Crystallography & Symmetry
patterns derived from their groups
Concept

Superlattice reflection — where it appears

A new reflection appearing between the parent's when a transition doubles the cell, at a position the enlarged cell's reciprocal lattice supplies. Its intensity grows as the square of the order parameter, so it measures the ordering directly.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

An antiphase boundary in p4. Where two antiphase states meet. Above the line the species alternate one way and below it the other, so at the boundary two cells of the same species sit next to one another and the ordering is out of step. This is a domain wall with no change of orientation across it: the crystal is not twinned, its lattice is undisturbed, and diffraction sees it only in the width of the superlattice reflections.

The domains a lost translation makes, which nothing optical can see

An ordering transition can leave the crystal class untouched and take away translations instead. The domains that result have the same orientation, the same shape and the same optical properties as each other, and where two of them meet the ordering is simply out of step — a boundary with no change of direction across it and no way to find it except by looking at the ordering itself.

applied · Domains p4m: freezing Γ3 leaves pmg. The same crystal three times. On the left, a pattern with the full symmetry of p4m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is pmg, of index 4 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

The cell a zone-boundary mode doubles

An order parameter that alternates from cell to cell keeps only half the translations, so the frozen structure has a cell twice as large and reflections that were never there before. The phases at such a wavevector are ±1, so the whole computation stays in exact integers.

space-groups · Landau How many superstructures a diffraction pattern tells apart. For two-species superstructures of the square lattice with one to twelve sites in the primitive cell: the number of structures, and the number of classes that the positions of their superlattice spots separate them into, the positions together with the composition, and every intensity. At twelve sites 2,012 structures give 167 sets of spot positions, 504 with the composition and 1,815 sets of intensities. At five, seven and eleven sites the positions distinguish only the cells; the intensities distinguish everything. The bar shows the share resolved by positions, dark, and by intensities, light.

The spots say the cell and little else

Of the 2,012 two-species superstructures of the square lattice with twelve sites in the cell, more than half light every superlattice spot their cell allows, so the positions of the spots say which cell and nothing more. Positions separate the 2,012 into 167 classes, the composition takes that to 504, and every intensity to 1,815. At a prime number of sites no spot is ever missing, because a sum of distinct p-th roots of unity cannot vanish.

operations · Counting When a three-species superstructure's spots can go dark. Every structure of three species on the square lattice with three to eight sites in its primitive cell, using all three species, and every superlattice spot of each, sorted by whether the spot is dark for every choice of scattering contrasts, dark at exactly one ratio of the two contrasts, or never dark; with the number of distinct ratios and their degree over the rationals. At the primes 3, 5 and 7 no spot is dark for all contrasts and no dark ratio is rational: none at three sites, quadratic ones at five and cubic at seven. At four and six sites every dark ratio is rational, at eight some are and some are not, and at six and eight some spots are dark for every contrast.

Three species hide a spot at the golden ratio

With two species and a prime number of sites in the cell, no superlattice spot can ever be missing, because a sum of distinct roots of unity of prime order vanishes only when it is all of them. With three species the structure factor weighs two sums against each other with two scattering contrasts, and the argument changes shape without breaking. At a prime size no spot is dark for every contrast and none at a rational ratio of them, but a spot can be dark at one irrational ratio: at five sites, a power of the golden ratio. It happens only in structures with a centre of symmetry.

operations · Counting Where an antiferromagnet's reflections land. Two cubic antiferromagnets, with their nuclear reflections as dots and their magnetic reflections as triangles, in units of the conventional reciprocal lattice. On the left, a simple cubic lattice with every neighbour opposite: the magnetic reflections sit at half-integer indices, in a layer the nuclear pattern never visits. On the right, a body-centred lattice with the corners up and the centres down: the magnetic reflections sit at integer indices with an odd sum — exactly the reflections the body centring had extinguished, lit up again by the moments. In both, no magnetic reflection falls on a nuclear one.

Magnetic reflections land where nuclear ones cannot

An antiferromagnet scatters neutrons into places its chemical structure leaves dark, and never into a place it lights. The reason is one line of algebra: the magnetic pattern is the nuclear pattern shifted by half a reciprocal-lattice vector. Each of a lattice's seven halvings is one such shift, and how many of them its symmetry keeps decides how many kinds of domain the magnet must form.

point-groups · Magnetic

Named alongside it

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

SuperstructureStructure factorAntiphase boundaryCyclotomic polynomialsDomain stateBurnside lemmaCentrosymmetryEnumerationEquivarianceGolden ratioHolohedryHomometry

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