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) structure, atoms are located at each of the corners of the cube and at the centers of each face of the cube.

### Step 2: Calculate the Number of Atoms per Unit Cell
1. **Atoms at the Corners**:
- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Total contribution from corner atoms = \( 8 \times \frac{1}{8} = 1 \) atom.

2. **Atoms at the Faces**:
- There are 6 faces in a cube.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.
- Total contribution from face-centered atoms = \( 6 \times \frac{1}{2} = 3 \) atoms.

3. **Total Atoms per Unit Cell**:
- Total = Atoms from corners + Atoms from faces = \( 1 + 3 = 4 \) atoms.

### Step 3: Determine the Number of Octahedral and Tetrahedral Holes
1. **Octahedral Holes**:
- The number of octahedral holes per unit cell is equal to the number of atoms per unit cell, which is \( n \).
- Since we found \( n = 4 \), the number of octahedral holes = 4.

2. **Tetrahedral Holes**:
- The number of tetrahedral holes per unit cell is given by \( 2n \).
- Therefore, the number of tetrahedral holes = \( 2 \times 4 = 8 \).

### Final Answer
- **Octahedral holes**: 4
- **Tetrahedral holes**: 8

Thus, in a face-centered cubic arrangement, there are **4 octahedral holes and 8 tetrahedral holes per unit cell**.

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