Reading a graph for dispersion relation of normal modes in 3D lattice

Question

I'm trying to interpret graphs of this type. In this example, dispersion relation $\omega = \omega(k)$ for some waves corresponding to lattice vibrations in GaAs are shown.

What is uncomprehensible to me is the abscissa that should represent the 3D wave vector $\vec{k}$ in some way. Units in these graphs are never displayed. Some points along the axis are marked with letters that represent "high symmetry points" or "critical points" in Brillouin zone of the 3D lattice and therefore specific value for $\vec{k}$.

QUESTIONS:

a) How are the "high symmetry points" chosen? Do they have particular properties?

b) Which value of $\vec{k}$ (modulus and direction) does correpond to a point on the horizontal axis of the example graph that is found between two high symmetry points?

Answer 1

Take a look at a Brillouin zone for zinc blende crystals. You’ll see that it’s a polyhedron. The “high-symmetry” points are just points such as the center of the polyhedron ($\Gamma$) or corners or centers of faces, etc. In the $\omega$ vs. $k$ plot, a region between labeled lines is taken to be a straight-line path within the 3D Brillouin zone between the two labeled points. So, you can quickly see from the graph how the frequency varies as $k$ goes from $\Gamma$ to $L$, for instance. This is easier than representing the complete $\omega(\mathbf{k})$ function,which would be a 4D plot.

This type of 2D plot is a simplification, but it’s useful because all the interesting things (such as maxima and minima of bands) happen at the high-symmetry points, or at least somewhere on a line between them.