Quasicrystals

On the morning of 8 April 1982, the materials scientist Dan Shechtman was staring at an electron-diffraction photograph of a rapidly cooled aluminium–manganese alloy, and it made no sense. Fire a beam of electrons through an ordinary crystal and the atoms scatter them into a pattern of sharp bright spots — a fingerprint of a regular, repeating atomic lattice. Shechtman had that: crisp, unmistakable spots. But the spots were arranged in a ring of ten, with a clean ten-fold (five-fold) rotational symmetry. He counted them again. He wrote in his notebook, "10 Fold???"

The problem was that ten-fold symmetry is supposed to be impossible. The crystallographic restriction proves that a periodic crystal can only carry 2-, 3-, 4- or 6-fold rotational symmetry — never 5-fold, never 10-fold — because no five-cornered tile can fill the plane by simple repetition. Yet the sharp spots screamed order. Shechtman had a material that was ordered and yet not periodic: a quasicrystal. This page teaches that one revolutionary idea — a solid whose atoms are the mathematics of Penrose tilings made real.

Worked example 1 — the paradox that made it a revolution

To feel why the result was so heretical, hold the two clues side by side. Everything hangs on reading each one correctly.

Clue A — the spots are sharp, so the material has long-range order. A diffraction pattern is, mathematically, the strength of the wave scattered off the atoms in every direction. If the atoms sat at random (like a glass or a liquid), the scattered waves would blur into a smooth, fuzzy halo — no spots. To get sharp spots, waves scattered from atoms far, far apart must arrive perfectly in step and reinforce. That only happens if the atomic positions are correlated over enormous distances: the material has genuine long-range order. So a quasicrystal is not amorphous, not random, not a glass.

Clue B — the symmetry is ten-fold, so the order is not periodic. The crystallographic restriction is airtight: a pattern that repeats by translation (shift it by a lattice vector and it lands on itself) simply cannot possess 5- or 10-fold symmetry. So if the diffraction genuinely shows ten spots evenly spaced by 360^\circ / 10 = 36^\circ, the underlying order cannot be built from a repeating unit cell. It is not periodic.

Put the two together and you are forced to an idea nobody had a name for in 1982: ordered but aperiodic. The atoms follow a strict long-range rule — a quasiperiodic one — that never exactly repeats, yet is deterministic enough to lock those scattered waves into sharp Bragg-like peaks. That is exactly the behaviour of a Penrose tiling: cover it and you find local five-fold motifs everywhere, matching rules that force order across the whole plane, and yet no translation ever carries the pattern onto itself.

A quasicrystal is a solid that is ordered but not periodic: its atoms have long-range quasiperiodic order, so they diffract into sharp spots, but the pattern never repeats by translation.
Because it is not periodic, it is free of the crystallographic restriction and can display the "forbidden" symmetries — 5-fold, 10-fold, 8-fold, 12-fold, and in 3-D the full icosahedral symmetry.
It is best understood as a physical Penrose-like tiling made of atoms; its sharp diffraction is the order it inherits from a higher-dimensional lattice via the cut-and-project method.

The picture that shocked the world

Here is the essence of what Shechtman saw, built up in three steps. A bright central spot is the straight-through beam; around it sit sharp diffraction spots. Notice that they fall on rings, ten to a ring, each spot exactly 36^\circ from its neighbour — the signature of ten-fold symmetry — while the rings themselves grow by factors of the golden ratio \varphi = \tfrac{1+\sqrt5}{2} \approx 1.618, the same number that governs the Penrose tiling. Sharp spots (order) arranged with a forbidden symmetry (not periodic): the whole paradox in one image.

Worked example 2 — real matter, real uses

Quasicrystals are not a laboratory curiosity confined to one odd alloy. Since 1982 hundreds have been made and characterised, and they turn out to be a genuine, stable phase of matter.

Al–Mn (the original). Shechtman's aluminium–manganese was metastable — quenched so fast the atoms froze mid-arrangement. It showed icosahedral symmetry, the 3-D analogue of the 2-D ten-fold pattern above.
Al–Cu–Fe and Al–Cu–Co (stable quasicrystals). Within a few years, alloys were found that form quasicrystals as their genuine equilibrium state — you can grow big, faceted single quasicrystals of them, just as you grow ordinary crystals.
Uses. Quasicrystalline surfaces are unusually hard, low-friction, poor at conducting heat, and resist sticking and corrosion — so they have been explored as non-stick, wear-resistant coatings (for instance on cookware and other surfaces) and as strengthening particles in alloys.
A natural quasicrystal. In 2009 a quasicrystal (icosahedrite, an Al–Cu–Fe mineral) was identified in a rock from the Khatyrka meteorite — proof that nature makes quasicrystals on her own, without a metallurgist's furnace.

All of this rests on the same mathematics you already know: the atoms sit at the vertices of an aperiodic tiling, and their sharp diffraction is the order inherited from a higher-dimensional periodic lattice, sliced and projected down into our space. If you want the machinery behind that sharpness, it lives in reciprocal-space diffraction and in the ordinary crystal lattices that quasicrystals so spectacularly break away from.

Shechtman's result was so heretical that his own colleagues turned on him. He was told to go re-read a crystallography textbook, and the head of his research group asked him to leave, saying he was bringing disgrace on the team. The most crushing blow came from the towering figure of twentieth-century chemistry, the double Nobel laureate Linus Pauling, who declared from the podium: "There are no quasicrystals, only quasi-scientists." Pauling insisted to the end of his life that Shechtman was simply misreading twinned ordinary crystals.

He was wrong, and Shechtman was right. As more stable quasicrystals were made and the diffraction was checked and re-checked, the community came around; in 1992 the International Union of Crystallography even rewrote its official definition of a crystal to make room for aperiodic order. In 2011, nearly three decades after that morning at the bench, Dan Shechtman was awarded the Nobel Prize in Chemistry — alone — "for the discovery of quasicrystals." A reminder that in science, a sharp diffraction photograph outranks the loudest authority.

The single most common mistake is to file a quasicrystal under "amorphous / random," or else to assume that "ordered" must mean "periodic." Both are wrong. A quasicrystal is not amorphous (a glass gives fuzzy halos, never sharp spots) and it is not a periodic crystal (a periodic crystal cannot have 5- or 10-fold symmetry). It sits in the third box almost everyone forgets exists: ordered but aperiodic — long-range quasiperiodic order.

That third box is precisely what makes the forbidden symmetries possible. Periodicity is what the crystallographic restriction rules out five-fold symmetry for; drop periodicity while keeping order, and the restriction no longer applies. So a quasicrystal can show its 5-, 10- or icosahedral symmetry and give sharp diffraction peaks at the same time — not despite being ordered, but because it is ordered in a way that never repeats.