X-rays reflecting from the lattice planes of a monochromator crystal: path difference, the Bragg angle, a rocking curve and the wavelength-angle relation.
The science
X-rays of wavelength lambda hit a crystal whose lattice planes are a distance d apart at a glancing angle theta measured from the planes. A ray reflected by the next plane travels an extra path 2 d sin(theta). When that path is a whole number of wavelengths the reflections add in phase: n lambda = 2 d sin(theta) (Bragg's law). The photon energy follows from E = h c / lambda, or E[keV] = 1.23984 / lambda[nm].
With N planes the kinematic intensity is I(theta) = [sin(N phi / 2) / (N sin(phi / 2))]^2 with phi = 4 pi d sin(theta) / lambda. The peaks sharpen as N grows, with width about lambda / (2 N d cos(theta)). A monochromator therefore picks a narrow energy band from the broad synchrotron spectrum and also passes harmonics n = 2, 3 ... at the same angle, which beamlines suppress with mirrors.
Simplifications: kinematic (single scattering) theory, no absorption or extinction (a real perfect crystal follows the Darwin curve and has a finite reflectivity), no polarisation factor and no structure factor, so forbidden reflections are not removed. Default spacing 0.314 nm is the silicon (111) value used by the 3D page. Checks below: the numerical peak position of the rocking curve against Bragg's law, and the energy-wavelength constant against CODATA values.