Multidimensional spacial eigenmode analysis. Page: 3 of 8
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obtain the number of neutrons produced at a cell k from those born at cellj [Ref. 6], [Ref. 7]. The
fission matrix is constructed by killingg the following matrix for all source regions
F(x -+ xI) --(xN x)1
O(xl -xN) ..(XN -xN )
where sourcere xdeecor). Once the entire fission matrix is constructed, the eigenvalues of this
matrix are obtained from an eigenvalue calculation routine [Ref. 8].
The result of a fission matrix analysis is the fission source in the source regions, which is
proportional to the flux. Fission matrix analysis has some drawbacks. First, we require many fixed-
source transport calculations to populate the fission matrix (an N2 calculation), and second, we do
not obtain the full fluxes in any nonsource (nonfissioning) regions. However, we are able to
simultaneously obtain several eigenvalues and eigenfunctions when we calculate the eigensystems.
III. Multi-Dimensional Eigensystems
To continue with past work in multi-dimensional systems where a code user would generally
attempt to use reflective boundary conditions to reduce the dominance ratio (DR) of the problem,
we examine infinite square and cylindrical systems in two dimensions. Note that an infinite cylinder
is usually a one-dimensional calculation with flux variations only allowed in the radial dimension.
Here, however, we use r-9 geometry and allow functional variation also in the azimuthal dimension.
Inserting the two dimensional difference operator into the neutron diffusion equation and
performing separation of variables, we obtain the familiar Sturm-Liouville equation, with the
solutions of the form
0,(r, 9) ~ .o~9 J,(Vr / R) ,(2)
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Parsons, Donald Kent & Kornreich, D. E. (Drew E.). Multidimensional spacial eigenmode analysis., article, January 1, 2003; United States. (https://digital.library.unt.edu/ark:/67531/metadc933229/m1/3/: accessed April 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.