Question

Calculate the efficiency of packing in case of a metal crystal for body-centred cubic

Answer 1

Body-centred cubic:

It can be observed from the above figure that the atom at the centre is in contact with the other two atoms diagonally arranged.

From ΔFED, we have:

b² = a² + a²

⇒ b² =  2a²

⇒ b = `sqrt2a`

 Again, from ΔAFD, we have:

c² = a² + b²

⇒ c² = a² + 2a²     (Since b² = 2a²)

⇒ c² = 3a²

⇒ `c = sqrt3a`

Let the radius of the atom be r.

Length of the body diagonal, c = 4π

⇒`sqrt3a = 4r`

⇒`a =(4r)/sqrt3`

or `r = (sqrt3a)/4`

Volume of the cube `a^3 = ((4r)/sqrt3)^3`

A body-centred cubic lattice contains 2 atoms.

 So, volume of the occupied cubic lattice `2pi4/3 r^3`

=`8/3pir^3`

:.Packing efficiency = `("Voulume occupied by two spheres in the unit cell")/"Total volume of unit"xx100%`

= `(8/3pir^3)/(4/(sqrt3)r)^3xx100%`

=`(8/3pir^3)/(64/(3sqrt3)r^3) xx 100%`

=68%