Q5
(a) An X-ray beam of wavelength ($\lambda_1$) undergoes a first order Bragg reflection at a Bragg angle of 30°. X-ray of wavelength 97 nm undergoes 3rd order reflection at a Bragg angle of 60°. Consider that the two beams are reflected from the same set of planes. Find the value of $\lambda_1$. (10 marks) (b) Using the expression for internal energy $U=3N\frac{\hbar\omega}{e^{\hbar\omega/k_BT}-1}$, show that Einstein specific heat capacity is given by; $C=3R\left(\frac{\hbar\omega}{k_BT}\right)^2\frac{e^{\hbar\omega/k_BT}}{\left(e^{\hbar\omega/k_BT}-1\right)^2}$. Also show that Einstein specific heat capacity given above is proportional to $e^{-\hbar\omega/k_BT}$ at very low temperature. (10 marks) (c) $\rho^\circ$ and $K^\circ$ mesons both decay mostly to $\pi^+$ and $\pi^-$. Explain why the mean lifetime of $\rho^\circ$ is shorter ($\sim$10$^{-23}$s) compared to the mean lifetime of $K^\circ$($\sim$10$^{-10}$s). (10 marks) (d) What are the properties of the particles made up of the following quarks ? (a) $u\bar{d}$ (b) $\bar{u}d$ (c) $dds$ (d) $uss$ (10 marks) (e) What are chain reactions ? What do you mean by critical size of the core in which chain reaction takes place ? (10 marks)
हिंदी में प्रश्न पढ़ें
(a) $\lambda_1$ तरंग-दैर्घ्य का एक एक्सरे किरणपुंज 30° के ब्रैग-कोण पर पहले क्रम के ब्रैग परावर्तन से गुजरता है । 97 nm तरंग-दैर्घ्य का एक्सरे 60° के ब्रैग कोण पर तृतीय क्रम के परावर्तन से गुजरता है । मान लीजिए कि दोनों किरणपुंज (बीम) एक ही तल से परावर्तित होते हैं तो $\lambda_1$ का मान ज्ञात कीजिए । (10 अंक) (b) आंतरिक ऊर्जा के व्यंजक $U=3N\frac{\hbar\omega}{e^{\hbar\omega/k_BT}-1}$ का इस्तेमाल करते हुए दिखाइए कि आइंस्टीन की विशिष्ट ऊष्मा धारिता निम्नलिखित द्वारा निर्धारित है; $C=3R\left(\frac{\hbar\omega}{k_BT}\right)^2\frac{e^{\hbar\omega/k_BT}}{(e^{\hbar\omega/k_BT}-1)^2}$। यह भी दिखाइए कि ऊपर दी गई आइंस्टीन की विशिष्ट ऊष्मा धारिता कम तापमान पर $e^{-\hbar\omega/k_BT}$ के समानुपाती होती है । (10 अंक) (c) $\rho^\circ$ और $K^\circ$ मेसॉन दोनों ही अधिकतर $\pi^+$ और $\pi^-$ में क्षय होते हैं । समझाइए कि $\rho^\circ$ का औसत जीवन काल ($\sim$10$^{-23}$s) $K^\circ$ के औसत जीवन काल ($\sim$10$^{-10}$s) की तुलना में छोटा क्यों है । (10 अंक) (d) निम्नलिखित क्वार्कों से बने हुए कणों के गुण क्या हैं ? (a) $u\bar{d}$ (b) $\bar{u}d$ (c) $dds$ (d) $uss$ (10 अंक) (e) श्रृंखला अभिक्रियाएँ क्या होती हैं ? कोर के क्रांतिक परिमाण से, जिसके भीतर श्रृंखला अभिक्रिया होती है, आप क्या अर्थ निकालते हैं ? (10 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve each sub-part systematically with clear physical reasoning. For (a), apply Bragg's law correctly with order considerations; for (b), derive the specific heat expression rigorously and apply low-T approximation; for (c), explain decay mechanisms using strong vs. weak interaction; for (d), identify hadron properties from quark content; for (e), define chain reaction concepts with criticality condition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e).
Key points expected
- For (a): Correct application of Bragg's law 2d sinθ = nλ to both cases, equating interplanar spacing d to find λ₁ = 0.168 nm or 1.68 Å
- For (b): Proper differentiation of U with respect to T to obtain C, and valid exponential approximation e^(ℏω/k_BT) >> 1 at low T showing C ∝ e^(-ℏω/k_BT)
- For (c): Explanation that ρ⁰ decays via strong interaction (resonance, ~10⁻²³s) while K⁰ decays via weak interaction (strangeness violation, ~10⁻¹⁰s), both to π⁺π⁻ final state
- For (d): Identification of (a) π⁺ (uđ), (b) π⁻ (ūd), (c) Ξ⁰ or neutron-like (dds = -1 charge, strangeness -1), (d) Ξ⁻ (uss: charge -1, strangeness -2, baryon number 1)
- For (e): Definition of chain reaction as self-sustaining neutron-induced fission sequence; critical size as minimum dimension where neutron multiplication factor k=1 (production = loss)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | Correctly identifies Bragg law application for (a), Einstein model assumptions for (b), strong vs. weak interaction selection rules for (c), quark quantum numbers (charge, strangeness, baryon number) for (d), and neutron economy in criticality for (e); no conceptual errors across any sub-part | Mostly correct concepts with minor errors in one sub-part, such as confusing Bragg angle with glancing angle, or incomplete explanation of why weak interaction dominates K⁰ decay | Major conceptual errors in multiple sub-parts, such as applying classical statistics to Einstein model, or confusing meson and baryon properties in quark analysis |
| Derivation rigour | 20% | 10 | Complete step-by-step derivation for (b) showing dU/dT transformation to C_V with proper chain rule application; valid low-T expansion with clear justification; algebraic manipulation in (a) shown explicitly | Derivation mostly correct but skips key intermediate steps or has minor algebraic errors; low-T approximation justified inadequately | Missing derivation steps, incorrect differentiation, or invalid mathematical approximations; no attempt at showing working for numerical or analytical parts |
| Diagram / FBD | 10% | 5 | Clear Bragg diffraction geometry diagram for (a) showing incident/reflected rays, crystal planes, and θ angle; energy level diagram for Einstein model in (b); Feynman diagram or decay schematic for (c); quark structure diagrams for (d); reactor core geometry for criticality in (e) | Diagrams present but poorly labeled or incomplete; missing one or two required diagrams from sub-parts | No diagrams provided where essential for physical understanding, or diagrams with fundamental errors in representation |
| Numerical accuracy | 25% | 12.5 | Precise calculation yielding λ₁ = 0.168 nm (or 1.68 Å) with correct unit conversion; proper handling of sin(30°)=0.5 and sin(60°)=√3/2; consistent significant figures; correct numerical evaluation of dimensionless ratios in (b) | Correct method but arithmetic error in final value, or unit confusion (nm vs. Å); minor calculation mistakes in algebraic manipulation | Incorrect order of magnitude result, wrong trigonometric values, or no numerical answer provided for calculational parts |
| Physical interpretation | 25% | 12.5 | Insightful explanation of why Einstein model fails at low T (exponential freeze-out vs. Debye T³ law), physical significance of characteristic temperature θ_E = ℏω/k_B; clear distinction between resonances and particles; connects critical size to Indian nuclear program (Dhruva, Apsara reactors) or global examples | Basic interpretation provided without depth; mentions physical significance but doesn't elaborate on limitations or extensions of models | Purely mathematical treatment with no physical insight; fails to explain what results mean in experimental or observational context |
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