How many waves a group permits

Every pattern in this collection has been written the same way: a motif, an orbit under a group, and the group rediscovered from the points. There is a second way to write a periodic pattern down — the one a diffraction experiment hands over — and a crystallographer uses it more often than the first.

A periodic function is a sum of waves — one per reciprocal lattice vector — and requiring the function to be invariant under a wallpaper group does two things to that sum. It ties the coefficients together within each orbit of vectors, so an orbit carries one amplitude rather than as many as it has members. And it kills some orbits outright, because an operation can carry a vector to itself while attaching a phase, and then the only coefficient satisfying the condition is zero.

The two conditions, from one requirement #

Let the density be $\rho(\mathbf{r}) = \sum c(\mathbf{h})\,e^{2\pi i\,\mathbf{h}\cdot\mathbf{r}}$ and require ρ(g r) = ρ(r) for each operation {M | t} of the group. Comparing coefficients gives one condition:

$$c(hM) = c(h)\, e^{2\pi i\, h \cdot t}$$

Everything follows from it. Within an orbit of reciprocal vectors under the point group, every coefficient is determined by any one of them, up to the phases the translation parts attach — so an orbit carries one complex amplitude.

And if some operation carries h to itself with h·t not a whole number, the condition reads $c(\mathbf{h}) = c(\mathbf{h})\,e^{2\pi i\,\mathbf{h}\cdot\mathbf{t}}$ with the exponential not equal to one, whose only solution is zero. That is a systematic absence, arrived at from the invariance of a function rather than from a diffraction experiment.

This collection derives the absences twice already — from the extinction conditions of a space group and from reading a group off its absences — and this is a third route with the same answers.

Which groups have absences, decided twice #

Running the orbit arithmetic over the seventeen gives four groups with extinct orbits: pg, pmg, pgg and p4g.

Those are exactly the groups carrying a glide, and the machinery decides that a second way — by classifying each operation and asking whether any is a glide — without being told the first answer. The two lists are identical.

The group cm deserves a note, because a reader expecting it in the list will not find it. Its glide lines are real and its operations, modulo the centred lattice, contain no glide: the centring translation makes the glide equal to a mirror times a lattice vector. This collection has the same finding from the pattern side, where the detector reports no glide for cm either, and it is the same fact seen twice.

Building a density, and asking what it is #

The construction is short. Take the first few surviving orbits, symmetrise a plane wave over each — average cos(2π(hM·r + h·t + φ)) over the group, which is the averaging projector applied to a wave — and add them with amplitudes.

Then take the level set: the grid points where the density is largest, a fixed share of them. That is a point set, and it goes to the detector.

The density can never have less symmetry than the group, since the waves are invariant by construction, and the machinery refuses a result that does. The interesting direction is the other one: at low resolution the density has more.

A density can be too symmetric #

A density built from one orbit of waves is a set of stripes, and stripes are more symmetric than almost anything. The detector says so: p1 built from a single orbit comes back with a group of order twenty-four rather than one, because the stripes have translations and mirrors nobody asked for.

Adding orbits removes the extra symmetry one piece at a time. The number of orbits at which the detected group finally equals the intended one is a measurement of how much information a group is, in waves, and it is not the group’s order — the seventeen do not sort by size here at all:

• one orbit suffices for cm, cmm, p4m and p6m;
• two for p1, p2, pm, pg, pmm, pmg, p3 and p31m;
• three for p4g and p3m1;
• four for pgg and p4;
• five for p6.

p6 is the striking row. Its first four orbits have stars that a mirror carries onto themselves, so a density built from them has mirrors the group does not; the fifth orbit is the first whose star is not mirror-symmetric, and it is the one that finally makes the density p6 rather than p6m.

The same lesson as the comma #

This is the site’s founding finding in another notation.

A pattern built from too symmetric a motif has the wrong group — a dot is not a comma, and a single dot verifies only eleven of the seventeen. A density built from too few waves has the wrong group for the same reason: not enough independent information has been supplied to distinguish the intended arrangement from a more symmetric one.

The two failures are the same failure. In the motif case the missing information is the asymmetry of the motif; in the wave case it is the resolution. And in both the detector is what catches it, because a picture of the wrong group looks exactly like a picture of the right one.

That is a practical warning for structure determination rather than a curiosity. A density map computed from too few reflections is more symmetric than the crystal, and reading its symmetry off gives a group too large — which is the commonest way of arriving at a wrong space group from real data, and a relative of the symmetry an average acquires.

The phases are half the information #

One step in the construction is easy to skip and changes every answer: the phase of each orbit’s amplitude.

A real density needs c(−h) to be the conjugate of c(h). If the group carries h to −h, the two conditions meet and the phase is pinned to one of two values half a turn apart — a centric reflection, which this collection meets in the phase-restriction essays. If no operation reverses h, the phase is free.

Taking every phase to be zero — which is what summing bare cosines does — builds a density with a centre of inversion nobody asked for. The first version of this computation did exactly that and reported p3 as p6m at every resolution, however many orbits were added. Restoring the phases fixed it.

And for the centric orbits, taking the same choice of the two allowed values for every orbit is a second version of the same error: it builds a density more symmetric than the group again, and p6, pgg and p4g each came back at twice their order until the signs were allowed to differ between orbits.

Both failures produced groups that were exactly twice the right size, which is the kind of wrong answer that looks like a result.

Two descriptions of one pattern #

The collection now has two complete descriptions of a periodic pattern, and it is worth setting out what each is good for.

As an orbit of a motif. The pattern is a set of points, the group is rediscovered from them, and every claim is exact integer arithmetic. This is the description the site is built on, and it is the right one for deciding whether a symmetry is present: the answer is yes or no, with no tolerance anywhere.

As a sum of waves. The pattern is a function, the group constrains its coefficients, and the description is naturally truncated — one keeps the waves up to a resolution. This is the description a diffraction experiment produces, since each measured reflection is one coefficient’s magnitude.

The second is lossy in a way the first is not, and the loss is exactly the resolution. That is the trade this essay measures: how much of the symmetry survives a truncation, and how many coefficients are needed before the two descriptions agree about the group.

An experiment has no choice about which description it gets. What it can choose is how far out to measure, and the answer to how far is far enough to see the symmetry is the count in the table above — in the model’s units rather than in ångströms, but with the same shape.

Which orbits matter most #

Not every orbit contributes equally to fixing the group, and which ones matter has a pattern.

An orbit whose star is symmetric under operations the group does not have contributes nothing towards distinguishing the group from a larger one: the wave it produces is invariant under the larger group too. The orbits that matter are those whose stars are not mirror-symmetric — for p6, the first of them is the fifth orbit, which is why five are needed.

That gives a rule of thumb with a computation behind it: the resolution needed to see a group’s symmetry is set by the first reciprocal vector whose star breaks the symmetry of the next larger group. It is not set by the group’s order and not by how many orbits are available.

The rule has an experimental reading. A crystal whose true symmetry is lower than its apparent one is distinguished from the higher-symmetry candidate by particular reflections — the ones whose stars differ between the two groups — and those may be weak, few, and far out. Which is why the wrong space group is usually the higher-symmetry one.

What an absence looks like as a function #

The extinctions have a reading here that the diffraction picture does not give directly, and it is worth a paragraph.

An extinct orbit’s symmetrised wave is not small: it is identically zero, at every point of the plane, checked at points chosen off any special position. The averaging that produces the symmetry-adapted function cancels the wave completely.

So a systematic absence is not a reflection whose intensity happens to vanish. It is a function that does not exist — there is no invariant density with a component at that wavevector, whatever the structure is — and the absence is a property of the group rather than of the atoms in it.

That is the sharpest statement of what makes systematic absences useful for identifying a group, and it is why they are systematic rather than accidental: an accidental absence is a structure’s arithmetic coincidence, and a systematic one is the non-existence of a function.

Counting the permitted waves #

Aside from the round trip, the orbit arithmetic answers a question of its own: how many independent invariant functions a group has up to a given resolution.

It is the number of surviving orbits, and it varies over the seventeen by a factor of nearly six at the same cutoff — eighty for p1 down to fourteen for p6m and p4m. That is the sense in which a high-symmetry group is a strong constraint on a function: at a given resolution, a p6m density has fourteen free amplitudes and a p1 density has eighty.

The same count is what makes a structure determination in a high-symmetry group easier: fewer parameters for the same data. This collection has that reading in the unknowns against the observations essay, where the ratio of the two decides whether a structure is determinable at all, and the orbit count is where the unknowns come from.

What is checked #

That the density has the group. Every density built from a group’s own waves is required to contain that group’s operations; a result missing any of them would mean the symmetrisation was wrong, and the machinery raises an error rather than reporting it.

That an extinct orbit’s wave is identically zero. Sampled at points off any special position, the symmetrised wave of a forbidden orbit comes back as zero to the last bits — which is the absence, seen as a function rather than as a missing spot.

That the two absence lists agree, one from the orbit arithmetic and one from classifying the operations.

And that one orbit is not enough, on thirteen of the seventeen — the negative result which stops the round trip from being a test that passes for free.

The count is the same count the diffraction essays make #

The number of surviving orbits inside a cutoff is computed here from the orbit arithmetic on reciprocal lattice vectors. It is a quantity the diffraction field of this collection computes too, by a completely different route, and the two must agree.

A unique reflection is an orbit of the reciprocal lattice under the Laue group, and counting them is what decides how many independent measurements an experiment makes. A permitted symmetrised wave is an orbit of the reciprocal lattice under the group, minus the extinct ones, and it is what decides how many independent numbers a density has.

Those are the same orbits, counted with the same short-orbit corrections. So the count of independent parameters of an invariant density and the count of unique reflections an experiment records are one number, and that is not a coincidence — it is the statement that a diffraction experiment measures exactly the coefficients the symmetry leaves free, with the phases as the part it does not get.

The agreement is worth having as a check because the two computations share nothing. One runs over lattice vectors under the point group and asks about stabilisers; the other runs over the group’s operations and asks which orbits the translation parts extinguish. A discrepancy would mean one of them had mishandled a short orbit or an absence, and both are easy to mishandle in the same silent way.

What the wave description does not know #

There is a constraint on a real density that this construction cannot express, and its absence is the reason the densities here are drawn as level sets rather than treated as structures.

An electron density is non-negative everywhere. Nothing in the symmetrisation imposes that: a sum of symmetry-adapted waves with arbitrary coefficients is an invariant function, and an invariant function goes negative wherever the coefficients happen to make it. Every density built here does, in the regions between its peaks.

That is not a defect of the arithmetic and it is a real constraint on the physics, and the gap between them is where a whole method lives. The phase relations are consequences of non-negativity and of the density being concentrated at atoms — two facts about the function that the group has nothing to say about — and they are what makes the phases recoverable at all.

So the two descriptions divide the information cleanly. Symmetry decides which coefficients may be non-zero and how they are tied together, exactly and in integers. Positivity and atomicity constrain what values the surviving coefficients may take, approximately and statistically. A structure is fixed by both, and neither on its own comes close.

What the pictures cannot show #

The density is drawn as a shading and read as a level set. The detector is handed the darkest share of a grid, not the function, and a different share would give a slightly different point set. The group it reports is stable across a range of shares, which is checked; the point set is not the density.

The arithmetic of the density is floating point. The orbits, the phases and the absences are exact rational computations, and the density is a sum of cosines. Its values are rounded before the level set is taken, which is safe because symmetry-related grid points agree to the last bit while distinct values differ by a great deal more — and it is stated here rather than left to be discovered.

And a resolution is not a number of reflections. The cutoff here is on the integer indices rather than on a physical resolution, which for a real measurement would depend on the wavelength and the cell. The count of orbits inside a cutoff is the right analogue of a data set’s size, and the correspondence is not exact.