Crystallography & Symmetry
patterns derived from their groups
Concept

Diffraction — where it appears

The scattering of radiation by a periodic or ordered structure into a pattern of sharp peaks. Nobody has seen a space group: crystals are read from where they scatter and, just as informatively, from where they do not.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

Order is not periodicity

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

aperiodic · Aperiodic What p3 scatters, and what the scattering shows. The structure on the left has point group 3, of order 3. The intensities it scatters, on the right, have point group 6, of order 6 — more symmetric than the thing that produced them. Reversing the sign of both indices conjugates every term in the sum and leaves the modulus alone, so a diffraction pattern always acquires a centre of symmetry, and in the plane a centre is a half turn. Both numbers are measured: the left from the operations, the right by testing each candidate against the computed intensities.

The symmetry diffraction adds

A diffraction pattern is always more symmetric than the crystal that made it. The extra symmetry is not a mistake in the experiment and no care removes it — it is a property of what a detector records, and it collapses the seventeen groups onto six.

diffraction · Accidental symmetry The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

point-groups · Diffraction 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern.

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

aperiodic · Monotile

Named alongside it

The objects these essays reach for when they reach for this one.

Point groupAperiodicityChiralityCut-and-projectFriedel lawLaue classStructure factorAccidental symmetryAnomalous dispersionAverage structureBragg peakCentrosymmetric

All concepts