A New Paradigm
for the Structure of Quasicrystals

Paul J. Steinhardt

Department of Physics
Princeton University
Princeton, NJ 08544 USA

Based on NATURE article
by P.J. Steinhardt (Penn) and H.-C. Jeong (Maryland)
Vol. 382, pages 433-5 (1996) (1996),

and recent experimental results reported in NATURE
P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, and A.P. Tsai
Vol. 396, 55-57 (1998)

Outline of Past Work on Quasicrystals

Ever since their discovery in 1984,[1] quasicrystals have posed a perplexing puzzle: Why do the atoms form a complex, quasiperiodic pattern rather than a regularly-repeating, crystal arrangement? Most explanations have been based on some analogy with tilings. For example, in the Penrose tiling picture, the notion has been that atoms arrange themselves into two types of clusters analogous to the obtuse and acute rhombic Penrose tiles and have interactions which force connections between clusters analogous to the Penrose matching rules for tiles.[2] In the original formulation of the random tiling picture,[3] two clusters are also needed corresponding to obtuse and acute rhombic tiles without matching rules.

The tiling models suggest that the conditions necessary to form quasicrystals are significantly more complex than the conditions for forming crystals. For example, the requirement of two types of cluster appears to be necessary to obtain quasiperiodicity. Yet, it is difficult to imagine energetics that permit two clusters in the just the right proportion in density (and exclude any other clusters), especially considering that most known quasicrystals are composed of metallic elements with central force potentials rather than rigid covalent bonding. In the case of the Penrose picture, there is the additional problem of finding energetics that impose the matching rules.

An important development has been the emergence of a new paradigm for the atomic structure of quasicrystals which simplifies the description of quasicrystal structure and suggests a simple thermodynamic mechanism for quasicrystal formation. In the new paradigm, quasicrystals are described in terms of a single, repeating cluster. The repeating cluster is analogous to the unit cell in periodic crystals. The novel feature is that the neighboring clusters ``overlap." Atoms in the overlap region are shared by the two clusters enabling the hypothetical surfaces that bound the clusters to interpenetrate. However, the sharing means that there is no duplication or crowding of atoms. The new picture does not have a simple interpretation in terms of tiling; the term ``covering" is more apropos.

The New Paradigm: The Quasi-unit cell Picture

A new picture of quasicrystals emerges in which the structure is determined entirely by a single repeating cluster which overlaps (shares atoms with) neighbor clusters according to simple energetics. We first discuss the the structure in terms of somewhat artificial overlap rules, which play the same role as Penrose edge-matching rules in forcing a unique structure isomorphic to Penrose tiling. We then discuss how the overlap rules may arise from physically plausible energetics.

In this paper, we will focus on the case of decagonal quasicrystals whose quasiperiodic layers have the same symmetry as two-dimensional Penrose tilings. The two-dimensional analogue for the overlapping cluster model consists of decagonal tiles which overlap to form a covering of the two-dimensional plane. Petra Gummelt[15] first conjectured that decagonal tiles with appropriately chosen overlap rules can force a perfect quasiperiodic tiling and she outlined a proof. Jeong and Steinhardt[16] provided a simple, alternative proof which makes clear the isomorphism to two-tile Penrose tilings.

In the new paradigm, the atomic structure of the decagonal phase is determined entirely by the atom decoration of the overlapping decagon tiles. The decagon tiles represent decagonal cluster columns in the three-dimensional quasicrystals structure. For the case of a perfect quasiperiodic structure, the overlapping cluster can be dubbed a ``quasi-unit cell," since it is analogous to the conventional unit cell in a perfect periodic crystal. However, an important difference is that the atomic decoration of the quasi-unit cell is constrained: the atom configuration inside the quasi-unit cell must have the property that neighboring clusters can share atoms without significant distortion of their atomic arrangements.

Figure 1: A quasiperiodic tiling can be forced using marked decagons shown in (a). Matching rules demand that two decagons may overlap only if shaded regions overlap. This permits two possibilities in which the overlapped area is either small (A -type) or large (B -type), as shown in (b)> If each decagon is inscribed with a large obtuse rhombus, as shown in (c), a tiling of overlapping decagons (d, left) is converted into a Penrose tiling (d, right).
\epsfxsize=5.0 in \epsfbox{stein1.ps}\end{center}\end{figure}

The decagonal quasi-unit cell is shown in Fig. 1(a) with a decoration consisting of kites and star-like shapes designed as a mnemonic for the overlap rules. To force a perfect quasiperiodic covering isomorphic to a Penrose tiling, the decagons are permitted to overlap only in two ways, A - or B -type, as shown in Fig. 1(b). With these overlap rules, kite regions always overlap kite regions and star-like regions always overlap star regions. The isomorphism between decagons and Penrose tilings can be realized by inscribing each decagon with a large Penrose obtuse rhombus tile marked with single- and double-arrows, as illustrated in Fig. 1(c). The decagon tiling in Fig. 1(d) is thereby converted into a Penrose tiling. (Spaces are left where acute Penrose rhombi can be inserted.)

The reduction of a Penrose tiling or decagonal phase to a single-repeating cluster is a remarkable simplification. In terms of atomic modeling, it means that the entire structure is defined by the atomic decoration of the quasi-unit cell, similar to the familiar case of periodic crystals. However, as an explanation for why quasicrystals form, a worrisome aspect is the overlap rules, which appear to require complex energetics. Hence, a second important discovery[16] for the new paradigm has been that it is possible to avoid matching or overlap rules altogether. Instead, a perfect Penrose tiling can arise simply by maximizing the density of some chosen atom cluster, C .

To illustrate the result, we idealize the discussion by considering arrangements of obtuse and acute rhombi, each representing some atom cluster. We assume no matching or overlaps rules. Without any further specifications, there are an infinite number of distinct tilings possible, including Penrose tiling, periodic tilings, and random arrangements. If there is no energetics to distinguish among the possibilities, the ground state is degenerate.

Figure 2: Given obtuse and acute rhombi and no matching rules, the Penrose tiling is configuration with the highest density of C -clusters. (a) Shows a C -cluster; if a decagon is circumscribed about the central 7 rhombi (dotted line), the decagons from an overlapping decagon tiling. (b) Shows the two kinds of overlaps between C -clusters which bring the centers of the C -clusters closest together. (The A -types have the same separation between centers.) If decagons are circumscribed about the central 7 rhombi of each C -cluster, the A - and B -type overlaps between C -clusters transform into the precisely the A - and B -types overlaps between decagons in the quasi-unit cell description.
\epsfxsize=4.0 in \epsfbox{stein2.jpg}\end{center}\end{figure}

In realistic models, it is natural to suppose that the tiles represent atom clusters and that some tile cluster C is low energy compared to the others. The degeneracy may, thereby, be broken. Jeong and Steinhardt[16] have shown that, for an appropriately chosen C cluster, the Penrose tiling emerges as the unique ground state. That is, if one imagines that the chosen cluster of tiles represents some energetically preferred atomic cluster, then minimizing the free energy would naturally maximize the cluster density. Jeong and Steinhardt have shown that the Penrose tiling is the unique configuration with the maximum density of C clusters.

An example is the cluster C shown in Fig. 2, although other choices are possible.[17] Two neighboring C 's can share tiles. The greatest overlaps correspond precisely to the A and B overlaps of the central decagonal region of C . It is clear from this that the C -cluster is inspired by the decagon covering described above, although the precise relationship is quite subtle. For example, if there are no explicit overlap rules to forbid certain arrangements for the C cluster, the hexagon tabs must introduced to prevent undesirable overlaps. The non-trivial result is that maximizing the density of C clusters automatically leads to a structure in which the C -clusters are in one-to-one correspondence with the quasi-unit cells in a single decagon tiling. In particular, of all possible arrangements of obtuse and acute tiles, the Penrose tiling is the unique arrangement of C in which every C has an A or B overlap with its neighbors. Jeong and Steinhardt have also shown that the ground state remains unique for a wide range of assignments of energies to clusters.

Recent Experiments

To progress beyond abstract studies of tilings, examples of real quasicrystals must be found which support the new paradigm. It would be particularly useful to find an example with a simple atomic decoration of the quasi-unit cell so that the correspondence is apparent. A promising system is the decagonal phase of AlNiCo , one of the most studied quasicrystals. High resolution lattice images reveal a network of overlapping decagonal clusters columns about 2 nm in diameter.[18], [19]

Figure 3: Superposition of a perfect decagon tiling on the high angle annular dark-field (HAADF) lattice image of water-quenched Al72Ni20Co8 obtained by the high angle annular dark field method by Saitoh et al. The overlay decagon tiling is shown separately in the following figure. Note the high degree of order in the lattice image and the near-perfect correspondence with the overlaid decagons.

A historic difficulty with AlNiCo has been disorder and superlattice effects which have confounded structural analysis. Early overlapping cluster models, such as Burkov's,[4] characterized the structure in terms of overlapping clusters with decagonal or pentagonal symmetry using overlap rules which produce random tilings. The random tiling picture seemed appropriate because of evidence of some diffuse scattering.

Recently, however, Tsai et al.[20] have found a simple decagonal phase in water-quenched Al72Ni20Co8 which exhibits no superlattice reflections and no diffuse streaks. Figure 3 shows a superposition of the high angle annular dark-field (HAADF) image for Al72Ni20Co8 obtained by Saitoh et al.[21] and a single decagon tiling. The bright spots in the HAADF image correspond to the positions of the transition metal atoms. Holding the image at an angle, one observes that the lattice image shows no detectable phason strain across nearly 15 nm. Hence, the image is isomorphic to a perfect Penrose tiling. The structure also appears to be composed of overlapping, decagonal clusters. See Figure 4. The HAADF image shows that the innermost ring of atoms inside the clusters has neither pentagonal nor decagonal symmetry. Rather, the structure breaks decagon symmetry in precisely the same sense that the overlap rules (see the superposed kite-shape decorations on decagon tiles). Using convergent beam electron diffraction, the space group was determined to be centrosymmetric P105/mmc using convergent beam electron diffraction.

Figure 4: An blow-up of a decagonal cluster in water-quenched Al72Ni20Co8 obtained by the high angle annular dark field method by Saitoh et al. The decagon is nearly 2 nm across. The overlay is a single decagon quasi-unit cell with decagon symmetry-breaking decoration to indicate overlap rules.

Figure 5: The single decagon tiling used to overlay the HAADF lattice image in Figure 3.

In Figure 3, the lattice image is superposed by a perfect single decagon tiling. The perfect tiling overlay is shown in Figure 5. The correspondence appears near-perfect across the image, with atomic decoration of each decagon and each kite-shape decoration within the decagon appearing to be identical. Jeong and Steinhardt have developed a simple calculational scheme for relating the atomic decoration of the quasi-unit cell to the stoichiometry and density.[23] Figure 6 is a candidate atomic decoration of the decagon unit cell that agrees with current measurements: the stoichiometry, Al72Ni21Co7 , and the density, 3.94 g/cm3 , lie within 1% of the measured values. The computed HREM lattice image agrees closely with the experimental image. This model is not equivalent to an atomic decoration of acute and obtuse Penrose tiles in which every acute tile is decorated equivalently and every obtuse tile is decorated identically. Hence, Al72Ni20Co8 appears to be an example for which the single decagon picture can be verified and explored in fine detail.

Figure 6: A candidate model (improved over the one appearing in Nature) for the atomic decoration of the decagonal quasi-unit cell for Al72Ni20Co8 . Large circles represent Ni (red) or Co (purple) and small circles represent Al . The structure has two distinct layers along the periodic c -axis. Solid circles represent c=0 and open circles represent c=1/2 .
\epsfxsize=3.3 in \epsfbox{stein6.eps}\end{center}\end{figure}
Figure 7: The computed HREM lattice image agrees closely with the experimental image; see Figure 7.
\epsfxsize=3.3 in \epsfbox{stein6.eps}\end{center}\end{figure}


A new paradigm emerges from recent mathematical discoveries about Penrose tilings.[15,16,17] The structure of perfect quasiperiodic tilings can be interpreted in terms of a single quasi-unit cell and matching rules can be replaced by simple energetics which favor the formation of some specific atom cluster.

In the new paradigm, the atomic structure of a quasicrystal can be totally characterized by the decoration of a single cluster, rather than two clusters as the Penrose tiles would suggest. The result simplifies the problem of specifying and of determining the atomic structure since the only degrees of freedom are the atom types and the atom positions within the quasi-unit cell. Jeong and Steinhardt have shown that every atomic decoration of the conventional Penrose tiling can be reinterpreted in terms of an atomic decoration of the quasi-unit cell, although the converse is not true.[23] Some decorations of the quasi-unit cell are not equivalent to decorating each obtuse tile identically and each acute tile identically. In this sense, the quasi-unit cell picture encompasses more possibilities.

In the new paradigm, the atomic decoration of the quasi-unit cell encodes the symmetry of the structure. In the past, the symmetry of the structure has been determined by appealing to reciprocal space (see Chapter 6 by Mermin in this volume) or to perp-space (see Chapter 3 by Janot and de Boissieu in this volume). These indirect techniques can be substituted by a a real-space description. That is, there is a well-defined correspondence between the atomic decoration of the quasi-unit cell in real space and the space group symmetry of the structure.[23]

The new paradigm implies a closer physical relationship between quasicrystals and crystals. Now it appears that both can be described in terms of the close-packing of a single cluster or unit cell. In a crystal, the unit cell packs edge-to-edge with its neighbors. Quasicrystals correspond to a generalization in which the quasi-unit cells overlap. In both cases, the formation of the particular structure appears to be explained by a low-energy atomic cluster, although the atomic arrangement in the case of quasicrystals is constrained to allow overlap. Hence, the new paradigm makes plausible why many materials form quasicrystals and, at the same time, explains why quasicrystals are less common than crystals.

The new paradigm requires a mechanism to explain how quasicrystals grow. If the quasicrystals are grown slowly, then thermodynamic relaxation to the ground state is possible. However, some of the most perfect quasicrystal samples, including AlNiCo , are formed by rapid quenching. In this volume, Socolar has described a scheme for solids equivalent to Penrose patterns based on obtuse and acute rhombi using vertex rules and stochastic growth similar to diffusion limited aggregation. This approach can be adapted to overlapping clusters. (Janot[13] has already suggested a similar mechanism for overlapping clusters, although his vertex rules allow random tilings as well as perfect quasiperiodic tilings.) If quasicrystals form due to a particular cluster being energetically favored, a simpler kinematic mechanism may be through local atomic rearrangement that increases the local density of the given atomic cluster.

The overlapping cluster picture may also account for other physical properties of quasicrystals. Janot[14] has suggested that the cluster picture can naturally explain the inelastic neutron scattering properties, and the electrical and thermal conductivity behavior. Finally, the new paradigm suggests a natural explanation of why quasicrystals form, shedding new light on an old mystery. In the new picture, the problem reduces to the behavior of small atom clusters. Perhaps total energy calculations based on a modest number of atoms may be used to understand why quasicrystals form and to predict new ones.

From future structural and kinematical studies of known quasicrystals, such as AlNiCo , these principles may be established providing a new understanding of and new control over the formation and structure of quasicrystals.

Outline of Previous Work on Quasicrystals by Penn-Princeton Group

1981 - Proposal of icosahedral short-range order in supercooled liquids and glasses
    M. Ronchetti, D.R. Nelson, P.J. Steinhardt {\it Phys. Rev. Lett.} 47, 1297-300 (1981).

1984 - Introduction of the concept of quasicrystals (short for "quasiperiodic crystals") as a new phase of solid matter

Explanation of Penrose tiles as 2D quasiperiodic patterns; 3D generalization to rhombohedral structure with icosahedral symmetry

Analysis of the icosahedral phase of AlMn (discovered by D. Shechtman, I. Blech, D. Gratias, J. Cahn) showing that the quasicrystal model explains its structural properties.
    D. Levine and P.J. Steinhardt Phys. Rev. Lett. 53: 2477-80 (1984).

1985 - Development of elasticity theory and theory of disclocations for quasicrystals
    D. Levine, T. Lubensky, S. Ostlund, S. Ramaswamy, P.J. Steinhardt and J. Toner, Phys. Rev. Lett. 54: 1520-3 (1985).

1986 - Three-dimensional Penrose tiling constructed from zonohedra
    J. Socolar and P.J. Steinhardt, Phys. Rev. B, 34, 617-647 (1986).

1988 - Construction of local rules that make it possible to construct perfect Penrose tilings and, by inference, perfect quasicrystals (local rules were once thought to be mathematically impossible); this theoretical work preceded discovery of near perfect quasicrystals in the laboratory
    G. Onoda, P.J. Steinhardt, D. DiVincenzo, and J. Socolar, Phys. Rev. Lett. 60, 2653-6 (1988).

1996 - New paradigm for quasicrystals -- the quasi-unit cell picture based on a single repeating unit; proof that the structure can be obtained by maximizing density of low-energy clusters
    P.J. Steinhardt and H.-C. Jeong, Nature 382, 433-5 (1996)

1998 - Empirical evidence for quasi-unit cell picture based on AlNiCo
    P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, and A.P. Tsai, Nature 396, 55-57 (1998).