💎Crystallography Unit 4 – Crystal Structures and Packing

Fundamental Concepts

• Crystallography studies the arrangement of atoms in crystalline solids
• Crystals exhibit long-range order and periodic atomic arrangements
• Unit cells are the smallest repeating units that make up a crystal structure
• Lattice points represent the locations of atoms or groups of atoms in a crystal
• Symmetry plays a crucial role in determining the properties of crystals
• Translational symmetry involves the repetition of a unit cell in three dimensions
• Point symmetry includes operations such as rotation, reflection, and inversion
  • Rotation symmetry occurs when a crystal appears unchanged after rotation around an axis
  • Reflection symmetry exists when a crystal remains unchanged upon reflection across a plane

Types of Crystal Systems

• There are seven distinct crystal systems based on the symmetry and shape of the unit cell
• Triclinic system has the lowest symmetry with no restrictions on cell parameters
• Monoclinic system has one angle (β) not equal to 90° and a ≠ b ≠ c
• Orthorhombic system has all angles equal to 90° and a ≠ b ≠ c
• Tetragonal system has all angles equal to 90° and a = b ≠ c
  • Tetragonal crystals often exhibit unique optical and electrical properties
• Trigonal (Rhombohedral) system has all angles equal and a = b = c
• Hexagonal system has all angles equal to 90°, a = b ≠ c, and γ = 120°
  • Graphite and many ceramics crystallize in the hexagonal system
• Cubic system has the highest symmetry with all angles equal to 90° and a = b = c
  • Examples of cubic crystals include sodium chloride (NaCl) and diamond

Unit Cells and Lattice Points

• A unit cell is the smallest repeating unit that can generate the entire crystal structure through translation
• Lattice points are mathematical points representing the locations of atoms or groups of atoms
• Primitive unit cells contain only one lattice point, typically at the cell corners
• Non-primitive unit cells may have additional lattice points at cell faces or inside the cell
• The Bravais lattices describe the 14 unique lattice types based on the seven crystal systems
  • Cubic system has three Bravais lattices: simple cubic, body-centered cubic, and face-centered cubic
  • Hexagonal system has one Bravais lattice: hexagonal close-packed
• Unit cell parameters (a, b, c, α, β, γ) define the size and shape of the unit cell
• Coordination number represents the number of nearest neighbors for each atom in a crystal structure

Symmetry Operations and Elements

• Symmetry operations are transformations that leave a crystal unchanged in appearance
• Symmetry elements are geometric entities (points, lines, or planes) about which symmetry operations are performed
• Identity operation (E) leaves the crystal unchanged and is present in all crystal systems
• Rotation axes (1, 2, 3, 4, 6) describe the number of times a crystal can be rotated to coincide with itself
  • Two-fold rotation axis (2) requires a 180° rotation to achieve self-coincidence
  • Six-fold rotation axis (6) requires a 60° rotation to achieve self-coincidence
• Mirror planes (m) reflect the crystal across a plane, resulting in an identical structure
• Inversion centers (i) transform each point (x, y, z) to (-x, -y, -z) through the center of symmetry
• Rotoinversion axes combine rotation and inversion operations
• Screw axes combine rotation and translation operations
  • 21 screw axis involves a 180° rotation followed by a translation of 1/2 the unit cell length

Close Packing and Coordination Numbers

• Close packing describes the most efficient arrangement of atoms in a crystal structure
• Hexagonal close packing (HCP) and cubic close packing (CCP) are common close-packed structures
  • HCP has an ABABAB... stacking sequence of close-packed planes
  • CCP, also known as face-centered cubic (FCC), has an ABCABCABC... stacking sequence
• Coordination number (CN) is the number of nearest neighbors for each atom in a crystal structure
  • In close-packed structures (HCP and CCP), the coordination number is 12
  • Octahedral voids in close-packed structures have a coordination number of 6
  • Tetrahedral voids in close-packed structures have a coordination number of 4
• Interstitial sites are empty spaces between atoms where smaller atoms can be accommodated
  • Octahedral sites are larger and can accommodate larger interstitial atoms compared to tetrahedral sites

Miller Indices and Crystal Planes

• Miller indices (hkl) are used to describe the orientation of crystal planes and directions
• Crystal planes are denoted by integers h, k, and l, which are the reciprocals of the intercepts on the x, y, and z axes, respectively
  • (100) plane intersects the x-axis at 1 and is parallel to the y and z axes
  • (111) plane intersects all three axes at 1
• Directions in crystals are denoted by [uvw], where u, v, and w are the components of the direction vector
  • [100] direction is parallel to the x-axis
  • [111] direction passes through the opposite corners of the unit cell
• Planes and directions with the same symmetry are denoted by {hkl} and , respectively
• Miller indices are crucial for understanding the arrangement of atoms and predicting crystal properties

Crystal Defects and Imperfections

• Real crystals contain various types of defects and imperfections that deviate from the perfect periodic arrangement
• Point defects are localized defects involving one or a few atoms
  • Vacancies are empty lattice sites where atoms are missing
  • Interstitials are atoms occupying non-lattice sites between regular atoms
  • Substitutional defects occur when an atom is replaced by an atom of a different type
• Line defects, or dislocations, are irregularities along a line in the crystal structure
  • Edge dislocations are caused by the insertion or removal of an extra half-plane of atoms
  • Screw dislocations result from the displacement of atoms in a spiral manner around the dislocation line
• Planar defects involve irregularities in the stacking sequence of atomic planes
  • Stacking faults occur when the regular stacking sequence is disrupted
  • Twin boundaries separate two mirror-image regions of the crystal
• Grain boundaries are interfaces between crystallites (grains) with different orientations in polycrystalline materials
• Defects can significantly influence the mechanical, electrical, and optical properties of crystals

Practical Applications in Materials Science

• Crystallography plays a vital role in understanding and designing materials with desired properties
• X-ray diffraction (XRD) is a powerful technique for determining crystal structures and identifying phases
  • Bragg's law ($nλ = 2d \sin θ$) relates the wavelength of X-rays to the interplanar spacing and diffraction angle
  • XRD patterns provide information about lattice parameters, crystal symmetry, and phase composition
• Electron microscopy techniques (SEM, TEM) enable the visualization of crystal structures and defects at high resolutions
• Structure-property relationships link the atomic arrangement to macroscopic material properties
  • Mechanical properties (strength, ductility) are influenced by the presence and motion of dislocations
  • Electrical properties (conductivity, semiconductivity) depend on the electronic band structure determined by the crystal structure
• Crystal engineering involves the design and synthesis of functional materials with tailored properties
  • Pharmaceutical compounds can be engineered to have desired solubility and bioavailability
  • Metal-organic frameworks (MOFs) are crystalline materials with tunable porosity for gas storage and catalysis applications
• Epitaxial growth techniques enable the fabrication of single-crystal thin films for electronic and optoelectronic devices
  • Molecular beam epitaxy (MBE) allows precise control over the growth of semiconductor heterostructures
  • Pulsed laser deposition (PLD) is used to grow complex oxide thin films with specific crystal orientations