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.