The plate that only fits when it is twinned

The plane a deformation leaves alone ends at an impasse. Two deformations can share an interface only if they differ by a rank-one term, which happens exactly when a certain symmetric matrix has one positive eigenvalue, one negative and one exactly zero. Every pair of variants of a cubic-to-tetragonal transition satisfies that. No single variant satisfies it against the undeformed parent, because its middle principal stretch is η₁ and η₁ is not one.

So a crystal that transforms cannot have an interface between the two phases. It does anyway. The way out is in the equation and it has been there all along.

It is worth registering how unusual the resolution is before it arrives. Most impasses in this collection are removed by finding a bigger group, a finer classification or a better invariant. This one is removed by noticing that a quantity in the equation was never required to be what it looked like: F is the deformation of a region, and a region can be inhomogeneous inside. Nothing about the compatibility condition changes; what changes is the set of things allowed to satisfy it, and that set turns out to be a one-parameter family rather than a finite list.

An average is a deformation too #

The compatibility condition is about the deformation on each side of the interface. Nothing in it says the deformation has to be a single variant.

Stack two variants in alternating layers, thin enough that whatever is looking at them — the interface, and the compatibility condition — sees only their average. If the two variants are U₁ and U₂, and the twinning equation between them is Q U₁ − U₂ = a ⊗ n, then a laminate with volume fraction λ of the first has average deformation

F(λ) = U₂ + λ a ⊗ n.

At λ = 0 this is U₂ and at λ = 1 it is Q U₁, so the two ends are the two variants and everything in between is something new. It is a one-parameter family of deformations, and the compatibility condition against the parent is one scalar equation, so there is every reason to expect solutions.

The curve is the whole argument in one picture. It is the compatibility condition, plotted as a function of the only free thing in the construction, and it dips below zero because the two variants stretch different axes: mixing them lets the average contract in a direction neither of them contracts in alone.

Two things about that family are worth being precise about, because the construction is easy to state loosely and get wrong.

The first is that F(λ) is not a deformation any point of the crystal experiences. Every point is in one lamella or the other and experiences U₂ or Q U₁; the average is what a region containing many lamellae undergoes, and it is a deformation of that region rather than of any material within it. The compatibility condition is applied to the average because the interface is flat over many lamellae — which is exactly the assumption that has to hold and is stated again below.

The second is that the rank-one form of the twin is what makes the average simple. Adding λ a ⊗ n to U₂ is adding a shear on a fixed plane, so the family is a straight line in the space of deformations rather than an arbitrary path. That is why the condition along it is one scalar equation with a small number of roots, and why the roots can be found by watching a sign change rather than by searching a surface.

There is also a question of why the curve dips rather than rising, and it is answerable without any algebra. Variant one stretches the x axis by η₃ and the other two by η₁; variant two stretches y by η₃ and the others by η₁. Their average, in the crude sense of averaging the two matrices, stretches both x and y by something between η₁ and η₃ and leaves z at η₁. So the average has two directions in which the stretch has moved towards one and one in which it has not, and the middle eigenvalue is the one that moves. The laminate average is not quite that crude average — it is the crude average plus a shear, which is what the rank-one term supplies — but the direction of the effect is the same and it is why mixing helps.

Where the fractions are #

The roots are not arbitrary numbers and they are not fitted to anything.

A prediction of a volume fraction from two lattice parameters is an unusual kind of statement and it is worth being clear about what it is. It does not say that a particular sample will have that fraction; it says that a sample which has a coherent interface with the parent phase must have it, because no other fraction satisfies the equation. Whether a sample has a coherent interface is a separate question about how it was cooled. The theory constrains rather than predicts, which is the same relation what symmetry decides about a material has to a measurement.

The two roots being complementary is not a coincidence and it is not quite a symmetry either. Exchanging the two variants sends λ to 1 − λ, and it also exchanges the two solutions of the twinning equation, so the pair of roots maps to itself. What that means physically is that a plate and its mirror image are both available and neither is preferred by the geometry — the choice between them, in a real transformation, is made by whatever asymmetry the sample has.

The fineness is the one idealisation in the construction and it should be named rather than buried. The average deformation is the deformation of the laminate only in the limit of infinitely thin layers; a laminate of finite thickness meets the parent across a surface that is planar on average and stepped in detail, and the steps carry an energy the theory does not compute. That is why real plates have twins that are thin and why they get thinner near the interface, and it is the point at which arithmetic stops and materials science begins.

There is a check available on the whole construction which costs nothing and is worth running. Each root is handed back to the general rank-one solver, which produces the plane and then verifies it by multiplying the decomposition out. So the volume fractions are found by one method — watching a scalar cross zero — and the planes are produced by another, and the second would fail loudly if the first had found a spurious root. A root found by bisection on a quantity that never actually reaches zero is the classic failure of this kind of computation, and it is caught here rather than assumed away.

The four orientations per pair are worth counting carefully, because the number recurs. Each root of the scalar equation gives an average deformation; each average deformation has two rank-one connections to the parent, as every compatible pair does; and there are two roots. So four. With three pairs of variants that is twelve plate orientations for the transition, before any of them is asked whether it is the one a given crystal forms. Two hundred and forty-seven descents counts the symmetry descents this collection knows about; the plate count is the same kind of arithmetic done one level further out, on the interfaces rather than on the states.

The planes are irrational #

The habit planes that come out are not planes anybody would have guessed.

This is the result that historically made the theory believable, and it is worth saying why. Habit planes had been measured for decades and reported as things like {3 10 15} — indices nobody would write down for any other purpose, differing between alloys, and drifting with composition. Every attempt to explain them as crystallographic planes failed, because they are not crystallographic planes. They are the solutions of a quadratic whose coefficients are lattice parameters, and lattice parameters are real numbers.

The law of rational indices is the reason that is surprising. Crystal faces have small integer indices, and the argument for it is that a face is a lattice plane and a lattice plane has integer indices by construction. A habit plane is not a lattice plane of either phase — it is a plane across which two different lattices are matched — and nothing forces it to be rational. The two facts sit next to each other and neither weakens the other.

Two further remarks about irrationality, since it is the part most likely to be misread.

The first is that “irrational” here is a statement about the numbers and not a claim of transcendence. The habit plane normal is an algebraic function of the two stretches, so for algebraic lattice parameters it is algebraic; what it is not is a ratio of small integers, and that is all the observation requires. The distinction matters because a plane with indices {3 10 15} is rational, and the honest statement is that the computed plane is close to that and not equal to it, with the difference far larger than the measurement’s precision.

The second is what makes it possible at all. Everything in the law of rational indices applies to planes of one lattice. The habit plane is defined by a relation between two lattices with no common sublattice, and a relation between two lattices is governed by real numbers rather than by integers. The same thing happens at an interface between two different crystals, where the coincidence conditions are approximate for the same reason.

The one inequality #

The construction can fail, and where it fails is a clean statement about the two numbers.

That identity was found by sweeping and is then asserted as an identity, which is the right order: the sweep is what suggested it and the assertion is what makes it a claim the machinery would fail on. It is also a much sharper condition than the obvious one. Straddling — one stretch above one and the other below — is necessary and it is not sufficient: η₁ = 1.01 with η₃ = 0.99 straddles, and 1.01² + 0.99² = 2.0002, so it fails by two parts in ten thousand and no plate exists.

The geometric content of η₁² + η₃² ≤ 2 is that the transition has to change the volume of a face by at least as much as it changes its shape. A transition that stretches one axis and contracts another by amounts that nearly cancel leaves every direction almost unchanged, and a laminate of two such variants has nowhere to go: averaging two nearly identical deformations gives another nearly identical one, and none of them contracts enough.

It is worth putting the inequality beside a real number. For the transition drawn throughout, η₁ = 1.0619 and η₃ = 0.9178, so the sum of squares is 1.970 and the condition is met with room to spare. A transition with a smaller tetragonal distortion moves the sum towards two from below and the two roots towards each other; at the boundary they merge into one, at a volume fraction of exactly one half, and beyond it there are none. So a material can be tuned out of having a habit plane by composition alone, without any change in symmetry — which is a strong statement, since symmetry is what usually decides whether a thing exists.

One further reading of the sweep is worth having because it says what kind of object the answer is. The quantity being plotted is a continuous function of λ with a single minimum, so the number of roots is zero, one or two and never anything else. A transition therefore has either no plate, one degenerate plate at exactly half and half, or two plates related by exchanging the variants. There is no transition with three volume fractions and none with a range of them. That is a strong structural statement and it comes from the shape of one curve rather than from any symmetry argument.

Six variants, and what changes #

Three variants make three pairs and every pair twins. Six variants make fifteen pairs and every pair twins as well, for the same reason — each pair is related by an operation of the parent’s symmetry, and that relation forces the middle eigenvalue.

What changes with the number of variants is not whether the pairs twin but how many distinct plates the material can form. Each pair that twins gives up to two volume fractions and each fraction gives two habit planes, so the count of possible plate orientations grows with the number of pairs — and it is that count, rather than any single plate, that decides whether a polycrystal can accommodate an arbitrary imposed shape change. How many domains a transition makes counts the variants; the count of plates is that number squared, roughly, and it is why some transitions give shape-memory behaviour and others give a brittle mess.

There is a second thing the shears decide, and it connects back to the fineness assumption. A wall carrying a shear of a tenth stores energy proportional to its area, so a laminate with more walls costs more; a laminate with fewer walls has thicker lamellae, and thick lamellae make the interface with the parent stepped rather than planar, which costs energy too. The equilibrium spacing is where those two costs balance, and it scales as the square root of the plate thickness. That is the one quantitative prediction in this whole area that needs an energy, and it is the reason the twin spacing is the observable that varies from sample to sample while the volume fraction does not.

One more consequence of the inequality deserves stating, because it is the practical one. η₁² + η₃² ≤ 2 is a condition on two numbers that a chemist can move. Alloying changes lattice parameters continuously, so a series of compositions sweeps the sum of squares through two, and the theory says that a plate exists on one side of that composition and not on the other — with the volume fractions merging at exactly one half as the boundary is approached. That is a prediction about a composition series made from crystallography alone, and it is testable without measuring an energy.

A last comparison, since the collection has the neighbouring cases. The walls a strain permits finds the planes between two variants and finds them by a linear condition; this finds the plane between a mixture of variants and the undeformed parent, and needs the exact one. The two answers do not contradict each other and they are not the same calculation: the first is about a wall inside the product phase, the second about the boundary of the product phase. A crystal in the middle of transforming has both at once, and the picture with both drawn — fine twins inside a plate, the plate meeting the parent on an irrational plane — is what a micrograph of martensite shows.

What this settles #

The chain is short and each link is an equation rather than a model. Two lattice parameters give three variants. Every pair of variants twins, by an argument from the parent’s symmetry. The laminate of a twinned pair has an average deformation with one free parameter. That parameter passes through the compatibility condition exactly when η₁² + η₃² ≤ 2, and where it does it gives two volume fractions and four habit planes, all irrational, all determined.

Nothing in that chain is an approximation except the assumption that the twins are fine compared with the plate. No energy is minimised, no modulus appears, and no experimental number is used other than the two lattice parameters. What comes out is the shape of a microstructure, which is not the sort of thing a purely geometrical argument usually produces.

The honest limits are two. The construction says which microstructures are compatible, not which one a given sample forms — that needs an energy, and the energy is where the material’s identity finally enters. And it treats each phase as homogeneously deformed at the scale of the interface, so it says nothing about the core of the interface itself, which is the same boundary the coincidence site lattice draws when it describes a grain boundary by the lattices on either side rather than by the atoms in it.