Stars as Laboratories for Fundamental Physics - MPP Theory Group

224 Chapter 6
particle species. Hence, a measure for the Coulomb interaction energy
is Z 2 α/a i , assuming the ions have charge Ze. One usually introduces
the parameter
Γ ≡ Z2 α
a i T = (k ia i ) 2
(6.68)
3
as a measure for how strongly the plasma is coupled, where ki 2 =
4πZ 2 α/T . For Γ ≪ 1 it is weakly coupled and approaches an ideal
Boltzmann gas.
The Debye structure factor of a one-component plasma can be written
as
|a i q| 2
S D (q) =
|a i q| 2 + 3Γ . (6.69)
This result applies even for large Γ if |a i q| ≪ 1. For Γ ≫ 1, the plasma
is strongly coupled, and for Γ > ∼ 178 the ions will arrange themselves
in a body centered cubic lattice (Slattery, Doolen, and DeWitt 1980,
1982).
In Fig. 6.9 I show S and S D as functions of a i q = |a i q| for Γ = 2, 10
and 100 where S was numerically determined (Hansen 1973; Galam and
Hansen 1976). The emerging periodicity for a strongly coupled plasma
is quite apparent. It is also clear that for Γ < ∼ 1 the Debye formula gives
a fair representation of the structure factor while for a strongly coupled
plasma it is completely misleading. The interior of white dwarfs is in
the regime of large Γ, and old white dwarfs are believed to crystallize.
(See Appendix D for an overview over the conditions relevant for stellar
plasmas.)
6.4.4 Screened Coulomb Scattering
Armed with these insights one may turn to the issue of Coulomb scattering
processes in a plasma. In the limit of nonrelativistic and essentially
static sources for the electric fields the relevant quantity entering the
matrix element is the Fourier component ρ(q) of the charge distribution
ρ(r) where q is the momentum transferred by the Coulomb field
to the sources. The squared matrix element thus involves the quantity
ρ(q)ρ ∗ (q) which is ρ(q)ρ(−q) because ρ(r) is real. Taking a statistical
average over all possible configurations of the charge distribution leads
to a rate proportional to
S(q) = ⟨ρ(q)ρ(−q)⟩. (6.70)
This is the static structure factor introduced earlier as a measure of the
correlation between the charged particles of the medium.