Crystallography & Symmetry
patterns derived from their groups
Concept

Phase problem — where it appears

That a detector records how much radiation arrives and not when, so half of every diffraction measurement is lost before it is written down. The half that is lost turns out to be the half that carries the structure, which is why a map made from intensities alone shows vectors and not atoms.

Named by 14 essays across one field — each of them below, with the objects they name alongside it.

A lattice and its reciprocal. The reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out.

The reciprocal lattice

Nobody has seen a space group. Crystals are read from where they scatter, and where they scatter is a second lattice in which long has become short and short has become long.

diffraction · Diffraction The phase problem. The same structure rebuilt from its diffraction three ways: with the true amplitudes and phases, with the phases scrambled, and with the amplitudes discarded but the phases kept. The atoms survive the loss of the amplitudes and do not survive the loss of the phases, which is what an experiment throws away.

The phase problem

A detector records how much light arrives and not when it arrives, so half of every diffraction measurement is thrown away before it is written down. The half that is lost turns out to be the half that carries the structure.

diffraction · Diffraction A structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.

The map that needs no phases

A diffraction experiment measures intensities and loses phases, so the electron density cannot be computed from it. One map can be: the transform of the intensities, whose peaks are not atoms but the vectors between them — every ordered pair, brought to a common origin.

diffraction · The Patterson function Every vector between every pair of atoms. The Patterson map of a four-atom structure: the transform of the intensities with every phase set to zero, so it is computable from a measurement and nothing else. Its peaks are not atoms but the vectors between them, and the ringed one is the strongest that is not the origin — taken here as an interatomic vector exactly as a crystallographer takes the vector between two heavy atoms. That it really is one of the structure's own vectors is checked rather than assumed: it lands within 0.0031 of a cell of a difference of two positions, and it sits 0.67 of a cell from the origin, well outside the skirt of the tall peak there. A candidate taken too close to the origin is the same peak seen again and the whole method fails quietly.

Solving from the vector set

A Patterson map contains a copy of the structure laid over every atom in turn. Shift it by one interatomic vector, take the pointwise minimum with itself, and the copies that fail to coincide are cut away — leaving the structure, together with its inverse, from a measurement that carries no phases at all.

diffraction · The Patterson function Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation.

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

diffraction · Friedel's law Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them.

Two structures, one Patterson

Eight arrangements of four atoms on a ring of eight sites, and only seven distinct sets of interatomic vectors between them. Two of the arrangements are genuinely different and no measurement of intensities can tell them apart — at any resolution, for ever.

diffraction · Homometry Where a homometric pair comes from. A set that factors as a sumset gives its own partner. If every point of A is a sum b + c with b in B and c in C, and every sum arises once, then reversing C produces a different set with the same vectors — because reversing a factor and reversing its conjugate cancel in the product that the vector set is. Both factors must be asymmetric, which is the constraint that decides where the construction can be used: a two-point set is its own reflection up to a translation, so the smallest useful factorisation is three points by three points, and the smallest structure it builds has nine atoms.

Where the pairs come from

A structure whose atoms are the sums of two smaller sets has a partner: reverse one factor and the interatomic vectors do not notice. The construction is Patterson's own, it explains why homometry exists, and the smallest structure it can build has nine atoms for a reason worth following.

diffraction · Homometry pmg: 80 of 80 restricted. The reflections of pmg inside a window of ±4, with the ones whose phase symmetry restricts to two values picked out. Every solid spot has a structure factor that must be real up to a fixed rotation — a sign, in effect — whatever the atoms turn out to be, and the pale ones have a phase symmetry says nothing about. The 4 palest spots carry two incompatible restrictions at once, which leaves them nothing to be but zero. Which spots these are was computed from the operations, before any structure existed.

The zones that behave as if there were a centre

The phase problem is usually stated as though symmetry had nothing to say about phases. It is true of most reflections and false of some, and which is decidable from the group alone: where an operation carries a reflection onto its own negative, the phase is confined to two values half a turn apart, computed from that operation's translation.

diffraction · Statistics ⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

Three phases that do not move when the origin does

A phase is a property of the description, not of the crystal: shift the origin and every one of them changes. A sum of three phases whose indices add to zero does not change, because the shifts cancel. That sum is the smallest thing about a structure that a diffraction experiment could in principle know, and it is not distributed at random.

diffraction · Direct methods A map from amplitudes alone, 0.59 grid steps out. The density after 150 cycles of flipping, with the atoms that produced the data drawn as rings — moved into the origin and the handedness the solution chose, because a phase set does not fix either and comparing without allowing for them measures the arbitrariness of the description. Every peak of the map is an atom and every atom has a peak. Nothing about the arrangement went into the calculation: the input was a list of amplitudes and a random set of phases.

The solver that knows no symmetry

Compute a map from amplitudes and random phases, reverse the sign of everything below a small threshold, transform back and keep the phases. Repeat. The structure appears — and so does its space group, which was never supplied.

diffraction · Direct methods Two kinds of atom, and more ambiguity rather than less. Exhaustive searches on rings of four sizes. The third column counts homometric groups when every atom is identical; the fourth counts them when each atom may be one of two kinds. The fourth is larger at every size, and at nine and ten sites the third is nothing at all — there is no pair of arrangements of four identical atoms that a diffraction experiment cannot separate, and there are six and four once the atoms may differ. Distinguishing the atoms adds information to the structure and adds ambiguity to the measurement.

When the atoms are not all the same

Every homometric pair found so far is a pair of point sets, where an atom is a point and counts once. Give the atoms different scattering powers and the ambiguity does not go away — it grows. On a ring of nine there is no pair of four identical atoms that diffraction cannot separate, and there are six once two kinds of atom are allowed.

diffraction · Homometry The same structure, mapped from intensities and from differences. Left, the ordinary Patterson map of the structure: 14762 interatomic vectors, a continuous field of overlapping peaks, and the two vectors between the anomalous scatterers — circled — nowhere among its strongest. Right, the map from squared Bijvoet differences over the same reflections: two peaks after the origin, and they are those two vectors. The difference map is 7381 times smaller a problem to read.

A map of the atoms that break the law

Feed a Patterson synthesis the differences between the two halves of each Friedel pair instead of the intensities, and the map that comes back holds the vectors between the anomalous scatterers and nothing else. Two atoms among a hundred and twenty-two: 14,762 vectors become two.

diffraction · The Patterson function The sign turns when the cross terms go weak. Every quartet among the strong reflections of a small structure, sorted by the mean of its three cross terms, with the mean cosine of the phase sum in each bin. Where the cross terms are strong the quartet behaves like a triplet and the cosine is near one. Where they are weakest it is -0.698 — negative — and 93 per cent of those quartets have a cosine below nought. The four reflections of the quartet are equally strong in every bin; what changes is three reflections that are not in the sum at all.

The relation that can say no

Every phase relation before this one pushes a sum towards zero, so none of them can contradict another — a phase set satisfying all of them badly is still satisfying them in the same direction. A quartet can be estimated at π instead, and it is when its three cross terms are weak, so the information arrives from the reflections nobody would have thought worth measuring.

diffraction · Direct methods An equality, with a known factor in it. Sayre's identity says the structure factor is proportional to the convolution of the structure factors with themselves — an equality, not a probability — with a factor that depends on the atoms and the resolution and on nothing else. Computed for a structure of Gaussian atoms, the ratio of the two sides falls with resolution exactly as predicted: the logarithm of the ratio is a straight line in the squared index, its slope is minus half the width in the atomic factor, and the ratio is very nearly real and positive, so the identity relates the phases and not only the amplitudes.

The relation that is an equality

Every relation of this kind so far is a probability — right nine times in ten, useless applied once and decisive applied ten thousand times. One is not. For a structure of equal, resolved atoms the squared density has peaks in the same places, so every structure factor is exactly a convolution of all the others, with a factor that depends only on the atom. It is exact, and on its own it is useless.

diffraction · Direct methods

Named alongside it

The objects these essays reach for when they reach for this one.

Structure factorThe Patterson functionDirect methodsInteratomic vectorAutocorrelationCentrosymmetryConvolutionFriedel lawHeavy atom methodHomometryMeasurementNormalised structure factor

All concepts