Crystallography & Symmetry
patterns derived from their groups
Concept

Euclidean normaliser — where it appears

The motions of space that carry a space group onto itself while keeping every length of its cell. Each one rewrites a crystal's description without changing the crystal, so the size of this group says how many coordinate lists one structure has. Which motions qualify depends on the cell as measured, not only on the group.

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

The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

The same pattern, described twice

Two coordinate lists for one structure can disagree in every number and describe exactly the same arrangement, because a group does not fix its own origin. The operations that may be applied to a description without changing what it describes are its normaliser, and they can be found by looking at pictures rather than at matrices.

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 The subgroups of index two, and what joins them. Every affine type of space group by crystal system: its subgroups of index two, 1813 in all; classes under the origin shifts alone, 1392; classes under the whole affine normaliser, 1159; and the fewest classes one cell can give, 1161. Best cell and affine agree in every system except the triclinic.

No cell joins the seven halves

Every identification the normaliser makes between Wyckoff positions is made by an isometry of some cell, and it was natural to expect the same of subgroups. It nearly is. Across all 219 types and their 1,813 subgroups of index two, one cell does everything the normaliser does, except in the two triclinic groups. P1 has seven ways to be halved, and its normaliser makes them all one. No cell can, because seven is the number the crystallographic restriction forbids.

operations · Normalisers

Named alongside it

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

NormaliserEquivalent originHolohedryMetric tensorConjugationCrystallographic restrictionDescriptionOrbitOrigin choiceSite symmetrySubgroupWyckoff positions

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