(a) What is the importance of study of deuteron ? Obtain the solution of Schrödinger equation for ground state of deuteron and show that deuteron is a loosely bound system. (20 marks)

(b) Show that in the nuclear shell model, the level spacing betwe

Question

(a) What is the importance of study of deuteron ? Obtain the solution of Schrödinger equation for ground state of deuteron and show that deuteron is a loosely bound system. (20 marks)

(b) Show that in the nuclear shell model, the level spacing between major oscillator shells is approximately $\hbar\omega$=41$A^{-1/3}$ MeV. (15 marks)

(c) How many types of neutrinos exist ? How do they differ in their masses ? (15 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction on nuclear structure studies in Indian context (Bhabha's contributions). For part (a), derive the Schrödinger solution using square well potential, showing binding energy ~2.2 MeV proves loose binding; allocate ~40% time. For (b), derive the oscillator spacing using nuclear radius relation R=r₀A^(1/3) and equating Fermi energy to ℏω; allocate ~30% time. For (c), enumerate three neutrino flavors with mass hierarchy and mention Indian experiments like INO; allocate ~30% time. Conclude with significance for nuclear astrophysics.
- Part (a): Importance of deuteron as only two-nucleon bound system, simplest test of nuclear forces; solution of radial Schrödinger equation for l=0 with finite square well; boundary conditions at r=R; obtain transcendental equation; show binding energy E_b≈2.224 MeV << typical nuclear energy scale (~8 MeV/nucleon) proving loose binding
- Part (a): Mention deuteron's large size (r_d≈4.3 fm), no excited bound states, electric quadrupole moment indicating non-spherical shape and tensor force importance
- Part (b): Derivation starting from 3D isotropic harmonic oscillator with Hamiltonian H=p²/2m + ½mω²r²; energy levels E_N=(N+3/2)ℏω where N=2(n-1)+l; major shells at N=0,1,2,3...
- Part (b): Relate ℏω to nuclear size using R=r₀A^(1/3) and Fermi momentum; equate ℏ²k_F²/2m to ℏω scaling; derive ℏω≈41A^(-1/3) MeV with r₀≈1.2 fm
- Part (c): Three types—electron neutrino (ν_e), muon neutrino (ν_μ), tau neutrino (ν_τ); mass differences from neutrino oscillation experiments; solar neutrino puzzle and KamLAND, Super-Kamiokande evidence
- Part (c): Mass hierarchy: normal (m₁<m₂<m₃) vs inverted; Δm²₂₁≈7.5×10⁻⁵ eV², |Δm²₃₁|≈2.5×10⁻³ eV²; mention upper limits (~eV scale) and cosmological constraints; Indian Neutrino Observatory relevance

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies deuteron as unique probe of nucleon-nucleon interaction; accurately states three neutrino flavors with proper mass-squared differences; correctly relates nuclear radius to mass number; no confusion between binding energy per nucleon vs total binding energy	Identifies basic importance of deuteron but misses uniqueness; lists three neutrinos but confuses mass differences with absolute masses; states formula for ℏω without clear physical basis; minor errors in energy scale comparisons	Confuses deuteron with other nuclei; states incorrect number of neutrino types; fundamental misunderstanding of shell model quantum numbers; incorrect binding energy values by order of magnitude
Derivation rigour	25%	12.5	Complete derivation for (a): radial equation, interior/exterior solutions, logarithmic derivative matching, transcendental equation, numerical estimate of binding energy; for (b): clear steps from oscillator Hamiltonian to energy formula, proper scaling argument with Fermi energy matching	Sets up Schrödinger equation correctly but skips boundary condition details; states ℏω formula with partial derivation; mentions oscillation probability formula for neutrinos without derivation; some steps implied rather than shown	Writes Schrödinger equation without attempting solution; merely quotes ℏω=41A^(-1/3) without any derivation; no mathematical treatment of neutrino masses; major gaps in logical flow
Diagram / FBD	15%	7.5	Clear square well potential diagram with V₀, R, E_b labeled; wavefunction plot showing exponential tail outside well (characteristic of weak binding); shell model energy level diagram showing oscillator shells with magic numbers; neutrino mass hierarchy diagram (normal vs inverted)	Sketch of potential well without proper labels; mentions wavefunction behavior verbally; simple list of magic numbers without diagram; no mass hierarchy visualization	No diagrams despite clear need for visualization; or incorrect diagrams (e.g., infinite well for deuteron); missing essential figures for all three parts
Numerical accuracy	20%	10	Correct binding energy E_b=2.224 MeV; correct deuteron radius ~4.3 fm; correct coefficient 41 MeV in ℏω formula with proper derivation of numerical factor; correct Δm² values (7.5×10⁻⁵ eV² and 2.5×10⁻³ eV²) with proper units and scientific notation	Binding energy approximately correct (~2 MeV); order of magnitude correct for ℏω coefficient; knows mass differences are small but gives approximate or unit-confused values; one significant numerical error	Binding energy off by factor of 10 or more; incorrect exponent in ℏω formula; confuses mass differences with absolute masses; multiple numerical errors or missing values entirely
Physical interpretation	20%	10	For (a): clearly explains why small E_b and large size mean 'loosely bound'—wavefunction extends far beyond force range, no excited states; for (b): explains why oscillator model works (nucleons move independently in mean field) and significance of magic numbers; for (c): connects neutrino masses to oscillation phenomenon, solar neutrino problem resolution, and cosmological implications; mentions Indian contributions (Bhabha, INO)	States deuteron is loosely bound without clear physical reasoning; describes shell model as 'atoms in nucleus' analogy; explains neutrino oscillation qualitatively; limited connection to experiments or applications	No interpretation of 'loosely bound' beyond repeating phrase; treats shell model as purely mathematical exercise; lists neutrino types without explaining mass significance; no physical insight or context

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