Crystallography & Symmetry
patterns derived from their groups
Concept

Burgers vector — where it appears

The amount by which a closed circuit through a lattice fails to close when it encloses a dislocation. It is a lattice vector and can be nothing else, so the defect is quantised by the lattice, and two such vectors related by the point group name one defect.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

A circuit that closes on the wrong point: (1, 0). A square lattice with one extra half-column, drawn as a graph: the rows above the core have one more site than the rows below, and the core is the site at the end of the extra column. The path is 4 steps east, 4 north, 4 west and 4 south — the same number out as back — and it ends one lattice vector from where it started. Every one of the 12 circuits in the survey that goes round the core fails by that vector, and all 10 that miss it close exactly.

The circuit that does not close

A defect in a crystal is usually introduced as a picture — an extra half-row of atoms, a wedge taken out. What makes a defect a crystallographic object rather than a drawing is a closure failure: walk a closed circuit through the lattice and come back to the wrong point, by an amount the lattice itself decides.

applied · Defects square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here.

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

applied · Defects 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.

DislocationCoincidence site latticeDisclinationGrain boundaryClosureCrystal growthDefectDsc latticeEnumerationThe Euler characteristicFrank formulaGrowth spiral

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