The total of tetrahedral and octahedral voids in the face centred unit cell is …………. .

Options

A. 6 , 6
B. 8 , 4
C. 10 , 12
D. 12 , 8

Explanation

To find the total number of tetrahedral and octahedral voids in a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step 1: Understand the FCC Unit Cell Structure
In a face-centered cubic (FCC) unit cell, atoms are located at:
- The corners of the cube (8 atoms)
- The centers of each of the six faces (6 atoms)

### Step 2: Calculate the Contribution of Atoms
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among 8 unit cells.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared between 2 unit cells.

**Calculation:**
- Contribution from corner atoms: \( 8 \times \frac{1}{8} = 1 \) atom
- Contribution from face-centered atoms: \( 6 \times \frac{1}{2} = 3 \) atoms

**Total number of atoms (Z) in FCC unit cell:**
\[ Z = 1 + 3 = 4 \]

### Step 3: Calculate the Number of Tetrahedral Voids
In an FCC unit cell, the number of tetrahedral voids is twice the number of atoms per unit cell (Z):
\[ \text{Tetrahedral voids} = 2 \times Z = 2 \times 4 = 8 \]

### Step 4: Calculate the Number of Octahedral Voids
The number of octahedral voids in an FCC unit cell is equal to the number of atoms per unit cell (Z):
\[ \text{Octahedral voids} = Z = 4 \]

### Step 5: Find the Total Number of Voids
Now, we can find the total number of tetrahedral and octahedral voids:
\[ \text{Total voids} = \text{Tetrahedral voids} + \text{Octahedral voids} = 8 + 4 = 12 \]

### Final Answer
The total number of tetrahedral and octahedral voids in the face-centered unit cell is **12**.

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