Crystallography & Symmetry
patterns derived from their groups
Concept

Zone boundary — where it appears

The set of wavevectors half a reciprocal lattice vector from the origin, where the phase from cell to cell is a sign rather than a general complex number. A mode there doubles the cell and keeps only half the translations.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

pgg at (1/2, 0): the operators multiply up to a sign. Every product of two Bloch operators of the little group of (1/2, 0) in pgg, against the operator of the product. They agree up to a scalar, and the scalar is +1 or −1: 4 of the 16 products come back with a minus sign. No rephasing removes them, and the search that says so tries every assignment of twelfth roots of unity to the operators. A representation that multiplies only up to this sign cannot be one-dimensional, because scalars commute and these operators do not. Each entry is an exponent modulo twelve, so the table is exact.

A glide sticks two levels together

The translation attached to a glide moves no wavevector at all. It comes back as a phase factor, and at the edge of the zone the factor is minus one — after which the operators of the little group no longer multiply the way the group does, and no rephasing repairs it.

space-groups · K symmetry 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 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 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.

Order parameterWavevectorEquivarianceGlide reflectionIrreducible representationSuperstructureAntiphase boundaryCouplingDegeneracyDipole momentFactor systemGlide plane

All concepts