Stars as Laboratories for Fundamental Physics - MPP Theory Group

222 Chapter 6
It depends on the physical circumstances which procedure is a better
approximation. If one considers bremsstrahlung processes with degenerate
electrons scattering off nondegenerate nuclei, the crossing time of
an electron of a region the size kS
−1 is short compared to the crossing
time of nuclei. Hence, the latter can be viewed as static, the probe sees
one configuration at a time, and one certainly should use the “square
first” procedure instead of Eq. (6.61) to account for screening. This is
achieved by the following consideration of correlation effects.
6.4.2 Correlations and Static Structure Factor
The screening of electric fields in a plasma is closely related to correlations
of the positions and motions of the charged particles. If a negative
test charge is known to be in a certain position, the probability of finding
an electron in the immediate neighborhood is less than average,
while the probability of finding a nucleus is larger than average. It is
this polarization of the surrounding plasma which screens a charge.
Take one particle of a given species to be the origin of a coordinate
system, and take their average number density to be n. The electrostatic
repulsion of the test charge causes a deviation of the surrounding
charges from the average density by an amount
S(r) = δ 3 (r) + n h(r), (6.62)
where h(r) measures the particle correlations. They vanish in an ideal
Boltzmann gas: h(r) = 0. The Fourier transform
∫
S(q) = d 3 r S(r) e −iq·r (6.63)
is the static structure factor of the electron distribution. In the absence
of correlations (h = 0) one has trivially S(q) = 1.
In order to make contact with Debye screening consider the Yukawa
potential of Eq. (6.57) which represents a charge density
ρ(r) = δ 3 (r) − k2 S
4π
e −k Sr
r
. (6.64)
The volume integral of ρ(r) vanishes, giving zero total charge, i.e. complete
screening at infinity. If one imagines that only one species of
charged particles is mobile on a uniform background of the opposite
charge, then Eq. (6.64) implies correlations between the mobile species
of n h(r) = −(kS/4πr) 2 e −kSr . As expected, Debye screening corresponds