The isotropic correlation function of plane figures: the triangle case

TL;DR

This paper derives an explicit algebraic expression for the isotropic correlation function of a triangular plane figure, which is useful for analyzing scattering data involving cylindrical particles with triangular cross-sections.

Contribution

It provides the first closed-form analytic expression for the correlation function of a triangle, considering different side and height configurations.

Findings

Explicit algebraic formulas for the triangle correlation function.

The formulas depend on the relative order of sides and heights.

Facilitates analysis of scattering from cylindrical particles with triangular cross-sections.

Abstract

The knowledge of the isotropic correlation function of a plane figure is useful to determine the correlation function of the cylinders having the plane figure as right-section and a given height as well as to analyze the out of plane intensity collected in grazing incidence small-angle scattering from a film formed by a particulate collection of these cylinders. The correlation function of plane polygons can always be determined in closed algebraic form. Here we report its analytic expression for the case of a triangle. The expressions take four different forms that depend on the relative order among the sides and the heights of the triangle.