The vacant space in B.C.C. unit cell is

Options

A. 0.32
B. 0.1
C. 0.23
D. 0.46

Answer

To determine the vacant space in a Body-Centered Cubic (BCC) unit cell, we can follow these steps:

### Step 1: Understand the Concept of Packing Efficiency
Packing efficiency is defined as the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell. It can be expressed mathematically as:

\[ \text{Packing Efficiency} = \frac{\text{Volume occupied by atoms}}{\text{Total volume of the unit cell}} \]

### Step 2: Identify the Parameters
For a BCC unit cell:
- The number of atoms per unit cell (Z) = 2 (1 atom at the center and 1/8 of an atom at each of the 8 corners)
- The volume of a single atom (assuming it is spherical) is given by:

\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]

### Step 3: Relate the Radius to the Edge Length
In a BCC structure, the relationship between the radius (r) of the atom and the edge length (a) of the unit cell is given by:

\[ r = \frac{\sqrt{3}}{4} a \]

### Step 4: Calculate the Volume Occupied by Atoms
Using the above relationship, the total volume occupied by atoms in the BCC unit cell can be calculated as:

\[ \text{Total volume occupied} = Z \times \text{Volume of one atom} = 2 \times \frac{4}{3} \pi r^3 \]

Substituting the value of r:

\[ = 2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3 \]

### Step 5: Calculate the Total Volume of the Unit Cell
The total volume of the unit cell is simply the cube of the edge length:

\[ \text{Total volume of the unit cell} = a^3 \]

### Step 6: Calculate the Packing Efficiency
Now, substituting the volume occupied by atoms and the total volume of the unit cell into the packing efficiency formula:

\[ \text{Packing Efficiency} = \frac{2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3}{a^3} \]

After simplifying, we find:

\[ \text{Packing Efficiency} = \frac{\sqrt{3} \pi}{8} \approx 0.68 \]

### Step 7: Calculate the Void Efficiency
The void efficiency is the fraction of the volume that is not occupied by the atoms, which can be calculated as:

\[ \text{Void Efficiency} = 1 - \text{Packing Efficiency} \]

Substituting the packing efficiency:

\[ \text{Void Efficiency} = 1 - 0.68 = 0.32 \]

### Conclusion
Thus, the vacant space in the BCC unit cell is 0.32.