Scattering functions of Platonic solids

Abstract

In this report the single-particle scattering properties of five Platonic solids, including tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron, are investigated in a systematic manner. For each given geometry, the Debye spatial autocorrelation function γ (r), pair distance distribution function (PDDF) p (r) and intraparticle structure factor (form factor) P (Q) are respectively calculated and compared to the corresponding scattering function of the spherical referential system. Based on our theoretical models, the empirical relationship between the dodecahedral and icosahedral structural characteristics and those of the equivalent spheres is found. Moreover, the single-particle scattering properties of the icosahedral and the spherical shells with the same volume are further investigated and the prospect of using different data analysis approaches to explore their structural difference is also presented and discussed.