Crystallography & Symmetry
patterns derived from their groups
Concept

Dislocation — where it appears

A line defect whose content is a lattice vector: a circuit taken round it closes one such vector away from where it began. The vector is quantised, so a dislocation is a topological object rather than a smudge in the arrangement.

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

A screw dislocation of Burgers vector 1, after 40 steps of growth. The height of a growing surface, light for low and dark for high, over a patch 25 cells across with a screw dislocation at its centre. Growth is an integer rule — a site rises when it has a neighbour a layer higher — and the only thing that makes this patch different from a flat one is a branch cut along which the comparison is offset by the Burgers vector. The step winds round the centre instead of running out: after 40 steps the centre has climbed 10 layers and the surface is still growing at 110 sites a step. The shading is normalised to the patch's own range, so the shape is the steady state the mechanism predicts and is the same at every step count; the numbers at the foot are what changes, and they are what the claim of unending growth is actually about.

The step that never runs out

A perfect crystal face cannot grow: an atom arriving on a flat plane touches it on one side and leaves again. Faces grow anyway, and the reason is a defect — a screw dislocation puts a step on the surface that winding round it never consumes.

applied · Growth 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 Σ5: three lattices in one picture. Two copies of the square lattice turned by 36.87 degrees against one another — one drawn pale, one drawn in the second colour — with the points they share ringed. The fine dots are the lattice generated by both together, the DSC lattice, which contains each crystal with index 5 exactly as the coincidences sit inside each crystal with index 5. Three lattices nested at the same index, and the middle one is the crystal.

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.

applied · Interfaces

Named alongside it

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

Burgers vectorCoincidence site latticeGrain boundaryCrystal growthDsc latticeFrank formulaGrowth spiralHermite normal formMisfitNucleationScrew dislocationStep

All concepts