The strain that arrives with the transition

A crystal cooled through a transition often changes shape. A cube becomes slightly tetragonal; a hexagonal cell becomes slightly oblique. Nobody squeezed it: the change of shape arrives with the change of symmetry, and its size is whatever the material’s energetics settle on.

Whether a shape change is possible is not a question about energetics at all. It is a subtraction between two counts, each of which is the number of independent strain components a class permits — and if the child permits more than the parent, a component that the parent held at zero is free to become non-zero, so the crystal distorts.

Strain as a thing the group acts on #

A strain in the plane is a symmetric two-by-two array: two extensions and a shear, three numbers. The group acts on it — an operation carries a distortion of the crystal to another distortion — and the action is e → MᵀeM with M the operation’s integer matrix, which is the transformation a change of basis makes to a metric.

That the strain is a change of the metric is the right way to hold it here, because it keeps the arithmetic exact and it explains the transformation law. This collection uses the same rule on a Gram matrix when reducing a lattice, and it is the same rule for the same reason: both objects measure lengths, and both transform by sandwiching between a matrix and its transpose.

So the space of strains is a three-dimensional space carrying a representation of the class, and the components a class permits are the ones it leaves alone — the invariants.

Counting them, twice #

The count is made by averaging, exactly as the invariant counting is: push each of the three basis strains through every operation, add the results, and take the rank of what comes back. That gives a basis for the permitted strains and not merely a number.

It is checked against a character sum. The character of the group acting on symmetric tensors is (tr M)² + tr(M²) over two, and its average over the group is the multiplicity of the trivial representation — the same count, by a route with no polynomials in it. The two agree on all ten classes.

The answers are worth having in front of a reader, because the pattern in them decides everything below:

• classes 1 and 2 permit three strains — every symmetric array is invariant, since the half-turn acts trivially on strains;
• m and 2mm permit two — the two extensions, with the shear forbidden;
• 4, 4mm, 3, 3m, 6 and 6mm permit one — a uniform dilation, and nothing else.

A rotation of order three or more forces a crystal to be isotropic in shape. That is the same fact the property table records at rank two, and it is why a hexagonal crystal has a cell shape that symmetry fixes completely while a rectangular one does not.

Both routes to the count are worth drawing together, because agreement between two computations that share nothing is a stronger statement than either of them alone. The averaging route works with polynomials and produces a basis; the character route works with traces and produces only a number; and the number is the same on every class. A disagreement anywhere would mean one of the two had a bug in it that no other check reaches, since neither is verified against a table.

Aizu’s criterion, as a subtraction #

A transition is ferroelastic when the child permits a strain the parent does not. That is Aizu’s criterion, and computed this way it is a difference of two integers.

4mm → 2mm goes from one permitted strain to two, so it is ferroelastic: the fourfold crystal was isotropic in shape and the twofold one is not, so the cell becomes rectangular. 6mm → 3m goes from one to one and is not: both are shape-isotropic, so the transition leaves the cell alone. 2mm → 2 goes from two to three, and the new component is the shear.

The spontaneous strain is the piece the child has and the parent has not. It is produced here rather than named: the child’s basis is reduced against the parent’s, and what is left over is the distortion. For 4mm → 2mm that is the difference of the two axial strains — the cell getting longer one way and shorter the other — with the uniform part removed, since the uniform part was permitted all along.

The strain that is not the order parameter #

A distinction has to be made here, because it decides how a transition behaves and it is easy to skate over.

Sometimes the strain is the order parameter: the transition happens because the crystal wants a different shape, the strain grows from zero, and the symmetry drops because the shape did. Such a transition is called proper ferroelastic.

More often the strain is a secondary quantity. Something else — an ordering, a tilt, a displacement — is the order parameter, and the strain comes along because the energy contains a term coupling the two. Then the strain is proportional to the square of the order parameter rather than to the parameter itself, so it grows more slowly near the transition, and its size is a fact about the coupling.

Symmetry decides which case applies, by asking whether the strain transforms as the same representation as the order parameter. If it does, the coupling is linear and the strain is the parameter’s twin; if not, the lowest coupling is quadratic. That is a character computation of the kind this ladder makes throughout, and it is the same computation that decides whether a polarisation appears in a transition that was about something else.

How many shapes, and how many domains #

The count of domains is the index of the child in the parent. The count of distinct shapes can be smaller, and where the two differ is worth naming.

The parent’s operations move the spontaneous strain about, and the number of distinct images is the number of orientation states that differ in shape. For 4mm → 2mm the index is two and the strain has two images, so the two domains differ in shape and a microscope sensitive to shape can tell them apart.

For a descent where the lost operations include translations — the zone-boundary case — some domains are related by a translation, which does not move a strain at all. Those domains have identical shapes and differ in something else: they are antiphase domains, and no measurement of shape can see the boundary between them.

So the shape count divides the domain count, and the ratio is exactly the number of domains a shape-sensitive measurement merges. That is a useful thing to know before choosing an instrument.

A worked descent, in numbers #

4mm → 2mm is the case to follow, because every quantity in it is small.

The parent permits one strain: the uniform dilation, whose array is the identity. The child permits two: the identity and the difference of the two axial extensions, whose array has entries +1 and −1 on the diagonal. So the spontaneous strain is that difference, and the crystal’s cell becomes longer along one axis and shorter along the other, keeping its area.

The parent’s operations move it. The fourfold rotation carries the array with +1, −1 to the array with −1, +1 — the same distortion turned through ninety degrees, which is the other domain. The mirrors leave both alone. So the orbit has two members, matching the index of two, and the crystal has two domains that differ in shape.

Their difference is the array with entries +2 and −2, whose metric trace on a square basis is zero. The directions in which that difference does nothing are the two diagonals, and those are where a wall between the domains can run — which is exactly what a micrograph of a ferroelastic crystal of this species shows.

The same descent in three dimensions is the standard ferroelastic example: 4/mmm → mmm, a tetragonal holohedry losing its fourfold axis to an orthorhombic one, index two, two domains named by the two cosets. Every quantity above survives the addition of a third axis unchanged in kind — the strain gains three more components, the counts run from one to six instead of one to three, and the subtraction that decides whether the transition is ferroelastic is the same subtraction. The plane version computed here is that arithmetic with one axis removed, which is why it can be checked completely.

Why the count is one for a threefold axis #

The most useful entry in the strain table is the one that says a class with a rotation of order three or more permits exactly one strain, and it is worth seeing why rather than reading it off.

A strain is a quadratic form. A quadratic form invariant under a rotation of order n is invariant under a group whose invariants at degree two are one-dimensional whenever n ≥ 3 — which is the Molien count at degree two, and the same statement as the averaging argument that a finite group of matrices preserves a metric.

So a crystal with a threefold, fourfold or sixfold axis has a cell whose shape is fixed by its symmetry: a hexagonal cell is hexagonal, and no distortion consistent with the symmetry can make it otherwise. The only strain it permits is the one that scales it.

That is why every ferroelastic descent in the table starts from such a class and ends below it, and why the shape change is such a reliable signature: it can only appear when a rotation of order three or more is lost, and that loss is exactly what a shape measurement is sensitive to.

The area does not change, and that is a theorem #

One property of the spontaneous strain holds in every case and is not obvious: the difference between two domains’ strains changes no area.

The reason is that the two domains are images of one another under an operation of the parent, and the parent leaves the area invariant — the determinant of every one of its matrices is ±1. So both domains have the same area change relative to the parent, and their difference has none.

Stating that correctly needs care about basis, and the care is the interesting part. The trace of the strain array is not the area change when the lattice basis is not orthonormal: on a hexagonal basis the two domain strains of a 6mm → m descent differ by an array whose diagonal sums to −2, and nothing is wrong. The invariant is tr(G⁻¹e) with G the metric, and the metric is obtained by averaging — the same trick that gives a finite group an invariant metric in the first place.

Computed that way the difference is exactly zero, in every pair of domains of every ferroelastic descent found here. It is the fact the next essay turns into a statement about walls.

What the shape change does not tell #

A shape change is the most visible consequence of a transition and the least informative about its cause.

It does not name the order parameter. Several different order parameters can produce the same spontaneous strain, since the strain is a secondary quantity coupled to whatever is happening. A measurement of the cell’s distortion cannot distinguish them.

It does not give the amplitude of anything but itself. The relation between the strain and the order parameter is through a coupling constant, so a strain of a given size implies an order parameter of a size that depends on a number symmetry does not fix.

And it can be zero for a transition that certainly happened. Every non-ferroelastic descent in the table above changes the symmetry with no change of shape at all, and a crystal that has undergone one looks, to a measurement of its cell, exactly as it did before. Those transitions are found by other means — a superlattice reflection, a mode softening, a property appearing — and the shape stays silent.

The strain a diffraction experiment reports #

A shape change is measured, in practice, as a change in the positions of Bragg reflections rather than by looking at a crystal. It is worth connecting the two, because the reciprocal-space version has a feature the real-space one does not.

A cell that lengthens along one axis and shortens along the other moves every reflection, and moves them by amounts proportional to their indices — so the effect is largest far from the origin, which is where a high-resolution measurement looks. That is the reciprocal of the real-space statement and follows from the reciprocal lattice relationship this collection derived early.

The feature is that a split appears. Two reflections that were equivalent under the parent’s symmetry — related by the fourfold axis, say — are no longer related once the axis has gone, and their positions separate. A single peak becomes two, and the separation is proportional to the spontaneous strain.

That is why the shape change is often the first thing seen in a powder pattern, and it is also why domains complicate the reading: a specimen containing both domains of a 4mm → 2mm descent produces both members of every split pair, in equal measure, and looks at first like a single phase with a larger cell.

What is measured and what is asserted #

Everything in this essay is a count or a basis, computed in exact rational arithmetic, and checked in two ways.

The strain counts are computed by averaging and confirmed by a character sum, on all ten classes.

The spontaneous strains are checked to be genuinely spontaneous: a strain the parent already permitted would give an orbit of one under the parent, and every spontaneous strain found here has an orbit of at least two, so it is moved by the operations the transition removed.

The area invariance is checked with the averaged metric on every pair of domains of every ferroelastic descent found by the mode census, and it is exactly zero rather than small.

What is not computed anywhere here is a magnitude. The size of a spontaneous strain is a fact about a material — a fraction of a per cent in most ferroelastics, larger in a few — and no symmetry argument reaches it. What symmetry reaches is which component is free to be non-zero, and that is the whole of the claim.

What the pictures cannot show #

The strain figures draw a distortion at a size chosen so that it can be seen, typically a fifth of a cell edge. Real spontaneous strains are two orders of magnitude smaller, and a figure drawn to scale would be indistinguishable from the undistorted cell.

Nor is any temperature dependence drawn. A spontaneous strain grows below the transition, in a way that depends on whether the coupling is linear or quadratic in the order parameter, and both of those behaviours are outside what is computed here.

And this is a plane argument. In three dimensions the strain has six components rather than three, the classes permit between one and six of them, and Aizu’s criterion produces the list of ferroelastic species — 94 of them — that the ferroics literature tabulates. The plane’s version has the same structure and a shorter list.

Which power of the order parameter the strain follows #

The distinction between a strain that is the order parameter and a strain that is dragged along by one has an arithmetic form, and the arithmetic decides how the shape change grows as the crystal is cooled.

The energy is a polynomial in the order parameter and the strain, and only the terms invariant under the parent’s operations may appear. Which terms those are is settled by the same character arithmetic that counted the permitted strains.

If the strain transforms like the order parameter, the product of the two is an invariant, and a term linear in each is permitted. Minimising then gives a strain proportional to the order parameter itself. The shape change grows as $\eta$, appears immediately below the transition, and is a proper ferroelastic transition.

If the strain transforms differently, no such term exists, and the lowest permitted coupling is between the strain and the square of the order parameter. The strain is then proportional to $\eta^2$ — it grows more slowly at first, and it is a secondary consequence rather than the thing that is happening.

The second case can also change the order of the transition. Eliminating the strain from the energy leaves a term in $\eta^4$ with a negative coefficient, since the crystal always lowers its energy by relaxing. If that negative contribution outweighs the positive quartic term already present, the transition becomes discontinuous — a jump in the order parameter rather than a smooth rise. A shape change that was merely a consequence has then decided the character of the transition it followed.

How each case announces itself #

Both readings are testable, and they are tested by two quite different instruments.

A proper ferroelastic transition softens an elastic constant. The crystal’s resistance to the strain that is about to appear falls to zero as the transition is approached, because the energy’s dependence on that strain is what is vanishing. Measure the speed of sound in the right direction — the elastic constants are what fix it — and the relevant one drops towards zero on approach and recovers below.

An improper one does not. The strain is dragged along by something else, the elastic constant has no reason to vanish, and what is seen instead is a step or a kink at the transition with the constant staying finite on both sides. So the acoustic measurement separates the two cases cleanly, without any structural work.

And the domains are visible in a microscope. A strain changes the crystal’s optical properties along with its shape, so two domains strained in different directions have different refractive indices along a given direction. Between crossed polarisers they extinguish at different angles and the domain pattern appears directly — which is the oldest way ferroelastic domains were studied and still the fastest.

That visibility is exactly the subtraction this essay computes. A transition whose child permits no strain the parent did not has domains that are optically identical and invisible by this means; a transition with a spontaneous strain has domains that differ in the one property a polarising microscope measures. The count in the table is a prediction about what a piece of glass and two filters will show.