Stars as Laboratories for Fundamental Physics - MPP Theory Group

196 Chapter 6
and density, most of the results relevant for particle physics in stars
predate the development of this formalism; they were based on the
old-fashioned tools of kinetic theory. Indeed, for simple issues of dispersion
or collective effects a kinetic approach seems often physically
more transparent while yielding identical results. At any rate, the following
discussion is based entirely on kinetic theory.
6.2 Particle Dispersion in Media
6.2.1 Refractive Index and Forward Scattering
How does one go about to calculate the all-important dispersion relation
for a given particle in a medium with known properties Usually
it is enough to follow the elementary approach of calculating the forward
scattering amplitude of the relevant field excitations with the constituents
of the background medium, an approach which has the added
advantage of physical transparency over a more formal procedure. 30
To begin, consider a scalar field Φ which may be viewed as representing
one of the photon or electron polarization states. If a plane
wave excitation of that field with a frequency ω and a wave vector k interacts
with a scatterer at location r = 0 an additional spherical wave
will be created. The asymptotic form of the original plus scattered
wave is
(
)
Φ(r, t) ∝ e −iωt e ik·r + f(ω, θ) eikr
, (6.1)
r
where k = |k|, r = |r|, and f is the scattering amplitude. It was assumed
that it has no azimuthal dependence, something that will always
apply on average for a collection of randomly oriented scatterers. The
differential scattering cross section is dσ/dΩ = |f(ω, θ)| 2 .
If there is a collection of scattering centers randomly distributed
in space, all of the individual scattered waves will interfere. However,
because of the random location of the scatterers, constructive and destructive
interference terms will average to zero. Thus the total cross
section of the ensemble is the (incoherent) sum of the individual ones.
In the forward direction, however, the scattered waves add up coherently
with each other and with the parent wave, leading to a phase
shift and thus to refraction. This is seen if one considers a plane wave in
the z-direction incident on an infinitesimally thin slab (thickness δa) at
30 The derivation below follows closely the exposition of Sakurai (1967).