The Effect of Random Geometry on the Criticality of a Multiplying System III: Three Dimensional Systems and Spherical Absorbers

A method has been developed for calculating the probability distribution of the multiplication factor in a system in which the fissile or absorbing elements are randomly distributed across the core and can have random material properties. It has practical applications to the storage of radioactive waste in drums in which fissile material is stored in a background matrix. The procedure is based upon the source-sink method of heterogeneous reactors developed by Feinberg, Galanin, Horning and Stewart in which the fuel element or absorber is replaced by a point sink of thermal neutrons and a point source of fast neutrons. The positions and material properties are sampled from a random distribution and a probability distribution is built up for the multiplication factor k_eff. Calculations are made for spheres in a cubic system and probability distributions, mean values and variances are obtained for 1, 2, 3, 5, 10 and 25 spheres in both water and graphite moderated systems. Some interesting fine structure is found in the probability distributions which is attributed to preferred symmetric groupings of the spheres in the lattice. We also examine the effect of small random movements of the spheres about their mean positions and in particular study the effect of anisotropy of motion, i.e. perpendicular to the plane and in the plane, on the mean value of the multiplication factor and the associated probability distributions. Some experimental results obtained by Lloyd on reactivity changes in random lattices are examined and qualitative agreement is obtained. A convenient form of the three dimensional Greens function for a rectangular box is developed which is especially useful for numerical purposes due to its rapid convergence properties.