Turn a lattice against itself and almost nothing lines up

Take a square lattice, make a copy, and turn the copy about one of its points. At almost every angle the two share exactly that point and nothing else: no other lattice point of the copy lands on a lattice point of the original, and the two arrays are as incommensurate as two arrays can be.

At 36.87°, one point in five lands exactly on a point of the other lattice. At 22.62°, one in thirteen. At 28.07°, one in seventeen.

Those shared points are themselves a lattice — a coincidence site lattice — and its index in the original is the number the whole subject is organised around. It is a sublattice in exactly the sense centring made precise, arrived at by intersection rather than by construction.

Why almost every angle shares nothing #

A rotation R maps a lattice point x to Rx, and that is a lattice point of the original exactly when Rx has integer coordinates. For that to happen for even one non-zero x, R must be a rational matrix — its entries must be ratios of integers — because it maps at least two independent integer vectors to integer vectors.

Rational rotation matrices are rare among all rotations in the sense that the rationals are rare among the reals — and the same rarity is what makes an aperiodic pattern’s spacings never quite commensurate. So a rotation picked at random has no coincidence beyond the origin, and that is the generic case rather than a special one.

The angles that do work in the plane come from a single construction. For coprime integers m and k, the rotation with cos θ = (m² − k²)/(m² + k²) and sin θ = 2mk/(m² + k²) is rational — it is the rotation whose half-angle has tangent k/m, and the Pythagorean triple (m² − k², 2mk, m² + k²) is why. Every rational rotation of the plane arises this way, so the list of candidate angles is a list of Pythagorean triples.

Σ, computed rather than quoted #

Write the rotation as A/N with A an integer matrix and N = m² + k². A point x belongs to both lattices when x is integral and R⁻¹x is too, which is the congruence

$$A^{\mathsf T}\mathbf{x} \equiv 0 \pmod N$$

That is a linear condition modulo N, its solutions form a sublattice, and the index of that sublattice is what metallurgy calls Σ. Computing it means counting the solutions — N² of them in the plane at most, and the ratio N²/(number of solutions) is the index.

The figures here compute Σ that way, from the definition, and then compare it against the formula everybody actually uses: Σ is the odd part of N, obtained by dividing out every factor of two. The two are required to agree on every relation in the enumeration, and a figure whose two answers disagreed would not be drawn at all rather than drawn with the number they happened to share.

That arrangement — a definition and a formula, computed separately and checked against each other — is the same second route this site puts under every count it makes. The definition is the honest one and the formula is the fast one, and neither is trusted alone.

Why a metallurgist cares #

A polycrystal is a mass of crystallites — grains — in different orientations, meeting at boundaries. Each grain is an ordinary crystal with an ordinary lattice, and what is at stake is entirely what happens where two of them meet. The energy of a boundary depends on how badly the two lattices fit across it, and a boundary is where most of the interesting things in a metal happen: corrosion starts there, atoms diffuse along there, cracks run there, and the boundaries are what stop a dislocation.

A general boundary has essentially no atom in register. Every atom near the interface is in a position that suits neither side, the bonding is strained everywhere, and the energy per unit area is high.

A boundary between grains related by a coincidence rotation is different. One atom in Σ is at a site both lattices agree on, so a fraction 1/Σ of the interface is unstrained, and the rest is accommodated by a periodic array of dislocations rather than by a general mess. The lower Σ is, the more of the boundary is in register and the cheaper it is — which is the coincidence-site-lattice model of grain boundaries, proposed by Kronberg and Wilson in 1949 after they found that recrystallised copper had far more boundaries at certain misorientations than chance would give.

There is a sanity check available on that claim, and the figures here make it. The number of shared points inside a drawn window is counted and compared with the window’s area divided by Σ — so the marked points are asserted to be as sparse as an index of Σ requires, rather than being marked because they looked right. A selection that happened to look like a lattice would fail that count.

The strongest case is Σ3. One atom in three is shared, which is a great many, and the resulting boundary energy in copper is roughly a twentieth of a general boundary’s. Σ3 boundaries are consequently everywhere in annealed face-centred cubic metals: they are the annealing twins visible in any micrograph of brass or of stainless steel, straight-sided and parallel-sided and abundant.

The sublattice, which this site already had #

There is nothing new about a sublattice here. Centring read as a sublattice made the point already: a body-centred lattice contains a primitive one of half the density, a face-centred lattice one of a quarter, and the relation between them is an index.

A coincidence site lattice is exactly that relation with a different origin for it. Instead of arising from a centring convention, the sublattice arises from intersecting a lattice with a rotated copy — and the index is a number that has to be computed rather than read off a letter.

What is genuinely new is the question. Centring asks which sublattices of a given lattice have the right symmetry to be worth a name. Coincidence asks which rotations produce a sublattice at all, and the answer turns out to depend on arithmetic that has nothing to do with crystallography: in the plane it is Pythagorean triples, and in three dimensions it is sums of four squares. Neither is a fact about crystals, which is the sort of thing the crystallographic restriction has already led this site to expect. That is the next rung.

The model’s limits, which are considerable #

The coincidence-site-lattice model is a geometric criterion for a physical quantity, and the connection between them is empirical.

Σ is a property of the misorientation alone. It says nothing about the plane the boundary lies in, and boundary energy depends strongly on that plane. A Σ3 boundary on the coherent twin plane is very cheap; the same Σ3 misorientation on a different plane is an incoherent twin boundary and costs several times as much. So Σ is one of two numbers and the model uses one.

Exact coincidence is a measure-zero condition. Real grains are never at exactly 36.87°. What saves the model is that a boundary near a coincidence orientation can accommodate the difference with dislocations, and the standard rule of thumb — the Brandon criterion — allows a deviation of about 15°/√Σ before the boundary stops counting as special. That is a fitted threshold, not a derived one.

The correlation with energy is loose. Low-Σ boundaries are cheap on average and there are exceptions in both directions. The model predicts a list of candidate special orientations and the actual energies have to be measured or computed.

So the honest statement is that the arithmetic is exact and its physical consequence is a tendency. This site can compute the list of coincidence rotations exactly, and cannot compute a boundary energy at all. Saying which is which is the whole discipline of the field, and it is permitted against present once more, and this is the sharpest case of it since near-symmetry: an exact geometric criterion sitting behind an inexact physical claim, with a fitted tolerance in between.

The dislocations that take up the difference #

The picture so far is a boundary where one atom in Σ is in register and the rest are not. What happens to the rest is worth a paragraph, because it is the mechanism the whole model depends on.

Between the coincidence points, the two lattices are out of step by an amount that grows and then resets. That is exactly the situation a dislocation describes: a line defect across which the lattice is displaced by one lattice vector. So a boundary near a coincidence orientation is built as flat patches of good fit separated by a periodic array of dislocations, one every time the mismatch has accumulated to a full lattice vector.

The spacing of those dislocations is set by how far the actual misorientation is from the exact coincidence one. Exactly at coincidence they are infinitely far apart — which is to say there are none, and the boundary is perfect. A degree away they are close together, and once they are close enough to overlap the description stops meaning anything and the boundary is a general one again.

That overlap condition is where the Brandon criterion comes from, and it is why the tolerance scales as one over the square root of Σ: a coincidence lattice of index Σ has a cell √Σ times larger in each direction, so its dislocation array can absorb √Σ times less deviation before the dislocations touch.

The scaling is easier to believe once a high index has been looked at. At Σ5 the shared points are close enough together that a window a few cells across holds several of them; at Σ25 the same window holds one or none, and the coincidence cell is five times larger in each direction than the crystal’s. A boundary built on the second relation has one atom in twenty-five in register, its accommodating dislocations are correspondingly further apart along the boundary, and correspondingly less deviation is needed before they run into each other.

The model is therefore not a claim that boundaries sit exactly at special angles. It is a claim that a boundary near one has a describable structure, and the arithmetic supplies the centres of those neighbourhoods.

The counts are small, which is the useful part #

There are seventeen distinct coincidence relations of the cubic lattice with index up to twenty-five. That is a short list, and its shortness is what makes the model usable.

A misorientation between two grains is three numbers — an axis and an angle, or three Euler angles — so the space of possible misorientations is three-dimensional and continuous. Reducing it to a short list of special orientations plus a tolerance is a large compression, and the compression is what lets a texture measurement be summarised as this fraction of the boundaries are Σ3, this fraction Σ9, this fraction general.

It also has a consequence for how a boundary is specified. Five numbers are needed in general — three for the misorientation and two for the plane the boundary lies in — which is a five-dimensional space and far too large to sample experimentally. The coincidence classification reduces the first three to a label, which leaves two, and two dimensions can be scanned. Almost everything known about the variation of boundary energy with orientation has been measured that way: fix Σ, vary the plane, and map the result.

That summary is the basis of grain boundary engineering: process a metal so as to raise the proportion of low-Σ boundaries, and its resistance to intergranular corrosion and creep improves measurably. The processing is thermomechanical and empirical; the classification it is aimed at is the one computed here.

Reading a micrograph as arithmetic #

There is a satisfying closure available here that is worth spelling out, because it takes the whole field back to where it started.

A micrograph of annealed brass shows straight parallel bands crossing many grains. A mineralogist looking at a thin section of calcite sees the same thing and calls them twin lamellae. A metallurgist calls them annealing twins. Both are describing regions related by a rotation of 60° about a ⟨111⟩ direction, both are describing a boundary at which one atom in three is shared, and both are describing a Σ3 coincidence.

The vocabulary diverged because the two fields grew up separately — one from mineral collecting and one from metal working — and the arithmetic did not. Twinning by reticular merohedry and a Σ3 coincidence boundary are the same object under two names, which is a fourth instance of this field’s standing observation and the one with the longest history behind each name.

What the coincidence framing adds to the twinning one is the index as a continuous handle: a twin is Σ3, and Σ5 and Σ7 and Σ9 are the same construction with fewer shared atoms and higher energy, forming a graded series that the twinning vocabulary has no word for. What the twinning framing adds to the coincidence one is the group: a twin law is a coset, and the coset structure says how many distinct laws there are, which the coincidence picture does not on its own.

The lattice that is as fine as the coincidence lattice is coarse #

The coincidence site lattice is the points the two grains agree on. There is a second lattice in the same picture — the points either grain would accept — and it is the one the boundary’s dislocations are made of.

Take every vector that translates one grain and leaves the pattern of coincidences unchanged, and every vector that does the same for the other. Together they generate a lattice finer than either grain’s, called the displacement shift complete lattice. Its vectors are the shifts one grain can be given relative to the other without changing the boundary’s structure — so they are exactly the Burgers vectors the boundary’s dislocations may have.

The two lattices are related by a clean piece of arithmetic. The coincidence lattice sits inside each grain’s lattice at index Σ, and each grain’s lattice sits inside the displacement lattice at index Σ. A high-Σ boundary therefore has a coarse coincidence lattice and a fine set of available Burgers vectors, in exactly the same proportion, and the two facts are one fact seen from opposite sides.

That explains something the coincidence picture alone does not. A high-Σ boundary has few atoms in register, which sounds like a reason for it to be expensive — and it also has very short Burgers vectors available, so the dislocations taking up any departure from the exact orientation are cheap. The two effects run against each other, which is part of why the correlation between Σ and boundary energy is a tendency rather than a rule.

What happens away from the special angles #

A boundary at exactly a coincidence orientation is a measure-zero case, and the model’s most useful extension is the one describing everything else.

Bollmann’s O-lattice generalises the coincidence lattice to any misorientation whatever. Instead of asking which lattice points the two grains share exactly, it asks which points of space are equivalent positions in both — allowing a point to correspond to a different lattice point of the second grain rather than the same one. That set exists for every rotation, it varies continuously with the angle, and it reduces to the coincidence lattice when the rotation is one of the special ones.

The cells of the O-lattice are the regions of good fit, and their walls are where the misfit is concentrated — which is to say the dislocations. So the spacing of a boundary’s dislocations is read off the O-lattice directly, at any angle, and the near-coincidence picture the essay describes is the special case where the O-lattice is coarse and the walls are far apart.

That is what turns the model from a list of special angles into a description of a continuum. The special angles are where the structure is simplest; the arithmetic runs everywhere, and it is the departure from a special angle rather than the angle itself that a micrograph shows.

Where the ladder goes next #

Every index in the list above is odd. That is not an accident of the range chosen and not a feature of the cubic system alone: it is a theorem, with a short proof, about rational orthogonal matrices.

The next rung proves it, enumerates the series properly, and then asks the same question in the plane — where the answer is a different list, governed by which numbers are sums of two squares, and where three of the indices that are perfectly ordinary in three dimensions do not exist at all.