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Quasicrystals: A fundamentally new phase of solid matter exhibits symmetries that areimpossible for ordinary crystalsAuthor(s): Paul Joseph SteinhardtReviewed work(s):Source: American Scientist, Vol. 74, No. 6 (November-December 1986), pp. 586-597Published by: Sigma Xi, The Scientific Research SocietyStable URL: http://www.jstor.org/stable/27854355 .Accessed: 20/12/2011 12:38

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Quasicrystals Paul Joseph Steinhardt

A fundamentally new phase of solid matter exhibits symmetries that are impossible for ordinary crystals

For nearly two centuries, physicists have believed that

every pure solid is either crystalline or glassy. Now it

appears that this view must be changed. A recent theory proposes a new phase of solid matter called a quasicrys tal, and the independent experimental discovery of a new metallic alloy appears to be an example of just such a phase. The breakthroughs suggest the possibility of a new class of materials with physical properties that

distinguish them from all previously known solids.

Perhaps the most basic principle of solid state

physics is that a solid is composed of atoms packed in a dense arrangement and that the ordering of the atomic

arrangement determines many properties of the solid. The atomic arrangement in a solid can be compared to a mosaic. Atoms or clusters of atoms appear in repeating motifs called unit cells, which are analogous to the tiles

Paul J. Steinhardt is Professor of Physics at the University of Pennsylvania. He received his Ph.D. from Harvard University and mas a Junior Fellow in

the Society of Fellows at Harvard. In addition to his work on quasicrystals and glassy materials, he has done extensive research in elementary particle

physics and cosmology, where he is known as one of the co-inventors of the

"new inflationary" model of the very early universe. Address: Department of

Physics, David Rittenhouse Laboratory, University of Pennsylvania,

Philadelphia, PA 19104-6396.

in a mosaic. The order in an arrangement of atoms (or in a mosaic) is determined by the way in which the unit cells (or tiles) are joined to form the complete structure.

Crystals have highly ordered atomic arrangements in which all the unit cells are identical, analogous to a trivial mosaic constructed from a single tile shape, as in

typical bathroom tiling (see Fig. 1). A single atomic duster or unit-cell shape is repeated periodically (with equal spacing between cells) to form the structure.

Crystals have positional order: given the position of one unit cell, the positions of all the other unit cells are determined. Crystals also have orientational order: given the orientation of one unit cell, the orientations of all the other unit cells are determined.

The orientational order can be characterized in terms of a rotational symmetry. A special set of discrete rotations leaves the orientations of the unit cells un

changed. According to the well-established, rigorous theorems of crystallography, only a small list of rotation al symmetries is possible for crystals: crystals can have twofold, threefold, fourfold, or sixfold axes of rotational

symmetry; no other possibilities, such as fivefold, seven

fold, or eightfold symmetry, are allowed. This corre

sponds to the observation that one can tile a bathroom wall using a single tile shape if the tiles are all rectangles,

586 American Scientist, Volume 74

triangles, squares, or hexagons, but not if they are all

pentagons. (A "crystal" tiling can also be constructed if the tile shape is a parallelogram, in which case the

analogous crystalline lattice exhibits no rotational sym metry.)

In contrast to crystals, glass has a highly disordered atomic arrangement. A glass is typically formed by rapidly cooling a vapor or liquid well below its freezing point until the atoms are frozen into a dense but random

arrangement. Window glass, the most common exam

ple, is formed by silicon and oxygen joined in a random network of covalent (directed) bonds. Physicists have succeeded in rapidly cooling various mixtures of metal atoms to form "metallic" glasses. In this case, there are no preferred directions for the bonds, and the atoms are

packed in a dense but random arrangement.

A new source of two-dimensional designs

Quasiperiodic structures contain two or more shapes used over and

over in a predictable but subtle sequence that never quite repeats. Once the principles are understood, it becomes possible to generate

any number of patterns with unusual types of symmetry in two,

three, or more dimensions. In two dimensions, it is convenient to

think of quasiperiodic structures as mosaics that tile a surface using

only a small number of different shapes. Some of the new tilings have, in fact, been patented for commercial use.

The tiling shown above has ninefold rotational symmetry and

is constructed from four tile shapes. On the facing page, the tiling on the left, constructed from three shapes, has sevenfold symmetry; the tiling on the right, constructed from two shapes, has eightfold

symmetry. The method of producing these tilings can be general ized to three dimensions, leading to quasicrystalline structures

with symmetries forbidden for ordinary crystals. (All color photo

graphs by the author, from computer-generated video displays.)

1986 November-December 587

A glass is analogous to a mosaic formed from an infinite number of tile shapes randomly joined together. The concept of a unit cell is normally not used in this case, since there is no well-defined scheme for dividing the atoms into infinitely many unit-cell shapes. The structure has neither positional nor orientational order: the position and orientation of one unit cell does not determine the position, orientation, or shape of others a distance away.

Recently, the possibility of a new class of ordered atomic structures has been proposed by Levine and Steinhardt (1984), based on a detailed study of a special two-dimensional tiling pattern discovered by Penrose

(1974) some ten years earlier. The new structures are

analogous to mosaics with more than one tile shape but

only a finite number of shapes; the opening pages of this article show several examples. Although the structures have positional order, the tiles are neither periodically nor randomly spaced; instead, they are quasiperiodically spaced. This means that, given the position of one tile, the positions of the other tiles are determined according to a predictable but subtle sequence which never quite repeats. The new structures also exhibit orientational order: each tile of a given shape is oriented along one of a

small, discrete set of special directions. The rotational

Shapes that can be joined to form periodic (crystal-like) tilings

Pentagons cannot be Random (glasslike) joined periodically tiling

symmetry is defined by the set of rotations which leaves the set of orientations for each of the different tile shapes unchanged. Because the new structures are highly or dered like crystals but are quasiperiodic instead of peri odic, they have been called quasiperiodic crystals, or

quasicrystals for short. The most striking property of quasicrystals is that,

because they are not periodic, they are not subject to the

rigorous theorems of traditional crystallography. Quasi crystals can have any rotational symmetry in any num ber of dimensions. In particular, quasicrystals can have fivefold symmetry axes even though this is impossible for crystals.

In a remarkable coincidence, just as these theoretical notions were being developed, Shechtman, Blech, Gra tias, and Cahn (1984) were independently studying a

puzzling new alloy of aluminum and manganese which,

they discovered, had fivefold symmetry axes. The new material was discovered accidentally as part of an exten sive survey to develop lighter and stronger aluminum

alloys. It was formed by a method known as melt

spinning, in which a hot liquid alloy mixture is sprayed onto a cold, spinning wheel so that the liquid rapidly solidifies. The alloys being studied by Shechtman and his colleagues cooled into long strips of metal. For an

appropriate mixture of aluminum and manganese, the

strips contained tiny "grains," about 10 \ across, within which appeared to be a homogeneous material.

To determine the atomic structure of the new mate rial, Shechtman and his colleagues used a technique called electron diffraction analysis. They aimed a beam of electrons at a single grain of the alloy and recorded the

pattern formed when the electrons scattered off the atoms in the material and struck a photographic plate. For a crystal, it is well known that the electrons scatter

coherently from the positionally ordered array of atoms to form a "diffraction pattern" of sharp spots on the

plate. The pattern of spots depends on the symmetry of the crystal and its orientation with respect to the electron beam. For a glass, the electrons scatter off an isotropie, disordered array of atoms to form a diffraction pattern of diffuse rings which is the same for all orientations.

Figure 1. The atomic structure of a solid can be compared to a

mosaic in which different tile shapes represent different atoms or

clusters of atoms. A crystal is like a mosaic constructed from a

single tile shape: each atom or cluster of atoms is repeated

periodically, that is, at regular intervals. A periodic arrangement of

atoms can exhibit several axes of rotational symmetry; a rotation

about one of these axes by certain specified angles produces structures in which the atoms or clusters have equivalent orientations. A structure that repeats itself after one-fourth of a full

rotation about an axis, for example, has fourfold rotational

symmetry. These degrees of symmetry correspond to the shapes of

the tiles in a mosaic. Only special symmetries are possible for

crystals (top three rows), corresponding to tilings constructed from

closely packed rectangles (twofold symmetry), triangles (threefold), squares (fourfold), hexagons (sixfold), or parallelograms (periodic but no rotational symmetry). A fivefold symmetry is not possible for periodic crystals, just as the analogous mosaic cannot be

constructed using pentagonal tiles without introducing unwanted

gaps (the diamond-shaped white areas, bottom left); the tiles cannot

fill the plane in a connected array. A glass is analogous to a mosaic

constructed from tiles with random shapes, positions, and

orientations (bottom right).

588 American Scientist, Volume 74

For the aluminum-manganese alloy, a pattern of

sharp spots was found which clearly indicated a fivefold

symmetry axis (Fig. 2). By rotating the sample in the electron beam, it was found that the material had many fivefold symmetry axes, as well as threefold and twofold

symmetry axes. By noting the angle between the sym metry axes, it could be shown that the material had a three-dimensional icosahedral symmetry, one of the

most familiar examples of a disallowed crystallographic symmetry.

The icosahedron is one of the five regular polyhedra that are referred to as Platonic solids. The word icosahe dron means "twenty faces"; the icosahedron has twenty identical triangular faces, thirty edges, and twelve verti ces (Fig. 3). The black pentagons on the surface of a soccer ball are centered on the vertices of an icosahe dron. Each of the vertices lies on one of six fivefold

symmetry axes which connect opposite vertices. Because of the fivefold symmetry axes, icosahedral symmetry is disallowed for crystals. In particular, icosahedra cannot be packed so as to fill space completely, just as penta gons cannot be joined to form a complete tiling of a

plane. As a result, there has been relatively little interest in icosahedra among solid state physicists until recently.

Although the diffraction patterns found for the new

alloys are clearly impossible for crystals, they correspond very closely with the theoretical computations of the diffraction pattern expected for an icosahedral quasicrys tal (Levine and Steinhardt 1984; Elser 1985; Kalugin et al. 1985; Duneau and Katz 1985). The correspondence has led to the suggestion that the new alloy may be the first

example of a quasicrystal (Levine and Steinhardt 1984). If this suggestion is correct, it implies the existence of a new kind of solid matter with many surprising physical properties.

Penrose tilings The icosahedral quasicrystal can be viewed as a three dimensional generalization of a curious mosaic pattern discovered by Roger Penrose. Penrose's goal had noth

ing to do with the analysis of atomic structures, nor was he aware that the tilings he constructed had a well defined quasiperiodic positional order. Instead, he was

Figure 2. A new alloy of aluminum and

manganese forms a solid with an icosahedral

symmetry, which is strictly forbidden for

crystals. (An icosahedron, as illustrated in

Figure 3, has fivefold symmetry axes.) The

photograph at the upper right reproduces an electron micrograph showing five-sided "snowflake" grains of the alloy. Beside the

photograph is a model of the atomic structure obtained by observing many samples at different angles and

reconstructing the three-dimensional view. Traditional crystallography cannot account for the structure of this alloy. One of the

most promising explanations is that the

grains of metal are a new state of matter called quasicrystals, in which the atoms or clusters of atoms are neither periodic, as in

crystals, nor randomly arranged, as in

glasses. The diffraction pattern formed by electrons scattering off the material in one of its fivefold symmetry planes (bottom right) should be compared with the pattern expected for an ideal icosahedral

quasicrystal {bottom left). (Photograph, model, and diffraction pattern courtesy of R.

Shaefer; scattering pattern for the ideal

quasicrystal from Levine and Steinhardt 1984.)

1986 November-December 589

interested in solving a fascinating puzzle concerning the tiling of a

plane. For many years, mathemati cians had been trying to find a set of tile shapes which could be fitted to

gether to fill a plane, but only in a

nonperiodic fashion (Gardner 1977). It is well known that triangles, for

example, can tile a plane periodically or nonperiodically, but here the goal was to find tile shapes that forced

only nonperiodic tilings. The first ex

ample of such a tiling was discovered

by Robert Berger in 1964, but it consisted of more than 20,000 shapes. Over the years, smaller sets of shapes were found, and Penrose became interested in finding even smaller sets.

In 1974, Penrose discovered that he was able to reduce the number of tile shapes to two. The tile shapes can take various forms, but they always produce mosa

ics, known as "Penrose tilings," with the same positional and orientational symmetries. The simplest choice of tile

shapes consists of the two rhombuses which form the

tiling shown in Figure 4. In the context of quasicrystals, the fat and skinny rhombuses should be thought of as the two unit-cell shapes that repeat to form the structure.

Each rhombus can be repeated by itself with equal spacings to produce a periodic structure. In order to force nonperiodicity, a set of matching rules which constrain the way two tiles can join together, edge to

edge, must be added. The rules can be imposed by defonning the edges of the tiles such as with small,

interlocking tabs, so that they can fit together only in certain ways. Alternatively, the tiles can be decorated

with the colored bands shown in Figure 4. Then, impos ing the rule that two tiles can be put together only when each band joins across each tile interface is sufficient to force nonperiodicity. In building a cluster of tiles out

ward, one can sometimes construct a region that cannot be filled according to the matching rules. The problem can be eliminated by a small rearrangement of the tiles, and the tiling can be continued to fill the plane.

In order to show that his match

ing rules forced nonperiodicity and that the tiling could be continued to

. infinite distances, Penrose showed I that the matching rules were in a

1 one-to-one correspondence with a I set of "deflation rules." The deflation W rules, shown in Figure 4, are rules for

f dividing each of the tile shapes into smaller units of similar shape. If each of the tiles in a cluster constructed

according to the matching rules is deflated in this way, one obtains a

new cluster containing more tiles in which the matching rules are still obeyed. By repeating the deflation over and over, the matching rules can be extended to a tiling consisting of an infinite number of units. By knowing the number of new units produced in each deflation, the ratio of fat to skinny tiles in an infinite tiling can be shown to be an irrational number, namely, the golden ratio, =

(1 + V5~)/2. The deflation rules also mean that the structure is self-similar: there exists a simple rule for dividing a tiling into a new tiling consisting of similar

shaped units scaled down by some constant factor. From the point of view of crystallography, the

striking feature of the Penrose tiling is that each edge of each unit cell is oriented parallel to an edge of a

pentagon. That is, each tile shape is oriented along one of a special, discrete set of directions corresponding to a fivefold symmetry, even though fivefold symmetry is

strictly disallowed for periodic crystals. The secret to the Penrose tiling, as it was eventually

realized (de Bruijn 1981; Levine and Steinhardt 1984), is that it is not simply nonperiodic. Instead, the tiling has a

well-defined, long-range positional order. To observe the positional order, it is useful to decorate the tile

shapes used in Figure 4 with the line segments shown in

Figure 5. (For clarity, the decorative bands that were added to the tiles in Figure 4 to illustrate matching rules have been removed.) Remarkably, when the tiles are then joined together to form the full Penrose tiling, the line segments automatically join to form five sets of

threefold symmetry fivefold symmetry twofold symmetry

Figure 3. The icosahedron is a regular polyhedron with twenty identical triangular faces, twelve vertices, and thirty edges. Ten axes

of symmetry connect the centers of triangular faces on opposite sides of the solid; each of these axes exhibits threefold rotational

symmetry, as can be seen by looking straight down on a triangular face (left). Six axes of symmetry connect the vertices on opposite

sides; each of these axes exhibits fivefold rotational symmetry, as

can be seen by looking straight down on a vertex (center). Each of

the thirty edges defines an axis of twofold rotational symmetry, as

can be seen by looking straight down on an edge (right). The black

pentagons on the surface of a soccer ball lie at the vertices of an

icosahedron (above).

590 American Scientist, Volume 74

Figure 4. About a decade before the

discovery of the alloy with fivefold symmetry shown in Figure 2, the

mathematician Roger Penrose devised tiling patterns, shown later to be quasiperiodic, with properties in two dimensions that are

completely analogous to the three dimensional icosahedral structure of the new

alloy. One of the simplest Penrose tilings can be constructed from the two rhombuses shown above. "Matching rules" for joining the tiles together can be imposed by decorating each tile with colored bands, as

shown, and then allowing tiles to be joined only if bands of the same color join continuously across the interface. The "deflation rules" shown below determine how to subdivide each tile shape into smaller units of similar shape. Given a

cluster of tiles that obey the matching rules, a scaled-down tiling obtained by deflation is

guaranteed also to obey the matching rules.

parallel lines that pass through the entire structure. If one draws a pentagon on the same plane as the tiles, it can be rotated so that each set of lines is parallel to one of the edges of the pentagon. The line-segment decoration

was discovered by Robert Ammann, and the parallel lines are sometimes referred to as Ammann lines.

The fact that the decorations join in such a special way is already a clear sign of long-range positional order. A closer examination reveals that the intervals between

neighboring parallel lines can be determined according to a well-defined and predictable sequence (de Bruijn 1981; Grunbaum and Shepherd, in press). First, each of the five sets of parallel lines is equivalent; that is, each has the same intervals between lines, in the same

sequence. For any one set, there are only two types of intervals between parallel lines: a long (L) and short (S) interval, where the ratio of L to S is the golden ratio, .

The sequence of intervals, ...SLSLLSLSL..., is also very special. It is neither periodic nor random; it is quasiperio dic.

One way of generating the sequence of intervals is

by iteratively applying a "substitution rule" to a shorter, finite sequence. In each application of the substitution rule, each L in the sequence is replaced by LS and each S in the sequence is replaced by L (see Fig. 6). By repeating this substitution over and over, a sequence of longer and

longer length can be obtained. The resulting sequence is referred to as a Fibonacci

sequence because the substitution rule was first studied

by Leonardo Fibonacci in the thirteenth century (Hog gatt 1969). He was not interested in quasicrystals, of course; instead, Fibonacci was considering an idealized

problem related to the progeny of rabbits. He imagined a rabbit population in which each month a new baby rabbit (S) would be born for each adult rabbit (L). Furthermore, each month every baby rabbit already in the population would grow into an adult. Each month the rabbit population underwent exactly the same trans formation as defined in the substitution rule for the Fibonacci sequence described above, and no rabbits ever died. Fibonacci showed that the ratio of adults (L) to babies (S) approached an irrational value after repeated substitutions and that the irrational value was precisely the golden ratio.

The substitution rule for the intervals between the Ammann lines is like a one-dimensional deflation rule and accounts directly for the self-similar properties of the Penrose tiling. The sequence of long and short intervals, which determines the positional order, is called quasi periodic because it can be obtained as a cross between two different periodic sequences, as shown by the

example of uncolored and colored markers in Figure 6.

(This example corresponds with the mathematical defini tion: a quasiperiodic function can be written as a sum of

periodic functions.) Quasiperiodicity is the key element necessary to

construct atomic structures with symmetries which are

impossible for crystals. For example, in Figure 5, one sees that the quasiperiodically spaced lines in the Am

mann decoration form a mesh in which the holes have a finite number of shapes and sizes. Such a lattice is a sensible template for an atomic structure in that we can

imagine filling each different kind of hole with a different atomic cluster. On the other hand, if the quasiperiodical

1986 November-December 591

Figure 5. If each tile shape used in Figure 4 is decorated with the white line segments shown at the right, the segments join to form five sets of continuous, parallel lines

that run throughout the

mosaic. For each set, there are

only two intervals, long (I) and short (5), between neighboring lines, and LI S is the golden ratio. These properties suggest that the Penrose tiling has some sort of long-range positional order. In fact the sequence of intervals between lines, ...SLSLLSLSL..., is

quasiperiodic, as demonstrated in Figure 6. Such a quasiperiodic order, neither periodic nor random, may explain the unique diffraction pattern shown in Figure 2. The lattice of parallel lines shown in this figure, sometimes called Ammann lines, has only a

finite number of sizes of holes, each of which could represent a

cluster of atoms. It is a plausible physical structure. By contrast, a

fivefold-symmetric mesh constructed from equally (periodically) spaced lines (below) produces a lattice that could not be the basis of a regular crystalline solid, because it has infinitely many sizes of holes.

ly spaced lines are replaced by equally (periodically) spaced lines, as in the lower part of Figure 5, the lines form a mesh with an infinite variety of holes in which the sizes of the holes can be arbitrarily small. Such a

template for an atomic structure is not sensible, since it

requires an infinite variety of atomic clusters some of which must be infinitesimally small, whereas even a

single atom has a finite size. The particular irrational ratio of spacings required to obtain a sensible lattice

depends on the symmetry; a ratio of is required for

pentagonal symmetry but a ratio of V2~ - 1 is required

for octagonal, or eightfold, symmetry. Once the role of quasiperiodicity in the Penrose

tiling becomes apparent, extensions to other disallowed

crystallographic symmetries can be envisaged. The pos sibility of generalizing the Penrose tiling pattern to three dimensions had been considered by several researchers in addition to Levine and Steinhardt, including Am

mann (unpubl.), Mackay (1981), Mosseli and Sadoc

(1983), Kramer (1982), and Kramer and Neri (1984). Today, there are a number of techniques for generating quasiperiodic tilings and lattices with any symmetry in two, three, or more dimensions. Determining the unit cell shapes is rather straightforward. The simplest choice is a set of rhombuses (in two dimensions) or rhombohe dra (in three dimensions) whose interior angles are determined by the desired orientational symmetry. (For example, the interior angles for the case of fivefold

symmetry are ir/5, 2tt/5, 3 /5, and 4W5.) However, the

shapes alone yield very little information, since the tiles can be joined (or the solids can be packed) periodically, quasiperiodically, or randomly.

The difficult task is to determine the joining or packing of the unit-cell shapes. One approach, undoubt

edly the most difficult, is to find a set of matching and inflation rules analogous to what Penrose discovered for the case of fivefold symmetry (Levine and Steinhardt 1984, 1986). A second, very elegant approach was dis covered by de Bruijn (1981) and applied to the case of fivefold symmetry (the Penrose tiles) and then to more

general symmetries by Elser (1985, 1986), Duneau and Katz (1985), Kalugin, Kitaev, and Levitov (1985), and Bak (1986). In this approach, the three-dimensional packing is obtained as a special projection along a strip in a

higher-dimensional periodic hypercubic lattice. A third

technique, also introduced by de Bruijn (1981) for the case of fivefold symmetry and then discussed for general symmetries by Kramer and Neri (1984), Socolar and his

colleagues (1985), and Gahler and Rhyner (1986), in volves a topological mapping technique sometimes re ferred to as a "generalized dual" transformation. The

techniques can be used to generate quasicrystal packings with any symmetry. The opening pages of this article show some examples in two dimensions. Figure 7 shows a cut, along a plane perpendicular to the fivefold symme try axis, through a three-dimensional icosahedral pack ing constructed from two rhombohedral shapes. From this cut, the structure is reminiscent of the two-dimen sional Penrose tiling shown in Figure 4.

The techniques generate a very wide class of tilings and packings, most of which are not as special as the

original Penrose tilings. For example, in addition to the

tiling shown in Figure 4, many other fivefold-symmetric tilings can be generated. Although the tilings are quasi periodic and fivefold symmetric, they cannot be generat

592 American Scientist, Volume 74

ed by a simple matching and inflation in which all tiles of the same shape have the same rules. (Other tilings may be consistent with a more general matching or inflation

procedure in which two tiles of the same shape may have different rules.)

It is a mathematical challenge and potentially inter

esting for physics to find the special three-dimensional

packings for other symmetries which can be forced by simple matching and inflation rules, like the original Penrose tilings. These special packings may be of physi cal interest if the matching rules can be imposed by some local interaction between atoms or clusters of atoms. It

must be emphasized, though, that we do not know if the structures of aluminum-manganese and related alloys are determined by analogy with such local matching-rule interactions or if instead they correspond to one of the

more general packing rules. Thus far, such special tilings for fivefold, eightfold,

and twelvefold symmetry in two dimensions, and icosa hedral packing in three dimensions, have been discov ered. (Whether or not such special tilings and packings exist for other symmetries is not known.) The icosahe dral case is particularly interesting and the most compli cated; Socolar and Steinhardt (1986) give a complete description of the matching rules, inflation rules, and

generalized Ammann decoration. As it turns out, although simple matching rules can

be found (Levine 1986) for the two rhombohedra shown in the cut-away solid of Figure 7, simple inflation rules cannot be found. To have both simple matching and

simple inflation rules, a different set containing the four different cell shapes shown in Figure 7 must be used. The shapes correspond to clumps of one, four, ten, and

twenty of the two rhombohedra. On the faces of the cells are shown markings which indicate the matching rules: two cells may be joined along a face only if they have the same marking in the same orientation. The reader is

invited to make copies of the templates, form the cell

shapes, and then try to build the icosahedral structure consistent with the matching rules.

A new phase of solid matter? The unique symmetry properties of quasicrystals ac count for their distinctive diffraction pattern, the pattern produced by scattering electrons off an ideal quasicrys talline solid. For a traditional crystallographer, one strik

ing feature is that the pattern consists of sharp spots, just as for a periodic crystal, but with a symmetry that is disallowed for crystals. The sharp spots are the sign of

positional order. The traditional crystallographer has

always associated positional order with periodicity, and therefore would expect only the usual symmetries al lowed for a periodic crystal. Instead, we now see that another kind of positional order?quasiperiodic order? allows the possibility of diffraction patterns with new

symmetries. Another striking feature is that the pattern of dif

fraction spots is dense; in particular, between any two

spots there are yet more spots. Only the brightest spots have been shown in Figure 2. In the diffraction patterns for a periodic cubic crystal, by contrast, there is an equal interval between spots along each symmetry direction, due to the fact that all the unit cells are equally separated in the atomic structure. In a quasicrystal, however, the unit cells are separated by at least two different spacing lengths whose ratio is an irrational number. It is straight forward to show that the diffraction spots should lie at all

possible integer combinations of at least two intervals whose ratio is likewise irrational. Allowing for positive and negative integer combinations, a dense set of spots should appear.

Thus, when Shechtman and his colleagues reported the very unusual diffraction pattern of an aluminum

?t?t. ?t ?tat ?t Substitution rule: every generation, each adult rabbit produces one

baby; each baby grows into an adult

Figure 6. The sequence of long (I) and short (S) intervals between

parallel lines shown in Figure 5 is a special sequence that can be

generated in at least two ways. In the first, each I interval is

iteratively replaced with the pair of intervals LS and each S interval with the single interval L. The mathematician Fibonacci originally studied the sequence in the thirteenth century. He posed a problem

involving "idealized" rabbits which multiply to form a ratio of adults (I) to babies (5) that approaches the golden mean after

Substitution rule: markers more than half exposed change color

infinitely many generations (left). With every generation, each adult rabbit has one baby (every ? is replaced by an LS) and each baby becomes an adult (every S is replaced by an I). This description of the sequence points out its self-similar character?the substitution rules are analogous to the deflation rules of the Penrose tiles. A second description (right) points out the quasiperiodic nature of the sequence: markers are periodically spaced one unit-length apart (the units are arbitrary). Next, a stencil with windows one unit-length wide, periodically (equally) spaced 2.618... unit-lengths apart (the

square of the golden mean) is superimposed. If the center of a

marker lies within a window, the marker is painted a color. The

resulting sequence of uncolored and colored markers is equivalent to the Fibonacci sequence of adult and baby rabbits, which is

equivalent to the sequence of L and 5 intervals between the Ammann lines in Figure 5.

1986 November-December 593

manganese alloy which consisted of a relatively dense

pattern of sharp spots with an icosahedral symmetry (1984), it was a clear signal to the theorists that the alloy

might be an example of the hypothetical phase they were studying?an icosahedral quasicrystal (Levine and Steinhardt 1984). If true, the alloy represented an exam

ple of a fundamentally new phase of solid matter, one of the most exciting discoveries in crystallography in the last fifty years. Yet, before this hypothesis is accepted, several alternative explanations need to be considered.

One possibility is that the alloy is really composed of

many ordinary periodic crystal pieces arranged in icosa hedral configurations. That is, although the whole clus ter has an icosahedral symmetry, it is really composed of

many separate, ordinary crystalline components in a

symmetric arrangement. "Multiple twinning" is the

crystallographer's term for this phenomenon. Multiple twinning is fairly common in crystal growth, and there are beautiful examples of icosahedral clusters of multi

ple-twinned gold. Shechtman and his colleagues were aware of this possibility and worked hard to find exam

ples of multiple twinning (see also Field and Fraser

1984). For example, they tried focusing the beam of their electron microscope on a very small section of the material with the hope of finding just one of the crystal line components, in which case the diffraction pattern

would have changed from one with icosahedral symme try to one showing a symmetry allowed for crystals.

However, using this and many other experimental tech

niques, neither they nor others since have found any direct evidence for such multiple twinning.

More and more complicated multiple-twinning models for the new alloy continue to be suggested, most

recently by the highly regarded Nobel laureate Linus

Pauling (1985). As the resolution of the experiments has

improved, though, it has become clear that multiple twinning alone is not able to explain the observations

(Bancel et al. 1986). The icosahedral order extends over such great distances that crystalline multiple-twin pieces of comparable size would easily have been seen in the electron microscope. It is still conceivable that there are

many tiny multiple twins in the samples; however, in this case one is still left with the mystery of how the

multiple twins align themselves to produce the long range order. If the order is due to a conspiratorial alignment of many tiny crystalline pieces, an explanation

Figure 7. Many quasiperiodic structures can be generated for which we do not have both simple matching and simple deflation rules such as those illustrated in Figure 4; some of these challenging

mathematical constructions may yet prove to have real physical counterparts. On the left is shown a cut through an icosahedral

quasicrystal along a plane perpendicular to the fivefold symmetry axis. This quasicrystal can be built with the two rhombohedra

shown in the drawing and is a three-dimensional analogue to the

Penrose tiling of Figure 4. However, although matching rules can be

found for the two rhombohedra, simple deflation rules cannot. Both

matching and deflation rules do exist for the four different unit-cell

shapes shown on the facing page; the cells correspond to clumps of

one, four, ten, and twenty of the two rhombohedra. Each cell of the same shape has the same matching rules and deflation rules. The

templates for these cells can be cut out and glued together (it helps to copy the page on thicker paper and to leave tabs for gluing). The

matching rules are imposed by allowing two cells to join together

only if the markings on the adjoining faces are identical and are

oriented the same way.

for the conspiracy is needed. A second explanation is that the alloy is an example

of an ordinary crystal, but one with a very large unit cell whose interior looks like a quasicrystal. For example, one of the icosahedral unit cells (a rhombohedron, say) can be deflated many times and then stacked periodically. The interior of the cell corresponds precisely to a quasi crystal and scatters electrons accordingly. Only scatter

ings that occur coherently over many such unit cells can be used to detect the ultimate crystallinity of the struc ture. A model such as this can never be eliminated per se, because the putative unit cell can always be made

larger and larger until it is beyond the resolution of

experiments. From current experiments (Bancel et al.

1986) it is known that such a unit cell, if it really exists, must contain at least 15,000 atoms, and future experi ments should push that limit upward. It would be very surprising to find such a large unit cell in a metal alloy. Furthermore, almost all the physical properties of the material would still be determined by the quasicrystal line substructure rather than the large-scale periodicity. In short, although it is impossible to eliminate such a

model altogether, the interest in it will fade if experi ments push the limits on the unit-cell size to larger and

larger values.

A third possibility is that the alloy is an example of a different kind of new phase which is known as an icosahedratic (Steinhardt et al. 1981, 1983). This term refers to a hypothetical phase which has icosahedral orientational order but no long-range positional order. The diffraction pattern of such a phase would have icosahedral symmetry, but the pattern would consist of

fuzzy streaks or broad spots rather than sharp spots because of the absence of positional order. Although the

spots in the electron diffraction pattern of the new alloy appear to be fairly sharp, more careful x-ray diffraction

experiments show that they are somewhat fuzzy (Bancel et al. 1985).

In terms of the quasicrystal model, the explanation for the fuzziness could be that the samples are strained or contain defects. Strain and defects produce a position al disorder and thus a fuzziness in the diffraction pattern for crystals or quasicrystals. Recently, evidence for such strain has also been found in electron microscope images and in careful measurements of the electron diffraction

pattern (Lubensky et al., in press). However, there

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remains the possibility that the alloy is an icosahedratic and not a quasicrystal at all. Shechtman and Blech (1985) and Stephens and Goldman (1986) have designed com

puter simulations of a special case of an icosahedratic structure in which icosahedra are randomly (not quasi periodically) packed vertex to vertex with a common orientation. This structure, which in some sense repre sents a quasicrystal with a very high degree of positional disorder, has a diffraction pattern that also closely resem bles the experimental patterns. Although the quasicrys tal model appears to be the simplest and most elegant explanation in many respects, the issue will not be settled unless a much more perfect quasicrystal is formed which has much longer-range positional order.

Challenges for the future The most immediate experimental challenge is to find a material and a technique to make larger and more perfect samples. The original aluminum-manganese alloy was formed by the very rapid cooling technique of melt

spinning. It is essentially impossible to keep the cooling rate and relative concentration of aluminum and manga nese constant over the length and thickness of the strip. In general, lack of control in the cooling process and the simultaneous growth of many grains produces strain and defects even in crystals and limits the size of the

grains. Since the original discovery, melt spinning and

several other rapid-cooling techniques have been used to

produce some twenty or thirty different metallic alloys which exhibit an icosahedral phase. However, all have been produced with rather small and imperfect grains. This poses a difficult problem for crystallographers. Although there have been some interesting speculations about the atomic structures of the new materials (Elser and Henley 1985; Guyot and Audier 1985; Proceedings of Les Houches Workshop 1986), some of the mathemat ical aspects of quasicrystalline structures make it much

more difficult to determine the atomic arrangement compared with that of a periodic crystal (Bak 1986; Socolar and Steinhardt 1986). Only with larger and more

perfect grains can a full battery of experimental tech

niques be applied that will reveal the detailed atomic

configurations. The problem with cooling liquid alloys more slowly

to obtain larger and more perfect grains has been that the material then forms crystals instead of quasicrystals; that

is, the lowest energy phase for these materials is the

crystalline phase, and only by cooling the liquid rapidly can it be trapped in the icosahedral phase. For different

materials the icosahedral phase has different energies with respect to the crystalline phase, and it is at least

theoretically possible that for some material the icosahe dral phase has lower energy than the crystalline phase (Alexander and McTague 1978; Bak 1985a; Kalugin et al.

1985; Mermin and Troian 1985; Jaric 1985; Sachdev and Nelson 1985). Unfortunately, there is no reliable theory for predicting which particular elements or relative mix tures will form the most stable samples of the icosahe dral phase. If a quasicrystalline material with an energy close to or lower than the energy of the crystalline phase can be found, a much slower cooling method can be

applied and, one hopes, a larger and more perfect

sample formed. The search is on! Recently, icosahedral

grains up to tenths of a millimeter in size have been

reported in a slowly cooled alloy of aluminum, lithium, and copper (Ryba and Bartges, unpubl.); however, ex tensive measurements of the physical properties have not been completed.

The possibility of a fascinating new phase of solid matter has captured the imagination of physicists throughout the world, and many new scientific results have been found. The subject is wonderfully attractive and stimulating because so many exciting questions remain to be answered. For experimentalists, in addition to determining whether the new alloys are indeed

quasicrystals, it is also important to determine how "robust," or common, this phenomenon is. Originally, the formation of an icosahedral phase appeared to be a rare event, but now more and more materials have been found which exhibit the phase. Perhaps even geological samples of more stable quasicrystal materials will be found.

Although experimentalists have been hampered by small and imperfect grains, some theoretical progress has been made in understanding the structural proper ties of quasicrystals. Theoretical arguments have been raised to explain why icosahedral quasicrystalline struc tures may be stable atomic arrangements (Alexander and

McTague 1978; Bak 1985a; Kalugin et al. 1985; Mermin and Troian 1985; Jaric 1985; Sachedev and Nelson 1985). Bak (1985b) and Levine and his colleagues (1985) have studied the effects of stresses on quasicrystals. The latter

group has also studied the nature of defects in quasicrys tals and has shown that the defects are much more difficult to remove (by heating or applying stress) than in

ordinary crystals. The result implies that quasicrystals will have unique elastic properties; in particular, they will generally be more brittle and difficult to deform

compared with ordinary crystals (Lubensky et al. 1985, 1986). However, little real progress has been made on thermal and vibrational behavior or on what may be the

most exciting aspects of quasicrystals?their electronic

properties. It is well known, though, that all these

properties of a solid depend critically on their symme tries, and since quasicrystals have special symmetries, new and intriguing phenomena are expected.

Finally, the discovery of quasicrystals may funda

mentally change our understanding of solid state struc tures. Quasicrystals appear to be a "missing link" be tween the highly ordered crystals and the highly disordered glasses. Previously there appeared to be little connection between them, but now it appears that they represent poles in a continuum of structural possibilities. This realization may be crucial in understanding the

physical properties of glasses, quasicrystals, and crys tals, and their relation to one another. In this way, the

discovery of quasicrystals may help pave the way to a "unified theory" of solid structure.

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"My old lab was cluttered with test tubes, bottles, flasks, paper.... Lord, how I miss it."