An analytical approach to fast neutron spectra by the modified Wigner approximation

For these several years there has been considerable interest in the application of continuous slowing down (CSD) theory to problems in Fast Reactor Analysis. In such applications it is very important how to redefine the moderating parameters and how to treat inelastic scatterings in a resolved region and in an unresolved region. Treating inelastic and elastic scattering separately Stacey expanded the total collision density in a two-term Taylor series and gave an accurate neutron spectrum for a representative fast reactor composition, while Dunn and Becker incorporated inelastic scatterings into their moderating parameters by using the multigroup inelastic scattering matrix. In this paper we extend the CSD theory to the space-dependent problem by assuming the factorized neutron flux so as to derive the modified diffusion equation. In order to treat analytically the neutron flux in a finite bulk medium it is desired that the overall moderating process is described by as few moderating parameters as possible which can be defined for any energy region and any composition of materials by the unified formalism. To satisfy this requirement we propose the modified Wigner approximation (MWA) which is the CSD theory of the Wigner-type and its moderating parameter xi(u)-circumflex is given iteratively by the simple definition. For rapid computations of our parameter xi(u) we use the separate-type synthetic kernels for elastic scattering and inelastic scatterings. For the space-dependent problem in a finite bulk medium an simple analytical formula is derived by solving the modified diffusion equation and is used to study the space-dependence of fast neutron fluxes and the leakage effects on fast neutron fluxes at various points. This analytical solution brings out the fine structure of the fast neutron spectrum in greater detail than comparable multigroup treatments and allows simple analyses of fast neutron time-of-flight spectra