The probability density function of the multiplication factor due to small, random displacements of fissile spheres

摘要

An analytical expression is obtained for the probability density function of the multiplication factor of an array of spheres when each sphere is displaced in a random fashion from its initial position. Two cases are considered: (1) spheres in an infinite background medium in which the total cross section in spheres and medium is the same, and (2) spheres in a void. In all cases we use integral transport theory and cast the problem into one involving average fluxes in the spheres which interact via collision probabilities. The statistical aspects of the problem are treated by first order perturbation theory and the general conclusion is that, when the number of spheres exceeds about 5, the reduced multiplication factor ξ = (k - k_0)/k_0, where k_0 is the unperturbed value, is given accurately by the Gaussian distribution P(ξ) = [1/(2π)~(1/2)σD_T] exp[-(ξ~2)/(2σ~2D_T~2)]. The partial standard deviation σ = 2δ/3~(1/2), <δ being the maximum movement of the sphere from its equilibrium position. D_T is a function of the system properties and geometry. Some numerical results are given to illustrate the magnitude of the effects and also the accuracy of diffusion theory for this type of problem is assessed. The overall accuracy of the perturbation method is assessed by an essentially exact result obtained using simulation, thereby enabling the range of perturbation theory to be investigated.