Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: The group acts on functions — page 2

A rotation carries an atom onto an atom, and it carries a density, a displacement or a wave onto another one. The second action is linear, so the group becomes a set of matrices, and questions that look analytic — which levels must coincide, how many independent components a property has, how a shell of neighbours splits — become counts of whole numbers.
4mm: 17 of 25 transitions forbidden. Every pair of irreducible representations of 4mm, with the number of times the identity occurs in the product of the two with the vector operator. A zero is a prohibition: the integral that would give the transition rate vanishes for every choice of functions carrying those representations, whatever the material is made of. A positive number is a permission and nothing more. Rows are final states and columns initial ones; the labels are the dimensions of the representations, so the twos are the degenerate levels. Every one of the 17 prohibitions here was checked again against explicit polynomials. What symmetry decides

What a group forbids to happen

Two levels and a thing that might carry a crystal from one to the other. Whether it can is one sum over the group — and a zero there is a prohibition that no material, no temperature and no intensity of light gets round.

 Where time reversal does something, and what. Every wavevector of every plane group at which time reversal changes the answer, with the square of each antiunitary operator, the unitary prediction, Herring's corrected prediction and the measured degeneracies. Case (b) is Kramers' theorem in a crystal with no spin, and it happens exactly where a glide's operator squares to −1. Case (c) is a representation being carried to a different one by the antiunitary operator, so the two become one level. Nine of these rows were open before the criterion was built — three the earlier census called unaccounted and six it could not reach at all. Into space

The degeneracy time reversal forces

A crystal with a glide has levels that stick together at the edge of its zone for a reason no character table contains. The operation responsible is antiunitary, it squares to minus one, and Kramers' theorem then applies to a model with no spin anywhere in it — which closes nine rows an earlier census in this collection had to leave open.

 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. What symmetry decides

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.

 How many constants a texture permits. Every one of Curie's seven groups against every property this collection computes, 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 symmetry decides

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.

 The kagome net's level that does not move. Three levels of the kagome net across the zone, one of them flat. The reason is drawn beside it: a state that alternates in sign round one hexagon and vanishes everywhere else is an exact eigenvector of the adjacency operator at −2, because every site outside the hexagon that touches it touches exactly two of its vertices and those two carry opposite signs. The check is integer arithmetic in a supercell of 27 sites, with a residual of exactly zero. A state confined to one hexagon has no wavevector, and a level made of such states cannot depend on one — which is what a flat line across a zone means. Symmetry at work

The level that does not move

Three levels cross the kagome net's zone and one of them is a horizontal line. The reason is a state that alternates in sign round a single hexagon and is exactly zero everywhere else — a solution with no wavevector in it at all, which is why no wavevector can move it.

 the kagome net: 34 of 144 wavevectors carry a mechanism. The zone of the kagome net, with a mark at every wavevector whose rigidity matrix drops rank — which is to say at every wavevector that carries a motion of the bars. There are few of them and they are isolated, so enlarging the cell adds mechanisms slowly. The ranks at the half-integer wavevectors are exact; the others are computed with a stated tolerance, because the matrix there has genuinely complex entries. Symmetry at work

A mechanism that is a wave

The framework essays found the kagome net's mechanism count growing with the cell it was looked for in, and recorded it as a finding without an explanation. Here is the explanation: the motions lie along lines in reciprocal space, and a larger cell samples a line at more places.

 Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character. Symmetry at work

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

 3m → 1: the two directions a wall between domains may take. The difference between the strains of two domains, sampled around a circle of directions: the first colour where that direction is stretched, the second where it is compressed. The two solid lines are the directions where it is neither, and those are the only orientations a straight wall between the two domains can take without straining itself — Sapriel's condition, one dimension down from the planes it is usually written for. There are exactly two, and that is not luck: the two domains are images of one another under the parent group, so their strains have the same area change and their difference changes no area at all. A form that changes no area takes both signs, and its zero set is a pair of directions. Symmetry at work

The walls a strain permits

Two domains of different shape can only meet along a line neither of them stretches. That condition is a quadratic in a direction, so a pair of domains has exactly two permissible walls — and the reason there are always two rather than sometimes none is that their strains differ by no area at all.

 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. Symmetry at work

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.

 One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves. Symmetry at work

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted. What symmetry decides

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

 A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large. What symmetry decides

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

 The table of marks of 4mm. Every conjugacy class of subgroup of 4mm, against every other. The entry is the number of cosets of the column's subgroup that the row's subgroup holds still. The first row is the identity, which fixes everything, so it is the size of each coset space; the last column is the whole group, whose only coset is fixed by everybody. What symmetry decides

The table that decides every action

Burnside's lemma counts orbits and stops there — two completely different actions with the same orbit count are indistinguishable to it. The object that settles the whole question is a square table whose entries count fixed cosets: lower triangular because a subgroup fixes no coset of anything smaller, positive on the diagonal because it fixes its own, and therefore invertible. Inverting it turns a list of fixed-point counts back into the orbits themselves.

 Ten chains, two phases. The Zak phase of the lower band of a two-site chain, as the ratio of the two hoppings is swept. Every value is exactly zero or exactly π and nothing lies between them, because the chain has an inversion centre and inversion maps the zone loop to itself reversed — which forces the phase to equal its own negative modulo a full turn. The switch happens where the two hoppings are equal, which is the one place the band gap closes and the phase belongs to no band. Into space

The phase a symmetry turns into a number

Carry a band's state once across the Brillouin zone and it returns with a phase. In a chain with an inversion centre that phase is exactly zero or exactly π and never anything else — and the two values turn out to be the two positions in the cell that an inversion centre fixes. Remove the centre and the phase moves continuously, which is what a quantisation claim has to be able to lose.

 Five strains, and which of them move the atoms. A honeycomb of harmonic bonds, strained five ways, with the internal coordinate minimised at fixed cell each time. The shuffle is how far the second atom moves away from where the strain alone would have put it. Which strains produce one is decided before any energy is computed: a strain that leaves the site's three-fold axis intact forces the shuffle to vanish, because the only vector a three-fold rotation of the plane fixes is the zero vector. The prediction and the measurement are in adjacent columns. What symmetry decides

The strain the atoms do not follow

The rule that makes an elastic constant computable — deform the cell and move every atom by the same map — is exact for a lattice with one atom in it and wrong for every other, and the reason is a site symmetry rather than a mechanical one. Which strains move the atoms inside the cell is decided by what survives of the site's own group, and a wrong answer here is a constant that is too stiff by a third.

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