Question

If a stands for the edge length of the cubic systems: simple cubic, body-centered cubic and face-centered cubic, then the ratio of radii of the spheres in these systems will respectively be :

• A. $\left(\frac{1}{2}\right)a:\left(\frac{\sqrt{3}}{2}\right)a:\left(\frac{\sqrt{2}}{2}\right)a$
• B. $1a:\sqrt{3}a:\sqrt{2}a$
• C. $\left(\frac{1}{2}\right)a:\left(\frac{\sqrt{3}}{4}\right)a:\left(\frac{1}{2\sqrt{2}}\right)a$
• D. $\left(\frac{1}{2}\right)a:\sqrt{3}a:\left(\frac{1}{\sqrt{2}}\right)a$

Answer 1

The correct option is C $\left(\frac{1}{2}\right)a:\left(\frac{\sqrt{3}}{4}\right)a:\left(\frac{1}{2\sqrt{2}}\right)a$

For simple cubic,
$r=\frac{a}{2}$
For body centered: $r=\frac{a\sqrt{3}}{4}$
For face centered: $r=\frac{a}{2\sqrt{2}}$
Where a = edge length, r = radius.
$\therefore$
Ratio of radii of the three will be:
$\left(\frac{1}{2}\right)a:\left(\frac{\sqrt{3}}{4}\right)a:\left(\frac{1}{2\sqrt{2}}\right)a$