The Atomic Packing Factor (APF) is a key concept in crystal structures. It measures how efficiently atoms are packed in a crystal lattice.

APF helps explain metal density, ductility, and mechanical properties, which are critical for selecting metals in engineering applications.

Definition

Atomic Packing Factor (APF)

APF is the fraction of the volume of a unit cell that is actually occupied by atoms.

$$\text{APF} = \frac{\text{Volume of atoms in the unit cell}}{\text{Volume of the unit cell}}$$

• Value ranges from 0 to 1 (or 0% to 100%).

• Higher APF → more closely packed atoms → higher density and strength.

Core Concept Explanation

1. Each atom is considered a hard sphere with radius $r$.

2. Total volume of atoms in unit cell = number of atoms × volume of one atom

$$V_{\text{atoms}} = n \times \frac{4}{3}\pi r^3$$

3. Volume of the unit cell depends on lattice type:

   • Cubic: $V_{\text{cell}} = a^3$

   • Hexagonal: $V_{\text{cell}} = \frac{3\sqrt{3}}{2} a^2 c$

4. APF = $V_{\text{atoms}} / V_{\text{cell}}$

APF for Common Crystal Structures

FCC and HCP are closest-packed structures, with maximum APF of 74%.

Importance in Engineering

• Determines metal density and weight of components.

• Helps predict ductility and mechanical properties:

  • Low APF (SC, BCC) → lower density, harder, less ductile

  • High APF (FCC, HCP) → higher density, more ductile

• Useful in alloy design and material selection.

Exam-Focused Points

• APF = fraction of unit cell volume occupied by atoms.

• Formula:

$$\text{APF} = \frac{n \times \frac{4}{3}\pi r^3}{V_{\text{unit cell}}}$$

• FCC and HCP → APF = 74% → closest-packed

• BCC → APF = 68% → moderately packed

• SC → APF = 52% → loosely packed

Common Exam Traps

• Confusing APF of BCC (68%) with FCC/HCP (74%)

• Using wrong lattice parameter formula for radius

• Forgetting the number of atoms per unit cell

• Mixing up cubic and hexagonal unit cell formulas

Example Competitive Exam Questions

What is Atomic Packing Factor (APF)?
Answer — Fraction of volume of unit cell occupied by atoms.

APF of FCC and HCP metals?
Answer — 74%

APF of BCC metal?
Answer — 68%

APF of Simple Cubic?
Answer — 52%

Formula to calculate APF?
Answer — APF = (Volume of atoms in unit cell) / (Volume of unit cell)

Quick Revision

• APF = fraction of unit cell volume occupied by atoms

• SC → 52%, BCC → 68%, FCC/HCP → 74%

• FCC/HCP = closest-packed

• Number of atoms per unit cell: SC=1, BCC=2, FCC=4, HCP=6

• APF explains density, ductility, and mechanical properties