Reuss, P. - Neutron Physics (EDP Sciences, 2008)

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NUCLEAR ENGINEERING

Neutron Physics

Paul ReussInstitut national des sciences et techniques nuclaires

17, avenue du HoggarParc dactivits de Courtabuf, BP 112

91944 Les Ulis Cedex A, France


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The author would like to thank Nova Traduction (K. Foster) and Chris Latham for thetranslation of his book.

Cover illustrations: Jules Horowitz (1921-1995), a highly talented physicist, founded the Frenchschool of neutron physics. In 2014, the Jules Horowitz reactor being built at Cadarache will becomethe main irradiation reactor in the world (100 MWth) for research on materials and nuclear fuels.In the background, the meshing for a neutron physics core calculation and in the foreground the

power distribution, result of this calculation. (Documents courtesy of CEA.)

Cover conception: Thierry Gourdin

Printed in France

ISBN: 978-2-7598-0041-4

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-casting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of

this publication or parts thereof is only permitted under the provisions of the French and GermanCopyright laws of March 11, 1957 and September 9, 1965, respectively. Violations fall under theprosecution act of the French and German Copyright Laws.

c EDP Sciences 2008


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Introduction to the NuclearEngineeringCollection

Within the French Atomic Energy Commission (CEA), the National Institute of NuclearScience and Technology (INSTN) is a higher education institution operating under the jointsupervision of the Ministries of Education and Industry. The purpose of the INSTN is tocontribute to disseminating the CEAs expertise through specialised courses and continuingeducation, not only on a national scale, but across Europe and worldwide.

This mission is focused on nuclear science and technology, and one of its main featuresis a Nuclear Engineering diploma. Bolstered by the CEAs efforts to build partnerships withuniversities and engineering schools, the INSTN has developed links with other higher ed-ucation institutions, leading to the organisation of more than twenty five jointly-sponsoredMasters graduate diplomas. There are also courses covering disciplines in the health sec-

tor: nuclear medicine, radiopharmacy, and training for hospital physicists.Continuous education is another important part of the INSTNs activities that relies onthe expertise developed within the CEA and by its partners in industry.

The Nuclear Engineering course (known as GA, an abbreviation of its French name)was first taught in 1954 at the CEA Saclay site, where the first experimental piles werebuilt. It has also been taught since 1976 at Cadarache, where fast neutron reactors weredeveloped. GA has been taught since 1958 at the School for the Military Applicationsof Atomic Energy (EAMEA), under the responsibility of the INSTN. Since its creation, theINSTN has awarded diplomas to over 4400 engineers who now work in major companiesor public-sector bodies in the French nuclear industry: CEA, EDF (the French electricity

board), AREVA, Cogema, Marine Nationale (the French navy), IRSN (French TSO). . . Manyforeign students from a variety of countries have also studied for this diploma.

There are two categories of student: civilian and military. Civilian students will obtain jobs in the design or operation of nuclear reactors for power plants or research estab-lishments, or in fuel processing facilities. They can aim to become expert consultants,analysing nuclear risks or assessing environmental impact. The EAMEA provides educa-tion for certain officers assigned to French nuclear submarines or the aircraft carrier.

The teaching faculty comprises CEA research scientists, experts from the Nuclear Safetyand Radiation Protection Institute (IRSN), and engineers working in industry (EDF, AREVA,

etc.). The main subjects are: nuclear physics and neutron physics, thermal hydraulics,nuclear materials, mechanics, radiological protection, nuclear instrumentation, operationand safety of Pressurised Water Reactors (PWR), nuclear reactor systems, and the nu-clear fuel cycle. These courses are taught over a six-month period, followed by a finalproject that rounds out the students training by applying it to an actual industrial situation.


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vi Neutron Physics

These projects take place in the CEAs research centres, companies in the nuclear industry(EDF, AREVA, etc.), and even abroad (USA, Canada, United Kingdom, etc.). A key featureof this programme is the emphasis on practical work carried out using the INSTN facilities

(ISIS training reactor, PWR simulators, radiochemistry laboratories, etc.).Even now that the nuclear industry has reached full maturity, the Nuclear Engineeringdiploma is still unique in the French educational system, and affirms its mission: to trainengineers who will have an in-depth, global vision of the science and the techniquesapplied in each phase of the life of nuclear installations from their design and constructionto their operation and, finally, their dismantling.

The INSTN has committed itself to publishing all the course materials in a collectionof books that will become valuable tools for students, and to publicise the contents of itscourses in French and other European higher education institutions. These books are pub-lished by EDP Sciences, an expert in the promotion of scientific knowledge, and are also

intended to be useful beyond the academic context as essential references for engineersand technicians in the industrial sector.

The European Nuclear Education Network (ENEN) fully supported INSTN, one of itfounder members, in publishing this book. For ENEN this book constitutes the first of a se-ries of textbooks intended for students and young professionals in Europe and worldwide,contributing to the creation of the European Educational Area.

Joseph SafiehNuclear Engineering Course Director

ENEN President


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Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x x i i i

Part I Fundamentals of neutron physics

Chapter 1: Introduction: general facts about nuclear energy1.1. A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1. Fermis pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2. The end of a long search... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3. ... and the beginning of a great adventure . . . . . . . . . . . . . . . . . . . . . . 6

1.2. Principle of a nuclear power plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3. Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4. Principle of chain reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5. Main moderators and coolants; types of reactor . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6. Monitoring and control of reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7. Nuclear fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8. Nuclear safety and radiation protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.9. Nuclear programmes: prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Exercises

Chapter 2: Nuclear physics for neutron physicists

A. Structure of matter and nuclear binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1. Structure of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1. The classical atomic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.2. Elements and isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.3. Nuclide notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.4. Stable and unstable nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.5. Pattern of stable nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


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2.2. Nuclear binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1. Mass defect and nuclear binding energy . . . . . . . . . . . . . . . . . . . . . . . . 292.2.2. Nuclear units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.3. Nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.4. Liquid drop model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.5. Magic numbers and the layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.6. Spin and parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.7. Excited levels of nuclei (isomeric states) . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.8. Other nuclear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3. Principle of release of nuclear energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.1. Nuclear recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2. Reaction energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.3. Principle of fusion and fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1. Regions of instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2. Main types of radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3. Law of radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.4. Examples of radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.5. Alpha instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4.6. Beta instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.7. Gamma instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.8. Radioactive series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.9. Radioactive series equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5. General information about nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.1. Spontaneous reactions and induced reactions . . . . . . . . . . . . . . . . . . . 472.5.2. Nuclear reaction examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5.3. Laws of conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.4. Cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5.5. Macroscopic cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6. Neutron reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.2. Scattering and real reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.6.3. Main reactions induced by neutrons in reactors . . . . . . . . . . . . . . . . . 522.6.4. Partial cross-sections and additivity of cross-sections . . . . . . . . . . . . . 532.6.5. Neutron cross-section curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.7. Why resonances? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.7.1. Resonant cross-sections: BreitWigner law . . . . . . . . . . . . . . . . . . . . . . 602.7.2. Resonant cross-sections: statistical aspects . . . . . . . . . . . . . . . . . . . . . . 642.7.3. Cross-sections in the thermal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.8. Neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.8.1. Spontaneous sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.8.2. Reactions induced by radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.8.3. Fusion reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.8.4. Spallation reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.9. Spontaneous fission and induced fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.9.1. The fission barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.9.2. Fission-related thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.9.3. Parity effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


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2.9.4. Quantum effects: tunnel effect and anti-tunnel effect . . . . . . . . . . . . 722.10. Fission products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.10.1. Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.10.2. Fission fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.10.3. Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.11. Measuring basic neutron physics data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.11.1. Neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.11.2. Detection of neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.11.3. Measurement of total cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.11.4. Measurement of partial cross-sections and number of neutrons

emitted per fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.11.5. Integral measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.12. Evaluation and libraries of nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.13. Processing of nuclear data for neutron physics codes . . . . . . . . . . . . . . . . . . . . . 81

Exercises

Chapter 3: Introduction to neutron physics

3.1. Neutronmatter interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.1.1. Cross-sections (review) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.1.2. Neutron density, neutron flux, reaction rate . . . . . . . . . . . . . . . . . . . . . 983.1.3. Concept of phase flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1.4. Concept of current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.1.5. Concept of opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1.6. The Boltzmann equation: a first approximation . . . . . . . . . . . . . . . . . . 102

3.2. General representation of a neutron population . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.1. Variables to introduce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.2. General concept of neutron flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.3. Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.4. Probabilistic and deterministic solutions of the Boltzmann

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.3. Neutron spectra and energy balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.3.1. Fast neutron reactors and thermal neutron reactors . . . . . . . . . . . . . . 1073.3.2. Neutron balances: the four-factor formula and variants . . . . . . . . . . 108

Exercises

Chapter 4: Point kinetics

4.1. Kinetics without delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.1.1. First approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.1.2. Chain reaction equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.1.3. Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.2. Kinetics with delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.2.1. Parameters of delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.2.2. Qualitative aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2.3. Chain reaction equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123


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4.2.4. Inhour equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.5. Low reactivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.2.6. High reactivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.7. The natural unit of reactivity: the dollar . . . . . . . . . . . . . . . . . . . . . 1264.2.8. Effective proportion of delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.9. Fast kinetics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.10. Slow kinetics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3. A few specific problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.1. Kinetics with source term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.2. Emergency shutdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.3. Reactivity window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3.4. Reactivity ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Exercises

Chapter 5: Diffusion equation

5.1. Establishing the diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1.1. Neutron balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1.2. Evaluating the current: Ficks law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.1.3. Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.1.4. Initial condition, boundary conditions, interface conditions . . . . . . 1445.1.5. External boundary: black body extrapolation distance;

extrapolated surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.1.6. Approach based on the integral equation . . . . . . . . . . . . . . . . . . . . . . . 1465.1.7. Conditions for validity of the diffusion approximation . . . . . . . . . . . . 1465.1.8. Transport correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.2. Example problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2.1. Kernels of the diffusion equation in a homogeneous, infinite

medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2.2. Generalisation: the Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.2.3. The albedo concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.2.4. Calculating the albedo of a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.2.5. Use of the albedo as boundary condition . . . . . . . . . . . . . . . . . . . . . . . 1525.2.6. Calculation of configurations described by a single space variable 1525.2.7. Example of configuration where flux is factorised . . . . . . . . . . . . . . . . 1525.2.8. Homogeneous bare reactor: eigenfunctions of the Laplace

operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.2.9. Steady-state problem: flux calculation by decomposition

on the eigenfunctions of the Laplace operator . . . . . . . . . . . . . . . . . . . 1555.2.10. Study of kinetics after injecting a burst of neutrons. . . . . . . . . . . . . . . 156

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Chapter 6: One-group/diffusion theory

6.1. A few problems in one-group/diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.1.1. Shape of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.1.2. Bare homogeneous spherical pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.1.3. Other homogeneous bare pile examples . . . . . . . . . . . . . . . . . . . . . . . . 1726.1.4. Interpretation of critical condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1.5. Reflector saving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1.6. Reflector saving calculation for a plate reactor . . . . . . . . . . . . . . . . 1756.1.7. Geometries described by a single space variable . . . . . . . . . . . . . . . . 1766.1.8. Example of a problem where flux is factorised . . . . . . . . . . . . . . . . . . . 177

6.2. Main numerical methods used for the diffusion equation . . . . . . . . . . . . . . . . . . 1776.2.1. Introduction to numerical processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.2.2. Source problem and critical problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.2.3. Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.2.4. Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.2.5. Nodal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.2.6. Synthesis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Exercises

Chapter 7: Neutron slowing down

7.1. Scattering collision laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.1.1. Elastic and inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.1.2. Laws of elastic collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.1.3. Laws of elastic and isotropic collision . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.1.4. Lethargy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.1.5. Evaluating the number of collisions required to slow down

a neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057.1.6. Comparison of the main moderators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.1.7. Laws of inelastic collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

7.1.8. Slowing down equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2087.1.9. First form of the slowing down equation . . . . . . . . . . . . . . . . . . . . . . . . 2097.1.10. Second form of the slowing-down equation . . . . . . . . . . . . . . . . . . . . . 210

7.2. Analysis of a few specific problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.2.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.2.2. Decay of the neutron spectrum by successive scattering events . . . 2117.2.3. Slowing down without absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.2.4. Slowing down in hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2157.2.5. Slowing down in the presence of resonance traps . . . . . . . . . . . . . . . 216

7.2.6. Slowing down in the presence of low, slowly varying absorption 2187.2.7. Space-energy coupling: Age Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Exercises


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Chapter 8: Resonant absorption of neutrons (physical aspects)

8.1. Resonant absorption in a homogeneous case by a purely absorbentmaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.1.1. Self-shielding in a homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.1.2. Interpretation of the resonance escape probability formula . . . . . . . 2348.1.3. Flux factorisation and the concept of effective cross-section . . . . . . 2358.1.4. Practical formula for the resonance escape probability

in a homogeneous situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2368.2. Slowing down by the absorbing material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.2.1. Equation for the fine structure of the flux in a homogeneoussituation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.2.2. Slowing-down models for resonant nuclei. . . . . . . . . . . . . . . . . . . . . . . 238

8.3. Resonant absorption in a heterogeneous situation . . . . . . . . . . . . . . . . . . . . . . . . 2398.3.1. Flux and fine structure equations in the heterogeneous case . . . . . 2398.3.2. Wigner and Bell-Wigner approximations: the concept

of homogeneous-heterogeneous Equivalence . . . . . . . . . . . . . . . . . . . . 2428.3.3. Fuel containing a mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2428.3.4. The Dancoff effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2448.3.5. Formula for the resonance escape probability in a heterogeneous

situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.4. Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8.4.1. Importance of the Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8.4.2. Origin of the Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2468.4.3. Doppler effect calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2478.5. Future prospects: problems that a resonant absorption theory will have

to solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.5.1. Evaluating the Bell factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.5.2. Continuous-multigroup equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.5.3. Allowing for complicated geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.5.4. Cases with several resonant nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.5.5. Definition and calculation of effective temperatures . . . . . . . . . . . . . 250

Exercises

Chapter 9: Thermalisation of neutrons

9.1. Qualitative aspects of thermalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.1.1. What distinguishes the thermal domain from the slowing-down

domain? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599.1.2. Maxwell spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2619.1.3. Principle of microreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2629.1.4. Scattering equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

9.1.5. Thermalisation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.2. Appearance and characterisation of the thermal spectrum . . . . . . . . . . . . . . . . . 2649.2.1. Difference between the thermal neutron spectrum

and the Maxwell spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2649.2.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266


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9.2.3. Average cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689.2.4. Processing a heterogeneous situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

9.3. Balance of thermal neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

9.3.1. General considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2699.3.2. Thermal utilisation factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2709.3.3. Reproduction factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.3.4. Optimum moderation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.3.5. Problem of using a boron solution in the primary circuit of PWRs 2749.3.6. Problem of using plutonium in PWRs . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Exercises

Chapter 10: Multigroup theory

10.1. Principle of multigroup theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.2. Implementation of multigroup theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.3. Examples of multigroup partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28710.4. Multigroup diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28910.5. Two-group theory calculation of a reflected cylindrical pile . . . . . . . . . . . . . . . 289

Exercises

Chapter 11: Poisoning by fission products

11.1. Fission products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29811.1.1. Equations governing fission products. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29811.1.2. Fission pseudo-products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.1.3. Concept of poisoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11.2. The xenon effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30111.2.1. The xenon 135 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30111.2.2. Simplified equations for the xenon 135 series . . . . . . . . . . . . . . . . . . . 30211.2.3. Reactor start-up, equilibrium, and shutdown . . . . . . . . . . . . . . . . . . . . 30211.2.4. Spatial instabilities due to xenon 135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

11.3. The samarium effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.3.1. Samarium 149 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.3.2. Simplified equations in the samarium 149 series . . . . . . . . . . . . . . . . 30511.3.3. Reactor start-up, equilibrium and shutdown . . . . . . . . . . . . . . . . . . . . . 306

Exercises

Chapter 12: Fuel evolution (heavy nuclei)

12.1. Evolution series and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.1.1. Evolution series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.1.2. Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32112.1.3. Neutron fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32212.1.4. Variation of cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32312.1.5. Burn-up and combustion rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324


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12.1.6. Example of heavy nucleus balance (pressurised water reactor) . . . 32512.2. Evolution of the multiplication factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

12.2.1. Evolution of the infinite multiplication factor . . . . . . . . . . . . . . . . . . . . 326

12.2.2. Multiple-frequency core management . . . . . . . . . . . . . . . . . . . . . . . . . . 32712.2.3. Other core management problems (pressurised water reactors) . . . 329

12.3. Conversion and recycling of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.3.1. Fissile, fertile, and sterile nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.3.2. Conversion factor and regeneration gain . . . . . . . . . . . . . . . . . . . . . . . . 33312.3.3. Recycling plutonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33412.3.4. Thorium-uranium 233 cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33612.3.5. Incineration of nuclear waste in reactors . . . . . . . . . . . . . . . . . . . . . . . . 336

Exercises

Chapter 13: Temperature effects

13.1. Counter-reaction loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34713.2. Definition of temperature coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34813.3. Physical effects that contribute to temperature coefficients . . . . . . . . . . . . . . . . 34913.4. Effects of temperature on reactivity in pressurised water reactors . . . . . . . . . . 35113.5. Overview of counter-reaction effects in pressurised water reactors . . . . . . . . . 35313.6. Temperature effects in other reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Exercises

Part II Elements of neutron physics calculations

Chapter 14: Boltzmann equation

14.1. The two forms of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37014.1.1. Incoming density, outgoing density, and neutron flux . . . . . . . . . . . . 370

14.1.2. Collision operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37114.1.3. Transport operator (integral form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37114.1.4. Transport operator (differential form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37214.1.5. Equivalence between the two forms of the transport operator . . . . . 37314.1.6. The two deterministic approaches to the Boltzmann equation . . . . 37414.1.7. Probabilistic approach to the Boltzmann equation . . . . . . . . . . . . . . . 374

14.2. Processing the collision operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37514.3. Treatment of the integral form of the transport operator . . . . . . . . . . . . . . . . . . . 376

14.3.1. Isotropic collision assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37614.3.2. Transport correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37714.3.3. First collision probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37814.3.4. Reciprocity and complementarity relationships between the first

collision probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38014.3.5. Probabilities involving a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381


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14.3.6. Reciprocity and complementarity relationshipsbetween probabilities involving a surface . . . . . . . . . . . . . . . . . . . . . . . 381

14.3.7. First collision probabilities for a homogeneous convex body . . . . . 382

14.3.8. Calculation of collision probabilities in x Geometry and x yGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

14.3.9. Calculation of probabilities on an infinite lattice of identical cells 38514.3.10. Cylindrisation of cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38714.3.11. Principle of multicell geometry calculations . . . . . . . . . . . . . . . . . . . 388

14.4. Handling the differential form of the transport operator . . . . . . . . . . . . . . . . . . . 39014.4.1. Handling the diffusion operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39014.4.2. Spherical harmonic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39214.4.3. Diffusion approximation and transport correction . . . . . . . . . . . . . . . 39414.4.4. Method of simplified spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . 396

14.4.5. Method of discrete ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39614.4.6. Ray effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39714.4.7. Handling the space variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

14.5. Concept of fundamental mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40214.5.1. Why is the fundamental mode of interest? . . . . . . . . . . . . . . . . . . . . . . . 40214.5.2. A few analytical solutions of the Boltzmann equation

in monokinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40214.5.3. Concept of fundamental mode in a homogeneous medium

in monokinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40514.5.4. Physical interpretation of the fundamental mode . . . . . . . . . . . . . . . . 40714.5.5. Existence andcalculation of the leakagecoefficient . . . . . . . . . . . . . . . 40714.5.6. Balance in fundamental mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40914.5.7. Generalisation to the spectrum case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41114.5.8. Concept of fundamental mode in a regular lattice . . . . . . . . . . . . . . . 412

14.6. Use of Monte Carlo techniques in neutron physics. . . . . . . . . . . . . . . . . . . . . . . . 41314.6.1. Outline of the Monte Carlo method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41314.6.2. Analogous simulation and non-analogous simulation . . . . . . . . . . . . 41314.6.3. Overview of sampling problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41514.6.4. Analogous simulation of a neutron path . . . . . . . . . . . . . . . . . . . . . . . . . 41714.6.5. Estimating the multiplication factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41714.6.6. Semi-analogous simulation of neutron paths . . . . . . . . . . . . . . . . . . . . 419

Exercises

Chapter 15: Theory of Resonant Absorption of Neutrons

15.1. Energy scales of different neutron physics problems . . . . . . . . . . . . . . . . . . . . . . . 45515.2. The heterogeneous-homogeneous equivalence: choice of Bell factor . . . . . . 456

15.2.1. Principle of pre-tabulations (review) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45615.2.2. Principle of heterogeneous-homogeneous equivalence . . . . . . . . . . 457

15.2.3. Definition of the simplified problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45715.2.4. Implementation with the narrow resonance model . . . . . . . . . . . . 45815.2.5. Implementation with the wide resonance model . . . . . . . . . . . . . . 45815.2.6. Examining the width of resonances: example of uranium 238 . . . . 45915.2.7. Macrogroup-by-macrogroup equivalence . . . . . . . . . . . . . . . . . . . . . . . 459


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15.3. Continuous-multigroup equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45915.3.1. Why calculate the real flux rather than the macroscopic flux? . . . . 45915.3.2. Principle of continuous-multigroup equivalence . . . . . . . . . . . . . . . . . 460

15.4. One-by-one handling of situations with several resonant nuclei . . . . . . . . . 46215.4.1. The need to handle situations with several resonant nuclei . . . . . . . 46215.4.2. Principle of one-by-one handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

15.5. Extensions of the Livolant-Jeanpierre theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46315.5.1. Allowing for capture in the moderator . . . . . . . . . . . . . . . . . . . . . . . . . . 46315.5.2. Self-shielding in the thermal domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46415.5.3. Other slowing-down models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46415.5.4. Handling geometries with several moderator zones . . . . . . . . . . . . . . 46515.5.5. Handling of cases with several resonant zones . . . . . . . . . . . . . . . . . . 46515.5.6. Treatment of the general case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46715.5.7. Problem of interference between resonances . . . . . . . . . . . . . . . . . . . . 467

15.6. Method of probability tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46815.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46815.6.2. Principle of a probability table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46815.6.3. Table of partial cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47015.6.4. Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47115.6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

15.7. Treatment of the Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47215.7.1. Calculating the Doppler widening of resonances . . . . . . . . . . . . . . . . 472

15.7.2. Allowing for temperature gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.8. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.8.1. Validation of self-shielding calculations . . . . . . . . . . . . . . . . . . . . . . . . . 47315.8.2. Problems still pending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Exercises

Chapter 16: Perturbation theory

16.1. Concept of adjoint flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48316.1.1. Importance in neutron physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

16.1.2. Mathematical definition of adjoint flux . . . . . . . . . . . . . . . . . . . . . . . . . . 48416.1.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48516.1.4. Physical definition of adjoint flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

16.2. Perturbation formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48616.2.1. Associated critical reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48616.2.2. Exact perturbation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48716.2.3. First-order perturbation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

16.3. Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48816.3.1. Plutonium 239 equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48816.3.2. Differential and integral efficiency of a control rod . . . . . . . . . . . . . . 49116.3.3. Error due to nuclear and technological uncertainties . . . . . . . . . . . . . 491

16.4. Generalised perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Exercises


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Chapter 17: Overview of the Calculation Scheme

17.1. Nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49817.2. Tabulation for the processing of resonance self-shielding . . . . . . . . . . . . . . . . . . 49817.3. Assembly calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49917.4. Reflector calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50217.5. Core calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50317.6. Problem of homogenisation and condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50517.7. Transport-diffusion equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50717.8. Generalisation: the concept of equivalence in neutron physics . . . . . . . . . . . . 50717.9. Evolution and counter-reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50917.10. Fast kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51017.11. Summary of the main approximations of the calculation scheme . . . . . . . . . . 511

17.12. Validation of calculation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51317.13. Qualification of calculation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51417.14. Tendency search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51417.15. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

Exercises

Chapter 18: Overview of core design problems

18.1. General elements of core design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53718.1.1. Reactor target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

18.1.2. Choice of reactor type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53818.1.3. Elements required for lattice sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53918.1.4. Elements for core sizing and choice of reflector . . . . . . . . . . . . . . . . . 540

18.2. General remarks about core control and management . . . . . . . . . . . . . . . . . . . . 54118.2.1. Control concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54218.2.2. Reactivity balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54418.2.3. Core management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54518.2.4. Plutonium recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

18.3. Prospects for nuclear energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

18.3.1. Strategic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54918.3.2. Use of nuclear fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54918.3.3. Multirecycling of plutonium in reactors . . . . . . . . . . . . . . . . . . . . . . . . . 55018.3.4. Next-Generation reactor types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55318.3.5. Hybrid reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55418.3.6. The problem of waste management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56018.3.7. Nuclear energy and sustainable development . . . . . . . . . . . . . . . . . . . 565

Exercises

Appendice A: Annotated Bibliography

A.1. Introduction to nuclear energy and context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576


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A.1.1. General texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576A.1.2. Historical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577A.1.3. Risks, safety and accidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

A.1.4. Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577A.1.5. Fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577A.1.6. Nuclear defence and risk of proliferation . . . . . . . . . . . . . . . . . . . . . . . . 578A.1.7. Nuclear fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578A.1.8. The Oklo phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578A.1.9. The world needs of energy and the possible

contribution of the nuclear energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578A.2. General works on reactor physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578A.3. Works by the present author on neutron physics . . . . . . . . . . . . . . . . . . . . . . . . . . 579A.4. Nuclear physics, nuclear data and qualification . . . . . . . . . . . . . . . . . . . . . . . . . . 580

A.4.1. General information on nuclear physics. . . . . . . . . . . . . . . . . . . . . . . . . 580A.4.2. Radioactivity and radiation protection . . . . . . . . . . . . . . . . . . . . . . . . . . 580A.4.3. Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580A.4.4. Nuclear Physics for Use in Neutron Physics . . . . . . . . . . . . . . . . . . . . . 580A.4.5. Measurement of nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581A.4.6. Compilation and processing of nuclear data . . . . . . . . . . . . . . . . . . . . . 581A.4.7. Integral measurements and their use for the qualification

of nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581A.4.8. General approach to the qualification of nuclear data . . . . . . . . . . . 582A.4.9. Database of integral experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

A.5. Reactor kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583A.6. Slowing down, thermalisation and resonant absorption of neutrons . . . . . . . . 583

A.6.1. General information on resonant absorption . . . . . . . . . . . . . . . . . . . . 583A.6.2. The method of effective reaction rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 583A.6.3. The method of probability tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584A.6.4. Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584A.6.5. Validation and qualification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585A.6.6. Thermalisation of neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

A.7. Processing the neutron spectrum multigroup theory . . . . . . . . . . . . . . . . . . . . . 586A.8. Transport operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

A.8.1. General publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587A.8.2. A few references about the French approach . . . . . . . . . . . . . . . . . . . . 587

A.9. Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588A.9.1. General works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588A.9.2. Examples of Monte Carlo codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A.9.3. A few French developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

A.10. Equivalence, homogenisation and calculation of reflectors . . . . . . . . . . . . . . . . 590A.10.1. Homogenisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590A.10.2. Reflector calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

A.11. Neutron leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591A.12. Perturbation calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592A.13. Mathematical methods and numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 592A.14. Software development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

A.14.1. Calculation of first collision probabilities . . . . . . . . . . . . . . . . . . . . . . . . 594


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A.14.2. The APOLLO 1 and APOLLO 2 codes . . . . . . . . . . . . . . . . . . . . . . . . . . 594A.14.3. Simplified transport core calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 594A.14.4. Deterministic calculations for radiation protection . . . . . . . . . . . . . . . 595

A.15. Validation of software and calculation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 595A.16. Design and applied research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

Appendice B: Physical tables and constants

B.1. Table of the elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599B.2. Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

B.2.1. Units used in nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599B.2.2. Main physical constants used in neutron physics . . . . . . . . . . . . . . . . 599

B.3. Selection of nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601B.3.1. Data concerning the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601B.3.2. Data concerning nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604B.3.3. Energy released by fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

Appendice C: Mathematical supplement

C.1. Dirac distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608C.1.1. Intuitive definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608C.1.2. Mathematical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608C.1.3. Definitions as Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608C.1.4. Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

C.2. Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609C.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609C.2.2. Any linear operator is integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609C.2.3. Adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610C.2.4. Eigen elements of an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

C.3. Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612C.3.1. Translation-invariant operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612C.3.2. Definitions of the Fourier transform of a function . . . . . . . . . . . . . . . . 613C.3.3. Fourier transform of a convolution product . . . . . . . . . . . . . . . . . . . . . . 613C.3.4. One-dimensional examples (convention a) . . . . . . . . . . . . . . . . . . . . . . 614

C.3.5. Fourier transforms in two- or three-dimensional space . . . . . . . . . . . 614C.3.6. Fourier transforms of symmetric functions (convention a) . . . . . . . . 615C.3.7. Poisson summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615C.3.8. Eigenvalues of translation-invariant operators. . . . . . . . . . . . . . . . . . . . 616C.3.9. Linear operators on an infinite, regular lattice . . . . . . . . . . . . . . . . . . . 616

C.4. Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617C.4.1. Rotation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617C.4.2. Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619C.4.3. Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621C.4.4. Rotation-invariant operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

C.5. Eigenfunctions of the Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623C.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623C.5.2. Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623C.5.3. Factorised general solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624


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C.5.4. Eigenfunctions of the Laplace operator vanishing at the surfaceof a domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

C.6. Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

C.6.1. Bessel equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626C.6.2. Expansions at the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627C.6.3. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630C.6.4. Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630C.6.5. Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631C.6.6. Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632C.6.7. Jn generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632C.6.8. Representations by a definite integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 632C.6.9. Addition formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632C.6.10. Complete basis of functions defined in the interval [0,1] . . . . . . . . . 633C.6.11. Fourier transform in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633C.6.12. Half-order functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633C.6.13. A few numerical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

C.7. Streaming operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634C.7.1. Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634C.7.2. Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634C.7.3. Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

C.8. Peierls operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635C.8.1. Three-Dimensional Geometry (x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . 636

C.8.2. Two-Dimensional Geometry (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636C.8.3. One-dimensional geometry (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636C.9. Integral exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

C.9.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637C.9.2. Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637C.9.3. Expansions at the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637C.9.4. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638C.9.5. Fourier cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

C.10. Bickley-Naylor functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639C.11. Quadrature formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

C.11.1. General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640C.11.2. Constant-step formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641C.11.3. Gauss formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

Appendice D: Handbook

D.1. Units and constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647D.1.1. Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647D.1.2. A few physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

D.2. Nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648D.2.1. Characteristics of a (non-relativistic) particle . . . . . . . . . . . . . . . . . . . . 648D.2.2. Constitution of an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648D.2.3. Binding energy in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649D.2.4. Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649D.2.5. Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649


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D.2.6. Reactions involving neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649D.3. Neutron diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

D.3.1. Multiplication factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

D.3.2. Neutron paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650D.3.3. Neutron population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650D.3.4. Transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650D.3.5. Diffusion approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651D.3.6. One-group neutron theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

D.4. Neutron spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652D.4.1. Infinite multiplication factor of thermal neutron reactors . . . . . . . . . 652D.4.2. Elastic slowing down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652D.4.3. Maxwell spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

D.5. Reactor kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

D.5.1. Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653D.5.2. Supercritical situation with prompt neutrons . . . . . . . . . . . . . . . . . . . . 653D.5.3. Evolution in the neighbourhood of criticality . . . . . . . . . . . . . . . . . . . . 654

D.6. Reactor operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654D.6.1. Temperature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654D.6.2. Poisoning by fission products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655D.6.3. Evolution of heavy nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

Index and glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657


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Foreword

This guide to neutron physics is intended to provide a basic knowledge of this area ofscience to Nuclear Engineering students.

This book is laid out according to the syllabus of the Nuclear Engineering diploma astaught at Saclay, Cadarache and Cherbourg.

Its contents are partly culled from my previous publications in the field of neutronphysics and partly inspired by the documents and photocopied teaching aids used bymy fellow instructors. I would therefore like to thank all of them for these very helpfulmaterials.

I strongly advise anyone who has had no instruction in nuclear engineering to startby reading my little book Lnergie nuclaire (Que sais-je ? No. 317, PUF, 2006), whichintroduces the basic concepts of energy and the structure of matter, and gives a descriptionof nuclear power plants and an analysis of the issues involved with this type of energy.

I can also recommend another book in the same series: La Neutronique (PUF, Quesais-je ? No. 3307, 1998), which introduces the same concepts as the present Guide, butin a more accessible way.

Finally, I wish to mention the book I co-wrote with Jean Bussac: Trait de neutronique(Hermann, 1978 and 1985). Although it is not very recent, it can still be a useful referencefor anyone seeking further information or additional details about the physical aspects. Inthe current volume, I have not reviewed certain analytical theories that were used at thetime: the fast fission factor theory, the ABH (Amouyal-Benoist-Horowitz) theory, the Cadil-hac secondary thermalisation model, etc. but which have now fallen out of use. Onthe other hand, I have given more detail about the techniques for solving the Boltzmannequation, although I confine myself to a discussion of the main principles because thisbook is intended more for those who use calculation software than for the specialists whodevelop the software.

This guide to neutron physics is the English translation of my books published by EDPSciences in the Gnie atomique series, Prcis de neutronique (2003) and (partly) Exercicesde neutronique (2004).

For the French version of this guide and the associated exercises, we used the CGS sys-tem of units as almost all the neutron physicists did since the very beginnings of neutronics


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and are still doing in their books, articles, reports or communications. Nevertheless, whenan English translation was decided by INSTN, it seemed to us important to follow therecommandations of the Academies and Universities, and the usage in the other branches

of physics, for the SI (international system of units). So we converted all the numerical ex-amples and applications into this last system. The main parameters which are concernedare the lengths (m or mm instead of cm), the macroscopic cross-sections (m1 instead ofcm1) and the fluxes (m2 s1 instead of cm2 s1). The mass burn-ups must be expressedin J kg1 (or more conveniently in TJ kg1 [i.e. 1012 Jkg1]) if this system is used. Howeverwe kept also the values with the usual units (MWd/t or GWd/t) in order to avoid the readermentally converting.

Acknowledgements: I would like to offer my most sincere thanks to Nova traduction whotranslated the Prcis and to Christopher Latham who greatly improved my own translation

of the exercices, as well as ENEN and INSTN who participated to the financial backing.Many thanks also to EDP Sciences particularly France Citrini for agreeing to publishthis book and for making improvements to its presentation. Last but not least, I express allmy gratitude to Laurent Turpin and Joseph Safieh (INSTN) who let me have the possibilityto finalize this project.

P. R.Gif-sur-Yvette, February 2008.


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About the Author

Paul Reuss is an alumnus of the cole Polytechnique and has a doctorate in physicalsciences.

He has spent his entire career at the French Atomic Energy Commission (CEA) at Saclay

and Fontenay-aux-Roses, dividing his time between research and development, teaching,and training.

His research focuses on improving, validating, and qualifying computer programs usedby engineers to design and monitor nuclear reactor cores, most notably CORGRAF (nat-ural uranium and graphite reactors) and APOLLO (all reactor types, particularly waterreactors). He has participated in some important developments, such as the physical studyof plutonium recycling in water reactors (the topic of his doctoral thesis), the generalisationof the theory of resonant neutron absorption, and trend research, i.e. the use of neutronphysics measurements performed on critical experiments and power reactors to gain in-creased knowledge about nuclear data. He has over a hundred technical publications to

his name.After taking the DEA graduate degree in nuclear reactor physics, Paul Reuss soon be-

came a lecturer, and finally the professor in charge of this DEA. He has also taught manyother courses. He is currently the coordinator in charge of neutron physics teaching for theNuclear Engineering diploma. In addition to the present work (Neutron Physics: A Guide),he is the author of several texts on neutron physics and nuclear physics; most notably theco-author with Jean Bussac of Trait de neutronique, which is considered to be the keyreference text for neutron physics students and specialists.

Paul Reuss has supervised the thesis work of about twenty doctoral candidates, and has

been a member of over a hundred thesis committees. His other education-related activitieshave included two years in charge of training at the Nuclear Safety and Protection Institute(formerly IPSN, now the IRSN), as well as many seminars given at the CEA and at lectricitede France on neutron physics, neutron transport theory, and neutron absorption; he alsowrote very complete course notes for all of the above.


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Books by Paul Reuss:

Trait de neutronique, Hermann, 1978 and 1985, 670 pages (with Jean Bussac). lments de physique nuclaire lusage du neutronicien, in the Enseignement series,

CEA/INSTN, 1981, 1987 and 1995, 91 pages.

lments de neutronique, in the Enseignement series, CEA/INSTN, 1986 and 1995,175 pages.

Clefs pour la neutronique des racteurs eau, in the Enseignement series,CEA/INSTN, 1990, 348 pages.

Lnergie nuclaire, in the Que sais-je ? series, No. 317, PUF, 1994, 1999 and 2006,

128 pages. La Neutronique, in the Que sais-je ? series, No. 3307, PUF, 1998, 128 pages. Prcis de neutronique, Gnie atomique series, EDP Sciences, 2003, 533 pages. Exercices de neutronique, Gnie atomique series, EDP Sciences, 2004, 334 pages. Lpope de lnergie nuclaire, une histoire scientifique et industrielle , Gnie atomique

series, EDP Sciences, 2007, 167 pages.


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Part I

FUNDAMENTALS

OF NEUTRON PHYSICS


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1Introduction: general

facts about nuclearenergy

1.1. A brief history

1.1.1. Fermis pile

The date was December 2nd, 1942, and for the very first time, Man created a fission chainreaction. The credit for this achievement goes to a Chicago team led by Enrico Fermi(19011954). On that day, the neutron population scattering in the pile amplified verygradually, even after the source was withdrawn. When the nuclear power level reachedabout half a watt, the cautious Fermi ordered the insertion of the cadmium control rod tostop the divergence.

What a shame that nobody had thought to invite a photographer for the occasion. Theevent was immortalised by a table and a drawing, reproduced below (see Figure 1.1).They show that the critical condition (the configuration allowing the chain reaction to beself-sustaining) was reached when 400 tonnes of graphite, 6 tonnes of uranium metal and37 tonnes of uranium oxide were piled up1 in a carefully planned arrangement.

Some of the main principles later to be applied in all reactors, both research reactorsand power plants, were already used in Fermis pile.

1/ Monitoring and control, symbolised by the two operators at the bottom: on the left,the operator monitoring the detector display represents the monitoring function. On

the right, the operator in charge of the cadmium control rod represents the controlfunction. Cadmium is an efficient neutron-capturing material. When the rod ispushed in, the number of neutrons captured by the cadmium increases. This reducesthe number of neutrons causing fission in the uranium. The chain reaction is thenstifled. Conversely, if the rod is pulled out slightly, more neutrons become availableto cause fission reactions. The chain reaction is then amplified. To control the systemaccording to requirements, the monitoring and control functions must talk to eachother (in this case, simply a verbal dialogue between the two operators).

2/ Safety depends first and foremost on good monitoring and control. It also requires

an emergency stop mechanism in the event of an incident. In this experiment, theemergency stop function is provided by an unseen operator located above the pile.

1 This explains the origin of the term atomic pile, which we often use to refer to a nuclear reactor. It is now aslightly archaic term.


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Figure 1.1. Fermis pile (courtesy of Argonne National Laboratory).

This person is armed with an axe, and on Fermis signal can cut the rope holdingan emergency cadmium control rod. The last line of defense consisted in a tank of

cadmium salt solution to release the solution into the pile.

3/ Radiation shielding is provided in this case by a detector hanging in front of the pileto measure the ambient radiation level. The signal passes through the cable run-ning along the ceiling to a display placed in view of Fermi himself, on the balcony.Fermi can thereby ensure that he and his colleagues do not run the risk of excessiveirradiation and can trigger the emergency stop if necessary.

1.1.2. The end of a long search...

The divergence of Fermis pile concluded half a century of very active research in nuclearphysics.

Nuclear physics is generally considered to have begun in 1896 with the discovery,almost by chance, of radioactivity by Henri Becquerel (18521908). Becquerel was in-trigued by some photographic plates that were clouded even though they had been keptin a drawer away from sunlight. (It turned out that they had been placed in the vicinity ofsome uranium samples.)

A brief chronology with a few milestones:1898: Discovery of polonium and radium by Marie Sklodowska (18671934) and her

husband Pierre Curie (18591906).1913: First atomic models developed by Ernest Rutherford (18711937) and Niels Bohr(18851962).

The very concept of the atom had already pervaded physics and chemistry for manyyears. It was thought of by Democritus (ca. 460-370 B.C.), and appears very clearly in the


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1 - Introduction: general facts about nuclear energy 5

work of such chemists as John Dalton (17661844) and Louis Joseph Gay-Lussac (17781850). It is the only model that makes the periodic table of the elements suggested in 1869by Dmitri Ivanovich Mendeleev (18341907) comprehensible.

1932: Discovery of the neutron by James Chadwick (18911974).1934: Artificial radioactivity discovered by Frederic Joliot (19001958) and his wife

Irene Curie (18971956): by the action of alpha particles on aluminium 27 (commonaluminium), a neutron and a phosphorus 30 are produced. The phosphorus takes two anda half minutes to disintegrate by beta radioactivity, as this experiment revealed.

19341938: Study of neutron-induced reactions.As soon as it was known how to create neutron radiation, particularly following the

work of Chadwick, nuclear physicists became interested in the reactions between theseparticles and the various elements in Mendeleevs table. Because the neutron has noelectrical charge, it can easily approach the nuclei of atoms, and experimental evidence

shows that neutrons are quite often captured. This process creates an isotope of the initialnucleus, which is sometimes radioactive and is transformed into another element by betadecay (these radioactive processes will be described in greater detail in the next chapter).Fermi in particular was interested in these reactions. He thought that by bombardinguranium element number 92, the last one in Mendeleevs table he should be able tocreate new, artificial elements and extend the list of known elements.

Experiments did indeed show that the reaction gave off radioactive products, but therewere clearly more of them than expected.

1938: Discovery of fission.Fermis experiment was repeated in other laboratories. It took four years for a cor-

rect explanation of the phenomenon to be found. When Otto Hahn (18791968) andFritz Strassmann discovered that barium was present among the reaction products, andbecause barium was an element with an intermediate mass, they concluded that the ura-nium nucleus had split in two after absorbing the neutron. The discovery of fission wasannounced by Lise Meitner (18781968), who calculated that a considerable quantity ofenergy must be released when this splitting occurs, which lends credence to the theory.

1939: Patents for an energy production device.As soon as the discovery of fission was announced, the experiment was repeated in

other laboratories, and obviously there was a great deal of activity in the field of nuclear

physics in the 1930s. In particular, Joliot and his colleagues Hans von Halban and LewKowarski (19071979) discovered the emission of secondary neutrons during fission, andmeasured the average number of secondary neutrons to be approximately three (whichturned out to be slightly optimistic, as subsequent measurements would show). Theyquickly realised that this should make a self-sustaining chain reaction possible, becauseeach fission would release neutrons that would in turn induce new fission reactions.

Along with their colleague Francis Perrin (19011992), who introduced the conceptof critical mass, they designed and submitted patent applications for a device that wouldlater be known as a nuclear reactor. The research team was disbanded within a few weeksof the German invasion, and these patents remained secret throughout the War.

Research continued on a very active basis in Great Britain, Canada, and the USA.Many feared that Nazi Germany might gain a decisive advantage by developing an atomicweapon2. This fear was most notably expressed in the letter sent by Albert Einstein

2 This fear turned out to be unfounded.


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(18791955) on the initiative of Leo Szilard (18981964) and Eugene Wigner (19021995)to President Roosevelt on 2 August 1939, and led the United States to undertake the gi-gantic Manhattan Project.

1945: Hiroshima and Nagasaki.This project, led by Robert Oppenheimer (19041967), explored the two possible

routes (which will be discussed later): uranium 235 and plutonium 239. Plutonium iselement 94, which Fermi was seeking but which was finally discovered by Glenn Seaborg(19121999) in 1940. The plutonium route led to the Trinity test at Alamogordo (NewMexico) on 16 July 1945, and then the bomb that was dropped on Nagasaki (9 Au-gust 1945). The uranium 235 route led to the bomb that was dropped on Hiroshima(6 August 1945).

1.1.3. ... and the beginning of a great adventure

There is no doubt that this original sin left an impression that affected public opinionabout the post-war developments in the field of nuclear energy. This was all the more truebecause the surrender of Japan did not spell the end of the arms race; on the contrary. Theparticipants in this race were the United States and some of the other developed countries,particularly the USSR. It only took a few years (1952 and 1953, respectively) for these twosuperpowers to develop an even more terrifying weapon: the fusion H-bomb.

Let us concentrate, however, on peaceful developments in the field of nuclear energy.

As soon as the war ended, most of the large industrialised nations took an interest in thisnew energy and, in some cases, undertook very ambitious programmes.Nuclear electricity was first produced in 1951 in Chicago in a small fast neutron reactor

called EBR-1.In France, the CEA (Atomic Energy Commission) was created by a decree signed by

Charles de Gaulle on 18 October 1945. Just three years later (15 December 1948), Frenchatomic scientists made the Zoe pile diverge (natural uranium and heavy water) at FortChatillon at Fontenay-aux-Roses (Hauts-de-Seine). After Fontenay, three other major nu-clear research centres were created: Saclay (Essonne), Grenoble (Isre), and Cadarache(Bouches-du-Rhne), as well as several centres for military applications. After Zoe, Francedeveloped the UNGG procedure (natural uranium, graphite, gas) with prototypes G1, G2,and G3, followed by six high-power reactors which have now been shut down. France wasalso interested in the heavy water procedure (Bre

Reuss, P. - Neutron Physics (EDP Sciences, 2008)

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