The effective number of atoms per unit cell in a face centred cube and body centred cube are :

Options

A. 4,2
B. 2,4
C. 14,9
D. 3,2

Explanation

To determine the effective number of atoms per unit cell in a face-centered cubic (FCC) and body-centered cubic (BCC) structure, we can follow these steps:

### Step 1: Understanding the Face-Centered Cubic (FCC) Structure
In an FCC unit cell:
- There are atoms located at each of the 8 corners of the cube.
- There is also one atom at the center of each of the 6 faces of the cube.

### Step 2: Calculating Contribution from Corner Atoms in FCC
Each corner atom is shared among 8 adjacent unit cells. Therefore, the contribution of one corner atom to the unit cell is:
\[ \text{Contribution from corner atoms} = \frac{1}{8} \times 8 = 1 \text{ atom} \]

### Step 3: Calculating Contribution from Face Atoms in FCC
Each face atom is shared between 2 unit cells. Therefore, the contribution of one face atom to the unit cell is:
\[ \text{Contribution from face atoms} = \frac{1}{2} \times 6 = 3 \text{ atoms} \]

### Step 4: Total Effective Number of Atoms in FCC
Adding the contributions from the corner and face atoms:
\[ \text{Total atoms in FCC} = 1 + 3 = 4 \text{ atoms} \]

### Step 5: Understanding the Body-Centered Cubic (BCC) Structure
In a BCC unit cell:
- There are atoms located at each of the 8 corners of the cube.
- There is also one atom located at the center of the cube.

### Step 6: Calculating Contribution from Corner Atoms in BCC
Similar to FCC, the contribution from the corner atoms is:
\[ \text{Contribution from corner atoms} = \frac{1}{8} \times 8 = 1 \text{ atom} \]

### Step 7: Calculating Contribution from Body Center Atom in BCC
The body-centered atom is not shared with any other unit cell, so its contribution is:
\[ \text{Contribution from body center atom} = 1 \text{ atom} \]

### Step 8: Total Effective Number of Atoms in BCC
Adding the contributions from the corner and body center atoms:
\[ \text{Total atoms in BCC} = 1 + 1 = 2 \text{ atoms} \]

### Final Answer
- The effective number of atoms per unit cell in a Face-Centered Cubic (FCC) structure is **4 atoms**.
- The effective number of atoms per unit cell in a Body-Centered Cubic (BCC) structure is **2 atoms**.

### Summary
- FCC: 4 atoms
- BCC: 2 atoms