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.