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Many Scatterers

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 reflection14.3 will convince you that in general:

\frac{d\sigma}{d\Omega} = \frac{k^4}{(4\pi\epsilon_0 E_0)^2...
...$m$}_j/c \right\} e^{i {\bf q} \cdot {\bf x}_j} \right\vert^2
\end{displaymath} (14.53)

{\bf q} = {\bf k}_0 - {\bf k} .
\end{displaymath} (14.54)

accomodates the relative phase difference between the field emitted by the scatterers at different locations. The geometry of this situation is pictured below.

Figure 14.2: Geometry of multiple scatterers. The relative phase of two sources depends on the projection of the difference in wave vectors onto the vector connecting the scatterers.

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:

\frac{d\sigma}{d\Omega} = \frac{d\sigma_0}{d\Omega} {\cal F}({\bf q})
\end{displaymath} (14.55)

where $\frac{d\sigma_0}{d\Omega}$ is the scattering cross-section of a single scatterer and the ${\cal F}({\bf q})$ is called a ``structure factor'':
$\displaystyle {\cal F}({\bf q})$ $\textstyle =$ $\displaystyle \left\vert \sum_j e^{i {\bf q} \cdot {\bf x}_j} \right\vert^2$ (14.56)
  $\textstyle =$ $\displaystyle \sum_{i,j} e^{i{\bf q} \cdot ({\bf x}_j - {\bf x}_i)}.$ (14.57)

This last expression is 1 on the diagonal $i = j$. 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 $N$ (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 $N^2$ 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.

next up previous contents
Next: Relativistic Electrodynamics Up: Optical Scattering Previous: Scattering from a Small   Contents
Robert G. Brown 2007-12-28