A small angle is a row of dislocations

Two crystals of one substance meeting at an angle have a boundary, and this collection has so far treated one as a coincidence problem. Turn a lattice against itself and ask which of its points land on points of the original: at almost every angle only one does, and at a discrete set of special angles a whole sublattice does. The indices are odd, the arithmetic is exact, and a boundary built on a coincidence orientation is cheap.

That is the right question for a large angle and the wrong one for a small angle, and the reason is worth stating carefully because both descriptions are correct.

At one degree the coincidence index is enormous or does not exist, so the coincidence account says the two lattices share essentially nothing. And yet a one-degree boundary is very nearly perfect crystal: a rotation of one degree moves an atom near the boundary by a hundredth of a lattice spacing, which is nothing. Both statements hold, and what reconciles them is that the mismatch, though never zero, stays small for a long way and then accumulates all at once.

The picture, and whose it is #

A small-angle boundary is not a region of disorder. It is good crystal interrupted by dislocations, spaced by whatever distance the misfit takes to accumulate to one Burgers vector.

That reading is Burgers’s and Bragg’s, from the late nineteen-thirties, and it was the argument that made dislocations respectable before anybody had seen one: a boundary made of them has a computable energy, that energy depends on the angle in a particular way, and the dependence matched what was measured on bicrystals. The dislocation went from a theorist’s device to a thing with a spacing.

The spacing follows from one line of arithmetic. Walking a distance D along the boundary, the two crystals’ lattice rows drift out of register at a rate set by the angle; when the drift reaches a whole lattice vector b, an extra half-plane has been inserted. So

$$D = \frac{b}{2\sin(\theta/2)}$$

which is Frank’s formula, and which for small angles is very nearly b/θ.

Measuring it rather than quoting it #

The formula is one line and this collection’s habit is not to take a line on trust. So the arrangement here is built and the spacing is measured.

Two half-crystals are constructed, each rotated by half the angle in opposite senses, so that the boundary plane is a mirror of the arrangement and no arbitrary choice about which crystal is the reference one enters. The atoms nearest the boundary on each side face each other in height exactly — that is what the mirror gives — so the misfit is not in where along the boundary they sit but in how far from it.

Then the gap between each facing pair is recorded along the length of the boundary, and the number of times it has to be corrected by a whole lattice spacing is counted. The distance per correction is the spacing.

What the agreement is worth #

The measured spacings match Frank’s formula to better than a part in twenty across the whole range from two degrees to fifteen, and at most angles to three decimal places.

That agreement is not a surprise and is not the point. The point is that a formula stated in one line, about a physical arrangement, has been checked against the arrangement — and the checking found the right answer only after the measurement was got right twice over. The first attempt counted the nearest partner across the boundary and found a spacing of one cell at every angle, because the atomic periodicity swamped the moiré. The second counted the misfit’s wraps and found a spurious dislocation at a vanishing angle, because a gap that sits on a whole number wobbles across it. The third — the total drift, unwrapped — is a difference of two numbers and has neither failure.

Recording that is worth more than the agreement, because the two failed measurements produced plausible numbers. A spacing of one cell at every angle looks like a result; a dislocation at a tenth of a degree looks like an edge effect. Neither announced itself as wrong.

Reading the spacing as a count #

There is a second way to read Frank’s formula that makes it obvious rather than derived, and it is worth having because the derivation above is a rate argument and rate arguments are easy to get backwards.

Count the lattice planes meeting a segment of the boundary from each side. Because the two crystals are tilted in opposite senses, one presents slightly more planes to a given length of boundary than the other — the difference over a length L being L · 2 sin(θ/2) / b, which is just the projected difference in plane spacings.

Planes cannot end in the middle of a crystal. Every plane that has no partner on the other side must terminate at the boundary, and a plane terminating at a boundary is an edge dislocation. So the number of dislocations in a length L is exactly the number of unmatched planes, and dividing gives the spacing.

That version has no rates in it and no accumulation: it is a count of planes, which is the sort of argument this collection prefers. It also makes the failure at large angles obvious — when the count per unit length approaches one per few cells, “unmatched plane” stops picking out anything localised.

Where the description stops meaning anything #

Frank’s picture is a row of dislocations with good crystal between them. It therefore says nothing once the spacing is comparable to the size of a dislocation itself, because there is no good crystal left to be between them.

That is a number rather than an opinion. Taking the width of a dislocation as three cells, the spacing falls to three cells at a shade over nineteen degrees, and above that the boundary is not a row of anything.

What takes over there is exactly the coincidence description this essay began by setting aside. At large angles the interesting question is which orientations put a large fraction of atoms into positions the other crystal also wants, and the answer is the arithmetic of coincidence indices. So the two accounts are not rivals but neighbours, meeting somewhere near twenty degrees, and each is silent where the other speaks.

The energy is also where the two descriptions can be compared quantitatively. A coincidence boundary at a special angle has an energy that dips below the Read–Shockley curve, because a large fraction of its atoms sit in positions both crystals want — which is the coincidence argument making an energetic statement rather than an arithmetic one. Those dips are measured in bicrystal experiments, and they are the reason the coincidence indices matter to a metallurgist rather than only to a number theorist.

The two accounts, side by side #

It is worth stating the pair explicitly because the contrast is the reason this rung exists.

The coincidence account is exact, integer, and about which angles are special. Its answer at a general angle is “nothing coincides”, which is true and unhelpful for a boundary a degree wide.

The dislocation account is approximate, geometric, and about how the misfit is accommodated. Its answer at a special angle is “the dislocations are so close together that the picture has failed”, which is true and unhelpful for a boundary at thirty-six degrees.

Between them they cover the range, and the crossover is where the exact description stops being informative and the approximate one starts. That shape of situation — an exact account and an approximate one, each valid where the other is useless — is unusual in this collection, whose habit is to prefer the exact everywhere, and it is worth having one clear case of the exact account being the less useful of the two.

What a boundary costs, and why the logarithm #

The energy of a small-angle boundary is not computed here — it needs an elastic modulus, which this collection does not have — and the shape of the answer follows from the geometry above and is worth stating.

Each dislocation carries an elastic field falling off as one over the distance, so its energy involves a logarithm of the range over which that field is felt. In an isolated dislocation the range is the crystal; in a row of them, each one’s field is screened by its neighbours at about the spacing, so the range is the spacing.

The energy per unit length of boundary is therefore the number of dislocations per unit length — which is proportional to θ — times the energy of each, which is a constant less a logarithm of θ. That gives the Read–Shockley form: an energy per unit area proportional to θ(A − ln θ), rising steeply from zero and flattening out.

The shape is the interesting part and it comes from the geometry rather than from the elasticity: the linear factor is the count of dislocations, and the logarithm is the screening. Both are visible in the pictures above, which is the reason a geometric account is worth having before an energetic one.

Why the misfit is a sawtooth and not a smear #

The shape of the measured curve carries the whole argument, and it is worth insisting on.

If a boundary at a small angle were a strip of disordered material, the misfit would rise near the boundary and fall away, with no particular structure along its length. What the measurement shows instead is a linear ramp that resets — the misfit accumulating steadily, and then being discharged in one place.

That is the signature of a localised accommodation. The strain does not spread; it is stored along the boundary and released at points, and the points are the dislocations. The energy of the boundary is then a sum over those points rather than an integral over a strip, which is what makes it computable and what makes it grow as the logarithm of the angle rather than linearly.

The Read–Shockley formula for that energy is the classical result and it is not derived here — it needs an elastic energy per dislocation, which is physics with a modulus in it. What is here is the geometry the formula is built on, measured.

The dislocations are the lattice’s own #

One thing this essay has quietly assumed is that the misfit accumulates to a lattice vector and not to some other amount, and it is worth making explicit because it is where the symmetry re-enters.

The extra half-plane inserted at each reset is a plane of the crystal, so the discontinuity it produces is a translation of the lattice — a Burgers vector, quantised, exactly as it is in the circuit essays. A boundary cannot accommodate its misfit in arbitrary instalments; it accommodates it in units the lattice supplies.

Which units are available is an enumeration this collection has already made: the distinct dislocations of a lattice are the orbits of its short vectors under the point group, and the cheapest is the shortest. So a tilt boundary in a given lattice uses a particular Burgers vector, its spacing is set by the length of that vector, and both are decided before any boundary is built.

What a tilt boundary is not #

Two boundaries this essay does not describe, and naming them keeps the claim honest.

A twist boundary. Rotating about an axis perpendicular to the boundary rather than lying in it gives a different arrangement — a grid of screw dislocations rather than a row of edge ones — and the spacing arithmetic is similar and not identical.

An asymmetric tilt boundary. Turning one crystal and leaving the other gives a boundary that is not a mirror of its two sides, and the two crystals then present different planes to the interface. The dislocation content is still computable, by Frank’s general formula rather than by the symmetric special case here.

Both are ordinary in real materials, and both would need the machinery here generalised rather than reused. The interfaces anchor has room for them.

Where the exactness went #

This is one of the few essays in the collection whose central number is a measurement rather than an integer identity, and the boundary is worth marking as sharply as the physics one.

The arrangement is exact: two lattices, an angle, atoms at exactly computed positions. The accumulation is exact: the gap between two facing atoms is a difference of two coordinates. What is measured is the number of times the accumulated gap crosses a lattice spacing over a finite patch, and that count depends on where the patch was cut, which is why it is reported as a mean with a spread rather than as a number.

The comparison with Frank’s formula is therefore a comparison of two approximate statements about a finite piece of an infinite arrangement. It is a good comparison, it agrees to a per cent, and it is not the kind of claim the coincidence essays make. Saying so is the difference between a collection that measures and one that only asserts.

The formula that covers every boundary #

The row of edge dislocations is the symmetric tilt case, and there is one expression covering it, the twist case and everything between — which is worth having because it shows that the three are not three constructions.

Take a vector v lying in the boundary plane, and let R be the rotation relating the two grains. Walk a circuit that goes along v in one grain and back along v in the other. The two paths do not close, and the amount by which they miss is

$$\mathbf{B} = (R - I)\,\mathbf{v}.$$

That is Frank’s formula, and it says the total Burgers vector crossed by any vector in the boundary is a linear function of that vector. The dislocation content of a boundary is therefore a single tensor, not a list — and which kind of boundary it is depends on how B sits relative to v rather than on a separate classification.

Read it in the essay’s case. For a symmetric tilt about an axis in the boundary, (R − I)v is perpendicular to v for v along the boundary, so the Burgers vectors are edge-like and lie in a row — which is the picture. For a twist boundary the rotation axis is normal to the boundary, (R − I)v lies in the boundary and parallel to nothing, and the content is screw-like in two directions at once, giving a grid.

And the small-angle expansion is one line of it. For small θ, (R − I)v ≈ θ (n̂ × v), so the content is proportional to θ — which is the spacing b/θ this essay measures, arriving as the leading term of a formula that also handles the cases the essay sets aside.

Where the description stops, with a number #

The essay says the dislocation account gives way to the coincidence one at large angles, and the crossover has a value that follows from the spacing alone.

The dislocations sit b/θ apart, with b a lattice vector. The description assumes they are separate objects, each with its own core and its own strain field, and that assumption fails when the spacing approaches the size of a core — a few lattice spacings.

Setting b/θ equal to about four lattice spacings gives θ of roughly a quarter of a radian, which is about fifteen degrees. That is the conventional boundary between low-angle and high-angle grain boundaries, it is quoted in every account of the subject, and it is not a measured constant: it is the angle at which the cores of a row of dislocations begin to touch.

Above it the row is not a row. There is no space between the dislocations for undisturbed crystal, so nothing distinguishes one from the next, and the boundary is better described as a structure in its own right — which is where the coincidence account takes over, describing the boundary by what the two lattices share rather than by what separates them.

That the two descriptions meet at a definite angle, and that the angle comes from a ratio of two lengths rather than from an energy, is the tidiest thing about this pair of essays.

And it stops at the other end too #

The failure at large angles is the one everybody quotes. There is a failure at small angles as well, of a different kind, and it is worth setting beside the first because the two together say what the description is actually a description of.

The spacing is b/θ, so it diverges. At a tenth of a degree the dislocations are about five hundred and seventy cells apart; at a hundredth, five thousand seven hundred. Nothing in the arithmetic objects to that — the formula is happy to report a spacing of a million — but the object it is describing has stopped being a row. A patch of crystal forty cells across, cut anywhere along such a boundary, contains no dislocation at all, and what a microscope sees there is a lattice plane bending very slightly. The misorientation is real and the structure carrying it is not local.

That is the source of the noise in the measurement above. At two degrees the patch holds only a handful of resets, and a count over a finite window with a handful of events in it has a spread of the order of one event, which is a large fraction of a handful. The agreement with Frank’s formula is worst at exactly the angles where the arithmetic is best behaved, and the reason is arithmetically dull: there is nothing much to count.

The two failures are not the same kind of failure, and the distinction matters. At the top the objects overlap, so the picture of separate dislocations with good crystal between them is describing something that is not there. At the bottom the objects are too rare to be a structure, so the picture is right about each dislocation and wrong to call the collection a boundary rather than a gentle curvature. A description bounded above by the size of its own objects and below by their spacing is the ordinary shape for a defect model, and saying where both bounds fall is more useful than quoting either.

Where this goes #

The nearest unbuilt thing is the twist boundary, and after it the general case: Frank’s formula in its full form, which gives the dislocation content of an arbitrary boundary from the rotation relating the two grains. That is more than one essay’s worth of work.

The nearer neighbour already here is the growth spiral, where the same defect appears alone rather than in a row and makes possible something a perfect crystal cannot do — and where, as here, what the symmetry supplies is the list of available Burgers vectors while the arrangement decides the rest.