Abstract
The goal of reactor physics is to maintain the reactor in a stable state. Nevertheless, the reactor must be brought to a stable state at full power, starting from a cold shutdown state. Further, a reactivity incident may modify the stable state thus achieved. Reactor kinetics studies the transient time phenomena and is the starting point of studies for reactivity accidents.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
To avoid any confusion with Greek letter, nu (ν) used in the production cross section and the velocity, exceptionally, we use the velocity in non-italics ν as for the remaining equations. Some references use the symbol Λfor the generation time.
- 2.
This analogy means that if all couples had twins at 20, this would be equivalent to have one child at 10, and a second one 10 years later. This example shows that the calculation of the population over a long period is identical in both cases, even if the basic hypothesis is inconsistent.
- 3.
R. B. Roberts, J. R. Hafstad, R. C. Meyer and P. Wang: The delayed neutron emission which accompanies fission of uranium and Thorium, Phys. Rev. 55 (664), March 1939.
- 4.
John Archibald Wheeler (1911–2008). American theoretical physicist. After receiving his PhD from John Hopkins University in 1933, he taught at the University of North Carolina, then, from 1938, at Princeton, becoming one of the youngest professors at this prestigious university. In 1939, together with Niels Bohr he proposed an explanation for nuclear fission. He contributed to the Manhattan Project at the Hanford plutonium-producing reactor site. He predicted xenon poisoning by which reactor start-up could be stopped. Wheeler worked on reactor physics between 1941 and 1945. On returning to Princeton, he worked on quantum gravitation and relativity, and it was he who coined the term “black hole”. He directed the theoretical physics center in Texas in 1976. In 1969 he was awarded the Franklin Medal for his fission theory.
- 5.
R. J. Tuttle: Delayed-neutron data for reactor-physics analysis, Nuclear science and engineering, 56, pp. 37–71 (1975).
- 6.
Lothar Wolfgang Nordheim (1899–1985), a German physicist of Jewish origin, emigrated to the United States in 1934 like many German physicists, due to the rise of Nazism. He taught at Duke University where he spent most of his career as of 1937. He was appointed head of the theoretical physics group of the Clinton laboratories, the ancestor of Oak Ridge National Laboratories in the Manhattan Project. He worked on kinetics problems and explosions in the wake of development of the hydrogen bomb. He also contributed to nuclear shell theory. The Fowler-Nordheim model establishes that the emission of electrons under an electric field at the surface of a metal follows a Fermi-Dirac distribution and obeys the tunneling effect. The latter has many practical applications.
- 7.
This may seem paradoxical in a thermal neutron equation, but it is indeed the prompt neutron lifetime, i.e. the mean time between the birth of a (prompt) neutron and the birth of (prompt) neutrons of the new generation appearing in the equation. The reader should bear in mind the obvious statement that a neutron can only be thermal if it was initially fast.
- 8.
There is (unfortunately) some confusion regarding notation in the references on neutron kinetics, and also on occasion, regarding the numerical values of prompt neutron lifetimes. The following table gives corresponding definitions from the better-known references.
Marguet
Keepin (1965, p. 166)
Hetrick (1993, p. 6)
Ott and Neuhold (1985, p. 24)
Reuss (2003, p. 119)
Prompt neutron lifetime in an infinite medium
\( {\ell}_0=\frac{1}{\rm{v}{\varSigma}_a} \)
\( {\ell}_{\infty }{\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( {\ell}_{\infty }{\displaystyle \begin{array}{c}\\ {}\end{array}} \)
Prompt neutron lifetime in a finite medium
\( {\ell}_{B_g}=\frac{\ell_0}{1+{L}^2{B}_g^2} \)
\( \ell {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( {\ell}_0{\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( \ell {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( \theta {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
Effective prompt neutron lifetime or generation time
\( \ell =\frac{\ell_{B_g}}{k_{eff}}=\frac{1}{\rm{v}\nu {\varSigma}_f} \)
\( \Lambda {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( \ell {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( \Lambda {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
\( \ell {\displaystyle \begin{array}{c}\\ {}\end{array}} \)
- 9.
D. R. Wyman, A. A. Harms, Kinetics of driven multiplying media, Nuclear Science and Engineering, 83, p. 483 (1983).
- 10.
R. C. Greenwood, K. D. Watts: Delayed neutron energy spectra of Br87, Br88, Br89, Br90, I137, I138, I139 and Te136, Nuclear Science and Engineering, Vol. 126, pp. 324–332 (1997).
- 11.
G. Keepin, Progress in Nuclear Energy, Vol. 1, Series 1, Pergamon Press, London (1956).
- 12.
S. Das : The importance of delayed neutrons in nuclear research—a review, Progress in Nuclear Energy, Vol. 28, No. 3, pp. 209–264 (1994). This import journal on delayed neutrons concludes this chapter perfectly. It contains several recent data (particularly on delayed spectra and β eff for several experiments).
- 13.
D. Saphier, D. Ilberg, S. Shalev, S. Yiftah: Evaluated delayed neutron spectra and their importance in reactor calculations, Nuclear Science and Engineering, 62, pp. 660–694 (1977).
- 14.
Véronique Zammit-Averlant : Validation intégrale des estimations du paramètre béta effectif pour les réacteurs MOX et incinérateurs [Integral validation of calculations of the effective beta parameter for MOX reactors and incinerators], thesis at the University of Aix-Marseille (1998).
- 15.
On the different calculation techniques related to reactivity measurements, see Joao Manoel Losada Moreira : Space-time analysis reactivity measurements, PhD at The University of Michigan, 1984.
- 16.
Here we have the exact form of the Weber equation, with the ¼ coefficient. However, (Hetrick 1993) chose to write it without this coefficient, resulting in modified values of α and κ.
- 17.
The first Hermite polynomials are: H 0(x) = 1, H 1(x) = 2x, H 2(x) = 4x 2 − 2, H 3(x) = 8x 3 − 12x. The Hermite functions are orthogonal in L2(ℜ): \( \underset{\Re }{\int \limits }{\varphi}_n(x)\kern0.1em {\varphi}_m(x)\kern0.1em dx=0\kern1em n\ne m \).
- 18.
Klaus Fuchs, a German-born British scientist who worked on the Manhattan Project. He was famously tried as a USSR spy and in 1950 he was sentenced to 14 years in prison. (The Rosenbergs were also convicted for espionage and sentenced to the electric chair, despite international protest). He was released after 9 years and immigrated to East Germany.
- 19.
Ajoy K. Ghatak: Non-linear prompt neutron kinetics in multi-group diffusion, theory, PhD thesis, Cornell University (1963). The Nordheim-Fuchs model is thoroughly described. The use of higher modes in plane geometry is worked out to its limits.
- 20.
J. Chernick: The dependence of reactor kinetics on temperature, Rapport BNL-173, 38 pages, 20 décembre 1951.
- 21.
R. T. Ackroyd, G. H. Hinchin, J. E. Mann, J. D. McCullen: Stability considerations in the design of fast reactors, Proc 2nd Conf. on peaceful uses of atomic energy, 1958, Vol 12 p. 230.
- 22.
Tomoaki Suzudo : Reactor noise analysis based on nonlinear dynamic theory—application for power oscillation, Nuclear Science and Engineering, 113 pp. 145–160 (1993).
- 23.
A.S. Thompson, B.R. Thompson , A model of reactor kinetics, Nuclear science and engineering, 100, pp. 783–88 (1988).
- 24.
Several such applications are described in: Progress in Nuclear Energy, Vol. 29, No 3–4 (1995).
- 25.
On Poisson distribution, see (Ross 1992).
- 26.
Jean Tachon: Etude neutronique d’une pile à neutrons thermiques au plutonium «Proserpine»: corrélations entre neutrons dans une réaction en chaîne [Neutron study of the Proserpine plutonium thermal neutron reactor: correlations between neutrons in a chain reaction], PhD thesis, Science Faculty of Paris, 1960.
- 27.
R. P. Feyman, F. de Hoffmann, H. Serber: Statistical fluctuations in the water boiler and the dispersion of neutrons emitted per fission, Los Alamos Scientific Laboratory report LA-101 (1944).
- 28.
Gregory D. Springs : The reactor noise threshold, Nuclear Science and Engineering, 116, pp. 67–72 (1994). The author precisely defines the limit that must not be exceeded in order for the measurement to be useful. For a critical PWR with a prompt neutron lifetime of 10−3 s, the limit of 0.00025 W is found. This can be measured (with difficulty!) in an industrial reactor (fission power is largely exceeded by residual power). For an over-critical stable reactor, this constraint is less strict. For a stable sub-critical reactor, the criterion depends on the source.
- 29.
Bruno Rossi (1905–1993) studied in Padua and Bologna. He received his PhD in Physics in 1937 and worked in Florence and Padua, before leaving Italy in 1938 for Denmark and subsequently moving to Great Britain. In the United States, he worked on the Manhattan project as co-director of the detectors group, and was in charge of instrumentation for physics testing of the atomic bomb. From 1946, he taught physics at MIT, where he became a worldwide expert in cosmic rays. At the end of his career, he was working on plasma physics and astrophysics.
- 30.
Gregory D. Springs : Two Rossi − α techniques for measuring the effective delayed neutron fraction, Nuclear Science and Engineering, 112, pp. 161–172 (1993).
- 31.
Andrey Andreyevitch Markov (1856–1922) was a Russian mathematician and a student of Tchebychev at the University of St Petersburg. He became a member of the science academy of the same city in 1886. He is the author of several major works in probability theory, particularly Markov inequality: if X is a real discrete random variable with values in R +, then P(X ≥ a) ≤ E(X)/a where E(X) is the expected value of X. More generally, when X is a random numerical variable and a strictly positive increasing function, then for any a > 0: P(X ≥ a) ≤ E(φ(X))/φ(a). A Markov process is a random finite state without past memory where the next state in the process depends solely on its present state. To simplify, only the present state and not past states, influences the future. It should be noted that his son, also named Andrey, was also a prominent Russian mathematician.
- 32.
Williams (1974) used a mean fission time defined by τ f = 1/(vΣ f ), which can be related to the generation time by \( {\tau}_f=\overline{\nu\;}\ell \). Here, the choice was made so as to avoid new definitions of the lifetime, which could prove confusing.
- 33.
E. D. Courant , P. R. Wallace , Fluctuations of the number of neutrons in a pile, Phys. Rev. Vol 72, December 1947.
- 34.
André Brillon: Le bruit neutronique [Neutron background noise], Bulletin de la Direction des Etudes et Recherches d’EDF, Epure n°17, pp. 3–13, Janvier 1988, from which the illustration is taken.
- 35.
On spectral analysis, the lecture of (Kay 1988) is very helpful.
- 36.
I. Pazsit, O. Glöckler: On the neutron noise diagnostics of pressurized water reactor control rod vibrations: III Application at a power plant, Nuclear science and engineering, 99, pp. 313–328 (1988).
Bibliography
Ernest J. Henley, Jeffery Lewins (Editors), Advances in Nuclear Science and Technology Volume 6, Academic Press, ISBN Library card number 62-13039, 1972, 239 pages. Especially the chapter Stability Analysis of nonlinear point reactor kinetics by Stig-Olof Londen.
Annual review of nuclear science Volume 2, Annual reviews, inc., Stanford, USA, 1953, 429 pages. Particularly, R.A. Alpher and R.C. Herman: The origin and abundance distribution of the elements, p 1–40.
Milton Abramowitz, Irene Stegun, Handbook of mathematical functions, Dover, USA, ISBN 0-486-61272-4, re-edition of the 1972 version, 1046 pages. This thick textbook constitutes a complete mathematical handbook. The Dover edition makes it accessible to everyone.
Ziya Akcasu, Gerald S. Lellouche, Louis M. Shotkin, Mathematical methods in nuclear reactor dynamics, Academic Press, USA, Number of Library of Congress 71-137637, 1971, 460 pages. This book discusses innovative academic problems such as allowing for gravity in transport problems, the Pontryagin stability criterion, the Lyapunov functions in reactor dynamics.
Milton Ash, Nuclear reactor kinetics, 2nd edition, McGraw Hill, USA, ISBN 0-07-002380-8, 1979, 445 pages.
George Bell, Samuel Glasstone, Nuclear reactor theory, Van Nostrand, USA, Library of Congress Card Number 73-122674, 1970, 619 pages. The Bell and Glasstone is an institution that has contributed in the education of generations of engineers.
Franck M. Callier, Charles A. Desoer, Linear system theory, Springer, New-York, USA, ISBN 0-387-97573-X, 1991, 509 pages. A good introduction to optimal control of linear systems.
M. Duquesne, R. Grégoire, M. Lefort, Travaux pratiques de physique nucléaire et de radiochimie [Practical work in nuclear physics and radiochemistry], Masson, Paris, 1960, 304 pages. Several experimental descriptions for radiation.
A. Erdélyi (Editor), Higher transcendental functions, volume 1, McGraw-Hill, New-York, USA, Library of Congress Card Number 53-5555, 1953, 302 pages. Gamma function, Whittaker function, and so on.
Samuel Glasstone, Milton C. Edlund, The elements of Nuclear reactor theory, Mac Millan, USA, 1972, 416 pages. It is the re-edition of the 1952 version published at Van Nostrand.
Samuel Glasstone, Alexander Sesonske, Nuclear reactor engineering tome 1 et 2, Chapman-Hall, USA, ISBN 0-412-98521-7 et 0-412-98531-4, 1994, 841 pages in two parts, 4th edition. This reference is essential for reactor physics and was successfully edited several times. Glasstone published several work on the subject.
Pierre Grivet, Austin Blaquière, Le bruit de fond, Masson, Paris, France, 1958, 495 pages. The chapter on background noise and its mathematical representation is useful for understanding neutron noise.
David L. Hetrick, Dynamics of nuclear reactors, American Nuclear Society, La grange Park, USA, ISBN 0-89448-453-2, 1993, 542 pages, To my knowledge, the best textbook on reactor kinetics with [Keepin, 1965]. He develops several types of problems starting from the point-reactor. Essential to the neutron physicist.
A. Hitchcock, Nuclear reactor stability, Temple Press, London, United Kingdom, 1960, 61 pages, in the collection nuclear engineering monographs. Very oriented on the graphite-gas reactor of Calder-Hall. Off-topic: I hope that the author was not named Alfred!
Harry H. Hummel, David Okrent, Reactivity coefficient in large fast power reactors, American Nuclear Society, USA, Library Congress Card Number 73-119000, 1970, 386 pages. Very complete. This book is a must for specialists in accidentology.
Gérard Iooss, Daniel D. Joseph, Elementary stability and bifurcation theory (2nd edition), Springer-Verlag, Berlin, ISBN 3-540-97069-1, 1995, 324 pages.
Théo Kahan, M. Gauzit, Physique et calcul des réacteurs nucléaires [Reactor physics and calculations], Dunod, Paris, 1957, 388 pages. Several elements on the dimensioning of the French graphite-gas reactors. Indeed, since France relied more on those reactors at that time, there is no mention of PWR.
Steven M. Kay, Modern spectral estimation, Prentice Hall, New-Jersey, USA, ISBN 0-13-598582-X, 1988, 543 pages. Several books exist on signal processing but this textbook is very well written. A soft floppy disk of 5 inches brings us several years back in time.
George Robert Keepin, Physics of nuclear kinetics, Addison-Wesley, Library of congress 64-20831, 1965, 435 pages. The first book entirely devoted to reactor kinetics, and one of the ten essential reference books to the reactor physicist. The six-group of delayed neutron are still used in most codes presently. Up to the publication of [Hetrick, 1993] some 20 years later, it was hard to by-pass this work!
Nordine Kerkar, Philippe Paulin, Exploitation des cœurs REP [Operating PWR cores], EDP Sciences-INSTN, Paris, ISBN 978-2-86883-3, 2008, 304 pages. It is a specialized work which presents the essential knowledge to operate PWR. The glossary is complete and gives an overview of the jargon with trigrams (thrre-letter abbreviations) which are often unfriendly to beginners.
John Lamarsh, Anthony J. Barrata, Introduction to nuclear engineering, Prentice Hall, ISBN 0-201-82498-1, 3rd edition, 2001, 783 pages. More of an engineering textbook where few physical or mathematical proofs are illustrated.
J. Lewins, Nuclear reactor kinetics and control, Pergamon Press, La Grange Park, USA, ISBN 0-08-021682-X, 1978, 264 pages. On an anecdotic note, this book contains a remarkable colored folded page of the control circuits of a four-loop PWR. The first chapter refers to reactor stability (transfer function, Nyquist diagram, Padé approximant
Robert V. Meghreblian, David K. Holmes, Reactor analysis, McGraw-Hill, New-York, Library of Congress Catalog Card Number 59-15469, 1960, 807 pages. This work is entirely devoted to reactor physics and neutron physics and is within the framework of this textbook. No technology or operating details. One of the best references on the subject.
C.W. Merriam III, Optimization theory and the design of feedback control system, McGraw-Hill, New-York, Library of Congress Catalog Card Number 63-19315, 1964, 315 pages. Especially on the Wiener-Hopf criteria and Gill method.
Raymond L. Murray, An Introduction to nuclear engineering, Prentice Hall, New-York, USA, Library of Congress Card Number 54-8207, 1954, 418 pages. More technological than physical.
Karl O. Ott, Robert J. Neuhold, Nuclear reactor dynamics, American Nuclear Society, USA, ISBN 0-89448-029-4, 1985, 362 pages.
Imre Pazsit, Lenard Pal, Neutron fluctuations: A Treatise on the Physics of Branching Processes, Elsevier, ISBN 978-0-08-045064-3, 2007, 360 pages.
Jacques Planchard, Méthodes mathématiques en neutronique [Mathematical method in neutron physics], Eyrolles, Paris, ISBN 0399-4198, 1995, 431 pages. Especially for theorems on the critical equation and for cases with neutron sources. Jacques Planchard unfortunately died in 2009 – he was an expert at EDF R&D.
M.M.R. Williams, N.J. McCormick (editors), Progress in nuclear energy Volume 15, Proceedings of the Specialist Meeting On Reactor Noise (SMORN IV), 15–19 October 1985, Dijon, Pergamon Press, ISBN 0-08-031648-4, 1985, 1002 pages. Compilation of articles on neutron noise with numerous applications. This compilation provides a review of theoretical approaches.
Paul Reuss, Précis de neutronique [Neutron physics], EDP Sciences, collection INSTN, Paris, ISBN 2-86883-637-2, 2003, essential elements from [Bussac and Reuss 1985], with a more modern typography but simplified content (directed towards students in a more didactic form). The only textbook of pure neutron physics still available in French.
Sheldon M. Ross, Applied probability models with optimization applications, Gauthier-Villars, Dover, New-York, USA, ISBN 0-486-67314-6, 1992, 198 pages. Distributions and counting.
Daniel Rozon, Introduction à la cinétique des réacteurs [Introduction to reactor kinetics], Editions de l’école polytechnique de Montréal, Canada, ISBN 2-253-00223-8, 1992, 413 pages, the most complete reference on kinetics in French. Essential for the accidentology of reactors. Rozon presents thoroughly the problem of the form function that is used in the point-kinetics model, which is not very much discussed in other references. However, the precision in the terms increases the names of variables (all the amplitudes must be described), thereby making it difficult to read: it is not obvious to recall that ζ(t) is the amplitude of the concentration of delayed neutrons. Besides, there is no index.
H. Soodak (editor), The reactor handbook, volume II1: Physics, Interscience Publishers, Library of Congress Card Number 60-11027, 1962, 2nd edition, 313 pages. It is a more complete version of the previous reference.
Weston M. Stacey, Nuclear reactor physics, John Wiley, USA, ISBN 0-471-39127-1, 2001, 707 pages.
Henry Tellier, Cinétique des réacteurs nucléaires [Nuclear reactor kinetics], INSTN/CEA, ISBN 2-7272-0167-2, 1993, 91 pages. One of the rare French references on the subject. This reference can be used as an introduction to [Rozon, 1992], which is more complete.
Joseph A. Thie, Power reactor noise, American Nuclear Society, La Grange Park, USA, ISBN 0-89448-025-1, 1981, 208 pages. Réedition de la version de 1963 publiée chez Rowman et Littlefield, Inc, New-York. One of the rare books on the subject with [Uhrig, 1970] and [Williams, 1974].
Robert E. Uhrig, Random noise techniques in nuclear reactor systems, Ronald Press Company, Library of Congress card number 71-110558, 1970, 490 pages. More details than [Thie, 1981], but less theoretical than [Williams, 1974]. This book has both theoretical and experimental aspects on processing neutron noise. A wise chapter on random noise precedes the purely neutronics part. The probabilistic approach of radioactivity in the second chapter is very instructive.
Lynn E. Weaver, System analysis of nuclear reactor dynamics, Rowman and Littlefield, New-York, USA, 1963. 285 pages. This book explains the feedback effects on reactor kinetics using transfer functions (Nyquist diagrams, impedance of a system, etc.) and thus requires some background knowledge on optimal control. The first chapter deals with transforms in the complex plane and the second chapter uses electrical analogies. Only in the third chapter is neutron kinetics discussed.
Lynn E. Weaver (coordinator), Reactor kinetics and control, proceedings of a symposium at the University of Arizona, march 1963. Division of technical information extension, Oak Ridge, Tennessee, USA, 1964. This compilation contains several articles (by Uhrig, Albrecht, Zivi and Wright,…) on neutron noise.
Lynn E. Weaver, Reactor dynamics and control, Elsevier, New-York, USA, 1968. Very much based on optimal control of which Weaver is an expert.
Joel Weisman, Elements of nuclear reactor design, Elsevier/North Holland, ISBN 0-444-41509-2, 1977, 466 pages. This book contains chapters on designing reactors where thermalhydraulic and thermal conduction are discussed. The section E deals with safety issues and chapter 14 discusses reactivity accidents with sufficient description of the Nordheim-Fuchs and Bethe-Tait models.
M.M.R. Williams, Random processes in nuclear reactors, Pergamon Press, 1974, 243 pages. This book on neutron noise completes [Thie, 1981] and [Uhrig, 1970] on theoretical aspects. The clear presentation of the Feynman and Rossi methods as well as the Langevin technique. In the 2000’s, Williams published theoretical work on the transport equation in random media (geometry with grains, etc.).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Marguet, S. (2017). Reactor Kinetics. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-59560-3_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59559-7
Online ISBN: 978-3-319-59560-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)