This article was downloaded by: [North Carolina State University]On: 05 November 2013, At: 01:58Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Contemporary PhysicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tcph20

QuasicrystalsDenis Gratias aa Centre d'Etudes de Chimie Metallurgique , Centre Nationalde la Recherche Scientifique , 15, rue Georges Urbain,94400, Vitry-sur-Seine, FrancePublished online: 20 Aug 2006.

To cite this article: Denis Gratias (1987) Quasicrystals, Contemporary Physics, 28:3, 219-239,DOI: 10.1080/00107518708219071

To link to this article: http://dx.doi.org/10.1080/00107518708219071

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information(the “Content”) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions andviews of the authors, and are not the views of or endorsed by Taylor & Francis. Theaccuracy of the Content should not be relied upon and should be independentlyverified with primary sources of information. Taylor and Francis shall not be liablefor any losses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly or indirectly inconnection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

CONTEMP. PHYS., 1987, VOL. 28, NO. 3, 219-239

Qu as i c r ys t a Is?

Denis Gratias, Centre &Etudes de Chimie Metallurgique, Centre National de la Recherche Scientijque, 15, rue Georges Urbain, 94400 Vitry-sur-Seine, France

ABSTRACT. The recent discovery of ametallic alloy that behaves like a crystal, but which is characterized by a ‘forbidden’ fivefold or icosahedral symmetry, has led to the redefinition of the crystalline state. These ‘quasi- crystals’ have since become the subject of intense study and debate. In this paper we describe the remarkable properties of quasicrystals and show how their structure can be described with the aid of concepts developed in other disciplines: the golden number, Penrose tiling, and the theory of irrational numbers.

1. Introduction On 12 November 1984, a short article published in the highly respected journal Physical Review Letters announced experimental evidence for a metallic alloy with exceptional properties (Shechtman et al. 1984). Studied by electron diffraction, this alloy apparently behaved like a crystal. Its diffraction pattern was made up of bright spots, well defined and regularly placed, just as for a crystal. However, the pattern was also characterized by ‘icosahedral’ symmetry which, in a crystal, is strictly forbidden by geometry.

Four research workers were involved in writing the 1984 article: Dany Shechtman, who made the original discovery, and Ian Blech of the Technicon Institute of Haifa (Israel), John W. Cahn of the National Bureau of Standards (Gaithersburg, U.S.A.) and myself, from the Centre for Studies in Chemistry and Metallurgy of CNRS at Vitry-sur- Seine (France).

We were all convinced that this strange discovery was going to be of great interest in the fields of solid-state physics and crystallography. We have not been disappointed: there have been more than two-hundred scientific publications devoted to these novel objects, that today are called ‘quasicrystals’. Within a few months, an elegant theoretical model of quasicrystals saw the light of day. It called upon mathematical techniques devised to describe non-periodic structures that are visually fascinating, of which the famous Penrose tilings are the archetypes. In less than a year, many other alloys were discovered and new types of symmetry demonstrated. So much so, that, far from being an exceptional arrangement, the quasi-crystalline state was found to be much more common than we had ever imagined.

The notion of a quasicrystal is of fundamental interest because it generalizes and completes the definition of a crystal. The theory that derives from it replaces the centuries’ old idea of ‘the unit repeated in a strictly periodic fashion in space’ by the key notion of long-range order. From it comes an extension of crystallography whose new riches have hardly begun to be explored. Its importance in the world of minerals ranks with the introduction of irrational numbers into the set of the rational numbers in mathematics.

What is a quasicrystal? What are its properties and how can one describe them? Nowadays there are many questions that one can start to answer, using arguments whose foundations are now well established.

t This article originally appeared in French in La Recherche, No. 178, June 1986, pp. 788-798. Copyright 0 La Recherche, June 1986.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

220 D. Gratias

2. The canonical concept of ‘three-dimensional wallpaper’ Like all solids, crystals are characterized by their chemical composition and their structure. The composition indicates the nature and relative proportions of the chemical species that constitute the material: the structure describes the arrangement of the atoms in space. The many structural states of solids can be classified with reference to two extreme cases: the crystalline state and the amorphous state. Surprisingly, both states can exist for a given chemical composition. Thus silica (SiO,) exists in its crystalline form as quartz and in its vitreous form as ordinary glass. The latter is amorphous: even though it consists of identical SiO, units, globally it does not possess a well-defined geometrical form. Quartz, by contrast, is made up of the strictly periodic repetition of the same unit parallel to itself (i.e. by translation). It is a kind of three- dimensional wallpaper pattern, whereas vitreous silica resembles a random pile of identical units. Within the crystal there exists what is called ‘long-range order’: the units align themselves in a strictly periodic fashion in relation to each other. In an amorphous structure, on the other hand, the units repeat themselves irregularly and are not generally equidistant from each other: a state of short-range order only.

The most spectacular manifestation of order (short or long range) arises from the phenomenon of diffraction. When one directs a beam of particles (photons, electrons, neutrons, etc.) onto a sample, particles are deflected from their original trajectories by interactions with the atoms of the sample. In the case of an amorphous material, the number of particles deflected in a particular direction varies continuously with the angle of deflection. The diffraction pattern obtained consists of a central spot, corresponding to the undefected particles, around which are several rings, correspond- ing to the mean distances most commonly found in the material. When the material is crystalline, the diffraction pattern takes on an even more spectacular appearance: the periodicity of the atomic positions causes the diffraction to become ‘coherent’ in a certain number of privileged directions characteristic of the crystal geometry. Along these privileged directions, the elementary scattering processes from the atoms combine to give rise to a strong perfectly localized intensity. In other directions, the elementary scattering processes cancel each other exactly. The result is a diffraction pattern comprising a series of regularly spaced discrete points (Bragg reflections), which are characteristic of the periodicity and symmetry of the material.

The global symmetry of the diffraction pattern indicates the geometric form of the elementary building block (called the ‘unit cell’) from which the crystal is constructed by simple juxtaposition of this building block parallel to itself. The angular separation of the Bragg reflections is inversely proportional to the dimensions of the unit cell and the intensity of the reflections allows the spatial structure of the crystal to be deduced by numerical simulation, that is, the nature and position of the minimum number of atoms needed to reproduce the whole crystal by translation.

Even though there is no geometrical restriction on the size of the unit cell, which in certain organic compounds can be as much as several tens of nanometres, there is a severe constraint on its symmetry properties. Indeed, insofar as the crystal results from the juxtaposition ad ~~~~~~~~ of a single unique cell (face to face, parallel to itself), this elementary building block must completely ‘pave’ the whole of the space without any gaps. Only very few three-dimensional shapes can bring about this perfectly fitting division of space: these are the polyhedra that possess twofold (rotation through 2 ~ / 2 ) , threefold (rotation through 2 ~ / 3 , fourfold (rotation through 2n/4) or sixfold (rotation through 2x/6) axes of symmetry, to the exclusion of all other rotation symmetries. In particular, the unit cell of a crystal lattice cannot under any circumstances be a

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystals 22 1

polyhedron having fivefold symmetry, i.e. an object that can be mapped onto itself by a rotation through 2z/5. Geometrically, it is impossible to completely fill space with such objects; just as in two dimensions it is impossible to completely pave, or tile, flat space with, for example, regular pentagons. From this derives the fundamental geometric theorem that a crystal (an object with spatial periodicity) cannot have a symmetry that possesses one or more axes of fivefold rotation. This, of course, does not preclude the unit itself (the arrangement of atoms within the primitive cell) from having fivefold symmetry. Crystalline cyclopentadiene, for example, is made up of carbon atoms arranged in pentagonal rings. It is only the elementary lattice itself that cannot under any circumstances have fivefold symmetry, and it is precisely the symmetry of the lattice that is indicated by the diffraction pattern. Thus, either an object is periodic and its diffraction pattern consequently shows intense sharp reflections with rotation symmetry of two, three-, four- or sixfold order, or the object is amorphous (vitreous), so that diffraction does not generate sharp reflections but a series of diffuse rings characteristic of the short-range order that exists in highly disordered materials.

3. A forbidden symmetry that will not go away Such were the canons of traditional crystallography that we had in mind as we wrote the original article (Shechtman et al. 1984) and which seemed to be flouted by the extraordinary alloy of aluminium and manganese in which Shechtman was the first to be interested. This alloy is solidified from the melt by ultrafast quenching. Its temperature is made to fall at a rate of about one-million Kelvin per second. Thc alloy so produced comes out in thc form of very brittle metallic ribbons (figure 1 (a)). Observed under the electron microscope, these ribbons reveal a remarkable morph- ology, made up of dendritic nodules a few microns across (figure 1 (b)). Each nodule is homogeneous with long-range orientational order as indicated by the diffraction patterns, which consist of well-defined sharp spots. However, these spots are distributed in a pattern characteristic of the symmetry of an icosahedron, a regular polyhedron with 20 faces which possesses, along with other polyhedra, the famous forbidden fivefold axes (figure 2). We were thus faced with an apparently paradoxical object: having crystal-like long-range order but possessing a symmetry incompatible with translational periodicity, the periodicity that is the very definition of the crystalline state!

Before accepting this unbelievable result, Dany Shechtman spent two years eliminating all possible accidental causes that might explain such a pattern within the context of conventional crystallography. The most obvious is an artefact, called a 'twin', that arises from multiple diffraction of electrons by a homogeneous mixture of microcrystals oriented at an angle of 72" to each other. Even by using the most refined analysis techniques to detect any possible microscopic twins, Shechtman was unable to eliminate the forbidden symmetry. Every time, the diffraction pattern (as shown in figure 2) obstinately exhibited fivefold symmetry.

Several groups around the world have studied this unusual alloy by means of high- resolution electron microscopy (figure 3). They have all verified the perfect homogen- eity of the material in which local fivefold symmetry persists in microscopic fragments of dimensions close to that of atoms (a few tens of nanometres). The hypothetical microscopic twins responsible for such a symmetry must therefore have dimensions comparable with interatomic distances, which, conceptually, deprives them of all physical meaning (for an atomic structure to form a crystal, it must extend at least over several unit cells).

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

222 D. Gratias

- 500 nm

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quusicrystals 223

w 100 pm (4

Figure 1. The discovery of the first quasicrystal was announced in November 1984. It was an alloy of aluminium and manganese obtained by squirting a jet of the molten alloy onto a metal wheel turning at high speed. On contact with the wheel, the mixture solidified very rapidly (cooling at a rate of about one-thousand degrees per millisecond) forming a ribbon of several tens of microns in thickness. Seen under the electron microscope this type of alloy exhibits a fascinating morphology: within a matrix of aluminium there develop dendritic nodules several microns in extent. Their property of simultaneously having long- range order and being non-periodic is characteristic of a quasi-crystalline phase.

Part (a): electron microscopy image of a rapidly solidified ribbon of Al-Mn alloy: the icosahedral phase appears as pentagonal flowers of a few microns in size surrounded by the later solidified aluminium-rich matrix. (Photograph K. Yu-Zhand and R. Portier.)

Part (b) shows macroscopic single quasicrystals obtained in A16.4Li2.7Cu,,,9 after slow casting. Note the triacontahedral shape of the crystals. The facets are orthogonal to the twofold axis. (Reprinted by permission from Nature, Vol. 324, No. 1, p. 49, copyright 0 1986 Macmillan Journals Limited.)

Finally, the development that led us to publish the discovery of this ‘impossible’ crystal depended on two decisive arguments. One, qualitative, due to Shechtman and Blech (1985), showed that one could reproduce a diffraction pattern very like the experimental one from a non-space-filling stack of elementary icosahedra; the other was purely mathematical. Without claiming to explain the pattern that had been found, it did have the merit of resolving its paradoxical character. Back in 1935, mathema- ticians Harold Bohr (1924-1926) and A. S. Besicovitch (1932) had shown that periodicity is not a necessary condition for obtaining coherent diffraction (well defined, sharp reflections), only the condition of ‘almost-periodicity’ is required. An object satisfying this condition can be ‘almost exactly’ superimposed on itself by an infinite number of translations. A detailed explanation of this idea makes use of the concepts of ‘dense’ and ‘relatively dense’ ensembles, a formalism beyond the scope of this article. What is important to realize is that neur periodicity is a weaker condition than strict

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

224 D. Gratias

Figure 2. Looked at along certain special directions, the quasi-crystalline nodules of figure 1 produce remarkable electron diffraction patterns. The pentagons of all sizes which appear on these are characteristic of the famous forbidden fivefold symmetry. By changing the orientation of the specimen, one can identify the different symmetries possessed by the diagram. One is thus led to attribute to it the symmetry of an icosahedron (a regular polygon having 20 faces). But such a symmetry is geometrically incompatible with crystalline periodicity. Although of perfect long-range order and local icosahedral symmetry, the quasi-crystal is thus an object whose structure cannot be periodic in space (Photograph: D. Shechtman).

periodicity. Because of this, it is not incompatible with any rotation symmetries, in particular, with fivefold rotation!

It is the same for ‘quasi-periodicity’, a special case of near periodicity. Technically speaking, the difference between these two concepts lies in the finite or infinite number of indices needed to describe, respectively, quasi- periodic functions or nearly periodic functions. The term ‘quasicrystal’ (derived from the notion of quasi-periodicity) was coined in December 1984 by Dov Levine and Paul J. Steinhardt (Levine and Steinhardt 1984) who were trying to simulate amorphous structures by computer using stacking icosahedra. However, instead of retaining the classical idea of an amorphous solid with short-range order, Levine and Steinhardt sought to construct a model that included long-range order. It was just as they were obtaining their first significant results that Shechtman’s discovery became known to confirm the validity of their approach. Their quasi-crystalline model justified the geometry of the patterns and, though only in part, the experimentally observed intensity distributions. For us there was no longer any doubt: no fundamental argument contradicted the existence of quasicrystals. The high- resolution electron microscopy observations that Richard Portier had made in the meantime at the Centre for Chemical and Metallurgical Studies at Vitry (Portier et at. 1985), together with the theoretical work, combined to demonstrate the reality of quasicrystals. But what physical significance was to be given to the existence of such objects? And how could one reconcile the well-established laws of crystallography with these unlikely crystals?

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystals 225

Figure 3. When one causes the diffracted beams of figure 2 to interfere with each other, one obtains information about the distribution of atoms inside the material. This plate shows how the atoms are arranged geometrically and that the icosahedral symmetry is not due to a special superposition of ordinary crystals (twins) but does indeed result from an original, aperiodic arrangement of the atoms. A homogeneous distribution of pentagons can be seen, which is repeated but non-periodic. The smallest of them have sizes of several tens of nanometres, the order of magnitude of interatomic distances. The figure shows AI,,Mn,,Si, (Photograph R. Portier, CECM/CNRS).

4. Long-range order: the fundamental key to the problem The fundamental key which allows an explanation of the apparent paradox of a ‘crystal’ exhibiting fivefold symmetry, which is incompatible with translational symmetry, lies in the redefinition of the very idea of a crystal. Long-range order is the essential characteristic of the crystalline state: an arrangement of atoms has long-range order if it can be constructed according to a non-random mathematical law. In this way, a ‘geometric memory’ grows inside the material: the position and property of an atom depend not only on its near neighbours (short-range order) but equally, and with a correlation that is just as strong, on the infinite array of atoms at greater and greater distances. This collective memory is measured mathematically by the so-called ‘autocorrelation function of the system’. Such a function represents the degree to which the system maps onto itself when it undergoes an arbitrary translation. Thus, for a crystal, all translations over a distance equal to the lattice spacing superimpose the solid exactly on itself. As a function of displacement, the autocorrelation function of the crystal is periodic and does not continue to decrease as the displacement tends to infinity. For an amorphous solid, on the other hand, as soon as one displaces the object with respect to itself, the number of superposable local configurations decreases and the autocorrelation function tends to zero as the translation parameters increase. In general, an object is said to have long-range order if the autocorrelation function does not tend to zero with translation over a greater distance.

Why introduce this function which seems only to introduce unnecessary compli- cations? Simply because the diffraction pattern of a material is nothing but the Fourier transformation of this autocorrelation function. Put another way, it is possible to go

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

226 D. Gratias

Figure 4. Can an object both have long-range order and be non-periodic? This figure presents the simplest example that confirms that one can. It comprises a strictly periodic linear chain of atoms (at the top). This chain is modulated by a displacement wave whose period is irrational with respect to that of the initial lattice. Each atom is thus displaced by a different amount, which thus breaks the periodicity of the chain. The modulated chain (at the bottom) nonetheless has perfect long-range order, in the sense that the positions of the atoms are defined by a mathematical law with no random component. The displacement of each of the atoms is displayed by the lines that link the periodic chain (top) with the modulated chain (bottom).

from one to the other by means of a Fourier transform, a classic mathematical operation often encountered in physics. This is the reason why a pattern made up of a denumerable collection of sharp reflections is in fact typical of a strictly, nearly or quasi-periodic autocorrelation function. To obtain a denumerable array of reflections does not therefore necessarily mean that the diffracting object is periodic; it simply means that it has long-range order. Two examples allow us to demonstrate the fundamental character of this distinction and to identify clearly the context in which the discovery of quasicrystals is to be set.

The first of these two examples is based on a linear chain of atoms having constant interatomic spacing a. Let us impose on this ‘one-dimensional crystal’ a static displacement wave of period l a and amplitude ECI. Thus each atom is slightly displaced from its initial position by an amount EU sin (271ilna), where n designates the nth atom considered. If il is an irrational number a crucial feature emerges: no atom is affected in the same way as any other (figure 4). A chain ‘modulated’ by a very simple mathematical function has thereby become strictly aperiodic: no finite translation can superpose it on itself! This chain is, however, a perfectly ordered object. Its lack of periodicity is in no way brought about by the introduction of random disorder, but simply by the superposition of two incommensurate periodicities (periodicities whose ratio is an irrational number). These strange objects actually exist. They are called incommensurate phases or materials in order to emphasize the irrational character of the periodicities that come into play. Besides the fact that they give coherent diffraction patterns, these structures have intriguing properties which make them the first representatives, although not so designated at the time of their discovery, of aperiodic or quasi-crystals.

It is always possible to perform an arbitrary displacement of the modulation waves without fundamentally changing the structure. It suffices to add an arbitrary phase angle to the argument of the sine function. In the general case (that is, when this phase angle is not a multiple of 271) these new modulated chains are all different. However, they all rigorously give rise to the same diffraction pattern! In addition, these structures (which cannot be distinguished from each other by diffraction) possess the same thermodynamic and geometrical properties. The second remarkable property of these structures arises from the very special character of their aperiodicity. This calls upon several ideas from the theory of numbers that nowadays one meets more and more often in the physics of solids (see below).

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystals 221

5. The theory of numbers, which until a few years ago was considered to be of purely mathematical interest, has now become a useful tool for physicists. Many models make explicit use of it to describe the best interpolation between continuous and discrete variables. When a physical system can take only a denumerable set of states and when an optimum is found for an irrational distribution of these states, the system chooses, in fact, a rational distribution as near as possible to the optimum. To describe this distribution is formally analogous to the mathematical problem of describing irrational numbers as the limits of a series of rational numbers.

Numbers can be divided into two large families: the rationals and the irrationals. Rational numbers are those which can be cast in the form of a ratio x = P / Q , where P and Q are integers. (They are the solutions of linear equations with integer coefficients: Q x - P = 0.) Irrational numbers can be defined as the limit of fractions having an infinite number of ‘tiers’ called ‘continued fractions’:

Digression on irrational numbers and the golden number

x = n , + l / ( n , + 1 / (nz+ l / n 3 + ...)))...)).

When one terminates the fraction at the nth stage, one gets the nth approximation P J Q , to the irrational number x . The approximation satisfies the relation IP,/P, - X I < 1/Qi . (Everyone knows the famous approximation 22/7 which comes from truncating the continued fraction that defines the number n: the fraction approximates to n to within 1/49.)

According to the nature of the sequence of characteristic integers (no, n,, n 2 . . .), one obtains irrational numbers of different types. If, for example, the sequences are periodic the numbers obtained are termed ‘algebraic’ in the sense that they are the solutions of polynomials with integral coefficients. In contrast, transcendental numbers are not the solutions of any polynomial with integral coefficients. The irrational numbers associated with quasi-crystalline structures are always algebraic numbers. Thus the ‘golden number’ 2 = 2 cos (.Is), which characterizes fivefold symmetry, is defined by the seriesni=(l , l , l , ...), in otherwordsbythefraction l + l / ( l + l / ( l + l / l ...)= 1618 . . . . It is the solution of the second degree equation: ~ ’ - 2 - 1 =O. This gives it the remarkable property that its powers can all be written as a linear combination with integral coefficients of unity and itself; thus 23 = 22 + 1; 24 = 32 + 2, etc. The convergence of the series formed from the golden number’s rational approximants is exceptionally slow (it goes as lid5 Q i ) , which is sometimes explained by saying that it is the irrational number ‘furthest’ from the rationals.

If the parameter A were a rational number, it could be written in the form p / q , where p and 4 are integers. The chain would then be strictly periodic. In the example we have chosen, it is irrational and the chain is thus aperiodic. But like all irrational numbers, it can be approximated by a sequence of rational numbers P,,/Qn (whose definition is given above). The remarkable fact is that if one translates the chain globally by a factor qna, one then obtains a new chain in which the atomic positions differ very little from the original. The chain is then ‘almost periodic’ in the sense that it is superposed on itself almost exactly, and the bigger Q , is (thus, the better the approximation to A) the better is the superposition. In the limit, the chain would be exactly superposed on itself for infinitely large values of the ‘approximants’ P , and Q,, i.e. for an infinite translation. In the final part of this article, we shall see how this curious property played a trick on one of the most eminent specialists in chemistry and crystallography.. . but let us not anticipate.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

228 D. Gratias

Figure 5. This hierarchical structure is another example of an object with long-range order that is non-periodic. It is constructed, by repetition, from equilateral triangles that point upwards. At each step, a new triangle is formed which is twice as large and contains the pattern of the preceding triangles. The first few iterations of this construction are shown here from left to right. Yet again, this object is perfectly ordered and gives rise to a coherent diffraction pattern. However, in contrast to the chain model of figure 4, this structure cannot serve as a model for ‘crystals’. It is essentially an object with ‘holes’, and so does not possess the fundamental property of homogeneity that characterizes crystalline solids.

Let us rather consider a second example of an ordered non-periodic structure with equally fascinating properties. It is constructed by iteration, starting with an equilateral triangle pointing upwards. At each iteration, a new triangle twice as large as the previous one is drawn (pointing upwards); in addition there are associated with each triangle from the preceding stage two identical triangles also pointing upwards (figure 5). Pursued to infinity, this construction gives rise to an object with empty spaces (gaps), that is to say, the pattern is non-periodic but perfectly ordered. Its diffraction pattern can be calculated by making explicit use of the iterative construction. As in the preceding example, this diffraction pattern consists of sharp denumerable reflections without any diffuseness due to random disorder. The essential difference between this structure and normal crystals is that the reflections form what is called a ‘dense ensemble’, that is to say, they are infinitely close to each other.

To sum up, these two examples clearly demonstrate that it is possible to construct objects that are non-periodic but which at the same time have long-range order; and also that their diffraction pattern is a denumerable collection of sharp reflections. With the discovery of the first quasicrystal it was thus legitimate to think that this alloy too had long-range order. The principal question then was to identify this long-range order or, to put it another way, to find a geometric means of describing it.

6. A fortunate set of circumstances It was in January 1985, and thanks to a happy combination of circumstances, that the research work directed towards the description of long-range order was able to make a decisive step forward. At a workshop on mathematical crystallography organized by Louis Michel and Marjorie Senechal at the Institute of Advanced Scientific Studies at Bures-sur-Yvette, Shechtman, Cahn and 1 had the good fortune to meet Michcl

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystals 220

Duneau and Andrk Katz. These two theoretical physicists from the Ecole Polytechnique in Palaiseau had devised a technique of transparent simplicity for constructing the aperiodic tilings of space (Katz and Duneau 1985). All the questions that our account had left unanswered were cleared up in the space of one hour in an impromptu contribution from Katz and Duneau, followed, at the next session, by a very full lecture on the subject. Experiment and theory united in a coherent pattern rich in implications. Models analogous to that of Duneau and Katz (1985) were developed essentially at the same time by Veit Elser (1985) from the Bell Telephone Laboratories, as well as by P. A. Kulagin, A. Y. Kitaev and L. S. Levitov (Kalugin et al. 1985) from the Lev Landau Institute in Moscow. In less than two months, the mathematical theory of quasicrystals had reached maturity.

At the beginning of their work, Duneau and Katz had, I am sure, no idea how we were going to exploit and 'hijack' their results. Their research was directed towards discovering a satisfactory method of describing the famous 'Penrose figures' which are the most celebrated examples of quasi-periodic tilings (figures 7 (c), (e)). In 1972, the English mathematician Roger Penrose (1979) succeeded in covering flat space with only two elementary polygons arranged in an essentially aperiodic way. (An amusing detail is that Penrose had not then published his discovery because he had filed a patent for his puzzles!) In its simplest form, a Penrose tiling is a non-random assembly of two types of lozenge-shaped figure, one with an internal angle of 36", the other of 72". This paving, a veritable nightmare for floor tilers, has fascinating geometrical properties which found their explanation in 1981 in a seminal paper by the Dutch mathematician De Bruijn (1981). The very peculiar visual impression that it produces arises from the fact that one sees a perspective view of cubes in an arrangement that Escher himself would not have spurned.

To make this impression more apparent one can construct a simpler Penrose figure using three simple parallelograms (figure 6). This figure is also quasi-periodic: no

Figure 6. The figures devised by the English mathematician, Roger Penrose (figure 7 (c)), are the archetype of quasi-periodic tilings. The visual impression they give is that of a stack of cubes, seen in perspective and truncated. A fraction of the cubes is taken away from the original stack. To make this impression more evident, this figure has been constructed with three elementary parallelograms. Mathematically, it is obtained from a stack of cubes that is strictly periodic in three dimensions. The stack is first truncated along a direction that is irrational in relation to the directions of the initial lattice, then projected onto a plane normal to this direction. The quasi-periodicity of the tiling so obtained follows directly from the irrational character of the chosen direction.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

230 D. Gratias

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Figu

re 7.

T

o ge

nera

te a

nd ge

nera

lize t

he fa

mou

s Pen

rose

figu

res,

Mic

hel D

unea

u an

d A

ndrt

Kat

z de

velo

ped a

rem

arka

bly

effe

ctiv

e tec

hniq

ue. T

he re

leva

nce

of th

is m

etho

d to

crys

tallo

grap

hy is

illu

stra

ted

by th

ese

exam

ples

of p

erio

dic

and

quas

i-per

iodi

c la

ttice

s. T

he fi

rst (

a) is

stric

tly p

erio

dic.

It is

a s

impl

e ge

nera

lizat

ion

of th

e he

xago

nal s

yste

m w

ell k

now

n to

crys

tallo

grap

hers

and

foun

d, fo

r exa

mpl

e, in

gra

phite

. The

seco

nd la

ttice

(b) is

the p

roto

type

of t

he

quas

i-per

iodi

c oct

agon

al sy

stem

. It is c

onst

ruct

ed fr

om a

loze

nge (

havi

ng an

inte

rnal

ang

le o

f45"

) and

a sq

uare

and

no

long

er h

as th

e st

rict p

erio

dici

ty of

th

e pr

evio

us s

yste

m. T

he th

ird la

ttice

(c) i

s a

deca

gona

l sy

stem

. Fro

m i

t one

can

rec

over

the

orig

inal

Pen

rose

figu

res.

It is

, in

two

dim

ensi

ons,

the

equi

vale

nt o

f th

e th

ree-

dim

ensi

onal

latti

ce o

bser

ved

in th

e fir

st q

uasi

-cry

stal

line

allo

ys o

f al

umin

ium

and

man

gane

se. F

inal

ly, t

he la

st s

yste

m (d

) is

duod

ecag

onal

. It h

as b

een

seen

in c

erta

in a

lloys

who

se d

iffra

ctio

n pat

tern

s th

us p

osse

ss a

twel

vefo

ld ax

is o

f sym

met

ry. L

ike

thos

e of

(b)

and

(c),

such

a

latti

ce p

osse

sses

no

sym

met

ry, a

lthou

gh it

was

cal

cula

ted

usin

g a

stric

tly p

erio

dic

stac

k of

hyp

ercu

bes i

n a

spac

e of

four

dim

ensi

ons.

Oth

er sy

stem

s of

orde

r 14

, 16,

18,

20,

etc

. can

als

o be

con

stru

cted

usi

ng t

he m

etho

d of

Dun

eau

and

Kat

z. T

his

clea

rly s

how

s th

e ric

hnes

s of

the

varie

ty o

f ne

w

h,

w

crys

tallo

grap

hic

phas

es th

at t

he n

otio

n of

the

qua

si-c

ryst

al h

as r

evea

led.

L

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

232 D. Grutius

displacement will superpose it exactly on itself. However, the reader will readily recognize a perspective view of a strictly periodic stacking of cubes in three- dimensional space. In fact, it is the very special way in which this stacking is truncated and then projected onto a plane that gives this projection its property of being quasi- periodic. The method consists of literally cutting a slice through the initial stacking at an angle whose slope is irrational (with respect to the three directions of the stack). All the cubes whose tops are included in this slice are retained. Their outline is then projected in a direction perpendicular to the cut. The result is shown in figure 6. This quasi-periodic tiling in a plane is thus just a two-dimensional projection of a truncated cubic network and is, initially, strictly periodic in three dimensions. The same is true of the Penrose patterns which are also plane projections of a slice cut in a similar way through a simple cubic network, but this time the initial network is in a space of five dimensions!

Although it is impossible for us to visualize this kind ofnetwork, it is not difficult, by way of compensation, to calculate the projections with a computer. It is this very simple idea, contained in the original work of De Bruijn, that underlies the formalism developed by Duneau and Katz. An aspect greatly appreciated by non-mathematicians is that the theory uses essentially linear algebra and a small amount of group theory, basic tools of a university undergraduate. Different methods have been developed which elaborate on this general theme: the cut and projection of a structure that is strictly periodic in a space of higher dimensions. These lead to algorithms that allow Penrose figures to be constructed with any kind of symmetry, and even figures that are strictly periodic when the cut follows an angle having rational slope (figure 7 (a)). One of the fundamental contributions of Duneau and Katz was to prove the quasi-periodicity of the tiling when the cut is in an irrational direction, although the idea of an eventual application of Penrose tiling to crystallography dates back to 1981. By means of laser diffraction of these tilings, Allan L. Mackay, a London crystallographer, had demonstrated their long-range order, unfortunately without complete mathematical support. He had nonetheless suggested their inclusion in the armoury of the crystallographer. All this illustrates a general phenomenon well known in the history of science: the same ideas germinate in different disciplines at the same time, but are expressed in the language appropriate to each. Then, suddenly, triggered by a striking experimental or theoretical event, they are brought together to give birth to a new concept of which each at some time justly claims to be the father!

The notion of quasi-periodicity, for example, originated from the work of the mathematician Claude Esclandon (1902). It was subsequently generalized to almost- periodic functions in 1924 by H. Bohr. As for the idea of using a space of higher dimension, this is to be found in most branches of physics whenever ‘hidden symmetries’ that can act as a source of simplification are suspected. In this way, certain amorphous metallic structures are today described very elegantly as Euclidean projections of simple tilings of curved spaces in four dimensions (Mosseri and Sadoc 1984).

In the same way, crystallographers, following the work of De Wolf, Janner and Janssen, (Janner and Janssen 1980), used Euclidean spaces of higher dimensions (greater than three) to describe incommensurate phases. The model of Duneau and Katz thus succeeded in generalizing to discrete structures (the collection of points in the network) what the crystallographers had developed for continuous functions. Recently, Per Bak (1985) has shown the equivalence of these two approaches, which thus make the incommensurable phases the first known examples of quasicrystals.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystals 233

7. The surprising mechanical strength of quasicrystals Many kinds of quasi-crystalline alloys have been discovered. Besides those of icosahedral symmetry (of order 5) there also exist those of decagonal symmetry (of order 10) (Bendersky 1985) and of dodecagonal symmetry (of order 12) (Tshima et a!. 1985). Far from being an exceptional arrangement, the quasi-crystalline state thus seems to be quite widespread, not only in rapidly quenched alloys but also in structures near equilibrium such as those discovered among the industrially interesting alloys of the type Al-Li-Cu and Al-Li-Cu-Mg by Pierre Sainfort and his co-workers at the Pechiney research centre at Voreppe (Sainfort et al. 1985). Indeed, methods of obtaining quasi-crystalline phases without recourse to ultrafast quenching are currently being investigated because this technique is not always easy to put into practice in industry.

The physical properties of all these materials have hardly begun to be explored. Those that depend on the periodicity are certainly changed when the arrangement is purely quasi-periodic. In particular, the mechanical strength of quasi-crystalline alloys is increased to a remarkable degree; the absence of periodicity makes the propagation of dislocations slower than in the case of the metals we are used to. The very notion of a dislocation becomes more complex and several new types of defect are likely to be revealed in the near future ( K l h a n et al. 1986). This property holds great industrial promise: effective control of the icosahedral phase would permit the manufacture of alloys that are light and of great mechanical strength by precipitating fine particles of the quasi-crystals in a metallic matrix of aluminium.

However, before decisive progress can be made in all these areas, it is important to know the actual atomic arrangement in these structures. Although it is clear that the gometrical skeleton is quasi-crystalline, there remains the task of specifying precisely the relative positions of the atoms in order to understand and predict the special properties of these alloys. Shechtman and Blech (1985) proposed a simple model of the random stacking of the icosahedra of aluminium atoms surrounding a central manganese atom. This model required only that the polyhedra be connected parallel to each other by a common edge. It was abandoned after spectroscopic studies had shown that neither the stoichiometry nor the symmetry of the atomic sites was as expected (Guyot and Audier 1985). In June 1985 a very appealing solution was proposed by Pierre Guyot and Marc Audier who noted a strong resemblance between a known crystalline phase (aAl-Mn-Si) and the quasi-crystalline phase. At the same time, Christopher Henley and Veit Elser demonstrated the quasi-periodic arrangement underlying this crystal structure (Henley 1985, Elser and Henley 1985).

Guyot and Audier’s model is basically a framework made up of double icosahedra (an internal icosahedron of aluminium atoms surrounded by a second icosahedron of manganese atoms) connected by octahedra of manganese atoms which supply the rigidity needed to establish long-range order. The centres of gravity of the icosahedra are situated for the most part on certain sites of a mathematical quasi-lattice; steric hindrance prevents a complete occupation of all possible sites. In order to satisfy the stoichiometry and to achieve the density of the quasicrystal, additional aluminium and manganese atoms have to be added to this framework. In particular, one can place aluminium atoms on the edges of the manganese icosahedra, thus defining a unit cell of 54 atoms (figure 8) already described in a different context by Allan L. Mackay (1962) as one of the generating cells of icosahedral symmetry. Analysis of the X-ray and neutron diffraction patterns suggests, moreover, that a small fraction of the manganese atoms are replaced by aluminium atoms and vice versa. Here too, a description of the atomic

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

234 D. Gratias

Figure 8. The atomic structure of the quasi-crystalline alloys (Al, SikMn consists of quasi- periodic stacks of fundamental units each made up of 42 atoms of aluminium and 12 of manganese. These units are the so-called Bravais motifs of the crystalline structure (Al, SikMn whose analysis was first made in 1966. They are constructed from a central icosahedron of aluminium, surrounded by a second icosahedron, which is twice as large, of manganese, on the sides of which are placed icosadodecahedra of aluminium. In the interstices between units are found the additional aluminium atoms. The units are connected to each other by octahedral bridges that link a triangle of manganese of one unit, head to tail, with a triangle of an adjacent unit. The large spheres are Mn atoms and the small spheres are A1 atoms.

structure in a space of six dimensions is (relatively) simple and only requires the introduction of a total of seven variational parameters (Cahn and Gratias 1986).

Shechtman’s alloy would thus be a partially ordered quasicrystal. During solidification, Mackay’s units would form and rapidly adhere to each other by means of rigid octahedral ‘bridges’. Such bridges can be built easily, and hence rapidly, between two neighbouring units provided that they are truly regular polyhedra, that is, polyhedra whose 20 faces are equilateral triangles all of the same size. In order to form an octahedral bridge of manganese, all that is required is that any two of these triangles (one in each unit) should come near enough to each other, pointing head to tail and parallel to each other. The long-range order that results from such an octahedral bridging is thus due entirely to a simple geometrical constraint between two adjacent units, as in the growth of an ordinary crystal. One can thus see that these phases can form rapidly and should be found in many other systems, for example, those which in theliquid state show a tendency to form units whose symmetry is incompatible with the translational invariance of the crystal.

Such a scenario shows that the growth of quasicrystals and of amorphous materials have much in common, since, depending on the type of connection between adjacent units, one or the other of these two states will be obtained. If the connection prohibits any orientational degree of freedom, the solid will be quasi-crystalline: the centres of the

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

elementary polyhedra, being identical and parallel, will distribute themselves on the points of a quasi-lattice whose symmetry will be that of the polyhedra in question. If, on the other hand, the connections allow orientational disorder, the material will then be amorphous, with short-range order only. The distinction between the crystalline, quasi-crystalline and amorphous states seems to become more and more tenuous. There is, however, a fundamental difference; quasi-periodicity and periodicity both define a crystalline state having long-range order, in contrast to the amorphous state which has only local order.

8. Take a crystal as an approximation All the experimental and theoretical facts about quasicrystals can thus be set in a framework that is perfectly coherent and very beguiling. However, in an article published in October 1985 in Nature, Linus Pauling (1985) announced that he had solved the problem of the icosahedral phases in a different way: by constructing a cubic structure (and so strictly periodic and crystalline). According to Pauling, the apparent icosahedral symmetry was due to the multiple twinning of cubic crystallites. Pauling based his argument entirely on a powder X-ray diffraction pattern that shows only the radial distribution of diffracted intensity. He managed, with good precision, to index this diagram, that is to say, to relate each diffraction ring to the geometric characteristics of the unit cell of a crystalline unit cell. However, this cell had to be of enormous size (26.74 A) with some two-thousand atoms, and the model is contradicted by high-resolution micrographs and by electron diffraction patterns, which show a distribution of spots different from that announced by Pauling (Cahn et al. 1986; see also Tarnowsky (1986)). This attempt by Pauling seems, however, to have been welcomed with a certain sense of relief by one part of the scientific community not directly concerned with this subject; a community which regarded ‘quasicrystals’ with scepticism, seeing in them at best interesting abstract constructions with no strong relevance to the structure of real minerals.

The controversy entered into by Pauling does give rise to a certain semantic interest but this could in any case have been started at the time of the discovery of incommensurate phases. To paraphrase Duneau and Katz, one can say that quasi- crystalline structures are ‘interpolations between crystalline structures’ just as an irrational number is an interpolation between two rational numbers. The appearance of one or more irrational numbers in the description of quasi-crystalline phases lends credence to this comparison. These irrationals are, for example, the parameters that determine the orientation of the cut used in applying the method of Duneau and Katz. Approximating these numbers by rational ones leads immediately to periodic structures! Thus, if one concedes that the quasicrystal is not strictly icosahedral (and it can always depart from this within the limits of experimental error) it is then possible to describe it, as accurately as one wishes, by a crystalline phase. The initial paradox disappears; the object reverts to being a crystal whose unit cell is certainly unusually large but which can, in particular, be cubic.

The theory of quasi-periodicity allows one to construct algorithms that simulate this situation. In the calculation of structures of fivefold symmetry for example, let us replace the ‘golden number’ z = (1 + J5)/2 by its rational approximants. This number, which owes its name to its fascinating mathematical properties (see 9;5) is in fact characteristic of fivefold symmetries. In particular, the framework on which the atoms are placed is invariant when one enlarges it by a factor equal to the cube of the golden number. Its approximants are obtained by iteration: at each step a new fraction is

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

236 D. Grutias

Figure 9. True quasi-peripdic structures can be approached by a series of periodic structures, in the same way that irrational numbers are the limit of a series of rational numbers. The approximation improves as the size of cell of the periodic structure increases. In this example are shown the first four periodic arrangements which converge, in proportion to the size of the unit cell, towards the octagonal structure of figure 7 (b). As one can see, these objects, although periodic, become more and more complex in the internal ordering of the motifs that contain the essential features of these geometric properties: the periodicity is now only a weak property of these structures.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quusicrystals 237

calculated which approximates more and more closely the irrational number. Starting with the fraction l / l , the numerator becomes the denominator of the next fraction whilst the sum of the numerator and the denominator becomes the numerator of the next fraction. Thus 131 gives (1 + l)/l=2/1, which becomes (2+ 1)/2=3/2, then (3 +2) /3 = 5/3, etc. At each step the algorithm of Duneau and Katz allows one to obtain crystalline structures that approximate the quasicrystal (figure 9). The higher the order of the approximant (for example 610/377), the more closely the calculated structure approximates a true quasicrystal. The counterpart of this is that the size of the primitive cell increases with the powers of the golden number: in the limit, the quasicrystal is a crystal whose unit cell is infinite! It is in this limit only that the diffraction pattern of the object shows exact icosahedral symmetry.

Before the appearance of Pauling’s article, I had constructed with John Cahn a series of these structures (Gratias and Cahn 1986) to show the remarkable convergence of the atomic sites to icosahedral symmetry. The size of Pauling’s unit cell would correspond to the fifth approximant of this series which would depart from exact symmetry by only one part in 400, which is of the order of magnitude of the experimental precision. The significant difference between Pauling’s model and this fifth approximant is that not only does the latter not require any kind of twinning to be introduced in order to explain the symmetry of the diffraction patterns, but also that the reflections this time have the right directions, all to within a few parts in one-thousand.

9. For complete thoroughness one last question remains to be answered. Since it is always possible to find a crystal structure that is compatible with the experimental results to within the limits of experimental error, how can we distinguish between this approximate structure and a true quasi-crystalline structure? How can we find ‘irrefutable experimental proof‘ that the new concept of the quasicrystal is really needed? To ask such a question is exactly the same as wishing to test the irrational nature of the number n by measuring experimentally the ratio of the circumference of a circle to its diameter. Since the result that one would obtain is necessarily finite, it is easy to approximate it by a rational number within the limits of the experimental precision of the measurement. However, no one doubts that n is indeed an irrational number! In the same way for quasicrystals, if one analyses the high-resolution electron micrographs, no periodicity in the arrangement of atoms can be discerned. The ultimate period of the crystalline network would be greater than the size of the areas observed! Once again, the solution to the problem raised by these new phases lies in the revision of the fundamental concepts of crystallography by abandoning strict periodicity as the fundamental property of crystals. To conclude, firstly it is now mathematically proven that it is possible to create structures with long-range order of any symmetry. From this comes the completion of the generalization of crystal- lography, started with the introduction of the incommensurate phases. Secondly, for fundamental physical reasons structures of high symmetry are generally favoured for certain values of the thermodynamic parameters, at the expense of structures of lower symmetry. In particular, periodicity is not a necessary condition for states of minimum energy. The concept of icosahedral structures is thus not in itself less probable than that of cubic or hexagonal structures. We are involved with an idealization of the real structures and one cannot experimentally be entirely certain that they possess the symmetry that one assigns to them. Finally, the theory of quasicrystals leads to the only descriDtion pertinent to the approximate structures referred to above. As the degree of

Good use of ‘irrefutable experimental proof’

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

238 D. Gratias

approximation increases, the number of sites in the primitive cell grows in size as the cube of the order of approximation to the irrational number being approximated. Thus, not far short of 2000 sites are needed to describe the fifth approximant of an icosahedral structure. Hence the motif (the flower that is periodically repeated in the wallpaper) becomes a gigantic bouquet whose internal arrangement contains the essence of the geometric properties of the crystal! Periodicity then becomes a property that is weak, or secondary, compared to the regimentation of the motif. Conventional crystallography is then inadequate to describe the intrinsic properties of the motif since it requires that the position of each atom be explicitly given in order to arrive at a complete description of the object. By contrast, the theory of quasicrystals simultaneously takes account of the arrangements within and between the motifs with the help of a limited number of simple elementary polyhedra.

In these circumstances, whether or not Shechtman’s alloy corresponds to the exact definition in the mathematical sense of the word of a limited quasi-periodic structure, is a question which experimentally cannot be answered irrefutably and which in any case is of no great importance. Even on the unlikely hypothesis that this alloy might one day be proved to be a true crystal with an enormous unit cell, the notion of the quasicrystal generalizes and considerably simplifies its description. That is the only fundamentally important point: the concept of the quasicrystal and of the quasicrystalline state are at one and the same time the most general and the best adapted to describe the properties of this class of structure.

Further reading Two basic works on crystallography FRIEDEL, G., 1964, LeFons de Crist$lographie (Paris: Blanchard). BURGER, M. J., 1956, Elementary Crystallography (New York: John Wiley).

On the theory of numbers HARDY, G. H., and WRIGHT, E. M., 1960, An Introduction to the Theory of Numbers (Oxford:

Clarendon Press).

General references on quasicrystals GRATIAS, D., and MICHEL, L. (editors), 1986, Workshop on Aperiodic Crystals, J . Phys., Puris,

JARIC, M. V. (editor), 1987, Aperiodic Crystals: Vol. 1 , Introduction to Quasicrystals (New York:

STEINHARDT, P. J.. and OSTLUND, 0. (editors), 1987, Selected Reprints on Quasicrystals (New

Colloq., 47, c3.

Academic Press).

York: Plenum).

References BAK, P., 1985, Phys. Rev. Lett., 54, 1517; 1985, Phys. Rev. B, 32, 5764. BENDERSKY, L., 1985, Phys. Rev. Lett., 55, 1461. BESICOVIC, A. S., 1932, Almost Periodic Functions (Cambridge University Press). BOHR, H., 1924, Acta Math., 45, 29; 1925, Ibid., 46, 101; 1926, Ibid., 46, 237. CAHN, J. W., and GRATIAS, D., 1986, J . Phys., Paris, Colloq., 47, C3-389. CAHN, J. W., GRATIAS, D., and SHECHTMAN, D., 1986, Nature, Lond., 319, 102. DE BRUIJN, N. G., 1981, Proc. Konink. Ned. Akad. Wetensch. A, 84, 39; Ibid., A, 84, 53. DUNEAU, M., and KATZ, A., 1985, Phys. Rev. Lett., 54, 2688. ELSER, V., 1985, Phys. Rev. Lett., 54, 1730; 1985, Phys. Rev. B, 32, 4892. ELSER, V., and HENLEY, C. L., 1985, Phys. Reu. Lett., 55, 2883. ESCLANDON, C., 1902, C. r. hebd. Stanc. Acad. Sci., Paris, 135,891. GRATIAS, D., and CAHN, J. W., 1986, Scripta Metall., 20, 1193. GUYOT, P., and AUDIER, M., 1985, Phil. Mag. B, 52, L15.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013

Quasicrystuls 239

HENLEY, C. L., 1985, J . non-crystalline Solids, 75, 91. HIRAGA, K., HIRABAYASHI, M., INOUE, A,, and MASUMOTO, T., 1985, J. phys. Soc., Japan, 54,4077.

ISHIMA, T., NISSEN, H.-U., and FUKANO, Y., 1985, Phys. Rev. Lett., 55, 511. JANNER, A., and JANSSEN, T., 1980, Physica BIC, 99, 334. KALUGIN, P. A., KITAEV, A. Yu, and LEVITOV, L. C., 1985, J E T P Lett., 41, 145; 1985, J. Phys.,

KLBMAN, M., GEFEN, Y., and PAVLOVITCH, A., 1986, EuroPhysics Lett., 1, 61. LEVINE, D., and STEINHARDT, P. J., 1984, Phys. Rev. Lett., 53, 2477. MACKAY, A. L., 1962, Acta Cryst., 15,916; 1981, Soviet Phys. Crystalloyr., 26, 517; 1982, Physitu

MOSSERI, R., and SADOC, J.-F., 1984, J . Phys., Paris, Lett., 45, 1827. PAULING, L., 1985, Nature, Lond., 317, 471. PENROSE, R., 1979, Math. Intell., 2, 32. PORTIER, R., SHECHTMAN, D., GRATIAS, D., and CAHN, J. W., 1985, J . Microsc. Spectrosc.

SAINFORT, P., DUBOST, B., and DUBUS, A., 1985, C. r. hebd. SCanc. Acad. Sci., Paris, 301, 689. SAINFORT, P., and DUBOST, B., 1986, J . Phys., Paris, Colloq., 471, C3-321. SHECHTMAN, D., BLECH, I., GRATIAS, D., and CAHN, J. W., 1984, Phys. Rev. Lett., 53, 1951. SHECHTMAN, D., and BLECH, I., 1985, MetafE. Trans. A, 16, 1005. TARNOWSKY, D., 1986, La Recherche, 174, 145.

4071.

Paris, Lett., 46, L601.

A, 114, 609.

Electron., 10, 107.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

01:

58 0

5 N

ovem

ber

2013