The Physics of Nuclear Reactors

Serge Marguet The Physics of Nuclear Reactors The Physics of Nuclear Reactors Serge Marguet The Physics of Nuclear Reactors Serge Marguet EDF-R&D - PERICLES Palaiseau, France ISBN 978-3-319-59559-7 ISBN 978-3-319-59560-3 DOI 10.1007/978-3-319-59560-3 (eBook) Library of Congress Control Number: 2017942959 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my parents Josette and Daniel who supported my studies. To my wife Agnès who continues to support me. . . ! To my children Hélène and Vincent, who excite my immune system! Foreword to the 2011 Edition In the context of a worldwide economic crisis, what better indication could there be than the publication of this textbook on reactor physics? Renunciation of the policy of adding further capacity to electrical production facilities in Europe, the immense energy requirements of countries such as China, India, and Brazil, and the growing international awareness that energy is a rare and expensive commodity are so many factors militating in favor of the proper use of the means of electricity production. Furthermore, the avowed official desire to combat climate change and reduce energy costs means that renewable energy and nuclear production are priorities. Nuclear energy is once again making vigorous strides forwards, as attested in France in particular by the commissioning of the EPR in 2012, 80 years after discovery of the neutron by James Chadwick in 1932. The brief history of civil nuclear power shows that it is not possible to harness this energy in the long run without flawless safety levels, in all places and at all times. Experts distinguish three safety functions: confinement of radioactive products, reliable removal of decay heat, and complete control over reactivity. To ensure that these conditions are met, the branches of neutron physics, fluid mechanics, thermal physics, materials physics, and chemistry have an essential role to play. Neutron physics is the science that describes and explains the behavior of neutrons in matter and the reactions they induce. In order to guarantee complete control of reactivity, a solid knowledge of neutron physics is indispensable in order to be able to define the proper measures to be taken, beginning with the reactor design stage and then throughout decades of operation. Achieving complete understanding of the complex physical phenomena that occur in a nuclear installation and displayed to the operation engineers via monitors and computers in the control rooms is of cardinal importance for the optimal operation of power reactors (440 in service in 2009, a number that will doubtless rise two- or threefold by 2030 or 2040), of experimental reactors for deepening our level of understanding, and of laboratories and facilities for fuel cycle studies that are set to increase worldwide. vii viii Foreword to the 2011 Edition This textbook addresses all aspects of neutron physics: experts, engineers, and students will find between its covers a host of scientific references that will enable them to acquire, maintain, and improve their skills. It is my hope that it will be used by the large community of engineers working in the service of peaceful use of nuclear energy, a lasting energy form, for the greater good of mankind. World Association of Nuclear Operators (WANO) Paris, France Laurent Stricker Chairman of Wano Foreword to the 2017 Edition The supply of clean, affordable, and reliable energy is a global challenge. The projected increase in populations, particularly in Africa and Asia, means that by 2035 global energy needs are predicted to increase by 50% over 2015 levels. The increasing evidence of man-made climate change has given greater prominence for low carbon technologies such as renewables, nuclear, and carbon capture and storage which should be developed and deployed widely. The Paris Agreement on climate change (2016) has shown governments’ resolve to reduce the world’s greenhouse gas emissions by accelerating the deployment of such technologies. Against this background, nuclear energy is a critically important part of the global energy mix. Today, nuclear power accounts for over 10% of the world’s electricity generation; over 440 reactors are operating around the world, delivering clean electricity to the grid. In 2017, there are more reactors being built than at any time during the previous 25 years. Sixty reactors are currently under construction in fourteen countries, including more than one third of these in China and others in countries which are new to nuclear, such as the United Arab Emirates. However in 2011, the Tōhoku earthquake and tsunami led to the nuclear accident at the Fukushima Daiichi plant which has, alongside previous nuclear accidents, emphasized the importance of nuclear safety and three crucial safety functions: confinement of radioactive products, reliable removal of decay heat, and complete control over reactivity. To ensure that these functions are delivered effectively, the branches of neutron physics, fluid dynamics, reactor physics, materials science, mechanical engineering, and chemistry/corrosion each have an essential role to play. Each of these technical fields is important in its own right as are the synergies between them. However, peculiar to nuclear science and engineering is the understanding of neutrons in materials and the reactions they induce. In order to ensure the control of the reactivity in operating reactor cores, a full understanding of neutron physics is required. Achieving a full understanding of the complex physical mechanisms that occur in a nuclear power reactor ensures both good design and safe operations. ix x Foreword to the 2017 Edition This textbook addresses all aspects of neutron physics creating a body of scientific knowledge and supporting references that will be invaluable to those learning about and responsible for nuclear reactor systems. I commend this textbook to students, engineers, experts, and reactor operators as a means to learn and maintain an up-to-date knowledge of such an important field of nuclear science. National Nuclear Laboratory London, UK Andrew H. Sherry Chief Scientist of NLL Acknowledgements This book would not have been possible without the friendly assistance of Paul Reuss: his advice was precious and his proofreading extremely scrupulous. If neutron physics is a kingdom, then Paul is undoubtedly one of its princes since his technical expertise is equaled only by his modesty, two areas that I should certainly work on personally! My first contact with Paul dates back to my very first day at EDF in a sense, namely September 1, 1987. My director had left two things for me on my empty desk: the output listing of the COCCINELLE code and the renowned neutron physics treatise by Jean Bussac and Paul Reuss, the bilious green 1985 edition that everyone in the department referred to simply as “Reuss”, in spite of Bussac’s co-authorship. On asking my colleagues about some physics question or other for the third time, I was told “It’s in Reuss”, and I realized that I might have to go through this thick textbook thoroughly. I do not feel that I wasted my time in so doing! Later on, I was lucky enough to attend Paul’s lectures as part of the Reactor Physics Master’s at Saclay. I always marveled at his crystal clear explanations. I also wish to thank my colleague and reactor physics expert, Michel Lam-Hime, for giving up (an enormous amount of) his personal time to proofread this book. My thoughts also go out to a few colleagues and close friends: Philippe Tétart, Patrick Erhard, and David Couyras, who were the stoic guinea pigs on whom I tested my craziest ideas, sometimes very late in the evening. Their judicious comments and remarks resulted in numerous improvements. Finally, I would like to thank JeanMichel Delbecq as well, my ex-Head of Department and member of the editorial committee at EDF, who believed in this project from the outset, Laurent Stricker, who did me the honor of writing the foreword to the 2011 edition, and Professor Andrew Sherry for the foreword to this edition. At least, I would like to thank Electricité de France for the funding of the English translation, especially Bertrand Bouriquet for his enthusiastic support to this project. It would be a shame to forget Ansar Calloo (EDF/R&D), whose translation is far beyond my poor level in English and Patrick Saunders (Novatrad) for his very xi xii Acknowledgements professional proofreading. Finally, the translation of the French manuscript was taken in good hands by Novatrad company. Finally, SPI-global (Saravanan and team!) made an amazing job increasing the global quality of this English version. Introduction Reactor physics is a young branch of science. It is widely held to have been born on December 2, 1942, in Chicago, with the news that “the Italian navigator has just landed in the New World”, a coded sentence informing all authorized persons that the Italian, Enrico Fermi, had succeeded in his prodigious feat of diverging a uranium and graphite pile. However, very soon after this incredible feat of which the general public was largely unaware, this young science struck terror into the world with the explosion of two US atomic bombs over the Japanese cities of Hiroshima and Nagasaki, on August 6 and 9, 1945. The Land of the Rising Sun was brought to its knees by nuclear fire. Humanity was led wincing into the atomic age, filled with a mixture of fear and fascination. Since then, public interest has never waned thanks to tireless efforts of popularization. The atomic age ushered in by scientists held out hope of unlimited energy, possibly free (according to the publications of the time), and to the end of war. Subsequent events unfortunately showed this dream to be an unrealizable utopia. The short supply on Earth of fissile material is already of great concern, and as for warfare. . . clearly when one war stops, another breaks out elsewhere on the planet. Reactor physics flourished in the 1950s and 1960s. The construction of large power reactors in the 1970s and 1980s led to intensive development of this technology, yielding sustainable technologies such as reactors running on natural uranium cooled with carbon gas and moderated by graphite, and pressurized water reactors, which form the backbone of the French reactor fleet. At the same time, the underlying physics became more stable: nuclear data were increasingly abundant and filled large databases, and the notation specific to the field of neutron physics became established. The knowledge of physicists was sustained through large-scale calculation codes, while generations of numerical physicists improved these codes thanks to breakthroughs in ever more elaborate and complex numerical methods. xiii xiv Introduction Popularization of the atomic age reached its peak in the 1950s (here we see a special edition of the French science magazine, “Science et Vie” [Science and Life] published in 1958, the Marguet collection, courtesy Science et Vie) In 1979, the accident at the Three Mile Island 2 reactor in the USA shook the scientific community; the reactor core was totally destroyed, despite such an accident being deemed impossible on account of all the precautionary measures taken. As a result of the almost complete lack of release of radioactive matter into the environment, awareness among the general public of the inherent dangers of the accident did not peak, but it created massive unease among scientists. Murphy’s Law had been confirmed once more. However, in 1986, the Chernobyl accident in Ukraine, in which most of the core was expelled into the environment, contaminating significant areas and with far-reaching effects throughout Europe, created a terrible shock. Civil nuclear energy programs were now perceived in the public mind as a huge threat to humanity, and hostility towards the nuclear technocrats gathered pace. In France, the combined action of the media and of the ecological parties, coupled with public defiance, led to decommissioning of the SuperPhénix fast-neutron reactor. The nuclear age stooped from splendor to misery, as the question of disposal of long-lasting nuclear waste materials scared the public, and scientists toiled without success to find a viable alternative to deep burial. By the end of the 1990s, as students showed disaffection with the so-called hard sciences, the popularity of nuclear engineering had sunk to a new low, with the Introduction xv Master’s in Reactor Physics almost being phased out in the mid-1990s owing to the lack of students and the poor prospects for renewal of the reactor fleet. However, in the 2000s, scientists showed beyond doubt that global warming was due to human activities and to the release of greenhouse gases into the atmosphere. Indeed, the oil crisis accompanying the wane of that particular fuel (total depletion within 40 years was being touted in 2008!) heralded a revival of the fortunes of nuclear energy; the latter does not produce greenhouse gases and it is assumed that the requisite natural resources will last ten times longer than oil, and plutonium could well prove to be the wildcard to replace uranium 235, which may run out before the advent of fusion. After a period of dwindling in human resources and loss of expertise over the years (in 2008, the French situation in terms of expertise in fast neutron reactors which had to be set up from scratch after the retirement of the SuperPhénix generation, is characteristic, and even perhaps a caricature of this state of affairs), know-how regarding reactor physics lay buried in the extensive computational codes. For this reason, it seemed to me timely to write this textbook amid the resurgence of interest in nuclear engineering to ensure renewal of the international reactor fleet, along with increasing energy demand. This textbook is thus addressed to students in higher studies, engineering students in nuclear energy and engineering, and engineers and research scientists at large who wish to review the founding notions of their professions. It is the culmination of 15 years of lectures given at the “Ecole Nationale Supérieure d’Ingénieurs” in Bourges (France), where I was able to observe (with great pleasure) the renewed interest of students in this particular field. I wanted this textbook to be both educational in terms of its content and convivial through its illustrations, and I hope that it will provide answers for beginners and knowledgeable readers alike. The textbook first sets out the minimum knowledge in nuclear physics required for an understanding of more advanced concepts. The subject itself has become a separate branch of science. It then examines neutron physics, which describes the intrinsic behavior of neutrons in matter, and then reactor physics, which is the art of making a pile critical in order to produce heat. The thermal hydraulics of the coolant material and the thermal physics of the nuclear fuel that are often associated with reactor physics will be explored in a separate textbook entitled “Physique des accidents dans les réacteurs nucléaires” [Physics of accidents in nuclear reactors]; these very important subject areas are too vast to be presented in the present volume. Indeed, as part of the generation that grew up with water reactors and because of my experience in this area, I have focused chiefly on Pressurized Water Reactors, which have flourished during my career at Électricité De France. Throughout the various chapters, I have done my utmost to review the history of this young science that is currently enjoying a revival. Contents Part I Neutronics 1 Fundamentals of Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Chemical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Avogadro’s Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Mass-Energy Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 The Electron Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 The Atomic Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Nuclear Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.1 Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.2 β Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.3 β+ Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.4 Electron Capture . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.5 γ Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.6 Internal Conversion . . . . . . . . . . . . . . . . . . . . . . . 1.13.7 (β ,n) Decay or Neutron Decay . . . . . . . . . . . . . . 1.13.8 Spontaneous Fission . . . . . . . . . . . . . . . . . . . . . . 1.14 Radioactive Decay Branches . . . . . . . . . . . . . . . . . . . . . . . 1.15 Heavy Nucleus Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 7 9 12 14 19 22 25 28 28 40 50 51 61 68 71 72 73 74 75 76 76 82 2 Interaction Between Neutrons and Matter . . . . . . . . . . . . . . . . . 2.1 Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Elastic Scattering on a Fixed Target . . . . . . . . . . . 2.1.2 Elastic Scattering on a Moving Target . . . . . . . . . . . . . 89 89 90 97 xvii xviii Contents 2.1.3 Moderator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . Transmutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 (n,γ) Neutron Capture or Radiative Capture . . . . . 2.2.3 (n,α) Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Other Forms of Capture . . . . . . . . . . . . . . . . . . . . 2.2.5 High-Energy Reactions . . . . . . . . . . . . . . . . . . . . 2.2.6 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Measurement of Cross Sections . . . . . . . . . . . . . . 2.5.3 Notion of Flux and Reaction Rate . . . . . . . . . . . . 2.5.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Fission Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Spontaneous Fission . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Neutrons Produced by Fission . . . . . . . . . . . . . . . 2.6.4 Prompt Fission Photons . . . . . . . . . . . . . . . . . . . . 2.6.5 Delayed Fission Neutrons . . . . . . . . . . . . . . . . . . Fission Products Resulting from Fission . . . . . . . . . . . . . . . 2.7.1 Direct Yield of an Isotope . . . . . . . . . . . . . . . . . . 2.7.2 Total Chain Yield . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Cumulative Yield of an Isotope . . . . . . . . . . . . . . 2.7.4 Slowing Down of Fission Products in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 100 103 104 105 106 106 107 107 110 110 111 111 113 114 116 128 131 133 134 142 143 147 149 150 151 151 Interaction of Electromagnetic Radiation and Charged Particles with Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 X-radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Interaction of Photons with Matter . . . . . . . . . . . . . . . . . . . 3.3.1 Attenuation of a Photon Beam . . . . . . . . . . . . . . . 3.3.2 Photon Transport . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Rayleigh-Thomson Scattering . . . . . . . . . . . . . . . 3.3.4 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Cumulative Effects . . . . . . . . . . . . . . . . . . . . . . . 3.3.8 Scattered Radiation and Build-Up Factors . . . . . . 3.3.9 Application of Photon Attenuation in Matter . . . . 3.3.10 Photoneutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.11 Photofission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Measuring Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 153 154 157 158 160 161 161 166 171 174 174 177 181 182 182 2.2 2.3 2.4 2.5 2.6 2.7 3 Contents 3.5 . . . . . . . . . . . . . . 184 186 187 189 189 190 190 193 198 204 205 205 206 207 Neutron Slowing-Down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Continuous-Energy Slowing-Down Theory . . . . . . . . . . . . . . 4.2.1 Elastic Collision with a Stationary Target . . . . . . . . 4.2.2 Collision Statistics . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Effect of the Motion of the Target Nucleus . . . . . . . 4.2.4 Transfer Probability as a Function of Angle . . . . . . 4.2.5 Isotropic Collision . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Continuous Slowing-Down Theory . . . . . . . . . . . . . . . . . . . . 4.3.1 Slowing Down by Non-Absorbing Hydrogen . . . . . 4.3.2 Taking into Account Absorption by Hydrogen . . . . 4.3.3 Taking Account of a Spectral Source . . . . . . . . . . . 4.3.4 Slowing Down by Targets Heavier Than Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Influence of the Fast Fission Spectrum . . . . . . . . . . 4.3.6 Mixture of Moderators . . . . . . . . . . . . . . . . . . . . . 4.4 Slowing Down in an Absorbing Medium . . . . . . . . . . . . . . . 4.4.1 Slowly Varying Absorption: The Greuling-Goertzel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Slowing Down in a Medium with a Resonant Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Inelastic Slowing-Down . . . . . . . . . . . . . . . . . . . . 4.4.4 The Qn Slowing-Down Approximation . . . . . . . . . . 211 211 214 215 223 226 227 230 231 237 246 247 268 271 275 Resonant Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cross Section Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . 5.1.2 Intermediate Nucleus Theory . . . . . . . . . . . . . . . . 5.1.3 Principle of Reciprocity . . . . . . . . . . . . . . . . . . . 281 281 281 282 285 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4 5 xix Interaction of Electrons with Matter . . . . . . . . . . . . . . . . . . 3.5.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Wilson Chamber . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Braking Radiation or Bremsstrahlung . . . . . . . . . . 3.5.5 Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cherenkov-Mallet Effect . . . . . . . . . . . . . . . . . . . . . . . . . . Charged Particles: Rutherford Diffusion . . . . . . . . . . . . . . . Transfer of Energy to Matter . . . . . . . . . . . . . . . . . . . . . . . Ion-Electron Pair Production by Ionization . . . . . . . . . . . . . Variation in Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Path Length in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biological Effects of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 248 256 259 260 265 xx Contents 5.2 Single-Level Breit-Wigner Formalism . . . . . . . . . . . . . . . . . 5.2.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Scattering Cross Section . . . . . . . . . . . . . . . . . . . . 5.2.3 Radiative Capture Cross Section . . . . . . . . . . . . . . 5.2.4 Fission Cross Section . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Absorption Cross Section . . . . . . . . . . . . . . . . . . . 5.2.6 Negative Resonances . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Distribution of Resonances . . . . . . . . . . . . . . . . . . 5.2.8 Resonant Absorption . . . . . . . . . . . . . . . . . . . . . . . Self-Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slowing-Down Through Resonances . . . . . . . . . . . . . . . . . . The Livolant-Jeanpierre Formalism . . . . . . . . . . . . . . . . . . . 5.5.1 Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . 5.5.2 Fine Structure Equation . . . . . . . . . . . . . . . . . . . . . 5.5.3 Tabulating Effective Cross Sections . . . . . . . . . . . . Modeling the Slowing-Down Operator Using the Resonant Isotope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Narrow Resonance Approximation . . . . . . . . . . . . . 5.6.2 Wide Resonance Approximation . . . . . . . . . . . . . . 5.6.3 Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 All Resonance Model (TR) . . . . . . . . . . . . . . . . . . Heterogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Two-Media Problem . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Accounting for Spatial Interaction . . . . . . . . . . . . . 5.7.3 Generalization to Several Self-Shielding Regions . . . . Accounting for Energy Interactions: Self-Shielding of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate Resonance Model in Flux Calculations . . . . . . . The Probability Table Method . . . . . . . . . . . . . . . . . . . . . . . 286 287 288 288 290 290 291 291 294 295 298 301 301 304 306 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 An Intuitive Analysis of the Doppler Effect . . . . . . . . . . . . . 6.2 Effective Interaction Cross Section with “Hot” Matter . . . . . . 6.2.1 Distribution of the Target Nuclei Velocities in Matter: The Free Gas Model . . . . . . . . . . . . . . . 6.2.2 Definition of the Effective Cross Section . . . . . . . . 6.2.3 Cross Section Inversely Proportional to Velocity . . . . 6.2.4 Constant Cross Section . . . . . . . . . . . . . . . . . . . . . 6.3 Generalized Doppler Broadening: Bethe-Placzek Formula . . . 6.4 Doppler Broadening of a Breit-Wigner Cross Section . . . . . . 6.4.1 Overview of the Breit-Wigner Formalism . . . . . . . . 6.4.2 Voigt’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Interference Function . . . . . . . . . . . . . . . . . . . . . . 6.5 Application to the Large Resonance of Uranium 238 . . . . . . . 333 333 334 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 6 308 308 309 310 311 313 313 317 320 322 323 326 335 336 337 337 341 345 345 347 353 354 Contents 6.6 Temperature Effect on Cross Sections . . . . . . . . . . . . . . . . . 6.6.1 First Voigt Function ψ . . . . . . . . . . . . . . . . . . . . . 6.6.2 Interference Function . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Asymptotic Numeric Evaluation . . . . . . . . . . . . . . 6.6.4 Derivatives of the Voigt Functions with Respect to Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.5 Some Mathematical Properties of Voigt Profiles . . . Effective Resonance Integral . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . 6.7.2 Heterogeneous Medium . . . . . . . . . . . . . . . . . . . . . 6.7.3 Analytical Calculation of a Broadened Resonance: The Campos-Martinez Model . . . . . . . . . . . . . . . . Effective Doppler Temperature . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Lattice Bonding Effects . . . . . . . . . . . . . . . . . . . . . 6.8.2 Heterogeneity Effects of the Temperature Field . . . . 356 357 358 359 Thermalization of Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Boltzmann Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Application to Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Neutron Flux Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Neutron Thermalization Equation . . . . . . . . . . . . . . . . . . . . . 7.6 Wigner-Wilkins Model: Free Proton Gas . . . . . . . . . . . . . . . 7.7 Asymptotic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Simplified Solution to Thermalization with Absorption . . . . . 7.9 Horowitz-Tretiakoff Model . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Case of Absorption Inversely Proportional to Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.3 Case of a Finite Reactor (with Leakage) . . . . . . . . . 7.9.4 Thermalization Equation for a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Heavy Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Cadilhac, Horowitz and Soulé Differential Model . . . . . . . . . 7.12 Application of the Cadilhac Model to Heterogeneous Media . . . . 7.13 Graphical Representation of Flux over the Energy Spectrum . . . . 7.14 True Moderators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15 Heating and Cooling by Scattering . . . . . . . . . . . . . . . . . . . . 7.16 Thermalized Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16.1 Calculation of Reaction Rate in a Pure Thermal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16.2 Definition of the Westcott Coefficient g(T) . . . . . . . 7.17 Calculation of the Reaction Rate in a True Thermal Spectrum . . . 387 387 388 392 395 397 401 404 408 412 412 6.7 6.8 7 xxi 362 363 364 364 367 374 378 378 380 419 419 420 421 422 426 431 432 434 437 440 441 446 xxii Contents 7.17.1 . 449 . . . . . . 453 456 457 458 460 461 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Setting Up the Boltzmann Equation . . . . . . . . . . . . . . . . . . . 8.1.1 Concept of Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Integro-Differential Transport Equation . . . . . . . . . . . . . 8.2.1 The Integro-Differential Transport Equation in Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Integro-Differential Equation in Steady-State . . . 8.3 Integral Form of the Boltzmann Equation . . . . . . . . . . . . . . . 8.3.1 Peierls Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Volume Integral Form . . . . . . . . . . . . . . . . . . 8.3.3 The First Collision Probability . . . . . . . . . . . . . . . . 8.3.4 1D Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Escape Probabilities . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 The Integral Equation in 2D . . . . . . . . . . . . . . . . . 8.3.7 Application to an Infinite Medium with a Fission Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.8 Graphical Solution to the Dispersion Equation . . . . 8.4 Third Form of the Transport Equation: the SurfaceIntegral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Placzek’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Flux Equation at the Interface . . . . . . . . . . . . . . . . 8.4.3 Application to the Milne Problem . . . . . . . . . . . . . 8.4.4 Second Complementarity Theorem . . . . . . . . . . . . 8.5 Concept of Characteristic Function . . . . . . . . . . . . . . . . . . . . 8.6 Fourier Transform of the Boltzmann Equation . . . . . . . . . . . 8.6.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Resolution Using Green’s Function . . . . . . . . . . . . 8.7 The 1D Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 General Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Lafore and Millot Method, Case Method . . . . . . . . 8.7.3 Perovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Asymptotic Solution for Diffusion . . . . . . . . . . . . . . . . . . . . 8.8.1 Exponential Relaxation of the Flux, Far from the Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 465 468 474 7.18 8 Westcott Formalism: Introduction of the Coefficients r and s . . . . . . . . . . . . . . . . . . . . . . . 7.17.2 Extension of the Model to Other Nuclides: The Linear Logarithmic Model . . . . . . . . . . . . . . 7.17.3 Progressive Junction at Epithermal Energy . . . . . . 7.17.4 Westcott Junction . . . . . . . . . . . . . . . . . . . . . . . . 7.17.5 Determination of Cut-Off Function . . . . . . . . . . . 7.17.6 Limits of the Westcott Formalism . . . . . . . . . . . . Application of the Westcott Formalism . . . . . . . . . . . . . . . . 474 475 507 507 510 512 522 524 538 539 540 543 544 546 547 548 549 553 553 555 559 559 562 571 572 572 Contents xxiii 8.8.2 8.9 9 Finding the Dispersion Equation from the Asymptotic Flux . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Critical Absorption Limiting the Asymptotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.4 Definition of a Diffusion Coefficient from the Transport Equation . . . . . . . . . . . . . . . . . . . . . . . The 3D Transport Equations . . . . . . . . . . . . . . . . . . . . . . . Computational Neutron Transport Methods . . . . . . . . . . . . . . . . 9.1 Discrete Ordinates Method Sn . . . . . . . . . . . . . . . . . . . . . . 9.2 Exact Sn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Legendre Polynomial Method . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Theory and Application to 1D Transport . . . . . . . 9.3.2 Multi-group 1D Transport and Diffusion Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 SPn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Interfaces Between Different Media . . . . . . . . . . . . . . . . . . 9.6 Spherical Harmonics Method . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 P1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Milne Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 DPn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Semi-infinite Plane: Albedo Problem . . . . . . . . . . . . . . . . . 9.9.1 Fundamentals of Discrete Eigenfunctions . . . . . . . 9.9.2 Ganapol Method by Laplace Transform . . . . . . . . 9.10 Bn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Tn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.12 Fn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.13 Cn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14 The SKn Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.15 Method of Characteristics (MOC) . . . . . . . . . . . . . . . . . . . . 9.15.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.15.2 Heterogeneous Geometries . . . . . . . . . . . . . . . . . 9.15.3 Characteristic Direction Probabilities (CDP) . . . . . 9.16 Even–Odd Formulation of the Transport Equation . . . . . . . . 9.16.1 Even–Odd Flux Equation . . . . . . . . . . . . . . . . . . 9.16.2 Variational Nodal Method of the Even–Odd Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.16.3 Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.17 Variational Method for Time-Dependent Problems . . . . . . . 9.18 Gauss-Seidel Method for Sources in Time-Dependent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.19 Probabilistic Approach: The Monte Carlo Method . . . . . . . . 9.19.1 Fundamental Concepts of the Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 . 582 . . 584 589 . . . . . 593 593 601 604 604 . . . . . . . . . . . . . . . . . . . . . . 619 623 628 630 630 638 640 643 646 646 652 657 667 670 670 675 677 677 679 684 686 687 . . . 691 694 697 . . 699 700 . 700 xxiv Contents 9.19.2 9.19.3 9.19.4 9.19.5 9.19.6 9.19.7 9.19.8 9.19.9 Application to Neutron Transport: A Simple 2D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Error . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Physical Quantities . . . . . . . . . . . . . Generalization, Biasing . . . . . . . . . . . . . . . . . . . . . Resonance Escape Probability Factor Calculation . . . . Midway Monte Carlo . . . . . . . . . . . . . . . . . . . . . . Quasi-Deterministic Approximation of the Importance Function . . . . . . . . . . . . . . . . . . Example of a Monte Carlo Calculation . . . . . . . . . . 705 713 713 714 716 719 723 726 Contents for Volume 2 Part II 10 Reactor Physics Diffusion Approximation in Neutron Physics . . . . . . . . . . . . . . . . 10.1 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Evaluation of the Neutron Diffusion Coefficient . . . 10.1.2 Discussion of the Hypotheses . . . . . . . . . . . . . . . . 10.1.3 The Diffusion Equation in a Force Field . . . . . . . . . 10.2 Boundary Conditions for a Medium Surrounded by a Vacuum in Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 P1 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Rulko’s Variational Approach . . . . . . . . . . . . . . . . 10.3 Boundary Conditions Between Any Two Media . . . . . . . . . . 10.3.1 Notion of a Reflector Albedo . . . . . . . . . . . . . . . . . 10.4 Diffusion Equation in Energy . . . . . . . . . . . . . . . . . . . . . . . . 10.5 One-Group Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 10.6 “Thermal Diffusion” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 “Thermal” Diffusion Equation . . . . . . . . . . . . . . . . 10.6.2 Interpretation of the Thermal Scattering Path . . . . . 10.6.3 Deriving the Four-Factor Formula . . . . . . . . . . . . . 10.7 Scattering of an Isotropic Source in a Non-Multiplying Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Point Source in an Infinite Scattering Medium . . . . 10.7.2 Anisotropic Point Source in Spherical Geometry . . . 10.7.3 Infinite Thin Rod Source in an Infinite Scattering Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.4 Infinite Plane Source in an Infinite Scattering Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.5 Infinite Plane Source in an Infinite Scattering Slab . . . 10.7.6 Uniform Source in an Infinite Scattering Slab . . . . . 10.7.7 Semi-infinite Slab Source . . . . . . . . . . . . . . . . . . . 731 731 731 736 741 743 744 745 749 750 751 753 755 755 757 759 759 760 763 769 771 773 775 776 xxv xxvi Contents for Volume 2 10.7.8 10.7.9 . 778 . . . . . 779 780 782 783 784 . . . . . . . . . 787 791 797 802 802 803 804 806 808 Nuclear Reactor Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Multiplication Factor of a Chain Reaction . . . . . . . . . . . . . . . 11.1.1 Deterministic Approach to Chain Reactions . . . . . . 11.1.2 Stochastic Approach to Chain Reaction . . . . . . . . . 11.2 “Four-factor” Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Detailed Analysis of the Four-factor Formula . . . . . 11.2.2 Technological Moderation Ratio Effect on the Four-factor Formula . . . . . . . . . . . . . . . . . . 11.3 Allowing for Leakages in a Finite Reactor . . . . . . . . . . . . . . 11.4 Two-group Multiplication Factor . . . . . . . . . . . . . . . . . . . . . 11.5 Multiplication Factor Through a Reaction Rate Balance . . . . 11.6 Reactivity Effects or Reactivity Difference . . . . . . . . . . . . . . 11.6.1 Comparison of the Effects on a UOX Fuel . . . . . . . 11.6.2 Reactivity Effect of Isotopic Change . . . . . . . . . . . 11.7 Calculation of Reactivity by Perturbation Theory Estimate . . . 11.8 Evolution of the Reactivity Along the Cycle . . . . . . . . . . . . . 815 815 815 816 821 822 827 828 829 835 840 841 842 845 847 Critical Homogeneous Reactor Theory . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Notion of Geometrical and Material Buckling . . . . . . . 12.3 Criticality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Notion of Critical Size: The Rod Model . . . . . . . . . . . . . . . 12.4.1 Analysis of Criticality . . . . . . . . . . . . . . . . . . . . . 12.4.2 Invariant Imbedding . . . . . . . . . . . . . . . . . . . . . . 12.5 Fundamental Mode for a Reactor with Simple Geometry . . . 12.5.1 Plane Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Parallelepiped . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 849 849 854 855 856 856 860 864 864 868 870 10.8 10.9 10.10 10.11 10.12 11 12 Extension to the Infinite Homogeneous Medium . . Expansion on the Eigenfunctions of the Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.10 Superposition of Flux Induced by Point Sources . . 10.7.11 Absorbing Slab in an Infinite Source Medium . . . . 10.7.12 Thin Absorbing Slabs, the Galanin Method . . . . . 10.7.13 Flux Transient . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of the Scattering Path of a Moderator by Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsed Neutron Method . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion in a Homogeneous Slab . . . . . . . . . . . . . . . . . . . Source Thermalization Transient in Diffusion Theory . . . . . 10.11.1 Infinite Medium . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.2 Finite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11.3 Expansion on Eigenfunctions . . . . . . . . . . . . . . . . 10.11.4 Case of a Pulsed Source . . . . . . . . . . . . . . . . . . . Polykinetic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents for Volume 2 12.5.4 Finite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.5 Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.6 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.7 Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.8 Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.9 Accounting for Singularities in 2D . . . . . . . . . . . . . 12.5.10 Anisotropic Point Source in a Multiplying Medium 12.5.11 Zero Flux Distance . . . . . . . . . . . . . . . . . . . . . . . . 12.5.12 Annular Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . Any Three-Dimensional Reactor . . . . . . . . . . . . . . . . . . . . . Fermi Age Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Overview of Slowing-Down . . . . . . . . . . . . . . . . . 12.7.3 Application to Neutron Diffusion . . . . . . . . . . . . . . 12.7.4 Relation Between Fermi Age and Time . . . . . . . . . 12.7.5 Link Between the Age Theory and Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.6 Two-Energy Group Equation in Fermi Age Theory . . . 12.7.7 Age-Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . Multi-Group Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactor Kinetics in One-Group Diffusion Theory with Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source Calculation: Extension to Multi-Group Conditions . . . 873 875 878 881 882 884 892 893 895 899 900 901 902 904 905 Neutron Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Some Mathematical Considerations on Reflectors . . . . . . . . . 13.2 Reflectors in Diffusion Theory . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Case of the Slab Reactor Surrounded by an Infinite Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Reflected Homogeneous Slab Reactor . . . . . . . . . . 13.2.3 Case of an Infinite Cylindrical Reactor Surrounded by an Infinite Reflector . . . . . . . . . . . . . . . . . . . . . 13.2.4 Case of an Infinite Cylindrical Reactor with a Finite Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Definition of Reflector Albedo . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Albedo Calculation for a Slab Reflector . . . . . . . . . 13.3.2 Albedo Calculation of a Cylindrical Reflector . . . . . 13.3.3 Albedo of a Spherical Reflector . . . . . . . . . . . . . . . 13.3.4 Albedo Calculation for the Upper Reflector of a Cylindrical Reactor . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Extrapolation and Null-flux Distances . . . . . . . . . . 13.3.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 13.4 Reflector Theory with Two Energy Groups . . . . . . . . . . . . . . 13.4.1 Slab Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 919 922 12.6 12.7 12.8 12.9 12.10 13 xxvii 907 909 912 912 914 916 922 926 928 934 939 941 942 942 943 944 947 947 948 xxviii Contents for Volume 2 13.4.2 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 14 Infinite Cylindrical Reactor with Reflector in Two Groups Without Up-Scattering . . . . . . . . . . . . 13.4.3 Flux Calculation in the Fuel . . . . . . . . . . . . . . . . . 13.4.4 Flux in the Reflector . . . . . . . . . . . . . . . . . . . . . . . Slab Reactor with Finite Reflector and Without Up-Scattering The Ackroyd “Magic Shell” Albedo Model . . . . . . . . . . . . . The Lefebvre-Lebigot Reflector Model . . . . . . . . . . . . . . . . . 13.7.1 “Equivalent” Reflectors Theory . . . . . . . . . . . . . . . 13.7.2 Calculation of Core Characteristics . . . . . . . . . . . . 13.7.3 Core/Reflector Operating Point . . . . . . . . . . . . . . . 13.7.4 Effect of Thermal-Hydraulic Feedbacks . . . . . . . . . 13.7.5 Calculation of Constants in the Mathematical Reflector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Albedo Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allowing for Up-Scattering . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion/Transport Correspondence . . . . . . . . . . . . . . . . . . . Reuss-Nisan Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mondot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized BETA Method . . . . . . . . . . . . . . . . . . . . . . . . . Absorption in the Reflector . . . . . . . . . . . . . . . . . . . . . . . . . Double-Differential Albedo . . . . . . . . . . . . . . . . . . . . . . . . . Heterogeneous Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Why Is Heterogeneity Desirable? . . . . . . . . . . . . . . . . . . . . . 14.2 Gurevich-Pomeranchuk Heterogeneous Resonant Absorption Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . 14.2.2 Effective Resonance Integral . . . . . . . . . . . . . . . . . 14.3 Modeling the Pin Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 First-Collision Probability . . . . . . . . . . . . . . . . . . . 14.3.2 The Amouyal-Benoist-Horowitz (A-B-H) Theory . . . 14.3.3 Multi-cell Approach in Two Dimensions . . . . . . . . 14.3.4 Carlvik Rational Approximation . . . . . . . . . . . . . . 14.3.5 Heterogeneity of the Isotopic Composition . . . . . . . 14.3.6 Shadowing Effect on the Resonance Integral . . . . . 14.3.7 Heterogeneous Pi , j Calculations for Fast Reactors with Perturbation Methods . . . . . . . . . . . . . . . . . . . 14.4 Transport-Diffusion Equivalence . . . . . . . . . . . . . . . . . . . . . 14.4.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Spatial Homogenization . . . . . . . . . . . . . . . . . . . . 14.4.3 Multi-group Approach . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Kavenoky-Hébert SPH Equivalence . . . . . . . . . . . . 14.4.5 Flux Reconstruction Between Different Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 950 952 955 957 959 960 965 967 969 970 971 972 977 978 984 986 987 988 991 991 993 993 998 999 1000 1002 1014 1032 1038 1038 1042 1045 1045 1047 1048 1049 1051 Contents for Volume 2 14.5 15 14.4.6 Spatial Homogenization with Leakage . . . . . . . . . 14.4.7 Equivalence for Slab Reactors . . . . . . . . . . . . . . . 14.4.8 Equivalence by Conservation of Reaction Rates . . Homogenization Theory in Diffusion . . . . . . . . . . . . . . . . . 14.5.1 Flux-Volume Homogenization . . . . . . . . . . . . . . . 14.5.2 Homogenization of Heterogeneous Neutron Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Average Flux Homogenization at the Boundary, Selengut Normalization . . . . . . . . . . . . . . . . . . . . 14.5.4 Pin Power Reconstruction . . . . . . . . . . . . . . . . . . 14.5.5 Discontinuity Factors . . . . . . . . . . . . . . . . . . . . . Fuel Cycle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Schematic Notation for Fuel Cycle Physics . . . . . . . . . . . . . 15.2 Disintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Neutron-Induced Reactions . . . . . . . . . . . . . . . . . . . . . . . . 15.4 The Bateman Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Heavy Nuclides . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.3 Activation Products . . . . . . . . . . . . . . . . . . . . . . . 15.5 Vectorial Form of the Bateman Equation . . . . . . . . . . . . . . 15.6 Calculation of Relevant Quantities for the Fuel Cycle . . . . . 15.6.1 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Burn-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.3 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.4 Calculation of Decay Heat . . . . . . . . . . . . . . . . . . 15.6.5 Photon γ and Neutron Dose Calculation . . . . . . . . 15.7 Isotopic Depletion Calculation . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Chain-Decay Process: Recurrence Relations . . . . . 15.7.2 Case of Heavy Nuclides . . . . . . . . . . . . . . . . . . . 15.7.3 Case of Fission Products . . . . . . . . . . . . . . . . . . . 15.7.4 Reference Composition of Some PWR Fuel . . . . . 15.8 Decay Chain Reduction Principle . . . . . . . . . . . . . . . . . . . . 15.8.1 Heavy Nuclide Chain for Reactivity Calculations of Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.2 Decay Chain Reduction . . . . . . . . . . . . . . . . . . . . 15.9 Activation: The Example of Control Rods . . . . . . . . . . . . . 15.10 Xenon Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10.1 Production of Xenon . . . . . . . . . . . . . . . . . . . . . . 15.10.2 Xenon Saturation . . . . . . . . . . . . . . . . . . . . . . . . 15.10.3 Xenon Poisoning After Reactor Shutdown . . . . . . 15.11 Samarium Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12 Gadolinium Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.13 The Industrial Fuel Cycle in France . . . . . . . . . . . . . . . . . . xxix . . . . . 1062 1067 1072 1076 1076 . 1077 . 1080 . 1082 . 1087 . . . . . . . . . . . . . . . . . . . . . 1091 1091 1092 1092 1092 1093 1095 1096 1097 1097 1097 1098 1104 1104 1115 1117 1118 1121 1122 1123 1124 . . . . . . . . . . 1126 1134 1137 1138 1138 1140 1142 1144 1145 1146 xxx 16 17 Contents for Volume 2 Neutronic Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Effect of Fuel Temperature on the Multiplication Factor . . . . 16.1.1 Fuel Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Doppler Effect on Reactor Behavior . . . . . . . . . . . 16.2 Moderator Temperature Effect . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Leakage and Absorber Effects . . . . . . . . . . . . . . . . 16.2.3 Pressure Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Graphite Moderator . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Neutron Spectrum Shift . . . . . . . . . . . . . . . . . . . . . 16.2.6 Void Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Boron Effect in Pressurized Water Reactors . . . . . . . . . . . . . 16.3.1 Differential Efficiency of Boron . . . . . . . . . . . . . . . 16.3.2 Boron Effect on the Moderator Differential Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Power Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Feedback Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 A Simple Model: Power Feedback . . . . . . . . . . . . . 16.5.2 An Advanced Feedback Model: The Lefebvre-Seban Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Historical Isotopic Correction . . . . . . . . . . . . . . . . . . . . . . . . Reactor Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Prompt Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Evolution of a Hypothetical Prompt Neutron Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Flux Calculation: Point Reactor Hypothesis . . . . . . 17.2 Delayed Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Delayed Neutron Fraction . . . . . . . . . . . . . . . . . . . 17.3 Effect of Delayed Neutrons on Reactor Kinetics . . . . . . . . . . 17.4 Neutron Kinetics Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Precursor Concentration . . . . . . . . . . . . . . . . . . . . 17.4.2 Point-Reactor Kinetics . . . . . . . . . . . . . . . . . . . . . 17.4.3 Mobile Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Nordheim Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 “Prompt Jump” Notion: Insertion of a Reactivity Step . . . . . . 17.7 Age Theory in the Kinetics Equation for Thermal Neutrons . . . . 17.8 Reduced Kinetics Equations . . . . . . . . . . . . . . . . . . . . . . . . . 17.9 Kinetics with an Imposed Neutron Source . . . . . . . . . . . . . . . 17.10 Delayed Neutron Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 17.11 First-Order Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.12 Numerical Reactimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.13 Practical Evaluation of Prompt Neutron Generation Time . . . 17.14 Main Causes of Reactivity Changes . . . . . . . . . . . . . . . . . . . 17.14.1 Increased Fissile Nuclei . . . . . . . . . . . . . . . . . . . . . 1153 1153 1153 1156 1158 1158 1160 1162 1163 1164 1165 1166 1166 1167 1168 1168 1171 1172 1183 1187 1187 1188 1193 1195 1199 1200 1203 1205 1206 1208 1208 1213 1215 1218 1220 1221 1229 1231 1234 1236 1236 Contents for Volume 2 17.15 17.16 17.17 17.18 17.19 17.20 17.21 17.22 17.23 17.24 17.25 17.26 17.27 18 xxxi 17.14.2 Increased Neutron Moderation . . . . . . . . . . . . . . . . 17.14.3 Decreased Neutron Capture . . . . . . . . . . . . . . . . . . Reactivity Accident: Insertion of Very High Reactivity Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.15.1 Analysis with One Group of Delayed Neutrons . . . . 17.15.2 Analysis of the Case of ρ >> β: The Reactivity Accident . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.15.3 Insertion of Low Reactivity 0  ρ << β . . . . . . . . Anti-reactivity Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactivity Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dropped Control Rod, Insertion of a Large Amount of Anti-reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactivity Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reactivity Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power Excursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.22.1 The Nordheim-Fuchs Model . . . . . . . . . . . . . . . . . 17.22.2 The Chernick Model . . . . . . . . . . . . . . . . . . . . . . . 17.22.3 The Bethe-Tait Model . . . . . . . . . . . . . . . . . . . . . . Subcritical Approach: Reactor Start-Up . . . . . . . . . . . . . . . . Reactor Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space-Time Xenon Oscillations . . . . . . . . . . . . . . . . . . . . . . Mechanical Kinetic Effects . . . . . . . . . . . . . . . . . . . . . . . . . Neutron Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.27.1 Noise Concept, Spectral Analysis . . . . . . . . . . . . . . 17.27.2 Neutron Correlations . . . . . . . . . . . . . . . . . . . . . . . 17.27.3 The Feynman-α Method . . . . . . . . . . . . . . . . . . . . 17.27.4 Delayed-Neutron Effect . . . . . . . . . . . . . . . . . . . . . 17.27.5 Application to Measurement of Void Fraction Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.27.6 Application to Detection of Vibrations . . . . . . . . . . Computation Methods in Diffusion Theory . . . . . . . . . . . . . . . . . 18.1 Calculation Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Multi-group Diffusion Equations . . . . . . . . . . . . . . . . . . . . 18.2.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 “1.5”-group Diffusion . . . . . . . . . . . . . . . . . . . . . 18.2.3 Adjoint Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 18.2.4 Taking into Account the Neutron Over-Production Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 The Power Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . 18.3.2 Matrix Representation . . . . . . . . . . . . . . . . . . . . . 18.3.3 Chebyshev Acceleration . . . . . . . . . . . . . . . . . . . 18.4 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . 1237 1237 1238 1238 1241 1243 1245 1246 1247 1249 1250 1254 1254 1255 1259 1262 1266 1267 1271 1276 1277 1278 1280 1287 1295 1296 1298 . . . . . . 1301 1301 1304 1304 1305 1305 . . . . . . 1307 1308 1308 1310 1312 1315 xxxii Contents for Volume 2 18.5 18.6 18.7 18.8 18.9 18.4.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 18.4.3 Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nodal Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.1 Nodal Method of Order 4 . . . . . . . . . . . . . . . . . . . 18.5.2 Quadratic Approximation of Transverse Leakage . . . 18.5.3 AFEN Method . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.2 Accounting for Boundary Conditions . . . . . . . . . . . Calculation of Control Rods . . . . . . . . . . . . . . . . . . . . . . . . . 18.8.1 Physical Effect of Rods . . . . . . . . . . . . . . . . . . . . . 18.8.2 Rod Worth: Perturbation Analysis . . . . . . . . . . . . . 18.8.3 Measuring Rod Efficiency in PWR . . . . . . . . . . . . . 18.8.4 Calculation of Rod Efficiency . . . . . . . . . . . . . . . . 18.8.5 Analytical Decomposition of the Rodded Domain . . Instrumentation Considerations . . . . . . . . . . . . . . . . . . . . . . 18.9.1 Modeling with Trace Quantities . . . . . . . . . . . . . . . 18.9.2 Modeling of the EPR Instrumentation: The KTM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315 1320 1321 1322 1324 1332 1335 1336 1340 1340 1342 1343 1344 1345 1348 1349 1353 1357 1357 1357 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367 Annex: Reactor Physics and Neutronic Codes at Electricité De France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431