Physics 2023 Paper II 50 marks Compulsory Derive

Q5

(a) How could you establish that $\nu_e$ and $\bar{\nu}_e$ are two different particles ? 10 marks (b) What is the age of a fossil that contains 6 g of carbon $^{14}C$ and has a decay rate of 27 decays per minute ? Given : Ratio $\frac{^{14}C}{^{12}C}$=$1.3\times10^{-13}$, Half life $(T_{1/2})$ of $^{14}C$ = 5730 yrs. 10 marks (c) Name the interactions via which the above nuclear decays occur : (i) $K^+ \longrightarrow \Pi^+ + \Pi^+ + \Pi^-$ (ii) $\Pi^+ + p \longrightarrow \Pi^+ + \Pi^+ + n$ (iii) $\Pi^+ + p \longrightarrow \Delta^{++} \longrightarrow \Pi^+ + p$ (iv) $\Sigma^\circ \longrightarrow \Lambda^\circ + \gamma$ (v) $\Sigma^+ \longrightarrow \Lambda^\circ + e^+ + \nu_e$ (vi) $K^- + p \longrightarrow K^+ + K^\circ + \Omega^-$ (vii) $\Pi^\circ \longrightarrow \gamma + e^+ + e^-$ (viii) $\Sigma^- \longrightarrow n + e^- + \bar{\nu}_e$ (ix) $\Lambda^\circ \longrightarrow p + e^- + \bar{\nu}_e$ (x) $e^+ + e^- \longrightarrow \gamma + \gamma$ 10 marks (d) Derive diffraction conditions using reciprocal lattice concept. What are these conditions known as ? 10 marks (e) Show that the Fermi level shifts upward, closer to the conduction band in an n-type semiconductor and shifts downward, closer to the valence band in a p-type semiconductor. 10 marks

हिंदी में प्रश्न पढ़ें

(a) आप किस प्रकार स्थापित करेंगे कि $\nu_e$ एवं $\bar{\nu}_e$ दो विभिन्न प्रकार के कण हैं ? 10 (b) जिस जीवाश्म में 6 g कार्बन $^{14}C$ है एवं उसकी क्षय दर 27 क्षय प्रति मिनट है, उसकी आयु क्या है ? दिया गया है : $\frac{^{14}C}{^{12}C}$ का अनुपात=$1.3\times10^{-13}$, $^{14}C$ की अर्ध-आयु $(T_{1/2})$ = 5730 वर्ष । 10 (c) उन अन्योन्य क्रियाओं को नामित कीजिए जिनके द्वारा निम्नलिखित नाभिकीय क्षय घटित होते हैं : (i) $K^+ \longrightarrow \Pi^+ + \Pi^+ + \Pi^-$ (ii) $\Pi^+ + p \longrightarrow \Pi^+ + \Pi^+ + n$ (iii) $\Pi^+ + p \longrightarrow \Delta^{++} \longrightarrow \Pi^+ + p$ (iv) $\Sigma^\circ \longrightarrow \Lambda^\circ + \gamma$ (v) $\Sigma^+ \longrightarrow \Lambda^\circ + e^+ + \nu_e$ (vi) $K^- + p \longrightarrow K^+ + K^\circ + \Omega^-$ (vii) $\Pi^\circ \longrightarrow \gamma + e^+ + e^-$ (viii) $\Sigma^- \longrightarrow n + e^- + \bar{\nu}_e$ (ix) $\Lambda^\circ \longrightarrow p + e^- + \bar{\nu}_e$ (x) $e^+ + e^- \longrightarrow \gamma + \gamma$ 10 (d) व्युत्क्रम जालक अवधारणा का उपयोग करके विवर्तन की शर्तों की व्युत्पत्ति कीजिये । इन शर्तों को किस रूप में जाना जाता है ? 10 (e) दर्शाइए कि फर्मी स्तर n-प्रकार के अर्धचालक में ऊपर की तरफ चालन बैंड के नजदीक विस्थापित होता है और p-प्रकार के अर्धचालक में नीचे की तरफ संयोजकता बैंड के नजदीक विस्थापित होता है । 10

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Directive word: Derive

This question asks you to derive. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

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How this answer will be evaluated

Approach

Begin with a brief introduction acknowledging the breadth from particle physics to solid state physics. For part (a), explain helicity, lepton number conservation, and scattering experiments; for (b), set up the radioactive decay equation and solve for age; for (c), classify all ten reactions by interaction type using conservation laws; for (d), derive Laue conditions from reciprocal lattice vectors; for (e), use charge neutrality and mass action law to show Fermi level shifts. Allocate time proportionally: ~15% each for (a), (b), (c), and 20% each for (d) and (e) due to derivations required.

Key points expected

  • (a) Distinguishing νₑ and ν̄ₑ: helicity differences, lepton number conservation (Lₑ = +1 vs −1), and experimental evidence from inverse beta decay (ν̄ₑ + p → n + e⁺) vs (νₑ + n → p + e⁻)
  • (b) Age calculation: determine initial ¹⁴C mass from given ratio, apply N = N₀e^(-λt) with λ = ln(2)/T₁/₂, relate activity A = λN, solve for t ≈ 11,400 years
  • (c) Interaction classification: (i) Weak, (ii) Strong, (iii) Strong (resonance formation), (iv) Electromagnetic, (v) Weak, (vi) Weak (strangeness changing), (vii) Electromagnetic, (viii) Weak, (ix) Weak, (x) Electromagnetic
  • (d) Reciprocal lattice derivation: define reciprocal basis vectors b₁, b₂, b₃, show scattering vector Δk = G (reciprocal lattice vector), derive Laue conditions k·G = G²/2 or equivalently 2k·G = G²
  • (e) Fermi level shifts: for n-type, n = N_D⁺ + nᵢ²/n, show E_F moves toward E_C; for p-type, p = N_A⁻ + nᵢ²/p, show E_F moves toward E_V using charge neutrality and mass action law

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies νₑ/ν̄ₑ distinction via lepton number and helicity; applies correct decay law for ¹⁴C dating; accurately classifies all ten particle reactions by dominant interaction; states Laue conditions correctly; properly explains Fermi level pinning and shift directions in doped semiconductorsIdentifies most concepts correctly but confuses weak vs electromagnetic in 1-2 reactions; minor errors in lepton number assignment; states final formulas without full conceptual justification for Fermi level shiftsFundamental errors such as claiming νₑ and ν̄ₑ are same particle, misclassifying strong interactions as weak, incorrect decay law application, or stating Fermi level moves opposite direction in doped semiconductors
Derivation rigour20%10Complete step-by-step derivation of Laue conditions from reciprocal lattice definition with clear vector algebra; rigorous derivation of Fermi level position using charge neutrality and mass action equations; explicit calculation steps for age determination with proper unit handlingStates key equations but skips intermediate algebraic steps; presents Laue conditions as given rather than derived; shows Fermi level results without explicit derivation from fundamental equationsNo derivations attempted; only final formulas stated without justification; missing critical steps like charge neutrality condition or reciprocal lattice vector definition
Diagram / FBD15%7.5Clear reciprocal lattice construction showing real and reciprocal space relationship; E-k diagram for semiconductors showing Fermi level shifts in n-type and p-type with donor/acceptor levels marked; scattering geometry diagram for Laue conditionBasic band diagram showing conduction and valence bands with Fermi level indicated; simple reciprocal lattice sketch without full construction details; missing one of the three expected diagramsNo diagrams provided where essential; or incorrect diagrams showing metallic band structure instead of semiconductor; no visualization of reciprocal space
Numerical accuracy20%10Correctly calculates: initial ¹⁴C atoms from 6g sample and ratio 1.3×10⁻¹³; decay constant λ = 1.21×10⁻⁴ yr⁻¹; converts 27 decays/min to proper units; arrives at age ≈ 11,400 years (accept 11,000-12,000 with proper significant figures)Correct formula setup but arithmetic errors in final calculation; incorrect unit conversion (e.g., minutes to years); correct order of magnitude but imprecise final answerMajor errors: uses wrong half-life formula, confuses activity with number of atoms, order of magnitude error in final age, or completely omits numerical part (b)
Physical interpretation25%12.5Explains why helicity measurement distinguishes neutrino types; interprets ¹⁴C age in archaeological context; explains why certain reactions proceed via weak interaction (strangeness/charm changing) vs strong (conservation of all quantum numbers); connects reciprocal lattice to Bragg's law physically; explains why Fermi level shift maintains charge neutralityStates physical interpretations without depth; mentions but does not explain selection rules; describes what happens without explaining why; limited connection between mathematical results and physical phenomenaNo physical interpretation provided; purely mathematical or formulaic response; fails to explain significance of any results in physical terms

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