55 Quasicrystals

The appearance of quasicrystals on the scene of materials has given a great thrust to these developments. 

Alan Mackay made the connection with crystallography [139], He designed a pattern of circles based on a quasi-lattice to model a possible atomic structure. An optical transformation then created a simulated diffraction pattern exhibiting local tenfold symmetry (see, in the Introduction). In this way, Mackay virtually predicted the existence of what was later to be known as quasicrystals, and issued a warning that such structures may be encountered but may stay unrecognized if unexpected  

The unique moment of discovery came in April 1982 when Dan Shechtman was doing some electron diffraction experiments on alloys, produced by very rapid cooling of molten metals. In the experiments with molten aluminum with added magnesium, cooled rapidly, he observed an electron diffraction pattern with tenfold symmetry (see, the pattern in the Introduction). It was as great a surprise as it could have been imagined for any well-trained crystallographer. Shechtman s surprise was recorded with three question marks in his lab notebook, 10-fold [140], 

The first report about Shechtman s seminal experiment did not appear until two and a half years after the experiment. The delay 

Independent of Mackay s predictions and Shechtman s experiments, there was another line of research by Steinhardt and Levine, leading to a model encompassing all the features of shechtmanite (the original quasiperiodic alloy was eventually named so) and other materials that are symmetric and icosahedral, but nonperiodic [143], It was a perfect timing that as soon as they built up their model and produced its simulated diffraction pattern, they could see its proof from a real experiment. 

By using slightly different words, approximants are translationally normal crystal compounds generally with large unit cells that contain condensed, highly symmetric building blocks such as dodecahedra and icosahedra and have compositions close to those of related quasicrystals. 

In a famous paper by Shechtman et al. (1984) electron diffraction patterns were shown of rapidly quenched and solidified aluminium-manganese alloys. Sharp diffraction peaks, suggesting long-range translational order, were observed with the presence however of five-fold symmetry (that is of a non-crystallographic symmetry see 3.6.1.1). By different orientation of the specimen five-fold axes (in 6 directions), three-fold axes (in 10 directions) and two-fold axes (in 15 directions) were identified with the subsequent observation of the existence also of an inver-sion centre the assignment of this phase to the icosahedral point group, m36, was defined. 

36° and 144°. The second figure is a tiling model of an octagonal phase. It is constructed by rhombi having angles of 45° (36078) and 135° and squares. 

In concluding this section in which some properties of modulated structures and of quasicrystals have been considered, we underline that the characteristics of these two types of structures do not coincide. Incommensurately modulated structures show main and satellite diffractions, an average structure and crystallographic point symmetry. The quasicrystals have no average structure, non-crystallographic point symmetry, and give one kind of diffraction only. 

1 Notes on the crystallography of quasi-periodic structures. A general way to face the problems related to the interpretation of quasi-periodic structures (modulated structures, quasicrystals) is based on the introduction and application of higher-dimensional crystallography (de Wolff 1974, 1977, Janner and Janssen 1980, Yamamoto 1982, 1996, Steurer 1995). 

The triacontahedron was discovered by Kepler and it was shown by Kowalewski [21] that it can be built from one acute and one obtuse rhombohedron (Fig 2.14). 

Ten of these rhombohedra form the triacontahedron. The faces of the polyhedra are golden rhombuses, i.e. the quotient between the lengths of the 

What kind of symmetry is this It is clearly not translational. (Is it fivefold ) 

Another class of transformations, known as conformal transformations, preserves angles but not lengths. One example is dilatation, which is repetition by scaling. The golden Fiboimaci sequence is an example of dilatation, as is the pattern of pentagons in Fig. 2.17. 

Conformal symmetry is very common in nature e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is non-Euclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. 

John Cahn was irate what he had been shown was manifest nonsense, and he was sure that a publication making such a claim would relegate both of them, Shechtman and Cahn, to the nether regions of demonstrated crankiness. It took two more years of experimental work, and a good deal of reading of earlier theoretical speculation, before Shechtman and Cahn, together with two French crystallogra- 

It took a long time before everyone accepted the reality of quasicrystallinity. No less a celebrity than Linus Pauling took a hard line, and published a paper in Nature (Pauling 1985) insisting, erroneously as was finally proved some time later, that the pattern was caused by an array of minute crystals in twinned arrangement. 

The fivefold symmetry discovered by Shechtman is modelled in terms of the stacking of icosahedra and the term icosahedral symmetry is sometimes used. 

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. 

Defects as described in Section 8.1 do not appear to form in these compounds. Each composition, no matter how close, chemically, it is to any other, appears to generate a unique and ordered structure, often with an enormous unit cell. Because of this, such structures are sometimes called infinitely adaptive structures . 

Experimentally, a perfect crystal gives a diffraction pattern consisting of sharp reflections or spots. This is the direct result of the translational order that characterises the crystalline state. The translational order allowed in a crystal (in classical crystallography) has been set out earlier in this book. To recapitulate, a crystal can only be built from a unit cell consistent with the seven crystal systems and the 14 Bravais 

As discussed earlier, the symmetry of the structure plays an important part in modifying the intensity of diffracted beams, one consequence of which is that the intensities of a pair of reflections hkl and hlcl are equal in magnitude. This will cause the diffraction pattern from a crystal to appear centrosymmetric even for crystals that lack a centre of symmetry and the point symmetry of any sharp diffraction pattern will belong to one of the 11 Laue classes, (see Section 4.7 and Chapter 6, especially Section 6.9). 

Since then, many other alloys that give rise to sharp diffraction patterns and which show five-fold, eight-fold, ten-fold and twelve-fold rotation symmetry have been discovered. The 

Quasicrystals can also be considered as three-dimensional analogues of Penrose tilings. 

As was mentioned in chapter 1, the discovery of five-fold and ten-fold rotational symmetry in certain metal alloys [5] was a shocking smprise. This is because perfect 

Our quasicrystal example in one dimension is called a Fibonacci sequence, and is based on a very simple construction consider two line segments, one long and one short, denoted by L and S, respectively. We use these to form an infinite, perfectly periodic solid in one dimension, as follows  

This sequence has a unit cell consisting of two parts, L and S we refer to it as a two-part unit cell. Suppose now that we change this sequence using the following rule we replace L by LS and S by L. The new sequence is 

It is easy to show that in the sequence with an -part unit cell, with n - oo, the ratio of L to 5 segments tends to the golden mean value, r = (1 -I- V5)/2. Using this formula, we can determine the position of the nth point on the infinite line (a point determines the beginning of a new segment, either long or short), as 

How can icosahedra, which we know cannot fill aU space, form solids with enough long-range order to diffract x-rays One of the simplest ways would be to place each icosahedron on a Bravais lattice site and fill in the space between them with other atoms. Actually, most quasicrystal structures are more complicated than this and will not be dealt with here. 

A unit cell of a crystal consists of a volume which contains the array of atoms that are repeated. The smallest possible unit cell that can form the structure is a primitive unit cell although is it often convenient to describe to solid using nonprimitive unit cells that have higher symmetries. A unit cell is described by three lattice vectors that define the directions the cell can be propagated to form the crystal lattice or framework of the structure. The basis of the crystal describes the positions of the atoms within the unit cell. The basis plus the lattice defines the structure. TTae set of all possible translations along the lattice vectors, forms the translation group. 

Directions relative to the lattice vectors are defined by the projections onto the lattice vectors on these axes converted to whole numbers with no common denominator and set in square brackets with no commas. Negative numbers are denoted by bars above the number. Equivalent families of vectors, such as directions along the lattice vectors, are denoted by pointed brackets. 

Families of equivalent planes, such as planes perpendicular to the lattice vectors, are denoted by curly brackets. For cubic lattices only, the vector perpendicular to a plane has the same indices as the Miller indices of the plane. 

Sometimes, particularly when describing higher order reflecting planes in XRD, it is necessary to define a specific plane. In this case, the Miller indices are not reduced to the nearest common denominator. 

---

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.)...

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. 

---