Icosahedral crystals

Nevertheless, it must be pointed out that Pauling s explanation has not been generally accepted. Indeed, alternative and widely accepted rationalizations invoking such concepts as Penrose tiling and icosahedral crystals in hyperspaces have been advanced to explain the anomalous existence of the quasicrystals [95-103]. 

By group theoretical methods it was shown (65,66) that by this distortion the icosahedr point group Ih is reduced to the subgroup Ds which is compatible with the space group R 3 m. This explains why this space group is preferred in icosahedral crystal structures. 

Fig. 8. Electron micrograph showing crystallization of icosahedral phase from glassy Pd—U—Si alloy.

Fullerenes are described in detail in Chapter 2 and therefore only a brief outline of their structure is presented here to provide a comparison with the other forms of carbon. The C o molecule, Buckminsterfullerene, was discovered in the mass spectrum of laser-ablated graphite in 1985 [37] and crystals of C o were fust isolated from soot formed from graphite arc electrodes in 1990 [38]. Although these events are relatively recent, the C o molecule has become one of the most widely-recognised molecular structures in science and in 1996 the codiscoverers Curl, Kroto and Smalley were awarded the Nobel prize for chemistry. Part of the appeal of this molecule lies in its beautiful icosahedral symmetry - a truncated icosahedron, or a molecular soccer ball, Fig. 4A. 

A pecuhar sohd phase, which has been discovered not too long ago [172], is the quasi-crystalline phase. Quasi-crystals are characterized by a fivefold or icosahedral symmetry which is not of crystallographic type and therefore was assumed to be forbidden. In addition to dislocations which also exist in normal crystals, quasi-crystals show new types of defects called phasons. Computer simulations of the growth of quasicrystals [173] are still somewhat scarce, but an increasing number of quasi-crystalline details are studied by simulations, including dislocations and phasons, anomalous self-diffusion, and crack propagation [174,175]. 

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.

Table 1 Analysis of X-ray diffraction pattern of icosahedrally twinned cubic crystals of MnAl6...

So-called Icosahedral and Decagonal Quasicrystals Are Twins of an 820-Atom Cubic Crystal... 

Whereas only hh0 can diffract with the beam along a fivefold axis of the icosatwin because of the tilt of the cubic crystals, any plane in the zone 110 of the decatwin can diffract because the seed orients the cubic crystals with this zone axis parallel to the fivefold axis of the twin. It is seen from Fig. 1 (b) that the meridional spots of decagonal FeAl are the same as for icosahedral MnAl6, representing orders of hh0, but that there are many more equatorial spots, eleven instead of three. Indices are assigned in Table II some spots involve double... 

FIG. 2. A complex of twenty Friauf polyhedra, with icosahedral symmetry (Samson, Ref. 23). This complex contains 104 atoms, if the central icosahedral position is not occupied. Most of the atoms show approximate icosahedral ligan-cy twenty atoms, at the centers of the Friauf polyhedra, have ligancy 15 or 16. The complex was first identified in Mg32(Al,Zn)<9. In the cubic crystals that form the icosatwins and decatwins these complexes are packed in such a way as to approximate an icosahedral arrangement of twelve complexes about a central one, the structure being similar to that of 0-W. 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

I conclude that the evidence in support of the proposal that the so-called icosahedral and decahedral quasicrystals are icosatwins and decatwins of cubic crystals is now convincingly strong. I point out that there is no reason to expect these alloys to have unusual physical properties. 

Fig. 4 Crystal structures of alS AgTlB (A = K,Rb,Cs), b NauKgTlisM (M = Mg,Zn,Cd,Hg), and c Mg2Zni 1, all of which contain centered icosahedral of the main group elements...

The Li centers of the Lii2 unit are located on the edges of the P8 framework, whereas the Li centers of the Li60 core are each located over a face. This situation led to different coordination numbers (2 vs 4) for the two types of Li centers, implying much shorter Li-P bonds for the external ones (A ca. 0.2 A). That some of the 12 As centers of the anionic icosahedral partial structure in 6a can be replaced by other anions has only recently been demonstrated (32). Thus, 7a crystallizes in the monoclinic space group P2i/n and possesses a slightly distorted anionic icosahedral framework, which is identical to that observed for 6a, but two of the (AsR)2- units are replaced by iodide centers (Fig. 9). 

A special group of particles that are often produced are the icosahedral (I5) and decahedral (D5) structures shown in Fig. 9. These particles have a fivefold symetry axis which is forbidden for infinite crystals. Yang (1 0) has described these particles using a non-Fcc model. The particles are composed by five (D5) and twenty (I5) tetrahedral units in twin relationship. However the units have a non-Fcc structure. The decahedral is composed by body-centered orthorhombic units and the icosahedral by rhombohedral... 

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