# Jackson EM Conundrum: Power Spectrum Polarization dependence for Thomson Scattering

I'm having some issues wrapping my head around what Jackson did for his derivation of nonrelativistic Thomson scattering. I'm using Gaussian units although I don't think there's a conflict here

Nonrelativistic electric field for an accelerated particle is given by (implicit retarded time)

$$\vec{E} = \frac{q}{rc} (\hat{n} \times (\hat{n} \times \dot{\vec{\beta}})$$

The power pattern (angular distribution of power) just follows from

$$\frac{dP}{d\Omega} = \frac{c}{4\pi}r^{2}|\vec{E}|^{2} = \frac{e^{2}}{4\pi c}|\hat{n} \times (\hat{n} \times \dot{\vec{\beta}})|^{2}$$

In the instance of Thomson scattering, we have an incident plane wave driving oscillations of an electron up and down along a fixed axis. Now I can solve this just fine and get the correct $$\sin^{2}\theta$$ dependence. My question rests more on what the hell Jackson is doing.

In equation 14.120 he'll just jump and say that

$$\frac{dP}{d\Omega} = \frac{e^{2}}{4\pi c^{3}}|\epsilon^{*} \cdot \dot{\vec{v}}|^{2}$$

Where he says that $$\epsilon$$ is the polarization state that the power is radiated into.

OK so I tried to figure this out which I imagined would be sort of straightforward. I presume that the vector $$\hat{n} \times (\hat{n} \times \hat{\beta})$$ is the polarization of radiated light, so simply working out the geometry should do it right? Using a triple product vector identity, I get

$$\hat{n} \times (\hat{n} \times \hat{\beta}) = \hat{\beta} - (\hat{\beta}\cdot \hat{n})\hat{n}$$

If I calculate the square magnitude as the original power pattern formula says, I get

$$1 - (\hat{\beta}\cdot \hat{n})^{2}$$

If assume I have an electric field incident along the $$\hat{z}$$ which drives oscillations in the $$z$$ direction, then following spherical geometry I get the $$\sin^{2} \theta$$ shape of the power pattern.

Trying Jackson's method, using $$\hat{n} \times (\hat{n} \times \hat{\beta})$$ as the polarization doesn't work! In fact, it gives the square of the previous answer. What the heck is going on?

EDIT: I'm starting to suspect that this is just awful notation for what Jackson is referring to. Does he mean that we are just projecting the vector amplitude down into a particular polarization basis and calculating the relative amplitudes that way?

So trying computing $$\hat\epsilon^*\cdot(\hat n\times(\hat n\times\dot{\vec\beta}))$$, remembering that $$\hat\epsilon^*\cdot\hat n=0$$.