Crystallography & Symmetry
patterns derived from their groups
Concept

Measurement — where it appears

A number obtained from a construction with a stated error or tolerance, as opposed to one decided exactly in integers. Every essay here that quotes one says which kind of number it is, because a measurement and a theorem carry different warranties.

Named by 22 essays across 6 fields — each of them below, with the objects they name alongside it.

Every property count, in every class. For each of these 32 classes, how many independent components a property may have: piezoelectric moduli from 18 down to 0, 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.

Permitted is not present

A symmetry argument says which components of a property may be non-zero. It is routinely read as saying they are — and the gap between the two is where every practical use of this table either works or quietly fails.

point-groups · Neumann's principle 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.

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.

point-groups · Properties Every group decided by a window of radius 1. For each of the seventeen, the radius at which a round window on the pattern admits exactly the group's own operations and no others — with the numbers it admits at each smaller radius beside it. Two opposite failures are visible. Most groups under-report at small radii, because an operation carrying points out of the window cannot be tested at all; cm over-reports, admitting operations the pattern does not have. The groups that take longest to settle are the ones distinguished by a glide, which moves a point half a cell before anything can be compared.

How much pattern is enough

Every claim here about a pattern's group is a claim about an infinite pattern. A reader sees a patch. Measuring what a finite window can decide gives a number — about one cell's radius — and two opposite ways of being wrong on the way there.

classification · Seventeen One patch of radius 2. A Penrose patch of 476 vertices, with the vertices within 2 edge lengths of one of them marked and the circle drawn. That marked set, written in coordinates relative to its centre, is what the census compares: two vertices have the same patch when their marked sets agree. Every vertex of the tiling is the centre of one such patch, and the question is how many different ones there are.

How many patches of each size

A periodic tiling has one kind of neighbourhood however far out you look. Random points have as many kinds as neighbourhoods. A Penrose tiling has a number in between that never stops growing and never catches up — and the count is a measurement rather than a theorem.

aperiodic · Complexity 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 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.

diffraction · Resolution 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.

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.

diffraction · Resolution (17, 5) and (23, 7) reduced in 3 steps. Lagrange's reduction, run on the basis (17, 5), (23, 7). Each step subtracts a whole multiple of the shorter vector from the longer and swaps them; after 3 steps neither can be shortened by the other and the pair is reduced. The faint arrows are the intermediate bases and the solid pair is the answer, of length 1.41. The procedure always terminates and always finds the shortest vector, and in the plane that is a theorem rather than a hope.

The shortest vector, and where it stops being easy

Two moves find the shortest vector of a plane lattice, and they always terminate. A crystal never needs more, because a crystal's lattice has two or three dimensions. In general the same question is NP-hard, the best polynomial procedure returns an answer that may be exponentially too long, and an entire branch of cryptography is built on the gap.

lattices · Lattice 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 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.

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.

aperiodic · Entropy Equilibrium and growth are different shapes. Two predictions for the habit of the same cubic crystal, computed through the same intersection of half-spaces. The equilibrium shape puts each face at a distance proportional to its surface energy, which is Wulff's construction; the growth shape puts it at a distance proportional to its growth rate, taken here from this site's own spacing rule. They differ — the equilibrium shape carries {111}, {110}, {100} and the growth shape {100} — and the difference is between two rules rather than between two pieces of code. A crystal on a bench has grown; a crystal annealed long enough has relaxed; the two look different and neither picture is wrong.

The fast faces are the ones that vanish

A crystal has two predicted shapes and they are not the same. One minimises surface energy and is what a crystal settles into; the other is what growth leaves behind, and in it a face that grows quickly grows itself out of existence.

applied · Growth 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.

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.

applied · Indexing 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 Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other.

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

applied · Packing 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 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 [001]: 8 of 24 faces. The 24 faces of the form {210} of a cubic crystal, stereographically projected, with one zone marked. The great circle is the set of directions perpendicular to the axis [001]; the 8 faces on it are the zone, and each is on it because the integer sum hu + kv + lw is exactly zero. No length and no angle enters that test. Faces of the lower hemisphere are drawn faintly.

A zone is a vanishing dot product

Look at a crystal and the obvious thing about it is that the faces run in bands — sets whose edges are all parallel. A face belongs to such a band exactly when three integers multiplied by three others sum to zero. No length enters, no angle enters, and that is why every index on a nineteenth-century mineral specimen is still the index used now.

applied · Forms 342 unlabelled spots, cell volume 52. A bag of 342 reflection positions with no indices on them, collected out to a bound of 3 on each index. Their pairwise differences generate the reciprocal lattice; a basis of that is taken by integer elimination and then reduced, and the reduced basis is printed. Its determinant is 52, which is the volume of the cell the reflections were computed from — so the cell has been recovered from positions alone, with no intensity used anywhere.

A cell from a bag of spots

A single-crystal experiment returns a list of directions with no labels on them. Recovering the cell is recovering the lattice those directions generate, and the whole of it is take differences, reduce, read the answer. What no quantity of data settles is whether the lattice found is the true one or a sublattice of it.

applied · Indexing 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 h = 1 anomaly moves with the length of the chain. For a size-coupled chain with ε = 0.03, the ratio of the second-order pair's tilt, divided by k², to the first-order tilt at h = 1, from plain sums over the sites and from sums with a smooth window, for 22 chain lengths from 10,000 to 480,000 sites. The plain ratio swings between -0.08 and 5.60; at 30,000 and 120,000 sites, the two lengths first tried, it sits near 0.31 both times. The windowed ratio is one half at every length.

The satellite beside the zero-order reflection

An earlier essay found that the second-order satellites of a modulated chain tilt half as far as the first-order ones at every reflection but the first, where they tilted a third as far, the same on chains of 30,000 and 120,000 sites. It left the departure unexplained and proposed a higher order of the expansion. The expansion was never wrong. The weak satellite at h = 1 sits a quarter of a reciprocal unit from the strongest reflection a chain has, and a plain sum over a finite chain carries that reflection's tail, which does not shrink steadily as the chain grows. It swings, and the two lengths tried happened to swing to the same place.

aperiodic · Modulation The second-order satellites alone cannot measure the curvature of a size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from six sets of satellites of a modulated chain of 30,000 sites with 2% intensity error, and the correlation between the errors in ε₂ and the linear coefficient ε₁. From second-order satellites alone the error is thousands of times ε₂ at one reflection and still a hundred times it at three, with the two coefficients correlated at 1.000. With the first-order satellites added, one reflection gives ε₂ to about a third of itself and three to about a tenth.

The second order cannot see a curved size effect alone

When the spacing between atoms depends on composition along a curve rather than a line, the displacement of a modulated chain gains a second harmonic, and the second-order satellites should measure it. They cannot do it on their own. At every reflection a second-order satellite sees the straight part and the curved part of the size effect in almost the same proportion, so adding reflections adds copies of one equation. The first-order satellites, which see only the straight part, are what separates them.

aperiodic · Modulation What each kind of satellite of a two-wave chain says about the curvature of the size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from seven sets of satellites of a chain modulated by two incommensurate waves, with 2% intensity error: with the two modulation amplitudes refined as well (upper bar of each pair) and with them known (lower bar). The sum or the difference satellites alone say almost nothing either way. Together, with the amplitudes known, they fix the curvature to under one per cent from a single reflection; with the amplitudes free they do not, and adding the first-order satellites, which fix the amplitudes, restores it.

Two waves measure the curvature one wave hides

A chain modulated by one wave cannot tell a curved size effect from a straight one at second order: every second-order satellite sees the two in the same proportion. Modulate it by two incommensurate waves and the curvature puts the product of the waves into the displacement, at their sum and at their difference. The satellites there see the curvature with opposite signs. Two equations of opposite sign separate two unknowns, and once the two waves' amplitudes are known, the mixed satellites of a single reflection fix the curvature to under one per cent.

aperiodic · Modulation At the sum of all three wavevectors the cubic and the quadratic paths nearly cancel. The intensity of the satellite at h + q₁ + q₂ + q₃ for the first four reflections, computed with only one size coefficient switched on at a time and with all three. The cubic coefficient alone gives about 3 × 10⁻⁷ at h = 1, the quadratic alone about 2 × 10⁻⁷, the linear alone less than 10⁻¹⁰. With all three the satellite is ten times weaker than with the cubic alone, because the cubic term's displacement and the quadratic term's pair harmonic times a first-order wave arrive with opposite signs.

Three waves measure the cube through the slope

A size effect that bends twice puts the product of three modulation waves into the displacement, at the sums and differences of all three wavevectors, and nowhere else does a cubic term reach so directly. The satellites there were expected to measure it. They see it, but the strongest of them is ten times weaker than the cube alone would make it, because the quadratic term arrives at the same place with the opposite sign, and against any realistic background they say nothing. The cube is measured instead by the strongest satellites of all, the first-order ones, because it makes each wave's slope depend on how strong the other two waves are.

aperiodic · Modulation

Named alongside it

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

Structure factorGolden ratioIncommensurateModulationResolutionSatellite reflectionInterplanar spacingLimiting spherePhase problemSystematic absenceAnisotropyAutocorrelation

All concepts