Chapter_03_avi_revRyan_1

Chapter 3: The Structure of Crystalline Solids

Issues to Address

  • Atoms and Solid Structures: Focus on how atoms assemble into solid structures, particularly metals.

  • Density and Structure: Explore how the density of a material relates to its structure.

  • Variation in Material Properties: Discuss when material properties may vary with the sample's orientation.

Outline of Chapter 3

  • Fundamental Concepts

  • Unit Cells

  • Metallic Crystal Structure

  • Density and Polymorphism

  • Crystal Systems

  • Point Coordinates, Crystallographic Planes, and Directions

  • Linear and Planar Densities

  • Crystalline and Nanocrystalline Materials

Energy and Packing

  • Types of Packing:

    • Non-Dense, Random Packing: Atoms arranged randomly.

    • Dense Regular Packing: Atoms arranged regularly, leading to lower energy configurations.

Atomic and Crystal Structure

  • Atomic Structure: Refers to the number of protons and neutrons in the nucleus and the arrangement of electrons.

  • Crystal Structure: Pertains to the arrangement of atoms, ions, or molecules in crystalline materials.

Materials and Packing

  • Crystalline Materials:

    • Atoms pack in periodic, three-dimensional arrays.

    • Examples: Metals, many ceramics, some polymers.

  • Non-Crystalline Materials:

    • Atoms lack periodic packing.

    • Typical in complex structures and materials cooled rapidly (e.g., amorphous SiO2).

Unit Cell

  • Definition: The smallest repeating structural unit of a crystalline solid.

  • Characteristics:

    • Defines the symmetry of a crystal structure.

    • Atoms in the crystal can be generated by translations of the unit cell.

  • Models: Hard-sphere model, reduced-sphere model.

Lattice

  • Lattice Definition: A three-dimensional array of points corresponding to atomic positions.

  • Lattice Points: Can contain atoms, molecules, or ions.

Metallic Crystals

  • Dense Packing:

    • Typically consist of one element, leading to similar atomic radii.

    • Metallic bonding is directional and lacks many restrictions compared to ionic or covalent bonds.

  • Common Crystal Structures: Simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HCP).

Simple Cubic Structure (SC)

  • Coordination Number (C.N.): 6 nearest neighbors.

  • Density: Rare due to poor packing; an example is polonium (Po).

Body Centered Cubic Structure (BCC)

  • Coordination Number: 8, with close-packed directions along cube diagonals.

  • APF (Atomic Packing Factor): 0.68.

Face Centered Cubic Structure (FCC)

  • Coordination Number: 12, with close-packed directions along face diagonals.

  • APF: 0.74, includes 4 atoms per unit cell.

Hexagonal Close-Packed Structure (HCP)

  • Coordination Number: 12, with ABAB stacking sequence.

  • APF: 0.74.

Alloys and Compounds

  • Similar Structures: Often have close-packed structures.

  • Example: Sodium chloride (NaCl).

Theoretical Density

  • Density Calculation:

    • Example of Copper: Theoretical density ( Cu = 8.89 g/cm³) compared with actual density ( Cu = 8.94 g/cm³).

Material Classes and Densities

  • Metals: Close-packing and large atomic mass.

  • Ceramics: Less dense packing and lighter elements.

  • Polymers: Poor packing and lighter elements (C, H, O).

  • Composites: Intermediate density values.

Single Crystals vs. Polycrystals

  • Single Crystals: Anisotropic properties (vary with direction).

  • Polycrystals: Isotropic properties if grains are randomly oriented, or anisotropic if textured.

Allotropy and Polymorphism

  • Allotropy: The ability of an element to exist in multiple crystal structures (e.g., iron).

    • Example: Iron transitions between bcc and fcc.

  • Polymorphism: Occurs when compounds exhibit multiple crystalline forms (e.g., carbon).

Crystal Systems

  • Seven Crystal Systems: Based on geometrical properties of the unit cell.

    • Cubic: a = b = c; α = β = γ = 90°

    • Tetragonal: a = b ≠ c; α = β = γ = 90°

    • Orthorhombic: a ≠ b ≠ c; α = β = γ = 90°

    • Rhombohedral: a = b = c; α = β = γ ≠ 90°

    • Hexagonal: a = b ≠ c; α = β = 90°, γ = 120°

    • Monoclinic: a ≠ b ≠ c; α = γ = 90°; β ≠ 90°

    • Triclinic: a ≠ b ≠ c; α ≠ β ≠ γ.

Point Coordinates and Crystallographic Directions

  • Point Coordinates: Specified using fractional lengths of unit cell dimensions.

  • Crystallographic Directions: Defined by vectors enclosed in square brackets (e.g., [uvw]).

Crystallographic Planes

  • Defined by Miller indices (hkl) except in the hexagonal system.

  • Family of planes represented as sets (e.g., {100}).

X-Ray Diffraction (XRD)

  • XRD Basics: Utilizes the diffraction of X-rays through crystal planes to determine structure.

  • Bragg's Law: Relates the angle of diffraction to interplanar spacing and wavelength.

Summary

  • Materials can form crystalline or amorphous structures, and their density can be predicted based on atomic properties and geometry.

  • Material properties exhibit anisotropic behavior in single crystals but may be isotropic in polycrystalline materials with random orientation.