Which faces are flat

The BFDH ranking predicts a crystal’s shape from its lattice: rank the faces by how far apart their planes lie, corrected for the translations the extinction rules reveal, and the faces at the top of the ranking are the ones a crystal shows. It gets the three cubic habits right — cube, octahedron, rhombic dodecahedron — and it does it without knowing a single thing about what is bonded to what.

That is a strength and a limitation in the same sentence. Two substances with the same lattice and completely different bonding get the same prediction, and the prediction is a fact about a set of planes rather than about a material.

Hartman and Perdok’s rule, from 1955, uses the other half of the information. A periodic bond chain is an uninterrupted chain of strong bonds repeating along a lattice direction, and a face is classified by how many non-parallel chains run inside it: two or more and the face is flat, exactly one and it is stepped, none and it is kinked.

Why a chain count is a growth argument #

The rule reads as a piece of bookkeeping and is not. It is an argument about how a face advances.

A kinked face has no chain in it. Every position along its surface is a site where an arriving atom can bond to what is already there, so the face advances one atom at a time with no barrier: it grows fast, and a fast face grows itself out of existence, because the crystal’s corners overtake it.

A flat face has two non-parallel chains in it. Once a layer has begun it can extend in two directions, so the layer completes quickly — but beginning a layer requires an atom to sit on a flat surface with nothing to bond into except the plane beneath, which is a nucleation event with a barrier. Flat faces therefore advance slowly, and slow faces survive.

A stepped face is between: one chain, so an arriving atom joins a growing edge and the face advances by a step without needing to nucleate a whole layer, but only in one direction.

That is the same shape of argument BFDH makes and it is not the same argument. BFDH says a widely-spaced layer grows slowly because a new layer is far from the last one and has little to attach to. Hartman–Perdok says a face grows slowly because a new layer has no chain to grow along. The first is a distance and the second is a connectivity, and the two are computed here from disjoint inputs.

The chains, which are the whole input #

The model here is stated rather than hidden, because the analysis is only as good as the bond list it is handed. Each structure is one atom per lattice point, bonded to its nearest neighbours: six of them for the primitive cubic lattice, twelve for the face-centred, eight for the body-centred. Nothing else is a bond.

That choice is the whole of the chemistry in this essay, and it is Hartman–Perdok’s own dependence made explicit. In a real structure “strong bond” means something the crystallographer decides — a covalent bond and not a van der Waals contact, or a metal–oxygen bond and not a hydrogen bond — and different decisions give different chains and different predictions. What counts as a bond is a question this collection has already had to answer once, for nets, and the answer is the same: it is an input.

A chain direction is a bond direction, taken up to sign, so the counts are three, six and four. A face (hkl) contains the direction [uvw] exactly when hu + kv + lw = 0, which is an integer test with no metric in it — and the classification is that test run over every chain.

The three answers, which are the three habits #

Run it on the primitive cubic structure. The chains are [100], [010] and [001]. The face (100) contains two of them, so it is flat. The face (110) contains only [001], so it is stepped. The face (111) contains none — every cube axis has a non-zero dot product with (111) — so it is kinked. One flat form, {100}, and the predicted habit is a cube.

The face-centred structure’s chains are the six face diagonals. Now (111) contains three of them and (100) contains two, so both are flat and (111) is flatter; (110) has one and (210) has none. The predicted habit leads on {111}: an octahedron, with cube faces subordinate.

The body-centred structure’s chains are the four body diagonals. (110) contains two of them and is the only flat form in the list; (100) and (111) contain none at all. A rhombic dodecahedron.

The face-centred case, walked through #

The face-centred structure is worth doing by hand, because it is the one where the classification has a tie-break in it and because the geometry is visible.

Its twelve nearest neighbours sit at the face centres of the surrounding cubes, at (±½, ±½, 0) and the two permutations of that — so the six chain directions are [110], [11̄0], [101], [101̄], [011] and [011̄], the face diagonals of the cube.

Ask which lie in the octahedral face (111). The test is h u + k v + l w = 0, so it is u + v + w = 0, and three of the six pass: [11̄0], [101̄] and [011̄]. Three chains, at sixty degrees to one another in the plane of the face — which is the close-packed layer, and the three chains are its three rows of touching atoms. That is exactly the picture a reader of close packing already has: (111) is the plane in which the spheres are packed hexagonally, and a chain is a row of them in contact.

Ask the same of the cube face (100). The test is u = 0, and two chains pass: [011] and [011̄]. Two chains at right angles, which are the rows of the square arrangement the atoms make on a (100) surface of a face-centred structure. Flat, but with one chain fewer than the octahedral face.

Ask it of (110). The test is u + v = 0, and only [11̄0] passes. Stepped.

And of (210): 2u + v = 0 has no solution among the six. Kinked.

So the ordering {111} then {100} then {110} then {210} comes out of counting, and it is the ordering every textbook gives for the morphological importance of a face-centred cubic metal. The tie-break — three chains beating two — is the rule’s own, and it is the same statement as “the close-packed plane is the flattest surface a close-packed structure has”.

One thing about that walk is worth keeping. Every step of it is an integer test. There is no metric in u + v + w = 0, no angle, no length, and no tolerance: the classification of a face is decided by whether a dot product of integers is zero, which is the same kind of decision as everything else on this site.

Two rules, no shared quantity #

The agreement is the point, and it is worth being precise about why it is not circular.

BFDH’s input is the metric and the extinction rules. Its calculation is a corrected interplanar spacing: take 1/√(hᵀG*h), then divide by the smallest integer multiplier that makes the reflection present. It never asks which atoms are near which.

Hartman–Perdok’s input is the bond list. Its calculation is a count of integer dot products that vanish. It never asks how far apart anything is.

The two share the list of (hkl) and nothing else, and on all three structures they put the same form first. That is the kind of agreement this collection collects: two computations of one quantity that could have disagreed, run rather than argued.

There is a reason underneath it, and having the reason does not make the check redundant. The extinction correction and the bond list are two views of the same centring. A face-centred lattice’s absences are the reason {111} wins the BFDH ranking, and its twelve nearest neighbours along the face diagonals are the reason {111} has three chains — and both are consequences of the centring vector. So the two rules are not independent in principle; they are independent in computation, which is what makes the agreement evidence that neither has a bug in it rather than evidence about crystals.

Where the rule’s discrimination actually lives #

Everything above uses chains of one bond repeated. That is not the only kind of chain, and it is not obviously the right cutoff.

A chain that alternates two different bonds is a perfectly good periodic bond chain: every step of it is a bond, and the path closes on a lattice point every repeat. In the face-centred structure, [001] is such a chain — no bond points along a cube axis, but (0, ½, ½) followed by (0, −½, ½) arrives at (0, 0, 1). There is no principled reason to exclude it.

Admit them, and the classification collapses. The primitive structure goes from three chains to nine and from one flat form to three; the face-centred from six to twenty-one and from two flat forms to four; the body-centred from four to thirteen and from one flat form to four. The kinked category empties completely on all three structures. By second order, every low-index face of every structure is flat or stepped, the three predictions are nearly the same list, and the rule has stopped distinguishing anything.

That is the honest reading of Hartman–Perdok as computed here, and it is the same shape as a finding this collection has made before about a figure of merit: the discriminating power of a criterion can live entirely in a term that looks like bookkeeping. Here it lives in the cutoff — a modelling choice, made by whoever runs the analysis, and not something the structure hands over.

The practising crystallographer’s version of that cutoff is “strong bond”, and it does the same work. Restrict the bond list to the genuinely strong interactions and the chains stay short and the classification discriminates; admit everything and every face is flat and the rule predicts a sphere.

What the rule was actually invented for #

The cubic elements are the wrong advertisement for Hartman–Perdok, and it is worth saying what the rule is for, because the case where it beats BFDH is the case where the two inputs come apart.

BFDH knows the lattice and its extinctions. Those are the same for every substance with that space group, so BFDH’s prediction is a property of a space group and a cell, and two minerals sharing them get the same predicted habit whatever their chemistry. That is often enough — the three cubic habits above are a real and useful result — and it is exactly wrong for a layered or chain-like structure, where the interesting thing about the material is that its strong bonds run in some directions and not others.

Take a structure whose strong bonds lie in sheets, with only weak contacts between the sheets. Its lattice can be perfectly ordinary and its BFDH ranking will report whatever the spacings report. Its chain analysis will find chains only inside the sheets, so the sheet plane is the one face with two chains in it and everything else is stepped or kinked — and the crystal grows as a plate, which is what such materials do. Graphite, the micas and the clays are that argument, and no ranking of interplanar spacings produces it, because the spacing of the sheet planes is a number about the lattice and the reason they are flat is a fact about the bonding.

That is the division of labour between the two rules. BFDH knows which faces are candidates; the chain analysis knows which of them the chemistry makes flat. The cubic elements are the case where the two agree because the bonds and the centring are the same fact seen twice, and a layered structure is the case where they do not, because a sheet’s bonds have nothing to do with the lattice’s absences.

This essay computes the agreeing case, because the agreeing case is the one where both computations can be run in full and compared. The disagreeing case needs a structure with two bond strengths in it, which needs a decision about where the cutoff between them lies — and that is exactly the choice the last section showed the whole answer depends on.

What the analysis refuses #

The fourth is the one that caught a real error while this was being built. A form is an orbit of faces under the point group, and every face of a form must get the same classification — the group permutes the bonds, so it permutes the chains, so it cannot leave one face of {111} flat and another kinked. The first version of the second-order chain search took sums of bonds without considering both signs, which produces a chain set the cubic group does not preserve, and the assertion failed immediately with (111) flat and (11̄1) kinked.

Nothing else would have caught it. The chain list looked plausible, every face got a classification, and the totals were unremarkable. What made it visible was requiring a symmetry the answer must have and did not.

The other thing a crystal shows, which neither rule predicts #

Both rules above predict a ranking of faces, and a ranking is not a crystal. Two things separate them and both are worth naming, because a reader who has followed this far will otherwise take the octahedron as a prediction about a specimen.

The first is that a habit is a set of relative growth rates, and the shape a crystal ends up with depends on the ratio of them rather than the order. A face that grows twice as fast as its neighbour survives, smaller; one that grows ten times as fast is gone. Neither rule gives a rate, so neither says whether the octahedron in front of a reader is an octahedron with tiny cube truncations or a cube with tiny octahedral ones — and mineral collections have both, of the same substance.

The second is that neither rule knows about the medium. The same substance grows as needles from one solvent and plates from another with the same lattice and the same bonds throughout, because a solvent molecule that adsorbs preferentially on one face slows that face and changes the ranking from outside. That is chemistry this collection does not touch, and it is the reason a morphology prediction is a starting point in practice rather than an answer.

What the two rules do settle is which faces are candidates, and that is a symmetry statement: a face appears only if it is in the list, and the list is decided by the lattice and the bonds. Everything after that is kinetics.

Where the exactness stops #

Computed here: the nearest-neighbour bond directions of the three cubic structures; the chains of order one and two; the classification of every form of index two or less by how many chains lie in it, as an integer dot product; the solid bounded by the flat faces; and, separately, the BFDH ranking of the same three structures from corrected interplanar spacings.

One atom per lattice point, and nearest neighbours only. Real structures have several atoms in the cell and bonds of several strengths, and both change the chain list. The analysis is unchanged in form and its input is not, which is the whole of the dependence this essay is about.

Chains up to two bonds. A chain of three or more alternating bonds is legitimate and is not computed; the trend across the two orders that are computed is that admitting more makes the classification weaker, so nothing here suggests the third order would restore it.

A classification is not a growth rate. F, S and K say which faces grow slowly, faster and fastest, and nothing about how fast. The rule has no supersaturation in it, no solvent, no temperature and no impurity — the same disclaimer the spacing ranking carries, and it applies here in full: the same substance grows as needles from one solvent and plates from another with the same bonds throughout.

Where the ladder goes next #

Back, to the rule this one is measured against: which faces a crystal shows, where the ranking is computed from spacings and the extinction correction turns out to be the whole prediction, and to the fast faces are the ones that vanish, where the same growth argument decides between two predicted shapes.

Sideways, to the object a chain is made of: what counts as a bond, where a structure becomes a net by a decision about which contacts are bonds, and to a structure with the distances thrown away, which is that decision taken to its conclusion.

Onward, to the defect that lets a flat face grow at all: the step that never runs out, where a screw dislocation supplies the step a flat face cannot nucleate, and which is the reason real crystals with flat faces grow at ordinary supersaturations.