Crystallography & Symmetry
patterns derived from their groups
Concept

Mechanical representation — where it appears

The representation carried by the displacements of a set of atoms, whose character counts the atoms each operation leaves in place. Multiplying that count by the trace of the operation's linear part and decomposing says which distortions those atoms can make.

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

p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set.

Which modes a site can carry

An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

operations · Fundamental domain 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

Named alongside it

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

Antiphase boundaryCharacterDisplacement modeEquivarianceFree parameterKlassengleicheOrbit-stabiliserOrder parameterSite symmetrySuperlattice reflectionSuperstructureWavevector

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