The structure of quasicrystals

Journal of Non-Crystalline Solids 156-158 (1993) 852-864 North-Holland

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Section 9. Quasicrystals

The structure of quasicrystals Christian J a n o t Institut Laue-Langevin, BP 156, 38042 Grenoble cddex 9, France

Quasicrystals are long range ordered structures in which atoms are not periodically distributed in physical space. These structures have been analyzed as (i) physical cut of structures which are periodic in higher dimensional spaces (quasicrystallography), (ii) modification of periodic approximants with large unit cell, (iii) random tilting model entropically stabilized or (iv) some sort of glassy structures. With the increasing perfection of quasicrystals, all these descriptions are converging to the quasicrystallography approach which gives a reasonable understanding of the atomic arrangement.

1. Introduction The discovery of new solids exhibiting symmetries forbidden for ordinary crystals was first reported by Shechtman et al. [1] in A1-Mn and AI-Mn-Si alloys. In the intervening eight years, hundreds of other compounds have been observed with quasicrystalline phases. Classically forbidden symmetries, namely five-fold, eightfold, ten-fold and twelve-fold, have been reported. Most of the quasicrystalline materials which are known now are Al-based binary or ternary metallic alloys, or analogous alloys with Ga or Ti playing the role of Al. During the early years of the field, there were some speculations that quasicrystals might be inherently disordered and unstable. The speculations proved wrong, though. There are now at least a dozen known compounds which are thought to be thermodynamically stable. In a few cases, the phase diagram has been worked out to some extent [2,3]. Also, many of the newer, thermodynamically stable quasicrystals have translational correlation lengths and facetting morphology (with icosahedral symmetry) which rival the best conventional metallic crystals. Correspondence to: Professor Ch. Janot, Institut Laue-Langevin, avenue des Martyrs, BP 156, F-38041 Grenoble c6dex, France. Tel: + 33 76 20 73 27. Telefax: 133 76 48 39 06.

Reproducible electronic characterizations were also very hard to make in the early years because of the poor quality of the samples. This too has changed, and reliable information is now available for the electrical conductivity, Hall coefficients, thermopower, etc. for many of the alloys [4-6]. At the time of the experimental discovery of icosahedral alloys, Levine and Steinhardt [7] were independently formulating their hypothesis of a new class of solids which they dubbed 'quasiperiodic crystals', or 'quasi-crystals'. Based on the published electron diffraction patterns displaying sharp peaks arranged with icosahedral symmetry, they proposed that the new alloys might be laboratory examples of quasicrystals and they outlined some basic mathematical and structural principles. At the same time, several alternative structural models have been proposed and explored, such as structures obtained by multiple-twinning of small crystallites [8] or by the random packing of icosahedral clusters ('icosahedral glass' models) [9]. Quasicrystallography has not yet achieve the level of structure refinement which is currently reported in regular crystallography. Also, the description of the structure, although very easy in its high-dimensional periodic image, is not that straightforward when one deals with the three-dimensional physical atomic arrangements. This

0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

Ch. Janot / The structure of quasicrystals

difficulty has stimulated many attempts toward atomic modeling, directly in real space, by approaching the quasiperiodic structures with periodic approximant crystals. So-called 'random tiling' models were also introduced in order to give an easy answer to quasicrystal growth problems and also to support the idea that quasicrystals phases may be favoured for entropic reasons [10]. Actually periodic approximants or random tiling structures may appear as more realistic models than perfect quasiperiodicity for experimental systems whose structures have not achieved perfection in their preparation procedures. In the present paper, the basic principles of quasiperiodicity are briefly introduced and illustrated with an example of structure determination.

2. Quasiperiodicity: the basic principles Quasicrystals are new types of solids which defy previous standard classifications. They are neither periodically ordered like ordinary crystals, nor are they disordered or amorphous solids. They have a well defined, discrete group symmetry, like crystals, but one which is explicitly incompatible with three-dimensional periodic translational order (e.g., exhibiting five-, eight-, ten- or twelvefold symmetry axes). Instead, quasicrystals possess a novel kind of translational order known as quasiperiodicity. Atomic order is best defined in terms of the Fourier transform of the mass density of the solid. In an ordinary crystal, this transform can be written as a Fourier series:

1 p ( r ) = -~ E p ( G )

exp(iG . r ) .

853

The a*s are said to span the reciprocal lattice. They are related to the 'basis' vector a i, which defines the unit cell of the crystal in physical space. In a quasicrystal, the Fourier transform of the mass density is again a Fourier series and the wavevectors in the Fourier sum also form a discrete reciprocal lattice. However, the number of integer linearly independent basis vectors required to 'span' the reciprocal lattice exceeds the spatial dimension and the point symmetry of the reciprocal lattice is incompatible with periodic translational order. For example, six basis vectors are required to span the reciprocal lattice for three-dimensional quasicrystals with icosahedral symmetry:

G = nla ~ + n2a ~ + naa ~ + n , a ~ + nsa ~ + n6a~, (3) where the a* can be selected to point along the five-fold axes of an icosahedron. Of course the vectors G, as given by eq. (3), must be expressed in a three-dimensional space. This expression is illustrated in fig. 1. In the cubic coordinate system, the a* vectors are of the form (+1, + r , 0) (and permutations) with r = (1 + ~ - ) / 2 = 2 cos36 ° = 1.618034... the golden mean. Thus, all the vectors G have cubic coordinates of the form (h + rh', k + r k ' , l + rl') with h,h', k,k', l,l' integral numbers (selected within extinction rules). This manifold is the simplest possible definition of a quasicrystal. The expression of the reciprocal vectors, G, in terms of their cubic

~s

(1)

G

The set of wavevectors, G, defines a discrete reciprocal lattice in which each wavevector in the sum can be written as an integer linear combination of three 'basis' vectors, a*: G --ha*l =kay" +/a~'.

x

(2)

The a* are integer linearly independent, which means that G = 0 if and only if all the h,k,l = O.

Fig. 1. Vertex reciprocal lattice vectors, a * , for a regular icosahedron. Cubic axes are also shown on which the a~' vectors have components of the form ( + 1, _+ r, 0).

854

Ch. Janot / The structure of quasicrystals

coordinates shows that, due to irrationality of ~-, the quasiperiodic structures may exhibit some unusual properties, when referred to regular crystals. (i) The G vectors form a dense pseudo-continuous set, and Bragg peaks are expected to show up 'everywhere' in the reciprocal space (h + ~'h' are not integral numbers and their fractional parts fill densely the interval [0,1]). (ii) There is not a 'smallest' basic G-value as a consequence of property 1; this lack will be a difficulty when indexing experimental diffraction patterns. (iii) Quasiperiodic structures will obey some inflation rules due again to the peculiar properties of the golden mean, ~-; for instance, if all h,h', k,k', l,I' integers are allowed by extinction rules, ~- inflated structures are identical within rescaling (note that T n + l = T n "b T n - l ) . (iv) The dense reciprocal space of a quasicrystal as described by eq. (3) may be given a periodic image in a high dimensional space; for instance, eq. (3) may be considered as the description of a six-dimensional periodic reciprocal lattice whose Fourier transform would generate a six-dimensional periodic mass density distribution. Projection and cut operation will then relate the physical three-dimensional description to its six-dimensional image. This is advocated further in the next section.

3. The hyperspace construction The most important mathematical notion for present thinking on quasi-crystals is certainly the hyperspace concept [11-13]. One application of this idea is in atomic modeling. Diffraction data for icosahedral quasicrystals are now successfully interpreted in terms of a three-dimensional slice through a periodic mass density function in six dimensions. Points of high density in three dimensions (i.e., atoms) are associated with threedimensional 'atomic surfaces' of high density in the six-dimensional space. A three-dimensional slice through the six-dimensional surface shows up as a point in three-dimensional real space. This rather hard-to-visualize description of the

t Fig. 2. A quasiperiodic chain as generated by a one-dimensional irrational cut of the decorated two-dimensional square lattice.

quasiperiodic structure is best illustrated by the toy model in fig. 2, in which a one-dimensional quasicrystal is produced by slicing a two-dimensional square lattice. First, the figure illustrates 'atomic surfaces', identified as line segments placed in each unit cell, which become atomic points when cut by the 'physical' space Rlpa rThe slope of Rlpa r is at an angle incommensurate with any lattice plane and results in a one-dimensional chain of atoms separated by short and long intervals occurring in a quasiperiodic sequence. As suggested by the label Rlpar, the physical space is also called parallel space while the complementary space is consequently dubbed perpendicular s p a c e , Rlperp. The atomic surfaces, Aperp , which become 'volumes' when real three-dimensional quasicrystals are concerned, must obey obvious requisites: (i) they have no 'thickness' in R p a r if they have to generate point-like atoms in physical slicing; (ii) they have the symmetry of the structure (e.g., icosahedral); (iii) they cannot intercept nor be closer to each other than a threshold distance related to physical atom closeness (the so-called hard core condition); (iv) they allow translational invariances of the quasiperiodic structure, parallel to both R par and R perp"

Ch. Janot / The structure of quasicrystals

Such a slicing procedure may be felt awkward or, at least, rather artificial. This procedure is actually a simple extension of what may be best understood when referred to simple modulated structures. An n-dim modulated structure can be viewed as the intersection of an (n + 1)-dim periodic structure with the n-dim physical space. To illustrate the point, let us consider, for instance, the simple case of an incommensurate structure resulting from the sinusoidal modulation of a periodic chain, periods of the chain and of the sine function being irrationally related. Position (or origin, or phase) of the modulation function with respect to the chain may be chosen arbitrarily. When this phase is changed, atom positions in the chain are 'rearranged' into a new configuration having the same total equilibrium free energy as before. A continuous shift of the modulation function with respect to the chain generates an infinite number of 'indistinguishable' incommensurate structures which can be visualized simultaneously by piling them up along an axis perpendicular to the chain (fig. 3) [14]. In the then-defined perpendicular space, a given point of the chain has 'equivalent' positions distributed on a periodic profile which is nothing else than the modulation function. The incommensurate structure, in all its 'indistinguishable varieties', then appears as a cut of a periodic high-dim (2-dim here) structure by the physical space. The

Rperp

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THE REGULARCHAIN ' ATOMPOSITIONS Fig. 3. Continuous phase shifts of the modulation function generate 'indistinguishable' structures. All these structures can be visualized when piled up together in a 'perpendicular' space. This is the foundation of the cut, or embedding, methods [14].

855

perpendicular space, as describing the phases of the modulation function, is called the phase space, or phason space. Any translation in the high-dim space with non-zero component only in the perpendicular space is called a 'phason mode' or phason for short. The term phonon is restricted to translations parallel to the physical space. Although all these concepts can easily be extended to quasicrystals, there are some important qualitative differences between incommensurate phases and quasicrystals. Most of the incommensurate structures correspond to 1-dim or 2-dim modulations and the overall symmetry is rather low. The associated high-dim space has then dimension four or five and the cut that generates the physical space has low symmetry. The relative locations of the atoms are distributed in space according to a modulation law which is a continuous function of the perpendicular space coordinates; the atomic surfaces in the high-dim space are then continuous surfaces. The existence of an average underlying periodic lattice also gives sense to the notions of average unit cell, average atomic distances, etc. Conversely, for quasicrystals the high-dim space may have very large dimensions. For instance, the associated high-dim space for icosahedral quasicrystals has minimal dimension six. Moreover, the overall symmetry is the highest possible in 3-dim. There is no underlying periodic lattice and, consequently, no 'average' parameters can be sensibly defined. The most dramatic difference comes from the atomic surfaces in the high-dim space, which are not continuous any more, but appear as piece-wise discontinuous objects, mostly parallel to the complementary space (see fig. 2). For the conservation of atoms with respect to the R perp variables, these atomic surfaces must connect in such a way that there is no 'annihilation/creation' of atoms under any Rperp translation (this is called the closeness condition). In reciprocal space, the Fourier components of the hyperspace square lattice are modulated by the Fourier transform Aperp(Gpew) of the atomic surface Aperp, which is a decaying function of the perpendicular component Gp~w of the Bragg vectors, G. As a mathematical property of the Fourier transform, the Fourier pattern of the sliced

Ch. Janet / The stm zture of quasicrystals

856

quasiperiodic structure is simply obtained by projection on R$, the reciprocal space associated to R par.This is shown in fig. 4, which also illustrates the indexing scheme and the dense character of the Bragg pattern. Experimentally, the hyperspace scheme can be used to specify a quasiperiodic structure using

diffraction data. Once the point group of symmetry is determined (e.g., icosahedral), Bragg peaks which have been measured are indexed with the high-dimensional Miller indices (e.g., six integral numbers for icosahedral quasicrystals). Figure NB) illustrates this indexing for the l-dim quasicrystals of fig. 2. Then the diffraction pattern is

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, 1

I

I

2

Gpar (2n/a

3

4

unit)

Fig. 4. Fourier pattern of the quasiperiodic chain of fig. 2, as projected from the reciprocal ‘lattice’ of the two-dimensional structure. Diffraction data usually give the one-dim pattern (4Bf which, after proper indexing, can be ‘lifted’ in the hyperspace Fourier pattern (4A). In principle, the hyperspace structure (fig. 2) is then deduced via inverse Fourier transform procedure and a cut by the physical space gives the final real quasiperiodic structure.

Ch. Janot / The structureof quasicrystals

'lifted' into its high-dimensional periodic image (fig. 4(A)). The high-dim periodic mass density is obtained, in principle, via Fourier transform procedures. A final relevant slicing generates the 3-dim quasiperiodic systems. This is quasicrystallography. Beyond the usual drawbacks of crystallography (truncation effects, background noises, phase reconstruction techniques, etc.), quasicrystallography has to deal with specific problems. Actually, refinement procedures cannot apply to the many parameters that would be needed to specify all the details (size and shape) of the atomic surfaces. No diffraction experiments will ever be precise enough for rendering the boundaries of the atomic object in R p~rp by inverse Fourier transform. Some of the details are gained when contrast variation measurements are feasible [15] but modeling is always required. Such modelings are necessarily achieved via some arbitrary choice of a finite number of parameters that are intended to characterize the atomic volume boundaries. These parameters may be a sphere radius in the simplest naive approach, or any other geometrical features of chosen polyhedra. Then, refinement procedures give the best fit model within the corresponding class of structure. There is obviously no guarantee that other classes of structure, where the atomic objects are differently parameterized, would not yield a crystallographic fit that is as good or even better. This may suggest that the famous residual parameters are less significant in quasicrystallography than in regular crystallography for validating a possible model, on the sole basis of diffraction data. However, we must realize that we are talking of the 'skin' of the atomic objects. This concerns only a rather small part of the entire structure. Let us consider, for instance, the trivial case in which a 'true' triacontahedral atomic object would have been 'mistaken' for a sphere of equal volume. It is easy to calculate that the differences induced into the resulting structures, due to nonequivalent 'skins' of the atomic objects, concern about 5% of the atom positions. Moreover, most of these 'wrong' positions can be eliminated by an ad hoc tailoring procedure because they generate too short unphysical pair distances. It has recently been demonstrated also [16] that severe

857

constraints should apply for the choice of the atomic volume shapes, as direct consequences from the already mentioned closeness and hard core conditions. This comes from the requisite that any uniform translation parallel to Rperp must initiate only atomic jumps so that initial and final states have close configurational energies. In the example of fig. 2, such conditions have only the consequence of fixing the length of the Aperp segments. When applied to icosahedral symmetry, these conditions require that the boundaries of the atomic surfaces be twofold planes [16]. There are actually eight basic polyhedra which fulfil the conditions. The possible, acceptable, atomic volumes to decorate the 6-dim cube must then be one of these polyhedra, or any z-scaling and/or intersection of them. Obviously, there are still a number of alternative solutions! The eight basic polyhedra are pictured in fig. 5.

4. A little more about the phason variables

The slice construction also illustrates the phason degree of freedom, which is another characteristic of quasiperiodic structures. This simply corresponds to translating the atomic surfaces or equivalently the plane Rlpa r in the Rlpe~p direction. Such an operation is called a phason displacement, and two quasicrystals related by such a phason displacement are said to be locally isomorphic. Two locally isomorphic structures are indistinguishable in any physical measurement. There are important types of imperfection which are unique to quasicrystals that have a natural description in the hyperspace picture. The atomic surfaces may be displaced in the perpendicular space in various ways. As a consequence the slicing space, Rpar, intercepts another set of atomic surfaces and the atomic arrangement in physical space is modified: the same tiles may be still obtained but the distribution of tiles is modified. This difference is at variance with the effect of displacements of the atomic surfaces in parallel (physical space) which does not modify the topology (tile distributions) but introduces tile distortions. The former (phason disorder or phason modes) is typical of quasicrystals while the

858

Ch. Janot / The structure of quasicrystals

latter (phonon disorder or phonon modes) is observed in both crystals and quasicrystals. They are termed respectively phason and phonon strains because the corresponding field strains, with respect to perfect quasicrystals, relax via the propagation of phasons (atomic jumps) and phonons (vibrations), respectively. Because phason strains may involve discontinuous atomic motions, the properties of phason strains, i.e., relaxation times, responses to applied stress, etc., can differ from conventional phonon strains. In particular, phason strain may be frozen into the quasilattice if the barriers to phason 'hops' are greater than kT. Another important distinction between phonon

~

2

~

4

@ Fig. 5. The eight basic polyhedra bounded by twofold planes in Rpero for the 6-dim structure of the icosahedral phase. (Redrawn from ref. [16].)

and phason strain is found in their effects on diffraction experiments. Broadley speaking, both types of strain field disrupt the positional ordering and lead to changes in the observed diffraction peaks. Depending on the nature of the strain distribution, peaks may be reduced in intensity by a 'static' Debye-Waller factor, shifted in position, or broadened. The nature o f these diffraction effects will be identical for equivalent phonon and phason fields. However, because the fields each couple to their own conjugate vectors, the diffraction effects will scale with Gpar (the physical scattering vector at a Bragg reflection) in the presence of phonon strains and with Gperp for phason strains. Thus conventional lattice strains produce a peak broadening which increases monotonically with Gpa r. Analogous phason strains will produce a peak broadening monotonic in Gperp" The phonon and phason strain diffraction effects are easy to distinguish because the perpendicular wavevector depends in a complicated way on the reciprocal lattice vector. Phason strain diffraction effects have a puzzling or seemingly random behaviour if plotted agains the scattering vector, but show smooth monotonic behaviour versus Gperp. In icosahedral quasicrystals a simple case occurs when one measures a series of peaks of equivalent symmetry with Gpa r vectors which scale by the golden mean. In a r-series, Gperp scales inversely with the Gpa r and one finds that phason strain peak broadening decreases monotonically with Gpar. An equivalent description of the phason disorder can conveniently be made by leaving the 6-dim lattice undistorted and artificially allowing the slicing space, Rpar, to rotate (with rescaling), curve, randomly undulate, or being torn rather than the flat slice discussed above. Schematic representations of phason fluctuations in the 2-dim to 1-dim slicing are shown in fig. 6. If the slice is randomly undulating (isotropic random fluctuations), the ensemble average of the slice will maintain an average orientation, or slope, equivalent to the perfect slice and, on average, the quasicrystal (say icosahedral) symmetry will be preserved. Randomness will be introduced into the cutting of atomic surfaces, inducing positional disorder in physical space.

Ch. Janot / The structure of quasicrystals

859

c)

Fig. 6. Examples of phason disorders in the 2-1 dim slicing scheme: (a) perfect slicing (/-phase); (b) bounded fluctuations (attenuated Bragg peaks and diffuse scattering); (c) unbounded fluctuations (broad peaks with various lineshapes); (d) 'faceting' of the slice along commensurate directions (microcrystals).

This particular disorder.is termed random phason strain. If the fluctuating slice remains within a bounded distance from the original flat cut, then the structure still exhibits long range translational

(a)

order. In this case, diffraction maxima will still occur at icosahedral positions but with intensities decreased by an effective quenched phason Debye-Waller factor. Additional diffuse scattering

iIi

(b)

Fig. 7. Illustration of a phason strain transformation of the square lattice. The cut structure is quasiperiodic in (a) and periodic m (b), without and with a uniform phason strain, respectively.

860

Ch. Janot / The structure of quasicrystals

will appear around the diffraction peaks. Unbounded fluctuations will convert the broadened peaks with widths that scale monotonically with Gp~rp and the translational order will be lost. Isotropic icosahedral glass model [9] in which the slice can tear as well as fluctuate also produces peak broadening monotonic with Gperp but with a different power law. Anisotropic fluctuations alter the average orientation of the slice and globally break the icosahedral symmetry. In this case, diffraction maxima will be shifted from their icosahedral positions. An important case is that of a purely linear phason strain which is achieved by simply tilting the slice. Small tilts into rational orientations result in projected structures which are crystalline approximants to the quasicrystal. The shifting of approximant-phase Bragg peaks from icosahedral positions is linear in Gpe~p. This shift is illustrated in fig. 7, again in the scheme of the 2-dim to 1-dim slicing. The periodic approximant structure is locally very similar to the quasicrystal. One may easily imagine that the quasicrystal structure might be derived from that of the well known periodic approximant submitted to a convenient 'inverse' uniform phason strain, after lifting its structure in a high-dim space.

5. Real quasicrystals There are basically three families of materials, where structures can be considered as exhibiting various levels of perfection in quasiperiodicity. These three families also correspond to successive historical events. Although they obviously had the merit to initiate the subject, AIMn 'quasicrystals' suffer from some major drawbacks. (i) They are not equilibrium compounds and are obtained via rapid quenching procedures. (ii) The diffraction peaks are rather sharp but not really Bragg-like and in fact exhibit widths which do not scale monotically either in Gpar or in Gperp but rather in a quadratic function of both. (iii) They cannot be grown as large single (quasi)crystals. Their average structure is quasiperiodic but may be described as an icosahedral glass, in which icosahedral clusters are ran-

(a)

(b)

e 2 ~ e5 e3 e4 Fig. 8. (a) Perfect 2-dim Penrose tiling by rhombuses with 36 ° and 72 ° acute angles and strict matching rules; (b) 'random tiling' with the same tiles.

domly aggregated within orientation constraints [9]. Holes necessarily form at the cost of density problems. The next prominent milestone in the still rather short history of quasicrystals was the discovery of the AI6Li3Cu icosahedral phase [17] which formed (i) as an equilibrium compound and (ii) as single grains approaching millimeter sizes and with perfectly faceted triacontahedral shape. Moreover, the diffraction peaks still had finite widths but scaled with Gper~ rather than with Gpar. The experimental observations were rather consistent with the quasicrystal model in which linear phason strains are quenched during solidification. The strained quasicrystal model then generated so-called 'random tiling' structures which also stimulated the interesting suggestion that the icosahedral phase is favoured for entropic reasons. In such models, clusters are aggregated still within some randomness but with the requirement that no holes would form (fig. 8). Later, quasicrystals of the AIFeCu system were obtained [18], followed by A1PdMn quasicrystals [19]. They form by regular slow casting procedures, they behave as stable phases, single grains with dodecahedral shapes are easily grown and the widths of the diffraction peaks are only limited by the instrument resolution. In short, they are apparently perfect (phason less) quasicrystals whose structure can actually be described within the quasicrystal model and the high dimensional

Ch. Janot / The structure of quasicrystals

periodic representation. Of course other models, including glass-like aggregates, random tiling, multiple twinning or large unit cell crystals, can be continually altered so as to be reasonably consistent with the data. However, more and more complex modifications are required when samples improve. The 'glass structure' must be artificially modified so that its intrinsic disorder does not dominate the peak widths and in fact finally vanishes; multiple twinnings must be taken at such a scale that quasiperiodicity is actually restored; random tilings have to be designed with complicated ordering algorithms; large unit cells in crystal models have to be so large that they would imply a physically implausible range of interaction. By contrast, data and theory continue to converge on the quasicrystal model with possible long wavelength phason strain a n d / o r defects. As an example of quasicrystallography, it may be interesting to report on the recent study of the structure of the perfect quasicrystal A1PdMn [20,21]. More details about quasicrystallography can be found in refs. [22,23]. The AIPdMn system is a very favourable case for diffraction studies. Beyond the perfection of quasiperiodicity and the possibility to grow single grains, it is also possible to play with contrast variation effects (contrast on Mn sites with neutrons, and neutrons versus Xrays for contrast on Pd sites). This quasicrystal exhibits the icosahedral symmetry, as deduced for instance from electron diffraction patterns. Indexing the diffraction peaks in neutron and X-ray patterns shows that the 6-dim reciprocal lattice is body centred, with indices all odd or all even [24-26]. Bragg reflections with odd indices are weak, which suggests that the reciprocal space may be best described by a primitive lattice with a set of superstructure reflections having half integer indices. Thus, the Bravais lattice in 6-dim direct space can be pictured as a primitive cubic (parameter a = 6.451 ,~), with two families of non-equivalent lattice nodes having even or odd parity. (This parity refers to that of the sum of the six corresponding indices.) Rather accurate information about the size of the different atomic surfaces may result from models using balls and spherical shells, even if,

861

obviously, the atomic surfaces have to be faceted volumes. Actually~ at small values of Q~r (i.e., smaller than 0.5 A-l), the Fourier transform of an atomic surface is mainly influenced by its size and does not depend on its precise shape. A fitting procedure allows an accurate definition of these volumes. Within density and chemical composition constraints, the best fit to low Qper data was obtained with six atomic surfaces defined as follows: (i) at even lattice nodes, a core of Mn (radius 0.83a, where a is the parameter of the 6-dim cube) surrounded by an intermediate shell of Pd (extending up to 1.26a) and an outer shell of Al (up to 1.55a), (ii) at odd lattice nodes, a core of Mn (radius 0.52a, ~" time smaller than the one on the even node) surrounded by a shell of Al (up to 1.64a, i.e., larger than the even one) but without palladium, (iii) at odd body centres, a sphere of Pd (radius 0.71a), (iv) at even body centres, a small sphere of Al (radius 0.3a) or an empty volume (the fit did not show differences between these two hypotheses since the Al volume involved is very small). Obviously, the proposed model does not obey the closeness and hard core conditions and, in particular, introduces unphysically short distances in real space due to the atomic surfaces being given a spherical shape. The next step in modeling therefore is to define faceted objects which would fit density, composition and the diffraction data. It is interesting to wonder how far from the 'true' solution is such a spherical model. All models must have a common 'hard core' which represent at least 80-90% of the total number of atoms. This is mainly due to the constraint imposed by composition and density. The 6-dim space is wide open but positioning atomic surfaces which reproduce the experimental atomic density and avoid short distances forces solutions in which atomic surfaces are in contact. This strongly restricts the degrees of freedom for modeling. Now, what are the 'basic' or 'hard core' features when generating the 3-dim structure as a

Ch. Janot / The structure of quasicrystals

862

visible• The densest planes are characterized" by large spacing. They are found perpendicular to a fivefold axis. Twofold planes are also visible but they are closer to each other and contain a smaller density of points. Two-dimensional cuts of the 3-dim structure, perpendicular to a fivefold axis and made at differing levels along this axis, are pictured in fig. 10. A systematic search of the very dense fivefold planes showed that only a finite number of different atomic planes has to be considered. Each type of plane is characterized by reasonably well defined local order and average Chemical composition. Some planes are corrugated and may be described as a dense layer sandwiched between two somewhat less dense layers 0.5 .~ apart.

cut through the 6-dim model? As for the AILiCu quasicrystal [27], we may seek local icosahedral clusters [28], or alternatively we can describe the structure in terms of dense atomic planes. Two types Of cluster are actually present in the structure, namely a pseudo-Mackay cluster type 1, with a large icosahedron of Mn + AI and an icosidodecahedron of Pd + AI, and a pseudoMackay cluster type 2, with a large icosahedron of Mn + Pd and an icosidodecahedron of AI. In both clusters, the inner small icosahedrort is replaced by fragments of an A1 dodecahedron. The existence of dense planes in the the 3-dim structure may be deduced from planar projections of the structure. When looking at such a projection at glancing angle (fig. 9), a series of lines corresponding to atomic planes is clearly •

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Ch. Janot / The structure of quasicrystals

6. Conclusion

The basics for structural studies of quasicrystals are now well established, but details of the atomic arrangements cannot be obtained from diffraction data. As far as properties are concerned, they have just emerged from their infancy. For a long time properties such as electrical resistivity, magnetism, vibrational density of state, mechanical parameters, etc., have been measured on poor quality quasicrystals. Unfortunately, the corrupting effects of quasiperiodicity departure are more dramatic than periodicity breakings in crystals. Thus, the resulting property data have been frustratingly untypical, ranging from bad crystal to amorphous-like behaviour. Presently, the experimental situation is being tremendously improved, although still lacking the proper use of the special selection rules that should result from long range quasiperiodic order.

7. Outlook

Last but not least, a coating of quasicrystals deposited by high temperature pulverization has ideal properties for a frying pan! (hardness, stability, thermal conductivity, non-toxicity, and non-sticking!) [29]. Sticking is related to wetting which is prevented if the surface tension of the coating is low. It is noteworthy that a low density of states at the Fermi level contributes significantly to lowering the surface tension. Thus quasicrystals seem to be a gift from the gods: they have unusual exciting structures, peculiar properties.., and they may be very useful in ordinary life!

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[3] P.A. Bancel, in: Quasicrystals: the State of the Art, ed. D.P. di Vincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 17. [4] T. Klein, C. Berger, D. Mayou and F. Cyrot-Lackmann, Phys. Rev. Lett. 66 (1991) 2907. [5] K. Kimura and S. Takeuchi, in: Quasicrystals: the State of the Art, ed. D.P. di Vincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 313. [6] B.D. Biggs, Y. Li and S.J. Poon, Phys. Rev. B43 (1991) 8747. [7] D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53 (1984) 2477. [8] L. Pauling, in: Extended Icosahedral Structures (Academic Press, 1989) p. 137. [9] P.W. Stevens and A.I. Goldman, Phys. Rev. Lett. 56 (1986) 1168. [10] C.L. Henley, in: Quasicrystals: the State of the Art, ed. D.P. di Vincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 429. [11] A. Janner and T. Janssen, Physica A99 (1979) 47. [12] P. Back, Phys. Rev. B32 (1985) 5763. [13] A. Katz and M. Duneau, Phys. Rev. Lett. 54 (1985) 2688. [14] R. Currat and T. Janssen, Solid State Phys. 41 (1988) 201. [15] Ch. Janot, M. de Boissieu, J.M. Dubois and J. Pannetier, J. Phys.: Condens. Matter 1 (1989) 1029. [16] M. Cornier-Quiquandon, H. Quivy, S. Lefebvre, E. Elkaim, G. Heger, A. Katz and D. Gratias, Phys. Rev. B44 (1991) 2071. [17] B. Dubost, J.M. Lang, M. Tanaka, P. Sainfort and M. Audier, Nature 324 (1986) 48. [18] A.P. Tsai, A. Inoue and T. Masumoto, Trans. JIM 29 (1988) 521. [19] A.P. Tsai, A. Inoue, Y. Yokoyama and T. Masumoto, Philos. Mag. Lett. 61 (1990) 9. [20] M. Boudard, M. de Boissieu, Ch. Janot, J.M. Dubois and C. Dong, Philos. Mag. Lett. 64 (1991) 197. [21] M. Boudard, M. de Boissieu, Ch. Janot, G. Heger, C. Beeli, H.U. Nissen, H. Vincent, R. Ibberson, M. Audier and J.M. Dubois, J. Phys: Condens Matter 4 (1992) 1. [22] Ch. Janot and M. de Boissieu, in: Quasicrystals: the State of the Art, ed. D.P. di Vincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 57. [23] Ch. Janot, Quasicrystals: a Primer (Oxford University, Oxford, 1992). [24] S. Ebalard and F. Spaepen, J. Mater. Res. 4 (1989) 39. [25] J. Devaud-Rzepski, A. Quivy, Y. Calvayrac, M. CornierQuiquandon and D. Gratias (1989) Philos. Mag. B60 855. [26] A.P. Tsai, A. Inoue and T. Masumoto, Philos. Mag. Lett. 62 (1990) 95. [27] M. de Boissieu, Ch. Janot, J.M. Dubois, M. Audier and B. Dubost, J. Phys.: Condens. Matter 3 (1991) 1. [28] M. Duneau and C. Oguey, J. Phys. (Paris) 50 1989) 135. [29] J.M. Dubois, S.S. Kang and J. Von Stebut, J. Mater. Sci. Lett. 10 (1991) 537.