Physics 2025 Paper II 50 marks Calculate

Q6

(a) The total binding energies of ¹⁵₈O, ¹⁶₈O and ¹⁷₈O are 111·96 MeV, 127·62 MeV and 131·76 MeV respectively. Determine the energy gap between 1p₁/₂ and 1d₅/₂ neutron shells for the nuclide whose mass number is close to 16. 15 marks (b) State the basic assumption of single-particle shell model. How do the centrifugal and spin-orbit terms remove the degeneracy of three-dimensional spherical harmonic oscillator? 10+10=20 marks (c) Explain the various leptonic family members. What is leptonic number conservation? Based on this conservation law, tell whether the following reactions are possible or not: (i) π⁻ → μ⁻ + ν̄ₜ (ii) n → p⁺ + e⁻ + ν̄ₑ 15 marks

हिंदी में प्रश्न पढ़ें

(a) ¹⁵₈O, ¹⁶₈O और ¹⁷₈O की कुल बंधन ऊर्जाएँ क्रमशः: 111·96 MeV, 127·62 MeV और 131·76 MeV हैं। उस न्यूक्लाइड के लिए 1p₁/₂ और 1d₅/₂ न्यूट्रॉन कोशों के बीच के ऊर्जा अंतराल का निर्धारण कीजिए, जिसकी द्रव्यमान संख्या 16 के करीब है। 15 (b) एकल-कण कोश मॉडल का मूल अभिगृहीत बताइए। अपकेन्द्रीय एवं प्रचक्रण-क्ष पद, त्रिविमीय गोलीय सरल आवर्ती दोलक की अपभ्रष्टता (डीजेनेरेसी) को किस प्रकार समाप्त कर देते हैं? 10+10=20 (c) लेप्टोनिक परिवार के विभिन्न सदस्यों की व्याख्या कीजिए। लेप्टोनिक संख्या संरक्षण क्या है? इस संरक्षण नियम के आधार पर बताइए कि निम्नलिखित अभिक्रियाएँ संभव हैं या नहीं: (i) π⁻ → μ⁻ + ν̄ₜ (ii) n → p⁺ + e⁻ + ν̄ₑ 15

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Approach

Begin with a clear statement of the shell model assumptions for part (b), then proceed to calculate the energy gap in part (a) using binding energy differences—this carries the highest marks (15) and requires careful identification of neutron shell transitions. Allocate approximately 35% effort to (a), 40% to (b) given its theoretical depth (20 marks), and 25% to (c). Structure as: (b) theoretical foundation → (a) numerical application → (c) particle physics application with conservation law verification.

Key points expected

  • Part (a): Correct identification that ¹⁶O has closed shells (Z=N=8), and that ¹⁵O has a 1d₅/₂ neutron hole while ¹⁷O has a 1d₅/₂ neutron particle; calculation of energy gap using BE(¹⁶O) - BE(¹⁵O) and BE(¹⁷O) - BE(¹⁶O) with proper averaging
  • Part (b): Statement of independent particle motion in a mean potential; explanation of how l(l+1)ħ²/2mr² centrifugal term lowers energy for higher l at same n, and spin-orbit coupling ξ(r)L·S splits j = l ± 1/2 states with inverted ordering for natural parity
  • Part (b): Clear derivation or explanation of the spin-orbit term origin from Dirac equation or phenomenological potential, showing how it creates the shell structure magic numbers 2, 8, 20, 28, 50, 82, 126
  • Part (c): Enumeration of three lepton families (e, μ, τ) with their neutrinos and antiparticles; definition of lepton number Lₑ, Lᵤ, Lₜ with L = +1 for leptons, -1 for antileptons, 0 for hadrons
  • Part (c)(i): Analysis showing π⁻ → μ⁻ + ν̄ₜ violates tau lepton number conservation (Lₜ: 0 → 0 + (-1)), hence forbidden; correct allowed decay is π⁻ → μ⁻ + ν̄ᵤ
  • Part (c)(ii): Verification that n → p⁺ + e⁻ + ν̄ₑ conserves baryon number, charge, and all lepton numbers (Lₑ: 0 → 0 + 1 + (-1) = 0), hence allowed as standard beta decay

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Demonstrates precise understanding of shell model magic numbers, correctly identifies ¹⁶O as doubly magic reference; accurately states independent particle assumption and mean field concept; correctly enumerates all six lepton flavors with proper quantum numbers; identifies all conservation laws in reactions (i) and (ii)Identifies basic shell structure but confuses particle/hole states or misstates magic numbers; states shell model assumption vaguely; lists leptons but misses quantum numbers or families; applies lepton conservation mechanically with minor errors in reaction analysisFundamental misconceptions about shell model (e.g., treats nucleons as strongly coupled); fails to identify ¹⁶O as closed shell reference; omits or garbles lepton families; completely misapplies conservation laws or ignores them
Derivation rigour20%10Shows complete derivation of spin-orbit splitting from L·S = [j(j+1)-l(l+1)-s(s+1)]ħ²/2; derives energy level ordering with proper mathematical justification; clearly shows how centrifugal barrier modifies radial wavefunctions and energy eigenvaluesStates spin-orbit formula without full derivation; mentions j = l ± 1/2 splitting qualitatively; describes centrifugal effect in words without mathematical expression; partial mathematical treatment with gaps in logicNo mathematical treatment of spin-orbit or centrifugal terms; purely descriptive answer; incorrect formulas or confused notation; fails to connect quantum numbers to energy level structure
Diagram / FBD15%7.5Draws clear energy level diagram showing 3D harmonic oscillator degeneracy, then splitting by centrifugal term, then further splitting by spin-orbit coupling with proper j-values and inverted parity ordering; labels 1p₁/₂, 1d₅/₂ explicitly for part (a) contextSketchy level diagram showing some splitting but missing key features like inverted order or proper j-labeling; diagram present but lacks clarity in showing the two-stage degeneracy removalNo diagram provided; or diagram completely wrong showing incorrect level ordering; confuses proton and neutron shells; mislabels quantum numbers
Numerical accuracy20%10Correctly calculates Sₙ(¹⁶O) = 127.62 - 111.96 = 15.66 MeV and Sₙ(¹⁷O) = 131.76 - 127.62 = 4.14 MeV; identifies 1p₁/₂ to 1d₅/₂ gap as approximately 11.52 MeV (or recognizes ¹⁵O has 1p₁/₂ hole giving gap ≈ 15.66 MeV with proper interpretation); shows clear arithmetic with unitsCorrect arithmetic but wrong interpretation of which energy difference corresponds to which shell transition; or calculates one separation energy correctly but makes error in second; final answer in wrong ballpark due to sign errorMajor calculation errors; uses wrong binding energy values; adds instead of subtracting; no units; completely nonsensical final answer (e.g., negative energy gap or MeV values orders of magnitude wrong)
Physical interpretation20%10Explains why energy gap measurement requires doubly magic nucleus as reference; connects large 1d₅/₂-1p₁/₂ gap to stability of ¹⁶O and anomalous binding of ¹⁷O; relates spin-orbit strength to nuclear surface; explains why reaction (i) is forbidden in terms of weak interaction selection rules and family conservationStates numerical result without explaining why ¹⁶O is special; mentions spin-orbit effect without physical insight about nuclear surface; states conservation law violations without explaining interaction physicsNo physical interpretation provided; numbers presented without context; fails to explain why certain decays occur or are suppressed; no connection between shell structure and nuclear properties

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