Crystallography & Symmetry
patterns derived from their groups
Concept

Wyckoff set — where it appears

A Wyckoff position together with every other position a group's normaliser carries it onto. One arrangement cannot be said to occupy one member rather than another, so a set rather than a position is what a structure determines, and two correct reports of one crystal may name different positions.

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

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 Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points.

Three of them, and they are equivalent

The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.

operations · Normalisers 1,694 positions, 1,182 sets on the general cell, and 1,128 on the best. Every affine type of space group by crystal system: its Wyckoff positions, the sets those fall into under the normaliser on the cell the group requires, on the best cell it may sit on, and under the whole affine normaliser. The total is 1694 positions, 1731 with enantiomorphs counted twice, 1182 sets on the general cell and 1128 on the best, which equals the affine 1128 in every system. The two columns differ only in the orthorhombic system and in one monoclinic type.

One cell reaches every Wyckoff set

In the plane, every identification the affine normaliser makes between two positions is made by an isometry of some cell, and the essay that found this warned against carrying it into space, where changes of basis can mix axes without limit. It carries. The 1,731 positions of the 230 groups were found from stabilisers and the sets from normalisers, and on one cell for each type the two partitions agree. Past the monoclinic system that is a one-line average. Below it, the infinite normaliser turns out to have nothing left to join.

operations · Normalisers

Named alongside it

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

NormaliserConjugationEquivalent originHolohedryMetric tensorSite symmetryWyckoff positionsConjugacy classEuclidean normaliserIndexKlassengleicheNormal subgroup

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