(a) What is vector atom model ? How the principal features of vector atom model were explained by Stern-Gerlach experiment ?…

Spectroscopy reveals that atomic and molecular energy levels are discrete and spatially quantised. The three phenomena asked here are linked by the quantum treatment of angular momentum and light–matter interaction.

Vector atom model and Stern–Gerlach. In the vector atom model, the orbital, spin and total angular momenta of an atom are represented by vectors. For an electron, |L| = √[l(l+1)]ħ and |S| = √[s(s+1)]ħ; for many-electron atoms the individual lᵢ and sᵢ couple to total L and S, and in LS coupling L and S precess about a resultant J, |J| = √[J(J+1)]ħ. The magnetic moment is not parallel to J because the orbital and spin gyromagnetic ratios differ; its projection on a field direction is quantised, m_J = -J,...,+J. The Stern–Gerlach experiment passed a collimated beam of neutral atoms through a strongly inhomogeneous field. The force F_z = μ_z ∂B_z/∂z separates atoms according to the allowed values of μ_z. A beam of silver atoms (4s¹, L=0, S=1/2, J=1/2) split into two components, not one continuous spread. This confirmed space quantisation of the magnetic moment and, because L=0, proved the existence of intrinsic electron spin. The two-beam result also showed that the allowed projections of angular momentum, and hence magnetic moment, are discrete rather than classical continuous orientations. It removed the limitation of the Bohr–Sommerfeld model, which could not account for spin or the observed fine-structure and anomalous Zeeman effects.

Lande g-factor and Zeeman splitting. In LS coupling the magnetic interaction energy is ΔE = μ_B g_J B m_J, where μ_B is the Bohr magneton and m_J = -J,...,+J. The Landé g-factor follows by projecting the orbital and spin magnetic moments on J, with g_L=1 and g_S≈2:

g_J = 1 + [J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)].

For carbon’s 2p3s ³P₁ level, L=1, S=1, J=1. Hence g_J = 1 + [1(2)+1(2)-1(2)]/[2·1·2] = 1 + 2/4 = 3/2. For B = 0.1 T, μ_B g_J B = (9.274×10⁻²⁴ J T⁻¹)(1.5)(0.1) = 1.39×10⁻²⁴ J per unit m_J. The three sublevels m_J = -1,0,+1 are shifted by -1.39×10⁻²⁴, 0 and +1.39×10⁻²⁴ J. Thus the adjacent splitting is 1.39×10⁻²⁴ J = 8.68×10⁻⁶ eV, and the total spread from m_J=-1 to +1 is 2.78×10⁻²⁴ J = 1.74×10⁻⁵ eV.

Raman effect, quantum theory and rotational structure. The Raman effect is inelastic scattering of monochromatic light by molecules. Most photons are Rayleigh scattered with unchanged frequency; a small fraction emerge at lower frequency (Stokes) or higher frequency (anti-Stokes). Quantum mechanically, an incident photon of energy hν₀ raises the molecule to a short-lived virtual state; it then emits a photon and returns to a real initial or final state. If the final molecular energy is higher, the scattered photon loses energy: hνₛ = hν₀ - ΔE (Stokes). If the molecule starts in an excited state and falls to a lower state, hν_as = hν₀ + ΔE (anti-Stokes). The Rayleigh line marks the incident frequency; Stokes lines lie to lower frequency and anti-Stokes lines to higher frequency. The scattering intensity is governed by the change in molecular polarizability, represented by a polarizability tensor, which also determines selection rules.

For a rigid rotor, E_J = B J(J+1), with B the rotational constant. Rotational Raman scattering is allowed when ΔJ = ±2, because the polarizability is a rank-2 tensor. For a Stokes transition J → J+2, the shift is E_J+2-E_J = (4J+6)B, giving lines at 6B, 10B, 14B,... from the Rayleigh line. The anti-Stokes branch J → J-2 gives the same set of shifts on the other side of the Rayleigh line. Successive lines are separated by 4B. In homonuclear diatomics, nuclear spin statistics modify the intensities: for N₂ the even-J lines are about twice as intense as odd-J lines, while for O₂ (I=0) only odd-J levels are populated in the ground state. Thus Rayleigh, Stokes and anti-Stokes lines, with the 6B,10B,... rotational pattern, provide direct evidence of quantised rotational levels. Together with Stern–Gerlach and Zeeman splitting, these phenomena demonstrate space quantisation and the quantum mechanical treatment of atomic and molecular systems.