Periodic graph (crystallography)

History

A crystal net is an infinite molecular model of a crystal. Similar models existed in Antiquity, notably the atomic theory associated with Democritus, which was criticized by Aristotle because such a theory entails a vacuum, which Aristotle believed nature abhors. Modern atomic theory traces back to Johannes Kepler and his work on geometric packing problems. Until the twentieth century, graph-like models of crystals focused on the positions of the (atomic) components, and these pre-20th century models were the focus of two controversies in chemistry and materials science.

The two controversies were (1) the controversy over Robert Boyle’s corpuscular theory of matter, which held that all material substances were composed of particles, and (2) the controversy over whether crystals were minerals or some kind of vegetative phenomenon.{{Citation |editor-last = Lima-de-Faria |editor-first = J.

However, despite the growing use of stick-and-ball molecular models, the use of graphical edges or line segments to represent chemical bonds in specific crystals have become popular more recently, and the publication of{{Cite book |author-link = Coxeter |doi-access = free

Basic formulation

A Euclidean graph in three-dimensional space is a pair (V, E), where V is a set of vertices (sometimes called points or nodes) and E is a set of edges (sometimes called bonds or spacers) where each edge joins two vertices. There is a tendency in the polyhedral and chemical literature to refer to geometric graphs as nets (contrast with polyhedral nets), and the nomenclature in the chemical literature differs from that of graph theory.{{Citation

Symmetries and periodicity

A symmetry of a Euclidean graph is an isometry of the underlying Euclidean space whose restriction to the graph is an automorphism; the symmetry group of the Euclidean graph is the group of its symmetries. A Euclidean graph in three-dimensional Euclidean space is periodic if there exist three linearly independent translations whose restrictions to the net are symmetries of the net. Often (and always, if one is dealing with a crystal net), the periodic net has finitely many orbits, and is thus uniformly discrete in that there exists a minimum distance between any two vertices.

The result is a three-dimensional periodic graph as a geometric object.

The resulting crystal net will induce a lattice of vectors so that given three vectors that generate the lattice, those three vectors will bound a unit cell, i.e. a parallelepiped which, placed anywhere in space, will enclose a fragment of the net that repeats in the directions of the three axes.

Symmetry and kinds of vertices and edges

Two vertices (or edges) of a periodic graph are symmetric if they are in the same orbit of the symmetry group of the graph; in other words, two vertices (or edges) are symmetric if there is a symmetry of the net that moves one onto the other. In chemistry, there is a tendency to refer to orbits of vertices or edges as “kinds” of vertices or edges, with the recognition that from any two vertices or any two edges (similarly oriented) of the same orbit, the geometric graph “looks the same”. Finite colorings of vertices and edges (where symmetries are to preserve colorings) may be employed.

The symmetry group of a crystal net will be a (group of restrictions of a) crystallographic space group, and many of the most common crystals are of very high symmetry, i.e. very few orbits. A crystal net is uninodal if it has one orbit of vertex (if the vertices were colored and the symmetries preserve colorings, this would require that a corresponding crystal have atoms of one element or molecular building blocks of one compound – but not vice versa, for it is possible to have a crystal of one element but with several orbits of vertices). Crystals with uninodal crystal nets include cubic diamond and some representations of quartz crystals. Uninodality corresponds with isogonality in geometry and vertex-transitivity in graph theory, and produces examples objective structures.{{Citation

Geometry of crystal nets

In the geometry of crystal nets, one can treat edges as line segments. For example, in a crystal net, it is presumed that edges do not “collide” in the sense that when treating them as line segments, they do not intersect. Several polyhedral constructions can be derived from crystal nets. For example, a vertex figure can be obtained by subdividing each edge (treated as a line segment) by the insertion of subdividing points, and then the vertex figure of a given vertex is the convex hull of the adjacent subdividing points (i.e., the convex polyhedron whose vertices are the adjacent subdividing points).

Another polyhedral construction is to determine the neighborhood of a vertex in the crystal net. One application is to define an energy function as a (possibly weighted) sum of squares of distances from vertices to their neighbors, and with respect to this energy function, the net is in equilibrium (with respect to this energy function) if each vertex is positioned at the centroid of its neighborhood,{{Citation |url-access = subscription |doi-access = free

Active areas of crystal design using crystal nets

It is conjectured{{Cite book

Historically, crystals were developed by experimentation, currently formalized as combinatorial chemistry, but one contemporary desideratum is the synthesis of materials designed in advance, and one proposal is to design crystals (the designs being crystal nets, perhaps represented as one unit cell of a crystal net) and then synthesize them from the design.{{Citation |hdl-access = free

One of the primary issues in annealing crystals is controlling the constituents, which can be difficult if the constituents are individual atoms, e.g., in zeolites, which are typically porous crystals primarily of silicon and oxygen and occasional impurities. Synthesis of a specific zeolite de novo from a novel crystal net design remains one of the major goals of contemporary research. There are similar efforts in sulfides and phosphates.

Control is more tractable if the constituents are molecular building blocks, i.e., stable molecules that can be readily induced to assemble in accordance with geometric restrictions. Typically, while there may be many species of constituents, there are two main classes: somewhat compact and often polyhedral secondary building units (SBUs), and linking or bridging building units. A popular class of examples are the Metal-Organic Frameworks (MOFs), in which (classically) the secondary building units are metal ions or clusters of ions and the linking building units are organic ligands. These SBUs and ligands are relatively controllable, and some new crystals have been synthesized using designs of novel nets.{{Citation

References

References

1. (1991). "Recognizing Properties of Periodic Graphs". DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4: Applied Geometry and Discrete Mathematics.