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 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 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.

 Which sites a smooth error can mislead. For the eighteen pairs of disorder models at three site symmetries, under a smooth scale error of degree two (five molecule sizes) and of degree four (three sizes): how many verdicts of perfect counting go the wrong way, how many pairs no data can decide, the largest correlation between the error and the two models' difference, and how many harmonics of the error's degree the site's group leaves unchanged. At the cubic site the degree-two error correlates with nothing, to rounding, and decides no pair wrongly; the degree-four error, which the cubic site keeps one harmonic of, reaches it. What a lattice forbids

A cubic site is blind to a degree-two error

A smooth scale error left uncorrected made perfect counting choose the wrong disorder model for eight of ninety pairs and sizes, and the obvious way to predict which is to correlate the error with the two models' difference before refining anything. That correlation predicts none of them. What it does predict, exactly, is which pairs can never go wrong: at a cubic site a scale error of degree two correlates with nothing and is absorbed by nothing, because a cubic group keeps no harmonic of degree two. A degree-four error, which it keeps one of, reaches the cubic site at once.

 The first-order prediction against the actual verdict, for every pair and size. The 85 decidable pairs of disorder models at five molecule sizes, each placed by the first-order prediction of the wrong model's misfit minus the right model's, made from the two error-free refinements and their Jacobians, and by the actual difference after refining against data with a 3% smooth error. Shaded quadrants are agreement. 77 agree. Of the 8 wrong verdicts, ringed, 4 are predicted; 4 right verdicts are predicted wrong. What a lattice forbids

The wrong model moves too far for first order

Which disorder models a smooth error will turn the wrong way was left as a calculation to make before any data exist: project the error onto the directions each refinement's parameters can move. Made, it gets the sign of the verdict right in seventy-seven of eighty-five cases, catches four of the eight wrong verdicts and raises four false alarms. Every case it gets wrong is either a near-tie or a refinement that absorbed more than twice, or less than half, of what its first-order directions allow.

 A structure that is its own negative, and one that is not. Two arrangements of two species on a four-by-four block with p4m symmetry, each with eight sites of each species, beside the arrangement that exchanging the species produces. On the left the exchanged arrangement is the original one moved by an operation of the group, so the two are one structure: it is self-complementary, and its balance of species is forced by a symmetry. On the right the exchanged arrangement is a genuinely different structure, and the balance is a coincidence of count. Counting structures with the species interchangeable merges the pair on the right into one and leaves the one on the left as it was, which is why the count does not halve. Operations

Swapping the species does not halve the count

Let the two species on a lattice be interchangeable and every structure pairs with its negative, so the count should halve. It does not, because some structures are their own negative: an operation of the group carries them onto the arrangement with the species exchanged. Those are counted by the operations whose every cycle is even, they all sit at exactly half composition, and read as spins they are the arrangements whose zero magnetisation a symmetry enforces.

 Four inversion centres, and a band's charge on each. A square crystal with four atoms in each cell, at the points a quarter of the way along both diagonals, and the four points of the cell that an inversion maps onto themselves: the corner, the midpoints of the two edges, and the centre. None of the atoms sits on one. For each of four versions of the crystal, differing only in which bonds are stronger, the ring marks where the lowest band's charge is centred — computed from the band's Berry phase and predicted from four parities. Each version puts it on a different inversion centre, and in every case on a point with no atom. Into space

Four signs say where the charge sits

In a plane crystal whose only symmetry beyond its lattice is a centre, a band's charge must sit on one of the cell's four inversion centres — and which one is decided by four signs: whether the band is even or odd under inversion at the four wavevectors inversion leaves alone. The centre can be a point no atom occupies, and when the four signs multiply to minus one there is no centre at all, only a charge that winds round the cell.

 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. What symmetry 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.

 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. What symmetry decides

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.

 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. What symmetry decides

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.

 One arm of a star is cheap, and the whole star costs the cube. For nine wavevectors of the primitive cubic zone, the number of cells in the smallest periodic box that holds one arm of the wavevector's star, and in the smallest that holds every arm, on a logarithmic axis found by enumerating every box of each size. The zone corner R is held whole by two cells. The face centre X needs two cells for one arm and eight for its three; the thirds along an axis or a body diagonal need three for one arm and twenty-seven for the star; the quarters need four and up to sixty-four. Every smallest box for a whole star has the full symmetry of the cube. The classification

A whole star needs a cubic box

A periodic box for a cubic crystal holds only the wavevectors its own lattice allows, and a box chosen to hold one wavevector usually holds one arm of its star and not the others. One arm of the face centre X fits in two cells; all three need eight. One arm of a third along an axis fits in three; the whole star needs twenty-seven. The smallest box that holds a whole star always keeps the cube's full symmetry, so a calculation either pays for the cube or reports a symmetry its box imposed.

 Every crystal class permits the Faraday effect. The thirty-two crystal classes, each with two counts: the number of independent components of the tensor that turns a magnetic field into an optical rotation, and the number of components of the natural gyration tensor. The first is never zero: 5 classes have 1, 11 classes have 2, 11 classes have 3, 3 classes have 5, 2 classes have 9. The second is zero in 17 classes, every one with an inversion centre or a mirror arrangement that cancels it. The Faraday count is the mean of the squared trace over the class, which is never below one because the isotropic part always survives. What symmetry decides

A field lets every class rotate light

Seventeen of the thirty-two classes forbid a crystal to rotate the plane of polarisation on its own. Put the same crystals in a magnetic field along the ray and not one of the thirty-two forbids it, the centrosymmetric ones included. The field supplies the one thing natural rotation needs a handed structure for, and time reversal is what tells the two rotations apart.

All themes · All essays