Question

Calculate the percentage of the occupied space in a face-centred cubic unit cell.

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

Answer 1

A face-centered cubic unit cell contains four atoms. If r is the radius of an atom.

The volume occupied by 4 atoms = `4 xx 4/3 pi r^3`

= `16/3 pi r^3`    ...(i)

In a face-centred cubic unit cell, the corner atoms touch the face-centred atom (Fig.). Suppose the unit cell edge length is a. Image shows the face diagonal.

AC = 4r    ...(ii)

The right-angled triangle ABC

AC = `sqrt((AB)^2 + (BC)^2)`

= `sqrt(a^2 + a^2)`

= `sqrt2 a`    ...(iii)

From eqs. (ii) and (iii), we have

`sqrt 2 a = 4 r`

`a = 4/sqrt2 r`    ...(iv)

The volume of the unit cell = a³

= `(4/sqrt2 r)^3`

= `(64 r^3)/(2 sqrt 2)`

Packing fraction = `"Volume occupied by atoms"/"Volume of unit cell"`

= `(16/3 pi r^3)/((64 r^3)/(2 sqrt 2))`

= `(sqrt 2 pi)/6`

= 0.74

Hence, the percentage of the occupied space = 74%