Spring 2025 Intro to Material Sci & Engr (ENGR-2210-001, ENGR-2210-002, ENGR-2210-003, ENGR-2210-004)

Introduction

  • Review of crystal structures: Simple Cubic and Body-Centered Cubic (BCC)

  • Importance of unit cell in crystallography

Simple Cubic Structure

Definition

  • Consists of atoms at each corner of the cube.

Key Assumptions

  • Pure metals with identical atoms and spherical shape.

  • Atoms are in contact with nearest neighbors.

Coordination Number

  • Coordination number is 6, indicating the number of nearest neighbors.

Equivalent Number of Atoms

  • Each corner atom is shared by 8 unit cells:

    • 1 atom contribution per unit cell: 8 corners x 1/8.

Radius and Lattice Parameter

  • Radius (r) relates to unit cell length (a): Formula: r = a/2.

Atomic Packing Fraction

    • Volume of sphere: V = (4/3)πr³

    • Total volume = a³, where a is the unit cell edge.

    • Atomic packing fraction = (Volume occupied by atoms / Total volume).

    • APF for simple cubic = 0.52 (52% packing efficiency, 48% void space).

Body-Centered Cubic (BCC)

Structure

  • Additional atom at the center of the cube, 8 corner atoms unchanged.

Coordination Number

  • Coordination number is 8, accounting for corners and center atom interactions.

Equivalent Number of Atoms

  • Calculation: 2 atoms per unit cell (8 corner atoms x 1/8 + 1 center atom).

Radius and Lattice Parameter

  • Atoms do not touch on edge; diagonal plane method used for calculation:

    • Based on geometry, derived as: r = (√3a)/4.

Atomic Packing Fraction

  • APF calculated as: Pi√3/8

    • Gives an APF of approximately 0.68 (68% packing efficiency).

Face-Centered Cubic (FCC)

Structure

  • Atoms are located at each corner and at the center of each face.

Coordination Number

  • Coordination number is 12. Atom touches 4 in the same plane, 4 in the layer above, and 4 in the layer below.

Equivalent Number of Atoms

  • 4 atoms per unit cell (8 corner atoms x 1/8 + 6 face-centered atoms x 1/2).

Radius and Lattice Parameter

  • Radius relation: r = (√2a)/4, where all corner atoms touch face-centered atoms on any given face.

Atomic Packing Fraction

  • Calculation involves:

    • Number of atoms = 4,

    • Volume of sphere = (4/3)πr³,

    • Total volume = a³,

    • Resulting APF = (Pi√2)/6 = 0.74 (approximately 74% packing efficiency).

Hexagonal Close-Packed (HCP)

Structure

  • Two-dimensional hexagonal base with alternating layers (ABAB structure).

Coordination Number

  • Same as FCC: Coordination number is 12.

Equivalent Number of Atoms

  • Total number of atoms = 6 per unit cell (based on corner and layered contributions).

Radius and Lattice Parameter

  • Radius to edge (a) relation: r = a/2.

Theoretical Density Calculation

  • Density formulas explored:

    • 𝜌 = nA/VcNa,

    • n = number of atoms per unit cell,

    • A = atomic weight,

    • Vc = volume of unit cell,

    • Na = Avogadro's number.

Numerical Example

Example Problem

  • Calculate theoretical density of Copper (FCC) given:

    • Atomic weight = 63.54 g/mol,

    • Atomic radius = 0.1278 nm,

    • Steps involve finding lattice parameter from radius and substituting values into density formula.

Conclusion

  • Importance of atomic arrangement in determining material properties.

  • Encourage students to understand calculations and physical implications for exams.