A crystal is made of particles X and Y.X form fcc packing and Y occupies all the octahedral voids . If all the particles along one body diagonal are removed then the formula of the crystal would de :

Options

A. `X_(4)Y_(3)`
B. `X_(5)Y_(5)`
C. `X_(5)Y_(4)`
D. None of these

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the structure of the crystal
The crystal is made up of two types of particles: X and Y. The X particles form a face-centered cubic (FCC) lattice, and the Y particles occupy all the octahedral voids in this structure.

### Step 2: Determine the number of X particles in the FCC lattice
In an FCC lattice:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) to the unit cell.
- There are 6 face-centered atoms, each contributing \( 1 \) to the unit cell.

Calculating the total contribution of X particles:
- Contribution from corner atoms: \( 8 \times \frac{1}{8} = 1 \)
- Contribution from face-centered atoms: \( 6 \times 1 = 6 \)

Total number of X particles in the unit cell:
\[ \text{Total X} = 1 + 6 = 7 \]

### Step 3: Determine the number of Y particles in the octahedral voids
In an FCC lattice, the octahedral voids are located:
- 1 at the body center (contributes 1)
- 12 at the edge centers, each contributing \( \frac{1}{4} \)

Calculating the total contribution of Y particles:
- Contribution from body center: \( 1 \)
- Contribution from edge centers: \( 12 \times \frac{1}{4} = 3 \)

Total number of Y particles in the unit cell:
\[ \text{Total Y} = 1 + 3 = 4 \]

### Step 4: Removing particles along one body diagonal
When we remove the particles along one body diagonal, we are removing:
- 2 X particles from the corners (each corner contributes \( \frac{1}{8} \))
- 1 Y particle from the body center

### Step 5: Calculate the effective number of X and Y particles after removal
After removing the particles:
- Remaining X particles:
- Original: 7
- Removed: 2
- Remaining: \( 7 - 2 = 5 \)

- Remaining Y particles:
- Original: 4
- Removed: 1
- Remaining: \( 4 - 1 = 3 \)

### Step 6: Determine the effective number of particles
Now we need to express the remaining particles in terms of effective contributions:
- For X:
- 5 X particles remain.
- For Y:
- 3 Y particles remain.

### Step 7: Write the formula of the crystal
The ratio of X to Y is:
\[ \text{Ratio} = \frac{5}{3} \]

To express this in a whole number ratio, we can multiply by 4 to avoid fractions:
- X: \( 5 \times 4 = 20 \)
- Y: \( 3 \times 4 = 12 \)

Thus, the formula of the crystal becomes:
\[ \text{Formula} = X_5Y_4 \]

### Final Answer
The formula of the crystal is \( X_5Y_4 \).

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