Question:
The fraction of the total volume occupied by atoms in a Simple Cubic (SC) structure is:
The fraction of the total volume occupied by atoms in a Simple Cubic (SC) structure is:
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Standard packing efficiencies to remember:
• SC = \( \frac{\pi}{6} \approx 0.52 \)
• BCC = \( \frac{\pi\sqrt{3}}{8} \approx 0.68 \)
• FCC = \( \frac{\pi}{3\sqrt{2}} \approx 0.74 \)
• SC = \( \frac{\pi}{6} \approx 0.52 \)
• BCC = \( \frac{\pi\sqrt{3}}{8} \approx 0.68 \)
• FCC = \( \frac{\pi}{3\sqrt{2}} \approx 0.74 \)
Updated On: Jun 10, 2026
- \( \frac{\pi}{6} \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{3\sqrt{2}} \)
- \( \frac{\pi}{3\sqrt{3}} \)
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The Correct Option is A
Solution and Explanation
Concept:
The packing fraction (or packing efficiency) is defined as the fraction of volume occupied by atoms in a crystal lattice:
\[
\text{Packing fraction} = \frac{\text{Volume occupied by atoms in unit cell}}{\text{Volume of unit cell}}
\]
For spherical atoms:
• Volume of one atom = \( \frac{4}{3}\pi r^3 \)
• Volume of unit cell = \( a^3 \)
• If \( z \) atoms are effectively present, then total atomic volume = \( z \cdot \frac{4}{3}\pi r^3 \)
Step 1: Number of atoms in Simple Cubic (SC) In SC structure:
• 8 corner atoms are present
• Each corner atom contributes \( \frac{1}{8} \) to the unit cell
So: \[ z = 8 \times \frac{1}{8} = 1 \]
Step 2: Relation between edge length and radius In SC structure, atoms touch along edges: \[ a = 2r \Rightarrow r = \frac{a}{2} \]
Step 3: Packing fraction calculation \[ f = \frac{z \cdot \frac{4}{3}\pi r^3}{a^3} \] Substitute values: \[ f = \frac{1 \cdot \frac{4}{3}\pi \left(\frac{a}{2}\right)^3}{a^3} \] \[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \] So: \[ f = \frac{\frac{4}{3}\pi \cdot \frac{a^3}{8}}{a^3} \] Cancel \( a^3 \): \[ f = \frac{4\pi}{24} \] \[ f = \frac{\pi}{6} \] Thus, packing fraction of Simple Cubic structure is: \[ \boxed{\frac{\pi}{6}} \]
• Volume of one atom = \( \frac{4}{3}\pi r^3 \)
• Volume of unit cell = \( a^3 \)
• If \( z \) atoms are effectively present, then total atomic volume = \( z \cdot \frac{4}{3}\pi r^3 \)
Step 1: Number of atoms in Simple Cubic (SC) In SC structure:
• 8 corner atoms are present
• Each corner atom contributes \( \frac{1}{8} \) to the unit cell
So: \[ z = 8 \times \frac{1}{8} = 1 \]
Step 2: Relation between edge length and radius In SC structure, atoms touch along edges: \[ a = 2r \Rightarrow r = \frac{a}{2} \]
Step 3: Packing fraction calculation \[ f = \frac{z \cdot \frac{4}{3}\pi r^3}{a^3} \] Substitute values: \[ f = \frac{1 \cdot \frac{4}{3}\pi \left(\frac{a}{2}\right)^3}{a^3} \] \[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \] So: \[ f = \frac{\frac{4}{3}\pi \cdot \frac{a^3}{8}}{a^3} \] Cancel \( a^3 \): \[ f = \frac{4\pi}{24} \] \[ f = \frac{\pi}{6} \] Thus, packing fraction of Simple Cubic structure is: \[ \boxed{\frac{\pi}{6}} \]
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