In sc, bcc and fcc the ratio of number of atoms per unit cell is given by

Options

A. `1:2:3`
B. `4:2:1`
C. `1:2:4`
D. `1:1:3`

Answer

To solve the problem of finding the ratio of the number of atoms per unit cell in Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) structures, we can follow these steps:

### Step 1: Simple Cubic (SC)
In a simple cubic unit cell:
- There are 8 corner atoms.
- Each corner atom is shared by 8 unit cells.

**Calculation:**
- Contribution from corner atoms = \( \frac{1}{8} \times 8 = 1 \) atom.

Thus, the number of atoms per unit cell in SC is **1**.

### Step 2: Body-Centered Cubic (BCC)
In a body-centered cubic unit cell:
- There are 8 corner atoms, each shared by 8 unit cells.
- There is 1 atom located at the center of the cube.

**Calculation:**
- Contribution from corner atoms = \( \frac{1}{8} \times 8 = 1 \) atom.
- Contribution from the body-centered atom = 1 atom.

Thus, the total number of atoms per unit cell in BCC is \( 1 + 1 = 2 \).

### Step 3: Face-Centered Cubic (FCC)
In a face-centered cubic unit cell:
- There are 8 corner atoms, each shared by 8 unit cells.
- There are 6 face-centered atoms, each shared by 2 unit cells.

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

Thus, the total number of atoms per unit cell in FCC is \( 1 + 3 = 4 \).

### Step 4: Ratio of Atoms per Unit Cell
Now we can summarize the number of atoms per unit cell for each structure:
- SC: 1 atom
- BCC: 2 atoms
- FCC: 4 atoms

The ratio of the number of atoms per unit cell in SC, BCC, and FCC is:
\[ 1 : 2 : 4 \]

### Final Answer
The ratio of the number of atoms per unit cell in SC, BCC, and FCC is **1 : 2 : 4**.

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