11.1 Crystal Structures and Lattices

Condensed matter physics explores the fascinating world of crystal structures and lattices. These arrangements of atoms form the backbone of solid materials, dictating their properties and behaviors.

Understanding crystal structures is key to grasping how materials work. From metals to semiconductors, the way atoms line up in crystals affects everything from strength to conductivity, shaping our modern world.

Crystal Structures and Characteristics

Types of Crystal Systems and Structures

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• Seven main crystal systems classify crystal structures
  • Cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic
  • Each system defined by unique symmetry and geometric properties
• Three most common crystal structures in metals
  • Body-centered cubic (BCC)
  • Face-centered cubic (FCC)
  • Hexagonal close-packed (HCP)
• Unit cell forms the smallest repeating unit of the crystal lattice
  • Defines the overall structure and symmetry of the crystal
  • Contains all necessary information to describe the entire crystal through repetition

Crystal Structure Characteristics and Visualization

• Coordination number indicates nearest neighbors for each atom in the lattice
  • Influences properties like atomic packing and material strength
  • Varies among different crystal structures (BCC: 8, FCC: 12, HCP: 12)
• Polymorphism allows solid materials to exist in multiple crystal structures
  • Depends on temperature and pressure conditions
  • Examples include carbon (diamond, graphite) and iron (BCC, FCC phases)
• Visualization methods for crystal structures
  • Space-filling models emphasize atomic size and packing
  • Ball-and-stick models highlight bonding between atoms
  • Polyhedral models showcase coordination environments and geometric relationships

Lattice in Crystalline Materials

Lattice Fundamentals and Bravais Lattices

• Lattice creates three-dimensional, periodic array of points representing atomic arrangement
  • Defines the repeating pattern in crystalline solids
  • Provides framework for understanding crystal symmetry and properties
• Primitive vectors form the basis for describing crystal structure
  • Three non-coplanar vectors define the unit cell
  • Allow mathematical description of atomic positions within the lattice
• 14 unique Bravais lattices arrange points in three-dimensional space
  • Represent all possible lattice types in crystals
  • Examples include simple cubic, body-centered cubic, and face-centered cubic

Lattice Parameters and Symmetry

• Lattice parameters describe unit cell geometry
  • Lengths of unit cell edges (a, b, c)
  • Angles between edges (α, β, γ)
  • Vary among different crystal systems and structures
• Symmetry operations applicable to crystal structure determined by lattice
  • Translations move the crystal to equivalent positions
  • Rotations and reflections preserve overall structure
  • Point group and space group classifications based on symmetry elements
• Reciprocal lattice aids in analyzing diffraction patterns
  • Mathematical construct describing crystal periodicity in reciprocal space
  • Useful for understanding X-ray diffraction and electron microscopy results

Crystal Structure vs Material Properties

Mechanical and Electrical Properties

• Crystal structure influences mechanical properties through atomic arrangement and bonding
  • Close-packed structures (FCC, HCP) often exhibit higher ductility
  • BCC structures typically show higher strength and lower ductility
  • Slip systems and dislocation movement affected by crystal structure
• Electrical conductivity relates to electron configuration and band structure
  • Metals with simple crystal structures (Cu, Ag, Au) show high conductivity
  • Semiconductors (Si, Ge) have more complex band structures due to crystal arrangement
  • Insulators (diamond) have large band gaps influenced by crystal bonding

Thermal, Optical, and Magnetic Properties

• Thermal properties affected by vibrational modes within crystal lattice
  • Thermal expansion coefficients vary among crystal structures
  • Heat capacity influenced by atomic bonding and lattice vibrations
  • Thermal conductivity related to phonon propagation through the crystal
• Optical properties depend on symmetry and atomic arrangement
  • Refraction and birefringence linked to crystal structure anisotropy
  • Examples include calcite (double refraction) and quartz (optical activity)
• Magnetic properties closely tied to crystal structure
  • Ferromagnetic materials (Fe, Ni, Co) exhibit specific crystal symmetries
  • Antiferromagnetic ordering depends on crystal lattice arrangement
  • Magnetocrystalline anisotropy influenced by crystal structure

Crystal Defects and Material Behavior

• Defects in crystal structures significantly alter material properties
  • Point defects (vacancies, interstitials) affect diffusion and electrical properties
  • Line defects (dislocations) influence mechanical strength and plasticity
  • Planar defects (grain boundaries, stacking faults) impact various material behaviors
• Defect interactions with crystal structure determine overall material performance
  • Solid solution strengthening relies on atomic size mismatch in crystal lattice
  • Precipitation hardening utilizes coherent precipitates within the crystal structure
  • Radiation damage creates complex defect structures, altering material properties

Packing Fraction and Atomic Density

Packing Fraction Calculation and Significance

• Packing fraction represents efficiency of space utilization in crystal structure
  • Calculated as ratio of volume occupied by atoms to total unit cell volume
  • Higher packing fraction indicates more efficient atomic arrangement
• FCC and HCP structures achieve highest possible packing fraction for equal spheres
  • Packing fraction approximately 0.74
  • Results in high density and close atomic packing
• BCC structure exhibits lower packing fraction of about 0.68
  • More open structure compared to FCC and HCP
  • Allows for easier atomic diffusion and phase transformations

Atomic Density and Structure Comparisons

• Atomic density defines number of atoms per unit volume in crystal structure
  • Typically expressed in atoms per cubic meter or atoms per cubic centimeter
  • Directly related to material density and packing efficiency
• Calculation of atomic density uses number of atoms per unit cell, atomic mass, and cell volume
  • $\text{Atomic Density} = \frac{\text{Number of atoms per unit cell}}{\text{Unit cell volume}}$
  • Considers both the crystal structure and the specific elements involved
• Comparison of packing fractions and atomic densities between structures explains property variations
  • Higher packing fraction often correlates with higher melting point (Cu vs. Fe)
  • Density differences between polymorphs (graphite vs. diamond)
  • Mechanical property variations among different crystal structures of the same element