Polyhedral Nets

A simple example of a polyhedral network or polynet, constructed from truncated octahedra and hexagonal prisms. The building process indicated produces a labyrinth. The labyrinth graph happens to be a diamond net, with truncated octahedron centres at the nodes and the prism axes along the edges. This particular example gives a representation of the zeolite framework FAU (faujasite). The edges of the structure form a 4-connected net; in faujasite Si and Al atoms are centered at the polyhedron vertices and the polyhedron edges correspond to the oxygen linkages. Note the large voids and channels in the structure, the characteristic feature of zeolites.

An icosahedron with octahedra on eight of its faces can be extended to a polynet with icosahedral nodes centered on a bcc lattice. The labyrinth graph is 8-connected. In Al 12 Mo aluminium atoms are centered at the vertices of this structure. Molybdenum atoms are at the centers of the icosahedra.

Alpha manganese Another intricate bcc structure is  -Mn. The structural units are centered truncated tetrahedra as in a Laves phase. They occur both as nodes at the bcc positions and also as the links between the nodes. Recall that, in the regular space filling of tetrahedra and truncated tetrahedra (equivalently: in a Laves phase) the hexagonal faces are shared by pairs of polyhedra related by inversion. In  -Mn this this does not occur. Instead, the contiguous polyhedra are related by reflection in hexagonal or in triangular faces. The pictures show two ways in which diamond type polynets can be built with Friaufs as both nodes and links. In  -Mn these are combined to give an 8-connected net...

Basic polynet structure of  -Mn. The centers of the white units are at the vertices and at the center of the cubic unit cell. The central atoms of all the polyhedra are 16-coordinated. The atoms at the centers of the nodal units (white) are have 12 neighbours at the polyhedron vertices and 4 in the centers of neighbouring polyhedra – as in a Laves phase. The centers of link polyhedra (purple) are coordinated to the 12 vertices and to 4 vertices of nearby polyhedra. Most of the polyhedron vertices are 12-coordinated. The net extended further. Sadoc, J. F. & Mosseri, R. Geometrical Frustration. Cambridge Univ. Press (1999).

How and why manganese atoms arrange themselves in such an intricate pattern is a mystery. Another manganese phase,  -Mn is just as intricate but quite different: A stereo pair of images illustrating the  -Mn structure. It has been identified by Nyman, Carroll & Hyde as a packing of helical rods of tetrahedra. Nyman, H., Carroll, C. E. & Hyde, B. G. Rectilinear rods of face-sharing tetrahedra and the structure of  -Mn. Z. Kristallogr. 196 (1991)

The structure of very many crystalline materials can be described in terms of polyhedral nets. Those with labyrinth graphs of diamond type are common. Left: unit cell of a pair of complementary diamond type nets (D-nets). In polynets with D-nets as labyrinth graphs the two complementary structures may or may not be equivalent. Moreover, the nodes of a D-net may be of two varieties in an alternating arrangement (as in the zinc blende ZnS). Right: A portion of a polynet of D-type. Inthis example the nodes and the links connecting them are octahedra. This structure occurs in the mineral pyrochlore. Polyhedral D-nets

The structure of the silicate  -crystobalite. A D-net of vertex-connected tetrahedra. Silicon atoms lie at tetrahedron centers, each coordinated to four oxygen atoms at the vertices. The tetrahedra in the uninodal tiling of space by tetrahedra and truncated tetrahedra form this pattern. The basic structure of pyrochlore can be described as a pair of complementary (interwoven) D-type polynets. One formed of octahedra as both nodes and links, the other formed of vertex-connected tetrahedra as in  - crystobalite. Nyman & Andersson have identified and described several materials with this basic structure. Nyman, H. & Andersson, S. the pyrochlore structure and its relatives. J. Solid State Chem. 26 (1978) A ‘pyrochlore’ unit. Four octahedra round a central octahedron.

A unit cell of the D-net of octahedra in the pyrochlore structure. Nodal octahedra omitted for clarity (and because in pyrochlore they lack a central atom). In the generalised pyrochlore geometry identified in a variety of materials, a D-net of stella quadrangulae replaces the D-net of tetrahedra.

Two interwoven D-nets. One of octahedra and one of vertex-connected stella quadrangulae. (The edge length of the tetrahedra is 4/5 the edge length of the octahedra.) The structure of W 2 Fe 3 C

The pictures above show a gamma brass cluster composed of four interpenetrating icosahedra with minimal distortion, and one with slightly more distortion in which the hexagonal regions indicated in grey have an exact inversion symmetry. These polyhedra can pack together with the grey hexagons in contact, to form a D-type polynet. In what follows, the cluster of four interpenetrating icosahedra will be referred to simply as a  -unit. Surprisingly, the generalised pyrochlore structure just described is a D-net of  - brass clusters!

Another view of the  -unit, emphasising its internal stella quadrangula. The tetrahedral cluster of five units that constitutes the node structure of the D-net.. The complementary D-net is the D-net of octahedra. The picture on the left shows how a pyrochlore unit and an icosahedron belonging to a  -unit fit together.

Four pyrochlore units attached to a  -unit. The generalised pyrochlore structure can be described as a space filling of octahedra and  -units. This structure occurs in a large number of complex alloys “of Ti 2 Ni type”. In Ti 2 Ni the nickel atoms are at the stella quadrangulae vertices, the rest are titanium. There are 96 atoms per cubic unit cell.

Nyman and Anderson have analysed and described the amazingly intricate structure of Mg 3 Cr 2 Al 18. The following pictures illustrate its subtle geometry. Nyman, H. & Andersson, S. the pyrochlore structure and its relatives. J. Solid State Chem. 26 (1978) A D-net in which the nodes are alternately pyrochlore units and truncated tetrahedra. Two interwoven (complementary) nets of pyrochlore units and truncated tetrahedra. Observe that they do not touch.

Left: the two interwoven D-nets, distinguished by color for greater clarity. Right: the gaps between close pairs of pyrochlore units are nearly exact regular icosahedra! Left: one of the D-nets and the related set of icosahedra. Right: The configuration of icosahedra with the D- nets removed for clarity. A new way of understanding the structure emerges...

The structure of Mg 3 Cr 2 Al 18 is a D-net of Kreiner-Franzen L- units linked by their vertices like the tetrahedra in a Laves phase. Recall the L-unit introduced by Kreiner & Franzen: a tetrahedron of vertex-sharing icosahedra. Kreiner, G. & Franzen, H. F. J. Alloys and Compounds 221 (1995)

NaZn 13 The stella quadrangula or “tetrahedral star”is a cluster of five tetrahedra made by placing a tetrahedron on each face of a tetrahedron. Two icosahedra can be joined by a stella quadrangula. Below, left: a network of icosahedra, centered on a primitive cubic lattice, linked in this way. The complement of this polynet is an array of snub cubes:

In NaZn 13 the icosahedra are 13-atom Zn clusters; the Na atoms are at the centers of the snub cubes. A large number of intermetallics have this kind of geometry. Haüssermann, U., Svensson, C. and Lidin, S. Tetrahedral stars as flexible basis clusters in sp-bonded intermetallic frameworks and the compound BaLi 7 Al 6 with the NaZn 13 structure. J. Am. Chem. Soc. 120 (1998)

Polynets with dodecahedra Pearce, P. Structure in Nature is a Strategy for Design. MIT press (1978). The dihedral angle of a regular dodecahedron 116.6° – close to 120 °. Hence three dodecahedra sharing an edge can be slightly deformed to bring them into face contact. The vertex angle in a regular pentagon is 108° – close to the tetrahedral coordination angle °. A tetrahedral cluster of four face-sharing ‘almost regular’ dodecahedra is possible: The structure can be extended to produce a D-net – a portion of which is shown on the right.

The complementary labyrinth of the D-net of dodecahedra is a D-net of hexakai- dodecahedra. This 16-faced polyhedron has 12 pentagonal faces and 4 hexagonal faces. It is one of the typical shapes of the Voronoi cells surrounding the atoms of a Frank- Kasper phase. The dodecahedra and 16-hedra in this space filling arrangement represent the Voronoi regions surrounding the atoms in a Laves phase. The 4- connected network of polyhedron vertices and edges is the zeolite framework MTN.

Voronoi cells of Frank-Kasper phases From left to right: Dodecahedron. 12 pentagons. Maximal symmetry icosahedral. 14-hedron. 12 pentagons and two hexagons. Maximal symmetry 12m2 15-hedron. 12 pentagons and 3 hexagons. Maximal symmetry 16-hedron. 12 pentagons and 4 hexagons. Maximal symmetry tetrahedral.

A space filling of dodecahedra and 14-hedra. Rods built of 14-hedra (grey) in three mutually perpendicular directions are packed together. The voids are dodecahedra (yellow) centered on a bcc lattice. A unit cell is outlined in green. The edges form a 4-connected network. The hydrogen bonds in chlorine hydrate form such a network. (The vertices correspond to water molecules; chlorine atoms are at centers of the 14- hedra. This polyhedral space-filling also represents the pattern made by the Voronoi regions of beta-tungsten (  -W).

A tiling of space with 12-, 14- and 15-hedra. Columns of 14-hedra run parallel to the hexagonal symmetry axis. Layers of 12- and 15-hedra alternate. Williams, R. The Geometrical Foundations of Natural Structure. Dover (1979).