quasicrystal

Microscopic images of quasicrystalline structures

Dan Shechtman, a researcher from Technion, a part of the Israel Institute of Technology, and his colleagues at the National Bureau of Standards (now the National Institute of Standards and Technology) in Gaithersburg, Md., discovered quasicrystals in 1984. A research program of the U.S. Air Force sponsored their investigation of the metallurgical properties of aluminum-iron and aluminum-manganese alloys. Shechtman and his coworkers mixed aluminum and manganese in a roughly six-to-one proportion and heated the mixture until it melted. The mixture was then rapidly cooled back into the solid state by dropping the liquid onto a cold spinning wheel, a process known as melt spinning. When the solidified alloy was examined using an electron microscope, a novel structure was revealed. It exhibited fivefold symmetry, which is forbidden in crystals, and long-range order, which is lacking in amorphous solids. Its order, therefore, was neither amorphous nor crystalline. Many other alloys with these same features have subsequently been produced.

The electron microscope has played a significant role in the investigation of quasicrystals. It is a versatile tool that can probe many important aspects of the structure of matter. Low-resolution scanning electron microscopy magnifies the shapes of individual grains. Symmetries of solid grains often reflect the internal symmetries of the underlying atomic positions. Grains of salt, for example, take cubical shapes consistent with the cubic symmetries of their crystal lattices. Quasicrystalline aluminum-copper-iron has been imaged using a scanning electron microscope, revealing the pentagonal dodecahedral shape of the grains. Its 12 faces are regular pentagons, with axes of fivefold rotational symmetry passing through them. That is to say, rotations about this axis by 72° leave the appearance of the grain unchanged. In a full 360° rotation the grain will repeat itself in appearance five times, once every 72°. There are also axes of twofold rotational symmetry passing through the edges and axes of threefold rotational symmetry passing through the vertices. This is also known as icosahedral symmetry because the icosahedron is the geometric dual of the pentagonal dodecahedron. At the centre of each face on an icosahedron, the dodecahedron places a vertex, and vice versa. The symmetry of a pentagonal dodecahedron or icosahedron is not among the symmetries of any crystal structure, yet this is the symmetry that was revealed in the electron microscope image of the aluminum-manganese alloy produced by Shechtman and his colleagues.

High-resolution electron microscopy magnifies to such a great degree that patterns of atomic positions may be determined. In ordinary crystals such a lattice image reveals regularly spaced rows of atoms. Regular spacing implies spatial periodicity in the placement of atoms. The angles between rows indicate rotational symmetries of the atomic positions. In a high-resolution electron microscope image of quasicrystalline aluminum-manganese-silicon, parallel rows occur in five sets, rotated from one another by 72°, confirming that the fivefold symmetry suggested by the shape of the pentagonal dodecahedron grain reflects a fivefold symmetry in the actual placement of atoms.

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 = (Square root of√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.

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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.

Quasiperiodicity

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.

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.