Crystallography & Symmetry
patterns derived from their groups
Generator

p3, single and twinned

p3, single and twinned
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.

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.

3 essays call twin-diffraction. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Symmetry at work

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.

 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. Symmetry at work

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.

 The twin fraction, recovered from a moment and nothing else (12 atoms). A structure of 12 atoms twinned at each of 6 fractions, with the second moment of its intensity distribution measured and the fraction solved back out of it. The recovery is within a few hundredths as far as thirty per cent — 4 rows here — and 4 of the 6 fractions get a number at all. Beyond thirty per cent the relation flattens: the derivative of 2α(1−α) vanishes at a half, the two roots meet, and a small error in the moment becomes a large one in the fraction. Where the sampled moment falls below 1.5 the quadratic has no real root and the estimate refuses rather than clamping, which is why a nearly perfect twin is the hard case in practice rather than the easy one. How it is known

A twin hides in the statistics

A twinned crystal scatters as two orientations at once and the detector cannot separate them. What arrives is a sum of two intensities — and adding two independent quantities narrows a distribution, which is a signature no model of the structure is needed to read.

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