Polyhedral units

No detailed structural studies have been carried out experimentally on T4 derivatives but a number of computational studies have been undertaken as part of the drive to understand the fundamental nature of silicate structures, many of which are made up of small polyhedral units. 

The intermediate-range order, beyond the short-range local order as characterized by the polyhedral units in covalent glasses, is of considerable interest. There is increasing evidence for the existence of fairly well-defined clusters of atoms, containing 10 or more atoms, in the structure of certain covalent glasses (e.g. Ge-Se alloys) vibrational spectroscopy as well as diffraction experiments indicate their presence. 

The secondary structure unit in zeolites A. X, and V is the truncated octahedron. These polyhedral units are linked in three-dimensional space through the four- or six-membered rings, The former linkage produces the zeolite A structure, and the latter the topology of zeolites X and Y and of the mineral faujasite. 

An enormous number of derivatives of these basic polyhedral units (and also larger ones) exist and a great deal of their chemistry is known, but much of the chemistry of these compounds is outside the scope of this book and space limitations prohibit a more complete discussion of them here. For more details, the reader should consult the references at the end of this chapter. 

For still larger clusters, particularly those which consist of several fused polyhedral units, the electron counting and structural predictions are much more complex. ... 

Three examples will suffice to demonstrate this information Figure 3 shows the polyhedral units in the synthetic zeolite Linde Type A, which link to provide a three-dimensional interconnecting array of channels, Figure 4 illustrates the essentially two-dimensional system of channels in the mordenite framework, and Figure 5 shows the major channels in synthetic zeolite Linde Type L arranged as parallel one-dimensional channels and shown as a stereo pair. Table 6 lists the Atlas notations for these structures with explanations, including the symbols used in Tables 2-5. 

In general, borates are structurally complex, since the boron atoms can be in 3 and/or 4 coordination and oligomer, ring, and chain polymers are all found (Christ and Clark, 1977 Wells, 1975). We shall not attempt to describe fully the complexity of these structures but will concentrate on the fundamental polyhedral units. The molecular geometric and electronic structures of these materials can be studied using many of the site-specific spectroscopies previously discussed. The bulk properties of the materials also change, of course, depending upon the molecular structure. 

Our polygon description of the structure of 2D dense random packings of hard disks parallels Bernal s description of three-dimensional (3D) dense random packings of hard spheres as space-filling arrays of elementary polyhedral units ( Bernal holes, or canonical polyhedra ) [2-5]. Bernal s approach to 3D liquid structure is discussed in more detail in Section IV.A. 

Boron Nitride, Metal Borides, and Related Spedes.— Alo.o6BeB3 05, Le. BeB3 belongs to the space group P6/mmm, and contains B12 icosahedra and other polyhedral units of Be and B atoms. The linkages between the polyhedra resemble those in /3-rhombohedral boron. Aluminium atoms occupy interstitial... 

Larger polyhedral units (> 24 tetrahedra) as discussed by Ref. 2 and 28 are not considered here because they are not believed to be important in determining mid-infrared spectral characteristics. 

See Table IV for definition. Ideal size and symmetry of polyhedral units D-4, 8 tetrahedra, cube, Tdj D-6, 12 tetrahedra, hexagonal prism, D6h Cancr. 18 tetrahedra, DshJ T.O., 24 tetrahedra, Tdj Gmel., 24 tetrahedra, D3h. c In cc/cc of crystal. 

E. M. Flanigen You have raised a difficult point in our interpretation which we have considered and discussed in the paper. Our present feeling is that the internal tetrahedron vibrations are highly coupled, and that it is by reason of the large pore volume which isolates polyhedral units, such as the double four rings in zeolite A, that they can be seen as distinct entities. We propose that that is why, in the less open structures with higher densities and lower pore volumes, you do not see the same kind of general features in the infrared pattern as you do in the most open frameworks. 

Figure 4.1-7 Polyhedral unit cell representation of SrFe3(P04)3 as viewed down the a axis showing the condensed Fe-O framework.

Figure 4.1-8 Polyhedral unit cell representation of NaBaFe4 (HP04)3(P04)3"H20 showing the stacking of layers which run parallel to the a axis. The sodium (small cross-hatched circles) and barium (large cross-hatched circles) atoms lie between the layers along with the isolated water molecules which are shown as open circles. The FeOe polyhedra are lined and the PO4 tetrahedra are dotted [90].

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