Crystallography & Symmetry
patterns derived from their groups
Concept

Stabiliser — where it appears

The operations that leave a given point exactly where it is, a subgroup whose order times the orbit's length is the order of the group. It is why symmetric arrangements have short orbits and why a count of them is an average rather than a division.

Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.

{111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone.

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

applied · Forms Why p4 cannot be drawn with dots. The same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.

The motif must be a comma

A dot is too symmetric to illustrate most wallpaper groups. Its orbit acquires mirrors nobody asked for, and the resulting figure is quietly of a different group from the one in its caption.

classification · Accidental symmetry Growing the p4 orbit. One motif, then more of the group's operations applied to it, until applying another produces nothing new. The pattern is the orbit; the drawing is only its shadow.

The orbit is the pattern

A wallpaper is not designed and then found to have symmetry. It is the set of places a group sends a single mark, and once that is taken literally the pattern can be grown, checked, and caught out.

operations · What symmetry is Special positions in p4m. Every point of a 12×12 grid inside the cell of p4m, drawn at a size set by how many operations fix it. 80 of the 144 are general — nothing but the identity leaves them alone, so their orbit is the full 8 points. The other 64 are special, and fall into 3 kinds: 60 points fixed by 2 operations, with orbits of 4; 2 points fixed by 4 operations, with orbits of 2; 2 points fixed by 8 operations, with orbits of 1.

The points a group treats differently

Almost every point of a cell has an orbit as long as the group. The exceptions are the points some operation leaves alone, and they are where atoms sit, where a structure's formula comes from, and where a careless motif destroys the group it was meant to illustrate.

operations · Fundamental domain A fundamental domain for p4m. One representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

The fundamental domain

The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.

operations · Fundamental domain The general positions of P2₁/c. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. 12 general positions, the orbit of a three-point asymmetric motif, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

One part in however many, and why it is never quite that

A crystal's contents are the asymmetric unit repeated by the group. The unit's volume is the cell's divided by the order of the group — except that it is always a little more, and the excess is exactly the special positions counted whole.

operations · Fundamental domain 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 The friezes inside the seventeen. Every plane group contains frieze groups: keep only the operations that map one lattice row onto itself and what is left is a group on a strip, which must be one of the seven. Across all seventeen plane groups and their principal directions, all seven frieze groups appear. The commonest is p2, in 9 of the 32 rows examined.

The friezes inside the seventeen

Take one lattice row of a wallpaper pattern and keep only the symmetries that leave that row where it is. What survives is a frieze group — and which of the seven it turns out to be is a fact about the plane group that its symbol does not state.

classification · Friezes What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

operations · Counting What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

operations · Counting The cell of 3.4.6.4, and the vertices in it. 3.4.6.4 drawn with the cell its own translations define. The lattice is hexagonal and the cell holds 6 vertexes, marked. Neither was chosen: the translations are the vertex-to-vertex vectors that carry every polygon of the patch onto a polygon of the patch, and the cell is the shortest independent pair of them. Expressed in that basis the vertices have coordinates that are exact and are not fractions — a vertex of this tiling sits at 1/(1 + √3) of a cell — which is why the detector that decides its group works in ℚ(√3) rather than in the rationals.

Eleven tilings, five groups

Hand each of the eleven uniform tilings to a detector that has never heard of tilings and ask what its symmetry is. Six of them answer p6m. Twelve of the seventeen wallpaper groups never appear at all — and the coordinates the question has to be asked in are not fractions.

classification · Tilings Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs.

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

restriction · Finite groups 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.

restriction · Local symmetry anisohedral: 2 orbits of congruent tiles. A tiling of the plane by 8 copies of one shape per cell of a lattice of index 64, drawn 1 cell across and 8 up, and coloured by which orbit of the tiling's own symmetry group each tile belongs to. The group has 4 operations per cell and 2 orbits: every tile is congruent to every other, and no motion of the whole pattern carries a tile of one colour to a tile of another. Congruence is a fact about the shapes; an orbit is a fact about the pattern, and they are different facts.

One shape, two kinds of tile

A tiling by copies of a single shape looks as though it must be homogeneous — every tile is congruent to every other, so what could distinguish them? The symmetry group can. There are shapes that tile the plane and admit no tiling whose group carries any tile to any other, and the smallest of them has eight cells.

classification · Isohedral Sublattices of index n, in space. How many sublattices a three-dimensional lattice has at each index, beside the plane's answer, with the Hermite enumeration and the coefficient of ζ(s)ζ(s−1)ζ(s−2) in separate columns. The two are computed by routines sharing no code, and a row where they disagreed would be a failure rather than a result. The last column counts the ones that survive every operation of the cubic group, and it is almost always empty.

The three that stay cubic

A lattice in space has far more sublattices than one in the plane — 651 of index sixteen against 31 — and almost none of them keeps the symmetry it came from. The ones that do exist at indices m³, twice m³ and four times m³, there is exactly one at each, and they are the primitive, face-centred and body-centred cubic lattices, arrived at by asking which sublattices keep a symmetry rather than by enumerating centrings.

lattices · Sublattices square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here.

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

applied · Defects 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 Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm.

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

operations · Normalisers

Named alongside it

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

OrbitSpecial positionSite symmetryAsymmetric unitMultiplicitySubgroupWyckoff positionsAccidental symmetryAverage structureEnumerationHolohedryIndex

All concepts