杏吧原创

Draw a triangle with angles that add up to more than 180掳. Impossible?
Well, only if you鈥檙e locked into Euclid鈥檚 world of geometry on a flat plane. But
draw it on the surface of a sphere and the old geometrical rules
change鈥攖he seemingly impossible becomes possible.

A simple change of perspective can work wonders, solving problems that have
baffled people for years. Such a change is taking place in materials science. In
1982, a class of alloys was discovered that broke all the old rules of
crystallography鈥攖he impossible symmetry of these 鈥渜uasicrystals鈥� stumped
the experts. That is, until two months ago, when a paper in Nature
finally showed how quasicrystals sidestep the rules.

The central idea of crystallography is that crystals are orderly arrangements
of identical building blocks, or unit cells, that fit together neatly like tiles
on a kitchen floor. For example, a unit cell might be a cube with atoms at each
corner, as in salt crystals.

Play around with tiles of different shapes, and you soon find that only
certain shapes can fit together without leaving gaps. With triangular, square,
or hexagonal tiles, it is easy to cover a floor completely. But with a
five-sided tile, you鈥檒l never find a way to cover the floor without leaving
gaps.

The same is true for the three-dimensional building blocks of crystals.
Shapes with fivefold symmetry, such as pentagonal prisms, simply cannot be
stacked together properly. Indeed, as early as 1848 a Frenchman called Auguste
Bravais worked out that there are just seven basic shapes that fit together
without leaving gaps. They can have two, three, four or sixfold
symmetry鈥攂ut not fivefold.

This dogma was unchallenged for more than 130 years. In the 20th century,
crystallographers confirmed it by firing beams of X-rays at their samples, and
working out the crystal symmetries from the diffraction patterns produced. Every
material fell into one of the traditional patterns.

Then in 1982 came a shock. Materials scientists were working on a new alloy
that looked like a crystal to the naked eye. But the diffraction patterns
clearly showed that it had fivefold and tenfold symmetry. How could that be?

There was already one promising way to explain these weird solids. Back in
the 1970s, the Oxford mathematician Roger Penrose had been fooling around with
ways to cover a surface with tiles. He found that if you used two types of tile,
a fat diamond and a thin diamond, you could build up patterns with quasiperiodic
order. Unlike a pattern made from identical building blocks, a quasiperiodic
pattern doesn鈥檛 repeat exactly over short distances. The Penrose tiling looks
cockeyed close up, yet over large distances the tiles tend to form repeating
patterns with fivefold order, just like quasicrystals.

But Penrose tiling has problems as a model for real quasicrystals. 鈥淔irst,
you need two kinds of tile or cluster, as opposed to just one for regular
crystals,鈥� says Paul Steinhardt, a physicist at Princeton University. But why
should there be two types when the rest of the crystalline world makes do with
one? Worse, these clusters are hard to put together properly. 鈥淚f you鈥檝e ever
played with these things, you start laying out the tiles and make a mistake and
have to rearrange them and go back again and again,鈥� says Steinhardt. How
could quasicrystals form so quickly if nature plays that game?

All this left things in a pretty mess. There was a feeling among researchers
that these things were alchemy. You throw together a bunch of metals, heat and
cool, and hey presto, the atoms find their way into an ordering that doesn鈥檛
make sense.

The beginnings of a breakthrough came in 1996, when Petra Gummelt, a
mathematician at the University of Greifswald in Germany, showed that ten-sided
tiles, decagons, could be the fundamental unit of quasiperiodic systems鈥攊f
they are allowed to overlap with each other. This is the crucial change of
perspective.

鈥淥verlapping is a really bad idea if you鈥檙e making bathroom tiles, where the
tiles all bump into each other,鈥� explains Steinhardt, 鈥渂ut for atomic structure,
it鈥檚 a beautiful idea. What鈥檚 really going on is that you have overlapping
atomic clusters that share atoms.鈥� When putting these things together, you have
to make sure that the internal patterns of atoms match wherever there is an
overlap鈥攕o a mathematical matching rule is still needed.

Perfect match

Later in 1996, Steinhardt and Hyeong-Chai Jeong of Sejong University in Seoul
were inspired by Gummelt鈥檚 work to prove rigorously that overlapping decagonal
tiles could be used to construct quasiperiodic patterns. They showed that
arrangements of a single decagonal tile were equivalent to a Penrose tiling
(see diagram). So, rather than having two types of building block,
quasicrystals really have just one, like periodic crystals
(鈥淐razy crystals鈥�, New 杏吧原创, 25 January 1997, p 32).

But Steinhardt and Jeong went further, and showed that decagons make more
sense than Penrose tiles. Using decagons, you can replace the mathematical rules
for how to overlap tiles with a more physical requirement: quasicrystals try to
minimise their energy. This shouldn鈥檛 be surprising鈥攁fter all, regular
crystals form because atoms try to find the configuration with lowest energy.
The trick with quasicrystals was working out how that physical imperative
resulted in the weird structure.

Steinhardt鈥檚 idea was that in a quasicrystal, atoms like to form low-energy
clusters. But these clusters can overlap, so the greater the overlap, the lower
the overall energy. As the quasicrystal grows, clusters jostle around trying to
overlap as much as possible. Penrose tiles, coming in two varieties, would have
a much harder time getting together in this way.

So at last there was a physical theory that could explain quasicrystals. But
was it right? In 1997, Steinhardt received some high-resolution electron
microscope pictures from An Pang Tsai at the National Research Institute for
Metals in Japan, showing some of the extraordinarily high-quality quasicrystals
that Tsai鈥檚 group had made with an aluminium-nickel-cobalt alloy. This alloy
(Al₇₂Ni₂₀Co₈) has the advantage that it is made of
stacked two-dimensional layers, like graphite. It is only quasiperiodic within
these 2D layers, so there鈥檚 no need to look at its full 3D structure.

Up close, the pictures all showed clear signs of Gummelt鈥檚 decagonal
structures, which Steinhardt calls 鈥渜uasi-unit-cells鈥�. Indeed, the two groups
found that they could overlay the micrograph with a quasiperiodic pattern of
Gummelt tiles that perfectly matched the pattern of atoms.

This pattern was also consistent with the density and composition of the
Al₇₂Ni₂₀Co₈ alloy. Steinhardt and Tsai did a
calculation in which they juggled the identities of the atoms at each lattice
position. They found an arrangement that matched the density and composition
of the alloy almost perfectly (see diagram).

And the final triumph for decagon tiles is that, when you look on the level
of atoms, the alloy doesn鈥檛 follow a Penrose pattern. 鈥淢athematically, we had
shown that every Penrose tile pattern is also a decagon tiling, but the converse
is not true,鈥� Steinhardt explains.

In other words, Gummelt鈥檚 tiles can create a large universe of quasiperiodic
patterns, but Penrose鈥檚 tiles only work for a subset of this. If you take some
decagon-tile arrangements and sketch in a pattern of Penrose tiles, you find
that not all the diamonds contain the same pattern of atoms鈥攚hich makes a
nonsense of the original idea of using identical tiles. And Tsai鈥檚 material
turns out to be an example of this.

The overlapping cluster model has breathed new life into quasicrystal
research. 鈥淲e need to redo quasicrystallography because people had abandoned the
unit cell,鈥� says Steinhardt. The next step will be to apply the model to 3D
quasicrystals, he says.

Understanding quasicrystals should make it easier to grow them, and to
predict which alloys will form quasicrystals. And this could be important, as
the special properties of quasicrystals are already finding uses.

For example, it may be possible to create stronger metal alloys. The Swedish
steel maker Sandvik is making surgical steel alloys with quasicrystal
inclusions. Like the gravel in concrete, the quasicrystals toughen the mixture
because they don鈥檛 have convenient fracture planes, and so fend off cracks that
would propagate along the regular planes of normal crystals.

That鈥檚 not all. A French cookware company, Sitram, is selling a frying pan
with a quasicrystal nonstick coating. The pentagonal surface structure makes the
quasicrystal slippery, perhaps because ordinary stuff can鈥檛 match itself up with
this unusual pattern, so you don鈥檛 get intimate bonding. And being metal, the
quasicrstal coating is much tougher and more temperature-resistant than plastic
coatings such as Teflon.

And never one to miss a high-tech beat, the military already has its eye on
quasicrystals. The US Air Force Office of Scientific Research wants researchers
to come up with weapons applications. There are also potential applications in
electronics, because quasicrystals have strange conductive properties that are
relatively insensitive to temperature鈥攑robably because electrons have a
harder time negotiating the quasiperiodic pathways than they do neat periodic
lattices.

So these disobedient structures look like becoming extremely useful.
Sometimes it helps to break the rules.