The zones that behave as if there were a centre

The phase problem is usually stated as a clean division: intensities are measured, phases are not, and the structure is in the phases. It is true of most reflections and false of some, and which is decidable from the group alone with no structure and no experiment.

Where the group contains an operation carrying a reflection onto its own negative, that reflection’s phase is confined to two values half a turn apart.

Not determined — confined. It is half the information, arriving free, and knowing which reflections have it is the difference between a phase problem with a continuum of unknowns and one with a great many binary choices.

Where the restriction comes from #

A symmetry operation makes one reflection’s structure factor into another’s. If the operation is (M, t), then

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

which is a statement about a pair of reflections and says nothing about either one alone.

But suppose hM = −h for some operation of the group. Then the relation is about h and −h — and those two are already related, because the electron density is real, which gives F(−h) = F(h)*. Putting the two together and writing F = |F| exp(iφ):

$$e^{-i\varphi} = e^{i\varphi}\,e^{-2\pi i\,h\cdot t},$$

so 2φ ≡ 2π h·t modulo 2π, and

$$\varphi = \pi\,h\cdot t \quad\text{or}\quad \pi\,h\cdot t + \pi.$$

Two values, half a turn apart, both computed from the translation part of the operation that produced the restriction. Such a reflection is called centric — it behaves as though the structure had a centre of symmetry, whether or not it has one.

Which reflections these are #

The answer varies from nothing to everything, and both extremes occur in the plane.

p1 has none. Its only operation is the identity, which carries h to h and not to −h.

p2 has all of them. Its half-turn is the two-dimensional inversion: the matrix is −I, so hM = −h for every reflection, and every phase in the whole pattern is restricted to two values.

Everything between has zones. pm and pg restrict a single row — the reflections perpendicular to the mirror direction. p3m1 restricts three rows and p31m six, which is a difference between two groups that share a point group and differ in how it sits on the lattice.

Reading the zones off the group #

The rule for finding them is short enough to apply by hand, and doing so once makes the figures legible.

A reflection h is centric when some operation’s matrix M satisfies hM = −h — with h treated as a row vector, because that is how indices transform. So the question is which operations have $-I$ among the matrices h can be sent to, and the answer is read off the point group.

• A half-turn in the plane has matrix $-I$ outright, so it makes every reflection centric.
• A mirror across the b axis has matrix diag(1, −1), so hM = (h, −k), and that equals −h only when h = 0. The centric zone is the 0k row.
• A four-fold has matrices $\pm I$ and the two quarter turns; the quarter turns send (h, k) to (−k, h), which is never −h unless both are zero. So p4’s centric reflections come from its half-turn alone — which is all of them.

The pattern is that the centric zones are decided by which operations are equivalent to an inversion on some subspace, and in three dimensions the same rule gives the familiar cases: a two-fold axis makes the reflections in its perpendicular plane centric, a mirror makes the row along its normal centric, and a centre makes everything centric.

The check: computed phases land where they must #

The restriction is a prediction about a structure that does not exist yet, so it is checked against one that does.

Atoms are placed in general positions from a fixed seed, the orbit under the group is generated, and structure factors are summed over the whole cell. Every centric reflection’s phase is then compared with the two values its group permits.

A thousand two hundred and forty-eight centric reflections across the seventeen, every one on a permitted phase to one part in a thousand million. Nothing about the atoms was chosen to make that happen.

The acentric phases in the same figure are the control. If they were also confined, the restriction would be saying nothing; they are not, and the measure of how unconfined they are is reported with them.

When two restrictions disagree, the answer is zero #

A reflection can be centric by way of two different operations, and then it carries two restrictions. Usually they agree — the two permitted pairs are the same pair.

When they do not, there is no complex number satisfying both except zero.

So a systematic absence is a phase restriction with no solution, which is a second derivation of a quantity this collection has computed several ways already. Thirty reflections in the window used here are of that kind, and all thirty are computed to have zero amplitude from the atoms — the two routes agreeing, as they must.

That is worth pausing on. Systematic absences are normally derived by summing the structure factor over the general position and finding the sum vanishes identically. The same reflections arrive here from asking what phase the group permits and finding the permissions inconsistent. Two arguments, one set of reflections, and the second explains why absences come in zones — because phase restrictions do.

The statistics that follow #

A centric reflection’s structure factor is a real number with a sign, once the fixed rotation is taken out. An acentric one is a complex number with a free phase. Two different distributions, and the difference is measurable without knowing any phase at all.

For many atoms in general positions the classic averages are

$$\langle |E^2 - 1| \rangle = 0.968 \;\text{(centric)}, \qquad 0.736 \;\text{(acentric)},$$

and the gap between them is what makes the presence of a centre a statistic rather than a guess.

The zones are where this test is sharpest. A crystallographer who suspects a centre can compare the statistics of the whole data set against the two limits; but a crystallographer who knows the group’s zones can compare a zone against the rest of the same crystal’s data, with the same scale factor, the same absorption and the same everything else. It is an internal control, and internal controls are worth a great deal.

What the restriction is worth #

Half of a phase is a great deal in a subject where phases are the whole difficulty, and it is worth being precise about how much.

A fully centrosymmetric structure — one of the eleven a diffraction pattern reports — has every reflection centric, and the phase problem becomes a sign problem: each reflection is plus or minus, and the search space is finite rather than continuous. That is why centrosymmetric structures were solved first historically, and why direct methods were developed for them before being extended.

An acentric structure with zones has the sign problem on those zones and the full problem everywhere else. The zones are typically a small fraction of the data — pm’s is one row out of many — but they are the reflections that anchor the origin, because a phase restricted to two values cannot be shifted continuously by moving the origin.

That last point is the practical one. Fixing the origin in a phase determination means choosing phases for a few reflections arbitrarily, and the reflections that may be chosen are constrained by exactly this arithmetic: which combinations of indices have phases that a change of origin can move, and which do not.

And the mechanism is visible in the sum itself, which is worth seeing once because it explains why the restriction is so robust. A structure factor is a sum of one contribution per atom in the cell, each a vector of length equal to that atom’s scattering power turned through an angle that depends on where the atom sits. For a reflection with no operation carrying it to its own negative, those contributions point wherever the atoms put them and the total lands wherever the chain ends. For a centric reflection the cell’s atoms come in pairs — an atom and its image under that operation — and the two contributions of a pair are mirror images in one fixed line. Their sum lies along the line exactly, for every pair, whatever the atom’s position and whatever its scattering power. So the total is on the line before the last atom has been added, and no arrangement of atoms can take it off.

That is a stronger statement than “the computed phases came out right”, and it is the one the figure above checks: not merely that the total lands on a permitted value, which four atoms might manage by accident, but that each pair does, one pair at a time. Move every atom to a new random position and recompute, and the centric phase is still on the same two values while the free one has moved — which is what it means for the restriction to belong to the group and not to the contents of the cell.

What is not being claimed #

A centric reflection is not a determined one. Two values is not one, and choosing between them is exactly as hard, per reflection, as the sign problem in a centrosymmetric structure. What symmetry supplies is that the choice is binary.

And the statistics are not a proof. The two limits assume many atoms, none dominant, all in general positions, and each of those fails somewhere: a heavy atom on a special position skews the distribution towards centric-looking values whether or not a centre is present, which is a standing trap in structure determination and the reason the test is quoted with its assumptions.

Nor does any of this see the difference between a crystal and its mirror image. Friedel’s law makes the intensities centrosymmetric regardless, and the phase restrictions computed here are on top of that — they say nothing about which enantiomorph is present, only about where the phases of a given one may sit.

The two relations are worth keeping apart, because they look alike and do different work. Friedel’s law relates a reflection to its own negative and holds for every crystal, since it follows from the density being a real number and from nothing else; it is what makes a diffraction pattern look centrosymmetric whether or not the crystal is. A phase restriction needs a second relation on top of that, supplied by an operation of the group that carries the reflection onto its own negative — and only some reflections have one. Where the two meet, the phase is pinned to a pair; where only Friedel’s law applies, the phase is free and the intensity is still symmetric. So a pattern that looks completely centrosymmetric may have almost no restricted phases in it, which is exactly the position an acentric crystal is in and exactly why the extra relation is worth computing separately.

Why the restriction survives things that break most arguments #

One property makes the restriction unusually robust, and it is worth stating because most symmetry-derived facts about a diffraction pattern are more fragile.

The derivation used two ingredients: the density is real, and the group contains an operation with hM = −h. Neither mentions the atoms. So the restriction holds whatever the structure is, however disordered, however badly refined, and whether or not the model is right. A phase computed from a wrong structure in the right space group still lands on a permitted value.

That makes the restriction a test of the space group rather than of the structure. A data set whose zonal phases refuse to behave has the wrong group, and no improvement in the model will fix it — which is a much cleaner diagnosis than most of what a refinement produces.

It also means the restriction survives averaging. A twinned crystal produces intensities averaged over two orientations, and averaging destroys most statistical signatures; it does not destroy this one, because each individual’s phases obey the restriction and the twin law relates the two individuals’ restrictions.

Who worked it out #

The restrictions were tabulated as the space groups were, and appear in the International Tables as a column of permitted phase values per reflection class. The systematic derivation is due to Buerger and to the direct-methods literature of the 1950s: Hauptman and Karle’s statistical approach uses the centric-acentric distinction constantly, because the probability relationships between phases have different forms in the two cases.

The intensity statistics are Wilson’s, from 1949, and were introduced for a different purpose entirely — to put measured intensities on an absolute scale — with the distinction between the two distributions falling out of the same analysis. The test for a centre came free from a calibration procedure, which is the kind of thing this subject does regularly and rarely by design.

What a zone is worth in practice, with numbers #

It is easy to say that half a phase is valuable and harder to say how much, so here is the accounting for a small structure.

Suppose a two-dimensional structure in pm with four hundred measured reflections, of which twenty are in the centric row. A phase determination has four hundred unknowns, each continuous. The twenty are binary, which is 2²⁰ ≈ a million combinations — an enormous number in isolation and a trivial one beside a four-hundred-dimensional continuum.

More usefully, the twenty pin the origin. In pm the origin may be moved along the mirror direction freely, and moving it multiplies each structure factor by a phase depending on the index — but a centric reflection’s phase may only jump between its two values, so a shift that is not a lattice fraction is forbidden by the zone. The zone therefore reduces the origin’s freedom from a continuum to a finite set, and the finite set is exactly the equivalent origins the group’s normaliser describes.

That is the connection worth carrying: the reflections whose phases symmetry restricts are the reflections a change of origin cannot move freely, and the two statements are the same fact seen from reciprocal space and from real space.

The reflections that carry no anomalous difference #

There is a practical use for the centric zones that follows from nothing but the restriction, and it is the one an experimenter meets first.

Friedel’s law breaks when an atom scatters anomalously, and the size of the break is what a phase determination lives on. Measuring it means comparing I(h) with I(−h) — a difference of a per cent or two between two numbers each measured with its own errors — so the first question about any such data set is whether the differences seen are signal or noise.

The centric reflections answer it. A centric reflection’s phase is confined to two values half a turn apart, so its structure factor lies on a line through the origin; F(h) and F(−h) are the same real number, and the Bijvoet difference is exactly zero whatever the anomalous scattering is. So the differences measured on the centric zones are pure measurement error, and their spread is a direct estimate of the noise against which the acentric differences must be judged.

That is an unusually clean control, and the reason is the one this whole essay has been about: the zero is predicted from the group and from the reality of the density, with no model of the structure and no assumption about which atoms scatter anomalously. It costs nothing to measure, because the reflections are in the data set already, and it is measured on the same crystal, in the same experiment, with the same scale and the same absorption. A control that shares every systematic error with the measurement it is controlling is worth more than a better control measured elsewhere.

That is an internal control of a rare kind: it needs no second crystal, no repeated measurement and no model, and it is available from the group alone before anything is known about the structure. A data set whose centric Bijvoet differences are as large as its acentric ones has no usable anomalous signal, and the conclusion is available in one line of arithmetic.

The same fact has a converse worth stating. An anomalous scatterer does not break the restriction on a centric reflection — the phase stays on its two permitted values — because the restriction came from the group and the group has not changed. What anomalous scattering breaks is the relation between h and −h in the acentric case, where the two are different reflections with independent phases. A reader who expects a symmetry-derived restriction to soften when the physics gets complicated has the direction of the argument backwards: the restriction is a statement about the operations, and no property of the atoms touches it.

Which phase the origin does not fix #

There is a second choice a phase determination has to make, it is not the origin, and the centric zones are where the difference between the two shows.

Fixing the origin removes the freedom to translate the structure. It does not remove the freedom to invert it: for a group with no centre, a structure and its mirror image satisfy every relation among amplitudes equally well, so a phase determination that has fixed its origin still has two solutions, related by negating every phase.

The centric reflections cannot break that tie. Negating a phase confined to 0 and π maps the pair to itself, so every centric reflection has the same value in both hands — which is precisely why the tie exists. What breaks it is choosing the sign of one acentric phase, and that choice is called fixing the enantiomorph. It is one further arbitrary decision, of exactly the same kind as the origin, and it must be made on a reflection the restriction does not touch.

So the accounting for a phase determination in an acentric group runs: a few reflections to fix the origin, chosen for their index parities; one more to fix the hand, chosen from outside the centric zones; and everything else determined relative to those. The centric zones are where half the answer is free and the choice is not available, which is a useful thing to know in both directions.

Where the ladder goes next #

Upwards, into what the statistics can be made to do. A twin, a heavy atom, a pseudo-translation and a genuine centre each leave a signature in the intensity distribution, and telling them apart is what a modern data-analysis programme spends its first minutes doing.

Sideways, into the other thing symmetry says about a reflection before any structure is known: that its amplitude is zero. The absence that fills itself in is about how far that stronger statement can be trusted — and the two essays together are the whole of what a group says about a diffraction pattern without any atoms in it.