Chapter 3 Crystal structures (1).pdf

Chapter 3: The Structure of Crystalline Solidsudy Theme Outcomes

  • Define unit cell shape and size in terms of axial lengths and inter-axial angles.

  • Determine coordination number and unit cells for BCC, FCC, BCT, and HCP.

  • Calculate lattice constants (a, c) of BCC, FCC, and HCP unit cells based on atomic sizes and vice versa.

  • Calculate atomic packing factor (APF) for BCC, FCC, and HCP crystals.

  • Distinguish between close-packed and non-close-packed directions and planes.

  • Sketch planes within unit cells, showing atomic arrangements to scale.

  • Explain the stacking sequence difference between FCC and HCP types.

  • Determine full density of materials from atomic mass, atomic radius, and crystal type.

  • Calculate planar densities of given planes and linear densities along directions for BCC, FCC, and HCP types.

3.1 The Space Lattice and Unit Cells

  • Crystal Structure: Regular three-dimensional arrangement of atoms.

  • Space Lattice: Intersection points in a 3D network representing atom or ion centers; each point has identical surroundings.

  • Unit Cell: Small, fully representative portion of the space lattice.

3.2 Principal Metallic Crystal Structures

  • Common Structures:

    • FCC (Face Centered Cubic)

    • BCC (Body Centered Cubic)

    • HCP (Hexagonal Close Packed)

    • BCT (Body Centered Tetragonal)

  • Metals exhibit these structures due to uniform atom sizes and electropositive nature.

Example Problem: Unit Cells in Iron

  • Given the unit cell size of Fe is 0.287 nm:

    • Convert 1 mm to nanometers:

      • 1 mm = 1 x 10^3 nm

    • Calculate number of unit cells fitting in 1 mm:

      • No. of unit cells = (1 x 10^3 nm) / (0.287 nm) = 3.48 x 10^3 unit cells.

Face Centered Cubic (FCC)

  • Coordination Number (CN): Number of neighboring atoms.

    • For FCC, CN = 12.

  • Atoms Per Unit Cell: 4 (1/2 of 8 corner atoms + 6 face-centered atoms).

  • Relationship: Can be determined via Pythagorean theorem:

    • a² = (4R)² where R is atom radius.

  • Atomic Packing Factor (APF):

    • APF = (Volume of atoms per unit cell) / (Volume of unit cell).

    • FCC APF = 0.74, with 4 atoms per unit cell.

    • Example metals: Al, Cu, Ni, Au.

Body Centered Cubic (BCC)

  • Contains one atom at the center of the cube.

  • Coordination Number: For BCC, CN = 8.

  • Atoms Per Unit Cell: 2 (1 central atom + 8 corner atoms each contributing 1/8).

  • Relationship: a = (4R)/√3.

  • Example calculation for Fe (BCC): R = 0.124 nm, calculate a and APF:

    • a = 0.286 nm, APF = 0.69.

Hexagonal Close Packed (HCP)

  • Structure consists of hexagonal close-packed arrangement with 6 equilateral triangles.

  • Coordination Number: CN = 12 for HCP.

  • Atoms Per Unit Cell: 6.

  • APF for HCP: 0.74.

    • Example metals: Cd, Zn, Mg.

3.3 Polymorphism or Allotropy

  • Transformation of materials from one crystal type to another due to temperature or pressure changes.

  • Examples:

    • Iron: transforms among different crystal types (α-Fe: BCC, β-Fe: FCC).

    • Carbon: transitions from graphite to diamond under high pressure.

3.4 Theoretical Density

  • Theoretical density can be calculated using:

    • Density (p) = (Mass of atoms/unit cell) / (Volume unit cell).

  • Example: Calculate theoretical density for BCC Fe.

3.5 Planar and Linear Densities

  • Planar Density (PD): Number of atoms in a plane per unit area.

  • Linear Density (LD): Number of atoms along a line per unit length.

  • Formulas:

    • PD = (Number of atoms on a plane) / (Area of plane).

    • LD = (Number of atomic diameters on line) / (Length of line).

3.6 Closed-packed Crystal Structures

  • FCC and HCP structures have an APF of 0.74 and define packing sequences:

    • FCC: ABCABC...

    • HCP: ABAB...

  • Differences:

    • FCC has multiple sets of closed-packed planes while HCP has one set.

    • FCC metals are often more ductile; HCP metals can be brittle.

  • BCC does not have a closed-packed structure.