Packing Efficiency of Solid with Atom X in FCC and Alternate Tetrahedral Voids

To find the packing efficiency of the given solid, we determine the number of atoms per unit cell, the relationship between the atomic radius and the unit cell edge length, and then calculate the fraction of the unit cell volume occupied by the atoms.

1. Number of Atoms per Unit Cell ($Z$)

• Atom $\text{X}$ occupies the FCC lattice sites (corners and face-centers): $\text{Contribution from corners} = 8 \times \frac{1}{8} = 1$ $\text{Contribution from face-centers} = 6 \times \frac{1}{2} = 3$ $\text{Total atoms at FCC sites} = 1 + 3 = 4$

• Atom $\text{X}$ also occupies alternate tetrahedral voids: $\text{Total tetrahedral voids in an FCC lattice} = 2 \times 4 = 8$ $\text{Number of alternate tetrahedral voids occupied} = \frac{8}{2} = 4$

Thus, the total number of atoms of $\text{X}$ per unit cell is: $Z = 4 + 4 = 8$

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2. Relationship Between Atomic Radius ($r$) and Edge Length ($a$)

The structure formed by atom $\text{X}$ is identical to the diamond cubic structure. The tetrahedral voids lie along the body diagonals at a distance of $\frac{\sqrt{3}a}{4}$ from the corners.

Since the atoms at the corners and the occupied tetrahedral voids touch each other along the body diagonal, the sum of their radii is equal to the distance between their centers: $2r = \frac{\sqrt{3}a}{4}$ $r = \frac{\sqrt{3}a}{8}$

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3. Calculation of Packing Efficiency

The packing efficiency ($\text{PE}$) is given by: $\text{Packing Efficiency} = \frac{\text{Volume occupied by } Z \text{ atoms}}{\text{Volume of the unit cell}} \times 100$

• Volume of 8 spherical atoms of radius $r$: $\text{Volume of atoms} = 8 \times \left(\frac{4}{3}\pi r^3\right) = \frac{32}{3}\pi \left(\frac{\sqrt{3}a}{8}\right)^3$

  $\text{Volume of atoms} = \frac{32}{3}\pi \times \frac{3\sqrt{3}a^3}{512} = \frac{\sqrt{3}\pi a^3}{16}$

• Total volume of the cubic unit cell = $a^3$

Substituting these into the packing efficiency formula: $\text{PE (\%)} = \frac{\frac{\sqrt{3}\pi a^3}{16}}{a^3} \times 100 = \frac{\sqrt{3}\pi}{16} \times 100$

Using the approximations $\sqrt{3} \approx 1.732$ and $\pi \approx 3.1416$: $\text{PE (\%)} \approx \frac{1.732 \times 3.1416}{16} \times 100 \approx \frac{5.4414}{16} \times 100 \approx 34.01\%$

The value closest to $34.01\%$ among the given options is $35\%$.

Correct Option: (B)