If ' a ' stands for the edge length of the cubic systems : simple...

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,

Options

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

Correct answer

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

Step-by-step solution

For simple cubic,

\begin{aligned} \therefore & a & =2 r \\ \therefore & r & =\frac{a}{2} \end{aligned}

For body centred cubic,

\begin{aligned} & a=\frac{4 r}{\sqrt{3}} \\ & r=\frac{\sqrt{3} a}{4} \end{aligned}

For face centred cubic,

\begin{aligned} & a=2 \sqrt{2} r \\ & r=\frac{a}{2 \sqrt{2}} \end{aligned}

Hence, the ratio of radii in simple cubic, body centred cubic and face centred cubic is

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

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