Absorption-fission and reaction matrices in the method of surface harmonics

Abstract

An important step in the calculation of nuclear reactors and reactor safety is the multigroup kinetic calculation and the preparation, based on this calculation, of the small-group characteristics of the cells. The VVI~R and RBMK reactors are calculated at this step using the UNIRASOS multirate program [1, 2] and the WIMS multigroup program [3], which, as a rule, employ the method of first-collision probabilities (in WIMS the discrete-ordinate method can also be used) for solving the kinetic equation inside cells.The multigroup calculation of a cell for determining K= with zero neutron current at the outer boundary of the cell is taken as the basis. The results of this calculation are used to prepare the small-group spatial and energy averages of the cross sections for absorption, fission, multiplication, and group transitions of neutrons. Since under real conditions neutrons flow into or out of a cell, the small-group cross sections must depend on the leakage. In the programs mentioned above the cross sections are corrected for neutron leakage. This is done with the help of the critical buckling as follows. First, a multigroup calculation of the cell is performed in order to determine 32=. Next, the critical buckling is found (assuming that Kef f = I is reached in the cell considered) for the homogenized cell in the multi-or small-group diffusion approximation, and for small systems (large values of B 2) with a hydrogen moderator in order to take into account more accurately the transport correction in the Bo-B 1 approximations. Next, the small-group cross sections of a cell are calculated by averaging them over the characteristic critical buckling function. The cell cross sections obtained in this manner take into account well the neutron leakage out of a cell in the asymptotic region, i.e., among identical cells and far from nonuniformities: strong absorbers or reflectors. Near such nonuniformities large computational errors in the neutron fields are observed in the calculation of a reactor.A more general and systematic approach for taking into account neutron-leakage effects in the cell characteristics has been developed on the basis of the method of surface harmonics (see, for example, [4, 5]).* In this method the distribution function is expanded in a series in the trial functions, ordered over all contributions introduced into this distribution function. The first trial function is a symmetric trial function ~g(co). The integral functionals (such as absorption, multiplication, fission, and so on) in the cell are determined mainly by this trial function. The trial function sog(co) is the neutron distribution function at the phase point o0 (,0 -r, fl, E) from a unit current of group-g neutrons flowing symmetrically into the cell.Absorption-Fission Matrix ~af in the Method of Surface Harmonics. We shall write out in the following form the matrix of effective absorption-fission cross sections t:, eff [9, 10], associated with the effective cross sections for absorption ~ag eft, fiSSiOn ~fgeff, and transition f...