Question

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,

• 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. $1a : \sqrt{3}a: \sqrt{2}a$

Answer 1

Answer: (A)

$\frac{1}{2} a: \frac{\sqrt{3}}{4} a:\frac{1}{2\sqrt{2}} a$

Answer 2

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:$a=2r \,or \, r=\frac{a}{2}$
For body centred cubic :$a=2 \sqrt{2}r \,or\, r=\frac{1}{2\sqrt{2}} a$
Thus the ratio of radii of spheres for these will be
simple: bcc: fcc
$=\frac{a}{2} :\frac{\sqrt{3}}{4} a: \frac{1}{2\sqrt{2}} a i.e.$
option (a) is correct answer.