White Paper Title: Crystallographic Topology of Nanomaterials.
Revision Date: March 7, 2001
Author: Carroll K. Johnson, Adjunct Professor, Department of Mathematics, University of Tennessee, Knoxville, TN 37996; and Consultant/Retiree, Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6197.
As an extension of our previous work in crystallographic topology, we plan to topologically characterize several families of crystalline nanotechnology materials and develop a framework for prediction of new nanomaterials. Subgoals include: extension of the crystallographic topology model to handle nanotechnology problems, organization of a conference to discuss the relevant mathematical and crystallographic problems, and writing a textbook or reference volume on crystallographic topology.
Our Crystallographic Topology project[i] had Laboratory Directed Research and Development funding at Oak Ridge National Laboratory during FY 94 - 96. The present author retired in 1996, and then continued the project[ii] informally without funding. The scientific progress made, its relevance to the expanding nanotechnology materials field, and the current emphasis on converting advanced mathematical results to productive applications, suggests a more aggressive development phase should be reinitiated. Structural crystallography provides a fruitful application area for recent advances in global analysis (Morse theory), Gaussian measures (Radon Nikodym derivatives), geometric topology (orbifolds and Heegaard splitting), and operator algebra (groupoids).
Our current research results in crystallographic topology are mainly exploratory with experimental verification of applicability through manual manipulation of examples. Fortunately, there are proper theoretical foundations on which to build this methodology. However, applied crystallography methods must also have a high level of computer automation to be viable in the contemporary crystallographic laboratory. Unlike numerical computing, computational topology is a young discipline with a very small archive of proven algorithms and subroutines.
Structural Crystallography using x-ray, neutron, or electron diffraction gives reliable and comprehensive analyses of atomic level geometry in solid-state materials ranging from simple metals to crystalline biological viruses. High precision crystal-structure analysis provides a positional 3-vector and Gaussian thermal motion 3x3 covariance matrix for each atom in a fundamental domain of the crystal lattice's unit cell. The fundamental domain is determined by the crystal's space group, which will be one of 230 infinite groups of rigid motion in Euclidean 3-space. Real crystals are always finite, imperfect, and more complex than the above description, but this classical model works remarkably well for the prevalent perfectly imperfect crystals used for most crystal structure analyses. Systematically imperfect crystals, which are often important in nanotechnology research, are discussed in a later section.
Crystallographic Topologyi ii
Crystallographic topology descriptions of a crystal structures use topological quantifiers describing environment rather than the usual interatomic distance and angle metric detail. Mathematical topology is the study of distortion invariant properties of spaces and objects, thus it is more than just lay-of-the-land topography. For example the surface of a donut, a coffee cup, and the Euclidean 3-orbifold for the simplest crystallographic plane group p1 (described in the orbifolds section) all have the topology of a 2-torus. The ability to distort an object or space drastically without changing its topology will be exploited in the section on orbifolds, but first we will describe some basic features of undistorted crystal-space topology.
Morse Function: We represent each atom in a crystal structure by its experimentally determined time-and-lattice averaged mean-square thermal displacement, called a Gaussian measure in stochastic analysis. The absolutely continuous components of motions for atom pairs may be related through a Radon-Nikodym derivative of one anisotropic Gaussian measure with respect to the second. This procedure also partitions space around each atom into a dented ball where each point of the partition has equal density contribution from two or more adjacent Gaussian measures. For the general anisotropic Gaussian measures, the dented ball is a fourth-degree function at the intersection of measure pairs, eight-degree at measure triples, twelfth-degree at measure quadruples, etc.
The ensemble of all atom Gaussian measures produces a total thermal-motion density function for the crystal, which may be described by its topological global analysis properties expressed as a Morse function. A Euclidean 3-space thermal-motion Morse function has non-degenerate critical points [i.e., points where the first derivative is zero, and the second derivative has non-zero eigenvalues with sign signature either -,-,- (peak); -,-,+ (pass); -,+,+ (pale), or +,+,+ (pit)]; and separatrix lines of minimum slope interconnecting topologically communicating critical point basins, which together form a critical-net graph representing the salient topological features of the stochastic thermal-motion Morse function.
The critical-net graph has peaks on the atomic sites, saddle-point passes between pairs of adjacent peaks, pits at minimal density points and saddle-point pales between pairs of adjacent pits. Atoms and strong bonds are always on peaks and near passes, respectively, but not all passes are along chemical bonds. We use thermal motion density rather than electron density for computational simplicity and to avoid quantum chemistry complications such as the lone-pair electron density peaks often found along chemical bonds in experimental (and quantum chemistry calculated) electron density functions.
Orbifold: The fundamental domains of a two-dimensional space group (i.e., one of the seventeen wall paper groups), have matching edges and may be wrapped up into a compact surface by overlaying all equivalent edge points to form the corresponding topological 2-surface (torus (p1), Klein bottle (pg), silvered edged annulus (pm), silvered edge Mobius band (cm) projective plane with singularities (pgg); or one of a series of silvered discal (e.g., p4mm) or spherical (e.g., p4) surfaces with singularities) called a Euclidean 2-orbifold. Singularities (which include points, lines, and silvered boundaries) arise from points, axes, and planes of symmetry in the parent plane group. The torus and Klein bottle orbifolds are also manifolds since they have no singularities.
Similarly, the 230 three-dimensional space groups fold up into 230 Euclidean 3-orbifolds, each of which is a closed topological 3-surface often with silvered boundary components arising from mirrors of symmetry, singularity lines from rotation axes, singularity points from inversion centers, and induced singularities where other symmetry elements intersect. Orbifold topology can be pictorially and mathematically characterized more succinctly than the parent groups since group symmetry related repetition has been removed by the "orbi-folding" quotient operation, expressed algebraically Q = E/G, where E is Euclidean 3-space, G is the space group, and Q is the Euclidean 3-orbifold. There are a number of quite unusual underlying topological spaces associated with the crystallographic 3-orbifolds. For example, space group P1bar's orbifold is a projective 3-space with eight-fold suspension from the eight different inversion centers of that space group, which provides an interesting visualization problem.
Orbifold Atlas: We have a partial compilation (the 36 cubic cases) of the 230 space-group orbifolds on our Crystallographic Topology web site[iii]. There are two identical singular set drawings for each orbifold with singular-set orbit and invariant lattice-complex labels on one drawing and the corresponding International Tables of Crystallography labels for Wyckoff special positions on the other. This information is also given algebraically in tabular form. Lattice complexes have a long crystallographic history but seem to have no counterpart in the mathematical literature. They provide a convenient notation for integrating global special position configuration information (such as D for diamond conformation site) with the local information of the conventional orbifold singular set notation.
An extension is planned to incorporate space group screw-and-glide groupoids[iv] into an orbifold to develop a generalized "orbifoldoid" which explicitly describes the topological properties of all symmetry elements of a space group rather than just the singularity orbits. Groupoid properties and additional crystallographic applications are discussed in the section on groupoids.
Morse function on orbifold (MFO): The critical net of the Morse function is superimposed onto the space group and distorted into two overlapping closed graphs by the orbi-folding transformation of the space group into a Euclidean 3-orbifold. The critical-net-on-orbifold crystal structure representation features are: (a) the group symmetry derived singular set (singularity points, lines, and surfaces); and (b) the crystal structure Morse function critical set (critical points and separatrix lines).
Feature sets a and b above are topologically interdependent and provide constraints for Morse function analysis and design; for example, all singularity points contain critical points but not vice versa. The symmetry singular set is a group-subgroup complex with the ratios of adjacent subgroup orders providing branching ratios for the associated topological critical set complex components; thus providing a concise crystal chemistry interpretation of the MFO.
Heegaard Splitting of MFO: A level (equal density) surface between the pass and pale critical points of a MFO is a Heegaard surface, which provides a Heegaard splitting of the MFO into two topological handlebodies, thus separating the peak and pass components in the (+) handlebody from the pale and pit components in the (-) handlebody. Corresponding level surfaces in Euclidean crystal space are usually hyperbolic and resemble the minimal surfaces ("soap bubble surfaces") used to outline chemical moieties in certain types of crystal structure drawings.
Modifications of MFO: Cut-and-paste modification of the topological critical set allows insertion, deletion, and rearrangement of critical set components to discover new crystal structures subject to: (1) topological "handlebody mechanics" and duality; (2) Euclidean space Poincare relationship relating numbers of peaks - passes + pales - pits = 0 in a unit cell, and Morse theory inequalities on subsets of critical points; (3) geometric Morse function properties such as the pass "wagon wheel" of pales around peak-pass-peak axes, and its topological dual for passes around pit-pale-pit axes; (4) space-group singular-set constraints; (5) space group / sub space group trees which progressively relax the singular set constraints; and (6) crystal chemistry properties such as atom pair reactivity, and archival molecular geometries.
Example MFO critical nets comparing the hexagonal close packed, graphite, and hexagonal diamond structures are shown in Fig. 5.3 on our web siteiii. All three are on the same space group orbifold and the later two are topological duals in which peaks and passes are interchanged with pits and pales.
Systematically imperfect crystals include quasicrystals, modulated crystals, and nanotechnology crystals in which a large variety of inclusions, imperfections, and sub-crystallite boundaries provide unusual physical and chemical properties. For small crystals and crystals with extensive voids and inclusions, the perfectly imperfect model can be amended to include crystallite interface effects, which brings us to groupoids.
Groupoid: A characteristic of a mathematical group is that any two members of the group's algebraic set can be multiplied to form another member of the set. On the other hand, a groupoid is an algebraic set which may have only a partially defined multiplication; that is, not all members of a groupoid algebraic set can be multiplied. Groups describe global homogeneous symmetry acting on a set of objects, but groupoids can in addition describe non-homogeneous symmetries for subsets of the object set. An established crystallographic groupoid application, used in order-disorder layered-structure crystal-structure analysis, utilizes one or more of the eighty 2-sided layer-group symmetries to describe the additional local symmetries between layer pairs within such a crystal, and any layer-to-layer disordering present as well.
Another crystallographic groupoid approach describes the joint properties of both the bulk internal and the boundary/interface symmetry[v] and structure. This should provide new opportunities in nanotechnology structure-activity modeling of finite crystals where surface properties are of fundamental interest. We are optimistic that much of our crystallographic topology model can be reformulated using such groupoids, but this is by no means a trivial task As a backup there is still much progress which can be made with the existing crystallographic topology tools since much of the nanotechnology progress to date seems to have required nothing past that available from conventional physical models.
… to be
[i] Crystallographic Topology and Its Applications by Carroll K. Johnson, Michael N. Burnett, and William D. Dunbar. Crystallographic Computing 7 Macromolecular Cystallographic Data, edited by P. E. Bourne and K. D. Watenpaugh, to be published by Oxford Press.
[ii] Crystallographic Topology 2: Overview and Work in Progress by Carroll K. Johnson, published in Trends in Mathematical Physics, AMS/IP Studies in Advanced Mathematics, edited by V. Alexiades and G. Siopsis, International Press, Cambridge MA, 1999.
[iii] Crystallographic Topology web site provides download copies of the two above publications at http://www.ornl.gov/ortep/topology/preprint.html, and related on-line information at http://www.ornl.gov/ortep/topology.html.
[v] Groupoids: Unifying Internal and External Symmetry by Alan Weinstein, Notices of the AMS, July 1996.