Which is correct order of void fraction?

Options

A. Simple cubic packing gt bcc gt fcc = hcp
B. Simple cubic packing gt bcc gt fcc gt hcp
C. Simple cubic packing lt bcc lt fcc = hcp
D. Simple cubic packing ltbcc ltfcc lthcp

Answer

To determine the correct order of void fractions for different types of crystal structures, we need to calculate the void fraction for each structure: Simple Cubic (SC), Body-Centered Cubic (BCC), Face-Centered Cubic (FCC), and Hexagonal Close Packing (HCP).

### Step-by-Step Solution:

1. **Understanding Void Fraction**:
- The void fraction is defined as the fraction of the volume of a unit cell that is not occupied by the atoms. It can be calculated using the formula:
\[ \text{Void Fraction} = 1 - \text{Packing Fraction} \]
- The packing fraction is the volume occupied by the atoms divided by the total volume of the unit cell.

2. **Calculating Void Fraction for Simple Cubic (SC)**:
- In a Simple Cubic structure, there is 1 atom per unit cell.
- The packing fraction for SC is approximately \(0.52\).
- Thus, the void fraction is:
\[ \text{Void Fraction}_{SC} = 1 - 0.52 = 0.48 \]

3. **Calculating Void Fraction for Body-Centered Cubic (BCC)**:
- In a Body-Centered Cubic structure, there are 2 atoms per unit cell.
- The packing fraction for BCC is approximately \(0.68\).
- Thus, the void fraction is:
\[ \text{Void Fraction}_{BCC} = 1 - 0.68 = 0.32 \]

4. **Calculating Void Fraction for Face-Centered Cubic (FCC)**:
- In a Face-Centered Cubic structure, there are 4 atoms per unit cell.
- The packing fraction for FCC is approximately \(0.74\).
- Thus, the void fraction is:
\[ \text{Void Fraction}_{FCC} = 1 - 0.74 = 0.26 \]

5. **Calculating Void Fraction for Hexagonal Close Packing (HCP)**:
- In a Hexagonal Close Packing structure, there are also 6 atoms per unit cell.
- The packing fraction for HCP is also approximately \(0.74\).
- Thus, the void fraction is:
\[ \text{Void Fraction}_{HCP} = 1 - 0.74 = 0.26 \]

6. **Comparing the Void Fractions**:
- Now, we can summarize the void fractions calculated:
- Simple Cubic: \(0.48\)
- Body-Centered Cubic: \(0.32\)
- Face-Centered Cubic: \(0.26\)
- Hexagonal Close Packing: \(0.26\)

7. **Correct Order of Void Fractions**:
- The correct order of void fractions from highest to lowest is:
\[ \text{SC} > \text{BCC} > \text{FCC} = \text{HCP} \]

### Final Answer:
The correct order of void fraction is:
**Simple Cubic (SC) > Body-Centered Cubic (BCC) > Face-Centered Cubic (FCC) = Hexagonal Close Packing (HCP)**.

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