Crystalline structures are deduced by the diffraction of known radiation incident on solid materials.

The three-dimensional regularity of unit cells in crystalline materials results in coherent scattering of the radiation has the wavelength l and the energy of the scattered radiation are the same as the incident radiation. The directions of the scattered beams are a function of the wavelength of the radiation and the specific interatomic spacing of the plane from which the radiation scatters. The intensity of the scattered beam depends on the position of each atom in the unit and on the orientation of the crystal relative to the direction of the incident X-ray beam. Those beams that scatter in a constructive manner result in allowed reflections, i.e., the intensity is a nonzero value. Those beams that scatter in a destructive manner result in unallowed reflections, i.e., the intensity has a minimal value. Each of the constructively scattered beams depends directly on the wavelength of the incident radiation.

Historical records show evidence of von Laue’s diffraction experiments before the publication of W.L. Bragg’s explanation of the diffraction of radiation by crystalline solids.

However, the inquisitive W.L. Bragg’s attempts to resolve an intellectual debate between his father W.H. Bragg and Laue resulted in the formulation of the famous Bragg’s Law. W.H. Bragg described the diffraction as billard-ball scattering called kinetic scattering. Laue argued that only optical rules determine the occurrences of diffraction (*i.e*., wave interactions only). Radiation acts as waves and the scattering depends on optical laws of scattering (e.g., Law of Reflectivity), and constructive interference (*i.e*., waves in phase add constructively and waves out of phase add destructively). These published rules even gained a name: The Laue Equations. Laue’s description of diffraction in 3-dimensional space required solving three equations with twelve unknowns. Even then, the derivation is arduous. W.H. Bragg’s rebuttal stated that any symmetrical elements present in the diffraction pattern reduces the number of unknowns and also states that only particle-particle interactions induce diffraction. The kinetic theory required that the particles scatter from atoms coherently, *i.e.,* the incident particle does not lose any energy during the scattering event. Even the earliest Laue experiment supported this account. W.L. Bragg utilized all of the above precepts to derive a simplistic description of coherent scattering.

nl = 2d_{hkl} sinq

[define terms of the equation in voiceover]

W.L. Bragg derived a simplistic description of coherent scattering from an array of periodic scattering sites, such as atoms in a crystalline solid. The scalar description of diffraction considers the case of monochromatic radiation impinging onto sheets of atoms spaced at a specific distance (*d*_{hkl}). The wavelength l of the radiation is smaller than the interatomic spacing *d*_{hkl}. Typically, in crystalline solids, we desire to measure atomic spacing on the order of the lattice constants, *e.g. *0.2-0.4 nm. Hence, we must use radiation with wavelengths less than 0.2 nm. This range of wavelengths includes those of X-rays and high-energy electrons. Based on this description of diffraction, we can conduct experiments to determine the distance between reflecting planes, crystallographic structure, coefficient of thermal expansion, texture, stress, and composition of thin films.