Crystallography & Symmetry
patterns derived from their groups
Series

Twinning — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

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

    Two orientations of one structure, grown together across a boundary the lattice runs straight through. The operation relating them cannot be a symmetry of the crystal, or there would be nothing to see, and it must be a symmetry of the lattice, or the boundary would be a crack — which leaves exactly a coset, and a short computable list.

    part 1 · applied
  2. Which classes can twin by merohedry, and how many ways. Every crystal class, with the number of twin laws its own lattice offers it. The index of the class in the point group of its lattice is the number of orientations available; 25 of the thirty-two have more than one, and the 7 holohedral classes have exactly one — their crystal already has every symmetry their lattice has, so there is nothing left over to twin by. The names along the right are the old mineralogical ones: hemihedral for half, tetartohedral for a quarter.

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

    The number of twin laws available to a crystal is the index of its class in the point group of its lattice, minus one. Doing that arithmetic for all thirty-two classes takes a moment and produces a census with a sharp edge on it — the seven classes that cannot twin this way are exactly the seven that already use everything their lattice has.

    part 2 · applied
  3. The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

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

    Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

    part 3 · applied
  4. p3, single and twinned. Left, the diffraction pattern of a single crystal of p3. Right, the same crystal twinned, with 50 per cent of it in one orientation. Not one spot has moved — the twin law is a symmetry of the lattice, so the two reciprocal lattices lie exactly on top of one another — and 72 of the 81 reflections drawn have changed intensity. At a fifty-fifty twin the pattern acquires the full symmetry of the lattice's point group and is indistinguishable from a crystal that genuinely has it.

    A merohedral twin moves no spot at all

    The twin law is a symmetry of the lattice, so the two individuals have reciprocal lattices lying exactly on top of one another. Nothing splits, nothing appears in a new place, and the only thing that changes is that pairs of intensities which were different have been averaged — which produces a diffraction pattern with a symmetry the crystal does not have and no sign that anything is wrong.

    part 3 · applied
  5. monoclinic: 9 twin laws, 1 of them exact. The twin laws of a monoclinic lattice: a two-fold about the row [uvw] paired with the plane (hkl) it is meant to be a mirror in, kept when the twin index is at most six and the obliquity at most six degrees, which are Friedel's own limits and are a convention rather than a theorem. 9 of the 9 are listed. The index n is how many lattice nodes there are per node the operation restores, computed from the integers and checked against the sublattice built from the plane and the row. The obliquity is the angle between the row and the plane's normal: 1 of these laws have none, and for those the operation restores a sublattice exactly.

    The index and the angle a twin misses by

    Whether a crystal will twin on a given operation is decided by its lattice, not by its structure. Two numbers decide it: how many lattice nodes there are per node the operation restores, and how far the operation is from being a symmetry at all. Both are computed from integers, and one of them is a fiction that has to be labelled as one.

    part 5 · applied

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