The index and the angle a twin misses by

A twin is a symmetry the crystal lacks: two orientations of one structure related by an operation that is not in the structure’s own group. Which operations a crystal will accept is not a question about the structure. It is a question about the lattice, and Friedel answered it in 1904 with two numbers.

Why the lattice decides #

A boundary between two orientations costs energy, and how much depends on how well the two lattices match across it. If the twin operation carries a sublattice of the crystal’s lattice onto itself, then that fraction of the lattice nodes on one side coincides with nodes on the other, and the atoms sitting on them are already where both orientations want them — the same accounting as a coincidence site lattice at a grain boundary, with a different name.

So the cheap twin operations are the ones that restore a coarse sublattice, and the two numbers measure exactly that:

• the twin index n, how many lattice nodes there are per restored node — the index of the shared sublattice;
• the obliquity ω, how far the operation is from being a symmetry at all.

Friedel’s empirical rule is that twinning is common when n ≤ 6 and ω ≤ 6°, and rare outside. That is a convention distilled from mineral specimens, not a theorem, and it is used here as one.

A twin law is a pair, not a direction #

There is a trap here that changed a calculation during the writing, and it is worth putting first because it inverts the whole picture.

It is tempting to think of a twin law as a direction: a two-fold about [uvw]. Then the obliquity would be the angle between that row and the plane perpendicular to it, and the question would be whether that plane is a lattice plane.

Every lattice row of every rational lattice has an exactly perpendicular lattice plane. With an integer Gram matrix G, the plane perpendicular to [uvw] has indices G·[uvw], and those are integers. So on this reading every obliquity is zero and the concept is empty.

The resolution is that a twin law pairs a row with a particular plane, and the pair is what a crystal chooses. A crystal grows on low-index planes; the exactly perpendicular plane to a low-index row is often a high-index one that no crystal grows on. So the operation the crystal actually uses is the two-fold about [uvw] treated as a mirror in a different, low-index plane — and the obliquity is the price of that substitution.

The index, computed twice #

The index is where a table would be consulted and a computation is available instead, so it is computed twice by routes that share nothing.

The integer route. The twin lattice is generated by the two-dimensional lattice of the plane (hkl) — every integer vector with h·m = 0 — together with the row [uvw]. Its index in the whole lattice is X = |uh + vk + wl|, which is arithmetic with no geometry in it. That sublattice is built here, by integer elimination over a box, and its determinant taken; it equals X for every one of the two thousand candidate pairs the search generates.

The rotation route. The two-fold about the row is a matrix; conjugating the lattice’s coordinates through it gives a map, and the vectors it returns to the lattice form a sublattice whose index can be taken directly. That route works only when the operation is exact — of which more below — and where it works it must agree.

The parity condition is the part a quoted table loses. When X is even, the two-fold restores one further coset of vectors, so the restored lattice is twice the twin lattice and the index is X/2 rather than X. Dropping that halving doubles every even index on the list and makes Friedel’s limit of six look like a limit of twelve. It is checked here as a refusal: an even X must not be reported as the index.

Checking on the rejected candidates is the point rather than thoroughness. The nine surviving laws are low-index by construction and low-index cases are where an arithmetic slip is least likely to show; the candidates with an index of forty are where a sign error or a missing halving would be visible. A check run only on the answers is a check run only where the answers are easiest.

The obliquity, and what it is the price of #

That identity is the whole reason pseudo-merohedral twinning is common. A monoclinic crystal with β = 90.2° has a lattice that is metrically almost orthorhombic, so an operation of the orthorhombic holohedry — a two-fold the crystal does not have — restores the lattice to within a fifth of a degree. The mismatch across a boundary built on it is a fifth of a degree of misorientation, which costs almost nothing.

A monoclinic crystal with β = 110° has no such option. The same operation misses by twenty degrees and the boundary would be a grain boundary rather than a twin.

So the propensity to twin is a property of the cell’s metric, and it is predictable from a cell measurement alone, before the structure is known. That is the practical content of Friedel’s theory and it is why the two numbers appear in every structure report of a twinned crystal.

Four kinds, and why the classification is a product #

The two numbers give four combinations and each has a name:

• n = 1, ω = 0 — twinning by merohedry. The operation restores the whole lattice exactly. It is available only when the crystal’s point group is a proper subgroup of its lattice’s holohedry, which is what merohedry means, and which is decided by comparing the crystal class with the holohedry that is its ceiling.
• n > 1, ω = 0 — reticular merohedry. Exact, but only on a sublattice. The cubic table above is entirely of this kind.
• n = 1, ω > 0 — pseudo-merohedry. The whole lattice is nearly restored. The monoclinic cell approaching ninety degrees is the standing example.
• n > 1, ω > 0 — reticular pseudo-merohedry. Both approximations at once, and the commonest kind in practice.

The three twin laws of quartz, read off the numbers #

Quartz is the standard specimen for this and its three laws are a small worked example of everything above.

The Dauphiné law is a two-fold about the c axis. Quartz’s point group is 32; the lattice’s holohedry is 6/mmm; a two-fold about c is in the second and not the first. Index one, obliquity zero — twinning by merohedry, and the two orientations have identical lattices in identical positions.

The Brazil law is an inversion, which is likewise in the holohedry and not in 32. Again index one and obliquity zero, and again merohedral — but the operation reverses handedness, so a Brazil twin joins a left-handed domain to a right-handed one where a Dauphiné twin does not.

The Japan law is a two-fold about a direction that is not a symmetry direction, with a mismatch of a fraction of a degree, and its index is larger than one. It is the reticular pseudo-merohedral case, and it produces the flattened V-shaped crystals that make quartz twinning visible to anybody.

All three are decided by the same two numbers, and the reason the first two are so much more common than the third is that index one costs less than index two. That is Friedel’s rule stated for one mineral, and the essay on quartz takes the operations apart in detail where this one takes the arithmetic apart.

Why the search is over pairs and how large it is #

The enumeration takes every lattice row with indices to two and every lattice plane with indices to two, forming two thousand one hundred and twenty-five pairs, and computes both numbers for each. That is a small search and its size is worth noting for what it says about the subject.

Twin laws are low-index objects. A law with indices beyond two or three has an index far outside Friedel’s limit, because X = |uh + vk + wl| grows with the indices. So the useful list is short — nine laws for the monoclinic lattice here, thirty-seven for the cubic — and a search over a slightly wider box adds nothing but rejected candidates.

That is a pleasant situation and it is not the usual one in this collection. Most enumerations here are bounded by an argument that the bound suffices; this one is bounded by the quantity being enumerated growing out of range on its own.

The exactness, and the one place it stops #

Everything above is integer arithmetic except one step, and it is worth isolating.

The lattice is given by a Gram matrix of integers. The twin operation is a rotation in Cartesian space, so the matrix deciding whether a vector returns to the lattice contains square roots. That test is therefore against a tolerance, stated as one part in a hundred million of a lattice unit, and it is the only tolerance in the calculation.

It is also where a wrong answer hid. An angle that is exactly zero comes back from a double-precision dot product as anything up to a hundred-thousandth of a degree, because the arccosine is flat at one. Testing exactness against a millionth of a degree therefore called a dozen exact operations inexact — and then reported that they restored a sublattice which, being inexact, nothing was supposed to restore. The threshold is a ten-thousandth of a degree, four orders below any obliquity a real twin law has, and the inconsistency is what found it.

What a twin does to the data #

The consequence a crystallographer meets first is in reciprocal space, and the two numbers predict it.

A twin of index one superimposes two complete reciprocal lattices exactly. Every reflection is a sum of two, and no amount of care in data collection separates them: the twin fraction has to be refined as a parameter. This is the case that ruins structure determinations quietly.

A twin of index n > 1 superimposes only one reflection in n. The rest are separable, the twin is visible in the pattern as a second lattice, and the situation is inconvenient rather than dangerous.

A twin of non-zero obliquity gives split reflections, by an angle proportional to ω — and, crucially, the splitting grows with the distance from the origin. So a pseudo-merohedral twin looks perfect at low resolution and splits at high, which is exactly the observation that identifies it. The same resolution-dependent tell distinguishes a twin from a genuine supercell, whose extra reflections do not split at all.

How many laws a lattice offers, and why the number tracks its symmetry #

The five lattice systems this enumeration reaches offer between nine and thirty-seven laws inside Friedel’s limits, and the exact ones among them run from all of them to none. That spread is worth reading in one place, because the quantity behind it is the same one that decided the dislocation counts a rung along.

A twin law is exact when the plane paired with the row is the plane exactly perpendicular to it — which, as the section above established, always exists as a lattice plane and is usually a high-index one. What decides whether the low-index plane a crystal actually grows on happens to be that plane is the lattice’s own symmetry: a high-symmetry metric puts low-index rows and low-index planes into perpendicular pairs, and a lopsided one does not.

So the exact fraction is a reading of the holohedry. The cubic lattice’s holohedry has forty-eight operations and every one of its thirty-seven laws is exact. The triclinic lattice’s has two, and none of its ten laws is. In between the fraction falls monotonically, and a crystallographer can predict the character of a mineral’s twinning from its cell before knowing anything about its structure.

The eight inexact hexagonal laws are the ones worth looking at, because they are where the substitution described earlier is visibly happening. Each pairs a low-index row with a low-index plane that is nearly, and not exactly, perpendicular to it; the exactly perpendicular plane exists, has higher indices, and is not a face any hexagonal crystal grows. The obliquity column is the angle between the two, and it is small for the same reason the substitution was worth making — a crystal will accept an operation that misses by a degree and will not build a face nobody grows on.

What is owned, and what is quoted #

Owned: the enumeration of row-and-plane pairs, the index by Friedel’s formula, the twin lattice constructed and its determinant taken, the restored sublattice computed from the rotation where the operation is exact, the obliquity, and the identity between obliquity and the cell’s departure from orthogonality.

Quoted: Friedel’s limits of six and six degrees, which are empirical; the observation that low-index planes are the ones crystals grow on, which is a growth statement; and every claim about how much a boundary costs, which is energetics and is not computed anywhere here. The same division as the dislocation count one rung along: the lattice says what is available, and everything else says what happens.

What the theory does not decide #

Friedel’s two numbers say which operations are cheap. Three things they do not say are worth listing, because the theory is often read as saying more than it does.

They do not say a crystal will twin. A lattice may offer a dozen laws inside the limits and a crystal grown from it may show none. Whether twinning happens depends on how the crystal nucleates and grows, and on whether a fluctuation at the growth front happens to start the second orientation — none of which is in the lattice. The cubic table above offers thirty-seven laws and most cubic minerals show at most one or two of them, which is the clearest statement of how loose the permission is.

They do not rank the laws that pass. Index three and index five are both inside the limit; nothing here says which of two permitted laws a crystal prefers, and the observed preference is often for the one whose boundary plane has a low index rather than for the one with the lower twin index.

And they say nothing about the structure. Two crystals with identical lattices and different structures have identical twin-law lists, and one may twin readily while the other does not — because what actually has to match across the boundary is the atoms, not the lattice nodes. Friedel’s theory is a necessary condition dressed as a prediction, and it is used because the necessary condition is cheap and the sufficient one does not exist.

Why the limits are six and six degrees #

Friedel’s two thresholds are quoted as though they were derived and they are empirical, and it is worth saying what each is standing in for, because the reasoning behind them is available even though the numbers are not.

The index measures how much of the lattice matches. A twin of index n restores one node in n, so the boundary between the two orientations has one atom in n sitting where both orientations want it and the rest displaced. The energy of the boundary rises with the fraction that does not match, so the cost climbs with n — and by the time only one node in six matches, the arrangement is not much better than an arbitrary boundary. That is a rationale for a limit rather than a derivation of six.

The obliquity measures how far the match has to be forced. An operation missing by ω means the two lattices, laid on one another, drift apart at a rate proportional to ω, so a coincidence exact at the boundary is lost within a distance of order the cell divided by ω in radians. At six degrees that is about ten cells; at a tenth of a degree it is six hundred. The limit is a statement about how large a coherent region can be before the mismatch has to be taken up by defects.

Both limits are soft, and the surveys say so. Twins are recorded outside them and many lattices offering laws well inside them never twin. The numbers are a shortlist rather than a criterion, which is the same status as every other empirical threshold in this collection — useful for saying where to look, and no evidence about any particular crystal.

The same two numbers, one scale up #

The arithmetic here is not specific to twins. It is the arithmetic of any boundary between two orientations of one lattice, and the case that has had far more attention is the general grain boundary.

Two grains of one material meet at an angle, and the fraction of lattice nodes they share is the coincidence site lattice, whose index is written Σ. That is the twin index under another name: Σ = 1 is a boundary across which the lattices coincide exactly, Σ = 3 is the coherent twin boundary of a face-centred cubic metal, and the whole classification of boundaries by Σ is the classification of twin laws by index applied to a pair of grains rather than to a crystal and its image.

The obliquity has a counterpart too. A real boundary is never at exactly the coincidence orientation, and the standard tolerance — how far from it a boundary may sit and still behave as a coincidence boundary — is taken to fall off as the inverse square root of Σ. The higher the index, the tighter the angle has to be, which is the same trade this page’s two numbers make: a coarse coincidence buys less, so less mismatch can be afforded on top of it.

That the two subjects share one arithmetic is worth noticing, because they do not share a literature. Twinning is mineralogy and crystallography; grain boundaries are metallurgy and materials science; and the index is the same integer, computed the same way, from the same lattice.

Where the ladder goes next #

Towards the measurement. A twin index and an obliquity predict which reflections overlap and by how much they split, and what a twin does to diffraction is where those predictions are tested against a pattern.

And towards the boundary itself. A twin is one orientation relationship; a general grain boundary is another, and the coincidence site lattice counts shared nodes for those in exactly the way the twin index does here. The two theories are the same arithmetic with different conventions, which is worth knowing before reading either literature.