Abstract
Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.
Highlights
The term ‘form factor’ has different meanings in science and in engineering
An important application domain is small-angle scattering by nanocrystals
We are concerned with the form factor of a geometric figure as defined in the physical sciences, namely the Fourier transform of the figure’s indicator function, called the ‘shape transform’ of the figure
Summary
We are concerned with the form factor of a geometric figure as defined in the physical sciences, namely the Fourier transform of the figure’s indicator function, called the ‘shape transform’ of the figure. We derive a numerically stable algorithm for computing the form factor of any polygon or polyhedron, as implemented in the GISAS software BornAgain (Pospelov et al, 2020). This algorithm was documented in a terse mathematical note (Wuttke, 2017). Derivations and results have been simplified, the material has been completely reorganized for better readability, and additional literature is taken into account
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