Seventeen groups, seven vector sets

The map that needs no phases is the one thing a diffraction experiment gives away for free. Transform the measured intensities with every phase set to zero and out comes a function whose peaks are not atoms but the vectors between them — every ordered pair, brought to a common origin.

The map at the head of this essay is one: peaks that are interatomic vectors rather than atoms, n² of them for n atoms, most piled on top of one another and the tallest of them the n atoms paired with themselves.

What symmetry does that map have? The usual answer is a rule to be applied: take the space group, add a centre of inversion, throw away the translation parts. That is correct, and it is a rule. This collection has a detector, so the question can be asked the other way round — build the vector set, hand it over with no indication of where it came from, and see what comes back.

The two answers agree, which they had better; what the detour buys is that the rule can be wrong and be found out. A rule applied is a rule believed. A construction handed to a detector that enumerates every operation the lattice permits and keeps the ones that work is a claim with a way of failing — and this collection’s standing complaint about pattern figures is that a picture of the wrong group looks exactly like a picture of the right one.

Two things happen, and they are not the same thing #

A centre appears, always. For every vector from atom A to atom B there is the vector from B to A, and it is the negative. So the vector set is its own point reflection whatever the structure is — no exceptions, no conditions, nothing to check. In the plane, a point reflection is a half turn, so every vector set has a two-fold rotation in it and none of the seventeen escapes.

That is the symmetry diffraction adds met in real space instead of reciprocal space. There the argument runs through Friedel’s law and the structure factor; here it is one sentence about subtraction.

Every translation part disappears. This is the second effect and it is the one worth the essay.

Take a glide: it reflects an atom across a line and slides it half a cell along. The image is a genuine atom of the pattern, so the vector between the two is a genuine peak of the map. Now ask what relates the peaks of the map to one another — and the answer has the slide taken out of it, because a difference of two positions is unmoved when both are shifted. The vector set of a glide-containing pattern has a mirror.

So the map’s group is always symmorphic: every operation of it passes through the origin, and the distinctions this collection spends whole essays on — the screw against the rotation, the glide against the mirror — are simply not present in it.

The two effects are logically independent and it is worth keeping them apart, because they are usually stated as one rule. The centre arrives from the pairing — from the fact that the map is built out of differences, which come in opposite pairs. The translation parts go from the differencing — from the fact that a difference does not see a shift applied to both of its terms. A construction with one property and not the other is easy to imagine: the set of all sums a + b, rather than differences, would lose the translations in a different way and would not be centrosymmetric at all.

What makes the Patterson the object it is, rather than one of several plausible maps, is that both effects arrive together and neither can be arranged away.

Seventeen onto seven #

Both effects together give a map from the seventeen plane groups to the groups their vector sets have, and it is not a rule applied here but a computation performed. For each group: build the orbit, take every ordered difference, reduce into the cell, hand the resulting point set to the routine that names a plane group from a set of points, and compare the answer with what the Laue class predicts.

Seven groups come back. p2, pmm, cmm, p4, p4m, p6 and p6m — and every one of them is symmorphic and contains a half turn, which is what the two effects above require.

Read the collapses and they are the two effects, separately. p1 becomes p2 — nothing but the centre. pg becomes pmm — the glide becomes a mirror and the centre arrives with it. p3 becomes p6 — the three-fold plus the centre is a six-fold, which is a collapse with no counterpart in three dimensions, where a three-fold plus an inversion is 3̄ and not a six-fold at all. p31m and p3m1 become one group, which is a merge this collection has an essay’s worth of care about in the other direction.

None of the seventeen is unchanged. Ten of them keep their operation count and seven gain, but even the ten are changed in kind: pmg’s four operations become four operations, and they are not the same four — a glide has become a mirror. The only way to have a vector set with the same group as the pattern is to be symmorphic and centrosymmetric already, and in the plane those are pmm, cmm, p4m, p6m, p2, p4 and p6 — the seven that appear as answers. The Patterson groups are exactly the fixed points of the collapse, which is a tidy way of saying what “symmorphic and centred” means without saying either word.

Six classes, seven groups #

The symmetry diffraction adds counts six, and that essay is right: six point groups survive, and a diffraction pattern’s rotational symmetry is one of six things. This essay counts seven, and it is counting something else.

The difference is one class. The point group 2mm sits on a rectangular lattice, where it gives pmm, and on a rhombic one, where it gives cmm. Same point group, two lattices, two plane groups. Every other Laue class is compatible with only one lattice type, so the counts agree everywhere else.

That is not a technicality; it is the difference between what a diffraction pattern shows and what the map is. A measurement of intensities reveals the point symmetry directly — the pattern of spots has it — and reveals the lattice type through where the spots are and which of them are missing. Six and seven are answers to those two questions.

And the split is not an artefact of the plane. The same thing happens in three dimensions, more often: the Laue class mmm is compatible with four Bravais lattices, so it produces four Patterson groups on its own. Eleven Laue classes give twenty-four Patterson groups, and the ratio is bigger than the plane’s because a lattice has more ways of being centred when there are three axes to centre between.

The same collapse happens in reciprocal space, and it is the same fact in the other language. Negating the indices conjugates the structure factor and leaves the intensity alone, so an experiment measuring intensities cannot separate a class from that class with a centre added: the thirty-two crystal classes collapse to the eleven Laue classes, and the seventeen plane groups to six point symmetries. That is Friedel’s law, and the centre the vector set acquires is its real-space form — the same statement about the same missing information, once as a symmetry of a map and once as an equality between two measured spots. Neither is a consequence of the other; both are consequences of the map being an autocorrelation and the intensity being a modulus squared, which are two ways of writing one convolution.

Which is why absences carry the load #

The consequence for practice follows immediately and is the reason structure determination is arranged the way it is.

A diffraction experiment measures intensities. From them a crystallographer gets the Laue symmetry — six possibilities in the plane, eleven in space — and that is everything the symmetry of the measurement contains. Every remaining distinction among the seventeen has to come from somewhere else, and the somewhere else is the reflections that are missing: a glide deletes alternate reflections along a row, exactly, whatever the atoms are doing.

So the workflow is: symmetry gives the class, absences give the translation parts, and the two together often still do not give one answer. What this essay adds is the reason the first step can never do more — not that it happens to be coarse, but that the map it reads is provably blind to exactly the operations absences exist to detect.

One more consequence is worth stating in the crystallographer’s own terms. The number of candidate groups a measurement leaves open is the size of the collapse’s fibre. In the plane, a pattern whose diffraction shows 2mm symmetry on a rectangular lattice could be any of pm, pg, pmm, pmg or pgg — five candidates, separated only by which reflections are missing. A pattern showing 6mm could be p3m1, p31m or p6m — three. And a pattern showing 4mm could be p4m or p4g — two, which is the pair a whole essay of this collection is about.

Those fibre sizes are not evenly distributed, and the largest is over the least symmetric class. That is the shape of the practical problem: the harder a structure is, the more candidates its measurement leaves, and the more work the absences have to do.

Where the extra symmetry is useful rather than a nuisance #

The vector set’s added symmetry is usually described as a loss, and for identifying the group it is. It is not a loss everywhere.

Where symmetry stacks the vectors is the same fact used the other way: because the map’s symmetry is higher than the structure’s, the vectors between symmetry-related atoms pile up on particular planes and lines rather than spreading through the map. The more symmetric the group, the more concentrated the pile — which is why a heavy atom in a group with several operations is easier to find than one in P1, where the vectors go everywhere.

There is a second use, less often stated. The vector set of a pattern is a pattern — a periodic point set with a group — so everything this collection knows about plane groups applies to it. Its own special positions are where vectors pile up; its own asymmetric unit is the part of the map a search need cover; its own fundamental domain is the fraction of the cell a computation has to visit. A map with p6m symmetry needs a twelfth of a cell examined and a map with p2 needs half of one, which is a factor of six in the size of a search and comes from nothing but the collapse above.

The map has more peaks than the pattern has atoms #

One number in the census is worth reading on its own, because it is where the difficulty of using a Patterson comes from.

A pattern with n atoms in a cell has n² ordered differences, and n of those are the atom-with-itself vector at the origin. So the map has n² − n + 1 peaks where the structure has n atoms, all crowded into the same cell — the origin peak carrying n times the weight of an ordinary one and telling nobody anything.

The added symmetry makes that worse and better at once. Worse, because a more symmetric map has more coincidences: distinct vectors that happen to be related by an operation land on top of one another. Better, because the coincidences are predictable, which is what a Harker section is. A crystallographer reading a Patterson is reading a map whose peaks outnumber the answer by a factor of the atom count, and whose symmetry is both the reason it is hard and the reason it can be done at all.

What this does not settle #

The plane’s seven are derived; space’s twenty-four are not. The same construction in three dimensions gives one Patterson group per arithmetic crystal class whose point group is a Laue class, and there are twenty-four of them. That number is from the International Tables. This collection does not enumerate the two hundred and thirty and cannot enumerate the arithmetic classes either, so twenty-four appears here as a count from a book with its provenance stated — the same treatment 230 gets everywhere else on this site.

The seven are the plane’s own answer and not a small version of space’s. p3’s vector set is p6, because a three-fold and a centre generate a six-fold in two dimensions; in three, a three-fold and an inversion generate 3̄, which is a rotoinversion and not a rotation of any order. The plane’s collapses are genuinely simpler rather than a scaled-down illustration, and an essay that carried them into three dimensions by analogy would get that one wrong immediately.

The map has weights and this essay ignored them. A Patterson peak has a height proportional to the product of the two atoms’ scattering, and the vector set used here is a set of positions with the multiplicities thrown away. That is the right object for a symmetry question — a symmetry either maps the set onto itself or does not — and the wrong one for solving a structure from it, where the heights are most of the information.

Nothing here is about a real measurement. The vector sets above are exact: every difference of an infinitely sharp, perfectly periodic orbit. A measured Patterson is a truncated Fourier sum with peaks of finite width, computed from intensities with errors on them, and its symmetry is a matter of degree rather than a fact — which is the tolerance problem this collection keeps at arm’s length by working on point sets. What is derived here is what the map’s symmetry is, not what an experiment would report it as.

And the collapse is about symmetry, not about information. Two structures with the same Patterson symmetry are not thereby indistinguishable: pm and pg have identical Patterson groups and completely different Patterson maps, and the absences separate them at a glance. What is lost is one particular route to the answer, and the essay is about that route’s ceiling rather than about the experiment’s.

The group collapses; the map does not go quiet #

Saying that every translation part disappears is a statement about the vector set’s symmetry group, and it is easy to read as the stronger claim that the map has forgotten the translation entirely. It has not, and the difference is the basis of a technique.

Follow one symmetry-related pair through the subtraction. In pm, with a mirror across the line $y = 0$, a point at $(x, y)$ has a partner at $(x, -y)$, and the vector between them is $(0, 2y)$. Whatever $y$ was, the first coordinate is zero — so every vector arising from that mirror lies on one line of the map.

In pg the same calculation lands somewhere else. The glide sends $(x, y)$ to $(x + \tfrac12, -y)$, so the vector between a point and its partner is $(\tfrac12, 2y)$. Again the first coordinate is fixed for every atom in the structure; it is simply fixed at a half instead of at zero.

These concentrations are the Harker lines, and in three dimensions the Harker sections. They are not symmetry of the map — the map’s group is the same symmorphic group either way — they are a pile-up of peaks on a particular line, and its position is the intrinsic translation of the operation that produced it, read directly off the picture.

So pm and pg give vector sets with the same group and different maps. A structure in pm shows a ridge of peaks along $u = 0$; a structure in pg shows the same ridge along $u = \tfrac12$. Both are consistent with the symmorphic group the census reports, because a group constrains where peaks may be and says nothing about where they crowd.

The general statement is worth keeping. The census above measures the largest group that carries the vector set onto itself, which is an upper bound on what the set gives away. What it gives away in practice is more, because a vector set carries a distribution as well as a symmetry, and the distribution remembers what the group forgot.

What genuinely cannot be recovered #

One thing on this page is a permanent loss rather than a bookkeeping one, and separating it from the rest is what makes the collapse useful instead of alarming.

The centre is added unconditionally. Every vector comes with its negative because subtraction is antisymmetric, so a vector set is centrosymmetric whether the structure was or not. No amount of looking at the map, and no cleverness about peak distributions, undoes that — unlike the translation parts, which leave the group and survive in the peak positions, the centre is manufactured rather than inherited.

Nor do the absences settle it. Systematic absences arise from translation parts — a glide or a screw removes a class of reflections — and a centre of symmetry has no translation part to remove anything. So the two most direct pieces of evidence a diffraction experiment offers are both silent on the single question of whether the structure has a centre. That is why space-group determination from absences alone regularly ends in a short list rather than an answer, and why the ambiguities on that list are almost always a centrosymmetric group beside its non-centrosymmetric subgroup.

The question is answered statistically instead. The intensities of a centrosymmetric structure are distributed differently from those of a non-centrosymmetric one: with a centre, each structure factor is a real number that can take either sign and is often near zero, while without one it is a complex number whose magnitude is rarely near zero. Collect enough reflections and the two distributions are distinguishable — many more very weak reflections in the centrosymmetric case, and a mean of $\big||E|^2 - 1\big|$ near 0.97 rather than near 0.74, once the magnitudes are normalised so that $\langle |E|^2 \rangle = 1$.

That test looks at the intensities before they are transformed, which is precisely why it works where the map cannot. The Patterson is an average — the transform of the intensities with all phase information discarded — and averaging is what imposed the centre in the first place. The distribution of the same numbers is not an average, so it survives.

Three sources of evidence, then, and each answers what the others cannot. The vector set’s group gives the Laue class. The absences give the translation parts, up to the pairs that share a signature. The intensity statistics give the centre. A space group determined without all three has been guessed at some step, and the step is usually this last one.