Crystallography & Symmetry
patterns derived from their groups
Concept

Limiting group — where it appears

One of the seven groups the symmetry of a uniform field can have, each with a continuous rotation axis or none at all. No crystal has one, and their use is as the symmetry of what is done to a crystal rather than of the crystal itself.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

The seven groups a field can have

Every group in this collection so far has been finite, because a lattice forbids the alternatives. A uniform field has no lattice: rotate it about its own axis through any angle at all and nothing has changed. There are exactly seven such groups, and they come out of the same closure argument that turns sixteen frieze candidates into seven.

point-groups · Curie What a crystal keeps of itself in a field. Each class, with what is left of it when a field is applied along the axis of its own setting. The residual is the intersection of the class with the field's own group, computed on matrices and matched against the thirty-two rather than named by hand. Where the residual is the class itself, the field takes nothing away — and for an electric field those are exactly the polar classes.

What a crystal keeps in a field

Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.

point-groups · Curie How many constants a texture permits. Every one of Curie's seven groups against every property in the census, as the number of independent components each permits. The counts are averages of a character over an infinite group, which is exact because the character is a trigonometric polynomial: the average is its constant term. A poled ceramic is the row ∞m, with one pyroelectric coefficient, two dielectric constants, three piezoelectric moduli and five elastic ones — and a zero for optical activity, which the mirrors forbid.

What a texture permits

A poled ceramic has no lattice, no cell and no class, and yet the number of piezoelectric moduli it may have is exactly three. The group is one of Curie's, the average over it is an integral, and the integral turns out to be a single Fourier coefficient — which is why the answer is exact and why a texture is indistinguishable from a hexagonal crystal until rank six.

point-groups · Curie The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups Which way a fibre of achiral crystals rotates light. For three achiral classes that permit a gyration tensor, every crystal direction that could be aligned along a fibre texture's axis, drawn on a disc as seen from above the z axis, and shaded by the sign of the rotation the texture then shows along its axis. In each class the rotation takes both signs, in regions separated by a cone on which it vanishes. For 4̅2m and mm2 that cone is exactly the class's mirror planes, where the texture inherits a mirror and must be inactive; for 4̅, which has no mirrors, the cone is where the rotation passes through zero between a right-handed texture and a left-handed one.

A fibre of achiral crystals can rotate light

Four achiral crystal classes permit optical rotation, of both signs in different directions. Grind such a crystal into a random powder and the rotation averages away completely. Align the grains along one direction and it comes back: the texture rotates light along its axis exactly as one grain does, half as much the other way across it — and the direction chosen for the alignment decides the texture's hand.

point-groups · Curie 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 Every fibre texture is a point in one plane. The two numbers that decide every coefficient of a polar fibre, the mean first and third Legendre polynomials of the grains' tilt, as a plane. A cone of grains at a single tilt is a point on the curve; any other texture is an average of such points and lies in the curve's convex hull. Spreads about the axis trace the inner path. For a grain with a large shear coefficient the largest d33 of any texture is at a point of the curve, the cone at 51°, 223 pC/N; the best spread about the axis reaches 168.

The best texture is a cone

Every fibre texture of a polar grain, however its orientations are distributed about the axis, reduces for piezoelectric purposes to two numbers, the averages of the first and third Legendre polynomials of the tilt. The textures fill a region of a plane bounded by one curve, and a coefficient is a straight line across that plane, so its best texture sits on the curve: a cone of grains at one angle. For a grain with a large shear coefficient the best cone lies at 51° and gives 223 pC/N, where the best spread about the axis gives 168.

point-groups · Curie

Named alongside it

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

Curie principleNeumann principleTexturePiezoelectricitySpherical harmonicsAxial vectorCrystal classPermissionPoint groupPolar vectorProperty tensorAnisotropy

All concepts