Crystallography & Symmetry
patterns derived from their groups
Concept

Piezoelectricity — where it appears

The electric polarisation a crystal develops under stress, possible in twenty of the twenty-one classes that lack a centre of inversion. The exception is the class 432, whose character sum cancels to nothing for a reason that is not about having too much symmetry.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

applied · Twinning What the inversion does to three sums in 4/mmm. The 16 operations of 4/mmm come in pairs — every operation together with its own negative, because the class contains the inversion — and each row pairs the two terms they contribute. For an even-rank polar property the pair is two equal bars: the inversion changes an even number of indices and the character cannot see it, so the average is whatever it was before the inversion was added. For an odd-rank polar property and for an axial one the pair is a bar and its reflection, and the sum is exactly zero — which is why piezoelectric moduli and gyration tensor are forbidden here outright rather than merely small. Neither statement is about this class: the cancellation is checked over all 11 centrosymmetric classes and all four odd or axial properties every time this figure is drawn.

Twenty of the twenty-one

Twenty-one crystal classes have no centre of symmetry, and twenty of them permit piezoelectricity. The exception is 432, which has twenty-four operations, no inversion, and a character sum that cancels to nothing — and the reason it fails is not that it has too much symmetry in any ordinary sense.

point-groups · Properties 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 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 What Kleinman's rule removes from the second harmonic, class by class. The 20 crystal classes that permit second-harmonic generation, each with its independent susceptibility components drawn as squares: those that survive when the tensor is made totally symmetric, and those Kleinman's rule removes, split into the traceless part of the gyration tensor and a polar vector. In all, 83 components become 48. Two classes, 422 and 622, lose everything; four, 6̅, 6̅2m, 23, 4̅3m, lose nothing.

Two classes go dark far from absorption

The second-harmonic tensor of a crystal has as many components as its piezoelectric tensor, and far from any absorption it gains a symmetry of its own that removes eight of the eighteen. What it removes is not arbitrary: it transforms as a polar vector plus the traceless part of the tensor that carries optical activity. So the classes that lose the most are the polar ones and the handed ones, and two of them, 422 and 622, lose everything.

point-groups · Properties

Named alongside it

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

CharacterLimiting groupNeumann principleProperty tensorSpherical harmonicsTextureCurie principleOptical activityAveraging projectorCentrosymmetricCosetCrystal class

All concepts