eigenfunctions;825
8.1.11.4;10.11.4 Case of a Pulsed Source;827
8.1.12;10.12 Polykinetic Diffusion;829
8.2;Chapter 11: Nuclear Reactor Reactivity;835
8.2.1;11.1 Multiplication Factor of a Chain Reaction;835
8.2.1.1;11.1.1 Deterministic Approach to Chain Reactions;835
8.2.1.2;11.1.2 Stochastic Approach to Chain Reaction;836
8.2.2;11.2 ``Four-factor´´ Formula;841
8.2.2.1;11.2.1 Detailed Analysis of the Four-factor Formula;842
8.2.2.1.1;11.2.1.1 Fuel Multiplication Factor ?;843
8.2.2.1.2;11.2.1.2 Fast Fission Factor ?;845
8.2.2.1.3;11.2.1.3 Neutron Slowing-down: Escape Probability Factor;846
8.2.2.1.4;11.2.1.4 Thermal Range: The Thermal Utilization Factor;847
8.2.2.2;11.2.2 Technological Moderation Ratio Effect on the Four-factor Formula;847
8.2.3;11.3 Allowing for Leakages in a Finite Reactor;848
8.2.4;11.4 Two-group Multiplication Factor;849
8.2.5;11.5 Multiplication Factor Through a Reaction Rate Balance;855
8.2.6;11.6 Reactivity Effects or Reactivity Difference;860
8.2.6.1;11.6.1 Comparison of the Effects on a UOX Fuel;861
8.2.6.2;11.6.2 Reactivity Effect of Isotopic Change;862
8.2.7;11.7 Calculation of Reactivity by Perturbation Theory Estimate;865
8.2.8;11.8 Evolution of the Reactivity Along the Cycle;867
8.3;Chapter 12: Critical Homogeneous Reactor Theory;835
8.3.1;12.1 Introduction;869
8.3.2;12.2 The Notion of Geometrical and Material Buckling;874
8.3.3;12.3 Criticality Condition;875
8.3.4;12.4 Notion of Critical Size: The Rod Model;876
8.3.4.1;12.4.1 Analysis of Criticality;876
8.3.4.2;12.4.2 Invariant Imbedding;880
8.3.5;12.5 Fundamental Mode for a Reactor with Simple Geometry;884
8.3.5.1;12.5.1 Plane Slab;884
8.3.5.2;12.5.2 Parallelepiped;888
8.3.5.3;12.5.3 Infinite Cylinder;890
8.3.5.4;12.5.4 Finite Cylinder;893
8.3.5.5;12.5.5 Disc;895
8.3.5.6;12.5.6 Sphere;898
8.3.5.7;12.5.7 Hemisphere;901
8.3.5.8;12.5.8 Polygon;902
8.3.5.9;12.5.9 Accounting for Singularities in 2D;904
8.3.5.10;12.5.10 Anisotropic Point Source in a Multiplying Medium;912
8.3.5.11;12.5.11 Zero Flux Distance;913
8.3.5.12;12.5.12 Annular Reactor;915
8.3.6;12.6 Any Three-Dimensional Reactor;919
8.3.7;12.7 Fermi Age Theory;920
8.3.7.1;12.7.1 History;921
8.3.7.2;12.7.2 Overview of Slowing-Down;922
8.3.7.3;12.7.3 Application to Neutron Diffusion;924
8.3.7.4;12.7.4 Relation Between Fermi Age and Time;925
8.3.7.5;12.7.5 Link Between the Age Theory and Diffusion Theory;927
8.3.7.6;12.7.6 Two-Energy Group Equation in Fermi Age Theory;929
8.3.7.7;12.7.7 Age-Diffusion Theory;932
8.3.8;12.8 Multi-Group Diffusion;932
8.3.9;12.9 Reactor Kinetics in One-Group Diffusion Theory with Source;934
8.3.10;12.10 Source Calculation: Extension to Multi-Group Conditions;936
8.4;Chapter 13: Neutron Reflectors;939
8.4.1;13.1 Some Mathematical Considerations on Reflectors;939
8.4.2;13.2 Reflectors in Diffusion Theory;942
8.4.2.1;13.2.1 Case of the Slab Reactor Surrounded by an Infinite Reflector;942
8.4.2.2;13.2.2 Reflected Homogeneous Slab Reactor;946
8.4.2.3;13.2.3 Case of an Infinite Cylindrical Reactor Surrounded by an Infinite Reflector;948
8.4.2.4;13.2.4 Case of an Infinite Cylindrical Reactor with a Finite Reflector;954
8.4.2.4.1;13.2.4.1 Monocinetic Calculation;954
8.4.2.4.2;13.2.4.2 Two-Energy Group Calculation;956
8.4.2.4.3;13.2.4.3 Flux in Two Dimensions;957
8.4.3;13.3 Definition of Reflector Albedo;959
8.4.3.1;13.3.1 Albedo Calculation for a Slab Reflector;961
8.4.3.2;13.3.2 Albedo Calculation of a Cylindrical Reflector;962
8.4.3.3;13.3.3 Albedo of a Spherical Reflector;962
8.4.3.4;13.3.4 Albedo Calculation for the Upper Reflector of a Cylindrical Reactor;963
8.4.3.5;13.3.5 Extrapolation and Null-flux Distances;964
8.4.3.6;13.3.6 Numerical Example;967
8.4.4;13.4 Reflector Theory with Two Energy Groups;967
8.4.4.1;13.4.1 Slab Reflector;968
8.4.4.2;13.4.2 Infinite Cylindrical Reactor with Reflector in Two Groups Without Up-Scattering;969
8.4.4.3;13.4.3 Flux Calculation in the Fuel;970
8.4.4.4;13.4.4 Flux in the Reflector;972
8.4.5;13.5 Slab Reactor with Finite Reflector and Without Up-Scattering;975
8.4.6;13.6 The Ackroyd ``Magic Shell´´ Albedo Model;977
8.4.7;13.7 The Lefebvre-Lebigot Reflector Model;979
8.4.7.1;13.7.1 ``Equivalent´´ Reflectors Theory;980
8.4.7.2;13.7.2 Calculation of Core Characteristics;985
8.4.7.3;13.7.3 Core/Reflector Operating Point;987
8.4.7.4;13.7.4 Effect of Thermal-Hydraulic Feedbacks;989
8.4.7.5;13.7.5 Calculation of Constants in the Mathematical Reflector;990
8.4.8;13.8 Albedo Matrix;991
8.4.9;13.9 Allowing for Up-Scattering;992
8.4.10;13.10 Diffusion/Transport Correspondence;997
8.4.11;13.11 Reuss-Nisan Model;998
8.4.12;13.12 Mondot Model;1004
8.4.13;13.13 Generalized BETA Method;1006
8.4.14;13.14 Absorption in the Reflector;1007
8.4.15;13.15 Double-Differential Albedo;1008
8.5;Chapter 14: Heterogeneous Reactors;1011
8.5.1;14.1 Why Is Heterogeneity Desirable?;1011
8.5.2;14.2 Gurevich-Pomeranchuk Heterogeneous Resonant Absorption Theory;1013
8.5.2.1;14.2.1 Theoretical Background;1013
8.5.2.2;14.2.2 Effective Resonance Integral;1018
8.5.3;14.3 Modeling the Pin Flux;1019
8.5.3.1;14.3.1 First-Collision Probability;1020
8.5.3.2;14.3.2 The Amouyal-Benoist-Horowitz (A-B-H) Theory;1022
8.5.3.2.1;14.3.2.1 Classical Thermal Utilization Factor Theory;1022
8.5.3.2.2;14.3.2.2 A-B-H Theory for the Thermal Utilization Factor;1025
8.5.3.3;14.3.3 Multi-cell Approach in Two Dimensions;1034
8.5.3.3.1;14.3.3.1 Context;1034
8.5.3.3.2;14.3.3.2 Dancoff-Ginsburg Factor;1036
8.5.3.3.3;14.3.3.3 The Dancoff Effect in Different Geometries: Shielding Problems;1040
8.5.3.3.4;14.3.3.4 Impact of the Dancoff Factor on Resonant Absorption;1049
8.5.3.3.5;14.3.3.5 Extension to Pin Lattices;1050
8.5.3.4;14.3.4 Carlvik Rational Approximation;1052
8.5.3.5;14.3.5 Heterogeneity of the Isotopic Composition;1058
8.5.3.6;14.3.6 Shadowing Effect on the Resonance Integral;1058
8.5.3.7;14.3.7 Heterogeneous Pi,j Calculations for Fast Reactors with Perturbation Methods;1062
8.5.4;14.4 Transport-Diffusion Equivalence;1065
8.5.4.1;14.4.1 Context;1065
8.5.4.2;14.4.2 Spatial Homogenization;1067
8.5.4.3;14.4.3 Multi-group Approach;1068
8.5.4.4;14.4.4 Kavenoky-Hébert SPH Equivalence;1069
8.5.4.5;14.4.5 Flux Reconstruction Between Different Operators;1071
8.5.4.5.1;14.4.5.1 Reflected Medium (Infinite);1074
8.5.4.5.1.1;14.4.5.1.1 Homogeneous B0 Model;1075
8.5.4.5.1.2;14.4.5.1.2 Homogeneous B1 Model;1078
8.5.4.5.2;14.4.5.2 Finite Medium;1079
8.5.4.6;14.4.6 Spatial Homogenization with Leakage;1082
8.5.4.7;14.4.7 Equivalence for Slab Reactors;1087
8.5.4.8;14.4.8 Equivalence by Conservation of Reaction Rates;1092
8.5.5;14.5 Homogenization Theory in Diffusion;1096
8.5.5.1;14.5.1 Flux-Volume Homogenization;1096
8.5.5.2;14.5.2 Homogenization of Heterogeneous Neutron Quantities;1097
8.5.5.3;14.5.3 Average Flux Homogenization at the Boundary, Selengut Normalization;1100
8.5.5.4;14.5.4 Pin Power Reconstruction;1102
8.5.5.4.1;14.5.4.1 Convolution with Pin Power Distribution;1102
8.5.5.4.2;14.5.4.2 Perturbation Approach: Rahnema Method;1103
8.5.5.5;14.5.5 Discontinuity Factors;1107
8.6;Chapter 15: Fuel Cycle Physics;1110
8.6.1;15.1 Schematic Notation for Fuel Cycle Physics;1110
8.6.2;15.2 Disintegration;1111
8.6.3;15.3 Neutron-Induced Reactions;1111
8.6.4;15.4 The Bateman Equations;1111
8.6.4.1;15.4.1 Heavy Nuclides;1112
8.6.4.2;15.4.2 Fission Products;1114
8.6.4.3;15.4.3 Activation Products;1115
8.6.4.3.1;15.4.3.1 Example of the Cobalt 60 Chain;1115
8.6.5;15.5 Vectorial Form of the Bateman Equation;1116
8.6.6;15.6 Calculation of Relevant Quantities for the Fuel Cycle;1116
8.6.6.1;15.6.1 Mass Balance;1116
8.6.6.2;15.6.2 Burn-up;1117
8.6.6.2.1;15.6.2.1 Thermal Burn-up;1117
8.6.6.2.2;15.6.2.2 Fission Burn-up (for Fast Neutron Reactors);1122
8.6.6.2.3;15.6.2.3 Fuel Depletion with Burn-up;1122
8.6.6.3;15.6.3 Activity;1123
8.6.6.4;15.6.4 Calculation of Decay Heat;1123
8.6.6.4.1;15.6.4.1 Summation Method;1124
8.6.6.4.2;15.6.4.2 Decay Heat Burst Function;1126
8.6.6.4.3;15.6.4.3 Elementary Value Curves;1128
8.6.6.4.4;15.6.4.4 Continuous Fission Curve Method;1130
8.6.6.4.4.1;15.6.4.4.1 Principle;1130
8.6.6.4.4.2;15.6.4.4.2 Capture Correction;1133
8.6.6.4.5;15.6.4.5 Calculation of Decay Sources and Their Spectrum;1133
8.6.6.5;15.6.5 Photon ? and Neutron Dose Calculation;1134
8.6.7;15.7 Isotopic Depletion Calculation;1136
8.6.7.1;15.7.1 Chain-Decay Process: Recurrence Relations;1137
8.6.7.2;15.7.2 Case of Heavy Nuclides;1140
8.6.7.3;15.7.3 Case of Fission Products;1141
8.6.7.4;15.7.4 Reference Composition of Some PWR Fuel;1142
8.6.8;15.8 Decay Chain Reduction Principle;1143
8.6.8.1;15.8.1 Heavy Nuclide Chain for Reactivity Calculations of Reactors;1145
8.6.8.1.1;15.8.1.1 Numerical Example: Plutonium Production in a Uranium Fuel Assembly;1147
8.6.8.1.2;15.8.1.2 Decay Chain for Heavy Nuclides and Fission Products from the SERMA79 Recommendation;1150
8.6.8.1.3;15.8.1.3 Decay Chain of Heavy Nuclides from the REL2005 Recommendation;1152
8.6.8.2;15.8.2 Decay Chain Reduction;1153
8.6.8.2.1;15.8.2.1 Chain Reduction;1153
8.6.9;15.9 Activation: The Example of Control Rods;1156
8.6.10;15.10 Xenon Physics;1157
8.6.10.1;15.10.1 Production of Xenon;1157
8.6.10.2;15.10.2 Xenon Saturation;1159
8.6.10.3;15.10.3 Xenon Poisoning After Reactor Shutdown;1161
8.6.11;15.11 Samarium Physics;1163
8.6.12;15.12 Gadolinium Physics;1164
8.6.13;15.13 The Industrial Fuel Cycle in France;1165
8.7;Chapter 16: Neutronic Feedback;1172
8.7.1;16.1 Effect of Fuel Temperature on the Multiplication Factor;1172
8.7.1.1;16.1.1 Fuel Doppler Effect;1172
8.7.1.2;16.1.2 Doppler Effect on Reactor Behavior;1175
8.7.2;16.2 Moderator Temperature Effect;1177
8.7.2.1;16.2.1 Definitions;1177
8.7.2.2;16.2.2 Leakage and Absorber Effects;1179
8.7.2.3;16.2.3 Pressure Effect;1181
8.7.2.4;16.2.4 Graphite Moderator;1182
8.7.2.5;16.2.5 Neutron Spectrum Shift;1183
8.7.2.6;16.2.6 Void Effect;1184
8.7.3;16.3 Boron Effect in Pressurized Water Reactors;1185
8.7.3.1;16.3.1 Differential Efficiency of Boron;1185
8.7.3.2;16.3.2 Boron Effect on the Moderator Differential Coefficient;1186
8.7.4;16.4 Power Coefficient;1187
8.7.5;16.5 Feedback Modeling;1187
8.7.5.1;16.5.1 A Simple Model: Power Feedback;1190
8.7.5.2;16.5.2 An Advanced Feedback Model: The Lefebvre-Seban Model;1191
8.7.6;16.6 Historical Isotopic Correction;1202
8.8;Chapter 17: Reactor Kinetics;1205
8.8.1;17.1 Prompt Neutrons;1205
8.8.1.1;17.1.1 Evolution of a Hypothetical Prompt Neutron Reactor;1206
8.8.1.2;17.1.2 Flux Calculation: Point Reactor Hypothesis;1211
8.8.2;17.2 Delayed Neutrons;1213
8.8.2.1;17.2.1 Delayed Neutron Fraction;1217
8.8.3;17.3 Effect of Delayed Neutrons on Reactor Kinetics;1218
8.8.4;17.4 Neutron Kinetics Equation;1221
8.8.4.1;17.4.1 Precursor Concentration;1223
8.8.4.2;17.4.2 Point-Reactor Kinetics;1224
8.8.4.3;17.4.3 Mobile Fuel;1226
8.8.5;17.5 Nordheim Equation;1226
8.8.6;17.6 ``Prompt Jump´´ Notion: Insertion of a Reactivity Step;1231
8.8.7;17.7 Age Theory in the Kinetics Equation for Thermal Neutrons;1233
8.8.8;17.8 Reduced Kinetics Equations;1236
8.8.9;17.9 Kinetics with an Imposed Neutron Source;1238
8.8.10;17.10 Delayed Neutron Spectrum;1239
8.8.11;17.11 First-Order Perturbations;1247
8.8.12;17.12 Numerical Reactimeter;1249
8.8.13;17.13 Practical Evaluation of Prompt Neutron Generation Time;1252
8.8.14;17.14 Main Causes of Reactivity Changes;1254
8.8.14.1;17.14.1 Increased Fissile Nuclei;1254
8.8.14.2;17.14.2 Increased Neutron Moderation;1255
8.8.14.3;17.14.3 Decreased Neutron Capture;1255
8.8.15;17.15 Reactivity Accident: Insertion of Very High Reactivity Value;1256
8.8.15.1;17.15.1 Analysis with One Group of Delayed Neutrons;1256
8.8.15.2;17.15.2 Analysis of the Case of ? ?: The Reactivity Accident;1259
8.8.15.3;17.15.3 Insertion of Low Reactivity 0 ? ?;1261
8.8.16;17.16 Anti-reactivity Insertion;1263
8.8.17;17.17 Overview of Cases;1264
8.8.18;17.18 Reactivity Step;1265
8.8.19;17.19 Dropped Control Rod, Insertion of a Large Amount of Anti-reactivity;1267
8.8.20;17.20 Reactivity R1268
8.8.21;17.21 Reactivity Transient;1272
8.8.22;17.22 Power Excursion;1272
8.8.22.1;17.22.1 The Nordheim-Fuchs Model;1273
8.8.22.2;17.22.2 The Chernick Model;1277
8.8.22.3;17.22.3 The Bethe-Tait Model;1280
8.8.23;17.23 Subcritical Approach: Reactor Start-Up;1284
8.8.24;17.24 Reactor Stability;1285
8.8.25;17.25 Space-Time Xenon Oscillations;1289
8.8.26;17.26 Mechanical Kinetic Effects;1294
8.8.27;17.27 Neutron Noise;1295
8.8.27.1;17.27.1 Noise Concept, Spectral Analysis;1296
8.8.27.2;17.27.2 Neutron Correlations;1298
8.8.27.3;17.27.3 The Feynman-? Method;1305
8.8.27.4;17.27.4 Delayed-Neutron Effect;1313
8.8.27.5;17.27.5 Application to Measurement of Void Fraction Instabilities;1314
8.8.27.6;17.27.6 Application to Detection of Vibrations;1316
8.9;Chapter 18: Computation Methods in Diffusion Theory;1319
8.9.1;18.1 Calculation Meshes;1319
8.9.2;18.2 Multi-group Diffusion Equations;1322
8.9.2.1;18.2.1 General Case;1322
8.9.2.2;18.2.2 ``1.5´´-group Diffusion;1323
8.9.2.3;18.2.3 Adjoint Diffusion;1323
8.9.2.4;18.2.4 Taking into Account the Neutron Over-Production Cross Sections;1325
8.9.3;18.3 The Power Iteration Method;1326
8.9.3.1;18.3.1 General Considerations;1326
8.9.3.2;18.3.2 Matrix Representation;1328
8.9.3.3;18.3.3 Chebyshev Acceleration;1330
8.9.4;18.4 Finite Difference Method;1333
8.9.4.1;18.4.1 Formalism;1333
8.9.4.2;18.4.2 Boundary Conditions;1338
8.9.4.3;18.4.3 Matrix Form;1339
8.9.5;18.5 Nodal Methods;1340
8.9.5.1;18.5.1 Nodal Method of Order 4;1342
8.9.5.2;18.5.2 Quadratic Approximation of Transverse Leakage;1350
8.9.5.3;18.5.3 AFEN Method;1353
8.9.6;18.6 Finite Element Method;1354
8.9.7;18.7 Variational Methods;1358
8.9.7.1;18.7.1 Principle;1358
8.9.7.2;18.7.2 Accounting for Boundary Conditions;1360
8.9.8;18.8 Calculation of Control Rods;1361
8.9.8.1;18.8.1 Physical Effect of Rods;1362
8.9.8.2;18.8.2 Rod Worth: Perturbation Analysis;1363
8.9.8.3;18.8.3 Measuring Rod Efficiency in PWR;1366
8.9.8.4;18.8.4 Calculation of Rod Efficiency;1367
8.9.8.5;18.8.5 Analytical Decomposition of the Rodded Domain;1371
8.9.9;18.9 Instrumentation Considerations;1375
8.9.9.1;18.9.1 Modeling with Trace Quantities;1375
8.9.9.2;18.9.2 Modeling of the EPR Instrumentation: The KTM Model;1375
8.9.9.2.1;18.9.2.1 Cross Sections in the KTM Model;1380
8.9.9.2.2;18.9.2.2 Fine Power Structures in the KTM Model;1380
9;Conclusion;1385
10;Annex: Reactor Physics and Neutronic Codes at Electricité De France;1387
11;Bibliography;1420
12;Index;1448
Introduction.- Basic concepts in nuclear physics.- Neutron interactions with matter.- Interactions of electromagnetic rays and charged particles with matter.- The slowing-down of neutrons.- The resonant absorption.- The Doppler Effect in nuclear fuel.- Neutron thermalization.- The Boltzmann equation.- Numerical methods in neutron transport theory.- The neutron diffusion theory.- Reactivity of a nuclear reactor.- The theory of the homogeneous critical reactor.- The theory of the nuclear reactor reflector.- The theory of the heterogeneous reactor.- Fuel cycle physics.- The nuclear feedbacks.- The kinetics of nuclear reactors.- Computation methods in diffusion theory.- Conclusions.- Appendices.- Reactor physics at Electricité de France.
