Abstract
The theoretical analysis of nuclear reactors began during World War II. The first steps in the race to acquire this knowledge took place in the United States with the construction of the Chicago Pile 1 (CP1) under the stands of the Chicago stadium. The simplest analysis consists in supposing that the pile is a homogeneous multiplying material for neutrons and the neutron flux is sought for under the criticality hypothesis. Mathematical calculations that are quite simple allow the study of the spatial flux distribution in a one-energy group theory.
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Notes
- 1.
Hermann Ludwig Ferdinand von Helmholtz (1821–1894). After starting out as a military doctor, then an anatomy professor, he became a professor of physics at the University of Berlin. He then worked on experiments in the field of electrophysiology (measurement of the nervous influx), optics and acoustics. He is famous for his definition of potential energy.
Hermann Von Helmholtz (Public domain) - 2.
For a quick primer on Bessel functions, refer to (Harper 1976, p. 186).
- 3.
Viktor Amaszaspovitch Ambartsumian, Reports from the Academy of Sciences of the USSR, 38, 299 (1943). Ambartsumian (1908–1996) was an Armenian astrophysicist. After completing his studies at the University of Leningrad, he worked at the Poulkovo observatory from 1928 to 1931. After the war, he founded the Byarakan observatory and wrote a book on theoretical astrophysics in 1952, in which he applied the invariance principle for radiation transfers that would be generalized under the form of invariant imbedding by Bellman. He was appointed president of the University of Erevan. His image was printed on an Armenian bank note in 1998. Note that his name is transliterated from Armenian in several forms: Ambarzumian or Hambardzumyan.
(The Marguet collection) - 4.
Ricard Ernest Bellman (1920–1984) was an American mathematician. After his studies at the Brooklyn college of the University of Wisconsin, he took part in the theoretical studies at Los Alamos on the Manhattan project. He obtained his PhD from Princeton in 1946 before teaching at Southern California University. Bellman is famous for inventing dynamic programming, an approach in the optimal control theory (Hamilton-Jacobi-Bellman equation). Bellman has published some forty books in the field of operational research and mathematics! His contribution to the nuclear field is the development of the invariant imbedding method used in radiative transfer problems. A good summary can be found in Invariant Imbedding Multipoint Boundary-Value Problems, Journal of Mathematical analysis and Applications 24, pp. 461–466 (1968).
- 5.
The albedo characterizes the reflection gain. It can be greater than 1 in a multiplicative medium. This notion will be discussed broadly in Chap. 13 on the reflector.
- 6.
Friedrich Wilhelm Bessel (1784–1846) was a German mathematician and physicist. He is known for the Bessel functions, despite the fact the latter were discovered by Daniel Bernoulli. At an early age, Bessel displayed a deep interest in astronomy and became the assistant of Johann Schöter at the Lilienthal observatory near Bremen, despite having no university degree. At 25, he headed the brand-new Konigsberg observatory. In 1811, he graduated with an honorary PhD from Gottingen University, thanks to a recommendation from Carl Gauss. While at Gottingen, Bessel undertook a great deal of work on star mapping and prepared precise parallax calculations. In 1844, his works on the deviation of the trajectory of Sirius led to the discovery of Sirius B (non-luminous). He corrected the simple variance formula (dividing by n−1 instead of n, since use of the average formula removes one degree of freedom).
(The Marguet collection) - 7.
For further explanations on the Bessel functions, see (Watson 1980).
- 8.
For the various types of boundary conditions, see (Wachspress 1966, p. 30).
- 9.
As done by K. Schwinkendorf and C. Eberle in: An angular leakage correction for modeling a hemisphere using one-dimensional spherical coordinates, Nuclear Science and Engineering 143, 47–60 (2003).
- 10.
M. Itagari , Y. Miyoshi : A geometric buckling expression for regular polygons II: analyses based on the multiple reciprocity boundary element method, Nuclear Technology, 103, 392–402, 1993.
- 11.
The Wielandt method is detailed in A. Daneri , M. Michelini , G; Toselli : Anisotropic diffusion calculations in generalized x,y-geometry, Proceedings of a seminar on numerical reactor calculations held in Vienna by the AIEA, 17–21 January 1972, pp. 487–496 (1972). Helmut Wielandt (1910–2001) was a German mathematician. After his studies at the University of Berlin in 1929, he worked as an assistant at the University of Tubingen, before attending the Kaiser Wilhelm Institute during the war. He worked on aerodynamics problems (eigenvalue calculations for operators that are not self-adjoint), for which he developed the algorithm that bears his name. As of 1952, he became the editor of the prestigious Mathematische Zeitschrift, while still working at the University of Tubingen.
Helmut Wielandt (University of Berlin) - 12.
Violaine Louvet : Etude numérique de problèmes de diffusion neutronique en présence de singularités [Numerical study of neutron diffusion problems with singularities], PhD paper at the University of Franche-Comté, 1998.
- 13.
Dan G. Cacuci: Two dimensional geometrical corner singularities in neutron diffusion: part 1: analysis, Nuclear Science and Engineering, 128, 1–16 (1998). Dan Cacuci. After his Ph. D. at the University of Columbia in 1978, he worked at Oak Ridge National Laboratory until 1988. He taught at several US universities (California, Illinois, Michigan, Virginia). From 1993 to 2004, he was an institute director at the nuclear center of Karlsruhe. He is also the editor of the famous journal Nuclear Science and Engineering since 1984, as well as the huge Handbook of Nuclear Engineering (3500 pages!) produced by Springer in 2010. Professor Cacuci is the recipient of several prizes, including the Wigner medal, the Seaborg medal and the Lawrence medal for his work in general on reactor physics.
- 14.
Introducing positive buckling \( {B}_z^2 \) has a physical meaning if the curvature \( \frac{\partial^2\varPhi }{\partial {z}^2} \) is negative. Conversely, the notation \( {B}_z^2 \) must be taken as a real value that may be negative.
Hermann Von Helmholtz (Public domain)
(The Marguet collection)
(The Marguet collection)
Helmut Wielandt (University of Berlin)Bibliography
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Marguet, S. (2017). Critical Homogeneous Reactor Theory. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_12
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