Chapter 11: X-Ray Diffraction: Geometry

The chapter distinguishes between two types of radiation produced: Bremsstrahlung, which forms a continuous spectrum across multiple wavelengths, and characteristic radiation such as K-alpha and K-beta lines that result from specific electronic transitions in the target material. Selection of monochromatic X-rays is accomplished through absorption edge filters and application of Beer's law, which enables isolation of precise wavelengths necessary for accurate structural analysis. The theoretical framework centers on Bragg's law, a foundational relationship that connects the diffraction angle, X-ray wavelength, and the spacing between successive crystal planes, thereby establishing the conditions under which constructive interference produces detectable diffracted beams. To understand when these diffraction conditions are satisfied, the chapter introduces the Ewald sphere construction in reciprocal space, a conceptual tool demonstrating that diffraction occurs only when a reciprocal lattice point intersects the sphere's surface, where the sphere's radius corresponds to the reciprocal of the wavelength. The discussion then addresses experimental methodologies tailored to different sample types. For polycrystalline materials, the powder diffraction technique employs either Bragg-Brentano geometry or the Debye-Scherrer camera configuration, both of which exploit random crystal grain orientations to satisfy diffraction conditions across multiple lattice planes, generating characteristic concentric ring patterns. For single-crystal samples, the Laue method utilizes polychromatic radiation with a stationary crystal to produce spot patterns distributed on conic sections such as ellipses and hyperbolas, enabling determination of crystal orientation through specialized analysis tools including the Greninger chart.