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 3 essays across 3 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 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.

Antiphase boundaryDomain stateSuperstructureEquivarianceHolohedryIndexIndex two subgroupKlassengleicheMechanical representationOrbit-stabiliserOrder disorderOrder parameter

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