Where a symmetric motif lifts the group

The motif must be a comma is a warning about drawing. Repeat a dot by the operations of a plane group and the pattern can come out more symmetric than the group, because a dot has every symmetry a point can have. So the careful draughtsman uses a comma, a mark with no symmetry at all, and the pattern then has exactly the group that made it. Near-symmetry, and the tolerance that is not here followed the same hazard into measured structures, where a motif that is almost symmetric makes a pattern that is almost of a larger group.

Neither essay asked the constructive question: which placements do it? Take a motif with some symmetry of its own, a point group $M$, and put it at a point of a plane group $G$. Sometimes the pattern’s group is larger than $G$ and sometimes it is exactly $G$. The answer depends on the plane group, on which Wyckoff position the motif occupies, on the motif’s own symmetry, and on how it is turned. This essay settles all four for every position of all seventeen groups.

The criterion that decides it is short. A symmetry of the motif survives into the pattern exactly when it normalises the group. From that follow a count of twenty-nine positions out of seventy-two where a motif can lift the group, and three things the count would not predict. General positions can be lifted in only four groups. At two sites the same motif lifts the group to different groups depending only on how it is turned. And ten placements, along seven of the lifts between groups, make the pattern’s cell smaller than the cell of the group that made it.

The criterion #

Put a motif with symmetry $M$ at a point $x$, so that every operation of $M$ fixes $x$ and carries the motif to itself. The pattern is the set of copies $G \cdot \text{motif}$. Take an operation $n$ of $M$. It carries the copy at $x$ to itself. It carries the copy at $g x$ to the copy at $n g x$. If $n$ normalises $G$, then $n g = g' n$ for some $g'$ in $G$, so $n g x = g' n x = g' x$, which is again a copy. So $n$ is a symmetry of the pattern.

If $n$ does not normalise $G$, some copy is sent where no copy is, and $n$ is not a symmetry. An operation of the motif that is a symmetry of the pattern must carry the orbit of $x$ to itself, and hence conjugate the orbit’s stabilisers into one another. Short of an accidental coincidence among copies, which a generic motif avoids, that forces it to normalise $G$.

So the pattern’s group is

$$G' = \langle\, G,\; M \cap N_x(G) \,\rangle ,$$

where $N_x(G)$ is the set of operations fixing $x$ that normalise $G$. Two consequences follow at once. A motif symmetry outside the lattice’s holohedry never lifts anything, because such an operation does not even carry the lattice to itself. A five-fold motif, or a mirror at an angle the lattice does not have, is simply wasted. And the most any motif at $x$ can do is decided by $x$ alone. It is $\langle G, N_x(G) \rangle$, reached by a motif that has at least the symmetry $N_x(G)$, and a motif of intermediate symmetry reaches the group in between.

The computation that follows applies the criterion literally. For each group, on the generic lattice of its type, each Wyckoff position is represented by one point. For a special position that is the special point. For a mirror line it is a generic rational point on the line, and for the general position a generic rational point anywhere. Each operation of the lattice’s holohedry is turned into the affine map fixing that point and tested for normalising the group, which is an exact test in rational arithmetic. The same site under two names used the normaliser to identify positions with each other. Here it decides which symmetry a motif can export.

Twenty-nine of seventy-two #

The seventeen groups have seventy-two Wyckoff positions between them, and twenty-nine can be lifted by a suitably symmetric motif. The distribution across kinds of position is uneven, and the imbalance is the first thing the census says.

Special points are usually liftable: twenty-two of thirty-seven. At a rotation centre of p4 the site symmetry is 4, and the normaliser operations fixing the point form 4mm. A motif with four mirrors there turns p4 into p4m. At a three-fold centre of p3 the site symmetry is 3 and the fixing normaliser is 6mm, the whole hexagonal holohedry. A motif there can make p6, p31m, p3m1 or p6m, depending on which part of 6mm it has.

Mirror lines almost never are: three of eighteen. A point on a mirror line of pmm has site symmetry $m$, and no operation fixing it adds anything, because the rectangular holohedry’s other mirror, through that point, does not normalise pmm. The three exceptions are the two mirror lines of pm and the one of cm. There a motif with a second mirror perpendicular to the line gives pmm or cmm.

General positions are liftable in exactly four groups: p1, pm, pg and cm, and in no others. A motif at a general point of p2, however symmetric, leaves the pattern p2. The reason is the normaliser’s shape, and the next section takes it up.

Three groups cannot be lifted at any position: p2, pmm and p6m. For p6m that is obvious, since it is the largest group its lattice allows. For p2 and pmm it is a statement about the generic metric. On a generic oblique lattice the only operations are $\pm 1$, which p2 already has at every point where it could use them. On a generic rectangular lattice the four operations of 2mm are already all present at pmm’s special points, and nowhere else can they be completed into normaliser elements.

Why a general point can be lifted in only four groups #

A general position has trivial site symmetry. So a motif there lifts the group only if some operation fixing a generic point, a point with no special relation to the group’s symmetry elements, normalises the group. For most groups nothing does. In p2, a half-turn about a point $x$ normalises the group only if $x$ lies on the lattice of half-turn centres of p2’s normaliser, which is a quarter of a cell apart, and a generic point does not.

The four exceptions have normalisers containing a continuous family. In p1 every translation normalises the group, so the half-turn about any point is a normaliser element: a motif with a two-fold centre anywhere makes p2. This is the most familiar case of all, because a centrosymmetric molecule crystallising in a triclinic cell always has the choice of giving the crystal an inversion centre. In pm, pg and cm, every translation along the mirror direction normalises the group. So a mirror perpendicular to the mirror lines, placed through any point at all, is a normaliser element. A motif with such a mirror at a general point gives pmm, pmg or cmm.

The middle panel is the least obvious of the three, and it is the reason the census distinguishes orientation. In pg a motif with a mirror placed at a general point, with its mirror along the glide lines, gives nothing new. The glide then relates the motif to a copy of itself and adds no symmetry. Turned so that its mirror crosses the glide lines, the same motif gives pmg, because a mirror perpendicular to a glide direction, through any point, normalises pg. The motif’s symmetry is the same in both placements. Only the direction of its mirror relative to the lattice changes.

The same motif, turned: two different groups #

That dependence on orientation is sharpest at two special sites, where a motif with a given symmetry at a given point can produce two different groups.

At a three-fold centre of p3 the fixing normaliser is 6mm, which contains two different 3m subgroups: the three mirrors along the lattice vectors and the three across them. A 3m motif belongs to one or the other depending on which way it is turned. With its mirrors along the lattice vectors the pattern is p31m, and turned thirty degrees it is p3m1. p3m1 and p31m separated these two groups by where their mirror lines pass through the three-fold centres. The census shows that one symmetric motif at one point can land in either, so that neither the site nor the motif’s point group decides which.

The square lattice has the same case one dimension of freedom lower. At a two-fold centre of p4 the fixing normaliser is 4mm, and its two 2mm subgroups, with mirrors along the axes or along the diagonals, give p4m and p4g respectively. A two-mirror motif at the half-turn of a four-fold pattern makes either of the square lattice’s mirror groups, depending on how it is turned.

Ten placements that shrink the cell #

The census turned up something the criterion does not advertise. At ten placements, along seven of the nineteen lifts between groups, the group generated by $G$ and the motif’s operations contains a translation that is not in $G$’s lattice. The lifted pattern then repeats on a smaller cell than the pattern the group made.

The cleanest case is pgg. Its two-fold centres have site symmetry 2, and the fixing normaliser there is 2mm. Put a motif with two mirrors at such a centre, and the pattern gains two pure mirror lines through that point. But pgg already has glide lines in both directions, and a mirror composed with a parallel glide is a translation. Here the translation is half a cell diagonal, $(\tfrac12, \tfrac12)$, a centring vector. The pattern is cmm, and its primitive cell is half the size of pgg’s. pmg does the same along one axis, becoming pmm on half the cell, and cmm’s own two-fold centres give pmm on half of cmm’s cell.

The square groups add the strangest version: lifts that return the same type on a smaller cell. A motif with a four-fold axis at a two-fold centre of p4 makes a pattern whose four-fold centres lie at every half-cell diagonal. That is p4 again, with a cell half the size, turned forty-five degrees. A motif with four mirrors at a 2mm site of p4m does the same for p4m. By the Hermann–Mauguin symbol, nothing has changed. By the cell, the pattern has twice as many repeats as the group that drew it.

The full list is short enough to give. Every one of the ten is at a two-fold centre, or at a four-fold or 2mm centre of p4g, and every one comes from a motif whose mirrors or four-fold axis, composed with an operation of the group that has the same linear part and a different translation, leave a translation behind. pmg with a 2mm motif at either of its two-fold centres gives pmm on half the cell. pgg with a 2mm motif at either of its two-fold centres gives cmm. cmm with a 2mm motif at the two-fold centres off its mirrors gives pmm. p4 at its two-fold centre gives p4 with a four-fold motif and p4m with a four-mirror one, both on half the cell. p4m at its 2mm site gives p4m again. p4g gives p4m at both its four-fold and its 2mm site. None of the ten is at a general position or on a mirror line. The four liftable general positions gain a two-fold or a mirror that the group has no copy of with the same linear part, so there is nothing for the new operation to compose with into a translation; that the mirror lines behave the same way is what the census found rather than what this argument proves.

That makes these lifts a trap of a sharper kind than the dot. A dot is at least recognised as a risk. A motif of two mirrors at a two-fold site of pgg looks like a deliberate, symmetric decoration of a pgg pattern. It produces a pattern whose true cell is half the one the draughtsman drew, and whose group is cmm, a different type. Naming each such group needed the lifted operations rewritten on the finer lattice and the finer lattice’s type read from the original metric. pgg’s rectangular lattice halved diagonally is a centred one, while p4’s square lattice halved diagonally is square again.

The whole map of lifts #

The picture at the head of this essay collects the census into nineteen lifts between groups. The oblique lattice has one lift, p1 to p2. The rectangular groups lift into pmm, pmg and, across to the centred lattice, cmm. The centred groups lift into cmm and, halving the cell, back to pmm. The square groups lift into p4g and p4m, and p3 lifts into four of the five hexagonal groups. p31m, p3m1 and p6 lift only to p6m.

The arrows run only upwards, from a group to a larger one on the same lattice or on a finer one, never sideways between two groups of the same order. That is a theorem rather than a pattern in the data: the lifted group contains the original, so it is either equal or larger. What the data add is which containments a motif can actually realise. The subgroup lattice of the seventeen has many more edges than nineteen. Most containments, such as p2 inside pmm, need a lattice or a position that a generic p2 does not have, and no motif placed on a p2 pattern supplies them.

What the census has to check against the patterns #

A computation that only generates groups could name them consistently and wrongly. So each lift is also checked against the pattern itself. A motif with exactly the lifting symmetry is built from a generic point near the site, repeated by the plane group’s own operations, and handed to the detector that recovers a pattern’s group from its points without being told which group made them.

In five cases across four lattices, the detected group and the generated group agree, name and order. The two refused claims are the two readings the old warning invites. The first is that a symmetric motif always lifts the group. At a general point of p2 no motif can, because no operation fixing a generic point normalises p2. The second is that any mirror in a motif can lift a group. A motif mirrored in the line along $(7, 2)$, repeated by p4, gives a pattern the detector finds to have exactly p4’s four operations per cell. That mirror is not a symmetry of the square lattice, so it cannot be a symmetry of the pattern.

What the census assumes #

The census uses the generic lattice of each type, and the convention matters. A p2 pattern drawn on a rectangular cell that happens to be rectangular by accident has a holohedry of 2mm, not just $\pm 1$. On that lattice a motif with mirrors at p2’s rotation centres lifts it to pmm. On a square cell a motif at the right point of pmm lifts it to p4m. Those lifts are real, but they are lifts of the metric as much as of the group, and they belong to the specialised lattices, not to the types. Near-symmetry showed how a nearly square cell makes that distinction a matter of measurement.

The census also takes a motif to be generic within its symmetry, with no accidental coincidences between copies at different sites. A motif large enough to overlap its neighbours, or placed so that two copies coincide, can produce a pattern with more symmetry by collision, the way a dot does. The criterion describes placement, not overlap.

In three dimensions everything here carries over and multiplies. A molecule sitting at a site of a space group lifts the space group exactly when its own symmetry operations normalise the group, and what a molecule gives up to sit in a crystal is the other half of the same exchange. There the molecule loses the symmetry its position cannot accommodate. Here the pattern gains the symmetry the normaliser can.

Still open: the census in space #

The plane has seventy-two positions and the census runs in a fraction of a second. The two hundred and thirty space groups have well over a thousand Wyckoff positions. For each of them the question is the same one: which operations of the lattice’s holohedry, made to fix the position, normalise the group? The criterion needs nothing new, and the normalisers are already known class by class from the census of the space groups over their classes. The interesting outputs would be the three-dimensional analogues of the two surprises here. One is the general positions that can be lifted, which should occur exactly in the groups with a polar direction. The other is the lifts that shrink the cell, where a molecule of higher symmetry makes a crystal’s true cell smaller than the one its group prescribes. That second case is the one crystallographers meet as a warning in structure solution, and the census that would count it has not been run.