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