4.1 Lattice types and Bravais lattices

Lattice types and Bravais lattices are key to understanding crystal structures. They describe how atoms or molecules arrange in 3D space, forming repeating patterns that define a crystal's shape and properties.

The 14 Bravais lattices represent all possible 3D lattice structures. They're grouped into seven crystal systems based on symmetry and centering types, helping us classify and study different materials' atomic arrangements.

Lattice Fundamentals

Lattice Structure and Points

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• Lattice forms the fundamental structure of crystalline materials consisting of a periodic arrangement of atoms or molecules in three-dimensional space
• Lattice points represent positions in space where atoms or molecules are located within the crystal structure
• Lattice points form a repeating pattern throughout the entire crystal, creating a regular and orderly arrangement
• Translational symmetry characterizes the lattice structure allowing the entire crystal to be generated by repeated translations of a basic unit cell

Lattice Parameters and Unit Cells

• Lattice parameters define the dimensions and angles of the unit cell (a, b, c, α, β, γ)
• Unit cell serves as the smallest repeating unit of the crystal structure containing all necessary information to describe the entire lattice
• Edge lengths (a, b, c) measure the distances between adjacent lattice points along the three principal directions
• Interaxial angles (α, β, γ) describe the angles between the edges of the unit cell
• Lattice parameters vary depending on the crystal system and determine the overall shape and symmetry of the crystal structure

Symmetry Operations and Crystal Systems

• Translational symmetry allows the lattice to be generated by repeatedly shifting the unit cell along its edges
• Rotational symmetry involves rotating the lattice around an axis by specific angles (60°, 90°, 120°, 180°)
• Mirror symmetry reflects the lattice across a plane, creating an identical arrangement on both sides
• Seven crystal systems (cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, trigonal) classify crystals based on their symmetry and lattice parameters
• Each crystal system has unique relationships between lattice parameters and specific symmetry elements

Bravais Lattices

Bravais Lattice Classification

• Bravais lattice describes the geometric arrangement of lattice points in three-dimensional space
• 14 Bravais lattices represent all possible unique lattice structures in three dimensions
• Bravais lattices are classified based on the seven crystal systems and centering types
• Each Bravais lattice possesses a distinct combination of symmetry elements and lattice parameters
• Bravais lattices serve as the foundation for understanding crystal structures and their properties

Primitive and Non-Primitive Lattices

• Primitive lattice contains only one lattice point per unit cell located at the corners of the cell
• Primitive lattices have the simplest arrangement of lattice points within the unit cell
• Non-primitive lattices include additional lattice points within the unit cell beyond the corner points
• Body-centered lattices have an extra lattice point at the center of the unit cell
• Face-centered lattices contain additional lattice points at the center of each face of the unit cell
• Side-centered lattices (base-centered) have extra lattice points on two opposite faces of the unit cell

Bravais Lattice Types and Examples

• Cubic system includes three Bravais lattices: simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC)
• Tetragonal system comprises two Bravais lattices: simple tetragonal (ST) and body-centered tetragonal (BCT)
• Orthorhombic system contains four Bravais lattices: simple orthorhombic, body-centered orthorhombic, base-centered orthorhombic, face-centered orthorhombic
• Monoclinic system has two Bravais lattices: simple monoclinic and base-centered monoclinic
• Triclinic system consists of one Bravais lattice: simple triclinic
• Hexagonal system includes one Bravais lattice: simple hexagonal
• Trigonal system comprises one Bravais lattice: simple trigonal (rhombohedral)