Question

The atomic packing factor (APF) for BCC structure is approximately

• A. 0.52
• B. 0.68
• C. 0.74
• D. 0.60

Hint:

Standard Atomic Packing Factors (APF):
- Simple Cubic (SC): \( \approx 0.52 \)
- Body-Centered Cubic (BCC): \( \approx 0.68 \)
- Face-Centered Cubic (FCC): \( \approx 0.74 \) (Maximum packing density for spheres)
- Hexagonal Close-Packed (HCP): \( \approx 0.74 \)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the Atomic Packing Factor (APF) of a Body-Centered Cubic (BCC) crystal structure.
The APF is the fraction of the volume of a unit cell that is occupied by solid hard spheres representing the atoms.

Step 2: Key Formula or Approach:
The Atomic Packing Factor is calculated using:
\[ \text{APF} = \frac{N_{\text{eff}} \times V_{\text{atom}}}{V_{\text{cell}}} \]
where:
\( N_{\text{eff}} \) is the effective number of atoms per unit cell.
\( V_{\text{atom}} \) is the volume of a single spherical atom (\( \frac{4}{3}\pi R^3 \)).
\( V_{\text{cell}} \) is the volume of the cubic unit cell (\( a^3 \)).

Step 3: Detailed Explanation:

• BCC Unit Cell Characteristics:
-

Effective number of atoms (\( N_{\text{eff}} \)):
\[ N_{\text{eff}} = \left(8 \text{ corner atoms} \times \frac{1}{8}\right) + \left(1 \text{ center atom} \times 1\right) = 2 \text{ atoms} \]
-

Relationship between lattice parameter \( a \) and atomic radius \( R \):
Atoms touch along the body diagonal of the cube:
\[ \text{Body Diagonal} = a\sqrt{3} = 4R \implies a = \frac{4R}{\sqrt{3}} \]

• APF Calculation:
Substitute the terms into the APF equation:
\[ \text{APF} = \frac{2 \times \left(\frac{4}{3}\pi R^3\right)}{\left(\frac{4R}{\sqrt{3}}\right)^3} \]
\[ \text{APF} = \frac{\frac{8}{3}\pi R^3}{\frac{64 R^3}{3\sqrt{3}}} = \frac{8\pi}{3} \times \frac{3\sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \approx 0.6802 \]
- This means that 68% of the BCC unit cell volume is occupied by atoms, while the remaining 32% is empty pore space.


Step 4: Final Answer:
The atomic packing factor for a BCC structure is approximately 0.68.
Therefore, the correct choice is option (B).