(a) How do Stokes lines appear in Raman spectrum as per classical and quantum theory of Raman effect ? 20 marks (b) What is Lamb…

Raman effect. In the classical picture, an incident field E=E0 cos ωt makes the molecular polarizability oscillate as α=α0+(∂α/∂Q)Q, with Q=Q0 cos ω_v t. The induced dipole p=αE contains frequencies ω, ω±ω_v. The ω term gives Rayleigh scattering; ω−ω_v gives Stokes lines, lower in frequency because the molecule gains vibrational energy; ω+ω_v gives anti-Stokes lines. The scattered intensity is proportional to (∂α/∂Q)², so only vibrations that modulate polarizability are seen. The classical result explains the frequency shifts but not the detailed intensity ratio; quantum theory supplies the populations and selection rules. C.V. Raman’s 1928 Calcutta experiment used a mercury arc, polarizers, a liquid sample and a spectrograph to observe these weak shifted lines.

Quantum mechanically, scattering is a second-order process through virtual intermediate states, not real absorption. The virtual states are off-resonant, so energy conservation is satisfied by the molecular vibrational change. The Kramers–Heisenberg dispersion formula gives the amplitude as a sum over states n of products of dipole matrix elements divided by energy denominators, with both emission-then-absorption and absorption-then-emission terms. Placzek’s polarizability theory reduces the intensity to the square of the relevant polarizability tensor component, α_ij (α_xx, α_xy, α_zz, etc.), so a vibration is Raman active only if α_ij changes with the normal coordinate; for a harmonic oscillator Δv=±1. In centrosymmetric molecules, modes that change dipole moment are IR active while modes that change polarizability are Raman active, giving mutual exclusion. The Stokes/anti-Stokes ratio is I_S/I_AS ≈ (ν_S/ν_AS)^4 (N_v+1)/N_v ≈ (ν_S/ν_AS)^4 exp(hν_v/kT), where N_v is the mean vibrational occupation; hence Stokes lines dominate at ordinary temperatures.

Lamb shift. Dirac theory predicts degeneracy of hydrogen 2S1/2 and 2P1/2, but Lamb and Retherford (1947) found 2S1/2 higher by about 1058 MHz. Bethe’s calculation treated the electron’s interaction with the quantized radiation field and introduced mass renormalization to remove the self-energy divergence. In second quantization, the Dirac and electromagnetic fields are expanded in creation and annihilation operators, and the interaction Hamiltonian is H_int=−e∫ψbar γ^μ A_μ ψ d³x. This allows the electron to emit and reabsorb a virtual photon, represented by the one-loop self-energy Feynman diagram; vacuum fluctuations and radiation reaction shift the energy levels. The one-loop correction is integrated over virtual photon momentum; its divergence is absorbed into the electron mass, while the finite remainder gives the shift. The bare mass and charge are divergent, so renormalization fixes them to observed values, leaving a finite Lamb shift that confirms QED.

EPR. Electron Paramagnetic Resonance detects transitions between electron spin sublevels in a magnetic field. The electron spin magnetic moment is μ=−g μ_B S/h; in B0 the m_s=±1/2 levels split. For an unpaired electron, H=μ_B g S·B/h, giving ΔE=gμ_BB and resonance hν=gμ_BB, or B0=hν/(gμ_B). Hyperfine coupling, H_hf=A I·S, splits the line and identifies nuclear spin and coupling constants; in solids the g-factor is anisotropic, giving orientation-dependent resonance. In powders, anisotropy produces broadened spectra; in single crystals, orientation can be resolved. EPR uses microwave frequencies and lower fields because the electron magnetic moment is much larger than the nuclear magneton.

| Feature | EPR | NMR | |---|---|---| | Magnetic moment | μ_B, ~658 μ_N | μ_N | | Typical field | ~0.3 T (X-band) | ~10 T | | Relaxation | shorter T1/T2, spin–orbit/dipolar | longer, weaker coupling | | Sample | paramagnetic centres, small/frozen | abundant nuclei, larger/high field | | Sensitivity | very sensitive to unpaired e⁻ | bulk nuclei, needs high concentration |

Applications include ESR dating of archaeological samples such as Bhimbetka rock paintings, where trapped unpaired electrons record accumulated radiation dose; detection of radical pairs in photosynthetic reaction centres; Gd³⁺ MRI contrast agents that shorten water-proton T1; and spin labels for protein structure through dipolar distance measurements. These applications show EPR’s value in archaeology, biochemistry, medicine and structural biology.

Thus Raman scattering, Lamb shift and EPR respectively reveal molecular polarizability, QED self-energy corrections and electron spin resonance; each connects a microscopic interaction to a measurable spectral shift.