FCC vs SC Packing Fraction

This section analyzes the Assertion (A) and Reason (R) regarding the packing efficiency and atom count in Face-Centered Cubic (FCC) and Simple Cubic (SC) crystal structures.

Assertion (A): Packing Fraction

Assertion (A) states: FCC lattice has a higher packing fraction than simple cubic lattice.

• Calculation for Simple Cubic (SC):
  • Atoms per unit cell = 1.
  • Relationship between atomic radius ($r$) and lattice constant ($a$): $a = 2r$, so $r = a/2$.
  • Packing Fraction (PF_SC) = $ \frac{\text{Volume of atoms}}{\text{Volume of unit cell}} = \frac{1 \times \frac{4}{3}\pi r^3}{a^3} = \frac{\frac{4}{3}\pi (\frac{a}{2})^3}{a^3} = \frac{\pi}{6} \approx 0.5236 $
• Calculation for Face-Centered Cubic (FCC):
  • Atoms per unit cell = 4.
  • Relationship between atomic radius ($r$) and lattice constant ($a$): The face diagonal is $4r = a\sqrt{2}$, so $r = \frac{a\sqrt{2}}{4}$.
  • Packing Fraction (PF_FCC) = $ \frac{4 \times \frac{4}{3}\pi r^3}{a^3} = \frac{\frac{16}{3}\pi (\frac{a\sqrt{2}}{4})^3}{a^3} = \frac{16\pi a^3 2\sqrt{2}}{3 \times 64 a^3} = \frac{\pi\sqrt{2}}{3} \approx 0.7405 $

Comparing the packing fractions, PF_FCC ($ \approx 0.7405 $) is significantly greater than PF_SC ($ \approx 0.5236 $). Thus, Assertion (A) is correct.

Reason (R): Atoms Per Unit Cell

Reason (R) states: FCC contains less atoms per unit cell than simple cubic.

• A Simple Cubic (SC) unit cell has atoms only at the 8 corners, contributing $8 \times \frac{1}{8} = 1$ atom per unit cell.
• A Face-Centered Cubic (FCC) unit cell has atoms at the 8 corners and the centers of the 6 faces, contributing $(8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4$ atoms per unit cell.

Since the FCC unit cell contains 4 atoms and the SC unit cell contains 1 atom, FCC contains more atoms per unit cell, not less. Therefore, Reason (R) is incorrect.

Select Correct Option

Based on the analysis:

• Assertion (A) is correct.
• Reason (R) is incorrect.

This matches the condition where (A) is correct but (R) is not correct.