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 find the formula of the compound formed by the atoms A, B, and C in the given cubic solid lattice, we will follow these steps:

### Step 1: Identify the positions of the atoms in the cubic lattice
- **Atom A**: Located at the body center of the cube.
- **Atom B**: Located at the corners and edge centers of the cube.
- **Atom C**: Located at the face centers of the cube.

### Step 2: Count the number of each type of atom in the unit cell before removal
- **Atom A**: There is 1 atom at the body center.
- **Atom B**:
- There are 8 corners, each contributing \( \frac{1}{8} \) of an atom, giving \( 8 \times \frac{1}{8} = 1 \) atom from corners.
- There are 12 edges, each contributing \( \frac{1}{4} \) of an atom, giving \( 12 \times \frac{1}{4} = 3 \) atoms from edges.
- Total for B = \( 1 + 3 = 4 \) atoms.
- **Atom C**: There are 6 face centers, each contributing \( \frac{1}{2} \) of an atom, giving \( 6 \times \frac{1}{2} = 3 \) atoms.

### Step 3: Total number of atoms before removal
- Total for A = 1
- Total for B = 4
- Total for C = 3

### Step 4: Identify and remove atoms from the specified axes
- **Two 4-fold axes**: Each 4-fold axis passes through the body center and two face centers.
- Removing atoms from these axes:
- From the body center (A): 1 atom is removed.
- From the face centers (C): 2 atoms are removed (1 from each of the two face centers on the 4-fold axes).

- **One 2-fold axis**: This axis passes through the edge centers.
- Removing atoms from this axis:
- From the edge centers (B): 2 atoms are removed (1 from each of the two edge centers on the 2-fold axis).

### Step 5: Count the remaining atoms after removal
- **Remaining A**: \( 1 - 1 = 0 \) (no A atoms left)
- **Remaining B**: \( 4 - 2 = 2 \) (2 B atoms left)
- **Remaining C**: \( 3 - 2 = 1 \) (1 C atom left)

### Step 6: Write the formula of the compound
The remaining atoms are 2 B atoms and 1 C atom. Therefore, the formula of the compound is:
\[ \text{B}_2\text{C} \]

### Summary
The final formula of the compound after the removal of specified atoms is:
\[ \text{B}_2\text{C} \]

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