Packing efficiency of fcc and bcc lattice

Packing Efficiency of FCC and BCC Lattice

Packing efficiency refers to the percentage of total space occupied by particles (atoms, ions, or molecules) in a unit cell of a crystal lattice. It is an important concept in solid-state chemistry that helps understand the density and arrangement of particles in different types of lattices.

1. Packing Efficiency in FCC (Face-Centered Cubic) Lattice

In an FCC structure, atoms are located at each corner and the centers of all the faces of the cube.

• Total atoms per unit cell: 4
• Relation between edge length (a) and radius (r): √2a = 4r ⇒ a = 2√2r

Formula for packing efficiency:

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

= [(4 × (4/3)πr³) / a³] × 100
= [(4 × (4/3)πr³) / (2√2r)³] × 100
= 74%

Conclusion: FCC structures have a high packing efficiency of 74%, making them densely packed. Examples: Copper, Aluminum, Silver.

2. Space Utilization in a Body-Centered Cubic (BCC) Arrangement

In a BCC structure, atoms are located at each corner and a single atom at the center of the cube.

• Total atoms per unit cell: 2
• Relation between edge length (a) and radius (r): √3a = 4r ⇒ a = 4r/√3

Formula for packing efficiency:

Packing Efficiency = [(2 × (4/3)πr³) / a³] × 100
= [(2 × (4/3)πr³) / (4r/√3)³] × 100
= 68%

Conclusion: BCC structures are less efficient than FCC, with a packing efficiency of 68%. Examples: Iron, Chromium, Tungsten.

Comparison Table

Conclusion

The packing efficiency indicates how tightly atoms are packed in a crystal lattice. FCC has a higher efficiency than BCC, which contributes to differences in density, stability, and physical properties of the materials.