Question

If the length of the body diagonal of a FCC unit cell is \( x \, \text{Å} \), the distance between two octahedral voids in the cell in \( \text{Å} \) is:

• A. \( \frac{x}{\sqrt{2}} \)
• B. \( \frac{x}{\sqrt{3}} \)
• C. \( \frac{x}{\sqrt{6}} \)
• D. \( \frac{x}{\sqrt{8}} \)

Hint:

In FCC crystals, the relationship between the body diagonal and the edge length is \( \text{Body Diagonal} = \sqrt{3} \cdot \text{Edge Length} \). This can be used to compute the distance between voids.

Answer 1

The Correct Option is C

In a face-centered cubic (FCC) unit cell, octahedral voids are located at the body center and edge centers. To calculate the distance between two octahedral voids: Step 1: The length of the body diagonal is: \[ x = \sqrt{3}a, \] where \( a \) is the edge length of the unit cell. Step 2: From the body diagonal length, the edge length \( a \) is: \[ a = \frac{x}{\sqrt{3}}. \] Step 3: The distance between two octahedral voids is along the body diagonal. Since they are located at the body center and edge centers, the distance is: \[ \text{Distance} = \frac{a}{\sqrt{2}}. \] Substitute \( a = \frac{x}{\sqrt{3}} \): \[ \text{Distance} = \frac{\frac{x}{\sqrt{3}}}{\sqrt{2}} = \frac{x}{\sqrt{3} \cdot \sqrt{2}} = \frac{x}{\sqrt{6}}. \] Final Answer: \[ \boxed{\frac{x}{\sqrt{6}}} \]