the joint UM-StL Physics & Astronomy/Center for Molecular Electronics'

include P. Fraundorf, W. Heffern and D. Witt

Space-filling dodecahedron-based structures can be built from two building blocks, namely the acute (A6) and oblate (O6) rhombohedrons whose faces are golden rhombi, themselves constructed (to first order) from eight 20-atom dodecahedra as shown in the figures below.

acute\prolate\A6 --|-- O6/oblate/obtuse

The above models of A6 and O6 are constructed from perfect dodecahedra. Close inspection will show some misfit (A6) and overlap (O6), which nonetheless allows for tetrahedrally coordinated atom construction. As you might imagine from the symmetric design, the residual strain is taken up internally, without breaking symmetry of the rhombohedra themselves. Hence the stability question involves energetics, not strain relief (at least to first order), for structures built up from this point.

A6 atom positions

The interior of the A6 rhombohedron, constructed from atoms each with 4 bonds in approximately tetrahedral coordination, contains 40 atoms: 24 atoms (36 half atoms and 6 whole atoms) sit on the edges of 4 hexagons on the surface of the two internal adjoining non-regular polydedra which show up as open spaces in the A6 model above. *These 24 atoms have one pair of bonds pulled into a 120 degree angle (compared to the tetrahedral 119.47 degrees), while the other pair are pulled into the pentagonal 108 degrees.* The hexagons are surrounded by 12 pentagons on these internal polyhedra, which also offer sites for 8 additional atoms on the internal polygons. The rhombohedron also contains 8 atoms bonded thereto, near but not on the periphery of the rhombohedron's interior. *These 16 atoms have opposing bond pairs pulled into a pentagonal 108 degree angle.* The models, of course, also include atoms beyond the 40 mentioned above. These complete the finite thickness golden rhombi which make up its faces, and hence are external to the building block itself.

A periodic structure made from this A6 rhombohedron would have a density of approximately 92% that of a diamond face-centered structure (like diamond or semiconductor silicon). We have not yet considered it's mechanical, thermal, electronic, and spectroscopic properties, or it's appearance in diffraction.

O6 atom positions

The interior of the O6 rhombohedron, also constructed from atoms each with 4 bonds in approximately tetrahedral coordination, contains only 24 atoms. These include 12 atoms (24 half atoms) with a 120 degree bond pair opposite a pentagonal 108 degree pair, 6 atoms (12 half atoms) with a square 90 degree bond pair opposite a pentagonal 108 degree pair, and 6 atoms on a central ring with opposing bond pairs pulled to 90 degrees. Hence this structure, energy calculations aside, appears more strained than the A6 rhombohedron.

A periodic structure made from this O6 rhombohedron would have a density of approximately 89% that of a diamond face-centered structure with the same average tetrahedral bond length. We have not yet considered it's mechanical, thermal, electronic, and spectroscopic properties, or it's appearance in diffraction.

The A6 and O6 building blocks themselves also can be combined to fill space aperiodically, in a quasicrystal structure. Given the usual ratio of __:__ between A6 and O6 building blocks in the quasilattice, the density of this material would be approximately 90% that of a diamond face-centered structure made from the same atoms. We have not yet considered it's mechanical, thermal, electronic, and spectroscopic properties, or it's appearance in diffraction.

Although there are many contributors, the person responsible for errors is P. Fraundorf. This site is hosted by the Department of Physics and Astronomy (and Center for Molecular Electronics) at UM-StL. Mindquilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter linked subset of pages" since 4/7/2005: .