What Shechtman measured

On the eighth of April 1982, Dan Shechtman put a rapidly cooled aluminium–manganese alloy into an electron microscope and got a diffraction pattern with sharp spots arranged in tenfold symmetry. His notebook entry for the day reads, in Hebrew, “10 fold???”.

The three question marks are the right response and they are worth understanding properly. The observation was not surprising because it violated a theorem. It was surprising because the standard explanation for such a pattern — the explanation everybody reached for, including Shechtman — turned out not to fit.

What a diffraction pattern says #

Two features of a pattern carry almost all the information, and they say different things.

Sharpness says order. A reflection is sharp when contributions from atoms far apart arrive in phase. That requires the positions of distant atoms to be correlated — long-range order — and a diffuse or smeared pattern is the signature of its absence. Sharp spots had always come from crystals, and the inference “sharp, therefore periodic” was so reliable that nobody had separated its two halves.

The arrangement says symmetry. The spots sit at the reciprocal lattice points, and their arrangement has the symmetry of the crystal’s own point group. Tenfold spots mean a tenfold rotation somewhere in the structure, which the crystallographic restriction forbids to any lattice.

Sharp and tenfold together were therefore a contradiction, given the unstated premise that sharp implies periodic. The premise was the thing that failed.

Twinning, and why it was the right first guess #

The standard explanation for a diffraction pattern with a forbidden symmetry is twinning: several crystallites grown together in different orientations, each perfectly ordinary, superimposing their patterns.

Five crystallites of a twofold structure at $72°$ to one another produce a tenfold pattern. Every spot is a legitimate reflection from one of the five, the symmetry is an artefact of the superposition, and nothing anywhere is aperiodic. It is a common phenomenon, it explains this kind of observation almost every time, and Linus Pauling maintained it publicly for a decade.

Shechtman ruled it out, and how he did it is the part of the story usually left out.

Dark-field imaging. In an electron microscope, a single reflection can be selected and used to form an image, which lights up only the regions of the sample contributing to that reflection. If five twins were producing the pattern, each spot would light up a different fifth of the sample. Every spot lit up the whole grain.

Convergent-beam diffraction. Shrinking the beam to a few nanometres and moving it around gave the same tenfold pattern everywhere, from regions far smaller than any plausible twin domain.

The symmetry was icosahedral, not merely tenfold. Tilting the sample and recording patterns along different directions revealed axes of orders five, three and two arranged as an icosahedron — a full three-dimensional point group, not a coincidence of one projection. Constructing that from twins requires an implausibly specific arrangement of many orientations.

The measurement, in other words, was a negative one. What made it convincing was not the tenfold pattern but the exclusion of the explanation that would have made the tenfold pattern ordinary.

What was actually wrong #

The theorem was untouched. The failure was in a step nobody had written down.

The reasoning everybody had been using ran: sharp diffraction → periodic structure → lattice → restriction → no tenfold. The first arrow is the false one. Sharp diffraction requires long-range order and does not require periodicity, and no proof of the implication had ever been offered because nobody had noticed it was an implication.

Separating the two ideas is what the episode forced, and it is the whole content of the distinction between order and repetition. A quasicrystal is completely determined, has long-range order, diffracts sharply, and has no repeating cell. Before 1982 that combination was not thought impossible; it was not thought about.

Why publication took two years #

The paper — Shechtman, Blech, Gratias and Cahn — appeared in Physical Review Letters in November 1984, thirty-one months after the observation.

The delay was not mysterious. An extraordinary claim with an ordinary explanation available needs the ordinary explanation excluded thoroughly, and Shechtman spent that time excluding it. He was also, by his own account, asked to leave his research group over the claim, and was told to reread the textbook on the subject.

The reception after publication was worse. Pauling — the most decorated chemist alive, twice a Nobel laureate — proposed a series of increasingly elaborate multiply-twinned cubic structures with very large cells, and maintained until his death in 1994 that there were no quasicrystals, only “quasi-scientists”. Each of his proposed structures was tested against the data and each failed, but the objection had to be answered each time.

Two things resolved it. Levine and Steinhardt published a theoretical model within weeks of the 1984 paper, coining the word quasicrystal and showing the diffraction pattern such a structure would give. And in 1987 samples were grown large and well ordered enough to produce sharp X-ray patterns rather than electron ones, with diffraction peaks too narrow for any twinning model to reproduce.

Shechtman received the Nobel Prize in Chemistry in 2011, alone.

What replaced the assumption #

The higher-dimensional description arrived quickly and is what the field now uses.

A quasicrystal is described as a slice of a periodic structure in more dimensions than three — six, for icosahedral symmetry — and its diffraction pattern is indexed by six integers rather than three. In that description every familiar tool comes back: the spots are reciprocal lattice points of a six-dimensional lattice, the absences follow the usual rules, and refinement proceeds much as it does for a crystal, with the extra parameter being the shape of the acceptance window.

That is the cut-and-project construction, and it had been published by de Bruijn in 1981 — before the measurement — for reasons entirely internal to mathematics.

The definitional consequence came in 1992, when the International Union of Crystallography replaced “crystal” as a periodic arrangement with “crystal” as any solid with an essentially discrete diffraction pattern. Periodicity moved from the definition to a special case. It is unusual for a measurement to change what a word means, and this is one of the clearer instances.

What the material turned out to be #

The diffraction pattern says where the atoms scatter and not where they are, so the structure took longer than the symmetry, and the answer is worth a paragraph because it is less exotic than the pattern suggests.

Icosahedral quasicrystals are built from clusters — icosahedral or nearly icosahedral shells of a few dozen atoms — arranged so that the cluster centres form a quasiperiodic point set. The clusters themselves are ordinary chemistry, of the kind found in perfectly periodic intermetallic compounds; what is unusual is only their arrangement. Several periodic phases in the same alloy systems, the approximants, are built from the same clusters arranged periodically, and their structures were solved first and used as a way in.

That relationship is exactly the rational-slope picture from the cut-and-project construction. An approximant is what comes out at a rational slope close to the golden one: a periodic structure with a large cell, locally almost indistinguishable from the quasicrystal, and much easier to solve. Working outward from the approximants is how the quasicrystal structures were determined.

What quasicrystals turned out to be good for #

The applications are a small and slightly odd list, and they follow from a single structural fact: a quasiperiodic lattice has no slip planes.

Hardness and low friction. Dislocations move easily along the close-packed planes of a periodic crystal, which is why metals are ductile. A quasicrystal has no such planes, so dislocations are pinned, and the material is hard and brittle with an unusually low coefficient of friction. Quasicrystalline coatings were sold commercially as non-stick surfaces for cookware.

Low thermal and electrical conductivity. Electron and phonon transport both rely on the extended states a periodic potential supports. A quasiperiodic potential supports states that are neither extended nor localised, and the result is a metal made entirely of good conductors that conducts poorly — aluminium, copper and iron in an icosahedral phase behave more like a semiconductor than like any of their constituents.

Precipitation hardening. The most economically significant use is the least glamorous: quasicrystalline particles precipitated in a steel matrix act as hardening agents, and are used in surgical instruments and in some maraging steels.

None of these is a large industry, and the honest summary is that quasicrystals matter far more for what they forced the subject to reconsider than for what has been built from them.

Where the exactness stops #

This site’s machinery decides nothing about any of this, and the boundary is sharper here than on any other page.

The integer round trip is two-dimensional and periodic. It has no lattice to work with in a quasicrystal, no finite holohedry to enumerate, and no exact coordinates to compare. Every quantity reported about an aperiodic pattern here is a measurement with its tolerance stated.

The tenfold figure is a computation, not a photograph. It is built from integer combinations of ten star vectors — the reciprocal-space description of a quasicrystal — and it produces the arrangement such a structure would scatter. What it is not is Shechtman’s pattern, or any measured pattern. The historical claims on this page are cited from the literature and are not verified by anything here.

And the material claim is a model. That a real alloy is a six-dimensional projection is supported by how well the indexing works and by the sharpness of the peaks, and it is not a mathematical fact. This site can say that the mathematics permits such a structure. Whether a given lump of aluminium–manganese is one is an experimental question.

The surprising part #

The most useful thing to take from the episode is not about crystals at all. It is that the load-bearing assumption was invisible because nobody had ever needed to state it.

“Sharp diffraction implies periodicity” was not a belief anybody held, examined and defended. It was a step so reliably true in every case anybody had met that it had never been separated into a premise and a conclusion. That is the characteristic form of an assumption that fails badly: not a claim under dispute, but a joint between two ideas that nobody had noticed was a joint.

The connection worth carrying is to what this site’s figures are built against. The accidental-symmetry hazard is the same shape of error, one level down: a pattern drawn with a dot has more symmetry than intended, the picture looks entirely right, and the failure is invisible precisely because nothing about it is unusual. In both cases the remedy is the same — state the step, and give it a test it could fail. What took the subject thirty-one months in 1982 was building the test.

The measurement that would have settled it sooner #

With hindsight there is a single measurement that separates a quasicrystal from a twinned crystal cleanly, and it is worth naming because it explains why the argument lasted as long as it did.

Peak width. A twinned crystal with a very large cell has diffraction peaks whose width is set by the size of the individual domains. A quasicrystal’s peaks are limited only by the coherence of the instrument, because the order extends across the whole grain. So a sufficiently good measurement of peak width distinguishes the two, and Pauling’s models all predicted peaks measurably broader than the ones eventually observed.

That measurement was not available in 1982. Electron diffraction, which is what Shechtman had, gives peak positions well and peak widths badly — the spots are broadened by the instrument and by the thinness of the sample, and the intrinsic width is not recoverable. Only X-ray diffraction from a large, well-ordered single grain gives the width honestly, and grains that good were not grown until 1987.

So the five years between the observation and its general acceptance were not five years of argument about interpretation. They were five years of metallurgy: the decisive experiment was known and the samples for it did not yet exist. That is a more ordinary and more useful account of a scientific controversy than the one usually told, in which a stubborn establishment eventually gives way.

The general shape is worth extracting. The claim that was doubted and the measurement that settled it were about different quantities — the symmetry of the pattern, and the width of its peaks — and identifying which measurement would decide the question was most of the work.

Who else was nearly there #

Two near-misses are worth recording, because they show how ready the ground was.

Alan Mackay, working in London, published in 1981 an optical diffraction pattern computed from a Penrose tiling — showing that such a pattern gives sharp tenfold spots — and asked in print whether any real material might do it. His paper predates Shechtman’s observation and describes exactly what Shechtman found.

And there were earlier electron diffraction patterns with forbidden symmetries in the literature from the 1970s, recorded, published and explained away as twinning by authors who had no reason to doubt the explanation. The observation was not rare. What was rare was the refusal to accept the standard account of it.

Ishimasa, Nissen and Fukano reported a twelvefold pattern in nickel–chromium in 1985, from work begun independently. Within five years quasicrystals had been found in dozens of alloy systems, and in 2009 a naturally occurring one — icosahedrite — was identified in a meteorite fragment from Kamchatka, having formed in the early solar system.

Where the ladder goes next #

The distinction the measurement forced is order without periodicity, and the theorem it did not break is the crystallographic restriction.

The mathematics that was waiting for it is cut and project and the dimension where five-fold is legal; the patterns it turned out to describe are Penrose tilings and their inflation hierarchy.

What the pictures here cannot show. The tenfold pattern on this page is computed from a mathematical model of a quasicrystal, and the entire argument of 1982 was about telling such a pattern apart from a superposition of five ordinary ones — which look, in a single image, indistinguishable. Nothing in a diffraction figure settles that question. It was settled by dark-field imaging and by moving the beam, and neither is a picture of the pattern.