In the calculation of packing efficiency of hcp arrangement,the height of the unit cell is taken as 4r√2/3. Why?

In the hexagonal closest-packed structure, a = b = 2r and c = 4 √2/3 r , where r is the atomic radius of the atom. The sides of the unit cell are perpendicular to the base, thus α = β = 90o

For a closest-packed structure, the atoms at the corners of base of the unit cell are in contact, thus a = b = 2 r. The height © of the unit cell, which is more challenging to calculate, is c = 2a √2/3 r = 4 √2/3 r.

Let the edge of hexagonal base =a

And the height of hexagon = h

And radius of sphere = r

The centre sphere of the first layer lies exactly over the void of 2^{nd} layer B.

The centre sphere and the spheres of 2^{nd} layer B are in touch

So, In Δ PQR (an equilateral triangle)

PR = 2r, Draw QS tangent at points

∴ In Δ QRS ∠QRS = 30°, SR = r

Cos30° = SR/QR

QR = r / √3/2 = 2r / √3

∴ PQ = √PR^{2} – QR^{2} = √4r^{2} – 4r^{2} / 3

h^{1} = √8r^{2} / 3 = 2 √2/3 r

∴ h = 2h^{1} = **4 √2/3 r**

Hence, in the calculation of packing efficiency of hcp arrangement,the height of the unit cell is taken as 4r√2/3.