Interfaces — the series · Crystallography & Symmetry

Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence.

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

part 1 · applied

Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel.

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

part 2 · applied

13 of one on 12 of the other. Two rows of atoms whose spacings are in the ratio 1.042. Every 12 cells of the substrate come to within 3.97 per cent of 13 cells of the film, so the two are nearly in register at those points and out of register between them. There is no exact coincidence anywhere, and there cannot be: exact coincidence needs the ratio to be rational, and no measured ratio is.

Two different lattices never coincide, and the question becomes how nearly

Grow one crystal on another and their spacings are in a ratio that no measurement ever makes rational, so exact coincidence is unavailable in principle. What is left is the best rational approximation inside a tolerable repeat — a quantity that jumps rather than drifts as the ratio changes, and whose acceptability is decided by elasticity rather than by arithmetic.

part 3 · applied

6°: a boundary with a dislocation every 9.5 cells. Two crystals of the same lattice, each turned by half of 6 degrees in opposite senses, meeting on the dashed line. Almost everywhere along it the atoms of one side face the atoms of the other at very nearly the right distance — the boundary is good crystal — and at the marked places the misfit has accumulated to a whole lattice vector and an extra half-plane has to be inserted. Those are the edge dislocations, and they are 9.5 cells apart against the 9.6 that Frank's formula gives.

A small angle is a row of dislocations

Turn one crystal a degree against another and the coincidence arithmetic says they share almost nothing. The boundary between them is nevertheless nearly perfect crystal, and both statements are true: the misfit stays small for a long way and then, all at once, needs an extra half-plane.

part 4 · applied

The dislocations a boundary allows

Two crystals meeting at a coincidence angle share one lattice and generate another. The second is where a boundary's own defects live, its shortest vector is one over the square root of the index, and a dislocation's energy is the square of that.

part 5 · applied

A 21.79° twist, and the cell its beat has. Two copies of the same lattice, one turned. The coarse pattern a reader sees is the beat between them, and the outlined cell is computed from the two lattices rather than measured off the picture: the moiré reciprocal lattice is the original acted on by (I − R), so the moiré cell is the original scaled by one over twice the sine of half the twist, and turned through a right angle plus half the twist.

A beat is not a period

Lay one lattice on another and turn it: the coarse pattern that appears has a spacing anyone can compute, a over twice the sine of half the twist, and it exists at every angle whatever. Whether the superposition actually repeats is a different question with a different answer — countably many angles say yes, and at most of those the true cell is larger than the beat by a definite factor. On a square net it always is.

part 6 · applied

No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records.

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

part 7 · applied