The chord-length probability density of the regular octahedron

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 [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.

The chord-length distribution of a polyhedron

The chord-length distribution function [γ''(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ''(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ₁], Δ₁ being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ''(r) can only show singularities of the forms |r - δ|^-n and |r - δ|^-m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

The correlation functions of the regular tetrahedron and octahedron

The expressions of the autocorrelation functions (CFs) of the regular tetrahedron and the regular octahedron are reported. They have an algebraic form that involves the arctangent function and rational functions of r and (a + br 2)1/2, a and b being appropriate integers and r a distance. The CF expressions make the numerical determination of the corresponding scattering intensities fast and accurate even in the presence of a size dispersion.