## Delta Function Potentials p.9

### Scattering from a finite crystal

You may want to review the material dealing with scattering.

We have an incident beam on the far right moving towards the
crystal: *I* exp(-*i kx*), the wave is partially transmitted
into the crystal, where it is knocked around by the nuclei.
Some of the wave emerges from the lhs
as the transmitted wave: *A* exp(-*i kx*). There
is also a reflected wave on the far right: *R* exp(+*i kx*).
The above represents a switch from our past set-up (in which waves came in
from the left) which we have instituted so we can use the machinery
set up to find bound states. Thus we "start" with the transmitted
wave (which consists of just one solution), connect it through
*N* nuclei, and find what the incident and reflected
waves must have been. Following our earlier work we have:

(*I*,*R*) = [*M*]^{N} · (*A*,0)

We might as well make *A*=1, and our reflection
probability (which,confusingly, we will also call *R*)
will then be: |*R*|^{2}/|*I*|^{2}.
We will, of course, use our eigenvectors to simplify calculation.

m={{(1-I/k)Exp[- I k a],-I/k},{I/k,(1+I/k)Exp[+ I k a]}}
{{l1,l2},{v1,v2}}=Eigensystem[m]
Solve[{1,0}==c v1+d v2,{c,d}]
c l1^n v1 + d l2^n v2 /. First[%]
ds[k_,a_,n_]= %
ref[k_,a_,n_]=Abs[Last[ds[k,a,n]]/First[ds[k,a,n]]]^2
ParametricPlot[{.5 k^2, ref[k,4,32]},{k,0,3}]

Here are some calculated reflection probabilities a
a function of the incident beam energy for
various crystal sizes:

Note that at some energies the crystal is highly reflective.
As the following picture shows, those highly reflective
energies correspond to the band gaps.

Mathematically this makes sense as the gaps are exactly where
the eigenvalues are not of magnitude 1. There the waves must
exponentially decay (or grow) in the crystal. In order to
transmit a wave with *A*=1, the incident wave must be huge
as the incident wave is exponentially damped in the crystal (and the
energy is reflected back).

Alternatively you can say that what is happening is Bragg scattering
of the incident wave. Inside the crystal the wavelength of the
wave is determined by *q* not *k*. The gaps are
then exactly the Bragg scattering condition.

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