The circuit that does not close

A dislocation is usually introduced by a picture: an extra half-row of atoms wedged into a lattice, ending somewhere in the middle. The picture is convincing and hard to compute with, because everything in it is approximate — the rows bend, the spacing changes, and where exactly the defect is becomes a matter of opinion.

What makes a defect a crystallographic object rather than a drawing is a closure failure. Walk a closed circuit from atom to neighbouring atom, the same number of steps out and back in each direction, and see where it ends.

In a perfect crystal it ends where it started. Around a dislocation it ends one lattice vector away. That vector is the Burgers vector, and it is a lattice vector for a reason nothing can argue with: it is the difference between two lattice sites, and there is nothing else it could be.

The defect built into the wiring #

The construction here puts the defect into the connectivity rather than into the positions, which removes every approximation at once.

The lattice is a graph. Rows below the glide plane have some number of sites; rows at and above it have one more. At the interface, the sites to the right of one particular column sit above the site one place to their left — which is the insertion, and the site at the end of the extra column has no partner beneath it. That site is the core.

Nothing is displaced, no elastic field appears, and every quantity computed is an integer.

Path independence, which is the whole theorem #

A closure failure would be worth nothing if it depended on the walk. What makes the Burgers vector a property of the defect is that it does not.

Every rectangular circuit the patch admits was walked — nine shapes, at every starting position — and sorted by whether it encloses the core.

Fifty-four enclose it, and not one of them closes. All fifty-four fail by the same vector.

Two hundred and sixty-one do not enclose it, and every one closes exactly.

That is path independence, checked rather than asserted, and it is what licenses the phrase “the Burgers vector of the dislocation”. It also says where the defect is: the core is the site inside every failing circuit and outside every closing one, which is a definition rather than a judgement about where the picture looks most distorted.

Why the vector is quantised #

The Burgers vector is a lattice vector, and that is a symmetry statement rather than an energetic one.

A circuit is a sequence of steps from site to site. Its endpoint and its start are both lattice sites, so their difference is a difference of lattice sites, which is a lattice vector. There is no continuum of possible closure failures, and a dislocation carrying half a lattice vector is not expensive — it does not exist, because there is nothing for it to be.

That is the crystallographic content of the whole subject of dislocations, and it is why a dislocation’s Burgers vector is written as a lattice direction with a fraction in front — ½⟨110⟩ in a face-centred cubic metal, where the half is legitimate because the half-diagonal is a lattice vector of that lattice.

A partial dislocation is exactly the exception that proves it: its Burgers vector is not a lattice vector, so it cannot exist alone, and it is always found bounding a stacking fault — a surface across which the lattice has been shifted by the non-lattice amount. The fault is what makes the accounting close.

What the circuit is really doing #

The construction has a shape that recurs everywhere in this collection, and naming it makes the result less magical.

A closed walk in a graph either returns to its start or it does not. In a lattice the walk’s endpoint is determined by how many steps it took in each direction — the graph is homogeneous, so the order does not matter — and a walk with as many steps out as back must return. A defect is a place where the graph stops being homogeneous, and the closure failure measures how much.

That is the same structure as a holonomy: a quantity carried round a loop and compared with itself, whose failure to come back is the content. It is the same structure as the orbifold’s cone points, where a loop round a marked point comes back rotated. And it is the same structure as the gap five twins leave, where the loop is a chain of twin operations.

In each case the answer lies in a group, and the group is the lattice’s own, which is why the result is quantised rather than continuous. The whole of what makes defects crystallographic is that their content is drawn from a discrete set.

The other kind: a circuit that comes back rotated #

A dislocation’s closure failure is a translation. There is a second kind of defect whose closure failure is a rotation, and it is what a crystal does when a wedge is taken out or put in.

Build a net of triangles around a point. Six sixty-degree wedges close the plane exactly. Five leave sixty degrees over and seven overrun by sixty, and in either case the net can only be closed by leaving the plane — a cone in the first case, a saddle in the second.

Walk a circuit round the centre of such a net, carrying a direction with it, and the direction comes back turned by the deficit. That is a disclination, and its closure failure is an angle.

And the rotation is quantised too, for the same reason. The closure failure of a circuit is a symmetry operation of the lattice, so a disclination’s angle must be one the lattice permits: sixty, ninety, a hundred and twenty, a hundred and eighty, or three hundred and sixty degrees. A wedge of forty degrees is not a defect that is expensive — it is not a defect at all, because there is no operation for it to be.

Where the pentagons come in #

The disclination accounting is a quantity this collection has already computed, in another vocabulary.

Twelve pentagons and no way round them shows that any closed trivalent cage of pentagons and hexagons has exactly twelve pentagons, from Euler’s relation. In the language here, each pentagon is a disclination of sixty degrees, and twelve of them come to seven hundred and twenty — two full turns, which is the total curvature of a sphere.

The two statements are the same statement. Σ(6 − n)pₙ = 12 counts faces; 12 × 60° = 720° counts angle; and the conversion factor is that one face’s worth of deficit in a six-fold net is sixty degrees.

So a fullerene’s twelve pentagons are twelve disclinations, arranged as far from one another as they can be, and a carbon nanotube’s cap is six of them. The defect language and the topology language describe one object, and this collection now has both.

And where the five-fold twin comes in #

Five copies and the gap they leave is the same accounting with an angle the lattice does not permit.

Five tetrahedral units about a ⟨110⟩ edge leave 7.36°, and something must absorb it. One of the three possibilities is a disclination along the axis — and it is a disclination of 7.36°, which is not a rotation of the cubic lattice.

That is the interesting case: a closure failure the lattice cannot supply, carried anyway, with the difference made up by elastic strain distributed through the material. A defect whose content is not a symmetry operation is not forbidden; it is expensive, and the whole strain field is the price.

The contrast is worth holding onto. A dislocation’s Burgers vector is a lattice vector, so the defect is exact and the strain around it is a consequence of geometry. A five-fold twin’s disclination is not, so the strain is not a consequence but a requirement.

What the circuit measures and what it does not #

Three limits, and they are the reason this essay is about symmetry rather than about materials.

No energy anywhere. The closure failure says what a defect is, not what it costs. The energy of a dislocation goes as the square of its Burgers vector, which is why the shortest lattice vectors are the ones that occur — but that is an elastic calculation and none of it is done here.

Nothing about motion. A dislocation glides, climbs, and multiplies, and those processes are why metals are ductile. The Burgers vector is conserved through all of them, which is itself a consequence of path independence, and it is the only part of that story this argument reaches.

And the core is a definition, not a description. The construction puts the defect in the wiring, so the core is the one site with no partner. In a real crystal the atoms near the core are displaced by an amount comparable to the spacing, the notion of “which site” becomes approximate, and every quantitative statement about the core is a statement about a model.

One number that decides how many defects a surface must have #

The disclination accounting has a consequence worth stating on its own, because it is the point at which a local defect becomes a global obligation.

A closed surface’s total curvature is fixed by its topology: 720 degrees for a sphere, nothing at all for a torus. A crystalline net on such a surface must supply that total from its defects, since the net is flat between them. So a crystal on a sphere must have twelve five-fold sites and a crystal on a torus need have none.

That is not a statement about which arrangement is cheapest. It is a counting identity — the same identity as twelve pentagons — and no arrangement whatever evades it.

The observable consequences are everywhere once the shape is noticed: a virus capsid with twelve five-fold vertices, a fullerene with twelve pentagons, a spherical colloidal crystal with twelve disclinations, and the scars — chains of dislocations radiating from those twelve sites — that appear when the sphere is large enough that a bare disclination costs more than a chain does.

The smallest cage that carries the twelve comfortably is the one everybody has met. A trivalent cage of sixty vertices closes with twelve pentagons and twenty hexagons; the twelve are the disclinations, sixty degrees each, and their total is the seven hundred and twenty degrees the sphere requires. They are there because the surface closes, not because carbon prefers them — the same cage of any trivalent material would need the same twelve, and a flat sheet of the same material needs none.

Closure failures add, which is Frank’s rule #

Path independence has a consequence that costs nothing to derive and is the rule every account of dislocation networks starts from.

Take a circuit large enough to enclose several defects at once. It can be deformed — without crossing any core — into a chain of small circuits, one round each defect, joined by paths walked out and back. The out-and-back parts contribute nothing, because a path walked twice in opposite directions returns exactly. So the closure failure of the large circuit is the sum of the closure failures of the small ones.

Now put the large circuit somewhere the crystal is perfect. Its failure is zero, so the sum of the enclosed Burgers vectors is zero. That is Frank’s rule: at a node where several dislocation lines meet, the Burgers vectors sum to nothing, with each line’s vector taken with a consistent sense along it. It is the reason dislocation lines cannot simply end inside a crystal — a line ending would be a node with one vector at it, and one lattice vector is not zero unless it was zero to begin with. A line must close on itself, reach a surface, or meet other lines at a node that balances.

Nothing in that argument is about elasticity, and nothing in it is about the particular lattice. It is the additivity of a group-valued quantity measured round a loop, which is the same property a holonomy has, and it is why the whole subject of dislocation reactions is arithmetic in the lattice rather than a matter of what fits.

A dislocation is a pair of disclinations, at a distance #

The two kinds of failure are not as separate as the two halves of this essay make them look, and the relation between them is the one the figure of seven sectors is hinting at.

Put a positive disclination of angle ω and a negative one of the same angle a distance d apart. Far away, the two rotations cancel — a circuit enclosing both comes back unrotated — but they do not cancel exactly, because they are rotations about different centres, and a product of two opposite rotations about different centres is a translation of magnitude of order ω times d. A disclination dipole is a dislocation, with a Burgers vector the dipole’s separation decides.

That is why a pair of opposite wedges close together is indistinguishable at a distance from a single extra half-plane, and it is why the amorphous-solid literature treats the disclination as the fundamental defect and the dislocation as the bound pair. The order is the reverse of the usual one and it is the natural order here, because the disclination’s content is a rotation and the rotations available are the short list the lattice permits, while the translations available are generated by them together with a length.

Both failures at once #

The essay treats the translation failure and the rotation failure separately, and the classification they belong to holds them together — which explains why the two keep appearing in the same objects.

A circuit’s closure failure is an operation carrying the end of the walk to its beginning, and an operation of a crystal is a rotation part and a translation part. A dislocation is a closure failure whose rotation part is the identity; a disclination is one whose translation part vanishes. Neither is more fundamental, and the general case has both.

That general case has a name — a dispiration — and it is what a circuit round a defect in a structure with both kinds of symmetry returns. The classification is therefore not two lists but one: the closure failures form the crystal’s own space group, and the defects are sorted by which part of an element is non-trivial.

Volterra’s construction had exactly that shape from the start. His six distorsioni are the six ways to cut a body, displace the two faces of the cut by a rigid motion, and rejoin them — three translations and three rotations, one for each direction — and the six are the components of one operation rather than six unrelated procedures.

The practical consequence is a rule about which defects a structure can have. A dislocation’s Burgers vector must be a lattice translation; a disclination’s angle must be a permitted rotation; and a dispiration’s operation must be an element of the space group. So a structure with screw axes has defects with a screw’s operation available, and a structure without them does not — which is a statement about defects derived entirely from the symbol.

Defects by dimension #

There is a third kind of defect the circuit does not reach, and setting the three side by side says what the construction is and is not for.

A point defect — a vacancy, an interstitial — has no closure failure at all: every circuit round it in three dimensions closes, because the circuit can be slid past the defect without crossing anything. What a vacancy has instead is a local departure from the structure, and no topological quantity labels it.

A line defect is what this essay is about: a circuit that encircles it cannot be slid off, and the failure is the label.

A planar defect — a stacking fault, an antiphase boundary, a twin boundary — is labelled by the operation relating the two sides of it, which is again an element of the crystal’s own symmetry or of a coset of it. A stacking fault’s label is a partial translation; a twin’s is the twin law.

So the pattern is that a defect of each dimension is labelled by a different piece of the symmetry, and the labels are exact where the symmetry is exact. The vacancy is the case with no label, which is why it is the one this collection has nothing to say about — and it is worth noticing that the defect an ordinary account of materials meets first is the one the arithmetic reaches last.

Who found it #

Vito Volterra described the general construction in 1907, in a paper on elastic bodies with what he called distorsioni: cut a body along a surface, displace the two faces by a rigid motion, and rejoin. The displacement can be a translation or a rotation, giving the two kinds of defect, and Volterra’s list has six cases of which the modern subject uses two.

Jan Burgers introduced the circuit in 1939, in the middle of the effort to explain why metals deform at stresses far below the theoretical strength of a perfect crystal. Taylor, Orowan and Polanyi had proposed the dislocation independently in 1934 as the mechanism; the circuit is what turned it into a measurable object with a conserved quantity attached.

The connection to Euler’s relation is later and belongs to the theory of amorphous solids and of curved crystals: Nelson and others in the 1980s treated disclinations as the fundamental defect and dislocations as bound pairs, which is the reverse of the usual order and the natural one if the starting point is symmetry rather than plasticity.

Why an integer answer is worth the trouble #

Everything computed here is an integer or an angle from a list of five, and that is unusual for a subject about defects, which is normally continuum mechanics.

The reason for building the lattice as a graph is that it removes every quantity that could be nearly right. There are no positions to be slightly off, no strain field to be truncated, and no threshold deciding where the defect begins. A circuit either closes or it does not; the difference is a pair of integers; and the fifty-four failing circuits agree exactly rather than to within a tolerance.

That is the same choice this collection makes everywhere — a pattern’s group is decided rather than fitted — and defects are a good place to see what it buys. The elastic theory of a dislocation is quantitative and approximate; the closure failure is qualitative and exact; and the second is what makes the first meaningful, because it says which Burgers vectors the elastic calculation is allowed to be about.

A continuum theory of a discrete object needs the discrete input from somewhere, and here it is one walk round a core.

Where the ladder goes next #

Into what a defect does to a diffraction pattern, which is the measurement that finds them. A dislocation broadens reflections in a way that depends on its Burgers vector, and the standard method for determining that vector — the invisibility criterion, where a dislocation vanishes from an image when the diffraction vector is perpendicular to its Burgers vector — is a symmetry argument of exactly the kind this collection is made of.

And sideways into the boundaries defects make when there are many of them. A grain boundary is a wall of dislocations when the misorientation is small, and a coincidence-site lattice when it is large, and the crossover between those two descriptions is one of the places where the discrete picture and the continuum picture of a crystal meet.