Systematic absences

A diffraction pattern is a field of spots. The spots carry the structure, and identifying which of them are missing — not weak, exactly zero — identifies the symmetry.

This is how symmetry is actually determined in practice, and it is worth appreciating how odd that is. A glide plane is invisible. Nobody has looked at one. What is seen is a gap in a field of spots, and the gap is diagnostic.

Why anything cancels #

The structure factor for reflection $(hk)$ is a sum over the atoms in one cell:

$$F(h,k) = \sum_j f_j \, e^{2\pi i (h x_j + k y_j)}.$$

Symmetry means atoms come in related pairs. If the structure has an operation mapping atom $j$ to atom $j'$, then both appear in the sum, and their contributions are related — which for certain $(h,k)$ means they are equal in magnitude and opposite in sign.

When that happens for every pair at once, the whole sum is zero, and it is exactly zero rather than small. The cancellation is a consequence of the symmetry rather than of any accident of the atom positions, so it holds for every structure with that symmetry regardless of what the atoms are or where in the asymmetric unit they sit.

That is the meaning of systematic: the absence is a property of the space group, not of the particular compound.

The glide, worked out #

Take a glide reflecting across the line $y = 0$ and sliding by half a cell along $x$. It maps an atom at $(x, y)$ to one at $(x + \tfrac12, -y)$.

Consider reflections with $k = 0$ — the row along the $h$ axis. The two atoms contribute

$$e^{2\pi i h x} + e^{2\pi i h (x + 1/2)} = e^{2\pi i h x}\left(1 + e^{\pi i h}\right).$$

The bracket is $1 + (-1)^h$, which is $2$ when $h$ is even and zero when $h$ is odd.

So every reflection $(h0)$ with $h$ odd is systematically absent. Half of one row of the diffraction pattern is gone, and the missing half is the glide’s signature.

The general rule is the same calculation with different indices. A glide with a half-cell slide along a direction removes the odd-index reflections in the row corresponding to that direction, and the condition for presence — $h$ even, in this case — is what the International Tables list for every space group.

What the absences tell, and how directly #

The identification runs the other way round in practice, and it is remarkably direct.

Observe the pattern. Note which rows have alternate reflections missing. Look up which space groups produce that set of conditions. In many cases the answer is unique; in others it narrows to two or three, which are then distinguished by other means.

The strength of the method is that it needs no phases. A systematically absent reflection has $|F| = 0$, and zero survives being squared into an intensity — so the absences are directly observable whatever the phase problem is doing. Space group determination therefore comes before structure solution, and it constrains everything that follows.

The four groups that show absences #

In the plane the arithmetic is small enough to enumerate, and the answer lines up exactly with a structural property established elsewhere.

Of the seventeen wallpaper groups, four contain a glide that is not merely a mirror composed with a translation already present: pg, pgg, pmg and p4g. Those four are precisely the non-symmorphic groups — the ones for which no choice of origin removes the translation part from every operation at once.

They are also precisely the four whose symbols contain a g, which is the notation doing its job. And they are precisely the four that show systematic absences.

Three descriptions — non-symmorphic, contains a g, shows absences — picking out the same four groups by three unrelated routes. That is a good check on all three, and it is the reason those four keep recurring in these essays: they are where the subject’s structure is most visible.

The same alignment holds in space, where seventy-three of the two hundred and thirty space groups are symmorphic and the remaining hundred and fifty-seven contain a glide or a screw. Every one of those hundred and fifty-seven announces itself in the diffraction pattern.

Screw axes do the same thing #

In three dimensions the glide has a partner, and it produces absences by identical arithmetic.

A screw axis is a rotation combined with a translation along the rotation axis — a $2_1$ axis rotates by $180°$ and slides by half a cell, a $3_1$ axis rotates by $120°$ and slides by a third, and so on. Like the glide, it is a symmetry although neither of its parts is.

A $2_1$ axis along $b$ maps $(x,y,z)$ to $(-x, y + \tfrac12, -z)$, and the same calculation as above shows that reflections $(0k0)$ with $k$ odd are absent. So a screw axis removes reflections along a line in reciprocal space, where a glide removes them across a plane.

That difference is what makes the two distinguishable. Absences confined to an axis indicate a screw; absences filling a zone indicate a glide; and the pattern of which axes and which zones identifies which.

The space group P2₁/c — the commonest in organic crystallography, accounting for roughly a third of all published organic structures — is identified by exactly two conditions: $(0k0)$ absent for odd $k$, from the screw axis, and $(h0\ell)$ absent for odd $\ell$, from the glide. Two rules, and the group is fixed.

An absence is not a weak reflection #

The distinction the whole method rests on deserves its own paragraph, because it is the one that goes wrong.

A systematic absence is an exact zero, forced by symmetry, holding for every structure in that space group regardless of composition. A weak reflection is a small number, arising because the atoms happen to be arranged so that contributions nearly cancel, and it depends entirely on the particular compound.

The two are different in kind and identical in appearance. A detector records a count; a systematic absence gives zero counts plus noise, and a very weak reflection gives a few counts plus noise. Distinguishing them requires long enough exposure that the noise floor falls below the weak reflection’s intensity, and there is no way to know in advance how long that is.

This is why the space-group reassignment literature exists. A structure published in a low-symmetry group because some reflections were taken as present, or in a high-symmetry group because some were taken as absent, refines perfectly well and reports bond lengths that are slightly wrong throughout. Richard Marsh spent much of a career finding such cases and publishing corrections, and the flow has not stopped.

Centring does it too #

There is a third source of absences, and it is the easiest to derive.

A centred cell contains lattice points at positions other than the corners — the centre of the cell, or the centres of faces. Those extra points are lattice translations, so every atom appears twice per conventional cell, offset by the centring vector.

For a cell centred at $(\tfrac12, \tfrac12)$, the pair contributes a factor $1 + e^{\pi i (h+k)}$, which vanishes when $h + k$ is odd. So half of all reflections are absent, in a checkerboard pattern.

That is a useful reminder of what a centred cell means. The conventional cell is larger than necessary, chosen for the convenience of having axes along the symmetry directions, and the systematic absences are the diffraction pattern pointing out the redundancy. Index the pattern on the primitive cell instead and the absences vanish — because they were never absences of scattering, only artefacts of describing the lattice with more points than it has.

Reading a pattern, in order #

The procedure a crystallographer follows is short, and it is worth laying out because it shows where absences sit in the sequence.

Index the reflections. Find the reciprocal basis on which every observed spot lands at integer coordinates. This gives the unit cell and the crystal system.

Read the diffraction symmetry. The point symmetry of the pattern is the crystal’s point symmetry, plus an apparent inversion centre from Friedel’s law. This narrows the point group to one of eleven Laue classes rather than the thirty-two crystal classes.

Read the absences. Which rows, which zones, which parity conditions. This narrows the space group, often to one.

Then, and only then, solve the structure. Recover phases, compute the electron density, place the atoms.

The order matters because each step constrains the next. Getting the space group wrong at step three means solving the structure in the wrong symmetry at step four, which is the failure described above and which does not announce itself.

It is also worth noticing what the sequence has in common with this site’s method. Steps one to three determine the symmetry without ever seeing an atom, from a pattern of positions and a pattern of gaps. That is the same posture the pattern figures take, arrived at by a completely different route.

The second independent route #

Here is why this essay exists on a site otherwise concerned with patterns rather than experiments.

Every pattern figure elsewhere on this site asserts its group directly from the point set: apply the group to a motif, hand the result to a detector that enumerates candidate operations and keeps the ones that work, and require the detected set to match the generating set exactly.

Every diffraction figure computes what that same point set would scatter: evaluate the structure factor over a range of indices, and see which reflections come out zero.

The two calculations share the atom positions and nothing else. One works with integer matrices and exact rationals in the lattice basis; the other with complex exponentials and floating-point sums. When they agree about which groups have glides, that agreement is evidence.

The assertion each figure carries is precisely that correspondence: a group with a glide must show absences, and a group without one must show none. It is checked for every group drawn, and a disagreement would stop the build.

That is a deliberately stronger arrangement than checking one calculation twice. An error inside a chain of reasoning tends to be invisible from within the chain and obvious from outside it — the same reason for having both an algebraic and a geometric proof of the crystallographic restriction.

What absences cannot settle #

Three limits, and the third is a genuine open difficulty rather than a caveat.

They cannot distinguish a structure from its mirror image. Friedel’s law makes intensities at $\mathbf{q}$ and $-\mathbf{q}$ equal, so the observed diffraction symmetry always appears centrosymmetric. Absences narrow the space group; separating enantiomorphic pairs — P3₁ from P3₂, say — requires anomalous scattering.

They cannot detect a symmetry that removes nothing. Symmorphic groups have no glides and no screws, so they produce no absences beyond any from centring. A pattern with no absences is compatible with several groups, and the choice among them has to be made by other means — usually by trying each and seeing which refines sensibly.

They can be faked. A reflection can be too weak to observe for ordinary reasons — light atoms, an unlucky arrangement, a poor sample — and a weak reflection mistaken for an absent one leads to the wrong space group. The resulting structure refines, gives plausible bond lengths, and is wrong. This is the commonest serious error in small-molecule crystallography, and it has a literature: a periodic stream of papers reassigns published structures to a different space group, usually one of higher symmetry.

The remedy is to treat “absent” as a claim requiring evidence rather than as an observation. That is the same discipline this site applies to pattern figures, and for the same reason: the failure is invisible, plausible and self-consistent.

How the figures here compute it #

The absence figures on this site evaluate the sum directly, and the description is short enough to give.

A wallpaper group is applied to a motif, giving a finite set of points in fractional coordinates — the same point sets the pattern figures draw. For each $(h,k)$ in a range, the sum $\sum_j e^{2\pi i (h x_j + k y_j)}$ is evaluated over those points and its magnitude taken. A magnitude below a small threshold is drawn as a cross; everything else is drawn as a spot sized by intensity.

The figure then counts the crosses and asserts the correspondence with the group’s structure: a glide-containing group must produce at least one, and a glide-free group must produce none. That is the check, and it is what makes these figures a route rather than an illustration.

Two limitations are worth naming. The scattering strengths are taken as equal, so these are patterns from identical atoms rather than a real compound. And the point set is finite, so the reflections have widths a real experiment would not — which is the same honest caveat that applies to the aperiodic figures.

Who worked out the rules #

Systematic absences were understood almost as soon as diffraction was, because they are the first thing an experimenter notices and demands an explanation.

The theory belongs to the 1920s and 1930s. Carl Hermann and Charles Mauguin’s notation, developed in the same period, was designed to make the connection legible: a g in a symbol announces a glide, and a glide makes a prediction about absences. The notation and the observable were built to correspond.

The International Tables for X-ray Crystallography, first published in 1935, tabulate the reflection conditions for every one of the two hundred and thirty space groups, and those tables remain the working reference. Reading a diffraction pattern’s absences and looking them up is a procedure that has been essentially unchanged for ninety years, which is unusual longevity for a scientific method and reflects that the underlying arithmetic is exact.

Where the ladder goes next #

The framework this sits in is the reciprocal lattice, and the notation it confirms is Hermann–Mauguin, whose g is a testable prediction rather than a label.

The motion being detected is the glide, which is the one of the four that has no fixed point and cannot be seen directly in a pattern.

And the case where the whole framework had to be extended is aperiodic order, where the reflections are sharp, dense, and indexed by more integers than there are dimensions.

What the pictures here cannot show. An absence is a claim that a quantity is exactly zero, and a drawing shows a cross where a spot is not. The exactness is a property of the arithmetic rather than of the picture, and a reader comparing two figures is comparing two computations’ outputs rather than two observations.