In a BCC unit cell, if half of the atoms per unit cell are removed, then percentage void is

Options

A. `68%`
B. `32%`
C. `34%`
D. `66%`

Answer

To solve the problem of finding the percentage void in a BCC unit cell after removing half of the atoms, we can follow these steps:

### Step 1: Determine the number of atoms in a BCC unit cell
In a Body-Centered Cubic (BCC) unit cell, there are:
- 1 atom at the center
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.

Thus, the total number of atoms per unit cell is:
\[ \text{Total atoms} = 1 + 8 \times \frac{1}{8} = 1 + 1 = 2 \text{ atoms} \]

### Step 2: Calculate the number of atoms remaining after removal
According to the problem, half of the atoms are removed from the BCC unit cell:
\[ \text{Remaining atoms} = \frac{2}{2} = 1 \text{ atom} \]

### Step 3: Calculate the volume of the unit cell
Let \( A \) be the edge length of the unit cell. The volume of the unit cell is given by:
\[ \text{Volume of unit cell} = A^3 \]

### Step 4: Calculate the volume of one atom
The radius \( r \) of the atom in a BCC lattice can be expressed in terms of the edge length \( A \):
\[ \sqrt{3}A = 4r \implies r = \frac{\sqrt{3}A}{4} \]
The volume of one atom is given by the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{\sqrt{3}A}{4}\right)^3 \]
Calculating this gives:
\[ \text{Volume of one atom} = \frac{4}{3} \pi \left(\frac{3\sqrt{3}A^3}{64}\right) = \frac{\sqrt{3}\pi A^3}{16} \]

### Step 5: Calculate the packing efficiency
The packing efficiency is defined as the ratio of the total volume of the atoms to the volume of the unit cell:
\[ \text{Packing efficiency} = \frac{\text{Number of atoms} \times \text{Volume of one atom}}{\text{Volume of unit cell}} \times 100 \]
Substituting the values:
\[ \text{Packing efficiency} = \frac{1 \times \frac{\sqrt{3}\pi A^3}{16}}{A^3} \times 100 = \frac{\sqrt{3}\pi}{16} \times 100 \]
Calculating this gives approximately:
\[ \text{Packing efficiency} \approx 33.99\% \approx 34\% \]

### Step 6: Calculate the percentage void
The percentage void is calculated as:
\[ \text{Percentage void} = 100\% - \text{Packing efficiency} \]
Substituting the packing efficiency:
\[ \text{Percentage void} = 100\% - 34\% = 66\% \]

### Final Answer
The percentage void in the BCC unit cell after removing half of the atoms is **66%**.

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