On the spectrum of discrete-ordinates neutron transport problems
DOI:
https://doi.org/10.15392/bjrs.v9i2.1662Keywords:
transport equation, discrete ordinates, spectral analysis, discrete eigenvalues, anisotropic scatteringAbstract
Over the last six decades, the discrete spectrum of the neutron transport operator has been widely studied. Important theoretical results can be found in the literature regarding the one-speed linear transport equation with anisotropic scattering. In this work, the discrete-ordinates (SN) transport problem with anisotropic scattering has been considered and the discrete spectrum results in multiplying media have been corroborated. The numerical results obtained for the dominant SN eigenvalues agreed with the ones for the analytic problem reported in the literature up to a triplet scattering order. A compact methodology to perform the spectral analysis to multigroup SN problems with high anisotropy order in the scattering and fission reactions is also presented in this paper.Downloads
References
REFERENCES
CASE, K. M. Elementary Solutions of the Transport Equation and Their Applications. Ann Phys, v. 9, p. 1–23, 1960.
ZWEIFEL, P. F. Kenneth Case and his Singular “Eigenfunctions”. Transport Theor Stat, v. 41, p. 406–417, 2012.
MCCORMICK, N. J.; KUSCER, I. Singular eigenfunction expansions in neutron transport theory. In: LEWINS, J.; HENLEY, E. J. Advances in Nuclear Science and Technology, Academic Press, New York, USA, 1973. p. 181–282.
SAHNI, D. C.; TURECI, R. G. Discrete Eigenvalues of Case Spectrum with Anisotropic Scattering. Nucl Sci Eng, v. 191, p. 121–135, 2018.
MODAK, R. S.; SAHNI, D. C.; PARANJAPE, S. D. Evaluation of higher k–eigenvalues of the neutron transport equation by S_N–method. Ann Nucl Energy, v. 22, p. 359–366, 1995.
MIKA, J. R. Neutron Transport with Anisotropic Scattering. Nucl Sci Eng, v. 11, p. 415–427, 1961.
GARCIA, R. D. M.; SIEWERT, C. E. On Discrete Spectrum Calculations in Radiative Transfer. J Quant Spectrosc Ra, v. 42, p. 385–394, 1989.
VAN DEN EYNDE, G. M.; BEAUWENS, R.; MUND, E. Calculating the Discrete Spectrum of the Transport Operator with Arbitrary Order Anisotropic Scattering. Transport Theor Stat, v. 36, p. 179–197, 2007.
PRINJA, A.K.; LARSEN, E.W. General principles of neutron transport. In: CACUCI, D.G. Handbook of nuclear engineering, 1st ed., New York, USA: Springer Science + Business Media, 2010. p.427–542.
MORATO, S.; BERNAL, A. ; Miró, R.; Roman, J. E. ; Verdú, G.; Calculation of lambda modes of the multi–group neutron transport equation using the discrete ordinates and Finite Difference Method. Ann Nucl Energy, v. 137, p. 107077, 2020.
ALGLIB PROJECT, ALGLIB–numerical analysis library 1999–2021. Available at Last accessed: 14 Dec. 2020.
DE ABREU, M. P.; ALVES FILHO, H.; BARROS, R. C. A numerical method for multigroup slab–geometry eigenvalue problems in transport theory with no spatial truncation error Transport Theor Stat, v. 25, p. 61–83, 1996.
BARROS, R. C.; ALVES FILHO, H.; ORELLANA E. T. V.; DA SILVA, F. C.; DO COUTO, N.; DOMINGUEZ, D. S.; HERNÁNDEZ, C. R. G. The application of spectral nodal methods to discrete ordinates and diffusion problems in Cartesian geometry for neutron multiplying system. Prog Nucl Energy, v. 42, p. 385–426, 2003.
MORAES, L. R. C.; ALVES FILHO, H.; BARROS, R. C. Calculation of Neutron Interior Source Distribution Within Subcritical Fission–Chain Reacting Systems for a Prescribed Power Density Generation. In: INTERNATIONAL NUCLEAR ATLANTIC CONFERENCE, 2017, Belo Horizonte, Brazil, 2017. Associação Brasileira de Energia Nuclear (ABEN), 2017. P. 1–14.
BARROS, R. C.; LARSEN, E. W. A numerical method for one–group slab–geometry discrete ordinates problems with no spatial truncation error. Nucl Sci Eng, v. 104, p. 199–208, 1990.
DE ABREU, M. P. Numerical methods for the generation of the spectrum of the multigroup slab–geometry discrete ordinates operator in neutron transport theory. Ann Nucl Energy, v. 29, p. 1837–1853, 2002.
SILVA , D. M.; LYDIA, E. J.; GUIDA, M. R.; ZANI, J. H.; ALVES FILHO, H.; BARROS, R. C. Analytical methods for computational modeling of fixed–source slab–geometry discrete ordinates transport problems: Response matrix and hybrid S_N. Prog Nucl Energy, v. 69, p. 77–84, 2013.
MENEZES, W. A.; ALVES FILHO, H.; BARROS, R.C. Spectral Green’s function nodal method for multigroup S_N problems with anisotropic scattering in slab–geometry non–multiplying media. Ann Nucl Energy, v. 64, p. 270–275, 2014.
OLIVA, A. M.; ALVES FILHO, H.; SILVA , D. M.; GARCIA, C. R. The spectral nodal method applied to multigroup S_N neutron transport problems in One–Dimensional geometry with Fixed–Source. Prog Nucl Energy, v. 105, p.106–113, 2018.
CURBELO, J. P.; DA SILVA, O. P.; BARROS, R. C. An adjoint technique applied to slab–geometry source–detector problems using the generalized spectral Green's function nodal method. J Comput Theor Transp, v. 47, p. 278–299, 2018.
EDWARDS, C. H.; PENNEY, D. E. Differential Equations & Linear Algebra, 6th ed. USA: Pearson, 2008.
Published
Issue
Section
License
Copyright (c) 2021 Brazilian Journal of Radiation Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
Licensing: The BJRS articles are licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
How to Cite
