In a diamond, each carbon atom is bonded to four other carbon atoms tetrahedrally. Alternate tetrahedral voids are occupied by carbon atoms. The number of carbon atoms per unit cell is:

Options

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

Answer

To determine the number of carbon atoms per unit cell in diamond, we can follow these steps:

### Step 1: Understand the Structure of Diamond
Diamond has a crystal structure where each carbon atom is bonded to four other carbon atoms in a tetrahedral arrangement. The carbon atoms form a face-centered cubic (FCC) lattice, and alternate tetrahedral voids are occupied by additional carbon atoms.

### Step 2: Calculate the Number of Carbon Atoms in the FCC Lattice
In a face-centered cubic (FCC) lattice:
- 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.

Calculating the total contribution:
- Contribution from corner atoms:
\[ 8 \text{ corners} \times \frac{1}{8} = 1 \text{ atom} \]
- Contribution from face-centered atoms:
\[ 6 \text{ faces} \times \frac{1}{2} = 3 \text{ atoms} \]
- Total carbon atoms in the FCC lattice:
\[ 1 + 3 = 4 \text{ atoms} \]

### Step 3: Calculate the Number of Tetrahedral Voids
The number of tetrahedral voids in an FCC lattice is given by:
\[ \text{Number of tetrahedral voids} = 2 \times \text{Number of atoms in FCC} \]
Thus, for 4 atoms in the FCC lattice:
\[ \text{Number of tetrahedral voids} = 2 \times 4 = 8 \text{ voids} \]

### Step 4: Determine the Number of Carbon Atoms in Tetrahedral Voids
Since alternate tetrahedral voids are occupied by carbon atoms, only half of the tetrahedral voids are filled:
\[ \text{Carbon atoms in tetrahedral voids} = \frac{1}{2} \times 8 = 4 \text{ atoms} \]

### Step 5: Calculate the Total Number of Carbon Atoms per Unit Cell
Now, we can sum the carbon atoms from the FCC lattice and the carbon atoms occupying the tetrahedral voids:
\[ \text{Total carbon atoms per unit cell} = \text{Atoms in FCC} + \text{Atoms in tetrahedral voids} = 4 + 4 = 8 \text{ atoms} \]

### Conclusion
The total number of carbon atoms per unit cell in diamond is **8**.

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