Question

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : FCC lattice has higher packing fraction than simple cubic lattice.
Reason (R) : FCC contains less atoms per unit cell than simple cubic.
In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Memorize the "Big Three" unit cell properties:
Simple Cubic: $Z=1$, APF = 0.52.
Body-Centered Cubic (BCC): $Z=2$, APF = 0.68.
Face-Centered Cubic (FCC): $Z=4$, APF = 0.74.

Answer 1

The Correct Option is C

Step 1: Concept:
This problem compares two fundamental crystallographic properties—Atomic Packing Factor (APF) and the number of atoms per unit cell ($Z$)—for Simple Cubic (SC) and Face-Centered Cubic (FCC) lattice systems.

Step 2: Key Formula or Approach:
- Packing Fraction (APF) is the volume of atoms divided by the volume of the unit cell.
- SC Packing Fraction $\approx 0.52$ (52%)
- FCC Packing Fraction $\approx 0.74$ (74%)
- Atoms per unit cell ($Z$):
- SC: 8 corners $\times (1/8) = 1$ atom.
- FCC: 8 corners $\times (1/8) + 6$ faces $\times (1/2) = 1 + 3 = 4$ atoms.

Step 3: Step-by-step Explanation:

• Evaluating Assertion (A): The packing fraction of an FCC lattice (0.74) is mathematically higher than that of a Simple Cubic lattice (0.52). FCC is a closest-packed structure. Therefore, the assertion is undeniably correct.

• Evaluating Reason (R): The reason claims that FCC contains less atoms per unit cell than simple cubic. As calculated above, FCC contains $Z=4$ atoms per cell, while SC contains only $Z=1$. Thus, FCC contains more atoms per unit cell, making the Reason statement explicitly incorrect.

Step 4: Final Answer:
Assertion (A) is correct, but Reason (R) is factually false. This matches option (C).