Mathematical Explorations in Modern X-ray Crystallography

In this thesis, we explore crystallography through a mathematical lens. We review the basics of crystallography with a mathematical focus, and expand into contributions on two specific areas of the crystallographic refinement process. The first of these is the detection of twin components within crystals using the effect of twinning on detected diffraction peaks. We focus on using the information of particularly underestimated peaks along with the lattice structure to intelligently search for the most viable twin laws. The second contribution concerns the use of non-spherical form factors in crystallographic refinement, and testing of the impact of setting the form factor derivative to zero within the least-squares refinement process. We utilise numerical differentiation to approximate this derivative more exactly, and evaluate the impact of these choices for modelling the derivative through three test molecules to find that, within the current bounds of uncertainty, modelling the form factor derivative as zero has insignificant impact on the results of refinement. Additional curiosities encountered within our investigations of crystallography are also documented, such as the implementation of extinction parameters.