The Second spatial moment of neutron distribution in hydrogenous moderators containing voids

Assuming that neutron energy decrement in collisions with oxygen nuclei in negligibly small compared with that in collisions with hydrogen nuclei, and that elastic scatterings by hydrogen nuclei are isotropic in the center of the mass system, the [stochastic method to calculate the second spatial moment of neutron distribution in a homogeneous medium, already reported, is generalized to cases of random distribution of the same voids in hydrogenous moderators. In general, the distribution law of free path length in such a system deviates from the exponential rule. The result for the migration area becomes the sum of two terms of an average exponential distribution and of the mean square deviation therefrom peculiar to the heterogeneous system. This mean square deviation may be calculated by one-group theories developed by several authors, each factor of this deviation being expressed as a sum of terms at each energy, and the approximations contained in their theories being rather good in isotropic systems. Thus we may conclude that one-group treatments reported by several authors may be generalized into multi-group problems without any approximation being added. Our numerical results show that the ratio of heterogeneous to homogeneous terms has a maximum at a void ratio which varies with the radius of the spheres, and the oblate deformation of spherical voids has no vital effect. (author)