The Physics

The idea is to investigate the spectrum (the distribution of the eigenvalues) as the ratio, $m/M$, of the two masses changes. Your program should list the eigenvalues in ascending order and then plot a graph of eigenvalue against position in the list. When $m=M$, the crystal is ``perfect'', the graph is smooth, and you should be able to work out all the eigenvalues analytically (easiest when using periodic boundary conditions). But when $m$ and $M$ begin to differ, the graph becomes a a ``devil's staircase'' with all sorts of fascinating fractal structure. Try to understand the behaviour at small $\omega$ (when the wavelength is long and the waves are ``acoustic'') and the limits as $m/M \rightarrow 0$ and $m/M \rightarrow \infty$.

Another interesting thing to do is to plot values of $m/M$ (on the $y$ axis) against the vibrational frequencies (on the $x$ axis). Choose a value of $m/M$, work out all the frequencies, and put a point on the graph for each. The graph now has a line of points parallel to the $x$ axis at the given $y$ value. Do this for a number of values of $m/M$ and see how the spectrum develops as $m/M$ changes. Again, you should try to understand the behaviour when the frequency tends to zero, and the large and small mass ratio limits.

If you have time it is interesting to investigate the fractal structure by focusing in on a single peak for a short sequence and investigating how it splits when you add another ``generation''. You should find that the behaviour is independent of the number of generations at which you start.