Crystallography & Symmetry
patterns derived from their groups
Concept

Property tensor — where it appears

The array of numbers relating one physical quantity to another in a crystal — a conductivity, a susceptibility, an elastic modulus. Symmetry fixes how many of its components are independent, and sometimes fixes the directions of its principal axes as well.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

How many components a property may have, at each rank and in each class. One row per plane point group, one column per rank of a fully symmetric property tensor, with the number of independent components in each cell — counted by averaging the tensor over the group index by index, and equal at every entry to the Molien coefficient of that degree. A symmetric property of rank r is a form of degree r, so Neumann's principle and the invariant ring are the same arithmetic in two notations. The last column is the elastic tensor, which is not fully symmetric — symmetric within each pair of indices and under exchanging the pairs — and its counts are not in the table to its left. A property with its own symmetries needs its own average, and that is why the elastic constants are not read off a degree.

The parts a property splits into

A symmetric property of rank r is a polynomial of degree r wearing indices, so the number of components a class permits it is a coefficient of an invariant ring's series. The elastic tensor is not a polynomial in disguise, and its counts are not in that table — which is the most useful thing about it.

point-groups · Neumann's principle Where the axes are free, they move. Five different invariant tensors of each of three classes, with the trace of each tensor's principal axes on the page. In the orthorhombic class every sample gives the same three directions: the axes are the two-fold axes and symmetry has fixed them. In the monoclinic class one direction is common to every sample and the other two rotate freely in the plane across it. In the triclinic class nothing is common at all. Each sample stands for a different material, or the same material at a different wavelength — which is what makes the middle picture the dispersion of the optic axes.

The axes a class pins down

A property tensor has a shape and an orientation, and symmetry treats them differently. Three principal directions fixed for ever in an orthorhombic crystal; one in a monoclinic one, with the other two turning as the wavelength changes.

point-groups · Optical 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 Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups Seven cubic antiferromagnets, and what a powder of each can say. Seven collinear antiferromagnets on the three cubic lattices, each with the size of its propagation vector's orbit, the number of the forty-eight cubic operations that keep the vector, the kind of every magnetic ring out to a fixed radius, summed from the actual magnetic reflections of one domain, and what a powder pattern can therefore read about the moments' direction. The two whose halving the cubic group keeps are blind in every ring. An orbit of three leaves every ring uniaxial about a cube axis, an orbit of four about a body diagonal, and the body-centred orbit of six leaves rings that read the direction. The cubic chemistry is the same in every row; the propagation vector decides.

Where a powder goes blind

A neutron sees only the part of a magnetic moment across its path, so each reflection knows which way the moments point. A powder adds up every reflection of a given length, and for a ring with cubic symmetry that sum comes out the same whichever way they point. But the ring's symmetry is not the chemistry's: it is the symmetry the propagation vector keeps, and most cubic antiferromagnets have lost enough of it to be read.

point-groups · Magnetic 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 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 How many resonances each crystal class needs for a second-order response. The 32 crystal classes sorted by the number of distinct oscillators a model needs before the class can have any second-order optical response, with the dimensions of the class's invariants among the cubic couplings: fully symmetric (one oscillator), mixed (two), alternating (three). The eleven centrosymmetric classes have none. Eighteen need one; four of those (23, 4̅3m, 6̅, 6̅2m) have no mixed invariant and obey Kleinman's rule at every frequency. 422 and 622 need two. 432 needs three.

A crystal in 422 needs two resonances

Kleinman's rule silences classes 422 and 622 far from absorption, and the obvious guess is that dispersion brings them back near it. In the simplest model of dispersion, one charged oscillator with an anharmonic potential, it does not: they stay dark at every frequency. Two oscillators with different resonances light them, with a response growing as the square of the frequency. 432 needs three, and then only mixes two different frequencies, growing as the sixth power. The number of resonances a class needs is a count of invariants, and it sets how fast the forbidden part can grow.

point-groups · Properties Photoelastic constants of the eleven Laue classes, split into four parts. The number of independent photoelastic constants each Laue class permits, from 36 for 1̅ to 3 for m3̅m, drawn as squares and split into the elastic tensor's part, which is symmetric under exchanging the two index pairs, and the extra fifteen-dimensional part, which divides into an axial vector, a second quadrupole and an octupole. The axial vector survives only in 1̅, 2/m, 4/m, 3̅ and 6/m; the octupole is what separates m3̅ from m3̅m; and in m3̅m nothing extra survives, so the two counts agree at three.

Fifteen constants an energy would have forbidden

A strain changes a crystal's refractive index through a fourth-rank tensor with the elastic tensor's index pairs and without its one remaining symmetry, because no energy is differentiated twice to make it. The missing symmetry is worth fifteen components. They split into an axial vector, a quadrupole and an octupole, and each does something the elastic count cannot: the octupole gives 23 and m3̅ a fourth constant, and the axial vector is a part of the tensor no turn of the axes removes.

point-groups · Neumann's principle The constants a turn of the axes removes, class by class and property by property. The thirteen crystal classes whose operations all commute with some turn of the axes, against six properties: pyroelectric, dielectric, optical activity, piezoelectric, elastic and photoelastic, and all six together. Each cell gives the printed number of constants, and where a turn removes some, the number left. Triclinic classes have three turns and lose up to three; the others have one and lose at most one. Classes 6 and 6/m lose nothing in any property, and the last column loses the turns once, however many properties could spend them.

A six-fold crystal hides its frame until rank six

Thirteen crystal classes can be turned without moving any of their operations, and the tables print each property's constants in one frame among the many such a turn allows. Spending the turn removes a constant only where a property has a part that turns with the frame, and an n-fold axis keeps such a part only at rank n or above. So a monoclinic crystal's free angle shows in its dielectric tensor, a tetragonal one's first in its elastic constants, and a hexagonal one's in nothing anybody measures. And a crystal has one frame, not one per property: the angle is spent once.

point-groups · Neumann's principle

Named alongside it

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

CharacterIndependent componentsLaue classNeumanns principleElastic constantsPiezoelectricityAveraging projectorGyration tensorIrreducible representationLimiting groupNeumann principleOptical activity

All concepts