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, we need to determine the ratio of percentage voids in hexagonal close-packed (hcp) and body-centered cubic (bcc) structures after halving the number of atoms per unit cell in both structures.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- In an hcp structure, the packing efficiency is 74%. Therefore, the percentage of voids is:
\[ \text{Percentage of voids in hcp} = 100\% - 74\% = 26\% \]
- In a bcc structure, the packing efficiency is 68%. Thus, the percentage of voids is:
\[ \text{Percentage of voids in bcc} = 100\% - 68\% = 32\% \]

2. **Halve the Number of Atoms**:
- For hcp, if the total number of atoms is halved, the new packing efficiency becomes:
\[ \text{New packing efficiency for hcp} = \frac{74\%}{2} = 37\% \]
Therefore, the new percentage of voids in hcp is:
\[ \text{New percentage of voids in hcp} = 100\% - 37\% = 63\% \]

- For bcc, halving the number of atoms gives us a new packing efficiency:
\[ \text{New packing efficiency for bcc} = \frac{68\%}{2} = 34\% \]
Thus, the new percentage of voids in bcc is:
\[ \text{New percentage of voids in bcc} = 100\% - 34\% = 66\% \]

3. **Calculate the Ratio of Percentage Voids**:
- Now, we need to find the ratio of the percentage voids in hcp to that in bcc:
\[ \text{Ratio of voids} = \frac{\text{Percentage voids in hcp}}{\text{Percentage voids in bcc}} = \frac{63\%}{66\%} \]
- Simplifying this ratio:
\[ \text{Ratio} = \frac{63}{66} = \frac{21}{22} \]

### Final Answer:
The ratio of the percentage voids in hcp and bcc structures after halving the number of atoms per unit cell is:
\[ \frac{21}{22} \]