Rayleigh vs Compton: Classical vs Quantum Theory

[Angelos K]

For certain frequencies both Rayleigh and Compton scattering are important. As I could not see the connection between the derivations of the assosciated formulae, I wondered what made a photon "decide" wether it should be subject to one or the other phaenomenon.

My reading is that Compton scattering assumes nothing but the electrons to be free, whilst Rayleigh requires them to be bound to the atom for a restoring oscillator force $$F_R=m\omega^2r$$ to act. So, if I send photons into matter the ones that hit (sufficiently) free targets will suffer Compton scattering, while the others obey Rayleigh's law (approximately)! Is that correct?

If so, the classical result for free targets (Thompson scattering) should be somehow connected to the Compton Effect.

 

[Valerie Park]

How?Thompson scattering is a type of Compton scattering, in which the electron is considered to be free. This means that the electron is not bound to an atom, so the restoring force of the oscillator does not act, and the photon and electron interact without the exchange of momentum. The result is that the scattered radiation is distributed uniformly in all directions, as opposed to being concentrated in certain directions as it would be with Rayleigh scattering.

 

[sir-pinski]

The difference between Rayleigh and Compton scattering can be attributed to the fundamental differences between classical and quantum theory. In classical theory, particles are treated as point-like objects with definite positions and velocities, and can be described by deterministic equations. On the other hand, in quantum theory, particles are described by wave functions that can exhibit both particle-like and wave-like behavior, and their behavior is governed by probabilistic equations.

Rayleigh scattering is a classical phenomenon that occurs when light interacts with particles much smaller than its wavelength, such as atoms or molecules. In this case, the electrons in the atom or molecule are treated as bound particles that oscillate in response to the incident electromagnetic wave. The restoring force acting on these electrons is proportional to their displacement, resulting in a scattered wave with the same frequency but with different amplitude and phase. This phenomenon is described by the Rayleigh scattering formula, which is derived from classical electrodynamics.

On the other hand, Compton scattering is a quantum phenomenon that occurs when a photon interacts with a free electron. In this case, the electron is treated as a free particle that can exchange energy and momentum with the incident photon. This exchange results in a scattered photon with a longer wavelength and lower energy, and a recoiling electron with higher energy and momentum. This phenomenon is described by the Compton scattering formula, which is derived from quantum electrodynamics.

The connection between the two phenomena lies in the fact that the classical result for free targets, known as Thomson scattering, is a special case of Compton scattering when the energy of the incident photon is much smaller than the rest energy of the electron. In this case, the energy exchange is negligible and the wavelength of the scattered photon remains the same. Therefore, the classical result can be seen as an approximation of the quantum result for low-energy photons.

In summary, the choice between Rayleigh and Compton scattering depends on the nature of the target and the energy of the incident photon. For free targets, Compton scattering is the dominant phenomenon, while for bound targets, Rayleigh scattering is more relevant. The classical result for free targets is connected to the quantum result for low-energy photons, providing a bridge between classical and quantum theory.