Carroll K. Johnson and Michael N. Burnett

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6197, USA [e-mail (, (].

Our Crystallographic Orbifold Atlas illustrates space-group topology by showing asymmetric units of space groups wrapped up to form closed spaces, called Euclidean 3-orbifolds, which have singular sets corresponding to the Wyckoff sites. The Gaussian density for a crystal structure, based on overlapping Gaussian density functions centered on atomic sites, has a critical-net representation with critical points joined by density gradient-flow separatrices. Crystal-structure critical nets, wrapped into the corresponding space-group orbifolds, form Crystal Orbifold Morse Functions (COMFs) with the singular set of the space group acting as a template for the critical net. COMFs provides a new approach for classifying both crystal structures and space groups.

For simple crystal structures, each component of the critical net, which includes (a) peaks, (b) passes, (c) pales, and (d) pits, as well as (ab), (bc), and (cd) separatrices, plus the (da) steepest gradient paths, corresponds to a classical crystallographic lattice complex. This geometric arrangement of lattice complexes provides the global characteristics needed to characterize and classify crystal structure families using only the asymmetric units of the unit cells wrapped up as COMFs. Morse functions on orbifolds have unique topological characteristics which currently are not well characterized in the mathematical topology literature.

Crystallographers have long bemoaned the fact that traditional space group nomenclature is more a hindrance than a help in classification requiring systematic symmetry breaking. We are trying to derive a more structurally related space-group classification based on the imbedding properties of a basis set of simple COMFs into space group orbifolds. This classification also will incorporate space-group/subgroup relationships as given by the color Shubnikov groups represented as color orbifolds.

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Research sponsored by the Laboratory Directed R&D Program of ORNL, managed by LMERC for U.S. DOE under contract DE-AC05-96OR22464.