A twin is a symmetry the lattice has and the crystal does not

A staurolite crystal from Brittany is a cross. Two prisms of the same mineral, interpenetrating at very nearly ninety degrees, forming a shape no single crystal of anything has.

A gypsum crystal from Sicily is a swallowtail: two blades meeting along a line, mirror images of one another about it. Aragonite grows in what look like hexagonal prisms and are three orthorhombic crystals fitted together at sixty degrees. Feldspar in almost every granite is finely striped, and the stripes are hundreds of alternating slabs of two orientations.

All of these are twins: two or more orientations of one structure, grown together in a definite relationship. The relationship is what makes them a subject rather than a curiosity, because it is never arbitrary and there are never many choices. It is a symmetry operation in the sense this site fixed in its first essay — a map that leaves something invariant — and the interesting question is what it leaves invariant.

The two conditions, and what is left after them #

An operation W relates the second orientation to the first. Two requirements pin down what W can be, and they push in opposite directions.

W must be a symmetry of the lattice. If it were not, the lattice points on one side of the boundary would not continue as lattice points on the other, and the junction would be a break rather than a join. What holds a twin together is that the two halves share their lattice exactly — one of the fourteen, the same one on both sides — so atoms in the boundary can belong to both, and the interface costs almost nothing.

W must not be a symmetry of the crystal. If it were, applying it would produce the same crystal in the same orientation, and there would be no second individual to see. A twin by an operation the crystal already has is not a twin; it is a crystal.

So W lies in the point group of the lattice and outside the point group of the crystal. The first is the holohedry, the ceiling on how symmetric a crystal on that lattice can be; the second is the crystal class. And a set defined as in the big group, not in the small one is a union of cosets. Cosets are the object conjugation and sameness had to introduce to say when two symmetries are the same symmetry; here they are the object that says when two twin laws are the same law.

Why the law is a coset and not an operation #

Take two operations W and W′ differing by a symmetry g of the crystal, so W′ = gW. Apply each to the structure. The second gives the same result as the first, because g moved the crystal onto itself before W did anything.

Two operations in the same coset produce the identical second orientation, so they are one twin law. Counting them separately is the standard way to over-count, and it is easy to do: a mineralogy text listing “twinning on {111}, {1̅11}, {11̅1} and {111̅}” in a cubic crystal is naming four planes and one law, because the crystal’s own three-fold rotations carry each of those planes onto the others.

The number of laws is therefore the number of non-trivial cosets, which is the index of the class in the holohedry minus one. That single sentence is the whole of the next rung, where it gets counted for all thirty-two classes.

What the boundary is, and the three ways it can run #

The composition surface — the interface between the two individuals — is a separate question from the law, and the two are routinely conflated in the older literature.

The law says how the two orientations are related. The composition surface says where they meet. One law can be realised on several surfaces, and the surface that is actually chosen is decided by energy rather than by symmetry.

The shapes that result are classified by how the individuals are arranged:

• A contact twin has the two individuals on either side of a plane, each a complete crystal in its own half-space. Gypsum’s swallowtail is one.
• A penetration twin has them interpenetrating, each apparently passing through the other, with an irregular internal surface. Staurolite’s cross and fluorite’s interpenetrating cubes are of this kind.
• A polysynthetic twin has many parallel lamellae alternating between the two orientations. Plagioclase feldspar does this so reliably that the resulting stripes are how it is identified under a microscope.

None of that is decided by the group. What the group decides is the short list of orientations available, and the crystal, the melt and the cooling history decide which of them appears and in what arrangement. That is permitted against present again, in a fourth place.

Why the boundary costs so little #

The energy argument is worth doing roughly, because it explains why twins are common rather than rare and why they appear during growth, during cooling and under stress alike.

An ordinary grain boundary between two randomly oriented crystallites is a mess. Almost no atom on one side is where the other side would want it; the bonding across the interface is strained in every direction; and the energy per unit area is a substantial fraction of the energy of a free surface.

A twin boundary is not like that. The two individuals share their lattice exactly, so every lattice point in the boundary plane belongs to both, and an atom sitting on one is satisfying both orientations at once. What differs is only the decoration of those points — the arrangement inside each cell — and that arrangement is the same structure seen from a different direction rather than a different structure.

For the best composition planes the mismatch is confined to second neighbours, and the boundary energy comes out an order of magnitude below a general grain boundary. That is why twins nucleate readily, why they survive annealing, and why some materials twin under stress as an alternative to slipping: a twin is a way of accommodating shear that costs a boundary and no dislocation.

It also explains the polysynthetic case. If one boundary is cheap, a hundred parallel boundaries are a hundred times cheap, which is still small compared with the strain they relieve. Plagioclase’s stripes and the deformation twins in calcite marble are both that arithmetic.

Twinning is a shortage of symmetry, not an excess #

The most useful way to hold the whole subject is backwards from the way it is usually met.

A crystal whose class is its lattice’s holohedry has no twin laws by this mechanism at all. There is nothing left over — every symmetry the lattice has, the crystal already uses. Twinning by merohedry is available exactly to crystals with less symmetry than their own lattice, and the amount of twinning available measures how much less.

That is a satisfying inversion. A twinned crystal presents more apparent symmetry than an untwinned one — an aragonite twin looks hexagonal and aragonite is orthorhombic — and the mechanism producing that appearance is the crystal having less symmetry than its lattice. The extra symmetry the observer sees is the lattice’s own, borrowed back by growing two orientations.

The plane figure is the cleanest place to watch this. Of the seventeen, eight have their lattice’s full point group and admit no twin law: p2 on an oblique lattice, pmm, pmg and pgg on a rectangular one, cmm on a rhombic one, p4m and p4g on a square one, p6m on a hexagonal one. Every other plane group has at least one law available, and p3 has three — the most of any, because its point group has order three inside a holohedry of order twelve.

The plane case, and what it shows that three dimensions cannot #

A plane twin fits on a page, and the three-dimensional case does not. That is worth exploiting, because two things about twins are hard to believe from a description and immediate from a picture.

The first is that the boundary is not damaged. Every cell in the drawing is a cell of the same lattice, and the line crosses cell edges without disturbing them. In a real twin the atoms within a few cells of the boundary are in positions that satisfy both orientations reasonably well, which is why the boundary energy is small and why twins are common.

The second is that the second individual is genuinely different. It is easy to look at a twin figure and suspect that the two halves are the same and the line decorative — which is the accidental-symmetry hazard in a new costume, since a figure whose two halves agreed would be a picture of no twin at all. The figure asserts otherwise before it draws: the image of the point set under the twin law is compared with the point set, and they must differ. An operation failing that test would be an operation already in the group, and the picture would be a picture of a crystal with a line drawn on it.

The named laws, and what naming them means #

Mineralogy names twin laws after where they were first described, and the names are worth reading as coset representatives.

The spinel law in cubic minerals is a reflection in {111}, or equivalently a rotation of sixty degrees about the perpendicular body diagonal — one law, two descriptions, one coset. It is why spinel and magnetite grow flattened triangular plates, and it is the same relationship that makes an annealing twin in copper.

The Carlsbad law in feldspar is a two-fold rotation about c. The Albite law is a reflection in (010). The Baveno and Manebach laws are two more, and a feldspar crystal often shows several at once, because feldspar’s class is very much smaller than its lattice’s holohedry.

The Japan law in quartz is a contact twin on (112̅2), giving two prisms at about 84°33′ — an angle that is not any simple fraction of a turn, and which is what it is because it is a lattice symmetry of a cell with quartz’s particular axial ratio.

Naming these was a nineteenth-century occupation and the names have stuck. What the coset reading adds is completeness: given a class and a lattice, the list of laws is finite, computable and closed, so a proposed twin law that is not on it is either a mis-indexing or something other than twinning.

Reading a twin off a picture #

There is a practical question hiding in all of this: given a specimen, how is the law identified?

The classical answer is by the re-entrant angle. A single crystal is convex — every one of its faces is a supporting plane of a convex solid, and its outline has no notches. A contact twin is two convex solids stuck together at an angle, so the outline has a notch in it wherever the two individuals meet, and that notch is a re-entrant angle. Finding one is proof of twinning and its size is a measurement of the law.

The second answer is by the false symmetry. An aragonite twin looks hexagonal; measure the angles carefully and they are not quite hexagonal, because aragonite’s orthorhombic cell has a b/a ratio near but not equal to √3. The departure is a few tenths of a degree, it is systematic, and it identifies both the twin and the law.

The third is by the diffraction pattern, and it is the subject of this anchor’s last rung. A twinned crystal’s reflections either coincide exactly with a single crystal’s or they do not, and which of the two happens is decided by whether the law is a symmetry of the lattice or only of a sublattice — which is to say, by exactly the distinction this essay opened with.

What the machinery does not decide #

Three things, and it is worth being explicit because this field’s earlier essays have already had to be.

Whether a crystal twins at all. That is a question about the energy of a boundary against the energy of the alternative, and about what happened while the crystal was growing. Symmetry supplies the menu and says nothing about the order.

Which composition surface it uses. Also energetics. A surface with many atoms in common between the two orientations costs less, and finding which surface those are is a calculation about the structure rather than about the group.

How much of the crystal is in each orientation. Nothing at all in the symmetry decides that, and the fraction is a fitted parameter in every structure refinement of a twinned crystal — which is where the trouble starts, three rungs along.

There is one more class of twin the coset argument does not reach at all: twinning by pseudo-merohedry, where the operation relating the two individuals is a symmetry of a lattice the crystal’s own lattice merely resembles. A monoclinic cell with β very near ninety degrees can twin as though it were orthorhombic, and the misfit is real but small. That is not decided by integers, it is measured, and it is the tolerance problem arriving in the middle of an exact subject. The last rung of this anchor is about exactly that.

Where the ladder goes next #

The claim so far is qualitative: a twin law is a coset, and the number of them is the index of the class in the holohedry, minus one.

The next rung does the arithmetic for all thirty-two classes, and two results come out of it that are worth stating in advance. Twenty-five of the thirty-two admit twinning by merohedry and seven do not, the seven being exactly the holohedral classes. And the number of laws available depends on the lattice, not only on the class — a trigonal crystal on a rhombohedral lattice and the same class on a hexagonal one have different answers, which is how quartz comes to have three twin laws where a table indexed by class alone would give it one.

Three ways a twin arrives #

The coset argument says which laws are available and nothing about how a crystal comes to use one. There are three routes, they leave different evidence, and separating them is what a petrographer does with a twin once it is identified.

A growth twin forms while the crystal is growing, when a layer nucleates in the wrong orientation and the crystal continues from it. It is a single event, so the boundary is usually one plane and the two individuals are of comparable size — the contact twins and penetration twins of the classical descriptions, and the staurolite cross.

A transformation twin forms on cooling through a phase transition, when the high-symmetry parent’s lost operations leave several equivalent choices and different regions choose differently. It is not one event but many, so the specimen is a mosaic of fine domains rather than two individuals, and the domains obey the compatibility conditions of the transition rather than growing on a favourable plane.

A deformation twin forms under stress, when part of the crystal shears into the twin orientation because doing so is cheaper than slipping. It is fast, it produces thin lamellae, and the shear is a specific one determined by the law rather than by anything about the boundary.

All three produce the same relation between the two individuals, since all three must use a law from the same coset list. What differs is the geometry: a plane, a mosaic, or a lamella, with the population and the shapes recording which mechanism was at work.

The twin that carries the deformation #

The third route deserves a sentence of its own, because it is a mechanism of plastic flow rather than a defect, and it is why some metals deform at all.

A crystal deforms by slip — planes of atoms sliding over one another — and slip needs enough independent slip systems to accommodate an arbitrary shape change. A hexagonal metal does not have enough: its easy slip is confined to the basal plane, which gives too few independent modes, and a polycrystal of it would be brittle.

Twinning supplies the missing modes. A region shearing into the twin orientation produces a definite strain, and it produces it in directions slip cannot reach — so magnesium, titanium, zinc and their relatives deform by a combination of slip and twinning, with the twin lamellae visible afterwards as bands.

That puts the coset arithmetic at the base of a mechanical property. Which laws are available is a symmetry question with an exact answer; which of them a stress can activate, and how much strain each carries, is not — the strain of a deformation twin is a geometric quantity computed from the law and the lattice, and whether it happens is a competition with slip that no group decides.