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 :
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 (PFSC) = $ \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 (PFFCC) = $ \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, PFFCC ($ \approx 0.7405 $) is significantly greater than PFSC ($ \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.
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