Structures of the cubic and rhombohedral high-pressure modifications of silicon as packing of the rod-like substructures determined by the algebraic geometry

Abstract

Tessellations by generating clusters are proposed for the high-pressure phases BC8 and R8 of silicon. The structures of both high-pressure phases are represented by the parallel packings of rods. The latter are stacks of distorted icosahedra, joined by a common triangular face and flattened out along the threefold symmetry axis. Along the rod axis there is an alternation of empty and double-centered icosahedra. The empty icosahedra are additionally distorted in the cubic BC8 phase by antiparallel rotation about the rod axis, while the double-centered icosahedra are distorted by that rotation in the rhombohedral R8 phase. A possible mechanism of the reversible BC8 <--> R8 transformation is proposed as the rotation about the rod axis of the common triangular face of the neighboring icosahedra, thus transforming between distorted and undistorted icosahedra. The graphs of the generating clusters for both the BC8 and R8 structures are determined by two subconfigurations of the same construction of the finite projective geometry.