Critical parameters and decay constants for one-speed neutrons in slabs and spheres with anisotropic scattering

C. Yíldíz

doi:10.1515/kern-2001-0107

Enjoy 40% off

academic books on De Gruyter Brill *

Article

Critical parameters and decay constants for one-speed neutrons in slabs and spheres with anisotropic scattering

C. Yíldíz

Published/Copyright: March 17, 2022

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Kerntechnik Volume 66 Issue 5-6

Abstract

Time-dependent, one-speed neutron transport equations with strong forward and backward scattering together with isotropic scattering are studied in homogeneous slabs and spheres. First, a simple formal equivalence between the transport equations for a critical and for a time-decaying system is established. Then, the transport equation is converted into a more conventional one. The FN method of solving the resulting transport equation is applied to the calculation of the critical parameters and decay constants for the fundamental mode of the flux distribution and one-speed neutrons in spheres and infinite slabs. Numerical results are given for a number of significant figures and compared with those already available in the literature.

Abstract

Untersucht werden zeitabhängige Transportgleichungen für Neutronen gleicher Geschwindigkeit mit starker Vorwärts- und Rückwärtsstreuung zusammen mit isotroper Streuung in homogenen Stäben und Kugeln. Ausgehend von der formalen Äquivalenz zwischen den Gleichungen für ein kritisches und ein zeitabhängiges System wird die Transportgleichung in eine eher konventionelle Form umgewandelt. Die Fn-Methode zur Lösung dieser resultierenden Transportgleichung wird angewendet für die Berechnung der kritischen Parameter und Zerfallskonstanten für Neutronen gleicher Geschwindigkeit in Kugeln und Stäben. Numerische Ergebnisse werden für eine Reihe signifikanter Fälle vorgestellt und mit Literaturdaten verglichen.

Acknowledgments

The author would like to express his gratitude to Professor Nils G. Sjöstrand, Department of Reactor Physics, Chalmers University of Technology, S-41296 Göteborg, Sweden, and Professor D. C. Sahni, Theoretical Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India, for sending several matters relating to this manuscript. This research was partially supported by the Istanbul Technical University Research Foundation under grant ITURF-1372.

References

1 Case, K. M.; Zweifel, P. F: Linear Transport Theory. Addison-Wesley, New-York, (1967)Search in Google Scholar

2 Kaper, H. G.; Lindeman, A. J.; Leaf, G. K.: Bench Values for the Slab and Sphere Criticality Problem in One-Group Neutron Transport Theory. 54 (1974) 9410.13182/NSE54-94Search in Google Scholar

3 Sahni, D. C.; Sjöstrand, N. G.: Criticality and Time Eigenvalues in One-Speed Neutron Transport. Progr. Nucl. Energy 23 (1990) 24110.1016/0149-1970(90)90004-OSearch in Google Scholar

4 Siewert, C. E.; Williams, M. R. R.: The Effect of Anisotropic Scattering Using on a Synthetic the Critical Kernel. Slab J. Problem Phys. D.: in Appl. Neutron Phys. Transport 11 (1977) Theory 203110.1088/0022-3727/10/15/007Search in Google Scholar

5 Williams, M. M. R.: The Energy-Dependent Backward-Forward Isotropic Scattering Model with Some Application to the Neutron Transport Equation. Ann. Nucl Energy 12 (1985) 16710.1016/0306-4549(85)90042-8Search in Google Scholar

6 Sahni, D. C.; Sjöstrand, N. G.; Garis, N. S.: Criticality and Time Eigenvalues for One-Speed Neutrons in a Slabs with forward and Backward Scattering. J. Phys. D.: Appl. Phys. 25 (1992) 138110.1088/0022-3727/25/10/001Search in Google Scholar

7 Carlvik, I.: Monoenergetic Critical Parameters and Decay Constans for Small Homogeneous Spheres and Thin Homogenous Slabs. Nucl. Sci. Eng. 31 (1968) 29510.13182/NSE31-295Search in Google Scholar

8 Sharma, A.; Sahni, D. C.: Evaluation of Fundamentale Decay Constants by the SN Method. Annals Nucl. Energy 20 (1993) 44910.1016/0306-4549(93)90086-5Search in Google Scholar

9 Sanchez, R.; McCormick, N. J.: A Review of Neutron Transport Approximations. Nucl. Sci. Eng. 80 (1982) 48110.13182/NSE80-04-481Search in Google Scholar

10 Tezcan, C.; Yíldíz, C.: The Criticality Problems with the Fn Methodfor the FBIS Model. Annals Nucl. Energy 13 (1986) 34510.1016/0306-4549(86)90091-5Search in Google Scholar

11 Sahni, D. C.; Sjöstrand, N. G.: Non-Monotonic Variation of the Criticality Factor with the Degree of Anisotropy in One-Speed Neutron Transport Transp. Theory Statist. Phys. 20 (1991) 33910.1080/00411459108203910Search in Google Scholar

12 Hartman, J.: A Method to Solve the Integral Transport Equation Employing a Spatial Legendre Expansion. EUR 533e, Joint Nuclear Research Centre, Ispra, (1975)Search in Google Scholar

13 Wosnicka, U; Sjöstrand, N. G.: On a Problem with Strong Forward and Backward Neutron Scattering in Spheres. Ann. Nucl. Engergy 24 (1997) 127710.1016/S0306-4549(97)00043-1Search in Google Scholar

14 Hemd, H.: The Integral Transform Method for Neutron Transport Problems. Nucl. Sci. Eng. 40 (1970) 22410.13182/NSE70-A19684Search in Google Scholar

15 Garis, N. S.; Pazsit, I.; Sahni, D. C.: Generalization of the Eigenvalue Two-Region Problem Systems. of the Progr. One-Speed Nucl. Neutron Energy 27 Transport (1992) 305 Equation in10.1016/0149-1970(92)90009-RSearch in Google Scholar

16 Sahni, D. C.; Sjöstrand, N. G.: Time Eigenvalue Spectrum for One_Speed Neutron Transport In Spheres with Strong ForwardBackward Scattering. Annals Nucl. Energy 26 (1999) 34710.1016/S0306-4549(98)00072-3Search in Google Scholar

17 Dahl, E. B.; Protopopescu, V.; Sjöstrand, N. G.: On the Relation Between Decay Constants and Critical Parameters in Monoenergetic Neutron Transport. Nucl. Sci. Eng. 83 (1983 ) 37410.13182/NSE83-A17570Search in Google Scholar

18 Hemd, H.; Kschwendt, H.: Solution of the Transport Integral Equation with Anisotropic Scattering. J. Comp. Phys. 10 (1972) 53410.1016/0021-9991(72)90051-4Search in Google Scholar

19 Dahl, E. B.; Sjöstrand, N. G.: Time-Eigenvalue Spectra for One-Speed Neutrons in Systems with Vacuum Boundary Conditions.Annals Nucl. Energy 16 (1989) 52710.1016/0306-4549(89)90004-2Search in Google Scholar

20 Dahl, E. B.: Special Eigenvalues of the Time-Dependent One-Speed Neutron Transport Equation for a Homogeneous Sphere with Anisotropic Scattering. Annals Nucl. Energy 12 (1985) 15310.1016/0306-4549(85)90092-1Search in Google Scholar

21 Garis, N. S.: Time eigenvalues for One-Speed Neutrons in Reflected Slabs and Spheres. Kerntechnik 56 (1991) 3310.1515/kern-1991-560114Search in Google Scholar

22 Garis, N. S.; Sjöstrand, N. G.: Approximations for Very-High-Order Eigenvalues in Neutrons Transport Theory. Kerntechnik 54 (1989) 18310.1515/kern-1989-540319Search in Google Scholar

23 Sahni, D. C.; Paranjape, S. D.; Kumar, V.: Time Eigenvalues of the One-Speed Neutron Transport Equation for Spherically Asymmetric Modes in a Homogeneous Sphere. Annals Nucl. Energy 18 (1991) 44310.1016/0306-4549(91)90088-FSearch in Google Scholar

24 Dahl, E. B.; Sjöstrand, N. G.: Eigenvalue Spectrum of Multiplying Slabs and Spheres for Monoenergetic Neutrons with Anisotropic Scattering. Nucl. Sci. Eng. 69 (1979) 11410.13182/NSE69-114Search in Google Scholar

25 Garis, N. S.: One-Speed Transport Eigenvalues for Reflected Slabs and Spheres. Nucl. Sci. Eng. 107 (1991) 34310.13182/NSE91-A23796Search in Google Scholar

26 Grandjean, P.; Siewert, C. E.: The Fn Method in Neutron-Transport Theory. Part II Applications and Numerical Results. Nucl. Sci. Eng. 69 (1979) 16110.13182/NSE79-A20608Search in Google Scholar

27 Siewert, C. E.; Benoist, P.: The FN Method in Theory. Part I Theory and Applications. Nucl. Neutron-Transport Sci. Eng. 69 (1979) 15610.13182/NSE79-1Search in Google Scholar

28 Yíldíz, C.: Influence of Anisotropic Scattering on the Size of the Time-Dependent Systems in Monoenergetic Neutron Transport. J. Phys. D.: Appl. Phys. 32 (1999) 31710.1088/0022-3727/32/3/020Search in Google Scholar

29 Yíldíz C.: Calculation of the higher order eigenvalues for a slab with forward-backward scattering using Fn method. Kerntecknik 65 (2000) 24010.1515/kern-2000-655-613Search in Google Scholar

30 Garcia, R. D. M.: A Review of the Fn method in Particle Transport Theory. Transp. Theory Statist. Phys. 14 (1985) 39110.1080/00411458508211686Search in Google Scholar

31 Garcia, R. D. M.; Siewert, C. E.; Thomas, J. R. Jr.: FN Method for Solving Transport Problems. Trans. Am. Nucl. Soc. 71 (1994) 212Search in Google Scholar

32 Garcia, R. D. M.; Siewert, C. E.: Multigroup Transport Theory. II.Numerical Results. Nucl Sci. Eng. 78 (1981) 31510.13182/NSE81-A21365Search in Google Scholar

33 Inönü, E.: A Theorem on Anisotropic Scattering. Transp. Theory Statist. Phys. 3 (1973) 13710.1080/00411457308205276Search in Google Scholar

34 Mitsis, G. J.: Transport Solutions to the Monoenergetic Critical Problem. Nucl. Sci. Eng. 17 (1963) 5510.13182/NSE63-A17210Search in Google Scholar

35 Wu, S.; Siewert, C. E.: One-Speed Transport Theory for Spherical Media with Internal Sources and Incident Radiation. J. Appl, Math. Phys. (ZAMP) 26 (1975) 63710.1007/BF01594038Search in Google Scholar

Received: 2001-04-05

Published Online: 2022-03-17

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/kern-2001-0107