Thomson scattering and radiating power

[Shinji83]

I have a question about Thomson scattering from an electron hit by an incident em plane wave.

The derivations that I have found all state the same thing.
You have an electron in the origin at rest as the initial condition, the incident plane wave is linearly polarized towards z with amplitude E₀.
The force on the electron is -e*E₀cos(ωt).
The quadratic mean value of the acceleration is obviously:
<a²>=e²E₀²/2m_e²

Now plugging that value of the acceleration in the Larmor's formula with obtain the average irradiated power by the electron:

P=e²*<a²>/6πε₀c³=(8/3)π*r_e²*I₀=σ_tI₀

Now my question about this derivation is the following.
If an accelerated electron emits radiation, when we calculate the acceleration value that we put inside the Larmor's formula we can't assume that the only force acting on the electron is -e*E₀cos(ωt).
We should model the radiating loss with a (usually viscous-like) damping force. We can't get the right acceleration value using a model without losses and the remembering that we have a loss later like it seems to me that authors do in this kind of derivation.
Shouldn't the approach be the same of what we use when we want to find the complex permittivity in a dielectric medium where molecules act like oscillating radiating dipoles?
In that case losses arising from both radiation scattering and interaction with other molecules are modeled using a damping force.

 

[Shinji83]

Up? :D
Maybe there's a better section for this question? I dunno.

 

[tech99]

My suggestion is that, by analogy from what we know of antennas, if we assume that the electron, by virtue of being able to move, has a radiation resistance, then half the incident power will be absorbed in this resistance and the other half will pass by unaltered. The absorbed energy will then be stored in local fields, the induction fields, and then re-radiated. The local fields arise because when an electron moves it builds a magnetic field, and this varying magnetic field will create a quadrature electric field. Notice that these are not radiated fields, because B and E are in quadrature, but they serve to give the electron a delayed push, or recoil, which is when it does the re-radiating.
In defining the incident power, we somehow need to define an antenna aperture for a single electron, but whatever it is, the principle should stand, because the same aperture applies to both "receiving" and "transmitting".
So by considering the "damping losses" caused by re-radiation, I believe you are correct, but the effect is that half the incident power is absorbed/re-radiated and half ignored.

 

[Shinji83]

My suggestion is that, by analogy from what we know of antennas, if we assume that the electron, by virtue of being able to move, has a radiation resistance, then half the incident power will be absorbed in this resistance and the other half will pass by unaltered. The absorbed energy will then be stored in local fields, the induction fields, and then re-radiated. The local fields arise because when an electron moves it builds a magnetic field, and this varying magnetic field will create a quadrature electric field. Notice that these are not radiated fields, because B and E are in quadrature, but they serve to give the electron a delayed push, or recoil, which is when it does the re-radiating.
In defining the incident power, we somehow need to define an antenna aperture for a single electron, but whatever it is, the principle should stand, because the same aperture applies to both "receiving" and "transmitting".
So by considering the "damping losses" caused by re-radiation, I believe you are correct, but the effect is that half the incident power is absorbed/re-radiated and half ignored.

Thank you for your answer. I was thinking about something along the same line.
The logic indeed works if we consider an electric dipole under a constant electric field.
Assuming that one of the charges is in a fixed position, if we don't consider any damping force, the system starts to oscillate around the equilibrium point of the forces (eE and -kx). So the power is a sinusoid as well with zero mean value.
That means that in half period the electric field gives energy to the dipole and in the other half the dipole gives it back to the electric field, so we can assume that this is re-radiated energy in phase with the incident field and overall no intensity loss in the incident wave occurs. So we don't need to necessarily include a damping force to justify the radiating losses because using this logic the energy given back to the electric field is already the radiating loss.
The problem is that if we consider a free electron there is no elastic force due to a dipole.
A static field E generates a constant force -eE on the electron, the acceleration is eE/m_e. Here there is no power oscillation since the power is Fv= m*a²*t.
But if the Larmor's formula tells us that there is an energy loss due to radiation that means that this energy must come from somewhere. If it's from the kinetic energy of the electron then the net acceleration on the electron is not what was calculated as if only the force acting on the electron was e*E, we need to introduce a damping force in the model.
The other option is that the electron moves under the force eE with a= eE/m_e, the electric field must provide energy to account for both the variation in time of the kinetic energy AND in addition as a separate process it must provide the energy that will be re-radiated. In this case it makes sense to use 'a' as the acceleration value in the Larmor's formula. I think this explanation might fit with the derivations I'm reading on different books.
So basically there is no need to introduce a damping force in the mechanical model to account for radiating losses, because that energy must provided by the applied field separately from the energy involved in the mechanical aspect. If the net field is attenuated or not will depend on the relative phases between the incident and re-radiated fields but that's an other issue.
Unfortunately it's a bit frustrating that there are no sources that state these things clearly (or at least I haven't been able to find them).

 

[tech99]

I will give more thought, but I am only an amateur scientist. I have been studying the relation of antenna near fields to the radiation process for some time, including measurements of B, E and Power Flux Density. It is very hard to find clear published information but I have obtained some very interesting results. I find I can distinguish between radiated and stored energy around the antenna.
The motion of the electron, with the restoring force, is described by the ideas about Plasmon Resonance, for instance, at the following URL:-

 

[tech99]

Correction
Sorry, my error, I wanted to give the URL given below about the restoring force at work with plasma oscillations (in a metal) ;-
For the case of an isolated electron in space, its inertia appears to come from the mass plus the action of building a magnetic field when it moves, and it would seem to have damping coming from radiation resistance.