Crystallography & Symmetry
patterns derived from their groups
Concept

Structure factor — where it appears

The sum over a cell's atoms of their scattering with a phase from each position, whose modulus is what a diffraction experiment records. Every systematic absence on this site is computed from that sum rather than looked up, which turns an extinction condition into a theorem.

Named by 32 essays across 8 fields — 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 What pg scatters. The diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen.

Systematic absences

The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.

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 What p3 scatters, and what the scattering shows. The structure on the left has point group 3, of order 3. The intensities it scatters, on the right, have point group 6, of order 6 — more symmetric than the thing that produced them. Reversing the sign of both indices conjugates every term in the sum and leaves the modulus alone, so a diffraction pattern always acquires a centre of symmetry, and in the plane a centre is a half turn. Both numbers are measured: the left from the operations, the right by testing each candidate against the computed intensities.

The symmetry diffraction adds

A diffraction pattern is always more symmetric than the crystal that made it. The extra symmetry is not a mistake in the experiment and no care removes it — it is a property of what a detector records, and it collapses the seventeen groups onto six.

diffraction · Accidental symmetry The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

point-groups · Diffraction The h0l layer of P2₁/c. The h0l reflections of P2₁/c out to 4 in each index, with each spot decided by summing the structure factor over the group's operations: 44 survive and 36 vanish identically, whatever the atoms are. The pattern of holes is the condition h0l: l even, read back off the spots rather than imposed on them.

The reflections that are not there

A screw axis and a glide plane leave no mark on the intensity of any reflection. What they do is delete some, exactly, for every possible arrangement of atoms — and the pattern of deletions is computed here from the sum a crystallographer writes down, rather than read from a table.

diffraction · Diffraction p3, single and twinned. Left, the diffraction pattern of a single crystal of p3. Right, the same crystal twinned, with 50 per cent of it in one orientation. Not one spot has moved — the twin law is a symmetry of the lattice, so the two reciprocal lattices lie exactly on top of one another — and 72 of the 81 reflections drawn have changed intensity. At a fifty-fifty twin the pattern acquires the full symmetry of the lattice's point group and is indistinguishable from a crystal that genuinely has it.

A merohedral twin moves no spot at all

The twin law is a symmetry of the lattice, so the two individuals have reciprocal lattices lying exactly on top of one another. Nothing splits, nothing appears in a new place, and the only thing that changes is that pairs of intensities which were different have been averaged — which produces a diffraction pattern with a symmetry the crystal does not have and no sign that anything is wrong.

applied · Twinning A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.

The satellites that need a second integer

A crystal whose atoms are displaced by a wave of the wrong wavelength has no unit cell at all, and diffracts to sharp spots anyway. Indexing them takes two integers per reflection instead of one — and the intensities of the extra spots are Bessel functions, which is a check the arithmetic can be made to pass.

aperiodic · Modulation What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

diffraction · Diffraction Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed.

The law that hides handedness

With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone — so every diffraction pattern is centrosymmetric whatever the crystal is. The escape is an imaginary component that the negation does not touch, and it is how the handedness of a molecule is measured.

diffraction · Friedel's law Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions.

The reflections a superlattice adds

Centring a lattice makes reflections vanish. Ordering two kinds of atom onto a sublattice makes new ones appear, exactly n − 1 of them per parent cell, and their intensity is a difference rather than a sum — which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons.

lattices · Sublattices 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 Observations per unknown, against resolution. Unique reflections divided by refinable parameters, for a triclinic cell, no angle a right angle, with three coordinates and six displacement parameters for every atom and one non-hydrogen atom per 18 ų, in a molecular crystal. The scale is logarithmic because the fall is a cube: 16.3 at 0.8 Å and 0.30 at 3 Å. The line at one is where a determination stops being over-determined, and it is crossed at about 2.0 Å.

The unknowns against the observations

A structure determination is a fit of some number of parameters to some number of measurements, and both counts can be worked out before any data exist. The ratio turns out not to depend on how large the crystal's cell is, or on how symmetric it is — only on the resolution, and on that as a cube.

diffraction · Resolution -2 0 -3 is forbidden and is reached 64 ways. A layer of the reciprocal lattice of Cc: pale spots are reflections the group extinguishes, solid ones are allowed. The path shows a detour — the beam diffracts once at -3 -3 -3 and again at 1 3 0, and the two together send it in exactly the direction a single reflection at -2 0 -3 would. That reflection is forbidden, so intensity arrives where the symmetry said none could. There are 64 such routes to this one spot inside this window alone.

The absence that fills itself in

A systematic absence is the strongest evidence this subject has: a whole zone of reflections cancelling exactly, for reasons of symmetry rather than of arithmetic accident. The exactness belongs to a model — that the beam scatters once. A beam that has already been diffracted can be diffracted again, and the two events together land where the group said nothing could.

diffraction · Diffraction 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 p4m from 3 orbits of waves — detected p4m. A density built as a sum of 3 symmetry-adapted waves of p4m, each of them the average of a plane wave over the group, shaded from light to dark across one cell. The level set of this density — the darkest points of it — was handed to the same detector the pattern figures use, and it reports p4m, which is exactly the group the waves were built from. The waves are invariant by construction, so the density can never have less symmetry than the group; the interesting direction is the other one.

How many waves a group permits

A pattern can be written as a sum of waves instead of as an orbit of a motif, and then the group ties the coefficients together and forbids some outright. Building a density from the permitted ones and handing it back to the detector closes the same loop through a different door — and at low resolution the density has symmetry the crystal has not.

classification · Seventeen B = 3.45 against 3.4, K = 0.37 against 0.37. The mean intensity of each resolution shell of a cell of 1194 reflections, divided by Σf² computed from the cell's content alone, and logged. The points fall on a line whose slope gives B = 3.45 against the 3.4 put in, and whose intercept gives a scale of 0.37 against 0.37 — both recovered before a single atom has been placed. The atoms here are independent, so the line is straight at every resolution; the shells below the cutoff are marked in the second colour.

The average that knows the atoms and not where they are

Square a structure factor and average it over a shell of reflections at one resolution. The cross terms — every one of which carries a fact about the arrangement — cancel, and what is left is a sum over the *content* of the cell with no position in it anywhere. A scale and a temperature factor come out of that before a single atom has been placed.

diffraction · Statistics One crystal, two rows: one streaks and one does not. Two rows of reflections from the same faulted crystal, at a fault rate of 0.05, each drawn against the same row from a perfect one. The upper row has h − k not divisible by three, so each layer contributes a different cube root of unity and the sequence of layers enters the sum: the sharp peaks collapse into a streak. The lower row has h − k divisible by three, the phase factor is one, every layer scatters in step, and the peaks are exactly as sharp as in the perfect crystal. The sorting is an integer condition — a reflection either can see the stacking or cannot, decided by h − k modulo three — which is the most direct evidence there is that the disorder is in the stacking and not in the layers. The profiles are averaged over 16 independently faulted crystals, because one crystal gives speckle rather than a diffuse profile.

The streaks a faulted stack makes

Close packing settles two directions and leaves the third to chance. A crystal that chooses wrongly now and then has a lattice in the plane of its layers and none across them — and its diffraction pattern says so, with some rows of spots as sharp as ever and others smeared into streaks, sorted by an integer condition.

diffraction · Diffraction The diffuse intensity of an alloy with α₁ = -0.46. The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the average structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it.

The average scatters sharply and the rest does not

A crystal whose lattice is perfect and whose occupation is not scatters in two parts: the average structure gives Bragg reflections, and the variance is spread over everything between them. The split is exact, the total is one unit per site whatever the disorder does, and an ensemble of n arrangements mislays exactly a fraction 1/n of it.

diffraction · Disorder 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 The cross, from the selection rule alone. The layer lines of a helix with the first maximum of each marked on both sides. Nothing here is a picture of a photograph: each mark is at the radius where the Bessel function of the lowest order the selection rule permits on that layer line first peaks, and that radius is proportional to the order. The order rises by one per layer line until the middle of the repeat, so the maxima lie on two straight lines through the origin — the X — and the larger marks are the layer lines that reach the axis.

What a thread scatters

A helix with ten subunits in a turn is not a screw axis a crystal may have, and nothing about its diffraction pattern is lawless. The pattern lies on layer lines, and on each one only certain angular orders may contribute — a selection rule as hard as any extinction condition. The lowest permitted order rises by one per layer line, a Bessel function of order n does nothing until its argument is about n, and the maxima therefore lie on two straight lines through the origin.

classification · Subperiodic 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 Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch.

The ice rule is a conservation law

Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

aperiodic · Entropy The average is the site's orbit, with occupancies. A molecule at a site of symmetry mmm keeping a subgroup of order two takes four orientations, and the average over them is the site group's orbit of each of the molecule's atoms, every image at one over the length of its own orbit. Atoms in general positions give eight images at an eighth each, and give the same eight whichever subgroup the molecule keeps. Atoms on a locus the model keeps give a shorter orbit at a higher occupancy, drawn larger and darker, and those are the only atoms that differ between models. The total scattering is the same for every model, so all of them agree exactly at zero scattering angle.

The molecule size that hides a disorder

Two disorder models with the same occupancy leave averaged structures that differ only in a handful of partial atoms. The difference is 20% in structure factors for a ten-atom molecule and 3% for a sixty-atom one — so the data choose between the models for a small molecule and stop choosing for a large one, and seven pairs are identical at any size.

restriction · Local symmetry 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 One change of setting, four rules. The four things a structure report contains and the rule each obeys under a change of setting with basis change P and origin shift p. The cell and the indices are multiplied by P; a coordinate is multiplied by its inverse, after the origin has been subtracted; and an operation is conjugated and then shifted by (W − I)p, a term the other three have no equivalent of. Applied to Pnma with the change below, the operations still close into 8, the orbit maps point for point, and every |F| is unchanged.

One matrix, four rules

Changing the setting of a structure is one matrix and one origin shift — and the cell, the coordinates, the indices and the operations each obey a different rule under it. Three of the four ways of getting it wrong still leave a closed group of the right order, so closure catches none of them.

space-groups · Settings Two symmetric structures a diffraction pattern cannot separate. Two arrangements of 6 atoms on a 6 × 6 torus, each invariant under the plane group p6m, drawn beside the Patterson they share. No translation and no inversion carries one onto the other, so they are different structures; every one of the thirty-six interatomic vector counts is the same, so every diffracted intensity is the same and no measurement at any resolution separates them. Of the 4 structures with this symmetry and this many atoms, there are only 3 Pattersons — so imposing the most symmetric of the seventeen plane groups has not removed the ambiguity.

Symmetry does not rescue a Patterson

Every homometric pair found so far sits on a bare ring with no operations imposed, and a real crystal sits in a space group. Impose one and the ambiguity does not go away: 12 of the 13 groups searched still have pairs, and at six atoms the hexagonal groups are indistinguishable two to three times as often as the general position.

diffraction · Homometry 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.

Phase problemSystematic absenceFriedel lawLaue classThe Patterson functionDirect methodsEnumerationMeasurementReciprocal latticeAnomalous scatteringCentrosymmetricCentrosymmetry

All concepts