Abstract
Neutrons reach thermal energy following successive collisions with moderator nuclides provided they are neither absorbed nor leak out of the reactor. At this point, they can either lose energy by collision or gain energy if the target nucleus is itself in thermal motion, and can thus transfer momentum to the neutrons. Hence, an equilibrium state is obtained whereby neutrons may be assimilated as a gas of weak density, in which velocities are characterized by the moderator temperature.
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Notes
- 1.
James Clerk Maxwell (1831–1879) was a Scottish physicist. He studied at the University of Edinburgh and took an early interest in the polarization of light. He completed his studies at Cambridge and taught as of 1860 at Aberdeen, then London, at King’s College, where he developed a theory of electromagnetism for which he is world famous. His measurements of the speed of electromagnetic waves—which he noted was very close to that of light—led him to postulate that light is a wave. The Maxwell equations are founding postulates of electromagnetism and relate electricity and magnetism, thereby constituting the first step towards unification of the various forces in physics. He died at 48 leaving behind him a significant body of work.
- 2.
Ludwig Eduard Boltzmann (1844–1906) was an Austrian physicist and a key figure in modern physics thanks to his work on statistical physics. After obtaining his PhD in 1866 on the kinetic theory of gases under the supervision of Jozef Stefan, he studied in Germany and then obtained the chair of physics at Graz. He established the famous equation of entropy in statistical thermodynamics S = k LogΩ. A staunch defender of the atomic theory, he suffered from the misunderstanding of his theories. He suffered from depression and committed suicide in 1906. His name is associated with the Stefan-Boltzmann law for the power emitted by a surface as well as the Boltzmann integro-differential equation in the kinetic theory of gases, which may be applied to neutrons. His name is immortalized in the Boltzmann constant k, one of the rare constants governing the behavior of the Universe.
- 3.
Mathematically, functional n(E) is different from n(v), but for the sake of clarity, the same terminology n is used. Its units are defined according to the context in which it is used.
- 4.
The density of thermal neutrons is of the order of 107 to 108 neutrons per cm3, which is negligible compared to the number of molecules per unit volume of the current moderator, even in gaseous form.
- 5.
R.C. Tolman, Proc. Natl. Acad. Am. 11 (1925), 436.
- 6.
R.H. Fowler, E.A. Milne, Proc. Nat. Acad. Am. 22 (1925), 400.
- 7.
V.V. Smelov, On the question of neutron thermalisation, Journal of Atomic Energy (USSR), 10, 1957.
- 8.
Ernest Wilkins Jr (1923–2011) was an African-American mathematician who was extremely precocious (he attended the University of Chicago at the age of 13). In 1942, he obtained his PhD in mathematics at the age of 19. He is in fact the 7th African-American to receive this distinguished title. At the same time, he obtained his Bachelor of Mechanical Engineering in 1942 at the University of New-York, which was followed by a Masters in the same field in 1960. He taught mathematics at the Tuskegee Institute from 1943 to 1944. Wilkins was associated with the Manhattan Project from 1944 to 1946. He taught mathematics and carried out his research at the renowned Metallurgical Laboratory of the University of Chicago under the supervision of Enrico Fermi. From 1946 to 1950, Wilkins was in charge of mathematical studies for the American Optical Company; from 1950 to 1955, he was Senior Mathematician for the Nuclear Development Corporation of America. Furthermore, he was also responsible of the department of physics and mathematics (1958–1959), before becoming its director (1960–1965). Dr. Wilkins also worked for the department of theoretical physics at General Atomic from 1960 to 1970. In 1970, he was Distinguished Professor of Applied Mathematical Physics at Howard University. From 1974 to 1975, he served as President of ANS. In 1976, he was elected to the National Academy of Engineering of the USA. Besides his work in pure mathematics, he also worked in the 1940s to 1950s on the slowing-down and thermalization of neutrons, and then on photon transport and radiation protection.
- 9.
Eugène Paul Wigner, J. Ernest Wilkins Jr: Effect of the temperature of the moderator on the speed distribution of neutrons with numerical calculations for H as a moderator, AECD 2275 (1944).
- 10.
Jules Horowitz (1921–1995) is considered by many as the father of reactor physics in France. A brilliant physicist, he joined the CEA in 1946 after graduating from Ecole Polytechnique, where his studies were interrupted by the war. He joined CEA at the same time as Michel Trocheris, Claude Bloch and Anatole Abragam, with the group being known collectively as the “Three Musketeers” (Lefebvre 2002, p33). He laid down the founding elements of reactor physics by calculating the Zoé pile on the advice of Lew Kowarski. During his work on experimental reactors as director of the atomic piles, his requirements led to the introduction of the first scientific computers in France at the department of mathematical physics, where they were developed by Albert Amouyal (Lefebvre 2002, p116). His scientific work has been re-edited in the collection of important actors at the CEA. The new materials test reactor presently being built at Cadarache bears his name.
- 11.
Oleg Tretiakoff, a former student at the Ecole Polytechnique (1950–1952), was an engineer in the Armaments Corps at the French Defense Ministry. He remained at the CEA throughout his whole career (1955–1981), retiring as Chief Weapons Engineer. In 1975, with his wife he invented a piezo-electric system that allows the decoding of text printed in Braille for the blind. He applied for a patent in 1977. Nowadays, he lives in Florida where he heads a company working on these concepts.
- 12.
J. Horowitz, O. Tretiakoff: Effective cross sections for thermal reactors, European American Nuclear Data Committee, EANDC (E) 14, 1960, referred to in the work of Jules Horowitz. See also Francis Vitton : Mesures d’indices de spectre à César [Measurement of the Cesar spectrum indices]. His PhD thesis at the University of Orsay (1967) describes the complete model.
- 13.
Michel Cadilhac: Méthodes théoriques pour l’étude de la thermalisation des neutrons dans les milieux infinis et homogènes [Theoretical methods for the study of neutron thermalization in infinite and homogeneous absorbing media], PhD thesis at the Faculty of Science of Paris (1964) and CEA technical report referenced as CEA R 2368. Michel Cadilhac (1934–) studied at the Ecole Normale Supérieure in Saint-Cloud before passing his aggrégation (competitive examination to become a lecturer). He joined the CEA, where he developed several theoretical notions related to thermalization (diffusion coefficient, spectrum, etc.). His thesis on the subject was presided by Jacques Yvon, assisted by Jules Horowitz and Austin Blaquière as examiners. It contained highly innovative ideas for the time and improved the Wigner-Wilkins model, making it one of the most important theses in French neutron physics. His work has been translated into English (Theoretical methods for the study of neutron thermalization in infinite and homogeneous absorbing media, Euratom translation EURAEC-1203, 1963). As a professor at the University of Marseille, he focused on the diffraction of electromagnetic waves in crystals.
(The Marguet collection)
- 14.
M. Cadilhac, J. Horowitz, J.L. Soulé: Some mathematical and physical remarks on neutron thermalization in infinite homogeneous systems, Conference on neutron thermalization, Brookhaven (April 1962).
- 15.
Jean-Pierre de Brion: Application du modèle secondaire à la thermalisation en milieu hétérogène [Application of the secondary model to thermalization in heterogeneous media], PhD thesis at the University of Paris, 1966.
Jean-Pierre de Brion (1936–1998). After his studies at Polytechnique (entry year 1957), he completed his military service in Algeria before joining the CEA around 1961–1962. In November 1964, he authored technical report SPM No 306: “Calcul de la densité neutronique dans une cellule hétérogène [Calculation of neutron density in a heterogeneous cell]”. He defended his thesis on March 2, 1966 with a jury composed of Jules Horowitz, and Austin Blaquière and Oleg Tretiakoff as examiners and Michel Livolant as an invited member. In 1970, he married Monique de Masfrand, with whom he had three children. He defended a higher doctorate in May 1973. He then left reactor physics for particle physics, in which he pursued an international career. After 2 years at Rochester in the United States, then at Brookhaven National Laboratory on Long Island, he joined CERN with Carlo Rubbia, and published several papers in the field of particle physics. Around 1985, he left particle physics and joined the Direction of Military Applications at Bruyères le Châtel for a more confidential career at BIII/PTN Service (work on the ELSA electron accelerator), before retiring in 1997. He died on August 18, 1998.
(Courtesy De Brion family)
- 16.
REMARK—For heavy nuclides, ξ ≈ 2/A.
- 17.
Coveyou , Bate and Osborn , in Reactor Physics Constants ANL-5800, p89, obtained from the Monte Carlo calculations for moderators of different masses.
- 18.
Carl H. Westcott (1912–1977) graduated from Cambridge in 1933 and started his career at the Cavendish Laboratory. He worked from 1937 to 1940 at the University of Aberdeen, then specialized in radar technology during the war. He co-authored a book on the subject in 1948. In 1944, he immigrated to Canada and became a professor at the McGill Institute at Montreal in 1949. In 1954, he joined the Chalk River center where he specialized in the processing of cross sections. Other than the formalism that bears his name, he was at the origin of the idea that a linear accelerator can simultaneously accelerate positive and negative ions, thereby significantly increasing the potential value of such machines.
In this photograph from 1934 with Ernest Rutherford in the foreground, Carl Westcott is the young man walking by in the background. (Public Domain)
- 19.
C. Wagemans, P. Schillebeeckx , A.J. Deruytter and R. Barthélémy : The Subthermal neutron induced fission cross sections of the common fissile isotopes and their impact on the Westcott g-factor, Nuclear data for science and technology (1988 Mito), 91–95 (1988).
- 20.
P. Cogné : Etude des caractéristiques nucléaires de l’empilement critique MARIUS [Study of the nuclear characteristics of the Marius critical pile], technical report CEA/SPM No 567, August 1959. For the general properties of graphite, see (Reynolds 1968).
- 21.
I.M. Cohen , M. Arrondo , M.A. Arribère, M.C. Fornaciari Iljadica: A method for fast determination of the α parameter in nuclear reactors, Nuclear Science and Engineering, 154, p110–117 (2006).
- 22.
C.H. Westcott: Effective cross section values for well-moderated thermal reactor spectra (3rd edition, corrected), CCRP-960, November 1960.
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Marguet, S. (2017). Thermalization of Neutrons. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_7
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