Nested polyhedra units: a geometrical concept for describing complicated cubic structures

A method for describing cubic giant cells is proposed. The structures may be regarded as arrangements of nested polyhedra units, isolated or linked, centred at sites of high point symmetry. Five basic types of nested polyhedra units, taken from well known cubic structures are defined: α-Mn (29 atoms), BCC (27), γ-brass (26), Ca₃Ag₈ (26) and Ti₂Ni (22) as well as variants of two of them, the non-centred α-Mn unit (28 atoms) and the centred Ca₃Ag₈ unit (27 atoms). All seven units have a cubo-octahedron as the outermost polyhedron. Sixteen structures are described in this paper, among which some have more than 400 atoms in the unit cell. This purely geometric description has the advantage that it can be easily obtained from the atom coordinates. It further makes structural similarities evident and simplifies the task of visualizing and memorizing complicated cubic structures.