The streaks a faulted stack makes

Two stackings of one density is this collection’s account of the freedom close packing leaves. The first layer of spheres is forced. The second is forced. The third is not — it can sit over the first or over the other set of hollows — and every layer after it faces the same free choice, with nothing in the geometry preferring either.

The periodic choices are the polytypes: …ABCABC… is cubic, …ABAB… is hexagonal, and there are more of them at every longer period. This essay is about the choices that are not periodic.

A real close-packed crystal makes the choice wrongly now and then. The result is a stacking sequence that is a word rather than a repeat — a sequence over three letters with no period at all — and a structure that has a lattice in two directions and none in the third.

A crystal with a lattice in two directions #

That last sentence is worth sitting with, because it is a case this collection’s machinery is not built for and a case that occurs in ordinary metals.

Every symmetry argument here has begun by assuming a lattice. The crystallographic restriction assumes one on its first line; the seventeen and the two hundred and thirty are classifications of groups containing one; the whole of the exactness this collection trades on comes from working in a lattice basis where the operations are integer matrices.

A faulted close-packed metal has a perfectly good two-dimensional lattice in the plane of its layers. Across them it has nothing: no translation carries the stack onto itself, because the sequence never repeats. It is not a quasicrystal — there is no substitution rule, no self-similarity, nothing determined about the sequence at all — and it is not amorphous. It is a crystal in two directions and a word in the third.

Which reflections can see the layers #

The diffraction sorts itself into two kinds, and which kind a reflection belongs to is decided by an integer.

The three registrations differ by in-plane shifts of nothing, a third along each of two directions, and two thirds. So a reflection at in-plane indices (h, k) picks up a phase of exp(2πi(h − k)/3) per unit of shift. When h − k is a multiple of three the factor is one: the reflection cannot tell the layers apart, every layer scatters in step, and the row of spots along that direction is as sharp as if the crystal were perfect. When it is not, the phase factor is a cube root of unity, the layer sequence enters the sum, and the spots along that row spread into a streak.

The arithmetic is worth doing once rather than quoted, because the shape of it explains why three registrations and not some other number. The three positions a layer may take differ by the in-plane vectors $(0,0)$, $(\tfrac13, \tfrac23)$ and $(\tfrac23, \tfrac13)$ in the hexagonal basis of a single layer. A reflection at in-plane indices $(h, k)$ multiplies a shift $(u, v)$ by $\exp(2\pi i(hu + kv))$, and putting the second registration in gives $\exp(2\pi i(h + 2k)/3)$, which is $\exp(2\pi i(h - k)/3)$ because $2k$ and $-k$ agree modulo three. The third registration gives the square of it. So the three registrations are the three cube roots of unity, and a reflection is blind to the stacking exactly when that root is one.

That also says how blind, and the answer is completely. There is no reflection that is slightly sensitive: the phase factor is either $1$, or one of the two primitive cube roots, and nothing in between is available. A row of reflections is therefore either untouched by the disorder or fully exposed to it, and a specimen showing intermediate broadening on a $h - k \equiv 0$ row is telling the metallurgist about something else — crystallite size, or strain, or an instrument.

So the same crystal shows sharp spots and smeared ones side by side, sorted by arithmetic. That is the most direct evidence there is that the disorder is in the stacking and not in the layers: disorder within a layer would broaden everything, and disorder between layers broadens exactly the reflections that are sensitive to the shift.

The sorting rule is exact. It is a statement about h − k modulo three and it has no measurement in it, which puts it in the same family as systematic absences — a condition on integer indices deciding what an experiment sees, before any intensity is computed.

What the profiles are, and what they are not #

The profiles themselves are measurements and are reported as such.

The intensity along a row is a Fourier sum over the layers of the stack: each layer contributes a cube root of unity for its registration times a phase factor for its height, and the sum is squared. A perfect stack makes every term conspire at certain heights and the sum reaches the number of layers; a faulted one makes them conspire only between one fault and the next.

One faulted crystal is not enough. A single sequence gives a profile full of speckle — sharp spikes at accidental coincidences of that particular sequence — and measuring a width on it measures the speckle. A real specimen contains an enormous number of independently faulted regions and the experiment sees their sum, so the honest calculation averages over crystals: several sequences at the same fault rate from different seeds, one profile each, added. The number of members is quoted with every width.

What a rod is #

The word streak has been doing a lot of work and it is worth saying what the object actually is.

A perfect three-dimensional crystal scatters at a discrete set of points — the reciprocal lattice — and nowhere else. A crystal that is periodic in two directions and not in the third scatters along rods: continuous lines in reciprocal space, perpendicular to the layers, passing through the two-dimensional reciprocal lattice of a layer. Where the stacking is periodic the intensity along a rod concentrates into sharp maxima at the heights the period demands, and those maxima are the ordinary reflections. Where it is not, the intensity is spread along the rod.

So the streaks are not a smearing of spots. They are what was always there, revealed: a rod is the natural object for a two-dimensionally periodic structure, and the sharp spots of a polytype are the special case where the rod happens to be concentrated.

That is the same relationship the whole of this collection has between a structure and its transform, run in one dimension fewer. A finite crystal scatters into small blobs rather than points, and the blob is the transform of the crystal’s shape; a two-dimensionally periodic stack is infinite in two directions and unbounded-but-aperiodic in the third, so its transform is a delta function in the two periodic directions and a continuous function in the third. Nothing about faulting is needed to produce a rod. What faulting decides is only how the intensity is distributed along one, and a perfect polytype puts all of it into a few points.

The practical consequence is that a rod has to be measured rather than integrated. A structure determination collects a reflection by summing everything inside a small box and calling the result an intensity, which is exactly right for a spot and throws away the whole of a rod’s information. Diffuse scattering is what that box discards, and the instruments built to record it — area detectors run without the background subtraction a structure refinement wants — are recording the shape of the rod rather than its total.

That reframing decides how the calculation is done. The profiles here are computed along a rod — intensity as a function of the height l, at fixed in-plane indices — rather than as a three-dimensional pattern, because the rod is where all the information is.

Why width at half height is the wrong number #

The width a reader expects is the width at half the peak’s height, and it is the wrong number to lean on here. The reason is worth recording because it is a general hazard in profile analysis.

At a high fault rate the profile stops having a clean maximum. The half-height walk outward from the peak runs into a neighbouring bump and stops early, reporting a width smaller than at a lower fault rate — which reads as the peak sharpening when it has in fact dissolved. The number is not noisy; it is systematically wrong in a direction that flatters the crystal.

Integral breadth has no such failure mode. It is the area of the profile divided by its height — the width of the rectangle with the same area and the same peak — and it is defined for any profile at all, including one with no clean maximum. It grows monotonically as the peak spreads, and it is what the table quotes.

That the constant of proportionality comes out near a half is a measurement rather than a derivation. The classical treatments of this problem, from Wilson and from Hendricks and Teller in the nineteen-forties, derive a coefficient from a Markov chain over the stacking sequence; nothing here derives one, and the number is reported as what was measured on a stack of four hundred layers averaged over sixteen crystals.

Two kinds of fault, and why one is drawn #

The sequences here are generated by a single rule: at each layer, take the intended next registration with probability one minus α and the other permitted one with probability α. The intended one is what perfect cubic packing would have chosen.

That is a deformation fault in the metallurgical sense — a single mistake in an otherwise cubic sequence, of the kind a slipping plane produces. There is a second kind, the growth fault or twin fault, in which the sequence reverses direction from that point on rather than making an isolated mistake, and it produces a different profile: the peaks shift as well as broaden, because a reversal alters the mean spacing between the heights the intensity concentrates at.

Only the first is computed here, and the essay says so rather than letting a reader assume that a fault is a fault. The classical analyses treat both, along with a third possibility in which the faults are correlated rather than independent, and distinguishing them from a measured profile is what makes the subject a subject rather than a formula.

The independence assumption is the strongest one in the calculation. A real deformed metal has faults on planes that interact, so their positions are correlated; the model here assumes each layer decides independently of every other, which is a Markov chain of the simplest possible kind.

The determinism this requires #

A small practical point deserves a paragraph, because it is a rule this collection follows and rarely has to state.

The faults are random, and a figure that is redrawn must come out the same twice. So the sequences are generated from a stated seed through a small linear congruential recurrence written out alongside the drawing rather than from whatever the machine offers as a random number, and the seed appears in every figure’s description. That is a poor source of randomness and an excellent source of the same randomness: the same crystal wherever it is drawn.

The check that goes with it is the obvious one and it is made every time the figures are drawn: the same seed gives the same word, and a different seed gives a different one. A recurrence that had quietly become deterministic in the wrong way — always returning the same sequence whatever the seed — would pass the first half of that and fail the second, and the figures do not appear if it does.

The entropy the freedom buys #

There is a thermodynamic side to the same freedom, and this collection has computed part of it elsewhere.

A stack of N layers has, in principle, two choices at each layer after the second, so 2^(N−2) sequences are available and the entropy per layer is log 2. Nothing in the geometry prefers any of them; the energy difference between cubic and hexagonal packing comes from interactions beyond nearest neighbours and is small. So at a high enough temperature a close-packed metal has no reason to be ordered along the stacking direction at all, and the observed structures are decided by a competition between a small energy and a substantial entropy.

That is a rather different account of a stacking fault from the mechanical one. In the mechanical picture a fault is damage — a plane that slipped. In the thermodynamic picture disorder is the default and order is what has to be explained. Cobalt, which transforms between the two stackings near 700 K and is heavily faulted in between, is the case where both accounts are needed at once.

Either way, the diffraction is the same and the arithmetic that sorts the reflections is the same. Which reflections can see the layers is a fact about indices, and it does not care why the layers are where they are.

Reading a fault rate off a pattern #

What the measurement is for, in practice, is running it backwards.

A metallurgist has a diffraction pattern from a cold-worked metal in which some peaks are broad and some are not. The sorting by h − k modulo three identifies the broadening as stacking disorder rather than as small crystallites or strain — both of which broaden every reflection — and the breadth of the affected peaks then gives the fault probability, which is a number about how the metal was deformed.

That inference has been standard since the nineteen-fifties and it is one of the earliest examples of diffuse scattering used quantitatively: the information is not in the sharp spots but in the shape of what lies between and around them, and it is information a structure determination throws away.

What this collection can and cannot say about it #

It can compute the sorting rule exactly, because it is arithmetic on indices.

It can compute profiles, with a stated stack height, a stated grid and a stated number of ensemble members, and it reports all three.

It cannot assign a space group, because there is not one: a structure with no period in one direction has no three-dimensional space group, and the honest description is a two-dimensional plane group in the layer plus a statement about the sequence. This is the same limit an incommensurate structure runs into from a different direction, and it is the boundary of everything the rest of this collection computes.

And it cannot decide whether a real specimen is faulted or is a long-period polytype. A polytype of period fifty and a faulted crystal with a fault every fifty layers produce similar patterns, and separating them needs either much better data or an argument about how the material grew. What diffraction cannot tell apart is the general version of that limit.

The one number that is exact #

Amid the measured profiles and stated seeds there is one exact statement, and it is the one the essay turns on.

h − k mod 3 is an integer. A reflection either can see the stacking or it cannot, and no experiment, no tolerance and no fault rate changes which. That is the same kind of statement as a systematic absence: a condition on indices, decided by arithmetic, true of the structure rather than of a measurement of it.

Everything else here is soft. The breadth is measured; the fault rate is a parameter; the ensemble size is a choice; the model of how faults occur is one of several. But the sorting is not, and it is the sorting that turns a broad peak from a nuisance into a diagnosis.

That division is the honest summary of what a collection like this can say about disorder. The exact half is small and it is the half that identifies the mechanism; the measured half is large and it is the half that quantifies it. Reporting them together, with the method attached to each, is the whole of the discipline.

Where this goes #

The thread this essay belongs to runs through every essay in this collection about a structure that is not simply a lattice with something on it: an object whose interesting content is a sequence. A net is a graph with the distances thrown away; an approximant is a chain whose word is a repeat; a faulted stack is a word that is not.

In each case the machinery that made this collection’s earlier claims exact — a lattice basis, integer matrices, rational translations — is unavailable, and what is left is a combinatorial description together with measurements made on it. That the measurements can still be sharp, and can still be sorted by an exact arithmetic condition, is the reason such structures are studied at all rather than filed as disordered.

The peaks move as well as broaden #

Broadening is the effect this essay measures, and it is not the only one a fault produces. The peaks also shift, the shift is a more robust measurement than the width, and it carries a piece of information the width does not.

A fault interrupts the sequence, and the interruption changes the average spacing between layers of a given registration. That average is what a peak’s position reports, so the affected peaks move — and they move in a direction and by an amount fixed by the reflection’s indices rather than by anything about the material.

The two effects sort the same reflections in different ways. Some of the streaking reflections shift towards higher angles and some towards lower, according to the sign of a term in their index arithmetic, and some do not shift at all. So a pattern from a faulted metal shows a characteristic pattern of relative displacements among its peaks, superposed on the broadening.

That gives a second and better route to the fault probability. A shift is a position and a width is a shape, and a position is measured far more reliably: it survives instrumental broadening, it survives the peak’s profile being non-Gaussian, and it does not need the deconvolution a width does.

The standard analysis — Warren’s — uses both, and the reason for using both is that they are affected differently by the two kinds of fault. A deformation fault and a growth fault produce different combinations of shift and broadening, so measuring one of each disentangles the two populations, which measuring either alone cannot.

Neither is computed here. The profiles on this page are computed from generated sequences with one kind of fault, so their peaks shift and the shift is not measured — which is worth stating, because a reader taking these figures for the whole of the effect would be missing the half a metallurgist actually uses.