Free-Free Absorption Coefficient: Intuitive Concept

[Ethelred]

I am trying to form some intuitive concept about the free-free absorption coefficient for a hydrogen plasma.
The standard expression (cf. e.g. Schwarzschild's Structure and Evolution of the Stars, 1958, p64) contains a reasonably familiar particle type cross-section component, but there is also a velocity term in the denominator.
This velocity term means that the transparency rises (absortpion goes down) as the temperature rises ( a \sqrt(T) term in the denominator).
This might seem reasonable, I think, if it meant, for example, that the Debye length decreased with this temperature term. Basically, then a photon would have less chance to see a proton with nearby electron pair with which to exchange some energy, because there would be less chance to find the electron in a suitable, interactable, state.
But the Debye length in a plasma increases with just that same \sqrt(T) dependence -- so this idea cannot be right.
Another possibility might be a relativistic foreshortening effect on the cross-section which will increase with increasing transverse velocity. But there is no mention of 'relativistic effect' when one comes across the alpha_ff formula.

The root question, I suppose, is, where does the alpha_ff formula come from? I have not found it sourced in the books at my disposal.

Any suggestions?

 

[Ethelred]

Ok. I think I can see a way to resolve this. Although in a 'free-free' state, a photon will "see" and respond with a proton-electron pair (a hydrogen plasma is visulaized here) if that electron is within the Debye length of the proton. This length decreases with decreasing temperature as we move outward from the star centre. These ionized 'atoms' then become effectively smaller, and liberate their potential energy into the photon stream. The number of photons per particle-pair then decreases as we go further into the star, below what would be expected from a Boltzmannian T^4 prediction. It rises as a T^(7/2) law -- as predicted by the Kramers' formula. Well, actually, the Kramers' formula is an opacity formula -- so it is the inverse of this -- but the physical justification can be the same.

I think this could be one way to visualize this simple formula by 'physical intuition', although it would be interesting to know if anyone else feels persuaded by this argument.

 

[sir-pinski]

The free-free absorption coefficient is a critical concept in understanding the behavior of a hydrogen plasma. It is a measure of how much energy a photon can transfer to the plasma particles as it passes through, and it is influenced by both the particle cross-section and the velocity of the particles. This can be a bit confusing at first, but it is important to remember that in a plasma, the particles are not stationary, but rather moving and interacting with each other.

As you mentioned, the velocity term in the denominator of the free-free absorption coefficient formula does indeed mean that as temperature (and therefore particle velocity) increases, the plasma becomes more transparent to photons. This can be thought of in terms of the Debye length, which is a measure of the distance over which charged particles can interact. As temperature increases, the Debye length also increases, meaning that there is a smaller chance for a photon to interact with a nearby electron and transfer energy. This is similar to your intuition about the electron being in a suitable state for interaction.

However, as you also pointed out, the Debye length itself increases with temperature, so this cannot fully explain the behavior of the free-free absorption coefficient. One possible explanation for this is that as temperature increases, the plasma becomes more ionized, meaning that there are more free electrons available for interaction with photons. This increase in the number of particles can counteract the increase in the Debye length and result in a decrease in the absorption coefficient.

Another factor to consider is the relativistic effects on the particle cross-section. As particles move at high velocities, their size appears to decrease due to relativistic foreshortening. This can result in a higher cross-section for interaction with photons, which could contribute to the temperature dependence of the absorption coefficient.

In terms of the origin of the alpha_ff formula, it is derived from the theory of thermal bremsstrahlung, which describes the emission of radiation by charged particles in a plasma. The formula takes into account the particle cross-section, velocity, and other factors such as the plasma density and temperature. It is a well-established formula in plasma physics and astrophysics, and its origins can be traced back to the work of physicists such as Schwarzschild.

Overall, the free-free absorption coefficient is a complex concept that is influenced by multiple factors. It is important to keep in mind that in a plasma, particles are constantly interacting and moving, and this can have a significant impact on the behavior of the plasma.