If the total number of atoms per unit cell in an hcp structure and a bcc structure gets halved, then ratio of percentage voids in hcp and bcc structures is

Options

A. `34/27`
B. `21/22`
C. `22/21`
D. `37/34`

Answer

To solve the problem of finding the ratio of percentage voids in hexagonal close-packed (hcp) and body-centered cubic (bcc) structures when the total number of atoms per unit cell is halved, we will follow these steps:

### Step 1: Understand the Structures
- **Hexagonal Close-Packed (hcp)**: In an hcp structure, there are 6 atoms per unit cell.
- **Body-Centered Cubic (bcc)**: In a bcc structure, there are 2 atoms per unit cell.

### Step 2: Halve the Number of Atoms
- If we halve the number of atoms in each structure:
- For hcp: \( \frac{6}{2} = 3 \) atoms per unit cell.
- For bcc: \( \frac{2}{2} = 1 \) atom per unit cell.

### Step 3: Calculate the Number of Voids
- **Tetrahedral Voids**: In a close-packed structure, the number of tetrahedral voids is given by \( 2n \), where \( n \) is the number of atoms.
- For hcp with 3 atoms: \( 2 \times 3 = 6 \) tetrahedral voids.

- **Octahedral Voids**: The number of octahedral voids is equal to the number of atoms, \( n \).
- For hcp with 3 atoms: \( 3 \) octahedral voids.

- **Total Voids in hcp**: \( 6 + 3 = 9 \) voids.

- **For bcc**: The number of tetrahedral voids is \( 2n \) and the number of octahedral voids is \( n - 1 \).
- For bcc with 1 atom:
- Tetrahedral voids: \( 2 \times 1 = 2 \).
- Octahedral voids: \( 1 - 1 = 0 \).

- **Total Voids in bcc**: \( 2 + 0 = 2 \) voids.

### Step 4: Calculate the Percentage of Voids
- **Percentage Voids in hcp**:
\[ \text{Percentage voids in hcp} = \left( \frac{\text{Total voids}}{\text{Total atoms + Total voids}} \right) \times 100 = \left( \frac{9}{3 + 9} \right) \times 100 = \left( \frac{9}{12} \right) \times 100 = 75\% \]

- **Percentage Voids in bcc**:
\[ \text{Percentage voids in bcc} = \left( \frac{\text{Total voids}}{\text{Total atoms + Total voids}} \right) \times 100 = \left( \frac{2}{1 + 2} \right) \times 100 = \left( \frac{2}{3} \right) \times 100 \approx 66.67\% \]

### Step 5: Calculate the Ratio of Percentage Voids
- The ratio of percentage voids in hcp to bcc:
\[ \text{Ratio} = \frac{\text{Percentage voids in hcp}}{\text{Percentage voids in bcc}} = \frac{75}{66.67} \approx 1.125 \]

### Final Answer
The ratio of percentage voids in hcp and bcc structures is approximately **1.125**.

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