The satellites that need a second integer

Take a perfectly ordinary crystal and displace every atom from its lattice site by a wave. If the wave’s period is a whole number of cells, the result is a superstructure with a larger cell and nothing conceptually new. If it is not — if the period is 4.74 cells, say, or 1/0.211 of one — then no cell of any size holds the structure. The displacement pattern never repeats.

Such a thing is called incommensurately modulated, and the first surprise is that it diffracts to sharp spots. Not a smear, not a halo: points, as sharp as the main reflections and in perfectly definite places. The second surprise is where they are. Around each ordinary reflection there appear extra ones — satellites — at positions that no cell explains, and indexing the pattern takes two integers per reflection where a crystal needs one.

This is the case the site’s aperiodic field has named twice in passing and never drawn: neither a crystal nor a quasicrystal, discovered forty years before quasicrystals, and the reason the machinery for describing them was already in place in 1982 when it was needed for something else.

The spectrum, computed from where the atoms are #

The diffraction calculation on this site has always been the same three lines: sum exp(2πiQ·x) over the atoms, take the modulus, square it. Nothing in that sum assumes periodicity — it runs over the atoms wherever they are.

Running it on the modulated chain gives a pattern with two kinds of peak. The main reflections sit at whole numbers, which is where the average structure would put them: an atom’s average position is its lattice site, so the average structure is the ordinary lattice and its reflections survive. Around each of them, at distances ±q, ±2q, ±3q, sit the satellites.

Sharp is not the same as periodic #

That sentence is the one the aperiodic field on this site exists to make, and this is its cleanest case.

A sharp diffraction peak means the atoms are correlated over long distances — that knowing where one atom is fixes where a distant one is. Periodicity is one way to achieve that and not the only one. A modulated structure achieves it with two ingredients, each perfectly determinate: a lattice, and a wave. Neither is random, so the positions are known however far one goes, and the diffraction is sharp.

What periodicity does buy is a description in three integers. Lose it and the reflections cannot be labelled with three whole numbers however the cell is chosen, and something else has to be found — which is the practical form the discovery took, and it was met as an indexing failure long before it was understood.

Two integers, and the count that shows it #

The indexing is where the argument becomes arithmetic rather than description.

Every peak in the computed spectrum can be written as h + m·q with h and m small integers: h counts the reflection of the average structure, m counts how many wavelengths of the modulation the reflection carries. With both integers available, every peak is accounted for. With only h, most of the pattern is unexplained.

The extra integer is not a fudge and it is not a fitted parameter. It is a count, of how many quanta of the modulation a reflection carries, and it is an integer for the same reason a Miller index is: a diffraction peak arises from a periodicity, and each satellite arises from the beating of two — the lattice’s and the wave’s.

The intensities are Bessel functions, which is a check #

Positions alone would leave the account descriptive. The intensities settle it, and they do so exactly.

A displacement that is sinusoidal enters the structure-factor sum as exp(2πih·A sin θ), and that expression has a classical expansion: exp(iz sin θ) = Σ J_m(z) exp(imθ), the generating identity of the Bessel functions. Term by term, that says the m-th satellite of the reflection h has amplitude J_m(2πhA) — so the intensities of the satellites are J_m(2πhA)², with nothing free to adjust.

Two consequences follow immediately and both are used in practice.

The amplitude of the modulation can be measured from intensity ratios. J₁/J₀ at small argument is roughly πhA, so the ratio of first satellite to main reflection gives A directly, and doing it at several h over-determines the answer. A displacement of a hundredth of a cell is measurable this way.

The first of those has a consequence for what counts as measurable. An ordinary refinement reports positions with an uncertainty; a modulated refinement reports the amplitude of a wave, and the amplitude can be far smaller than the uncertainty on any individual atom, because it is determined by a ratio of intensities across many reflections rather than by one. Displacements of a few hundredths of an ångström are routinely reported this way — quantities no single atomic position could resolve, extracted from the systematic pattern in a set of weak reflections. Systematic absences do the same trick with zero intensity: a claim about a whole class of reflections is stronger than a claim about any one.

And satellites of order m grow with h. The argument of the Bessel function is 2πhA, so high-angle reflections have strong satellites where low-angle ones have almost none. That is the signature that distinguishes a modulation from a superstructure with a similar-looking pattern — and it is visible in the figure below, where a Bessel zero puts a main reflection almost out.

Displacement is not the only kind #

The modulation above moves atoms. Two other kinds are common and produce the same shape of pattern.

Occupational modulation leaves the sites where they are and varies how much of an atom is on each — a composition wave, in an alloy where two species alternate with a period incommensurate with the lattice. When such a wave happens to be commensurate it is an ordinary superstructure and the satellites land on rational positions, which is the case the next rung is about. Anisotropic-displacement modulation varies the size of the thermal ellipsoid rather than the position.

All three produce satellites at the same places, because the positions of the satellites depend only on q. What differs is the intensities, and telling the three apart is a refinement question rather than an indexing one. So the position of a satellite says there is a modulation and how long its wave is; the intensity says what the wave is doing.

The average structure, and what a refinement that ignores the satellites reports #

There is a practical failure mode here that is worth naming, because it was how several of these materials announced themselves.

Ignore the satellites — index only the main reflections, refine only them — and what comes back is the average structure: the atoms at their lattice sites, with the modulation absorbed into the one parameter that can absorb it, the temperature factor. The refinement converges. The residual is acceptable. And the reported thermal ellipsoid is enormous and elongated along the modulation direction, because the refinement is describing a spread of positions as though it were vibration.

That is a well-formed answer to the wrong question, and it is the same shape of failure this site has met in near-symmetry and in what a powder pattern loses: a procedure that cannot express the truth returns the closest thing it can express, with no indication that it has done so. The tell is the ellipsoid, and the fix is to index the extra spots.

Where these actually occur #

Incommensurate phases are common, unglamorous and much older than quasicrystals.

Sodium nitrite going through its ferroelectric transition passes through an incommensurate phase over a range of about a degree. Sodium carbonate has one at room temperature. Charge-density-wave materials — one-dimensional metals in which the conduction electrons form a periodic pattern of their own — modulate the lattice at a wavevector fixed by the electron count, which is generally incommensurate. Many minerals, plagioclase feldspars among them, show satellite reflections that were photographed for decades before anybody could refine them.

The wavevector in all of these is set by something physical: an electronic instability, a composition, a competition between two interaction lengths. Nothing about the lattice prefers a particular q, which is precisely why the value is generally irrational — a rational value would be a coincidence, and coincidences are rare.

Who found them, and why the timing matters #

Satellite reflections were photographed in the 1920s and 1930s and called Gitterstörungen — lattice disturbances — which is a name that says nobody knew what they were. Through the 1940s and 1950s they accumulated: alloys, minerals, ferroelectrics, all showing extra spots that no cell explained and that were too sharp to be disorder.

The description that works arrived in the 1970s, from Pim de Wolff, Aloysio Janner and Ted Janssen, who proposed treating the structure as periodic in a space of more than three dimensions and taking the physical structure as a section through it. By the end of the decade the superspace groups had been enumerated, refinement software existed, and incommensurate structures were being solved as a matter of routine.

The timing is the part worth noticing. When Shechtman’s tenfold pattern appeared in 1982, the machinery for aperiodic-but-ordered structures had been in place for a decade and was being used weekly — and it took years for the connection to be made, because the two communities did not overlap. What incommensurate structures had that quasicrystals do not is a lattice underneath the modulation; what they share is the idea that sharpness comes from long-range order rather than from repetition, and the idea was already written down.

What separates this from a quasicrystal #

The two are constantly confused and the distinction is sharp.

A modulated structure has a lattice. Its average structure is an ordinary crystal, its main reflections are that crystal’s, and the satellites are a perturbation of it — one that can be made arbitrarily small by turning the amplitude down, at which point the material becomes an ordinary crystal continuously.

A quasicrystal has no lattice and no average structure. Its diffraction pattern has no distinguished subset of main reflections; the peaks fill reciprocal space densely; and there is no parameter that can be turned down to make it periodic. Its point symmetry is one no lattice may have, which the modulated case never is.

Where the exactness stops #

Three limits, and the first is the honest one about the word incommensurate.

No measurement establishes irrationality. A wavevector measured as 0.2110 ± 0.0002 is consistent with 0.211, with 19/90, and with an irrational number. What is measurable is that q is not a simple fraction, and what is inferred is that it varies continuously with temperature or composition — a rational value would have to be locked, and a value that drifts is not. So “incommensurate” is a statement about behaviour, not about a number, and the same limit applies to every claim of irrationality this site makes about a real material.

The chain here is one-dimensional and finite. Two hundred atoms give peaks of width about a two-hundredth, which is why the satellite comparison is made on the mean of a pair. A real crystal has 10⁸ cells and peaks limited by the instrument, and the arithmetic is the same with the ripple far smaller.

And the round trip does not run. Every periodic figure on this site hands its point set to a detector that rediscovers the group; a modulated structure has no space group in three dimensions for a detector to find. What is verified here instead is the pair of independent calculations — sum and Bessel series — and that is a weaker guarantee, honestly a different kind of one, and it is named rather than glossed.

The phase of the modulation is free, and nothing measures it #

There is a degree of freedom in a modulated structure that has no analogue in an ordinary crystal, and it is worth naming here because it is the reason the description needs a second integer rather than a second lattice.

The modulation is a wave, and a wave has a phase. Slide the whole modulation along the chain by any amount — change the offset in the sine — and the structure changes: every atom moves. The diffraction pattern does not. Every satellite keeps its position, because positions depend on q alone, and every satellite keeps its intensity, because the Bessel amplitudes depend on the modulation’s amplitude and not on where its crest happens to fall.

In a commensurate structure that freedom would not exist. If q were a rational fraction, only finitely many offsets give distinct structures, and they generally differ in energy — one of them is the arrangement the crystal takes. When q is irrational the offsets form a continuum, every one of them is a legitimate structure, and the energy of the ideal infinite crystal is the same for all of them.

That continuous freedom is a phason, and it is the modulated structure’s own version of the degree of freedom an ordinary crystal has not. Two consequences follow. A refinement of a modulated structure has to fix the offset by convention, exactly as an ordinary refinement fixes an origin, because it is not determined by the data. And the freedom is a real physical variable rather than a bookkeeping one — a slow sliding of the modulation is a mode the material actually has, and in a real, finite, defective crystal it is pinned rather than free, which is where the difference between the ideal description and the material begins.

Locking in #

The wavevector is set by something physical, and physical quantities change with temperature — which produces the behaviour these materials are best known for and which the arithmetic on this page does not predict.

As a modulated phase is cooled, q typically drifts, and then at some temperature it stops drifting and sits at a simple rational value: a lock-in transition. Below it the structure is commensurate, has a genuine unit cell — a supercell of the original — and the satellites become ordinary superlattice reflections indexed on three integers again.

Why it happens is easy to state and hard to compute. A commensurate structure has only finitely many distinct offsets, so the crystal can choose the one of lowest energy; an incommensurate one cannot, and the energy it gains by locking is what the drift is trading against. The result is that q as a function of temperature is not a smooth curve but a sequence of plateaux at rational values, with the widest plateaux at the simplest fractions — the devil’s staircase, which is the same arithmetic of continued fractions that three gaps and never four is about, arriving here as a phase diagram. That the plateaux exist at every rational, and that their widths fall off as steeply as they do, is derived for a model chain in every fraction holds a window, where it turns on the repulsion being convex and on nothing else about it.

None of that is decidable by the machinery on this page, and the boundary is the site’s usual one. Where the satellites sit, given q, is exact. How strong they are, given the amplitude, is exact. What q is, and what it does when the crystal is cooled, is a measurement about a material.

The two paragraphs above are the same fact seen from opposite ends. An irrational wavevector buys a continuum of structures that the measurement cannot distinguish and the energy does not separate; a rational one buys finitely many, which the energy does separate and the crystal chooses between. What sits between them is not a small gap: it is the difference between a structure with a unit cell and one without, and the material moves across it at a definite temperature.

Where the ladder goes next #

The description so far is a lattice plus a wave, which is two objects. There is a way to make it one: add a dimension, in which the structure is periodic again, and take the physical structure as a cut through it. That is superspace, it is the reason the quasicrystal problem was solvable the moment it appeared, and it is the next rung.