The theme: No origin removes it — page 3
Into space
Two operations make rock salt's group, and Pmmm needs six
Every space group that its own rotations, screws, mirrors, glides and inversions can build is built by a handful of them, and how small a handful is now known for all 212. A quotient of each group says how many it cannot do without, a search finds that many for 198, and for the other fourteen a finite quotient proves that one more is needed. Rock salt's group needs two operations, Pmmm six, and P312 needs four where its abelian quotient allows two.
How it is known
The lead is wherever it is taken out
Charge flipping solves a cell of nineteen carbons and one lead once the lead's own density is taken out before each flip and put back after. The obvious worry was precision: a lead subtracted a fraction of a grid step from where it really is leaves a residual larger than any carbon. The residual is there, larger than a carbon at a tenth of a step. It does not matter. Subtracted two steps away, or half the cell away, the lead still lets the solver succeed at the same rate, because the solver builds the whole structure round wherever it is put.
Operations
Where the plane had places, space has pitches
Conjugacy in a plane group came down to one lattice, and it counted four families of half-turn to a cell, three of three-fold rotation, two of quarter-turn and one of sixth. In space the same lattice gives the same four numbers for every one of the 219 space-group types. What changes is what the families are. In a centred lattice, and along some directions of a primitive one, they are not rotations in different places but screws of different pitch, and a right-hand screw and a left-hand one in the same coset are one symmetry exactly when the group contains a reflection.
Into space
Two mirrors never make a crystal
Sixty-four space groups are generated by two of their own operations. Which two kinds can do it is a finite question, and it has been settled for all sixty-four and all fifteen pairs of kinds. A screw and a glide make thirty-two of them. Two mirrors make none, and neither does a rotation with a mirror, for reasons that need no computation: two operations that undo themselves generate a dihedral group, and an axis and a plane always share a point or a direction. Two plain rotations make six groups, every one of them cubic and without a mirror.
Operations
Thirteen sites obey Friedel's law without a centre
At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.
How it is known
Two leads need their vector to half a step
With one heavy atom in a cell with no symmetry, charge flipping did not care where the atom was subtracted: it rebuilt the structure round whatever place it was given. With two heavy atoms that freedom runs out, as predicted, because no translation can put both under wrong references at once. Subtract one of them a whole grid step from where it belongs and the solver never succeeds. But the tolerance is five times looser than the estimate it was predicted from: up to a quarter of a step the rate does not move, at half a step it halves, and the residual the solver absorbs is three carbons' worth, not one.
Into space
Three mirrors never make a crystal either
Ninety-three space groups need three of their own operations to generate them, and which three kinds can do it is a finite question with 3,255 cases. All but two are now decided. Three mirrors make none of the ninety-three, as a theorem about reflection groups requires. A rotation with two mirrors makes exactly three, all cubic. Three plain rotations make thirteen, every one without a mirror. One group can be made thirty ways and three can be made only one.
Into space
A screw keeps a line and a glide keeps a plane
The census of the seventeen zones left two predictions for space. A transition at the zone boundary should meet a gradient term where one screw or glide is active, along that operation's direction, and a second should remove it. Across all 219 types, the first half holds without exception in the monoclinic groups and the direction is wrong: a glide's term fills its plane. The second half fails from the orthorhombic groups on. And the zone centre, empty in the plane, is not empty in space.
Operations
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.
Into space
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.