Q8
(a) Sketch the dc load line for the circuit shown. (10 marks) (b) A solid contains a dilute concentration of Nd³⁺ ions, each of which possess three 4f electrons. Assuming that there are 10²⁵ m⁻³ of these ions, calculate the magnetic susceptibility of the sample at 1K. (20 marks) (c) Explain the phenomenon of internal conversion and define the internal conversion coefficient. Discuss under what conditions the internal conversion process becomes important. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) दिखाये गये परिपथ के लिए डी.सी. भार रेखा को रेखांकित कीजिए । (10 अंक) (b) एक ठोस में Nd³⁺ आयनों की तनु सांद्रता होती है जिनमें से प्रत्येक में तीन 4f इलेक्ट्रॉन होते हैं । यह मानते हुए कि यह आयन 10²⁵ m⁻³ हैं, 1K पर नमूने की चुंबकीय प्रवृत्ति की गणना कीजिए । (20 अंक) (c) आंतरिक रूपांतरण की घटना की व्याख्या कीजिए और अभ्यंतर रूपांतरण गुणांक को परिभाषित कीजिए । चर्चा कीजिए कि किन परिस्थितियों में आंतरिक रूपांतरण प्रक्रिया महत्वपूर्ण हो जाती है । (20 अंक)
Directive word: Calculate
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How this answer will be evaluated
Approach
Begin with the directive 'calculate' for part (b) which carries the highest marks (20), then address 'sketch' for part (a) (10 marks) and 'explain/discuss' for part (c) (20 marks). Allocate approximately 35% time to part (b) for rigorous derivation of magnetic susceptibility using Hund's rules and Curie law, 25% to part (a) for accurate load line construction with proper intercepts, and 40% to part (c) for comprehensive explanation of internal conversion with coefficient definition and conditions. Structure as: (a) circuit analysis → load line sketch; (b) quantum mechanical derivation → numerical substitution; (c) phenomenon explanation → coefficient definition → conditions discussion.
Key points expected
- Part (a): Correct identification of DC load line endpoints (cut-off and saturation points) from circuit parameters, proper axes labeling (V_CE vs I_C), and accurate linear plot showing Q-point placement
- Part (b): Application of Hund's rules to determine total angular momentum J for Nd³⁺ (4f³ configuration), calculation of Landé g-factor, derivation of effective magnetic moment, and correct substitution into Curie law formula χ = μ₀Nμ_eff²/(3k_BT) with proper unit handling
- Part (b): Correct numerical evaluation yielding χ ≈ 4.5 × 10⁻³ (dimensionless SI) or equivalent, showing all intermediate steps for μ_eff, μ_B conversion, and temperature dependence
- Part (c): Clear explanation of internal conversion as radiationless nuclear de-excitation with energy transfer to orbital electron, proper definition of α_IC = N_e/N_γ (conversion electrons to gamma photons ratio)
- Part (c): Discussion of conditions favoring internal conversion: high Z nuclei, low transition energy, large change in nuclear angular momentum (ΔL ≥ 2), and Mössbauer spectroscopy relevance for Indian nuclear physics research
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | For (a): correctly identifies load line as intersection of transistor characteristic with circuit constraint; for (b): applies Hund's rules accurately to 4f³ configuration yielding J=9/2, L=6, S=3/2; for (c): distinguishes internal conversion from Auger effect and beta decay with correct energy conservation | Identifies basic load line concept and performs susceptibility calculation with minor errors in Hund's rules application; defines internal conversion but confuses with similar phenomena | Misidentifies load line as AC load line or confuses with bias line; applies classical instead of quantum treatment for magnetic moment; describes internal conversion as gamma emission |
| Derivation rigour | 20% | 10 | For (b): complete derivation from Russell-Saunders coupling through Landé g-factor to Curie law with explicit μ_eff = g_J√[J(J+1)]μ_B substitution; for (c): derives α_IC from transition matrix elements showing E0 and M0 transitions | Shows main derivation steps for susceptibility with some skipped algebraic steps; states Curie law without full derivation; gives definition without derivation for coefficient | Jumps to final formula without derivation; plugs numbers without showing intermediate steps; fails to define coefficient mathematically |
| Diagram / FBD | 15% | 7.5 | For (a): neat load line with labeled axes (V_CE horizontal, I_C vertical), marked V_CC and V_CC/R_C intercepts, Q-point indication, and transistor output characteristics family shown; for (c): clear energy level diagram showing nuclear transition and electron ejection process | Sketch shows basic load line with intercepts but missing labels or incorrect axes orientation; minimal or no diagram for part (c) | No diagram for part (a) or completely incorrect sketch; diagram missing essential features like intercepts or showing AC load line instead |
| Numerical accuracy | 25% | 12.5 | For (b): precise calculation with g_J = 8/11, μ_eff = 3.62 μ_B, and final χ = 4.47 × 10⁻³ (or 0.355 in CGS emu/cm³) with correct powers of 10, proper constants (μ₀, k_B, N_A), and unit consistency throughout | Correct order of magnitude for susceptibility with minor arithmetic errors or incorrect g-factor; proper handling of ion density conversion | Order of magnitude error in final answer; confuses SI and CGS units; incorrect substitution of temperature or density values; missing μ₀ or using wrong Boltzmann constant |
| Physical interpretation | 20% | 10 | For (a): explains load line as operating limit set by supply voltage and load resistance; for (b): interprets paramagnetic susceptibility temperature dependence and compares with experimental values for Nd compounds; for (c): explains why IC dominates in heavy nuclei like those studied at BARC/Trombay and relevance to Mössbauer spectroscopy calibration | States physical meaning of load line slope and intercepts; notes inverse temperature dependence; mentions high Z favoring IC without detailed explanation | No physical interpretation provided; treats all parts as purely mathematical exercises; fails to connect internal conversion to any practical applications |
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