Question

What type of crystal structure from following has 52.36% packing efficiency?

• A. FCC
• B. BCC
• C. Hexagonal cubic
• D. Simple cubic

Hint:

Remember the three milestone numbers for cubic cell packing efficiencies: Simple Cubic = 52.4

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to identify the specific crystal lattice arrangement that exhibits a packing efficiency of exactly 52.36

Step 2: Key Formula or Approach:
Packing efficiency is defined as the percentage of total space filled by the constituent particles inside a unit cell: $$ \text{Packing Efficiency} = \frac{\text{Volume occupied by atoms in unit cell}}{\text{Total volume of unit cell}} \times 100 $$

Step 3: Detailed Explanation:
Let's review the standard packing efficiencies for common cubic lattice systems:

• Face-Centered Cubic (FCC) / Cubic Close Packed (CCP): Contains 4 effective atoms per unit cell. Its packing efficiency is the highest among cubic systems at approximately 74

• Body-Centered Cubic (BCC): Contains 2 effective atoms per unit cell. Its packing efficiency evaluates to approximately 68

• Simple Cubic (SC): Contains only 1 effective atom per unit cell. The edge length $a$ relates to atomic radius $r$ via $a = 2r$. $$ \text{Packing Efficiency} = \frac{1 \times \frac{4}{3}\pi r^3}{(2r)^3} \times 100 = \frac{\pi}{6} \times 100 \approx 52.36\% $$


Step 4: Final Answer:
A packing efficiency of 52.36% corresponds uniquely to the simple cubic structure, matching option (D).