Neutron flux in cells of finite, square lattices

The purpose of this paper is to determine the multigroup neutron flux in cells of finite, regular, unreflected, square lattices. The dependence on buckling and on cell squareness is shown explicitly for the moderator spectrum only. The corresponding fuel spectrum at each lattice site may be obtained readily from the condition of flux continuity.Heterogeneous theory of the source-sink type forms the starting point of the present theory. It is shown that each integration constant appearing in the heterogeneous solution for a finite periodic lattice separates into a product of two constants, one of which is a purely geometrical factor of position xi, yi of the i'th element, while the other is a purely physical constant depending on the physical characteristics of the lattice alone. Knowledge of the position dependence allows the flux in a square cell of a finite system to be determined for all multipole orders by means of summation techniques. The physical constants are obtained from multipole moderator-to-fuel boundary conditions. These conditions are expressed in terms of response coefficients and result in a set of equations from which the position dependence is again eliminated by means of summation techniques. The result is a set of simultaneous equations for the physical constants alone which can be solved once the criticality condition is satisfied. The number of these equations is independent of the number of elements in the lattice and equals twice the number of multipoles times the number of energy groups.