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 be respectively

• A. \( \frac{1}{2} \sqrt{3} \, a \)
• B. \( \frac{\sqrt{3}}{4} \, a \)
• C. \( \frac{\sqrt{2}}{4} \, a \)
• D. \( \frac{a}{\sqrt{2}} \)

Hint:

The ratio of the radius of the sphere to the edge length in different cubic structures depends on the arrangement of atoms in the unit cell.

Answer 1

The Correct Option is C

Step 1: Understand the relationship between the edge length and radius.
For simple cubic, body-centered cubic, and face-centered cubic, the ratio of radius to edge length differs based on the arrangement of the atoms in the unit cell.
Step 2: Conclusion.
For these arrangements, the radius to edge length ratio is \( \frac{\sqrt{2}}{4} \) for the face-centered cubic structure.
Final Answer: \[ \boxed{\frac{\sqrt{2}}{4} \, a} \]