(a) What is vector atom model ? How the principal features of vector atom model were explained by Stern-Gerlach experiment ? (5+10=15 marks)

(b) What is Lande's g factor ? Evaluate the Lande's g factor for the ³P₁ level in the 2p3s configuration of

Question

(a) What is vector atom model ? How the principal features of vector atom model were explained by Stern-Gerlach experiment ? (5+10=15 marks)

(b) What is Lande's g factor ? Evaluate the Lande's g factor for the ³P₁ level in the 2p3s configuration of the ⁶C atom. Also calculate the splitting of the level when the atom is placed in an external magnetic field of 0·1 tesla. (5+5+5=15 marks)

(c) What is Raman effect ? Explain Quantum theory of Raman effect and Rotational Structure of a Raman spectrum. (5+10+5=20 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction linking atomic structure to spectroscopic phenomena. For part (a), spend ~30% time explaining vector atom model concepts and Stern-Gerlach experimental validation. For part (b), allocate ~30% to deriving Landé g-factor with proper LS coupling for ³P₁ state and calculating Zeeman splitting. For part (c), dedicate ~40% to quantum theory of Raman effect with rotational energy level diagrams. Conclude by emphasizing how these three phenomena collectively demonstrate space quantization and quantum mechanical treatment of atomic systems.
- Part (a): Vector atom model with orbital (L), spin (S) and total (J) angular momentum vectors; Stern-Gerlach experiment showing space quantization and electron spin discovery through Ag beam splitting
- Part (a): Explanation of how Stern-Gerlach results confirmed quantization of magnetic moment and existence of spin angular momentum, resolving Bohr-Sommerfeld model limitations
- Part (b): Landé g-factor formula derivation for LS coupling: g_J = 1 + [J(J+1) + S(S+1) - L(L+1)]/[2J(J+1)]; calculation for ³P₁ (L=1, S=1, J=1) yielding g_J = 3/2
- Part (b): Zeeman energy shift ΔE = μ_B·g_J·B·m_J with m_J = -1,0,+1; numerical evaluation for B=0.1 T giving splitting of ±9.27×10⁻²⁴ J or ±5.79×10⁻⁵ eV
- Part (c): Raman effect as inelastic light scattering with Stokes/anti-Stokes lines; quantum theory involving virtual energy levels and polarizability tensor
- Part (c): Rotational Raman spectrum showing ΔJ = ±2 selection rule, spacing of 4B, and alternating intensity due to nuclear spin statistics (relevant for homonuclear molecules like N₂, O₂)
- Part (c): Distinction between Rayleigh, Stokes and anti-Stokes lines with energy level diagram showing rotational transitions

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	22%	11	Accurately defines vector atom model with proper vector coupling; correctly identifies Stern-Gerlach significance for electron spin; states Landé formula with correct quantum numbers; explains Raman effect as polarizability modulation; rotational selection rules ΔJ=±2 stated correctly	Basic definitions present but confusion between LS and jj coupling; Stern-Gerlach described without spin implication; Raman confused with fluorescence; rotational selection rule stated as ΔJ=±1	Fundamental errors: treats vector model as Bohr model extension; Stern-Gerlach attributed to orbital quantization only; Raman described as elastic scattering; incorrect quantum numbers for ³P₁ state
Derivation rigour	20%	10	Complete derivation of Landé g-factor from μ_J·J/J² projection; explicit calculation with L=1, S=1, J=1 substituting into formula; Zeeman energy shift derived from ΔE = g_Jμ_BBm_J; quantum Raman theory with perturbation treatment of polarizability	Landé formula stated without derivation; direct substitution for g_J shown; Zeeman splitting formula used without m_J enumeration; Raman theory descriptive without matrix element treatment	No derivation attempts; incorrect formula used (g_s=2 for orbital); arithmetic errors in substitution; missing steps in energy level calculation; classical wave theory for Raman only
Diagram / FBD	18%	9	Stern-Gerlach apparatus diagram with magnet geometry and beam splitting; vector diagram showing L, S, J precession and associated magnetic moments; Zeeman splitting diagram for ³P₁ showing three components; Raman energy level diagram with virtual states and rotational levels (J=0,1,2,3)	At least two clear diagrams present; Stern-Gerlach sketch without field gradient indication; vector coupling shown but not labeled; Raman diagram missing virtual level distinction	No diagrams or very rough sketches; confusing vector orientations; missing energy level diagrams for Raman rotational structure; diagrams contradict text description
Numerical accuracy	20%	10	Correct g_J = 3/2 = 1.5 for ³P₁; proper unit conversion in Zeeman splitting; ΔE = 1.5 × 9.274×10⁻²⁴ J/T × 0.1 T = 1.39×10⁻²⁴ J per m_J; total splitting 2.78×10⁻²⁴ J or 1.74×10⁻⁵ eV; proper significant figures	Correct g_J value but arithmetic error in final splitting; correct formula with μ_B value error; units confused between J and eV; order of magnitude correct	Incorrect g_J calculation (e.g., using wrong J value); missing μ_B factor; B-field value misused; no numerical evaluation attempted; answer off by orders of magnitude
Physical interpretation	20%	10	Connects Stern-Gerlach to electron spin discovery and exclusion principle foundation; explains anomalous Zeeman effect significance for ³P₁; relates Raman rotational structure to molecular bond lengths via B = ℏ/4πcI; discusses Indian contributions (C.V. Raman, 1930 Nobel) and applications in atmospheric physics, materials characterization	Basic interpretation of each phenomenon separately; mentions Raman discovery but no application context; Zeeman effect described without normal vs anomalous distinction; no connection between parts	No physical insight; fails to explain why space quantization matters; no mention of spectroscopic applications; Raman described without relevance to molecular structure determination

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