Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 7

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
Observations per unknown, against resolution. Unique reflections divided by refinable parameters, for a triclinic cell, no angle a right angle, with three coordinates and six displacement parameters for every atom and one non-hydrogen atom per 18 ų, in a molecular crystal. The scale is logarithmic because the fall is a cube: 16.3 at 0.8 Å and 0.30 at 3 Å. The line at one is where a determination stops being over-determined, and it is crossed at about 2.0 Å. How it is known

The unknowns against the observations

A structure determination is a fit of some number of parameters to some number of measurements, and both counts can be worked out before any data exist. The ratio turns out not to depend on how large the crystal's cell is, or on how symmetric it is — only on the resolution, and on that as a cube.

 21 points, 3 gap lengths. The first 21 multiples of 377/610, marked on a circle of circumference one, together with the point at zero. The 22 gaps between neighbours take 3 distinct lengths — 13/610 (1 of them), 21/610 (9 of them), 34/610 (12 of them). The largest is the sum of the other two: 13 + 21 = 34. Every quantity here is a whole number over the denominator, so nothing is measured. Order without repetition

Three gaps, and never four

Mark the points α, 2α, 3α round a circle of circumference one. They look scattered. The gaps between neighbouring points do not: for every angle and every number of points there are at most three distinct gap lengths, and when there are three the largest is the sum of the other two. That is where a chain with exactly two tile lengths comes from.

 One orbit of p4m, two of p4. The general position of p4m — 8 points in a cell, all equivalent under that group — with each point coloured by which orbit of p4 it belongs to. Losing half the operations does not move a single point; it changes which of them are related, and the one orbit becomes 2. An atom sitting on this position in the parent becomes 2 crystallographically distinct atoms in the child, which may then be different elements, or move independently, or order. Operations

What a position becomes on the way down

Cool a crystal through a transition and it loses operations. Nothing moves — and one crystallographic site becomes two, which is how an ordering transition finds somewhere to put a second kind of atom.

 How many different lattices share a determinant. One bar per determinant: the number of inequivalent integral lattices whose metric has that determinant, which is the class number of the corresponding discriminant. Area does not decide shape — at determinant 1 and 2 there is one lattice each, and by 11 there are four — and the count does not grow steadily either. Each bar is computed twice: once by enumerating the reduced forms directly, and once by reducing every form in a box and collecting the distinct results, which is a search followed by an algorithm rather than a search over answers. The two agree at every bar. Lattices

How many lattices share a determinant

Area does not decide shape. The number of inequivalent lattices whose metric has a given determinant is a class number, computed by enumerating reduced forms — and checked by reducing every form in a box and counting what comes back distinct.

 heesch-two: surrounded 2 times. A shape that tiles nothing, with the rings of copies it does accept: the seed in the first colour and 2 coronas of 7 and 16 copies round it. The search that built this finished, so the shape's Heesch number inside this box is exactly 2, and it cost 3,097 placements. Every cell touching a tile of one ring, corners included, is covered by the next. The classification

Surrounded twice over, and covering nothing

A shape that tiles the plane can be surrounded by copies of itself for ever. A shape that tiles nothing cannot be surrounded for ever — but it can be surrounded once, and sometimes twice, and the number of times is a measurement of how much local success a global impossibility permits.

 -2 0 -3 is forbidden and is reached 64 ways. A layer of the reciprocal lattice of Cc: pale spots are reflections the group extinguishes, solid ones are allowed. The path shows a detour — the beam diffracts once at -3 -3 -3 and again at 1 3 0, and the two together send it in exactly the direction a single reflection at -2 0 -3 would. That reflection is forbidden, so intensity arrives where the symmetry said none could. There are 64 such routes to this one spot inside this window alone. How it is known

The absence that fills itself in

A systematic absence is the strongest evidence this subject has: a whole zone of reflections cancelling exactly, for reasons of symmetry rather than of arithmetic accident. The exactness belongs to a model — that the beam scatters once. A beam that has already been diffracted can be diffracted again, and the two events together land where the group said nothing could.

 The whole space of plane lattices, and its corner. Every plane lattice appears exactly once in this picture. Scaling changes no density, so the leading coefficient is fixed at one; reduction then confines the other two to 0 ≤ b ≤ 1 ≤ c, and every lattice has exactly one reduced form. The curves are the levels of constant density, which are parabolas — a density d needs 4c − b² to equal (π/2d)². They crowd toward the corner b = c = 1, which is the hexagonal lattice at π/√12 ≈ 0.9069; the square lattice sits on the left edge at π/4 ≈ 0.7854. The picture is a search over a region rather than over a list, which is what makes the answer a decision: there is nowhere else for a lattice to be. Symmetry at work

The densest lattice in the plane

Which arrangement of equal discs covers the most floor is a question about infinitely many lattices, and reduction turns it into a question about a two-parameter region with a corner. The answer is at the corner, and the argument finishes.

 two chains, periods 1 and 1.62. Two interpenetrating chains of atoms with periods 1 and 1.62, whose ratio is irrational, so no length is a whole number of both. Each chain is displaced from its own lattice by a wave with the other's period — the short ticks show each atom's displacement from where an unmodulated chain would put it — which is what makes this one crystal rather than two side by side. Nothing here is a unit cell: any length chosen contains a whole number of one chain's atoms and a fractional number of the other's. Order without repetition

Two lattices, one crystal, and no cell at all

A modulated crystal has a lattice and a wave running through it. A composite has neither host nor guest: two interpenetrating substructures with periods that share no common multiple, each modulating the other. Every reflection needs an index from both, and the unit cell a diffractometer reports belongs to whichever half scattered harder.

 What a crystal keeps of itself in a field. Each class, with what is left of it when a field is applied along the axis of its own setting. The residual is the intersection of the class with the field's own group, computed on matrices and matched against the thirty-two rather than named by hand. Where the residual is the class itself, the field takes nothing away — and for an electric field those are exactly the polar classes. What symmetry decides

What a crystal keeps in a field

Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.

 p4: (1/2, 0) has a star of 2. The first Brillouin zone of the square lattice, with the reciprocal lattice points at its corners and centre, and the whole star of the wavevector (1/2, 0) under p4. The star has 2 members and the little group — the operations that leave the wavevector where it is, modulo the reciprocal lattice — has order 2. The two multiply to the order of the point group, which is the orbit–stabiliser theorem and is checked rather than displayed. Each member is drawn at whichever of its equivalent copies lies nearest the origin, because that is where a reader expects a wavevector to be. Into space

The star of a wavevector

A plane group acts on the plane, and this collection has spent two hundred essays watching it. It also acts on the reciprocal lattice, where the action is different in one decisive way: the translations move no wavevector at all, and come back instead as a phase.

 11 nets, and one accounting. Every plane net folds onto a torus when its own translations are divided out, and a torus has Euler characteristic zero — so the quotient's vertices, edges and faces satisfy n − e + f = 0 and the number of faces is not something to be counted off a drawing but e − n. Dividing through gives one over the mean face size plus one over the mean degree equal to a half, which is the same relation that forbids a plane tiling by pentagons, reached here with no geometry in it at all. It holds for every net in the table. The classification

Every net folds onto a torus

Divide a plane net by its own translations and the quotient is a finite graph drawn on a doughnut. A doughnut has Euler characteristic zero, so the number of faces is not something to count — it is forced, and with it a relation between how many edges meet at a vertex and how many bound a face.

 9 approximants, period 2 to 89. The approximants of the Fibonacci chain: the n-th Fibonacci word taken as a unit cell and repeated. Each is a perfectly ordinary periodic crystal — it has a lattice, a cell and a space group — and each has the composition of the quasicrystal to the accuracy a ratio of Fibonacci numbers can manage, since its long and short tiles are consecutive Fibonacci numbers and their ratio is a convergent of the golden ratio. The last column is where the approximant stops agreeing with the infinite chain letter for letter: always past its own period, because the infinite word begins with every finite Fibonacci word, and never for ever. At order 9 the error in the composition is -3.87e-4, and it falls by a factor of τ² at every step up the sequence. Order without repetition

The crystal you get by rounding τ off

Everything aperiodic about a Fibonacci chain comes from one irrational number in the slope of a cut. Replace it by a fraction and the whole construction survives: the same lattice, the same strip, the same rule, and a chain that is periodic — agreeing with the quasicrystal for a length that grows with the denominator.

 Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other. Symmetry at work

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

 Every plane lattice, shaded by Σ|v|^(−4). The region every plane lattice is one point of, with each point shaded by the sum of the inverse powers of the lengths of that lattice's own vectors, at equal cell area — dark where the sum is small. The square lattice is the ringed point on the vertical axis and the hexagonal one is at the corners, which are the same lattice on two bases. The minimum is at the corner, and it is at the corner at every exponent tried. That is not the same statement as the densest packing, which is decided by the shortest vector alone: this sum counts every shell, and there was no reason in advance for the two questions to have the same answer. Lattices

The lattice that minimises a sum

Packing discs asks about the shortest vector alone. Summing an inverse power over every vector of a lattice asks about all of them at once, and there was no reason in advance for the two questions to have the same answer. They do — at every exponent, and the measurement says by how much and where it cannot say.

 6mm: 6 irreducible characters on 6 classes. The character table of the plane point group 6mm, constructed rather than quoted. The columns are its 6 conjugacy classes, with the number of operations in each; the rows are its 6 irreducible representations, of dimensions 1, 1, 1, 1, 2, 2. The dimensions satisfy 1² + 1² + 1² + 1² + 2² + 2² = 12, which is the order of the group, and that identity together with the count of classes is what proves the list complete. Every entry is an exact element of the twelfth cyclotomic ring; the decimals shown are the numerical value of an integer vector, not a computed approximation. What symmetry decides

What a group does to a function

A symmetry operation moves points about, and two hundred essays here have watched it do so. It also acts on everything defined over those points — a density, a displacement, a wave — and that action is linear, which turns a group of motions into a set of matrices and every question about it into arithmetic.

 A gap of exactly 1.00. The two folded bands with the ordering switched on. The faint curves are the same bands before it, crossing at the boundary of the reduced zone; the ordering couples them there and separates them by exactly twice its own strength. The gap appears at the wavevector where the superlattice's extra reflections appear, and for the same reason: both are the Fourier component of the potential at that wavevector. Lattices

A bigger cell, a smaller zone

Ordering two kinds of atom onto a sublattice adds reflections to the diffraction pattern and opens a gap in the levels. It is one fact told twice: the same Fourier component of the potential, at the same wavevector, doing the same thing.

 The ten plane classes, and the dimensions they permit. Every crystallographic point group of the plane, with one block per irreducible representation and each block as wide as its dimension. Nine of the ten have only one-dimensional representations; 4mm, 3m and 6mm carry a two-dimensional one, drawn in the measured colour. Nothing is wider than two, and the sum of the squares of the widths in each row is the order of that group — the identity that says the row is complete. What symmetry decides

How large a degeneracy may be

Symmetry can force two things to have the same value, and in a crystal it can force three. It can never force five, and the reason is a sum of squares — the same kind of arithmetic that forbids a five-fold axis, arriving at a question about levels rather than about rotations.

 3 whole-number solutions: (6, 3), (4, 4), (3, 6). Every pair of whole numbers from three to 12, with the mean face size across and the mean degree down. A square in the first colour is a pair satisfying one over p plus one over q equals a half exactly — the flat case, where a periodic net is possible — and there are 3 of them: 6 and 3, 4 and 4, 3 and 6. The lighter squares above and to the left have a sum greater than a half, which is a closed polyhedron rather than a plane tiling; the ones below and to the right have a sum less than a half and belong to a surface of negative curvature. The plane is the boundary between them and it is thin. The classification

Three answers in whole numbers

One over the face size plus one over the degree equals a half. Ask for whole numbers and there are exactly three answers, which are the three nets everybody has drawn since childhood — and the pairs on either side of them are a closed polyhedron and a plane the plane has no room for.

 ⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them. How it is known

Three phases that do not move when the origin does

A phase is a property of the description, not of the crystal: shift the origin and every one of them changes. A sum of three phases whose indices add to zero does not change, because the shifts cancel. That sum is the smallest thing about a structure that a diffraction experiment could in principle know, and it is not distributed at random.

 p3m1 at (0, 0): the levels the little group requires, and the ones measured. The levels of the p3m1 model at (0, 0), with degenerate ones drawn thick. The little group there has order 6, and its characters predict levels of dimensions 1, 1, 2, 2. The measurement is 1, 2, 2, 1, and the account is "unitary". The values are numerical and the multiplicities are read at a stated gap; the prediction they are compared against is exact. Into space

Where two levels must meet

At most wavevectors nothing of a crystal's symmetry survives, and its levels are as unconstrained as any operator's. At a handful of them a whole point group survives, and where that group has a two-dimensional representation, two levels are obliged to coincide — before anything about the material is known.

 Sublattices of index n, in space. How many sublattices a three-dimensional lattice has at each index, beside the plane's answer, with the Hermite enumeration and the coefficient of ζ(s)ζ(s−1)ζ(s−2) in separate columns. The two are computed by routines sharing no code, and a row where they disagreed would be a failure rather than a result. The last column counts the ones that survive every operation of the cubic group, and it is almost always empty. Lattices

The three that stay cubic

A lattice in space has far more sublattices than one in the plane — 651 of index sixteen against 31 — and almost none of them keeps the symmetry it came from. The ones that do exist at indices m³, twice m³ and four times m³, there is exactly one at each, and they are the primitive, face-centred and body-centred cubic lattices, arrived at by asking which sublattices keep a symmetry rather than by enumerating centrings.

 pgg at (1/2, 0): the operators multiply up to a sign. Every product of two Bloch operators of the little group of (1/2, 0) in pgg, against the operator of the product. They agree up to a scalar, and the scalar is +1 or −1: 4 of the 16 products come back with a minus sign. No rephasing removes them, and the search that says so tries every assignment of twelfth roots of unity to the operators. A representation that multiplies only up to this sign cannot be one-dimensional, because scalars commute and these operators do not. Each entry is an exponent modulo twelve, so the table is exact. Into space

A glide sticks two levels together

The translation attached to a glide moves no wavevector at all. It comes back as a phase factor, and at the edge of the zone the factor is minus one — after which the operators of the little group no longer multiply the way the group does, and no rephasing repairs it.

 A wavevector of thirds, and the boxes that cannot see it. Which sizes of box can carry the wavevector at the corner of a hexagonal zone. The characters of the box's translation group are its wavevectors, and there are exactly N² of them — the fractions with denominator dividing N. A wavevector of thirds is therefore present in a box of three, six, nine or twelve cells and absent from one of two, four or five: not approximated badly, not resolved coarsely, absent. A mechanism or a level living there is invisible to such a calculation, and that is the practical content of a mechanism count depending on the cell it was looked for in. The classification

Crystallography in a box

A calculation over a crystal is not performed on a crystal. It is performed on a finite block with its edges glued, and the block has a symmetry group of its own — finite, complete in one direction and missing something decisive in the other.

 square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here. Symmetry at work

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

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