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Packing Efficiency: hcp And ccp Lattice

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Packing Efficiency: hcp And ccp Lattice - Lesson Summary

Hexagonal close packing (hcp):
In this arrangement, the spheres are closely packed in successive layers in the ABABAB type of arrangement.

Each unit cell has 17 spheres with radius “r” and edge length of unit cell “2r.”

Number of atoms per unit cell:

Contribution of Corner Spheres to Unit Cell = 12 x 1/6 = 2
Contribution of Face Corner Spheres to Unit Cell = 2 x 1/2 = 1
Contribution of Spheres Inside the Unit Cell = 3 x 1 = 3
Contribution of All Types of Spheres to Unit cell = 6

Calculation of the volume of an HCP unit cell:

Volume  = Base Area x height

Base Area = 6 x √3/4 x ( 2r ) 2
Height of Unit Cell = 4r x √(2/3)
Total Volume = 6√3 r 2 x 4r √(2/3)
                    = 24 x √2 x r 3

Calculation of Packing fraction in HCP:

 Packing Fraction = Volume Occupied by Spheres / Volume of Unit Cell

          V (spheres) = 6 x 4/3 x π x r 3
                        = 8 π r 3
Packing Fraction = ( 8 π r 3 )/(24√2 r 3)
                         = π /3√2
                         = 0.74
                         = 74% 

Cubic close packing (ccp) :
Edge length of a unit cell be “a =2r,” and the radius of each sphere be “r”. In this arrangement, each unit cell has eight spheres at the eight corners, and six spheres at the six face centres.

Number of atoms per unit cell:

Contribution of Corner Spheres to Unit Cell = 8 x 1/8 = 1
Contribution of Face Corner Spheres to Unit Cell = 6 x 1/2 = 3

Total Contribution = 1 + 3 = 4               

Radius (r) = √2/4 .a

Formula for packing efficiency = volume occupied by the spheres per unit cell / volume of a unit cell
                  
Volume = 4 x 4/3 x π x r 3
               
= 4 x 4/3 x π x (√2/4 . a) 3

Packing Efficiency in CCP Arrangement = 0.7406 = 74.06%

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