The absence that fills itself in

A systematic absence is the strongest evidence in this subject. A glide plane makes a whole zone of reflections cancel, and cancel is the word that matters: the sum defining the structure factor comes to zero for reasons of symmetry rather than of arithmetic accident, so a zero measured there is a statement about the group and not about where the atoms happen to be.

Reading a space group off its absences rests on that entirely.

The exactness is a property of a model, and the model has one assumption in it that is never stated in the derivation: the beam scatters once.

What a second scattering event does #

A diffracted beam is a beam. If it happens to satisfy the diffraction condition for another set of planes, it diffracts again — and the direction it ends up in is set by the sum of the two scattering vectors.

So if h₁ and h₂ are both allowed reflections, a beam can reach the direction of h₁ + h₂ by two successive events, without any single reflection at that index occurring at all. If h₁ + h₂ happens to be forbidden, intensity arrives where the symmetry said none could.

Renninger measured this in 1937 on diamond, at the 222 reflection, which is forbidden for the diamond structure and which he could not make go away. The effect is called Umweganregung — excitation by a detour — and every crystallographer has met a weak “forbidden” reflection that would not behave.

The arithmetic of which absences are reachable is exact, and it separates the two kinds of absence completely.

Centring absences are safe #

A centred lattice extinguishes every reflection failing an integral condition: h + k odd for a C-centred cell, h + k + l odd for a body-centred one, mixed parities for a face-centred one.

Those conditions are linear, modulo an integer. So the allowed reflections form a group — add any two of them and the sum satisfies the same condition — and the allowed set is closed under addition.

A detour cannot leave a closed set, so no pair of allowed reflections can add to a centring absence, and double diffraction can never produce one.

Glide and screw absences are not #

A glide plane’s condition holds on a zone — 0kl with k odd, say — and a screw axis’s on a row. Neither defines a subgroup: two general reflections lying off the zone can perfectly well add to one on it.

And in practice they always do. Across every group built here, every single glide or screw absence inside the window is reachable, usually by dozens of routes.

So half of what a diffraction pattern says about a space group is safe from a second scattering event and half is not — and it is the half that names the glides and the screws, which is the half that distinguishes most of the two hundred and thirty from one another.

One group with both kinds #

The sharpest case is a group whose lattice is centred and which has a glide, because then the two kinds of absence sit in the same pattern and look identical.

A crystallographer looking at that pattern sees zeros. Nothing on the film says which are safe. The one that reappears when the crystal is turned is the one from the glide, and knowing which to expect requires exactly the arithmetic above.

How the effect is recognised #

The detour needs a third beam to be in position, and that is what makes it identifiable.

For the two-step path to exist, the intermediate reflection h₁ must itself be satisfying the diffraction condition — which happens only at particular orientations of the crystal about the scattering vector of the forbidden reflection. Rotate the crystal about that axis and the intermediate beam comes into position and out again, so the forbidden intensity appears and vanishes as the crystal turns.

A genuine reflection does not do that. Its intensity is constant under rotation about its own scattering vector, because the geometry of a single scattering event does not depend on that angle.

So the test is an azimuthal scan, and the pattern of peaks it produces — a Renninger scan — is a fingerprint of the intermediate reflections available at that orientation. It is used deliberately: the positions of the peaks measure the ratios of lattice parameters very precisely, and the shape of a peak carries the phase relationship between the three structure factors involved. The effect that spoils an absence is also the only routine method for measuring a triplet phase directly.

What is computed here, and what is not #

The reachability is arithmetic on indices and is exact: for every group in the library, every absent reflection inside a window is tested against every pair of allowed ones, the pairs are counted, and the closure of each lattice’s allowed set is checked directly rather than argued for.

No intensity is predicted anywhere. How strong a detour reflection is depends on the two intermediate structure factors, on the crystal’s perfection, on its thickness and on its orientation, and none of that is modelled here. A perfect crystal shows the effect strongly; a mosaic one weakly; and the dynamical theory that computes it is a different subject.

What is owned is the statement about which reflections can be reached, which is a fact about the group’s absence conditions and about nothing else.

Why the two kinds of condition differ at all #

It is worth saying where the asymmetry comes from, because it looks like an accident of how the conditions are written and is not.

A lattice condition comes from translations. The centring vector is a translation of the crystal, so its effect on the structure factor is a phase factor multiplying the whole sum — and the reflections that survive are those where the factor is one, which is a condition of the form h·t being a whole number. That is linear in h, so its solutions are a subgroup.

A glide or screw condition comes from an operation with a linear part. Its effect is to relate the structure factor at h to the one at hM, and the relation only forces a cancellation where hM = h — that is, on the zone or row the operation leaves fixed. The condition therefore applies to part of reciprocal space rather than to all of it, and a set defined only on a zone is not closed under addition, since adding two reflections generally leaves the zone.

So the split is between conditions coming from translations and conditions coming from operations, which is the same split as intrinsic against locative seen from the other side. Everything about how absences behave follows from which of the two produced them.

What it costs a determination #

The practical consequence is that an extinction symbol is a shortlist rather than an answer, and the two reasons for that are worth separating.

The first reason is old and well known. Several space groups share an extinction symbol, because absences see the operations’ translation parts and not their linear parts, so a group and its centrosymmetric relative are often indistinguishable. That is a limit of the method and not a defect of the data.

The second is this essay’s. A weak reflection observed where a glide forbids one may be a detour, a genuine violation of the assumed symmetry, or a badly measured background — and the three have to be told apart before the space group can be assigned. The azimuthal test settles the first; the second is settled by whether the violations are systematic; and the third is what most of them turn out to be.

And the first reason has a sharp instance in the group drawn above. Cc and C2/c differ by a centre of symmetry, and adding that centre changes not one absence. Run the same census on C2/c and the numbers come back identical to Cc’s at every window — the same lattice absences from the C-centring, none of them reachable, and the same glide absences, every one of them reachable. That is not a coincidence of the window: an inversion centre contributes no translation of its own, so it adds no condition of the first kind, and its linear part is −I, which fixes no zone and therefore adds no condition of the second kind either. A centre is invisible to absences by construction, and C2/c is the commonest centrosymmetric space group an organic molecule adopts.

So a determination that reaches C2/c has not read it off the zeros. What separates it from Cc is the distribution of the intensities rather than which of them vanish — a centrosymmetric structure has more very weak and more very strong reflections than an acentric one, which is a statistical statement about the whole data set rather than a statement about any reflection. The absences narrow the two hundred and thirty to a handful and then stop, and the last step is made by an argument of a completely different kind.

Where the same structure appears #

The distinction between a condition that defines a subgroup and one that does not turns up throughout this subject, and it is worth naming.

Lattice conditions are group conditions. Centring, the reciprocal lattice itself, and the sublattice a superstructure introduces all define subgroups of the reciprocal lattice, and everything closed under addition inherits the same immunity.

Operation conditions are not. A glide’s zone, a screw’s row, and the absences a modulated structure produces are conditions on part of reciprocal space, and no closure protects them.

The same split decides which absences survive twinning, which survive disorder, and which survive a second scattering event, because all three are asking whether the set of surviving reflections is closed under some operation. A closed set is robust and an unclosed one is not, and that sentence is doing all of the work in three separate arguments.

The three-beam condition, geometrically #

The reachability arithmetic says which detours exist. Whether one is active is a separate, geometric question, and the two should not be confused.

A detour needs the intermediate reflection to be diffracting at the same time, which means both h₁ and h — the forbidden one — must lie on the Ewald sphere simultaneously. Two reciprocal lattice points on a sphere through the origin is one condition too many for a general orientation, so it happens at isolated orientations: rotate the crystal about h’s own scattering vector and the intermediate point sweeps through the sphere at particular azimuths.

So a forbidden reflection is dark at most orientations and bright at a few, and the few are computable from the lattice alone. That is why the azimuthal scan is diagnostic and why the peaks in it can be indexed: each corresponds to a particular intermediate reflection coming into position.

The number of routes computed here is therefore an upper bound on how much trouble a given absence can be in — a reflection reachable sixty ways has sixty azimuths at which it can light up, and one reachable twice has two.

Six groups are tabulated together above for the sake of the comparison, and reading down the column is the quickest way to see the split this essay turns on. P2₁ has one condition, on the 0k0 row; Pc has one, on the h0l zone; P2₁/c has both of those and nothing else — three groups whose every absence is of the second kind, and therefore reachable. C2 is the opposite case: its only condition is the C-centring rule on all of hkl, of the first kind, and not one of its absences can be filled in. Cc has one of each, which is what makes it the figure above. Fddd has four — the F-centring rule, and a d-glide condition on each of the three principal zones — so it extinguishes more reflections than any other group here and puts the smallest fraction of them at risk.

And that ordering is the reverse of how useful the absences are. C2’s zeros are the trustworthy ones and they say only that the lattice is C-centred; P2₁/c’s are the fragile ones and they are what names the group. The information and the risk are carried by the same conditions, which is the awkward shape of this whole subject.

Who found it, and what happened next #

Moritz Renninger published the effect in 1937, having found intensity at the 222 reflection of diamond, which the diamond structure forbids. He established that it came from double diffraction by showing that it depended on the azimuth — the crystal’s rotation about the scattering vector — which no single-scattering effect can.

The theory belongs to the dynamical treatment of diffraction, which had been developed by Darwin, Ewald and Laue between 1914 and 1931 and which treats the beam and the crystal as a coupled system rather than as a perturbation. Renninger’s effect is the simplest thing in that theory that the kinematic approximation cannot produce at all.

The modern use is the reverse of the original problem. Because a three-beam interaction involves the phases of all three structure factors, the shape of a Renninger peak carries phase information — and this is one of the few experimental routes to a phase rather than an intensity, which is the quantity the phase problem says a diffraction experiment does not measure. It is difficult, it is slow, and it works.

What a reader should take from this about absences generally #

The habit this collection tries to build is that a claim comes with the conditions under which it holds, and systematic absences are a good case for practising it.

Within the kinematic model, an absence is exact. Not small, not usually zero — exactly zero, for reasons that do not depend on the atoms. That is a real and unusual kind of statement, and it is why absences are worth so much.

Outside the model, an absence is a strong expectation. A second scattering event, a twin, a trace of a second phase, or a detector’s tail can each put counts where the model says none. The zeros that survive all of those are the ones a lattice condition produces.

And the model is not a bad one. Double diffraction is a small effect in a mosaic crystal, and structure determination has proceeded successfully for a century on the kinematic approximation. The point is not that absences are unreliable but that their reliability has a structure — and the structure is computable, in one line, from whether the condition is linear.

In electron diffraction the exception is the rule #

Everything above treats double diffraction as a correction to a good approximation. There is an instrument where it is not a correction, and the difference is worth stating because it changes what an absence is evidence for.

Electrons interact with matter perhaps four orders of magnitude more strongly than X-rays do. A specimen thin enough to transmit them is still thick enough for a diffracted beam to be diffracted again several times over, so multiple scattering is the normal condition rather than a nuisance, and forbidden reflections appear in electron diffraction patterns routinely and strongly.

The consequence for space-group determination is severe. A pattern showing intensity where a glide forbids it says nothing at all in electron diffraction, because the detour is always available; the kinematic reading of absences, which is the whole of the method for X-rays, is simply unavailable. For decades that made electron diffraction a technique for lattices and orientations rather than for space groups.

It also inverts the usual reliability ordering. The centring absences are still safe — the closure argument above knows nothing about how many scattering events occurred, only that indices add — so a lattice type is readable from an electron pattern and a glide is not. That is the same split as before, with one half of it now carrying all the weight.

What the dynamical theory gives back #

The repair is a fine one, and it is worth telling because it is the multiple scattering itself that supplies it.

When a converging beam is used, each reflection becomes a disc rather than a spot, and the intensity across that disc varies with the direction the beam took through the crystal. A dynamically forbidden reflection — one absent kinematically and reached by detours — is then not uniformly bright: the detours reaching it come in pairs whose paths are related by the very screw or glide that forbade the reflection, and along particular directions the two members of a pair cancel exactly.

The result is a pair of dark lines across the disc, in a characteristic cross or bar depending on the operation. They are the Gjønnes–Moodie lines, and their presence is evidence of a screw axis or a glide plane that survives the dynamical scattering rather than being destroyed by it.

So the effect that destroyed the absence restores the information in another form. The cancellation is exact for the same reason the kinematic absence was exact — a half-translation making two contributions equal and opposite — and it holds to all orders of scattering because the pairing is a symmetry of the paths rather than a property of any one of them. Convergent-beam electron diffraction determines space groups routinely on that basis, and it does so on crystals far too small to mount on a diffractometer.

Where the ladder goes next #

Two directions.

Into what else the kinematic model gets wrong. Extinction — the weakening of strong reflections in a good crystal, because the beam is depleted before it reaches the far side — is the other standard failure, and it biases exactly the reflections a structure determination weights most heavily.

And into the other half of what symmetry says about a reflection before any structure is known. An absence is symmetry forcing an amplitude to zero; there is a weaker statement, applying to many more reflections, in which symmetry forces the phase to one of two values without saying anything about the amplitude at all. That is the zones that behave as if there were a centre, and it is the half of the phase problem that comes free.

The two sections together make one point about instruments. Which absences survive a second scattering event is decided by the arithmetic of the indices and is the same for every technique; how often a second scattering event happens is decided by how strongly the radiation interacts and is not. So the same theorem reads as a footnote for X-rays and as the central difficulty for electrons, and the repair in each case is found by asking what the multiple scattering itself preserves.