In silicon crystal, `Si` atoms from fcc arrangement where `4` out `8 TVs` are alos occupied by `Si` atoms. `Z_(eff)` of unit cell is

Options

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

Explanation

To find the effective number of silicon atoms (Z_eff) in the unit cell of a silicon crystal with a face-centered cubic (FCC) arrangement, we can follow these steps:

### Step-by-Step Solution:

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

2. **Calculating the Number of Atoms in the FCC Unit Cell**:
- Total contribution from corner atoms:
\[ 8 \text{ corners} \times \frac{1}{8} = 1 \text{ atom} \]
- Total contribution from face-centered atoms:
\[ 6 \text{ faces} \times \frac{1}{2} = 3 \text{ atoms} \]
- Therefore, the total number of silicon atoms in the FCC unit cell is:
\[ 1 + 3 = 4 \text{ atoms} \]

3. **Considering the Tetrahedral Voids**:
- In an FCC structure, there are 8 tetrahedral voids.
- According to the problem, 4 out of these 8 tetrahedral voids are occupied by silicon atoms.

4. **Calculating the Total Number of Silicon Atoms**:
- The total number of silicon atoms in the unit cell is the sum of the atoms in the FCC arrangement and those occupying the tetrahedral voids:
\[ \text{Total silicon atoms} = \text{Atoms in FCC} + \text{Atoms in tetrahedral voids} \]
- Thus, we have:
\[ \text{Total silicon atoms} = 4 + 4 = 8 \]

5. **Conclusion**:
- Therefore, the effective number of silicon atoms (Z_eff) in the unit cell is:
\[ Z_{\text{eff}} = 8 \]

### Final Answer:
The effective number of silicon atoms (Z_eff) in the unit cell is **8** (Option D).

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