Question

Arrange the following in decreasing order of packing fraction :
A. Face centered cubic structure
B. Body centered cubic structure
C. Simple cubic
D. Diamond cubic structure
Choose the correct answer from the options given below :

• A. D, C, B, A
• B. B, A, D, C
• C. A, B, C, D
• D. D, B, A, C

Hint:

Even though diamond is the hardest natural material, its atomic structure is actually the least densely packed among common lattices (only 34% full space). Hardness comes from the immense strength of its covalent $sp^3$ bonds, not packing density!

Answer 1

The Correct Option is C

Step 1: Concept:
The question asks to arrange four common crystal lattice structures in decreasing order (highest to lowest) of their Atomic Packing Fraction (APF), which represents the percentage of volume in a unit cell actually occupied by hard-sphere atoms.

Step 2: Key Formula or Approach:
The APF values for standard crystallographic structures must be memorized:
- Face-Centered Cubic (FCC): This is a closest-packed structure. APF = $\frac{\pi\sqrt{2}}{6} \approx 0.74$ (74%).
- Body-Centered Cubic (BCC): Not perfectly close-packed, but dense. APF = $\frac{\pi\sqrt{3}}{8} \approx 0.68$ (68%).
- Simple Cubic (SC): Highly inefficient packing, rare in nature (Polonium). APF = $\frac{\pi}{6} \approx 0.52$ (52%).
- Diamond Cubic: Consists of two interpenetrating FCC lattices (like Carbon). Because atoms are covalently bonded in a rigid tetrahedral geometry, it is very open and porous. APF = $\frac{\pi\sqrt{3}}{16} \approx 0.34$ (34%).

Step 3: Step-by-step Explanation:

• Assign the APF values to the given letters:
A. FCC = 0.74
B. BCC = 0.68
C. Simple Cubic = 0.52
D. Diamond Cubic = 0.34

• Sort the values in decreasing order (highest first):
$0.74 > 0.68 > 0.52 > 0.34$

• Map the sorted values back to the letters:
A $>$ B $>$ C $>$ D.

Step 4: Final Answer:
The correctly arranged sequence is A, B, C, D, which matches option (C).