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Form factor of any polyhedron: a general compact formula and its singularities
Author(s) -
Croset Bernard
Publication year - 2017
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s1600576717010147
Subject(s) - gravitational singularity , polyhedron , mathematics , pure mathematics , perpendicular , mathematical analysis , general theory , geometry , physics , mathematical economics
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.