Question

Answer the following in brief.

Calculate the packing efficiency of metal crystal that has simple cubic structure.

Answer 1

Packing efficiency of metal crystal in the simple cubic lattice:

• Step 1: Radius of sphere:
  In the simple cubic unit cell, particles (spheres) are at the corners and touch each other along the edge.
  A face of a simple cubic unit cell is shown in the figure.
  From the figure, we can find that
  a = 2r or r = `"a"/2`       ......(1)
  where, ‘r’ is the radius of atom and ‘a’ is the length of the unit cell edge.
• Step 2: Volume of sphere:
  Volume of a sphere = `4/3π"r"^3`.
  Substitution for r from equation (1) gives:
  Volume of one particle = `4/3π ("a"/2)^3` = `(pi"a"^3)/6`         ....(2)
• Step 3: Total volume of particles:
  Because a simple cubic unit cell contains only one particle, the volume occupied by the particle in unit cell =`(pi"a"^3)/6`
• Step 4: Packing efficiency:
  Packing efficiency = `"Volume occupied by particles in unit cell"/"Total volume of a unit cell" xx 100`
  `= (pi"a"^3//6)/"a"^3 xx 100 = (100 pi)/6 = (100 xx 3.142)/6 = 52.36 %`

Thus, in a simple cubic lattice, 52.36 % of total space is occupied by particles and 47.64% is empty space, that is, void volume.