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

(a) Vector atom model and the Stern-Gerlach experiment

In the vector atom model, the orbital, spin and total angular momenta of an electron or atom are treated as quantised vectors. For one electron |L| = √(l(l+1))ħ and |S| = √(s(s+1))ħ. In a many-electron atom with LS coupling, the l_i add to L and the s_i to S, and L and S precess about their resultant J, |J| = √(J(J+1))ħ. Features: (1) the magnitudes are quantised as above; (2) space quantisation, in which the component along a field direction is J_z = m_Jħ with m_J = −J, …, +J, giving 2J + 1 orientations; (3) the associated magnetic moment is μ = −g_J μ_B J/ħ, and it is not parallel to J because the orbital and spin g-factors differ (1 and 2). Stern-Gerlach experiment: a collimated beam of neutral silver atoms from an oven passes through a strongly inhomogeneous magnetic field and strikes a plate. Each atom feels the force F_z = μ_z ∂B_z/∂z. A classical continuous distribution of orientations would give a smeared band, but the beam split into two distinct components. Hence μ_z, and so the angular-momentum component, takes only discrete values: space quantisation is real. A silver atom (5s¹, L = 0) has no orbital moment, so the two spots (m_J = ±½) were later understood as arising from electron spin. An inhomogeneous field is essential, since a uniform field only produces a torque and no net force. The Bohr model, which predicted no such splitting, could not account for this result.

(b) Landé g-factor of ³P₁ and its Zeeman splitting

The Landé g-factor is the factor that relates the magnetic moment of an atom in a given LS-coupled level to its total angular momentum, μ_J = −g_J μ_B J/ħ. It follows by projecting the orbital (g_L = 1) and spin (g_S ≈ 2) moments on J: g_J = 1 + [J(J+1) + S(S+1) − L(L+1)]/[2J(J+1)]. For ³P₁ of carbon (2p3s): 2S + 1 = 3 gives S = 1, P gives L = 1, and the subscript gives J = 1. g_J = 1 + (2 + 2 − 2)/(2 × 2) = 3/2. In B = 0.1 T the energy shifts are ΔE = g_J m_J μ_B B, with μ_B B = 9.274×10⁻²⁴ × 0.1 = 9.274×10⁻²⁵ J. Then g_J μ_B B = 1.391×10⁻²⁴ J = 8.68×10⁻⁶ eV. The level splits into three equally spaced sublevels, m_J = −1, 0, +1, at −1.39×10⁻²⁴ J, 0 and +1.39×10⁻²⁴ J. The adjacent splitting is 1.39×10⁻²⁴ J (8.7 μeV), and the total spread is 2.78×10⁻²⁴ J (17.4 μeV).

(c) Raman effect, its quantum theory and rotational structure

Raman effect: when monochromatic light is scattered by a molecule, a small fraction of the scattered light has frequencies different from the incident one, ν₀ ± ν_m, where ν_m is a molecular transition frequency. This inelastic scattering (observed by Raman and Krishnan in 1928) is distinct from Compton scattering, which concerns free electrons at X-ray energies. Quantum theory: an incident photon hν₀ takes the molecule to a short-lived virtual state, and a photon is re-emitted at once. If the molecule ends in a higher level, the photon loses energy, hν_s = hν₀ − ΔE (Stokes line, lower frequency). If it was already excited and falls to a lower level, hν_as = hν₀ + ΔE (anti-Stokes line). The unshifted Rayleigh line lies between them, and anti-Stokes lines are weaker because fewer molecules start in excited levels. A transition is Raman-active only if the molecular polarizability changes during it. Rotational structure: for a rigid rotor, E_J = BJ(J+1). The polarizability is a rank-2 tensor, so the selection rule is ΔJ = ±2 (for linear molecules, including homonuclear ones, which have no rotational infrared spectrum). For J → J + 2 the shift is B[(J+2)(J+3) − J(J+1)] = (4J + 6)B, giving Stokes lines at 6B, 10B, 14B, … from the Rayleigh line, with anti-Stokes lines symmetrically on the other side. Adjacent lines are 4B apart, and B (hence the bond length) is obtained from the spacing. In homonuclear molecules, nuclear-spin statistics alter the intensities: in N₂ the even-J lines are twice as strong as the odd-J ones, and in ¹⁶O₂ only odd J occur, so the spacing is 8B.