Structural chirality measurements and computation of handedness in periodic solids

We compare the various chirality measures most widely used in the literature to quantify chiral symmetry in extended solids, i.e., the continuous chirality measure and the Hausdorff distance. By studying these functions in an algebraically tractable case, we can evaluate their strengths and weaknesses when applied to more complex crystals. Going beyond those classical calculations, we propose a different method to quantify the handedness of a crystal based on a pseudoscalar function, i.e., the helicity during a soft phonon mode driven displacive phase transition from an achiral structure. This quantity, borrowed from hydrodynamics, can be computed from the eigenvector carrying the system from the high-symmetry nonchiral phase to the low-symmetry chiral phase. Different model systems like ${\mathrm{K}}_{3}{\mathrm{NiO}}_{2}$, ${\mathrm{CsCuCl}}_{3}$, and ${\mathrm{MgTi}}_{2}{\mathrm{O}}_{4}$ are used as test cases where we show the superior interest of using helicity to quantify chirality in displacive chiral transitions together with the handedness distinction.