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- Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Seitenzahl: 964
Veröffentlichungsjahr: 2018
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CONTENTS
Cover
Title Page
Copyright
Dedication
Preface
Preface to Second Edition
Preface to Third Edition
Part 1: Basic Reactor Physics
Chapter 1: Neutron–Nuclear Reactions
1.1 Neutron-Induced Nuclear Fission
1.2 Neutron Capture
1.3 Neutron Elastic Scattering
1.4 Summary of Cross Section Data
1.5 Evaluated Nuclear Data Files
1.6 Elastic Scattering Kinematics
References
Further Readings
Problems
Chapter 2: Neutron Chain Fission Reactors
2.1 Neutron Chain Fission Reactions
2.2 Criticality
2.3 Time Dependence of a Neutron Fission Chain Assembly
2.4 Classification of Nuclear Reactors
References
Further Readings
Problems
Chapter 3: Neutron Diffusion and Transport Theory
3.1 Derivation of One-Speed Diffusion Theory
3.2 Solutions of the Neutron Diffusion Equation in Nonmultiplying Media
3.3 Diffusion Kernels and Distributed Sources in a Homogeneous Medium
3.4 Albedo Boundary Condition
3.5 Neutron Diffusion and Migration Lengths
3.6 Bare Homogeneous Reactor
3.7 Reflected Reactor
3.8 Homogenization of a Heterogeneous Fuel–Moderator Assembly
3.9 Control Rods
3.10 Numerical Solution of Diffusion Equation
3.11 Nodal Approximation
3.12 Transport Methods
References
Further Readings
Problems
Chapter 4: Neutron Energy Distribution
4.1 Analytical Solutions in an Infinite Medium
4.2 Multigroup Calculation of Neutron Energy Distribution in an Infinite Medium
4.3 Resonance Absorption
4.4 Multigroup Diffusion Theory
References
Further Reading
Problems
Chapter 5: Nuclear Reactor Dynamics
5.1 Delayed Fission Neutrons
5.2 Point Kinetics Equations
5.3 Period–Reactivity Relations
5.4 Approximate Solutions of the Point Neutron Kinetics Equations
5.5 Delayed Neutron Kernel and Zero-Power Transfer Function
5.6 Experimental Determination of Neutron Kinetics Parameters
5.7 Reactivity Feedback
5.8 Perturbation Theory Evaluation of Reactivity Temperature Coefficients
5.9 Reactor Stability
5.10 Measurement of Reactor Transfer Functions
5.11 Reactor Transients with Feedback
5.12 Reactor Fast Excursions
5.13 Numerical Methods
References
Further Reading
Problems
Chapter 6: Fuel Burnup
6.1 Changes in Fuel Composition
6.2 Samarium and Xenon
6.3 Fertile-to-Fissile Conversion and Breeding
6.4 Simple Model of Fuel Depletion
6.5 Fuel Reprocessing and Recycling
6.6 Radioactive Waste
6.7 Burning Surplus Weapons-Grade Uranium and Plutonium
6.8 Utilization of Uranium Energy Content
6.9 Transmutation of Spent Nuclear Fuel
6.10 Closing the Nuclear Fuel Cycle
References
Further Reading
Problems
Chapter 7: Nuclear Power Reactors
7.1 Pressurized Water Reactors
7.2 Boiling Water Reactors
7.3 Pressure Tube Heavy Water–Moderated Reactors
7.4 Pressure Tube Graphite-Moderated Reactors
7.5 Graphite-Moderated Gas-Cooled Reactors
7.6 Liquid Metal Fast Reactors
7.7 Other Power Reactors
7.8 Characteristics of Power Reactors
7.9 Advanced Generation-III Reactors
7.10 Advanced Generation-IV Reactors
7.11 Advanced Subcritical Reactors
7.12 Nuclear Reactor Analysis
7.13 Interaction of Reactor Physics and Reactor Thermal Hydraulics
References
Further Readings
Problems
Chapter 8: Reactor Safety
8.1 Elements of Reactor Safety
8.2 Reactor Safety Analysis
8.3 Quantitative Risk Assessment
8.4 Reactor Accidents
8.5 Passive Safety
References
Further Readings
Problems
Part 2: Advanced Reactor Physics
Chapter 9: Neutron Transport Theory
9.1 Neutron Transport Equation
9.2 Integral Transport Theory
9.3 Collision Probability Methods
9.4 Interface Current Methods in Slab Geometry
9.5 Multidimensional Interface Current Methods
9.6 Spherical Harmonics (PL) Methods in One-Dimensional Geometries
9.7 Multidimensional Spherical Harmonics (PL) Transport Theory
9.8 Discrete Ordinates Methods in One-Dimensional Slab Geometry
9.9 Discrete Ordinates Methods in One-Dimensional Spherical Geometry
9.10 Multidimensional Discrete Ordinates Methods
9.11 Even-Parity Transport Formulation
9.12 Monte Carlo Methods
References
Further Readings
Problems
Chapter 10: Neutron Slowing Down
10.1 Elastic Scattering Transfer Function
10.2 P1 and B1 Slowing-Down Equations
10.3 Diffusion Theory
10.4 Continuous Slowing-Down Theory
10.5 Multigroup Discrete Ordinates Transport Theory
References
Further Reading
Problems
Chapter 11: Resonance Absorption
11.1 Resonance Cross Sections
11.2 Widely Spaced Single-Level Resonances in a Heterogeneous Fuel–Moderator Lattice
11.3 Calculation of First-Flight Escape Probabilities
11.4 Unresolved Resonances
11.5 Multiband Treatment of Spatially Dependent Self-Shielding
11.6 Resonance Cross Section Representations *
References
Further Reading
Problems
Chapter 12: Neutron Thermalization
12.1 Double Differential Scattering Cross Section for Thermal Neutrons
12.2 Neutron Scattering from a Monatomic Maxwellian Gas
12.3 Thermal Neutron Scattering from Bound Nuclei
12.4 Calculation of the Thermal Neutron Spectra in Homogeneous Media
12.5 Calculation of Thermal Neutron Energy Spectra in Heterogeneous Lattices
12.6 Pulsed Neutron Thermalization
References
Further Readings
Problems
Chapter 13: Perturbation and Variational Methods
13.1 Perturbation Theory Reactivity Estimate
13.2 Adjoint Operators and Importance Function
13.3 Variational/Generalized Perturbation Reactivity Estimate
13.4 Variational/Generalized Perturbation Theory Estimates of Reaction Rate Ratios in Critical Reactors
13.5 Variational/Generalized Perturbation Theory Estimates of Reaction Rates
13.6 Variational Theory
13.7 Variational Estimate of Intermediate Resonance Integral
13.8 Heterogeneity Reactivity Effects
13.9 Variational Derivation of Approximate Equations
13.10 Variational Even-Parity Transport Approximations
13.11 Boundary Perturbation Theory
References
Further Reading
Problems
Chapter 14: Homogenization
14.1 Equivalent Homogenized Cross Sections
14.2 ABH Collision Probability Method
14.3 Blackness Theory
14.4 Fuel Assembly Transport Calculations
14.5 Homogenization Theory
14.6 Equivalence Homogenization Theory
14.7 Multiscale Expansion Homogenization Theory
14.8 Flux Detail Reconstruction
References
Further Readings
Problems
Chapter 15: Nodal and Synthesis Methods
15.1 General Nodal Formalism
15.2 Conventional Nodal Methods
15.3 Transverse Integrated Nodal Diffusion Theory Methods
15.4 Transverse Integrated Nodal Integral Transport Theory Models
15.5 Transverse Integrated Nodal Discrete Ordinates Method
15.6 Finite-Element Coarse-Mesh Methods
15.7 Variational Discrete Ordinates Nodal Method
15.8 Variational Principle for Multigroup Diffusion Theory
15.9 Single-Channel Spatial Synthesis
15.10 Multichannel Spatial Synthesis
15.11 Spectral Synthesis
References
Further Readings
Problems
Chapter 16: Space–Time Neutron Kinetics
16.1 Flux Tilts and Delayed Neutron Holdback
16.2 Spatially Dependent Point Kinetics
16.3 Time Integration of the Spatial Neutron Flux Distribution
16.4 Stability
16.5 Xenon Spatial Oscillations
16.6 Stochastic Kinetics
References
Further Readings
Problems
Appendix A: Physical Constants and Nuclear Data*
Appendix B: Some Useful Mathematical Formulas
Appendix C: Step Functions, Delta Functions, and Other Functions
C.1 Introduction
C.2 Properties of the Dirac δ-Function
Further Readings
Appendix D: Some Properties of Special Functions
References
Appendix E: Introduction to Matrices and Matrix Algebra
E.1 Some Definitions
E.2 Matrix Algebra
Appendix F: Introduction to Laplace Transforms
F.1 Motivation
F.2 “Cookbook” Laplace Transforms
References
Index
End User License Agreement
List of Tables
Table A.1
Table A.2
Table A.3
Table A.4
Table F.1
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table P4.5
Table P4.11
Table P4.12
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 5.5
Table 6.1
Table 6.2
Table 6.3
Table 6.4
Table 6.5
Table 6.6
Table 6.7
Table 6.8
Table 6.9
Table 6.10
Table 6.11
Table 6.12
Table 7.1
Table 7.2
Table 8.1
Table 8.2
Table 8.3
Table 9.1
Table 9.2
Table 9.3
Table 11.1
Table 11.2
Table 11.3
Table 11.4
Table 12.1
Table 13.1
Table 13.2
Table 15.1
List of Illustrations
Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig. 1.4
Fig. 1.5
Fig. 1.6
Fig. 1.7
Fig. 1.8
Fig. 1.9
Fig. 1.10
Fig. 1.11
Fig. 1.12
Fig. 1.13
Fig. 1.14
Fig. 1.15
Fig. 1.16
Fig. 1.17
Fig. 1.18
Fig. 1.19
Fig. 1.20
Fig. 1.21
Fig. 1.22
Fig. 1.23
Fig. 1.24
Fig. 1.25
Fig. 1.26
Fig. 1.27
Fig. 1.28
Fig. 1.29
Fig. 1.30
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4
Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
Fig. 3.5
Fig. 3.6
Fig. 3.7
Fig. 3.8
Fig. 3.9
Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5
Fig. 4.6
Fig. 4.7
Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Fig. 5.6
Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7
Fig. 6.8
Fig. 6.9
Fig. 6.10
Fig. 6.11
Fig. 6.12
Fig. 6.13
Fig. 6.14
Fig. 6.15
Fig. 6.16
Fig. 6.17
Fig. 6.18
Fig. 6.19
Fig. 6.20
Fig. 7.1
Fig. 7.2
Fig. 7.3
Fig. 7.4
Fig. 7.5
Fig. 7.6
Fig. 7.7
Fig. 7.8
Fig. 7.9
Fig. 7.10
Fig. 7.11
Fig. 7.12
Fig. 7.13
Fig. 7.14
Fig. 7.15
Fig. 7.16
Fig. 7.17
Fig. 7.18
Fig. 7.19
Fig. 7.20
Fig. 7.21
Fig. 7.22
Fig. 7.23
Fig. 8.1
Fig. 8.2
Fig. 8.3
Fig. 8.4
Fig. 8.5
Fig. 8.6
Fig. 8.7
Fig. 9.1
Fig. 9.2
Fig. 9.3
Fig. 9.4
Fig. 9.5
Fig. 9.6
Fig. 9.7
Fig. 9.8
Fig. 9.9
Fig. 9.10
Fig. 9.11
Fig. 9.12
Fig. 9.13
Fig. 9.14
Fig. 9.15
Fig. 9.16
Fig. 9.17
Fig. 9.18
Fig. 9.19
Fig. 9.20
Fig. 9.21
Fig. 9.22
Fig. 9.23
Fig. 9.24
Fig. 9.25
Fig. 10.1
Fig. 10.2
Fig. 10.3
Fig. 11.1
Fig. 11.2
Fig. 11.3
Fig. 11.4
Fig. 11.5
Fig. 11.6
Fig. 12.1
Fig. 12.2
Fig. 12.3
Fig. 12.4
Fig. 12.5
Fig. 12.6
Fig. 12.7
Fig. 13.1
Fig. 13.2
Fig. 14.1
Fig. 14.2
Fig. 14.3
Fig. 14.4
Fig. 14.5
Fig. 14.6
Fig. 15.1
Fig. 15.2
Fig. 15.3
Fig. 15.4
Fig. 15.5
Fig. 15.6
Fig. 15.7
Fig. 15.8
Fig. 15.9
Fig. 15.10
Fig. 16.1
Guide
Cover
Table of Contents
Preface
Part 1
Chapter 1
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Weston M. Stacey
Nuclear Reactor Physics
3rd, Revised Edition
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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To Penny, Helen, Billy, and Lucia
Preface
Nuclear reactor physics is the physics of neutron fission chain reacting systems. It encompasses those applications of nuclear physics and radiation transport and interaction with matter that determine the behavior of nuclear reactors. As such, it is both an applied physics discipline and the core discipline of the field of nuclear engineering.
As a distinct applied physics discipline, nuclear reactor physics originated in the middle of the twentieth century in the wartime convergence of international physics efforts in the Manhattan Project. It developed vigorously for roughly the next third of the century in various government, industrial, and university R&D and design efforts worldwide. Nuclear reactor physics is now a relatively mature discipline, in that the basic physical principles governing the behavior of nuclear reactors are well understood, most of the basic nuclear data needed for nuclear reactor analysis have been measured and evaluated, and the computational methodology is highly developed and validated. It is now possible to accurately predict the physics behavior of existing nuclear reactor types under normal operating conditions. Moreover, the basic physical concepts, nuclear data, and computational methodology needed to develop an understanding of new variants of existing reactor types or of new reactor types exist for the most part.
As the core discipline of nuclear engineering, nuclear reactor physics is fundamental to the major international nuclear power undertaking. As of 2000, there are 434 central station nuclear power reactors operating worldwide to produce 350,442 MWe of electrical power. This is a substantial fraction of the world's electrical power (e.g., more than 80% of the electricity produced in France and more than 20% of the electricity produced in the United States). The world's electrical power requirements will continue to increase, particularly as the less developed countries strive to modernize, and nuclear power is the only proven technology for meeting these growing electricity requirements without dramatically increasing the already unacceptable levels of greenhouse gas emission into the atmosphere.
Nuclear reactors have additional uses other than central station electricity production. There are more than 100 naval propulsion reactors in the U.S. fleet (plus others in foreign fleets). Nuclear reactors are also employed for basic neutron physics research, for materials testing, for radiation therapy, for the production of radioisotopes for medical, industrial, and national security applications, and as mobile power sources for remote stations. In the future, nuclear reactors may power deep space missions. Thus, nuclear reactor physics is a discipline important to the present and future well-being of the world.
This book is intended as both a textbook and a comprehensive reference on nuclear reactor physics. The basic physical principles, nuclear data, and computational methodology needed to understand the physics of nuclear reactors are developed and applied to explain the static and dynamic behavior of nuclear reactors in Part 1. This development is at a level that should be accessible to seniors in physics or engineering (i.e., requiring a mathematical knowledge only through ordinary and partial differential equations and Laplace transforms and an undergraduate-level knowledge of atomic and nuclear physics). Mastery of the material presented in Part 1 provides an understanding of the physics of nuclear reactors sufficient for nuclear engineering graduates at the B.S. and M.S. levels, for most practicing nuclear engineers, and for others interested in acquiring a broad working knowledge of nuclear reactor physics.
The material in Part 1 was developed in the process of teaching undergraduate and first-year graduate courses in nuclear reactor physics at Georgia Tech for a number of years. The emphasis in the presentation is on conveying the basic physical concepts and their application to explain nuclear reactor behavior, using the simplest mathematical description that will suffice to illustrate the physics. Numerous examples are included to illustrate the step-by-step procedures for carrying out the calculations discussed in the text. Problems at the end of each chapter have been chosen to provide physical insight and to extend the material discussed in the text, while providing practice in making calculations; they are intended as an integral part of the textbook. Part 1 is suitable for an undergraduate semester-length course in nuclear reactor physics; the material in Part 1 is also suitable for a semester-length first-year graduate course, perhaps with selective augmentation from Part 2.
The purpose of Part 2 is to augment Part 1 to provide a comprehensive, detailed, and advanced development of the principal topics of nuclear reactor physics. There is an emphasis in Part 2 on the theoretical bases for the advanced computational methods of reactor physics. This material provides a comprehensive, though necessarily abridged, reference work on advanced nuclear reactor physics and the theoretical bases for its computational methods. Although the material stops short of descriptions of specific reactor physics codes, it provides the basis for understanding the code manuals. There is more than enough material in Part 2 for a semester-length advanced graduate course in nuclear reactor physics. The treatment is necessarily somewhat more mathematically intense than in Part 1.
Part 2 is intended primarily for those who are or would become specialists in nuclear reactor physics and reactor physics computations. Mastery of this material provides the background for creating the new physics concepts necessary for developing new reactor types and for understanding and extending the computational methods in existing reactor physics codes (i.e., the stock-in-trade for the professional reactor physicist). Moreover, the extensive treatment of neutron transport computational methods also provides an important component of the background necessary for specialists in radiation shielding, for specialists in the applications of neutrons and photons in medicine and industry, and for specialists in neutron, photon, and neutral atom transport in industrial, astrophysical, and thermonuclear plasmas.
Any book of this scope owes much to many people besides the author, and this one is no exception. The elements of the subject of reactor physics were developed by many talented people over the past half-century, and the references can only begin to recognize their contributions. In this regard, I note the special contribution of R.N. Hwang, who helped prepare certain sections on resonance theory. The selection and organization of material has benefited from the example of previous authors of textbooks on reactor physics. The feedback from a generation of students has assisted in shaping the organization and presentation. Several people (C. Nickens, B. Crumbly, S. Bennett-Boyd) supported the evolution of the manuscript through at least three drafts, and several other people at Wiley transformed the manuscript into a book. I am grateful to all of these people, for without them there would be no book.
WESTON M. STACEYAtlanta, GeorgiaOctober 2000
Preface to Second Edition
This second edition differs from the original in two important ways. First, a section on neutron transport methods has been added in Chapter 3 to provide an introduction to that subject in the first section of the book on basic reactor physics that is intended as the text for an advanced undergraduate course. My original intention was to use diffusion theory to introduce reactor physics, without getting into the mathematical complexities of transport theory. I think this works reasonably well from a pedagogical point of view, but it has the disadvantage of sending BS graduates into the workplace without an exposure to transport theory. So, a short section on transport methods in slab geometry was added at the end of the diffusion theory chapter to provide an introduction.
Second, there has been a resurgence in interest and activity in the improvement of reactor designs and in the development of new reactor concepts that are more inherently safe, better utilize the uranium resources, discharge less long-lived waste and are more resistant to the diversion of fuels to other uses. A section has been added in Chapter 7 on the improved Generation-III designs that will be coming online over the next decade or so, and a few sections have been added in Chapters 6 and 7 on the new reactor concepts being developed under the Generation-IV and Advanced Fuel Cycle Initiatives with the objective of closing the nuclear fuel cycle.
The text was amplified for the sake of explication in a few places, some additional homework problems were included, and numerous typos, omissions and other errors that slipped through the final proof-reading of the first edition were corrected. I am grateful to colleagues, students and particularly the translators preparing a Russian edition of the book for calling several such mistakes to my attention.
Otherwise, the structure and context of the book remains unchanged. The first eight chapters on basic reactor physics provide the text for a first course in reactor physics at the advanced undergraduate or graduate level. The second eight chapters on advanced reactor physics provide a text suitable for graduate courses on neutron transport theory and reactor physics.
I hope that this second edition will serve to introduce to the field the new generation of scientists and engineers who will carry forward the emerging resurgence of nuclear power to meet the growing energy needs of mankind in a safe, economical, environmentally sustainable and proliferation-resistant way.
WESTON M. STACEYAtlanta, GeorgiaMay 2006
Preface to Third Edition
Nuclear reactor physics is that branch of applied nuclear physics that describes the physical behavior of nuclear reactors; as such it is the core discipline of the field of nuclear power engineering. More specifically, nuclear reactor physics describes the physics of the neutron chain fission reaction in a nuclear reactor, which depends on the transport and interaction with matter of fission neutrons and their progeny within a reactor. This physics determines the time-dependent behavior of the neutron distribution in a nuclear reactor and its dependence on the composition and configuration of the materials making up the reactor, which in turn determines the time dependence of the nuclear power level and distribution within the reactor and the change over time in material composition within the reactor due to neutron interactions.
The field of nuclear reactor physics originated in the work of Enrico Fermi, Eugene Wigner, Walter Zinn, and others who designed the Chicago Pile and the Hanford plutonium production reactors in the WWII Manhattan Project in the early 1940s. The early post-war development of the field took place in reactor development programs at what became the Argonne and Oak Ridge National Laboratories, in the naval reactor development Bettis and Knolls Atomic Power Laboratories, and in similar reactor development laboratories abroad (USSR, Canada, France, England, Germany, Japan). Industrial contributions to the field began in the second half of the twentieth century at Westinghouse, General Electric, Combustion Engineering, and Babcock & Wilcox in the United States and subsequently other firms in Europe and Japan, and more recently in China. (The author was privileged to have known and worked with some of the pioneers of the field at the Knolls and Argonne Laboratories.)
This book evolved from lecture notes developed for undergraduate and graduate courses in reactor physics and a graduate course in the related subject of neutron transport theory developed at Georgia Tech in the 1990s. By that time the field had advanced beyond the "bibles" of the field - ANL-5800: Reactor Physics Constants and Naval Reactors Physics Handbook, and the great early texts on the subject by Weinberg & Wigner, Glasstone & Edlund, Meghreblian & Holmes, LaMarsh, Henry, and Duderstadt & Hamilton were becoming dated, and a lot of new theory had been developed.
My intention in organizing this book was that the first eight chapters would constitute a comprehensive first course in nuclear reactor physics at an advanced undergraduate level. The student would be expected to have some familiarity with the concepts of number densities, cross sections, particle fluxes, radioactivity, and so on going into the course and could expect to come out of the course with a basic knowledge of nuclear reactor physics that would prepare him or her to go on to advanced study of the subject or to take an entry-level job in the nuclear power industry.
Chapters 9-16 are intended for people who are at least acquainted with the material in Chapters 1-8 and want to prepare themselves for advanced nuclear reactor analysis or the development of methods for analyzing new types of nuclear reactors. There is enough material for a graduate course in neutron transport theory (Chapter 9) and more than enough other material (Chapters 10-16) for a graduate course in nuclear reactor physics, and that is the way I have taught it, but of course other arrangements are possible.
Nuclear reactor physics is a math-intensive subject. Understanding of the material in this book would be greatly enhanced by a familiarity with solution of PDEs, by separation of variables and eigenfunction expansion, and by a familiarity with Laplace and Fourier transform methods for the solution of differential equations. This material is usually covered in an advanced undergraduate course in engineering mathematics.
The world certainly needs nuclear power. The climatic threat of continued reliance on fossil fuels and the questionable credibility of deployment of reliable, large-scale baseline solar or wind power plants is authoritatively documented in Burton Richter's Beyond Smoke and Mirrors. So, rational maintenance of our standard of living in the developed world and its extension to the remainder of the planet would seem to be dependent on expansion of nuclear power. Improved versions of present reactors and many new variants of reactors are being proposed, which means that many new reactor physics methods must be developed in order to analyze their likely performance. A major purpose of this book is to educate the people who will make these developments and analyze these reactors.
The first and second editions of this book have met with some success, and have been translated into Russian and Chinese. The translators have been familiar with the subject matter, which has resulted in some good questions, and of course they have found some typos and a mistake or two. Similarly, colleagues and readers have identified a few places where a fuller description would be useful. This third edition benefits from their work, which I gratefully acknowledge.
Finally, no book exists without the efforts of the people who produce the physcial product. Martin Preuss stayed after me for a number of years to prepare this third edition, and Stephanie Volk ably edited it at Wiley-VCH. Abhishek Sarkari at Thomson Digital led the copy editors who were essential in pulling together the rather complex final product. I am sincerely grateful to all these people.
WESTON M. STACEYAtlanta, GeorgiaNovember, 2017
Part 1Basic Reactor Physics
1Neutron–Nuclear Reactions
The physics of nuclear reactors is determined by the transport of neutrons and their interaction with matter within a reactor. The basic neutron nucleus reactions of importance in nuclear reactors and the nuclear data used in reactor physics calculations are described in this chapter.
1.1 Neutron-Induced Nuclear Fission
Stable Nuclides
Short-range attractive nuclear forces acting among nucleons (neutrons and protons) are stronger than the Coulomb repulsive forces acting among protons at distances on the order of the nuclear radius (R ≈ 1.25 × 10−13 A1/3 cm) in a stable nucleus. These forces are such that the ratio of the atomic mass A (the number of neutrons plus protons) to the atomic number Z (the number of protons) increases with Z; in other words, the stable nuclides become increasingly neutron-rich with increasing Z, as illustrated in Fig. 1.1. The various nuclear species are referred to as nuclides, and nuclides with the same atomic number are referred to as isotopes of the element corresponding to Z. We use the notation (e.g., ) to identify nuclides.
Fig. 1.1 Nuclear stability curve. (With permission from Ref. [1]. Copyright 1996, McGraw-Hill.)
Binding Energy
The actual mass of an atomic nucleus is not the sum of the masses (mp) of the Z protons and the masses (mn) of A − Z neutrons of which it is composed. The stable nuclides have a mass defect:
This mass defect is conceptually thought of as having been converted to energy (E = Δc2) at the time that the nucleus was formed, putting the nucleus into a negative energy state. The amount of externally supplied energy that would have to be converted to mass in disassembling a nucleus into its separate nucleons is known as the binding energy of the nucleus, BE = Δc2. The binding energy per nucleon (BE/A) is shown in Fig. 1.2.
Fig. 1.2 Binding energy per nucleon. (With permission from Ref. [1]. Copyright 1996, McGraw-Hill.)
Any process that results in nuclides being converted to other nuclides with more binding energy per nucleon will result in the conversion of mass into energy. The combination of low A nuclides to form higher A nuclides with a higher BE/A value is the basis for the fusion process for the release of nuclear energy. The splitting of very high A nuclides to form intermediate-A nuclides with a higher BE/A value is the basis of the fission process for the release of nuclear energy.
Threshold External Energy for Fission
The probability of any nuclide undergoing fission (reconfiguring its A nucleons into two nuclides of lower A) can become quite large if a sufficient amount of external energy is supplied to excite the nucleus. The minimum, or threshold, amount of such excitation energy required to cause fission with high probability depends on the nuclear structure and is quite large for nuclides with Z < 90. For nuclides with Z > 90, the threshold energy is about 4–6 MeV for even-A nuclides, and generally is much lower for odd-A nuclides. Certain of the heavier nuclides (e.g., and ) exhibit significant spontaneous fission even in the absence of any externally supplied excitation energy.
Neutron-Induced Fission
When a neutron is absorbed into a heavy nucleus (A, Z) to form a compound nucleus (A + 1, Z), the BE/A value is lower for the compound nucleus than for the original nucleus. For some nuclides (e.g., , , , ), this reduction in BE/A value is sufficient that the compound nucleus will undergo fission, with high probability, even if the neutron has very low energy. Such nuclides are referred to as fissile; that is, they can be caused to undergo fission by the absorption of a low-energy neutron. If the neutron had kinetic energy prior to being absorbed into a nucleus, this energy is transformed into additional excitation energy of the compound nucleus. All nuclides with Z > 90 will undergo fission with high probability when a neutron with kinetic energy in excess of about 1 MeV is absorbed. Nuclides such as , , and will undergo fission with neutrons with energy of about 1 MeV or higher, with high probability.
Neutron Fission Cross Sections
The probability of a nuclear reaction, in this case fission, taking place can be expressed in terms of a quantity σ that expresses the probable reaction rate per unit area normal to the neutron motion for n neutrons traveling with speed v, a distance dx in a material with N nuclides per unit volume:
The units of σ are area that gives rise to the concept of σ as a cross-sectional area presented to the neutron by the nucleus, for a particular reaction process, and to the designation of σ as a cross section. Cross sections are usually on the order of 10−24 cm2, and this unit is referred to as a barn, for historical reasons.
The fission cross section, σf, is a measure of the probability that a neutron and a nucleus interact to form a compound nucleus that then undergoes fission. The probability that a compound nucleus will be formed is greatly enhanced if the relative energy of the neutron and the original nucleus, plus the reduction in the nuclear binding energy, corresponds to the difference in energy of the ground state and an excited state of the compound nucleus, so that the energetics are just right for formation of a compound nucleus in an excited state. The first excited states of the compound nuclei resulting from neutron absorption by the odd-A fissile nuclides are generally lower lying (nearer to the ground state) than the first excited states of the compound nuclei resulting from neutron absorption by the heavy even-A nuclides, which accounts for the odd-A nuclides having much larger absorption and fission cross sections for low-energy neutrons than do the even-A nuclides.
Fission cross sections for some of the principal fissile nuclides of interest for nuclear reactors are shown in Figs. 1.3–1.5. The resonance structure corresponds to the formation of excited states of the compound nuclei, the lowest lying of which are at less than 1 eV. The nature of the resonance cross section can be shown to give rise to a 1/E1/2 or 1/υ dependence of the cross section at off-resonance neutron energies below and above the resonance range, as is evident in these figures. The fission cross sections are largest in the thermal energy region E < ∼1 eV. The thermal fission cross section for is larger than that of or .
Fig. 1.3 Fission cross sections for . (From www.nndc.bnl.gov/.)
Fig. 1.4 Fission cross sections for . (From www.nndc.bnl.gov/.)
Fig. 1.5 Fission cross sections for . (From www.nndc.bnl.gov/.)
Fission cross sections for and are shown in Figs. 1.6 and 1.7. Except for resonances, the fission cross section is insignificant below about 1 MeV, above which it is about 1 barn. The fission cross sections for these and other even-A heavy mass nuclides are compared in Fig. 1.8, without the resonance structure.
Fig. 1.6 Fission cross sections for . (From www.nndc.bnl.gov/.)
Fig. 1.7 Fission cross sections for . (From www.nndc.bnl.gov/.)
Fig. 1.8 Fission cross sections for principal nonfissile heavy mass nuclides. (With permission from Ref. [2]. Copyright 1963, Argonne National Laboratory.)
Products of the Fission Reaction
A wide range of nuclides are formed by the fission of heavy mass nuclides, but the distribution of these fission fragments is sharply peaked in the mass ranges 90 < A < 100 and 135 < A < 145, as shown in Fig. 1.9. With reference to the curvature of the trajectory of the stable isotopes on the n versus p plot of Fig. 1.1, most of these fission fragments are above the stable isotopes (i.e., are neutron rich) and will decay, usually by β-decay (electron emission), which transmutes the fission fragment nuclide (A, Z) to (A, Z + 1), or sometimes by neutron emission, which transmutes the fission fragment nuclide (A, Z) to (A − 1, Z), in both instances toward the range of stable isotopes. Sometimes several decay steps are necessary to reach a stable isotope.
Fig. 1.9 Yield versus mass number for fission. (From Ref. [2].)
Usually either two or three neutrons will be emitted promptly in the fission event, and there is a probability of one or more neutrons being emitted subsequently upon the decay of neutron-rich fission fragments over the next second or so. The number of neutrons, which are emitted in the fission process, ν, on average, depends on the fissioning nuclide and on the energy of the neutron inducing fission, as shown in Fig. 1.10.
Fig. 1.10 Average number of neutrons emitted per fission. (With permission from Ref. [3]. Copyright 1976, Wiley.)
Energy Release
The majority of the nuclear energy created by the conversion of mass to energy in the fission event (207 MeV for ) is in the form of the kinetic energy (168 MeV) of the recoiling fission fragments. The range of these massive, highly charged particles in the fuel element is a fraction of a millimeter, so that the recoil energy is effectively deposited as heat at the point of fission. Another 5 MeV is in the form of kinetic energy of prompt neutrons released in the fission event, distributed in energy as shown in Fig. 1.11, with a most likely energy of 0.7 MeV (for ). This energy is deposited in the surrounding material within 10–100 cm as the neutron diffuses, slows down by scattering collisions with nuclei, and is finally absorbed. A fraction of these neutron absorption events result in neutron capture followed by gamma emission, producing on average about 7 MeV in the form of energetic capture gammas per fission. This secondary capture gamma energy is transferred as heat to the surrounding material over a range of 10–100 cm by gamma interactions.
Fig. 1.11 Fission spectrum for thermal neutron-induced fission in . (With permission from Ref. [3]. Copyright 1976, Wiley.)
There is also on average about 7 MeV of fission energy directly released as gamma rays in the fission event, which is deposited as heat within the surrounding 10–100 cm. The remaining 20 MeV of fission energy is in the form of kinetic energy of electrons (8 MeV) and neutrinos (12 MeV) from the decay of the fission fragments. The electron energy is deposited, essentially in the fuel element, within about 1 mm of the fission fragment, but since neutrinos rarely interact with matter, the neutrino energy is lost. Although the kinetic energy of the neutrons emitted by the decay of fission products is almost as great as that of the prompt fission neutrons, there are so few delayed neutrons from fission product decay that their contribution to the fission energy distribution is negligible. This fission energy distribution for is summarized in Table 1.1. The recoverable energy released from fission by thermal and fission spectrum neutrons is given in Table 1.2.
Table 1.1 Fission Energy Release
Form
Energy (MeV)
Range
Kinetic energy fission products
168
< mm
Kinetic energy prompt gammas
7
10–100 cm
Kinetic energy prompt neutrons
5
10–100 cm
Kinetic energy capture gammas
7
10–100 cm
Decay of fission products
Kinetic energy electrons
8
∼mm
Kinetic energy neutrinos
12
∞
Table 1.2 Recoverable Energy from Fission
Source: Data from Ref. [3]; used with permission of Wiley.
Isotope
Thermal Neutron
Fission Neutron
190.0
–
192.9
–
198.5
–
200.3
–
–
184.2
–
188.9
–
191.4
–
193.9
–
193.6
–
196.9
–
196.9
–
200.0
Thus, in total, about 200 MeV per fission of heat energy is produced. One Watt of heat energy then corresponds to the fission of 3.1 × 1010 nuclei per second. Since 1 g of any fissile nuclide contains about 2.5 × 1021 nuclei, the fissioning of 1 g of fissile material produces about 1 megawatt-day (MWd) of heat energy. Because some fissile nuclei will also be transmuted by neutron capture, the amount of fissile material destroyed is greater than the amount fissioned.
1.2 Neutron Capture
Radiative Capture
When a neutron is absorbed by a nucleus to form a compound nucleus, a number of reactions are possible, in addition to fission, in the heavy nuclides. We have already mentioned radiative capture, in which the compound nucleus decays by the emission of a gamma ray, and we now consider this process in more detail. An energy-level diagram for the compound nucleus formation and decay associated with the prominent resonance for incident neutron energies of about 6.67 eV is shown in Fig. 1.12. The energy in the center-of-mass (CM) system of an incident neutron with energy EL in the laboratory system is Ec = [A/(1 + A)]EL. The reduction in binding energy due to the absorbed neutron is ΔEB. If Ec + ΔEB is close to an excited energy level of the compound nucleus, the probability for compound nucleus formation is greatly enhanced. The excited compound nucleus will generally decay by emission of one or more gamma rays, the combined energy of which is equal to the difference in the excited- and ground-state energy levels of the compound nucleus.
Fig. 1.12 Energy-level diagram for compound nucleus formation. (With permission from Ref. [3]. Copyright 1976, Wiley.)
Radiative capture cross sections, denoted σγ, for some nuclei of interest for nuclear reactors are shown in Figs. 1.13–1.21. The resonance nature of the cross sections over certain ranges corresponds to the discrete excited states of the compound nucleus that is formed upon neutron capture. These excited states correspond to neutron energies in the range of a fraction of an eV to 103 eV for the fissile nuclides, generally correspond to neutron energies of 10–104 eV for even-A heavy mass nuclides (with the notable exception of thermal resonance), and correspond to much higher neutron energies for the lower mass nuclides. The 1/υ “off-resonance” cross section dependence is apparent.
Fig. 1.13 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.14 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.15 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.16 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.17 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.18 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.19 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.20 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.21 Radiative capture cross section for . (From www.nndc.bnl.gov/.)
The Breit–Wigner single-level resonance formula for the neutron capture cross section is
where E0 is the energy (in the CM) system at which the resonance peak occurs (i.e., Ec + ΔEB matches the energy of an excited state of the compound nucleus), Γ is the full width at half-maximum of the resonance, σ0 is the maximum value of the total cross section (at E0), and Γγ is the radiative capture width (Γγ/Γ is the probability that the compound nucleus, once formed, will decay by gamma emission). The fission resonance cross section can be represented by a similar expression with the fission width Γf, defined such that Γf/Γ is the probability that the compound nucleus, once formed, will decay by fission.
Equation (1.3) represents the cross section describing the interaction of a neutron and nucleus with relative (CM) energy Ec. However, the nuclei in a material are distributed in energy (approximately a Maxwellian distribution characterized by the temperature of the material). What is needed is a cross section averaged over the motion of the nuclei:
where E and E′ are the neutron and nuclei energies, respectively, in the laboratory system, and fmax(E′) is the Maxwellian energy distribution:
Using Eqs. (1.3) and (1.5), Eq. (1.4) becomes
where
A is the atomic mass (amu) of the nuclei, and
Neutron Emission
When the compound nucleus formed by neutron capture decays by the emission of one neutron, leaving the nucleus in an excited state which subsequently undergoes further decays, the event is referred to as inelastic scattering and the cross section is denoted σin. Since the nucleus is left in an excited state, the energy of the emitted neutron can be considerably less than the energy of the incident neutron. If the compound nucleus decays by the emission of two or more neutrons, the events are referred to as n − 2n, n − 3n, and so on, and the cross sections are denoted σn,2n, σn,3n, and so on. Increasingly higher incident neutron energies are required to provide enough excitation energy for single, double, triple, and so on neutron emission. Inelastic scattering is the most important of these events in nuclear reactors, but it is most important for neutrons 1 MeV and higher in energy.
1.3 Neutron Elastic Scattering
Elastic scattering may take place via compound nucleus formation followed by the emission of a neutron that returns the compound nucleus to the ground state of the original nucleus. In such a resonance elastic scattering event, the kinetic energy of the original neutron–nuclear system is conserved. The neutron and the nucleus may also interact without neutron absorption and the formation of a compound nucleus, which is referred to as potential scattering. Although quantum mechanical (s-wave) in nature, the latter event may be visualized and treated as a classical hard-sphere scattering event, away from resonance energies. Near resonance energies, there is quantum mechanical interference between the potential and resonance scattering, which is constructive just above and destructive just below the resonance energy.
The single-level Breit–Wigner form of the scattering cross section, modified to include potential and interference scattering, is
where (Γn/Γ) is the probability that, once formed, the compound nucleus decays to the ground state of the original nucleus by neutron emission, R ≃ 1.25 × 10−13 A1/3 cm is the nuclear radius, and λ0 is the reduced neutron wavelength.
Averaging over a Maxwellian distribution of nuclear motion yields the scattering cross section for neutron laboratory energy E and material temperature T:
where
The elastic scattering cross sections for a number of nuclides of interest in nuclear reactors are shown in Figs. 1.22–1.26. In general, the elastic scattering cross section is almost constant in energy below the neutron energies corresponding to the excited states of the compound nucleus. The destructive interference effects just below the resonance energy are very evident in Fig. 1.26.
Fig. 1.22 Elastic scattering cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.23 Elastic scattering cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.24 Elastic scattering cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.25 Elastic scattering cross section for . (From www.nndc.bnl.gov/.)
Fig. 1.26 Elastic scattering cross section for . (From www.nndc.bnl.gov/.)
The energy dependence of the carbon scattering cross section is extended to very low neutron energies in Fig. 1.27 to illustrate another phenomenon. At sufficiently small neutron energy, the neutron wavelength
becomes comparable to the interatomic spacing, and the neutron interacts not with a single nucleus but with an aggregate of bound nuclei. If the material has a regular structure, as graphite does, the neutron will be diffracted and the energy dependence of the cross section will reflect the neutron energies corresponding to multiples of interatomic spacing. For sufficiently small energies, diffraction becomes impossible and the cross section is once again insensitive to neutron energy.
Fig. 1.27 Total scattering cross section of . (With permission from Ref. [3]. Copyright 1976, Wiley.)
1.4 Summary of Cross Section Data
Low-Energy Cross Sections
The low-energy total cross sections for several nuclides of interest in nuclear reactors are plotted in Fig. 1.28. Gadolinium is sometimes used as a “burnable poison,” and xenon and samarium are fission products with large thermal cross sections.
Fig. 1.28 Low-energy absorption (fission + capture) cross sections for several important nuclides. (With permission from Ref. [3]. Copyright 1976, Wiley.)
Spectrum-Averaged Cross Sections
Table 1.3 summarizes the cross section data for a number of important nuclides in nuclear reactors. The first three columns give fission, radiative capture, and elastic scattering cross sections averaged over a Maxwellian distribution with T = 0.0253 eV, corresponding to a representative thermal energy spectrum. The next two columns give the infinite dilution fission and radiative capture resonance integrals, which are averages of the respective resonance cross sections over a 1/E spectrum typical of the resonance energy region in the limit of an infinitely dilute concentration of the resonance absorber. The final five columns give cross sections averaged over the fission spectrum.
Example 1.1 Calculation of Macroscopic Cross Section
The macroscopic cross section Σ = Nσ, where N is the number density. The number density is related to the density ρ and atomic number A by N = (ρ/A) N0, where N0 = 6.022 × 1023 is Avogadro's number, the number of atoms in a mole. For a mixture of isotopes with volume fractions υi, the macroscopic cross section is Σ = Σiυi(ρ/A)iN0σi; for example, for a 50:50 vol% mixture of carbon and , the macroscopic thermal absorption cross section is Σa = 0.5(ρC/AC)N0σaC + 0.5(ρU/AU)N0σaU = 0.5(1.60 g/cm3 per 12 g/mol)(6.022 × 1023 atom/mol)(0.003 × 10−24 cm2) + 0.5(18.9 g/cm3 per 238 g/mol)(6.022 × 1023 atom/mol)(2.4 × 10−24 cm2) = 0.0575 cm−1.
Table 1.3 Spectrum-Averaged Thermal, Resonance, and Fast Neutron Cross Sections (barns)
Source: Data from www.nndc.bnl.gov/.
Nuclide
Thermal Cross Section
Resonance Cross Section
Fission Spectrum Cross Section
σ
f
σ
γ
σ
el
