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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?

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As the footnote on page 665 below your first equation explains

As noted in Chapter 9, in writing angular distributions of radiation we always exhibit the polarization explicitly by writing the absolute square of a vector that is proportion to the electric field. If the angular distribution for some particular polarization is desired, it can be obtained by taking the scalar product of the vector with the appropriate polarization vector before squaring.

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

(To me, it seems that Jackson should have written "implicitly" rather than "explicitly".)

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