Crystallography & Symmetry
patterns derived from their groups
Concept

Space group — where it appears

The symmetry group of a three-dimensional periodic structure, of which there are two hundred and thirty. The factor of thirteen over the seventeen plane groups comes from an operation the plane cannot hold: a translation no choice of origin removes.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

The step a flat surface has no room for

Seventeen becomes two hundred and thirty, and the factor of thirteen is not a matter of having more directions to work in. It comes from an operation the plane cannot hold — a translation that no choice of origin will remove.

space-groups · Space groups 2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

space-groups · Space groups P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

One group, three symbols

P2₁/c, P2₁/a and P2₁/n are the same space group written on three choices of axes, and the literature contains all three as though they were different. Deriving a symbol from a group's own operations shows why — and found an entry on this site that had been carried under another setting's name for two phases.

space-groups · Settings One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours.

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

applied · Packing How many close packings there are of each period. Every cyclic sequence over three letters with no two adjacent alike is a close packing, and two sequences describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions. Counting the classes that remain gives 1 of period 2, 1 of period 3, 1 of period 4, 1 of period 5, and 38 altogether up to period 10. Period two is hexagonal close packing and period three is cubic; everything above them is a polytype, equally dense and equally close packed, and silicon carbide has been found in more than two hundred of them. Nothing in the geometry chooses. What chooses is an energy difference of a few thousandths of an electron volt per atom, and this site computes no energies.

How many polytypes there are

One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.

applied · Packing P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra.

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

space-groups · Space groups Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

space-groups · Flat space p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

The groups a single hand may sit in

A protein is built from one enantiomer of every amino acid, and a crystal of it contains nothing else. That single fact deletes most of the classification at a stroke: any operation reversing orientation would put the other hand in the same crystal. The criterion is one line of arithmetic, and in the plane the enumeration is complete — five of the seventeen.

space-groups · Chirality Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other.

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

applied · Packing One local configuration, and a number of structures that doubles. How many kinds of adjacent pair a close-packed stack has, how many kinds of triple, and how many stackings of each period there are. The first column never moves: every pair of layers is congruent to every other, at every period, which is what an order-disorder family is. The last two agree with 2ⁿ + 2(−1)ⁿ, which is the chromatic polynomial of a ring of n layers at three colours, because a stacking of period n is exactly a proper three-colouring of that ring. The gap between the first column and the last is the whole subject.

A stack with no space group

Every pair of layers in a close-packed stack is congruent to every other pair, and the number of stacks doubles with every layer added. A family whose local configuration is completely determined and whose global structure is not determined at all has no single symmetry group — what it has is a set of operations that compose only when their ends match, which is a groupoid.

applied · Packing Which indices a screw axis contains itself at. The fifteen kinds of axis a space group may have, against the index of the sublattice taken along the axis. A filled cell is an index at which the axis contains a copy of its own kind; the darker cells are the indices at which what comes back is the mirror image instead. A pure rotation axis is filled everywhere and a screw is not, and which indices a screw loses is decided by one congruence rather than by any geometry.

A screw that contains its own mirror image

No operation of a crystal turns a right-handed screw axis into a left-handed one — that is what makes the eleven enantiomorphic pairs pairs. And yet a 4₁ axis contains copies of 4₃ as subgroups, at every index congruent to three modulo four. One congruence decides both which indices are possible and which hand comes back, and it is the same congruence for all fifteen kinds of axis.

space-groups · Isomorphic subgroups One change of setting, four rules. The four things a structure report contains and the rule each obeys under a change of setting with basis change P and origin shift p. The cell and the indices are multiplied by P; a coordinate is multiplied by its inverse, after the origin has been subtracted; and an operation is conjugated and then shifted by (W − I)p, a term the other three have no equivalent of. Applied to Pnma with the change below, the operations still close into 8, the orbit maps point for point, and every |F| is unchanged.

One matrix, four rules

Changing the setting of a structure is one matrix and one origin shift — and the cell, the coordinates, the indices and the operations each obey a different rule under it. Three of the four ways of getting it wrong still leave a closed group of the right order, so closure catches none of them.

space-groups · Settings P4₁: the lattices, as an ideal across and a multiple along. Every sublattice the point group of P4₁ carries to itself, indexed by the norm of the ideal it uses across the axis and by the multiple it takes along it. The entry is the space group that sits on it: the parent's own type in one colour, a different type in the other, and a dash where no group with the parent's point group survives at all. A dot marks a lattice that is maximal — one whose step is a single prime, across or along, with nothing between it and the whole. The rows and columns are two divisibility orders and the table is their product, which is the shape the plane's answer predicted.

An ideal across and a prime along

In the plane a copy of a group inside itself grows by a prime ideal, and the maximal indices are the norms of the primes of a ring. In space with one principal axis there are two directions to grow in, and the question the plane left was whether the two constraints multiply. They do not — and the place they fail is an index the plane calls maximal, because the step in between carries the group's mirror image.

space-groups · Isomorphic subgroups A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does.

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

operations · Composition

Named alongside it

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

Screw axisEnantiomorphClose packingGlide planeHandednessPolytypeStacking sequenceChange of basisChiralityCosetFixed pointIndex

All concepts