The packaging fration for a body centred cubic structure is

Options

A. 0.42
B. 0.53
C. 0.68
D. 0.82

Answer

To find the packing fraction for a body-centered cubic (BCC) structure, we can follow these steps:

### Step 1: Understand the Structure
A body-centered cubic (BCC) structure consists of atoms located at each of the eight corners of a cube and one atom at the center of the cube.

### Step 2: Determine the Number of Atoms (Z)
In a BCC unit cell, there are 2 atoms per unit cell. This is because each corner atom is shared among eight unit cells (1/8 of an atom from each corner) and the center atom is fully contained within the unit cell. Therefore:
\[ Z = 2 \]

### Step 3: Calculate the Volume of Atoms in the Unit Cell
The volume occupied by the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
For BCC, the effective radius of the atom can be derived from the relationship between the radius (r) and the edge length (a) of the cube. In a BCC structure:
\[ a = \frac{4r}{\sqrt{3}} \]
Thus, the volume of the two atoms in the unit cell is:
\[ V_{\text{atoms}} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 4: Calculate the Volume of the Unit Cell
The volume of the unit cell (V_cell) is given by:
\[ V_{\text{cell}} = a^3 \]
Substituting for a:
\[ V_{\text{cell}} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 5: Calculate the Packing Fraction
The packing fraction (PF) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell:
\[ PF = \frac{V_{\text{atoms}}}{V_{\text{cell}}} \]
Substituting the volumes we calculated:
\[ PF = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]
This simplifies to:
\[ PF = \frac{8 \pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \]

### Step 6: Calculate the Numerical Value
Using the approximate value of \(\pi \approx 3.14\):
\[ PF \approx \frac{3.14 \times 1.732}{8} \approx \frac{5.441}{8} \approx 0.68 \]

### Conclusion
Thus, the packing fraction for a body-centered cubic structure is approximately **0.68**.