Neutron density fluctuations due to randomly distributed sources in finite media

We formulate a theory which describes the effect on the neutron flux of sources in the form of lumps randomly distributed in a finite medium. One-speed diffusion theory is used, but the method can readily be extended to energy-dependent transport theory. The mean flux, the covariance and a new statistical indicator, the mean flux difference between regions, are calculated for an infinite and semi-infinite medium. Numerical results are obtained using Markov and Gaussian models for the random geometry of the lumps. We conclude that the statistical anisotropy of the medium can play a significant role and that, even though we may be considering what is ostensibly a one-dimensional problem in the x-direction, it is necessary to incorporate the medium properties in the other two directions via the correlation lengths λy and λz. We also note that there can be marked differences in the average fluxes in the various source regions which can deviate considerably from the conventional average flux. These results may have practical implications in the design of radioactive waste drums in which sources of neutrons (or gamma rays) are encased randomly in a background matrix. Conventional averaging may lead to a significant underestimate of the surface flux and current. The theory neglects the associated random material properties but should be a good approximation if the volume fraction of the source material is small. In the Appendix, we develop a random geometry model for the source lumps and derive the associated covariance function.