Crystallography & Symmetry
patterns derived from their groups
Concept

Bragg peak — where it appears

A sharp maximum in a diffraction pattern, whose intensity grows as the square of the number of scatterers because every contribution is in phase. That growth is the test that separates it from a broad maximum, and it needs no threshold to apply.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

Which inflation factors a tiling may have. Every distinct inflation factor produced by a two-letter substitution whose matrix has entries up to 4, plotted against its algebraic conjugate. The two grey lines are the unit circle, which in the quadratic case is the pair of values ±1. A factor whose conjugate lies strictly inside is a Pisot number and the chain it grows has sharp Bragg peaks; 39 of the 77 factors here lie outside and cannot. The golden ratio is the smallest of them all, which is the arithmetic reason it turns up in every quasicrystal anybody has drawn.

Which inflation factors exist

A tiling grown by substitution has an inflation factor, and it is an eigenvalue of an integer matrix — so it is an algebraic integer, and sharp diffraction demands that its conjugates be small. That condition is an inequality between two integers, and it explains why the golden ratio turns up in every quasicrystal anybody has drawn.

aperiodic · Aperiodic A peak that grows, and not fast enough. The strongest peak of three chains, divided by the square of the number of letters, as each chain is lengthened. A Bragg reflection is a sum of terms in phase, so its intensity grows as the square of the count and this number settles: the Fibonacci chain and the period-doubling chain both do, at exponents of about two. The Thue–Morse chain does neither — its strongest peak grows, so it is not diffuse scattering, and it grows more slowly than the square, so it is not a Bragg peak. The fitted exponents are printed beside each curve and no threshold enters the comparison.

Neither a peak nor a bump

A chain whose strongest reflection grows as the length to the power one and a half. A Bragg peak grows as the square and a diffuse bump grows as the length itself, so this is neither — and the essay that ruled out the first possibility could only say so by quoting a theorem.

aperiodic · Complexity A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern.

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

aperiodic · Monotile Diffuse scattering gathered under every Bragg peak. The one-phonon thermal diffuse scattering of a square lattice with springs to nearest and next-nearest neighbours, over the reciprocal plane from minus 2.5 to 2.5 in each index, with the Bragg reflections marked. The intensity rises as one over the squared distance to every reflection, because the acoustic phonons' frequencies vanish there, and it grows with the square of the scattering vector, so the lobes are strongest far from the origin. The lobes are not round: a small step from (1,0) along k gives 2.7 times the intensity of the same step along h, because it probes transverse phonons, which are softer, and that ratio is what an experiment reads the elastic constants from.

A warm plane crystal has no Bragg peaks

Thermal motion is disorder that moves, and in space it costs a crystal a little of each Bragg peak and gathers it into diffuse scattering underneath. In a plane it costs more than a little. Computed exactly from the phonons of a harmonic square lattice, an atom's mean-square wandering grows by the same amount every time the crystal doubles in size, so no Debye–Waller factor survives, and every Bragg peak fades as a power of the crystal's size — a power proportional to the temperature times the square of the reflection's distance from the origin.

diffraction · Disorder 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 The error in the count, against tiles chosen at random. For clusters from about twenty-five hats to nearly two trillion, the distance of the reflected count from the exact share of all hats. For the hat it stays under one half. For independent random choice at the same share the expected error grows as the square root of the size, past 400,000 at the largest cluster.

The count of reflected hats is never one out

In a hat tiling one tile in about eight is reflected, and that ratio was only ever quoted here. Four clusters and a table of small integers give it exactly: seven plus three root five over two, the fourth power of the golden ratio. The same table says something the ratio does not. Grow a cluster to two trillion hats and its reflected count is still within half a tile of the exact share, where the same share chosen at random would be out by nearly half a million.

aperiodic · Monotile

Named alongside it

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

Average structureDiffuse scatteringDisorderAperiodicityDebye waller factorLong-range orderMonotilePhononSubstitutionThermal diffuse scatteringAlgebraic integerCoincidence condition

All concepts