Continuous Neutron Slowing Down Theory Applied to Resonances

Abstract

Neutronic formalisms that discretize the neutron slowing down equations in large numerical intervals currently account for the bulk effect of resonances in a given interval by the narrow resonance approximation (NRA). The NRA reduces the original problem to an efficient numerical formalism through two assumptions: resonance narrowness with respect to the scattering bands in the slowing down equations and resonance narrowness with respect to the numerical intervals. Resonances at low energies are narrow neither with respect to the slowing down ranges nor with respect to the numerical intervals, which are usually of a fixed lethargy width. Thus, there are resonances to which the NRA is not applicable. To stay away from the NRA, the continuous slowing down (CSD) theory of Stacey was invoked. The theory is based on a linear expansion in lethargy of the collision density in integrals of the slowing down equations and had notable success in various problems. Applying CSD theory to the assessment of bulk resonance effects raises the problem of obtaining efficient quadratures for integrals involved in the definition of the so-called ''moderating parameter.'' The problem was solved by two approximations: (a) the integrals were simplified through a rationale, such that the correct integrals weremore » reproduced for very narrow or very wide resonances, and (b) the temperature-broadened resonant line shapes were replaced by nonbroadened line shapes to enable analytical integration. The replacement was made in such a way that the integrated capture and scattering probabilities in each resonance were preserved. The resulting formalism is more accurate than the narrow-resonance formalisms and is equally as efficient.« less