Crystallography & Symmetry
patterns derived from their groups
Concept

Bragg's law — where it appears

The condition that a reflection is observed when the wavelength is twice the spacing times the sine of the angle, which caps what is reachable. It is why a copper source cannot measure past about eight tenths of an ångström however long the experiment runs.

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 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

Named alongside it

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

Reciprocal latticeThe Fourier transformFriedel lawInterplanar spacingLaue classLimiting sphereMeasurementMultiplicityPhase problemResolutionStructure factorUnique reflections

All concepts