Crystallography & Symmetry
patterns derived from their groups
Series

Local symmetry — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A 5-fold cluster in a crystal that has no 5-fold axis. A cluster of 10 points with an exact 5-fold axis at the centre of each cell, repeated by the lattice. Two measurements, on the same points. The cluster is carried onto itself by a turn of 72° to within 2e-16 of a cell — exact, as far as the arithmetic goes. The pattern is not: applying the same turn about a lattice point sends some atoms 1.19 of a cell from the nearest atom, which is most of the way across it. Both are true at once. The axis is a symmetry of the contents of one cell and not of the crystal, which is what non-crystallographic symmetry means and why a virus with a sixty-fold capsid can crystallise in an ordinary space group.

    A fivefold axis in an ordinary crystal

    A virus with sixty-fold symmetry crystallises in a space group that has none of it. The restriction forbids a fivefold axis to the lattice and says nothing about what sits inside one cell — so the axis is exact, the crystal genuinely lacks it, and both statements are measurable on the same set of atoms.

    part 1 · restriction
  2. p2 in one cell, p4 on average. On the left, a molecule in one orientation at a site whose symmetry is larger than its own: the arrangement has 2 operations and the detector says p2. On the right, the average over the 2 orientations the site offers, which is what a diffraction experiment measures because different cells choose differently and nothing prefers one choice. The average has 4 operations — it is p4 — and every atom in it is present in half of the cells. Both groups are detected from the point sets rather than assumed, and the difference between them is the reason a refined structure can have symmetry no molecule in the crystal has.

    The symmetry of an average

    A diffraction experiment measures an average over some 10²⁰ unit cells, and the average of several orientations is more symmetric than any of them. So a refined structure can carry symmetry that no molecule in the crystal has — including, in the worst case, a centre of inversion in a crystal built entirely of one hand.

    part 2 · restriction
  3. The most of an icosahedron a crystal can keep. Every subgroup of the sixty rotations of an icosahedron, found by closure, with the crystallographic ones marked — those whose rotation orders are all among the 1, 2, 3, 4 and 6 that a three-dimensional lattice admits. The largest is 23, of order 12, at index 5; everything containing a fivefold axis is refused. So a crystal containing an icosahedral molecule may fix a twelfth of the molecule's own symmetry and no more, and the remaining 5 orientations have to be related by something other than the site's symmetry.

    The most of an icosahedron a crystal can keep

    C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.

    part 3 · restriction
  4. 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis.

    Five copies, and the gap they leave

    Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

    part 4 · restriction
  5. orders 5 and 7 reach a site of symmetry 1 and no more. A molecule whose only symmetry is one n-fold axis, and the highest site symmetry it may occupy in any of the 45 space groups this site builds. The site's symmetry has to be a subgroup of the molecule's, so the site's order must divide n and the site group must be cyclic. Orders 1, 2, 3, 4 and 6 reach a site of their own order. Orders 5 and 7 reach one, because no site symmetry in any space group contains an operation of order five or seven — the orders available are 1, 2, 3, 4, 6, computed by asking every operation of every group whether it moves a point. A five-fold molecule keeps its axis; the crystal simply has no use for it.

    What a molecule gives up to sit in a crystal

    A molecule brings its own symmetry. A crystal offers sites with symmetries of their own, and the two have to be compatible — the site's symmetry must be a subgroup of the molecule's. So a molecule may always keep more than its site offers, and a molecule with a five-fold axis may sit only where the crystal offers nothing at all.

    part 5 · restriction
  6. 19 site symmetries, and the counts each can impose. Every distinct site symmetry across the space groups this site builds, named by the multiset of its operation types, with the orientation counts a disordered molecule there may take. The counts are the indices of the site group's subgroups, computed by closing subsets under multiplication rather than looked up. Nearly every row offers every divisor of its order. One does not: a site of order twelve whose group is the tetrahedral rotation group refuses an orientation count of two, because that group has no subgroup of order six.

    How many orientations a disorder needs

    A molecule at a site with more symmetry than it has resolves the contradiction by occupying several orientations at once. How many is not fitted: it is the index of the molecule's symmetry in the site's, so the occupancy is the reciprocal of a whole number — and one site in the census refuses a divisor of its own order.

    part 6 · restriction
  7. Disorder models against occupancies, site by site. Every distinct site symmetry in the space groups built here — 19 of them — with its order, its number of subgroups, the number of distinct disorder models, which are the subgroups up to conjugacy by the operations of the site symmetry, and the number of different occupancies those models can have. The last column is the largest number of models that share one occupancy. In all, 162 models share far fewer occupancies; the most crowded is 4/mmm, where 11 different models all give an occupancy of 1/4. At every site, the classes of operation a model keeps separate it from every other model with the same occupancy.

    The occupancy does not name the disorder

    A molecule disordered on a special position takes a number of orientations fixed by a group index, and its occupancy is the reciprocal. Many different disorders share one occupancy — eleven at a single kind of tetragonal site — and what separates them is which of the site's operations the molecule keeps, which the averaged structure records and the occupancy does not.

    part 7 · restriction
  8. The average is the site's orbit, with occupancies. A molecule at a site of symmetry mmm keeping a subgroup of order two takes four orientations, and the average over them is the site group's orbit of each of the molecule's atoms, every image at one over the length of its own orbit. Atoms in general positions give eight images at an eighth each, and give the same eight whichever subgroup the molecule keeps. Atoms on a locus the model keeps give a shorter orbit at a higher occupancy, drawn larger and darker, and those are the only atoms that differ between models. The total scattering is the same for every model, so all of them agree exactly at zero scattering angle.

    The molecule size that hides a disorder

    Two disorder models with the same occupancy leave averaged structures that differ only in a handful of partial atoms. The difference is 20% in structure factors for a ten-atom molecule and 3% for a sixty-atom one — so the data choose between the models for a small molecule and stop choosing for a large one, and seven pairs are identical at any size.

    part 8 · restriction
  9. What refinement takes back from a wrong disorder model. Every directed pair of disorder models sharing an occupancy, 462 in all: data made from one model's averaged structure, the other model refined against them with positions on its own loci, two-component displacement parameters and fixed occupancies. Across is the residual between the two ideal structures; up is the residual after refinement. Points on the dashed line lost nothing to refinement; the median point lost 58 per cent of its residual. Every point on the floor, where the wrong model fits exactly, is one whose loci contain the true model's.

    What refinement takes back from a wrong model

    Two disorder models sharing an occupancy leave averaged structures typically a sixth apart. Refine the wrong one against the right one's data and it takes back more than half of that — all of it wherever its loci contain the true ones. Yet the data still choose, even for a molecule of sixty-four atoms, because a likelihood summed over every reflection sees what no single reflection can, and the wrong model's partial atoms give it away through the size of their displacement parameters, not their shape.

    part 9 · restriction
  10. Better counting makes an uncorrected smooth error choose the wrong model. Eighteen pairs of disorder models, each decided by the likelihood ratio at the 0.1 per cent level, at sixty-four and a hundred and twenty-eight atoms, as the random error falls from 3% to 1% to 0.3% of the mean amplitude. In each group, the first bar has random error only, the second adds a smooth scale error of 3% across reciprocal space, and the third adds the same error with a degree-two scale surface refined alongside the structure. With random error only, better counting decides more pairs correctly. With the uncorrected smooth error it stops helping and the number decided for the wrong model grows, to three at a hundred and twenty-eight atoms. With the surface refined, the result is the random-only one in every group.

    Better counting, and the wrong model

    A smooth error across reciprocal space, as large as the random one, hardly changes which disorder model the data choose: it enters both models' residuals and cancels from their difference. What it changes is where better data lead. With random error alone, counting harder decides more pairs correctly. With the smooth error left in, counting harder stops helping and starts deciding pairs for the model that did not make the data — at the 0.1% level, and more of them the better the counting.

    part 10 · restriction
  11. Which sites a smooth error can mislead. For the eighteen pairs of disorder models at three site symmetries, under a smooth scale error of degree two (five molecule sizes) and of degree four (three sizes): how many verdicts of perfect counting go the wrong way, how many pairs no data can decide, the largest correlation between the error and the two models' difference, and how many harmonics of the error's degree the site's group leaves unchanged. At the cubic site the degree-two error correlates with nothing, to rounding, and decides no pair wrongly; the degree-four error, which the cubic site keeps one harmonic of, reaches it.

    A cubic site is blind to a degree-two error

    A smooth scale error left uncorrected made perfect counting choose the wrong disorder model for eight of ninety pairs and sizes, and the obvious way to predict which is to correlate the error with the two models' difference before refining anything. That correlation predicts none of them. What it does predict, exactly, is which pairs can never go wrong: at a cubic site a scale error of degree two correlates with nothing and is absorbed by nothing, because a cubic group keeps no harmonic of degree two. A degree-four error, which it keeps one of, reaches the cubic site at once.

    part 11 · restriction
  12. The first-order prediction against the actual verdict, for every pair and size. The 85 decidable pairs of disorder models at five molecule sizes, each placed by the first-order prediction of the wrong model's misfit minus the right model's, made from the two error-free refinements and their Jacobians, and by the actual difference after refining against data with a 3% smooth error. Shaded quadrants are agreement. 77 agree. Of the 8 wrong verdicts, ringed, 4 are predicted; 4 right verdicts are predicted wrong.

    The wrong model moves too far for first order

    Which disorder models a smooth error will turn the wrong way was left as a calculation to make before any data exist: project the error onto the directions each refinement's parameters can move. Made, it gets the sign of the verdict right in seventy-seven of eighty-five cases, catches four of the eight wrong verdicts and raises four false alarms. Every case it gets wrong is either a near-tie or a refinement that absorbed more than twice, or less than half, of what its first-order directions allow.

    part 12 · restriction

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