Crystal systems and lattices are the building blocks of crystallography. They categorize crystals based on symmetry and geometry, with seven systems ranging from highly symmetric cubic to low-symmetry triclinic. Understanding these structures is key to grasping crystal properties and behavior.

Bravais lattices describe how atoms arrange in 3D space. There are 14 unique lattices, including primitive, body-centered, and face-centered types. These arrangements determine crystal packing efficiency and influence physical properties, making them crucial for material science and engineering.

Crystal Systems and Lattices

Fundamental Crystallographic Structures

• Seven crystal systems form the basis of crystallography categorizing crystals based on their symmetry and geometric properties
• Cubic system exhibits highest symmetry with equal edge lengths and right angles between all faces
• Tetragonal system features two equal axes and one different axis, all at right angles
• Orthorhombic system has three unequal axes intersecting at right angles
• Hexagonal system characterized by four axes, three equal coplanar axes at 120° and a fourth axis perpendicular to this plane

Bravais Lattices and Lattice Types

• Fourteen Bravais lattices represent unique ways atoms or molecules can be arranged in three-dimensional space
• Primitive lattice contains lattice points only at the corners of the unit cell
• Body-centered lattice includes an additional lattice point at the center of the unit cell
• Face-centered lattice features extra lattice points at the center of each face of the unit cell
• Centered lattices (body-centered and face-centered) increase the packing efficiency of atoms within the crystal structure

Advanced Lattice Concepts

• Side-centered lattices exist in certain systems, with additional lattice points on two opposite faces
• Rhombohedral system can be described as a special case of the hexagonal system with specific axial ratios
• Monoclinic system has three unequal axes with one oblique angle
• Triclinic system represents the lowest symmetry with three unequal axes and three oblique angles

Unit Cell Properties

Fundamental Unit Cell Characteristics

• Lattice points represent positions in the crystal structure where atoms, ions, or molecules are located
• Unit cell serves as the smallest repeating unit of the crystal structure, defining its overall symmetry and properties
• Translational symmetry allows the unit cell to be repeated in three dimensions to form the complete crystal structure
• Edge lengths of the unit cell (a, b, c) determine the dimensions and shape of the crystal structure

Geometric Parameters of Unit Cells

• Unit cell dimensions include the lengths of the three edges (a, b, c) measured in angstroms or nanometers
• Axial angles (α, β, γ) describe the angles between the edges of the unit cell
• Alpha (α) represents the angle between the b and c axes
• Beta (β) denotes the angle between the a and c axes
• Gamma (γ) indicates the angle between the a and b axes
• Relationship between unit cell parameters and crystal system symmetry determines the constraints on cell dimensions and angles

Advanced Unit Cell Concepts

• Primitive unit cells contain the minimum number of lattice points necessary to define the crystal structure
• Conventional unit cells may include additional lattice points to better represent the symmetry of the crystal
• Miller indices (hkl) describe planes and directions within the crystal structure relative to the unit cell axes
• Reciprocal lattice concept relates the real-space lattice to the diffraction pattern observed in X-ray crystallography