Crystallography & Symmetry
patterns derived from their groups
Concept

Satellite reflection — where it appears

A diffraction spot at a position that is not a reciprocal lattice vector, sitting beside a main reflection at a displacement given by the modulation wavevector. Its index is a second integer counting how many multiples of that wavevector away it lies, and its intensity measures the modulation's amplitude.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.

The satellites that need a second integer

A crystal whose atoms are displaced by a wave of the wrong wavelength has no unit cell at all, and diffracts to sharp spots anyway. Indexing them takes two integers per reflection instead of one — and the intensities of the extra spots are Bessel functions, which is a check the arithmetic can be made to pass.

aperiodic · Modulation 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

aperiodic · Modulation

Named alongside it

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

Arithmetic classBessel functionCohomologyGroup extensionIncommensurateIndexingModulated structureModulationOrigin shiftPlane groupPoint groupReciprocal lattice

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