Algebraic approximations of a polyhedron correlation function stemming from its chord-length distribution

An algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ ′ ′ (r), its chord-length distribution (CLD), considering first, within the subinterval [Di − 1,   Di] of the full range of distances, a polynomial in the two variables (r − Di − 1)1/2 and (Di − r)1/2 such that its expansions around r = Di − 1 and r = Di simultaneously coincide with the left and right expansions of γ ′ ′ (r) around Di − 1 and Di up to the terms O(r − Di − 1)K/2 and O(Di − r)K/2, respectively. Then, for each i, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end-points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q's, the asymptotic behaviour of the exact form factor up to the term O[q − (K/2+4)]. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.

Source: Acta Crystallographica Section A - December 8, 2020 Category: Chemistry Authors: Ciccariello, S. Tags: small-angle scattering polyhedra chord-length distribution correlation functions asymptotic behaviour short communications Source Type: research