Physics of Nuclear Reactors

Description

Physics of Nuclear Reactors presents a comprehensive analysis of nuclear reactor physics. Editors P. Mohanakrishnan, Om Pal Singh, and Kannan Umasankari and a team of expert contributors combine their knowledge to guide the reader through a toolkit of methods for solving transport equations, understanding the physics of reactor design principles, and developing reactor safety strategies. The inclusion of experimental and operational reactor physics makes this a unique reference for those working and researching nuclear power and the fuel cycle in existing power generation sites and experimental facilities. The book also includes radiation physics, shielding techniques and an analysis of shield design, neutron monitoring and core operations.

Those involved in the development and operation of nuclear reactors and the fuel cycle will gain a thorough understanding of all elements of nuclear reactor physics, thus enabling them to apply the analysis and solution methods provided to their own work and research. This book looks to future reactors in development and analyzes their status and challenges before providing possible worked-through solutions.

Cover image: Kaiga Atomic Power Station Units 1 – 4, Karnataka, India. In 2018, Unit 1 of the Kaiga Station surpassed the world record of continuous operation, at 962 days. Image courtesy of DAE, India.

Includes methods for solving neutron transport problems, nuclear cross-section data and solutions of transport theory
Dedicates a chapter to reactor safety that covers mitigation, probabilistic safety assessment and uncertainty analysis
Covers experimental and operational physics with details on noise analysis and failed fuel detection

Front Cover 2
Physics of Nuclear Reactors 5
Copyright 6
Contents 7
Contributors 13
Foreword 15
Preface 17
Acknowledgments 21
Chapter 1: Introduction 23
- 1.1. Introduction 23
- 1.2. The atom and its nucleus 23
  - 1.2.1. The Bohr model of the atom 25
  - 1.2.2. The atomic energy levels 26
  - 1.2.3. Ionization 27
- 1.3. The nucleus 28

Front Cover 2
Physics of Nuclear Reactors 5
Copyright 6
Contents 7
Contributors 13
Foreword 15
Preface 17
Acknowledgments 21
Chapter 1: Introduction 23
- 1.1. Introduction 23
- 1.2. The atom and its nucleus 23
  - 1.2.1. The Bohr model of the atom 25
  - 1.2.2. The atomic energy levels 26
  - 1.2.3. Ionization 27
- 1.3. The nucleus 28
  - 1.3.2. Nuclear charge distribution 29
  - 1.3.1. Nuclear dimensions and density 29
  - 1.3.3. Representation of a nucleus 30
  - 1.3.4. Isotopes, isotones, isobars 31
- 1.4. Nuclide number density 32
- 1.5. Nuclear stability 32
- 1.6. Binding energy 34
  - 1.6.1. Binding energy per nucleon (BE/A) 36
  - 1.6.2. Binding energy of the last nucleon 37
  - 1.6.3. The drip lines 38
- 1.7. Radioactivity 38
  - 1.7.1. Alpha decay 39
  - 1.7.2. Beta decay 40
  - 1.7.3. Positron decay or β+ decay 41
  - 1.7.4. Electron capture 41
  - 1.7.5. Gamma decay 42
  - 1.7.7. Mean life 44
  - 1.7.6. Rate of radioactive decay 44
  - 1.7.8. Half-life 45
  - 1.7.9. Units to express radioactivity emission 45
- 1.8. Nuclear models 46
  - 1.8.1. The liquid drop model 46
  - 1.8.2. The shell model 49
- 1.9. Nuclear energy levels 50
  - 1.9.1. Bound and excited energy levels 50
- 1.10. Nuclear reactions 51
  - 1.10.1. Q-value of nuclear reactions 52
- 1.12. Neutron elastic scattering 53
  - 1.12.1. Kinematics of neutron elastic scattering 54
  - 1.12.2. The moderator 58
- 1.11. Neutron interactions 53
- 1.13. Neutron inelastic scattering 59
  - 1.13.1. Discrete and continuum inelastic scattering 59
  - 1.13.2. Kinematics of neutron inelastic scattering 60
- 1.14. Radiative capture 63
- 1.16. Fission 64
- 1.17. Probabilities of nuclear reactions 64
  - 1.17.1. Reaction rate and microscopic cross-section 66
  - 1.17.2. Flux, macroscopic cross-section, mean-free-path 67
  - 1.17.3. Total, capture, and absorption cross-sections 69
- 1.15. (n, x) reaction 64
- 1.18. Variation of cross-section with energy 70
- 1.19. Compound nucleus reaction versus direct reaction 71
  - 1.19.1. Resonances 75
  - 1.19.2. Resolved and unresolved resonances 77
- 1.20. Cross-section data representation and resonance formalism 79
- 1.21. Fission mechanism 80
  - 1.21.1. Chance fissions 83
  - 1.21.3. Fission products 84
  - 1.21.2. Fission cross-sections 84
  - 1.21.4. Asymmetric fission 86
  - 1.21.5. Instability of the fission fragments 87
- 1.22. Fission neutrons 89
  - 1.22.1. Prompt and delayed 89
  - 1.22.2. Fission neutron energies 90
- 1.23. Energy released in fission 92
- 1.24. Neutron chain reaction and generations 93
  - 1.24.1. Fission chain reaction 93
  - 1.24.2. Neutron generations and multiplication 93
- 1.25. Concept of a nuclear reactor 95
  - 1.25.1. Criticality, critical size, and critical mass 96
  - 1.25.2. Basic design considerations 98
- 1.26. Thermal reactor and fast reactor 99
  - 1.26.1. Energy region and fissile enrichment 100
  - 1.26.2. Energy distribution of neutrons in a reactor 104
- 1.27. Spatial dependence of the neutrons in a reactor 104
- 1.28. Basic physics of reactor design 106
- 1.30. Summary 107
- 1.31. Exercise problems 107
- 1.29. The reactor-power 107
- Further reading 108
- References 108
Chapter 2: Nuclear data 109
- 2.1. Introduction: Nuclear data and its importance 109
  - 2.1.1. Application areas of nuclear data 111
- 2.2. Nuclear data production 113
  - 2.2.1. Measurement of cross-sections 113
  - 2.2.2. Prediction of cross-sections 113
  - 2.2.3. Measurement of total, partial, and differential cross-sections 114
  - 2.2.4. Total cross-section estimation 116
  - 2.2.5. Partial cross-section estimation 117
  - 2.2.6. Use of cross-section standards 118
  - 2.2.7. EXFOR and CINDA 119
  - 2.2.8. Reaction model principles and cross-section predictions 119
  - 2.2.9. The spherical optical model 122
  - 2.2.10. The compound nucleus model 124
  - 2.2.11. Nuclear level densities 125
  - 2.2.12. The components of the potential 126
  - 2.2.14. Validation of predicted data 127
  - 2.2.13. Reaction model codes 127
  - 2.2.15. Error and co-variance data 128
- 2.3. Nuclear data evaluation 128
  - 2.3.1. Evaluated nuclear data files 131
  - 2.3.2. The ENDF format 133
  - 2.3.3. Point-data representation in ENDF 134
  - 2.3.4. Resonance parameter data in ENDF 136
  - 2.3.5. The SLBW formalism 137
  - 2.3.6. The MLBW formalism 142
  - 2.3.7. The RM formalism 142
  - 2.3.8. Other RRR formalisms 143
  - 2.3.9. The unresolved resonance region 144
- 2.4. Nuclear data processing 147
  - 2.4.1. Linearization 148
  - 2.4.2. Resonance reconstruction 149
  - 2.4.3. Doppler broadening 150
  - 2.4.4. Calculation of Doppler broadened resonance cross-sections 151
  - 2.4.5. Cross-section multigrouping 156
  - 2.4.6. Weighting function and self-shielding 159
  - 2.4.7. Problem-independent multigroup cross-section set 162
  - 2.4.8. Obtaining effective cross-sections in the resonance region 165
  - 2.4.9. Self-shielding in the URR 166
  - 2.4.10. Effect of Doppler-broadening on the reaction rate 167
  - 2.4.11. Group cross-sections in the resonance region for thermal neutron reactors 168
  - 2.4.12. Group-to-group transfer cross-sections 169
  - 2.4.14. Transport cross-sections 170
  - 2.4.12.1. Thermal scattering cross-sections 170
  - 2.4.13. Production cross-sections for particles other than neutrons 170
  - 2.4.15. Multigrouping the cross-sections from a linearized grid 171
  - 2.4.16. Cross-sections for shielding applications 171
  - 2.4.17. Displacement cross-sections 172
  - 2.4.18. Nuclear data processing codes 172
  - 2.4.19. Multigroup data validation 173
- 2.5. Summary 173
- 2.6. Exercise problems 174
- References 175
- Further reading 176
Chapter 3: Types of nuclear reactors 177
- 3.1. Introduction 177
- 3.2. Classifications of nuclear reactors 177
  - 3.2.1. Based on materials used in reactors 178
  - 3.2.2. Other classifications 178
- 3.3. Thermal neutron reactors 180
  - 3.3.1. Graphite moderated and water cooled reactors 180
  - 3.3.2. RBMK reactors 180
  - 3.3.3. Gas-cooled graphite-moderated reactors 182
  - 3.3.4. Boiling water reactors 183
  - 3.3.5. Pressurized water reactors 185
    - 3.3.5.1. Russian pressurized water reactors 186
  - 3.3.6. Pressured heavy water reactors 188
- 3.4. Fast neutron reactors 190
  - 3.4.1. Breeding 191
  - 3.4.2. Sodium-cooled fast neutron reactors 193
- 3.5. Future energy systems 195
- 3.6. Fusion reactor systems 196
  - 3.6.1. Lawson criteria 196
  - 3.6.2. Inertial confinement 198
  - 3.6.3. Magnetic confinement 198
  - 3.6.4. Early fusion devices 199
- 3.7. Fission-fusion hybrids 200
- 3.8. Particle accelerators [11] 200
  - 3.8.1. LINACs and tandem accelerators 201
  - 3.8.2. Cyclotrons 202
  - 3.8.4. Large hadron collider at CERN [13] 203
  - 3.8.3. Synchrotron radiation sources 203
- 3.9. Accelerator-driven subcritical systems (ADSS) 204
  - 3.9.1. Operational aspects of accelerator-driven system and its interface with reactor 206
  - 3.9.2. Overview of ADSS projects around the world 207
  - 3.9.5. Other concepts 208
  - 3.9.3. The CERN energy amplifier 208
  - 3.9.4. The Los Alamos waste transmuter 208
- 3.10. One way coupled booster reactor concept in BARC 209
- 3.11. More topics of interest 210
- 3.12. Exercise problems 210
- References 212
- Further reading 213
Chapter 4: Homogeneous reactor and neutron diffusion equation 215
- 4.1. Introduction 215
- 4.2. Neutron density and flux 216
- 4.3. Neutron continuity equation 217
- 4.4. Transport cross-section 219
- 4.5. Fick's law of neutron diffusion 221
  - 4.5.1. Physical interpretation 225
  - 4.5.2. Fick's law limitations 226
- 4.6. Diffusion equation 228
  - 4.6.1. Boundary conditions for the steady-state diffusion equation 229
- 4.7. Neutron diffusion in nonmultiplying media 232
  - 4.7.1. A point source in an infinite medium 232
  - 4.7.2. Line source in an infinite media 233
  - 4.7.3. Plane source in a finite medium (bare slab) 235
  - 4.7.4. Physical meaning of diffusion length 236
- 4.8. Neutron diffusion in a multiplying media 238
  - 4.8.1. Time-dependent flux in a slab reactor 239
  - 4.8.2. Finite bare cylindrical reactor 242
- 4.9. Reflected infinite slab reactor 248
  - 4.9.1. Reflector savings 250
- 4.10. Neutron life cycle in a thermal reactor 251
- 4.11. Slowing down equation 255
  - 4.11.1. Moderation of neutrons in a medium having A>1 255
  - 4.11.2. Slowing down in an infinite medium of moderator and resonant absorber 259
- 4.12. Neutron thermalization 264
  - 4.12.1. Monoatomic gas as moderator 266
  - 4.12.2. Bound moderator atoms in liquids and solids 266
    - 4.12.2.1. Water 267
    - 4.12.2.2. Heavy water 269
    - 4.12.2.3. Graphite 269
  - 4.12.3. Maxwellian distribution of neutron flux 269
  - 4.12.4. Reaction rate in a Maxwellian spectrum 270
  - 4.12.5. Comparison of PWR spectrum and FBR spectrum 271
- 4.13. Two-group diffusion theory 271
- 4.14. Multigroup diffusion equation 274
- 4.15. Numerical solution of the diffusion equation in 1-D slab geometry 276
  - 4.15.1. Center mesh finite differencing scheme 277
    - 4.15.1.1. Boundary conditions 279
  - 4.15.2. Corner mesh finite differencing scheme 280
- 4.17. Exercise problems 281
- 4.16. Summary and more topics of interest 281
- References 283
- Further reading 284
Chapter 5: Methods of solving neutron transport equation 285
- 5.1. Introduction 285
- 5.3. Neutron-nucleus interaction cross-sections 286
  - 5.3.1. Differential cross-section 287
- 5.2. Assumptions in the neutron transport theory 286
- 5.4. The neutron transport equation 287
  - 5.4.1. Formulation of the neutron transport problem 288
  - 5.4.3. Boundary conditions 291
  - 5.4.2. Interface conditions 291
  - 5.4.4. Time-independent transport equation: Fixed source problem 292
  - 5.4.6. The eigenvalues of neutron transport equations and their relations 293
  - 5.4.5. Time-independent transport equation: k-eigenvalue problem 293
- 5.5. Solution to the neutron transport equation 294
  - 5.5.1. Multigroup approximation: Energy discretization 295
  - 5.5.2. The PN method 295
  - 5.5.3. P1 approximation 296
  - 5.5.4. The discrete ordinate method 297
- 5.6. Numerical solution to the neutron transport equation 299
  - 5.6.1. The spatial discretization and sweeping scheme 299
  - 5.6.2. Limit on the size of spatial meshes and negative flux issue 301
  - 5.6.3. Source iteration method for multidimensions 302
  - 5.6.4. Discrete ordinate quadrature sets 303
    - 5.6.4.1. The double PN quadratures 304
  - 5.6.5. Discrete ordinate quadrature sets for multidimensional problems 304
    - 5.6.5.1. Level symmetric quadrature set 305
  - 5.6.6. The Ray effect 305
- 5.7. The synthetic acceleration schemes 306
  - 5.7.1. The corrective equation 307
  - 5.7.3. The transport synthetic acceleration 308
  - 5.7.2. The diffusion synthetic acceleration 308
- 5.8. The integral form of transport equation 308
- 5.9. Solution to the integral transport equation 311
  - 5.9.1. Collision probability method 312
    - 5.9.1.1. Collision probability for slab geometry 313
    - 5.9.1.2. Collision probability for annular geometry 315
    - 5.9.1.3. Boundary condition 317
  - 5.9.2. Interface current method 319
    - 5.9.2.1. Discretized flux equation 320
    - 5.9.2.2. Properties of the collision probability matrices 323
    - 5.9.2.3. Calculation of collision probabilities in two-dimensional geometry 324
      - Region-to-region collision probabilities 325
      - Region-to-surface escape probability 326
      - Surface-to-surface transmission probability 327
    - 5.9.2.4. Computation of collision probability integrals 328
    - 5.9.2.5. Use of boundary condition and solution to Collision Probability equations 329
  - 5.10. Method of characteristics approach 331
    - 5.10.1. MOC solution to one-dimensional slab geometry 333
    - 5.10.2. MOC solution for two-dimensional geometry 336
    - 5.10.3. Neutron tracking and quadrature weights 337
      - 5.10.3.1. Tracking of neutron trajectories 337
      - 5.10.3.2. Quadrature weights for polar and azimuthal angles 339
    - 5.10.4. Solution to the multigroup MOC equation 340
  - 5.11. Monte Carlo neutron transport 341
    - 5.11.1. Validity of Monte Carlo estimates 342
      - 5.11.1.3. The central limit theorem 342
      - 5.11.1.1. Chebyshev inequality 342
      - 5.11.1.2. Law of large numbers 342
    - 5.11.2. Random variable 343
      - 5.11.2.1. Uniform random variable 343
      - 5.11.2.2. Sampling from continuous distribution 344
    - 5.11.3. Simulating neutron transport with Monte Carlo method 344
      - 5.11.3.1. Simulating a neutron's history 345
      - 5.11.3.2. Sampling of source neutrons 346
      - 5.11.3.3. Sampling for neutron direction cosine from isotropic distribution 346
      - 5.11.3.4. Sampling for type of collision 347
    - 5.11.4. Tally computation 347
      - 5.11.4.1. Collision estimator 348
    - 5.12. More topics of interest 349
    - 5.13. Exercise problems 349
    - References 351
  - Chapter 6: Fuel burnup, fuel management, and fuel cycle physics 353
    - 6.2. Fuel burnup and its effects 353
      - 6.2.1. Energy production 354
      - 6.2.2. Changes in fuel composition and fission product buildup 354
        
        6.2.2.1. Units for measuring fuel burnup 356
      - 6.2.3. Fission product buildup 357
      - 6.2.4. Reactivity changes, effect of burnable poison, and absorber burnup 359
      - 6.2.5. Changes in kinetic parameters 361
    - 6.1. Introduction 353
    - 6.3. Bateman equation and solutions 362
      - 6.3.1. Solution to the Bateman equations 365
        
        6.3.1.1. Analytical methods 366

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