Packing Efficiency and Void Calculation in Crystal Structures

Number of Tetrahedral and Octahedral Voids

• Number of tetrahedral voids is 2N, where N is the number of particles.
• Number of octahedral voids is N.

Packing Efficiency

• Packing efficiency is the fraction or percentage of the total space occupied by the spheres (particles).
• It is a measure of how tightly particles are packed together.
• Formula: Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100

Packing Efficiency in Simple Cubic Lattice

• In simple cubic unit cell, particles are at the corners and touch each other along the edge.
• Radius of sphere (r) = a/2, where a is the length of unit cell edge.
• Volume of sphere = (4/3π)(r³)
• Volume occupied by particles in unit cell = (4/3π)(r³) = πa³/6
• Total volume of unit cell = a³
• Packing efficiency = (πa³/6 / a³) × 100 = 52.4%

Packing Efficiency in Face-Centred Cubic Lattice (FCC or CCP or HCP Lattice)

• Radius of particle (r) = a/√2, where a is the length of unit cell edge.
• Volume of sphere = (4/3π)(r³) = πa³/8
• Volume occupied by particles in unit cell = 4 × πa³/8 = πa³/2
• Total volume of unit cell = a³
• Packing efficiency = (πa³/2 / a³) × 100 = 74%

Comparison of Packing Efficiency in Different Lattices

• Face-centred cubic lattice (FCC or CCP or HCP) has the most efficient packing of particles, with a packing efficiency of 74%.
• Simple cubic lattice has the least efficient packing of particles, with a packing efficiency of 52.4%.

Coordination Number and Packing Efficiency in Various Cubic Systems

• Coordination number is the number of neighbouring spheres that surround a sphere in a lattice.
• Table 1.4 shows the coordination number and packing efficiency in simple cubic, body-centred cubic, and face-centred cubic lattices.