Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 221
In the limit of degenerate electrons the integral is also easily solved
and leads to the familiar Thomas-Fermi wave number 33 (Jancovici 1962)
kS 2 = kTF 2 = 4α π E Fp F = 3ω2 P
. (6.60)
vF
2
However, kD 2 always exceeds kTF 2 so that in a medium of degenerate
electrons and nondegenerate ions the main screening effect is from the
latter. Recall that the Fermi momentum is related to the electron
density by n e = p 3 F/3π 2 and the Fermi energy is E F = (p 2 F + m 2 e) 1/2 .
To compare the Thomas-Fermi with the Debye scale take the nonrelativistic
limit (k TF /k D ) 2 = 3 T/(E 2 F − m e ). This is much less than 1
or the medium would not be degenerate whence k TF ≪ k D . Therefore,
if the electrons are degenerate and the ions nondegenerate, a test
charge is mostly screened by the polarization of the ion “fluid” because
the electrons form a “stiff” background. Unfortunately, one often finds
calculations in the literature which include screening by the electrons
(screening scale k TF ) but ignore the ions. The resulting error need not
be large because the screening scale typically appears logarithmically
in the final answer (see below).
Screening effects in Coulomb processes are often found to be implemented
by a modified Coulomb propagator
1
|q| 4 → 1
(q 2 + k 2 S) 2 , (6.61)
where q is the momentum transfer carried by the intermediate photon.
This substitution arises if one considers Coulomb scattering from a
Yukawa-like charge distribution. It corresponds to the substitution
Eq. (6.57), i.e. to a single charge with an exponential screening cloud.
This picture is appropriate if the Coulomb scattering process itself is so
slow that the charged particles move around and rearrange themselves
so much that the probe, indeed, sees an average screening cloud.
In the opposite limit, a given probe sees a certain configuration, a
different probe a different one, etc., and one has to average over all of
these possibilities. In this case one needs to square the matrix element
first, and then take an average over different medium configurations.
For Eq. (6.61) one averages first, obtains an average scattering amplitude
or matrix element, and squares afterward.
33 For a textbook derivation from a Thomas-Fermi model see Shapiro and Teukolsky
(1983). Note that they work in the nonrelativistic limit: their E F = p 2 F /2m e.