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 2 essays across one field — 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

Named alongside it

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

Bragg's lawBragg reflectionClusteringCorrelation functionDiffuse scatteringEnsembleOccupational disorderPhase problemReciprocal latticeShort-range orderStructure factorSuperstructure

All concepts