The Physics of Nuclear Reactors (eBook)

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.

Foreword to the 2011 Edition 6
Foreword to the 2017 Edition 8
Acknowledgements 10
Introduction 12
Contents 15
Contents for Volume 2 23
Part I: Neutronics 31
Chapter 1: Fundamentals of Nuclear Physics 32
1.1 Chemical Elements 32
1.2 Molecules 36
1.3 Isotopes 38
1.4 Atoms 41
1.5 Avogadro´s Number 43
1.6 Mass-Energy Equivalence 48
1.7 Neutrons 51
1.8 Electrons 54
1.9 Protons 57
1.10 The Electron Cloud 57
1.11 The Atomic Nucleus 69
1.12 Nuclear Spin 79
1.13 Radioactivity 80
1.13.1 Alpha Decay 90
1.13.2 ?- Decay 97
1.13.3 ?+ Radioactivity 100
1.13.4 Electron Capture 101
1.13.5 ? Radioactivity 102
1.13.6 Internal Conversion 103
1.13.7 (?-,n) Decay or Neutron Decay 104
1.13.8 Spontaneous Fission 105
1.14 Radioactive Decay Branches 105
1.15 Heavy Nucleus Chains 111
Chapter 2: Interaction Between Neutrons and Matter 117
2.1 Neutron Scattering 117
2.1.1 Elastic Scattering on a Fixed Target 118
2.1.2 Elastic Scattering on a Moving Target 125
2.1.3 Moderator 127
2.1.4 Inelastic Scattering 128
2.2 Transmutations 131
2.2.1 Absorption 132
2.2.2 (n,?) Neutron Capture or Radiative Capture 133
2.2.3 (n,?) Capture 134
2.2.4 Other Forms of Capture 134
2.2.5 High-Energy Reactions 135
2.2.6 Energy Balance 135
2.3 Fission 138
2.4 Fusion 138
2.5 Cross Sections 139
2.5.1 Basic Definitions 139
2.5.2 Measurement of Cross Sections 141
2.5.3 Notion of Flux and Reaction Rate 142
2.5.4 Resonance 144
2.6 Nuclear Fission 156
2.6.1 Fission Energy 159
2.6.2 Spontaneous Fission 161
2.6.3 Neutrons Produced by Fission 162
2.6.3.1 Theoretical Fission Spectrum 166
2.6.3.2 Average Energy of Fission Neutrons 169
2.6.4 Prompt Fission Photons 170
2.6.5 Delayed Fission Neutrons 171
2.7 Fission Products Resulting from Fission 175
2.7.1 Direct Yield of an Isotope 177
2.7.2 Total Chain Yield 178
2.7.3 Cumulative Yield of an Isotope 179
2.7.4 Slowing Down of Fission Products in Matter 179
Chapter 3: Interaction of Electromagnetic Radiation and Charged Particles with Matter 180
3.1 Electromagnetic Radiation 180
3.2 X-radiation 181
3.3 Interaction of Photons with Matter 184
3.3.1 Attenuation of a Photon Beam 185
3.3.2 Photon Transport 187
3.3.3 Rayleigh-Thomson Scattering 188
3.3.4 Photoelectric Effect 188
3.3.5 Compton Effect 193
3.3.6 Pair Production 198
3.3.7 Cumulative Effects 201
3.3.8 Scattered Radiation and Build-Up Factors 201
3.3.9 Application of Photon Attenuation in Matter 204
3.3.10 Photoneutrons 208
3.3.11 Photofission 209
3.4 Measuring Radiation 209
3.5 Interaction of Electrons with Matter 211
3.5.1 Ionization 213
3.5.2 Wilson Chamber 214
3.5.3 Excitation 216
3.5.4 Braking Radiation or Bremsstrahlung 216
3.5.5 Annihilation 217
3.6 Cherenkov-Mallet Effect 217
3.7 Charged Particles: Rutherford Diffusion 220
3.8 Transfer of Energy to Matter 225
3.9 Ion-Electron Pair Production by Ionization 231
3.10 Variation in Charge 232
3.11 Fission Products 232
3.12 Path Length in Matter 233
3.13 Biological Effects of Radiation 234
Chapter 4: Neutron Slowing-Down 238
4.1 Historical Background 238
4.2 Continuous-Energy Slowing-Down Theory 241
4.2.1 Elastic Collision with a Stationary Target 242
4.2.2 Collision Statistics 250
4.2.3 Effect of the Motion of the Target Nucleus 253
4.2.4 Transfer Probability as a Function of Angle 254
4.2.5 Isotropic Collision 257
4.3 Continuous Slowing-Down Theory 258
4.3.1 Slowing Down by Non-Absorbing Hydrogen 264
4.3.1.1 Neutronic Definition of the Napier´s Constant 271
4.3.2 Taking into Account Absorption by Hydrogen 273
4.3.3 Taking Account of a Spectral Source 274
4.3.4 Slowing Down by Targets Heavier Than Hydrogen 275
4.3.5 Influence of the Fast Fission Spectrum 283
4.3.6 Mixture of Moderators 286
4.4 Slowing Down in an Absorbing Medium 287
4.4.1 Slowly Varying Absorption: The Greuling-Goertzel Model 292
4.4.2 Slowing Down in a Medium with a Resonant Cross Section 295
4.4.3 Inelastic Slowing-Down 298
4.4.4 The Qn Slowing-Down Approximation 302
Chapter 5: Resonant Absorption 307
5.1 Cross Section Model 307
5.1.1 Historical Background 307
5.1.2 Intermediate Nucleus Theory 308
5.1.3 Principle of Reciprocity 311
5.2 Single-Level Breit-Wigner Formalism 312
5.2.1 Total Cross Section 313
5.2.2 Scattering Cross Section 314
5.2.3 Radiative Capture Cross Section 314
5.2.4 Fission Cross Section 316
5.2.5 Absorption Cross Section 316
5.2.6 Negative Resonances 317
5.2.7 Distribution of Resonances 317
5.2.8 Resonant Absorption 320
5.3 Self-Shielding 321
5.4 Slowing-Down Through Resonances 324
5.5 The Livolant-Jeanpierre Formalism 327
5.5.1 Homogeneous Medium 327
5.5.2 Fine Structure Equation 330
5.5.3 Tabulating Effective Cross Sections 332
5.6 Modeling the Slowing-Down Operator Using the Resonant Isotope 334
5.6.1 Narrow Resonance Approximation 334
5.6.2 Wide Resonance Approximation 335
5.6.3 Statistical Approach 336
5.6.4 All Resonance Model (TR) 337
5.7 Heterogeneous Medium 339
5.7.1 Two-Media Problem 339
5.7.2 Accounting for Spatial Interaction 343
5.7.3 Generalization to Several Self-Shielding Regions 346
5.8 Accounting for Energy Interactions: Self-Shielding of Mixtures 348
5.9 Intermediate Resonance Model in Flux Calculations 349
5.10 The Probability Table Method 352
Chapter 6: Doppler Effect 358
6.1 An Intuitive Analysis of the Doppler Effect 358
6.2 Effective Interaction Cross Section with ``Hot´´ Matter 359
6.2.1 Distribution of the Target Nuclei Velocities in Matter: The Free Gas Model 360
6.2.2 Definition of the Effective Cross Section 361
6.2.3 Cross Section Inversely Proportional to Velocity 362
6.2.4 Constant Cross Section 362
6.3 Generalized Doppler Broadening: Bethe-Placzek Formula 366
6.4 Doppler Broadening of a Breit-Wigner Cross Section 370
6.4.1 Overview of the Breit-Wigner Formalism 370
6.4.2 Voigt´s Formula 372
6.4.3 Interference Function 378
6.5 Application to the Large Resonance of Uranium 238 379
6.6 Temperature Effect on Cross Sections 381
6.6.1 First Voigt Function ? 382
6.6.2 Interference Function 383
6.6.3 Asymptotic Numeric Evaluation 384
6.6.4 Derivatives of the Voigt Functions with Respect to Energy 387
6.6.5 Some Mathematical Properties of Voigt Profiles 388
6.7 Effective Resonance Integral 389
6.7.1 Homogeneous Medium 389
6.7.2 Heterogeneous Medium 392
6.7.3 Analytical Calculation of a Broadened Resonance: The Campos-Martinez Model 399
6.8 Effective Doppler Temperature 403
6.8.1 Lattice Bonding Effects 403
6.8.2 Heterogeneity Effects of the Temperature Field 405
Chapter 7: Thermalization of Neutrons 358
7.1 Historical Background 411
7.2 Boltzmann Theory of Gases 412
7.3 Application to Neutrons 416
7.4 Neutron Flux Spectrum 419
7.5 Neutron Thermalization Equation 421
7.6 Wigner-Wilkins Model: Free Proton Gas 425
7.7 Asymptotic Spectrum 428
7.8 Simplified Solution to Thermalization with Absorption 432
7.9 Horowitz-Tretiakoff Model 436
7.9.1 Principle 436
7.9.2 Case of Absorption Inversely Proportional to Speed 443
7.9.3 Case of a Finite Reactor (with Leakage) 443
7.9.4 Thermalization Equation for a Homogeneous Medium 444
7.10 Heavy Gas Model 445
7.11 Cadilhac, Horowitz and Soulé Differential Model 446
7.12 Application of the Cadilhac Model to Heterogeneous Media 450
7.13 Graphical Representation of Flux over the Energy Spectrum 455
7.14 True Moderators 456
7.15 Heating and Cooling by Scattering 458
7.16 Thermalized Absorption 461
7.16.1 Calculation of Reaction Rate in a Pure Thermal Spectrum 464
7.16.2 Definition of the Westcott Coefficient g(T) 465
7.17 Calculation of the Reaction Rate in a True Thermal Spectrum 470
7.17.1 Westcott Formalism: Introduction of the Coefficients r and s 473
7.17.2 Extension of the Model to Other Nuclides: The Linear Logarithmic Model 477
7.17.3 Progressive Junction at Epithermal Energy 480
7.17.4 Westcott Junction 481
7.17.5 Determination of Cut-Off Function 482
7.17.6 Limits of the Westcott Formalism 484
7.18 Application of the Westcott Formalism 485
Chapter 8: The Boltzmann Equation 488
8.1 Setting Up the Boltzmann Equation 488
8.1.1 Concept of Flux 491
8.2 The Integro-Differential Transport Equation 497
8.2.1 The Integro-Differential Transport Equation in Kinetics 497
8.2.2 The Integro-Differential Equation in Steady-State 498
8.2.2.1 Setting Up the Integro-Differential Form 498
8.2.2.2 The Eigenvalue Problem 502
8.2.2.3 Solutions of the Transport Equation for Simple Cases 505
8.2.2.4 Adjoint Transport Theory 510
8.2.2.4.1 The Adjoint Integro-Differential Equation 510
8.2.2.4.2 The Adjoint Equation for the Computation of Neutron Multiplication 514
8.2.2.4.3 Neutron Importance 516
8.2.2.4.4 Perturbation Theory Approach to the Subcritical Flux with Source 517
8.2.2.5 The Critical Reactor Eigenvalue Problem 520
8.2.2.6 Uncollided Flux 521
8.2.2.6.1 Green´s Functions, Uncollided Neutron Flux 521
8.2.2.6.2 Uncollided Flux from a Point Source 522
8.2.2.6.3 Uncollided Flux from a Plane Source 524
8.2.2.6.4 Uncollided Flux from an Isotropic Line Source 525
8.2.2.6.5 Using Homogeneous Green´s Functions 526
8.3 Integral Form of the Boltzmann Equation 530
8.3.1 Peierls Operator 530
8.3.2 The Volume Integral Form 533
8.3.3 The First Collision Probability 535
8.3.3.1 Definition of the First Collision Probability 535
8.3.3.2 Calculating First Collision Probabilities 539
8.3.3.3 Dirac´s Chord Method 543
8.3.4 1D Geometry 545
8.3.5 Escape Probabilities 547
8.3.5.1 Escape Probability from a Slab 547
8.3.5.2 Escape Probability from a Sphere 549
8.3.5.3 Internal Escape Probability from a Hollow Sphere 550
8.3.5.4 Escape Probability from a Cylinder 551
8.3.5.5 Concept of Opacity 553
8.3.5.6 Multiple Collision Escape Probability 553
8.3.5.7 Escape Probability in Transient States 555
8.3.5.8 Interface Current Method 557
8.3.6 The Integral Equation in 2D 561
8.3.7 Application to an Infinite Medium with a Fission Source 562
8.3.8 Graphical Solution to the Dispersion Equation 563
8.4 Third Form of the Transport Equation: the Surface-Integral Form 566
8.4.1 Placzek´s Lemma 567
8.4.2 Flux Equation at the Interface 569
8.4.3 Application to the Milne Problem 570
8.4.4 Second Complementarity Theorem 571
8.5 Concept of Characteristic Function 572
8.6 Fourier Transform of the Boltzmann Equation 576
8.6.1 Formalism 576
8.6.2 Resolution Using Green´s Function 578
8.7 The 1D Transport Equation 582
8.7.1 General Points 582
8.7.2 Lafore and Millot Method, Case Method 585
8.7.2.1 Historical Overview 585
8.7.2.2 Theory 590
8.7.3 Perovich Method 594
8.8 Asymptotic Solution for Diffusion 595
8.8.1 Exponential Relaxation of the Flux, Far from the Source 595
8.8.2 Finding the Dispersion Equation from the Asymptotic Flux 603
8.8.3 Critical Absorption Limiting the Asymptotic Solution 605
8.8.4 Definition of a Diffusion Coefficient from the Transport Equation 607
8.9 The 3D Transport Equations 612
Chapter 9: Computational Neutron Transport Methods 616
9.1 Discrete Ordinates Method Sn 616
9.2 Exact Sn Method 624
9.3 Legendre Polynomial Method 627
9.3.1 Theory and Application to 1D Transport 627
9.3.2 Multi-group 1D Transport and Diffusion Equivalence 642
9.4 SPn Method 646
9.5 Interfaces Between Different Media 651
9.6 Spherical Harmonics Method 653
9.6.1 Principle 653
9.6.2 P1 Approximation 661
9.7 Milne Problem 663
9.8 DPn Method 666
9.9 Semi-infinite Plane: Albedo Problem 669
9.9.1 Fundamentals of Discrete Eigenfunctions 669
9.9.2 Ganapol Method by Laplace Transform 675
9.10 Bn Method 680
9.11 Tn Method 690
9.12 Fn Method 693
9.13 Cn Method 693
9.14 The SKn Method 698
9.15 Method of Characteristics (MOC) 700
9.15.1 Principle 700
9.15.2 Heterogeneous Geometries 702
9.15.3 Characteristic Direction Probabilities (CDP) 707
9.16 Even-Odd Formulation of the Transport Equation 709
9.16.1 Even-Odd Flux Equation 710
9.16.2 Variational Nodal Method of the Even-Odd Formulation 714
9.16.3 Ritz Method 717
9.17 Variational Method for Time-Dependent Problems 720
9.18 Gauss-Seidel Method for Sources in Time-Dependent Problems 722
9.19 Probabilistic Approach: The Monte Carlo Method 723
9.19.1 Fundamental Concepts of the Monte Carlo Method 723
9.19.2 Application to Neutron Transport: A Simple 2D Case 728
9.19.3 Statistical Error 736
9.19.4 Calculation of Physical Quantities 736
9.19.5 Generalization, Biasing 737
9.19.6 Resonance Escape Probability Factor Calculation 739
9.19.7 Midway Monte Carlo 742
9.19.8 Quasi-Deterministic Approximation of the Importance Function 746
9.19.9 Example of a Monte Carlo Calculation 749
Part II: Reactor Physics 751
Chapter 10: Diffusion Approximation in Neutron Physics 616
10.1 Fick´s Law 752
10.1.1 Evaluation of the Neutron Diffusion Coefficient 752
10.1.2 Discussion of the Hypotheses 757
10.1.3 The Diffusion Equation in a Force Field 762
10.2 Boundary Conditions for a Medium Surrounded by a Vacuum in Diffusion Theory 764
10.2.1 P1 Approximation 765
10.2.2 Rulko´s Variational Approach 766
10.3 Boundary Conditions Between Any Two Media 770
10.3.1 Notion of a Reflector Albedo 771
10.4 Diffusion Equation in Energy 772
10.5 One-Group Diffusion Equation 774
10.6 ``Thermal Diffusion´´ 776
10.6.1 ``Thermal´´ Diffusion Equation 776
10.6.2 Interpretation of the Thermal Scattering Path 778
10.6.3 Deriving the Four-Factor Formula 780
10.7 Scattering of an Isotropic Source in a Non-Multiplying Medium 780
10.7.1 Point Source in an Infinite Scattering Medium 781
10.7.2 Anisotropic Point Source in Spherical Geometry 784
10.7.3 Infinite Thin Rod Source in an Infinite Scattering Medium 790
10.7.4 Infinite Plane Source in an Infinite Scattering Medium 792
10.7.5 Infinite Plane Source in an Infinite Scattering Slab 794
10.7.6 Uniform Source in an Infinite Scattering Slab 796
10.7.7 Semi-infinite Slab Source 797
10.7.8 Extension to the Infinite Homogeneous Medium 799
10.7.9 Expansion on the Eigenfunctions of the Laplacian Operator 800
10.7.10 Superposition of Flux Induced by Point Sources 801
10.7.11 Absorbing Slab in an Infinite Source Medium 803
10.7.12 Thin Absorbing Slabs, the Galanin Method 804
10.7.13 Flux Transient 805
10.8 Measurement of the Scattering Path of a Moderator by Attenuation 808
10.9 Pulsed Neutron Method 812
10.10 Diffusion in a Homogeneous Slab 818
10.11 Source Thermalization Transient in Diffusion Theory 823
10.11.1 Infinite Medium 823
10.11.2 Finite Medium 824
10.11.3 Expansion on Eigenfunctions 825
10.11.4 Case of a Pulsed Source 827
10.12 Polykinetic Diffusion 829
Chapter 11: Nuclear Reactor Reactivity 835
11.1 Multiplication Factor of a Chain Reaction 835
11.1.1 Deterministic Approach to Chain Reactions 835
11.1.2 Stochastic Approach to Chain Reaction 836
11.2 ``Four-factor´´ Formula 841
11.2.1 Detailed Analysis of the Four-factor Formula 842
11.2.1.1 Fuel Multiplication Factor ? 843
11.2.1.2 Fast Fission Factor ? 845
11.2.1.3 Neutron Slowing-down: Escape Probability Factor 846
11.2.1.4 Thermal Range: The Thermal Utilization Factor 847
11.2.2 Technological Moderation Ratio Effect on the Four-factor Formula 847
11.3 Allowing for Leakages in a Finite Reactor 848
11.4 Two-group Multiplication Factor 849
11.5 Multiplication Factor Through a Reaction Rate Balance 855
11.6 Reactivity Effects or Reactivity Difference 860
11.6.1 Comparison of the Effects on a UOX Fuel 861
11.6.2 Reactivity Effect of Isotopic Change 862
11.7 Calculation of Reactivity by Perturbation Theory Estimate 865
11.8 Evolution of the Reactivity Along the Cycle 867
Chapter 12: Critical Homogeneous Reactor Theory 835
12.1 Introduction 869
12.2 The Notion of Geometrical and Material Buckling 874
12.3 Criticality Condition 875
12.4 Notion of Critical Size: The Rod Model 876
12.4.1 Analysis of Criticality 876
12.4.2 Invariant Imbedding 880
12.5 Fundamental Mode for a Reactor with Simple Geometry 884
12.5.1 Plane Slab 884
12.5.2 Parallelepiped 888
12.5.3 Infinite Cylinder 890
12.5.4 Finite Cylinder 893
12.5.5 Disc 895
12.5.6 Sphere 898
12.5.7 Hemisphere 901
12.5.8 Polygon 902
12.5.9 Accounting for Singularities in 2D 904
12.5.10 Anisotropic Point Source in a Multiplying Medium 912
12.5.11 Zero Flux Distance 913
12.5.12 Annular Reactor 915
12.6 Any Three-Dimensional Reactor 919
12.7 Fermi Age Theory 920
12.7.1 History 921
12.7.2 Overview of Slowing-Down 922
12.7.3 Application to Neutron Diffusion 924
12.7.4 Relation Between Fermi Age and Time 925
12.7.5 Link Between the Age Theory and Diffusion Theory 927
12.7.6 Two-Energy Group Equation in Fermi Age Theory 929
12.7.7 Age-Diffusion Theory 932
12.8 Multi-Group Diffusion 932
12.9 Reactor Kinetics in One-Group Diffusion Theory with Source 934
12.10 Source Calculation: Extension to Multi-Group Conditions 936
Chapter 13: Neutron Reflectors 939
13.1 Some Mathematical Considerations on Reflectors 939
13.2 Reflectors in Diffusion Theory 942
13.2.1 Case of the Slab Reactor Surrounded by an Infinite Reflector 942
13.2.2 Reflected Homogeneous Slab Reactor 946
13.2.3 Case of an Infinite Cylindrical Reactor Surrounded by an Infinite Reflector 948
13.2.4 Case of an Infinite Cylindrical Reactor with a Finite Reflector 954
13.2.4.1 Monocinetic Calculation 954
13.2.4.2 Two-Energy Group Calculation 956
13.2.4.3 Flux in Two Dimensions 957
13.3 Definition of Reflector Albedo 959
13.3.1 Albedo Calculation for a Slab Reflector 961
13.3.2 Albedo Calculation of a Cylindrical Reflector 962
13.3.3 Albedo of a Spherical Reflector 962
13.3.4 Albedo Calculation for the Upper Reflector of a Cylindrical Reactor 963
13.3.5 Extrapolation and Null-flux Distances 964
13.3.6 Numerical Example 967
13.4 Reflector Theory with Two Energy Groups 967
13.4.1 Slab Reflector 968
13.4.2 Infinite Cylindrical Reactor with Reflector in Two Groups Without Up-Scattering 969
13.4.3 Flux Calculation in the Fuel 970
13.4.4 Flux in the Reflector 972
13.5 Slab Reactor with Finite Reflector and Without Up-Scattering 975
13.6 The Ackroyd ``Magic Shell´´ Albedo Model 977
13.7 The Lefebvre-Lebigot Reflector Model 979
13.7.1 ``Equivalent´´ Reflectors Theory 980
13.7.2 Calculation of Core Characteristics 985
13.7.3 Core/Reflector Operating Point 987
13.7.4 Effect of Thermal-Hydraulic Feedbacks 989
13.7.5 Calculation of Constants in the Mathematical Reflector 990
13.8 Albedo Matrix 991
13.9 Allowing for Up-Scattering 992
13.10 Diffusion/Transport Correspondence 997
13.11 Reuss-Nisan Model 998
13.12 Mondot Model 1004
13.13 Generalized BETA Method 1006
13.14 Absorption in the Reflector 1007
13.15 Double-Differential Albedo 1008
Chapter 14: Heterogeneous Reactors 1011
14.1 Why Is Heterogeneity Desirable? 1011
14.2 Gurevich-Pomeranchuk Heterogeneous Resonant Absorption Theory 1013
14.2.1 Theoretical Background 1013
14.2.2 Effective Resonance Integral 1018
14.3 Modeling the Pin Flux 1019
14.3.1 First-Collision Probability 1020
14.3.2 The Amouyal-Benoist-Horowitz (A-B-H) Theory 1022
14.3.2.1 Classical Thermal Utilization Factor Theory 1022
14.3.2.2 A-B-H Theory for the Thermal Utilization Factor 1025
14.3.3 Multi-cell Approach in Two Dimensions 1034
14.3.3.1 Context 1034
14.3.3.2 Dancoff-Ginsburg Factor 1036
14.3.3.3 The Dancoff Effect in Different Geometries: Shielding Problems 1040
14.3.3.4 Impact of the Dancoff Factor on Resonant Absorption 1049
14.3.3.5 Extension to Pin Lattices 1050
14.3.4 Carlvik Rational Approximation 1052
14.3.5 Heterogeneity of the Isotopic Composition 1058
14.3.6 Shadowing Effect on the Resonance Integral 1058
14.3.7 Heterogeneous Pi,j Calculations for Fast Reactors with Perturbation Methods 1062
14.4 Transport-Diffusion Equivalence 1065
14.4.1 Context 1065
14.4.2 Spatial Homogenization 1067
14.4.3 Multi-group Approach 1068
14.4.4 Kavenoky-Hébert SPH Equivalence 1069
14.4.5 Flux Reconstruction Between Different Operators 1071
14.4.5.1 Reflected Medium (Infinite) 1074
14.4.5.1.1 Homogeneous B0 Model 1075
14.4.5.1.2 Homogeneous B1 Model 1078
14.4.5.2 Finite Medium 1079
14.4.6 Spatial Homogenization with Leakage 1082
14.4.7 Equivalence for Slab Reactors 1087
14.4.8 Equivalence by Conservation of Reaction Rates 1092
14.5 Homogenization Theory in Diffusion 1096
14.5.1 Flux-Volume Homogenization 1096
14.5.2 Homogenization of Heterogeneous Neutron Quantities 1097
14.5.3 Average Flux Homogenization at the Boundary, Selengut Normalization 1100
14.5.4 Pin Power Reconstruction 1102
14.5.4.1 Convolution with Pin Power Distribution 1102
14.5.4.2 Perturbation Approach: Rahnema Method 1103
14.5.5 Discontinuity Factors 1107
Chapter 15: Fuel Cycle Physics 1110
15.1 Schematic Notation for Fuel Cycle Physics 1110
15.2 Disintegration 1111
15.3 Neutron-Induced Reactions 1111
15.4 The Bateman Equations 1111
15.4.1 Heavy Nuclides 1112
15.4.2 Fission Products 1114
15.4.3 Activation Products 1115
15.4.3.1 Example of the Cobalt 60 Chain 1115
15.5 Vectorial Form of the Bateman Equation 1116
15.6 Calculation of Relevant Quantities for the Fuel Cycle 1116
15.6.1 Mass Balance 1116
15.6.2 Burn-up 1117
15.6.2.1 Thermal Burn-up 1117
15.6.2.2 Fission Burn-up (for Fast Neutron Reactors) 1122
15.6.2.3 Fuel Depletion with Burn-up 1122
15.6.3 Activity 1123
15.6.4 Calculation of Decay Heat 1123
15.6.4.1 Summation Method 1124
15.6.4.2 Decay Heat Burst Function 1126
15.6.4.3 Elementary Value Curves 1128
15.6.4.4 Continuous Fission Curve Method 1130
15.6.4.4.1 Principle 1130
15.6.4.4.2 Capture Correction 1133
15.6.4.5 Calculation of Decay Sources and Their Spectrum 1133
15.6.5 Photon ? and Neutron Dose Calculation 1134
15.7 Isotopic Depletion Calculation 1136
15.7.1 Chain-Decay Process: Recurrence Relations 1137
15.7.2 Case of Heavy Nuclides 1140
15.7.3 Case of Fission Products 1141
15.7.4 Reference Composition of Some PWR Fuel 1142
15.8 Decay Chain Reduction Principle 1143
15.8.1 Heavy Nuclide Chain for Reactivity Calculations of Reactors 1145
15.8.1.1 Numerical Example: Plutonium Production in a Uranium Fuel Assembly 1147
15.8.1.2 Decay Chain for Heavy Nuclides and Fission Products from the SERMA79 Recommendation 1150
15.8.1.3 Decay Chain of Heavy Nuclides from the REL2005 Recommendation 1152
15.8.2 Decay Chain Reduction 1153
15.8.2.1 Chain Reduction 1153
15.9 Activation: The Example of Control Rods 1156
15.10 Xenon Physics 1157
15.10.1 Production of Xenon 1157
15.10.2 Xenon Saturation 1159
15.10.3 Xenon Poisoning After Reactor Shutdown 1161
15.11 Samarium Physics 1163
15.12 Gadolinium Physics 1164
15.13 The Industrial Fuel Cycle in France 1165
Chapter 16: Neutronic Feedback 1172
16.1 Effect of Fuel Temperature on the Multiplication Factor 1172
16.1.1 Fuel Doppler Effect 1172
16.1.2 Doppler Effect on Reactor Behavior 1175
16.2 Moderator Temperature Effect 1177
16.2.1 Definitions 1177
16.2.2 Leakage and Absorber Effects 1179
16.2.3 Pressure Effect 1181
16.2.4 Graphite Moderator 1182
16.2.5 Neutron Spectrum Shift 1183
16.2.6 Void Effect 1184
16.3 Boron Effect in Pressurized Water Reactors 1185
16.3.1 Differential Efficiency of Boron 1185
16.3.2 Boron Effect on the Moderator Differential Coefficient 1186
16.4 Power Coefficient 1187
16.5 Feedback Modeling 1187
16.5.1 A Simple Model: Power Feedback 1190
16.5.2 An Advanced Feedback Model: The Lefebvre-Seban Model 1191
16.6 Historical Isotopic Correction 1202
Chapter 17: Reactor Kinetics 1205
17.1 Prompt Neutrons 1205
17.1.1 Evolution of a Hypothetical Prompt Neutron Reactor 1206
17.1.2 Flux Calculation: Point Reactor Hypothesis 1211
17.2 Delayed Neutrons 1213
17.2.1 Delayed Neutron Fraction 1217
17.3 Effect of Delayed Neutrons on Reactor Kinetics 1218
17.4 Neutron Kinetics Equation 1221
17.4.1 Precursor Concentration 1223
17.4.2 Point-Reactor Kinetics 1224
17.4.3 Mobile Fuel 1226
17.5 Nordheim Equation 1226
17.6 ``Prompt Jump´´ Notion: Insertion of a Reactivity Step 1231
17.7 Age Theory in the Kinetics Equation for Thermal Neutrons 1233
17.8 Reduced Kinetics Equations 1236
17.9 Kinetics with an Imposed Neutron Source 1238
17.10 Delayed Neutron Spectrum 1239
17.11 First-Order Perturbations 1247
17.12 Numerical Reactimeter 1249
17.13 Practical Evaluation of Prompt Neutron Generation Time 1252
17.14 Main Causes of Reactivity Changes 1254
17.14.1 Increased Fissile Nuclei 1254
17.14.2 Increased Neutron Moderation 1255
17.14.3 Decreased Neutron Capture 1255
17.15 Reactivity Accident: Insertion of Very High Reactivity Value 1256
17.15.1 Analysis with One Group of Delayed Neutrons 1256
17.15.2 Analysis of the Case of ? ?: The Reactivity Accident 1259
17.15.3 Insertion of Low Reactivity 0 ? ? 1261
17.16 Anti-reactivity Insertion 1263
17.17 Overview of Cases 1264
17.18 Reactivity Step 1265
17.19 Dropped Control Rod, Insertion of a Large Amount of Anti-reactivity 1267
17.20 Reactivity Ramp 1268
17.21 Reactivity Transient 1272
17.22 Power Excursion 1272
17.22.1 The Nordheim-Fuchs Model 1273
17.22.2 The Chernick Model 1277
17.22.3 The Bethe-Tait Model 1280
17.23 Subcritical Approach: Reactor Start-Up 1284
17.24 Reactor Stability 1285
17.25 Space-Time Xenon Oscillations 1289
17.26 Mechanical Kinetic Effects 1294
17.27 Neutron Noise 1295
17.27.1 Noise Concept, Spectral Analysis 1296
17.27.2 Neutron Correlations 1298
17.27.3 The Feynman-? Method 1305
17.27.4 Delayed-Neutron Effect 1313
17.27.5 Application to Measurement of Void Fraction Instabilities 1314
17.27.6 Application to Detection of Vibrations 1316
Chapter 18: Computation Methods in Diffusion Theory 1319
18.1 Calculation Meshes 1319
18.2 Multi-group Diffusion Equations 1322
18.2.1 General Case 1322
18.2.2 ``1.5´´-group Diffusion 1323
18.2.3 Adjoint Diffusion 1323
18.2.4 Taking into Account the Neutron Over-Production Cross Sections 1325
18.3 The Power Iteration Method 1326
18.3.1 General Considerations 1326
18.3.2 Matrix Representation 1328
18.3.3 Chebyshev Acceleration 1330
18.4 Finite Difference Method 1333
18.4.1 Formalism 1333
18.4.2 Boundary Conditions 1338
18.4.3 Matrix Form 1339
18.5 Nodal Methods 1340
18.5.1 Nodal Method of Order 4 1342
18.5.2 Quadratic Approximation of Transverse Leakage 1350
18.5.3 AFEN Method 1353
18.6 Finite Element Method 1354
18.7 Variational Methods 1358
18.7.1 Principle 1358
18.7.2 Accounting for Boundary Conditions 1360
18.8 Calculation of Control Rods 1361
18.8.1 Physical Effect of Rods 1362
18.8.2 Rod Worth: Perturbation Analysis 1363
18.8.3 Measuring Rod Efficiency in PWR 1366
18.8.4 Calculation of Rod Efficiency 1367
18.8.5 Analytical Decomposition of the Rodded Domain 1371
18.9 Instrumentation Considerations 1375
18.9.1 Modeling with Trace Quantities 1375
18.9.2 Modeling of the EPR Instrumentation: The KTM Model 1375
18.9.2.1 Cross Sections in the KTM Model 1380
18.9.2.2 Fine Power Structures in the KTM Model 1380
Conclusion 1385
Annex: Reactor Physics and Neutronic Codes at Electricité De France 1387
Bibliography 1420
Index 1448