The number of atom per unit in a simple cubic, face - centered cubic and body - centered cubic are ….respectively

Options

A. 1,4,2
B. 1,2,4
C. 8,14,9
D. 8,4,2

Answer

To find the number of atoms per unit cell in simple cubic, face-centered cubic (FCC), and body-centered cubic (BCC) structures, we can follow these steps:

### Step 1: Simple Cubic Unit Cell
1. **Identify the structure**: In a simple cubic unit cell, atoms are located only at the corners.
2. **Count the corners**: There are 8 corners in a cubic unit cell.
3. **Calculate contribution from corner atoms**: Each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell because it is shared with 7 other unit cells.
4. **Total contribution from corner atoms**:
\[ \text{Total atoms} = 8 \text{ corners} \times \frac{1}{8} = 1 \]
Therefore, the total number of atoms in a simple cubic unit cell is **1**.

### Step 2: Face-Centered Cubic Unit Cell
1. **Identify the structure**: In a face-centered cubic unit cell, atoms are located at the corners and at the center of each face.
2. **Count the corners and faces**: There are 8 corners and 6 faces.
3. **Calculate contribution from corner atoms**:
\[ \text{Contribution from corners} = 8 \text{ corners} \times \frac{1}{8} = 1 \]
4. **Calculate contribution from face atoms**: Each face atom contributes \( \frac{1}{2} \) of its volume to the unit cell because it is shared with another unit cell.
\[ \text{Contribution from faces} = 6 \text{ faces} \times \frac{1}{2} = 3 \]
5. **Total contribution**:
\[ \text{Total atoms} = 1 + 3 = 4 \]
Therefore, the total number of atoms in a face-centered cubic unit cell is **4**.

### Step 3: Body-Centered Cubic Unit Cell
1. **Identify the structure**: In a body-centered cubic unit cell, atoms are located at the corners and one atom is at the center of the cube.
2. **Count the corners and center**: There are 8 corners and 1 center.
3. **Calculate contribution from corner atoms**:
\[ \text{Contribution from corners} = 8 \text{ corners} \times \frac{1}{8} = 1 \]
4. **Calculate contribution from the body-centered atom**: The atom at the center contributes fully to the unit cell.
\[ \text{Contribution from center} = 1 \]
5. **Total contribution**:
\[ \text{Total atoms} = 1 + 1 = 2 \]
Therefore, the total number of atoms in a body-centered cubic unit cell is **2**.

### Final Result
The total number of atoms per unit cell in simple cubic, face-centered cubic, and body-centered cubic structures are **1, 4, and 2**, respectively.

### Summary
- Simple Cubic: **1 atom**
- Face-Centered Cubic: **4 atoms**
- Body-Centered Cubic: **2 atoms**