Crystallography & Symmetry
patterns derived from their groups
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The theme: From the diffraction back — page 4

Nobody has seen a space group. They are inferred from where a crystal scatters and, just as informatively, from where it does not.
What refinement takes back from a wrong disorder model. Every directed pair of disorder models sharing an occupancy, 462 in all: data made from one model's averaged structure, the other model refined against them with positions on its own loci, two-component displacement parameters and fixed occupancies. Across is the residual between the two ideal structures; up is the residual after refinement. Points on the dashed line lost nothing to refinement; the median point lost 58 per cent of its residual. Every point on the floor, where the wrong model fits exactly, is one whose loci contain the true model's. What a lattice forbids

What refinement takes back from a wrong model

Two disorder models sharing an occupancy leave averaged structures typically a sixth apart. Refine the wrong one against the right one's data and it takes back more than half of that — all of it wherever its loci contain the true ones. Yet the data still choose, even for a molecule of sixty-four atoms, because a likelihood summed over every reflection sees what no single reflection can, and the wrong model's partial atoms give it away through the size of their displacement parameters, not their shape.

 Better counting makes an uncorrected smooth error choose the wrong model. Eighteen pairs of disorder models, each decided by the likelihood ratio at the 0.1 per cent level, at sixty-four and a hundred and twenty-eight atoms, as the random error falls from 3% to 1% to 0.3% of the mean amplitude. In each group, the first bar has random error only, the second adds a smooth scale error of 3% across reciprocal space, and the third adds the same error with a degree-two scale surface refined alongside the structure. With random error only, better counting decides more pairs correctly. With the uncorrected smooth error it stops helping and the number decided for the wrong model grows, to three at a hundred and twenty-eight atoms. With the surface refined, the result is the random-only one in every group. What a lattice forbids

Better counting, and the wrong model

A smooth error across reciprocal space, as large as the random one, hardly changes which disorder model the data choose: it enters both models' residuals and cancels from their difference. What it changes is where better data lead. With random error alone, counting harder decides more pairs correctly. With the smooth error left in, counting harder stops helping and starts deciding pairs for the model that did not make the data — at the 0.1% level, and more of them the better the counting.

 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. How it is known

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.

 Which sites a smooth error can mislead. For the eighteen pairs of disorder models at three site symmetries, under a smooth scale error of degree two (five molecule sizes) and of degree four (three sizes): how many verdicts of perfect counting go the wrong way, how many pairs no data can decide, the largest correlation between the error and the two models' difference, and how many harmonics of the error's degree the site's group leaves unchanged. At the cubic site the degree-two error correlates with nothing, to rounding, and decides no pair wrongly; the degree-four error, which the cubic site keeps one harmonic of, reaches it. What a lattice forbids

A cubic site is blind to a degree-two error

A smooth scale error left uncorrected made perfect counting choose the wrong disorder model for eight of ninety pairs and sizes, and the obvious way to predict which is to correlate the error with the two models' difference before refining anything. That correlation predicts none of them. What it does predict, exactly, is which pairs can never go wrong: at a cubic site a scale error of degree two correlates with nothing and is absorbed by nothing, because a cubic group keeps no harmonic of degree two. A degree-four error, which it keeps one of, reaches the cubic site at once.

 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. How it is known

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.

 The first-order prediction against the actual verdict, for every pair and size. The 85 decidable pairs of disorder models at five molecule sizes, each placed by the first-order prediction of the wrong model's misfit minus the right model's, made from the two error-free refinements and their Jacobians, and by the actual difference after refining against data with a 3% smooth error. Shaded quadrants are agreement. 77 agree. Of the 8 wrong verdicts, ringed, 4 are predicted; 4 right verdicts are predicted wrong. What a lattice forbids

The wrong model moves too far for first order

Which disorder models a smooth error will turn the wrong way was left as a calculation to make before any data exist: project the error onto the directions each refinement's parameters can move. Made, it gets the sign of the verdict right in seventy-seven of eighty-five cases, catches four of the eight wrong verdicts and raises four false alarms. Every case it gets wrong is either a near-tie or a refinement that absorbed more than twice, or less than half, of what its first-order directions allow.

 Three ways to fill a lattice with a wave. Three occupation waves at the same irrational wavevector, each drawn as its profile over one period, the occupancy of a site as a function of where the wave stands there, and as the first thirty sites of the chain it produces, each filled in proportion to its occupancy. A cosine fills every site partly. A crenel of width 0.3 fills three sites in ten completely and leaves the rest empty. A crenel of width 1/φ fills sites exactly where the Fibonacci chain has its long letter, so the third row is that chain written as two species on a lattice. Order without repetition

An occupation wave is its own Fourier series

A modulated crystal whose atoms stay put while a wave decides how much of an atom sits on each site diffracts to satellites, and the satellite of each order is simply the matching Fourier coefficient of the wave's profile. A cosine stops at first order and can never exceed a quarter of a site; a step makes satellites of every order, skips exactly the orders its width divides, and makes the strongest first-order satellite any wave can. And a step as wide as its own wavevector is the Fibonacci chain.

 Two waves that are each symmetric enough make a lopsided pair. The first-order satellite amplitudes at h − q and h + q around the main reflections h = 1 to 4, for three chains: an occupation wave on the lattice, whose pairs are equal; the strain wave alone, the displacement the size effect produces with every site equally full, whose satellites are weak and grow with h; and both together. Together the pairs are unequal, the low side stronger, by an amount that grows from one reflection to the next — an imbalance neither wave has on its own, from a displacement of less than a hundredth of a spacing. Order without repetition

The strain wave that tips the satellites

An occupation wave puts equal satellites on both sides of every main reflection, and a strain wave a hundredth of a lattice spacing high is enough to tip them. Real composition waves drag such a strain with them, because atoms of different sizes want different spacings, and the tilt that results is not a nuisance. To first order it does not depend on how strong the occupation wave is at all. It reads the size mismatch directly, and its sign says which species is the larger.

 Second-order satellites that only the strain makes. Satellite intensities on a logarithmic scale around the first four main reflections of a chain whose occupation is a pure cosine wave, with no strain and with the strain a size factor of 0.1 brings. Without strain the second-order satellites are absent, down at the chain's numerical floor; with it they appear at between a ten-thousandth and a hundredth of the first order's intensity, larger on the outer side of each reflection and growing with its order. Order without repetition

The second order reads the strain again

An occupation wave with a pure cosine profile has no second harmonic, so on its own it makes no second-order satellites. Strain the chain by the size difference of its atoms and they appear. Their strength measures the square of the displacement and cannot see its phase, where the tilt of the first-order pair measures the displacement times the cosine of its phase. So the two give the size factor twice, and the two readings agree only if the strain sits where a size effect puts it.

 Four inversion centres, and a band's charge on each. A square crystal with four atoms in each cell, at the points a quarter of the way along both diagonals, and the four points of the cell that an inversion maps onto themselves: the corner, the midpoints of the two edges, and the centre. None of the atoms sits on one. For each of four versions of the crystal, differing only in which bonds are stronger, the ring marks where the lowest band's charge is centred — computed from the band's Berry phase and predicted from four parities. Each version puts it on a different inversion centre, and in every case on a point with no atom. Into space

Four signs say where the charge sits

In a plane crystal whose only symmetry beyond its lattice is a centre, a band's charge must sit on one of the cell's four inversion centres — and which one is decided by four signs: whether the band is even or odd under inversion at the four wavevectors inversion leaves alone. The centre can be a point no atom occupies, and when the four signs multiply to minus one there is no centre at all, only a charge that winds round the cell.

 How many superstructures a diffraction pattern tells apart. For two-species superstructures of the square lattice with one to twelve sites in the primitive cell: the number of structures, and the number of classes that the positions of their superlattice spots separate them into, the positions together with the composition, and every intensity. At twelve sites 2,012 structures give 167 sets of spot positions, 504 with the composition and 1,815 sets of intensities. At five, seven and eleven sites the positions distinguish only the cells; the intensities distinguish everything. The bar shows the share resolved by positions, dark, and by intensities, light. Operations

The spots say the cell and little else

Of the 2,012 two-species superstructures of the square lattice with twelve sites in the cell, more than half light every superlattice spot their cell allows, so the positions of the spots say which cell and nothing more. Positions separate the 2,012 into 167 classes, the composition takes that to 504, and every intensity to 1,815. At a prime number of sites no spot is ever missing, because a sum of distinct p-th roots of unity cannot vanish.

 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. Order without repetition

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.

 I222 and I2₁2₁2₁: the same axes, apart. The two-fold rotation axes of the two space groups on the body-centred class of 222, drawn in one conventional cell seen down its third edge: axes along that edge as diamonds, axes along the first edge as solid lines and along the second as dashed lines, with the height of each written beside it. Both groups have pure two-fold rotation axes in all three directions. In the symmorphic group some axes of different directions cross, at the origin among other places; in the other none does, because the heights of the axes in each direction are a quarter out of step with the positions of the axes in the others. The classification

Two groups no single operation reveals

A space group is non-symmorphic when no choice of origin removes all of its translations at once, and every textbook example has a screw or a glide to show for it. Two of the two hundred and thirty do not need one. Every operation of I2₁2₁2₁, taken on its own, is also a plain rotation about some axis; so is every operation of I2₁3. What makes them non-symmorphic is that their rotation axes never meet — and no diffraction extinction can tell them from I222 and I23.

 When a three-species superstructure's spots can go dark. Every structure of three species on the square lattice with three to eight sites in its primitive cell, using all three species, and every superlattice spot of each, sorted by whether the spot is dark for every choice of scattering contrasts, dark at exactly one ratio of the two contrasts, or never dark; with the number of distinct ratios and their degree over the rationals. At the primes 3, 5 and 7 no spot is dark for all contrasts and no dark ratio is rational: none at three sites, quadratic ones at five and cubic at seven. At four and six sites every dark ratio is rational, at eight some are and some are not, and at six and eight some spots are dark for every contrast. Operations

Three species hide a spot at the golden ratio

With two species and a prime number of sites in the cell, no superlattice spot can ever be missing, because a sum of distinct roots of unity of prime order vanishes only when it is all of them. With three species the structure factor weighs two sums against each other with two scattering contrasts, and the argument changes shape without breaking. At a prime size no spot is dark for every contrast and none at a rational ratio of them, but a spot can be dark at one irrational ratio: at five sites, a power of the golden ratio. It happens only in structures with a centre of symmetry.

 Where an antiferromagnet's reflections land. Two cubic antiferromagnets, with their nuclear reflections as dots and their magnetic reflections as triangles, in units of the conventional reciprocal lattice. On the left, a simple cubic lattice with every neighbour opposite: the magnetic reflections sit at half-integer indices, in a layer the nuclear pattern never visits. On the right, a body-centred lattice with the corners up and the centres down: the magnetic reflections sit at integer indices with an odd sum — exactly the reflections the body centring had extinguished, lit up again by the moments. In both, no magnetic reflection falls on a nuclear one. What symmetry decides

Magnetic reflections land where nuclear ones cannot

An antiferromagnet scatters neutrons into places its chemical structure leaves dark, and never into a place it lights. The reason is one line of algebra: the magnetic pattern is the nuclear pattern shifted by half a reciprocal-lattice vector. Each of a lattice's seven halvings is one such shift, and how many of them its symmetry keeps decides how many kinds of domain the magnet must form.

 Seven cubic antiferromagnets, and what a powder of each can say. Seven collinear antiferromagnets on the three cubic lattices, each with the size of its propagation vector's orbit, the number of the forty-eight cubic operations that keep the vector, the kind of every magnetic ring out to a fixed radius, summed from the actual magnetic reflections of one domain, and what a powder pattern can therefore read about the moments' direction. The two whose halving the cubic group keeps are blind in every ring. An orbit of three leaves every ring uniaxial about a cube axis, an orbit of four about a body diagonal, and the body-centred orbit of six leaves rings that read the direction. The cubic chemistry is the same in every row; the propagation vector decides. What symmetry decides

Where a powder goes blind

A neutron sees only the part of a magnetic moment across its path, so each reflection knows which way the moments point. A powder adds up every reflection of a given length, and for a ring with cubic symmetry that sum comes out the same whichever way they point. But the ring's symmetry is not the chemistry's: it is the symmetry the propagation vector keeps, and most cubic antiferromagnets have lost enough of it to be read.

 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. Order without repetition

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.

 Heavier atoms spoil the line, and then mend it. Above, the scatter of Sayre's ratio about its fitted line, averaged over four random cells, as a quarter of the atoms is made heavier, for cells of eight and of twelve atoms. Below, one minus the square of ρ, the correlation between the density's transform and its square's, computed from the composition alone. The spread of atomic numbers grows without limit along this axis, but ρ does not keep falling: it is lowest near a weight of three and climbs back towards one as the heavy atoms come to dominate, because a cell ruled by a few equal heavy atoms is nearly a cell of equal atoms. The scatter rises and falls with it. At equal weight what scatter remains comes from atoms that overlap, more of them in the larger cell. How it is known

Unequal atoms break the equality, and very unequal ones mend it

Sayre's identity is exact for a cell of equal atoms. Give a quarter of them more weight and the line it predicts scatters, reflection by reflection, by exactly the difference between weighting atoms by Z and by Z squared. Make them heavier still and the scatter shrinks again, because a cell ruled by a few heavy atoms is nearly a cell of equal ones. The scatter measures the cell, but only if the phases are already known.

 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. Order without repetition

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.

 Which cells the flipping solver finds. For four cells of twenty atoms — all carbon, a quarter sulphur, one sulphur, one lead — the share of random starting phase sets that charge flipping takes to the right structure within 250 cycles, against the threshold below which density has its sign reversed. Each point pools several random arrangements of atoms with several starts each, and the atoms fall off with angle as real atoms do. The cells with sulphur are solved more often than carbon alone, the quarter-sulphur cell most of all, each at a threshold near one standard deviation; the lead cell is never solved at any threshold. How it is known

The heavy atom the flipping solver cannot see past

Sayre's identity degrades as a cell's atoms become unequal, most of all for a quarter of heavy atoms three times the weight of the rest, and the obvious prediction is that a solver built on the same sparseness should struggle there too. It does not. Charge flipping solves a cell of carbon with sulphur in it more often than carbon alone, and never solves a cell with one lead atom — because a threshold stated in standard deviations belongs to whichever atom owns the map's variance.

 Two hundred and thirty groups on a hundred and twenty-two diffraction symbols. Every space group placed in a column for its diffraction symbol — Bravais lattice, Laue group and systematic absences — with the columns grouped by lattice. There are 122 symbols. 50 of them name one group; 44 are shared by two, 23 by three, 2 by four and 3 by five. The classification

Every ambiguity in a diffraction pattern, sorted by its cause

A diffraction pattern shows a lattice, the symmetry of its intensities and the reflections that are missing, and the 230 space groups show 122 different combinations of the three. Fifty of those name a single group. For the rest, each pair of groups that share one can be sorted by why they do, and there are five reasons. Four of them are ambiguities some further measurement resolves. The fifth, two groups with the same operations in the same places except for where their axes cross, happens exactly twice, and nothing but a solved structure decides it.

 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. Order without repetition

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.

 Diffuse scattering gathered under every Bragg peak. The one-phonon thermal diffuse scattering of a square lattice with springs to nearest and next-nearest neighbours, over the reciprocal plane from minus 2.5 to 2.5 in each index, with the Bragg reflections marked. The intensity rises as one over the squared distance to every reflection, because the acoustic phonons' frequencies vanish there, and it grows with the square of the scattering vector, so the lobes are strongest far from the origin. The lobes are not round: a small step from (1,0) along k gives 2.7 times the intensity of the same step along h, because it probes transverse phonons, which are softer, and that ratio is what an experiment reads the elastic constants from. How it is known

A warm plane crystal has no Bragg peaks

Thermal motion is disorder that moves, and in space it costs a crystal a little of each Bragg peak and gathers it into diffuse scattering underneath. In a plane it costs more than a little. Computed exactly from the phonons of a harmonic square lattice, an atom's mean-square wandering grows by the same amount every time the crystal doubles in size, so no Debye–Waller factor survives, and every Bragg peak fades as a power of the crystal's size — a power proportional to the temperature times the square of the reflection's distance from the origin.

 The worst edge pair is the two most anomalous wavelengths. For each pair of the four wavelengths, and for the peak, inflection and high remote together, the median error of the phase difference between the whole structure and its anomalous atoms, recovered from both Friedel mates at each wavelength with a 1% error on every intensity. Beside each, the spread of f′ across the chosen wavelengths. The pairs that include a remote wavelength and one on the edge phase to seven or eight degrees; the peak and the inflection together, the two with the largest anomalous effect, to eighteen; the two remotes to fifty-five. How it is known

The two most anomalous wavelengths are the wrong pair

A MAD experiment measures both Friedel mates at two wavelengths near an absorption edge and solves for a phase. The obvious choice is the two wavelengths where the anomalous effects are largest, the peak of f″ and the dip of f′. Of all the pairs that use the edge, it is the worst. The sine of the phase is carried by f″, but the cosine is carried by how much f′ changes between the two wavelengths, and two points on the same edge barely change it.

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