Quartz has exactly three twin laws, and its lattice is why

Quartz is in class 32: a three-fold axis along c, three two-fold axes perpendicular to it, six operations in all. No mirrors, no inversion, no six-fold — which is why a quartz crystal is optically active, piezoelectric, and comes in two hands.

Its lattice is hexagonal, whose point group 6/mmm has twenty-four operations — the holohedry that is the ceiling on anything sitting on that lattice.

Six into twenty-four is four. So class 32 occupies one coset of four, three cosets are left over, and quartz has exactly three twin laws. Not approximately three, not three of the ones commonly seen — three, with nothing else possible by this mechanism.

The three, named #

Every operation in a block gives the same second orientation, so it is enough to name one representative of each.

Dauphiné. A two-fold rotation about c. Quartz’s own two-folds are perpendicular to c, so this one is not among them. It carries the structure onto itself with the three-fold intact, so the handedness is unchanged — a Dauphiné twin of right-handed quartz is right-handed on both sides. What it does change is the sense of the two-fold axes, and with them the sign of the piezoelectric response. Named after the French region where the twins were first described in the eighteenth century, and known in the trade as the electrical twin for exactly that reason.

Brazil. The inversion. It reverses handedness: a Brazil twin has right-handed quartz on one side of the boundary and left-handed on the other, and the plane of polarisation is rotated one way in one individual and the other way in the other. Known as the optical twin. Because it is the inversion, it is the Class I law, and every diffracted intensity is exactly the same as an untwinned crystal’s — which is the eleven Laue classes doing their damage in a new place.

Combined. The product of the other two, which is a mirror perpendicular to c. It reverses handedness and reverses the two-folds, so it does both. Sometimes called the Leydolt twin.

Those three names come from mineralogy, and they are the three cosets. The correspondence is complete: there is no fourth named quartz twin of this kind, and no coset without a name.

The lattice is doing the work, and a table indexed by class would not know #

Quartz is trigonal, and a trigonal crystal has a choice of lattice that no other system offers. It may sit on a rhombohedral lattice, whose point group has twelve operations, or on a hexagonal one, whose point group has twenty-four.

Quartz sits on the hexagonal one — its space groups are P3₁21 and P3₂21, an enantiomorphic pair differing only in the sense of a screw, and the P says primitive hexagonal. The distinction between the two settings is the same one the rhombohedral setting is about. So the index is 24/6 = 4 and there are three laws.

Had it sat on a rhombohedral lattice, the index would be 12/6 = 2 and there would be one law. That law would be the inversion, and Dauphiné twinning would not exist.

So two-thirds of quartz’s twinning is a consequence of which lattice it is on, not of which class it is in. A reference table indexed by crystal class alone gets this wrong in the direction that loses laws, silently, and the loss is exactly the twin that matters industrially.

Dauphiné twinning is a phase transition, not a growth accident #

The Dauphiné law has a second life, and it is the cleanest bridge in this field between twinning and the next anchor.

Quartz has a transition at 573 °C. Above it, β-quartz is in class 622; below it, α-quartz is in class 32. The atoms move very little — the framework of tetrahedra rotates slightly — and the symmetry drops by exactly the six operations that make 622 twice the size of 32. That is a descent of symmetry of index two, and the whole of the next anchor is about what such descents leave behind.

Now count. The index of 32 in 622 is two. A crystal cooling through that transition has two equally good ways to become α-quartz, and different parts of it choose differently. The regions that made the same choice are domains, and the operation relating two of them is the operation that was lost: a two-fold about c.

That is the Dauphiné law. A Dauphiné twin is not a growth feature at all — it is the domain structure left behind by a phase transition, and it can be produced in a previously untwinned crystal by heating it past 573 °C and cooling it, or by applying stress, which biases one domain state over the other and moves the walls.

The two derivations agree, and it is worth being clear about why they had to. The Dauphiné law is a coset of 32 in 6/mmm, and it is also a coset of 32 in 622. Those are different containing groups; what makes it the same law is that 622 is inside 6/mmm, so a coset of the smaller containment sits inside a coset of the larger one. A transformation twin is a merohedral twin whose law happens to be available from an intermediate group as well.

Brazil twinning is not like that. Its law is the inversion, which is not in 622 — β-quartz is no more centrosymmetric than α-quartz is — so no transition through 622 can produce it. Brazil twins are grown, not cooled into, and that is a prediction the arithmetic makes before anyone looks at a crystal.

What each law costs #

The three laws have three different consequences for anyone trying to use quartz, and the differences are exactly the properties the point-group field computed.

Dauphiné twinning ruins a piezoelectric device. The two orientations have opposite signs of the piezoelectric coefficient along the two-fold axes, so a crystal that is half one and half the other has an average response somewhere between full and zero. A resonator cut from Dauphiné-twinned quartz has the wrong frequency response and no way to correct it. This is why oscillator blanks are inspected for twinning, why they are cut from carefully selected regions, and why cultured quartz is grown under conditions that suppress it.

Brazil twinning ruins an optical component. The two individuals rotate polarised light in opposite senses, so a plate cut across a Brazil twin has regions of opposite rotation and is useless for anything depending on it. It is visible between crossed polarisers as sharply bounded patches, which is how it is detected.

Both are invisible to the shape. A quartz crystal twinned on any of the three laws still has the outward form of an ordinary quartz crystal, because every one of the three is a symmetry of the lattice and the faces are lattice planes. The re-entrant angle that gives away a contact twin does not appear, because these are not contact twins — the individuals interpenetrate on irregular surfaces inside the crystal.

The cut, and why the arithmetic reaches a factory #

A quartz oscillator is a thin plate cut from a crystal at a carefully chosen orientation, driven electrically at its mechanical resonance. The orientation is chosen so that the temperature coefficient of the resonant frequency vanishes to first order at room temperature, and the famous ones — the AT cut at about 35°15′ to the optic axis, the BT cut at about −49° — are named by that angle.

Everything about that design assumes the crystal is a single individual. The relevant piezoelectric coefficients are components of a third-rank tensor, and a third-rank tensor is exactly what an inversion kills, which is why quartz is usable at all. Dauphiné twinning reverses the sign of those components in the twinned region, so the two halves of a twinned plate drive each other in opposition.

The consequence is quantitative and unpleasant. A plate that is a fraction f in one orientation and 1 − f in the other has an effective coefficient proportional to 2f − 1, so a half-and-half crystal is not a poor oscillator; it is not an oscillator. And because Dauphiné twinning is a transformation twin, a plate can acquire it after manufacture — from a mechanical shock, from a thermal excursion, from stress in a mount — and the failure looks like a drift rather than a break.

The remedy is to work far from the transition and to select material. Cultured quartz is grown below 573 °C so that it never passes through the transition at all, which removes the mechanism rather than managing it.

The plane analogue, drawn #

The plane group p3 stands in the same relation to its lattice that quartz’s class does to its own, and it fits on a page.

p3 has three operations modulo translations; its hexagonal lattice’s point group has twelve; the index is four; there are three twin laws. One is a rotation, two are reflections, and they play the parts of Dauphiné, Brazil and the combination.

How a mineralogist finds each of the three #

None of the three shows on the outside of a crystal, so all three are found by a property rather than by a shape, and the three methods are different from one another.

Brazil twinning is found between crossed polarisers on a plate cut perpendicular to c. The two individuals rotate the plane of polarisation in opposite senses, so they extinguish at different settings and the boundary appears as a sharp edge between regions of different colour. It is the easiest of the three to see and the least consequential for most purposes.

Dauphiné twinning is found by etching. A polished surface treated with hydrofluoric acid dissolves at different rates in the two orientations, because the etch pits have the site symmetry of the surface and that symmetry is oriented oppositely on either side of a wall. The result is a visible pattern of patches with irregular boundaries — irregular because a Dauphiné wall is not a plane, having no crystallographic reason to be one.

Combined twinning is found by doing both and finding both.

That the three tests are three different kinds of measurement is not an accident of technique. Each law is detected by the property it reverses, and which properties a class has to reverse is settled by Neumann’s principle before any of them is measured.

What is asserted while this is computed #

Three claims, each of which could fail.

The class is a subgroup of the lattice’s point group. Not assumed — checked, matrix by matrix, before any coset is formed. Class 32 is written on hexagonal axes and 6/mmm is written on the same axes, so the comparison is meaningful; a class written on a different setting would fail this rather than produce a wrong count. Deriving a symbol from a group’s own directions had to solve the same problem, and solved it the same way.

The cosets partition the holohedry exactly once. Every operation is placed in a block and the total is compared with the order of the holohedry. A division cannot detect a mistake here and a covering count can.

The block containing the inversion is identified rather than assumed to exist. It does exist for every non-centrosymmetric class, and the figure finds it by looking rather than by knowing, which means the marking is a result and not a label.

Where the exactness stops #

Nothing in this essay decides how much quartz is twinned, or where the walls are, or whether a particular crystal is twinned at all. Those depend on the growth conditions, on the stress history and on the cooling rate, and symmetry supplies none of them.

Nor does it decide the width of a domain wall. A Dauphiné wall in quartz is a few unit cells thick and its structure is a question about interatomic forces. The group says the two states exist and are related by a two-fold; it says nothing about what happens in the region between them.

And there is a fourth kind of quartz twin the whole argument misses: the Japan law, a contact twin at about 84°33′ on (112̅2). That angle is not a symmetry of the hexagonal lattice, so it is not one of the three cosets. It is a symmetry of a sublattice — twinning by reticular merohedry, which the last anchor of this field takes up — and it is common enough to be collected.

The count, in general #

The arithmetic that gives quartz three is one division, and writing it in general says which crystals are exposed to this kind of twinning and how badly.

A class of order g sitting on a lattice whose holohedry has order h decomposes that holohedry into h/g cosets, one of which is the class itself. So the number of twin laws by merohedry is h/g − 1, and it is decided entirely by two integers.

Reading the extremes is instructive. A class that is its lattice’s holohedry has index one and no merohedral twin law at all — the holohedry is the ceiling and there is nothing above it to borrow an operation from. A class of low order on a lattice of high symmetry has many: class 3 on a hexagonal lattice has order three against twenty-four, giving seven laws.

The general shape is therefore that the more symmetry a crystal gives up relative to its lattice, the more ways it can twin, and the quantity measuring that is the index rather than either order on its own. Quartz’s three is a middling value: it keeps six of the twenty-four available operations, so three of the four cosets are laws.

That also says where merohedral twinning cannot occur. A holohedral crystal — one whose class is the full symmetry of its lattice — is safe from it entirely, whatever else it does, which is a prediction available from the class and the lattice type alone and before any crystal is grown.

Boundaries that can be moved #

There is a practical difference between the three laws that the coset arithmetic does not see, and it matters more to anybody using quartz than most of what is above.

A Dauphiné boundary separates two orientations related by a rotation about the axis both share. The atoms on either side of it are displaced only slightly, and the wall between them can move: applying a stress makes one orientation slightly more favourable than the other, and the boundary sweeps through the crystal. Heating helps. So Dauphiné twinning can be substantially removed from a quartz plate by treatment, and industrially it is.

A Brazil boundary separates two hands. Converting one into the other would mean reversing the sense of every screw in a region, which is a reconstruction of the structure rather than a displacement of it. No stress moves a Brazil boundary, and no annealing removes one: the twinning is settled when the crystal grows and is permanent.

That asymmetry follows from what the two operations are. Dauphiné’s is a rotation, so both orientations are the same hand and the transformation between them is small. Brazil’s is an inversion, so the two orientations are enantiomorphs and nothing continuous connects them.

The classification predicts the distinction without predicting the remedy. That one law is removable and the other is not follows from the operations being proper and improper; how much stress, at what temperature, for how long, is a measurement about a material.

Where the ladder goes next #

Two of the three laws are invisible to a shape and one of them is invisible to diffraction as well. The next rung is the one that makes the second half of that sentence precise: what a twin actually does to a diffraction pattern, why a merohedral twin moves no spot at all, and why a structure solved from a twinned crystal without noticing comes out wrong in a way that looks perfectly reasonable.