(a) (i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume that sodium has one free electron per atom and its density is 0·97 gm/cm³.

(ii) Draw the energy level diagram and mathematical exp

Question

(a) (i) Using free electron theory of metals, calculate the Fermi energy level of sodium atom at absolute zero. Assume that sodium has one free electron per atom and its density is 0·97 gm/cm³.

(ii) Draw the energy level diagram and mathematical expressions for the following :

I. Eₙ of an electron confined in a one-dimensional box

II. Linear harmonic oscillator

Make a qualitative comparison of the above two cases. 10+10=20 marks

(b) Show that for a diatomic molecule with two nuclei of mass 'M' separated by a distance 'a', the rotational energy of nuclear motion is lower than electronic energy by a factor of $\frac{m_e}{M}$. 15 marks

(c) Differentiate between L-S coupling and J-J coupling.

What are the possible orientations of $\vec{J}$ for the $J = \frac{3}{2}$ and $J = \frac{1}{2}$ states that correspond to $l = 1$? 5+10=15 marks

UPSC Answer Check · Accepted Answer

Begin with a brief introduction linking free electron theory to metallic bonding in Indian context (e.g., copper in IIT-Kharagpur research). For part (a)(i), calculate Fermi energy using n = ρN_A/M with proper unit conversions; for (a)(ii), draw two separate energy level diagrams with equations E_n = n²h²/8mL² and E_n = (n+½)hν, then compare spacing and zero-point energy. Spend ~40% time on (a) due to 20 marks. For (b), derive the ratio m_e/M using moment of inertia I = Ma²/2 and rotational energy E_rot = ħ²/2I versus electronic energy ~ħ²/ma². For (c), tabulate L-S vs J-J coupling with examples (light atoms like Na vs heavy atoms like Pb), then apply vector model for J = 3/2, 1/2 with m_J values and spatial quantization angles. Conclude with significance for spectroscopic studies in Indian atomic research centres.
- Part (a)(i): Correct calculation of electron number density n = ρN_A/M for sodium (M = 23 g/mol, ρ = 0.97 g/cm³), then E_F = (ħ²/2m)(3π²n)^(2/3) yielding ~3.1-3.2 eV
- Part (a)(ii): Energy level diagram for particle in 1D box showing E_n ∝ n² with non-uniform spacing and ground state at n=1; diagram for LHO showing E_n ∝ (n+½) with uniform spacing and zero-point energy
- Part (b): Derivation showing E_rot/E_el ~ (m_e/M) using I = μa² ≈ Ma²/2 for identical nuclei, with E_rot = ħ²l(l+1)/2I and E_el ~ ħ²/ma²
- Part (c): Clear differentiation table showing L-S coupling (light atoms, Hund's rules, total L and S first) vs J-J coupling (heavy atoms, individual j-j coupling first)
- Part (c) continued: For l=1, s=½: j=3/2, 1/2; possible J values with 2J+1 orientations; m_J = ±3/2, ±1/2 for J=3/2 and m_J = ±1/2 for J=1/2 with angle cosθ = m_J/√[J(J+1)]

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies Fermi energy as chemical potential at T=0; distinguishes between particle-in-box (infinite well, nodes at boundaries) and LHO (parabolic potential, nodes inside); correctly identifies Born-Oppenheimer approximation basis for part (b); accurately describes L-S coupling for light atoms (Z<30) and J-J for heavy atoms (Z>80) with proper examples like Na vs U	Basic correct formulas but confuses boundary conditions; vague on why rotational energy is smaller; mixes up coupling schemes without clear atomic number criteria	Fundamental errors like treating Fermi energy as average energy; wrong potential shapes in diagrams; claims rotational energy exceeds electronic energy; cannot distinguish coupling schemes
Derivation rigour	20%	10	Step-by-step derivation for (a)(i) with explicit substitution of n = (0.97×6.022×10²³)/23 cm⁻³; for (b) clearly shows μ = M/2, derives E_rot = ħ²/I ~ ħ²/Ma² and E_el ~ ħ²/ma², takes ratio; for (c) applies angular momentum addition rules with Clebsch-Gordan logic	Correct final formulas but skips intermediate steps; assumes reduced mass without explanation; states J values without showing j=l±½ derivation	Missing crucial steps or algebraic errors; incorrect reduced mass; wrong angular momentum addition; no derivation for orientation angles
Diagram / FBD	15%	7.5	Two clear, labeled diagrams: (1) 1D box with infinite walls, energy levels E₁<E₂<E₃... with spacing increasing as 3,5,7... in units of h²/8mL²; (2) LHO with parabolic potential, equally spaced levels, ground state at ½hν above minimum; includes wavefunction sketches showing nodes	Rough sketches without proper labeling; missing wavefunction shapes; incorrect spacing pattern shown	No diagrams or completely wrong representations; single diagram for both cases; missing energy level structure
Numerical accuracy	20%	10	Precise calculation: n = 2.54×10²² cm⁻³ = 2.54×10²⁸ m⁻³; k_F = (3π²n)^(1/3) = 9.0×10⁹ m⁻¹; E_F = ħ²k_F²/2m = 3.15 eV or ~5.0×10⁻¹⁹ J; correct unit conversions throughout; proper significant figures	Correct order of magnitude (~3 eV) but arithmetic errors; wrong powers of 10; unit confusion between eV and J	Order of magnitude wrong; fundamental errors like using atomic mass unit directly; no numerical answer
Physical interpretation	25%	12.5	Explains Fermi energy as highest occupied state at T=0 with relevance to sodium's metallic properties; contrasts 1D box (no zero-point, confinement) vs LHO (zero-point energy, binding); explains m_e/M ~10⁻⁴⁻10⁻⁵ as basis for Born-Oppenheimer separation; interprets J orientations via spatial quantization with θ angles (54.7°, 125.3° for J=3/2; 90° for J=1/2 m_J=±½)	States facts without physical insight; mentions zero-point energy without significance; lists angles without geometric interpretation	No physical interpretation; purely mathematical treatment; cannot explain why results matter for spectroscopy or molecular structure

Q4

Directive word: Calculate

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Physics 2024 Paper II