The denominators are much larger than which is of the order of the electron's kinetic energy, so we can ignore the second two terms. (Even if the intermediate and final states have unbound electrons, the initial state wave function will keep these terms small.)

This scattering

The only dependence is on polarization.
This is a good time to take a look at the meaning of the **polarization vectors** we've been carrying around in the calculation
and at the lack of any wave-vectors for the initial and final state.
A look back at the calculation shows that we calculated the transition rate from a state with one photon with
wave-vector
and polarization
to a final state with polarization
.
We have **integrated over the final state wave vector magnitude, subject to the delta function** giving energy conservation,
but, we have **not integrated over final state photon direction** yet, as indicated by the
.
There is no explicit angular dependence but there is some
**hidden in the dot product between initial and final polarization vectors, both of which must be transverse**
to the direction of propagation.
We are ready to compute four different differential cross sections
corresponding to two initial polarizations times two final state photon polarizations.
Alternatively, we average and/or sum, if we so choose.

In the high energy approximation we have made, there is no dependence on the state of the atoms,
so we are **free to choose our coordinate system** any way we want.
Set the **z-axis to be along the direction of the initial photon**
and set the x-axis so that the **scattered photon is in the x-z plane** (
).
The scattered photon is at an angle
to the initial photon direction and at
.
A reasonable set of **initial state polarization vectors** is

Pick to be in the

From these, we can compute any cross section we want. For example,

Even if the initial state is unpolarized, the **final state can be polarized**.
For example, for
, all of the above dot products are zero except
.
That means only the initial photons polarized along the y direction will scatter and that
the scattered photon is **100% polarized transverse to the scattering plane**(really just the same polarization as the initial state).
The angular distribution could also be used to deduce the polarization of the initial state if a large ensemble of
initial state photons were available.

For a **definite initial state polarization** (at an angle
to the scattering plane, the component along
is
and along
is
.
If we don't observe final state polarization we sum
and have

For **atoms with more than one electron**, this cross section will grow as
.

Jim Branson 2013-04-22