Abstract
Since the probability of neutron-matter interaction increases as neutron velocity decreases, slowing neutrons down increases the probability of fission. This is the goal of slowing down in thermal reactors. The field of neutronics owes much to Enrico Fermi , who introduced the mathematical formalism, thanks to his background as a mathematician. He also designed the logarithmic model for neutron slowing down used in setting up the world’s first atomic pile in Chicago in 1942.
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Notes
- 1.
Emilio Gino Segrè (1905–1989) was an Italian physicist who started his career under Fermi, and who fled the fascist regime of Mussolini. He is the co-discoverer of the anti-proton. His significant works in theoretical particle physics earned him the Nobel Prize in 1959.
- 2.
Edoardo Amaldi (1908–1989) worked under Fermi after his engineering studies at Rome University. He defended his PhD thesis in 1929 on the Raman Effect of benzene. In 1937, he replaced Corbino as Professor of Experimental Physics at Rome University. Throughout the dark period of war, he tried to ensure that the Roman physics school remained. He worked on spectroscopy and detectors. A conference at La Sapienza University paid tribute to him in 2008.
- 3.
Franco Rasetti (1901–2001) followed the career of Fermi from Florence to Roma. He became the first assistant to Corbino, Professor of Experimental Physics. His specialization in experimental physics complemented Fermi’s theoretical skills. In 1939, he left Italy for Quebec, where he became a world specialist on molecular spectroscopy. Although invited by his Italian friends Fermi and Segrè to join the Manhattan project in 1939, he vehemently refused on ethical grounds. His exceptional career is described in Franco Rasetti, physicien et naturaliste (Il a dit Non à la bombe) [Franco Rasetti, physicist and naturalist: the man who said “No” to the bomb] by D. Ouellet, Ed. Guérin (2000).
- 4.
Oscar d’Agostino (1901–1975) became a Doctor in Chemistry in 1926. After his work with Fermi, he joined the Italian National Scientific Research Center.
- 5.
Bruno Pontecorvo (1913–1993) was one of Fermi’s youngest assistants. Like many Italian Jews, he fled fascism in his home country. Moving to the United States, already in 1942, while working for a petrol prospection company, he was seeking to develop a principle based on neutron emission. In 1948, he became a British citizen. In 1950, at the height of the “Cold War”, he sensationally defected to the East with his family in unexpected and unexplained circumstances, despite having never worked on the bomb project. He was warmly welcomed In the USSR, where he specialized in high-energy physics until the end of his life.
- 6.
- 7.
In transport theory, angle ψ is usually denoted by θ 0 and its cosine by μ 0. However, we differentiate here between this variable and the polar angle θ for the sake of clarity.
- 8.
The specific slowing-down power is expressed by some authors as the moderation ratio, which risks being confused with the technological moderation ratio if the latter term is shortened.
- 9.
- 10.
S.A. Pozzi , I. Pazsit: Neutron slowing down in a detector with absorption, Nuclear Science and Engineering, 154, p367–373 (2006). After obtaining his PhD at the Roland Eötvös University in Budapest in 1975, Imre Pazsit, taught nuclear engineering at the University of Michigan, and then at the University of Chalmers in Sweden. He has written numerous articles dealing in particular with neutron noise, as well as a treatise on neutron fluctuations.
(Courtesy Pazsit) - 11.
In Masakuni Narita, Koïchi Narita: Average number of collisions necessary for slowing down of neutrons, Journal of Nuclear Science and Technology, Vol. 26 No 9, pp819–825 (1989).
- 12.
R.L. Murray, Nuclear Science and Engineering 128, p329–330 (1998)
- 13.
The notation R(E) is preferred here instead of the more common F(E) (Glasstone and Edlund 1972; GA Vol. 1, 1967, p213), since it provides a mnemonic (R(E) for slowing-down Reaction Rate).
- 14.
\( \frac{d}{du}\left(\underset{-\infty }{\overset{u}{\int \limits }}f\left(u,v\right) dv\right)=f\left(u,u\right)+\underset{-\infty }{\overset{u}{\int \limits }}\frac{df\left(u,v\right)}{du} dv \)
- 15.
E. Amaldi, E. Fermi, Physical Review. 50, p899 (1936).
- 16.
In strict mathematical terms, the function R s (u) does not correspond to the same functional R s (E) where the variable E = E 0 e −u has been changed. We have opted against overloading the text with new variables. The context in which the notations appear is thus essential for a correct understanding of quantities and their units.
- 17.
G. Placzek, Phys. Rev., 69, p423 (1946).
- 18.
John Von Neumann (1903–1957) was a Hungarian-born mathematician. As a child, he displayed prodigious mathematical skills. After his PhD in 1926 in Budapest, he left for Germany to work with the major physicists of the time. In 1930, he immigrated to the United States, where he worked at Princeton with Albert Einstein. He developed applied mathematics, then a thriving field with the advent of information technology, and he contributed significantly to numerical analysis and game theory, as well as economic analysis. He worked on the Manhattan Project, where his exposure to radiation is assumed to be the cause of his fatal bone cancer. (Public domain)
- 19.
In practice, this is due to the fact that the flux results from a non-scattering flux added to a first-scattering flux, second-scattering flux, and so on to infinity in a non-absorbing medium, although the number of scattering events is obviously finite in a realistic medium.
- 20.
Eugene T. Greuling studied at Duke University and obtained his PhD in 1942 from the University of Indiana (Theoretical Half-Lives of Forbidden beta-Transition), with Emil Konopinski as his doctoral advisor—himself a specialist of β disintegration. As a specialist in nuclear physics, Greuling wrote his first article on the computation of permitted transitions for disintegrations using the tensor method as early as 1942 (Phys. Rev. 61–588 (1942). He was the doctoral advisor of Paul Frederick Zweifel (born 1929) in 1954 (Capture-Positron Branching Ratios), the well-known author of several books on neutronics [among which (Case and Zweifel 1967)]. He contributed to the Manhattan Project at Los Alamos, then Oak Ridge. Greuling later taught physics at Duke University.
- 21.
Gerald Goertzel (1920–2002) obtained his PhD in theoretical physics from New York University. He was proficient in both numerical analysis and nuclear engineering. He was a member of the Manhattan Project, and later of the Nuclear Development Corporation of America. In addition, he developed medical instruments for Sage Instruments throughout part of his career. He then spent 28 years until his retirement at the research division of IBM devising algorithms for data compression. He designed the Goertzel algorithm, which helps identify the dominant cycles of a noise signal by computing the Fourier coefficients of multi-frequency sinusoidal signals inside the noise.
- 22.
Martine Pujol (1941–1995) completed her Master’s Degree in Reactor Physics in 1964–1965. She worked on her PhD at the CEA/Department of Pile Studies/Department of Mathematical Physics of Saclay, where she met Jean Bussac, Head of the Department, Oleg Tretiakoff, Michel Cadilhac, and Paul Reuss, then a young engineer. She later left the CEA to teach in Senegal, then at the University of Orleans. In 1971, she became a scientific journalist under the pen-name Martine Barrère for the scientific newspaper “La Recherche”, where she worked in the “Science and Politics” section. She worked on several documentaries on nuclear power, the Soufrière volcano and AIDS. From 1979 to 1981, she presided over the Scientific Journalist Association. She left La Recherche in 1990 due to disagreements and worked as a freelance journalist for Le Monde amongst others. She taught scientific journalism at the University of Paris VII and contributed significantly to subjects dealing with Science or Consciousness. She died of cancer in 1995.
(Courtesy La Recherche) - 23.
Jean Bussac (1929–). After his studies at the illustrious “Ecole Polytechnique” (promotion 1948), he was recruited in September 1951 to the « Commissariat à l’Energie Atomique » by Jacques Yvon au (CEA) where he participated in the developments of the mathematical theories applicable to nuclear reactors. In August, 1952, he participated in the International Seminar of physics of Les Houches in French Savoy (around thirty young scientists, some French people of whom future Nobel price Pierre-Gilles de Gennes). Appointed in 1962 head of the department of mathematical physics of the CEA, which he will lead at the top of the excellence in the field of the reactor physics, he went in 1972 to Oklo (Gabon) to study the natural reactor of uranium which produced fission there are 2 billion years. Having been the principal private secretary of the High-commissioner of the CEA, he went to be the Director of Research for Nuclear Safety before retiring in 1984. He was also named Associated Professor at the University of Paris-XI and at the “Conservatoire National des Arts et Métiers”. Writing en 1978 avec Paul Reuss the famous Treatise of Neutronics, republished several times, he also has written for the Encyclopedia Universalis with Jules Horowitz the article Nuclear reactors published in 1980 et he participated to the collective work L’aventure de l’Atome (the Adventure of Atom) published in 1992.
(Courtesy Jean Bussac) - 24.
Martine Pujol: Modèle approché pour la diffusion inélastique des neutrons rapides [Approximated model for inelastic scattering of fast neutrons], PhD thesis at the Faculty of Science, University of Orsay, 1968.
- 25.
Michel Cadhilac, Martine Pujol: A simple model for the inelastic scattering of fast neutrons, Journal of Nuclear Energy, Vol. 21, pp58–63, 1967.
- 26.
Brigitte Rocca-Volmerange: After her PhD at the CEA in 1976, this French physicist turned to astrophysics. Today, she works at the Institut d’Astrophysique de Paris, where she specializes in the evolution model of galaxies. She teaches at the University of Paris XI in this field. Furthermore, she is also Vice-President of the association Femmes et sciences. [Women and Science]
- 27.
Brigitte Rocca-Volmerange: Approximation Q n du ralentissement dans les réacteurs nucléaires à neutrons rapides [Qn approximation of slowing-down in fast-neutron nuclear reactors], PhD thesis at the University of Orsay, 1975.
(Courtesy Pazsit)
(Courtesy La Recherche)
(Courtesy Jean Bussac)Bibliography
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.
Alain Baur, Protection contre les rayonnements : aspects physiques et méthodes de calcul [Radiation shielding: physical aspects and calculation methods], CEA/INSTN, ISBN 2-7272-0102-8, 1985, 287 pages. This is an excellent book on radiation physics and one of the most complete references for theoretical aspects [Métivier, 2006].
K.H. Bekurts, Karl Wirtz, Neutron physics, Springer Verlag, 1964, 444 pages. It is a more complete version of [Wirtz and Bekurts, 1958] in German. It is extremely clear with modern notations. There is also a significant chapter on detector calculations and on experimental methods.
Carlo Bernardini and Luisa Bonolis (editeurs), Enrico Fermi, his work and legacy, Springer, New-York, ISBN 3-540-22141-7, 2004, 410 pages. Collection of articles by scientists on the work of Enrico Fermi. Contains many anecdotes.
Kenneth M. Case, Paul F. Zweifel, Linear transport theory, Addison Wesley, USA, 1967. It is an international reference on transport theory. Impossible to talk about transport without reading this book!
Gustav Doetsch, Introduction à l’utilisation pratique de la transformation de Laplace [Introduction to the practical use of Lapalce transform], Gauthier-Villars. Paris, 1959, 198 pages. Table with lots of transforms for applications to differential equations.
Lawrence Dresner, Resonance absorption in nuclear reactors, International series of monograph in nuclear energy, Vol. 4, Pergamon press, USA, Library of Congress card number 60-14191, 1960, 132 pages. This volume is a classical text for resonance absorption. Crystal clear, it is THE reference for chapter 5.
Jeol H. Ferziger, Paul F. Zweifel, The theory of neutron slowing-down in nuclear reactor, MIT press, Massachussets, USA, Library of Congress Card Number 66-13806, 1966, 307 pages. In my opinion, the best textbook on slowing-down. I recommend the notations.
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.
Walter Greiner, Ludwig Neise, Horst Stöcker, Thermodynamique et mécanique statistique [Thermodynamics and statistical mechanics], Springer, Heidelberg, Allemagne, ISBN 3-540-66166-2, 1999, 532 pages. Very didactic.
Les génies de la science (revue), Fermi, un physicien dans la tourmente, Les génies de la science, Pour la science, Belin, 2001, 98 pages. Each issue of “Génie de la science” is dedicated to a famous scientist. This one is a very interesting and fully illustrated issue on Fermi, mostly unknown to the general public.
James G. Holbrook, Laplace transforms for electronic engineers, Pergamon Press, Oxford, United Kingdom, Library of Congress Card Number 59-12607, 1969, 2nd edition, 347 pages. The applications are mostly in the fields of electricity or networking. There is a very complete table of transforms for real networks.
Wilfred Kaplan, Operational methods for linear systems, Addison-Wesley, Library of Congress Card Number 62-9401, 1962, 577 pages. The Fourier and Laplace transforms are discussed thoroughly.
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.
Jacques Ligou, Installations nucléaires []Nuclear installations], Presses Polytechniques Universitaires Romandes, ISBN 2-88074-011-8, 1982, 2nd edition, 430 pages. Indeed, a very technology-oriented book, although it contains some theoretical background on fusion (chapter 2) and reactor physics (chapter 4).
Robert M. Mayo, Introduction to nuclear concepts for engineers, American Nuclear Society, La Grange Park, Illinois, USA, ISBN 0-89448-454-0, 1998, 361 pages. This book is more than just a good introduction.
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.
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.
Paul Reuss, Neutron Physics, EDP Sciences, collection INSTN, Paris, ISBN 978-2-7598-0041-4, 2008, 669 pages. Translation of the precis with numerous exercises.
P. Silvennoinen, Reactor core fuel management, Pergamon Press, Oxford, United Kingdom, ISBN 0-08-019853-8, 1976, 257 pages. Curiously, this book hardly deals with fuel management.
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.
B.J. Starkey, Laplace transforms for electrical engineers, Wireless Engineer, London, United Kingdom, 1954, 279 pages. Theory + tables.
Richard Stephenson, Introduction to nuclear engineering, McGraw-Hill, USA, Library of Congress Card Number 53-10623, 1954, 387 pages. Published at that time in the Chemical Engineering Series, thereby showing that reactor physics was not yet a true branch of science, this book covers field completely. Several diagrams of the first reactors are available from declassified data from the USA. There is a complete chapter on radiation shielding, showing that this problem was already central to engineering problems.
J.H. Tait, Neutron transport theory, Longmans, London, United Kingdom, 1964, 142 pages. The name of Tait, a physicist from Harwell, is known through the Bethe-Tait model for evaluating the energy dissipated during a reactivity accident. This book contains innovative approaches for slowing-down and transport solutions. Theoretical slowing-down problems are discussed (Placzek transients, the age equation, slowing-down in absorbing medium, etc.). One of the rare books dedicated exclusively to neutron transport.
Alan E. Walter, Albert B. Reynolds, Fast Breeder Reactors, Pergamon Press, ISBN 0-08-025982-0, 1981, 852 pages. This is an excellent summary with several chapters on technology and some on pure physics. Chapter II (Neutronics) should be noted for cross sections and setting them into multigroup structures, as well as problems on fast spectrum (sodium effect, void coefficient, change of spectrum, and so on).
Alvin M. Weinberg, Eugene P. Wigner, The physical theory of neutron chain reactors, Chicago press, library of congress 58-8507, 1958. This book is a monument in the history of reactor physics. It was published for the second conference of Atoms for peace in Geneva where the first data on the nuclear industry was declassified.
David Vernon Widder, The Laplace transform, Princeton University Press, USA, 1952, 3rd edition, 406 pages. Very complete in the proofs but no summary tables.
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Marguet, S. (2017). Neutron Slowing-Down. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_4
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