Crystal systems and Bravais lattices are the building blocks of solid-state structures. They describe how atoms or molecules arrange themselves in crystals, forming repeating patterns that define a material's properties.

Understanding these concepts is crucial for predicting and explaining a material's behavior. From simple cubic to complex triclinic systems, these arrangements impact everything from a crystal's shape to its physical and chemical characteristics.

Lattice Fundamentals

Basic Building Blocks of Crystal Structures

• Unit cell forms the basic repeating structural unit of a crystal lattice
• Lattice points represent the positions of atoms or molecules in a crystal structure
• Primitive cell contains the minimum number of lattice points to define the crystal structure
• Crystal systems categorize crystals based on their symmetry and geometric properties
• Bravais lattices describe 14 unique three-dimensional lattice arrangements

Crystal System Classification

• Seven crystal systems classify all possible lattice structures based on unit cell geometry
• Each crystal system defined by specific relationships between lattice parameters (a, b, c) and angles (α, β, γ)
• Cubic system characterized by equal edge lengths and right angles (a = b = c, α = β = γ = 90°)
• Hexagonal system features two equal basal edges and a unique vertical axis (a = b ≠ c, α = β = 90°, γ = 120°)
• Tetragonal system has a unique c-axis length (a = b ≠ c, α = β = γ = 90°)

Bravais Lattice Configurations

• 14 Bravais lattices represent all possible three-dimensional lattice arrangements
• Primitive lattices contain lattice points only at cell corners
• Body-centered lattices include an additional lattice point at the center of the unit cell
• Face-centered lattices have extra lattice points at the center of each face
• Base-centered lattices feature additional lattice points on two opposite faces
• Combination of crystal systems and centering types yields the 14 Bravais lattices

Crystal System Types

Cubic System Characteristics

• Highest symmetry among all crystal systems
• Three equal axes intersect at right angles (a = b = c, α = β = γ = 90°)
• Three Bravais lattices: simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC)
• SC structure exemplified by polonium
• BCC structure found in many metals (iron, chromium, tungsten)
• FCC structure common in metals and ionic compounds (copper, aluminum, sodium chloride)

Hexagonal and Tetragonal Systems

• Hexagonal system features two equal basal axes and a unique vertical axis (a = b ≠ c, α = β = 90°, γ = 120°)
• Only one Bravais lattice for hexagonal system: simple hexagonal
• Hexagonal close-packed (HCP) structure common in metals (magnesium, zinc, cobalt)
• Tetragonal system characterized by two equal axes and one unique axis (a = b ≠ c, α = β = γ = 90°)
• Two Bravais lattices for tetragonal system: simple tetragonal and body-centered tetragonal
• Tetragonal structures found in minerals (rutile) and superconductors (mercury barium calcium copper oxide)

Lower Symmetry Crystal Systems

• Orthorhombic system has three unequal axes at right angles (a ≠ b ≠ c, α = β = γ = 90°)
• Four Bravais lattices for orthorhombic system: simple, body-centered, base-centered, face-centered
• Orthorhombic structures occur in minerals (aragonite) and organic compounds (sulfur)
• Monoclinic system features three unequal axes with one non-right angle (a ≠ b ≠ c, α = γ = 90° ≠ β)
• Two Bravais lattices for monoclinic system: simple and base-centered
• Monoclinic structures found in minerals (gypsum) and organic compounds (sucrose)
• Triclinic system has lowest symmetry with three unequal axes and no right angles (a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°)
• Only one Bravais lattice for triclinic system: simple triclinic
• Triclinic structures occur in minerals (feldspar) and complex organic molecules

Advanced Lattice Concepts

Miller Indices and Crystallographic Planes

• Miller indices (h, k, l) describe planes and directions in crystal lattices
• Intercepts of planes with crystal axes determine Miller indices
• Negative indices denoted with a bar above the number
• Low-index planes often have important physical and chemical properties
• Miller-Bravais indices used for hexagonal systems include an extra index (h k i l)
• Examples of important planes: (100) cube face, (111) close-packed plane in FCC structures

Symmetry Operations in Crystals

• Symmetry operations leave crystal structure unchanged after their application
• Translation symmetry involves moving the entire structure by a lattice vector
• Rotational symmetry includes 2-fold, 3-fold, 4-fold, and 6-fold rotations
• Mirror symmetry reflects the structure across a plane
• Inversion symmetry inverts the structure through a point
• Glide planes combine translation and mirror symmetry
• Screw axes combine translation and rotational symmetry
• 32 crystallographic point groups describe all possible combinations of symmetry elements

Close-Packed Structures and Coordination

• Close-packed structures maximize atomic packing efficiency
• Two common close-packed arrangements: hexagonal close-packed (HCP) and cubic close-packed (CCP)
• CCP structure identical to face-centered cubic (FCC) lattice
• Stacking sequences: ABAB... for HCP, ABCABC... for CCP
• Coordination number describes the number of nearest neighbors for each atom
• Close-packed structures have coordination number of 12
• Packing efficiency calculated as the ratio of atomic volume to total unit cell volume
• FCC and HCP structures have highest packing efficiency of ~74%