Which faces a crystal shows

The law of rational indices says the faces of a crystal have small integer indices, and a form is an orbit of a face says which faces have to appear together once one does. Neither says which faces appear, and that is the question a mineralogist answers by looking.

The oldest answer is Bravais’s, sharpened by Friedel and then by Donnay and Harker in 1937, and it is one line: the faces that appear are those with the largest interplanar spacing. A layer of planes far apart grows slowly, because a new layer has less to attach to; a slow face survives on the finished crystal while fast ones grow themselves out of existence. So rank the (hkl) by d, and the ranking is the morphology.

The correction is the prediction #

Stated as above the rule cannot work, and seeing why is the whole point.

The three cubic lattices have the same metric and the same list of indices. Rank the faces of any of them by spacing alone and {100} wins in every case, because d₁₀₀ = a is the largest spacing there is. Uncorrected, the rule predicts a cube for rock salt, for fluorite and for garnet, and it predicts nothing about anything.

What separates them is that the spacing is not decided by the lattice alone. A systematic absence means the planes of that family are interleaved by a translation the indices do not show, so the true spacing is a half or a third of the apparent one — and which families that happens to is decided by the centring, the screws and the glides.

So the corrected spacing divides d by the smallest multiple of the indices that is not absent, and the absences come from the same extinction machinery systematic absences is built on. Nothing about which lattices or which glides is written into the rule; it asks the group.

The three cubic lattices, worked #

Primitive. Nothing is absent. d₁₀₀ = 1, d₁₁₀ = 0.707, d₁₁₁ = 0.577, and {100} wins. The habit is a cube — halite, galena, fluorite’s cleavage aside, and pyrite.

Face-centred. Reflections are present only when h, k and l are all even or all odd. So (100) is absent and its first present multiple is (200): the true spacing is a/2 = 0.5. (110) is absent too, giving 0.354. (111) is present as it stands, at 0.577 — and it wins. The habit is an octahedron: diamond, fluorite, spinel, magnetite.

Body-centred. Reflections are present only when h + k + l is even. (100) is absent, giving 0.5; (110) is present, at 0.707 — and it wins. The habit is a rhombic dodecahedron: garnet, and the sodalite group.

Three habits from one metric. The only input that differs is an extinction rule, which is a statement about a diffraction pattern — so a morphological prediction is being made from an experiment nobody performed on the crystal.

That is the connection Friedel made in 1907 and it is a genuinely surprising one. The shape of a mineral specimen and the pattern of missing reflections in its X-ray photograph are the same fact.

What a screw axis does #

The correction is not only about centring. A screw axis produces axial absences and a glide plane produces zonal ones, and both move faces down the ranking.

Comparing P4 with P4₁ isolates it: same lattice, same metric, one screw. The 4₁ makes (00l) absent unless l is a multiple of four, so the basal family’s true spacing is a quarter of what its indices suggest, and the basal face falls from first in the ranking to tenth.

Nothing about the lattice changed. A translation inside the cell — an operation with no consequence at all for the shape of the repeat — moved a face down the list of what a crystal should show. This is the sharpest available demonstration that morphology carries information about the space group and not only about the lattice, which is what Donnay and Harker added to Bravais’s rule in 1937 and why the rule has four names attached to it.

What the ranking looks like for a real group #

The cubic cases are the memorable ones and the general case is less tidy, which is worth showing rather than hiding.

In a cubic metric the spacings cluster, because all three axes are equal: 1, 0.707, 0.577, 0.447 for the first four forms, with each form carrying many faces. The predicted solid is a highly symmetric polyhedron and small differences in ranking produce large differences in shape.

In an orthorhombic metric they spread out. Pnma with a 1 : 1.3 : 1.7 cell has its largest spacings on {100} and {010}, and the ranking runs down through a dozen forms before the spacings become comparable. The predicted habit is a prism — elongated along the shortest axis, which is the direction whose planes are most closely spaced and therefore the fastest-growing direction.

That is the general rule stated backwards and it is the one a chemist uses: a crystal is long along its short axis. A cell with one short axis grows as a needle along it; a cell with one long axis grows as a plate perpendicular to it. Both follow from spacings and neither needs a computation.

Where the exactness stops, and it is a long list #

The rule is a ranking derived from a lattice and a group, and everything that decides a real crystal’s shape is outside it.

No growth rate, no supersaturation, no solvent, no impurity, no temperature. The same substance grows as needles from one solvent and plates from another with this ranking unchanged throughout. Sodium chloride grows as cubes from water and as octahedra from water containing urea — the urea adsorbs on the {111} faces and slows them, which is a chemical fact the arithmetic cannot see.

And no energy anywhere. This is the standing boundary of every essay in this field: the arithmetic here states which faces are candidates and in what order, and it predicts that nothing will happen. A rule that ranks faces by an attachment energy — Hartman and Perdok’s periodic bond chain theory, from 1955 — does better on real crystals and needs a structure and a force field, neither of which is a symmetry object.

What the rule does supply is the list a chemist starts from. It is right about the common habits of most simple ionic and molecular crystals, it costs nothing to compute, and it is exact in the sense that matters here: given the group and the metric, the ranking is a computation with no adjustable parameter in it.

The polyhedra drawn here are also a choice. Placing each face at a distance inversely proportional to its spacing is the standard way of turning a ranking into a shape, and it is a convention rather than a consequence — a different monotone function of d gives the same ordering of faces and a different solid. The forms that appear are the prediction; the exact proportions are illustration.

Reading a habit backwards #

The rule runs in reverse, and that is how it was used for a century before diffraction.

A crystal showing a well-developed octahedron and small cube faces says that {111} has the larger corrected spacing, which for a cubic metric means the lattice is face-centred. A dodecahedral habit points at body-centring. Morphology was a lattice-determination method, and Bravais’s rule is what made it one.

The inference is weak in a way worth stating: many things can suppress a face, so seeing {111} dominant is evidence for F-centring and not proof of it. But it is evidence available from a specimen in a drawer, and it was the only evidence available before 1912.

There is one more reversal that still matters. A habit inconsistent with a determined structure is a signal that something is wrong — a misindexed cell, an unrecognised twin, or a structure solved in a group of too high symmetry. A crystal whose refined space group predicts a habit nobody has ever seen for that compound is worth a second look, and the check costs a minute.

The other half of a form, and why it is needed here #

A ranking of forms is not yet a shape, and the missing half is the open-and-closed distinction.

A closed form bounds a solid on its own — an octahedron does, a cube does. An open form does not: a prism is four or six faces round an axis with the ends unbounded, and a pedion is a single face. A crystal bounded only by open forms is not a crystal that grows infinitely; it is a crystal whose actual boundary needs a second form to close it.

So the morphology a ranking predicts is the first few forms together, not the first one — and how many are needed is decided by whether they close. The cubic examples above close at one form each, which is why they are the memorable cases. The orthorhombic example does not: {100} alone is two parallel faces, and a shape needs at least three forms.

That test is an integer half-space computation with no metric in it, and this site already makes it: 113 kinds of form across the thirty-two classes, of which 43 are closed. Combining that count with this ranking is what turns a list of spacings into a polyhedron, and it is the reason the figures here take the top two forms rather than the top one.

How much difference the second form makes is worth reading off the picture, because it is not the same in every row and the reason is arithmetic rather than crystallography. In a cubic metric the distance at which a form’s planes are placed is proportional to the square root of the sum of the squares of its indices, and that is exactly the distance from the centre to the corresponding corner of the cube. So {110} and {111} arrive tangent to a cube built on {100} and take nothing off it, which is why the primitive prediction is a cube however many forms are added to it. The face-centred case is different only because the extinction correction has moved {100} inwards: halved, it now reaches inside the octahedron and truncates its corners. The second form matters exactly where the correction has moved something, which is the growth rule’s content arriving in the shape rather than in the ranking.

Why the widest spacing is the one that survives #

The rule ranks faces by interplanar spacing and takes the largest, and that direction of the inequality is not obvious. A face that grows fast ought to be the prominent one, and it is the opposite.

The resolution is that a fast-growing face grows itself out of existence. Picture a crystal as a polyhedron whose faces advance outwards at their own rates. A face advancing quickly sweeps forward until its neighbours meet in front of it and it disappears; a face advancing slowly is overtaken by nothing and ends up bounding the whole solid. The faces a finished crystal shows are its slowest ones, and every question about habit is therefore a question about which faces grow slowly.

That is where the spacing enters. A layer of thickness d added to a face is bound to the crystal beneath it by whatever reaches across the gap, and a large d means a large gap — fewer and weaker bonds holding the new layer on, so the layer is harder to complete and the face advances slowly. Small spacings mean densely connected layers, fast growth, and a face that vanishes. So the ranking by decreasing spacing is a ranking by increasing growth rate turned round, and the correction for systematic absences is the statement that what matters is the true repeat across the face rather than the one the indices suggest.

The argument is qualitative and that is exactly its limit. It says the order and not the rates, it uses no bond energies, and it assumes the layer-by-layer picture — so it fails where growth is not layer-by-layer, which is what attachment-energy methods were introduced to handle. What it retains is that the ordering is derived from the lattice and the group alone: two substances with the same group and metric get the same prediction, and if their habits differ, the difference is chemistry rather than symmetry. That is a falsifiable statement, and it is the useful thing about a rule with no energies in it.

Who found it, and when #

Bravais stated the rule in 1866, in the form: the important faces are those with the highest reticular density, which is the same statement as the largest spacing, since density in a plane and spacing between planes are reciprocal.

Friedel added the extinction correction in 1907, having noticed that centred lattices produce habits Bravais’s rule got wrong — and doing it before the diffraction experiment that would have explained why, since von Laue’s photograph is 1912. Friedel was correcting a morphological rule with an argument about lattice planes; that the same correction is the extinction rule is a coincidence in the history and an identity in the mathematics.

Donnay and Harker extended it to screw axes and glide planes in 1937, which is where the rule acquires the full space group rather than just the lattice. Their paper is titled A new law of crystal morphology extending the law of Bravais, and the extension is exactly the axial and zonal absences.

Hartman and Perdok replaced it in 1955 with periodic bond chain theory, which ranks faces by attachment energy rather than by spacing and is better where the two disagree. BFDH survives because it needs no structure — a cell and a space group are enough — which is exactly the case where a prediction is most useful.

Why a ranking rather than a shape #

The rule is often quoted as though it predicted a polyhedron, and it does not; it predicts an order. The distinction is worth keeping because it is where the rule is strong and where it is weak.

The order is a consequence. Given a lattice and a group, the corrected spacings are computed and sorted, and nothing about the sort is adjustable. Two people running the rule on the same structure get the same list in the same order.

The shape is an interpretation. Turning the list into a solid needs a rule for how far each face sits from the centre, and taking that distance inversely proportional to the spacing is the standard choice and is a choice. A different monotone function gives the same faces in the same order and a visibly different polyhedron, with the same forms present and different relative sizes.

So the defensible statement is which forms appear and in what order of importance, and the drawings here are illustrations of that statement rather than predictions of a specimen’s proportions. A real crystal’s proportions depend on how long it grew in which direction, which is a rate and therefore outside this arithmetic entirely.

Cleavage is a different question with the same flavour #

It is worth separating habit from cleavage, because both are about planes and they are decided by different things.

Habit is about growth, and BFDH ranks the faces a crystal grows. Cleavage is about breaking, and a crystal cleaves on the planes across which the bonding is weakest — which is a statement about what is between the layers rather than about how far apart they are.

The two often agree, because widely spaced planes tend to be weakly bonded to each other, and that is why mica cleaves on the same planes it grows as sheets. They can also disagree completely: fluorite grows as cubes when the lattice is face-centred, and cleaves on {111}, which is the octahedral form the ranking puts first. So a fluorite crystal is a cube that breaks into octahedra, and both facts follow from the same lattice by different arguments.

What symmetry supplies in each case is the same thing: the orbit. A cleavage plane belongs to a form, so a crystal that cleaves one way cleaves in every symmetry-related direction — what a cleave leaves is that argument, made from the sectional layer group rather than from spacings. The strength is chemistry; the multiplicity is symmetry.

Where the ladder goes next #

This rung establishes a prediction from a lattice and a group. Two rungs sit above it, and only one belongs here.

The one that does is the relation between habit and the form arithmetic already computed: a ranked list of forms is a list of orbits under the class, so the number of faces on the predicted solid is a sum of orbit sizes, and whether the solid closes is the open-and-closed test a form is an orbit of a face already makes. Combining the two gives the full predicted morphology — which forms, how many faces each, and whether they bound a solid at all.

The one that does not is attachment energy, and everything downstream of it: growth rates, kinetic roughening, solvent effects and impurity poisoning. Every one of those needs a number this site does not compute, and the boundary is the same one the whole applied field is held to — the arithmetic here states a permission and a ranking, and never predicts that anything will happen.