Question

Calculate the efficiency of packing in case of a metal crystal for face-centred cubic (with the assumptions that atoms are touching each other).

Answer 1

Face-centred cubic

Let the edge length of the unit cell be ‘a’ and the length of the face diagonal AC be b.

From ΔABC, we have:

AC² = BC² + AB²

⇒ b² = a² + a²

⇒ b² = 2a²

⇒ `b =  sqrt2a`

Let r be the radius of the atom.

Now, from the figure, it can be observed that:

b = 4r

⇒ `sqrt2a=4r`

⇒a = 2sqrt2`

Now volume of the cube , `a^3 = (2sqrt2r)^3`

We know that the number of atoms per unit cell is 4.

So, volume of the occupied unit cell  = `4pi 4/3 r^3`

∴ Packing efficiency = `("Volume occupied by four sphere in the units")/"Total volume of the unit cells"xx100%`

= `(4pi4/3 r^3)/((2sqrt2r)^3)xx100%`

= `(16/3pir^3)/(16sqrt2r^3) xx 100%`

= 74%