Q6
(a) Show that for a specific value (n, l), there exists a large degeneracy relative to the energy characterized by the quantum number (N). Find the shell closures and the magic numbers predicted by harmonic oscillator potential. (15 marks) (b) Considering atoms hard, uniform spheres, find the number of atoms per unit cell and packing fraction for simple cubic, bcc and fcc structures. (15 marks) (c) Write down the Weizsäcker semi-empirical mass formula and explain each term. Explain why ₉₂²³⁸U nuclide is an α-emitter and not a β⁻-emitter ? (10+10 marks)
हिंदी में प्रश्न पढ़ें
(a) दर्शाइए कि एक विशिष्ट स्तर (n, l) के लिए क्वांटम संख्या (N) की विशेषता वाली ऊर्जा के सापेक्ष एक बड़ी अपभ्रष्टता (डि-जनरेसी) होती है। आवर्ती दोलक विभव द्वारा अनुमानित शेल क्लोजर और जादुई संख्याओं को ज्ञात कीजिए। (15 अंक) (b) परमाणुओं को कठोर, एकसमान गोले मानते हुए साधारण घन, बीसीसी और एफसीसी संरचनाओं के लिए प्रति एकक कोष्ठिका (सेल) परमाणुओं की संख्या और संकुलन गुणांक ज्ञात कीजिए। (15 अंक) (c) वाइज़ैकर के अर्ध अनुभवसिद्ध द्रव्यमान सूत्र को लिखिए और प्रत्येक पद की व्याख्या कीजिए। समझाइए कि क्यों ₉₂²³⁸U एक α-उत्सर्जक है और क्यों यह β⁻-उत्सर्जक नहीं है ? (10+10 अंक)
Directive word: Derive
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How this answer will be evaluated
Approach
Begin with a concise introduction linking nuclear structure, crystallography and nuclear stability. For part (a), derive the degeneracy formula g_N = (N+1)(N+2)/2 and identify magic numbers 2, 8, 20, 28, 50, 82, 126 using N = 2(n-1)+l. For part (b), calculate packing fractions π/6, π√3/8 and π√2/6 with clear unit cell diagrams. For part (c), state the full Weizsäcker formula with all seven terms, then apply Q-value calculations to show ²³⁸U prefers α-decay over β⁻ due to higher binding energy per nucleon and Coulomb barrier considerations. Allocate ~35% effort to (a), ~30% to (b), and ~35% to (c) including the comparative analysis.
Key points expected
- Part (a): Derivation of total quantum number N = 2(n-1) + l and proof that degeneracy g_N = (N+1)(N+2)/2 for 3D isotropic harmonic oscillator
- Part (a): Enumeration of shell closures at N = 0,1,2,3,4,5,6 leading to magic numbers 2, 8, 20, 40, 70, 112, 168; correction to observed magic numbers 2, 8, 20, 28, 50, 82, 126 with spin-orbit coupling mention
- Part (b): Calculation of atoms per unit cell (1, 2, 4) and packing fractions (π/6 ≈ 0.52, π√3/8 ≈ 0.68, π√2/6 ≈ 0.74) for SC, BCC, FCC with proper geometric derivation involving atomic radius and lattice parameter relations
- Part (c): Complete Weizsäcker formula with volume, surface, Coulomb, asymmetry, pairing, shell correction and pairing terms with correct coefficients; physical explanation of each term
- Part (c): Q-value calculation for α-decay (Q_α ≈ 4.27 MeV, spontaneous) versus β⁻-decay energetics; explanation using N/Z ratio, Coulomb repulsion dominance, and liquid drop model predictions for heavy nuclei (A>210)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 22% | 11 | Correctly identifies N = 2(n-1)+l as the total oscillator quantum number; accurately states all seven terms of Weizsäcker formula with proper signs; correctly explains why ²³⁸U (Z=92, N=146) lies above stability line and α-decay reduces Coulomb energy | Identifies basic quantum numbers but confuses n, l, N relationships; writes semi-empirical formula with 5-6 terms with minor coefficient errors; gives generic explanation for α-emission without specific Q-value comparison | Confuses principal quantum number with radial nodes; omits Coulomb or asymmetry terms in mass formula; claims β⁻-emission is impossible without energy calculation or misidentifies decay mode |
| Derivation rigour | 24% | 12 | Complete step-by-step derivation of g_N = Σ(2l+1) over l = 0,2,4...N or 1,3,5...N; rigorous geometric derivation of packing fractions using atomic radius relations (r = a/2, r = a√3/4, r = a√2/4); shows summation yields closed form | States degeneracy formula without full summation; derives one packing fraction correctly but uses proportional reasoning for others; skips algebraic steps in geometric calculations | States results without derivation; uses incorrect geometric relationships (e.g., body diagonal confusion); no mathematical justification for magic number sequence |
| Diagram / FBD | 16% | 8 | Clear 3D unit cell diagrams for SC, BCC, FCC with atoms shown as hard spheres touching along edges, body diagonal and face diagonal respectively; includes nuclear energy level diagram showing shell filling with spin-orbit splitting for magic numbers | 2D representations of unit cells with partial 3D indication; omits contact geometry or shows incorrect touching directions; textual description of energy levels without diagram | No diagrams; or incorrect diagrams showing overlapping spheres, wrong coordination numbers; confusing sketches without labels |
| Numerical accuracy | 20% | 10 | Exact values: g_N calculations for N=0-6; packing fractions to 3 decimal places (0.524, 0.680, 0.740); correct Q_α for ²³⁸U → ²³⁴Th + α using mass tables or binding energies; accurate magic number identification | Correct formulas with arithmetic errors in final decimals; approximate packing fractions (0.52, 0.68, 0.74); correct order of stability but wrong Q-value magnitude | Order-of-magnitude errors; incorrect formulas leading to wrong values; confuses atomic mass with mass number in calculations |
| Physical interpretation | 18% | 9 | Explains why harmonic oscillator alone fails for higher magic numbers (28,50,82,126) requiring spin-orbit coupling (Maria Goeppert Mayer); connects packing efficiency to metallic properties (Fe BCC vs Cu FCC); relates α-decay to tunneling through Coulomb barrier and Gamow factor | Mentions spin-orbit coupling as correction without elaboration; states FCC is close-packed without explaining ductility connection; gives energy release argument without barrier penetration concept | No physical insight; treats all terms as purely mathematical; fails to connect nuclear structure to observables like nuclear stability or material properties |
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