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 $\gamma\pp(r)$, 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 left and the right expansions of $\gamma\pp(r)$ around $D_{i-1}$ and $D_i$ up to the terms $O\big(r-D_{i-1}\big)^{K/2}$ and $O\big(D_i-r\big)^{K/2}$, 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.