Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: From the diffraction back — page 2

Nobody has seen a space group. They are inferred from where a crystal scatters and, just as informatively, from where it does not.
Sliding the window catches different points. The periodic lattice that cut-and-project starts from, with the strip drawn at two positions 0.21 apart, which is 15 per cent of the window's width. Most lattice points are caught by both; a few are caught by one and not the other, and those are the whole difference between two quasicrystals. The slope has not changed, so the density, the two tile lengths and the ratio of their frequencies are identical — the offset is a parameter with no energy attached to it, which is what makes a phason a degree of freedom rather than a defect. Order without repetition

The freedom a crystal has not

Slide the window of a cut-and-project construction and the tiling changes — different tiles in different places — while its density, its tile ratio and its diffraction pattern do not. That parameter is a phason, it costs nothing, and no local measurement whatever can determine where it sits.

 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. Lattices

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.

 Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as. What symmetry decides

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

 The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell. Lattices

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

 N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does. How it is known

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

 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. How it is known

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.

 Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off. Symmetry at work

Which faces a crystal shows

Rock salt grows as cubes, fluorite as octahedra, garnet as dodecahedra. All three have cubic lattices and the same list of possible faces, and what separates them is which reflections are systematically absent — a rule about diffraction predicting a shape a mineralogist can hold.

 The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry. Lattices

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

 Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092. How it is known

How many reflections there are

The reciprocal lattice is infinite and a measurement is not. A wavelength cuts a sphere out of it, the number of points inside is the sphere's volume times the cell's, and symmetry then removes all but a fraction — though never quite the fraction a division would give.

 Three atoms, four resolutions. A one-dimensional Fourier synthesis of the same three atoms, cut at four different resolutions. Nothing is approximate except the edge: every amplitude and every phase used is exact, and the only information withheld is the reflections outside the sphere. The peaks broaden as the cut-off comes in, and beside every peak sits a negative ripple that the coarsest map cannot distinguish from a real absence of density. Both effects are the transform of the sphere rather than anything about the structure. How it is known

As sharp as the sphere is wide

A map made from a truncated sum is not a blurred picture of the structure. It is the structure convolved with the transform of the sphere — so peaks acquire a width proportional to the resolution, and a negative ripple of twenty-two per cent that no improvement in the data ever reduces.

 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. How it is known

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.

 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. How it is known

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.

 pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice. How it is known

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

 Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not. Order without repetition

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

 7 cells explain the lines; one of them is right. A line list from a face-centred cubic cell of 5.64 Å, with a realistic error added, handed to a sweep over every cubic cell between 2 and 12 Å in all three centrings. 7 distinct cells explain every line within the tolerance, and each is a genuine solution rather than a numerical accident. The true cell comes top by de Wolff's figure of merit — the last Q over twice the mean discrepancy times the number of lines the candidate says should have been visible — which punishes a candidate for predicting lines nobody saw. That is the whole of what makes indexing decidable in practice: not the arithmetic, which has many answers, but a criterion for preferring one. Symmetry at work

Indexing a powder pattern

A powder pattern is a list of numbers and a cell is six. Getting the second from the first is the first step of every powder study and the one that fails — because the arithmetic has many answers, and choosing between them is a ranking rather than a deduction.

 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. How it is known

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.

 -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. How it is known

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.

 two chains, periods 1 and 1.62. Two interpenetrating chains of atoms with periods 1 and 1.62, whose ratio is irrational, so no length is a whole number of both. Each chain is displaced from its own lattice by a wave with the other's period — the short ticks show each atom's displacement from where an unmodulated chain would put it — which is what makes this one crystal rather than two side by side. Nothing here is a unit cell: any length chosen contains a whole number of one chain's atoms and a fractional number of the other's. Order without repetition

Two lattices, one crystal, and no cell at all

A modulated crystal has a lattice and a wave running through it. A composite has neither host nor guest: two interpenetrating substructures with periods that share no common multiple, each modulating the other. Every reflection needs an index from both, and the unit cell a diffractometer reports belongs to whichever half scattered harder.

 p4: (1/2, 0) has a star of 2. The first Brillouin zone of the square lattice, with the reciprocal lattice points at its corners and centre, and the whole star of the wavevector (1/2, 0) under p4. The star has 2 members and the little group — the operations that leave the wavevector where it is, modulo the reciprocal lattice — has order 2. The two multiply to the order of the point group, which is the orbit–stabiliser theorem and is checked rather than displayed. Each member is drawn at whichever of its equivalent copies lies nearest the origin, because that is where a reader expects a wavevector to be. Into space

The star of a wavevector

A plane group acts on the plane, and this collection has spent two hundred essays watching it. It also acts on the reciprocal lattice, where the action is different in one decisive way: the translations move no wavevector at all, and come back instead as a phase.

 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. How it is known

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.

 A gap of exactly 1.00. The two folded bands with the ordering switched on. The faint curves are the same bands before it, crossing at the boundary of the reduced zone; the ordering couples them there and separates them by exactly twice its own strength. The gap appears at the wavevector where the superlattice's extra reflections appear, and for the same reason: both are the Fourier component of the potential at that wavevector. Lattices

A bigger cell, a smaller zone

Ordering two kinds of atom onto a sublattice adds reflections to the diffraction pattern and opens a gap in the levels. It is one fact told twice: the same Fourier component of the potential, at the same wavevector, doing the same thing.

 ⟨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. How it is known

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.

 the most consistent answer has 100 per cent of the signs. 60 runs of the sign procedure from 60 different random starts, each plotted at its self-consistency — a figure computed without any knowledge of the answer — against the fraction of its signs that are in fact right. Throwing out the 1 run that reached the uniform solution — every sign the same, perfectly consistent and physically a single peak — the highest consistency belongs to a run with 100 per cent of the signs right. The ranking works here, and the reason it works is that the cell is small. Nothing in the plot's horizontal axis knows the answer, which is the only reason a procedure of this kind is a procedure at all. How it is known

The formula that has the answer already

The tangent formula rebuilds each phase from all the others, and the true phase set is very nearly a fixed point of it — hand it the answer and it hands the answer back. Start it anywhere else and it does not arrive. Having a fixed point and finding it are different problems, and the second is where the subject spent twenty years.

 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. The classification

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.

All themes · All essays