Theme

The theme: No origin removes it

An operation's translation splits in two: a part that belongs to the operation, and a part that only records where somebody put the origin. Almost every argument about space groups is about telling them apart, and the first half of the split is the whole difference between a rotation and a screw.

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.

Into space

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.

The translation of every operation, split in two. Every operation of P2, P2₁, Pm and Pc other than the identity, with its translation split into the intrinsic part — one n-th of the sum of the operation applied to itself n times, which no choice of origin can remove — and the location part, which is only a statement about where the origin was put. 2 of the 4 operations shown have a non-zero intrinsic part, and those are exactly the screws and the glides.

Into space

The half of a translation that is not a choice

Every operation's translation splits in two — a part that belongs to the operation and a part that only records where somebody put the origin. Almost everything peculiar about space groups is a consequence of that split, including why there are two hundred and thirty rather than seventy-three.

Into space

Sixteen candidates, ten groups

One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact.

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.

Into space

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.

The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. 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.

Into space

Turning and climbing at once

A rotation has a fixed point and a screw has none. That sounds like a small difference and it is the reason a space group is not a point group with extra letters, the reason two hundred and thirty is not seventy-three, and the reason a helix can be a crystal.

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.

Operations

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.

The five kinds of glide plane. All five glide letters: a, sliding by a/2; b, sliding by b/2; c, sliding by c/2; n, sliding by (a+b)/2, (b+c)/2 or (a+c)/2; d, sliding by (a+b)/4, (b+c)/4 or (a+c)/4. 3 of those drawn are axial — the slide is half of a single cell vector lying in the plane. A glide's square is a pure translation, so its slide can only be half of a lattice vector, and the quarter-cell d glide exists only where centring has already made the half-diagonal a lattice vector.

Into space

Reflect, then slide by half of something

A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.

$What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.$ How it is known

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

The symmetry elements of Ccmm. Space group Ccmm, number 63, projected down c on a C-centred orthorhombic cell. The symmetry elements drawn: 4 2-fold rotation axes, 6 glide planes, 12 2₁ screw axes, 2 mirror planes, 8 inversion centres. 2 kind(s) of element in this group have no line or mark in a projection down c — an axis at an angle to the page, or a plane lying parallel to it — and are not in the picture.

Into space

The operations nobody put in

A group is not a list of generators. Compose two of them and something arrives that neither contained — a screw where there were only mirrors, a glide where there was only a mirror and a centring vector — and in three dimensions most of a group's operations get there this way.

P2₁2₁2₁: the sections its symmetry forces. The Patterson cell of P2₁2₁2₁ with the sections marked. Each operation (M, t) sends an atom at x to Mx + t, so the vector between them is (I − M)x − t; where I − M is singular that vector cannot leave a plane, and the plane's equation comes from the left null space in integers. This group has 3 such operations, giving the sections w = 0.5, u = 0.5, v = 0.5. A heavy atom's vector to its own image is somewhere on one of them, which is what made structure solution possible before computers: a plane can be searched by eye and a volume cannot.

How it is known

Where symmetry stacks the vectors

A Patterson map of a real structure is a blur with thousands of overlapping peaks. A screw axis rescues it: the vectors between symmetry-related atoms cannot leave a plane, so the search for a heavy atom is a search of a section rather than of a volume — and which plane it is falls out of the operation's matrix in integers.

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.

Into space

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.

Fddd has two published origins. The two conventions, computed from the operations. The International Tables place the origin at the point of highest site symmetry, and also at a centre of inversion, and for this group those are different points — so the group is printed twice, with every coordinate in the second table shifted by (-0.125, 0.125, -0.125) from the first. A structure published on one and read on the other has every atom in the wrong place by that vector, the refinement fails in a way that looks like bad data, and nothing in the symbol says which was used.

Into space

Two origins for one group

The International Tables place the origin at the point of highest site symmetry, and also at a centre of inversion. For twenty-four of the two hundred and thirty those are different points, so the group is printed twice with every coordinate shifted — and nothing in the symbol says which table a structure was written against.

Each arithmetic class holds exactly one symmorphic group. The arithmetic crystal classes this site enumerates in full, with the number of space groups each produces and the number of those that are symmorphic — that is, that have an origin at which every operation's translation part vanishes. The right-hand column is one in every row, over 25 groups in all, and the figure asserts it rather than reporting it. That is the bijection behind the number 73: there are seventy-three arithmetic crystal classes in three dimensions and seventy-three symmorphic space groups, and the correspondence is this one, class by class.

Into space

One symmorphic group per class

Every arithmetic crystal class holds exactly one space group in which some origin clears every translation part at once. That bijection is why there are seventy-three symmorphic space groups and seventy-three arithmetic classes, and it is checked here class by class rather than counted.

The symmetry of a cut through Pnma. Every height in one cell, and the number of operations of Pnma that map the plane at that height to itself. The answer is 2 almost everywhere and rises to 4 at the special heights, where the plane group named above the spike is what a reader looking down at that surface would see. The rule is two conditions and no more: the operation must not tilt the plane, and the plane must come back to its own height — so a twofold axis lying in the plane survives at two heights per cell and a screw axis along the normal survives nowhere.

The classification

What a cleave leaves

A surface is a crystal that has been cut, and the symmetry it presents is what the space group leaves of itself on that plane. Two conditions decide it — the plane must not tilt, and it must come back to its own height — and the answer changes with where the cut was made.

Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused.

Into space

The plane that carries two glides

A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop it.

Pnma has 6 names. The group Pnma with its three axes relabelled in each of the six possible ways. Every row is the same group, and each row is checked as it is drawn: the change of basis has determinant one, conjugating back gives the original operations exactly, and the census of screws, glides, mirrors and rotations is unchanged — and 6 different symbols come out. Each symbol is derived from the conjugated operations, not looked up, by the same routine that has to reproduce the symbol every group was entered under.

Into space

Six ways to name one group

Pnma is also Pmnb, Pbnm, Pcmn, Pmcn and Pnam. Nothing about the crystal changes between those six; what changes is which axis was called a. In an orthorhombic group the axes are inequivalent and unlabelled, and naming them is a choice made six ways.

Operations

What is left when the order is forgotten

Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate.

P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn.

Operations

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

Operations

One crystal, and sixteen coordinate lists

Two structure reports can disagree in every number and describe the same arrangement of atoms, because a space group does not fix its own origin or its own axes. How many genuinely different lists there are is the index of the group in its Euclidean normaliser — a number, computable, and the thing a structural database has to divide out before it can say two entries are the same compound.

Three atoms, four resolutions. A one-dimensional Fourier synthesis of the same three atoms, cut at four different resolutions. Nothing is approximate except the edge: every amplitude and every phase used is exact, and the only information withheld is the reflections outside the sphere. The peaks broaden as the cut-off comes in, and beside every peak sits a negative ripple that the coarsest map cannot distinguish from a real absence of density. Both effects are the transform of the sphere rather than anything about the structure.

How it is known

As sharp as the sphere is wide

A map made from a truncated sum is not a blurred picture of the structure. It is the structure convolved with the transform of the sphere — so peaks acquire a width proportional to the resolution, and a negative ripple of twenty-two per cent that no improvement in the data ever reduces.

75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it.

The classification

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

p = 2: 1, 3, 6, 12, 24 vertices at each distance. Every sublattice of index a power of 2, up to scale, joined when one contains the other with index 2. From the whole lattice there are 3 ways down, because a sublattice of index 2 is a line over the field of 2 elements and there are 3 of those; from each of those there are 3 again, one of which is the way back. So the counts are 1, 3, 6, 12, 24 — that is (2 + 1)·2^(k−1) — and the graph has no cycles, both of which are checked on every vertex whose whole neighbourhood was grown rather than read off the picture. The object is the Bruhat–Tits tree of the p-adic plane, and it is what the set of sublattices is rather than how many there are.

Lattices

Every way down, and no way round

There are as many sublattices of a given index as the index has divisors, and counting them is where that essay stopped. This one asks what they are to each other, and the answer is a shape: an infinite tree in which every vertex has exactly p + 1 neighbours and no path ever comes back.

15 classes may rotate light, 11 of them chiral. A crystal is chiral when its point group contains no improper operation, and there are 11 such classes. A crystal may rotate the plane of polarisation when its class permits a non-zero gyration tensor, and there are 15. The four in the difference — 4̅, m, 4̅2m, mm2 — are achiral and may still rotate light, which is why the two words are not synonyms. In each of the four, symmetry forces the tensor to be traceless, so the rotation changes sign with direction and cancels in any average over directions.

What symmetry decides

Fifteen may rotate light, and eleven are chiral

Optical rotation and handedness are treated as the same thing and are not. Eleven crystal classes are chiral; fifteen permit a crystal to rotate the plane of polarisation; and the four in between are the reason quartz and sodium chlorate are the examples everybody uses.

pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

How it is known

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

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