A procedure is under development to derive families of topologically transmuted crystal structures using the methods of 3-manifold topology [1].

For a known starting crystal structure, Radon Nikodym derivatives [2] for pairs of thermal ellipsoids are used in a maximum likelihood calculation to find peak, pass, pale, and pit critical points and their topological connecting paths. This provides a 3-D critical net representation [2] of the Morse function global thermal motion density. We then add representations for the rotational axes and mirrors of the space group, and calculate all intersections (within an asymmetric unit of the unit cell) with the Morse function Heegaard surface, which is a constant density surface partitioning (passes + peaks) and (pales + pits) into two disjoint sets [3].

The asymmetric unit is topologically cut out and wrapped up to superimpose all symmetry equivalent faces of the asymmetric unit, thus producing a "Heegaard critical net on orbifold" closed space representation of the crystal structure, space group, and Heegaard surface. The Heegaard surface is used in Heegaard splitting [1,3] of the space group Euclidean 3 orbifold [2] and critical net into a pair of handlebody 3-orbifolds [4] with a shared hyperbolic 2-orbifold boundary between them. The peaks and passes of the critical net are in the (+) handlebody and the remainder in the (-).

To derive other crystal structures, one can transmute the above Heegaard surface and critical net to other forms using some of the many innovative approaches to 3-manifold classification. A simple example is to interchange Haken quadrilateral normal surfaces [1,3] in the F-43m tetrahedral 3-orbifold [2,4], thus transmuting the ZnS and NaCl structure types.

[1] K. Johannson, Topology and Combinatorics of 3-Manifolds, Springer (1995). [2] C. K. Johnson and M. N. Burnett, Crystallographic Topology Web Site: http://www.ornl.gov/ortep/topology.html. [3] M. Scharlemann, "Heegaard Splitting of Compact 3-Manifolds" in Handbook of Geometric Topology, Elsevier, in press. [4] B. Zimmermann, Michigan Math. J. 43, 593-610 (1996).
*Operated by Lockheed Martin Energy Research Corp. for the U.S. Dept. of Energy under Contract No. DE-AC05-96OR22464.