The Solid State - Result Question 23
####23. If ’ $a$ ’ stands for the edge length of the cubic systems : simple cubic, body centred cubic and face centred cubic, then the ratio of radii of the spheres in these systems will be respectively,
[2008]
(a) $\frac{1}{2} a: \frac{\sqrt{3}}{4} a: \frac{1}{2 \sqrt{2}} a$
(b) $\frac{1}{2} a: \sqrt{3} a: \frac{1}{\sqrt{2}} a$
(c) $\frac{1}{2} a: \frac{\sqrt{3}}{2} a: \frac{\sqrt{3}}{2} a$
(d) $a: \sqrt{3} a: \sqrt{2} a$
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Solution:
- (a) Following generalization can be easily derived for various types of lattice arrangements in cubic cells between the edge length $(a)$ of the cell and $r$ the radius of the sphere.
For simple cubic : $a=2 r$ or $r=\frac{a}{2}$
For body centred cubic :
$a=\frac{4}{\sqrt{3}} r$ or $r=\frac{\sqrt{3}}{4} a$
For face centred cubic :
$a=2 \sqrt{2} r$ or $r=\frac{1}{2 \sqrt{2}} a$
Thus the ratio of radii of spheres will be simple : bcc : fcc
$=\frac{a}{2}: \frac{\sqrt{3}}{4} a: \frac{1}{2 \sqrt{2}} a$
