Suppose that a plane wave of monochromatic electromagnetic radiation is
incident on a free particle of charge
and mass
. The particle will
experience a force from this field, and will accelerate. As it accelerates,
it will emit radiation in different directions, *dispersing* the incident
beam.

For a non-relativistic particle accelerated by a force we can see that: dPd&Omega#Omega; = e^216&pi#pi;^2&epsi#epsilon;_0 1c^3 ^&ast#ast;·.^2 (where for a particular polarization perpendicular to the plane of and ).

The (leading order) acceleration is due to the plane wave electric field
with polarization
, wave vector
, and Newton's Law:
. = em E_0 _0 e^i_0 ·-&omega#omega;t
If the charge moves much less than one wavelength during a cycle (true
for all but the lightest particles and strongest fields) then
|.|_av = 12 Re (. ·
.^&ast#ast;)
Thus the average power flux distribution is
( dPd&Omega#Omega; )_av & = & c32 &pi#pi;^2
&epsi#epsilon;_0 E_0^2 ( e^2mc^2 )^2
|^&ast#ast;·_0|^2

& = & {e^24 &pi#pi;&epsi#epsilon;_0 mc^2}^2
&epsi#epsilon;_0 c E_0^22 |^&ast#ast;·_0|^2

This is clearly of the same general form as the scattering expressions
we described and derived earlier. Since the result contains
it
makes sense to divide out the incident intensity and thus obtain a
*differential* cross section that works for all but the strongest
fields. We thus divide out the time-averaged flux of the Poynting
vector of the incident plane wave:
I = &epsi#epsilon;_0 c E_0^22
hence
d&sigma#sigma;d&Omega#Omega; = {e^24 &pi#pi;&epsi#epsilon;_0 mc^2}^2
|^&ast#ast;·_0|^2

If we let the plane wave be incident along the
axis, let
form
an angle
with that axis, and pick two polarization directions
in and perpendicular to the
plane (as before), and average
over polarizations then this dot product yields:
d&sigma#sigma;d&Omega#Omega; = {e^24 &pi#pi;&epsi#epsilon;_0
mc^2}^2 12 (1 + ^2 &thetas#theta;).
as it did back in our earlier work on scattering, but now for a *point* particle.

This is the **Thomson formula** for scattering of radiation by free
charge. It works for X-rays for electrons or
-rays for
protons. It does not work when the photon momentum and the recoil of
the charged particle cannot be neglected. The integral of this,
&sigma#sigma;_T = 8 &pi#pi;3 {e^24 &pi#pi;&epsi#epsilon;_0
mc^2}^2
is called the **Thomson cross-section**. It is
m
for electrons.

The quantity in parentheses has the units of length. If the total
``mass-energy'' of the electron were due to its charge being
concentrated in a ball, then this would be the close order of the radius
of that ball; it is called the **classical electron radius**. This
number crops up quite frequently, so you should remember it. What it
tells us is that even *point* particles have a *finite*
scattering cross-section that appears in this limit to be independent of
the wavelength of the light scattered.

However, this is not really true if you recall the approximations made
- this expression will fail if the wavelength is on the same order as
the classical radius, which is precisely where pair production becomes a
significant process quantum mechanically. In quantum mechanics, if the
energy of the incident *photon*
for the
*electron*, significant momentum is transferred to the electron by
the collision and the energy of the scattered photon cannot be equal to
the energy of the incident photon. Whatever a photon is ...

We can actually fix that without too much difficulty, deriving the
Compton scattering formula (which takes over from Thomson in this
limit). This formula adds a wavelength/angle dependence to Thomson's
general result and yields the *Klien-Nishina* formula, but this is
beyond our scope in this course to derive or discuss in further detail.

We are almost finished with our study of electrodynamics. Our final object of study will be to to try to address the following observation:

Accelerated charges radiate. Radiation accelerates charge. Energy must be conserved. These three things have not been consistently maintained in our treatments. We study one, then the other, and require the third to be true in only part of the dynamics.

What is missing is *radiation reaction*. As charges accelerate, they
radiate. This radiation carries energy away from the system. This, then
means that a *counterforce* must be exerted on the charges when we try to
accelerate them that *damps* charge oscillations.

At last the folly of our ways is apparent. Our blind insistence that
only *retarded* fields are meaningful (so that we can imagine the
fields to be zero up to some time and then start moving a charge, which
subsequently radiates) has left us with only one charge that can produce
the field that produces the force that damps applied external forces --
the charge itself that is radiating. No other charge produces a field
that can act on this charge ``in time''. We have invented the most
sublime of violations of Newton's laws - an object that lift's itself
up by its own bootstraps, an *Aristotelian* object that might even
be able to come to rest *on its own* in the absence of external
forces.

Clearly we must investigate radiation reaction as a *self-force*
acting on an electron due to its own radiation field, and see if it is
possible to salvage anything like a Newtonian description of even
classical dynamics. We already know that Larmor radiation plus stable
atoms spells trouble for Newton, but Newton still *works*
classically, doesn't it?

Let's take a look. Uh-oh, you say. Wasn't the, well, wasn't *everything* singular on a point charge? Won't we get infinities at
every turn? How will we realize finite results from infinite fields,
potentials, self-energies, and so on?

*Yes!* I cry with glee. *That's* the *problem*. Finally we
will learn how to take a singular field, a singular charge, and infinite
energy, and make a physically realized (almost) radiation reaction force
out of it.