Physics 2022 Paper II 50 marks Explain

Q7

(a) Explain the drawbacks of Einstein's theory of specific heat and how it was overcome by Debye. 20 marks (b) A neutron and a proton can undergo radiative capture at rest: n + p → d + γ Find the energy of the photon emitted in this capture. Is the recoil of the deuteron important? 15 marks (c) Compare the dependence of resistance on temperature of a superconductor with that of a normal conductor. Describe briefly the formation of Cooper pairs. 15 marks

हिंदी में प्रश्न पढ़ें

(a) आइंस्टीन के विशिष्ट ऊष्मा सिद्धांत की कमियों की व्याख्या कीजिए और यह भी समझाइए कि कैसे डिबाय के द्वारा इसे दूर किया गया था। 20 (b) दर्शाइए कि एक न्यूट्रॉन और एक प्रोटॉन विराम अवस्था में विकिरणी प्रग्रहण कर सकते हैं: n + p → d + γ इसे प्रग्रहण में उत्सर्जित फोटॉन की ऊर्जा प्राप्त कीजिए। क्या ड्यूटरॉन का प्रतिक्षेप महत्वपूर्ण है? 15 (c) एक अतिचालक के तापक्रम पर एक सामान्य चालक के साथ प्रतिरोध की निर्भरता की तुलना कीजिए। कूपर-युग्मों के निर्माण का संक्षेप में वर्णन कीजिए। 15

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Directive word: Explain

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How this answer will be evaluated

Approach

Begin with a brief introduction acknowledging the historical development of quantum theories of specific heat. For part (a), spend approximately 40% of effort explaining Einstein's assumption of independent oscillators and Debye's improvement using a continuous spectrum of phonon modes with the Debye cutoff. For part (b), allocate 30% to derive the photon energy using mass-energy equivalence and binding energy of deuteron (~2.224 MeV), then assess recoil energy significance. For part (c), use remaining 30% to contrast resistance-temperature curves and explain Cooper pair formation via electron-phonon interaction. Conclude by noting the unifying theme of quantum effects in condensed matter.

Key points expected

  • Part (a): Einstein's theory assumes all atoms vibrate with same frequency ν_E, leading to C_V → 0 exponentially at low T instead of T³ law; Debye treats solid as continuous elastic medium with cutoff frequency ν_D, introducing Debye temperature Θ_D and density of states g(ν) ∝ ν²
  • Part (a): Debye's integral expression for specific heat and its correct prediction of C_V ∝ T³ at T << Θ_D and Dulong-Petit law at T >> Θ_D; mention limitations at intermediate T
  • Part (b): Application of mass-energy conservation: E_γ = [m_n + m_p - m_d]c² = B_d (binding energy of deuteron) ≈ 2.224 MeV; recoil energy E_recoil = E_γ²/(2m_dc²) ≈ 1.3 keV is negligible (~0.06%)
  • Part (c): Normal conductor: resistance increases with T due to enhanced phonon scattering (ρ ∝ T for T > Θ_D, ρ ∝ T⁵ at low T); superconductor: zero resistance below T_c with sharp transition
  • Part (c): Cooper pair formation via attractive interaction mediated by phonon exchange; electrons with opposite k and spin form bound state with energy gap; coherence length and BCS theory essence

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness22%11Accurately identifies Einstein's independent oscillator assumption vs Debye's phonon continuum; correctly states deuteron binding energy and BCS pairing mechanism; no confusion between bosonic phonons and fermionic electronsBasic understanding of Einstein and Debye differences; correct photon energy formula but may miss recoil significance; general awareness of Cooper pairs without mechanism clarityConfuses Einstein with Debye assumptions; treats recoil as significant or ignores it; describes superconductivity as merely 'zero resistance' without quantum basis
Derivation rigour20%10Full derivation of Debye specific heat integral with proper limits; explicit calculation of recoil energy showing E_recoil/E_γ << 1; clear energy gap equation for Cooper pairsStates Debye formula without full integration; calculates photon energy correctly but recoil analysis incomplete; mentions Cooper pair binding qualitativelyNo mathematical derivation; incorrect energy conservation; no mention of phonon-mediated attraction matrix element
Diagram / FBD16%8Sketch of phonon dispersion relation with Debye cutoff; C_V vs T/Θ showing Einstein, Debye, and experimental curves; resistance-temperature comparison plot with T_c marked; Feynman diagram or schematic of Cooper pair formationOne or two relevant plots (e.g., C_V vs T curves only); basic resistance comparison without proper scalingNo diagrams or irrelevant sketches; confused labeling of axes
Numerical accuracy20%10Precise values: Θ_D for typical solids (e.g., Cu: 343 K), deuteron binding energy 2.224 MeV, recoil energy ~1.3 keV, typical T_c values (Nb: 9.2 K, YBCO: 92 K); correct order-of-magnitude estimatesApproximate photon energy correct; order of magnitude for recoil acceptable; generic T_c mentioned without specificityWrong energy scale (eV instead of MeV); significant calculation errors; no numerical values provided
Physical interpretation22%11Explains why Einstein fails at low T (neglects collective modes) and Debye succeeds; justifies recoil neglect via momentum conservation; connects Cooper pairing to macroscopic quantum coherence and Meissner effect; references Indian contributions (e.g., S.N. Bose's legacy in quantum statistics, BCS theory impact)General explanation of improvements; basic recoil argument; standard description of superconductivityNo physical insight; mere formula listing; fails to connect microscopics to macroscopic phenomena

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