How many dislocations a lattice has

Walk a closed circuit through a lattice round a defect and come back to the wrong point. The amount by which the walk misses is the Burgers vector, and it is a lattice vector — not approximately, not usually, but as a consequence of the circuit being made of lattice steps and the closure failure being a difference of lattice points.

That settles what a dislocation is. It leaves open how many there are, and that question has a clean answer.

Two vectors, one defect #

A Burgers vector names a dislocation. It does not name it uniquely, because a lattice looks the same from several directions.

Take the square lattice and a dislocation whose Burgers vector is (1,0). Rotate the whole crystal by a quarter turn and the same defect has Burgers vector (0,1). Nothing physical distinguishes them: the crystal is unchanged by the rotation, so a defect and its rotated copy are the same defect described from a turned position.

So the number of kinds of dislocation is the number of orbits of lattice vectors under the lattice’s own automorphism group, and orbits are countable.

Which ones a crystal actually has #

Not every orbit is a dislocation a crystal will support, and the reason is energy — the one place in this essay where physics enters.

Frank’s rule: the elastic energy of a dislocation is proportional to |b|². It comes from elasticity theory, it is not derived here, and it is the only thing here that is not arithmetic. Everything else follows the habit this site keeps everywhere: a claim given a test it could fail.

Given it, everything else is arithmetic again. A dislocation with Burgers vector b splits into two with vectors b₁ and b₂ whenever b = b₁ + b₂ and |b₁|² + |b₂|² < |b|². Both b₁ and b₂ must be lattice vectors, so the search over splittings is finite and the comparison is between integers. Finding the shortest of them is the same problem as reducing a basis, and in two and three dimensions it is easy for the same reason.

The two conditions do different jobs and it is worth separating them. That the total falls is the energy statement, and it is the half that comes from elasticity. That b₁ and b₂ are lattice vectors is the symmetry statement, and it is the half that decides what is on the list at all. Relax the second and the arithmetic keeps working and stops describing the lattice: the cheapest way to write any vector as a sum is to halve it, so every dislocation would split, for ever, and there would be no stable defect of any kind.

The shortest vector never splits, for a reason that needs no computation: there is nothing shorter to split it into. That is asserted for every lattice here rather than assumed, and it is the kind of assertion worth writing precisely because it cannot fail — an assertion that could not fail is a check on the machinery rather than on the mathematics.

Symmetry and how many kinds #

Set the lattices side by side and a pattern appears that is worth stating carefully, because the obvious version of it is wrong.

More symmetry means larger orbits. The cubic lattice’s six shortest vectors are one orbit; the triclinic lattice’s two shortest are one orbit, and every orbit it has contains exactly two vectors, since its only automorphisms are the identity and the inversion.

More symmetry does not straightforwardly mean fewer kinds, and it would be easy to write that it does. The count of orbits depends on how far out the enumeration reaches, and the reach here is set as a multiple of each lattice’s own shortest vector — which is the right bound and the wrong statistic to compare. The number that compares properly is the average orbit size, and that rises with the symmetry every time.

That distinction cost a wrong assertion during the writing: the square lattice and a genuinely oblique one both gave three orbits at the same relative reach, and the claim that the square has fewer had to be replaced by the claim about orbit sizes, which is what the symmetry actually controls.

How the automorphisms are found #

The count of orbits needs the automorphism group of the lattice, and it is worth saying how that is obtained, because it is the same trick the rest of the site uses and it is short.

A lattice automorphism is an integer matrix preserving the Gram matrix, so each column of it is an image of a basis vector and must have the same squared length as the vector it replaces. Squared lengths are integers, so the candidate columns can be listed by length before any matrix is assembled — and then a matrix is built column by column, with each new column required to have the right inner product with the ones already placed.

That is a search with almost no branching, because the length condition eliminates nearly everything at the first step. The square lattice has eight automorphisms, the hexagonal twelve, the cubic forty-eight, and every one of them comes out of the search rather than from a table.

The check that it is right is the crystallographic restriction itself: the orders of the matrices found are 1, 2, 3, 4 and 6 and nothing else, which is a consequence of the same integrality and is verified across this collection in half a dozen places.

What “the same defect” is really claiming #

The identification of a vector with its orbit deserves one more sentence, because it is a modelling decision rather than a theorem.

Two dislocations with Burgers vectors in one orbit are related by an operation of the lattice. If the crystal’s own group is smaller than the lattice’s — as it is whenever the structure has less symmetry than its lattice, which is most of the time — then the two defects may be genuinely different in the crystal even though the lattice cannot tell them apart.

So the count here is an upper bound on how much identification is legitimate. For a structure whose point group equals its lattice’s holohedry, it is exact. For anything less symmetric, some of the orbits split further, and the splitting is decided by the structure rather than by the lattice.

That is exactly the distinction between a lattice and the pattern on it that this collection makes everywhere, arriving in a place where it has a physical consequence: two slip systems that a lattice calls identical and a crystal treats differently.

The lattice a real metal has #

Two cases from the physical literature, both of which the arithmetic here reaches and neither of which it derives.

Face-centred cubic metals slip on a/2⟨110⟩. That is the shortest lattice vector of the fcc lattice, and by Frank’s rule it is the cheapest dislocation. Copper, aluminium, nickel and austenitic steels all do it.

Body-centred cubic metals slip on a/2⟨111⟩, which is the shortest vector of that lattice. Iron at room temperature, tungsten, molybdenum. Same rule, different lattice, different answer — and the answer is read off the lattice rather than the metal.

The important case is the one the arithmetic here explicitly cannot reach. In fcc, the a/2⟨110⟩ dislocation dissociates into two partial dislocations with vectors a/6⟨112⟩, whose squared lengths sum to less than the parent’s. But a/6⟨112⟩ is not a lattice vector: it is a sixth of a face diagonal, and the lattice contains no such translation. A circuit round a partial does not close up to a lattice vector; it closes up to a stacking fault, which is a two-dimensional defect the partial trails behind it.

So the enumeration here — over lattice vectors, and only lattice vectors — is by construction blind to partials. That is not a bug to be fixed. It is the boundary of what a purely lattice-theoretic account of dislocations can say, and the two-dimensional defect that appears the moment it is crossed is a different object requiring different machinery.

The other closure failure #

A circuit can fail to close in a second way, and the lattice quantises that one too.

A disclination is a wedge of material taken out or put in, so a circuit round it comes back rotated rather than displaced. The rotation must be one the lattice permits, and the permitted rotations are exactly the ones the crystallographic restriction allows — so a wedge of sixty degrees is possible in a hexagonal net and a wedge of forty degrees is not. The same restriction that forbids a five-fold axis forbids a seventy-two degree wedge.

The two defects are the two parts of a lattice’s symmetry. A dislocation is quantised by the translation subgroup; a disclination by the point group. Every defect a lattice permits is a failure of one or the other to close, and the amount of the failure is an element of the corresponding group. Nothing else is available.

And in both cases the failure does not depend on the circuit. Walk a larger loop round the same dislocation, or a differently shaped one, and it fails to close by the same lattice vector — which is what makes the Burgers vector a property of the defect rather than of the walk, and is established where the circuit is. The reason is the same as everywhere else here: the failure is a lattice vector, lattice vectors are discrete, and a quantity that varies continuously with the path but can only take discrete values does not vary at all. Deform the path a little and the answer would have to jump; nothing is available for it to jump by, so it holds.

That is what “topological” means in this context, and it is a stronger statement than it looks. It is not that the defect is hard to remove; it is that no continuous rearrangement of the crystal can remove it, because such a rearrangement would have to carry the circuit’s failure to zero through values the lattice does not contain. A dislocation can only leave a crystal by reaching its surface or by meeting another dislocation whose Burgers vector cancels its own — and both of those are events rather than relaxations.

The count in three dimensions #

The plane lattices make the argument visible; the lattices a crystal actually has are three-dimensional, and the counts there are worth reading off.

Primitive cubic, with forty-eight automorphisms, has three kinds of dislocation out to three times its shortest squared length: ⟨100⟩ with six vectors in the orbit, ⟨110⟩ with twelve, and ⟨111⟩ with eight. All three are stable — none splits into a pair of shorter lattice vectors — and the twenty-six vectors between them are three defects.

Triclinic, with two automorphisms, has four orbits over the same reach and every one of them has two members. Eight vectors, four defects, and no economy anywhere.

Hexagonal, with twenty-four, has two orbits over that reach, one of which splits — so a hexagonal lattice has essentially one cheap dislocation, which is why a hexagonal metal’s slip behaviour is so much more anisotropic than a cubic one’s.

That last sentence is the boundary of what this arithmetic supports. It says the lattice offers one cheap Burgers vector rather than several; it does not say which planes the dislocation moves on, and the fact that hexagonal metals are hard to deform is about the planes rather than about the vectors.

Why this is a symmetry statement and not a physics one #

The whole of this essay’s arithmetic uses two facts about a lattice: which vectors it contains, and which operations map it to itself. It uses no elastic constants, no core model and no temperature.

What comes out is a classification, not a prediction. Which dislocations exist is decided by the lattice. Which of them a particular crystal contains, at what density, moving how fast, is decided by everything else, and this essay says nothing about it.

Even Frank’s rule is used here more weakly than it is usually stated. The rule as an energy statement gives a number in joules per metre once the elastic constants are supplied; what this essay uses is only the ordering it induces on lattice vectors, which needs no constants at all because they cancel in the comparison. So a splitting reported here is a claim that one arrangement costs less than another and never a claim about how much less, and the ratio between two entries in the table is not an energy ratio in any real material — the anisotropy of the elastic constants alone would spoil it.

That is a habit worth naming, because it recurs across this collection’s applied field: take from the physics the least that is needed, and take it as an ordering rather than as a quantity. A symmetry argument that reaches for a number has usually stopped being a symmetry argument.

That division is worth keeping sharp because the literature does not always keep it. A statement like “fcc metals slip on close-packed planes in close-packed directions” contains one part that is a fact about the fcc lattice — the shortest vector lies in the close-packed direction — and one part that is a fact about how atoms shear past one another. Only the first is here.

Two vectors, two defects, one line #

A dislocation is a line in the crystal, and a Burgers vector is a vector, and the relation between the two directions is worth stating because it is the standard classification.

When the Burgers vector is perpendicular to the line, the dislocation is an edge: an extra half-plane of atoms, ending on the line. When it is parallel, the dislocation is a screw: the lattice planes around the line join into one helical surface, which is exactly the screw axis this collection builds as a symmetry operation, appearing here as a defect instead.

The arithmetic above says nothing about which, because the line direction is not a lattice quantity in the way the Burgers vector is. A given Burgers vector can belong to an edge, a screw, or anything between, and the same defect changes character along its length as the line curves.

That is a real limitation and it is the same one as everywhere else here: the lattice decides what is available and the crystal decides what happens. What the lattice does decide is that the two extremes are the same short vector, which is why the edge and screw forms of one dislocation cost comparably and why a dislocation loop can be part one and part the other.

The general classification, of which this is one case #

The argument on this page counts Burgers vectors and identifies the ones a symmetry relates. There is a general theory of which that is an instance, and naming it explains two features of the count that otherwise look like coincidences.

A defect is classified by what a circuit round it does in the space of possible local states. For a crystal that space is the set of positions the pattern can be translated to, which is one unit cell with its opposite faces identified — a torus. A circuit round a line defect traces a loop in that space, and two circuits describe the same defect exactly when their loops can be deformed into one another. So line defects are classified by the loops of a torus, which is the lattice.

That is where the quantisation comes from, and it is why the answer is a lattice vector rather than any vector: the classification is by a homotopy group, and the homotopy group of a torus is ℤ³. It is also why the count above is a count of orbits — the space of local states carries the point group’s action too, and two loops related by it describe the same defect physically.

The same machinery gives the other defects by changing which homotopy group is asked. Domain walls are the components of the space of states: a structure whose ordering can be done two ways has two components, hence walls. Point defects in three dimensions are its spheres. And disclinations come from the rotational part of the symmetry rather than the translational, which is the same statement as a circuit can come back rotated made in the classification’s vocabulary.

Why two dislocations simply add #

One property of the crystalline case is so convenient that it is easy to take for its own definition, and it fails in materials near enough to be worth naming.

The loops of a torus commute: going round one loop and then another gives the same class as doing them in the opposite order. So two dislocations’ Burgers vectors add, the sum is all that matters, and a dislocation and its negative annihilate whenever they meet. Every statement in this essay about combining and splitting Burgers vectors rests on that, and it is the reason the arithmetic is arithmetic rather than a word problem.

It is not universal. In a material whose local states form a space with a non-abelian fundamental group — a biaxial liquid crystal is the standard example — the class of a pair of defects depends on the order in which the circuit takes them, and two defect lines can be entangled in a way no motion undoes. Passing one through another changes both. There is no Burgers vector to add, because the labels do not add.

That is worth knowing here precisely because a crystal never shows it. The convenience of the crystalline case comes from the local states being a torus, and a torus is about as simple as such a space can be. The classification is more general than the arithmetic, and the arithmetic is what a lattice happens to buy.

Where the ladder goes next #

Sideways, to the other way a crystal accommodates an orientation it does not have: a twin, where two orientations of one structure share a sublattice rather than meeting at a defect line. The index of that sublattice and the angle by which the fit misses are the two numbers a twin law carries, and both are computed from the lattice alone in the same spirit as the count here.

Down, into the physics this essay stops short of: Frank’s rule is quoted and not derived, partials are named and not enumerated, and the stacking fault they bound is a two-dimensional defect that the arithmetic of lattice vectors cannot see. Every one of those is a real boundary rather than an omission.