Crystallography & Symmetry
patterns derived from their groups
Concept

Occupancy — where it appears

The fraction of a crystallographic site that is actually filled, averaged over the crystal. A refinement reports it as a number between zero and one, and questions about connectivity, conduction and rigidity depend on the arrangement it averages over rather than on the number itself.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

restriction · Local symmetry One crossing or the other, and never both. Two events on a square patch of lattice: a path of occupied sites crossing from left to right, and a path of vacant sites crossing from top to bottom. On the triangular lattice exactly one of them happens in every configuration tested — the claim is combinatorial rather than statistical, so one counterexample would end it. On the square lattice both can fail at once, and do, in more than a quarter of the configurations. That difference is the whole of what follows.

The threshold a symmetry pins down

Occupy sites at random and somewhere the occupied ones first join up across the crystal. For almost every lattice that occupancy is known only to a few digits. For the triangular lattice it is exactly a half, and the reason is that on a lattice whose faces are all triangles an occupied path and a vacant path cannot slip past each other — a statement about one configuration at a time, with no probability in it.

diffraction · Disorder 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.

restriction · Local symmetry 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.

restriction · Local symmetry A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern.

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

aperiodic · Monotile 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.

restriction · Local symmetry

Named alongside it

The objects these essays reach for when they reach for this one.

Average structureDisorderSite symmetryOrbitSpecial positionStabiliserIndexSubgroupAccidental symmetryAperiodicityBragg peakConfigurational entropy

All concepts