Question:
A cubic lattice has A atoms at the body center, B atoms at the corners and C atoms at half of the face centers. The formula of the lattice is:
A cubic lattice has A atoms at the body center, B atoms at the corners and C atoms at half of the face centers. The formula of the lattice is:
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For unit-cell calculations:
\[
\text{Corner atom contribution}=\frac{1}{8}
\]
\[
\text{Face-centered atom contribution}=\frac{1}{2}
\]
\[
\text{Body-centered atom contribution}=1
\]
Always calculate the effective number of atoms first and then convert them into the simplest whole-number ratio.
Updated On: Jun 26, 2026
- \(ABC_2\)
- \(AB_2C_4\)
- \(A_2B_2C_3\)
- \(ABC_3\)
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The Correct Option is C
Solution and Explanation
Step 1: Calculate the contribution of A atoms.
A atoms are present at the body center.
A body-centered atom belongs completely to the unit cell.
Therefore, \[ \text{Number of A atoms} = 1\times 1 = 1 \]
Step 2: Calculate the contribution of B atoms.
B atoms are present at the \(8\) corners of the cube.
Each corner atom contributes \[ \frac{1}{8} \] to the unit cell.
Therefore, \[ \text{Number of B atoms} = 8\times \frac{1}{8} = 1 \]
Step 3: Calculate the contribution of C atoms.
There are \(6\) face centers in a cube.
The question states that C atoms occupy half of the face centers.
Hence, C atoms are present at \[ \frac{6}{2}=3 \] face centers.
Each face-centered atom contributes \[ \frac{1}{2} \] to the unit cell.
Therefore, \[ \text{Number of C atoms} = 3\times \frac{1}{2} = \frac{3}{2} \]
Step 4: Determine the simplest whole-number ratio.
Thus, the effective number of atoms is \[ A:B:C = 1:1:\frac{3}{2} \] Multiplying throughout by \(2\), \[ A:B:C = 2:2:3 \] Therefore, the formula of the lattice is \[ A_2B_2C_3 \]
Step 5: Final conclusion.
Hence, the formula of the cubic lattice is \[ \boxed{A_2B_2C_3} \] Therefore, the correct option is \[ \boxed{(3)} \]
A atoms are present at the body center.
A body-centered atom belongs completely to the unit cell.
Therefore, \[ \text{Number of A atoms} = 1\times 1 = 1 \]
Step 2: Calculate the contribution of B atoms.
B atoms are present at the \(8\) corners of the cube.
Each corner atom contributes \[ \frac{1}{8} \] to the unit cell.
Therefore, \[ \text{Number of B atoms} = 8\times \frac{1}{8} = 1 \]
Step 3: Calculate the contribution of C atoms.
There are \(6\) face centers in a cube.
The question states that C atoms occupy half of the face centers.
Hence, C atoms are present at \[ \frac{6}{2}=3 \] face centers.
Each face-centered atom contributes \[ \frac{1}{2} \] to the unit cell.
Therefore, \[ \text{Number of C atoms} = 3\times \frac{1}{2} = \frac{3}{2} \]
Step 4: Determine the simplest whole-number ratio.
Thus, the effective number of atoms is \[ A:B:C = 1:1:\frac{3}{2} \] Multiplying throughout by \(2\), \[ A:B:C = 2:2:3 \] Therefore, the formula of the lattice is \[ A_2B_2C_3 \]
Step 5: Final conclusion.
Hence, the formula of the cubic lattice is \[ \boxed{A_2B_2C_3} \] Therefore, the correct option is \[ \boxed{(3)} \]
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