It is, however, worthwhile to spend a moment considering a collections of
identical scatterers at fixed spatial positions. Each scatterer then acts
identically, but is scattering an electromagnetic field with its own
(spatially dependent) phase at a given moment of time. The scattered fields
then propagate freely, recombine, and form a total EM field that is measured
by the detector. In order to evaluate the total differential cross-section
we must sum the field amplitudes times the appropriate phases, project out the
desired polarization moments, and *then* square.

A moment of quiet reflection^{16.3} will convince you that
in general:
d&sigma#sigma;d&Omega#Omega; = k^4(4&pi#pi;&epsi#epsilon;_0 E_0)^2
&sum#sum;_j { ^&ast#ast;·_j + (×^&ast#ast;)
·_j/c } e^i **q** ·**x**_j ^2
where

(16.2) |

accomodates the

In all directions but the forward direction, this depends on the distribution of scatterers and the nature of each scatterer. If we imagine all the scatterers to be alike (and assume that we are far from the collection) then this expression simplifies:

(16.3) |

where is the scattering cross-section of a single scatterer and the is called a ``structure factor'':

(16.4) | |||

(16.5) |

This last expression is 1 on the diagonal . If the (e.g.) atoms are uniformly but randomly distributed, the sum of the off-diagonal terms averages to zero and the total sum goes to (the number of atoms). This is an incoherent superposition and the scattered intensitities add with negligible interference.

If the atoms are instead on a regular lattice, then ``Bragg'' scattering
results. There will exist certain values of **q** that match the spacing
between planes in such a way that whole rows of the matrix are 1. In those
direction/wavelength combinations, the scattered intensity is of order
and hence is much brighter. The scattered power distribution thus has bright
spots in is corresponding to these directions, where constructive interference
in the scattered waves occurs.

**Structure factor** sums occur in many branches of physics. If you
think about it for a moment, you can easily see that it is possible to
do a structure factor sum using the Green's function expansions you have
studied. In electrodynamics and quantum multiple scattering theory
these sums appear frequently in association with spatially fixed
structures (like crystal lattices or molecules). In field theory,
lattice sums are sometimes used as a discretized approximation for the
continuum, and ``lattice gauge'' type field theories result. In these
theories, our ability to do the structure factor sums is used to
construct the Green's functions rather than the other way around.
Either way, you should be familiar with the term and should think about
the ways you might approach evaluating such a sum.

We are now done with our discussion of scattering from objects per se.
It is well worth your while to *read J10.2 on your own*. I have
given you the semi-quantitative argument for the blue sky; this section
puts our simple treatment on firmer ground. It also derives the *perturbation theory of scattering* (using the Born approximation), and
discusses a number of interesting current research topics (such as
critical opalescence). I will probably assign one problem out of this
section to help you out. However, perturbative scattering is easier to
understand, and more useful, in the context of (scalar) quantum theory
and so I will skip this section, expecting that you will see enough of
it there.

You should also read J10.3. This presents *one* way to derive the
Rayleigh expansion for a (scalar) plane wave in terms of free spherical
waves (there are several). However, it goes further and addresses
expansions of e.g. circularly polarized plane waves in terms of vector
spherical harmonics! Lord knows why this is stuck off in this one
section all by itself - I need to put the equivalent result for
expansion in terms of Hansen solutions (which of course will be much
more natural and will precompute most of the annoying parts of the
algebra for us) in the sections on the Hansen functions and VSHs where
it belongs, as it will actually be much simpler to understand there.

J10.4 redoes scattering from a sphere ``right'' in terms of VSHs, and
again, if we wished to pursue this we would need to redo this in terms
of Hansen functions to keep it simple. The primary advantage of reading
this chapter is that it defines the *partial wave phase shifts* of
scattering from a sphere, quantities that are in use in *precisely
the same context* in quantum scattering theory in e.g. nuclear physics.
SO, if you plan to go into nuclear physics you are well advised to read
*this* chapter as well and work through it.

However, we cannot do this at this time because we had to go back and
redo J7 and J8. Besides, we're doubtless a bit bored with multipoles
and want to become excited again. We will therefore now move on to one
of my favorite topics, *relativity theory*.