Chapter 3: What Is a Crystal Structure?

A crystal structure comprises two essential components: a lattice, which is a three-dimensional mathematical grid defining the translational symmetry of the entire crystal, and a motif, which is the repeating unit of atoms or molecules positioned at each lattice point. Although atoms undergo thermal vibration around their equilibrium positions, lattice points represent the time-averaged locations of atoms in the crystal. The lattice geometry is completely determined by three basis vectors labeled a, b, and c, where any lattice point can be accessed through integer linear combinations of these vectors. Six lattice parameters fully describe the lattice: the three vector lengths and the three inter-axial angles. The chapter progresses systematically from two-dimensional systems, where five distinct Bravais nets emerge from combining four fundamental geometries, to three dimensions where seven crystal systems form the complete classification: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. These systems are distinguished by specific mathematical constraints on their lattice parameters. When centering operations such as body-centering, face-centering, and base-centering are applied to these seven systems, fourteen distinct three-dimensional Bravais lattices result, providing a complete enumeration of all possible lattice types in nature. The chapter further differentiates between conventional unit cells and primitive unit cells, where the primitive cell contains exactly one lattice point and forms the smallest possible repeating unit. The Wigner-Seitz cell is presented as an alternative geometric construction that defines the primitive cell as the region encompassing all points closer to one lattice point than to any neighboring lattice point, ensuring the cell preserves the full symmetry properties of the underlying lattice structure.