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New approach to the neutron self-shielding factor with multiple scattering
Abstract
A formalism is developed to calculate the self-shielding factor of probes in a neutron field taking scattering into account in many-collision approximation. The collision probability is written as an integral over the probe volume of the first-collision probability density weighted with a function which represents the number of collisions the neutron suffers inside the probe. For this function, an integral equation was written which has the advantage over the traditional integral transport equation of being independent of the external neutron flux density. It is determined entirely by the collision properties and geometry of the probe. A procedure is proposed to obtain a fast convergence of this integral. The method is applied to thin foils.
... In another limit, d § d ! 1, which corresponds to a thick absorber, the absorption cross-section is § a = § a d § d : Expressions similar to this formula appear when considering the self-shielding e®ect, when the inner parts of the absorber are partly protected from the incoming neutrons by outer absorbing layers, and neutron scattering is also taken into account (cf. [1] Sec.11.2, [4], [5]). 4 E® ective absorption cross-section § ef f a of the heterogeneous system The e®ective absorption cross-section § eff a of the heterogeneous system in Case A is given by the formula: ...
The structure of a heterogeneous system influences diffusion of thermal neutrons. The thermal-neutron absorption in grained
media is considered in the paper. A simple theory is presented for a two-component medium treated as grains embedded in the
matrix or as a system built of two types of grains (of strongly differing absorption cross-sections). A grain parameter is
defined as the ratio of the effective macroscopic absorption cross-section of the heterogeneous medium to the absorption cross-section
of the corresponding homogeneous medium (consisting of the same components in the same proportions). The grain parameter depends
on the ratio of the absorption cross-sections and contributions of the components and on the size of grains. The theoretical
approach has been verified in experiments on prepared dedicated models which have kept required geometrical and physical conditions
(silver grains distributed regularly in Plexiglas). The effective absorption cross-sections have been measured and compared
with the results of calculations. A very good agreement has been observed. In certain cases the differences between the absorption
in the heterogeneous and homogeneous media are very significant. A validity of an extension of the theoretical model on natural,
two-component, heterogeneous mixtures has been tested experimentally. Aqueous solutions of boric acid have been used as the
strongly absorbing component. Fine- and coarse-grained pure silicon has been used as the second component with well-defined
thermal-neutron parameters. Small and large grains of diabase have been used as the second natural component. The theoretical
predictions have been confirmed in these experiments.
A recently developed theoretical formalism for the calculation of multiple-scattering neutron self-shielding factors G(E) as a function of the incident neutron energy E is applied to some of the main resonances of Mn, W, Cu, Au and In. Tables of the self-shielding factors of these dosimetrical materials are presented for thin foils of different thicknesses. Two special cases are analysed separately: the narrow resonance and the interfering resonances from different isotopes of the same material.
The thermal neutron absorption is considered in a medium which contains highly absorbing grains. The theoretical approach is based on a definition of the effective macroscopic absorption cross-section for such a medium. The parameter of the grain effect is defined as the ratio of the effective cross-section to the cross-section of a corresponding homogeneous material. It is generally obtained as a function of the ratio of the absorption cross-sections of components, of the grain size, and of the contributions of the components. A further analysis is performed for the grain size effect when the two other parameters are fixed. The presented approach is applied to an interpretation of experiments on Plexiglas models containing silver grains. A good agreement of the experimental and theoretical results is obtained when relevant effective energy-dependent cross-sections are weighted by the thermal neutron energy distribution.
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