The extra dimension that makes it periodic

A modulated structure is described by two things: a lattice, and a wave whose period does not fit it. Two objects, neither of which is the structure, and a diffraction pattern that needs an extra integer per reflection.

There is a way to make it one object. Add a dimension. In four dimensions the lattice and the wave become a single periodic pattern with an ordinary unit cell, and the physical structure is what a three-dimensional slice through it meets. Every awkward feature of the description turns into an ordinary feature of a periodic crystal seen at an angle — the extra integer becomes a fourth Miller index, the wave becomes a coordinate, and the group becomes a superspace group of which there are as ordinary a list as the two hundred and thirty.

The construction is cut-and-project, which this collection built for quasicrystals, with one difference. That difference turns out to be the entire distinction between an incommensurate crystal and a quasicrystal, and it is visible in the picture.

Why an extra dimension helps at all #

The trick is worth stating in general before it is applied, because it is one of the most useful moves in the subject.

A structure that is not periodic in d dimensions can sometimes be represented as a periodic structure in d + n dimensions, cut along a subspace at an irrational orientation. The reason it can is that the reflections of the original structure need d + n integers — which is to say they are the projection of a (d + n)-dimensional reciprocal lattice. And a reciprocal lattice in d + n dimensions has a direct lattice in d + n dimensions behind it, so the structure that produced those reflections must be a section through something periodic up there.

That is not a metaphor: the arithmetic is a Fourier transform and it works in either direction. Count the integers a diffraction pattern needs, and that is the dimension of the space in which the structure is periodic.

For a modulated crystal the count is 3 + 1 in the simple case — three lattice indices and one satellite order — so the structure is periodic in four dimensions. For an icosahedral quasicrystal it is six. For the Fibonacci chain it is two, which is why that example can be drawn in full.

Curves, not intervals #

Here is the difference this essay exists for.

In the quasicrystal construction the atomic surfaces are line segments with ends — a window, a strip, an acceptance domain. Moving the cut brings some segments into contact and takes others out of it, so different cuts meet different numbers of surfaces and the resulting point sets differ: not merely displaced, but containing different atoms. That is what makes a quasicrystal aperiodic in a strong sense; there is no average lattice, because the points themselves come and go.

In the modulated construction the atomic surfaces are curves that cross every cut. Each one is a graph — one x for each y — so a line meets it exactly once however it is placed. Every cut therefore gives the same number of atoms, one per cell, with the same average positions. The structure has an average lattice, it is a perturbation of an ordinary crystal, and turning the amplitude down returns it continuously to one.

That single topological difference — a segment with ends against a curve without — is the whole distinction, and everything else follows from it.

The fourth coordinate is a phase, and it is not observable #

If every cut gives an equally good structure, what distinguishes them?

Nothing measurable. Shifting the cut perpendicular to the physical space slides the modulation wave along the chain — the atoms move, but the structure that results is the same structure with the wave’s phase changed. In an infinite crystal there is no experiment that distinguishes one phase from another: the diffraction pattern is identical, since intensities depend on the amplitudes and not on where the wave started.

So the fourth coordinate is a degree of freedom the material genuinely has and no measurement can pin down. That has a name in the physics — the phason — and it is not merely bookkeeping: a slow variation of the phase across a real crystal is a real defect, it costs almost no energy in a truly incommensurate phase, and it broadens satellite reflections in ways that are measured.

This is the strongest argument for taking superspace as a description rather than a device. If the extra coordinate had no physical consequence it would be a convenient fiction. It has one, and the fiction is the idea that the structure was ever only three-dimensional.

There is a caution about the picture worth stating, because a two-dimensional drawing of a four-dimensional idea invites one particular misreading. The vertical direction in these figures is not a direction in the material. Nothing is stacked; no atom sits above another; the material is the row of points along the bottom and nothing else. The extra coordinate is a bookkeeping dimension that happens to have physical consequences, which is an unusual combination and the reason the subject took a decade to be accepted.

The right way to read the height of an atom on its curve is as the phase of the wave at that atom — a number between zero and one saying how far through the modulation cycle that particular site is. Read that way the picture contains no extra space at all: it is a plot of phase against position, with the curves saying how displacement depends on phase.

What is a symmetry, up there #

A superspace group is the same kind of object as a space group, in four dimensions and with one restriction.

Its operations are integer matrices acting on the four coordinates together with translations — the round trip’s own definition of a symmetry, a dimension up. The restriction is that the physical three-dimensional subspace must be preserved: an operation may not tip the cut into a different orientation, since that would describe a different material. So the matrices are block-structured, with a three-dimensional part acting on the physical space and a one-dimensional part acting on the phase.

The consequence a crystallographer sees is a fourth Miller index and a longer symbol. Extinction conditions follow exactly as they do in three dimensions, computed from the same kind of sum, and they now apply to satellites as well as to main reflections. Superspace groups were enumerated in the 1970s and 1980s and the tables exist; this site does not derive them, for the same reason it does not derive the 230.

The rational case, where the cut lands on a repeat #

The distinction between incommensurate and merely large-celled is a distinction about one number, and the picture makes it a distinction about the slope of a line.

If q is rational — p/r — the cut has rational slope, it passes through the higher-dimensional lattice periodically, and the resulting structure repeats after r cells. It is a superstructure: an ordinary crystal, with a cell r times as large, whose “satellites” are perfectly ordinary reflections of that larger cell. If q is irrational the cut never closes, and no cell of any size holds the structure.

There is a second reading of the same figure that a metallurgist will recognise. The best rational approximations to an irrational number are its continued-fraction convergents, and those are exactly the rows marked in the first colour — the same objects that decide which repeat an epitaxial layer chooses, and the same reason the golden ratio is the hardest number to lock. A modulated crystal choosing a commensurate period and an overlayer choosing a coincidence repeat are solving one arithmetic problem in two subjects.

Locking in, which is a real transition and not a limit #

Many incommensurate phases end by becoming commensurate as the temperature falls: q drifts with temperature, approaches a simple fraction, and at some point jumps to it and stays. That is a lock-in transition, it is first-order, and it happens because a commensurate structure has an energy term an incommensurate one does not.

The mechanism is worth one sentence because it is a nice piece of physics that symmetry alone cannot supply. In a commensurate structure the wave has a definite phase relative to the lattice — the phason freedom is gone, because only certain phases are equivalent — and the structure can lower its energy by choosing the best one. In an incommensurate structure every phase is equivalent, so there is nothing to choose and nothing to gain. The gain is what locks the wave in, and it competes with the elastic cost of moving q off its natural value.

Which fraction wins is decided by that competition, and the competition has a name and a literature: the frustration between a length the electrons want and a length the lattice offers is the same problem as a Frenkel–Kontorova chain, and the sequence of locked states it produces as the parameter is swept is a devil’s staircase. Every step of that staircase is one row of the shortlist above, and the width of the step is the energy the locking gains. The arithmetic here supplies the shortlist — the convergents, in order of how little q has to move — and stops, because the shortlist says which fractions are near and not which of them is cheap. The step widths are derived separately, for a chain of particles with a stated convex repulsion, in every fraction holds a window: there every rational locks over an interval of positive width, the ordering of the widths is the Farey ordering rather than the ordering of the convergents, and the widths depend on the range of the interaction while the set of locked fractions does not. What is still not computed anywhere here is a material’s energy — the repulsion in that model is postulated, and the essays that name a real energy say that it is being named rather than derived.

Where the exactness stops #

Three limits.

The construction is exact and the material is not. The atomic surfaces above are smooth sine curves because the modulation was defined that way. A real modulation is a periodic function of the phase and can have harmonics, discontinuities, and a shape that has to be refined; the description handles all of it, with more parameters, and the picture stops being one curve.

Irrationality is never measured, only inferred from behaviour, which is the same limit the previous rung ends on. A wavevector that varies continuously with temperature is behaving incommensurately; one measured at a single temperature is a number with an error bar.

A superspace group is not visible in the physical structure. What a measurement sees is three-dimensional: an average structure with a space group, and satellites with extinction conditions. The four-dimensional group is inferred from those, and two different superspace groups can produce the same three-dimensional evidence — which is the same underdetermination ordinary space groups have, one dimension up and with fewer people having met it.

And no detector runs up there. This site’s round trip rediscovers a group from a point set in two and three dimensions. There is no four-dimensional detector here, so the superspace figures are generated from a stated construction and verified by the agreement of two calculations rather than by rediscovery. That is the same limit the three-dimensional machinery removed when it was written, and it is worth recording that the removal has not happened in four.

The reason this mattered in 1984 #

There is a closing observation which is the whole reason this pair of essays sits in the aperiodic field rather than in the diffraction one.

When quasicrystals appeared, the objection was that a sharp diffraction pattern implies a lattice. The answer already existed: incommensurate crystallography had spent a decade describing sharp patterns from structures with no three-dimensional cell, using higher-dimensional periodicity, and had the mathematics, the software and the tables to prove it. The quasicrystal case was the same construction with a different acceptance domain — segments instead of curves, six dimensions instead of four — and the community that had built it recognised the shape of the problem immediately.

Two subjects, one construction, and a decade of separation because nobody had asked whether they were the same question. That is the pattern this site’s applied field is built on, arriving in the field that ought to have been proof against it.

How many extra dimensions, and what gets classified up there #

One extra dimension is the simple case, and it is worth saying what happens when the structure needs more, because the counting is the same counting and the answers are tabulated.

The rule is one extra dimension per independent modulation wavevector. A structure with a single wave needs 3 + 1. A structure modulated by two waves whose wavevectors are independent over the rationals needs 3 + 2, and three such waves need 3 + 3. Beyond that nothing occurs in practice, so the classification stops at six dimensions — and the quasicrystals, which need 6 for the icosahedral case, are the same construction reached from a different direction.

Those higher-dimensional groups are enumerated exactly as the two hundred and thirty were. The (3+1)-dimensional superspace groups number 775, they were derived in the 1980s by the same extension arithmetic that produces the space groups, and they are what a modern refinement of a modulated structure reports. The symbol is a space-group symbol with the modulation wavevector and one extra position attached, which is a notation for the same information the extra index carries.

That is the sense in which the description is ordinary crystallography rather than an analogy. There is a lattice, a point group, a set of extensions, a list, a symbol and a table of systematic absences — every apparatus the three-dimensional subject has, in four dimensions, with the same derivations. What is missing is only the ability to draw it.

The trick is older than the crystals #

The move of representing an aperiodic object as a slice through a periodic one in more dimensions did not arrive with these materials, and knowing where it came from makes it look less like a device.

A function that is a sum of cosines with incommensurate frequencies is not periodic. It is the restriction to a line of a function that is periodic in as many variables as there are frequencies — put each frequency on its own axis, take the obvious periodic function of several variables, and read it along a line whose direction is the frequency vector. Harald Bohr built the theory of almost periodic functions on that observation in the 1920s, and the Diophantine approximation that decides how nearly the function repeats is the same arithmetic that decides how nearly a modulated crystal does.

The same picture is standard in dynamics, where a quasiperiodic orbit is a line of irrational slope winding round a torus, dense and never closing. That is the figure at the top of this page with the atoms removed: the cut is the line, the torus is the periodic structure, and the fact that the line never repeats is the fact that the crystal has no cell.

So the superspace description is a third instance of one idea rather than a crystallographic invention, and it is worth saying so for the reason the three-gap theorem’s rediscoveries were worth saying. One irrational number generating a one-dimensional sequence turns up in number theory, in analysis, in dynamics and in crystallography, and only the last of them calls the result a crystal.

Where the ladder goes next #

The obvious next rung is composite structures — two interpenetrating subsystems with incommensurate periods, where each is a modulation of the other and the superspace description holds both at once. It is the case where the higher-dimensional picture stops being a convenience and becomes the only description that is symmetric between the two halves.

Both observations point the same way. The extra dimension is not an invention to rescue an awkward case; it is what the description of an aperiodic object looks like whenever the aperiodicity comes from a finite number of incommensurate periods. Where it stops working is where that condition fails — a structure that is aperiodic for some other reason, a random arrangement or a glass, has no finite superspace to be periodic in, and no amount of adding dimensions produces one.