See biography by Sir Kerr Grant (1952).
When X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as the Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.
The interference is constructive when the phase shift is a multiple to 2π; this condition can be expressed by Bragg's law:
According to the 2θ deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences
Although the misleading common opinion reigns that Bragg's Law measures atomic distances in real space, it does not. Furthermore, the term demonstrates that it measures the number of wavelengths fitting between two rows of atoms, thus measuring reciprocal distances. Max von Laue had interpreted this correctly in a vector form, the Laue equation
where is a reciprocal lattice vector and and are the wave vectors of the incident and the diffracted beams.
Together with the condition for elastic scattering and the introduction of the scattering angle this leads equivalently to Bragg's equation.
The concept of reciprocal lattice is the Fourier space of a crystal lattice and necessary for a full mathematical description of wave mechanics.
There will be a path difference between the 'ray' that gets reflected along AC' and the ray that gets transmitted, then reflected along AB and BC paths respectively. This path difference is: