Rayleigh Scattering Frequency Law

Worked Example

Let's calculate the cross section ratio for a harmonically bound electron.

Given parameters:

• Frequency ratio r = ω / \omega_0 = 0.2
• Damping ratio g = \gamma_0 / \omega_0 = 0.1

1. Calculate the Rayleigh approximation: r^4 = 0.2^4 = 0.0016
2. Calculate the full response denominator: (1 - r²)² + g² r² = (1 - 0.2²)² + 0.1² · 0.2² = 0.96² + 0.0004 = 0.9216 + 0.0004 = 0.922
3. Calculate the full cross section ratio: r^4 / 0.922 = 0.0016 / 0.922 ≈ 0.0017353579175704993

Experiment

You can use the interactive module to explore the relationship between the full response and the Rayleigh approximation across different frequency regimes.

1. Set the Frequency ratio to 0.1 and observe the close match between the full response and the approximation.
2. Increase the Frequency ratio to 1 (resonance) and observe the massive divergence caused by the damping term dominating the denominator.
3. Change the Damping ratio to see how it affects the peak height at resonance without significantly changing the low-frequency behavior.

Explore this in the module: Rayleigh Scattering Frequency Law

Limitations

This classical model provides a qualitative understanding of scattering but has several limitations:

• It assumes a single resonance frequency for bound electrons, whereas real atoms and molecules have multiple complex transitions.
• It is a low-frequency limit; as frequencies approach X-ray energies (Compton scattering), quantum mechanical effects and recoil must be considered.
• The model does not apply directly to large dielectric spheres (which require Mie theory) or complex atmospheric phenomena beyond basic molecular scattering.

References

For more detailed mathematical derivations and physical context, consult the following source:

• Fitzpatrick, R. "Rayleigh Scattering," Classical Electromagnetism