The ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic structure are x: y : z. The numerical value of sum of x, y and z is _______ .

Explanation

To solve the problem, we need to determine the effective number of atoms present in three types of crystal structures: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). We will denote the effective number of atoms in these structures as x, y, and z respectively.

### Step 1: Determine the effective 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, and each corner atom is shared by 8 adjacent unit cells.

- Effective number of atoms in SC:
\[ \text{Effective atoms} = \frac{1}{8} \times 8 = 1 \]
Thus, we have \( x = 1 \).

### Step 2: Determine the effective number of atoms in Body-Centered Cubic (BCC)
In a body-centered cubic structure, there are atoms at each corner of the cube and one additional atom in the center of the cube.

- Effective number of atoms in BCC:
\[ \text{Effective atoms} = \left(\frac{1}{8} \times 8\right) + 1 = 1 + 1 = 2 \]
Thus, we have \( y = 2 \).

### Step 3: Determine the effective number of atoms in Face-Centered Cubic (FCC)
In a face-centered cubic structure, atoms are located at each corner and at the center of each face. Each corner atom is shared among 8 unit cells, and each face atom is shared between 2 unit cells.

- Effective number of atoms in FCC:
\[ \text{Effective atoms} = \left(\frac{1}{8} \times 8\right) + \left(\frac{1}{2} \times 6\right) = 1 + 3 = 4 \]
Thus, we have \( z = 4 \).

### Step 4: Write the ratio of the number of atoms
Now we have:
- \( x = 1 \) (for SC)
- \( y = 2 \) (for BCC)
- \( z = 4 \) (for FCC)

The ratio of the number of atoms is:
\[ x : y : z = 1 : 2 : 4 \]

### Step 5: Calculate the sum of x, y, and z
Now, we can find the sum:
\[ x + y + z = 1 + 2 + 4 = 7 \]

### Final Answer
The numerical value of the sum of \( x, y, \) and \( z \) is **7**.

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