(a) (b)Figure 9: a) Dispersion relation for photons in vacuum (potential is zero!); b) Density of states in
terms ofωfor photons (potential is nonzero!).
5.2 Empty Lattice Approximation (Photons)
What happens to the dispersion relationship of photons when there is a crystal instead of vacuum?
In the Empty Lattice Approximation a crystal lattice is considered, but no potential. Therefore
diffraction is possible. The dispersion relation in vacuum is seen in fig. 9(a).
The slope is the speed of light. If we have a crystal, we can reach diffraction at a certain point.
The first time we do that is whenkreaches πa. Then we are on this plane which is the half way to
the first neighboring point in reciprocal space. In a crystal the dispersion relation is bending over
near the brillouin zone boundary and then going on for higher frequencies. With photons it happens
similar to what happens with electrons. Close to the diffraction condition a gap will open in the
dispersion relationship if a nonzero potential is considered (with the Empty Lattice Approximation no
gap occurs, because the potential is assumed to be zero), so there is a gap of frequencies where there
are no propagating waves. If you shoot light at a crystal with these frequencies, it would get reflected
back out. Other frequencies will propagate through.
Because of the gap in the dispersion relationship for a nonzero potential, it also opens a gap in the
density of states. In terms ofkthe density of states is distributed equally. The possible values ofkare
just the ones you can put inside with periodic boundary conditions. In terms ofω(how many states
are there in a particular range ofω) it is first constant followed by a peak resulting of the dispersion
relation’ bending over. After that there is a gap in the density of states and another peak (as seen in
fig. 9(b)).
TheEmpty Lattice Approximationcan be used to guess what the dispersion relationship looks
like. So we knowwhere diffraction is going to occurandwhat the slope is. This enables us to
tell where the gap occurs in terms of frequency.