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Mathematical formulae for neutron self-shielding properties of media in an isotropic neutron field

The complexity of the neutron transport phenomenon throws its shadows on every physical system wherever neutron is produced or used. In the current study, an ab initio derivation of the neutron self-shielding factor to solve the problem of the decrease of the neutron flux as it penetrates into a material placed in an isotropic neutron field. We have employed the theory of steady-state neutron transport, starting from Stuart's formula. Simple formulae were derived based on the integral cross-section parameters that could be adopted by the user according to various variables, such as the neutron flux distribution and geometry of the simulation at hand. The concluded formulae of the self-shielding factors comprise an inverted sigmoid function normalized with a weight representing the ratio between the macroscopic total and scattering cross-sections of the medium. The general convex volume geometries are reduced to a set of chord lengths, while the neutron interactions probabilities within the volume are parameterized to the epithermal and thermal neutron energies. The arguments of the inverted-sigmoid function were derived from a simplified version of neutron transport formulation. Accordingly, the obtained general formulae were successful in giving the values of the experimental neutron self-shielding factor for different elements and different geometries.