Crystallography & Symmetry
patterns derived from their groups
Concept

Misfit — where it appears

The amount by which two lattices meeting at an interface fail to register with one another. At a small-angle boundary it accumulates linearly along the interface and is discharged at dislocations, giving a sawtooth rather than a smear.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

applied · Interfaces 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.

applied · Interfaces

Named alongside it

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

Coincidence site latticeBurgers vectorContinued fractionDislocationEpitaxyFrank formulaGolden ratioGrain boundaryIncommensurateTilt boundaryTolerance

All concepts