Algebraic Crystallography

This chapter describes crystallography as an experimental science, characterizes crystal structures using X-ray diffraction, and includes vector algebra in orthogonal and nonorthogonal coordinate systems. In this context, crystallography is treated as the multidisciplinary science of crystals (crystalline solids) that deals with atomic structures responsible for properties of engineering materials. Knowledge of X-ray diffraction and reciprocal lattice (three-dimensional parallelepiped) defined by a scattering vector ghkl is essential to complement the analysis of crystallography and related atomic structure of a specimen surface. This includes the Ewald sphere methodology, Bragg’s law, and reciprocal lattice as used in X-ray and electron diffraction. The main objectives of this chapter include basic knowledge for describing, understanding, and characterizing crystalline solids with respect to crystallography and vector algebra. In essence, vector algebra in crystallography defines the vector coordinates related to the dot product or cross-product of crystallographic directions. It should be mentioned that this chapter includes basic vector algebra and excludes advanced concepts of algebraic crystallography and the structure of space groups, which are made from 32 crystallographic point groups associated with seven crystal lattice systems and 14 Bravais lattices.