## SYSTEMATIC CRYSTAL CHEMISTRY VIA CRITICAL NETS ON ORBIFOLDS

### Carroll K. Johnson and Michael N. Burnett

Chemical Sciences Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831-6197, USA
[e-mail (ckj@ornl.gov), (mnb@ornl.gov)].

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.

Please visit our WWW site at
http://www.ornl.gov/ortep/topology.html

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