QB_PH_SEC7_001: Crystal Structure & Lattice Dynamics

The reciprocal lattice of a Body-Centered Cubic (BCC) real space lattice of lattice parameter $a$ is

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Real-Space BCC Primitive Vectors**: $\vec{a}_1 = \frac{a}{2}(\hat{x} + \hat{y} - \hat{z}), \quad \vec{a}_2 = \frac{a}{2}(-\hat{x} + \hat{y} + \hat{z}), \quad \vec{a}_3 = \frac{a}{2}(\hat{x} - \hat{y} + \hat{z})$ 2. **Reciprocal Basis Calculation**: Primitive volume $\Omega = |\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)| = a^3/2$. $\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\Omega} = \frac{2\pi}{a}(\hat{x} + \hat{y})$ $\vec{b}_2 = \frac{2\pi}{a}(\hat{y} + \hat{z}), \quad \vec{b}_3 = \frac{2\pi}{a}(\hat{z} + \hat{x})$ 3. **Lattice Identification**: These vectors define the primitive translation vectors of a Face-Centered Cubic (FCC) lattice whose conventional unit cell length is $A' = 2 \times (2\pi / a) = 4\pi/a$. 4. **Conclusion**: Reciprocal lattice is FCC with cube edge $4\pi / a$. Option (A) is correct.