What is the packing efficiency in fcc unit cell?

The correct answer is option 2. 74%.

Face centered cubic unit cell (FCC)

Let the unit cell edge length be ‘a’ and face diagonal \(AC = b\).

In \(\Delta ABC\)

\(AC^2 = b^2 = BC^2 + AB^2 = a^2 + a^2 = 2a^2\)

or \(b = \sqrt{2}a\)

If \(r\) is the radius of the sphere, we find

\(b = 4r = \sqrt{2}a\)

or \(a = \frac{4r}{\sqrt{2}}\) 

\(a = 2\sqrt{2}r\)

we can also write,

\(r =\frac{a}{2\sqrt{2}}\)

We know that each unit cell in ccp structure i.e., FCC unit cell has effectively \(4\) spheres.

Total volume of four spheres is equal to \(4 \times \frac{4}{3}\pi r^3\) and volume of the cube is \(a^3\) or \((2\sqrt{2}r)^3\).

Therefore,

Packing efficiency = \(\frac{\text{Volume occupied by four spheres in the unit cell}}{\text{Total volume of the unit cell}} \times 100%\)

Packing efficiency = \(\frac{4 × \frac{4}{3}\pi r^3}{(2\sqrt{2}r)^3} x 100% = 74%\)