Alternative titles: quasi-periodic crystal; quasicrystalline solid



Structure and symmetry

Microscopic images of quasicrystalline structures




Translational periodicity and symmetry

Fivefold symmetry axes are forbidden in ordinary crystals, while other axes, such as sixfold axes, are allowed. The reason is that translational periodicity, which is characteristic of crystal lattices, cannot be present in structures with fivefold symmetry. Figures 1 and 2 can be used to illustrate this concept. The triangular array of atoms in Figure 1 has axes of sixfold rotational symmetry passing through each atomic position. The arrows represent translational symmetries of this crystalline structure. That is, if the entire array of atoms is displaced along one of these arrows, say the one labeled a, all new atomic positions coincide with the locations of other atoms prior to the displacement. Such a displacement of atoms that leaves atomic positions invariant is called a symmetry of the crystal. In Figure 1, if two different symmetries are combined such that the structure is first displaced along arrow a and then along arrow b, the net result is equivalent to a displacement along arrow c, which itself must be a symmetry of the structure. Again, atomic sites coincide before and after the displacement. Repeated displacements along the same arrow demonstrate the translational periodicity of the crystal.

The atomic arrangement shown in Figure 2 exhibits fivefold rotational symmetry but lacks the translational symmetries that must be present in a crystalline structure. The arrows (other than arrow c) represent displacements that leave the arrangement invariant. Assume they are the shortest such displacements. Now, as before, consider the combinations of two symmetries a and b with the net result c. The length of c is smaller than either a or b by a factor τ = (5 + 1)/2, which is known as the golden mean. The new atomic position, outlined with a dotted line, does not coincide with a previous atomic position, indicating that the structure does not exhibit translational periodicity. Therefore, an array of atoms may not simultaneously display fivefold rotational symmetry and translational periodicity, for, if it did, there would be no lower limit to the spacing between atoms.

In fact, the compatibility of translational periodicity with sixfold rotational symmetry (as shown in Figure 1) is a remarkable accident, for translational periodicity is not possible with most rotational symmetries. The only allowed symmetry axes in periodic crystals are twofold, threefold, fourfold, and sixfold. All others are forbidden owing to the lack of minimum interatomic separation. In particular, fivefold, eightfold, tenfold, and twelvefold axes cannot exist in crystals. These symmetries are mentioned in particular because they have been reported in quasicrystalline alloys.

Since a high-resolution electron microscope image of aluminum-manganese-silicon quasicrystal clearly reveals an axis of fivefold symmetry, it may be concluded that the arrangement of atoms lacks translational periodicity. That, in itself, is no great surprise, for many materials lack translational periodicity. Amorphous metals, for example, are frequently produced by the same melt-spinning process that was employed in the discovery of quasicrystals. Amorphous metals have no discrete rotational symmetries, however, and high-resolution electron microscope images reveal no rows of atoms. The arrangement of atoms in a quasicrystal displays a property called long-range order, which is lacking in amorphous metals. Long-range order permits rows of atoms to span the image and maintains agreement of row orientations. Ordinary crystal structures, such as that of Figure 1, display long-range order. Strict rules govern the relative placement of atoms at remote locations in solids with long-range order.

Electron diffraction confirms the presence of long-range order in both crystals and quasicrystals. Quantum mechanics predicts that particles such as electrons move through space as if they were waves, in the same manner that light travels. When light waves strike a diffraction grating, they are diffracted. White light breaks up into a rainbow, while monochromatic light breaks up into discrete sharp spots. Similarly, when electrons strike evenly spaced rows of atoms within a crystalline solid, they break up into a set of bright spots known as Bragg diffraction peaks. Symmetrical arrangements of spots reveal axes of rotational symmetry in the crystal, and spacings between the discrete spots relate inversely to translational periodicities. Amorphous metals contain only diffuse rings in their diffraction patterns since long-range coherence in atomic positions is required to achieve sharp diffraction spots.

The original electron diffraction pattern of quasicrystalline aluminum-manganese published by Shechtman and his coworkers is shown in Figure 3. Rings of 10 bright spots indicate axes of fivefold symmetry, and rings of six bright spots indicate axes of threefold symmetry. The twofold symmetry axes are self-evident. The angles between these axes, indicated on the figure, agree with the geometry of the icosahedron. The very existence of spots at all indicates long-range order in atomic positions. Recalling the earlier result that fivefold symmetry axes are forbidden in crystalline materials, a paradox is presented by quasicrystals. They have long-range order in their atomic positions, but they must lack spatial periodicity.


Dov Levine and Paul Steinhardt, physicists at the University of Pennsylvania, proposed a resolution of this apparent conflict. They suggested that the translational order of atoms in quasicrystalline alloys might be quasiperiodic rather than periodic. Quasiperiodic patterns share certain characteristics with periodic patterns. In particular, both are deterministic—that is, rules exist that specify the entire pattern. These rules create long-range order. Both periodic and quasiperiodic patterns have diffraction patterns consisting entirely of Bragg peaks. The difference between quasiperiodicity and periodicity is that a quasiperiodic pattern never repeats itself. There are no translational symmetries, and, consequently, there is no minimum spacing between Bragg peaks. Although the peaks are discrete, they fill the diffraction pattern densely.

The most well-known quasiperiodic pattern may be the Fibonacci sequence, discovered during the Middle Ages in the course of studies conducted on rabbit reproduction. Consider the following rules for birth and maturation of rabbits. Start with a single mature rabbit (denoted by the symbol L for large) and a baby rabbit (denoted by S for small). In each generation every L rabbit gives birth to a new S rabbit, while each preexisting S rabbit matures into an L rabbit. A table of rabbit sequences may be established as follows. Start with an L and an S side by side along a line. Replace the L with LS and the S with L to obtain LSL and repeat the procedure as shown in Table 1. The numbers of rabbits present after each generation are the Fibonacci numbers. The population grows exponentially over time, with the population of each generation approaching τ (the golden mean) multiplied by the population of the previous generation. The sequence of L and S symbols forms a quasiperiodic pattern. It has no subunit that repeats itself periodically. In contrast, a periodic sequence such as LSLLSLLSLLSLLSL . . . has a fundamental unit (LSL) that is precisely repeated at equal intervals. In crystallography such a repeated unit is called a unit cell. Quasiperiodic sequences have no unit cell of finite size. Any portion of the Fibonacci sequence is repeated infinitely often, but at intervals that are not periodic. These intervals themselves form a Fibonacci sequence.

Fibonacci sequences of rabbits
generation sequence mature rabbits babies
1 LS 1 1
2 LSL 2 1
3 LSLLS 3 2

An example of a two-dimensional pattern that combines fivefold rotational symmetry with quasiperiodic translational order is the Penrose pattern, discovered by the English mathematical physicist Roger Penrose and shown in Figure 4. The diffraction pattern of such a sequence closely resembles the fivefold symmetric patterns of Figure 3. The rhombic tiles are arranged in sets of parallel rows; the shaded tiles represent one such set, or family. Five families of parallel rows are present in the figure, with 72° angles between the families, although only one of the five has been shaded. Within a family the spacings between rows are either large (L) or small (S), as labeled in the margin. The ratio of widths of the large rows to the small rows is equal to the golden mean τ, and the quasiperiodic sequence of large and small follows the Fibonacci sequence.

Levine’s and Steinhardt’s proposal that quasicrystals possess quasiperiodic translational order can be examined in terms of a high-resolution electron micrograph. The rows of bright spots are separated by small and large intervals. As in the Penrose pattern, the length of the large interval divided by the length of the small one equals the golden mean, and the sequence of large and small reproduces the Fibonacci sequence. Levine’s and Steinhardt’s proposal appears consistent with the electron diffraction results. The origin of the name quasicrystals arises from the fact that these materials have quasiperiodic translational order, as opposed to the periodic order of ordinary crystals.

Symmetries observed in quasicrystals

Figure 3 represents quasicrystals with the symmetry of an icosahedron. Icosahedral quasicrystals occur in many intermetallic compounds, including aluminum-copper-iron, aluminum-manganese-palladium, aluminum-magnesium-zinc, and aluminum-copper-lithium. Other crystallographically forbidden symmetries have been observed as well. These include decagonal symmetry, which exhibits tenfold rotational symmetry within two-dimensional atomic layers but ordinary translational periodicity perpendicular to these layers. Decagonal symmetry has been found in the compounds aluminum-copper-cobalt and aluminum-nickel-cobalt. Structures that are periodic in two dimensions but follow a Fibonacci sequence in the remaining third dimension occur in aluminum-copper-nickel.

All the compounds named thus far contain aluminum. Indeed, it appears that aluminum is unusually prone to quasicrystal formation, but there do exist icosahedral quasicrystals without it. Some, like gallium-magnesium-zinc, simply substitute the chemically similar element gallium for aluminum. Others, like titanium-manganese, appear chemically unrelated to aluminum-based compounds. Furthermore, some quasicrystals such as chromium-nickel-silicon and vanadium-nickel-silicon display octagonal and dodecagonal structures with eightfold or twelvefold symmetry, respectively, within layers and translational periodicity perpendicular to the layers.

The origin of quasicrystalline order

The origin of quasicrystalline order remains in question. No proven explanation clarifies why a material favours crystallographically forbidden rotational symmetry and translational quasiperiodicity when at nearby compositions it forms more conventional crystal structures. The American chemist Linus Pauling noted that these related crystalline structures frequently contain icosahedral motifs within their unit cells, which are then repeated periodically. Pauling proposed that quasicrystals are really ordinary crystalline materials caught out of equilibrium by a type of crystal defect called twinning, in which unit cells are attached at angles defined by these icosahedral motifs. While this may be a reasonable model for rapidly cooled alloys such as Shechtman’s original aluminum-manganese, other compounds, such as aluminum-copper-iron, possess quasicrystalline structures in thermodynamic equilibrium. These quasicrystals can be grown slowly and carefully using techniques for growth of high-quality conventional crystals. The more slowly the quasicrystal grows, the more perfect will be its rotational symmetry and quasiperiodicity. Measuring the sharpness of diffraction pattern spots shows perfect ordering on length scales of at least 30,000 angstroms in these carefully prepared quasicrystals. Twinning cannot account for such long-range order.

Levine and Steinhardt proposed that matching rules, such as those Penrose discovered to determine proper placement of his tiles to fill the plane quasiperiodically, may force the atoms into predefined, low-energy locations. Such a mechanism cannot be the complete explanation, though, since the compound forms ordinary crystalline structures at nearby compositions and temperatures. Indeed, it appears that, when quasicrystals are thermodynamically stable phases, it is only over a limited range of temperatures close to the melting point. At lower temperatures they transform into ordinary crystal structures. Thermodynamics predicts that the stable structure is the one that minimizes the free energy, defined as the ordinary energy minus the product of the temperature and the entropy. It is likely that entropy (a measure of fluctuations around an ideal structure) must be considered in addition to energy to explain stability of quasicrystals.


Elastic properties

Along with their novel structures and symmetries, quasicrystals are expected to exhibit unusual properties. Both their elastic and their electronic behaviour distinguish quasicrystals from ordinary crystalline metals. Elastic response may be studied by measuring the speed of sound waves propagating through the metal. Sound speeds usually vary depending on the direction of propagation relative to axes of high rotational symmetry. Because the icosahedron has such high symmetry—it is closer to a sphere than is, for instance, a cube—the sound speeds turn out to be independent of the direction of propagation. Longitudinal sound waves (with displacements parallel to the direction of propagation) have speeds different from transverse waves (with displacements perpendicular to the direction of propagation), as is the case for all matter. Because the sound speeds do not depend on direction of propagation, only two elastic constants are required to specify acoustic properties of icosahedral quasicrystals. In contrast, cubic crystals require three elastic constants, and lower-symmetry crystals require up to 21 constants.

As a consequence of the translational quasiperiodicity, there exists a second type of elastic deformation beyond the ordinary sound wave, or phonon. Known as phasons, these elastic deformations correspond to rearrangements of the relative atomic positions. Removal of a phason requires adjusting positions of all atoms within a row of atoms in a quasicrystalline structure. At low temperatures motion of atoms within the solid is difficult, and phason strain may be easily frozen into the quasicrystal, limiting its perfection. At high temperatures, close to the melting point, phasons continually fluctuate, and atoms jump from place to place.

Electric properties

The electric properties of quasicrystals have proved to be rather unusual. Unlike their constituent elements, which tend to be good electrical conductors, quasicrystals conduct electricity poorly. For alloys of aluminum-copper-ruthenium these conductivities differ by as much as a factor of 100. As the perfection of the quasicrystalline order grows, the conductivity drops. Such behaviour is consistent with the appearance of a gap in the electronic density of states at the Fermi surface, which is the energy level separating filled electronic states from empty ones. Since it is only Fermi-surface electrons that carry current, a vanishingly small density of such electronic states leads to low electrical conductivities in semiconductors and insulators. Such a gap in the density of states may also play a role in explaining the formation of quasicrystalline structures. This is known as the Hume-Rothery rule for alloy formation. Since the Fermi-surface electrons are the highest-energy electrons, diminishing the number of such electrons may lower the overall energy.

Mechanical properties

The mechanical properties of quasicrystals are especially significant because the desire to develop a material that exhibited these properties motivated the investigators who discovered quasicrystals. Mechanical properties also relate to their first potential practical applications. Quasicrystals are exceptionally brittle. They have few dislocations, and those present have low mobility. Since metals bend by creating and moving dislocations, the near absence of dislocation motion causes brittleness. On the positive side, the difficulty of moving dislocations makes quasicrystals extremely hard. They strongly resist deformation. This makes them excellent candidates for high-strength surface coatings. Indeed, the first successful application of quasicrystals was as a surface treatment for aluminum frying pans.

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