💎Crystallography Unit 11 – Crystallography in Materials Science

Fundamentals of Crystallography

• Crystallography studies the arrangement of atoms in crystalline solids
• Crystals exhibit long-range order and periodic atomic structure
• Unit cell represents the smallest repeating unit that defines the crystal structure
  • Characterized by lattice parameters (lengths and angles)
  • Primitive unit cell contains only one lattice point
• Bravais lattices describe the 14 unique arrangements of lattice points in 3D space
  • Includes cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, and rhombohedral lattices
• Coordination number indicates the number of nearest neighbors an atom has in a crystal structure (e.g., 12 in face-centered cubic)
• Atomic packing factor quantifies the fraction of space occupied by atoms in a unit cell
  • Highest packing efficiency achieved in close-packed structures (hexagonal and cubic close-packed)
• Interplanar spacing ($d$) is the distance between adjacent parallel planes in a crystal

Crystal Systems and Lattices

• Crystal systems classify crystals based on the symmetry of their unit cells
• Seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal, and hexagonal
  • Each system has distinct relationships between lattice parameters and angles
• Four lattice types: primitive (P), body-centered (I), face-centered (F), and base-centered (C)
  • Lattice types describe the arrangement of lattice points within the unit cell
• Combining crystal systems and lattice types yields the 14 Bravais lattices
  • Example: Cubic system has three Bravais lattices (simple cubic, body-centered cubic, face-centered cubic)
• Space groups incorporate symmetry elements (translations, rotations, reflections) to fully describe the symmetry of a crystal structure
  • 230 unique space groups exist in 3D

Symmetry Operations and Point Groups

• Symmetry operations are geometric transformations that leave the appearance of a crystal unchanged
• Translation symmetry involves the repetition of a motif by a specific distance (lattice vector)
• Rotation symmetry occurs when a crystal appears identical after rotation about an axis by a specific angle
  • Rotation axes can be 2-fold, 3-fold, 4-fold, or 6-fold
• Reflection symmetry exists when a crystal is unchanged by reflection across a mirror plane
• Inversion symmetry is present if a crystal remains unchanged when inverted through a point
• Point groups categorize crystals based on their combination of rotation, reflection, and inversion symmetries
  • 32 crystallographic point groups in 3D
• Schoenflies notation is used to represent point groups (e.g., $C_{2v}$, $D_{4h}$)
  • Indicates the primary rotation axis and additional symmetry elements

Miller Indices and Crystal Planes

• Miller indices ($hkl$) uniquely identify planes and directions in a crystal
• Planes are denoted by integers $h$, $k$, and $l$, which are the reciprocals of the intercepts on the $x$, $y$, and $z$ axes
  • Negative indices are written with a bar above the number (e.g., $\bar{1}$)
• Directions are represented by square brackets $[uvw]$ and are perpendicular to the corresponding ($hkl$) plane
• Families of planes with similar symmetry are denoted by curly brackets ${hkl}$
• Interplanar spacing ($d_{hkl}$) depends on the Miller indices and lattice parameters
  • Calculated using the quadratic form for each crystal system
• Structure factor ($F_{hkl}$) quantifies the amplitude and phase of the diffracted wave from a given ($hkl$) plane
  • Depends on the atomic positions and scattering factors

X-ray Diffraction Techniques

• X-ray diffraction (XRD) probes the atomic structure of crystalline materials
• Bragg's law relates the wavelength ($\lambda$), interplanar spacing ($d$), and diffraction angle ($\theta$): $2d\sin\theta = n\lambda$
  • Constructive interference occurs when Bragg's law is satisfied
• Powder XRD uses a polycrystalline sample with randomly oriented grains
  • Diffraction pattern consists of concentric rings corresponding to different ($hkl$) planes
• Single-crystal XRD provides more detailed structural information
  • Diffraction pattern is a series of spots related to the reciprocal lattice
• Laue method uses a polychromatic X-ray beam to obtain diffraction patterns quickly
• Rotating crystal method involves rotating the crystal during exposure to monochromatic X-rays
• Synchrotron radiation sources offer high-intensity, tunable X-rays for advanced diffraction studies

Crystal Structure Determination

• Crystal structure determination aims to elucidate the atomic positions within the unit cell
• Diffraction pattern provides information about the lattice parameters, symmetry, and intensity of reflections
• Intensity of diffracted beams depends on the structure factor ($F_{hkl}$) and atomic scattering factors
• Phase problem arises because diffraction experiments only measure intensities, not phases
  • Solved using methods like Patterson function, direct methods, or molecular replacement
• Fourier synthesis reconstructs the electron density map from the structure factors and phases
• Refinement process optimizes the atomic positions, occupancies, and thermal parameters to minimize the difference between observed and calculated intensities
  • Figures of merit (R-factors) assess the quality of the refined structure
• Rietveld refinement is used for powder XRD data to refine the crystal structure and quantify phase fractions

Defects and Imperfections in Crystals

• Real crystals contain various types of defects and imperfections
• Point defects are localized irregularities in the crystal lattice
  • Vacancies are missing atoms from lattice sites
  • Interstitials are extra atoms occupying non-lattice sites
  • Substitutional impurities replace host atoms in the lattice
• Line defects, or dislocations, are irregularities in the lattice along a line
  • Edge dislocations form when an extra half-plane of atoms is inserted
  • Screw dislocations result from a helical shift in the lattice
• Planar defects extend in two dimensions
  • Grain boundaries separate regions with different crystallographic orientations
  • Stacking faults occur when the regular stacking sequence of planes is disrupted
• Volume defects include pores, cracks, and inclusions
• Defects influence various properties of materials, such as mechanical strength, electrical conductivity, and diffusion

Applications in Materials Science

• Crystallography plays a crucial role in understanding and designing materials
• Structure-property relationships link the atomic arrangement to macroscopic properties
  • Example: Perovskite structure in ferroelectric materials enables polarization switching
• Phase diagrams represent the stability of different crystal structures as a function of composition, temperature, and pressure
  • Used to design alloys and optimize processing conditions
• Epitaxial growth techniques (molecular beam epitaxy, pulsed laser deposition) rely on crystal structure matching to grow thin films
  • Applications in semiconductor devices, superconductors, and functional oxides
• Texture analysis quantifies the preferred orientation of polycrystalline materials
  • Important for anisotropic properties (e.g., mechanical strength, magnetic anisotropy)
• Crystallography guides the development of advanced materials
  • High-entropy alloys with multiple principal elements in a single crystal structure
  • Metal-organic frameworks with tunable porosity and functionality
  • Nanostructured materials with size-dependent properties