Structural chirality measurements and computation of handedness in periodic solids

TL;DR

The paper critically compares structural chirality measures in extended solids, showing that traditional scalar metrics like the continuous chirality measure (CCM) and Hausdorff distance depend on reference choices and cannot distinguish enantiomers. It introduces helicity, a pseudoscalar handedness descriptor drawn from hydrodynamics, and demonstrates its ability to encode both the magnitude and sign of handed distortions during displacive transitions in several materials, including K$_3$NiO$_2$-like, CsCuCl$_3$, and MgTi$_2$O$_4$. While helicity reliably flags handedness in enantiomorphic space groups, the authors acknowledge limitations for chiral-connected non-enantiomorphic groups and emphasize the need for complementary measures or reciprocal-space formulations. Overall, the work provides a practical, sign-sensitive descriptor for crystal handedness and highlights the nuanced differences between chirality and handedness, with potential implications for high-throughput chiral materials screening and linking structural distortions to optical activity.

Abstract

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, the Hausdorff distance, and the angular momentum. 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 new method to quantify the handedness of a crystal based on a pseudoscalar function, i.e., the helicity. This quantity, borrowed from hydrodynamics, can be computed from the eigenvector carrying the system from the high-symmetry non-chiral phase to the low-symmetry chiral phase. Different model systems like K$_3$NiO$_2$, CsCuCl$_3$ and MgTi$_2$O$_4$ are used as test cases where we show the superior interest of using helicity to quantify chirality together with the handedness distinction.

Figures (5)

• Figure 1: Schematic view of the unit cell in the prototypical cubic ABO$_3$ perovskite structure together with their dominant instabilities without periodic boundary conditions. (a) Non-polar cubic centrosymmetric phase. (b) Non-polar rotation of the oxygen octahedra. (c) The off-centering motion of the B atom. Grey, blue, and red balls represent A, B, and O atoms, respectively. Yellow arrows indicate the direction for the non-zero displacements.
• Figure 2: Schematic representation of the K$_3$NiO$_2$ crystal structure where atoms occupy the $P4_2/mnm$ high symmetry positions. Arrows indicate the direction of the atom displacements that bring the system to the $P4_32_12$ chiral phase. Purple, grey, and red balls represent K, Ni, and O atoms
• Figure 3: Comparison of the evolution of the different chiral measures as a function of the amplitude of the chiral distortion $\eta$ in Na$_3$AuO$_2$. Positive (negative) values of $\eta$ correspond to the condensation of the modes towards the $P4_12_12$ ($P4_32_12$) phase. Red dots, blue squares, and green triangles correspond to CCM, Hausdorff, and Helicity measures. The values of the different measures have been normalized to display a value of $1$ at the optimal amplitude of the chiral distortion ($\eta=1$).
• Figure 4: Schematic representation of the CsCuCl$_3$ where atoms occupy the high symmetry positions. Arrows indicate the direction of the displacements into the $P6_122$ chiral phase. Grey, blue, and green balls represent Cs, Cu, and Cl atoms.
• Figure 5: Schematic representation of the MgTi$_2$O$_4$. Atoms occupy the high symmetry undistorted positions in a (a) $P4_12_12$ or (b) $P4_32_12$ representation. Arrows indicate the direction of the displacements into their respective chiral phases. Orange, blue and red balls represent Mg, Ti and O atoms respectively.