In a simple cubic, body-centred cubic and face-centred cubic structure, the ratio of the number of atoms present is respectively

Options

A. `8:1:6`
B. `1:2:4`
C. `4:2:1`
D. `4:2:3`

Explanation

To find the ratio of the number of atoms present in a Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) structure, we will calculate the number of atoms per unit cell for each of these structures step by step.

### Step 1: Calculate the number of atoms in Simple Cubic (SC)
In a Simple Cubic structure, atoms are located only at the corners of the cube.

- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.

Thus, the total number of atoms (Z) in a Simple Cubic unit cell is:
\[ Z_{SC} = 8 \times \frac{1}{8} = 1 \]

### Step 2: Calculate the number of atoms in Body-Centered Cubic (BCC)
In a Body-Centered Cubic structure, atoms are located at the corners and one atom is located at the center of the cube.

- There are still 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Additionally, there is 1 atom at the center contributing 1 full atom.

Thus, the total number of atoms (Z) in a Body-Centered Cubic unit cell is:
\[ Z_{BCC} = 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \]

### Step 3: Calculate the number of atoms in Face-Centered Cubic (FCC)
In a Face-Centered Cubic structure, atoms are located at the corners and the centers of each face of the cube.

- There are 8 corners in a cube, with each corner atom contributing \( \frac{1}{8} \).
- There are 6 faces, and each face has 1 atom at its center contributing \( \frac{1}{2} \) of an atom to the unit cell.

Thus, the total number of atoms (Z) in a Face-Centered Cubic unit cell is:
\[ Z_{FCC} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 4: Find the ratio of the number of atoms
Now that we have the number of atoms per unit cell for each structure:
- Simple Cubic (SC): 1 atom
- Body-Centered Cubic (BCC): 2 atoms
- Face-Centered Cubic (FCC): 4 atoms

The ratio of the number of atoms in SC : BCC : FCC is:
\[ 1 : 2 : 4 \]

### Final Answer
The ratio of the number of atoms present in Simple Cubic, Body-Centered Cubic, and Face-Centered Cubic structures is:
\[ \text{1 : 2 : 4} \]