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Packing Efficiency: bcc And Simple Cubic Lattice

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Packing Efficiency: bcc And Simple Cubic Lattice - Lesson Summary

In this arrangement, the particles are present at the corners and the body centre lattice points.

Each unit cell has 9 spheres with radius “r.” The edge length of a unit cell is
“a” = “2r.”

Among these, 8 spheres are present at the corners, and 1 sphere at the body centre of the unit cell.

Number atoms per unit cell:
Contribution of corner atoms =8*1/8
                                           =1
Contribution of body center atom =1*1=1
Contribution of sphere =1+1 = 2
Radius r =√3/4*a

Formula for packing efficiency = volume occupied by the spheres per unit cell / volume of a unit cell

Volume of Two Spheres  = 2 x 4/3 x π x (√3/4 . a) 3
Volume of Unit cell = a 3

Packing Efficiency = ( 2 x 4/3 x π x [√3/4 . a] 3 )/a 3
                            = ( 2 x 4/3 x π x [√3/4 ] 3. a 3 )/ a 3
                            = ( 2 x 4/3 x π x 3√3 )/ (4 x 4 x 4)
                            = 2 x 3.14 x 1.732 / 16
                            = 0.6805
                            = 68.05%

Packing efficiency in a BCC arrangement is equal to 68.05%

Simple cubic lattice arrangement: 
In this arrangement, the particles are present only at the corner lattice points of a unit cell.

                 

Each unit cell has 8 spheres with radius “r.” The edge length of a unit cell is
  “a” = “2r.”
All these 8 spheres are present at the corners of a unit cell.

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