The Archimedean truncated octahedron. II. Crystal structures with geometric units of symmetry 43m

Abstract

All geometric units in crystal structures of space groups

The Archimedean truncated octahedron. III. Crystal structures with geometric units of symmetrym3m

“…Thus polyhedra of m3m symmetry can be found in 2,3m geometric units and the reverse is not true. Theoretically, the cube and the octahedron can be present in geometric units of symmetry lower than m3m, but only octahedron and truncated octahedron are found in real z~3m geometric units as seen in the previous survey (Chieh, 1980). The cube and the small rhombicuboctahedron are present in m3m geometric units, for example those of Ge7Ir 3, Sb2TI 7, PdlTSe15 and C6Cr23.…”

“…The cuboctahedron, which is related to both the cube and the octahedron, is present in geometric units of both m3m and ~,3m symmetries; those in the latter are slightly distorted. Few 43m geometric units having more than one atom in the Fm3m space group consist of or start (from the center) with a tetrahedron, but many for space groups I43m, P43m, F43m and Fd3m begin with a tetrahedron (Chieh, 1980). In contrast, m3m geometric units consisting of more than one atom usually start with an octahedron or a cube.…”

“…The geometric units A and B for space group Fm3m have symmetry m3m whereas the unit C belongs to i~ili°'°'°i 3~3m, which has been discussed in part II (Chieh, 1980). …”

“…The application of this_concept to crystal structures with geometric units of 43m symmetry was given in part II (Chieh, 1980). Only geometric units in Im3m, Pm3m and Fm3m have m3m symmetry; they have one, two and three types of geometric units respectively.…”

Comments on the relationship betweenThe Archimedean truncated octahedron, and packing of geometric units in cubic crystal structuresby C. Chieh andOn the choice of origins in the description of space groupsby H. Burzlaff & H. Zimmermann

“…Representations of crystal structures were accomplished by listing the geometric units along the body diagonal of the unit cell. Examples in the application of this concept have been given by Chieh (1980Chieh ( , 1982. This method uses the advantage of origins with points of high symmetry of some space groups.…”