1.17 Which of the following lattices has the highest packing efficiency(i) simple cubic(ii) body-centred cubic and(iii) hexagonal close-packed lattice?

D Devendra Khairwa

(i) Simple cubic:-  In a simple cubic lattice the atoms are located only on the corners of the cube.

Thus, the edge length or side of the cube ‘a’, and the radius of each particle, r are related as a = 2r

Volume of cubic unit cell = $(2r)^3$ = $8r^3$

And  Volume of 1 atom :

$=\frac{4}{3}\Pi r^3$

$Packing\ efficiency = \frac{Volume\ of\ one\ atom}{Volume\ of\ cubic\ unit\ cell}\times100 \%$

$= \frac{\frac{4}{3}\Pi r^3}{8\Pi r^3}\times100 \% = \frac{\Pi }{6}\times100\%$

$= 52.4\%$

(ii) Body centred cubic:-  In body centred cubic, we have atoms at all corners and at body centre.

Clearly, the atom at the centre will be in touch with the other two atoms diagonally arranged.

$b = \sqrt{2}a$ ;                      and           $c = \sqrt{3}a$

Also, the length of body diagonal is equal to 4r.

$\sqrt{3}a = 4r$

$a = \frac{4r}{ \sqrt{3}}$

The volume of the cube              :

$= a^3 = \left ( \frac{4r}{ \sqrt{3}} \right )^3$

In BCC, a total number of atoms is 2.

$Packing\ efficiency = \frac{Volume\ of\ one\ atom}{Volume\ of\ cubic\ unit\ cell}\times100 \%$

$= \frac{2\times(\frac{4}{3}\Pi r^3)}{(\frac{4}{\sqrt{3}}r)^3}\times100 \%$

$= 68\%$

(iii) Hexagonal close-packed:-    We know that both types of (hcp and ccp) are equally efficient. We also know that the packing efficiency of ccp is 74 percent.

(i) Simple cubic = 52.4%

(ii) Body centred cubic=68%

(iii) Hexagonal close-packed=74%

Thus among all, packing efficiency of hcp is the highest.

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