In Face-Centred Cubic Structures the value of Z is?

Options

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

Answer

To determine the value of Z in a Face-Centred Cubic (FCC) structure, we need to analyze the arrangement of atoms within the unit cell. Here’s a step-by-step solution:

### Step 1: Understand the FCC Structure
In a Face-Centred Cubic structure, atoms are located at each of the corners and the centers of each face of the cube.

### Step 2: Count the Atoms at the Corners
In an FCC unit cell, there are 8 corner atoms. Each corner atom contributes only a fraction of itself to the unit cell because it is shared with adjacent unit cells.

### Step 3: Calculate the Contribution from Corner Atoms
Each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell. Therefore, the total contribution from the corner atoms is:
\[ \text{Contribution from corners} = 8 \text{ atoms} \times \frac{1}{8} = 1 \text{ atom} \]

### Step 4: Count the Atoms on the Faces
In addition to the corner atoms, there are 6 face-centered atoms in the FCC structure, one on each face of the cube.

### Step 5: Calculate the Contribution from Face Atoms
Each face atom contributes \( \frac{1}{2} \) of its volume to the unit cell since it is shared between two adjacent unit cells. Therefore, the total contribution from the face atoms is:
\[ \text{Contribution from faces} = 6 \text{ atoms} \times \frac{1}{2} = 3 \text{ atoms} \]

### Step 6: Calculate the Total Number of Atoms (Z)
Now, we can find the total number of atoms (Z) in the FCC unit cell by adding the contributions from the corner and face atoms:
\[ Z = \text{Contribution from corners} + \text{Contribution from faces} = 1 + 3 = 4 \]

### Conclusion
Thus, the value of Z in a Face-Centred Cubic structure is:
\[ \boxed{4} \]

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