Bragg diffraction and crystal structure determination

A crystal's planes of atoms are spaced at roughly the same scale as an X-ray wavelength, so waves reflected from successive planes interfere, and they reinforce only at angles where the extra path travelled by the deeper reflection is a whole number of wavelengths — Bragg's law. That turns a diffraction pattern into a measurement: each sharp spot or ring fixes one plane spacing, the set of spacings identifies the lattice, and their relative intensities carry where the atoms sit within the unit cell. It is how the structures of salt, metals, minerals, DNA and essentially every protein of known shape were determined.

Practise this concept

Think of a crystal as a stack of parallel, equally spaced planes of atoms, and send in a beam of X-rays. A little of the beam scatters from the first plane, a little more from the second, and so on down through thousands of planes. The wave from the second plane has travelled further than the wave from the first, by an amount that depends on the angle, and unless that extra path is a whole number of wavelengths the contributions from successive planes fall out of step and cancel. Only at the special angles where they stay in step does anything reach the detector, and the resulting pattern is a set of sharp peaks against almost nothing.

Writing the extra path out gives Bragg's law: it equals twice the plane spacing times the sine of the angle between the beam and the planes, and it must equal a whole number of wavelengths. Two conventions catch people out. The angle is measured from the planes, not from the normal as in ordinary reflection, so a detector sitting at twice theta corresponds to a Bragg angle of theta. And because a sine cannot exceed one, no diffraction happens at all unless the wavelength is shorter than twice the spacing — with atomic planes a few tenths of a nanometre apart, that rules out visible light entirely and is why the technique waited for X-rays.

Turning peaks into a structure has two halves. The peak positions give the set of plane spacings, and those spacings, with their directions, fix the size and shape of the unit cell. Where the atoms sit inside the cell is a separate question, answered by the relative intensities: a heavy atom scatters more strongly, and atoms arranged so that their scattered waves cancel can extinguish a reflection that the lattice alone would allow. The awkward part is that a detector measures intensity and so loses the phase of each reflection — the phase problem, which made early protein structures years of work and which modern methods sidestep rather than solve. The same physics with neutrons or electrons in place of X-rays is the everyday toolkit of materials science, and it is what produced the photograph from which the double helix was read.

Key ideas

Requires: Crystal structure and lattices, Interference and diffraction, X-rays and their production

Formulas

n\lambda = 2d\sin\theta

Symbol	Name	Unit
$lambda$	wavelength	m (metre)
$d$	distance	m (metre)
$theta$	angle	deg (degree)
$n$	order	1 (dimensionless)

Common misconceptions

Measuring the Bragg angle from the normal to the planes, the way angles are measured in ordinary reflection and refraction. Bragg's angle is taken from the planes themselves, not from the normal, which is why a peak recorded at a detector angle of two theta corresponds to a Bragg angle of theta. Using the other convention makes every spacing come out wrong by a factor that looks like a calibration error.
Assuming visible light would do the same job if the optics were good enough. The condition contains a sine, so a wavelength longer than twice the plane spacing gives no solution at any angle — with atomic planes a few tenths of a nanometre apart, light at five hundred nanometres cannot diffract from them at all. This is a hard limit on the wavelength, not a limit on lens quality, and it is why X-rays, neutrons or electrons are used instead.
Treating a diffraction pattern as a magnified image of the crystal, so that reading off the structure is a matter of looking closely enough. The pattern lives in reciprocal space: a spot's position gives a spacing and a direction, while where the atoms sit inside the unit cell is encoded in the intensities. Recovering the structure means fitting or transforming, and because a detector records intensity the phase of each reflection is lost — the phase problem, which is a real obstacle rather than a technicality.

Resources

Articles

Interactive

HyperPhysics — Bragg's law, with the geometry drawn outen

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

X-ray structural analysis ILOFY.02.020