YOU HAVE just moved into a new house, and you are choosing tiles to lay on the kitchen floor. Like most people, you don’t want an untidy mixture of shapes, but a single tile shape that repeats again and again. Triangles, rectangles or even hexagons will do: any one of these shapes can be used to cover the floor in a repeating pattern. But there are some shapes that you will not be able to use. Take pentagons, say, and no matter how hard you try to fit them together they will always leave annoying gaps.
The same phenomenon crops up when you start to think about how atoms are arranged in three dimensions in a solid. Most solids fall into one of two classes: crystals and glasses. In a glass, the atoms are arranged completely randomly, so the problem of regular patterns does not arise. But the atoms of a crystal sit in neat, periodic structures, just like three-dimensional kitchen tiles. It’s easy enough to find crystals built around threefold, fourfold or sixfold symmetry-with the atoms arranged in the 3D equivalents of triangles, squares and hexagons. But you will never find a crystal based on fivefold symmetry.
This notion is embodied in a set of mathematical rules that were developed nearly 150 years ago by the French physicist Auguste Bravais, and has long been firmly fixed in the minds of physicists. So people were stunned when Dan Shechtman, now at the Technion in Israel, and his colleagues at the National Bureau of Standards in Gaithersburg, Maryland announced just over a decade ago that they had found a strange new solid alloy that broke these rules. Their alloy of aluminium and manganese looked very much like a crystal, but its atoms were arranged with fivefold symmetry for six different axes. This arrangement flies in the face of Bravais’s rules. Yet, since then, many more of these strange “quasicrystal” solids have been found.
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As well as being a cause for scientific curiosity, quasicrystals seem to have novel electronic and mechanical properties. Researchers have high hopes that they could turn out to be invaluable in a host of practical applications-perhaps as exceptionally hard or slippery materials, or components in electronic devices. This has given extra impetus to physicists who are scrambling to understand why quasicrystals form, and so perhaps predict which as yet unknown alloy will have this strange new form. Last year, an explanation emerged that could resolve the mystery.
Quasicrystals are a sort of halfway house between crystals and glasses. Their atoms are neither periodically nor randomly spaced. Instead, they have a complex, predictable pattern of spacings, which mathematicians call “quasiperiodic”. While ordinary crystals are built up from identical clusters of atoms that are equally spaced, quasicrystals are composed of clusters that sit at two distinct distances from their neighbours. Divide the number of times the long-distance separations occur in a crystal by the number of the short distances and you get an irrational number-one that cannot be expressed as a ratio of integers. This is characteristic of their strange almost-order.
It’s hard to picture, but fortunately much of the mathematical groundwork was done more than twenty years ago. Back in 1974, the Oxford mathematician Roger Penrose was interested in whether it is possible find a set of tiles that completely covers a surface with patterns that do not repeat periodically. The solution he hit on involved tiles with two different diamond shapes: one fat and one thin. To prevent the tiles from joining together in periodic patterns, he developed a set of rules for joining neighbouring tiles together. Although these “matching rules” are complicated, one way to represent them simply is to paint a pattern of ribbons on the tiles and say that two tiles can join together only if the ribbons match at the interface (see Diagram).
Even before the first quasicrystal was discovered, Dov Levine, now at the Technion in Israel, and I were already working on extending Penrose’s concept into three dimensions. We had found that the Penrose pattern is quasiperiodic, and discovered that by packing together specially shaped “bricks” we could make an analogous, three-dimensional structure with a symmetry that is forbidden according to the conventional Bravais rules. We conjectured that real solids might exist, in which each brick is replaced by a cluster of atoms. We dubbed the hypothetical solids “quasicrystals”. We even worked out how the solid would scatter or diffract electron beams to give a characteristic pattern of spots. To our astonishment, when Shechtman presentedhis peculiar new alloy of aluminium and manganese, the diffraction pattern looked almost identical.
Awkward alloy
So does this mean that quasicrystals are simply Penrose tilings in three dimensions? Some awkward facts got in the way of this explanation. According to our model, the spots in the diffraction pattern should be as sharp as for an ordinary crystal, and arranged with the normally forbidden fivefold symmetry. But in all the early quasicrystal alloys, including aluminium-manganese, the peaks were not quite as sharp as they should have been, and were arranged very close to, but not quite perfectly at, fivefold symmetric positions. Some people began to doubt whether the Penrose structure was physically possible, and began to work on alternative models that involved a disordered atomic structure.
But within a few years, new quasicrystalline materials were found in which the quasiperiodic order is as perfect as the periodic order found in the best crystal alloys. They produced spectacularly sharp diffraction patterns, with spots in exactly the right places, and other tests showed that they had virtually no disorder in their structures. The weight of experimental evidence was swinging back in favour of Penrose tiling.
But the problems did not end there. Many theorists still rejected the idea that Penrose tiling could explain how quasicrystals form, because it seemed to require atoms to behave quite differently from the way they do in ordinary crystals. The building block of an ordinary crystal is a cluster of atoms called the unit cell. The unit cells that go to make up a particular crystal are identical to each other, and stack together like toy bricks. But in the Penrose tiling picture, the atoms would have to organise themselves into two different types of cluster, with just the right ratio of densities to act as the building blocks of the structure.
Moreover, the Penrose matching rules are complicated, permitting tiles to join only at certain angles and orientations, so that very complicated atomic forces would be needed to make the clusters join together in just the right way. All this seems thoroughly improbable, not least because all the quasicrystals discovered so far are metal alloys, with simple atomic forces. So we were left with a curious situation: the more that experiments supported the idea that Penrose tiling was responsible for quasicrystals, the more theoretical ideas seemed to call it into question.
With these concerns in mind, Hyeong-Chai Jeong at the University of Maryland in College Park and I came up last year with a new explanation of why quasicrystals form. We were inspired by the German mathematician Petra Gummelt, who suggested last year that it might be possible to construct a quasiperiodic tiling using only a single shape of tile, plus matching rules for how adjacent tiles join. The tile is a decagon (see Diagrams ), but the key difference between this and previous approaches is that neighbouring tiles can overlap, as long as the overlaid regions match in colour according to one of the two templates shown in the diagram.
First we provided a simple proof of Gummelt’s claim. We showed that, if each decagon is decorated with a fat diamond, the decagon tiling looks just like a Penrose tiling, with spaces left automatically for the thin diamonds. What’s more, we showed that the overlap rules are equivalent to the Penrose matching rules. So we established that Gummelt’s tiling really is quasiperiodic, and is mathematically equivalent to a Penrose tiling.
The result forces us to revise our ideas about quasicrystals. It now seems that you only need one atomic cluster as the basic building block for quasicrystals, just as for periodic crystals. According to this picture, the main thing distinguishing quasicrystals from crystals is that for quasicrystals the repeating clusters are allowed to “overlap”. This probably means that two neighbouring clusters share atoms, rather than actually interpenetrating. Experimental evidence neatly bears out this idea. From our understanding of the atomic structures of the materials involved, we have known for nearly a decade that most quasicrystals are composed of clusters that share atoms. Many researchers, ourselves included, believed that these overlapping clusters must be an important part of the picture. Now this notion is on firm mathematical footing.
But this still leaves another problem. Even if nature only needs one basic building block, rather than two, how does it enforce the matching rules? When a liquid cools to form a periodic crystal there are no matching rules. Instead it is energy that dictates what happens: the atoms simply shake about, arranging and rearranging themselves until they find the structure with the lowest possible energy for a given volume. A good way to identify the lowest-energy structure is to work out which of the clusters has the lowest possible energy. (This energy is usually negative). The structure with the lowest energy per unit volume will then be the one that packs such cells together as densely as possible. This very often produces a crystal, since the densest way to pack most clusters is in a side-by-side periodic crystalline arrangement.
Jeong and I decided to see whether a similar approach could work with quasicrystals. To study the general principle, we first tried out the idea on tilings rather than atomic models. So we returned to the fat and thin rhombi, but this time imagined constructing tilings without imposing any matching rules at all. If you repeat this over and over, you will end up with many different arrangements, some periodic, some random, and a tiny fraction, Penrose tilings.
Nature’s choice
The next question is why, if so few of the patterns end up as Penrose tilings, should nature ever choose just these? To try and answer this, we looked for a cluster of tiles that would represent the lowest-energy cluster of atoms. When we worked out the energy for each of the different tiling arrangements, we found that those with the most clusters had the lowest energy. And if you pick your cluster right, the lowest-energy solid, with the highest density of low-energy clusters, is sometimes a Penrose tiling.
But what is special about the clusters that lead to a Penrose tiling rather than a crystal? We found that the clusters that work can all share tiles with neighbouring clusters. In other words, a quasiperiodic packing, where clusters overlap, allows them to pack together more closely than in any periodic packing, where they sit side-by-side. The sharing is very similar to the overlap of decagons in Gummelt’s matching rule. But it is not forced by some rule, it just emerges naturally as a way to get the lowest-energy structure. We were surprised to discover that Penrose tiling can give the most efficient sharing for some choices of clusters.
This picture suggests that quasicrystals have more in common with ordinary periodic crystals than anyone had realised. Whether a crystal or quasicrystal forms just depends on the nature of the lowest-energy cluster of atoms for the material’s elements. What matters is how the clusters can stack, and whether neighbouring clusters can reduce their energy by sharing atoms.
These speculations can be tested. Theorists know enough about the atoms that make up some crystals to be able to calculate the energy and stability of the low-energy clusters and to work out if cluster overlap is possible. They may be able to suggest chemical mixes to enhance quasicrystal formation, which could be tested experimentally.
The question is not just academic. Time and again, when physicists have cooked up a new material, someone has found a useful new application for it. The properties of quasicrystals are being explored by researchers in France, Germany, Japan, Canada and the US. Quasicrystals could well turn out to be harder than ordinary crystals, because there are no neat planes of periodically spaced atoms along which cracks and defects can propagate. Experiments by Alan Goldman and his co-workers at Ames Laboratory and Iowa State University are already bearing this out. The researchers propose using a quasicrystalline alloy to coat surfaces subject to abrasive environments, such as pipelines carrying corrosive, gritty fluids.
Some quasicrystals also have slippery surfaces, which may allow them to replace other nonstick materials on kitchen utensils or other tools. Also, quasicrystals have unusually high electrical resistivity at low temperatures compared to crystalline metallic alloys. In ordinary metals, electrons encounter regular rows of atoms, which they can pass along without scattering. This makes it easier for them to pass through the crystal, and decreases the resistance to current. But because quasicrystals do not have this regular arrangement, the electrons find it much more difficult to pass. Although it is not clear yet what this could be used for, quasicrystals might find applications as electrical components, perhaps in electronic devices.
These are tantalising possibilities. But before we can put quasicrystals to work, we need to be able to control their chemical composition, and perhaps work out which new alloys are most likely to end up as quasicrystals. A better understanding of what led to the formation of the quasicrystals we already know about is a crucial first step.
