Stars as Laboratories for Fundamental Physics - MPP Theory Group

226 Chapter 6
Without correlations, the squared matrix element involves |q| −4
from the Coulomb propagator. In order to account for screening effects
one should substitute
|q| −4 → |q| −4 S(q) (6.71)
which implies
1
|q| → 1
4 q 2 (q 2 + kS)
2
(6.72)
in the weak-screening limit (Debye screening).
The difference in a scattering cross section implied by Eq. (6.72)
relative to (6.61) is easily illustrated. Observe that a cross section
involving a Coulomb divergence is typically of the form
∫ +1
−1
dx
(1 − x) f(x)
(1 − x) 2 , (6.73)
where x is the cosine of the scattering angle of the probe. Here, f(x) is
a slowly varying function which embodies the details of the scattering
or bremsstrahlung process. If this function is taken to be a constant,
the two screening prescriptions amount to the two integrals
∫ +1
−1
∫ +1
−1
dx
dx
( )
1
2 + κ
2
(1 − x + κ 2 ) = log ,
κ 2
( )
(1 − x)
2 + κ
2
(1 − x + κ 2 ) = log − 2
2 κ 2 2 + κ , (6.74)
2
where κ 2 ≡ kS/2p 2 2 is the screening scale expressed in units of the initialstate
momentum of the probe. Usually, it far exceeds the screening scale
whence κ 2 ≪ 1. Then Eq. (6.72) yields a cross section proportional to
log(4p 2 /kS) 2 while Eq. (6.61) gives [log(4p 2 /kS) 2 − 1].
Thus, if one is only interested in a rough estimate, either screening
prescription and any reasonable screening scale yield about the same
result. For an accurate calculation, however, one needs to identify
the dominant source of screening (for example, the nondegenerate ions
in a degenerate plasma and not the electrons), and the appropriate
moderation of the Coulomb propagator, usually Eq. (6.72).