Twenty-five of the thirty-two can twin, and seven cannot

A crystal class sits inside the point group of its lattice — inside the holohedry that is its ceiling. How far inside is a number — the index — and it is the only quantity needed to count the twin laws available.

If the class has order n and the holohedry has order N, the class occupies one of N/n cosets and the other N/n − 1 are the twin laws. That is the whole computation, and running it over all thirty-two classes takes less time than reading the result.

Twenty-five of the thirty-two admit twinning by merohedry. Seven do not. The seven are exactly the holohedral classes, one per system, and the reason is immediate: for them the index is one, the class fills its lattice’s whole point group, and there is nothing left over to be a law.

The old words, which mean exactly this #

The vocabulary is Haüy’s and Groth’s and it is index arithmetic under other names.

A holohedral class has index one: it has all of the lattice’s symmetry, and holos is the whole of it. A hemihedral class has index two — half. Tetartohedral is a quarter, index four; ogdohedral is an eighth, index eight, and it takes a lattice the class’s own system does not imply before that one is reachable at all.

Those names were invented to describe faces. A hemihedral crystal shows half the faces the holohedral form would have, because its class produces an orbit half the size — which is the same orbit count this field’s first anchor was about, seen from the outside of a crystal instead of from inside a group. The tetrahedron is the hemihedral form of the octahedron; the pyritohedron is the hemihedral form of the tetrahexahedron, which is why pyrite’s twelve pentagons identify its class.

So the merohedry of a class is one number doing two jobs. It says how much of the lattice’s symmetry the crystal is missing, which is a statement about faces; and it says how many twin laws are available, which is a statement about orientations. They are the same number because they are the same index.

The census, read #

The distribution is worth reading off rather than merely computing.

Seven at index one — 1̅, 2/m, mmm, 4/mmm, 3̅m, 6/mmm and m3̅m. No merohedral twinning at all, and these are the only seven, one per lattice system.

Nineteen at index two, which is most of the thirty-two. These are the classes missing exactly half their lattice’s symmetry, and they include almost every common structural class: 222, mm2, 4/m, 422, 4mm, 4̅2m, 32, 3m, 3̅, 6/m, 622, 6mm, 6̅2m, m3̅, 432, 4̅3m, and the three lowest — 1, 2 and m.

Class 1 arriving on that list is the pleasing case. A crystal with no symmetry whatever, on a triclinic lattice whose only operations are the identity and the inversion, has index two and exactly one twin law, and that law is the inversion.

Six at index four, the tetartohedral ones: 4, 4̅, 3, 6, 6̅ and 23. Each has a quarter of its lattice’s symmetry and three laws available.

And nothing else. Only the indices 1, 2 and 4 occur when each class is placed on the lattice its own system implies. No class comes out at three, or six, or eight — and the absence of odd indices above one has a one-line reason rather than being a coincidence of small numbers. If a class G is non-centrosymmetric, then G together with its inversion-images is a group of order 2·|G| sitting inside the holohedry, so 2·|G| divides |H| and the index is even. A centrosymmetric class could in principle have an odd index and none of the four non-holohedral ones does.

The word ogdohedral therefore describes nothing on this list. It is reserved for a case the list cannot show, because the list assumed each class sits on the lattice its system implies — and one family of classes has a choice.

How the index is computed here, and what is checked #

The index could be obtained by dividing two numbers from a table. It is not, and the reason is the one this site gives every time it refuses a division.

The holohedry is built rather than looked up. For six of the seven lattice systems it is the set of integer matrices preserving that system’s metric, which is how the fourteen Bravais lattices were enumerated. For the seventh — a rhombohedral lattice described on hexagonal axes, which is what the trigonal classes are written against — it is derived instead as the operations of 6/mmm that permute the two R-centring vectors among themselves. That comes out at order twelve and derives as the symbol 3̅m, and the agreement is the check that the construction was right rather than merely plausible.

Then the cosets are formed by partitioning, not by dividing. Every operation of the holohedry is placed in exactly one block, the blocks are counted, and two things are asserted: that the number of blocks equals the index, and that the blocks between them cover the holohedry exactly once. A division cannot fail. A partition can, and if the class had somehow not been a subgroup — if a matrix had been in the class and not in the holohedry — the covering count would come out wrong and the figure would not be drawn.

That the class is a subgroup of its lattice’s point group is itself asserted, because it is the assumption the whole coset argument rests on and it is easy to break by describing a group in the wrong basis. A trigonal class written on cubic axes is a perfectly valid group of matrices and is not a subgroup of anything in the hexagonal family, and the containment check is what catches that rather than a wrong number appearing at the end.

Class I and Class II, and a twin that cannot be seen #

Among the cosets of a non-centrosymmetric class there is always one containing the inversion, and it is special in a way that costs structural crystallographers a great deal of time.

A twin whose law lies in that coset relates the structure to its own inverse. Under Friedel’s law the intensity at (hkl) equals the intensity at (h̅k̅l̅) for every structure, so the two individuals of such a twin scatter identically at every reflection. The twinning changes nothing in the measured amplitudes at all — not slightly, not approximately, but exactly.

That is the distinction between Class I twinning, where the law is equivalent to the inversion modulo the crystal’s own symmetry, and Class II, where it is not. A Class I twin is invisible to an ordinary diffraction experiment and shows up only in the anomalous signal; a Class II twin superposes reflections that genuinely differ and can be detected in the intensity statistics.

Every non-centrosymmetric class has exactly one Class I law, because the inversion lies in exactly one coset. So of the twenty-five classes that can twin by merohedry, the twenty-one non-centrosymmetric ones each have one law that is undetectable this way, and the four centrosymmetric non-holohedral ones — 4/m, 3̅, 6/m and m3̅ — have none, because for them the inversion is already inside the class.

Even the group with no symmetry has a twin law #

Class 1 has one operation. Its lattice — triclinic, no constraints on anything — has two, the identity and the inversion. So the index is two, there is one twin law, and it is the inversion.

This is worth pausing on because it is counter-intuitive in both directions. A crystal with no symmetry at all can still be twinned, and the twin is the one nobody can see, because a Class I law in class 1 is the only law there is.

The plane analogue is exact and can be drawn. Plane group p1 — the group with nothing in it, and the one a single dot cannot even illustrate — has one operation modulo translations; its oblique lattice’s point group has two, the identity and the half-turn; index two, one law, and that law is the half-turn.

Where the arithmetic needs care #

The index is a property of the class and its lattice, and it is the second half that a table indexed by class alone gets wrong.

Most systems offer only one lattice, so the distinction does not arise. The trigonal classes offer two: a trigonal crystal can sit on a rhombohedral lattice, whose point group is 3̅m with twelve operations, or on a hexagonal one, whose point group is 6/mmm with twenty-four. The index of class 3 is four in the first case and eight in the second, and eight is where the word ogdohedral is used.

So a trigonal crystal’s twin laws cannot be looked up from its class. They have to be computed from its class and its lattice together, and the difference between the two answers is a factor of two in the number of orientations available. The next rung is quartz, where the difference is the whole story.

What the seven have in common besides the arithmetic #

The seven holohedral classes are the ones that cannot twin this way, and it is worth asking what else is true of them, because the answer is a small piece of evidence about how often this mechanism matters.

Every one of the seven is centrosymmetric — a holohedry contains the inversion, so a class equal to its holohedry does too. So none of the seven can be piezoelectric, none can be polar, none can be optically active, and none has an enantiomorph. The classes that cannot twin by merohedry are exactly the classes that are least interesting for every other reason in this field.

That is not a coincidence and it is not deep either. Both statements are consequences of having a lot of symmetry: a class with everything its lattice permits has no cosets left over and no property that an inversion would forbid. The classes worth building a device out of are, without exception, classes that can twin — and since a twin averages a property over two orientations, twinning is a standing hazard for exactly the materials whose properties are worth having.

Quartz oscillators are cut from untwinned quartz for that reason, and the twinning that has to be avoided is a specific one of the three laws its class permits.

Beyond merohedry: when a sublattice does the work #

There is a second mechanism, and it enlarges the subject considerably.

The argument so far required the twin law to be a symmetry of the lattice. Weaken that: require it only to be a symmetry of a sublattice — a subset of the lattice points forming a lattice of their own, with a fraction of the density. The boundary is then coherent for those points and slightly wrong for the rest, and if the fraction is high enough it is a perfectly good twin.

That is twinning by reticular merohedry, and the fraction is the index of the sublattice, which metallurgy writes Σ. The spinel law in cubic crystals is the case everybody meets: a rotation of sixty degrees about a body diagonal is not a symmetry of the cubic lattice, and it is a symmetry of a sublattice containing one point in three. Σ = 3, one atom in three is in perfect register across the boundary, and the boundary energy is correspondingly tiny — which is why annealing twins in copper and brass are everywhere.

The whole interfaces anchor of this field is that computation. It is the same coset arithmetic with the lattice replaced by a sublattice, the index of the sublattice becomes the thing to enumerate, and the answer is a short list of special angles that comes out of number theory rather than out of crystallography.

What is settled and what is not #

The census settles which laws are available. It says nothing at all about which are used.

Quartz has three laws available and uses all three routinely. Calcite has one and uses it constantly. Many crystals with several laws available are never observed twinned at all, because the boundary energy for their structure is high or because nothing during growth ever nucleated a second orientation.

That gap is the same one this field has now met four times: permitted is not present, for forms, for physical properties, for twin laws. The pattern is consistent enough to be worth naming as a habit — a symmetry argument produces an upper bound on what can happen, always, and never a prediction that it will.

What it does give, and what makes the census worth computing, is the negative. A crystal reported as twinned on a law not in its coset list is a crystal that has been mis-indexed, or is not twinned, or is not in the class it was assigned. The list is closed and finite, so the refusal is available, and a closed list that can reject something is worth a great deal more than an open one that cannot.

The group a twin law generates #

A twin law is a coset, and a coset is not a group. What is a group is the class together with the law, and naming it gives the twin an object of its own rather than only a relation.

Take the class H and one of its twin laws, represented by an operation g. The set H ∪ gH is a group exactly when g² lies in H, which happens whenever the coset has order two in the quotient — and for an index-two class it always does. That group is the twin point group, it has order 2n, and it is the symmetry of the twinned crystal considered as one object.

The distinction matters when the index is four. There the quotient has three non-trivial cosets and they are not independent: two of them compose to the third, so a crystal twinned on two laws is automatically twinned on the third, and the twin point group generated by any two is the whole holohedry.

That is why quartz’s three laws behave the way they do. Its index is four, its three cosets are the Dauphiné law, the Brazil law and their product, and a crystal showing two of them shows all three — which is exactly what the combined law is, and why it is not a fourth possibility.

So the coset arithmetic gives more than a count. It gives the composition table, and the composition table says which combinations of laws a single specimen can carry.

The twin that looks like a better crystal #

There is a specific and expensive failure this census makes possible to state, and it is the reason merohedral twinning is feared rather than merely noted.

A crystal twinned by merohedry with equal volumes of the two orientations produces intensities that are the average of the two — and the average has the symmetry of the twin point group, which is larger than the crystal’s own. So the diffraction pattern reports a higher symmetry than the crystal has, exactly and not approximately.

A determination that takes that report at face value assigns the higher class, refines in a group with too many operations, and produces a structure that is the average of the two orientations — a well-formed answer to the wrong question, with atoms at positions no molecule occupies.

Nothing in the symmetry catches it, because the symmetry is what has been falsified. What catches it is the intensity statistics — the distribution is narrowed by the averaging, in a way the essay on that subject makes quantitative — and chemistry, since the averaged structure usually contains something absurd.

The census on this page is what says which classes are exposed. A holohedral class cannot be twinned by merohedry, so it cannot suffer this; and a class of index four has three laws available and three ways to be misread. Knowing the index is knowing how much room there is for the error, before any data are collected.

Where the ladder goes next #

Two rungs of this anchor have been about counting. The third is a single mineral, and it is the one where every piece of the arithmetic so far shows up at once.

Quartz is class 32 on a hexagonal lattice. Its index is four, so it has three twin laws, and each of the three has a name, a history and a distinct consequence: one that swaps left-handed for right-handed quartz, one that reverses the sense of a screw without changing the handedness, and one that is the composition of the other two. One of the three is Class I and invisible to diffraction. And all three would be missing from a table that read quartz’s lattice off its class.