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?

(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 =  =

And  Volume of 1 atom :

(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.

;                      and

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

The volume of the cube              :

In BCC, a total number of atoms is 2.

(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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