3.2 14 Bravais lattices and their properties

Bravais lattices are the backbone of crystal structures, defining how atoms arrange in 3D space. There are 14 unique lattices, each with specific symmetry and centering. These lattices fall into seven crystal systems, from simple cubic to complex triclinic.

Understanding Bravais lattices is key to grasping crystal properties and behavior. They help predict how crystals form, grow, and interact with light and other forces. This knowledge is crucial for fields like materials science, geology, and even drug design.

Bravais Lattices and Crystal Systems

Fundamental Concepts and Definitions

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• 14 Bravais lattices represent unique three-dimensional arrangements of lattice points describing all possible crystal structures
• Seven crystal systems define specific relationships between lattice parameters and angles
  • Triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal, and hexagonal
• Distribution of Bravais lattices among crystal systems
  • Triclinic (1), monoclinic (2), orthorhombic (4), tetragonal (2), cubic (3), trigonal (1), hexagonal (1)
• Each Bravais lattice characterized by unique combination of lattice centering and crystal system symmetry
• Primitive cell contains exactly one lattice point, fundamental to understanding differences between Bravais lattices within a crystal system

Crystal Systems and Their Characteristics

• Triclinic system
  • Least symmetrical, no restrictions on cell parameters ($a \neq b \neq c, \alpha \neq \beta \neq \gamma$)
  • Example minerals (quartz, feldspar)
• Monoclinic system
  • One two-fold rotation axis or mirror plane ($a \neq b \neq c, \alpha = \gamma = 90°, \beta \neq 90°$)
  • Example minerals (gypsum, mica)
• Orthorhombic system
  • Three mutually perpendicular two-fold rotation axes ($a \neq b \neq c, \alpha = \beta = \gamma = 90°$)
  • Example minerals (topaz, aragonite)
• Tetragonal system
  • One four-fold rotation axis ($a = b \neq c, \alpha = \beta = \gamma = 90°$)
  • Example minerals (rutile, zircon)
• Cubic system
  • Four three-fold rotation axes ($a = b = c, \alpha = \beta = \gamma = 90°$)
  • Example minerals (halite, diamond)
• Trigonal system
  • One three-fold rotation axis ($a = b = c, \alpha = \beta = \gamma \neq 90°$)
  • Example minerals (calcite, corundum)
• Hexagonal system
  • One six-fold rotation axis ($a = b \neq c, \alpha = \beta = 90°, \gamma = 120°$)
  • Example minerals (beryl, apatite)

Primitive vs. Centered Lattices

Types of Lattice Centering

• Primitive lattices (P) have lattice points only at unit cell corners, representing minimum volume to generate entire crystal structure through translation
• Body-centered lattices (I) have additional lattice points at unit cell center and corners
• Face-centered lattices (F) have lattice points at center of each unit cell face and corners
• Base-centered lattices (C) have lattice points at center of two opposite unit cell faces, typically perpendicular to unique axis

Characteristics and Implications of Lattice Centering

• Centering choice affects lattice symmetry and number of atoms per unit cell, influencing physical and chemical properties of crystal
• Non-primitive lattices (I, F, C) often describable by smaller primitive cells, but conventional unit cell chosen to best represent crystal structure symmetry
• Volume relationships between primitive and non-primitive unit cells
  • $V(I) = 2V(P)$
  • $V(F) = 4V(P)$
  • $V(C) = 2V(P)$
• Packing efficiency varies among centering types
  • Face-centered cubic (FCC) most efficient (74% space filled)
  • Body-centered cubic (BCC) less efficient (68% space filled)
  • Simple cubic least efficient (52% space filled)

Symmetry Elements and Space Groups

Symmetry Operations and Notation

• Bravais lattices possess characteristic symmetry elements (rotations, reflections, inversions, rotoinversions)
• Hermann-Mauguin notation describes space group symmetry, incorporating lattice type and symmetry operations
• Point group symmetry describes rotational and reflectional lattice symmetry
• Space group symmetry includes translational symmetry elements
• International Tables for Crystallography provide standardized descriptions of 230 space groups derived from 14 Bravais lattices and 32 crystallographic point groups

Advanced Symmetry Concepts

• Systematic absences in diffraction patterns directly related to Bravais lattice centering and symmetry elements
• Screw axes and glide planes in space groups introduce additional symmetry analysis complexity, affecting diffraction conditions
• Understanding Bravais lattice symmetry crucial for predicting physical properties and interpreting experimental crystallography data
• Symmetry operations combined with translations generate space groups
  • 21 screw axis combines two-fold rotation with translation of 1/2 unit cell length
  • Glide plane combines mirror reflection with translation parallel to reflection plane

Predicting Bravais Lattices from Crystal Data

Analysis of Unit Cell Parameters

• Unit cell parameters ($a, b, c, \alpha, \beta, \gamma$) provide crucial information about crystal system and potential Bravais lattice types
• Equality or inequality of lattice parameters and interaxial angles narrows down possible crystal systems and Bravais lattices
• Atomic positions within unit cell, especially those related by symmetry operations, indicate centering presence and help distinguish between primitive and non-primitive lattices

Experimental Techniques and Data Interpretation

• Systematic absences in diffraction data used to identify lattice centering and certain symmetry elements
• Crystal morphology and physical properties analysis provides additional clues about underlying Bravais lattice
• Coordination number and packing efficiency of atoms in structure suggest certain Bravais lattice types, particularly in metallic and ionic crystals (NaCl, CsCl structures)
• Advanced techniques for complex structures
  • Group theory applications in crystallography
  • Computational crystallography methods (structure refinement algorithms)
• X-ray diffraction patterns reveal information about lattice spacing and symmetry
  • Bragg's law ($n\lambda = 2d\sin\theta$) relates diffraction angle to interplanar spacing