To solve the problem, we will follow these steps:
### Step 1: Identify the arrangement of atoms in the FCC lattice
In a face-centered cubic (FCC) lattice:
- **A atoms** are located at the corners.
- **B atoms** are located at the face centers.
- **C atoms** are located in the octahedral voids.
- **D atoms** are located in half of the tetrahedral voids.
### Step 2: Count the contributions of each type of atom in the unit cell
1. **A atoms (corners)**: There are 8 corners, each contributing \( \frac{1}{8} \). Therefore, total contribution of A:
\[
8 \times \frac{1}{8} = 1 \text{ A atom}
\]
2. **B atoms (face centers)**: There are 6 face centers, each contributing \( \frac{1}{2} \). Therefore, total contribution of B:
\[
6 \times \frac{1}{2} = 3 \text{ B atoms}
\]
3. **C atoms (octahedral voids)**: There are 4 octahedral voids, contributing 1 atom each. Therefore, total contribution of C:
\[
4 \text{ C atoms}
\]
4. **D atoms (tetrahedral voids)**: There are 8 tetrahedral voids, but only half are occupied, so the contribution is:
\[
\frac{8}{2} = 4 \text{ D atoms}
\]
### Step 3: Write the initial formula of the compound
Without any defects, the formula of the compound is:
\[
A_1 B_3 C_4 D_4
\]
### Step 4: Analyze the defect in the crystal
The problem states that all particles along one body diagonal of each unit cell are missing. A body diagonal in an FCC unit cell includes:
- 1 corner atom (A)
- 1 face-centered atom (B)
- 1 octahedral void atom (C)
- 1 tetrahedral void atom (D)
### Step 5: Calculate the remaining atoms after the defect
1. **A atoms**: 1 corner atom is missing, so:
\[
1 - \frac{1}{4} = \frac{3}{4} \text{ A atoms remaining}
\]
2. **B atoms**: No B atoms are missing, so:
\[
3 \text{ B atoms remaining}
\]
3. **C atoms**: 1 C atom is missing, so:
\[
4 - 1 = 3 \text{ C atoms remaining}
\]
4. **D atoms**: 1 D atom is missing, so:
\[
4 - 1 = 3 \text{ D atoms remaining}
\]
### Step 6: Write the new formula of the compound
The new formula after accounting for the defect is:
\[
\frac{3}{4} A, 3 B, 3 C, 3 D
\]
### Step 7: Eliminate the fractions
To eliminate the fraction, multiply all coefficients by 4:
\[
3 A, 12 B, 12 C, 12 D
\]
### Step 8: Simplify the formula
Now, divide all coefficients by 3 to simplify:
\[
A_1 B_4 C_4 D_4
\]
### Final Answer
The resulting formula of the compound is:
\[
\text{A}_1 \text{B}_4 \text{C}_4 \text{D}_4
\]
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