The reflections that are not there

Nobody has seen a space group. The diagrams in this field are pictures of what a structure would look like to an eye that could see it; what a diffraction experiment actually produces is a list of directions in which a crystal scatters, and how strongly.

Everything about the symmetry has to be inferred from that list — including, decisively, from the directions in which the crystal scatters nothing at all. Those are the systematic absences, and they are the only route to the operations that carry a translation. A screw axis and a glide plane change no intensity that an experiment can compare against anything; what they do is delete reflections, exactly, and the deletions are the signature.

Where an absence comes from #

The structure factor is the sum a crystallographer writes down: over every atom in the cell, a complex exponential of 2πi times the dot product of the reflection indices with the atom’s position. Its magnitude is what an experiment measures.

Now consider a crystal whose atoms sit at the orbit of one general position under a space group. Every atom is an image of the first under some operation, so the sum runs over the group’s operations instead of over the atoms:

F(h) = Σⱼ exp(2πi h·(Mⱼx + tⱼ))

Split the exponential. The operation’s linear part acts on h and the translation contributes a phase that does not depend on x at all:

F(h) = Σⱼ exp(2πi h·tⱼ) · exp(2πi (hMⱼ)·x)

Now group the terms by the value of hMⱼ. Terms with the same hM share the same function of x, and the functions for different hM are independent — no linear combination of them vanishes identically unless every coefficient does. So F vanishes for every x exactly when each group’s coefficient is zero, and each coefficient is a sum of exp(2πi h·t).

That is the whole derivation and it is worth noticing what it does not need. It says nothing about what the atoms are, how many there are, or where the first one sits. The absence is a property of the group, and it holds for every crystal in that group without exception.

Why the test is exact #

The coefficients are sums of complex exponentials, and a naive implementation would add floating-point numbers and compare the result against a small threshold — which would make an exact claim depend on a tolerance, and this site does not do that.

It does not have to. Every translation in a space group has a denominator dividing twelve. Halves come from mirrors and two-folds, thirds from three-folds, quarters from four-folds and the d glide, sixths from six-folds; the least common multiple is twelve. So h·t is always a twelfth of an integer, and each coefficient is a sum of twelfth roots of unity.

A sum of twelfth roots of unity is zero exactly when the polynomial recording how many of each root appear is divisible by the twelfth cyclotomic polynomial — because that polynomial is the minimal polynomial of a primitive twelfth root over the rationals, so if the root is a zero of the sum, the minimal polynomial must divide it.

Integer polynomial division. No floating point, no tolerance, and the same cyclotomic machinery this site already built for the restriction in higher dimensions, reused.

Three layers of condition, and what each layer means #

The conditions organise themselves and the organisation is the crystallography.

General conditions apply to every reflection hkl and come from centring. A C-centred lattice extinguishes every reflection with h + k odd; a body-centred one, every reflection with h + k + l odd; a face-centred one, everything without h, k and l all the same parity. The reason is direct: the centring translation is an operation with M = I, so it contributes a term to every coefficient, and the term is −1 exactly when the phase is a half.

Zonal conditions apply to a plane of reflections — h0l, hk0, 0kl — and come from glide planes. A c glide perpendicular to b kills h0l with l odd. The plane of reflections is the one perpendicular to the glide’s own normal, which is why a glide announces itself in a zone.

Axial conditions apply to a row — h00, 0k0, 00l — and come from screw axes. A 2₁ screw along b kills 0k0 with k odd. A 4₁ along c kills every 00l with l not divisible by four.

The reduction, and why it matters #

Deriving the conditions produces more of them than the Tables print, and the excess is a real problem rather than untidiness.

A C-centred group extinguishes h + k odd. That single statement implies “h0l with h even”, “hk0 with h+k even”, “0kl with k even”, “h00 with h even” and “0k0 with k even” — five more true statements, all consequences. Printing seven conditions suggests seven independent facts about the group when there is one.

So the machinery visits rules from the most general class to the most specific and keeps only those that kill something nothing above them has already accounted for. That ordering is not a tidy-up: it is the crystallographic reading, in which the general condition is the centring, the zonal ones are the glides and the axial ones are the screws, and each layer is read after the one above has been discounted.

One further ordering is needed and it is the one that catches people. Within a class, the rules have to be visited strongest first. P4₁ extinguishes every 00l whose index is not a multiple of four — so “00l with l even” is also a true statement about it, and visiting the weaker rule first keeps both and prints two conditions where the Tables print one. Sorting by how many reflections each rule kills puts “divisible by four” first, after which “even” adds nothing.

What the plane already showed #

This site derived the two-dimensional version of all of it in its first phase, and the absences essay there is worth reading alongside this one because the argument is identical and half the size.

In the plane there is one kind of glide and one kind of centring, so there are two mechanisms rather than three, and the conditions read h0 with h even, or hk with h + k even. The derivation is the same sum with two indices instead of three.

What is genuinely new in space is the axial condition, which the plane cannot have. An axial condition comes from a screw axis, a screw axis needs a direction to climb in, and the plane has none. So a two-dimensional pattern’s absences come in two layers — general and zonal — where a three-dimensional one has three, and the extra layer is the one that reports the operations that only exist in space.

What this is good for #

The practical use is direct and it is how space groups are actually assigned.

Collect a data set. Note which reflections are absent. Look up which groups produce that pattern of absences. In favourable cases the answer is unique and the space group is determined before a single atom has been located.

The Laue class — which is what the intensities give — narrows the possibilities to a crystal system and a point-group class. The absences then narrow it further, and between them they often leave one group.

Often, and not always. Roughly a third of the 230 are uniquely determined by their diffraction symbol; the rest share one with at least one other group, and telling those apart needs something else. That is the next essay, and its subject is the specific ways the experiment runs out.

Where the conditions come from historically #

The rules were found the other way round from the derivation above, and knowing that explains why they are usually presented as a table.

Systematic absences were noticed in the 1920s as an empirical regularity — certain classes of reflection were missing from certain crystals, reproducibly — and the connection to screw axes and glide planes was made afterwards, once the space groups were understood well enough to be tested against. So the tables in the International Tables are, historically, a record of a correspondence rather than a derivation of one.

That order of discovery is worth keeping in mind because it explains a feature of the literature. The conditions are printed as data, group by group, and a reader who wants to know why P2₁ extinguishes 0k0 for odd k will usually find the fact rather than the argument. The argument is four lines and it is the one at the top of this essay.

Two independent routes to one fact #

The thing worth stopping on is that this calculation and the rest of this collection share almost nothing.

Every pattern figure on this site asserts a group from a point set in real space: generate an orbit, forget the group, enumerate the operations a lattice permits, compare. The arithmetic is integer matrices acting on rational coordinates.

This calculation asserts the same group’s consequences from a sum over its operations in reciprocal space: no point set, no orbit, no detector. The arithmetic is roots of unity and polynomial division.

The two share the operation list and nothing else, and they agree — on every group tested, against the Tables, on every build. That is not a proof of anything, and it is the strongest kind of evidence a computation of this sort can offer: two implementations of two different arguments, written to answer different questions, arriving at the same answer.

It is also how the subject is actually done. Nobody sees a space group. The pattern figures in this collection are pictures of what a structure would look like to an eye that could see it, and the diffraction calculation is what the experiment produces. That the two agree is the reason anybody believes the pictures.

Reading a condition backwards #

The use a reader will make of all this is the reverse of the derivation, so it is worth doing once in that direction.

An observed pattern shows 0k0 absent for k odd and nothing else. What produces it? An axial condition on the b axis with a factor of two, which is a 2₁ screw along b. No other operation produces that and nothing else, so the crystal has a 2₁ along b and no glide and no centring — which narrows it to P2₁ and P2₁/m, the two monoclinic groups with a screw and no glide.

Now suppose the pattern shows 0k0 absent for k odd and h0l absent for l odd. Two conditions: the screw along b, and a c glide perpendicular to b. That is P2₁/c, and it is the only group with both.

The reason the reverse reading works at all is that the conditions are independent evidence. A screw and a glide produce conditions in different classes — a row and a zone — so their evidence does not overlap, and a group with more intrinsic translations is better determined than one with fewer. The best-determined groups in the whole classification are the ones with the most screws and glides, and the worst are the ones with none.

Why twelfth roots are enough #

The exact test works with twelfth roots of unity, and that is not a convenient choice of resolution — it is the whole set of phases a space group can produce.

Every translation part of every operation of every space group is a vector whose components are fractions with denominator 1, 2, 3, 4 or 6. That is not an observation about the tables; it follows from the same annihilation argument that bounds any extension of a finite group — the order of the point group kills the translation classes — together with the orders a lattice permits.

The least common multiple of those denominators is twelve. So h·t is always a multiple of a twelfth, exp(2πi h·t) is always a twelfth root of unity, and a coefficient in the derivation is always a sum of twelfth roots. The arithmetic is therefore exact in a fixed finite ring, and every question about a coefficient is a question about integers.

That is a stronger position than it sounds. A test that worked to fifteen decimal places would be right on every case anybody tried and would still be a test with a threshold in it; a test that reduces to integer polynomial division has no threshold at all, and a claim that a whole class of reflections is absent for every possible structure is exactly the kind of claim that should not rest on one.

Which sums of roots can vanish #

There is a general theorem about the vanishing sums the derivation runs into, and it is worth knowing because it says which absence conditions are possible before any group is examined.

A sum of n-th roots of unity can be zero in only one way, structurally: it must be built out of complete sets. Take a prime p dividing n; the p roots evenly spaced round the circle sum to zero, and so does any rotation of that set. Every vanishing sum of n-th roots is a non-negative integer combination of such rotated p-gons, one prime at a time — which is the Lam–Leung theorem, with earlier partial forms due to Conway and Jones.

The consequence for twelfth roots is a short list. Twelve is divisible by two and three, so a vanishing sum has length a non-negative combination of two and three — length two, three, four, five and upwards, but never one. So an absence condition never comes from a single operation cancelling by itself; it always comes from a set of operations whose phases form complete pairs or complete triples.

That is visible in every condition in the table. A centring absence is a pair — the identity and the centring translation, out of phase by π. A glide’s zonal condition is a pair. A three-fold screw’s axial condition is a triple, and a six-fold screw’s is a pair of triples or a triple of pairs depending on the reflection.

So the shape of the absence conditions is fixed by the arithmetic of twelve, and a proposed condition that would need a vanishing sum of some other length is not merely absent from the tables — it cannot exist.

Where the exactness stops #

Three limits, and the first is the largest.

An absence in the calculation is exact; an absence in an experiment is a threshold. This computation says a reflection vanishes identically for every arrangement of atoms in the group. An experiment says a detector recorded nothing above the noise. Those are different claims, and the gap between them is where real space-group assignment gets difficult: a weak reflection that should be absent may be present because of multiple scattering, because of a small deviation from the assumed symmetry, or because the crystal is twinned. Every crystallographer has assigned a group from absences and been wrong.

The derivation assumes a general position. The sum runs over the orbit of one point in general position, which is the right assumption because a real structure’s atoms are almost all general. An atom on a special position contributes a shorter orbit and can produce additional absences that are properties of that structure rather than of its group — those are not systematic absences and are not predicted here.

Only conditions in the catalogue can be reported. The tested list holds twenty-four candidate rules covering the centrings, the axial conditions to a factor of six, and the zonal ones to a factor of four. The machinery tests a fixed list of candidate rules; a group whose condition is not expressible in that list would have its strongest expressible rule reported instead. That failure mode is real and was hit while building this: without the quarter conditions in the list, Fddd’s zonal rules were reported as an axial one — true, weaker than what the group does, and with nothing to indicate the shortfall. The list now carries them, and the general lesson is that a catalogue that cannot express a rule reports the strongest rule it can express and looks right.