Crystallography & Symmetry
patterns derived from their groups
Concept

The Fourier transform — where it appears

The pairing between a periodic function and its coefficients on the reciprocal lattice. It turns a correlation function into a diffuse intensity and a density into a set of structure factors, with nothing lost in either direction.

Named by 7 essays across 2 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 How much one site knows about another, by separation. The Warren–Cowley parameters: the average of the product of the occupations of two sites a given vector apart, over every pair in every configuration. The value at the origin is exactly one — a site always agrees with itself — and it falls away with distance, alternating in sign where the alloy prefers unlike neighbours. These numbers are the whole of what the diffuse scattering measures: its intensity at a wavevector is their Fourier transform, computed here separately and agreeing to the last bits of the arithmetic. Nothing about them requires the crystal to be ordered, and their falling away is what short-range order means.

The order a diffuse pattern measures

Where a diffuse maximum sits says what the crystal is trying to become, and its shape is the Fourier transform of how much each site knows about its neighbours. The correlations are a small array of numbers, the intensity is their transform, and neither route to the other loses anything.

diffraction · Disorder 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.

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.

aperiodic · Modulation 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.

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.

aperiodic · Modulation 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 Almost no homometric pair on the torus is a reversed factor. For ten plane groups at several numbers of atoms on a 6 × 6 torus, the structures, the pairs of structures sharing one Patterson, how many of those pairs reversing one factor of a sumset explains, and how many are a patchwork of Galois classes. Factor reversal explains 2 of 3522; the patchwork explains all 3522.

A homometric twin is the structure moved class by class

On a ring, every homometric pair a small search found came from one construction: factor the structure as a sumset and reverse one factor. Imposing a plane group on a torus made homometry more common, and the obvious guess was that symmetry makes structures easier to factor. Tested pair by pair, factor reversal explains two of 3,522 homometric pairs on a 6 × 6 torus, and none at all at six atoms. What explains all 3,522 is a different mechanism: the partner is the structure's own Fourier transform moved independently on each Galois class of characters, by a shift, a sign or an inversion.

diffraction · Homometry

Named alongside it

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

IncommensurateModulationSatellite reflectionStructure factorBessel functionGolden ratioMeasurementOccupancyPhase problemSuperspaceBragg's lawBragg reflection

All concepts