Two lattices, one crystal, and no cell at all

A crystal is periodic. That statement has survived every complication this collection has put to it: a modulated structure is periodic in a higher-dimensional space, a quasicrystal is a slice of a periodic lattice in six dimensions, and in each case the cell reappears once the right space is found.

Here is a case where the cell does not reappear in three dimensions and there was never one to begin with, in a material anybody can buy.

A composite crystal is two substructures with incommensurate periods, occupying the same volume. The mercury chain compound Hg₃₋δAsF₆ is the standard example: chains of mercury atoms run through channels in a host of AsF₆ octahedra, and the mercury spacing along a channel is not a rational multiple of the host’s repeat. The misfit layer sulphides — (LaS)ₓNbS₂ and its relatives — are the other family, with two kinds of layer stacked alternately and mismatched in the plane.

Why this is not a modulated crystal #

The distinction takes one sentence and it matters for everything below.

A modulated structure has a basic structure, which is periodic, and a wave applied to it. Remove the wave and a crystal remains. The wavevector q is incommensurate with the lattice, and the reflections are the lattice’s own plus satellites at h + mq.

A composite has no basic structure at all. Take away subsystem one and subsystem two is left, which is also a crystal; take away subsystem two and subsystem one is left. Neither is a modulation of the other, and there is no third structure of which both are perturbations. Nothing here is a superstructure, which is the other thing extra reflections usually mean. The two are peers.

What makes a composite look like a modulated structure — and what makes the literature describe both in the same language — is that each subsystem is modulated, by the other. The mercury chains feel the host’s periodicity and take up its rhythm; the host feels the chains. So a composite is a pair of modulated structures whose modulations are each other’s lattices.

The diffraction needs two integers, and so does everything #

Where the difference becomes a measurement is in reciprocal space.

Four indices for a three-dimensional crystal. That is the same arithmetic as the modulated case, and in superspace it has the same explanation: the structure is a three-dimensional cut through a four-dimensional periodic object, and four indices are what a four-dimensional reciprocal lattice needs. Where the composite differs is that the fourth basis vector is not a modulation wavevector added to a lattice — it is one of the two lattices, and which one gets called the fourth is a choice with nothing behind it.

What the two subsystems do to each other #

The modulations are not decoration and leaving them out changes the diffraction, so it is worth being explicit about what they are.

Subsystem one sits at multiples of a₁. It is embedded in a medium whose density varies with period a₂, so the potential each atom sits in has that period, and each atom is displaced by an amount depending on where it falls within the other subsystem’s cell. The displacement is therefore a function of x mod a₂ — a wave of the other’s period, sampled at this one’s positions.

Subsystem two does the same in reverse. The result is a pair of coupled modulations, each with the other’s wavevector, and there is no order in which to introduce them: setting either amplitude to zero leaves the other’s modulation intact.

In reciprocal space that coupling is what puts intensity on the mixed reflections. Without it, subsystem one would scatter only at multiples of 1/a₁ and subsystem two only at multiples of 1/a₂, and the pattern would be two independent crystals superimposed. The peaks at h/a₁ + k/a₂ with both indices non-zero exist because each subsystem carries the other’s periodicity, and their intensities measure how strongly the two are coupled.

That is a genuinely useful measurement and it is what such structures are studied for: the amplitude of the mixed reflections is the interaction between the subsystems, read directly.

What a single-lattice procedure makes of it #

The practical consequence is worth a computation, because it is what actually happens at an instrument.

This is a real failure mode and it has a real history. Composite structures were solved as “superstructures with disorder” for years before superspace made the alternative describable, and the tell was always the same: a set of strong reflections nothing accounted for, at positions that were not rational multiples of anything. The same tell as a misindexed powder pattern, and with the same cure: widen the description rather than discard the data.

The count below the figure is the other half. Lowering the intensity threshold by a factor of ten roughly quadruples the number of peaks, and there is no threshold at which the count stops rising — the set of positions h/a₁ + k/a₂ is dense on the line. What makes a diffraction pattern look like a row of spots rather than a smear is not that the positions are discrete; it is that only finitely many of them are above any given intensity.

Dense, and yet a pattern #

That last point deserves its own paragraph, since it is the thing about incommensurate structures that is hardest to hold onto.

A periodic crystal’s reflections sit on a lattice: discrete, and separated by a fixed minimum distance. A composite’s sit on a module — the set of integer combinations of two incommensurate basis vectors — which is dense. Between any two peaks there are infinitely many more.

They are still sharp. Each peak is a delta function, not a broadened bump; the structure has genuine long-range order, and a measurement with fine enough resolution finds each one exactly where the two integers say. What falls away as the indices grow is the intensity, and it falls fast enough that any experiment sees a finite list.

And underneath both sits an arithmetic this collection has already built. The position of the nth atom of one chain, measured inside the other chain’s cell, is the fractional part of n·a₁/a₂ — which is a rotation of a circle by an irrational angle, exactly the object the three-gap theorem is about. So the local environments along a composite are not merely various: they take three families of spacing at every length of run, the largest being the sum of the other two, and which counts of atoms give two families rather than three is the continued fraction of the period ratio. A composite is a quasiperiodic sequence with atoms on it, and every statement that theorem makes about gaps is a statement about this structure’s contacts.

The control that says the effect is real #

An argument that a pattern cannot be indexed on one lattice is worth nothing unless the same procedure indexes a pattern that can.

So the whole calculation is run again with the period ratio set to three halves. Everything else is identical: the same two chains, the same modulation amplitudes, the same peak-finding, the same search over single periods. The best single period then indexes every one of the strongest peaks, because the structure has a common period of three units and is an ordinary crystal with a large cell.

The difference between the two runs is one number in one place, and it is the number that decides whether a unit cell exists. That is as close as a computation gets to isolating a cause.

How a composite is told from the alternatives #

Extra reflections are common and most of them are not composites. A working list of what else they can be, and how each is ruled out, is the useful part of this subject:

• A supercell. The extra peaks sit at rational fractions of the main ones, so multiplying the cell by a small integer accounts for everything. Test: are the positions rational multiples? Here they are not, at any denominator the data can resolve.
• Twinning. Two orientations of one structure superimpose two copies of one lattice. Test: is there an operation carrying one set of peaks onto the other? For a composite there is none, because the two sets have different spacings rather than different orientations. Twinning by merohedry is the case that looks most like this and it always has a lattice.
• A second phase. Two crystals in the beam give two lattices, which is exactly what a composite gives. Test: the mixed reflections. Two independent crystals produce no intensity at h/a₁ + k/a₂ with both indices non-zero, and a composite does — so the third colour in the figures above is the whole distinction, and it is a measurement rather than an inference.
• An incommensurately modulated structure. Test: is one set of reflections systematically weaker? If the satellites are an order of magnitude down on the main reflections there is a basic structure; if the two sets are comparable there is not.

The last two are the ones that need care, and both are settled by intensities rather than positions — which is a change of instrument as much as a change of argument, since positions are cheap to measure well and intensities are not.

Both are also measured here rather than asserted, and the two measurements are the same figure read twice. Set the modulation amplitudes of both chains to zero, leaving two independent lattices with the same two periods in the same beam, and among the forty strongest reflections there are no mixed ones at all. Restore the modulation and twenty-one of the forty are mixed. So the third family is the interaction between the subsystems and not an artefact of adding two diffraction patterns together — which is the second-phase test, run on the structure it is meant to exclude.

The modulated-structure test comes out of the same sorting. The mixed reflections here have a median intensity in the same order as the first chain’s own — the higher of the two, in fact — where a modulated structure’s satellites fall away as the square of a small amplitude and land one or two orders down. There is no strong-and-weak split to read, so there is no basic structure to nominate, and a refinement that nominates one anyway is choosing rather than finding.

Superspace, and what it buys #

The description that works treats the structure as a cut. Build a four-dimensional periodic object whose three-dimensional slice is the composite; the two subsystems become two sets of hypersurfaces at different slopes, and the modulations become the shapes of those surfaces.

What that buys is a group. A composite has a superspace group in exactly the sense a modulated structure does, its reflections have systematic absences, and the absences are read the same way. The classification does not break; it moves up a dimension and carries on.

What it does not buy is a unit cell in three dimensions. The cut of a lattice is not a lattice, and no choice of slice produces one. So the honest three-dimensional statement is that the structure has long-range order, a discrete diffraction module of rank four, and no translational period — which is precisely the definition a quasicrystal satisfies, and which the International Union of Crystallography adopted in 1992 when it redefined the word crystal, and composites are usually left out of that word only by convention.

The construction in its simplest form is a periodic lattice in two dimensions and a line of irrational slope cut through it — the picture cut and project is built on. Composites, modulated structures and quasicrystals are three things that one picture does, and what distinguishes them is what the higher-dimensional object looks like and nothing else. A modulated structure is a lattice of smooth wavy hypersurfaces; a composite is two sets of hypersurfaces at two slopes, each waving with the other’s period; a quasicrystal is a lattice of flat, bounded windows. The cut is the same operation in all three, the three-dimensional results look nothing alike, and the classification that separates them lives entirely upstairs.

What is owned here #

The arithmetic of two incommensurate periods, the positions and relative intensities computed from two chains built to a stated rule, the indexing search and its failure, and the rational control. All of it is one-dimensional and none of it is a real material.

Not owned: any claim about why mercury chains sit in AsF₆ channels, any energetics, any statement about how such a structure grows or what holds it together. The compounds are named because they are the reason the arithmetic is worth doing; nothing about them is derived here.

What holds the two together #

The arithmetic here is silent about why such a structure exists at all, and the silence is worth marking rather than passing over.

Nothing in the geometry requires two subsystems to interpenetrate. A crystal of chains in channels could equally have the chains commensurate with the host, at a slight cost in bond length, and many do. What makes a composite is that the cost of stretching one subsystem to fit the other exceeds the cost of the mismatch — a comparison of energies, decided by chemistry, and not computed anywhere in this essay.

What the arithmetic does say is what the mismatch costs in description. Once the two periods are incommensurate there is no cell, no Bravais lattice in three dimensions, and no space group in the ordinary sense; the classification has to move up a dimension or give up. That is a statement about the description rather than about the material, and it is the reason such structures were reported as disordered for years before superspace made them writable.

The two halves are worth keeping apart because a reader meeting the phrase incommensurate composite usually meets it as a claim about the crystal, and it is at least as much a claim about what a crystallographer is able to write down.

A formula that is not a ratio of small integers #

There is a consequence of the mismatch that reaches a place symmetry usually does not, and it is the one worth telling a chemist about.

A compound’s formula counts atoms per formula unit, and counting atoms means counting them in a cell. A composite has no cell. What it has is two subsystems whose densities along the shared direction are 1/a₁ and 1/a₂, so the ratio of the two atom counts is the ratio of the two periods — and that ratio is irrational by hypothesis.

So the composition is not a ratio of small whole numbers. The mercury chain compound is written Hg₃₋δAsF₆ with a δ of a fraction, and the fraction is not a defect concentration, a vacancy population or an impurity: it is the period ratio, arrived at geometrically, and it is as exact as the structure is. A sample with a different δ would be a different structure rather than a less pure one.

That is a genuine exception to the law of definite proportions, and it is worth being careful about how large an exception. Berthollide compounds — non-stoichiometric oxides and sulphides — are the usual counterexample and they are a different phenomenon: there the composition varies continuously over a range and the structure absorbs it in vacancies and interstitials. Here the composition is fixed and irrational, which is a stranger thing, because a definite composition that is not a ratio of integers was not among the possibilities the law was framed to allow or to forbid.

The two subsystems can slide #

The phase freedom of a modulated structure has a sharper form here, and it is a physical degree of freedom rather than a convention.

Slide one subsystem along the shared direction relative to the other. In a commensurate structure that has a finite number of distinct results, one of which is the arrangement of lowest energy. In an incommensurate composite the offsets form a continuum, and — for the ideal, infinite, rigid case — every one of them gives the same energy, because the two subsystems never come into registry anywhere and so no offset is preferred.

That means the relative position of the two subsystems is not a structural parameter at all: it is not determined by the diffraction, and it is not determined by the energy either. What is determined is everything about each subsystem separately and the modulations they impose on each other, and those are the quantities a refinement returns.

The freedom is also a mode. A structure whose subsystems can slide at no cost has a low-energy excitation corresponding to sliding, and in a real crystal — finite, defective, and with the subsystems not perfectly rigid — the sliding is pinned rather than free. The gap between the ideal freedom and the pinned reality is where the interesting behaviour of these materials sits, and none of it is visible in the arithmetic on this page, which describes the ideal.

Where the ladder goes next #

The obvious direction is up in dimension, to the misfit layer compounds where the mismatch is two-dimensional and the two subsystems share one axis and not the others. The arithmetic is the same with more indices; the classification is superspace groups of rank five and above, built the way the extra dimension is built for a single modulation.

The other direction is the one the whole subject of aperiodic order circles. A composite is the plainest possible statement that a local period is not a global one: walk along either chain and it repeats perfectly, and there is no walk that repeats the crystal. The answer to how far does a local fact reach is, here, exactly as far as its own subsystem and no further.

The two sections above are the same fact reaching two different audiences. A chemist meets the mismatch as a composition that will not reduce to whole numbers; a physicist meets it as a coordinate the structure does not fix and the energy does not choose. Both are consequences of there being no common multiple of the two periods, and neither is available in a description that insists on a cell — which is the argument for the higher-dimensional account rather than a taste for it.

It is also the reason the rational control matters as much as the irrational case. A ratio of three halves produces two chains, one cell, one formula with whole numbers in it, and no sliding freedom whatever — every one of the three consequences reversed at once, from a single change to a single number. That is what makes the three consequences one consequence rather than three observations that happen to co-occur.