Bragg's law
/ brag /
How can we possibly know that atoms sit on a regular grid we can never see with our eyes? Shine X-rays — whose wavelength is about the size of an atomic spacing — at a crystal. The regular sheets of atoms reflect them, and at just the right angles the reflections add up into a bright beam. Bragg's law is the simple rule that predicts those special angles.
The law is n times lambda = 2 times d times sin(theta): n is a whole number (the order), lambda the X-ray wavelength, d the spacing between adjacent atomic planes, and theta the angle between the incoming beam and the planes. Waves bouncing off successive planes travel an extra distance of 2 d sin(theta); when that extra path equals a whole number of wavelengths, the waves interfere constructively and produce a bright diffraction peak. Measure the peak angles, and you can solve for d and reconstruct the lattice.
This is the backbone of X-ray diffraction (XRD), the everyday method for identifying crystal structures, lattice parameters, and phases — from mapping the structure of DNA to checking which phase formed inside a heat-treated steel. It is named for William Henry Bragg and his son William Lawrence Bragg, who shared the 1915 Nobel Prize in Physics.
Bragg's law in action: with copper X-rays of wavelength lambda = 0.154 nm striking iron's (110) planes spaced d = 0.203 nm, first-order diffraction (n = 1) peaks where sin(theta) = lambda/(2d) = 0.379, that is theta of about 22.3 degrees. Measuring that angle back-calculates the spacing.
n lambda = 2 d sin(theta): peak angle in, plane spacing out.
Bragg's law gives only the directions of the diffracted beams (hence the plane spacings); the peak intensities, which reveal which atoms sit where, need the fuller structure-factor analysis. And a missing peak does not mean missing planes — that spacing may simply be extinct by symmetry.