Crystallography & Symmetry
patterns derived from their groups
Concept

Morphology — where it appears

Which faces a crystal actually shows and how large each is, a fact about how the specimen grew rather than about its symmetry. Two crystals of one mineral can look nothing alike and still have exactly the same angles between corresponding faces.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off.

Which faces a crystal shows

Rock salt grows as cubes, fluorite as octahedra, garnet as dodecahedra. All three have cubic lattices and the same list of possible faces, and what separates them is which reflections are systematically absent — a rule about diffraction predicting a shape a mineralogist can hold.

applied · Growth Equilibrium and growth are different shapes. Two predictions for the habit of the same cubic crystal, computed through the same intersection of half-spaces. The equilibrium shape puts each face at a distance proportional to its surface energy, which is Wulff's construction; the growth shape puts it at a distance proportional to its growth rate, taken here from this site's own spacing rule. They differ — the equilibrium shape carries {111}, {110}, {100} and the growth shape {100} — and the difference is between two rules rather than between two pieces of code. A crystal on a bench has grown; a crystal annealed long enough has relaxed; the two look different and neither picture is wrong.

The fast faces are the ones that vanish

A crystal has two predicted shapes and they are not the same. One minimises surface energy and is what a crystal settles into; the other is what growth leaves behind, and in it a face that grows quickly grows itself out of existence.

applied · Growth [001]: 8 of 24 faces. The 24 faces of the form {210} of a cubic crystal, stereographically projected, with one zone marked. The great circle is the set of directions perpendicular to the axis [001]; the 8 faces on it are the zone, and each is on it because the integer sum hu + kv + lw is exactly zero. No length and no angle enters that test. Faces of the lower hemisphere are drawn faintly.

A zone is a vanishing dot product

Look at a crystal and the obvious thing about it is that the faces run in bands — sets whose edges are all parallel. A face belongs to such a band exactly when three integers multiplied by three others sum to zero. No length enters, no angle enters, and that is why every index on a nineteenth-century mineral specimen is still the index used now.

applied · Forms Cube, octahedron, rhombic dodecahedron — from connectivity alone. Every form of index two or less, classified by how many chains lie inside it: two or more and the face is flat, exactly one and it is stepped, none and it is kinked. The number beside each flat form is how many chains it contains, which is the rule's own tie-break — a face with three chains is flatter than one with two. Nothing about interplanar spacing enters, and the three structures are told apart by their bonds.

Which faces are flat

Bravais ranks a crystal's faces by how far apart their planes lie. Hartman and Perdok classify them by how many uninterrupted chains of bonds run inside them, which uses no spacing at all — and on the three cubic structures the two rules put the same face first every time. Then the second rule's power turns out to live entirely in where the chain list is cut off.

applied · Growth

Named alongside it

The objects these essays reach for when they reach for this one.

Crystal formBfdhCrystal habitInterplanar spacingMiller indicesMeasurementSystematic absenceBroken-bondCentringClose packingCoordination numberCrystal net

All concepts