Algebraic Approximations of a Polyhedron Correlation Function Stemming from its Chord Length Distribution

TL;DR

This paper introduces an algebraic method to approximate the correlation function of polyhedra using polynomial expansions of the chord-length distribution, enabling accurate Fourier transform asymptotics for shapes like cubes and tetrahedra.

Contribution

The paper presents a novel algebraic approximation technique for polyhedron correlation functions based on polynomial expansions of the chord-length distribution, improving asymptotic Fourier transform accuracy.

Findings

Accurate algebraic approximations for cube, tetrahedron, and octahedron correlation functions.

The method reproduces the asymptotic behavior of the form factor at large q.

The approximation's accuracy improves with higher polynomial order K.

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.