Question

If 'a' stands for the edge length of the cubic systems - The ratio of radii in simple cubic, body centered cubic and face centered cubic unit cells is

• A. \(1a:\sqrt3a:\sqrt2a\)
• B. \(\frac{1}{2}a:\frac{\sqrt3}{4}a:\frac{1}{2\sqrt2}a\)
• C. \(\frac{1}{2}a:\frac{\sqrt3}{2}a:\frac{\sqrt2}{2}a\)
• D. \(\frac{1}{2}a:\sqrt3a:\frac{1}{\sqrt2}a\)

Answer 1

The Correct Option is B

1. Simple cubic (SC): 

Atoms touch along the edges.

\[ 2r = a \]

Therefore, radius for SC:

\[ r = \frac{a}{2} \]

1. Body-centered cubic (BCC):

Atoms touch along the body diagonal of the cube.

\[ 4r = \sqrt{3}a \]

Therefore, radius for BCC:

\[ r = \frac{\sqrt{3}a}{4} \]

1. Face-centered cubic (FCC):

Atoms touch along the face diagonal.

\[ 4r = \sqrt{2}a \]

Therefore, radius for FCC:

\[ r = \frac{\sqrt{2}a}{4} = \frac{a}{2\sqrt{2}} \]

Combining these results:

Simple cubic : Body-centered cubic : Face-centered cubic

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

Conclusion:

The correct ratio is:

\( \frac{1}{2}a : \frac{\sqrt{3}}{4}a : \frac{1}{2\sqrt{2}}a \)

Correct Answer: Option (B)