Triangular faces

Figure 16.4 The division of the surface of an icosahedron into asymmetric units, (a) One triangular face is divided into three asymmetric units into which an object is placed. These are related by the threefold symmetry axis.

Fig. 4. Three possible geometries for arranging the 72 atoms of the second layer the atoms above the pentagons of C q are shaded. The structure on the upper left can be transformed into the more evenly distributed arrangement of atoms on the upper right by 26° turns of the caps around the five-fold axes. From this, the structure on the bottom can be obtained by rotating each triangular face of atoms by 19°.

Fig. 5. Proposed arrangements of the atoms in the first four layers of an alkaline earth metal around a C o molecule the atoms at the icosahedral vertices are drawn in black and one of the triangular faces of atoms has been shaded in each layer. Note the spiral of atoms (dark grey) in the fourth layer.

It should be pointed out again that these layers would, of course, contain identical numbers of atoms if the triangular faces had not been rotated and, thus, the Ih-symmetry had been preserved[7]. The reason for preferring the arrangement with 1-symmetry (which can still be called icosahedral) is that it leads to higher coordination of the atoms at the borders between the triangular faces. 

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

Similar possibilities arise for 10-atom clusters. Thus, dimerization of the c/oso-CtBj claster l,5-Me2C2B3Et3 (56) by means of K metal then I2 in thf yields the classical adaniantane derivative Me4C4B6Et6 (f) when this is heated to 160° the mdd-tetracaibadecaborane cluster (g) is obtained rapidly and quantitatively. It will be noted that in (f) all four C atoms are 4-coordinate and all six B atoms are 3-coordinate, whereas in (g) the three C atoms in the C3 triangular face are 5-coordinate while the boron atoms are variously 4, 5 or 6 coordinate. 

A geodesic structure with all triangular faces is determinate. 

A geodesic structure with any non-triangular faces is underdetermined, since adding the struts needed to make all faces triangles will add constraints. 

Holes of the Al, A2 and A3 types lie outside the triangular faces of the central icosahedron of the Bg4 unit. Each hole is surrounded by 12 neighboring B atoms. The Al hole accommodates smaller atoms and not Sc or Zr. Silicon and Ge atoms can be accommodated in the A2 hole. There is no evidence for the accommodation of any atoms in the A3, FI or F2 holes. 

The most remarkable feature in the Mni4Al5,5+xGe3.x structure is noted in the second shell structure that surrounds the [Mn Al9Ge] clusters (Fig. 12.2b). The top half of the shell manifests a dome structure made of all triangular faces, while the ill-shaped bottom half is formed with randomly fused triangular and... 

The Lewis stmeture of SFg, shown in Figure 9-24a. indicates that sulfur has six S—F bonds and no lone pairs. The molecular geometry that keeps the six fluorine atoms as far apart as possible is octahedral in shape, as Figure 9-24Z) shows. Figure 9-24c shows that an octahedron has eight triangular faces. 

Views of sulfur hexafluoride (a) Lewis stmeture (b) ball-and-stick model (c) ball-and-stick model showing the triangular faces of the octahedron. 

Fig. 6.1 The fundamental structural unit found in the Chevrel phases (cluster MoeXg full circles Mo atoms) displayed in three ways to emphasize different views of the connectivity. In (a) an octahedron of molybdenums (Mo-Mo = 2.7 A) is encased in a cube of chalcogens (Mo-S 2.45 or Mo-Se 2.6 A). Scheme (b) exhibits the same cluster as consisting of an octahedron with its triangular faces capped by chalcogenides. In (c), the cluster has been reoriented so that a threefold axis is vertical. (Reproduced from [10])...

Completely closed, convex, single-shell clusters are called closo clusters their atoms form a polyhedron. If the polyhedron has only triangular faces, it is also called a delta-hedron. Depending on the number of available electrons, we can distinguish four general bonding types for closo clusters ... 

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