Table of Contents Page

CHAPTER 1 INTRODUCTION 1

1.1 Background 1
1.2 Historical Approach 2
1.3 Neutron Density 5 1.4 Reaction Rates 6 1.5 Design Approximations 11 1.6 Reactor Design Implications 12

CHAPTER 2 INTERACTIONS OF NEUTRONS
WITH MATTER 17

2.1 Compound Nucleus - Cross Sections 17
2.2 Nuclear Systematics of Naturally Occurring
Isotopes 23 2.3 Level Widths and Partial Cross Sections 25 2.3 Cross Section for Formulation of the Compound
Nucleus 26 2.5 Reaction Probabilities 28
2.6 Kinematics in the Center-of-Mass System 30 2.7 Relationships Between the Scattering
Angles in the LAB and CM Systems 36

CHAPTER 3 NUCLEAR FISSION 45

3.1 Binding Energy 45
3.2 Liquid Drop Model of Fission 48
3.3 Fission Product Yields 55
3.4 Fission Neutron Spectrum 57 3.5 Prompt Neutrons 59
3.6 Delayed Neutrons 62
3.7 Energy Production in Fission 64

CHAPTER 4 DERIVATION OF THE NEUTRON
DIFFUSION EQUATION 69

4.1 Fick's Law Derivation of the Diffusion Equation 69
4.2* The One-Speed Transport Equation 74
4.3* One-Dimensional, One-Speed Transport Equation 80
4.4* Derivation of the P1 Equations 86
4.5 The P1 Diffusion Theory Approximation 89
4.6 Comments on the P1 Equations of Diffusion Theory 92
4.7* Alternate Derivation of the Diffusion Equation 94

CHAPTER 5 ANALYTIC SOLUTIONS TO THE
ONE-SPEED DIFFUSION EQUATION 99

5.1 Partial Currents and the Extrapolated Boundary
Condition 99 5.2 Source Plane Boundary Conditions 107
5.3 Two Region Planar Problem 111
5.4 Point and Line Sources 113
5.5 Solution to the Inhomogeneous Source Problem 120
5.6* General Derivation of the Green's Function
Superposition Integral 123 5.7 Diffusion Length 126
5.8 Critical Reactors 128
5.9 Homogeneous Bare Critical Slab Reactor 131
5.10 Two Region Slab Reactor 135
5.11 One-Dimensional Bare Homogeneous Cylindrical Reactor 138
5.12 Bare Homogeneous Spherical Reactor 141
5.13 Comments on Multi-region One-Dimensional
Reactors 145 5.14 Multidimensional Reactors 149
5.15* Diffusion Length Experiment 152

CHAPTER 6 FEW-GROUP EQUATIONS AND
NUMERICAL SOLUTION METHODS 165

6.1 Energy-Dependent Diffusion Equation 165
6.2 Few-Group Diffusion Equations 167
6.3 Finite Difference Few-Group Equations in
One Dimension 172 6.4 Specification of Boundary Conditions 180
6.5 Direct Solution of the Source Problem 182
6.6* Iterative Solution of the Source Problem 187
6.7 Iterative Solution of the Critical Reactor
Problem 193 6.8* Convergence of the Outer Iteration to the
Fundamental Mode Solution 198 6.9 Qualitative Comparison Between One-Speed
and Two-Group Solutions 202

CHAPTER 7 PERTURBATION THEORY 211

7.1 Adjoint Equations 211
7.2 Matrix Form of the Adjoint Equations 217
7.3 First-Order Perturbation Theory 220
7.4* Perturbation Effects on Higher Harmonic
Modes 224 7.5 Control Rod Worth 226
7.6* Inhomogeneous Case 230

CHAPTER 8 REACTOR KINETICS 239

8.1 Multigroup Reactor Kinetics Equation 240
8.2 Interpretation of Spatial Kinetics Results 246
8.3 Point Kinetics Equations 249
8.4 Results for One Group of Delayed Neutrons 255
8.5 Experimental Measurement of Control Rod Worth 261
8.6* "Point" Kinetics Equations Derivation 263
8.7 Method of Obtaining and 267

CHAPTER 9 POISONING, TEMPERATURE EFFECTS,
AND DEPLETION IN REACTORS 275

9.1 The Fission Product Xenon-135 276
9.2 Xenon Poisoning 277
9.3 General Transient Solution for Buildup
of Xenon and Iodine 279 9.4 Spatial Xenon Transients 283
9.5 Xenon-Induced Spatial Power Oscillations 284
9.6 Samarium Poisoning 289
9.7 Temperature Effects on Reactivity 293
9.8 Effect of Thermal Feedback on Reactor Kinetics 297
9.9 Depletion 308

CHAPTER 10 NEUTRON MODERATION 321

10.1 Scattering Collisions 322
10.2 Slowing Down Problem 329
10.3 Slowing Down in Hydrogen With No Absorption 330
10.4 The Slowing Down Density in Hydrogen,
No Absorption 333 10.5 Moderation for A > 1, No Absorption 335
10.6 Asymptotic Slowing Down Theory, A > 1,
No Absorption 339 10.7* Slowing Down in Hydrogen With Absorption 343
10.8* Special Cases of Slowing Down With
Absorption, A > 1 347 10.9 Final Comments 353

CHAPTER 11 SLOWING DOWN IN THE PRESENCE
OF RESONANCES 357

11.1 Resonance Cross Sections 358
11.2 Doppler Effect 365
11.3 Slowing Down in the Presence of Resonances 373
11.4 Homogeneous-Medium Resonance Absorption 376
11.5 Homogeneous Resonance Integrals 385
11.6* Heterogeneous System Resonance Absorption 391
11.7 Reactor Design Implications of
Heterogeneous Lumping 407 11.8 The GAM Multigroup Slowing Down Equations 412

CHAPTER 12 THERMALIZATION 429

12.1 Neutron Balance Equation 430
12.2* Monatomic Gas Moderator Cross Sections 432
12.3 Solution to the Monatomic Gas
Thermalization Problem 443 12.4 General Case Solution 446
12.5 Spatial Effects in the Thermalization
Problem 451

APPENDIX A. THE SINGULARITY FUNCTIONS 461

APPENDIX B. THE LAPLACE TRANSFORM 465

APPENDIX C. MATRIX RELATIONSHIPS 473

APPENDIX D. SPHERICAL HARMONICS 479

APPENDIX E. DIFFERENTIATION OF AN
INTEGRAL 483

INDEX 485