Crystallography & Symmetry
patterns derived from their groups
Concept

Polar class — where it appears

A crystal class fixing at least one direction, and therefore able to carry a built-in polarisation. The permission is what symmetry supplies; whether a crystal in such a class actually has a dipole is a question about the material.

Named by 7 essays across 5 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 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 rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither.

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

restriction · Finite groups A thread's two signs, and the four kinds of operation. Every operation of a rod group carries the axis to itself, so it does two independent things: it keeps or reverses the direction along the thread, by a sign σ, and it keeps or reverses the handedness of the plane across the thread, by the determinant of a 2 × 2 matrix. The determinant in space is the product, so the shaded cells are the proper operations — a turn or screw about the axis, and a half-turn crossing it — and the unshaded ones are the improper. A rod group is chiral when all of its operations sit on the shaded diagonal, and polar along its axis when all of them sit on the top row. The two conditions pick out different diagonals of the same square, which is why neither implies the other.

A thread's hand is not a choice

A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.

space-groups · Chirality Two hundred and twelve of the 219 are made of their own operations. The 219 affine space-group types divided by which of their own operations generate them, lattice translations included. 110 are generated both by their screws and glides alone and by their rotations, mirrors and inversions alone; 67 by their screws and glides only; 29 by their point operations only; 6 only by the two kinds together; and 7 by nothing they contain — P1, P2, Pm, Pmm2, P4, P3, P6 — which need a translation given outright.

Seven space groups cannot make their own translations

Every space group contains a lattice of translations, and for all but seven of the 219 types those translations are products of the group's other operations. A screw done twice climbs its axis and a glide done twice slides along its plane, so the non-symmorphic operations make lattices by themselves. The seven that cannot are exactly the seven plane groups with no glide, repeated along a line they all fix, and the reason is a difference between a glide line and a glide plane.

space-groups · Screws and glides A looser fibre with more longitudinal response. The three piezoelectric coefficients of a polar fibre of grains with d33 = 90, d31 = −33 and d15 = 560 pC/N, as the tilt spread grows from nothing to a uniform one, with the degree-one and degree-three parts of d33 dashed. The degree-one part is 265 and the degree-three part -175; the negative octupole decays faster, so d33 rises from 90 to 168 at a spread of 30° before falling to nothing.

A looser fibre, more piezoelectric

Spread the grains of a fibre about its axis and every property decays, but not at one rate: each spherical harmonic a property is built from is multiplied by its own average, and the higher the degree the faster it goes. A property made of two degrees therefore changes shape as the fibre loosens. For a polar grain whose shear coefficient dwarfs its longitudinal one, the fibre's longitudinal response nearly doubles before it falls; for another, a shear coefficient changes sign.

point-groups · Curie Two positions in three can be lifted, and more than one in three can lose their cell. All 1694 Wyckoff positions of the 219 affine types by crystal system. On 1132 a motif with more symmetry than its site can raise the group of the crystal, and on 606 of those, across 126 types, some such motif makes the true cell smaller than the group's.

In space a motif can lift two positions in three

A molecule with more symmetry than its site can raise the group of the whole crystal, or not, and which one is decided by the site. In the plane twenty-nine positions of seventy-two allow it. In space it is 1,132 of 1,694, and 606 of them allow a lift that shrinks the cell, the case a structure solution meets as a warning. General positions can be lifted only in polar groups, as predicted, but eleven polar groups refuse: a mirror turns their screws over.

classification · Accidental symmetry

Named alongside it

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

Screw axisChiralityEnantiomorphHelixOrder parameterPoint groupRod groupAccidental symmetryArithmetic classCouplingCrystallographic restrictionCurie principle

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