Question

A cubic lattice has atoms of A at the body centre, atoms of B at the corners of the cube and atoms C at all the face centres. What is its formula?

• A. \(ABC_3\)
• B. \(ABC_2\)
• C. \(AB_2C\)
• D. \(A_2BC_3\)

Hint:

Remember the contributions of atoms in a unit cell: \[ \text{Corner atom}=\frac{1}{8} \] \[ \text{Face-centred atom}=\frac{1}{2} \] \[ \text{Body-centred atom}=1 \] These values are frequently used to determine the formula of crystalline solids.

Answer 1

The Correct Option is A

Step 1: Calculate the contribution of atom A.
Atom A is present at the body centre of the cube.
A body-centred atom belongs completely to the unit cell.
Therefore, \[ \text{Number of A atoms per unit cell}=1 \]

Step 2: Calculate the contribution of atom B.
Atoms B are present at the \(8\) corners of the cube.
Each corner atom is shared by \(8\) unit cells.
Hence, contribution of one corner atom is \[ \frac{1}{8} \] Therefore, \[ \text{Number of B atoms} = 8\times\frac{1}{8} = 1 \]

Step 3: Calculate the contribution of atom C.
Atoms C are present at all \(6\) face centres.
Each face-centred atom is shared by \(2\) adjacent unit cells.
Hence, contribution of one face-centred atom is \[ \frac{1}{2} \] Therefore, \[ \text{Number of C atoms} = 6\times\frac{1}{2} = 3 \]

Step 4: Determine the empirical formula.
Thus, the number of atoms per unit cell are: \[ A:B:C = 1:1:3 \] Hence, the formula is \[ \boxed{ABC_3} \]

Step 5: Final conclusion.
Therefore, the formula of the cubic lattice is \[ \boxed{ABC_3} \] Hence, the correct option is \[ \boxed{(1)} \]