The walls a strain permits

A crystal that has changed shape has done so in patches — one domain per choice the order parameter made — and the patches meet. Where two domains of different shape share a boundary, the boundary is under a constraint that neither domain is: it belongs to both, so it must have the same length measured in either.

That constraint decides the orientation. Not the position of the wall, not its energy, not whether it moves — only which directions it may run in — and it decides it completely, by a quadratic equation with rational coefficients.

The condition, in one line #

Let the two domains carry strains e₁ and e₂, and let t be a direction lying in the wall. Domain one stretches that direction by tᵀe₁t and domain two by tᵀe₂t; the wall belongs to both, so those must agree:

$$t^{\mathsf T}(e_1 - e_2)\,t = 0$$

Writing the difference as Δ and the direction as (x, y), that is Δ₁₁x² + 2Δ₁₂xy + Δ₂₂y² = 0 — a quadratic in the direction, homogeneous, so what it constrains is the ratio x : y and not the length.

A homogeneous quadratic in two variables has two roots, one, or none, according to the sign of its discriminant. So a pair of domains has two permissible walls, or one, or none at all, and which is a property of the pair.

Sapriel wrote the three-dimensional version of this down in 1975, where the same condition constrains a plane rather than a line and gives a pair of planes for each domain pair. The plane version is the same statement one dimension down, and it can be drawn.

Why the answer is always two #

Running the condition over every pair of domains of every ferroelastic descent the mode census produced gives two walls in every case. Not sometimes two: always.

That is not luck and it is not a property of the small sample. Two domains are images of one another under an operation of the parent group, and every operation of the parent leaves area alone — its matrix has determinant ±1. So the two strains change the area by the same amount, and their difference changes it by none.

A quadratic form that changes no area is one whose eigenvalues sum to zero, so unless it is identically zero it has one of each sign. It is indefinite, its discriminant is positive, and the equation has two real roots.

So the two walls are a theorem rather than a tally, and the theorem’s content is that domains related by a symmetry cannot differ by a distortion that stretches every direction. Something has to be left alone, and the directions left alone are the walls.

Getting the area right on a lattice basis #

The argument above needs the area change, and the area change is not the trace of the strain array unless the basis is orthonormal.

On a hexagonal basis, two domains of a 6mm → m descent differ by an array whose diagonal entries sum to −2. Nothing is wrong: the invariant quantity is tr(G⁻¹Δ) with G the metric of the lattice, and it is that combination which is zero. Computing it needs a metric the class leaves alone, which is got by averaging MᵀM over the group — the same trick, in the same collection, for the third time.

That is the sort of correction which is easy to skip and which changes the answer. A routine that tested the plain trace would report the hexagonal descents as violating a theorem that they satisfy, and the natural response — concluding the theorem is false in the hexagonal case — would be exactly wrong.

The site’s habit applies: the arithmetic is checked in a basis where it is exact, and the basis-dependence is named rather than assumed away.

Where the walls actually point #

The directions are the roots of a rational quadratic, which means they are rational when the discriminant is a perfect square and irrational otherwise.

For 4mm → 2mm on a square lattice the two walls are along the diagonals, [1 1] and [1 −1]. That is the familiar picture of a ferroelastic twin: the domains are rectangles elongated along the two axes, and they meet along a diagonal at forty-five degrees to both.

For descents on a hexagonal lattice the walls come out along lattice rows too, but not at the angles a reader used to the square case expects, because the basis is oblique. And for a general pair the roots need not be rational at all — the wall then runs along a direction that is not a lattice row, which has a physical consequence: a wall along an irrational direction cannot be atomically flat, and must be stepped.

That last point is where this argument meets the small-angle boundary essay. A boundary that cannot be coherent is accommodated by defects, and the spacing of those defects is set by how far the required direction is from a rational one.

Two domains, and what happens with more #

Every pair has two walls. A crystal with more than two domains has more than one pair, and the walls of different pairs are different directions — so the number of distinct permitted orientations in a specimen grows as the square of the index of the descent rather than in step with it.

The multiplication is not quite as free as that, and the correction is worth making because it is easy to state the count wrongly. A descent of index four has six pairs of domain states, but it need not have twelve wall directions: a descent can produce fewer distinct shapes than it does states, and two states of the same shape differ by no strain at all, so their difference vanishes identically and every direction satisfies the condition. 4mm → m is that case — four states, two shapes, and two of the six pairs excluded — so it has eight permitted walls and not twelve. The right statement is that the count grows as the square of the number of distinct shapes, and the number of shapes divides the index rather than equalling it.

What the condition constrains is each wall separately; nothing here says how the walls arrange themselves into a pattern, which is a question about energy, elastic interaction between walls and the history of the specimen — the same boundary between what symmetry permits and what a material does that every essay in this field runs into.

There is one arrangement the condition does forbid, and it is worth stating because it is the useful negative. Three domains meeting along a line — a triple junction — require all three pairwise walls to be compatible at that line, and the three conditions are generally inconsistent. So a triple junction is under strain even when each of its three walls is individually permissible, which is why such junctions are where a ferroelastic crystal cracks.

A wall is a twin boundary #

The domains of a ferroelastic transition are related by an operation the parent had and the child has not, which is exactly the definition of a twin law.

So the walls computed here are twin boundaries, and the condition on them is the compatibility condition a mineralogist would call the requirement that the composition plane be unstrained. The two literatures are describing one object: the ferroics literature comes at it from the transition and calls the result a domain wall; the mineralogy comes at it from the finished crystal and calls it a twin.

The coincidence-site essays in this collection describe a third case with the same geometry — two grains of the same phase in different orientations, meeting along a boundary — and there the condition is that a sublattice be shared rather than that a direction be unstretched. The three conditions are different, and the difference is exactly what the two crystals have in common at the boundary: a lattice, a direction, or a strain.

The same condition, in three familiar places #

The requirement that a boundary be unstretched is not particular to ferroelastics, and setting the cases side by side shows what varies.

A ferroelastic wall. Two domains of one phase, related by a lost symmetry, differing by a spontaneous strain. The condition is tᵀΔt = 0, and the answer is two directions.

A martensitic interface. Two phases with different cells, related by a lattice correspondence rather than by a symmetry. The strains no longer have equal area change, since the transition changes volume, so Δ need not be indefinite — and the condition can genuinely have no solution. What happens then is that the interface becomes a fine mixture of two variants whose average strain does satisfy the condition, which is the whole mechanism behind the microstructures those materials show.

A twin in a grown crystal. Two orientations of one structure sharing a lattice, related by an operation the lattice has and the crystal lacks. The condition there is not about strain at all: the two orientations have identical cells, and what is required is that a plane of the lattice be shared, which this collection computes as an index and an obliquity.

The three are separated by what the two sides of the boundary have in common. Equal strains and a shared lattice make the condition trivial; different cells make it insoluble; a symmetry relation makes it soluble with exactly two answers.

The condition in the domain’s own coordinates #

There is a second way to state the same condition, and it is the one a reader who has met twinning will recognise.

Instead of asking which directions have equal length in the two domains, ask what operation carries one domain to the other. It is the lost symmetry — a mirror, a rotation — and a wall is compatible when it is a mirror plane of that operation’s action on the strain: the two domains are mirror images across the wall.

For 4mm → 2mm the lost operation is the fourfold rotation, and the two permitted walls are the two diagonals, which are the fixed lines of the two diagonal mirrors of the parent. The pattern generalises: the permitted walls are the fixed lines of the parent operations that exchange the two domains.

That statement and the quadratic give the same answers, and each is easier in a different case. The quadratic needs no knowledge of which operation relates the domains and works when several do; the fixed-line reading is immediate when the relating operation is a single mirror. Both are computed here and required to agree.

Counting walls across the descents #

Gathering the results over every ferroelastic descent the mode census produced gives five descents, thirty-eight pairs of domain states between them, and nine distinct pairs of shapes. The tally is short: two permitted wall directions for every pair whose two shapes differ, which is thirty of the thirty-eight pairs of states, and none at all for the remaining eight, which are pairs of states carrying the same shape.

The rational ones are worth separating, because a wall along a lattice row can be atomically flat and a wall along an irrational direction cannot. Of the descents found here, the square-lattice cases give walls along the diagonals — rational, and flat. The hexagonal cases give walls along directions whose ratio involves the metric, and among them some come out rational in the hexagonal basis while others do not.

There is no general principle in the sample that predicts which. It depends on the discriminant of a particular quadratic with particular entries, and the discriminant being a perfect square is an arithmetic accident rather than a symmetry fact — which is itself worth knowing, because it means the question of whether a domain wall can be coherent is not answerable from the symmetry alone.

What the condition does not decide #

Not the position. A wall may lie anywhere along its permitted direction, and where it sits is decided by how the domains grew.

Not the energy. Two directions are permitted and a real crystal usually shows both; which is more common in a specimen depends on the energy of the wall, which depends on how the atoms actually arrange themselves in it, which is a question this argument cannot reach.

Not the thickness. The condition treats the wall as a line. A real wall has a width over which the order parameter turns from one domain’s value to the other’s, and the width is set by a competition between the energy cost of the gradient and the cost of being between the two states.

Not whether the boundary is a plane of atoms at all, which for an irrational direction it cannot be: the wall is then stepped, and the steps are dislocations in the ordinary sense.

And not whether the wall moves. Ferroelastic domains are usually mobile under stress, which is the property they are named for and used for. That mobility is entirely outside this computation.

When there is no undistorted direction at all #

The two-wall theorem rests on the two strains differing by no area, which holds because the domains are related by a symmetry. Where that hypothesis fails there may be no permitted wall, and the case is common enough to have a whole theory attached.

Two different phases are not related by a symmetry. Their cells differ in shape and in volume, so the difference of their strains has a non-zero trace, its two eigenvalues can have the same sign, and the quadratic then has no real roots: every direction is stretched, and no unstrained interface exists.

A crystal in that situation cannot simply put a wall somewhere, and what it does instead is the content of the crystallographic theory of martensite. The transformation strain is combined with a second deformation that leaves the lattice unchanged — a slip, or a fine-scale internal twinning — chosen so that the average of the two has an undistorted plane even though neither does.

That is a remarkable arrangement and it is forced. The internal twinning has no purpose of its own: its spacing and its proportion are set entirely by the requirement that the average deformation leave some plane unstretched, and a martensite plate is therefore full of fine twins whose existence is a consequence of a geometric condition rather than of anything about the two phases’ energies.

So the essay’s condition is not merely a constraint on where a wall may go. Where it cannot be satisfied, it dictates the interior structure of the product.

The plane that comes out irrational #

There is a consequence of that construction which is the best-known observation in the subject, and it explains the essay’s remark about rational and irrational directions.

The undistorted plane produced by the averaging is called the habit plane of the martensite plate, and it is the solution of the same quadratic condition with the averaged strain in it. The averaged strain depends continuously on the twin proportion, so the habit plane’s direction does too — and there is no reason whatever for it to come out along a lattice row.

It does not. Martensite habit planes in steel are famously irrational, quoted as approximately (2 5 9) or (3 10 15) and varying with composition, and the irrationality was a puzzle for decades before the averaging argument explained it. A plane whose indices vary continuously with the carbon content is not a lattice plane, and no crystallographic argument that produced only rational answers could ever have accounted for one.

That is the sharpest form of this essay’s own division. The condition is exact and the directions it returns need not be rational, because a quadratic with integer coefficients has irrational roots whenever its discriminant is not a square — and where the coefficients are themselves measured rather than integer, rationality is not even a possibility to be checked.

What is checked #

Three things, and the third is the one that keeps the others honest.

The wall directions satisfy the condition. Each root is substituted back into the quadratic and required to give zero, on every pair of every ferroelastic descent found.

The area change of the difference is zero, computed with the averaged metric rather than with the plain trace, on every pair.

And the count can be zero. A test that always returns two would be a test that had stopped asking, so the same routine is handed a pair of strains whose difference is definite — which no pair of domains can be — and it reports no permissible wall. That refusal is what makes the two-wall result a result rather than a property of the code.

What the pictures cannot show #

The wall figures draw the strain difference as a distortion of a cell at a size chosen to be visible — a fifth of a cell edge, against a real spontaneous strain of a fraction of a per cent. At the true size the two domains would be indistinguishable on the page, and their permitted wall directions would be exactly the same, since the condition is homogeneous in the strain and does not care about its magnitude.

The other thing not drawn is the wall itself. What is computed is the set of directions a wall may take; the figures draw those directions as lines through a circle of sampled directions, which is a picture of the condition rather than of a boundary. A drawing of an actual wall would need the atoms in it, and the atoms in a wall are exactly what this argument does not determine.