Settings — the series · Crystallography & Symmetry

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.

part 1 · space-groups

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.

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.

part 2 · space-groups

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.

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.

part 3 · space-groups

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.

part 4 · space-groups

530 printed rows, 499 descriptions, by crystal system. For each crystal system: the number of space group types, the number of distinct descriptions up to an origin shift, the number of rows in the printed list, and the difference. All thirty-one extra rows are monoclinic or orthorhombic.

Five hundred and thirty rows, four hundred and ninety-nine descriptions

The International Tables list 530 settings of the 230 space groups, and that number has long been quoted as a measure of how much naming a group can take. Counted from the operations, a setting being a description up to a shift of origin, the answer is 499. Thirty-one printed rows name a description already on the list, all of them in six types, and the rule that gives a group its second origin finds twenty-three of the twenty-four and misses one.

part 5 · space-groups