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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.