Subtopic: Crystal Lattices, Band Theory & Superconductivity
Question Statement:
Apart from the acoustic modes, 9 optical modes are identified from the
measurements of phonon dispersions of a solid with chemical formula AnBm, where
A and B denote the atomic species, and n and m are integers. Which of the following
combination of n and m is/are possible?
Options:
(A) n = 1, m = 1
(B) n = 2, m = 2
(C) n = 3, m = 1
(D) n = 4, m = 4
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.54)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Solid State Physics** within *Section 7: Solid State Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Solid State Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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