197
Views
4
CrossRef citations to date
0
Altmetric
Article

Eigenvalue Formulations for the PN Approximation to the Neutron Transport Equation

, , , &

References

  • Balay, S., S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, et al. 2020. PETSc users manual. Technical Report ANL-95/11 - Revision 3.13, Argonne National Laboratory.
  • Bell, G. I., and S. Glasstone. 1970. Nuclear reactor theory. Van Nostrand Reinhold.
  • Byerly, W. E. 1959. An elementary treatise on Fourier’s series, and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics. New York: Dover.
  • Cacuci, D. G., Y. Ronen, Z. Shayer, J. J. Wagschal, and Y. Yeivin. 1982. Eigenvalue-dependent neutron energy spectra: Definitions, analyses, and applications. Nucl. Sci. Eng. 81 (3):432–42.
  • Case, K. M., and P. L. Zweifel. 1967. Linear transport theory. Reading: Addison-Wesley.
  • Cercignani, C. 1988. The Boltzmann equation and its applications. New York: Springer.
  • Chentre, N., P. Saracco, S. Dulla, and P. Ravetto. 2019a. On the prompt time eigenvalue estimation for subcritical multiplying systems. Ann. Nucl. Energy 132:172–80.
  • Chentre, N., P. Saracco, S. Dulla, and P. Ravetto. 2019b. On fick’s law in asymptotic transport theory. Eur. Phys. J. Plus 134 (10):1–15.
  • Dahl, E. B., V. Protopopescu, and N. G. Sjöstrand. 1983. On the relation between decay constants and critical parameters in monoenergetic neutron transport. Nucl. Sci. Eng. 83 (3):374–9.
  • Davison, B. 1958. Neutron transport theory. Oxford: University Press.
  • Deo, K., P. D. Krishnani, and R. S. Modak. 2014. Development of one-dimensional neutron transport theory code based on method of characteristics. Report BARC/2014/E/015, Bhabha Atomic Research Centre, Mumbai.
  • Duderstadt, J. J., and L. J. Hamilton. 1976. Nuclear reactor analysis. New York: Wiley.
  • Dulla, S., P. Ravetto, and P. Saracco. 2018. The time eigenvalue spectrum for nuclear reactors in multi-group diffusion theory. Eur. Phys. J. Plus 133:1–24.
  • Ferziger, J. H., and M. Peric. 1999. Computational methods for fluid dynamics. Berlin: Springer.
  • Frank, W., and P. von Brentano. 1994. Classical analogy to quantum mechanical level repulsion. Am. J. Phys. 62 (8):706–9.
  • Hernandez, V., J. E. Roman, and V. Vidal. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31 (3):351–62.
  • Lathouwers, D. 2003. Iterative computation of time-eigenvalues of the neutron transport equation. Ann. Nucl. Energy 30 (17):1793–806.
  • Mark, J. C. 1944. The spherical harmonics method, Part I. Atomic Energy Report 92 National Research Council of Canada.
  • Mark, J. C. 1945. The spherical harmonics method, Part II. Atomic Energy Report 97 National Research Council of Canada.
  • Marshak, R. E. 1947. Note on the spherical harmonics method as applied to the Milne problem for a sphere. Phys. Rev. 71 (7):443–6.
  • McClarren, R. G. 2019. Calculating time eigenvalues of the neutron transport equation with dynamic mode decomposition. Nucl. Sci. Eng. 193 (8):854–67.
  • Meghreblian, R. V., and D. K. Holmes. 1960. Reactor analysis. New York: McGraw-Hill.
  • Modak, R. S., and A. Gupta. 2003. A simple scheme for the direct evaluation of time-eigenvalues of neutron transport equation. Ann. Nucl. Energy 30 (2):211–22.
  • Modak, R. S., D. C. Sahni, and S. D. Paranjape. 1995. Evaluation of higher k-eigenvalues of the neutron transport equation by SN method. Ann. Nucl. Energy 22 (6):359–66.
  • Protopopescu, V. 1983. Eigenvalue problems for the Boltzmann operator in various formulations. In Advances in Nuclear Science and Technology, ed. J. Lewins and M. Becker, vol. 15, 1–54. Boston: Springer.
  • Ronen, Y., D. Shvarts, and J. J. Wagschal. 1976. A comparison of some eigenvalues in reactor theory. Nucl. Sci. Eng. 60 (1):97–101.
  • Sahni, D. C., and N. G. Sjöstrand. 1990. Criticality and time eigenvalues in one-speed neutron transport. Prog. Nucl. Energy 23 (3):241–89.
  • Sahni, D. C., N. S. Garis, and N. G. Sjöstrand. 1995. Spectrum of one-speed neutron transport operator with reflective boundary conditions in slab geometry. Transp. Theory Stat. Phys. 24 (4–5):629–56.
  • Sanchez, R., D. Tomatis, I. Zmijarevic, and H. G. Joo. 2017. Analysis of alpha modes in multigroup diffusion. Nuclear Engineering and Technology 49 (6):1259–68.
  • Saracco, P., S. Dulla, and P. Ravetto. 2012. On the spectrum of the multigroup diffusion equations. Prog. Nucl. Energy 59:86–95.
  • Sood, A., R. A. Forster, and D. K. Parsons. 2003. Analytical benchmark test set for criticality code verification. Prog. Nucl. Energy 42 (1):55–106.
  • Sorensen, D. C., R. B. Lehoucq, and C. Yang. 1998. ARPACK users guide: Solution of large scale eigenvalue problems by implicitly restarted Arnoldi methods. Philadelphia: SIAM.
  • Velarde, G., C. Ahnert, and J. M. Aragonés. 1978. Analysis of the eigenvalue equations in k, λ, γ, and α applied to some fast- and thermal-neutron systems. Nucl. Sci. Eng. 66 (3):284–94.
  • Zoia, A., E. Brun, and F. Malvagi. 2014. Alpha eigenvalue calculations with TRIPOLI-4®. Ann. Nucl. Energy 63:276–84.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.