Crystallography & Symmetry
patterns derived from their groups
Series

Modulation — the series

13 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.

    The satellites that need a second integer

    A crystal whose atoms are displaced by a wave of the wrong wavelength has no unit cell at all, and diffracts to sharp spots anyway. Indexing them takes two integers per reflection instead of one — and the intensities of the extra spots are Bessel functions, which is a check the arithmetic can be made to pass.

    part 1 · aperiodic
  2. The chain as a cut through a periodic pattern. A periodic pattern in two dimensions: one atomic surface through each lattice point, drawn as the curve x = n + A·sin(2πy). The physical chain is the cut along the line y = qx with q = 0.211, and the atoms are where that line meets the curves — plotted along the bottom. Every cut meets every curve exactly once, so every cut gives a chain with the same 9 atoms and the same lattice, differently displaced. That is the difference from cut-and-project, where the atomic surfaces are intervals with ends and moving the cut adds and removes points: here the extra coordinate is a phase, and shifting it is a symmetry of the material rather than a different material.

    The extra dimension that makes it periodic

    A structure with no cell in three dimensions can be a slice through one that has a cell in four. The construction is cut-and-project with continuous atomic surfaces instead of intervals — and that single difference is what separates an incommensurate crystal from a quasicrystal.

    part 2 · aperiodic
  3. 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.

    part 3 · aperiodic
  4. 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

    Superspace groups in the plane

    A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

    part 4 · aperiodic
  5. 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.

    part 5 · aperiodic
  6. 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.

    part 6 · aperiodic
  7. Three ways to fill a lattice with a wave. Three occupation waves at the same irrational wavevector, each drawn as its profile over one period, the occupancy of a site as a function of where the wave stands there, and as the first thirty sites of the chain it produces, each filled in proportion to its occupancy. A cosine fills every site partly. A crenel of width 0.3 fills three sites in ten completely and leaves the rest empty. A crenel of width 1/φ fills sites exactly where the Fibonacci chain has its long letter, so the third row is that chain written as two species on a lattice.

    An occupation wave is its own Fourier series

    A modulated crystal whose atoms stay put while a wave decides how much of an atom sits on each site diffracts to satellites, and the satellite of each order is simply the matching Fourier coefficient of the wave's profile. A cosine stops at first order and can never exceed a quarter of a site; a step makes satellites of every order, skips exactly the orders its width divides, and makes the strongest first-order satellite any wave can. And a step as wide as its own wavevector is the Fibonacci chain.

    part 7 · aperiodic
  8. Two waves that are each symmetric enough make a lopsided pair. The first-order satellite amplitudes at h − q and h + q around the main reflections h = 1 to 4, for three chains: an occupation wave on the lattice, whose pairs are equal; the strain wave alone, the displacement the size effect produces with every site equally full, whose satellites are weak and grow with h; and both together. Together the pairs are unequal, the low side stronger, by an amount that grows from one reflection to the next — an imbalance neither wave has on its own, from a displacement of less than a hundredth of a spacing.

    The strain wave that tips the satellites

    An occupation wave puts equal satellites on both sides of every main reflection, and a strain wave a hundredth of a lattice spacing high is enough to tip them. Real composition waves drag such a strain with them, because atoms of different sizes want different spacings, and the tilt that results is not a nuisance. To first order it does not depend on how strong the occupation wave is at all. It reads the size mismatch directly, and its sign says which species is the larger.

    part 8 · aperiodic
  9. Second-order satellites that only the strain makes. Satellite intensities on a logarithmic scale around the first four main reflections of a chain whose occupation is a pure cosine wave, with no strain and with the strain a size factor of 0.1 brings. Without strain the second-order satellites are absent, down at the chain's numerical floor; with it they appear at between a ten-thousandth and a hundredth of the first order's intensity, larger on the outer side of each reflection and growing with its order.

    The second order reads the strain again

    An occupation wave with a pure cosine profile has no second harmonic, so on its own it makes no second-order satellites. Strain the chain by the size difference of its atoms and they appear. Their strength measures the square of the displacement and cannot see its phase, where the tilt of the first-order pair measures the displacement times the cosine of its phase. So the two give the size factor twice, and the two readings agree only if the strain sits where a size effect puts it.

    part 9 · aperiodic
  10. The h = 1 anomaly moves with the length of the chain. For a size-coupled chain with ε = 0.03, the ratio of the second-order pair's tilt, divided by k², to the first-order tilt at h = 1, from plain sums over the sites and from sums with a smooth window, for 22 chain lengths from 10,000 to 480,000 sites. The plain ratio swings between -0.08 and 5.60; at 30,000 and 120,000 sites, the two lengths first tried, it sits near 0.31 both times. The windowed ratio is one half at every length.

    The satellite beside the zero-order reflection

    An earlier essay found that the second-order satellites of a modulated chain tilt half as far as the first-order ones at every reflection but the first, where they tilted a third as far, the same on chains of 30,000 and 120,000 sites. It left the departure unexplained and proposed a higher order of the expansion. The expansion was never wrong. The weak satellite at h = 1 sits a quarter of a reciprocal unit from the strongest reflection a chain has, and a plain sum over a finite chain carries that reflection's tail, which does not shrink steadily as the chain grows. It swings, and the two lengths tried happened to swing to the same place.

    part 10 · aperiodic
  11. The second-order satellites alone cannot measure the curvature of a size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from six sets of satellites of a modulated chain of 30,000 sites with 2% intensity error, and the correlation between the errors in ε₂ and the linear coefficient ε₁. From second-order satellites alone the error is thousands of times ε₂ at one reflection and still a hundred times it at three, with the two coefficients correlated at 1.000. With the first-order satellites added, one reflection gives ε₂ to about a third of itself and three to about a tenth.

    The second order cannot see a curved size effect alone

    When the spacing between atoms depends on composition along a curve rather than a line, the displacement of a modulated chain gains a second harmonic, and the second-order satellites should measure it. They cannot do it on their own. At every reflection a second-order satellite sees the straight part and the curved part of the size effect in almost the same proportion, so adding reflections adds copies of one equation. The first-order satellites, which see only the straight part, are what separates them.

    part 11 · aperiodic
  12. What each kind of satellite of a two-wave chain says about the curvature of the size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from seven sets of satellites of a chain modulated by two incommensurate waves, with 2% intensity error: with the two modulation amplitudes refined as well (upper bar of each pair) and with them known (lower bar). The sum or the difference satellites alone say almost nothing either way. Together, with the amplitudes known, they fix the curvature to under one per cent from a single reflection; with the amplitudes free they do not, and adding the first-order satellites, which fix the amplitudes, restores it.

    Two waves measure the curvature one wave hides

    A chain modulated by one wave cannot tell a curved size effect from a straight one at second order: every second-order satellite sees the two in the same proportion. Modulate it by two incommensurate waves and the curvature puts the product of the waves into the displacement, at their sum and at their difference. The satellites there see the curvature with opposite signs. Two equations of opposite sign separate two unknowns, and once the two waves' amplitudes are known, the mixed satellites of a single reflection fix the curvature to under one per cent.

    part 12 · aperiodic
  13. At the sum of all three wavevectors the cubic and the quadratic paths nearly cancel. The intensity of the satellite at h + q₁ + q₂ + q₃ for the first four reflections, computed with only one size coefficient switched on at a time and with all three. The cubic coefficient alone gives about 3 × 10⁻⁷ at h = 1, the quadratic alone about 2 × 10⁻⁷, the linear alone less than 10⁻¹⁰. With all three the satellite is ten times weaker than with the cubic alone, because the cubic term's displacement and the quadratic term's pair harmonic times a first-order wave arrive with opposite signs.

    Three waves measure the cube through the slope

    A size effect that bends twice puts the product of three modulation waves into the displacement, at the sums and differences of all three wavevectors, and nowhere else does a cubic term reach so directly. The satellites there were expected to measure it. They see it, but the strongest of them is ten times weaker than the cube alone would make it, because the quadratic term arrives at the same place with the opposite sign, and against any realistic background they say nothing. The cube is measured instead by the strongest satellites of all, the first-order ones, because it makes each wave's slope depend on how strong the other two waves are.

    part 13 · aperiodic

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