Annu. Rev. Maler. Sci. 1991. 21:437-<i2 Copyright © /991 by Annual Reviews Inc. All rights reserved

QUASICRYSTALS

s. Ranganathan and K. Chattopadhyay

Centre for Advanced Study, Department of Metallurgy, Indian Institute of Science, Bangalore 560 012, India

KEY WORDS: icosahedral phases, electron microscopy, twins, rational approx­imants, Penrose tilings

INTRODUCTION

The imposition of translational symmetry on crystals restricts the allowed rotational axes of symmetry to two-, three-, four-, and sixfold axes. In 1984 this situation was dramatically changed by the discovery of a new ordered phase described by Shechtman et al (1). They showed that in rapidly solidified AI-Mn alloys a new phase appeared with its electron diffraction patterns displaying icosahedral symmetry, including the crys­tallographically disallowed fivefold symmetry. This finding was explained by the introduction of quasilattices, which possess quasiperiodic trans­lational order and icosahedral orientational order (2).

Since the original observation of Shechtman et al (I), the subject of quasicrystals (QC) has expanded in several directions. A number of new alloy systems displaying icosahedral symmetry has been identified. Ordered QC exhibiting a high degree of perfection and stability have been discovered. Other noncrystallographic symmetries such as decagonal, octagonal, and dodecagonal have been observed. There are now over 2000 publications, a bibliography (3), several reviews (4-7), edited volumes (8-10), and conference proceedings ( I I -IS) covering the field.

Even though the quasicrystalline model is favored, alternative models have also been advanced. These include en tropic quasicrystals, icosahedral glass, rational approximants and hypertwins. In this review we emphasize the interpretation based on the quasicrystalline model in a detailed fashion and include the other models in a comparative fashion. As electron microscopy has played a stellar role in these investigations, our focus is on electron diffraction and microscopic studies. In the concluding section, a unified approach is given. It is startling to note that while the five different models are based on different physical principles, they can all be derived

437 0084-6600/91/0801-0437$02.00

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438 RANGANATHAN & CHATTOPADHYAY

by an appeal to higher dimensional crystallography. The predictions of the different models come very close to experimental observations such that experimental distinction among them will often be difficult, if not impossible.

THREE-DIMENSIONAL QUASICRYSTALS

Following the observation of Shechtman et al (1), the icosahedral quasi­crystalline phase has been observed in a large number of aluminum­transition metal (AI-TM) binary and multicomponent systems (6, 16). In a classic investigation, Ramachandrarao & Sastry (17) emphasized the role of icosahedral coordination in crystalline alloys for the ultimate synthesis of quasicrystals by nonequilibrium processing and demonstrated its val­idity by synthesizing Mg-AI-Zn quasicrystals. Independent similar reason­ing led to the synthesis of Ti based QC by Kuo and his co-workers (18). Kelton et al (19) extended this work to Ti-Mn QC. Icosahedral QC have also been observed in the ternary system Pd-U-Si (20). Recently the occur­rence of this phase has been reported in iron-based systems as well (21). Although, in general, rapid solidification favors the formation of QC, it is by no means an essential requirement for synthesizing them. Stable icosahedral QC have been observed in Al-Cu-Li (22), Ga-Mg-Zn (23), and AI-Cu-Fe (24) systems under normal solidification conditions.

It is now clear that icosahedral QC can be treated as a phase with predominant icosahedral order similar to that of the Frank-Kasper phases, but having a quasiperiodic arrangement of atoms. Liquid to quasi­crystalline phase transformation is first order (1) and generally accounts for the majority of quasicrystalline transformations. QC can also be syn­thesized through other kinetic routes. For example, condensation from vapor (25), solidification under high pressure (26), de vitrification of glasses (27), solid state precipitation (28, 29), and interdiffusion (30) can lead to the formation of QC. In principle, all the techniques that are used for the synthesis of metastable crystalline and noncrystalline phases have the potential to produce qua sicrystals .

Figure 1 summarizes the different kinetic routes that can yield quasi­crystals. The formation of this phase depends upon the competition that the material faces from other phases during the process of nucleation and growth. This in turn is influenced by the process parameters. The observed range of existence of icosahedral phase in different systems in most cases reflects the kinetic competition with other phases rather than the thermo­dynamics dictated phase field of the metastable phase diagram. The fact that QC can be represented in a metastable phase diagram, however, indicates that a quasicrystal can form as a different transformation

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QUASICRYSTALS 439

CRYSTAL.� ______ ���� _______ � GLASS devdriflCalion

solid state amorphization

c Q iii c 0 0 0. "0 .. E "0 :; :0 0

r 0. 0 >

III '" c: ,,. e 41 III E III

.9 '"

VAPOUR condensat ion

------------- LIQUID � evaporation

Figure 1 A schematic diagram exhibiting different kinetic routes for transformations involv­ing a quasicrystalline phase.

product. For example it can appear as a primary phase from liquid (31), as an eutectic (32), or as a peritectic product (33). As indicated earlier, QC may occur as a stable equilibrium phase in some systems (22-24).

Transmission electron microscopy (TEM) has played a central role in the early study of these phases. The electron diffraction patterns (Figure 2) from these phases indicate icosahedral symmetry of the reciprocal space with prominent reflections arranged along five-, three-, and twofold sym­metry directions (1, 34). The point group symmetry of the patterns is m3 5. The spots along the twofold directions (even parity directions) can be obtained by r inflation (r = 1.618 . . . ) . r is the golden mean and is an irrational number arising from the Fibonacci series. One can reach the next spot in a given row of the diffraction pattern by multiplying the distance from the center to the spot by the number r. In the case of five­and threefold directions (odd parity directions) this factor is r3, which reflects the symmetry of the primitive icosahedron in reciprocal space. Quasicrystals exhibiting this feature are known as primitive icosahedral QC. Interestingly for certain quasicrystals, r inflation is valid also for odd parity directions. These QC are face-centered and body-centered and will be discussed below.

The morphology of the quasicrystalline phase depends upon the growth

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440 RANGANATHAN & CHATTOPADHYAY

Figure 2 Selected area electron diffraction patterns from the prominent zones of a single quasicrystal arranged along a unit Moebius triangle of a stereogram.

conditions and is similar to that of other crystalline stable and metastable phases. Additionally, certain distinctive features are worth noting. Bright field transmission electron micrographs often exhibit a characteristic speckle contrast. The origin of this contrast is not fully understood. The experiments to date tend to relate the chemical inhomogeneities that may have been frozen in during processing with this contrast (35). When quasi­crystals form as a primary phase from a liquid, they often evolve into shapes reflecting the inherent point group symmetry. Chattopadhyay et al (36) were the first to recognize that QC can exhibit faceted growth

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QUASICRYSTALS 441

morphologies. For example, the shape of the initially discovered AI­Mn quasicrystal is a pentagonal dodecahedron (37, 38), while AI-Cu-Li quasicrystal appears as a triacontahedron (39). Similar shapes were also found during solid state precipitation (28, 29, 40). Recently an icosi­dodecahedron shape was seen in an AI-Pd-Mn quasi crystal (41). The icosahedral quasicrystalline phase can grow in a twinned configuration (42, 43). The contrast in certain magnesium-based quasicrystals gives a sectorial appearance (44). In addition, growth morphologies involving icosahedral phase occasionally yield novel microstructures like the banded structure in AI-Mn alloys (45). Figure 3 shows select examples of a variety of observed morphologies of quasi crystalline phases.

Although a large number of primitive quasicrystals have been discovered with the signature of ,3 inflation along the odd parity directions, the details of the diffraction patterns from these phases (particularly the spot intensities) often differ significantly, thus indicating different atomic arrangements (46). Understanding the atomic structure from the diffrac­tion data remains one of the most challenging tasks for quasi crystal researchers. Developments in the field indicate the necessity of higher dimensional approaches to solve the problem of the structure of quasi­crystals. Various investigators have shown (47-51) that a truly quasi­periodic icosahedral quasicrystal can be regarded as a cut through a six­dimensional periodic space defined by a primitive hypercube lattice. Any reciprocal lattice vector [Q6] in six-dimensional space is given by

1.

where a is the hypercube parameter and et is the unit vector along the six­dimensional cube edge. On projection into three-dimensional physical space (real space) with icosahedral symmetry, the physical space vector is given by

2.

where P is the projection operator given by the matrix

J5

J5 - 1 - 1

J5 -1 -1 P = 1/J2· -1 J5 I -1

3.

-1 -1 J5

-1 -1 J5

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442 RANGANATHAN & CHATTOPADHYAY

Figure 3 Select examples of quasicrystalline morphologies: (top) Pentagonal snowflake

shape of AI-Mn quasicrystal embedded in an aluminum matrix. (Middle) Faceted quasi­crystals with dodecahedral morphology in Al-Fe-Cu alloy. (Bottom) Banded structure involv­ing the quasicrystalline phase in rapidly solidified AI-Mn alloys.

Defining the projection of the hypercube edge as the quasilattice constant, we get aR = a/.J2 where aR is the quasi lattice constant (47). The diffraction patterns can now be indexed in terms of six projected spanning vectors of the six-dimensional reciprocal lattice which, in this case, are identical to the icosahedral vertex vectors. Three schemes of indexing exist (52-54) that are based on the same principle of indexing with different scaling factors and modes of representation.

From the measurement of quasi lattice constants of all known icosa­hedral quasicrystals, it is clear that these can be classified basically into

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QUASICRYSTALS 443

two classes. All Al-TM quasicrystals have quasilattice constants � 0.46 nm, while for the Mg-AI-Zn and Al-Cu-Li quasicrystals the values are ",0.52 nm (55). Normalizing with atomic diameter, the values are'" 1.65-1.75 for Al-TM systems and �2.00 for Mg- and Li-containing quasi­crystals (4, 56). The Ti-based quasicrystals appear to fall into Al-TM class. The class of Pd-U -Si QC is uncertain.

The spot intensities in the diffraction patterns clearly bring out the differences between these two classes (46). It is now well established that the underlying atomic motif plays an important role in the atomic arrange­ments of quasicrystals. A simple introduction of two different motifs derived from the crystalline counterparts (Figure 4) (57, 58), namely the

oc:( on '" . oc:( � ':J/ ""

(0)

• Mn

o Ails; ( ouler)

n A.11 Si -' (inner)

MACKAY ICOSAHEDRON: Mn12(AI/S; )42

011.19

• Al /Zn (ouler)

<.) All Zn (inner)

(b) PAULING TRIACONTAHEDRON: 11.1920 (AljZn)24 Figure 4 Schematic diagram of the two prominent motifs that play an important role in quasicrystalline structures: (a) Mackay icosahedron and (b) Pauling triacontahedron.

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Mackay icosahedron for the AI-TM class of quasicrystals and the Pauling triacontahedron for Mg-AI-Zn class of quasicrystals, can largely account for the intensity differences (59). Following this approach, detailed cluster models have been developed to explain the atomic structure of icosahedral quasicrystals (60, 61). Attempts are also made to arrive at the actual structure from the experimental intensities by the hyperdimensional approach (62, 63).

ORDERED QUASICRYSTALS

Nelson & Sachdev (64), during the development of a density wave picture of quasicrystals, discussed the possibility of body-centered and face-cen­tered six-dimensional cubic lattices and computed the possible diffraction patterns using a Landau generation scheme. The possibility of such struc­tures was briefly mentioned by Chattopadhyay et al (36). Using a group theoretical approach, it was shown that only three types of Bravais lattices, namely primitive, body-centered, and face-centered, are possible for higher dimensional cubes except for the case of four dimensions where only simple and body-centered lattices are possible (65). The centered Bravais lattices can be disordered or may be related through an order-disorder trans­formation to a primitive cell. In the latter case, the projection of the ordered hypercube will yield an ordered quasicrystal.

The first experimental evidence of the possibility of an ordering reaction in a quasicrystal results from the work of Mukhopadhyay et al (66). These investigators reported the occurrence of diffuse intensities, which violate the r3 inflation along the odd parity direction. The diffuse arcs and the spots are arranged according to a r inflation scheme. This is precisely the diffraction pattern one expects from a face-centered quasicrystal whose hypercube reciprocal space will have body-centered symmetry. Soon after this observation, a stable ternary quasicrystal AI-Cu-Fe was discovered (24), which was shown to exhibit perfect face-centered ordering (67, 68). Figure 5a,b shows twofold patterns of a disordered and an ordered quasi­crystal. The corresponding computed pattern is shown in Figure 5c. High resolution electron micrographs of this phase show a high degree of order (69). Although the AI-Cu-Fe quasicrystal always appears in an ordered configuration, the possible existence of primitive to face-centered ordering reaction in quasicrystals (70) has been experimentally established for binary Al-Mn (71) and ternary AI-Cu-Mn and AI-Cu-Cr quasicrystals (72). The diffraction patterns could be successfully indexed by doubling the primitive quasilattice constant analogous to the crystalline ordering. Such an ordering is expected to give rise to domain boundaries of the type

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QUASICRYSTALS 445

F(qure 5 Twofold patterns from (a) primitive disordered and (b) face-centered ordered quasicrystals. A computed pattern from the ordered quasicrystal is shown in (c). The small stars are superlattice reflections, which are absent in the disordered quasicrystal.

[100000] expressed in terms of a primitive unit cell (73). The domain structures have been recently observed on imaging with the superlattice reflections (72, 73). The variety of electron diffraction patterns from face­centered icosahedral QC in AI-Cu-Fe has been examined and contrasted with that of primitive icosahedral QC (74). The ordered quasicrystals in the AI-Cu-Fe system is stable at higher temperature and is shown to fragment into an aggregate of a rhombohedral crystalline phase on cooling (75).

These observations have led Henley (56) to surmize that all AI-TM QC are ordered and face-centered icosahedral QC. Under suitable conditions they may be disordered and will appear as primitive icosahedral QC. It was also argued that Mg-based QC will not show such an order/disorder phenomenon. There are indications, however, that QC in AI-Li-Cu alloys have a propensity for ordering (76). Ti-based QC belong to the same class as AI-TM QC. The electron diffraction patterns from the former show arcs and diffuse intensity more prominently than the latter (77). They do not seem to evolve into an ordered QC, however, because the arcs and diffuse intensity are quite stable. It is intriguing to speculate that the order/disorder transformation in six-dimensional space mimics the ordered

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446 RANGANATHAN & CHATTOPADHYAY

and disordered structures found in a-AI-Mn-Si and a-AI-Fe-Si (S. Ran­ganathan, N. K. Mukhopadhyay, K. Chattopadhyay, submitted).

TWO-DIMENSIONAL QUASICRYSTALS

In 1978 Sastry et al (78) reported the existence of an unknown complex crystalline phase in a slowly cooled Al6oMnllNi4 alloy. Chattopadhyay et al (36, 79) were the first to report its occurrence in a rapidly solidified AI-14% Mn alloy and identify it as a quasicrystal with two-dimensional quasiperiodicity and one-dimensional periodicity. Figure 6 shows the characteristic electron diffraction pattern where the periodicity in the X­direction and quasiperiodicity in the V-direction are evident. Bendersky (80) independently arrived at the same conclusion and established its point group as lO/mmm using the convergent beam electron diffraction. As this quasicrystalline phase has a unique tenfold rotational axis of symmetry (Figure 6), it has been christened a decagonal quasicrystal. Subsequent studies showed that the space group P*105/mcm could be inferred for the decagonal phase in AI-Pd alloy (81). This space group incorporates a screw axis as well as a glide plane.

Two-dimensional quasicrystals with eight- and twelvefold rotational axes of symmetry, termed octagonal and dodecagonal quasicrystals, have been discovered subsequently. The octagonal quasicrystal occurs in rapidly

Figure 6 The stereogram indicates prominent zone axes for the decagonal phase. Patterns for the tenfold (A) and the twofold zone axes (F, G) are shown.

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QUASICRYSTALS 447

solidified V-Ni-Si and Cr-Ni-Si alloys (82). The dodecagonal quasicrystal has been observed in small particles of Ni-Cr (83) and in rapidly solidified V -Ni-Si alloys (84). Just as r is the characteristic length scale for icosahedral and decagonal phases, the quadratic irrationals J2 and J3 are related to octagonal and dodecagonal symmetry. Sasisekharan (85) has outlined a new method for the generation of quasiperiodic structures with n-fold axes. Levitov (86) has shown on the basis of energetics that only quadratic irrationals are allowed in quasicrystals and consequently non­crystallographic n-fold rotational axes are restricted to n = 5(pentagonal), 8(octagonal), lO(decagonal), and I 2 (dodecagonal). So far no pentagonal QC has been observed. The octagonal and dOdeCagonal QC are of rela­tively rare occurrcncc.

Decagonal Quasicrystals It is now well established that there is only one decagonal quasilattice (65). The decagonal phase is built up by stacking identical planes with equal spacing between them. In principle the decagonal phase can be indexed on the basis of five vectors. In order to exploit the full symmetry of the phase, however, a six index scheme is more convenient. Two possible schemes exist: the first helps to highlight the close relationship with the icosahedral quasicrystal and employs oblique axes (31,87, 88). The second system uses a basal set of five vectors placed 72° apart and an orthogonal c vector along the decagonal axis (89, 90). This system is similar to the Miller­Bravais indexing used for the hexagonal crystals. We follow the latter convention in this article.

It is of interest to derive the complete variety of electron diffraction patterns for the decagonal phase. Fitzgerald et al (91) extended the stereo­graphic analysis posited by Thangaraj et al (31). Figure 6 shows a stereo­gram in which zone axes are displayed as important intersections of relvec­tors, which correspond to the diffraction patterns (92). Daulton et al (93) have obtained the complete set of patterns predicted by Figure 6. The x­ray studies on single crystals of the decagonal phase by Steurer & Mayer (94), who established the point group symmetry as lO/mmm, are pertinent.

Decagonal quasicrystals form in a number of aluminum-transition metal alloys. The binary alloys are with manganese, iron, cobalt, nickel, ruthenium, rhodium, palladium, osmium, iridium, and platinum. While in all of these alloys decagonal quasi crystals form directly from the melt, in AI-Mn and AI-Ru they can also appear as a transformation product of the icosahedral phase. The discovery of the decagonal phase in AI-Fe is of special interest because the intense streaks in the patterns are parallel to the decagonal axis unlike in the AI-Mn and other decagonal phases where the streaks are parallel to the decagonal plane (95). The ternary

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alloys include AI-Mn-Ni, AI-Cr-Si, AI-Ni-Si, AI-Ge-Ni, as well as AI-Cu­Co (96).

AI-Cu-Co is of special interest. A decagonal quasi crystal was found in Al6sCu2oCo IS by He et al (97), the result of either slow solidification or long annealing at 800°C. Single crystals have been grown to mm size, particularly when Si is added to the alloy. The decagonal quasicrystal shows a decaprismatic morphology as reported by Kortan et al (98) and He et al (99). The electron diffraction patterns are also sharp, which indicates the high perfection of the phase. It also appears to be an equi­librium phase. Recently the atomic configuration of this decagonal phase has been imaged by the scanning tunneling microscope. The images appear to support the perfect tiling model (100). HREM studies ( lO 1) on the same alloy, however, were able to sample a larger region. The observed structure was more consistent with a random tiling than a perfect Penrose tiling. These tiling schemes are described below.

An important discovery relates to the variable periodicity along the decagonal axis, as was shown independently by Li & Kuo (102) and Menon et al ( lO3). Four periodicities of 0.4, 0.8, 1.2, and 1.6 nm have been shown to exist in AI-Co alloys. These correspond to repeating sequence of 2, 4, 6, and 8 decagonal planes. As 0.4 nm appears to be a basic repeating unit, these phases may be termed as T2, T4, T6, and Ts (in general as T 2n where n refers to the repeating unit). They can be viewed as a superstructure of the fundamental repeating unit (l00). This phenomenon appears to resemble polytypism and would predict a number of additional periodicities (102). MandaI & Lele (104) have considered projection from six-dimensional space to generate the decagonal structure. By extending this, it is predicted that there will be exactly seven distinct structures of the decagonal phase corresponding to n = 1 to 7 (105).

ONE�DIMENSIONAL QUASICRYSTALS

Two types of one-dimensional quasicrystals have been reported. The first type, which may be termed as a trigonal QC, occurs as the incommensurate limit of the vacancy ordered phases, as first pointed out by Chattopadhyay et al (106). The second type, which has been christened a digonal quasi­crystal by R. K. Mandai (private communication), was discovered by He et al (107) as a transformation product of the decagonal QC. The trigonal quasicrystal displays quasiperiodicity along a single threefold direction and periodicity along all other directions. In a similar fashion, the digonal quasicrystal shows a quasi periodicity along a single twofold axis and a periodicity along other directions. Since these phases occur as co-existing phases with other quasicrystals and possess orientation relationships with

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QUASICRYSTALS 449

them, further studies of these phases will enhance our understanding of quasicrystals.

In addition to these two cases, one-dimensional quasiperiodicity is of interest in artificially grown materials through molecular beam epitaxy (108). The electronic states in such one-dimensional QC are a matter of intense theoretical and practical interest. Sutton (109) discussed the struc­ture of grain boundaries. For irrational orientations of the boundary it is appealing to consider the structural units of the boundary arranged in a quasiperiodic fashion. Rivier (110) has also discussed the structure of grain boundaries in terms of quasiperiodic units.

Trigonal Quasicrystals Vacancy-ordered phases (yOP) , known as " phases, occur in AI-Cu-Ni alloys and were studied by Lu & Chang by X-rays (111) and by van Sande et al by TEM (112). After the discovery of QC, Chattopadhyay et al (106) re-examined the structure of these phases. Noting the close similarity in the arrangement of spots and their intensities with those of decagonal and icosahedral phases, they showed that the vacancy-ordered phases can be considered as a series of rational approximants converging to one­dimensional QC.

The" phases can be described with reference to a slightly deformed CsCI type structure, in which the cubes have become rhombohedra, thereby introducing a unique [I I I] axis, which becomes the rhombohedral axis of the resulting structure. The most important section of the reciprocal space is the [ lTO]CsCI zone with [111]* relvector. This reciprocal direction directly reveals the stacking sequence along the threefold axis and is divided into 2, 3, 5, 8, and 13 spots for "2, "3, "5, "8, and "i3 VOP. van Tendeloo et al (113) identified" 2i and" 34 phases. In addition they found" i8, "3 b and" 38 phases, which were demonstrated to belong to a new Fibonacci series. They have also argued that the limiting structure corresponds to an irrational modulated vector, and VOPs can thus be considered as rational approximants of one-dimensional QC.

Digonal Quasicrystals An interesting point concerns the two periodic axes showing twofold symmetry in the digonal phase (107). One of them is derived from the tenfold axis of the decagonal phase and inherits the periodicity of the decagonal phase. Thus for the T 6 phase the diffracted spot corresponding to a direct space periodicity of 0.21 nm is divided into six parts. Along the twofold axis perpendicular to that, the direct space periodicity of 0.3 nm is divided into five parts. Zhang & Kuo (114) conjecture that for the

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digonal phase corresponding to T 8, the division will be in eight parts. If the theory from R. K. Mandai & S. Lele (unpublished) is followed, then we may expect seven digonal phases corresponding to the seven decagonal phases. The former will show divisions of 13, 8, 5, 3, 2, 1, and 1 cor­responding to the range T 2 to T 14. Experimental observation of the deca­gonal phase T 1 0, T 12, and T 14 as well as the corresponding digonal phases will be rewarding.

VARIATIONS ON TILING SCHEMES

In our understanding of quasiperiodicity, Penrose tilings ( liS) play an important role. For the sake of simplicity we use a two-dimensional case. Figure 7a shows an acute and an obtuse rhombus arranged according to strict matching rules. It has been shown that such a deterministic tiling is perfect and will have the lowest energy. Nevertheless, there are serious questions as to how quasicrystals grow in reality.

Elser (116) was the first to draw attention to the possibility of local violation of the matching rules. In such instances we may expect the stabilization of the quasicrystalline configuration by an increase in entropy. The random tiling model or en tropic quasicrystal uses the same elementary tiles, but the matching rules are violated in some places (Figure 7b). The unique restriction implics that the neighboring vertices are linked by bonds along symmetry directions occurring with equal frequency (117, 118).

An alternative approach for the structural model of icosahedral phase was first suggcsted by Shechtman & Blech (119) almost simultaneously with the discovery of icosahedral phase ( I ). They considered that the ico­sahedra were joined in a random fashion, albeit with the preservation of the orientational order. This model was further developed by Stephens & Goldman (120), who coined the term icosahedral glass. Starting from an initial seed, icosahedra are successively attached to a randomly chosen vertex, provided each new icosahedron does not overlap with another already present in the array (Figure 7c). The structure factor of the vertex­sharing icosahedral packing model is in good agreement with the observed width and intensity of the diffraction patterns. Random-packing models are inherently disordered, thus leading to a finite width for the peaks. Stephens (121) reviewed this model and made a detailed comparison of high resolution electron micrographs with images provided by the model. It is fascinating that even such a random array can grow and give rise to a form with facets. Chidambaram et al (122) have used positron annihilation technique to study the crystallization of QC. Their results lend support to the icosahedral glass model.

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QUASICRYSTALS 451

Figure 7 (a) Perfect Penrose tiling and (b) random tiling made of the same acute and obtuse rhombuses (courtesy, F. Spaepen); (c) icosahedral glass model (courtesy, P. W. Stephens & A. I. Goldman).

DEFECTS IN QUASICRYSTALS

The atomic arrangement in three-dimensional quasicrystals is ordered and is periodic in a six-dimensional framework. Thus the normal defects observed in crystalline structures can be extended to the hypothetical six­dimensional periodic structure. The projection of these defects in three­dimensional real space yields the possible description of defects in quasi­crystals. The projected defects, however, will have components in three­dimensional perpendicular space orthonormal to the real space. Therefore the character of the defects is different from that of the normal crystalline

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452 RANGANATHAN & CHATTOPADHYAY

counterpart. For example, the phonon defects in crystalline materials, which reflect the displacement of atoms from lattice sites, yield phonons and, in the six-dimensional framework, a ph as on defect in real space. Due to orthonormal components, phasons are less mobile. Their movement involves a diffusive rearrangement of atoms. The theory of hydrodynamic modes of deformation of quasicrystals is now well developed (123-125). Kim et al (126) documented the deviation from strict icosahedral symmetry in rapidly solidified AI-Cu-Li-Mg QC. The unusual peak broadening as well as shift and shape change of the diffraction spots often observed in quasicrystalline diffraction patterns can be quantitatively analyzed by the phason strain (127). The present excitement following the discovery of the ordered quasicrystals is due to the fact the phasons are mobile in these quasicrystals (128).

The concept of dislocations can be extended to quasicrystals in a similar fashion (129, 130), and these defects are actually observed (131, 132). The task of experimental characterization of these defects has just begun. The conditions for the invisibility of these defects include an orthonormal component. A contrast theory for imaging such dislocations using higher dimensional formalism has been attempted (133). A description of low angle boundaries involving these defects has also been proposed (134). Direct lattice imaging has provided evidence for the quasicrystalline dis­locations and boundaries (135). Antiphase domains have been observed in AI-Cu-Fe QC (68, 73). There is no doubt that the study of defects in quasicrystals will increasingly find attention among the researchers in the near future.

RATIONAL APPROXIMANTS

Even though pure metals tend to crystallize in extremely simple structures, alloying can lead to the formation of intermetallic compounds with com­plicated structures. In his classic book, Pauling (136) has drawn attention to the frequent occurrence of an icosahedral motif in several alloy phases. An extensive review of these structures has been given by Shoemaker & Shoemaker (137).

The occurrence of an icosahedral motif leads to the modulation of diffraction patterns leading to the appearance of pseudo-icosahedral sym­metry. Two crystal structures have drawn particular attention in this context: ct-AI-Mn-Si and T-Mg32 (ZnAI)48. Elser & Henley analyzed the occurrence of Penrose rhombohedra in them, and they showed that ct-AI­Mn-Si and T-Mg32 (ZnAI)48 structures in particular can be developed by a periodic stacking of oblate and prolate rhombohedra (57, 58). It is possible to derive these structures from higher dimensional space by pro-

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QUASI CRYSTALS 453

jection on a rational plane. Effectively it involves replacing the golden mean T by its rational approximant p/q where p and q are consecutive integers in the Fibonacci series. It is clear that as p/q approaches T the unit cell size becomes larger and larger. Experimental distinction between the quasicrystal and rational approximant will become progressively difficult (138). Even though preliminary field-ion investigations were in favor of the quasi crystalline model, Fowler et al (139) found close agreement with the simulated images from a 3/2 rational approximant. 3/2 rational approx­imants have been reported in Mg-Zn-Al alloys (140) and Mg-Zn-Ga alloys (141). In a remarkable set of experiments Levine et al (142) analyzed the occurrence of rational approximants in a Ti-Mn-Si system. They reported a structure analogous to that of ct-Al-Mn-Si . This is a significant contribution because it helps place Ti-based QC in the same class as AI-TM QC. It is also clear that for the AI-Cu-Fe icosahedral phase, a rhombohedral crystalline approximant can be obtained (75).

Pauling (143) indicated the possibility of clusters larger than those pre­sent in ct-AI-Mn-Si and T-Mg-Zn-Al alloys by showing that multiply icosahedral complexes can be formed. It will be fruitful to gather experi­mental evidence for these new postulated structures.

Several approximants for the decagonal phase have been identified. The long known Al13Fe4 and the little studied Al13Mn4 and Al60Mn II Ni4 are particularly appealing. Kumar (144) drew attention to the similarity in the structural units in Al13Fe4 and the decagonal phase. van Tendeloo et al (145) carried out an extensive study of quasicrystals and crystals in the A160MnllNi4 ternary alloy. Under slow solidification conditions this alloy gives rise to crystals with diffraction patterns bearing a close resemblance to those from the decagonal phase. A third candidate is an A13Mn ortho­rhombic phase studied by Fitzgerald et al (91). Daulton et al (93) extended this study to obtain the Kikuchi band maps and showed excellent agree­ment in the patterns from Al3Mn and the decagonal phase. van Tendeloo et al (146) showed that in an AI-Cu-Mn alloy similar phases occur and are twinned as well, thus leading to striking similarity in patterns (Figure 8). Reyes-Gasga & Jose-Yacaman (147) discussed the occurrence of quasi­crystalline phases and rational approximants to the decagonal phase obtained from vapor-deposited AI-Mn thin films.

For the digonal phase, All3Fe4 appears to be a rational approximant, while the r phases or VOP are to be regarded as a series of rational approximants to the trigonal QC. In a similar fashion, rational approx­imants have been postulated for the octagonal phase and the dodecagonal phase (148-150).

Gratias (151) demonstrated that the variation of the orientation of the strip and projection space in the strip-projection formalism can lead to

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454 RANG ANA THAN & CHATTOPADHYAY

Figure 8 High resolution electron micrograph from Al-Cu-Mn alloy, where AI3Mn, the rational approximant to the decagonal phase, is present (courtesy, G. van Tendeloo).

different structures. Quasiperiodic structures are obtained when the orien­tations of the strip and the projection space are both irrational. Kulkarni (152) and Torres et al (153) have considered the case when the orientation of the projection space is rational and that of the strip is irrational. This gives a crystalline lattice with a quasiperiodic superJattice.

HYPERTWINS

Until the concept of quasiperiodicity was developed, observations of diffraction patterns with noncrystallographic symmetry were invariably ascribed to twinning. Thus, it is not surprising that Field & Fraser (154) in an almost simultaneous publication announcing the discovery of quasi­crystals (1) suggested twinning as the cause for the observed fivefold symmetry. Pauling (ISS) postulated the existence of multiple-ordered twins, termed them icosatwins and decatwins, and suggested new crys­talline structures with large unit cells. For example, Pauling interpreted (156) the X-ray diffraction patterns of rapidly solidified Mg-Zn-AI alloy reported by Rajasekharan et al (157) in terms of a large unit cell structure. In a variation of these twinning possibilities, Anantharaman (158)

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QUASICRYSTALS 455

advanced the formation of unit cells with tetragonal and orthorhombic symmetry. All these theories assumed rational twinning elements.

In a remarkable paper Bendersky et al (159) suggested the possibility of irrational twinning elements and called them hypertwins. They reported on the occurrence of polycrystalline aggregates with overall icosahedral symmetry in rapidly solidified Al-Mn-Fe-Si alloys. The crystals occur in five variants and are obtained by a fivefold rotation around an irrational [1 rO] axis. This orientation, however, leaves the icosahedral motif invariant. Thus in the entire aggregate the icosahedral motifs in all of the crystals are parallel. The twin boundaries that separate the variants disrupt the translational symmetry in the aggregate, but allow the uninterrupted propagation of the orientational order. Thus the aggregate offive variants exhibits non-crystallographic symmetry. Bendersky et al (159) supported their postulate with experimental electron diffraction patterns showing five-, three-, and twofold symmetry.

It is pertinent to point out that similar observations have been reported earlier in AI-Mn-Si alloys by Koskenmaki et al (160). They, however, did not offer any detailed interpretation. While his earlier descriptions of twinning seemed to vary, Pauling (156) in a review of his own work described the same twinning mode as that advocated by Bendersky et al (159).

Lalla et al (161) made a contribution to this field through their study of hyper twins in a rapidly solidified AI-Mn-Ge alloy. They emphasized the pseudo-icosahedral symmetry of the twinned aggregates. They addressed the question relating to the strict parallelism of motifs across the twin boundaries. This is possible only if the icosahedral motif within the cubic unit cell is undistorted. The same problem that is basic to quasicrystals appears to trouble the strict icosahedral symmetry claimed by Bendersky et al (159). As Shoemaker & Shoemaker (137), among others, have shown, the local icosahedral motifs occurring in crystals have only approximate icosahedral symmetry, the ideal icosahedral symmetry being broken by the requirements of the crystal lattice and the associated physical forces that must distort the icosahedron at least very slightly. Thus the icosahedral motif in different variants will only be approximately parallel. When the twin size is very small, the deviation may not be as noticeable. As the twin size increases, the deviation from strict icosahedral symmetry will be more noticeable in the electron diffraction patterns.

As discussed above, the variety of electron diffraction patterns from icosahedral quasicrystals has been elucidated by Chattopadhyay et al (34). It has been possible to obtain patterns at equivalent zone axes for the crystalline aggregates in AI-Fe-Mn-Si (R. K. Mandai, unpublished) and Al-Mn-Cr-Si (Figure 9; A. K. Srivastava, unpublished). Bendersky et al

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456 RANGANATHAN & CHATTOPADHYAY

Figure 9 (Left) Twofold (top), threefold (middle), and fivefold (bottom) diffraction patterns from hypertwins in a Al-Mn-Cr-Si alloy (courtesy, A. K. Srivastava). (Right) Corresponding computed patterns (cour­tesy, L. A. Bendersky).

(159) have demonstrated that specific crystalline aggregates can show icosahedral, decagonal, octagonal, and dodecagonal quasicrystalline sym­metry. It is possible to extend this argument to demonstrate that quasi­crystalline aggregates of a lower symmetry can display a higher symmetry in the composite pattern.

UNIFIED VIEWPOINT OF QUASICRYSTALLINE

AND RELATED PHASES

Figure 10 shows the relationship among the various quasicrystalline and related crystalline phases. This picture is incomplete since current and future investigations may uncover additional possibilities.

Cahn et al (54) were the first to emphasize the group-subgroup relation between crystals and quasicrystals. Ishii (162) has shown that the modu­lated structures of icosahedral quasicrystals can be regarded as lower symmetry phases where the global icosahedral symmetry is broken because of mode locking of phason degrees of freedom. Group theory can be used in an elegant fashion to derive all possible modes. The maximal subgroups are Ds (pentagonal symmetry 3m), D3 (trigonal symmetry 3m), and T (tetrahedral symmetry m3) achieved by suitable locking of phason fields.

Mai et al (163) derived the anisotropic linear phason strain matrices,

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Decagonal QC

10/mmm

AI-Cu-Co:\ -'lj

1 Pentagonal QC

8m

-' Dlgonal QC

222

AI-Cu-CO'�-D • 1

1 RAS

[AluFe, AhMn AI.oM" "NI 4

Icosahedral ac

- -

mS5

QUASICRYSTALS 457

AI-Mn-SI AI-Cu-Fe Mg-Zn-AI

/j�----Cubic RAS

rn3 a-AI-Mn-SI

T-Mg-Zn-AI

Hypertwlna - -

mSS

AI (Mn,Fe,cr)

(SI,Ge)

�

Trigonal RAS

3m

AI-Cu-Fe

Trigonal QC

3m 'teo

� VOP

-

Sm

'ts 't"s 'l:a

I

I B2 C8CI I F�qure 10 Symmetry breaking leading to lower dimensional quasicrystals and crystals from the icosahedral quasicrystalline phase. Some typical examples are included.

which convert the quasicrystalline state to the crystalline state. This appears to be equivalent to a rotation of the strip in six-dimensional hyperspace (164). The third possibility is to consider the distortion of a hypercubic crystal. The axial lengths are changed, but the axes remain orthogonal. MandaI & Lele 004, 105; unpublished work) have followed this route to illumine the relationship among the phases shown in Figure 10. This is equivalent to a distortion of the spanning vectors of the icosa-

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458 RANGANATHAN & CHATTOPADHYAY

hedron, and is an elegant and powerful approach since all the phases­crystalline as well as quasicrystalline-can be indexed on the same six index notations.

Ranganathan and co-workers (34, 74, 92, 140; unpublished work) have shown that the stereographic approach can be used to delineate the Kikuchi maps corresponding to all of these possibilities. The predicted diffraction zone axes again serve to bring out the relationship.

The intimate relationship among these phases is brought out in real experimental systems such as AI-Cu-M, where M is a transition metal (165). Table I depicts a systematic change in the observed phases for rapidly solidified, conventionally solidified, and annealed conditions. AI­Cu-Ti and AI-Cu-V give rise to an amorphous phase. In the latter alloy, a quasi crystal is produced on annealing. AI-Cu-Cr and AI-Cu-Mn give rise to icosahedral QC, which transform to ordered QC on annealing. AI­Cu-Fe, AI-Cu-Ru, and AI-Cu-Os give rise to stable icosahedral QC. AI­Cu-Co and AI-Cu-Rh give rise to stable decagonal phases. In AI-Cu-Co, it is also possible to produce icosahedral and digonal QC. AI-Cu-Ni gives rise to trigonal QC. It is possible to explain the occurrence of these phases on the basis of the electron/atom ratio (165).

The study of quasicrystals has opened new vistas in materials science. The importance of motif, its ability to swamp the effects of the lattice, the types of close-packed clusters of metallic atoms, and the ways in which

Table 1 Quasicrystals and related crystals in Al-Cu-M ternary alloys

M, Rapidly As cast or transition solidified annealed

metal alloys alloys ---�-

Ti Glass Crystal V Glass Face-centered

icosahedral QC Cr Primitive F ace-centered

icosahedral QC icosahedral QC Mn Primitive F ace-centered

icosahedral QC icosahedral QC Fe Face-centered StableQC

icosahedral QC CO Decagonal QC Stable

decagonal QC; digonal QC

Ni Trigonal QC Vacancy-ordered phases

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QUASICRYSTALS 459

such clusters can pack together offer fresh insights into the formidable problem of space filling. Intensive studies of complex crystalline com­pounds with thousands of atoms per unit cell have been a dividend. Several unresolved issues in this field are an invitation for further research.

ACKNOWLEDGMENTS

We wish to acknowledge stimulating discussions with Professors J. W. Cahn, K. H. Kuo, and S. Lele and research contributions of Dr. N. Thangaraj, Dr. N. K. Mukhopadhyay, Mr. A. Singh, and Mr. A . K. Srivastava. We thank Professors C. N. R. Rao, P. Rama Rao, and B. B. Rath for encouragement. The financial support from the Department of Science of Technology, Government of India and the Office of Naval Research, USA under the Indo-US cooperative research program is grate­fully acknowledged.

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Annual Review of Materials Science Volume 21,1991

CONTENTS

EXPERIMENTAL AND THEORETICAL METHODS

Electron Spin Resonance (ESR) Microscopy in Materials Science, Motoji Ikeya 45

NMR Methods for Solid Polymers, H. W. Spiess 131

Metastable Growth of Diamond-like Phases, John C. Angus, Yak in Wang, and Mahendra Sunkara 221

The Characterization of Polymer Interfaces, T. P. Russell 249

The Adhesion Between Polymers, H. R. Brown 463

Study of Sol-Gel Chemical Reaction Kinetics by NMR, Roger A. Assink and Bruce D. Kay 491

Air/Liquid Interfaces and Adsorbed Molecular Monolayers Studied with Nonlinear Optical Techniques, Viola Vogel and Y. R. Shen 515

PREPARATION, PROCESSING, AND STRUCTURAL CHANGES

Containerless Undercooling and Solidification of Pure Metals, D. M. Herlach 23

Microstructure and Mechanical Properties of Electroless Copper Deposits, S. Nakahara and Y. Okinaka 93

Atomic Layer Epitaxy of I1I-V Electronic Materials, A. Usui and H. Watanabe 185

Self-Heating Synthesis of Materials, J. B. Holt and S. D. Dunmead 305

Molecular Films, J. D. Swalen 373

Chemical Processes Applied to Reactive Extrusion of Polymers, S. Bruce Brown 409

Growth and Characterization of Diamond Thin Films, Robert J. Nemanich 535

PROPERTIES AND PHENOMENA

Nanophase Materials, R. W. Siegel 559

SPECIAL MATERIALS

Defects in Hydrogenated Amorphous Silicon, K. Winer

VI

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CONTENTS (continued) Vll

Crystal Chemistry of Oligophosphates, M. T. Averbuch-Pouchot and A. Durif 65

Molecular Composites and Self-Reinforced Liquid Crystalline Polymer Blends, G. T. Pawlikowski, D. Dutta, and R. A. Weiss 159

The Palladium-Hydrogen System, Ted B. Flanagan and W. A. Oates 269

Thermodynamic Considerations in Superconducting Oxides, R. Beyers and B. T. Ahn 335

STRUCTURE

Quasicrystals, S. Ranganathan and K. Chattopadhyay 437

INDEXES

Subject Index 579 Cumulative Index of Contributing Authors, Volumes 17-21 589 Cumulative Index of Chapter Titles, Volumes 17-21 591

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