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Lecture_2
## CRYSTAL STRUCTURES
### Lecture Overview
- Course: Mechanics of Materials (4MA00)
- Institution: Eindhoven University of Technology (TU/e)
---
## LENGTH SCALES
- Hierarchical scale of sizes:
- 10-18 - 10-15: Electron, proton, neutron
- 10-12: Atom
- 10-9: Crystals
- Up to 100: Products and larger structures
---
## FROM ATOMS TO SOLID
- Exploration of how atoms assemble into solid structures, focusing on metals.
- **Density Dependence**: Material density relies on atomic arrangement and packing efficiency.
- **Directions and Planes**: Importance of indicating specific directions and planes in atomic structures.
---
## PACKING SPHERES
- Discussion of atomic packing and arrangement in solids.
---
## ATOMIC MODELS
- Historical Models:
- **Bohr model** and **Quantum mechanical model** previously discussed.
- Current model:
- Atoms are considered as spheres (Atomic hard sphere model).
- Nearest neighbors in contact, influencing bonding and structure.
---
## STRUCTURE OF MATERIALS
- **Molecular Materials**:
- Comprised of covalently bonded atoms.
- Interactions: Van der Waals forces.
- **Non-Molecular Materials**:
- **Metals**: Examples include Fe, Al, Au.
- **Ceramics**: Examples include Al2O3, NaCl.
- Other forms include water, gases, and polymers (CnHm).
---
## ATOMIC PACKING
- Short Range Order: Atoms with similar structures in close vicinity.
- Long Range Order:
- **Amorphous Materials**: Randomly structured, lacking long-range order but may have short-range order (e.g., glass).
- **Crystalline Materials**: Atoms organized in 3D periodic arrays.
- Common in metals, ceramics, and semi-crystalline polymers.
---
## ENERGY AND PACKING
- Packing influences energy levels:
- Dense, regular structures typically exhibit lower energy states.
- Factors include bond lengths and energies associated with atomic arrangements.
---
## EXAMPLE: SELF-ORGANIZATION IN CRYSTALLINE STRUCTURE
- Illustration of atomic arrangements and forces at play in crystal formation.
- Key equation of motion illustrated: mv = Σ F.
---
## METALLIC CRYSTALS
- Characteristics:
- Metals typically exhibit simple crystal structures and are densely packed.
- Reasons for this packing include uniform atomic radii and non-directional metallic bonding.
---
## CRYSTALS
- Defined by long-range order and translation symmetry.
- Examples of symmetry in patterns comparable to M.C. Escher's artworks.
---
## SYMMETRY IN CRYSTALS
- Types of symmetry: Square, Rectangular, Centered Rectangular, Hexagonal, Oblique.
---
## UNIT CELL GEOMETRY
- Key parameters:
- Edge lengths (a, b, c) and angles (α, β, γ) define unit cells.
- Unit cell acts as the building block through repetition in crystal structure.
---
## CRYSTAL LATTICE
- Fundamental design:
- Unit cell as a repetitive volume.
- Lattice describes the arrangement of atoms within the crystal framework.
---
## SIMPLE CUBIC STRUCTURE (SC)
- Characteristics:
- Cubic unit cell with a = b = c and angles α = β = γ = 90°.
- Atom locations: Corners of the unit cell;
- Rare due to inefficient packing.
- Coordination number = 6 (nearest neighbors).
---
## ATOMIC PACKING FACTOR (APF) FOR SC
- Definition:
- APF = Volume of atoms in unit cell / Volume of unit cell.
---
## BODY CENTERED CUBIC (BCC)
- Coordination number = 8.
- Atom positions at corners and center of the unit cell.
- Close packed directions along cube diagonals.
---
## ATOMIC PACKING FACTOR: BCC
- Similar analysis as SC, with different arrangements.
---
## FACE CENTERED CUBIC (FCC)
- Coordination number = 12.
- Atom positions: Corners and centers of faces of the unit cell.
- Close packed directions along face diagonals.
---
## ATOMIC PACKING FACTOR: FCC
- Presented using density calculations and geometry of the atomic arrangement.
---
## HEXAGONAL CLOSE PACKED (HCP)
- Characteristics:
- Specific properties of HCP structures discussed including packing factors and coordination numbers.
---
## COMPARISON OF CRYSTAL STRUCTURES
| Structure | Unit Cell | Atoms per Unit Cell | Coordination Number | Atomic Packing Factor |
|-----------|-----------|---------------------|---------------------|-----------------------|
| BCC | Cubic | 2 | 8 | N/A |
| FCC | Cubic | 4 | 12 | 0.74 |
| HCP | Hexagonal | 6 | 12 | 0.74 |
---
## CANNON BALL STACKING EXAMPLE
- Analysis on stacking arrangements of spheres and proofs related to packing efficiency (Kepler conjecture).
---
## CRYSTAL STRUCTURES EXAMPLES
- Specific examples related to various metals and their crystalline structures:
- FCC: Cu, Al, Ag, Au;
- BCC: Fe, Cr, W;
- HCP: Mg, Ti, Zn.
---
## ADDITIONAL UNIT CELL GEOMETRIES
- Discussion on lattice geometries including:
- Tetragonal: Example - Tin, TiO2.
- Orthorhombic: Examples with practical implications.
- Monoclinic and Triclinic structures with specific material examples.
---
## POLYMORPHISM AND ALLOTROPY
- Explanation of materials existing in multiple crystal structures:
- Allotropic forms in iron parameterized by temperature and structural changes.
---
## THEORETICAL DENSITY EXPLANATION
- Worked example of density calculation for Copper FCC:
- Mass density = Atomic weight / Crystal structure volume calculations.
---
## VOLUME DENSITY and PLANAR DENSITY
- Describes relationships and calculations pertaining to material density based on atomic arrangements.
---
## CLOSE PACKING SUMMARY
- Overview of close packing strategies between FCC and HCP, including stacking sequences and importance in deformation mechanisms.
---
## PLANES AND DIRECTIONS IN CRYSTALLINE ARRANGEMENTS
- Discussions on how to denote crystallographic points, directions and planes utilizing Miller indices.
---
## MILLER INDICES
- Explanation and systematic approach to indexing specific planes (hkl) and directions (uvw) within crystal lattices.
---
## CRYSTALLOGRAPHIC PLANES & DIRECTIONS
- Detailed examples using cubic and hexagonal lattices to explain crystallographic coordination.
---
## SINGLE CRYSTAL VS. POLYCRYSTAL
- Contrasting structural attributes and mechanical behaviors of single crystal and polycrystalline materials.
---
## POLYCRYSTALS
- Characteristics and typical sizes observed in engineering materials.
---
## SINGLE CRYSTALS & EXAMPLES
- Specific examples highlighting unique properties and applications of materials like diamond and turbine blades in engineering.
---
## OTHER ORDERED ATOMIC STRUCTURES
- Overview of modern materials like graphene and carbon nanotubes, their structures, and properties.
---
## STUDY GUIDE
- Reference to Callister textbook for relevant sections and exercises for further study in preparation for assessments.
### Lecture Overview
- Course: Mechanics of Materials (4MA00)
- Institution: Eindhoven University of Technology (TU/e)
---
## LENGTH SCALES
- Hierarchical scale of sizes:
- 10-18 - 10-15: Electron, proton, neutron
- 10-12: Atom
- 10-9: Crystals
- Up to 100: Products and larger structures
---
## FROM ATOMS TO SOLID
- Exploration of how atoms assemble into solid structures, focusing on metals.
- **Density Dependence**: Material density relies on atomic arrangement and packing efficiency.
- **Directions and Planes**: Importance of indicating specific directions and planes in atomic structures.
---
## PACKING SPHERES
- Discussion of atomic packing and arrangement in solids.
---
## ATOMIC MODELS
- Historical Models:
- **Bohr model** and **Quantum mechanical model** previously discussed.
- Current model:
- Atoms are considered as spheres (Atomic hard sphere model).
- Nearest neighbors in contact, influencing bonding and structure.
---
## STRUCTURE OF MATERIALS
- **Molecular Materials**:
- Comprised of covalently bonded atoms.
- Interactions: Van der Waals forces.
- **Non-Molecular Materials**:
- **Metals**: Examples include Fe, Al, Au.
- **Ceramics**: Examples include Al2O3, NaCl.
- Other forms include water, gases, and polymers (CnHm).
---
## ATOMIC PACKING
- Short Range Order: Atoms with similar structures in close vicinity.
- Long Range Order:
- **Amorphous Materials**: Randomly structured, lacking long-range order but may have short-range order (e.g., glass).
- **Crystalline Materials**: Atoms organized in 3D periodic arrays.
- Common in metals, ceramics, and semi-crystalline polymers.
---
## ENERGY AND PACKING
- Packing influences energy levels:
- Dense, regular structures typically exhibit lower energy states.
- Factors include bond lengths and energies associated with atomic arrangements.
---
## EXAMPLE: SELF-ORGANIZATION IN CRYSTALLINE STRUCTURE
- Illustration of atomic arrangements and forces at play in crystal formation.
- Key equation of motion illustrated: mv = Σ F.
---
## METALLIC CRYSTALS
- Characteristics:
- Metals typically exhibit simple crystal structures and are densely packed.
- Reasons for this packing include uniform atomic radii and non-directional metallic bonding.
---
## CRYSTALS
- Defined by long-range order and translation symmetry.
- Examples of symmetry in patterns comparable to M.C. Escher's artworks.
---
## SYMMETRY IN CRYSTALS
- Types of symmetry: Square, Rectangular, Centered Rectangular, Hexagonal, Oblique.
---
## UNIT CELL GEOMETRY
- Key parameters:
- Edge lengths (a, b, c) and angles (α, β, γ) define unit cells.
- Unit cell acts as the building block through repetition in crystal structure.
---
## CRYSTAL LATTICE
- Fundamental design:
- Unit cell as a repetitive volume.
- Lattice describes the arrangement of atoms within the crystal framework.
---
## SIMPLE CUBIC STRUCTURE (SC)
- Characteristics:
- Cubic unit cell with a = b = c and angles α = β = γ = 90°.
- Atom locations: Corners of the unit cell;
- Rare due to inefficient packing.
- Coordination number = 6 (nearest neighbors).
---
## ATOMIC PACKING FACTOR (APF) FOR SC
- Definition:
- APF = Volume of atoms in unit cell / Volume of unit cell.
---
## BODY CENTERED CUBIC (BCC)
- Coordination number = 8.
- Atom positions at corners and center of the unit cell.
- Close packed directions along cube diagonals.
---
## ATOMIC PACKING FACTOR: BCC
- Similar analysis as SC, with different arrangements.
---
## FACE CENTERED CUBIC (FCC)
- Coordination number = 12.
- Atom positions: Corners and centers of faces of the unit cell.
- Close packed directions along face diagonals.
---
## ATOMIC PACKING FACTOR: FCC
- Presented using density calculations and geometry of the atomic arrangement.
---
## HEXAGONAL CLOSE PACKED (HCP)
- Characteristics:
- Specific properties of HCP structures discussed including packing factors and coordination numbers.
---
## COMPARISON OF CRYSTAL STRUCTURES
| Structure | Unit Cell | Atoms per Unit Cell | Coordination Number | Atomic Packing Factor |
|-----------|-----------|---------------------|---------------------|-----------------------|
| BCC | Cubic | 2 | 8 | N/A |
| FCC | Cubic | 4 | 12 | 0.74 |
| HCP | Hexagonal | 6 | 12 | 0.74 |
---
## CANNON BALL STACKING EXAMPLE
- Analysis on stacking arrangements of spheres and proofs related to packing efficiency (Kepler conjecture).
---
## CRYSTAL STRUCTURES EXAMPLES
- Specific examples related to various metals and their crystalline structures:
- FCC: Cu, Al, Ag, Au;
- BCC: Fe, Cr, W;
- HCP: Mg, Ti, Zn.
---
## ADDITIONAL UNIT CELL GEOMETRIES
- Discussion on lattice geometries including:
- Tetragonal: Example - Tin, TiO2.
- Orthorhombic: Examples with practical implications.
- Monoclinic and Triclinic structures with specific material examples.
---
## POLYMORPHISM AND ALLOTROPY
- Explanation of materials existing in multiple crystal structures:
- Allotropic forms in iron parameterized by temperature and structural changes.
---
## THEORETICAL DENSITY EXPLANATION
- Worked example of density calculation for Copper FCC:
- Mass density = Atomic weight / Crystal structure volume calculations.
---
## VOLUME DENSITY and PLANAR DENSITY
- Describes relationships and calculations pertaining to material density based on atomic arrangements.
---
## CLOSE PACKING SUMMARY
- Overview of close packing strategies between FCC and HCP, including stacking sequences and importance in deformation mechanisms.
---
## PLANES AND DIRECTIONS IN CRYSTALLINE ARRANGEMENTS
- Discussions on how to denote crystallographic points, directions and planes utilizing Miller indices.
---
## MILLER INDICES
- Explanation and systematic approach to indexing specific planes (hkl) and directions (uvw) within crystal lattices.
---
## CRYSTALLOGRAPHIC PLANES & DIRECTIONS
- Detailed examples using cubic and hexagonal lattices to explain crystallographic coordination.
---
## SINGLE CRYSTAL VS. POLYCRYSTAL
- Contrasting structural attributes and mechanical behaviors of single crystal and polycrystalline materials.
---
## POLYCRYSTALS
- Characteristics and typical sizes observed in engineering materials.
---
## SINGLE CRYSTALS & EXAMPLES
- Specific examples highlighting unique properties and applications of materials like diamond and turbine blades in engineering.
---
## OTHER ORDERED ATOMIC STRUCTURES
- Overview of modern materials like graphene and carbon nanotubes, their structures, and properties.
---
## STUDY GUIDE
- Reference to Callister textbook for relevant sections and exercises for further study in preparation for assessments.
0.0(0)
Explore Top NotesNoteStudied by 43 peopleNoteStudied by 14 peopleNoteStudied by 14 peopleNoteStudied by 17 peopleNoteStudied by 4 peopleNoteStudied by 230 people
Chapter 11: Sound
NoteStudied by 43 people5.0(1)
Princeton Review AP Calculus BC, Chapter 5: Differentiation: Composite, Inverse, and Implicit Functions
NoteStudied by 14 people5.0(2)
Chp 3 Ethnography: Studying Culture
NoteStudied by 14 people5.0(1)
03 tissues & integument
NoteStudied by 17 people4.5(2)
Chapter 10: Global Change
NoteStudied by 4 people5.0(1)
World War 1
NoteStudied by 230 people5.0(4)
