Empty space left in HCP in three dimension is

Options

A. 0.76
B. 0.74
C. 0.68
D. 0.26

Answer

To find the empty space left in Hexagonal Close Packing (HCP) in three dimensions, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding HCP Structure**:
- In HCP, the arrangement of atoms is such that there are 4 atoms per unit cell. This is denoted by \( Z = 4 \).

2. **Calculating the Volume of Atoms**:
- The volume of a single atom can be calculated using the formula for the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
- Therefore, the total volume of 4 atoms in the unit cell is:
\[ V_{\text{total atoms}} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

3. **Calculating the Volume of the Unit Cell**:
- The edge length \( a \) of the hexagonal unit cell is related to the radius \( r \) of the atoms. For HCP, the relationship is:
\[ a = 2\sqrt{2}r \]
- The volume of the hexagonal unit cell can be calculated using the formula for the volume of a hexagonal prism:
\[ V_{\text{cell}} = \frac{3\sqrt{3}}{2} a^2 c \]
- Here, \( c \) is the height of the unit cell. For HCP, \( c \) is related to \( a \) as \( c = \frac{4}{\sqrt{2}} a \).

4. **Calculating Packing Fraction**:
- The packing fraction (PF) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell:
\[ \text{Packing Fraction} = \frac{V_{\text{total atoms}}}{V_{\text{cell}}} \]
- After substituting the values, we find that the packing fraction for HCP is approximately \( 0.7404 \).

5. **Calculating Empty Space**:
- The empty space in the unit cell can be calculated as:
\[ \text{Empty Space} = 1 - \text{Packing Fraction} \]
- Substituting the packing fraction value:
\[ \text{Empty Space} = 1 - 0.7404 = 0.2596 \approx 0.26 \]

6. **Final Result**:
- The empty space left in HCP in three dimensions is approximately \( 0.26 \) or \( 26\% \).