Question

Calculate the percentage efficiency of packing in case of body centered cubic (BCC) structure.

Hint:

Remember the efficiencies for common structures to verify your work: - Simple Cubic: 52.4% - BCC: 68% - FCC/HCP: 74%

Answer 1

Concept: Packing efficiency is the percentage of total space filled by the particles (atoms) in a crystal lattice. Packing Efficiency = Volume occupied by atoms in unit cellTotal volume of unit cell × 100
In a BCC unit cell, the atoms touch along the body diagonal. Let the edge length of the cube be a and the radius of the atom be r. The length of the body diagonal is √(3)a. According to the geometry of the spheres along the diagonal: √(3)a = 4r a = 4r√(3)
• Number of atoms (z): In BCC, z = 8 × (1)/(8) (corners) + 1 (body center) = 2. • Volume of 2 atoms: 2 × (4)/(3)π r³ = (8)/(3)π r³. • Volume of unit cell (a³): ( 4r√(3) )³ = 64r³3√(3).
Efficiency = (8)/(3)π r³64r³3√(3) × 100 = π √(3)8 × 100 Efficiency = (3.142 × 1.732)/(8) × 100 ≈ 68%