Subtopic: Bravais Lattices, Miller Indices & Reciprocal Lattice
Question Statement:
Consider a crystal that has a basis of one atom. Its primitive vectors are a1⃗⃗⃗⃗ = aî ,
a2⃗⃗⃗⃗ = aĵ , a3⃗⃗⃗⃗ =
a
2 (î + ĵ + k̂), where î, ĵ, k̂ are the unit vectors in the x, y and z
directions of the Cartesian coordinate system and a is a positive constant. Which
one of the following is the correct option regarding the type of the Bravais lattice?
Options:
(A) It is BCC and the volume of the primitive cell is
a3
2
(B) It is FCC and the volume of the primitive cell is
a3
4
(C) It is BCC and the volume of the primitive cell is
a3
8
(D) It is FCC and the volume of the primitive cell is a3
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.12)**
**1. Crystallography & Primitive Cell Volume**:
- **Given Primitive Translation Vectors**:
$\vec{a}_1 = a\hat{i}, \qquad \vec{a}_2 = a\hat{j}, \qquad \vec{a}_3 = \frac{a}{2}(\hat{i} + \hat{j} + \hat{k})$
- Objective: Calculate primitive cell volume $V_p$ and identify the crystal Bravais lattice.
**2. Step-by-Step Mathematical Derivation**:
- Step 1: Compute cross product $\vec{a}_2 \times \vec{a}_3$:
$\vec{a}_2 \times \vec{a}_3 = (a\hat{j}) \times \left[\frac{a}{2}\hat{i} + \frac{a}{2}\hat{j} + \frac{a}{2}\hat{k}\right]$
Using standard basis cross products:
$\hat{j} \times \hat{i} = -\hat{k}, \quad \hat{j} \times \hat{j} = 0, \quad \hat{j} \times \hat{k} = \hat{i}$
$\vec{a}_2 \times \vec{a}_3 = \frac{a^2}{2}(-\hat{k} + \hat{i}) = \frac{a^2}{2}(\hat{i} - \hat{k})$
- Step 2: Compute scalar triple product $V_p = |\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)|$:
$V_p = \left|(a\hat{i}) \cdot \frac{a^2}{2}(\hat{i} - \hat{k})\right| = \frac{a^3}{2}$
- Step 3: Lattice Identification:
- The vector $\vec{a}_3 = \frac{a}{2}\hat{i} + \frac{a}{2}\hat{j} + \frac{a}{2}\hat{k}$ connects the origin $(0, 0, 0)$ to the center of the conventional cubic cell $(a/2, a/2, a/2)$.
- This vector geometry defines the Body-Centered Cubic (BCC) Bravais lattice with primitive volume $\frac{a^3}{2}$.
**3. Official Verification**:
The verified official IIT answer is **Option (A)** (`It is BCC and the volume of the primitive cell is a3/2`).
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