Question

The atomic packing factor (APF) of FCC structure is approximately

• A. 0.52
• B. 0.68
• C. 0.74
• D. 0.9

Hint:

Both FCC (Face-Centered Cubic) and HCP (Hexagonal Close-Packed) structures are close-packed structures with the same maximum packing density of 0.74 and coordination number of 12.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks for the Atomic Packing Factor (APF) of a Face-Centered Cubic (FCC) crystal structure.
The APF represents the efficiency of atomic packing within a crystal lattice.

Step 2: Key Formula or Approach:
The Atomic Packing Factor is calculated as the ratio of the total volume of solid spheres within a unit cell to the total volume of the unit cell itself:
\[ \text{APF} = \frac{N_{\text{eff}} \times V_{\text{atom}}}{V_{\text{cell}}} \]
where:
\( N_{\text{eff}} \) is the effective number of atoms in the 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:

• FCC Unit Cell Properties:
-

Effective number of atoms (\( N_{\text{eff}} \)):
\[ N_{\text{eff}} = \left(8 \text{ corners} \times \frac{1}{8}\right) + \left(6 \text{ faces} \times \frac{1}{2}\right) = 1 + 3 = 4 \text{ atoms} \]
-

Lattice parameter \( a \) in terms of atomic radius \( R \):
In FCC, atoms touch along the face diagonal of the cube:
\[ \text{Face Diagonal} = a\sqrt{2} = 4R \implies a = \frac{4R}{\sqrt{2}} = 2\sqrt{2}R \]

• APF Derivation:
Substitute these values into the APF equation:
\[ \text{APF} = \frac{4 \times \left(\frac{4}{3}\pi R^3\right)}{(2\sqrt{2}R)^3} \]
\[ \text{APF} = \frac{\frac{16}{3}\pi R^3}{16\sqrt{2}R^3} = \frac{\pi}{3\sqrt{2}} \approx 0.7405 \]
- This indicates that approximately 74% of the FCC unit cell volume is occupied by solid atomic spheres, which represents the maximum possible packing density for equal-sized hard spheres.


Step 4: Final Answer:
The atomic packing factor of an FCC structure is approximately 0.74.
Therefore, the correct choice is option (C).