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.