QB_PH_SEC7_024: Crystal Structure & Lattice Dynamics

What is the volume (in $\text{Å}^3$) of the first Brillouin zone of a simple cubic crystal having real-space lattice constant $a = 2.0\text{ Å}$? Take $\pi = 3.1416$. (Round off to the nearest integer)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Real-Space Unit Cell Volume**: For simple cubic lattice of edge $a = 2.0\text{ Å}$: $V_{\text{real}} = a^3 = (2.0\text{ Å})^3 = 8.0\text{ Å}^3$ 2. **Reciprocal Lattice Primitive Unit Cell**: Primitive reciprocal vectors are $\vec{b}_1 = \frac{2\pi}{a}\hat{x}, \vec{b}_2 = \frac{2\pi}{a}\hat{y}, \vec{b}_3 = \frac{2\pi}{a}\hat{z}$. 3. **First Brillouin Zone Volume Relation**: $V_{\text{BZ}} = |\vec{b}_1 \cdot (\vec{b}_2 \times \vec{b}_3)| = \left(\frac{2\pi}{a}\right)^3 = \frac{8\pi^3}{a^3}$ 4. **Numerical Evaluation**: $V_{\text{BZ}} = \frac{8 \pi^3}{8} = \pi^3 \approx (3.14159)^3 = 31.0063\text{ Å}^{-3}$ 5. **Nearest Integer**: 31.