Crystallography & Symmetry
patterns derived from their groups
Concept

Lifshitz invariant — where it appears

A term in an energy linear in a gradient of the order parameter and antisymmetric in its components. Where one exists the uniform state is not the best one and the crystal prefers a modulation whose period symmetry does not fix.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Which order parameters carry a cubic invariant, and therefore cannot grow from zero. Every order parameter of every plane class, with the number of independent cubic invariants it admits. The count is the degree-three coefficient of the Molien series of the representation's image — the same computation the invariant-ring figures make for a different reason — and Landau's condition is that it be zero. Where it is not, a free energy in that order parameter has a term of odd degree, which puts its minimum away from zero the moment the quadratic coefficient does anything at all, so the parameter jumps rather than growing. In the plane exactly two order parameters carry one, and both are the two-dimensional representation of a class with a threefold axis and no sixfold.

The cubic term that forbids a continuous change

A crystal may lose a symmetry gradually only if the quantity measuring the loss admits no cubic invariant. Whether it does is the third coefficient of a Molien series — so a question about how a material changes is answered by counting polynomials.

point-groups · Invariants Eight of the thirty-two classes allow a gradient term at the zone centre. The thirty-two crystal classes, marked by whether any order parameter at the centre of the zone has a term linear in its gradient. In the plane none does. In space eight do: 3, 4, 6, 32, 422, 622, 23 and 432. All eight are among the eleven classes with no operation that reverses hand, and the other three, 1, 2 and 222, have only one-dimensional order parameters.

A screw keeps a line and a glide keeps a plane

The census of the seventeen zones left two predictions for space. A transition at the zone boundary should meet a gradient term where one screw or glide is active, along that operation's direction, and a second should remove it. Across all 219 types, the first half holds without exception in the monoclinic groups and the direction is wrong: a glide's term fills its plane. The second half fails from the orthorhombic groups on. And the zone centre, empty in the plane, is not empty in space.

space-groups · Landau

Named alongside it

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

Order parameterLandau conditionCubic invariantDomain stateFirst order transitionGlide reflectionInvariant polynomialIrreducible representationIsotropy subgroupKernelMolien seriesPhase transition

All concepts