Question

In which of the following pair is most efficient packing is present?

• A. hcp and ccp
• B. hcp and bcc
• C. bcc and ccp
• D. bcc and simple cubic cell

Answer 1

The correct option is A

hcp and ccp

The explanation for the correct answer

Correct Answer: Option (A) HCP and CCP

Packing fraction

• Packing efficiency or packing fraction is the fraction of volume occupied by a constituent particle in a crystal structure.
• It is a dimensionless quantity and always less than unity.

HCP or hexagonal closed packing and Cubic closed packing

HCP has 6 atoms per unit cell,

Lattice constant a = 2r

Packing fraction= $\frac{{\mathrm{N}}_{\mathrm{Atoms}}{\mathrm{V}}_{\mathrm{Atoms}}}{{\mathrm{V}}_{\mathrm{Unit}\mathrm{cell}}}$

The volume of the unit cell for HCP is a hexagonal prism with an atom on each vertex, and 3 atoms in the centre.

$$\begin{gathered} {\mathrm{V}}_{\mathrm{Unit}\mathrm{cell}}=3\sqrt{3}/2\times \left(2{\mathrm{r}}^2\right)\times (\sqrt{2/3}\times 4\mathrm{r}) \\ \frac{6\times 4/3{\mathrm{\pi r}}^3}{3\sqrt{3}/2\times \left(2{\mathrm{r}}^2\right)\times (\sqrt{2/3}\times 4\mathrm{r})} \\ =0.74 \\ (\mathrm{or}\mathrm{c}/\mathrm{a}\mathrm{ratio}=1.633), \end{gathered}$$

Atomic Packing Factor APF = 74%.

Simple Cubic Packing

Hence, option a is correct.

The explanation for incorrect answers

(B) BCC has a packing fraction of 68%. Hence, option (B) is incorrect.

(C) Simple Cubic has a packing fraction of 52%. Hence, option (C) is incorrect.

(D) Option(D) is incorrect as the packing efficiency of BCC and SCP is less than HCP.