Jan. 15, 2001
Publications by Carroll K. Johnson et al. ::
BibTeX references.
Quote from Dr. Johnson:
"Based on the postulate that one should change directions every
five years to avoid stagnation,"...
Web links:
by Carroll
K. Johnson,
published in TRENDS IN MATHEMATICAL PHYSICS, AMS/IP Studies in Advanced
Mathematics,
edited by V. Alexiades and G. Siopsis, American Mathematical Society,
Providence RI,
and International Press, Cambridge MA, 1999.
Proceedings of conference held October 14-17, 1998 at the University of Tennessee, Knoxville.
This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vector valued Gaussian measures with the Radon-Nikodym property. Intended crystallographic benefits include new methods for visualization of space groups and crystal structures, analysis of the thermal motion patterns seen in ORTEP drawings, and a classification scheme for crystal structures based on their Heegaard splitting properties.
Carroll K. Johnson, Michael N. Burnett and William D. Dunbar
Presented at the IUCr sponsored Macromolecular Crystallography
Computing School held at Western Washington University in Bellingham,
Washington USA on August 17-23, 1996.
To be published in "Crystallographic Computing 7: Macromolecular
Crystallographic Data",
P. E. Bourne & K. D. Watenpaugh, eds., IUCr Crystallographic
Symposia, Oxford University Press, 2001+
Web link: http://citeseer.nj.nec.com/117300.html
Geometric topology and structural crystallography concepts are combined to define a new area we call Structural Crystallographic Topology, which may be of interest to both crystallographers and mathematicians. In this paper, we represent crystallographic symmetry groups by orbifolds and crystal structures by Morse functions. The Morse function uses mildly overlapping Gaussian thermal-motion probability density functions centered on atomic sites to form a critical net with peak, pass, pale, and pit critical points joined into a graph by density gradient-flow separatrices. Critical net crystal structure drawings can be made with the ORTEP-III graphics program.
An orbifold consists of an underlying topological space with an embedded singular set that represents the Wyckoff sites of the crystallographic group. An orbifold for a point group, plane group, or space group is derived by gluing together equivalent edges or faces of a crystallographic asymmetric unit.
The critical-net-on-orbifold model incorporates the classical invariant lattice complexes of crystallography and allows concise quotient-space topological illustrations to be drawn without the repetition that is characteristic of normal crystal drawings.
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