Find the sum of number of atoms present in a simple cubic, body centered cubic and face centered cubic structure.

Explanation

To find the sum 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 in each type of cubic structure 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.
- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared by 8 adjacent unit cells.

**Calculation:**
\[ \text{Total contribution from corners} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 2: Calculate the number of atoms in Body-Centered Cubic (BCC)
- In a body-centered cubic structure, there are atoms at the corners and one atom at the center of the cube.
- Again, there are 8 corner atoms, and each contributes \( \frac{1}{8} \).
- The atom at the center contributes fully as it is not shared with any other unit cell.

**Calculation:**
\[ \text{Total contribution from corners} = 8 \times \frac{1}{8} = 1 \text{ atom} \]
\[ \text{Total contribution from center} = 1 \text{ atom} \]
\[ \text{Total number of atoms in BCC} = 1 + 1 = 2 \text{ atoms} \]

### Step 3: Calculate the number of atoms in Face-Centered Cubic (FCC)
- In a face-centered cubic structure, there are atoms at the corners and at the centers of each face.
- There are 8 corner atoms, each contributing \( \frac{1}{8} \).
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) since each face atom is shared between 2 unit cells.

**Calculation:**
\[ \text{Total contribution from corners} = 8 \times \frac{1}{8} = 1 \text{ atom} \]
\[ \text{Total contribution from face centers} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]
\[ \text{Total number of atoms in FCC} = 1 + 3 = 4 \text{ atoms} \]

### Step 4: Sum the number of atoms from all structures
Now we add the total number of atoms from each structure:
\[ \text{Total number of atoms} = \text{Atoms in SC} + \text{Atoms in BCC} + \text{Atoms in FCC} \]
\[ = 1 + 2 + 4 = 7 \]

### Final Answer:
The sum of the number of atoms present in a simple cubic, body-centered cubic, and face-centered cubic structure is **7**.

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