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

Abstract

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 [D i−1, D i ] of the full range of distances, a polynomial in the two variables (r − D i−1)1/2 and (D i − r)1/2 such that its expansions around r = D i−1 and r = D i simultaneously coincide with the left and right expansions of γ′′(r) around D i−1 and D i up to the terms O(r − D i−1) K/2 and O(D i − 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.