You are looking at a Reticular Chemistry Structure Resource constructed especially to be useful in the design of new structures of crystalline materials and in the analysis of old ones (cf. the discussion of reticular chemistry by Yaghi, O. M. et al. Nature 2003, 423, 705-714).

The symbol is either a three-letter symbol pqr or a three letter symbol with an extension as in pqr-a. It is not necessary to know the meanings of the extensions but it can be useful, and clicking on the "symbol" link on the "nets" search page will provide an explanation of the extensions.

Much of the data (structures with embed type not 1) have been entered by hand, so there doubtlessly are errors; please let us know if you identify any.

The structures are nets (special kinds of graphs) and to use the data one needs to know the edges ("bonds"). For "embed type" 1 (most of the data) the shortest distances are the edges. For other structures it is generally sufficient to put a vertex of valence 2 at the midpoints of the edges (these data are given). Structures where this won't work (one has to explicitly use the symmetry) include elv, mab, tcb, tcc, ten.

At the end of this page are references to some of our recent work relevant to the database.

Credits

The origin of this project was a National Science Foundation small grant for exploratory research (SGER) DMR-0243082 to Omar Yaghi and Michael O'Keeffe who collected the data. Maxim Peskov prepared much of the tiling data. The current realization is by Olaf Delgado-Friedrichs based on an earlier version by Stuart Ramsden. Software and other support is being generously provided by Vladislav Blatov, Stephen Hyde and Davide M. Proserpio who have corrected numerous errors and contributed in many other ways.

Users familiar with the Atlas of Zeolite Structure Types (see "links") will recognize our debt to that valuable work.

See the references at the end for the Fischer-Koch sphere packings.

Embeddings of nets ^[10]

Usually in describing a net, parameters are given appropriate to a maximum-symmetry embedding. For space groups with two origins in the International Tables we always use the second one (origin at a center od symmetry). Ideally also the shortest distances are all equal (for convenience unity in the units of the lattice parameters) and correspond to the edges of the net. This ideal situation is not always possible however, as can seen by considering a generic infinite net without symmetry. Let the average coordination number be Z, then there are Z/2 edges per vertex. Each vertex has three coordinates (x, y, z), so if Z > 6 there are more constraints (edge lengths) than degrees of freedom (coordinates) and there is no solution for equal edges possible. In fact, as the equations for edge lengths are non-linear in coordinates, there is no guarantee that there is a solution even for Z <= 6. Of course we are not dealing with generic structures, but rather with translational and, usually, other symmetries, so the problem is quite complicated. There are several cases to consider:

1. For the simple nets most common in crystal chemistry the maximum symmetry embedding is a sphere packing. There are three cases:
   1. The coordinates and unit cell shape are fixed by symmetry in the maximum-symmetry embedding.
   2. In other cases there are still degrees of freedom remaining. It is common in this case to report a structure that is a minimum of density (maximum cell volume) subject to the constraint of equal edge lengths which can be taken as unity.
   3. Again there are free parameters, but the minimum density configuration corresponds to a structure with more contacts. This is almost the same as 2a (not exactly as minimum density may not be maximum symmetry).
2. The maximum symmetry embedding corresponds to a sphere packing with more contacts than the coordination number, n, but n of those contacts correspond to edges; there are two cases:
   1. There is an embedding as an n-coordinated sphere packing of lower symmetry in which the n edges correspond to edges.
   2. There is not an embedding as above (strictly we should say that such an embedding cannot be found, proving one does not exist might be difficult). This case is more common than might be supposed.
3. The maximum symmetry embedding is a sphere packing with fewer contacts than the coordination number, n; but the n shortest distances correspond to the edges of the n-coordinated net; there are three cases:
   1. There is a lower-symmetry embedding with equal edge lengths and edges still corresponding to shortest distances between vertices.
   2. There is no embedding with equal edge lengths.
   3. There is an embedding with shortest distances equal to edges, but they cannot all be made equal and remain shortest distances.
4. There is no embedding with shortest distances corresponding to edges. This is in fact the case for the vast majority of nets. Until recently they were not recognized in crystal chemistry, as normally bonds (corresponding to edges of nets) are formed to nearest neighbors, but, with flexible linkers instead of bonds as the edges of nets, they are beginning to appear (see e.g. tcb). Most catenated nets (see e.g. pcu-c) are like this.
5. There may be no faithful embedding in the maximum symmetry conformation because the symmetry will require edges to intersect. There is always a lower-symmetry embedding - all graphs have a faithful embedding in 3-dimensional Euclidean space. These lower symmetry embeddings may be any one of the types described above in 1-4 (but of course not of maximum symmetry), and there may be several equally "good". These structures can be recognized by the fact that: (a) They have symbol abc-z and there is no entry abc. (b) They have embed type 5. The maximum symmetry for these nets is given in the same box as the embed type.

Collisions

In computing properties of nets, for example in the program Systre, it is convenient to use center-of-mass (barycentric) coordinates (all vertices assigned the same mass - see reference [4]). For some exceptional structures one may find that two vertices then have identical coordinates. Such structures are identified by an asterisk in the embed type (e.g. *1b). Systre won't always recognize these structures. Systre can be obtained from gavrog.org.

Density

The density is the number of vertices per unit volume for the given embedding. For structures of embed, type 1 or 2 (all edges of equal length and no shorter intervertex distance shorter) on can calculate the space-filling fraction of equal spheres in contact for that structure by multiplying the density by π/6.

Genus

A net can be considered a periodic surface of genus g if one imagines the edges inflated to have finite width. If there are e edges and v vertices in the primitive cell then g = 1 + e - v. See the link for "density" on the search page.

Tiling data

Data for tilings are for the so-called "natural" tilings (reference [2] and [13]). The D-symbols are what were originally called "Delaney" symbols by Dress and "Delaney-Dress" symbols by subsequent workers. The economical one-line encoding is due to Delgado-Friedrichs so "D-symbol" seems appropriate. The size of the D-symbol is what is known as "flag transitivity" to mathematicians. One may use the search to find the only regular tiling (D-symbol size = 1) and all natural tilings with size up to 4. Regular nets are defined independently of tilings but are the only structures with natural tilings that have one kind of vertex, edge, face and tile (transitivity 1111 - see reference [2]).

Layers and polyhedra

Layers are three dimensional 2-periodic structures and their symmetries are layer groups. Strictly 2-dimensional 2-periodic structures have symmetries that are plane groups. Structures that are not 2-dimensional have the keyword "layer" We give also a 3-dimensional space group which can be used for entering the structure in a crystal-drawing program such as CrystalMaker. Where possible, coordinates are given for unit edge length.

Notice that for 2-dimensional structures, vertex symbols denote the polygons meeting at that vertex in cyclic order (compare 3^3.4^2 and 3^2.4.3.4); for layers vertex symbols are the same as for nets.

See reference [12] below for an account of plane nets in crystal chemistry. For nets named OKHnn, nn is the net number of that work.

Polyhedra are 0-periodic, but of course 3-dimensional. We use the term to include generalized polyhedra ("cages") such as the adamantane cage (see ada). The coordinates of the vertices are cartesian coordinates but can be used with the given space group to again draw as if it were part of a crystal structure. To do this set the unit cell of the given space group to d d d 90.0 90.0 90.0 with d some convenient number (say 10.0) and divide the given coordinates by d and use them as crystal coordinates. One will then get the polyhedron centered on 0, 0, 0.

Notice that for polyhedra with non-crystallographic symmetries more than one set of coordinates may be given for a single vertex type - see e.g. dod with icosahedral symmetry but described with m-3 symmetry - the single vertex type splits into V1a and V1b. Hexagonal polyhedra are described using an orthorhombic space group - see e.g. hxb. A square antiprism is another example of a polyhedron with non-crystallographic symmetry - see sap.

For convenience the coordinates of the centers of faces are also given.

In the strict definition adopted by many mathematicians polyhedra have a planar 3-connected graph and can be realized as a strictly convex solid. We refer to structures that do not have this property as cages for the purpose of the database (generally elsewhere the term "cage" includes polyhedra as in "sodalite cage" or "faujasite cage", but these structures are polyhedra sensu stricto).

Notice that generally we give vertices for equal edge; some polyhedra do not then have planar faces (see e.g. the faujasite cage fac which is given as it appears in the crystal structure). For Archimedean polyhedra (one kind of vertex) and the full symmetry the requirement of equal edges uniquely defines the embedding; for their duals (Catalan polyhedra with one kind of face) there are generally more degrees of freedom and there appears to be no definitive embedding agreed upon - the most pleasing have unequal edges. [Exceptions are rdo and trc which are the duals of the two quasiregular (one kind of edge) Archimedean polyhedra cbo and ido]

Simple polyhedra have exactly three faces meeting at a vertex; simplicial polyhedra have all triangular faces. The dual of a simple polyhedron is simplicial and vice versa.

References

References give in the database are to descriptions of nets, and not, except incidentally, to occurrences in crystals. [references to occurrences will be added later]

Some of our work relevant to 3-periodic nets and tilings:

1. Crystal Structures I: Patterns and Symmetry. M. O'Keeffe, B. G. Hyde. Min Soc. Amer., Washington D.C. (available at http://www.public.asu.edu/~rosebudx/okeeffe.htm)
2. Three-periodic nets and tilings: regular nets. O. Delgado-Friedrichs, M. O'Keeffe, O. M. Yaghi, Acta Crystallogr. A 59, 22-27 (2003).
3. Reticular synthesis and the design of new materials. O. M. Yaghi, M. O'Keeffe, N. W. Ockwig, H. K. Chae, M. Eddadoudi, J. Kim. Nature 423, 705-714 (2003).
4. Identification and symmetry computation for crystal nets. O. Delgado-Friedrichs, M. O'Keeffe, Acta Crystallogr. A 59 351-360 (2003). (This gives an informal account of how Systre works)
5. Three-periodic nets and tilings: semiregular nets. O. Delgado-Friedrichs, M. O'Keeffe, O. M. Yaghi, Acta Crystallogr. A 59, 515-525 (2003).
6. Three-periodic nets and tilings: minimal nets. C. Bonneau, O. Delgado-Friedrichs, M. O'Keeffe, O. M. Yaghi, Acta Crystallogr. A 60, 517-520 (2004).
7. Crystal nets as graphs: terminology and definitions. O. Delgado-Friedrichs, M. O'Keeffe, J. Solid State Chem. 178, 2480-2485 (2005).
8. Reticular chemistry: occurrence and taxonomy of nets and grammar for the design of frameworks. N. W. Ockwig, O. Delgado-Friedrichs, M. O'Keeffe & O. M. Yaghi. Accts. Chem. Res. 38, 176-182 (2005).
9. Three-periodic nets and tilings: face transitive tilings and edge transitive nets . O. Delgado-Friedrichs & M. O'Keeffe. Acta Crystallogr. A63, 344-357 (2007).
10. What do we know about 3-periodic nets? O. Delgado-Friedrichs, M. D. Foster, M. O'Keeffe, D. M. Proserpio, M. M. J. Treacy & O. M. Yaghi, J. Solid State Chem. 178, 2533-2554 (2005).
11. Taxonomy of periodic nets and the design of materials. O. Delgado-Friedrichs, M. O'Keeffe & O. M. Yaghi, Phys. Chem. Chem. Phys. 9, 1035-1943 (2007).
12. Plane nets in crystal chemistry. M. O'Keeffe & B. G. Hyde. Phil. Trans. Roy. Soc. A295, 553-623 (1980).
13. Three-periodic nets and tilings: natural tilings for nets . V. A. Blatov, O. Delgado-Friedrichs, M. O'Keeffe & D. M. Proserpio, Acta Crystallogr. A 63, 418-425 (2007).

A recent book that uses the RCSR system:

14. Molecule-Based materials. The Structural Network Approach . L. Öhrstrom & K. Larsson. Elsevier (2005).

The classics (note that "three-dimensional" here means "three-periodic"):

15. Three-dimensional nets and polyhedra. A. F. Wells. Wiley (1977).
16. Further studies of three-dimensional nets. A. F. Wells. ACA Monograph 8. American Crystallographic Association (1979).
17. An enormously valuable set of data on homogenous sphere packings come from Werner Fischer and Elke Koch and collaborators (3-coordinated: E. Koch and W. Fischer, Z. Kristallogr. 210, 407-414 (1995). Cubic: W. Fischer, Acta Cryst. A60, 246-249 (2004). Tetragonal: W. Fischer,Acta Cryst. A61, 435-441 (2005). Hexagonal: H. Sowa and E. Koch, Acta Cryst. A62, 379-399 (2006). Orthorhombic: H. Sowa, E. Koch & W, Fischer, Acta Cryst. A63, 354-364 (2007). Triclinic: W. Fischer & E. Koch, Acta Cryst. A58, 509-513 (2003)). The latest paper (by H. Sowa) is in Acta cryst. A70, 2014. These papers give references to earlier work.