The step that never runs out

A crystal face is a plane of atoms with nothing sticking out of it. An atom arriving on such a face is held on one side only and is far more likely to leave again than to stay. An atom arriving at a step on the face is held on two sides and stays. So growth happens at steps, and a face with no steps does not grow.

That is a problem, because faces do grow. Finishing a layer removes the step that built it, and the next layer has to begin with a two-dimensional island — a patch of atoms all of which are in the unfavourable position at once. The rate that predicts is far below what crystals actually manage at small supersaturations, by many orders of magnitude, and the discrepancy was one of the standing puzzles of crystal growth until 1949.

Three surfaces, one rule #

The whole argument is a comparison, and it is worth setting up as one. A height field on a patch of lattice; a rule that advances a site when some neighbour stands at least one layer higher; three starting conditions.

A flat face. Every site has the same height, so no site has a higher neighbour, so nothing advances. Ever. The count of sites that grew in forty steps is zero, and it would be zero after forty thousand.

An island. One layer’s worth of atoms in a disc on an otherwise flat face. Its edge is a step, so the sites next to it advance, and the island spreads. After fourteen steps it has covered the patch, the step has been consumed, and growth stops for the same reason the flat face never started.

A screw dislocation. A branch cut across which the comparison of heights is offset by the Burgers vector — one layer, in the simplest case. The surface is then not a stack of layers but a single helicoid, and the step running out from the dislocation cannot be grown away: advancing it moves it round the dislocation.

What a screw dislocation is, from the surface’s side #

The circuit essays introduced a dislocation as a circuit that fails to close: walk a loop through the lattice around the defect and arrive one lattice vector away from where the walk began. The failure is quantised — it has to be a lattice vector — and the vector is the Burgers vector.

Seen from a surface the same defect is a branch. The height of the surface is single-valued everywhere except along one line running out from the dislocation, and across that line it jumps by the Burgers vector. Walk a circuit round the dislocation on the surface, adding up the changes in height, and the total is not zero: it is the Burgers vector, exactly.

That is checked here rather than assumed. A ring of sites at a fixed distance is walked in angular order, the changes in height are summed with the branch’s offset included, and the total comes back as 1 for a simple dislocation and 2 for a double one.

Why the step turns rather than vanishing #

Here is the mechanism in one paragraph, and it is the paragraph Frank wrote.

Atoms attach along the step. The step advances. On a flat face an advancing step eventually reaches the edge of the layer and disappears; here it is anchored at the dislocation, which is a point it cannot leave. So the far end of the step advances and the near end pivots, and the step sweeps round the dislocation like a hand round a clock face.

After one full turn the surface has grown by one layer everywhere — and the step is back where it started, with the same length, ready to sweep again. Growth therefore continues at a steady rate for as long as material arrives, with no nucleation ever required.

The step also gets longer as it turns, because the outer parts have further to go than the inner ones, and it winds up into a spiral. That is the shape the mechanism is named after and the shape that confirmed it: Griffin saw growth spirals on beryl in 1950, a year after Frank’s paper, and they turned up afterwards on silicon carbide, on cadmium iodide and on almost every crystal anybody examined with the right microscope.

The arithmetic of the branch, in full #

The whole of the dislocation, in this automaton, is three lines of bookkeeping, and they are worth setting out because they are all there is.

The surface is an integer height on each site. The branch is a half-line running out from the centre — here, the positive x axis. A bond crossing the branch upward carries −b and one crossing downward carries +b; every other bond carries nothing.

The growth rule compares h(neighbour) + offset with h(site) and advances the site when the first is larger. That is the only place the offset appears, and it is the only difference between the third run and the first.

Everything else — the spiral, the steady rate, the circuit that climbs, the doubling — is a consequence of those three lines. Nothing was drawn, nothing was fitted, and the automaton has no parameters other than the patch size, the number of steps and the integer b.

The doubling, and what it says about the count #

A dislocation of Burgers vector two puts two steps on the surface — the circuit climbs two layers, so two separate step lines have to leave the dislocation — and they interleave into a double spiral.

Its growth rate is twice the single one, measured here as the height at the centre after a fixed number of turns: ten layers against twenty. That is the cleanest quantitative statement the automaton makes, and it is exactly the sort of thing an integer rule can settle without any physics: the number of steps is the Burgers vector, growth is proportional to the number of steps, so the rate is proportional to the Burgers vector.

What the automaton is and is not #

The rule here — advance when a neighbour is higher — is a caricature of attachment and detachment, and the caricature is deliberate. It contains no temperature, no supersaturation, no diffusion length, no energy, and no rate constant.

What it contains is the geometry: which sites are at a step, what happens to the step when they advance, and whether the step survives. Those are the parts of the argument Frank’s mechanism actually turns on, and they are decidable in integers.

What it leaves out is everything that decides how fast. The Burton–Cabrera–Frank theory, which followed in 1951, computes a growth rate against supersaturation and finds the crossover between the nucleation regime and the spiral one; the spacing between successive turns of a real spiral is set by the critical nucleus radius, a thermodynamic quantity. The spiral drawn here has a spacing set by the lattice and by nothing else, and the difference is stated because a reader could otherwise take a picture for a prediction.

The division is the usual one in this collection and it is worth naming here rather than leaving to be inferred. What is computed exactly is a statement about permission: a step exists, it cannot be consumed, and a circuit around the defect fails to close by a lattice vector. What is not computed is any rate, any temperature, any supersaturation, or any energy — and every quantitative claim in the literature on spiral growth is about one of those. A reader who takes the constant rate in the figure below for a growth rate would be reading a count of sites per iteration of an automaton as though it were microns per hour, and the two have nothing in common but the word.

Which faces this happens on #

The growth essays established that a crystal’s shape is decided by the slowest-growing faces: a face that grows quickly runs out of itself and disappears from the habit, and the survivors are the sluggish ones.

The spiral mechanism sharpens that. A face with a dislocation emerging on it grows steadily and at low supersaturation; a face without one has to nucleate every layer and hardly grows at all. So the presence or absence of a dislocation on a given face can decide the habit of the crystal, and two crystals of the same substance grown side by side can take different shapes because of where their defects happen to be.

That is an unusual kind of statement for this collection. Nearly every other explanation here is a consequence of the group — a form is an orbit, a twin law is a coset — and this one is a consequence of an accident of growth history that no symmetry decides at all.

Why the flat case had to be computed too #

An automaton that produces spirals is convincing and proves nothing on its own, because a rule can produce spirals for reasons having nothing to do with the argument. The flat and island runs are what make the demonstration a demonstration.

The flat face grows by nothing at all, forever, which is the premise the whole mechanism exists to escape. If the rule had allowed even slow growth on a perfect face, the dislocation would be an accelerant rather than an enabler and Frank’s argument would be beside the point.

The island grows and then stops, having advanced exactly one layer’s worth of sites and no more. That is the nucleation regime in miniature: growth requires a step, a step can be supplied by an island, and an island supplies exactly one layer of growth before it is used up.

Together they make the third run’s steadiness mean something. The same rule, the same patch, the same number of steps, and the only difference is one integer in a branch condition.

It is worth being exact about which integer. The rule advances a site when some neighbour is a layer higher than it, and the comparison between two neighbours is ordinary subtraction of their heights — except across one cut running from the dislocation to the edge of the patch, where the comparison is offset by the Burgers vector. That offset is the entire difference between the flat run and the spiral one. It is not an extra rule about growth, not a term added to an energy, and not a special case for sites near the centre; it is a single integer added to one subtraction, and out of it comes a surface that never stops advancing while an otherwise identical surface never starts.

And that integer is exactly what a Burgers circuit measures. Walk a closed loop of sites around the dislocation, adding up the height change at each step, and the total is the offset rather than zero — which is the same statement as saying the loop failed to close. So the thing put into the rule and the thing measured coming out of it are the same number, and the circuit figure is the measurement of the input rather than an illustration beside it. A defect defined by a failure to close is a topological object: no local rearrangement of the surface can remove it, because every circuit around it would have to change its total at once, and each circuit is measuring a lattice vector that has nowhere continuous to go.

Where the symmetry is, in an essay about an accident #

A reader arriving from the rest of this collection may reasonably ask what a growth automaton is doing among the groups. Three answers, in increasing order of how much they matter.

The lattice decides which dislocations exist. A Burgers vector is a lattice vector, and the distinct dislocations of a lattice are the orbits of its short vectors under the point group. So the defect that makes growth possible is itself an object the symmetry enumerates, even though which one a given crystal happens to contain is not.

The face decides how fast the spiral turns. The steps of a spiral run along directions the lattice provides, and the faces that survive are the slow ones, so the habit that results is still decided by the same arithmetic of rational indices as before — with the dislocation deciding which of the slow faces get to grow at all.

And the mechanism explains a fact the symmetry cannot. Two crystals of one substance, grown together, can have different habits. Nothing in a point group distinguishes them; the presence of a dislocation on one face of one of them does. That is a real limit on what the group-theoretic account of crystal habit can reach, and it is better stated than skirted.

The habit this belongs to #

Frank’s argument is a member of a small family that this collection keeps meeting: a defect makes possible what a perfect crystal forbids. It is worth collecting the family, because its members are scattered across three fields here and are recognisably one idea.

A perfect crystal cannot grow at low supersaturation; a screw dislocation lets it. A perfect crystal cannot deform except by shearing whole planes at once, which needs enormous stress; an edge dislocation lets it deform at a thousandth of that, by moving one row at a time. A perfect structure has no way to accommodate a small change of composition; a dislocation or a stacking fault does.

In each case the defect is a topological object — its content is a lattice vector that a circuit fails to close by — and in each case what it enables is not a small correction but the difference between happening and not happening. That is worth stating in a collection otherwise about perfect symmetry: much of what crystals do is decided by where they are not perfect.

The family has a shape and it is the same shape every time. A perfect structure is over-constrained: every atom is in the position every other atom requires, so any change has to be made everywhere at once and costs accordingly. A defect is a place where that requirement fails, and the failure is what admits a change that is local. Growth needs a place where a new atom is bound on two sides rather than one; deformation needs a place where a row can move without the rows beside it moving; a change of composition needs a place with room in it. In every case the defect supplies a boundary, and it is boundaries that cheap processes happen at.

What the symmetry contributes is the list of defects that can exist, which is not nothing and is not a prediction. The Burgers vector of a dislocation is a lattice vector, so the lattice enumerates the possibilities and their relative energies by the $|b|^2$ rule; the stacking faults a close-packed metal admits are the ways its layers may sit; a twin law is a coset. What no group decides is which of the permitted defects a particular crystal contains, and that is decided by its history.

The paradox Frank was answering #

The automaton’s flat run is the premise stated as a computation, and the reason it needed answering was a measurement rather than an argument. It is worth setting out, because it is one of the cleanest cases in this collection of a theory being refuted by a factor rather than by a sign.

Growth on a perfect face has to begin by forming an island, and an island of n atoms has a perimeter to pay for. Below a critical size the perimeter costs more than the interior gains, so small islands dissolve and only islands above the critical size grow. The rate at which such islands appear falls off exponentially as the supersaturation drops, and the classical calculation puts the crossover at a supersaturation of some tens of per cent.

Crystals were observed growing at a supersaturation of about one per cent, at rates the exponential predicts to be smaller by many orders of magnitude — not a discrepancy in a coefficient but a prediction of nothing against an observation of something. That is Volmer’s paradox, and it stood for two decades.

Frank’s resolution removes the nucleation step entirely. A face carrying a screw dislocation never needs to form an island, because it always has a step; there is no critical size to exceed and no exponential to overcome. The face grows at any supersaturation whatever, and the flat run above is what the alternative looks like.

What the spiral predicts that the automaton does not #

The mechanism as an automaton gives geometry and no rate. The rate is what Burton, Cabrera and Frank supplied in 1951, and its form is the strongest evidence for the mechanism, because it is not the form anything else predicts.

The spiral cannot wind arbitrarily tightly. A step that curves too sharply is a piece of an island smaller than the critical size and dissolves rather than advancing, so the spiral’s innermost turn has a radius fixed by the same critical nucleus the flat face could not form. That sets the spacing between successive turns, and the spacing sets the rate: a smaller critical radius packs the turns more tightly, so more steps sweep the face per unit time.

The critical radius goes inversely as the supersaturation, and the rate at which each step advances goes directly as it. The two multiply, and the growth rate comes out proportional to the square of the supersaturation at low values, crossing over to a linear dependence at high ones. That is a parabolic law where classical nucleation gives an exponential, and the two are not confusable in a measurement over any range at all.

And the spirals were then seen. Growth spirals were observed on real crystal faces within a few years of the prediction, with step heights measurable by optical interference — and on silicon carbide the step heights came out equal to the repeat of the polytype the crystal had grown in, which was the observation that turned a mechanism into a standard tool. A dislocation whose Burgers vector spans several layers imposes its own stacking on everything the spiral sweeps, which is a proposal about why a substance with many possible stackings grows a particular one.

The shape of Frank’s argument is worth extracting, because it is not the shape most explanations in this field have. He did not propose a faster version of the mechanism that was failing; he proposed that the mechanism was never used, and that the growth being measured had a different origin entirely. That is a stronger claim and a more falsifiable one — it predicts the wrong exponent for the old mechanism, the right one for the new, and a visible feature on the surface that nobody had looked for. All three were then checked, which is why a proposal about defects displaced a calculation about thermodynamics rather than being appended to it.

Where this goes #

The next thing to ask of this machinery is what a pair of dislocations does, since real crystals have many and their spirals interact: two of opposite sign annihilate into a closed loop of step, which grows out and disappears, while two of the same sign make a compound spiral. That is more than one essay’s worth of work.

The nearer neighbour is the small-angle boundary, where dislocations arrive not one at a time but in rows, and where the spacing between them is decided by arithmetic as exactly as the spiral’s turning is here.