Three atoms A, B and C crystallize in a cubic solid lattice where A atoms are present at the body centre, B atoms are present at the edge centre as well as at the corners of the cube and C atoms are present at the face centres of the cube. Now if all the atoms are removed from the two 4-fold axis and the one 2-fold axis passing through the cube, then the formula of the compound is

Options

A. `B_7 C_2`
B. A B_2 C_7
C. ABC_2
D. `A_5 C_2`

Answer

To solve the problem, we need to determine the number of atoms of A, B, and C present in the cubic lattice after removing the specified atoms. Let's break it down step by step.

### Step 1: Identify the positions of the atoms
- **A atoms** are located at the body center of the cube.
- **B atoms** are located at the edge centers and corners of the cube.
- **C atoms** are located at the face centers of the cube.

### Step 2: Calculate the number of A atoms
- In a cubic unit cell, there is **1 A atom** at the body center.
- Since no A atoms are removed from the body center, the total number of A atoms remains **1**.

### Step 3: Calculate the number of B atoms
- **B atoms** are located at the edge centers and corners:
- There are **12 edges** in a cube, and each edge contributes **1/4** of an atom (since each edge atom is shared by 4 unit cells).
\[ \text{Total from edges} = 12 \times \frac{1}{4} = 3 \]
- There are **8 corners** in a cube, and each corner contributes **1/8** of an atom (since each corner atom is shared by 8 unit cells).
\[ \text{Total from corners} = 8 \times \frac{1}{8} = 1 \]
- Therefore, the total number of B atoms before removal is:
\[ \text{Total B} = 3 + 1 = 4 \]

### Step 4: Calculate the number of C atoms
- **C atoms** are located at the face centers:
- There are **6 faces** in a cube, and each face contributes **1/2** of an atom (since each face atom is shared by 2 unit cells).
\[ \text{Total C} = 6 \times \frac{1}{2} = 3 \]

### Step 5: Remove atoms from specified axes
- **Removing atoms from the two 4-fold axes**:
- Each 4-fold axis passes through the face centers. Since there are **2 face centers** along these axes, we remove **2 C atoms**.
- **Removing atoms from the one 2-fold axis**:
- The 2-fold axis passes through the body center, removing **1 A atom**.

### Step 6: Calculate remaining atoms after removal
- Remaining A atoms:
\[ \text{Remaining A} = 1 - 1 = 0 \]
- Remaining B atoms:
- No B atoms are removed, so remaining B atoms are still **4**.
- Remaining C atoms:
\[ \text{Remaining C} = 3 - 2 = 1 \]

### Step 7: Write the formula of the compound
- The remaining atoms can be summarized as:
- A: 0
- B: 4
- C: 1
- Therefore, the formula of the compound is:
\[ \text{Formula} = B_4C_1 \]

### Final Answer
The formula of the compound is \( B_4C \).