Question

Find the total volume of a unit cell given the radius of the atom.

• A. \( 8r^3 \)
• B. \( \frac{64r^3}{3\sqrt{3}} \)
• C. \( \frac{32r^3}{\sqrt{2}} \)
• D. \( \frac{16r^3}{\sqrt{3}} \)

Hint:

The edge length \( a \) is related to the atomic radius \( r \) and depends on the type of cubic unit cell. Use \( a = 2r \) for simple cubic, \( a = \frac{4r}{\sqrt{3}} \) for BCC, and \( a = \frac{4r}{\sqrt{2}} \) for FCC.

Answer 1

The Correct Option is B

Unit cell volume is type-dependent.

The volume of a cubic unit cell is given by: \[ \text{Volume} = a^3, \] where \( a \) denotes the edge length. 
For a simple cubic structure, \( a = 2r \). 
For a body-centered cubic (BCC) structure, \( a = \frac{4r}{\sqrt{3}} \).

For a face-centered cubic (FCC) structure, \( a = \frac{4r}{\sqrt{2}} \). Applying the BCC formula: \[ \text{Volume} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}. \]