Packing efficiency of different types of unit cells

Packing Efficiency of Different Types of Unit Cells

Packing efficiency is a measure of how efficiently the constituent particles (atoms, ions, or molecules) are packed in a crystal lattice. It is defined as the percentage of the total space occupied by the particles in a unit cell.

Formula:

Packing Efficiency (%) = (Volume occupied by particles in the unit cell / Total volume of unit cell) × 100

1. Simple Cubic Unit Cell (SC)

• Particles are present only at the 8 corners of the cube.
• Each corner particle is shared by 8 adjacent unit cells.
• Number of atoms per unit cell: 1
• Atomic radius (r): a = 2r
• Packing efficiency = (Volume occupied by one atom ÷ Total volume of the unit cell) × 100
  = (Volume of 1 atom / Volume of cube) × 100
  = [(4/3)πr³ × 1] / (2r)³ × 100
  = (4.19r³ / 8r³) × 100
  ≈ 52.4%

2. Body-Centered Cubic Unit Cell (BCC)

• Atoms are located at all 8 corners and 1 atom is positioned at the center of the cube.
• Number of atoms per unit cell: 2
• Atomic radius (r): √3a = 4r
• Packing efficiency:
  = [(2 × (4/3)πr³) / a³] × 100
  = [(8.38r³) / ( (4r/√3)³ ) ] × 100
  ≈ 68%

3. Face-Centered Cubic Unit Cell (FCC) or Cubic Close Packed (CCP)

• Atoms occupy all 8 corners of the cube and the centers of each of its 6 faces.
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• Number of atoms per unit cell: 4
• Atomic radius (r): √2a = 4r
• Packing efficiency:
  = [(4 × (4/3)πr³) / a³] × 100
  = [(16.76r³) / ( (4r/√2)³ ) ] × 100
  ≈ 74% (Highest for cubic systems)

Summary Table:

The packing efficiency is maximum for the FCC unit cell (74%), making it the most densely packed structure among the cubic systems, followed by BCC and SC.