Which is / are correct statement ?

Options

A. Packing fraction in 2D-hcp is 0.785
B. Packing fraction in AAA….. Is 0.52
C. Packing fraction in ABAB…… is 0.74
D. Packing fraction in ABCABC……. Is 0.26

Answer

To determine which statements regarding packing fractions are correct, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Statement A: Packing fraction in 2D HCP is 0.785.**
- **Analysis:** The packing fraction for 2D hexagonal close packing (HCP) is indeed 0.785. This is calculated based on the arrangement of circles in a hexagonal pattern.
- **Conclusion:** This statement is **correct**.

2. **Statement B: Packing fraction in AAA type is 0.52.**
- **Analysis:** The AAA type packing refers to a simple cubic arrangement where atoms are located only at the corners of the cube. Each corner atom contributes 1/8 to the unit cell, and with 8 corners, the total contribution is 1 atom (Z = 1). The packing fraction is calculated as:
\[ \text{Packing fraction} = \frac{Z \times \text{Volume of one atom}}{\text{Volume of the unit cell}} = \frac{1 \times \frac{4}{3} \pi r^3}{a^3} \]
where \(a = 2r\). Substituting \(a\) gives us:
\[ \text{Packing fraction} = \frac{1 \times \frac{4}{3} \pi r^3}{(2r)^3} = \frac{1 \times \frac{4}{3} \pi r^3}{8r^3} = \frac{4 \pi}{24} = 0.52. \]
- **Conclusion:** This statement is **correct**.

3. **Statement C: Packing fraction in AB-AB is 0.74.**
- **Analysis:** The AB-AB type packing refers to hexagonal close packing (HCP). In HCP, there are 6 atoms per unit cell, and the packing fraction is calculated to be 0.74.
- **Conclusion:** This statement is **correct**.

4. **Statement D: Packing fraction in ABC-ABC is 0.6.**
- **Analysis:** The ABC-ABC type packing refers to face-centered cubic (FCC) arrangement. In FCC, the atoms at the corners and face centers give a total of 4 atoms per unit cell (Z = 4). The packing fraction is calculated as:
\[ \text{Packing fraction} = \frac{Z \times \text{Volume of one atom}}{\text{Volume of the unit cell}} = \frac{4 \times \frac{4}{3} \pi r^3}{a^3} \]
where \(a = \frac{4r}{\sqrt{2}}\). Substituting gives:
\[ \text{Packing fraction} = \frac{4 \times \frac{4}{3} \pi r^3}{\left(\frac{4r}{\sqrt{2}}\right)^3} = \frac{4 \times \frac{4}{3} \pi r^3}{\frac{64r^3}{8}} = \frac{4 \times \frac{4}{3} \pi}{8} = 0.74. \]
The statement claims it is 0.6, which is incorrect; 0.6 is actually the void fraction.
- **Conclusion:** This statement is **incorrect**.

### Final Conclusion:
The correct statements are **A, B, and C**.