Subtopic: Bragg's Law & X-ray Diffraction (Laue and Powder)
Question Statement:
In an X-Ray diffraction experiment on a solid with FCC structure, five diffraction
peaks corresponding to (111), (200), (220), (311) and (222) planes are observed
using 1.54 Å X-rays. On using 3 Å X-rays on the same solid, the number of
observed peaks will be ________________
Official Answer Key:
1 to 1
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.62)**
**1. Crystallography Principles & Bragg's Law**:
- In an FCC lattice, allowed diffraction reflections require $(h, k, l)$ to be unmixed (all odd or all even).
- Ordered allowed reflections:
1. $(111) \implies h^2 + k^2 + l^2 = 3$
2. $(200) \implies h^2 + k^2 + l^2 = 4$
3. $(220) \implies h^2 + k^2 + l^2 = 8$
4. $(311) \implies h^2 + k^2 + l^2 = 11$
5. $(222) \implies h^2 + k^2 + l^2 = 12$
- Bragg's condition:
$\lambda = 2d_{hkl}\sin\theta = \frac{2a}{\sqrt{h^2+k^2+l^2}}\sin\theta \implies \sin\theta = \frac{\lambda\sqrt{h^2+k^2+l^2}}{2a} \le 1$
**2. Detailed Step-by-Step Derivation**:
- For $\lambda_1 = 1.54$ Å, exactly 5 peaks are observed. Hence $(222)$ is observable:
$\frac{1.54\sqrt{12}}{2a} \le 1 \implies a \ge \frac{1.54 \times 3.4641}{2} \approx 2.667 \text{ \AA}$
- For $\lambda_2 = 3$ Å on the same crystal:
- For $(111)$: $\sin\theta = \frac{3\sqrt{3}}{2a} \approx \frac{5.196}{2(2.667)} \approx 0.974 \le 1$ (Observable!).
- For $(200)$: $\sin\theta = \frac{3\sqrt{4}}{2a} = \frac{6}{2(2.667)} \approx 1.12 > 1$ (Mathematically impossible, exceeds unity).
- Therefore, only the $(111)$ peak can be observed.
- Total number of observed peaks $= 1$.
**3. Official Verification**:
The verified official IIT Master Answer range is **1 to 1**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.