Unit – IV B

Density and Packing Efficiency of Unit Cell

  • In whatever way the constituent particles atoms, molecules or ions are packed, there is always some free space in the form of voids.
  • The fraction of the total space in unit cell occupied by the constituent particles is called packing fraction.
    Mathematically
  • Packing efficiency is the percentage of total space filled by the constituent particles in the unit cell.

Packing efficiency = Packing Factor x 100

  • A vacant space not occupied by the constituent particles in the unit cell is called void space.

Fraction of void space = 1 – Packing Fraction
% Void space = 100 – Packing efficiency.



Packing Efficiency of Simple Cubic Crystal Lattice (SCC):

  • In a simple cubic lattice the atoms are located only on the corners of the cube.
  • The particles touch each other along the edge as shown. Thus, the edge length or side of the cube ‘a’, and the radius of each particle, r are related as    a = 2r. Thus radius of an atom is half the side of the simple cubic unit cell.
  • The volume of the cubic unit cell  = a3 = (2r)3 = 8r3
  • Since a simple cubic unit cell contains only 1 atom The volume of the occupied space

  • Now, packing efficiency

  • The packing efficiency of simple cubic cell is 52.4 %. Thus 47.6 % volume is empty space(void space). i.e. almost half the space is empty. Hence the simple cubic crystalline solid is loosely bonded.


Packing Efficiency of Body Centred Cubic Crystal Lattice (BCC):

  • In a body centred cubic lattice the eight atoms are located on the eight corners of the cube and one at the centre of the cube.
  • Let the edge length or side of the cube ‘a’, and the radius of each particle be r. The particles along body diagonal touch each other.

∴  AD = r  + 2r + r = 4r   ……. (1)

  • Thus radius of an atom is  3/4 times the side of the body centred cubic unit cell.
  • The volume of the cubic unit cell

  • Since a body centred cubic unit cell contains 2 atoms The volume of the occupied space
  • Now, packing efficiency

  • The packing efficiency of body centred cubic cell is 68 %. Thus 32 % volume is empty space(void space).

Density and Packing Efficiency of Unit Cell

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