Neither a peak nor a bump

Which inflation factors exist settles one half of a question and states the other half as a quotation. The half it settles: a substitution whose inflation factor has a large algebraic conjugate cannot produce sharp diffraction, and the test is an inequality between two integers. The half it quotes: the converse is false, and the standing counterexample is the Thue–Morse substitution — factor two, trivially admissible, and no Bragg peaks at all.

That sentence has been carrying a theorem the site could not compute. What follows is what it can.

The measurement, and why it needs no threshold #

A Bragg reflection is a sum of terms that are all in phase. Its amplitude is proportional to the number of scatterers and its intensity to the square, so lengthening the chain by a factor of four multiplies the strongest peak by sixteen. Anything with phases that do not conspire behaves like a random walk instead: the intensity grows as the number of terms, not its square.

So the question does this chain diffract sharply has an answer with no threshold in it. Build the chain at several lengths, find the strongest peak, and fit an exponent to intensity against length. Two is a reflection; one is a bump. The Fibonacci chain fits 1.996 and the period-doubling chain fits 2.007, which is the answer this site expects from both.

The Thue–Morse chain fits 1.554.

That number is the essay. It is not two, so there is nothing in this pattern that a diffractometer would record as a spot; and it is not one, so there is nothing that could be called a broad background either. The peaks grow, and they do not grow fast enough.

The weighting deserves a note, because it is a choice and it has a reason. The two letters of a Thue–Morse chain occupy a perfectly periodic set of positions — this is a substitution of constant length, so the chain is a lattice with two kinds of scatterer on it — and the average scatterer therefore diffracts exactly like a crystal. Giving the two letters weights of plus and minus one subtracts that average away, leaving only what the ordering contributes. Without it, the interesting measurement would be sitting underneath a set of ordinary Bragg peaks with nothing to do with the sequence.

What lies between the two answers #

A diffraction pattern is a measure, and measures come in three kinds. A pure point measure is a sum of spikes with weights: that is Bragg scattering, and it is what every periodic crystal and every quasicrystal on this site produces. An absolutely continuous measure is an ordinary function spread over the whole line: that is diffuse scattering. Between them sits a third kind that has no everyday name — singular continuous — a measure with no spikes at all, that is nevertheless concentrated on a set of zero total length.

The Thue–Morse chain has one. Its diffraction is concentrated on a set of measure zero and carries no atoms, and the finite-length signature of that is precisely the exponent above: features that are sharpening indefinitely, at every stage, without ever becoming spikes.

This collection has met the same trichotomy once before, in a different quantity. A spectrum that is a Cantor set measures the energies a wave in a Fibonacci chain may have, and the answer there is also a set of measure zero with structure at every scale. The two are different objects — one is where a wave may sit, the other is where scattering goes — and it is worth keeping them apart while noticing that aperiodic order produces the same kind of thing in both.

The condition that decides it, when it decides anything #

There is a criterion that settles the pure-point question for substitutions of the kind Thue–Morse is, and it is a finite search rather than an analysis.

For a substitution of constant length — every letter replaced by a word of the same length — write out the substitute of each letter, one row per letter, and look down the columns. If some column holds the same letter in every row, the substitution is said to have a coincidence, and Dekking’s theorem says the resulting sequence has pure point diffraction.

The period-doubling substitution — A to AB, B to AA — has a coincidence in its first column at the first level, so it has a pure point spectrum, and the measurement above agrees: exponent 2.007. Thue–Morse — A to AB, B to BA — has none at level one, none at level two, and none at any level searched.

And the absence is not a failed search. The two substitutes differ in every position, so if σⁿ(A) and σⁿ(B) differ in every position then so do σⁿ⁺¹(A) and σⁿ⁺¹(B): each letter of the first is replaced by the complement of what the corresponding letter of the second is replaced by. The induction is one line and the machinery here checks it holds to ten levels, out to words of a thousand and twenty-four letters, because an induction that has never been evaluated is an induction nobody has tested.

That distinction — a bounded search that says nothing versus a bounded search backed by a reason that says everything — is the same one the undecidability of tiling forces on every claim of this kind, and it is worth having a case where the reason is available and short.

Complexity does not predict sharpness #

Here is the finding that is genuinely surprising, and it falls straight out of putting two of this site’s measurements beside each other.

The complexity function counts how many distinct blocks of each length a chain contains, and it is the standard measure of how complicated an aperiodic sequence is. The Fibonacci chain achieves the smallest complexity an aperiodic sequence can have: n + 1 blocks of length n, and no aperiodic chain manages fewer. Thue–Morse is considerably richer — 2, 4, 6, 10, 12, 16, 20, 22 — growing about three and a third times as fast.

So the simplest aperiodic chain there is diffracts to sharp peaks, and a more complicated one diffracts to nothing sharp whatsoever. The number that measures how many local configurations a sequence has does not measure how ordered it is in the sense a diffractometer records. Two chains can be built by the same kind of rule, with inflation factors that pass the same test, and land on opposite sides of the only question an experiment can ask.

And the value does not track sharpness either, which is worth stating plainly because it is the obvious next guess and the figure above refutes it. Of the three substitution chains measured on this page, the two whose diffraction is pure point are the Fibonacci chain, at balance one, and the period-doubling chain, at balance three. The one whose diffraction is not is Thue–Morse, at balance two. The property that is supposed to be doing the separating sits between the two it is supposed to separate from, so no threshold on it can work, and this is not a matter of the threshold being hard to place — the ordering is the wrong way round.

That leaves the coincidence condition as the thing that actually decides, and the reason is worth having. Balance, complexity and the window counts are all measurements of what a finite window sees, and they are measurements of the sequence. Whether the scattered waves come into a fixed phase relation is a question about how the substitution acts on the whole chain, and Dekking’s condition is a finite test of exactly that — it reads the substitution’s own columns and never looks at the sequence at all. Two chains can agree on everything a window can measure and part company on the only question a diffractometer asks.

The sequence is older than the subject #

The Thue–Morse sequence was not invented to be a counterexample in diffraction theory. Axel Thue introduced it in 1906 to answer a question about words: is there an infinite sequence over two letters containing no overlap — no block of the form xyxyx, a repetition with one letter over?

Thue’s answer was this sequence, and his proof is a page. Marston Morse rediscovered it in 1921 in a completely different subject — geodesics on surfaces of negative curvature — and it has been rediscovered several times since, which is the usual fate of an object that is the simplest thing with a property. Its use in crystallography came sixty years later still, when semiconductor superlattices could be grown layer by layer to any prescribed sequence and somebody grew this one.

The forty years between Thue’s construction and any physical question about it is worth noticing on a site whose subject is a physical one. Penrose’s rhombs had a similar gap and a similar ending; so did the six-dimensional lattices that index an icosahedral quasicrystal.

What an experiment would report #

A measurement is made on a finite sample, and the third kind of measure is the one a finite sample is worst at recognising.

A crystal a millimetre across holds some ten million cells along an edge, so its reflections are sharp to about one part in ten million and a diffractometer records spots. A Thue–Morse superlattice grown to a thousand layers has features whose normalised height is a thirtieth of a Bragg peak’s and whose widths are still falling; on a plot with a realistic background they read as broad maxima at a third and two thirds, which is exactly how the earliest measurements on such superlattices were described.

The distinction that matters is therefore not visible in one pattern. It is visible in a sequence of patterns from samples of different sizes, which is the measurement made here and not one an experiment normally makes. That is a general fact about this classification rather than a fact about this sequence: a peak, a bump and a singular feature look alike at any single length, and only their behaviour as the length changes separates them.

Where the substitution has been built #

Sequences of this kind stopped being purely mathematical in the mid-1980s, when molecular beam epitaxy made it possible to lay down semiconductor layers one at a time in any prescribed order. Fibonacci superlattices came first, in 1985, precisely because a quasicrystal had just been measured in a metal and a one-dimensional analogue was easy to build; Thue–Morse superlattices followed within a few years, for the reason this essay is about — they are the accessible example of order that is neither periodic nor quasiperiodic.

What is grown is not a chain of scatterers in the sense used here but a stack of layers of two thicknesses, and the quantity measured is usually the reflection of light or of X-rays at small angles. The predictions are the same ones: features at a third and two thirds, heights that do not scale as the square of the number of layers, and a hierarchy of smaller features between them that becomes visible only when enough layers are grown.

That is as far as this collection follows the physics, and the boundary is the usual one. Nothing here computes what a layer stack does to a wave — the Cantor spectrum essay is the one that takes up wave propagation in a chain, with its own machinery and its own limits.

Why the exponent is the right instrument #

An exponent fitted to four points is a modest-looking thing to hang a classification on, so it is worth saying what makes it the right measurement rather than a convenient one.

Every alternative is a threshold in disguise. Count the peaks above some fraction of the strongest requires a fraction. Measure the width and call it sharp below some value requires a value, and the width of every feature falls as the chain lengthens whatever kind of measure it belongs to. Look at the picture is worse: at any single length a Bragg peak, a singular feature and a broad bump all look like bumps of some width, and the eye has no access to the only thing that separates them.

What separates them is a limit, and the exponent is the finite-length shadow of that limit. Intensity against length is a power law with an exponent of two for a pure point measure and one for an absolutely continuous one, and those two numbers are not conventions — they are what adding N terms in phase and adding N terms at random respectively produce. A measured exponent of 1.55 is therefore a positive statement about which of the two mechanisms is operating, and it is one neither picture nor threshold could have made.

The same logic runs through this collection wherever a limit has to be measured on a finite object: the average is the same wherever it is taken fits an exponent to the convergence of a window frequency and gets one on the chain and a half on a shuffle; how much pattern is enough measures how large a window has to be before a group is decided. In each case the quantity that has a name in the mathematics is a limit, and the quantity a measurement can produce is a rate of approach to it. Reporting the rate, with the lengths it was measured over, is the honest form.

Where the exactness stops #

Measured here: the strongest normalised peak of three chains at four lengths each, and the exponent fitted to it; the coincidence condition run to eight levels on two substitutions; the letterwise complementarity of the Thue–Morse substitutes to ten levels; the absence of an overlap in two thousand letters, with a chain that has one as the control; and the complexity and balance of both chains.

Not measured, and quoted: that the Thue–Morse diffraction measure is singular continuous. An exponent fitted over four lengths is a number with an error bar, and no finite computation distinguishes an exponent of exactly 1.55 from one that is slowly approaching 2. The theorem is Kakutani’s, in the form Bombieri and Taylor’s work made available to this subject, and it is a statement about a limit.

And one more absence, which is the honest version of the essay’s title. The exponent measured here is a property of the strongest feature at the lengths tested, and the Thue–Morse measure’s scaling is not the same everywhere in reciprocal space. A single number stands in for a whole spectrum of local exponents. That is a real simplification and it is the price of a measurement that fits in one figure.

Where the ladder goes next #

Back, to the criterion this essay is the exception to. Which inflation factors exist is the necessary condition, its proof, and the sentence this essay was written to discharge.

Sideways, to the chains that do diffract sharply and to what makes them. The smallest quasicrystal computes the Fibonacci chain three ways and requires them to agree; the average is the same wherever it is taken measures the convergence that balance buys.

And upward, to what the distinction is worth. Order is not periodicity is the essay about the word crystal being redefined in 1992 around sharp diffraction; a chain like this one is precisely a structure that is completely ordered, generated by a rule with no randomness in it anywhere, and outside the definition.

What a finite sample would show #

The exponent is a statement about a limit, and every real measurement is made on something finite. It is worth being explicit about what that means, because the two things a reader might take from this essay are not the same.

A finite Thue–Morse chain has peaks. At any given length the scattering curve rises to sharp maxima at particular wavevectors, and they are the largest features in the curve. Nothing about the picture says “no Bragg peaks”; it says the maxima grow more slowly than the square of the length.

The distinguishing quantity is therefore a ratio, not a shape. Divide by the square of the number of scatterers and a Bragg peak keeps its height while these maxima fall away as the inverse square root of the length. Doubling the chain sends the ratio down by a factor of about 1.4, which is why the measurement here has to be made at two lengths and compared rather than made once and inspected.

An experiment cannot make the comparison. A crystal is whatever size it is; there is no second sample twice as long with the same structure. So the classification into pure point, singular continuous and absolutely continuous is a mathematical property of an idealised infinite object, and a diffraction pattern from a finite one is compatible with all three.

What an experiment does see is a width. A pattern’s peaks have a width set by the sample size and by the instrument, and a singular continuous spectrum shows up as peaks that never sharpen with better resolution — the intensity is spread over a set with no isolated points, so improving the instrument reveals more structure rather than a narrower line. That is the practical signature, and it needs an instrument good enough to be sure the limit is the sample’s and not the machine’s.

Where such chains are found #

The Thue–Morse sequence is not only a counterexample. It has been built, deliberately, and the reason is worth knowing.

A layered material can be given any sequence at all. Deposit two materials alternately under computer control and the order of the layers is whatever the operator specifies — periodic, Fibonacci, Thue–Morse, random. The result is a one-dimensional crystal whose sequence was chosen rather than found.

So the three spectral types can be realised in matter. A periodic stack diffracts to sharp peaks, a Fibonacci stack to sharp peaks needing two indices, and a Thue–Morse stack to the intermediate object this essay measures. All three have been grown and measured, in X-rays and in visible light, since the layer thicknesses can be tuned to whichever wavelength is convenient.

And the optical version is the more useful one. A stack of transparent layers reflects light in bands decided by the same transform, so the spectral type of the sequence becomes the structure of the material’s reflection spectrum — which makes the distinction between a peak, a bump and neither into something that can be seen rather than computed.