The packing fraction in a face - centred cubic cell of crystals is

Options

A. `(sqrt3)/(8)a`
B. `(pi)/(6)`
C. `(sqrt2)/(6)pi`
D. `(1)/(2sqrt2)pi`

Answer

To find the packing fraction in a face-centered cubic (FCC) cell, we will follow these steps:

### Step 1: Determine the edge length of the FCC unit cell.
In a face-centered cubic structure, the relationship between the edge length \( A \) and the radius \( r \) of the atoms is given by:
\[ A = 2\sqrt{2}r \]

### Step 2: Calculate the volume of a single atom.
The volume \( V \) of a single atom (considered as a sphere) is calculated using the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 3: Determine the number of atoms per unit cell.
In a face-centered cubic unit cell, there are 4 atoms per unit cell. This is because there are 8 corner atoms (each shared by 8 unit cells) contributing \( \frac{1}{8} \) of an atom each, and 6 face-centered atoms (each shared by 2 unit cells) contributing \( \frac{1}{2} \) of an atom each:
\[ Z = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 4: Calculate the total volume occupied by the atoms in the unit cell.
The total volume occupied by the atoms in the unit cell is:
\[ \text{Total volume of atoms} = Z \times V = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

### Step 5: Calculate the volume of the unit cell.
The volume of the unit cell is given by:
\[ \text{Volume of unit cell} = A^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \]

### Step 6: Calculate the packing fraction.
The packing fraction (or packing efficiency) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell:
\[ \text{Packing fraction} = \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} = \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} \]
Simplifying this gives:
\[ \text{Packing fraction} = \frac{16\pi}{48\sqrt{2}} = \frac{\pi}{3\sqrt{2}} \approx 0.74 \]

### Final Answer:
Thus, the packing fraction in a face-centered cubic cell is:
\[ \text{Packing fraction} = \frac{\pi}{3\sqrt{2}} \approx 0.74 \]