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, we need to determine the percentage void in a body-centered cubic (BCC) unit cell after removing half of the atoms. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the number of atoms in a BCC unit cell
In a BCC unit cell, the number of atoms (Z) is 2. This includes:
- 1 atom at the body center
- 8 corner atoms, each contributing 1/8 of an atom (8 × 1/8 = 1)

Thus, the total number of atoms in a BCC unit cell is:
\[ Z = 1 + 1 = 2 \]

**Hint:** Remember that in a BCC structure, the contribution of corner atoms must be calculated based on their fractional occupancy.

### Step 2: Calculate the remaining atoms after removal
If half of the atoms are removed from the BCC unit cell:
\[ \text{Remaining atoms} = \frac{Z}{2} = \frac{2}{2} = 1 \]

**Hint:** Always ensure to calculate the remaining quantity after a specified reduction.

### Step 3: Calculate the volume occupied by the remaining atom
The volume occupied by one atom (considering it as a sphere) is given by the formula:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]

So, the total volume occupied by the remaining atom is:
\[ \text{Volume occupied} = 1 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]

**Hint:** Use the formula for the volume of a sphere to find the space occupied by the atom.

### Step 4: Calculate the total volume of the unit cell
The volume of the BCC unit cell can be expressed in terms of the edge length (a). The relationship between the edge length and the radius (r) of the atom in a BCC structure is:
\[ a\sqrt{3} = 4r \]
Thus, the edge length \( a \) can be calculated as:
\[ a = \frac{4r}{\sqrt{3}} \]

The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

**Hint:** Remember to cube the edge length to find the volume of the unit cell.

### Step 5: Calculate the packing efficiency
The packing efficiency (PE) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell:
\[ \text{Packing Efficiency} = \frac{\text{Volume occupied by atoms}}{\text{Total volume of unit cell}} \]

Substituting the values:
\[ \text{PE} = \frac{\frac{4}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]

This simplifies to:
\[ \text{PE} = \frac{4\pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{16} \]

### Step 6: Convert packing efficiency to percentage
To express packing efficiency as a percentage:
\[ \text{Packing Efficiency (in percentage)} = \left(\frac{\pi \sqrt{3}}{16}\right) \times 100 \]

Calculating this gives approximately:
\[ \text{Packing Efficiency} \approx 33.9\% \]

### Step 7: Calculate the percentage void
The percentage void is calculated as:
\[ \text{Percentage void} = 100\% - \text{Packing Efficiency} \]
\[ \text{Percentage void} = 100\% - 33.9\% \approx 66.1\% \]

Thus, the final answer is:
**Percentage void = 66.1%**