Let the radius of atom X be 'r'.
1. Number of atoms X in the unit cell:
- Atom X forms an fcc lattice. In an fcc lattice, the number of atoms per unit cell (Zfcc) is 4. (Calculated as 8 corners × 1/8 atom/corner + 6 faces × 1/2 atom/face = 1 + 3 = 4 atoms).
- In an fcc lattice, there are 8 tetrahedral voids (TVs) per unit cell (2 × Zfcc = 2 × 4 = 8).
- Atom X occupies alternate tetrahedral voids. This means half of the TVs are occupied by atom X.
Number of atoms X in TVs = 8 / 2 = 4 atoms. - Therefore, the total number of atoms X per unit cell (Ztotal) = Atoms from fcc sites + Atoms from TVs = 4 + 4 = 8 atoms.
2. Relationship between edge length (a) and atomic radius (r):
- Since the same atom X (with radius r) occupies both the fcc lattice sites and the tetrahedral voids, the lattice must adjust to accommodate these atoms.
- A tetrahedral void is located at a distance of
(√3 a) / 4 from a corner atom (where 'a' is the edge length of the unit cell). - If an atom X (radius r) is at the corner and another atom X (radius r) is in the tetrahedral void, and they touch each other, then the distance between their centers is
r + r = 2r. - So, we have the relation:
2r = (√3 a) / 4. - From this, the edge length of the unit cell 'a' can be expressed in terms of 'r':
a = (8r) / √3.
3. Packing Efficiency (PE):
Packing Efficiency is defined as the ratio of the volume occupied by atoms in the unit cell to the total volume of the unit cell.
PE = (Volume occupied by atoms in the unit cell) / (Volume of the unit cell)
- Volume occupied by atoms = Ztotal × Volume of one atom
= 8 × (4/3)πr³ = (32/3)πr³ - Volume of the unit cell = a³
= ((8r) / √3)³ = (8³r³) / (√3)³ = (512r³) / (3√3)
Now, substituting these into the PE formula:
PE = [ (32/3)πr³ ] / [ (512r³) / (3√3) ]
PE = (32πr³ / 3) × (3√3 / 512r³)
The r³ terms and the 3s cancel out:
PE = (32π√3) / 512
PE = (π√3) / 16
4. Numerical Calculation:
Using the approximate values: π ≈ 3.14159 and √3 ≈ 1.732:
PE ≈ (3.14159 × 1.732) / 16
PE ≈ 5.44130468 / 16
PE ≈ 0.34708
To express this as a percentage:
PE % ≈ 0.34708 × 100% = 34.708%
The Correct Option is B (35)