The freedom a crystal has not

A crystal translated by half a cell is a different arrangement of atoms in space and the same crystal: the operation that moved it is not a symmetry, but the object it produced is congruent to the original and has the same everything. There is no parameter to vary — a lattice’s degrees of freedom are the lattice’s, and a rigid translation is not a new structure.

A quasicrystal has a parameter, and it is a genuinely new thing.

Cut and project builds an aperiodic chain by slicing a two-dimensional lattice with a strip of irrational slope and keeping the shadow of what falls inside. The construction has an input nobody needs to mention while building one chain: where the strip sits. Slide it perpendicular to itself, without turning it, and the lattice points that fall inside change.

What changes, and what does not #

The chain that comes out is different: the sequence of long and short tiles is not the same sequence. What is identical is everything a physicist would use to characterise it — the density of points, the two tile lengths, the ratio of their frequencies, the inflation factor, and the diffraction pattern.

That combination has no analogue in a periodic crystal. Two different arrangements of the same atoms with the same density, the same local environments in the same proportions, and the same diffraction pattern, related by a continuous parameter with no energy attached to it. The parameter is called a phason, by analogy with a phonon: a phonon is a displacement in ordinary space, a phason is a displacement in the perpendicular space the extra dimension that makes it periodic is about.

The difference is a set of flips #

The differences are not scattered damage. They are transpositions: an L and an S swap, leaving the number of each unchanged and moving one point of the chain by the difference of the two lengths. Each one is a phason flip, and it is the elementary excitation of the degree of freedom.

That a flip costs nothing is a statement about the matching rules as much as about the window. The rules are local conditions on how decorated tiles may meet, a legal tiling is one that satisfies them everywhere, and both chains here are legal — a flip exchanges two tiles for two others in a way the rules still permit. So the rearrangement is not a violation absorbed by some tolerance; it is a move within the set of legal tilings, and the existence of a continuum of them is what the freedom is.

Counting them makes the picture quantitative. At a window offset of a fifth, a chain of two hundred and seventy tiles has forty-odd flips relative to the unshifted one — about one tile in six — and the count grows with the offset — in proportion while the shift is small, and more slowly once it approaches half the window, where a point leaving one edge is met by a point entering the other. That is not a small perturbation of the structure; it is a substantial rearrangement, which makes the invariance of everything else more surprising rather than less.

Locally indistinguishable, which is stronger than similar #

Take any patch of one chain — any eight consecutive tiles, say — and look for it in the other. It is there. Take forty patches and every one of them is there.

This is not an approximation or a statistical statement. The two chains have exactly the same set of finite subwords, occurring at exactly the same frequencies, which is the property called local indistinguishability: no measurement confined to a finite region can tell which of the two chains it is looking at, however large the region, provided it is finite.

The reason is visible in the construction. A patch of the chain corresponds to a run of lattice points whose perpendicular coordinates lie in a sub-interval of the window; sliding the window slides which sub-interval, and since the perpendicular coordinates are equidistributed, every sub-interval recurs. So every patch recurs, in both chains, at the same rate.

That is the sharpest way to say what the freedom is. It is not that the two chains are alike. It is that no local experiment can be devised to tell them apart, and a global one would have to see the whole infinite chain at once.

The window is a choice with two edges #

Two details of the construction decide everything above, and both are worth stating because a reader building one will get them wrong otherwise.

The window’s width is not free. It is the projection of one unit cell of the higher lattice onto the perpendicular direction, and that value is forced: a wider window catches too many points and produces three tile lengths, a narrower one leaves gaps and the chain is no longer of constant density. The Fibonacci chain is where that is worked out, and the same figure that draws the chain draws what happens either side of the correct width.

The window’s edges belong to somebody. A lattice point whose perpendicular coordinate lands exactly on an edge is in or out according to which convention the code uses, and for almost every offset no point does — the coordinates are irrational multiples of the spacing. But there is a countable set of singular offsets at which a point lands on an edge, and at those the two conventions give chains differing in one tile. Those are the offsets at which the chain has extra symmetry: the singular Fibonacci chain is the one with a mirror in it, and it is the exception rather than the rule.

The count of offsets that give genuinely distinct chains is therefore uncountable, and the count that give chains with any symmetry at all is countable. A quasicrystal picked at random from its own family is asymmetric, and the symmetric members are a measure-zero set — which is the opposite of the situation in a periodic family, where the symmetric member is the generic one.

The diffraction cannot see it either #

The peak positions are identical, which is immediate: both chains are projections of the same lattice through the same slope, so the reciprocal structure is the same lattice’s, and only the phases of the structure factors change. And a phase change is exactly what a diffraction experiment throws away — which is the standing subject of the phase problem, arriving here as a consolation rather than a loss.

The intensities are computed rather than argued. On chains of a few hundred points the strong reflections agree to about a tenth of a percent between the two offsets, and the disagreement falls as the chains lengthen. So a diffraction experiment on a quasicrystal cannot determine where its window sits, any more than it can determine the phase of a structure factor, and for the same reason.

A tilt is a different thing entirely #

Sliding the strip is free. Tilting it is not, and the difference is what makes phasons measurable at all.

A change of slope is a uniform phason strain. It is not a symmetry of the construction and it is visible in the diffraction pattern: every reflection moves, by an amount proportional to its perpendicular index — the second integer that the satellites that need a second integer introduces — so the pattern is sheared in reciprocal space rather than left alone. Peaks with large perpendicular indices move a long way; ones with small perpendicular indices barely move at all.

That linear dependence on an index nobody can see directly is the signature experimentalists look for. A quasicrystal grown quickly and left unannealed has peaks displaced in exactly that pattern, and annealing it moves them back. The displacement measures a strain of the phason field, and it is one of the few quantities in the subject that is both a property of the sample and directly readable off a photograph.

So the honest statement about the freedom is two-sided. A constant phason shift is undetectable by any measurement. A gradient of it is detectable, and is measured routinely.

How many chains there are #

A last piece of counting, because it says what kind of object the family is.

Offsets differing by the perpendicular projection of a lattice vector give the same chain shifted along, so the genuinely distinct offsets form a circle rather than a line — the window’s width, with its ends identified. Every point of that circle gives a chain, distinct points give distinct chains, and all of them are locally indistinguishable from one another.

So the family of Fibonacci chains is a continuum, parameterised by a circle, on which no local measurement is a function. That is a strange object and it is the right one: the collection is what mathematicians call the hull of the tiling, the circle is its transversal, and the fact that the hull is connected while every member is rigid is exactly what makes quasiperiodic order a different thing from periodic order.

A periodic chain’s hull is also a circle — the translations modulo the period — and there the members are distinguishable, by where the atoms are relative to an origin. The difference is that a translation of a periodic chain moves it; a phason shift of a quasiperiodic one rearranges it. One is a rigid motion and the other is not, and only the second produces a genuinely new arrangement with the same everything.

Where the flips go when the crystal is warm #

The energy cost of a phason flip is not zero in a real material — the ideal construction has no energy in it at all, and a real quasicrystal is made of atoms with interactions. What is true is that the cost is small, because the two configurations either side of a flip have the same local environments in nearly the same proportions.

Two consequences follow, and both are experimental facts rather than parts of this computation.

Phason flips are the atomic mechanism of diffusion in quasicrystals, in the material what Shechtman measured turned out to be. An atom hops a short distance and the tiling rearranges locally; the same move in a periodic crystal would require a vacancy, and in a quasicrystal it does not, because the structure has a nearby configuration to fall into. Diffusion in icosahedral alloys is anomalously fast at low temperature for this reason.

Frozen phason disorder broadens peaks. A quasicrystal with random flips scattered through it has a diffraction pattern whose reflections are broadened by an amount growing with the perpendicular index, which is the same signature as a uniform strain smeared out. Distinguishing frozen disorder from strain is done by whether the peaks shift or merely broaden.

Neither statement is derived here. This essay computes what the ideal construction does; the material’s behaviour is physics, and the site’s habit is to name the boundary rather than blur it.

What a phason is not #

Three confusions worth heading off, because the word gets used loosely.

A phason is not a phonon. A phonon displaces atoms in physical space and costs elastic energy proportional to the square of a strain. A phason rearranges which atoms are where, in the perpendicular space, and in the ideal construction costs nothing at all. Both appear as extra branches in the excitation spectrum of a real quasicrystal, and the phason branches are the ones with no restoring force in the long-wavelength limit.

A phason is not disorder. A chain at any offset is a perfect quasicrystal — sharp peaks, exact self-similarity, the same tile frequencies. Frozen random flips are disorder, and their signature is broadening rather than a shift. The distinction matters because a quasicrystal with a large uniform phason strain is perfectly ordered and looks bad, while one with a little random flipping is imperfect and looks fine.

A phason is not a modulation. The satellites that need a second integer describes a periodic structure displaced by a wave, whose extra freedom is the phase of the wave — and that freedom is also called a phason, correctly, because it is the same higher-dimensional translation seen in the incommensurately modulated case. The two are the same object; the modulated case is the one where the physical space contains a lattice and the quasiperiodic case is the one where it does not.

Why the parameter exists at all, in one sentence #

The construction has a symmetry that a periodic one has not, and the freedom is its shadow.

A periodic crystal’s positions are a lattice: a discrete set closed under translation, so the only continuous parameters are the rigid motions of the whole thing. A quasicrystal’s positions are a projection of a higher-dimensional lattice, and that lattice has translations in directions the projection does not see — the perpendicular directions. Translating the higher lattice perpendicular to the cut is the same as translating the cut, and it produces a chain that is not the same chain.

So the degrees of freedom of a d-dimensional quasicrystal are those of a lattice in a higher dimension: d translations that move it in space, and the rest that rearrange it in place. The phason count is the number of extra dimensions the description needed, which is the tidiest thing about the superspace picture and the strongest argument for it.

What is exact here, and what is measured #

The claims in this essay divide cleanly, and the division is worth stating because two of them are exact and two are not.

Exact. The peak positions of two chains at different offsets are identical: both are projections of the same lattice, and the reciprocal lattice does not depend on where the cut is. The tile frequencies are identical, for the same reason.

Exact, on the construction. Every finite patch of one chain occurs in the other. Checked here on forty patches of eight tiles in each direction, which is a demonstration rather than a proof — the proof is the equidistribution of the perpendicular coordinates.

Measured. The intensities agree to a tenth of a percent on the strong reflections and to a few percent on weak ones, at chain lengths of a few hundred points. The residue is finite-size and shrinks with length; the numbers are reported rather than asserted, because a finite chain is not the object the exact statement is about.

Measured. The peak shifts under a tilt are of order a few percent of a reciprocal unit for a slope change of two percent, and their proportionality to the perpendicular index is read off the figure rather than fitted.

Where the ladder goes next #

The quasicrystals anchor now has four rungs: what Shechtman measured, the Fibonacci chain, icosahedral symmetry and the phason. The rung above is the one this essay keeps naming and refusing: a detector above three dimensions. Every periodic structure on this site is round-tripped — generated from a group and rediscovered from the drawing — and the superspace figures are the first whose group cannot be, because there is no detector in four dimensions here.

Building one would let the modulated and quasiperiodic structures make the trip the periodic ones make, and it would turn the phason from a construction into a symmetry statement: the phason shift is an element of the higher group that acts trivially on the physical space, and saying so precisely needs the higher group to exist in the code rather than in the prose.

A gradient in the parameter is visible #

The offset costs nothing when it is uniform, and that is the whole of what has been shown. Let it vary from place to place and the situation changes completely — which is what makes the parameter a physical field rather than a bookkeeping choice.

A uniform offset shifts the window; a gradient tilts it. If the offset varies linearly along the chain, the strip is no longer parallel to the line it was cut along: it is sheared, in the perpendicular direction, by an amount proportional to position.

A tilted strip selects different lattice points, and it does so systematically rather than locally. The resulting chain is not any of the chains a uniform offset produces, and it is not locally indistinguishable from them.

The diffraction sees it, and sees it in a characteristic way. Every peak of a quasicrystal is labelled by more integers than there are dimensions, and the extra ones amount to a coordinate in the perpendicular direction. A uniform offset multiplies each peak by a phase and leaves the intensities alone. A gradient moves the peaks, by an amount proportional to that perpendicular coordinate — so peaks with small perpendicular indices barely move and those with large ones shift visibly.

That pattern of shifts was the decisive test. In the years after the first icosahedral alloys were reported, several models proposed that the patterns came from twinned ordinary crystals or from randomly stacked icosahedral clusters rather than from a genuinely quasiperiodic structure. Measuring the peak positions and finding shifts scaling with the perpendicular index — and finding them vanish in well-annealed samples — is what distinguished a quasicrystal with some phason strain in it from something else entirely.

The parameter has its own dynamics #

Because the offset costs no energy when uniform, it behaves like a hydrodynamic variable, and its dynamics are unlike anything a periodic crystal has.

A phonon propagates and a phason diffuses. A long-wavelength distortion of the positions travels as a sound wave; a long-wavelength variation of the offset relaxes instead, by atoms hopping between the two positions a flip exchanges. The relaxation is slow, and it becomes slower as the wavelength grows.

So the flips are real atomic jumps. The rearrangement drawn above as a bookkeeping difference between two chains is, in a material, an atom moving from one site to another — which means phason relaxation is a diffusion process with an activation energy, frozen at low temperature and active at high.

And frozen disorder in the parameter scatters. A quasicrystal cooled with random phason fluctuations in it has diffuse intensity around its peaks, with a shape set by how those fluctuations are correlated — the same reading a diffuse pattern gets in an ordinary crystal, applied to a degree of freedom no ordinary crystal has.