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.

Intercluster Exchange-Stabilized Novel Complex Colloidal χc Phase

“…First, colloidal clusters, which allow the combination of properties of individual colloids, can result in some new properties not present in the individual colloidal building blocks and then may lead to new potential applications in photonics, − catalysis, and bioanalysis . Second, the crystal structure of the χ c phase could also be presented as packing of complex tetrahedral helices, which may make it a potential candidate for use in structural color palettes, nano-optics and nanopatterning. − Moreover, the dynamic exchange of particles in the χ c phase may be valuable for applications in which lattice rigidity and mass transport need to be coupled . Therefore, this concept and design principle based on hierarchical self-assembly of soft Janus colloids offers an easily extendable framework to construct complex colloidal cluster crystals with multiple functions.…”