How many octahedral and tetrahedral holes are present per unit cell in a face centred cubic arrangement of atoms ?

Options

A. 8,4
B. 1,2
C. 4,8
D. 2,1

Answer

To determine the number of octahedral and tetrahedral holes present per unit cell in a face-centered cubic (FCC) arrangement of atoms, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) unit cell, there are atoms located at the corners and the centers of each face of the cube.

### Step 2: Calculate the Number of Atoms per Unit Cell
- **Corner Atoms**: There are 8 corner atoms in a cube. Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among 8 adjacent unit cells.
\[ \text{Contribution from corner atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

- **Face Atoms**: There are 6 faces in a cube, and each face atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared between 2 adjacent unit cells.
\[ \text{Contribution from face atoms} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

- **Total Atoms in FCC**:
\[ \text{Total atoms per unit cell} = 1 + 3 = 4 \text{ atoms} \]
Thus, \( n = 4 \).

### Step 3: Calculate the Number of Octahedral Holes
The number of octahedral holes in a unit cell is equal to the number of atoms per unit cell (n).
\[ \text{Number of octahedral holes} = n = 4 \]

### Step 4: Calculate the Number of Tetrahedral Holes
The number of tetrahedral holes is twice the number of atoms per unit cell.
\[ \text{Number of tetrahedral holes} = 2n = 2 \times 4 = 8 \]

### Final Answer
In a face-centered cubic arrangement:
- Number of octahedral holes = 4
- Number of tetrahedral holes = 8

### Summary
Thus, the answer is:
- **Octahedral holes**: 4
- **Tetrahedral holes**: 8