The ratio of closed packed atoms to tetrahedral holes in cubic close packing is `:`

Options

A. `1:1`
B. `1:2`
C. `1:3`
D. `2:1`

Answer

To find the ratio of closed packed atoms to tetrahedral holes in cubic close packing (CCP), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure**:
- Cubic close packing (CCP) is also known as face-centered cubic (FCC) structure. In this arrangement, atoms are located at the corners and the centers of each face of the cube.

2. **Calculate the Effective Number of Atoms (z)**:
- In an FCC structure:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell.
- Therefore, the effective number of atoms (z) in FCC is calculated as follows:
\[ z = (8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4 \]

3. **Determine the Number of Tetrahedral Holes**:
- In a cubic close packing structure, the number of tetrahedral holes is twice the effective number of atoms.
- Thus, the number of tetrahedral holes is:
\[ \text{Number of tetrahedral holes} = 2 \times z = 2 \times 4 = 8 \]

4. **Calculate the Ratio**:
- Now, we can find the ratio of closed packed atoms to tetrahedral holes:
\[ \text{Ratio} = \frac{\text{Closed packed atoms}}{\text{Tetrahedral holes}} = \frac{z}{\text{Number of tetrahedral holes}} = \frac{4}{8} \]
- Simplifying this ratio gives:
\[ \text{Ratio} = \frac{1}{2} \]

5. **Final Answer**:
- Therefore, the ratio of closed packed atoms to tetrahedral holes in cubic close packing is \( 1:2 \).