The dislocations a boundary allows

Turn a lattice against itself and almost nothing lines up. At almost every angle two copies of one lattice share the point they were turned about and no other; at a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation is cheap, because a fraction of the atoms are already where both crystals want them.

That essay computes what the two crystals share. There is a second lattice in the same picture, it is the one the boundary’s own defects live in, and it is what the pair generate.

Three lattices, nested #

Write L for the crystal’s lattice and L′ for the rotated copy. Two lattices can be combined in two ways and both are lattices.

The intersection L ∩ L′ is the coincidence site lattice, the CSL. It is a sublattice of each of index Σ, and it is what the earlier essay is about.

The sum L + L′ — the set of all sums of a vector from one and a vector from the other — is the coarsest lattice containing both. It is called the DSC lattice, for displacement shift complete, and both crystals are sublattices of it.

The indices go the same way in both directions, and that is the first thing to check rather than to quote:

CSL ⊂ L ⊂ DSC, with index Σ at each step. So the DSC lattice is Σ times finer than the crystal, exactly as the coincidences are Σ times coarser, and the three together are a nested triple with one number in it.

The computation is short and stays in whole numbers. A coincidence rotation of the square lattice is an integer matrix divided by N = m² + n², so multiplying everything by N puts the sum lattice in reach of an ordinary Hermite normal form: it is generated by the columns of N times the identity together with the columns of the rotation’s numerator, and dividing the result by N gives the DSC basis. The coincidence lattice comes from the other side — the vectors whose image under the adjugate vanishes modulo N — and the two indices are then read off two determinants.

Why a boundary’s dislocations are not the crystal’s #

The reason the DSC lattice matters is the reason it was defined, and it is a statement about what may go wrong at a boundary.

A dislocation in a crystal is a closure failure: walk a circuit that would close in a perfect lattice, and come back to the wrong point. The amount by which it misses is a lattice vector, because a translation that is not a lattice vector would not leave the crystal looking the same — the defect would be visible far away instead of being a line.

A boundary is a different object. What has to be preserved at a distance is not one crystal but the pair, and the translations that leave the pair looking the same are those that carry the coincidence pattern onto itself. Shift one grain by a vector of the DSC lattice, and every point of that grain moves onto another possible position of the same grain relative to the other: the two crystals are in the same relationship they were, and the boundary is the boundary it was.

So a boundary dislocation’s Burgers vector is a DSC vector, and DSC vectors are shorter than lattice vectors by a factor that the table above pins down exactly: one over the square root of Σ. At Σ5 the shortest is 0.447 of a lattice spacing, at Σ13 it is 0.277, at Σ25 it is a fifth.

The energy of a dislocation goes as the square of its Burgers vector, so those defects cost one part in Σ of what an ordinary dislocation costs. A high-index boundary can therefore accommodate a change of misorientation with defects too cheap to notice, which is why the coincidence orientations are not knife-edge conditions in practice: a boundary a degree or two away from an exact Σ5 relation is an exact Σ5 boundary with a sparse array of very cheap dislocations in it, and it keeps most of the energy advantage.

The same accommodation, one rung down #

That last paragraph is the general form of a mechanism this collection has already computed in the special case where Σ is one.

A small angle is a row of dislocations: two crystals misoriented by a degree share nothing by the coincidence arithmetic, and the boundary between them is nevertheless nearly perfect crystal, because the misfit is taken up by ordinary lattice dislocations spaced a Burgers vector over the angle apart. Frank’s formula gives the spacing, and the boundary energy rises as the angle times the logarithm of it.

A misorientation near an exact coincidence does the same thing in the DSC lattice: the deviation from the exact relation is taken up by an array of boundary dislocations, spaced by the DSC Burgers vector over the deviation angle. Everything about Frank’s construction carries over with the lattice changed. That is why a plot of boundary energy against angle has cusps at the coincidence orientations — each cusp is a Frank formula in its own DSC lattice, starting from zero at the exact relation.

What is exact here and what is a model #

The lattice arithmetic is exact and the physics attached to it is not, and the boundary between them is worth marking.

Exact: the three lattices, their bases, the two indices, and the length of the shortest DSC vector. All of it is integer arithmetic on the coincidence matrix, with no lengths chosen and no tolerance anywhere. The agreement between the measured shortest vector and one over the root of Σ is to machine precision across every relation up to the bound, which makes it a statement rather than a trend.

A model: that a dislocation’s energy goes as the square of its Burgers vector. That is elastic theory, it is a good approximation away from the core, and it is what turns a lattice fact into an energy argument. Nothing in this collection computes it.

And what a plane cannot show. Real grain boundaries are surfaces in three dimensions and their DSC lattices are three-dimensional; the plane case here has the same structure and none of the difficulties, in particular none of the question of which plane the boundary lies in, which is a second parameter with as much influence as the misorientation. Every coincidence index is odd is where the difference between the plane’s arithmetic and the cubic case is set out; the same caution applies to everything above.

Reading the numbers in the table #

Three features of the census repay attention, because each of them is a statement rather than a column.

The shortest DSC vector is exactly one over the root of Σ, and not approximately. The DSC lattice of a square-lattice coincidence is itself square — it is the crystal’s lattice scaled and turned — and its determinant is one over Σ, so its shortest vector is the square root of that. The agreement between the measured value and the closed form is to fifteen figures at every index, which is the difference between a fitted relation and an identity that happened to be measured.

The angles are not evenly spaced and they are not arbitrary. Σ5 sits at 36.87 degrees, Σ13 at 22.62, Σ17 at 28.07, Σ25 at 16.26. Each is twice the arctangent of a ratio of two whole numbers, and the index is the odd part of the sum of their squares — which is why the available indices are exactly the odd numbers expressible as sums of two squares, and why the plane’s list is so much sparser than the cubic one.

And the same index can occur at more than one angle. The census keeps the smallest angle for each index, which is a choice; the full list has several relations at some indices, and they are genuinely different boundaries with the same coincidence density and different DSC lattices in different orientations. A table with one row per Σ is a summary, and this one says so.

Why the DSC lattice is complete #

The name carries the definition and it is worth unpacking, because it says something the sum-of-two-lattices description does not.

Displacement shift complete means: the set of displacements of one grain against the other that leave the coincidence pattern complete — unchanged as a whole, though individual atoms have moved. It is not obvious in advance that this set is a lattice, nor that it is the sum of the two crystal lattices, and both facts are the content of the construction.

The first follows from the coincidence pattern being a lattice: displacements that preserve it form a group under addition, and a discrete group of translations of the plane is a lattice — which is exactly the dichotomy this collection uses to justify the word lattice everywhere else. The second is a short argument about generators.

A useful consequence is that the DSC lattice does not depend on which crystal is called the first. It is symmetric in the two grains, which a defect at their shared boundary had better be.

Why the sum lattice is a lattice at all #

The construction assumes something that is worth checking rather than assuming: that the set of sums of vectors from two lattices is itself a lattice.

Closure under addition and negation is immediate, so the sum is a group of translations. What is not immediate is that it is discrete — and a subgroup of the plane’s translations that is not discrete is not a lattice at all but a dense set, which would make the whole construction meaningless. Discrete, or dense, and nothing between is the essay about that dichotomy, and it is the reason the question has to be asked.

For a coincidence relation the answer is yes, and the reason is exactly the coincidence. The rotation is an integer matrix divided by N, so every vector of the rotated lattice has coordinates that are multiples of 1/N in the original basis; the sum lattice therefore sits inside the lattice scaled by 1/N, which is discrete, and a subgroup of a lattice is a lattice.

At a general angle it is not. Turn a lattice by an angle whose tangent is irrational and the sum of the two lattices is dense in the plane: there are no coincidences, and the set of displacements preserving a pattern of coincidences that does not exist is everything. So the DSC lattice exists precisely when the CSL does, at the same discrete set of angles, and the three-lattice picture is a feature of special orientations rather than a general fact about pairs of crystals. That is the same discreteness threshold epitaxy runs into from the other direction, where two different lattices never coincide exactly and the question becomes how nearly.

What the third lattice is not #

Two things the DSC lattice is easy to confuse with are worth separating, because both are lattices in the same picture and neither is this one.

It is not the reciprocal of the coincidence lattice. The CSL is a sublattice of the crystal of index Σ and the DSC is a superlattice of index Σ, so the two are related by an index and the arithmetic looks reciprocal — but the relation is not duality. A dual lattice lives in a different space, built one point per family of rows at the inverse of the spacing, and the DSC lives in the same space as the crystals it was built from. That the determinants happen to be reciprocal is a consequence of the indices, not of a construction.

And it is not a smaller unit cell. A superlattice of index Σ contains the crystal’s lattice, so a crystallographer meeting it for the first time reasonably asks what structure sits on it. The answer is: none. The DSC lattice is a lattice of displacements rather than of atoms — the shifts of one grain against the other that leave the bicrystal’s pattern of coincidences unchanged — and no atom sits at a general DSC point in either crystal. It is the lattice a defect’s Burgers vector is drawn from, and nothing else.

Keeping those apart is what makes the sentence “a boundary dislocation has a Burgers vector Σ times shorter than a lattice vector” mean something rather than sound impossible. Nothing has moved by a fraction of a lattice vector inside either crystal. What has moved by a fraction of a lattice vector is one crystal relative to the other, and that is a quantity with no meaning inside either of them.

Where the exactness stops #

Computed here: the CSL basis, the DSC basis and the two indices for every coincidence relation of the square lattice up to index twenty-five; the shortest DSC vector in each case, compared with one over the square root of the index; and the trivial relation as a control, where the two crystals coincide, the DSC lattice is the crystal’s own, and the shortest vector is a full lattice spacing.

Quoted: the elastic energy of a dislocation, and the cusped shape of the energy against misorientation. Neither is arithmetic.

Not attempted: three dimensions, the choice of boundary plane, the atomic structure of the boundary core, and whether a given boundary actually adopts the coincidence orientation rather than merely being able to.

Who found it, and when #

The coincidence site lattice is Kronberg and Wilson’s, from a 1949 study of recrystallised copper. The DSC lattice came out of the electron microscopy of boundaries in the 1960s, when the dislocations at grain boundaries could first be imaged and turned out to have Burgers vectors much shorter than a lattice vector; Bollmann’s Crystal Defects and Crystalline Interfaces of 1970 is where the construction was set out in general.

The name is Bollmann’s and it is a description rather than an acronym anybody now expands in conversation. What the microscopists needed was a rule for which Burgers vectors could occur at a boundary of a given misorientation, and the rule is: the DSC lattice, which is a lattice sum, computed once per relation.

Where the ladder goes next #

Back, to what the two crystals share: the coincidence site lattice, the angles at which it exists, and why every index is odd.

Sideways, to the defect this rung is about. The circuit that does not close is the Burgers construction in a perfect crystal, and how many dislocations a lattice has counts the distinct ones as orbits of the point group — a count that could be made in the DSC lattice instead, and would give a different and larger answer.

And outward, to the interface where the two lattices are not copies of one another at all: epitaxy is a measurement, where exact coincidence is unavailable in principle and what is left is a best rational approximation.

Which boundaries a metallurgist wants #

The arithmetic here says that a low-index coincidence has a coarse DSC lattice and therefore cheap boundary defects, and that has turned into a way of designing materials rather than only of describing them.

Low-Σ boundaries have low energy. A boundary at an exact coincidence has many atoms in common between the two crystals, and the accommodation it needs is carried by DSC dislocations that cost a fraction of an ordinary one. Boundaries at general angles have neither advantage.

The extreme case is Σ3. In a face-centred cubic metal the coherent twin boundary is a Σ3 relation, with one atom in three shared, and its energy is roughly a twentieth of a general boundary’s. Twin boundaries are common in copper, brass and austenitic steels for that reason — they cost so little that they form during growth and during annealing.

And low-energy boundaries resist the things boundaries fail at. Corrosion, embrittlement, and crack propagation all proceed preferentially along boundaries, and preferentially along the high-energy ones. Increasing the proportion of low-Σ boundaries in a material — by cycles of deformation and annealing chosen to promote twinning — improves resistance to all three without changing the composition. That practice is called grain-boundary engineering, and the arithmetic on this page is what it is engineering towards.

How near an exact coincidence has to be is a rule of thumb. A real boundary is never at the exact angle, and the misorientation is accommodated by DSC dislocations whose spacing falls as the deviation grows. When the spacing approaches the DSC lattice’s own scale the description stops meaning anything. The usual criterion allows a deviation of about fifteen degrees divided by the square root of Σ — so a Σ3 boundary counts as coincident within nine degrees and a Σ25 within three.

That criterion is empirical, which is exactly the boundary this essay’s last section draws. The lattices, the indices and the DSC vectors are exact; how far from exact a real boundary may be and still behave like a special one is a fitted number, and it should be quoted as one.