Abstract
As discussed in previous chapters, the neutron interacts with the target nucleus as an incident wave. Indeed, the interaction probabilities at certain energies have a peaked shape due to quantum physics considerations. From the very inception of nuclear physics, more precise knowledge of cross sections as a function of energy and their modeling in a resonance zone were deemed essential. This is the goal of resonant absorption theory.
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Notes
- 1.
P.L. Kapur, R.E. Peirls, Proceedings of the Royal Society (London), A166, 277 (1938).
- 2.
Gregory Breit (1899–1981) was an American physicist of Russian origin. After working as Paul Ehrenfest’s assistant at the University of Leide, he left for the United States, where he was a professor at a number of prestigious universities. His work on the Dirac equation led to a relativistic wave theory named after him. His work on resonance with Wigner led to the forms of Breit-Wigner cross sections. At the outbreak of war, Breit was the leader of the team of scientists that worked on the atomic bomb. Due to differences of opinion, he was replaced on the Manhattan Project by Robert Oppenheimer.
Eugen Paul (Jenő Pálin Hungarian) Wigner (1902–1995) was a Hungarian physicist who immigrated to the United States. He joined the faculty of Princeton University and became a US citizen in 1937. The same year, he introduced the concept of total isospin tuples vector for nucleon systems to improve understanding of nuclear reactions (Mathieu 1991, p. 234). In 1939, he was part of the group of five scientists (that included Albert Einstein) who warned President Roosevelt of the potential military use of atomic energy by Germany. During the Second World War, he contributed to the design of plutonium reactors and worked on the Manhattan Project. In 1963 he was awarded the Nobel Prize for Physics along with Jensen and Maria Goeppert-Mayer for the discovery of the symmetry principle. He belongs to the highly renowned Hungarian School of Physicists, along with Leo Szilard, Edward Teller and John von Neumann. Furthermore, he was one of the first to propose the use of water as a moderator and a coolant in reactors.
- 3.
For details on the Heisenberg uncertainty principle, see (De Broglie 1982).
- 4.
G. Breit and E.P. Wigner: Capture of slow neutrons, Phys. Rev. 49, p. 519 (1936).
- 5.
g N = 2I N + 1 is the statistical weight of the target nucleus, and g n is that of the neutron (\( {g}_n=2\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1=2 \)).
- 6.
More precisely, the potential cross section is equal to σ pot = 4π λ 2 g sin2 δ 1, where δ 1 = R/λ − Arctg(R/λ) is the neutron wavelength (not to be confused with the decay constant) and is the statistical weight of the compound nucleus.
- 7.
L. Erradi : Etude des effets de température dans les réseaux caractéristiques des réacteurs nucléaires de la filière à eau ordinaire[‘Study of the effects of temperature in representative lattices of light water reactors’], PhD, Orsay, 1982.
- 8.
On the distribution of resonances, Tran Quoc Thuong provides a very good reference : Analyse statistique des distributions des largeurs réduites partielles[‘Statistical analysis of the distributions of reduced partial widths’], Doctoral thesis, Université de Paris-VII (1973).
- 9.
C.E. Porter, R.G. Thomas: Fluctuations of nuclear reaction widths, Phys. Rev. Vol. 104 n°2 pp. 483–491 (October 1956).
- 10.
A very good summary is provided in : Théorie de l’absorption résonnante des neutrons[‘Theory of resonant neutron absorption’], Technical report CEA-N-2679 (1991).
- 11.
Igor Illich Bondarenko (1926–1964) was a Soviet physicist. After reading physics at the University of Moscow in 1950, he worked for the Institute of Physics and Energy in Obninsk. He studied the measurement of cross sections in the fast reactor framework. He finished his career as the Vice-Director of the Institute of Physics and Energy of the State Committee for the Control of Soviet Nuclear Energy. He was awarded the Lenin Prize in 1960. He was at the origin of the idea of the Fast Pulse Reactor which was built at Dubna. This reactor produced power peaks for a short time interval for experimental purposes. His work on cross sections was published in 1958 and was widely read in the west. It led to allowed comparison of calculations using the American evaluations. His cross sections were weighted in the resonance zone by self-shielding factors. Furthermore, his 26-group energy structure was used at EDF for the computation of power reactors.
(Public domain) - 12.
This formalism is described in many references, but the work of Alain Santamarina is most suited for its clarity: Calcul de l’absorption résonnante des neutrons par les isotopes de l’Uranium et du Plutonium dans un réacteur nucléaire[‘Calculation of the resonant absorption of neutrons by isotopes of uranium and plutonium in a nuclear reactor’], PhD., Orsay, 1973. Alain Santamarina (b. 1948) spent his entire career at the CEA after obtaining his Masters in reactor physics in 1971, followed by his PhD. A world-renowned specialist in nuclear data evaluations and neutron calculation schemes, he was at the head of the EOLE and MINERVE experimental reactors from 1984 to 1988. He subsequently became director of research at CEA. His recent works at Cadarache (France) led to the set-up of a new energy mesh structure with 281 groups for APOLLO2.
Alain Santamarina in 2011 (Courtesy Santamarina) - 13.
Michel Livolant (b. 1938). After his studies at the Ecole Polytechnique, he completed the second Master’s degree course in reactor physics to be given at Saclay in 1963. His original theoretical works are on self-shielding theory. His rapid career path at CEA saw him nominated head of the “Institut de Protection et de Sûreté Nucléaire”, which would later become the IRSN.
Françoise Jeanpierre -Gantenbein (b. 1944) obtained her PhD in physics in 1969 after a BSc in physics at University of Orsay (1966) and a Master’s in reactor physics in 1967. She began her career at the CEA/Service de Physique Mathématique at Saclay from 1967 to 1971. Later, she worked on seismic mechanics. From 1988 to 1995, she was head of the Seismic Studies laboratory where the European TAMARIS facility is located. She became a senior expert in seismic questions at CEA. She coordinated research programs in safety from 1998 until retiring in 2007.
- 14.
Historically, the notation was 0 for the resonant nucleus and 1 for the light moderator. We preferred to use uniform notations with c for fuel (combustible), composed of U for the heavy resonant nucleus (referring to Uranium, the main fuel component), and m for mixed moderator and fuel, which is surrounded by an external moderator M.
- 15.
For the sake of clarity, the models for the slowing-down operator are grouped in a later section so as to introduce the Livolant-Jeanpierre model first, along with the relevant notations.
- 16.
Mireille Coste-Delclaux (b. 1956). After her studies in the mathematics section of the most famous “Ecole Normale Supérieure” in Fontenay (France), she passed the French “aggrégation” examination in mathematics, minoring in numerical analysis. She worked at the CISI Company, where she was involved in scientific programming languages. She later joined CEA/SERMA, specializing in problems dealing with nuclear data and self-shielding. Her PhD thesis at the CNAM: “Modélisation du phénomène d’autoprotection dans le code de transport multigroupe APOLLO2” [‘Modelling of self-shielding using APOLLO2 multi-group transport code’] (2006) is the most complete reference on self-shielding in French, and it is a masterpiece of scientific work that would merit to be published in a textbook form. She has been a senior expert at CEA since 2005 and teaches neutronics at ENSTA, a French engineer school.
- 17.
J. Chernick: The theory of uranium water lattices, Proceedings of the U.N. Conference on Peaceful Uses of Atomic Energy, Geneva, P603 (1955).
- 18.
This concept will be described in more details in the chapter on the Boltzmann equation, and extended in the chapter on the heterogeneous reactor. The calculation is not detailed here. It is presented thoroughly in Wolfgang Rothenstein : Collision probabilities and resonance integrals for lattices, Nuclear Science and Engineering, 7, pp. 162–171 (1960).
- 19.
George Irving Bell (1926–2000) was an American physicist. After obtaining his BSc in physics at Harvard University, he studied theoretical physics with Hans Bethe and obtained his PhD in 1951. He then worked for the theoretical physics division of Los Alamos in New Mexico. As a member of the T division, he worked on neutron transport and on the first thermonuclear bomb. In the 1960s, he worked on biology and immunology and indeed founded the theoretical biology and biophysics group in 1974. Similarly, in 1988, he was pivotal in the setting up of the Center for Human Genome Studies. In addition, he was very keen on mountaineering, and attempted the first ascent of the famous K2 (the attempt was thwarted by terrible weather conditions). However, in 1960, he took part in the first ascent of Mount Masherbrum.
- 20.
The complete work of Mireille Coste-Delclaux on self-shielding is found in her PhD thesis: Modélisation du phénomène d’autoprotection dans le code de transport multigroupe APOLLO2[‘Modelling of self-shielding using the APOLLO2 multi-group transport code’], University of Orsay, (2006), edited by the CEA under reference number CEA-R-6114, 667 pages, which is the most complete French-language reference work on self-shielding.
- 21.
M.N. Nicolaev, V.F. Khokhlov: A system of subgroup constants, Atomizdat, 1967
- 22.
A. Hébert , M. Coste : Computing moment-based probability tables for self-shielding calculations in lattice codes, Nuclear Science and Engineering, 142, pp. 245–257 (2002)
- 23.
T. Ushio , T. Takeda : The characteristics and subgroup methods in square light water reactor cell calculations, Nuclear Science and Engineering, 143, pp. 61–80 (2003)
- 24.
O.K. Bouhelal and P. Ribon: Tables de probabilités non statistiques, description des effets du ralentissement [‘Non-statistical tables: description of the effect of slowing’], Proc. Physor, Marseille, France, pIX-1 à IX-13, 1990. Pierre Ribon (b. 1932) graduated with an engineering degree from Arts et Métiers and ESE. After joining the CEA in 1956, in 1969 he defended his doctoral thesis,“L’étude de quelques propriétés des noyaux excités des noyaux composés, formés par l’interaction de neutrons lents avec le rhodium 103, le xénon, le gadolinium et le thorium 232” [‘The study of certain properties of excited nuclei of composite nuclei formed through the interaction of slow neutrons with rhodium 103, xenon, gadolinium and thorium 232’]. He is a renowned expert on nuclear data applied to reactor physics and although retired, he still maintains the CALENDF code package, the French equivalent of the American NJOY code.
(Courtesy Ribon) - 25.
Vladimirov (1979) is a good reference work on distribution theory.
- 26.
Alain Aggery : Calculs de référence avec un maillage multigroupe fin sur des assemblages critiques par APOLLO2[‘Reference calculations with fine multi-group mesh for critical assemblies using APOLLO2’], PhD thesis at the University of Aix-Marseille (1999). A detailed description of the probability table method is presented in this thesis.
- 27.
A. Khairallah, J. Recolin: Calcul de l’autoprotection résonnante dans les cellules complexes par la méthode des sous-groupes[‘Calculation of resonant self-shielding for complex cells using the subgroup method’], Proceedings of a seminar on numerical reactor calculations held in Vienna by the AIEA, 17–21 January 1972, pp. 305–317 (1972).
- 28.
Olivier Bouland : Amélioration du calcul de l’autoprotection des résonances résolues par un traitement quasi-exact du ralentissement des neutrons [‘Improved calculation of self-shielding of resonances solved by quasi-exact approach of the slowing-down of neutrons’], PhD thesis, University of d’Orsay (1994).
(Public domain)
Alain Santamarina in 2011 (Courtesy Santamarina)
(Courtesy Ribon)Bibliography
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Igor Il’ich Bondarenko (editor), Group constants for nuclear reactor calculations, Consultant Bureau, New-York. Library of Congress Card Number 64-23252, 1964, 151 pages. This book compiling 26 energy-group cross-sections had a huge success in its time, because of the English translation from Russian (surprising in a context of Cold War!). The condensation flux spectrum, inversely proportional to the total cross section in order to take into account the flux decrease in the resonance domain, is called « Bondarenko spectrum ». This group of sections is a seminal multigroup library and was used in EDF in the 60s, thanks to a French translation. The frequent use of 26-groups calculation in reactor physics is a heritage of Bondarenko procedure.
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Marguet, S. (2017). Resonant Absorption. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_5
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