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

Abstract

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, the problem of the decrease of the neutron flux as it penetrates into a material placed in an isotropic neutron field, has been attempted employing the theory of steady-state neutron transport, starting from Stuart's formula. Simple formulae were derived based on the integral cross-section parameters, which can 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 a weight representing the ratio between the total and scattering macroscopic 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 experimental values of the neutron self-shielding factor for different elements and different geometries. The results were validated by comparing them to the experimental data as well as a variety of data extracted from the literature.

Experimental Measurement of Infinite Dilution Thermal Neutron Self-shielding Factor

“…Based on the properties of the dimensional quantity of the average chord length ( ̄ ), which has the greatest impact on the medium's self-shielding proprieties [25], the values of thermal neutron self-shielding factors for thick sheets of indium, gold, zinc, and cylinders of mercury were determined experimentally relative to the thin foils made of the same materials. The reaction rates per atom ( RR( ̄ ) = 𝜙 t 𝜎 g ) were determined after the neutron irradiation of each sample, individually, using the following relations: where is the reaction rate with the parameters specified for sample having larger dimension, and is the reaction rate with the parameters specified for sample having smallest possible dimension.…”

“…( 4) derived in Ref. [25]. These isotopes had accurate data for scattering and absorption cross-section together with the limited number of their isotopes (1 in the case of gold and 2 in the case of indium in which the most abundant is used as the monitored reaction), see Table 2.…”

“…To make a comparison with models, we had used our ab initio [25] in which we proposed that the thermal neutron self-shielding factor can be calculated from the following equation that have a sigmoid shape.…”

“…However, most researchers treat the problem with the body that has the m) is the cross-section of formation of the metastable state 116m In (g) is the cross-section of formation of the ground state 116g In minimal volume in a class of convex bodies having the same dimensions [35]. We will use the simple formulation of the average neutronchord length based on the fact that the trajectories of the incident isotropic neutrons traverse different lengths within the body due to scattering are comparable [25], as long as we want to avoid the rigorous derivations of the average neutronchord length (cf. Refs.…”

“…There is an interplay between neutron absorption in the sample and the overall neutron flux [13]. Predominantly, the correlation between neutron self-shield factors and the set of parameters involved in the calculation of its value had been studied by several scientists [14][15][16][17][18][19][20][21][22][23][24][25], who gave dimensionless variables to identify and encompass the physical and geometric varieties of the sample's geometries to attain a universal formula for self-shielding. In this study, neutron self-shielding factors for Indium and gold samples were determined experimentally.…”