Scattering of light: Kramers-Heisenberg, Rayleigh, Thomson

SI units throughout; where a four-vector appears the metric is (+, −, −, −). Quoted from Unit 11: for an electron of charge −e in the Coulomb gauge the coupling is (e/m)A⋅p + (e²/2m)A², with A = Σₖₛ√(ħ/(2ε₀ωₖV))(aₖₛ + aₖₛ†)êₖₛ once the dipole approximation sets e^(±ik⋅r) to 1. Scattering takes |a; kê⟩ to |b; k′ê′⟩, destroying one photon and creating another. A² contains aₖ′†aₖ and aₖaₖ′†, so it does this in one step, at first order, with matrix element (e²ħ/(2mε₀V√(ωω′)))⟨b|a⟩ê⋅ê′, derived. A⋅p changes photon number by one and must act twice. It can absorb first, through |n; 0⟩ with energy denominator Eₐ + ħω − Eₙ, or emit first, through |n; kê, k′ê′⟩ with Eₐ − ħω′ − Eₙ. Each second-order amplitude also carries e², so A² is not a higher-order correction, and the emit-first path, whose intermediate state never conserves energy, is not small away from resonance.

Add the three amplitudes and apply the golden rule, which Quantum Mechanics II derived for second-order amplitudes too, with Vω′² dΩ/((2π)³ħc³) final photon states per unit energy and an incident flux of one photon per volume V at speed c. Every V cancels, the couplings collect into (e²/(4πε₀mc²))² = rₑ², and dσ/dΩ = rₑ²(ω′/ω)|bracket|², derived, with the bracket of the first formula card. Dimensions: ⟨p⟩²/(m × energy) is (kg m s⁻¹)²/(kg × kg m² s⁻²) = 1, so dσ/dΩ is an area per steradian. When b = a the scattering is elastic, ω′ = ω, and the A² term ⟨b|a⟩ê⋅ê′ is present. When b ≠ a it vanishes and ω′ = ω − (E_b − Eₐ)/ħ: Raman scattering, Stokes when b lies above a and anti-Stokes when below. Checked against measurement: nitrogen's vibrational spacing, 2331 cm⁻¹, moves 532 nm light to 607.3 nm, the wavelength at which Raman lidars detect atmospheric nitrogen.

Take b = a and use ⟨n|p|a⟩ = imωₙₐ⟨n|r|a⟩, quoted from Quantum Mechanics II. With X₁ = ⟨a|ê′⋅r|n⟩⟨n|ê⋅r|a⟩ and X₂ the other order, the A⋅p part of the bracket is −(m/ħ)Σₙωₙₐ²[X₁/(ωₙₐ − ω) + X₂/(ωₙₐ + ω)]. Split each fraction: ωₙₐ²/(ωₙₐ ∓ ω) = ωₙₐ ± ω + ω²/(ωₙₐ ∓ ω). The first pieces give −(m/ħ)Σₙωₙₐ(X₁ + X₂) = −ê⋅ê′ by the commutator sum rule, cancelling the A² term exactly. The second give −(mω/ħ)⟨a|[ê′⋅r, ê⋅r]|a⟩ = 0, since positions commute. What survives, with no approximation, is the bracket −(mω²/ħ)Σₙ[X₁/(ωₙₐ − ω) + X₂/(ωₙₐ + ω)]: an amplitude of order ω², so σ ∝ ω⁴, derived. For an S state, X₁ and X₂ summed over each level's sublevels are (ê⋅ê′)|⟨n|z|a⟩|², and σ = σ_T ω⁴[Σₙfₙₐ/(ωₙₐ² − ω²)]². As ω → 0 the sum is m/e² times the static polarisability α, which gives Rayleigh's (8π/3)(ω/c)⁴(α/4πε₀)².

For hydrogen's ground state Quantum Mechanics II gave α/4πε₀ = 4.5a₀³, quoted here. With Σf = 1 the Rayleigh limit reads σ_R = σ_T(ħω/12.83 eV)⁴: the atom scatters like one oscillator of unit strength at 12.83 eV, above Lyman-α's 10.20 eV because the 2p line carries f = 0.416 and supplies only 66% of α. At 1064 nm that is 6.8 × 10⁻⁵ σ_T, and the next term, of relative order ω², adds 2%. Dimensions: (ω/c)⁴ is m⁻⁴ and (α/4πε₀)² is m⁶, an area. Limit: ω → 0 gives zero, as it must for a bound charge that a static field can only displace. Measurement: nitrogen's refractive index at 532 nm, n − 1 = 2.99 × 10⁻⁴ at 0 °C and 1 atm, gives α/4πε₀ = (n − 1)/(2πN) = 1.77 × 10⁻³⁰ m³ and σ_R = 5.1 × 10⁻³¹ m²; the molecule's anisotropy adds 3%, quoted, and cavity ring-down measurements of the scattering itself agree within a few per cent. The λ⁻⁴ makes 450 nm light scatter 4.4 times more strongly than 650 nm: the blue of the sky.

When ħω is far above every Eₙ − Eₐ that carries oscillator strength, each A⋅p denominator is close to ∓ħω, the two time orders cancel because ⟨a|[ê′⋅p, ê⋅p]|a⟩ = 0, and what remains falls as (ωₙₐ/ω)². In the S-state form Σₙfₙₐ/(ωₙₐ² − ω²) → −1/ω² by the Thomas-Reiche-Kuhn rule, so σ → σ_T. Only A² survives: dσ/dΩ = rₑ²|ê⋅ê′|², which summed over ê′ is rₑ²sin²Θ, Θ the angle between ê and the scattered direction, and over all directions gives (8π/3)rₑ² = 6.652 × 10⁻²⁹ m², derived; rₑ² is an area, as it must be. Unpolarised light scatters as rₑ²(1 + cos²θ)/2 and is fully polarised at 90°. A free electron at rest has no A⋅p term in the dipole approximation, so this is its entire cross-section. Measurement: Thomson's value gives 0.200 cm² g⁻¹ for carbon, six electrons per 12.0 g mol⁻¹, the mass scattering coefficient Barkla measured for hard X-rays in light elements and used to count about half the mass number of electrons per atom. The window is binding ≪ ħω ≪ mc²; beyond it recoil and the Klein-Nishina formula take over, named here only.

As ω → ωₙₐ the absorb-first denominator Eₙ − Eₐ − ħω vanishes and the bracket diverges, because second-order perturbation theory has let the intermediate state live forever. Topics 12.06 and 12.07 showed that it decays, its amplitude carrying Eₙ − iħΓ/2, quoted here. Near an S to P line at ω₀ the third card becomes σ = (8π/3)rₑ²(fω₀/2)²/((ω − ω₀)² + Γ²/4), a Lorentzian of full width Γ peaking at (8π/3)rₑ²(fω₀/Γ)². If the upper level decays only back to a, Γ is topic 12.02's A, which written through f is 2frₑω₀²/(3c). Then f cancels: σ₀ = 3λ²/(2π), derived, for every such line. Lyman-α at 121.6 nm gives 7.06 × 10⁻¹⁵ m², 10¹⁴ times σ_T and 8 × 10⁵ times πa₀². Dimensions: λ² is an area. Limit: weaken the line and the peak does not fall, because the width narrows with it. Measurement: the same A gives the 2p lifetime, 1.60 ns, as measured. Over frequency the line holds ∫σ dν = σ₀Γ/4 = πrₑcf, the classical absorption area, since with Γ = A every photon absorbed is re-emitted: it is set by f alone, and broadening that conserves photons, such as the Doppler shifts of a warm gas, spreads it without changing it.