Structural chirality of β -Mn

Abstract

A theoretical study of Bragg diffraction by an enantiomorphic pair of structures should apply to patterns gathered on beta-Mn using resonance enhanced x-ray diffraction. The chiral polymorph of manganese, and structurally related compounds, will be more convenient than trigonal crystals of low-quartz, tellurium and berlinite used in previous structural studies of this type, because cubic symmetry removes some complications of correction for absorption and macroscopic birefringence. Intensity of a Bragg spot engaged by circular polarization in the primary beam of photons Y(h, k, l) is proposed as a chiral signature of the illuminated material. The partial intensity requires a knowledge of scattering amplitudes in all four channels of polarization, which are reported as functions of an azimuthal angle (rotation of the crystal about the axis of the reflection vector). Unlike trigonal chiral crystals, Y(0, 0, l) = 0 for beta-Mn and it is (h, k, 0), and symmetry related Bragg spots, that epitomize structural chirality. Specifically, reflection vectors (h, k, 0) with odd Miller indices and the chiral axis of beta-Mn mesh in terms of helicity, with the corresponding Y(h, k, 0) equal in magnitude and opposite in sign for partners in the enantiomorphic pair. Dependence of Y(h, k, 0) on the azimuthal angle does not mirror the dyad or tetrad axes of rotation symmetry in the cubic crystal structure.