CRITICAL POLYHEDRA ON ORBIFOLDS: TOPOLOGY IN CRYSTAL STRUCTURE
Michael N. Burnett and Carroll K. Johnson
Chemical & Analytical Sciences Division, Oak Ridge National Laboratory,
P.O. Box 2008, Oak Ridge, TN 37831-6197.
Orbifold and critical point theory may be combined to assist in systematic
enumeration of higher-symmetry candidate crystal structures. The representation
of crystal structures by critical polyhedra on orbifolds is discussed in an
accompanying abstract. A small neighborhood of any point in a Euclidean 3-
orbifold, Q, is isomorphic to an elliptic 2-orbifold, which is derived from the
Wyckoff point group for the corresponding neighborhood in the parent space
group for Q. Mirrors of the space group will appear as the boundary of Q, but
the main orbifold structure is determined by the one-dimensional singular locus
arising from n-fold rotation axes (n=2,3,4,6). An
edge of the singular locus graph within the orbifold interior is denoted
Snn (S for sphere), while an edge on the silvered boundary is
Dn'n' (D for disk), with ' indicating that an
element lies in a mirror. Intersections of axes produce vertices on the
singular locus graph which will be trivalent for interior intersections
Sn22, S332,- and S432. Real projective plane points arising from
-(RP), -(RP2), and -(RP3)
inversion centers also occur within Q and have 0, 1, and 1 incident edges,
respectively. Vertices on the silvered boundary D are Dn, D22',
D23', D32', Dn'2'2', D3'3'2', and D4'3'2'
with 1, 2, or 3 incident edges indicated by the number of digits in the symbol.
For a crystal structure's asymmetric unit on Q, every fixed point (vertex) of
Q's singular locus will contain a critical point from the asymmetric unit but
not vice-versa. A connecting edge in the critical polyhedra may traverse only
one elliptic 2-orbifold zone of Q. Thus, each peak, pass, pale, and pit vertex
and each canonical edge of the critical polyhedra is represented as an elliptic
2-orbifold, which in turn has a covering multiplicity corresponding to the
order of the associated isometry group. Ratios of adjacent region multiplicity
factors describe the branching ratios of the crystal structure. A rich set of
combinatorial topology relationships is inherent in this representation, which
should lead to improved algorithms for predicting crystal structures. Our
progress toward this goal will be described.
Ref: Thurston, W.P., The Geometry and Topology of Three-Manifolds, 1979,
The Geometry Center, firstname.lastname@example.org.
Research sponsored by the Laboratory Directed R & D Program of ORNL,
managed for the US DOE by Martin Marietta Energy Syst., Inc., under contract