Quasicrystal

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

Figure Bl.8.6. An electron diffraction pattern looking down the fivefold synnnetry axis of a quasicrystal. Because Friedel s law introduces a centre of synnnetry, the synnnetry of the pattern is tenfold. (Courtesy of L Bendersky.)...

A little later (Bendersky 1985, Chattopadhyay et al. 1985) decagonal (tenfold) symmetry was discovered in other Al-transition metal compounds quasiperiodic layers are stacked periodically in the third dimension. Since then, one or other of these forms of quasicrystal have been identified in many different compositions. A detailed review of the decagonal type is by Ranganathan et al. (1997). 

A good, accessible overview of quasicrystals, written only a few years after their discovery, is by Ranganathan (1990) Indian metallurgists played a major part in the early research. Many other published reviews require considerable mathematical sophistication before they can be understood by the reader. 

Interest in physical properties of quasicrystals is growing. Thus, a recent comment (Thiel and Dubois 2000) analyses the implications of the fact that decagonal quasicrystals have very much higher electrical resistivity, by orders of magnitude, than do their constituent metals, and moreover that resistivity decreases with rising temperature. For one thing, it seems that the concentration of highly mobile free electrons is much lower in such quasicrystals than in normal metals. 

For the first 15 years after the discovery, quasicrystals were studied purely as a compelling scientific issue. Just recently, applications have begun to appear. 

The many papers in this proceedings are partitioned into very abstruse theoretical analyses of structure and stability of quasicrystals on the one hand, and practical studies of surface structures, mechanical properties and potential applications. The subject shows signs of becoming as deeply divided between theorists and practical investigators, out of touch with each other, as magnetism became in the preceding century. 

Ranganathan, S. (1990) Quasicrystals, in Supplementary Volume 2 of the Encyclopedia of Materials Science and Engineering ed. Cahn, R.W. (Pergamon press, Oxford) p. 1205. 

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]. 

R. Mikulla, F. ICrul, P. Gumbsch, H.-R. Trebin. Numerical simulations of dislocation motion and crack propagation in quasicrystals. In A. Goldmann,... 

Ch. Dilger, R. Mikulla, J. Roth, H.-R. Trebin. Simulation of shear stress in icosahedral quasicrystals. Phil Mag A 75 425, 1997. 

The activated character of the dependence tj(T) shown in (1.123) is often considered as a feature suggesting a quasicrystal model of the liquid. Data taken from liquid-vapour co-existence curves are frequently analysed in coordinates In tj from 1/T in order to determine t/o- The point that tj(n, T) is a function not only of the temperature T, but also of the density n is ignored. The density along the co-existence curve is... 

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

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. 

Clustering in Condensed Lithium Ternary Phases A Way Towards Quasicrystals I 143... 

Stadnik, Z.M., Ed. Physical Properties of Quasicrystals Springer, New York, 1999. 

During the nearly ten years which have passed since the appearance of the " Shechtman paper" a large amount of both experimental and theoretical research has been carried out on quasiperiodic structures. For more material about quasicrystals we refer to a paper in La Recherche by the French collaborator in the Sheehtman team [6], to a thesis by Dulea [7J, and to a survey paper with a large number of referenees [8]. 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

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