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Crystallographic Topology and Its Applications * Carroll K 4 Crystallographic Topology and Its Applications * Carroll K. Johnson and Michael N. Burnett Chemical and Analytical Sciences Division, Oak Ridge National Laboratory Oak Ridge, TN 37831-6197, USA [email protected] http://www. ornl.gov/ortep/topology. html William D. Dunbar Division of Natural Sciences & Mathematics, Simon’s Rock College Great Barrington, MA 01230, USA wdunbar @plato. simons- rock.edu http://www. simons- rock.edd- wdunbar Abstract two disciplines. Since both topology and crystallography have many subdisciplines, there are a number of quite Geometric topology and structural crystallography different intersection regions that can be called crystall o- concepts are combined to define a new area we call graphic topology; but we will confine this discussion to Structural Crystallographic Topology, which may be of one well delineated subarea. interest to both crystallographers and mathematicians. The structural crystallography of interest involves the In this paper, we represent crystallographic symmetry group theory required to describe symmetric arrangements groups by orbifolds and crystal structures by Morse func- of atoms in crystals and a classification of the simplest arrangements as lattice complexes. The geometric topol- tions. The Morse @fiction uses mildly overlapping Gau s- sian thermal-motion probability density functions cen- ogy of interest is the topological properties of crystallo- tered on atomic sites to form a critical net with peak, pass, graphic groups, represented as orbifolds, and the Morse pale, and pit critical points joined into a graph by density theory global analysis of critical points in symmetric gradient-flow separatrices. Critical net crystal structure functions. Here we are taking the liberty of calling global drawings can be made with the ORTEP-111 graphics pro- analysis part of topology. gram. Our basic approach is that of geometric crystallogra- phers who find the pictorial reasoning of geometric topol- An orbifold consists an underlying topological of ogy intriguing. From a mathematical perspective, one can space with an embedded singular set that represents the reformulate the subject using algebraic topology concepts Wyckoff sites of the crystallographic group. An orbifold such as cohomology, which we seldom mention in this for a point group, plane group, or space group is derived by gluing together equivalent edges or faces a crystal- Paper. of The International Tables for Crystallography (ITCr), lographic asymmetric unit. Volume A: Space-Group Symmetry‘ is the chief source The critical-net-on-orbifold model incorporates the for the crystallographic material in the following discus- classical invariant lattice complexes of crystallography sion. It is our hope that the discipline of “Crystallographic and allows concise quotient-space topological illustra- Topology” will mature in completeness and usefulness to tions to be drawn without the repetition that is character- justify the addition of this subject to the ITCr series at istic of normal crystal structure draw ings. some future time. There are a number of crystallographic and topologi- 1. Introduction cal concepts that lead to the following mappings of struc- tural crystallography onto geometric topology. Only the first two of the three mapping series are discussed here. For our purpose we will say that crystallography is the study of atoms in crystals, topology is the study of Crystallographic Groups Spherical and Euclidean Orbi- distortion-invariant properties of mathematical objects, + folds and crystallographic topology is an intersection of those * Research sponsored by the Laboratory Directed Research and Development Program of the Oak Ridge National Lab o- ratory, managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under Contract No. DE-AC05-960R22464. “me submitted manuscript has been authored by a contractor Of the U.S. Government under contract No. DE- AC05-960R22464. Accordingly, the US. Government retains a nonex- clusive, royalty-free license to publish or reproduce the published form of this contribution, or allow 1 others to do so, for US. Government ~~ purposes.” DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. DISCLAIMER report was prepared as an account of work sponsored by an agency of the ThisUnited States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any cific commercial product, process, or service by trade name, trademark, manufac-spe- turer, or otherwise not necessarily constitute or imply its endorsement, mm- mendation, or favoringdoes by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. f Crystal Structures -+ Morse Functions + Critical Nets + through small perturbations.’*’ We have so far not found a Critical Nets on Orbifolds + Lattice Complexes on Criti- true degenerate critical point in a valid crystal structure cal Nets on Orbifolds and have a working hypothesis that all crystal structures are Morse functions, which are named after Marston Crystal Chemistry + Convolution of Chemical Motif Morse’ and have no degenerate critical points. Critical Nets onto Orbifold Singular Sets Critical points occur where the first derivative of the global density is zero. The second derivative at that point 1.1 Organization is a 3 x 3 symmetric matrix, which has a non-zero deter- minant only if the critical point is non-degenerate. The signs of the three eigenvalues of the second derivative Sect. 1 provides an overview and illustrates a simple critical net, orbifold, and critical net on orbifold based on matrix specify the types of critical points, which we term the sodium chloride crystal structure. Following a review peak (-,-,-), pass (+,-,-), pale (+,+,-) and pit (+,+,+). A of relevant orbifold references, Sect. 2 continues to illus- degenerate critical point will have a singular second de- trate and classify the 32 spherical 2-orbifolds derived from rivative matrix with one or more zero or nearly zero ei- the crystallographic point groups and shows how spherical genvalues. 2-orbifolds can be used as construction elements to build The critical points are best described as representing the singular sets of Euclidean 3-orbifolds. Sect. 2 also 0-, 1-, 2-, and 3-dimensional cells in a topological Morse illustrates basic topology surfaces, derivation of all 17 function CW complex (i.e., C for closure finite, W for Euclidean 2-orbifolds from crystallographic drawings of weak topology). We use a “critical net” representation that the plane groups, and example derivations of Euclidean 3- has unique topological “separatrices” joining the critical orbifolds by lifting base Euclidean 2-orbifolds. Some of point nodes into a graph. We denote the peak, pass, pale, the singular sets of the polar space group orbifolds are and pit critical points with the numbers 0, 1, 2, and 3, re- illustrated since polar space groups are the ones of chief spectively. The most gradual down-density paths from a interest to biological crystallographers. peak to a pit follow the sequence peak + pass + pale -+ Sect. 3 describes the Morse functions used and shows pit. These paths, shown by the separatrices (i.e., additional critical net examples, using ORTEP illustra- “connection links”) in Figs. 1.1 and 1.2, are topologically tions, and summarizes their characteristics. Sect. 4 illus- unique. This uniqueness arises because: (a) the pass and trates the derivation of critical nets on orbifolds, their pale critical points each have one unique eigenvector con- presentation in linearized form, and the derivation of a necting to the separatrices going to one peak and one pit, symmetry-breaking family of cubic lattice complexes on respectively, and (b) there are two-dimensional hyper- orbifolds. The crystallographic lattice complex model as planes connecting to the two remaining eigenvectors of modified for critical nets on orbifolds is discussed in Sect. each pass and pale and these non-parallel hyperplanes 5. Sect. 6 summarizes the current status of crystallo- intersect each other locally to form the pass-pale separa- graphic topology and the future developments required to trices. make it a productive subfield of contemporary crystallog- We postulate that there are no bifurcated (forked) raphy. The Appendix shows a grouplsubgroup graph for separatrices or degenerate critical points in the crystallo- the cubic space groups. graphic critical nets of interest here. In experimentally A Crystallographic Orbifold Atlas (in preparation) derived crystallographic macromolecule electron-density will eventually provide a full tabulation of those topologi- functions, this will not be the case because of critical point cal properties of crystallographic orbifolds that seem po- merging caused by inadequate resolution experimental tentially useful to crystallographers. We have basic results data and lattice-averaged static disorder. Theoretical covering most of the space groups, but at present we have quantum chemistry and high precision x-ray structure
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