Name: Hexagonal Close Packing in 2D and 3D
Uploaded: 2019-07-19
Duration: 5 min 29 s
Description: When the rows of spheres are stacked over each other, a 2D closed packed structure is produced. Let’s see how the stacking is done to produce a hexagonal clo...

Question

Calculate the packing efficiency in Face Centred Cubic (FCC) lattice.

Accepted Answer

The actual volume occupied by a face centred cubic (FCC) lattice is $74 %$ . This can be proved by deriving its packing efficiency. The following figure illustrating the face-centred cubic unit cell.

Suppose the edge length of the unit cell= a , and the radius of the sphere=r

It is clear from the figure that there are $8$ spheres at the corners and $6$ spheres at the faces.

No. of spheres in unit cell = $8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4$

Consider $∆ ABC$ . Since it is right-angled triangle,

${AC}^{2} = {AB}^{2} + {BC}^{2} = a^{2} + a^{2} = 2 a^{2}, AC = \sqrt{2 a}$

But, as evident from the figure, $AC = 4 r or \sqrt{2 a} = 4 r or a = \frac{4 r}{\sqrt{2}} = 2 \sqrt{2} r$

Volume of a unit cell= $a^{3} = {(2 \sqrt{2} r)}^{3} = 16 \sqrt{2} r^{3}$

Total volume of four spheres= $4 \times \frac{4}{3} {πr}^{3} = \frac{16}{3} {πr}^{3}$

Packing efficiency= $\frac{Volume of four spheres in the unit cell}{Total volume of unit cell} \times 100$

= $\frac{\frac{16}{3} {πr}^{3}}{16 \sqrt{2} r^{3}} \times 100 = \frac{π}{3 \times \sqrt{2}} \times 100 = \frac{3.142}{3 \times 1.414} \times 100 = 74 %$

Calculate the packing efficiency in Face Centred Cubic (FCC) lattice.

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