In which of the following crystal structure the void efficiency is `32%`?

Options

A. simple cube
B. face centered cube
C. hexagonal close packing
D. body centered cube

Answer

To determine which crystal structure has a void efficiency of 32%, we need to analyze the packing efficiency of various crystal structures. Here's a step-by-step breakdown of the solution:

### Step 1: Understand Packing Efficiency
Packing efficiency (η) is defined as the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules). It can be calculated using the formula:

\[ \eta = \frac{Z \times V_{\text{atom}}}{a^3} \]

where:
- \( Z \) = number of atoms per unit cell
- \( V_{\text{atom}} \) = volume of one atom
- \( a \) = edge length of the unit cell

### Step 2: Identify the Crystal Structures
We will analyze the following crystal structures:
1. Simple Cubic (SCC)
2. Body-Centered Cubic (BCC)
3. Face-Centered Cubic (FCC)
4. Hexagonal Close Packing (HCP)

### Step 3: Calculate Packing Efficiency for Each Structure

#### a. Simple Cubic (SCC)
- Number of atoms per unit cell (Z) = 1
- Packing efficiency (η) = \( \frac{\pi}{6} \approx 0.52 \) or 52%
- Void efficiency = \( 100\% - 52\% = 48\% \)

#### b. Body-Centered Cubic (BCC)
- Number of atoms per unit cell (Z) = 2
- Packing efficiency (η) = \( \frac{\sqrt{3} \pi}{8} \approx 0.68 \) or 68%
- Void efficiency = \( 100\% - 68\% = 32\% \)

#### c. Face-Centered Cubic (FCC)
- Number of atoms per unit cell (Z) = 4
- Packing efficiency (η) = \( \frac{\pi}{3 \sqrt{2}} \approx 0.74 \) or 74%
- Void efficiency = \( 100\% - 74\% = 26\% \)

#### d. Hexagonal Close Packing (HCP)
- Number of atoms per unit cell (Z) = 6
- Packing efficiency (η) is similar to FCC, approximately 74%
- Void efficiency = \( 100\% - 74\% = 26\% \)

### Step 4: Conclusion
From the calculations above, we find that:
- SCC has a void efficiency of 48%
- BCC has a void efficiency of 32%
- FCC has a void efficiency of 26%
- HCP has a void efficiency of 26%

Thus, the crystal structure with a void efficiency of 32% is **Body-Centered Cubic (BCC)**.

### Final Answer
The correct answer is **Body-Centered Cubic (BCC)**.

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