Crystallography & Symmetry
patterns derived from their groups
Concept

Interplanar spacing — where it appears

The distance between successive planes of a family, corrected downwards when the family's reflections are systematically absent. That correction is what turns a list of possible faces into a prediction of which faces a crystal shows.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Building the reciprocal lattice from spacings. Each family of lattice rows has a spacing, and each contributes one reciprocal point: perpendicular to the rows, at the inverse of the spacing. The points built that way were compared against the algebraic definition and agree exactly.

The dual lattice, as a construction

The reciprocal lattice is usually introduced as a formula and then used as a fact. Building it instead — one point per family of lattice rows, at the inverse of the spacing — makes every property it has obvious rather than memorable.

lattices · Lattice 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 Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092.

How many reflections there are

The reciprocal lattice is infinite and a measurement is not. A wavelength cuts a sphere out of it, the number of points inside is the sphere's volume times the cell's, and symmetry then removes all but a fraction — though never quite the fraction a division would give.

diffraction · Resolution 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 7 cells explain the lines; one of them is right. A line list from a face-centred cubic cell of 5.64 Å, with a realistic error added, handed to a sweep over every cubic cell between 2 and 12 Å in all three centrings. 7 distinct cells explain every line within the tolerance, and each is a genuine solution rather than a numerical accident. The true cell comes top by de Wolff's figure of merit — the last Q over twice the mean discrepancy times the number of lines the candidate says should have been visible — which punishes a candidate for predicting lines nobody saw. That is the whole of what makes indexing decidable in practice: not the arithmetic, which has many answers, but a criterion for preferring one.

Indexing a powder pattern

A powder pattern is a list of numbers and a cell is six. Getting the second from the first is the first step of every powder study and the one that fails — because the arithmetic has many answers, and choosing between them is a ranking rather than a deduction.

applied · Indexing 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.

BfdhCrystal formCrystal habitMeasurementMiller indicesMorphologySystematic absenceReciprocal latticeAccidental overlapBragg's lawBroken-bondCentred lattice

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