Stars as Laboratories for Fundamental Physics - MPP Theory Group

198 Chapter 6
only refers to scattering in the forward direction, but that all properties
of the wave and the scatterer are left unchanged. If the medium particles
have a distribution of momenta, spins, etc. the forward scattering
amplitude must be averaged over those quantities, and different species
of medium particles must be summed over.
For a practical calculation it helps to recall that dσ/dΩ = |f(θ)| 2
so that |f 0 | is the square root of the forward differential cross section.
For example, the Thomson cross section for photons interacting with
nonrelativistic electrons is dσ/dΩ = (α/m e ) 2 |ϵ · ϵ ′ | 2 with the polarization
vectors ϵ and ϵ ′ of the initial- and final-state photon. Forward
scattering implies |ϵ · ϵ ′ | 2 = 1 so that |f 0 | = α/m e . The dispersion relation
is then ω 2 = k 2 + ωP 2 with the plasma frequency ωP 2 = 4πα n e /m e .
Of course, the absolute sign of f 0 has to be derived from some other
information—for photon dispersion see Sect. 6.3.
The forward scattering amplitude and the refractive index are generally
complex numbers. Physically it is evident that in a medium the
intensity of a beam is depleted as e −z/l . The mean free path is given by
l −1 = σnv where σ is the total scattering cross section, n is the number
density of scatterers, and v is the velocity of propagation. Thus the
amplitude of a plane wave varies as e ikz−z/2l . Moreover, the derivation
of the refractive index indicates that the amplitude varies according to
e inrefrωz , yielding k = Re n refr ω and (2l) −1 = Im n refr ω. For relativistic
propagation (v = 1) the last equation implies σ(ω) = (4π/ω) Im f 0 (ω),
a relationship known as the optical theorem.
For the applications discussed in this book specific interaction models
between the propagating particles and the medium will be assumed
so that it is usually straightforward to calculate the dispersion relation
according to Eq. (6.5). One should keep in mind, however, that n refr as
a function of ω has a number of general properties, independently of the
interaction model. For example, its real and imaginary part are connected
by the Kramers-Kronig relations (Sakurai 1967; Jackson 1975).
6.2.2 Particle Momentum and Velocity
The four-vector (ω, k) which governs the spatial and temporal behavior
of a plane wave can be time-like (ω 2 − k 2 > 0) as for massive particles
in vacuum, it can be light-like (ω 2 − k 2 = 0) as for photons in vacuum,
or it can be space-like (ω 2 − k 2 < 0) as for visible light in water or air.
Because the quantized excitations of such field modes are interpreted
as particles, E = ¯hω is the particle’s energy. (I have temporarily restored
¯h even though it is 1 in natural units.) Similarly one may be