The wall has a group of its own

The walls a strain permits settles where a boundary between two ferroelastic domains may lie: neither domain may be stretched along the boundary, that condition is a quadratic in the direction, and a pair of domains whose shapes differ therefore has exactly two permissible walls — never one and never none, because their strains differ by no area at all. The qualification is doing work and a later section returns to it: two domain states can carry the same shape, and then the quadratic vanishes identically and there is no ferroelastic wall between them.

A wall is an object in its own right, and an object has a symmetry group. This one is not a plane group, and the reason is geometric rather than subtle: a wall is periodic along its length and bounded across it.

Periodic along, bounded across #

Seven friezes is one of this collection’s earliest essays and it presents the classification as a warm-up: the same argument as the seventeen, on a strip instead of a plane, short enough to check by hand. Hop, step, sidle, spinning hop, and three more.

A domain wall is exactly that object. Along the wall the crystal repeats, because the lattice does; across it there is nothing to repeat — one domain on one side, another on the other, and no third thing further out. So the symmetry group of the whole arrangement, restricted to what fixes the wall, is a group of motions of a strip, and there are seven of those.

That is the essay in one sentence, and the rest is which of the seven, and why the counting is not what it first looks like.

Two kinds of operation belong to a wall #

An operation belongs to the wall’s group when it carries the whole arrangement onto itself — the wall and both domains. Two kinds qualify.

Operations of the child group that fix the wall line. These leave each domain exactly where it is, since the child is what each domain retains, and they are the obvious half.

Operations of the parent, outside the child, that fix the wall line and exchange the two domains. These are the interesting half. A wall is unchanged by having its two sides swapped: the boundary is in the same place, the arrangement looks the same, and the operation is a symmetry of it even though it is a symmetry of neither domain.

So a wall is usually more symmetric than either domain, which is not the direction one expects a defect to go. The reason is the same accounting a twin law is: an operation the individual lacks, made available by an interface between two individuals. A twin law is a coset representative; so is a domain-exchanging wall operation, and in a ferroelastic crystal the two are the same object seen twice.

What comes out #

Running the computation over every ferroelastic descent gives five of the seven types, and which five is informative.

p2mm appears wherever the parent has a mirror in a permitted wall direction and a two-fold across it — the descents from 4mm and 6mm, where the walls sit along the mirror lines the parent already has.

p2 appears where the parent has rotations and no mirrors: the descents from 4 and 6. The wall’s group has the half-turn and nothing else, which is a wall with no reflection symmetry at all and a definite handedness across it.

p1m1 and p11m appear as a pair, and the pairing is the neatest thing in the census. In a descent from 3m the two permitted walls are of different types: one has a mirror across the wall and the other a mirror in it. A table with one entry per descent would have hidden that, which is why the census keeps both.

p1 appears where nothing at all is fixed — the descent from 3, whose parent has three-fold rotations that move every line.

The half-turn is not evidence #

One entry in the census needed a control before it could be believed, and stating it is the honest version of the whole computation.

A half-turn maps every line through the origin to itself. So in any class containing the half-turn — which in the plane is 2, 2mm, 4, 4mm, 6 and 6mm — a wall at a completely general angle already has p2, and reporting that as a symmetry of the wall would be reporting a fact about lines rather than about crystals.

The check is a wall at a general angle in a class without a half-turn: class m gives p1, the trivial frieze, which is what a wall in no special direction should give. And the same general angle in 4mm gives p2, for the reason above. Both are computed, and the second is the one that says the first was a test.

Where the wall’s own periodicity gives out #

There is a real limit on the classification and it comes from the wall directions themselves.

The permitted directions are the roots of a quadratic, and a root need not be a rational direction in the lattice. A wall along an irrational direction is not periodic at all — no translation of the crystal carries it onto itself — so it has no frieze group in the strict sense. It has a point symmetry, which is what is computed here, and the frieze label is a statement about the periodic case.

That is not a technicality: it is the same phenomenon epitaxy is a measurement is about, where two lattices with an irrational ratio never coincide and the question becomes how nearly. A wall in an irrational direction is a wall that is nearly periodic over some distance and then not, and real crystals resolve it by facetting — the wall breaks into segments along nearby rational directions, each of which does have a frieze group.

The census reports whether the discriminant is a square, which is the test for a rational direction, and it is worth reading beside the frieze column.

What a wall’s group is worth #

A group attached to an interface is not decoration; three consequences follow directly.

What may sit in the wall. A wall’s group constrains the properties of the material inside it in exactly the way a crystal class constrains the properties of a crystal — Neumann’s principle applied to a strip rather than to a plane. A wall whose group lacks a centre may be polar even when both domains are not, which is the mechanism behind the ferroelectricity that has been found in the walls of some non-ferroelectric crystals.

How walls meet. Where three walls meet at a line, the groups have to be compatible, and the compatibility is a group-theoretic condition that rules out most junctions. That is why domain patterns in a ferroelastic crystal look the way they do: not an arbitrary tangle, but a small number of permitted vertex types.

And what moves them. A wall’s group says which distortions couple to it, so it says which fields move a wall and which do not. A wall whose group contains an operation reversing a field direction cannot be driven by that field, and the argument is a selection rule of exactly the kind a character sum settles.

The wall’s group as a coset computation #

The two kinds of operation in a wall’s group are best written as a single object, and doing so makes the counting mechanical.

Let G be the parent class and H the child, so a domain is a coset gH and there are [G : H] of them — which is how many domains a transition makes. Fix two domains, H and gH, and a wall direction d. The wall’s group is

{ x in G : x fixes the line d, and x either fixes both domains or exchanges them }

and the two conditions can be checked independently. Fixing the line is a condition on the linear part alone; fixing or exchanging the domains is membership in H or in the coset gH, which is a lookup.

So a wall’s group is the intersection of two subgroups of G — the stabiliser of the line, and the union H ∪ gH where that union happens to be a group — and the arithmetic is a pair of set operations rather than a search. What makes the answer interesting is that neither factor is a wall’s group on its own: the stabiliser of the line ignores the domains, and H ∪ gH ignores the geometry.

The census reports both halves separately, which is why it prints how many operations fix each domain and how many swap them. A wall whose group is entirely of the first kind is a wall between two domains that happen to abut; one with operations of the second kind is a genuine twin boundary, related by an operation the crystal has lost.

Why the two walls of a pair can differ #

The 3m descents give a wall of type p1m1 and a wall of type p11m from the same pair of domains, and the asymmetry is worth explaining because it looks at first like an error.

The two permitted directions come from one quadratic and they are not equivalent to one another: they are two different lines, and the parent’s operations treat them differently. In class 3m the mirror lines run in three directions and the roots of the strain quadratic are one mirror line and one line perpendicular to a mirror. A wall along a mirror line has the mirror in it; a wall perpendicular to one has the mirror across it. The first is p11m and the second is p1m1, and the two are genuinely different friezes — the strip’s two positions are never interchangeable, because no rotation of a strip carries along onto across.

That is a real physical difference and not a labelling one. A wall with a mirror in it looks the same from both ends; a wall with a mirror across it looks the same from both sides. Which of the two a boundary has decides whether it can carry a polarisation along its length, and a crystal offering both kinds has two populations of wall with different properties — which is exactly what is observed in the ferroelastics where walls have been examined individually.

How many walls a descent has, which is not two per pair #

It is tempting to count the walls of a descent in one line. The domain states are the cosets of the child in the parent, so a descent of index four has four of them and six pairs; every pair has two permitted walls; six sixes are twelve. That count is wrong, and the way it is wrong is instructive.

The wall condition is a statement about strains, and two domain states need not have different strains. A descent from 4mm to m has four states and only two distinct shapes: the states come in pairs carrying the same spontaneous strain, related by an operation of the parent that leaves the strain alone while moving something else. For such a pair the difference of the two strains is not a quadratic form with two roots — it is zero. Every direction satisfies the condition equally, none is picked out, and there is no ferroelastic wall between those two domains at all.

So four of the six pairs have two walls each and two of them have none, and the descent has eight walls. The two excluded pairs are not a defect in the counting; they are a real physical case, and it is the case antiphase domains is entirely about — two regions that differ by something no measurement of shape can reach. Their boundary is still a domain wall and still has a symmetry group, but it is not classified by this argument, because this argument only ever sees the strain.

The same subtraction runs through the whole census. A 6mm to m descent has six states, three shapes and fifteen pairs, of which three share a shape: twenty-four walls rather than thirty. What the strain condition delivers is not a wall for every pair but a wall for every pair the strain can tell apart.

Where the exactness stops #

Computed here: for each ferroelastic descent produced by the mode census, the two permitted wall directions from the strain condition; the operations of the parent class fixing each direction; which of those lie in the child and which exchange the domains; and the frieze type read off the presence of a half-turn, a mirror across the wall and a mirror in it.

Point operations only. A glide is not distinguished from a mirror here, because only linear parts are used — so the types reported are the ones a point-symmetry argument reaches, and separating p11g from p11m needs the translation parts and a decision about where the wall sits inside the cell. That is a real limitation and the census says so rather than reporting a type it cannot see.

And the plane again. Real ferroelastic walls are planes in three dimensions and their symmetry groups are the eighty layer groups rather than the seven friezes — the object a layer is not a wallpaper is about. Everything above has the same structure with a different list at the end.

What the wall is made of #

Nothing above says anything about the structure inside a wall, and it is worth marking the boundary because the symmetry argument is often read as saying more than it does.

A wall is not a mathematical line. It is a region, a few unit cells thick in a good case, across which the structure changes continuously from one domain’s arrangement to the other’s. The symmetry computed here is the symmetry of that region taken as a whole, and it constrains the structure inside without determining it: the wall’s group says which arrangements are possible, and which one occurs is decided by energy.

Two consequences of that gap are worth stating.

The centre of the wall may have a symmetry the wall does not. Halfway across, the structure is neither domain, and it may sit at a configuration with extra symmetry — often the parent’s, restored locally, which is why a wall is sometimes described as a thin slab of the high-temperature phase. That is a plausible picture rather than a theorem, and whether it happens is a question about the free energy.

And the wall’s thickness is not a symmetry quantity at all. It is set by the competition between the energy cost of the gradient and the cost of being between the two minima, and it comes out of a Landau expansion rather than a group. An order parameter is a representation is where that expansion is set up; the thickness is the one number in this subject that symmetry has nothing whatever to say about.

Who found it, and when #

The wall orientations are Sapriel’s, from 1975, and the group-theoretic description of the walls themselves followed within a few years — Janovec’s work of the late 1970s set out the classification of domain pairs and their walls in terms of cosets, which is the accounting used above.

The physics that made it matter is much more recent. Domain walls were treated as defects to be eliminated for most of a century; the discovery in the 2000s that a wall can have properties the bulk does not — conducting walls in insulating crystals, polar walls in non-polar ones — turned the wall into a functional object, and its symmetry group into the thing that says what it may do.

A wall is a subperiodic object, and so is everything at a surface #

The move this essay makes — a two-dimensional argument giving a one-dimensional group — is one this collection has made twice before, and the three cases together are a small family worth noticing.

A layer is not a wallpaper takes a sheet that repeats in two directions and lives in three, and finds eighty layer groups rather than seventeen plane ones. Seventy-five ways to be a thread does the same for an object periodic in one direction and embedded in three. Both are subperiodic groups: periodic in fewer directions than the space they sit in.

A domain wall in the plane is the two-dimensional member of that family — periodic in one direction, embedded in two — and the answer is the seven friezes, which is why the frieze essay written as a warm-up turns out to be load-bearing here.

The general statement is worth having: whenever an object is periodic in k directions inside n, its symmetry group is one of a finite list that depends on the pair, and the lists get long quickly. Two in two is seventeen; one in two is seven; two in three is eighty; one in three is seventy-five. Every interface, every surface and every edge in this subject has a group from one of those lists, and which list is decided by counting directions rather than by anything about the material.

Where the ladder goes next #

Back, to where a wall may lie: the walls a strain permits, and the reason there are always exactly two.

Sideways, to the classification this rung uses: seven friezes, and why sixteen become seven for the closure argument that produces the list.

And to the count of the domains themselves: how many domains a transition makes is an index, which is the number of walls’ worth of pairs there are to classify.

The wall’s group wants a colour #

The census above reports one of the seven types for each wall, and in doing so it throws away a distinction the coset computation had already made. Recovering it costs nothing and sharpens every entry.

The two kinds of operation are not interchangeable. Some leave each domain where it is; the others swap them. An operation of the second kind still carries the arrangement onto itself, so it belongs in the group — but a physical quantity that distinguishes the domains, the spontaneous strain most obviously, is reversed by it and preserved by the first kind.

That is exactly an antisymmetry group: attach a colour to each domain, and the wall’s symmetry is a group of operations each of which either keeps the colours or exchanges them. The colour-preserving operations form a subgroup of index two whenever any exchanging operation exists at all, since composing two exchanges preserves.

So a coloured wall type is a frieze group together with a choice of index-two subgroup, and the plain frieze type in the census is what remains after the colour is forgotten. Two walls can therefore share a row here and behave differently under a field: the one whose mirror exchanges the domains is reversed by that mirror, and the one whose mirror preserves them is not.

The distinction is Shubnikov’s, and it is the same device magnetic groups use for time reversal — one operation with two possible meanings, decided by what is being carried along.