Form factor of any polyhedron and its singularities derived from a projection method

Abstract

An analytical and general form factor for any polyhedron is derived on the basis of a projection method, in terms of the vertex coordinates and topology of the polyhedron. An integral over the polyhedron equals the sum of the signed integrals over a set of dissected tetrahedra by defining a sign function, and a general tetrahedral form factor is established by defining a projection method. All possible singularities present in the formula are discussed in detail. Using a MATLAB implementation, illustrative examples are discussed to verify the accuracy and generality of the method. The use of the scalar product operation and the sign function in this work allows a general and neat formula to be obtained for any polyhedron, including convex and concave polyhedra. The formulas and discussions presented here will be useful for the characterization of nanoparticles using small-angle scattering techniques.