Crystallography & Symmetry
patterns derived from their groups
Concept

Texture — where it appears

The distribution of crystal orientations in an aggregate, such as a poled ceramic, a drawn fibre or a rolled sheet. Its symmetry is one of Curie's limiting groups, and a property of the aggregate is the average of the grains' property over the distribution.

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

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 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.

Limiting groupNeumann principlePiezoelectricitySpherical harmonicsCurie principleProperty tensorAveraging projectorAxial tensorCharacterChiralityGyration tensorLegendre polynomial

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