The ratio of number of atoms present in a simple cubic body centred cubic and face centred cubic structure are, respectively :

Options

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

Explanation

To solve the problem of finding the ratio of the number of atoms present in simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) structures, we can follow these steps:

### Step 1: Determine the number of atoms in a Simple Cubic (SC) structure
In a simple cubic structure, atoms are located only at the corners of the cube. There are 8 corners in a cube, and each corner atom is shared by 8 adjacent cubes. Therefore, the contribution of each corner atom to the unit cell is:

\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \]

Thus, the total number of atoms in a simple cubic structure is **1**.

### Step 2: Determine the number of atoms in a Body-Centered Cubic (BCC) structure
In a body-centered cubic structure, atoms are located at the corners and one atom is located at the center of the cube. The contribution from the corner atoms is the same as in the simple cubic structure:

\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \]

Additionally, there is 1 atom at the center, which contributes fully to the unit cell:

\[ \text{Contribution from body center} = 1 \]

Thus, the total number of atoms in a body-centered cubic structure is:

\[ 1 + 1 = 2 \]

### Step 3: Determine the number of atoms in a Face-Centered Cubic (FCC) structure
In a face-centered cubic structure, atoms are located at the corners and at the centers of the faces of the cube. The contribution from the corner atoms is again:

\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \]

There are 6 faces, and each face has an atom that is shared between 2 adjacent cubes. Therefore, the contribution from the face-centered atoms is:

\[ \text{Contribution from faces} = 6 \times \frac{1}{2} = 3 \]

Thus, the total number of atoms in a face-centered cubic structure is:

\[ 1 + 3 = 4 \]

### Step 4: Calculate the ratio of the number of atoms
Now we can summarize the total number of atoms in 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 a simple cubic, body-centered cubic, and face-centered cubic structure is **1 : 2 : 4**.

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