Q6
(a) Establish the Rutherford's scattering cross section formula for α-particle by considering the standard assumptions and symbols. 20 marks (b) By assuming the nucleus as a cubical box of length equal to the nuclear diameter 10⁻¹² cm, calculate the kinetic energy of the highest level occupied nucleon of iron-56 nucleus. 15 marks (c) What do you understand by nuclear forces ? Explain meson theory of exchange forces. 5+10=15 marks
हिंदी में प्रश्न पढ़ें
(a) मानक अभिधारणाओं एवं प्रतीकों को लेकर α-कणों के लिए रदरफोर्ड के प्रकीर्णन परिछेत्र के सूत्र को स्थापित कीजिए । 20 (b) नाभिक को नाभिक व्यास 10⁻¹² cm के समतुल्य लम्बाई का एक घनीय बॉक्स मान कर उच्चतम स्तर पर अधिष्ठित आयरन-56 नाभिक के न्यूक्लिऑन की गतिज ऊर्जा की गणना कीजिये । 15 (c) नाभिकीय बलों से आप क्या समझते हैं ? विनिमय बलों के मेसॉन सिद्धांत की व्याख्या कीजिये । 5+10=15
Directive word: Derive
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How this answer will be evaluated
Approach
Derive the Rutherford differential cross-section formula in part (a) by setting up the scattering geometry, applying Coulomb's law, and integrating to obtain the solid angle dependence. For part (b), apply the particle-in-a-box quantum mechanical model with appropriate boundary conditions to calculate the Fermi energy for Fe-56 nucleons. In part (c), define nuclear forces with their key characteristics, then explain Yukawa's meson exchange theory with mass-energy relation and range estimation. Allocate approximately 40% effort to (a), 30% to (b), and 30% to (c) based on mark distribution.
Key points expected
- Part (a): Coulomb scattering geometry with impact parameter b, scattering angle θ, and hyperbolic trajectory; derivation of relation b = (kZe²/2E)cot(θ/2); differential cross-section dσ/dΩ = (kZe²/4E)²cosec⁴(θ/2); assumptions: point charge nucleus, single scattering, non-relativistic α-particles, neglect of electron screening
- Part (a): Integration over solid angle to show total cross-section divergence, physical significance of Rutherford formula validation through Geiger-Marsden experiments at Manchester
- Part (b): Particle in a 3D cubic box energy levels E = (h²/8mL²)(nx²+ny²+nz²); nuclear diameter L = 10⁻¹² cm; neutron and proton as spin-½ fermions with degeneracy g=2; Fermi momentum and energy calculation for A=56, Z=26, N=30
- Part (b): Numerical computation with ħc = 197 MeV-fm, nucleon mass ≈ 938 MeV/c², proper unit conversion from cm to fm; comparison with empirical nuclear Fermi energy ~38 MeV
- Part (c): Nuclear force characteristics: short-range (~1-3 fm), spin-dependent, charge-independent, saturated, non-central tensor component; contrast with Coulomb force
- Part (c): Yukawa meson theory: virtual particle exchange, uncertainty principle ΔE·Δt ≈ ħ, meson mass mπ from range R ≈ ħ/mπc ≈ 1.4 fm giving mπ ≈ 140 MeV/c²; prediction of π-meson later discovered in cosmic rays by Powell (1950 Nobel Prize)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | All three parts demonstrate flawless conceptual foundation: correct Coulomb potential treatment in (a), proper Fermi-Dirac statistics application in (b), accurate distinction between Yukawa potential and Coulomb potential in (c) with correct meson mass-range relation | Minor conceptual gaps such as incorrect degeneracy factor in (b), confusion between virtual and real mesons in (c), or incomplete assumptions list in (a); overall physics direction remains correct | Fundamental misconceptions: treating nuclear force as Coulomb-type in (c), classical statistics for nucleons in (b), or incorrect scattering geometry in (a) |
| Derivation rigour | 25% | 12.5 | Part (a) shows complete step-by-step derivation from trajectory equation through angular momentum conservation to final cosec⁴(θ/2) law with all intermediate steps; part (b) explicitly solves Schrödinger equation with boundary conditions; part (c) derives Yukawa potential from field theory or energy-time uncertainty | Correct final formulas with gaps in derivation steps; missing explicit integration limits in (a) or direct quoting of Fermi energy formula without box derivation in (b) | Final formulas stated without derivation; or incorrect derivation with algebraic errors; missing essential steps like momentum conservation or energy quantization condition |
| Diagram / FBD | 15% | 7.5 | Clear hyperbolic trajectory diagram for (a) showing nucleus at focus, impact parameter b, scattering angle θ, and asymptotes; 3D box with quantum numbers labeled for (b); Feynman-style diagram or nucleon-meson exchange illustration for (c) | At least one relevant diagram present (typically for part a) with correct labels but missing some elements; or adequate textual description substituting for missing diagrams in other parts | No diagrams despite clear requirement in (a); or completely incorrect diagrams showing elliptical instead of hyperbolic orbits |
| Numerical accuracy | 20% | 10 | Part (b) shows explicit unit conversion (10⁻¹² cm = 10⁻¹⁴ m = 10⁵ fm), correct nucleon count with proper spin degeneracy, accurate arithmetic leading to E_F ≈ 30-40 MeV range; physical constants used correctly with proper powers of 10 | Correct formula substitution with minor arithmetic errors or unit conversion mistakes; final answer within order of magnitude but significantly off from expected ~35 MeV | Major numerical errors: wrong box dimension, failure to account for proton-neutron separate filling, or answer orders of magnitude incorrect (keV or GeV instead of MeV) |
| Physical interpretation | 20% | 10 | Part (a) discusses experimental validation via Geiger-Marsden data and deviation at large angles indicating nuclear size; part (b) relates Fermi energy to nuclear stability and symmetry energy; part (c) connects Yukawa theory to subsequent π-meson discovery and modern QCD context; Indian contribution via Bhabha-Heitler theory or Saha ionization mentioned where relevant | Brief mention of experimental verification without quantitative comparison; generic statement about 'strong force holds nucleus together' without depth | No physical interpretation provided; purely mathematical treatment without connecting to observable phenomena or historical development |
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