Form factor of any polyhedron: a general compact formula and its singularities

A general and compact formula is established for the form factor of any polyhedron, which involves only the apex coordinates and the apex connections. For large diffusion vector q, the form factor behaves like q⁻³ for generic directions, but it exhibits q⁻² singularities in the directions perpendicular to the edges and q⁻¹ singularities in the directions normal to the faces. General results are established for these singularities. Using a Python implementation, illustrative examples are discussed. The generality of the formula and of its singularities are likely to be important for any discussion of scattering from polyhedral particles.