In space a motif can lift two positions in three

Where a symmetric motif lifts the group settled a question about the seventeen plane groups. The motif must be a comma warned of the hazard, and that essay counted it. Put a motif with some symmetry of its own at a point of a group and repeat it, and the pattern may have more symmetry than the group that made it. The criterion is one line, and it uses the same object as the same pattern, described twice: a symmetry of the motif survives into the pattern exactly when it normalises the group. Counted over every position, twenty-nine of the seventy-two can be lifted. General positions can be lifted in only four groups, and ten placements shrink the cell.

It ended by naming the same census in space, and by making two predictions for it. General positions that can be lifted should occur exactly in the groups with a polar direction. And there should be lifts that shrink the cell, where a molecule of higher symmetry makes a crystal’s true cell smaller than the one its group prescribes, the warning a crystallographer meets in structure solution. The essay said the two hundred and thirty groups have “well over a thousand” positions and that nobody had counted.

With the positions and normalisers of all 219 types in hand, the count takes a couple of minutes. The second prediction is true on a scale that was not expected: a shrinking lift is available at more than one position in three. The first prediction is half right. Polar is necessary, without exception. It is not sufficient, and the groups that refuse are worth seeing, because they refuse for a reason a plane group cannot show.

What is being asked of each position #

The candidate symmetries a motif can bring are the operations of the holohedry of the lattice, the point group of the general cell of its type. A motif symmetry outside that, a five-fold axis say, can lift nothing, because no lattice-preserving motion has that linear part. Each candidate is made into the affine map that fixes the motif’s centre $x$. The map is tested against the group: if conjugating by it returns the group’s operations, it normalises the group and the motif’s symmetry survives. If not, some copy of the motif lands where no copy is.

The care needed is in choosing $x$. A position, in the sense of the points a group treats differently, may be a point, a line or a plane, and a lift must be a property of the whole position rather than of one grid point. A point on a mirror plane at a coordinate of exactly $\tfrac14$ can satisfy arithmetic that a generic point of that mirror does not. So an operation counts only if it normalises at several sampled points of one connected piece of the locus, all with exactly the same stabiliser. The first version of this census took the first point it found in each position, and it reported lifted general positions in 128 non-polar groups, since a group’s first free point can be its own origin. Choosing better points brought that to ten, including several cubic groups. All of them were grid arithmetic, and sampling a whole connected piece of each locus removed the last ten.

A lift is then read off by closing the group with the new operation. If the closure contains a translation the group did not have, the lift shrinks the cell.

The picture at the head of this essay is the census. Two positions in three can be lifted. The share runs from one in ten in the triclinic system, where the only candidate beyond the identity is a centre, to three in four in the tetragonal system.

Comparing with the plane is instructive because the two share a criterion and nothing else. The share of positions that can be lifted goes from forty per cent to sixty-seven. The share where the cell can shrink goes from fourteen per cent to thirty-six. Space has a third axis to exchange, more holohedry to draw candidates from and, crucially, many more groups with screws and glides, and a screw or a glide next to a matching pure operation is exactly what generates a new translation.

A lift that halves the cell #

The easiest case to read is the smallest. The group $P2/c$ has a two-fold axis along $b$, a glide normal to $b$ translating by half of $c$, and inversion centres at height $z = \tfrac14$. Put a molecule on one of those centres, which is a Wyckoff position every monoclinic crystallographer has met and the case what a molecule gives up reads from the other side, and suppose the molecule has a mirror of its own, normal to $b$ and through its centre.

That mirror normalises the group. So it survives into the crystal: the pattern now has a pure mirror normal to $b$ as well as the glide normal to $b$. Composing the two leaves a pure translation of half of $c$. The crystal’s true $c$ axis is half the length the group says it is, and every reflection with odd $l$ is absent for a reason no glide rule in the group accounts for. A refinement in $P2/c$ on that cell will proceed, and it will not converge on anything sensible, because it is describing one atom twice.

That is the shape of the warning, and the census says how common it is. Six hundred and six positions allow it, across 126 of the 219 types. Near-symmetric molecules on such sites are the translation that is nearly there, where no reflection quite vanishes and a statistical test answers wrongly, and near-symmetry and the tolerance is what becomes of the exact statement as the molecule’s mirror stops being exact.

The largest counts by class are $mmm$ at 160 positions and $4/mmm$ at 151, the two classes with the most glides beside mirrors.

In a holohedral group every lift halves something #

That pattern has a sharper form, and it is the one result here the plane essay gave no hint of. Take a group whose point group is the whole holohedry of its lattice: $2/m$, $mmm$, $4/mmm$, $6/mmm$, $m\bar3m$. Every linear part the lattice permits is already in the group. So a motif that lifts such a group cannot be bringing a new linear part, because there are none left. It can only be bringing a linear part the group already has, attached to a different translation.

Two operations with the same linear part and different translations compose to a pure translation, and that translation cannot have been in the group, or the new operation would have been in the group too. So in a holohedral group every lift shrinks the cell, with no exceptions possible. In these five classes the census finds 388 positions that can be lifted, and 388 that can shrink the cell. There is not one lift in any of them that keeps the group’s translations. $P2/c$ above is the smallest instance: its mirror was already present as a glide, and the only thing the motif could add was the difference between them.

Read the other way, that identifies where a lift can raise a crystal’s symmetry without changing its cell at all. It requires a point group with room left in the holohedry. Leaving aside $\bar3m$, which is holohedral only on a rhombohedral lattice, the other classes hold 898 positions, of which 709 can be lifted and only 205 can shrink the cell, and that gap is the room. In the holohedral classes the same logic leaves some groups with nothing to reveal at all. $P\bar1$’s nine positions are never lifted, and neither is any position of $Pmmm$, $P6/mmm$ or $Pm\bar3m$. Being symmorphic does not decide that, since $P4/mmm$ and $Fm\bar3m$ are symmorphic too and have three lifts each, every one of which shrinks the cell.

A crystallographer choosing a space group can use this directly. Finding a site with a higher-symmetry molecule in a group whose class is holohedral is not an invitation to raise the class. It is a reason to look at the cell.

Who can be lifted #

The pattern by site symmetry is not the simple decline one would guess. Two-fold sites are lifted at 70 per cent, three-fold sites at 89 per cent, and sites of order twenty-four at 85 per cent. Only the two ends of the table differ. Sites of order sixteen and forty-eight are never lifted: forty-eight is the entire holohedry, so nothing is left, and at sixteen the remaining holohedry always fails to normalise the group, in all nine cases.

The general position is at the other end, and it is low for an opposite reason. It has no site symmetry at all, so everything is a candidate, and yet only 53 of 219 can be lifted. A general point is generic, and that is a stringent demand.

Polar is necessary #

Why should a generic point be lifted at all? The argument takes a few lines. An operation $n$ fixing a point $x$ is the affine map that sends $y$ to $n(y - x) + x$, which is $n$ with a shift of $(I - n)x$. If it is to normalise the group for every $x$ in an open region, then $(I - n)x$ must be an allowed shift for every such $x$. So the image of $I - n$ must lie among the directions along which the group’s origin can slide freely.

Those directions are exactly the polar ones. A group has a free origin along a direction only if nothing in it reverses or rotates that direction, which is the definition of polar. So a general position can be lifted only in a polar group, and the census confirms it. Not one of the 155 non-polar types has one, and that is the prediction’s first half, exact and now proved.

The argument says more than the prediction did. In a group with a single polar axis the free directions are one line, and the image of $I - n$ must lie on it. The only non-trivial holohedry element with that property is the mirror across the axis. So a lifted general position in a uniaxial polar group is always a motif with a mirror normal to the polar axis, and the crystal it makes has that mirror too. $P4$ becomes $P4/m$. $Pmm2$ becomes $Pmmm$.

Polar is not sufficient #

And that one candidate can fail, which the prediction did not foresee.

Eleven of the sixty-four polar types refuse. They are $P4_1$, $I4_1$, $I4_1md$, $I4_1cd$, $P3_1$, $P6_1$, $P6_2$, $R3$, $R3m$, $R3c$ and $Fdd2$, and the reason is the same for all of them. The only candidate is the mirror across the axis, and that mirror does not carry the group to itself.

A mirror across a screw axis reverses the screw’s sense. A four-one screw rises a quarter of the cell with each quarter-turn; in a mirror across the axis it falls a quarter with each quarter-turn, which is a four-three. So the mirror carries $P4_1$ to $P4_3$, its enantiomorph, and the group is not normalised. The same holds for three-one and for six-one and six-two. It does not hold for four-two, or six-three, since a screw of half a turn is its own mirror image, and so $P4_2$ and $P6_3$ accept the lift. On a rhombohedral lattice the mirror across the axis turns the obverse centring into the reverse, which is a different lattice, whatever the screws do. $Fdd2$’s diamond glides translate by a quarter along the axis, and the mirror sends that to minus a quarter, which $F$ centring does not recover.

The refusal is about the sense of a screw, in groups with no reflection to reverse it. In $P4_1$, $P3_1$, $P6_1$ and $P6_2$ a mirror would turn the group into its enantiomorph. $I4_1$ and its two relatives do contain both senses of screw, but on different axes, so the mirror would have to come with a sideways shift, and a motif’s own mirror at a generic point supplies none. No plane group has a screw, so the plane’s four lifted groups could not have shown this. Polar there was both necessary and sufficient, and that coincidence is why the prediction was stated the way it was.

There is one more point about $P1$. Its free directions are all of space, so every holohedry element is a candidate, but its holohedry is only a centre. A motif with a centre at a generic point lifts $P1$ to $P\bar1$, which is the best known of all these warnings: a structure reported in $P1$ whose molecule has a centre is usually a structure in $P\bar1$ with a mistake in it.

What this does not show #

A position that can be lifted is not a crystal that has been. Everything here is about a single orbit, one motif at one position. A real structure has several independent atoms, and all of them must carry the extra symmetry for the group to rise. The census says where a single-site structure can be mis-assigned and where it cannot; it does not estimate how often a real refinement is affected.

Exact symmetry is assumed. Whether a molecule has a mirror is a tolerance question when measured, and a molecule very nearly satisfying a lift gives a very nearly absent set of reflections rather than an absent one. That is a different and more common problem than the one counted.

Only one operation at a time is closed here. A motif may bring several, and the group it lifts to can then be larger than any single closure. The census of shrinking lifts is unaffected, since a translation appears from one, but the size of the largest possible lift is not reported. Nor is the lifted group named: the census of which type goes to which, the graph the plane essay drew, has not been constructed for space.

The diagram of a screw and its mirror image is a cartoon of one case. It shows the sense reversed on a four-one axis, and the computation that decides each of the eleven is the comparison of operation lists.

What had to hold, and what had to be refused #

The positions and normalisers come from the same lists that reproduce 1,731 and 1,128. A motif candidate is accepted only when conjugation returns the group’s list exactly, and generic points are tested in numbers so that no value of a coordinate can pass for a property of a position.

Known cases have to come out, and in both directions. $P1$ goes to order two under a centre; $P\bar1$ must not be lifted at a general point, since a second centre through a generic point would add a generic translation. The contrast is the point of two of the tests. $P4$ must accept what $P4_1$ refuses, on the same lattice with the same point group. If those two came out alike, the explanation given above would be wrong.

The first version’s error is the reason for the sampling. Ten non-polar groups with lifted general positions would have been a surprising result, and it went against an argument that says it cannot happen. It came from too few grid points in each position. The argument and the computation had to meet, and they met only when the computation took a position to be more than one point.

Still open: the graph, and the lifts that are not exact #

The plane essay drew the whole graph of which group lifts to which. That graph for space would be large, but it is a loop over what is computed here, plus a way to name a closed group among the 230, and the answer would say which space groups are reachable from which by a molecule’s symmetry and which are not reachable at all. Some groups can only be arrived at, in the sense that no motif on any lower group produces them, and those would be the crystals that cannot be mistaken from below.

The other question is the one structure solution actually faces. A lift with a molecule that only nearly has the symmetry leaves superlattice reflections at some small fraction of the strong ones, and whether those can be detected above background depends on the position and the extra operation. A table saying, for each of the 606 positions, how weak the odd reflections become for a given departure from exactness would turn a warning into an instrument. It has not been built.