Crystallography & Symmetry
patterns derived from their groups
Concept

Order parameter — where it appears

The quantity that is zero in a crystal's high-symmetry phase and non-zero in the low-symmetry one, whose components the parent group mixes among themselves. It carries a representation, and the symmetry that survives is what leaves its value alone.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

A gap of exactly 1.00. The two folded bands with the ordering switched on. The faint curves are the same bands before it, crossing at the boundary of the reduced zone; the ordering couples them there and separates them by exactly twice its own strength. The gap appears at the wavevector where the superlattice's extra reflections appear, and for the same reason: both are the Fourier component of the potential at that wavevector.

A bigger cell, a smaller zone

Ordering two kinds of atom onto a sublattice adds reflections to the diffraction pattern and opens a gap in the levels. It is one fact told twice: the same Fourier component of the potential, at the same wavevector, doing the same thing.

lattices · Wigner–Seitz cells 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 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 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 Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character.

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

applied · Domains A mode with no dipole, landing in a phase that may have one. Two marks per row: the first is filled when the mode itself carries a dipole — the displacements, weighted by charge, summing to something other than zero — and the second when the class of the phase it produces permits a polarisation at all. A row with the first empty and the second filled is an improper case: nothing about the transition was about becoming polar, and the phase that results may be polar anyway, so a polarisation appears as a side effect at second order in an order parameter that is about something else. The zone-boundary rows are where these occur; at the zone centre in the plane there are none, because the only two-dimensional order parameters available there are the polarisation itself.

The polarisation nobody asked for

A mode whose displacements cancel exactly can still leave a phase whose class permits a polarisation. The crystal then becomes polar as a side effect of a transition that was about something else — and in the plane, at the zone centre, the arithmetic says this cannot happen at all.

applied · Domains The kinds of line defect each breaking allows. The eleven proper crystal classes, each with the order of the binary group that covers it, the number of conjugacy classes of that group other than the identity — which is the number of kinds of line defect — and whether the group commutes. 6 of the eleven do not, and in those media two defect lines cannot pass through each other without leaving a third line behind.

The defect that needs two laps

Which defects a medium can have is not a fact about the medium. It is a fact about the space its order parameter lives in, and for a rotational symmetry broken down to a point group that space has a fundamental group twice the size of the point group. The kinds of line defect are its conjugacy classes — and in six of the eleven cases they do not commute, which means two defect lines cannot pass through each other.

applied · Defects Seven fields and the number that counts each one. The degree of each field, computed by triangulating the sphere drawn round the defect, mapping every vertex, and adding the signed areas of the image triangles. The total is 4π times the degree, and the integral column is that total divided by 4π before rounding. Each is read on three successively finer meshes and required to give the same integer on all three, because a mesh too coarse for its field does not produce a noisy answer — it produces a confident wrong one.

The point defect whose charge has no sign

A line defect is read on a loop; a point defect is read on a sphere, and the number that comes off the sphere is a degree. In a nematic that degree is an integer whose sign depends on a choice nobody can make — and the media where no such number exists at all are exactly the ones whose residual symmetry is a crystal class.

applied · Defects Where a gradient term is allowed at a special point. For each of the seventeen plane groups and each special point of its zone — the centre, the edge midpoints, the corner of a rectangular or square zone and the corner K of a hexagonal one — every physically irreducible order parameter, by its real dimension, marked where it admits a Lifshitz invariant. They occur at 10 points in 7 groups: the edges of the glide groups pg, pmg, pgg and p4g, and the corner K of p31m, p6 and p6m. Nowhere else.

Where a gradient term survives at the edge of the zone

A transition that freezes a mode at the edge of the zone doubles the cell, but only if the energy has no term linear in the mode's gradient. That condition was stated here for the zone boundary and never computed. Computed for every order parameter at every special point of the seventeen zones, it fails in exactly ten places: the edge points of the four glide groups where one glide acts and no second glide crosses it, and the zone corner of three hexagonal groups that have no glide at all.

space-groups · Landau 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.

Zone boundaryEquivarianceLandau conditionLifshitz invariantWavevectorCouplingDefectDisclinationDomain stateGlide reflectionHomotopyIrreducible representation

All concepts