The fast faces are the ones that vanish

Which faces a crystal shows makes a prediction: rank the faces by interplanar spacing, correct for the extinctions the lattice centring imposes, and the widely spaced planes are the ones a reader sees on the specimen. That rule works, and it is a rule about growth rates.

There is a second prediction, from a completely different question, and it usually gives a different answer.

The two questions #

Equilibrium. Of all the shapes a given volume of crystal could take, which has the least total surface energy? A liquid drop answers sphere, because its surface energy is the same in every direction. A crystal’s is not, and the answer is a polyhedron.

Wulff’s answer, from 1901: put each face at a distance from a common centre proportional to its surface energy, and take the intersection of the half-spaces. A face with a high energy sits far out, is cut away by its cheaper neighbours before it reaches the surface, and does not appear at all.

The rule looks arbitrary until the arithmetic behind it is written out, and it is short enough to be worth writing. A polyhedron built by intersecting half-spaces at distances hᵢ has volume ⅓ Σ hᵢ Aᵢ, where Aᵢ is the area of the face at distance hᵢ — every face together with the centre makes a pyramid of that volume, and the pyramids fill the solid. Its surface energy is Σ γᵢ Aᵢ. So minimising Σ γᵢ Aᵢ at fixed ⅓ Σ hᵢ Aᵢ is minimising one weighted sum of the same areas against another, and the two agree exactly when hᵢ is proportional to γᵢ. The construction is not a heuristic that happens to work; it is what the constraint reduces to once the volume is written in terms of the distances rather than in terms of anything else.

That also says why a face can be absent rather than small. The half-spaces are intersected, so a plane whose distance exceeds what its neighbours already cut off contributes no area at all — it has Aᵢ = 0 and drops out of both sums. A shape’s list of forms is therefore a discrete answer coming out of a continuous minimisation, which is why a small change in the bonding can add a whole form at once.

Growth. A crystal in a supersaturated solution is nowhere near equilibrium. Its faces advance at rates set by how readily material attaches, and the shape that results follows a rule with the same form and a different meaning: a face’s distance from the centre grows in proportion to its rate. So a fast face runs away from the centre and is cut off by its slow neighbours — the fast faces are the ones that vanish, and the surviving habit is made of the slowest.

The two constructions are the same geometry over different numbers. That is what makes them comparable and it is also the trap: a picture of one is indistinguishable from a picture of the other, and the caption is where the difference lives.

Where the energies come from #

A table of surface energies would be a table, and this site does not run on tables. So the energies are computed from the simplest model that is not one.

Cleaving a crystal along a plane breaks the bonds that cross it. Count them per unit area of the exposed surface, and that count is the energy, up to the strength of a bond. For a simple cubic lattice with nearest-neighbour bonds only, the count for (hkl) is |h| + |k| + |l| divided by the square root of h² + k² + l² — one bond direction crossed per index, over the area of the plane’s own mesh.

One parameter turns it into a sequence. Second neighbours run along the face diagonals, and a plane cuts a different number of them; adding them with an adjustable strength makes {110} and {111} relatively cheaper and they appear as truncations.

Which is the lesson worth taking from the model rather than from its numbers. The habit is a consequence of bonding, not of symmetry. Every shape in that sequence has the full cubic symmetry — they are all unions of complete forms, as a form is an orbit requires — and the classification cannot distinguish them at all. Five classes grow the same cube makes the same point from the other side.

Both shapes are unions of forms, and that is forced #

Neither construction was told about the point group, and both produce shapes with its full symmetry. That is worth pausing on because it is a small theorem rather than a coincidence.

A symmetry operation of the crystal carries a face to a face with the same energy, because it carries the whole structure to itself and the two surfaces are indistinguishable. So faces in one orbit — one form — get the same distance in either construction, and either the whole form appears or none of it does.

That is why the shapes above can be described by naming forms rather than faces, and why the counts are 6, 8, 12 rather than arbitrary numbers. It is also why the habit is such weak evidence about the class: the construction respects the symmetry it was given, so a shape can tell a reader the lattice symmetry it was built with, and cannot distinguish two classes sharing it.

Testing the theorem instead of quoting it #

Wulff’s construction is a theorem: it produces the shape of least surface energy at fixed volume. That is checkable, and checking it is what turns the construction from an assertion into a measurement.

The subtle part is which energies the comparison uses. A perturbed shape is built from wrong distances and must be evaluated with the right energies — the true γ of each face that actually reaches the surface, times its area. Reading the energy off the distances the shape was built from would report every shape as optimal, since it would be measuring the shape against its own definition.

That distinction is the whole test, and it is the same shape of care the phasing refusals needed: a check whose threshold is generous enough tests the threshold rather than the claim.

And the test has to be able to fail. A perturbation that changes nothing — a form already cut away at both distances — gives a rise of exactly zero, and those are reported separately rather than counted as successes. What the check rests on is the perturbations that do change the shape: every one of them raises the energy, and if any had lowered it the construction or the energy model would have been wrong with nothing in the picture to say so.

A face too expensive to appear is absent, not small. Giving {111} five times its energy removes it from the shape entirely rather than shrinking it — the construction has a hard cut-off, and a face is either on the surface or cut away by its neighbours. That is the property that makes the shape a prediction rather than a description: the model says which forms occur, not merely how big they are.

Where the two shapes disagree, and why #

The disagreement is not a defect in either rule. They answer different questions and a real crystal is somewhere between them, depending on how it was made:

• A crystal grown quickly from solution is a growth shape: the faces present are the slow ones and the fast ones have gone.
• A crystal annealed at temperature for long enough relaxes towards the equilibrium shape, because surface diffusion lets material move to where it lowers the energy. Small particles do this readily and large ones essentially never.
• A crystal grown slowly in near-equilibrium conditions is somewhere in between, and which faces it shows may depend on the solvent, on impurities that adsorb preferentially on one face and slow it down, and on the supersaturation.

And a twin is a third possibility. A crystal that twinned during growth presents re-entrant angles where the two orientations meet, and material attaches at a re-entrant corner far more readily than on a flat face — so a twinned crystal grows in a direction the untwinned one does not, and takes a shape neither construction predicts. A twin is a symmetry the lattice has and the crystal does not, and the plate-like habits of many minerals are the visible consequence of one.

The last of those is the standard way a habit is engineered. An additive that sticks to one face slows its growth, and slowing a face makes it larger in the final shape — the opposite of the intuition that a fast-growing face should dominate. That inversion is the content of the growth rule and it is worth stating twice: making a face grow more slowly makes it bigger.

And the inversion is why the two rules can be told apart on a specimen at all. Both constructions put a face at a distance and intersect; what differs is what the distance is proportional to. So the same additive that makes a face larger in a growth shape would, if it lowered that face’s surface energy instead, make it smaller in an equilibrium one — a low-energy face sits close in, and sitting close in is what makes a face wide. The two rules therefore respond to the same intervention in opposite directions, and an experiment that changes one quantity and watches which way the habit moves is deciding which rule the crystal is obeying. That is a better test than comparing a photograph with a picture, because the two shapes are drawn by the same geometry and a single picture never says which numbers went into it.

Why a small crystal equilibrates and a large one does not #

The two rules are not equally relevant at every size, and the reason is a competition between two quantities that scale differently.

Reaching the equilibrium shape means moving material. A crystal that grew into the wrong shape can only relax by transporting atoms from one face to another, over the surface or through the medium, and the distance they must travel is the size of the crystal. The energy to be gained is a surface term, going as the square of the size; the material to be moved goes as the cube.

So small crystals relax and large ones do not. A nanoparticle a few tens of atoms across takes its equilibrium shape readily and is routinely observed doing so; a centimetre of quartz will keep the shape it grew into for geological time. The equilibrium shape is the right prediction for a catalyst particle and the wrong one for a museum specimen, and the growth shape is the other way round.

The same competition sets the other famous size effect. A small crystal’s surface is a larger fraction of it, so its solubility is higher — the Gibbs–Thomson effect — and in a population of crystals the small ones dissolve while the large ones grow, which is Ostwald ripening. The habit and the ripening are the same surface term seen twice.

Where the exactness stops #

The construction is exact and the model is a model. The intersection of half-spaces, the volumes and the areas are computed geometry, and the check that no perturbation lowers the energy is a numerical statement about that geometry. The surface energies are a broken-bond count with one adjustable parameter, and nothing about them is fitted to any real crystal.

The face set is bounded. Faces with indices up to two in each direction are considered, which is thirteen forms. A high-index face has a high energy in this model and would be cut away, so the bound is unlikely to be doing harm — but “unlikely” is the right word, and a model with strongly directional bonding could put a high-index face on the shape. The honest form of the caveat is that the bound has not been tested: widening it and finding the shape unchanged would settle the question, and until that is done what the construction establishes is the least-energy shape among the orientations it was offered rather than among all of them.

Growth rates are taken from the spacing rule. The BFDH rule is a proxy for a growth rate, not a measurement of one; real rates are measured face by face in a growth cell and depend on temperature, supersaturation and solvent. So the right-hand shape is this site’s growth prediction rather than an observation.

Neither shape depends on the space group’s translations, only on the point symmetry and the lattice centring. That is why both constructions produce unions of complete forms, and why the shape does not name the class: several classes share a habit, and the habit was never enough to identify a crystal.

Who found it, and when #

Georg Wulff stated the construction in 1901, in a paper on the growth and dissolution of crystals. He gave it as a rule and the proof came later — Dinghas in 1944, and a general treatment by Herring in 1951 that also gave the modern statement in terms of the γ-plot, the surface whose radius in each direction is the surface energy for that orientation.

The idea is older than the construction. Curie in 1885 had argued that a crystal at equilibrium minimises its total surface energy, and Gibbs’s treatment of surface thermodynamics is from the same decade. Wulff’s contribution is the geometry: the statement that the minimising shape is the intersection of half-spaces at distances proportional to the energies, which is what makes it computable.

The broken-bond model is Kossel’s and Stranski’s, from the 1920s, and it is the origin of the picture of a growing crystal as a surface with steps and kinks where atoms attach. The count of broken bonds per unit area is the crudest form of it and is still the first thing anyone computes.

Herring’s theorem is the reason the construction has teeth: an orientation whose γ-plot lies above the tangent construction is unstable and breaks up into a hill-and-valley structure of the orientations that are stable. So the faces missing from a Wulff shape are not merely small — a surface cut at that orientation will facet itself into others, which is observable and observed.

The temperature at which a face stops being a face #

Both constructions produce polyhedra with flat faces and sharp edges, and real equilibrium shapes are often partly rounded. The rounding is not a failure of the model; it is a transition with a temperature, and it is worth naming because it says when the flat-faced picture applies.

A flat face costs a crystal something. Adding a step to it raises the energy — that is the whole of the growth argument — but a step also has entropy, because it can wander. At low temperature the energy wins and the face stays flat; above some temperature the entropy wins, steps proliferate at no net cost, and the surface becomes rough.

Above that temperature the face is no longer a facet at all. It has no sharp edge against its neighbours, it grows without needing a step source, and in the equilibrium shape it appears as a curved region rather than as a plane. The transition happens at a different temperature for each face, generally lowest for the faces with the smallest energy anisotropy, so a crystal warmed through a range loses its facets one form at a time.

That explains a feature of real equilibrium shapes the construction as stated cannot produce: a shape with some flat regions and some curved ones, the flat ones being the faces still below their roughening temperature. Wulff’s construction handles it, since a direction with no cusp in the energy simply contributes a curved patch — but only if the energies used are the ones at the temperature in question, which the broken-bond count is not.

Reading the energies off the shape #

The construction runs one way here, from energies to a shape, and its most useful experimental application runs the other way.

Surface energies are hard to measure. There is no direct way to weigh one, and calculations of them disagree by tens of per cent. An equilibrium shape, however, is a picture of their ratios: the distance of each face from the centre is proportional to its energy, so measuring the distances on an equilibrated particle gives every ratio at once.

That is one of the few routes to the numbers, and it is used — small particles are annealed until they stop changing shape, imaged, and the ratios read off. What it cannot give is the absolute scale, since scaling every energy scales the shape and not its proportions, so one independent measurement is still needed to fix the unit.

And it requires the particle to have equilibrated, which is the condition the section above says holds for small particles and fails for large ones. A shape read this way from a crystal that grew rather than relaxed reports growth rates dressed as energies, which is a systematic error with no signature in the data.

Where this ladder goes #

Two rungs of this anchor now make two predictions about the same specimen, from two rules with the same geometry.

The growth rule ranks faces by corrected spacing, and its content is the extinction correction: without it, all three cubic lattices predict a cube, and with it they predict a cube, an octahedron and a rhombic dodecahedron.

The equilibrium rule ranks faces by surface energy, and its content is the bonding: with nearest neighbours only, every cubic crystal is a cube, and the truncations that make real habits come from the longer-range interactions.

Both are also worth setting against the third prediction this field makes about shape: the law of rational indices, which says only that the faces have small whole-number indices and says nothing about which. It is the weakest of the three and the only one that is a theorem about lattices rather than a model of a process.

Between them they say why habit is such poor evidence about symmetry, which is the practical conclusion the field arrived at in the nineteenth century by other means. The angles belong to the substance and the shape to the specimen: the angles between faces are fixed by the lattice and are what Steno’s law is about; which faces are present, and how large, is a fact about how this particular crystal was made.