Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

==== - Resonant cross-section (spherical/dipole geometry): ====

 σ0=3λ22π (peak on resonance).\boxed{\ \sigma_0 = \frac{3\lambda^2}{2\pi}\ }\quad\text{(peak on resonance)}. σ0=2π3λ2 (peak on resonance).
Event rate = photon flux Φ=I/(ℏω)\Phi = I/(\hbar\omega)Φ=I/(ℏω) times σ\sigmaσ:
R≈Φ σ(ω)R \approx \Phi\,\sigma(\omega)R≈Φσ(ω). The λ2\lambda^2λ2 area and the 3/2π3/2\pi3/2π factor come from integrating a dipole pattern over 4π4\pi4π—just the spherical geometry you emphasize.
* Einstein A (spontaneous rate) has the 13π\tfrac{1}{3\pi}3π1 from angular averaging of the dipole over the sphere and the ω3/c3\omega^3/c^3ω3/c3 from photon mode density in kkk-space (again a geometric count of directions).
* Elastic (Rayleigh) scattering far off resonance Using the Rabi frequency Ω=∣d⋅E∣/ℏ\Omega = |\mathbf d\cdot\mathbf E|/\hbarΩ=∣d⋅E∣/ℏ and I=12cε0E2I=\tfrac12 c\varepsilon_0 E^2I=21cε0E2:

 Rsc≈ΓΩ24Δ2=ΓIIsat11+(2Δ/Γ)2 .\boxed{\ R_{\rm sc} \approx \Gamma \frac{\Omega^2}{4\Delta^2}
= \Gamma \frac{I}{I_{\rm sat}} \frac{1}{1+(2\Delta/\Gamma)^2}\ }. Rsc≈Γ4Δ2Ω2=ΓIsatI1+(2Δ/Γ)21 .
So even pure scattering (no net energy change) produces dephasing proportional to intensity and to the dipole geometry.