Numerically stable form factor of any polygon and polyhedron

Area of Science:

• Computational geometry
• Mathematical physics
• Materials science

Background:

• Calculating form factors for complex shapes is computationally challenging.
• Existing methods may suffer from precision loss due to singularities.

Purpose of the Study:

• Derive coordinate-free expressions for form factors of polygons and polyhedra.
• Address and mitigate precision loss issues near singularities.

Main Methods:

• Application of the divergence theorem and Stokes's theorem.
• Detailed analysis of removable singularities.
• Utilizing series expansions to manage cancellations.

Main Results:

• Development of coordinate-free form factor expressions.
• Identification and characterization of all apparent singularities as removable.
• Demonstration of precision preservation via series expansions.

Conclusions:

• The derived coordinate-free method offers a robust way to compute form factors.
• Series expansions effectively overcome precision limitations associated with singularities.
• This work has direct applications in small-angle scattering by nanocrystals.