Crystallography & Symmetry
patterns derived from their groups
Concept

Long-range order — where it appears

Structure that persists over arbitrarily large distances, which sharp diffraction peaks report and periodicity is only one way to have. Separating it from repetition is what the discovery of quasicrystals forced.

Named by 13 essays across 2 fields — each of them below, with the objects they name alongside it.

A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

Order is not periodicity

For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten.

aperiodic · Aperiodic The substitution, 6 generations. The rule "every long tile becomes a long and a short, every short tile becomes a long", applied 6 times from a single tile. Each generation is as long as the previous two together, so the tile counts are Fibonacci numbers — 13 long and 8 short at the last row — and their ratio is 1.62500 against the golden ratio's 1.61803. The sequence never repeats and every finite piece of it recurs infinitely often, which is order without periodicity in its smallest form.

The smallest quasicrystal

Two tile lengths on a line, in the golden ratio, in a sequence that never repeats. Three completely different constructions produce it, they are required here to agree, and its diffraction needs two integers per peak where a periodic chain needs one.

aperiodic · Quasicrystals The Fibonacci chain: p(n) = n + 1. The number of distinct windows of each length in the Fibonacci chain, measured by sliding a window along 46,368 tiles. Every count is checked against the same count on half the chain, and only lengths where the two agree are drawn — a factor count on a finite word is otherwise a lower bound wearing the clothes of an answer.

n plus one, and no fewer

Slide a window along a chain and count what it can show. A periodic chain runs out of new views; an aperiodic one never does; and the fewest an aperiodic chain can manage is one more than the window's length — which is exactly what the Fibonacci chain manages.

aperiodic · Complexity Every window returns within 3.0 n. For each window length, the largest distance between two consecutive occurrences of the same window, measured over 46,368 tiles. The gaps are Fibonacci numbers, and the ratio to the window length stays below 3.00 — the chain is linearly repetitive. That is a strong statement of uniformity: there is no stretch of the chain, however far out, in which a given patch fails to occur within a bounded multiple of its own size.

Every patch comes back

A chain that never repeats still repeats everything in it. Every block of tiles occurs again, and again, within a bounded multiple of its own length — and how large that multiple is turns out to be a fact about the continued fraction of a slope.

aperiodic · Complexity One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are.

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

aperiodic · Complexity Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.

How many arrangements one rule allows

Every count in this collection so far has been a count of symmetries, or of orbits under one. Here is a different count: the arrangements a purely local rule permits on a fixed lattice, with no symmetry quotient anywhere in it. The answers are enormous, they are exact, and the useful quantity is not the number but its growth per site.

aperiodic · Entropy Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not.

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

aperiodic · Entropy 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.

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.

aperiodic · Modulation One crossing or the other, and never both. Two events on a square patch of lattice: a path of occupied sites crossing from left to right, and a path of vacant sites crossing from top to bottom. On the triangular lattice exactly one of them happens in every configuration tested — the claim is combinatorial rather than statistical, so one counterexample would end it. On the square lattice both can fail at once, and do, in more than a quarter of the configurations. That difference is the whole of what follows.

The threshold a symmetry pins down

Occupy sites at random and somewhere the occupied ones first join up across the crystal. For almost every lattice that occupancy is known only to a few digits. For the triangular lattice it is exactly a half, and the reason is that on a lattice whose faces are all triangles an occupied path and a vacant path cannot slip past each other — a statement about one configuration at a time, with no probability in it.

diffraction · Disorder Every rational holds a window, and there is nothing in between. The ground state density of a chain of particles with a convex repulsion, against the chemical potential that sets how many of them there are. Every density with denominator up to 24 is a flat step of positive width — 177 of them — and the steps with the simplest fractions are the widest: a half takes 19 per cent of the whole range on its own. The risers between them are not smooth stretches; they are where the densities with larger denominators sit, and a finer computation fills them with more steps. What is left after every rational has taken its window is the irrational densities, which are the genuinely incommensurate ground states and have no width at all.

Every fraction holds a window

Three essays here name the devil's staircase and none computes one. A chain of particles with any convex repulsion has a ground state at every rational density holding an interval of chemical potential to itself — 709 of them computed, the widest taking 19% of the axis and the narrowest two parts in a million million — and the incommensurate densities are what is left over.

aperiodic · Modulation How many locked phases an experiment of a given resolution can see. For three convex repulsions, the number of steps of the staircase wider than a given share of the chemical-potential axis, as that share is made smaller by factors of ten. A resolution of one part in a million shows 199 steps for the short exponential, 829 for the long one and 41 for the inverse sixth power. Every count is exactly the number of fractions with denominator up to some bound, because every step with one denominator has one width. The short exponential and the inverse power cross: the power law shows fewer steps at every practical resolution and more at one part in a hundred thousand million.

An experiment sees a Farey sequence

A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.

aperiodic · Modulation Along the rod above a reflection, a free sheet loses its peak. The Bragg intensity per site squared of a warm square-lattice sheet at the in-plane reflection (1, 0) and at heights 0.1, 0.2 and 0.3 along the rod above it, against the sheet's size, on logarithmic axes. In the plane the peak falls as a power of the size whatever the sheet does out of it. Above the plane a free sheet's peak collapses faster and faster, to 2.3e-2 at size 128 and height 0.3; under tension every height falls as a power, to 0.38.

A flapping sheet keeps its peaks in the plane

A warm sheet one atom thick wanders out of its plane far more than within it: a free sheet with only bending stiffness has a mean-square height that grows with its area. That wandering cannot touch a reflection whose scattering vector lies in the plane, which fades exactly as it would if the sheet were held flat. Every reflection with a component along the normal loses its peak faster than any power of the size. Tension puts it back on a power law.

diffraction · Disorder Every stacking reflection of a warm stack of sheets falls as a power of its size. The Bragg peak per site squared per layer squared of the stacking reflections (0, 0, l), l = 1 to 5, of a stack of flapping square-lattice sheets with bending stiffness 1 and coupling 0.1 between neighbouring layers, at kT/k₁ = 0.02, against the stack's width, with the number of layers growing as the width squared. Each order falls as a fixed power of the size, and the powers grow as the square of the order: 0.20, 0.80, 1.79 and 3.18 for the first four. The fifth falls as the fourth power, as fast as the crystal's volume grows, which is what a reflection with no peak at all does.

A stack lends each sheet a tension

A single warm sheet wanders so far out of its plane that every reflection with a component along its normal is lost. Stack the sheets and couple each to its neighbours, and inside the stack each sheet wanders only as a sheet under tension does, logarithmically. The stacking reflections survive as peaks that fall as powers of the crystal's size, and the power grows as the square of the order, so the fifth order of the stack computed here has no peak at all. A stack of finitely many sheets keeps even the first only out to a width growing as the square root of its thickness.

diffraction · Disorder

Named alongside it

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

Cut-and-projectDisorderQuasiperiodicEnumerationFactor complexityThe Fibonacci chainIncommensurateLocal rulesModulationSturmianAperiodicityBalance

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