## CRYSTAL STRUCTURE TOPOLOGY ILLUSTRATIONS: CRITICAL POLYHEDRA IN EUCLIDEAN
AND ORBIFOLD SPACES

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

Chemical
& Analytical Sciences Division, Oak Ridge National Laboratory, P.O. Box
2008, Oak Ridge, TN 37831-6197.

The canonical critical point network within a crystal forms a unique directed
graph connecting pits (polyhedra) to pales (faces) to passes (edges) to peaks
(atoms) by the gentlest up-density path, and may be represented by ORTEP
`critical polyhedra drawings' (see figure) with spheres, prolate ellipsoids,
oblate ellipsoids and smaller spheres for peaks, passes, pales, and pits,
(i.e., 0,1,2,3) respectively, connected along canonical paths. Alternately,
critical point networks embedded into orbifolds (defined below) provide a
more concise depiction of
crystal structure topology. Example illustrations of NaCl and FCC structures in
Fm3m, and HCP and hexagonal diamond in P6_{3}/mmc, and BCC in Im3m will
be compared. The (0,1,2,3)'s are on Wyckoff sites (a+b,e,d,c), (a,d,f,c+b),
(c,g+h,b+d+k,a+f), (f,c+g,a+h,b+d), and (a,c,d,b), respectively. These critical
point sets may differ from quantum chemistry results which may have additional
peaks not on atomic centers.

Each space group produces a unique orbifold by folding up the space group's
geometrical representation to overlay all symmetry-equivalent points. The
mirrors, rotation axes, and symmetry points become the singular set of the
orbifold. Mirrors become a silvered boundary on the orbifold and rotation axes
become knots, links, or graphs with point symmetry elements at the vertices.
There is a 1:1 correlation between Wyckoff sites and singular set components
except that a 3-fold axis or its intersecting elements may be continued through
the intersection. The cubic Fm3m yields a tetrahedral orbifold bounded by
mirrors. The asymmetric unit polyhedra for P6_{3}/mmc and Im3m have
mirrors on all but one face, where each has a 2-fold axis bisecting a glide
face. Folding around these 2-folds produces closed silvered boundaries.
Orbifolds may become an everyday working concept for 21st century
crystallographers.

Research sponsored by the Laboratory Directed R&D Program of ORNL, managed
for the U.S. DOE by Martin Marietta Energy Sys., Inc., under cont.
DE-AC05-84-OR21400.