Crystal structures of alpha and beta modifications of Mn as packing of tetrahedral helices extracted from a four-dimensional {3, 3, 5} polytope

Abstract

The crystal structures of both α- and β-Mn modifications have been presented as packing of tetrahedral helices extracted from four-dimensional {3, 3, 5} polytope construction. Presentation of the β-Mn structure as a primitive cubic arrangement formed by double tetrahedral helices around a central tetrahedral Coxeter–Boerdijk helix (tetrahelix) enables the inclusion in the structure description not only all atoms but also all tetrahedra; these tetrahedra are not accounted for in the preceding models for the β-Mn structure. The tetrahelix periodicity arising by minimal deformations of tetrahedra edges is equal to eight tetrahedra and coinciding with the lattice periods of both modifications. The linear substructure of α-Mn crystal consists of four tetrahelices which join to each other by edges around the common twofold axis. The α-Mn structure has been presented as primitive cubic arrangement constructed from such rods.