Crystallography & Symmetry
patterns derived from their groups
Concept

Symmetry breaking — where it appears

The replacement of a symmetry group by a subgroup, whether by cooling through a transition or by applying a field. What is lost is counted by the index, and what is left decides which properties become permitted.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

Domains of a subgroup

A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

operations · Fundamental domain What a crystal keeps of itself in a field. Each class, with what is left of it when a field is applied along the axis of its own setting. The residual is the intersection of the class with the field's own group, computed on matrices and matched against the thirty-two rather than named by hand. Where the residual is the class itself, the field takes nothing away — and for an electric field those are exactly the polar classes.

What a crystal keeps in a field

Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.

point-groups · Curie p6m: freezing Γ2 leaves p31m. The same crystal three times. On the left, a pattern with the full symmetry of p6m. 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 p31m, of index 2 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.

An order parameter is a representation

The quantity that measures a lost symmetry is not a number the physics chooses freely: the parent group mixes its components, so it carries a representation, and the symmetry that survives is what leaves its value alone. Every prediction here is checked by moving the atoms and asking the detector.

space-groups · Landau 4mm: which phase depends on where the order parameter points. The plane of a two-dimensional order parameter, with each sampled direction marked by the symmetry that survives when the parameter points that way. The directions along which some operation is preserved are drawn large with a spoke to the centre; the general directions, where nothing survives, are the small faint marks between them. One representation, several phases — and the symmetry does not say which of them a crystal takes. That is decided by terms in an energy, which no symmetry argument supplies: what symmetry supplies is the list a material must choose from.

Which way the order parameter points

A two-component order parameter has a direction as well as a size, and the symmetry that survives depends on where it points. One representation therefore offers several low-symmetry phases — and symmetry, having produced the list, has nothing to say about which one a crystal takes.

space-groups · Landau

Named alongside it

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

Domain stateIsotropy subgroupKernelOrder parameterSubgroupTwo-dimensional representationAsymmetric unitAxial vectorClassificationCosetCrystal classCurie principle

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