For a covalently bonded solid consisting of ions of mass m, the binding potential
can be assumed to be given by
U(r) = - 𝜖 (
r
r0
) e
- r
r0 ,
where 𝜖 and r0 are positive constants. What is the Einstein frequency of the
solid in Hz ?
Options:
(A) 1
2𝜋 √
𝜖e
mr0
2
(B) 1
2𝜋 √
𝜖
mer0
2
(C) 1
2𝜋 √ 2𝜖
mer0
2
(D) 1
2𝜋 √
𝜖e
2mr0
2
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.47)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle 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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